Matrix Form Research Articles

Modelling and monitoring multi-relational networks with ordinal information

Network relationships can be widely seen among entities in various fields such as social networks, supply networks and Internet of Things (IoT). Sometimes abnormal events such as cyber-attacks occur to cause an abrupt increase or decrease in the traffic of networks. Many anomaly detection methods have been developed to identify such abnormal events in networks. In recent years, statistical process control (SPC) has attracted more and more attention in network anomaly detection. However, many of the existing statistical models regard the interaction between two nodes in unweighted directed networks as a binary variable, i.e. presence and absence of contacts, which fails to reflect the intensity level of interactions. This article proposes a new model to describe the dyadic interactions with several ordinal levels and introduces special quantities to incorporate the ordinal information into the model. The model can be expressed in a matrix form to enable easy parameter estimation and derivation of a quadratic monitoring statistic. Numerous simulation studies show that the proposed methods detect anomalies in multi-relational networks more quickly than existing monitoring methods. A case study exhibits the implementation and superiority of the proposed method.

A randomly generated Majorana neutrino mass matrix using adaptive Monte Carlo method

A randomly generated complex symmetric matrix using the adaptive Monte Carlo method is taken as a general form of Majorana neutrino mass matrix, which is diagonalized by the use of eigenvectors. We extract all the neutrino oscillation parameters, i.e. two mass-squared differences ([Formula: see text] and [Formula: see text]), three mixing angles ([Formula: see text], [Formula: see text], [Formula: see text]) and three phases, i.e. one Dirac CP violating phase ([Formula: see text]) and two Majorana phases ([Formula: see text] and [Formula: see text]). The Charge-Parity (CP) violating phases are extracted from the mixing matrix constructed with the eigenvectors of the Hermitian matrix formed by the complex symmetric mass matrix. All the latest neutrino oscillation parameters within 3[Formula: see text] bound are only allowed in Normal Hierarchy (NH) consistent with the latest Planck cosmological upper bound, [Formula: see text][Formula: see text]eV. Three cases of one-zero texture i.e. [Formula: see text], [Formula: see text], [Formula: see text] and two cases of two-zero texture, i.e. [Formula: see text] and [Formula: see text] are allowed in normal hierarchy whereas none of zero texture is allowed in inverted hierarchy. We also study effective neutrino masses [Formula: see text] in tritium beta decay and [Formula: see text] in neutrinoless double beta decay and calculate numerically the allowed range of mass elements of Majorana mass matrix.

The PCovR biplot: a graphical tool for principal covariates regression

Biplots are useful tools because they provide a visual representation of both individuals and variables simultaneously, making it easier to explore relationships and patterns within multidimensional datasets. This paper extends their use to examine the relationship between a set of predictors X and a set of response variables Y using Principal Covariates Regression analysis (PCovR). The PCovR biplot provides a simultaneous graphical representation of individuals, predictor variables and response variables. It also provides the ability to examine the relationship between both types of variables in the form of the regression coefficient matrix.

Extended matrix modeling of integrated energy systems considering network dynamic characteristics and source–load uncertainty

A refined and efficient modeling approach is indispensable for enhancing the timeliness of scheduling strategies for the integrated energy system (IES). Non-flowing variables such as pressure and temperature introduced by energy devices, however, worsen the complexity while improving the completeness for models. Nonlinear and uncertain variables associated with network dynamic characteristics and source–load uncertainty in the systems seriously postpone the modeling processes. An extended matrix modeling approach is proposed in this paper to tackle such issues. New nodes, branches, definition rules and topological matrices are first presented to create an extended matrix model that is compatible with both flowing and non-flowing variables. To boost the modeling efficiency, adaptive piecewise linearization (APL) with extended matrix form and modified Taylor expansion are applied to approximate various nonlinear characteristics. The convex approximation based on Kullback–Leibler (KL) divergence and the strong duality theory characterized by Wasserstein distance are adopted to deal with uncertain variables, which adapt to combine with the above matrix model. A linearized matrix modeling framework applicable to highly automated by computers is thus established. The distributionally robust optimization (DRO) scheduling strategy is verified in a standard test system with the proposed modeling. The effect of dynamic characteristics and uncertain factors on the strategy is discussed. The results show that the extended matrix model simulates the response for real physical systems more effectively, which exhibits the better performances in modeling, objectives and scheduling strategies.

Verifiable privacy-preserving semantic retrieval scheme in the edge computing

Edge computing, with its characteristics of low latency and low transmission costs, addresses the storage and computation challenges arising from the surge in network edge traffic. It enables users to leverage nearby edge servers for data outsourcing and retrieval. However, data outsourcing poses risks to data privacy. Although searchable encryption is proposed to secure search of outsourced data, existing schemes generally cannot meet the requirements of semantic search, and they also exhibit security risks and incur high search costs. In addition, edge servers may engage in malicious activities such as data tampering or forgery. Therefore, we propose a verifiable privacy-preserving semantic retrieval scheme named VPSR suitable for edge computing environments. We utilize the Doc2Vec method to extract text feature vectors and then convert them into matrix form to reduce storage space requirements for indexes, queries, and keys. We encrypt matrices using an improved secure k-nearest neighbor (kNN) algorithm based on learning with errors (LWE) and calculate text similarity by solving the Hadamard product between matrices. Additionally, we design an aggregable signature scheme and offload part of the result verification tasks to edge servers. Security and performance analysis results demonstrate that the VPSR scheme is suitable for edge computing environments with high encryption and search efficiency and low storage cost while ensuring security.

A force-density framework for flexible multi-body dynamic analysis of clustered tensegrity structures

This paper develops a versatile and effective force-density framework for the flexible multi-body dynamic analysis of clustered tensegrity structures. In this framework, the force density is selected as the basic variable instead of force, and the clustered tensegrity structure is mathematically described in a vector and matrix form, encompassing topology, geometry, material, and force properties. A non-negative variable is defined as an indicator of the member stress state, and a complementary function is constructed to address the discontinuity issues that arise from the unidirectional axial stiffness of cables. Dynamic formulas are established within this force-density framework, with nodal coordinates selected as generalized parameters and formulations constructed in a matrix form. A complementary framework is established as an alternative for solving the dynamic equations, transforming the isolated steps of Newton’s iteration and cable state judgment (slack or tension) into a unified one, bringing more potential for improving solving efficiency. Numerical simulations are carried out to validate the approach, demonstrating that it effectively reveals the dynamic oscillation, tension changes, and cable slack behavior of clustered tensegrity structures during shape control. Comparative studies highlight the advantage of computational efficiency. The method proposed in this paper provides a robust mathematical model for studying clustered tensegrity structures, particularly regarding the shape control of deployable, active, and intelligent structures, aiding in understanding dynamic oscillation, tension changes, and cable slack behavior during their deformation. The methods can also be applied to cable net structures and other prestressed pin-jointed systems.

Rogue waves and nonzero background solutions for the Gross–Pitaevskii equation with a parabolic potential

In this paper an integrable Gross–Pitaevskii equation with a parabolic potential and a gain term is investigated. Its solutions with a nonzero background are derived. These solutions are constructed by using biliearization reduction approach and connections between the nonlinear Schrödinger equation and the Gross–Pitaevskii equation. The solutions are presented in double-Wronskian form and are classified in terms of canonical forms of a certain matrix. Various breathers and rogue waves are analyzed and illustrated.

Analysis of diode mixers using the method of node potentials in generalized matrix form in the frequency domain. Part 2. Isolation between ports, miscondition effect, noise level

A method for theoretical analysis of “input-output” and “local oscillator-output” decouplings of three types of diode frequency converters is presented: balanced, double balanced, triple balanced. For two operating modes of the local oscillator – “non-intensive” and “intensive” – the dependences of the “input-output” decoupling of the balanced mixer on the load conductivity and on the amplitude of the local oscillator voltage were obtained. Theoretical analysis and modeling were carried out. It is shown that the error between the calculated results and the simulation results does not exceed 3 dB. Expressions are obtained for errors introduced by the technological spread of diode parameters, which make it possible to estimate the maximum achievable values of the mixer characteristics (transmission coefficient and port isolation). A method for analyzing the noise properties of mixers is presented, the output noise spectra are calculated for each of the circuit elements (input resistance, diodes and output resistance), and analytical expressions for noise coefficients are obtained. Theoretical noise figure estimates are confirmed by simulation results with an accuracy of 1 dB.

Dynamic substructuring for visco-elastic materials using a POD-based model reduction method: application to laminated panels

Dynamic substructuring is now a well established and powerful tool for engineers used to analyze the dynamic response of mechanical systems. The technique is particularly favored for the simulation of mechanical vibrations allowing different aspects to be studied. It consists in splitting the global structure into several substructures from which reduced models are built. In Finite Element software, these reduced models, also called superelements, are generally built using the concept of Component Mode Synthesis. Difficulties arise when structures made of frequency-dependent materials are considered since classical modal reduction algorithms are inapplicable as they are because of the frequency-dependence of the stiffness matrix. Based on the Balanced Truncation approach, a novel methodology for the construction of a superelement for the dynamic analysis of elastic structures made with viscoelastic materials is presented. The methodology takes advantage of the Golla-Hughes-McTavish rheological model (GHM) before reducing the system via the Balanced Proper Orthogonal Decomposition (BPOD). The coupling of the superelement with a host structure is achieved via the use of Lagrange multipliers. The reduced system takes the form of a constant matrix of very small size compared to the original FE model. The method is applied to laminated panels such as an automobile windscreen.

On the Various Energy Forms of the Join of Complete Graphs

The concept of graph energy, first introduced in 1978, has been a focal point of extensive research within the field of graph theory, leading to the publication of numerous articles. Graph energy, originally associated with the eigenvalues of the adjacency matrix of a graph, has since been extended to various other matrices. These include the maximum degree matrix, Randić matrix, sum-connectivity matrix, and the first and second Zagreb matrices, among others. In this paper, we focus on calculating the energy of several such matrices for the join graph of complete graphs, denoted as J m ( K n ) . Specifically, we compute the energies for the maximum degree matrix, Randić matrix, sum-connectivity matrix, first Zagreb matrix, second Zagreb matrix, reverse first Zagreb matrix, and reverse second Zagreb matrix for J m ( K n ) . Our results provide new insights into the structural properties of the join graph and contribute to the broader understanding of the mathematical characteristics of graph energy for different matrix representations. This work extends the scope of graph energy research by considering these alternative matrix forms, offering a deeper exploration into the algebraic and spectral properties of graph energy in the context of join graphs.

Do Perineuronal Nets Stabilize the Engram of a Synaptic Circuit?

Perineuronal nets (PNNs), a specialized form of extra cellular matrix (ECM), surround numerous neurons in the CNS and allow synaptic connectivity through holes in its structure. We hypothesize that PNNs serve as gatekeepers that guard and protect synaptic territory and thus may stabilize an engram circuit. We present high-resolution and 3D EM images of PNN-engulfed neurons in mice brains, showing that synapses occupy the PNN holes and that invasion of other cellular components is rare. PNN constituents in mice brains are long-lived and can be eroded faster in an enriched environment, while synaptic proteins have a high turnover rate. Preventing PNN erosion by using pharmacological inhibition of PNN-modifying proteases or matrix metalloproteases 9 (MMP9) knockout mice allowed normal fear memory acquisition but diminished long-term memory stabilization, supporting the above hypothesis.

An AC Z-bus-based distribution factors for contingency analysis of AC–DC networks

The AC–DC networks are susceptible to deliberate and unforeseen contingencies, much like any other traditional power network. Due to the complex form of the unified impedance matrix derived for the provided network, the classic Z-bus-based contingency analysis approach cannot comprehend the AC–DC network. The distribution factors thus obtained from a unified complex Z-bus lead to an indication of the flow of complex current for DC lines. Therefore, an alternative method to discover post-contingency consequences is looked for. A streamlined approach for visualizing an equivalent AC network for the given AC–DC network has been proposed in this work. The AC-equivalent network takes the original AC–DC network and recreates it using two parallel AC networks to get the requisite distribution factors. The ultimate current/power distributions of the original AC–DC network can be determined by applying the superposition rule to the parallel networks, by the linear characteristics of the suggested technique. With this, the standard AC-contingency analysis method to complete the steady-state contingency analysis for the AC–DC network can be utilized. The proposed approach has been validated by the encouraging outcomes achieved for AC–DC networks with 10, 13, and 66 buses.

A uniformly convergent analysis for multiple scale parabolic singularly perturbed convection-diffusion coupled systems: Optimal accuracy with less computational time

This study addresses time-dependent multiple-scale reaction-convection-diffusion initial boundary value systems characterized by strong coupling in the reaction matrix and weak coupling in the convection terms for a locally optimal accurate solution. The discrete problem, which typically loses its tridiagonal structure, expands its bandwidth to four in such coupled systems, resulting in a substantial computational load. Our objective is to mitigate this computational burden through a splitting approach that transforms the non-tridiagonal matrix into a tridiagonal form while maintaining the consistency, local optimal accuracy in space, and stability of the numerical scheme. We employ equidistributed non-uniform grids, guided by a carefully chosen monitor function, to approximate the continuous space domain. The discretization strategy targets local optimal linear accuracy across space and time on the domain's interior points. In addition, we have also provided the global convergence analysis of the present splitting approach, mathematically. The mathematical evidence is also obtained from the numerical experiments by comparing the splitting approach (either diagonal or triangular forms) of the reaction matrix to its coupled form. The results strongly confirm the effectiveness of this approach in delivering uniform linear accuracy, based on the present problem discretizations while significantly reducing the computational costs.

Globally maximizing the ratio of two generalized quadratic matrix form functions over the Stiefel manifold

Globally maximizing the ratio of two generalized quadratic matrix form functions over the Stiefel manifold

Matrix-qubit algorithm for semantic analysis of probabilistic data

The paper presents a method for semantic data analysis by means of complex-valued matrix decomposition. The method is based on the quantum model of contextual decision-making, according to which observable probabilities are generated by qubit states, representing subjective meaning of the contexts relative to the basis decision. In the simplest three-context case, one of these qubits is decomposed to superposition of the remaining two, mathematically encoding semantic relations between the three contexts. For use in data analysis this model is translated to the matrix form, in which rows and columns correspond to the contexts and instances of experiment. The observable real-valued data then emerge from a complex-valued amplitude matrix, decomposed to a product of a real basis matrix and complex-valued matrix of superposition coefficients. This decomposition reveals stable process-semantic relations between the considered contexts, not captured by other methods of analysis. As a result, the data are approximated with higher precision and fewer parameters than singular and non-negative matrix decompositions, truncated to the same dimension. The model is experimentally approved in descriptive and prognostic regimes. The result opens prospects for development of nature-like computational architectures on novel logical grounds.

Analytical equation for camera imaging with refractive interfaces

Camera imaging through refractive interfaces is a crucial issue in photogrammetric measurements. Most past studies adopted numerical optimization algorithms based on refractive ray tracing procedures. In these studies, the camera and interface parameters are usually calculated iteratively with numerical optimization algorithms. Inappropriate initial values can cause iterations to diverge. Meanwhile, these iterations cannot efficiently reveal the accurate nature of refractive imaging. Therefore, obtaining camera calibration results that are both flexible and physically interpretable continues to be challenging. Consequently, in this study, we modeled refractive imaging by employing ray transfer matrix analysis. Subsequently, we deduced an analytical refractive imaging (ARI) equation that explicitly describes the refractive geometry in a matrix form. Although this equation is built upon the paraxial approximation, we executed a numerical experiment that shows that the developed analytical equation can accurately illustrate refractive imaging with a considerable object distance and a slightly tilted angle of the flat interface. This ARI equation can be used to define the expansion center and the normal vector of the flat interface. Finally, we also propose a flexible measurement method to determine the orientation of the flat interface, wherein the orientation can be measured rather than calculated by iterative procedures.

Short-time and long-time dynamics of the zeolite films in the light of alternative diffusion models

Analysis of the dynamics of diffusion of two mixtures of hydrogen and heavier components through a flat zeolite film was carried out. The concepts of short-time and long-time dynamics were introduced. They derive from the observation of temporal trajectories obtained by applying abrupt perturbations of concentrations on one side of the membrane.An alternative description of the diffusion dynamics was proposed compared to the commonly accepted model that employs a matrix of thermodynamic factors Γ. In this way, the difficulty of analytical determination of this matrix for more complex adsorption equilibria was eliminated.The model equations were expressed in a matrix form, which enables its easy extension to any number of components. The proposed research methodology was illustrated using an example of diffusion of two selected mixtures, i.e. {H2, n-C4H10} and {H2, CO2} in silicalite-1. These gaseous mixtures have applications in specific technological processes.The two models were used to perform quantitative benchmarking of short-time and long-time dynamics of flat zeolite membranes. The assumed boundary conditions correspond to a semi-batch Wicke-Kallenbach diffusion cell. It was demonstrated that the phenomena occurring in short-time and long-time diffusion represent two different dynamic features important from a scientific and practical point of view.A quantitative measure for degree of confinement was proposed based on dynamic adsorption-desorption hysteresis.

Irregular array optimization for beamforming with a polar coordinate-based partition coding approach

Abstract An innovative irregular array configuration optimization method for enhancing beamforming is introduced. This study presents partition coding to optimize sensor positioning and quantity of a non-uniform concentric circular array. This novel approach transcends traditional techniques by integrating structural partitioning and performance optimization to quantify the array’s geometry-performance correlation. Sensor candidate positions are mapped in polar coordinates, with each configuration translated into a sensor position matrix form. A significant innovation lies in the adaptation of the partition coding genetic algorithm to enhance the encoding of candidate positions and to refine crossover and mutation operations, underpinned by an elite retention strategy for selecting the optimal array configuration. Both simulation and experimental results substantiate the method’s effectiveness, achieving high-resolution acoustic mapping with commendably low computational complexity.

Improving data interpretability with new differential sample variance gene set tests.

Gene set analysis methods have played a major role in generating biological interpretations from omics data such as gene expression datasets. However, most methods focus on detecting homogenous pattern changes in mean expression and methods detecting pattern changes in variance remain poorly explored. While a few studies attempted to use gene-level variance analysis, such approach remains under-utilized. When comparing two phenotypes, gene sets with distinct changes in subgroups under one phenotype are overlooked by available methods although they reflect meaningful biological differences between two phenotypes. Multivariate sample-level variance analysis methods are needed to detect such pattern changes. We use ranking schemes based on minimum spanning tree to generalize the Cramer-Von Mises and Anderson-Darling univariate statistics into multivariate gene set analysis methods to detect differential sample variance or mean. We characterize these methods in addition to two methods developed earlier using simulation results with different parameters. We apply the developed methods to microarray gene expression dataset of prednisolone-resistant and prednisolone-sensitive children diagnosed with B-lineage acute lymphoblastic leukemia and bulk RNA-sequencing gene expression dataset of benign hyperplastic polyps and potentially malignant sessile serrated adenoma/polyps. One or both of the two compared phenotypes in each of these datasets have distinct molecular subtypes that contribute to heterogeneous differences. Our results show that methods designed to detect differential sample variance are able to detect specific hallmark signaling pathways associated with the two compared phenotypes as documented in available literature. The results in this study demonstrate the usefulness of methods designed to detect differential sample variance in providing biological interpretations when biologically relevant but heterogeneous changes between two phenotypes are prevalent in specific signaling pathways. Software implementation of the developed methods is available with detailed documentation from Bioconductor package GSAR. The available methods are applicable to gene expression datasets in a normalized matrix form and could be used with other omics datasets in a normalized matrix form with available collection of feature sets.

Local-time formula for dissipation in solid ionic electrolytes

When ions move through solids, they interact with the solid's constituent atoms and cause them to vibrate around their equilibrium points. This vibration, in turn, modifies the potential landscape through which the mobile ions travel. Because the present-time potential depends on past interactions, the coupling is inherently nonlocal in time, making its numerical and analytical treatment challenging. For sufficiently slow-moving ions, we linearize the phonon spectrum to show that these nonlocal effects can be ignored, giving rise to a draglike force. Unlike the more familiar drag coefficient in liquids, the drag takes on a matrix form due to the crystalline structure of the framework. We numerically simulate trajectories and dissipation rates using both the time-local and nonlocal formulas to validate our simplification. The time-local formula dramatically reduces the computational cost of calculating the motion of a mobile particle through a crystalline framework and clearly connects the properties of the material to the drag experienced by the particle. Published by the American Physical Society 2024