Symmetric Matrix Research Articles

On the Laplacian index of tadpole graphs

Abstract In this article, we study the Laplacian index of tadpole graphs, which are unicyclic graphs formed by adding an edge between a cycle C k {C}_{k} and a path P n {P}_{n} . Using two different approaches, we show that their Laplacian index converges to Δ 2 Δ − 1 = 9 2 \frac{{\Delta }^{2}}{\Delta -1}=\frac{9}{2} as n , k → ∞ n,k\to \infty , where Δ = 3 \Delta =3 is the maximum degree of the graph. This limit is known as the Hoffman’s limit for the Laplacian matrix. The first technique is a linear time algorithm presented in [R. Braga, V. Rodrigues, and R. Silva, Locating eigenvalues of a symmetric matrix whose graph is unicyclic, Trends in Comput. Appl. Math. 22 (2021), no. 4, 659–674] that diagonalizes the matrix preserving its inertia. Here, we adapt this algorithm to the Laplacian index of a tadpole graph. The second method is to apply a formula presented in [V. Trevisan and E. R. Oliveira, Applications of rational difference equations to spectra graph theory, J. Difference Equ. Appl. 27 (2021), 1024–1051] for solving rational difference equations that appear when applying the diagonalization algorithm in some cases.

Error bounds for the approximation of matrix functions with rational Krylov methods

AbstractWe obtain an expression for the error in the approximation of and with rational Krylov methods, where is a symmetric matrix, is a vector and the function admits an integral representation. The error expression is obtained by linking the matrix function error with the error in the approximate solution of shifted linear systems using the same rational Krylov subspace, and it can be exploited to derive both a priori and a posteriori error bounds. The error bounds are a generalization of the ones given in Chen et al. for the Lanczos method for matrix functions. A technique that we employ in the rational Krylov context can also be applied to refine the bounds for the Lanczos case.

Robust speckle covariance matrix estimation of sea clutter based on spectral symmetry

Speckle covariance matrix estimation plays an important role in adaptive target detection of maritime radars. Spatial heterogeneity of sea clutter incurs insufficient secondary data in estimation, which degrades detection performance. It is an effective way of estimation improvement to exploit the Doppler spectrum knowledge of sea clutter for structural dimension reduction of the speckle covariance matrix. This paper proposes a speckle covariance matrix estimation method based on the spectral symmetry knowledge of sea clutter, indicating that the spectrum of sea clutter is symmetric with respect to the Doppler offset. First of all, the spectral symmetry of sea clutter is analyzed by using the measured data from several opened databases and the results show that most of the data excluding sea spikes have symmetric Doppler spectra. Secondly, the spectral symmetry constrains the speckle covariance matrix into the Hadamard product of a special rank-one Toeplitz matrix and a real positive definite symmetric matrix. The degrees of freedom are reduced to almost half of that of a Hermitian matrix and thus fewer secondary data are required. Thirdly, a robust iterative maximum likelihood estimator is proposed to estimate the speckle covariance matrix with structural constraint and is free of the clutter texture distribution. The performance of the estimator is verified by the measured data and the results show that it is superior to traditional estimators, particularly in fewer secondary data.

A low-rank ODE for spectral clustering stability

Spectral clustering is a well-known technique which identifies k clusters in an undirected graph, with n vertices and weight matrix W∈Rn×n, by exploiting its graph Laplacian L(W). In particular, the k clusters can be identified by the knowledge of the eigenvectors associated with the k−1 smallest non zero eigenvalues of L(W), say λ2,…,λk (recall that λ1=0). Identifying k is an essential task of a clustering algorithm, since if λk+1 is close to λk the reliability of the method is reduced. The k-th spectral gap λk+1−λk is often considered as a stability indicator. This difference can be seen as an unstructured distance between L(W) and an arbitrary symmetric matrix L⋆ with vanishing k-th spectral gap. A more appropriate structured distance to ambiguity such that L⋆ represents the Laplacian of a graph has been proposed in Andreotti et al. (2021) [2]. This is defined as the minimal distance between L(W) and Laplacians of graphs with the same vertices and edges, but with weights that are perturbed such that the k-th spectral gap vanishes. In this article we consider a slightly different approach, still based on the reformulation of the problem into the minimization of a suitable functional in the eigenvalues. After determining the gradient system associated with this functional, we introduce a low-rank projected system, suggested by the underlying low-rank structure of the extremizers of the problem. The integration of this low-rank system, requires both a moderate computational effort and a memory requirement, as it is shown in some illustrative numerical examples.

A Matrix Approach to Vertex-Degree-Based Topological Indices

A VDB (vertex-degree-based) topological index over a set of digraphs H is a function φ:H→R, defined for each H∈H as φH=12∑uv∈Eφdu+dv−, where E is the arc set of H, du+ and dv− denote the out-degree and in-degree of vertices u and v respectively, and φij=f(i,j) for an appropriate real symmetric bivariate function f. It is our goal in this article to introduce a new approach where we base the concept of VDB topological index on the space of real matrices instead of the space of symmetric real functions of two variables. We represent a digraph H by the p×p matrix αH, where αHij is the number of arcs uv such that du+=i and dv−=j, and p is the maximum value of the in-degrees and out-degrees of H. By fixing a p×p matrix φ, a VDB topological index of H is defined as the trace of the matrix φTα(H). We show that this definition coincides with the previous one when φ is a symmetric matrix. This approach allows considering nonsymmetric matrices, which extends the concept of a VDB topological index to nonsymmetric bivariate functions.

Orthogonal-symplectic matrices and their parametric representation

The method of computing the parametric representation of an even orthogonal symplectic matrix is considered. The dimension of the family of such matrices is calculated. The general structure of matrices of small even dimensions up to 8 is discussed in detail. Theorem on the structure of a skew symmetric matrix generating a generic orthogonal symplectic matrix is proven. The problem of constructing an orthogonal symplectic matrix of dimension 4 by a given vector is solved. The application of this transformation to the study of families of periodic solutions to an autonomous Hamiltonian system with two degrees of freedom is discussed.

On the impact of spatial covariance matrix ordering on tile low‐rank estimation of Matérn parameters

AbstractSpatial statistical modeling involves processing an symmetric positive definite covariance matrix, where denotes the number of locations. However, when is large, processing this covariance matrix using traditional methods becomes prohibitive. Thus, coupling parallel processing with approximation can be an elegant solution by relying on parallel solvers that deal with the matrix as a set of small tiles instead of the full structure. The approximation can also be performed at the tile level for better compression and faster execution. The tile low‐rank (TLR) approximation has recently been used to compress the covariance matrix, which mainly relies on ordering the matrix elements, which can impact the compression quality and the efficiency of the underlying solvers. This work investigates the accuracy and performance of location‐based ordering algorithms. We highlight the pros and cons of each ordering algorithm and give practitioners hints on carefully choosing the ordering algorithm for TLR approximation. We assess the quality of the compression and the accuracy of the statistical parameter estimates of the Matérn covariance function using TLR approximation under various ordering algorithms and settings of correlations through simulations on irregular grids. Our conclusions are supported by an application to daily soil moisture data in the Mississippi Basin area.

Cost-effective one-time configuration for bridge-linked thermoelectric generator array using INFO optimizer

One of the key mechanisms of enhancing the power output of Thermoelectric Generators (TEG) under nonuniform temperature distribution (NUTD) is to restructure their arrays to disperse the NUTD profiles. Existing research works in this area primarily propose dynamic reconfiguration techniques of TEG arrays as a potential solution to maximize their output powers while taking no notice of the highly associated costs of their implementation, i.e., sensors, switches, and maintenance, in addition to the complexity of operation and limited service lifetime of switches. To alleviate these drawbacks, this paper introduces an optimization model for a one-time configuration (OTC) of TEG arrays to maximize their output powers while accommodating multiple NUTD profiles simultaneously. Using the INFO optimizer, the introduced optimization model yields the OTC layout for both symmetrical and asymmetrical TEG arrays across various NUTD scenarios. Given its superior performance, the optimization model is applied to the bridge-linked (BL) TEG array and compared to other configurations such as series (S), series-parallel (SP), and standard BL. The results of the proposed OTC model are compared with (1) standard, non-optimized SP and BL configurations and (2) different dynamic-based reconfiguration techniques. Further, the performance of the proposed model is tested and validated experimentally in the laboratory for the available symmetrical hardware setup, 5 × 5, TEG array under varying NUTD profiles. Compared to dynamic reconfigurations, the results show that the proposed OTC technique provides a partially optimal and cost-effective solution to enhance the output power of TEG arrays under different NUTD. Specifically, for 9×9 and 10×15 TEG arrays, the OTC saves approximately 1,498.5 ($) and 2750 ($) compared to dynamic reconfigurations, respectively. Moreover, BL-based OTC enhances the TEG array harvested power by maximum values of 1.7532%, 2.43%, and 3.156% compared with the standard S, SP, and BL configurations, respectively, based on the hardware setup. Throughout the detailed comparison between the proposed OTC and dynamic reconfiguration techniques, this paper contributes to a deeper understanding of the trade-offs between these configuration techniques and addresses a critical gap in the current research landscape of enhancing the output powers of TEG arrays.

Statistical physics of principal minors: Cavity approach.

Determinants are useful to represent the state of an interacting system of (effectively) repulsive and independent elements, like fermions in a quantum system and training samples in a learning problem. A computationally challenging problem is to compute the sum of powers of principal minors of a matrix which is relevant to the study of critical behaviors in quantum fermionic systems and finding a subset of maximally informative training data for a learning algorithm. Specifically, principal minors of positive square matrices can be considered as statistical weights of a random point process on the set of the matrix indices. The probability of each subset of the indices is in general proportional to a positive power of the determinant of the associated submatrix. We use Gaussian representation of the determinants for symmetric and positive matrices to estimate the partition function (or free energy) and the entropy of principal minors within the Bethe approximation. The results are expected to be asymptotically exact for diagonally dominant matrices with locally treelike structures. We consider the Laplacian matrix of random regular graphs of degree K=2,3,4 and exactly characterize the structure of the relevant minors in a mean-field model of such matrices. No (finite-temperature) phase transition is observed in this class of diagonally dominant matrices by increasing the positive power of the principal minors, which here plays the role of an inverse temperature.

Analysis of stochastic probing methods for estimating the trace of functions of sparse symmetric matrices

We consider the problem of estimating the trace of a matrix function f ( A ) f(A) . In certain situations, in particular if f ( A ) f(A) cannot be well approximated by a low-rank matrix, combining probing methods based on graph colorings with stochastic trace estimation techniques can yield accurate approximations at moderate cost. So far, such methods have not been thoroughly analyzed, though, but were rather used as efficient heuristics by practitioners. In this manuscript, we perform a detailed analysis of stochastic probing methods and, in particular, expose conditions under which the expected approximation error in the stochastic probing method scales more favorably with the dimension of the matrix than the error in non-stochastic probing. Extending results from Aune, Simpson, and Eidsvik [Stat. Comput. 24 (2014), pp. 247–263], we also characterize situations in which using just one stochastic vector is always—not only in expectation—better than the deterministic probing method. Several numerical experiments illustrate our theory and compare with existing methods.

Chess Game Rook‐Icon Movement‐Based Photovoltaic Array Reconfiguration for Higher Shade Dispersion and Improved Global Power Peaks Under Partial Shading Conditions: An Experimental Study

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, , , , and compared to existing reconfiguration methodologies.

Differentiable microstructures design via anisotropic thermal diffusion

We propose a novel method to design differentiable microstructures. Central to our algorithm is a new representation of the mapping from the parameters to microstructures, formulated as the anisotropic thermal diffusion. A metric field governs the anisotropic diffusion. The metric associated with each point is represented as a 2 × 2 symmetric positive definite matrix that becomes the design variable. To alleviate the difficulties caused by symmetric positive definite constraints, we perform the singular value decomposition of the metric matrix so that the design variable includes a rotation angle and a diagonal matrix. Then, the positive definiteness is converted to requiring the two diagonal entries of the diagonal matrix to be positive, which is easier to deal with. The effectiveness of our algorithm is demonstrated through evaluations and comparisons over various examples.

On the longest chain of faces of the completely positive and copositive cones

We consider a wide class of closed convex cones K in the space of real n×n symmetric matrices and establish the existence of a chain of faces of K, the length of which is maximized at n(n+1)2+1. Examples of such cones include, but are not limited to, the completely positive and the copositive cones. Using this chain, we prove that the distance to polyhedrality of any closed convex cone K that is sandwiched between the completely positive cone and the doubly nonnegative cone of order n≥2, as well as its dual, is at least n(n+1)2−2, which is also the worst-case scenario.

Analytical quality by design-based development of a capillary electrophoresis method for Omeprazole impurity profiling

Omeprazole (OME) is a proton pump inhibitor used to treat gastroesophageal reflux disease associated conditions. The current study presents an Analytical Quality by Design-based approach for the development of a CE method for OME impurity profiling. The scouting experiments suggested the selection of solvent modified Micellar ElectroKinetic Chromatography operative mode using a pseudostationary phase composed of sodium dodecyl sulfate (SDS) micelles and n-butanol as organic modifier in borate buffer. A symmetric three-level screening matrix 37//16 was used to evaluate the effect of Critical Method Parameters, including Background Electrolyte composition and instrumental settings, on Critical Method Attributes (critical resolution values, OME peak width and analysis time). The analytical procedure was optimized using Response Surface Methodology through a Central Composite Orthogonal Design. Risk of failure maps made it possible to define the Method Operable Design Region, within which the following optimized conditions were selected: 72 mM borate buffer pH 10.0, 96 mM SDS, 1.45 %v/v n-butanol, capillary temperature 21 °C, applied voltage 25 kV. The method was validated according to ICH guidelines and robustness was evaluated using a Plackett-Burman design. The developed procedure enables the simultaneous determination of OME and seven related impurities, and has been successfully applied to the analysis of pharmaceutical formulations.

Does the quasi-static polarizability have principal axes?

Beyond strict statics, the dipole polarizability tensor is a complex symmetric matrix. Such matrices may not be diagonalizable by an orthogonal similarity transformation (a rigid rotation of the reference frame). In this paper, we provide examples of polarizability tensors that have no real principal axes and discuss the conditions under which this counter-intuitive phenomenon can occur.

Optimizing Resource Allocation in M/M/1/N Queues with Feedback, Discouraged Arrivals, and Reneging for Enhanced Service Delivery

This article presents a novel computational approach for analyzing M/M/1/N queues with feedback, discouraged arrivals, and reneging, under the first-come, first-served (FCFS) discipline. We calculate explicit transient state probabilities and represent results using symmetric tridiagonal matrix eigenvalues. Through numerical simulations, we validate our method, providing practical insights for optimizing resource allocation. Our study contributes to both theory and application, advancing queueing theory and aiding decision-makers in improving service quality and resource management.

Topology of surface electromyogram signals: hand gesture decoding on Riemannian manifolds

Objective. Decoding gestures from the upper limb using noninvasive surface electromyogram (sEMG) signals is of keen interest for the rehabilitation of amputees, artificial supernumerary limb augmentation, gestural control of computers, and virtual/augmented realities. We show that sEMG signals recorded across an array of sensor electrodes in multiple spatial locations around the forearm evince a rich geometric pattern of global motor unit (MU) activity that can be leveraged to distinguish different hand gestures. Approach. We demonstrate a simple technique to analyze spatial patterns of muscle MU activity within a temporal window and show that distinct gestures can be classified in both supervised and unsupervised manners. Specifically, we construct symmetric positive definite covariance matrices to represent the spatial distribution of MU activity in a time window of interest, calculated as pairwise covariance of electrical signals measured across different electrodes. Main results. This allows us to understand and manipulate multivariate sEMG timeseries on a more natural subspace—the Riemannian manifold. Furthermore, it directly addresses signal variability across individuals and sessions, which remains a major challenge in the field. sEMG signals measured at a single electrode lack contextual information such as how various anatomical and physiological factors influence the signals and how their combined effect alters the evident interaction among neighboring muscles. Significance. As we show here, analyzing spatial patterns using covariance matrices on Riemannian manifolds allows us to robustly model complex interactions across spatially distributed MUs and provides a flexible and transparent framework to quantify differences in sEMG signals across individuals. The proposed method is novel in the study of sEMG signals and its performance exceeds the current benchmarks while being computationally efficient.

A reliable GTR-PLC approach for power enhancement and online monitoring of solar PV arrays during partial shading

The PV system encounters severe power loss due to the existence of unavoidable partial shading that is mainly overcome by reconfiguration techniques. Besides the advantage of power enhancement, reconfiguration techniques deal with drawbacks such as limited to symmetrical arrays, higher switches and sensors requirements, lower convergence to optimal solution, and improper monitoring. In this paper, a gorilla troop reconfiguration-power line communication (GTR-PLC) approach is proposed for performance enhancement and monitoring of PV arrays during partial shading. The proposed approach uses a considerably lower switch count along with a low-cost simple architecture for loss reduction and smooth operation of the PV array during partial shading. The proposed approach is validated for 5 × 5 and 9 × 9 arrays under time-domain dynamic shading scenarios in simulation whereas real-time hardware setup-based analysis is conducted for 2 × 3 and 12 × 4 arrays using numerous artificial shading cases. The performance of the proposed system is compared with twelve existing static and dynamic techniques and found to have zero misleading power loss with an average power increment of 38.37 %. Also, a power extraction efficiency of 98.99 % and an average calculation time of 0.09 s in the proposed system during partial shading is recorded.

Lp maximal regularity for vector-valued Schrödinger operators

In this paper we consider the vector-valued Schrödinger operator −Δ+V, where the potential term V is a matrix-valued function whose entries belong to Lloc1(Rd) and, for every x∈Rd, V(x) is a symmetric and nonnegative definite matrix, with non positive off-diagonal terms and with eigenvalues comparable each other. For this class of potential terms we obtain maximal inequality in L1(Rd,Rm). Assuming further that the minimal eigenvalue of V belongs to some reverse Hölder class of order q∈(1,∞)∪{∞}, we obtain maximal inequality in Lp(Rd,Rm), for p in between 1 and some q, and generation results.

An efficient image encryption model based on 6D hyperchaotic system and symmetric matrix for color and gray images

The security of images is one of the predominant pivotal aspects in the mammoth and still expanding digital domain. Due to chaotic system properties i.e. randomness and unpredictability is very appropriate to encrypt the images. In this research article, we construct an encryption model via 6D hyperchaotic map and a symmetric matrix for both color and grayscale images. We utilize the 6D hyperchaotic map in the confusion stage to change the pixel location and the symmetric matrix is used for changing the pixel value in the diffusion step for each RGB channel extraction from plain or original image. The image encryption model is checked over differential attacks (NPCR and UACI). Histogram analysis, correlation coefficients, and entropy analysis are also performed as statistical attacks. In conclusion, the image pixels are uniformly distributed, and the average entropy value are 7.9992 and 7.9973 for color and grayscale images, subsequently. The average NPCR and UACI for color images are 99.5956 and 33.4061, correspondingly, while the values for grayscale images are 99.5934 and 33.3054, respectively. These values are in the vicinity of optimal ranges. The suggested scheme's great efficiency and the proposed algorithm's resilience to a wide range of cryptanalytic attacks are implied by experimental results, statistical analysis, and differential attacks.