Form Of Matrices Research Articles

Symplectic eigenvalues of positive-semidefinite matrices and the trace minimization theorem

Symplectic eigenvalues are conventionally defined for symmetric positive-definite matrices via Williamson's diagonal form. Many properties of standard eigenvalues, including the trace minimization theorem, have been extended to the case of symplectic eigenvalues. In this note, we will generalize Williamson's diagonal form for symmetric positive-definite matrices to the case of symmetric positive-semidefinite matrices, which allows us to define symplectic eigenvalues, and prove the trace minimization theorem in the new setting.

On the recursive equivalence to Smith form of multivariate polynomial matrices

Abstract The Smith form of an $n$-$D$ polynomial matrix plays an important role in many areas of mathematics and engineering. In this paper, we investigate the recursive equivalence problem of $n$-dimensional polynomial matrices, i.e. if diag$(1,B)$ is equivalent to diag$(1,1,C)$, is B equivalent to diag$(1,C)$? We give a negative answer to this question by explicitly constructing a four-dimensional polynomial matrix which is not equivalent to its Smith form.

Ambiguous Strategy Choice in an Economic Game D(2,2) Describing a Competition

This study examines in detail the game known as the “two-bar dilemma” in the general form of the payoff matrices of the two player direct competitors. This dilemma is exclusively in the field of economics and competition. Its familiarity is therefore useful for readers who are concerned with decision making in economics in a competitive environment. All conclusions are the result of mathematical proofs and not of specific numbers specifically selected for this purpose and therefore the conclusions are generalized over all possible variants of payment matrices. The game has been shown to be much more complex and rich in possibilities than the widely known “prisoner’s dilemma”, to which the “two-bar dilemma” game was wrongly considered a complete analogue. It is explained why these two games cannot be analogous and why the game “dilemma of the two bars” is a more complex case. It has been shown that it is possible to have two Nash equilibria, which is not possible in the Prisoner’s Dilemma.

A probabilistic linguistic opinion dynamics method based on the DeGroot model for emergency decision-making in response to COVID-19

Emergency decision-making entails a multi-criteria problem with a short period and urgent events, which creates difficulties for decision makers to undertake an optimal decision. To ensure the validity and rationality of decision results, the probabilistic linguistic term set is adopted to represent the evaluation information of experts because it can assign different probabilities or importance to different linguistic terms, which is closely related to human cognition. In addition, to portray the dynamic changes in the emergency decision-making process, this study develops a new dynamics method based on the DeGroot model with probabilistic linguistic information. First, to simulate the transition matrix of probabilistic linguistic opinions, the basic operational rules are defined based on the transformation function and expectation function. Next, three forms of influence matrices incorporating similarity, self-persistence, and authority degrees are constructed, and the consensus conditions of the models are discussed. Then, considering the social networks and incomplete trust relationships between experts, a fourth trust-based influence matrix is devised. A case study of emergency decision-making for assessing response plans to COVID-19 is performed to verify the feasibility and effectiveness of the dynamic method. Furthermore, a sensitivity analysis is conducted. Finally, comparisons with classical methods are performed to illustrate the superiorities of the proposed algorithms.

Interface dislocations and grain boundary disconnections using Smith normal bicrystallography

The CSL/DSCL model for interfaces in crystalline materials offers a unified framework to study interface dislocations in phase boundaries and disconnections in grain boundaries. The model relies on the existence of a coincidence relation between the two lattices that meet at an interface. The model’s ability to quantitatively predict the thermodynamics and kinetics of interfaces has been demonstrated for a limited set of symmetric tilt grain boundaries (STGBs) in cubic materials and twin boundaries. However, the lack of a general framework of interface defects prevents its applicability to arbitrary rational boundaries. In this paper, we present a mathematical framework based on the Smith normal form (SNF) for integer matrices to study the bicrystallography of rational phase and grain boundaries. One of the main results of the paper is a constructive proof of the invariance of the coincident site lattice (CSL) under discrete relative displacements of the parent lattices (of possibly different kind) by a displacement shift complete lattice (DSCL) vector. In addition, we obtain necessary and sufficient conditions on two lattices, related by not only rotations but also lattice distortions, for the existence of a coincidence relation. We first apply these results to explore coincidence relations in arbitrary phase boundaries, and study interface dislocations. In particular, we demonstrate the framework for a phase boundary formed by a strained hexagonal lattice and a square lattice. As a second application, we show how to enumerate all possible (geometric) disconnection modes in arbitrary rational grain boundaries, including glide and non-glide modes in both STGBs and asymmetric-tilt grain boundaries (ATGBs). The constructive nature of the framework lends itself to an algorithmic implementation based exclusively on integer matrix algebra to construct all interfaces that admit CSLs up to a prescribed size, and determine disconnection modes in grain boundaries. We demonstrate the use of SNF bicrystallography on selected bicrystal misorientation axes/angles in face-centered cubic (fcc), body-centered cubic (bcc), and hexagonal (hex) lattices.

Neurotoxic effects of environmental contaminants—measurements, mechanistic insight, and environmental relevance

Pollution is a significant and growing concern for any population regardless of age because these environmental contaminants exhibit different neurodegenerative effects on persons of different ages. These environmental contaminants are the products of human welfare projects like industry, automobile exhaust, clinical and research laboratory extrudes, and agricultural chemicals. These contaminants are found in various forms in environmental matrices like nanoparticles, particulate matter, lipophilic vaporized toxicants, and ultrafine particulate matter. Because of their small size, they can easily cross blood-brain barriers or use different cellular mechanisms for assistance. Other than this, these contaminants cause an innate immune response in different cells of the central nervous system and cause neurotoxicity. Considering the above critiques and current needs, this review summarizes different protective strategies based on bioactive compounds present in plants. Various bioactive compounds from medicinal plants with neuroprotective capacities are discussed with relevant examples. Many in vitro studies on clinical trials have shown promising outcomes using plant-based bioactive compounds against neurological disorders.

MODEL IMPLEMENTASI LOGIKA FUZZY UNTUK PENILAIAN KINERJA KARYAWAN IT SUPPORT

Human resources are the most valuable asset in a company. Employee performance is one of the company's benchmarks in assessing the productivity of these assets. Methods in measuring employee performance can be done in various forms, one of which is using the fuzzy logic method. Fuzzy Mamdani is a fuzzy inference method that provides rules that must be passed before carrying out the process of determining the final value. There are three main criteria that will be analyzed in this study, namely attendance, quality of work and communication. Each of these criteria has a value obtained from the average value of several elements. The final result after going through the employee data analysis process is the average weighted value. Conclusions that can be drawn as the basis for employee performance and the results can be seen from the order of values, starting from the highest to the lowest. MATLAB or what we call it (Matrix Laboratory) is a program for analyzing and computing numerical data, and MATLAB is also an advanced mathematical programming language, which was formed on the premise of using the properties and forms of matrices. Graphical User Interface (GUI) is a visual media that makes users give certain commands on the computer without typing the command, but using the available images.

Concurrent product-process-supply chain strategy formulation for small medium enterprises

Small-medium enterprises (SMEs) have the potentials to translate ideas into innovative products in a quick and efficient manner. Therefore, they should be supported to transform into well-established companies. However, SMEs often lack knowledge regarding operation and management. This study proposes three-dimensional concurrent engineering (3DCE) approach to help SMEs accelerate their growth through differentiation and efficiency, leading to high-quality yet low-cost products with efficient delivery. This study aimed to develop a strategic formulation framework incorporating the 3DCE concept in product functionality, concurrent process design, and appropriate supply chain solutions. The framework has been developed and implemented in food and beverage SME. Strategy development follows the traditional approach, including environmental analysis, strategy formulation, and strategy mapping. However, the result of environmental analysis extracted in the form of SWOT and Grand Strategy matrices are specifically piled into concurrent strategic theme covering the concurrent product, process, and supply chain aspects. Predominantly, product features covering the product innovation and functionality and process features covering segmentation-based and cross-functional process designs were tailored with innovative supply chain solutions based on the segmented model. The detailed framework and its implementation are highlighted accordingly.

Algorithms and Tools for Securing and Protecting Academic Data in the Democratic Republic of Congo

This paper deals with the implementation of Algorithms and tools for the security of academic data protection in the Democratic Republic of Congo. It consists principally in implementing two algorithms and two distinct tools to secure data and in this particular case, academic data of higher and university education in the Democratic Republic of Congo. The design of algorithms meets the approach that any researcher in data encryption must use during the development of a computer system. Briefly, these algorithms are steps to follow to encrypt information in any programming language. These algorithms are based on symmetric and asymmetric encryptions, the first one uses Christopher Hill's algorithm, which uses texts in the form of matrices before they are encrypted and RSA as one of the asymmetric algorithms, which uses the prime numbers that we have encoded on more than 512 bits. As for tools, we have developed them in php which is only a programming language taken as an example because it is impossible to use all of them. The tools implemented are based on the algorithms of Caesar, Christopher Hill and RSA showing how the encryption operations are carried out thanks to graphical interfaces. They are only tools for pedagogical reasons to help students and other researchers learn how to use developed algorithms. We have not developed them for pleasure but rather used them in any information system, which would prevent and limit unauthorized access to computer systems. They will not be used only for the management of academic fees but for any other information system, which explains and shows the complexity of the tools developed. We have not been able to solve the problems of versions for the developed prototype, because if there is a new version later, some functions may be obsolete, which would constitute the limitation of these tools. This work targets primarily the Ministry of Higher Education and Universities, which will make these results their own and implement them in order to solve the problem of intrusions, unauthorized access to developers and researchers who will use tools already made instead of thinking about their development. We are trying to demonstrate the steps and the methodology that allowed us to reach our results, in the following lines.

МАТЕМАТИЧЕСКОЕ ОПИСАНИЕ ПРОЦЕССА МЕХАНОАКТИВАЦИИ MoS2 и МУНТ ДЛЯ МОТОРНОГО ТОПЛИВА И МАСЛА

It is possible to improve the properties of motor fuel and oil by using a new generation of additives that can improve a number of functional indicators. Increasing the efficiency of additives can be achieved through both the use of combinations of different types of additives that provide different properties, and the use of new effects realized during the transition to a nanoscale form. Thus, one of the nanoscale materials that can be used in combination with molybdenum disulfide (MoS2) are multi-walled carbon nanotubes (MWCNTs). The introduction of MoS2 and MWCNTs into motor fuel and oil is advisable to organize using a combined method of dispersion and mechanical activation, which can be implemented in devices with a vortex electromagnetic field. Such devices allow mechanical action by moving the grinding bodies in an alternating electromagnetic field. This breaks up agglomerates into MWCNTs and reduces MoS2, which increases the effectiveness of these additives and reduces their concentration in motor fuel and oil. The paper presents a mathematical description of the process of mechanical activation. The grinding of a dispersed material is presented in the form of matrices and the equations of motion of a ferromagnetic particle in a vortex layer are given. The principle of operation of the vortex layer apparatus (VLA) is described and the electromagnetic processes in the stator windings of the VLA are considered. An analysis of the VLA thermal control device with an implicitly pole inductor was carried out. The developed mathematical description of the process of mechanical activation of motor fuel and oil makes it possible to take into account a number of factors that affect the operating modes of the installation for mechanical activation.

The number of standard forms of matrices over imaginary Euclidean quadratic rings with respect to the $(z,k)$–equivalence

The $(z,k)$--equivalence of matrices over imaginary Euclideanquadratic rings is investigated. The classes of matrices overthese rings are selected for which the standard form with respectto $(z,k)$--equivalence is uniquely defined and equal to the Smithnormal form. It is established that the number of standard formsover imaginary Euclidean quadratic rings is finite. Bounds for anumber of standard forms are established.

Enforcing Stationarity through the Prior in Vector Autoregressions

Stationarity is a very common assumption in time series analysis. A vector autoregressive process is stable if and only if the roots of its characteristic equation lie outside the unit circle, constraining the autoregressive coefficient matrices to lie in the stationary region. However, the stationary region has a highly complex geometry which impedes specification of a prior distribution. In this work, an unconstrained reparameterization of a stationary vector autoregression is presented. The new parameters are partial autocorrelation matrices, which are interpretable, and can be transformed bijectively to the space of unconstrained square matrices through a simple mapping of their singular values. This transformation preserves various structural forms of the partial autocorrelation matrices and readily facilitates specification of a prior. Properties of this prior are described along with an important special case which is exchangeable with respect to the order of the elements in the observation vector. Posterior inference and computation are described and implemented using Hamiltonian Monte Carlo via Stan. The prior and inferential procedures are illustrated with an application to a macroeconomic time series which highlights the benefits of enforcing stationarity and encouraging shrinkage toward a sensible parametric structure. Supplementary materials for this article are available online.

The Impact of High-Speed Rails on Urban Consumption—From the Perspective of “Local-Adjacent” Effect

Under the background of the rapid development of high-speed rail (HSR), the urban spatial distance has been significantly compressed. Exploring the impact of high-speed railways on urban consumption from a spatial perspective is conducive to further understanding and clarifying the spatial economic effects caused by the opening of high-speed railways and providing a scientific basis for the future spatial layout of high-speed railways. Based on the panel data of prefecture-level cities from 2008 to 2018, the spatial spillover effect of high-speed railways on urban consumption is explored and decomposed by building a spatial econometric model. The results show that urban consumption shows significant spatial correlation, and the conclusion is still valid under three different forms of spatial weight matrices. In addition, the high-speed railway network has a positive effect on local urban consumption and acts on cities in neighboring areas through a spatial spillover effect. Further analysis shows that the spillover effect has an effective radius of 100–500 km, which is inverted U-shaped with distance, and 300 km from the central city is the optimal radius.

Generation of Accessible Sets in the Dynamical Modeling of Quantum Network Systems

In this paper, we consider the dynamical modeling of a class of quantum network systems consisting of qubits. Qubit probes are employed to measure a set of selected nodes of the quantum network systems. For a variety of applications, a state space model is a useful way to model the system dynamics. To construct a state space model for a quantum network system, the major task is to find an accessible set containing all of the operators coupled to the measurement operators. This paper focuses on the generation of a proper accessible set for a given system and measurement scheme. We provide analytic results on simplifying the process of generating accessible sets for systems with a time-independent Hamiltonian. Since the order of elements in the accessible set determines the form of state space matrices, guidance is provided to effectively arrange the ordering of elements in the state vector. Defining a system state according to the accessible set, one can develop a state space model with a special pattern inherited from the system structure. As a demonstration, we specifically consider a typical 1D-chain system with several common measurements, and employ the proposed method to determine its accessible set.

The explanation of two semi-recursive multibody methods for educational purpose

In this paper, two often-employed alternatives to the semi-recursive method are described and compared in terms of velocity transformation. The text offers a simplified way to understand the theory behind the two semi-recursive approaches. Focusing on planar cases, particular attention is given to reference and joint coordinates. Consequently, a multibody modeling analyst that is familiar with global formulations can easily extend his/her knowledge to the semi-recursive approach.Using semi-recursive methods, the open loops are formulated with a reduced set of coordinates, and the constraint equations are avoided in the dynamic equations of motion. Accordingly, computationally efficient forms of matrices and vectors will be generated to represent the dynamic equations of motion, which leads to enhanced numerical efficiency. The difference between the two studied alternatives is the definition of the reference point (i.e., the origin of the body frame) used to define the reference coordinates of the body. The reference point could be either (1) rigidly attached to the moving body for semi-recursive I or (2) coincident with the origin of the global frame for semi-recursive II. The latter leads to simple expressions of velocity transformation.

Long short-term cognitive networks

In this paper, we present a recurrent neural system named long short-term cognitive networks (LSTCNs) as a generalization of the short-term cognitive network (STCN) model. Such a generalization is motivated by the difficulty of forecasting very long time series efficiently. The LSTCN model can be defined as a collection of STCN blocks, each processing a specific time patch of the (multivariate) time series being modeled. In this neural ensemble, each block passes information to the subsequent one in the form of weight matrices representing the prior knowledge. As a second contribution, we propose a deterministic learning algorithm to compute the learnable weights while preserving the prior knowledge resulting from previous learning processes. As a third contribution, we introduce a feature influence score as a proxy to explain the forecasting process in multivariate time series. The simulations using three case studies show that our neural system reports small forecasting errors while being significantly faster than state-of-the-art recurrent models.

A Discontinuous Galerkin Method for Approximating the Stationary Distribution of Stochastic Fluid-Fluid Processes

The stochastic fluid-fluid model (SFFM) is a Markov process {(X_t,Y_t,varphi _t),tge 0}, where {varphi _t,{tge 0}} is a continuous-time Markov chain, the first fluid, {X_t,tge 0}, is a classical stochastic fluid process driven by {varphi _t,tge 0}, and the second fluid, {Y_t,tge 0}, is driven by the pair {(X_t,varphi _t),tge 0}. Operator-analytic expressions for the stationary distribution of the SFFM, in terms of the infinitesimal generator of the process {(X_t,varphi _t),tge 0}, are known. However, these operator-analytic expressions do not lend themselves to direct computation. In this paper the discontinuous Galerkin (DG) method is used to construct approximations to these operators, in the form of finite dimensional matrices, to enable computation. The DG approximations are used to construct approximations to the stationary distribution of the SFFM, and results are verified by simulation. The numerics demonstrate that the DG scheme can have a superior rate of convergence compared to other methods.

Sampled-data distributed state estimation with multiple transmission channels under denial-of-service attacks

This paper is concerned with the problem of sampled-data distributed state estimation for a linear continuous-time system with multiple transmission channels under denial-of-service attacks. To solve this problem, sampled-data distributed observers, which only have access to their respective measurable output variables and the information obtained from their neighbors, are designed, where a decomposition form of the coefficient matrices of the system and a time interval division-based method are proposed to deal with the complex interaction of the sampled-data schemes and the attack schemes among different transmission channels. Different from the previous results, the transmission channels from the sensors to the remote observers do not share the same sampled-data scheme and are allowed to be compromised by different adversaries. An example is provided to validate the effectiveness of the proposed method.

DivCom: A Tool for Systematic Partition of Groups of Microbial Profiles Into Intrinsic Subclusters and Distance-Based Subgroup Comparisons.

When analyzing microbiome data, one of the main objectives is to effectively compare the microbial profiles of samples belonging to different groups. Beta diversity measures the level of similarity among samples, usually in the form of dissimilarity matrices. The use of suitable statistical tests in conjunction with those matrices typically provides us with all the necessary information to evaluate the overall similarity of groups of microbial communities. However, in some cases, this approach can lead us to deceptive conclusions, mainly due to the uneven dispersions of the groups and the existence of unique or unexpected substructures in the dataset. To address these issues, we developed divide and compare (DivCom), an automated tool for advanced beta diversity analysis. DivCom reveals the inner structure of groups by dividing their samples into the appropriate number of clusters and then compares the distances of every profile to the centers of these clusters. This information can be used for determining the existing interrelation of the groups. The proposed methodology and the developed tool were assessed by comparing the response of anemic patients with or without inflammatory bowel disease to different iron replacement therapies. DivCom generated results that revealed the inner structure of the dataset, evaluated the relationship among the clusters, and assessed the effect of the treatments. The DivCom tool is freely available at: https://github.com/Lagkouvardos/DivCom.

Uncovering shape signatures of resting-state functional connectivity by geometric deep learning on Riemannian manifold.

Functional neural activities manifest geometric patterns, as evidenced by the evolving network topology of functional connectivities (FC) even in the resting state. In this work, we propose a novel manifold‐based geometric neural network for functional brain networks (called “Geo‐Net4Net” for short) to learn the intrinsic low‐dimensional feature representations of resting‐state brain networks on the Riemannian manifold. This tool allows us to answer the scientific question of how the spontaneous fluctuation of FC supports behavior and cognition. We deploy a set of positive maps and rectified linear unit (ReLU) layers to uncover the intrinsic low‐dimensional feature representations of functional brain networks on the Riemannian manifold taking advantage of the symmetric positive‐definite (SPD) form of the correlation matrices. Due to the lack of well‐defined ground truth in the resting state, existing learning‐based methods are limited to unsupervised methodologies. To go beyond this boundary, we propose to self‐supervise the feature representation learning of resting‐state functional networks by leveraging the task‐based counterparts occurring before and after the underlying resting state. With this extra heuristic, our Geo‐Net4Net allows us to establish a more reasonable understanding of resting‐state FCs by capturing the geometric patterns (aka. spectral/shape signature) associated with resting states on the Riemannian manifold. We have conducted extensive experiments on both simulated data and task‐based functional resonance magnetic imaging (fMRI) data from the Human Connectome Project (HCP) database, where our Geo‐Net4Net not only achieves more accurate change detection results than other state‐of‐the‐art counterpart methods but also yields ubiquitous geometric patterns that manifest putative insights into brain function.