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We consider the nonlinear elliptic system $$\begin{align*} \left\{ \begin{array}{ll} u \in W_{0}^{\frac{N}{N-1}}(\Omega): & -\textrm{div}(M(x)\nabla u) + u = -\textrm{div}(u\,M(x)\,\nabla\psi) + f(x),\\ \displaystyle \psi \in W_0^{1,2}(\Omega): & -\textrm{div}(M(x)\nabla \psi) + \psi = R(u) + E(x)\,\nabla\psi, \end{array} \right. \end{align*}$$ { u ∈ W 0 N N − 1 (Ω) : − div (M (x) ∇u) + u = − div (u M (x) ∇ψ) + f (x) , ψ ∈ W 0 1 , 2 (Ω) : − div (M (x) ∇ψ) + ψ = R (u) + E (x) ∇ψ , where Ω is a bounded, open subset of $ \mathbb {R}^N $ R N , $ N \geq ~3 $ N ≥ 3 ; $ M(x) $ M (x) is a coercive, symmetric matrix with $ L^{\infty }(\Omega) $ L ∞ (Ω) coefficients; $ f(x) $ f (x) and $ E(x) $ E (x) belong to some Lebesgue space, and $ R(s) $ R (s) is a continuous function such that $$\begin{align*} 0 \leq R(s)\leq |s|^{\theta}, \quad \mbox{for}\ \theta { \lt } \frac{2}{N}. \end{align*}$$ 0 ≤ R (s) ≤ | s | θ , for θ < 2 N. Using a duality technique, we prove existence of at least a weak solution $ (u,\psi) $ (u , ψ). Moreover, if N=3 or N=4, we prove under stronger assumptions on $ f(x) $ f (x) and $ E(x) $ E (x) that the solution u belongs to $ W_0^{1,2}(\Omega) $ W 0 1 , 2 (Ω). [ABSTRACT FROM AUTHOR]

Motivated by the inverse formula of the distance matrix of a tree and the Moore–Penrose inverse of a circum-Euclidean distance matrix (CEDM), in this paper, we study a general real square matrix M whose Moore–Penrose inverse can be expressed as the sum of a Laplacian-like matrix L and a rank one matrix. In particular, for a symmetric hollow matrix M, under an assumption, we show that M is a Euclidean distance matrix if and only if L is positive semidefinite. Based on this, we obtain a new characterization for CEDMs involving their Moore–Penrose inverses. As an application, we show that the distance matrices of block graphs and odd-cycle-clique graphs are CEDMs. Finally, we establish an interlacing property between the eigenvalues of a Euclidean distance matrix M (including the singular case) and its associated Laplacian-like matrix L, which generalizes the interlacing property proved for the distance matrices of trees. [ABSTRACT FROM AUTHOR]

It is known that the discrete matrix Riccati equation has the order preserving property under some assumptions. In this paper we formulate and prove the converse statement for the case when the dimensions of the matrices are 2 x 2 and the order preserving property holds for all such symmetric matrices. [ABSTRACT FROM AUTHOR]

Matrix factorization has demonstrated outstanding performance in machine learning. Recently, graph-based matrix factorization has gained widespread attention. However, graph-based methods are only suitable for handling small amounts of data. This paper proposes a fast semi-supervised learning method using only matrix factorization, which considers both global and local information. By introducing bipartite graphs into symmetric matrix factorization, the technique can handle large datasets effectively. It is worth noting that by utilizing tag information, the proposed symmetric matrix factorization becomes convex and unconstrained, i.e., the non-convex problem min x (1 − x 2) 2 is transformed into a convex problem. This allows it to be optimized quickly using state-of-the-art unconstrained optimization algorithms. The computational complexity of the proposed method is O (n m d) , which is much lower than that of the original symmetric matrix factorization, which is O (n 2 d) , and even lower than that of other anchor-based methods, which is O (n m d + m 2 n + m 3) , where n represents the number of samples, d represents the number of features, and m ≪ n represents the number of anchors. The experimental results on multiple public datasets indicate that the proposed method achieves higher performance in less time. [ABSTRACT FROM AUTHOR]

In this article, we investigate a class of non-smooth semidefinite multiobjective programming problems with inequality and equality constraints (in short, NSMPP). We establish the convex separation theorem for the space of symmetric matrices. Employing the properties of the convexificators, we establish Fritz John (in short, FJ)-type necessary optimality conditions for NSMPP. Subsequently, we introduce a generalized version of Abadie constraint qualification (in short, NSMPP-ACQ) for the considered problem, NSMPP. Employing NSMPP-ACQ, we establish strong Karush-Kuhn-Tucker (in short, KKT)-type necessary optimality conditions for NSMPP. Moreover, we establish sufficient optimality conditions for NSMPP under generalized convexity assumptions. In addition to this, we introduce the generalized versions of various other constraint qualifications, namely Kuhn-Tucker constraint qualification (in short, NSMPP-KTCQ), Zangwill constraint qualification (in short, NSMPP-ZCQ), basic constraint qualification (in short, NSMPP-BCQ), and Mangasarian-Fromovitz constraint qualification (in short, NSMPP-MFCQ), for the considered problem NSMPP and derive the interrelationships among them. Several illustrative examples are furnished to demonstrate the significance of the established results. [ABSTRACT FROM AUTHOR]

Sensitivity of eigenvectors and eigenvalues of symmetric matrix estimates to the removal of a single observation have been well documented in the literature. However, a complicating factor can exist in that the rank of the eigenvalues may change due to the removal of an observation, and with that so too does the perceived importance of the corresponding eigenvector. We refer to this problem as "switching of eigenvalues". Since there is not enough information in the new eigenvalues, post observation removal, to indicate that this has happened, how do we know that this switching has occurred? In this article, we show that approximations to the eigenvalues can be used to help determine when switching may have occurred. We then discuss possible actions researchers can take based on this knowledge, for example making better choices when it comes to deciding how many principal components should be retained and adjustments to approximate influence diagnostics that perform poorly when switching has occurred. Our results are easily applied to any eigenvalue problem involving symmetric matrix estimators. We highlight our approach with application to two real data examples. [ABSTRACT FROM AUTHOR]

This paper establishes a symmetric matrix based on the coopetition relationships among global automobile enterprises, and uses Ucinet software to obtain the visualization chart of the coopetition network of global automobile enterprises. Then, the multiple moderating effect method and two-stage least squares method are adopt to reveal the interaction between structural hole and centrality on enterprise market performance, and how the ego-network characteristics affect this interaction. The results show that structural hole and centrality have a positive interaction on market performance. Compared with high ego-network stability, the positive interaction between structural hole and centrality on enterprise market performance is stronger under the condition of low ego-network stability. And compared with low ego-network density, the positive interaction between structural hole and centrality on market performance is stronger under the condition of high ego-network density. These new findings further strengthen the connection between the overall network and ego-network. [ABSTRACT FROM AUTHOR]

Linear relationships, expressing the electrochemical properties of molecules as functions of structure, give insight into the associated electrochemical process and are a tool for prediction. Many biological activities rely on water-based dissociation, making electrochemical properties a bridge between structure and activity. Motivated by a previous study, a replica is made here on a different dataset in order to validate/invalidate the previously reported results. There are several methods for obtaining structure-based descriptors. Some of the methods have been devised to account for molecular topology, some to account for molecular geometry, and others to account for both. Two methods are involved here to derive structure-based descriptors and further obtain structure–property relationships (FMPI and ChPE). In order to express structure descriptors, both FMPI and ChPE express first the topology of the molecule, using the heavy atoms identity matrix and the heavy atoms adjacency matrix, both square symmetric matrices in the belief that symmetry is one major factor of molecular stability. A set of 2,6-dimethyl-1,4-dihydropyridine derivatives with oxidation peak potentials and coulometrically determined number of electrons experimental data is subjected to the search for structure–activity relationships. Even if the 2,6-dimethyl-1,4-dihydropyridine is a symmetric compound (of Cs point group), their derivatives are generally not symmetric (9 out of 24 are asymmetric). The dataset is subjected to descriptive and inferential statistics in order to filter out the most relevant structure–activity relationship. The geometry is built using three levels of theory (one from molecular mechanics and two others from density functionals, of which one accounts for the interaction with water as solvent). One challenge of picking one out of two reported measured values is dealt with by calculating the likelihood associated with the two choices. Relevant structure–activity models are extracted and discussed. The use of in vivo (in water, SM8 model) models in geometry optimization (from MMFF94 and B3LYP and to M06 + Water SM8) results in a precision gain, but this is, in most of the cases, not statistically significant, and this can be considered a negative result. [ABSTRACT FROM AUTHOR]

This study presents a bi-objective optimization model for the Green Vehicle-Routing Problem in cold chain logistics, with a focus on symmetric distance matrices, aiming to minimize total costs, including carbon emissions, while maximizing customer satisfaction. To address this complex challenge, we developed a Stage-Specific Multi-Objective Five-Element Cycle Optimization algorithm (MOFECO-SS), which dynamically adjusts optimization strategies across different stages of the process, thereby enhancing overall efficiency. Extensive comparative analyses with existing algorithms demonstrate that MOFECO-SS consistently outperforms in solving the multi-objective optimization model, particularly in reducing total costs and carbon emissions while maintaining high levels of customer satisfaction. The symmetric nature of the distance matrix further aids in achieving balanced and optimized route planning. The results highlight that MOFECO-SS offers decision-makers flexible route planning options that balance cost efficiency with environmental sustainability, ultimately improving the effectiveness of cold chain logistics operations. [ABSTRACT FROM AUTHOR]

Summary: Pfaffians are less familiar functions on skew-symmetric matrices, and yet Donald Knuth claims they are more fundamental than determinants. We explore connections between determinants and Pfaffians. Along the way, the reader will encounter combinatorial proofs of identities for both matrix functions and play with permutations, matchings, Reidemeister moves, and game theory. [ABSTRACT FROM AUTHOR]

Accurate full‐parameter construction of the Jones matrix is of great significance for encoding multiple functions in a single metasurface. Supercells composed of several elements are used for this purpose through coherent superposition. But all the elements in the reported studies should be lossless with near‐unity transmission, which limits the material system and restricts the Jones matrix construction in a unitary manner. Here a non‐unitary matrix decomposition method is proposed to build the supercell using a pair of 0°‐oriented rods and a pair of 45°‐oriented rods for on‐demand design of arbitrary symmetric Jones matrices. This method is applicable for lossy metasurfaces and offers straightforward physical guidance for structure design without optimization. The accuracy of the on‐demand Jones matrix generation is demonstrated by several 3D‐printed alumina metalenses with 3‐, 4‐, and 6‐channel polarization‐multiplexed focusing. The study opens a new pathway for non‐silicon meta‐optics and holds great potential for dynamic or switchable applications. [ABSTRACT FROM AUTHOR]

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This paper introduces a novel two-sample test for a broad class of orthogonally invariant positive definite symmetric matrix distributions. Our test is the first of its kind, and we derive its asymptotic distribution. To estimate the test power, we use a warp-speed bootstrap method and consider the most common matrix distributions. We provide several real data examples, including the data for main cryptocurrencies and stock data of major US companies. The real data examples demonstrate the applicability of our test in the context closely related to algorithmic trading. The popularity of matrix distributions in many applications and the need for such a test in the literature are reconciled by our findings. [ABSTRACT FROM AUTHOR]

We study a variant of the Sherrington–Kirkpatrick (S–K) spin glass model with external field, where the random symmetric couplings matrix does not consist of i.i.d. entries but is instead orthogonally invariant in law. For sufficiently high temperature, we prove a replica-symmetric formula for the first-order limit of the model free energy. Our analysis is an adaptation of a conditional second-moment-method argument previously introduced by Bolthausen for studying the high-temperature regime of the S–K model, where one conditions on the iterates of an Approximate Message Passing (AMP) algorithm for solving the TAP equations for the model magnetization. We apply this method using a memory-free version of AMP that is tailored to the orthogonally invariant structure of the model couplings. [ABSTRACT FROM AUTHOR]

The elements of the bidiagonal decomposition (BD) of a totally positive (TP) collocation matrix can be expressed in terms of symmetric functions of the nodes. Making use of this result, and studying the relation between Wronskian and collocation matrices of a given TP basis of functions, we can express the entries of the BD of Wronskian matrices as the values of certain symmetric functions evaluated at a single node. Moreover, in the case of polynomial bases, we obtain compact formulae for the entries of the BD of their Wronskian matrices. Interesting examples illustrate the applications of the obtained formulae. [ABSTRACT FROM AUTHOR]

This article aims to present novel identities for elementary and complete symmetric polynomials and explore their applications, particularly to generalized Vandermonde and special tri-diagonal matrices. It also extends existing results on Jacobi polynomials P (α,β) n (x) and introduces an explicit formula based on the zeros of P (α,β) n−1 (x). Several illustrative examples are included. [ABSTRACT FROM AUTHOR]

This paper focuses on algorithms for the truncated Takagi factorization of complex symmetric matrices. The problem is formulated as a Riemannian optimization problem on a complex Stiefel manifold and then is converted into a real Riemannian optimization problem on the intersection of the real Stiefel manifold and the quasi-symplectic set. The steepest descent, the Riemannian nonmonotone conjugate gradient, Newton, and hybrid methods are used for solving the problem and they are compared in their performance for the optimization task. Numerical experiments are provided to illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a new kernel approximation is constructed based on the basic principle of kernel approximation of meshless symmetric smooth particle hydrodynamics (SSPH) method. The first derivative is calculated by constructing a symmetric matrix through Taylor series expansion. Replacing the derivative of Newton's method, the SSPH-Newton iterative method for solving nonlinear equations is proposed. The advantage of this method is that it does not need to evaluate the derivative and overcomes the shortcomings of Newton's method. The quadratic convergence of the new method is proved. Numerical and application examples show that the method has the same accuracy as Newton method, the efficiency index is higher than Newton method, and the calculation amount is lower than Newton method. [ABSTRACT FROM AUTHOR]

A numerical method for evolving the nonlinear Schrödinger equation on a coarse spatial grid is developed. This trains a neural network to generate the optimal stencil weights to discretize the second derivative of solutions to the nonlinear Schrödinger equation. The neural network is embedded in a symmetric matrix to control the scheme's eigenvalues, ensuring stability. The machine-learned method can outperform both its parent finite difference method and a Fourier spectral method. The trained scheme has the same asymptotic operation cost as its parent finite difference method after training. Unlike traditional methods, the performance depends on how close the initial data are to the training set. [ABSTRACT FROM AUTHOR]

The symmetric nonnegative inverse eigenvalue problem is to determine when a multiset of n real numbers is the spectrum of a symmetric nonnegative $ n \times n $ n × n matrix. The problem is solved only for $ n\leq 4 $ n ≤ 4. In this article, the authors examine the set of possible spectra for nonnegative symmetric $ 6\times 6 $ 6 × 6 matrices, dividing the set into points that are realizable, points that are unrealizable by known necessary conditions, and points that satisfy all known necessary conditions, but their realizability remains unknown. [ABSTRACT FROM AUTHOR]

Event extraction is a complex and challenging task in the field of information extraction. It aims to identify event types, triggers, and argument information from the text. In recent years, overlapping event extraction has attracted the attention of researchers because of its higher challenge and practicability, and some work has carried out in-depth research on overlapping event extraction and achieved remarkable results. But these works (1) ignore the role of ontology knowledge in event extraction; (2) use the same semantic encoding for multi-stage models, lacking consideration for the independent characteristics of extraction tasks such as event types, triggers, and arguments; and (3) face issues in the training process of multi-stage models, such as error cascading and slow convergence. To address the above issues, we propose an ontology-guided and scheduled-sampling approach for overlapping event extraction, termed as OGSS. First, we design a symmetric matrix for event ontology knowledge representation and integrate it into the semantic encoding process, infusing ontology knowledge into event extraction. Second, for extraction targets such as event types, triggers, and arguments, we process the semantic encoding according to the characteristics of each extraction target, obtaining semantic representations tailored for each subtask. Finally, we view multi-stage predictions as sequential outputs of a joint model, using a scheduled sampling strategy between subtasks to effectively mitigate the cascading propagation of errors during training and accelerate model convergence. We conduct extensive experiments on the FewFc event extraction benchmark dataset. The results show that OGSS achieves significant improvements in overlapping event extraction tasks compared to previous methods. [ABSTRACT FROM AUTHOR]

In this paper, we employ a generalization of the Boas–Bellman inequality for inner products, as developed by Mitrinović–Pečarić–Fink, to derive several upper bounds for the 2 p -th power with p ≥ 1 of the numerical radius of the off-diagonal operator matrix 0 A B * 0 for any bounded linear operators A and B on a complex Hilbert space H. While the general matrix is not symmetric, a special case arises when B = A * , where the matrix becomes symmetric. This symmetry plays a crucial role in the derivation of our bounds, illustrating the importance of symmetric structures in operator theory. [ABSTRACT FROM AUTHOR]

Let A be drawn uniformly at random from the set of all n\times n symmetric matrices with entries in \{-1,1\}. We show that \[ \mathbb {P}(\det (A) = 0) \leqslant e^{-cn}, \] where c>0 is an absolute constant, thereby resolving a long-standing conjecture. [ABSTRACT FROM AUTHOR]

We determine the distribution of the sandpile group (also known as the Jacobian) of the Erdős–Rényi random graph G(n,q) along with its canonical duality pairing as n tends to infinity, fully resolving a conjecture due to Clancy, Leake, and Payne [Exp. Math. 24 (2015), pp. 1–7] and generalizing the result by Wood [J. Amer. Math. Soc. 30 (2017), pp. 915–958] on the groups. In particular, we show that a finite abelian p-group G equipped with a perfect symmetric pairing \delta appears as the Sylow p-part of the sandpile group and its pairing with frequency inversely proportional to |G| |\operatorname {Aut}(G,\delta)|, where \operatorname {Aut}(G,\delta) is the set of automorphisms of G preserving the pairing \delta. While this distribution is related to the Cohen–Lenstra distribution, the two distributions are not the same on account of the additional algebraic data of the pairing. The proof utilizes the moment method: we first compute a complete set of moments for our random variable (the average number of epimorphisms from our random object to a fixed object in the category of interest) and then show the moments determine the distribution. To obtain the moments, we prove a universality result for the moments of cokernels of random symmetric integral matrices whose dual groups are equipped with symmetric pairings that is strong enough to handle both the dependence in the diagonal entries and the additional data of the pairing. We then apply results due to Sawin and Wood [ The moment problem for random objects in a category , preprint] to show that these moments determine a unique distribution. [ABSTRACT FROM AUTHOR]

This article focuses on model-based clustering of subjects based on the shared relationships of subject-specific networks and covariates in scenarios when there are differences in the relationship between networks and covariates for different groups of subjects. It is also of interest to identify the network nodes significantly associated with each covariate in each cluster of subjects. To address these methodological questions, we propose a novel nonparametric Bayesian mixture modeling framework with an undirected network response and scalar predictors. The symmetric matrix coefficients corresponding to the scalar predictors of interest in each mixture component involve low-rankness and group sparsity within the low-rank structure. While the low-rank structure in the network coefficients adds parsimony and computational efficiency, the group sparsity within the low-rank structure enables drawing inference on network nodes and cells significantly associated with each scalar predictor. Being a principled Bayesian mixture modeling framework, our approach allows model-based identification of the number of clusters, offers clustering uncertainty in terms of the co-clustering matrix and presents precise characterization of uncertainty in identifying network nodes significantly related to a predictor in each cluster. Empirical results in various simulation scenarios illustrate substantial inferential gains of the proposed framework in comparison with competitors. Analysis of a real brain connectome dataset using the proposed method provides interesting insights into the brain regions of interest (ROIs) significantly related to creative achievement in each cluster of subjects. shows the convergence rate for the posterior predictive density of the proposed model, additional simulation examples with model misspecification, full conditional distributions to run the Markov chain Monte Carlo (MCMC) algorithm and also presents traceplots for various model parameters to demonstrate convergence of the MCMC algorithm. [ABSTRACT FROM AUTHOR]

Solar photovoltaic (PV) system generates low power because of nonuniform sun irradiation compared to standard test conditions. Partial shading conditions (PSCs) can arise from various factors, with passing clouds, trees, telecommunication towers, bird droppings, and similar elements being among the most prevalent causes. Under the shading scenario, multiple global and local maxima power points exist on the P–V curves. Herein, the chess game rules for the "Rook" icon are followed to attain the optimal placement of integer numbers (1–9) in the 9 × 9 size game puzzle matrix. The integer number‐based game puzzles such as Su‐Do‐Ku, symmetric matrix, and proposed Rook's movement‐game theory (RMGT) to rearrange the PV modules in arrays are investigated and compared with the standard arrangements such as series‐parallel, total‐cross‐tied during PSCs. The suggested RMGT puzzle‐based arrangement is able to achieve higher power at global maximum power point, performance enhancement, fill factor, and low power loss. As a result, the experimental study proves the superiority of RMGT configuration compared to conventional methodologies under the shading scenarios. The proposed RMGT‐based PV array is tested and obtained the performance higher side as, 63.27ImVm$\left(63.27 I\right)_{m}V_{m}$, 64.17ImVm$\left(64.17 I\right)_{m}V_{m}$, 58.95ImVm$\left(58.95 I\right)_{m}V_{m}$, and 76.77ImVm$76.77I_{m}V_{m}$ compared to existing reconfiguration methodologies. [ABSTRACT FROM AUTHOR]

We propose a robust randomized indicator method for the reliable detection of eigenvalue existence within an interval for symmetric matrices A. An indicator tells the eigenvalue existence based on some statistical norm estimators for a spectral projector. Previous work on eigenvalue indicators relies on a threshold which is empirically chosen, thus often resulting in under or over detection. In this paper, we use rigorous statistical analysis to guide the design of a robust indicator. Multiple randomized estimators for a contour integral operator in terms of A are analyzed. In particular, when A has eigenvalues inside a given interval, we show that the failure probability (for the estimators to return very small estimates) is extremely low. This enables to design a robust rejection indicator based on the control of the failure probability. We also give a prototype framework to illustrate how the indicator method may be applied numerically for eigenvalue detection and may potentially serve as a new way to design randomized symmetric eigenvalue solvers. Unlike previous indicator methods that only detect eigenvalue existence, the framework also provides a way to find eigenvectors with little extra cost by reusing computations from indicator evaluations. Extensive numerical tests show the reliability of the eigenvalue detection in multiple aspects. [ABSTRACT FROM AUTHOR]

In the present paper, we define and study a new family of holomorphic functions which involve the Bazilevič functions and the λ -pseudo-starlike functions. We establish coefficient estimates for the first four determinants of the symmetric Toeplitz matrices T 2 (2) , T 2 (3) , T 3 (1) and T 3 (2) for the functions in this family. Further, we investigate several special cases and consequences of our results. [ABSTRACT FROM AUTHOR]

Motivated by applications in matrix constructions used in the inverse eigenvalue problem for graphs, we study a concept of genericity for eigenvectors associated with a given spectrum and a connected graph. This concept generalizes the established notion of a nowhere-zero eigenbasis. Given any simple graph G on n vertices and any spectrum with no multiple eigenvalues, we show that the family of eigenbases for symmetric matrices with this graph and spectrum is generic, strengthening a result of Monfared and Shader. We illustrate applications of this result by constructing new achievable ordered multiplicity lists for partial joins of graphs and providing several families of joins of graphs that are realizable by a matrix with only two distinct eigenvalues. [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]

We propose and illustrate a general framework in which spatial autocorrelation is measured by the Frobenius product of two kernels, a feature kernel and a spatial kernel. The resulting autocorrelation index δ$$ \delta $$ generalizes Moran's index in the weighted, multivariate setting, where regions, differing in importance, are characterized by multivariate features. Spatial kernels can traditionally be obtained from a matrix of spatial weights, or directly from geographical distances. In the former case, the Markov transition matrix defined by row‐normalized spatial weights must be made compatible with the regional weights, as well as reversible. Equivalently, space is specified by a symmetric exchange matrix containing the joint probabilities to select a pair of regions. Four original weight‐compatible constructions, based upon the binary adjacency matrix, are presented and analyzed. Weighted multidimensional scaling on kernels yields a low‐dimensional visualization of both the feature and the spatial configurations. The expected values of the first four moments of δ$$ \delta $$ under the null hypothesis of absence of spatial autocorrelation can be exactly computed under a new approach, invariant orthogonal integration, thus permitting to test the significance of δ$$ \delta $$ beyond the normal approximation, which only involves its expectation and expected variance. Various illustrations are provided, investigating the spatial autocorrelation of political and social features among French departments. [ABSTRACT FROM AUTHOR]

The idea of Kernel and k-Kernel Symmetric (k-KS) Intuitionistic Fuzzy Matrices (IFM) are introduced with an example. We present some basic results of kernel symmetric matrices. We show that k-symmetric implies k-Kernel symmetric but the converse need not be true. The equivalent relations between kernel symmetric, k-kernel symmetric and Moore-Penrose inverse of IFM are explained with numerical results. [ABSTRACT FROM AUTHOR]

We construct first examples of Fano varieties with torsion in their third cohomology group. The examples are constructed as double covers of linear sections of rank loci of symmetric matrices, and can be seen as higher-dimensional analogues of the Artin–Mumford threefold. As an application, we answer a question of Voisin on the coniveau and strong coniveau filtrations of rationally connected varieties. [ABSTRACT FROM AUTHOR]

Symmetric non-negative matrix factorization (NMF) can naturally capture the embedded clustering structure in the graph representation. It is an important method for linear and nonlinear data clustering applications. However, it is sensitive to the initialization of variables, and the quality of the initialization matrix greatly affects the clustering performance. In semi-supervised clustering, it faces the challenge of learning a more discriminative representation from limited labeled data. This paper introduces a constrained propagation self-adaptive self-supervised non-negative matrix factorization clustering algorithm (CPS³NMF) to solve the above problems. The algorithm propagates finite constraints to unconstrained data points, constructing a similarity matrix imbued with constraint information. The resultant similarity matrix serves the role of a non-negative symmetric matrix decomposition in SNMF and functions as graph regularization for the assignment matrix, fully utilizing the limited constraint information to preserve the geometrical structure of data space. Concurrently, leveraging the sensitivity of initial features in SNMF, the algorithm employs adaptively learned weights to rank the quality of multiple initial matrices. By integrating results from multiple clustering attempts, it progressively enhances the performance of semi-supervised clustering. Experiments on 6 public datasets show that the proposed CPS³NMF algorithm outperforms other state-of-the-art algorithms, proving its effectiveness in semi-supervised clustering. [ABSTRACT FROM AUTHOR]

A new interpretation and applications of the 'Diophantine' and factorization properties of finite orthogonal polynomials in the Askey scheme are explored. The corresponding twelve polynomials are the (q-)Racah, (dual, q-)Hahn, Krawtchouk and five types of q-Krawtchouk. These (q-)hypergeometric polynomials, defined only for the degrees of $ 0,1,\ldots,N $ 0 , 1 , ... , N , constitute the main part of the eigenvectors of N + 1-dimensional tri-diagonal real symmetric matrices, which correspond to the difference equations governing the polynomials. The monic versions of these polynomials all exhibit the 'Diophantine' and factorization properties at higher degrees than N. This simply means that these higher degree polynomials are zero-norm 'eigenvectors' of the N + 1-dimensional tri-diagonal real symmetric matrices. A new type of multi-indexed orthogonal polynomials belonging to these twelve polynomials could be introduced by using the higher degree polynomials as the seed solutions of the multiple Darboux transformations for the corresponding matrix eigenvalue problems. The shape-invariance properties of the simplest type of the multi-indexed polynomials are demonstrated. The explicit transformation formulas are presented. [ABSTRACT FROM AUTHOR]

Several types of linear and nonlinear singularly perturbed time-delay differential systems are considered. Asymptotic stability of the linear systems and asymptotic stability of the trivial solution of the nonlinear systems, valid for any sufficiently small value of the parameter of singular perturbation, are analyzed. For the stability analysis in the linear case, a partial exact slow–fast decomposition of the original system and an application of the Symmetric Matrix Riccati Equation method are proposed. Such an analysis yields parameter-free conditions, providing the asymptotic stability of the considered linear singularly perturbed time-delay differential systems for any sufficiently small value of the parameter of singular perturbation. Using the asymptotic stability results for the considered linear systems and the method of asymptotic stability in the first approximation, parameter-free conditions, guaranteeing the asymptotic stability of the trivial solution to the considered nonlinear systems for any sufficiently small value of the parameter of singular perturbation, are derived. Illustrative examples are presented. [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]

Understanding the eigenvalue distribution of sequence Toeplitz matrices has advanced significantly in recent years. Notable contributors include Bogoya, Grudsky, Böttcher, and Maximenko, who have derived precise asymptotic expansions for these eigenvalues under certain conditions related to the generating function as the matrix size increases. Building on this foundation, the Stefano Serra-Capizzano conjectured that, under certain assumptions about Ω and Φ, a similar expansion may hold for the eigenvalues of a sequence of preconditioned Toeplitz matrices T-1n (Φ)Tn(Ω), given a monotonic ratio r=Ω/Φ. In contrast to current eigenvalue solvers, this work presents a novel method for efficiently calculating the eigenvalues of a sequence of large preconditioned banded symmetric Toeplitz matrices (PBST). Our algorithm uses a higher-order spline fitting extrapolation technique to gather spectral data from a smaller sequence of PBST matrices in order to forecast the spectrum of bigger matrices. [ABSTRACT FROM AUTHOR]

A necessary and sufficient condition for sum of secondary range symmetric matrices to be secondary range symmetric is given here. Also, the concept of parallel summable secondary range symmetric matrices and some of its characterizations are being explored. [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]

We study the problem of approximating the eigenspectrum of a symmetric matrix A ∈ R n × n with bounded entries (i.e., ‖ A ‖ ∞ ≤ 1 ). We present a simple sublinear time algorithm that approximates all eigenvalues of A up to additive error ± ϵ n using those of a randomly sampled O ~ log 3 n ϵ 3 × O ~ log 3 n ϵ 3 principal submatrix. Our result can be viewed as a concentration bound on the complete eigenspectrum of a random submatrix, significantly extending known bounds on just the singular values (the magnitudes of the eigenvalues). We give improved error bounds of ± ϵ nnz (A) and ± ϵ ‖ A ‖ F when the rows of A can be sampled with probabilities proportional to their sparsities or their squared ℓ 2 norms respectively. Here nnz (A) is the number of non-zero entries in A and ‖ A ‖ F is its Frobenius norm. Even for the strictly easier problems of approximating the singular values or testing the existence of large negative eigenvalues (Bakshi, Chepurko, and Jayaram, FOCS '20), our results are the first that take advantage of non-uniform sampling to give improved error bounds. From a technical perspective, our results require several new eigenvalue concentration and perturbation bounds for matrices with bounded entries. Our non-uniform sampling bounds require a new algorithmic approach, which judiciously zeroes out entries of a randomly sampled submatrix to reduce variance, before computing the eigenvalues of that submatrix as estimates for those of A . We complement our theoretical results with numerical simulations, which demonstrate the effectiveness of our algorithms in practice. [ABSTRACT FROM AUTHOR]

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 contemplate addressing nonlinear problems involving complex symmetric Jacobian matrices. Firstly, we establish a parameter-free method called modified Newton–CAPRESB (MN–CAPRESB) method by harnessing the modified Newton method as the outer iteration and the CAPRESB (Chebyshev accelerated preconditioned square block) method as the inner iteration. Secondly, the local and semilocal convergence theorems of MN–CAPRESB method are proved under some conditions. Eventually, the numerical experiments of two kinds of complex nonlinear equations are presented to validate the feasibility of MN–CAPRESB method compared to other existing iteration methods. [ABSTRACT FROM AUTHOR]

Let J be a symmetric Jacobi matrix associated with some Toda lattice. We find conditions for Jacobi matrix J to admit factorization J = LU (or J = il£) with L (or £) and U ( or il) being lower and upper triangular two-diagonal matrices, respectively. In this case, the Darboux transformation of J is the symmetric Jacobi matrix J^' = UL (or = £il), which is associated with another Toda lattice. In addition, we found explicit transformation formulas for orthogonal polynomials, m-functions and Toda lattices associated with the Jacobi matrices and their Darboux transformations. [ABSTRACT FROM AUTHOR]