Form Of Matrices Research Articles

The cotype zeta function of [formula omitted

We give an asymptotic formula for the number of sublattices Λ⊆Zd of index at most X for which Zd/Λ has rank at most m, answering a question of Nguyen and Shparlinski. We compare this result to work of Stanley and Wang on Smith normal forms of random integral matrices and discuss connections to the Cohen–Lenstra heuristics. Our arguments are based on Petrogradsky’s formulas for the cotype zeta function of Zd, a multivariable generalization of the subgroup growth zeta function of Zd.

Primitive recursive ordered fields and some applications

We establish primitive recursive (PR) versions of some known facts about computable ordered fields of reals and computable reals, and apply them to prove primitive recursiveness of several important problems in linear algebra and analysis. One of the central results of this paper is a partial PR analogue of Ershov–Madison’s theorem about real closures of computable ordered fields. It allows us, in particular, to obtain PR root-finding algorithms in the PR real and algebraic closures of PR fields with a certain property (PR splitting). We also relate the corresponding fields to the PR reals, as well as introduce and study the notion of a PR metric space. It enables us to derive sufficient conditions for PR computing of normal forms of matrices and solution operators of symmetric hyperbolic systems of PDEs. The methods represent a mix of symbolic and approximate algorithms.

Surjektifitas Pemetaan Eksponensial untuk Grup Lie Heisenberg yang Diperumum

The Heisenberg Lie Group is the most frequently used model for studying the representation theory of Lie groups. This Lie group is modular-noncompact and its Lie algebra is nilpotent. The elements of Heisenberg Lie group and algebra can be expressed in the form of matrices of size 3×3. Another specialty is also inherited by its three-dimensional Lie algebra and is called the Lie Heisenberg algebra. The Heisenberg Lie Group whose Lie Algebra is extended to the dimension 2n+1 is called the generalized Heisenberg Lie group and it is denoted by H whose Lie algebra is h_n. In this study, the surjectiveness of exponential mapping for H was studied with respect to h_n=⟨x ̅,y ̅,z ̅⟩ whose Lie bracket is given by [X_i,Y_i ]=Z. The purpose of this research is to prove the characterization of the Lie subgroup with respect to h_n. In this study, the results were obtained that if ⟨x ̅,y ̅ ⟩=:V⊆h_n a subspace and a set {e^(x_i ) e^(x_j ) ┤| x_i,x_j∈V }=:L⊆H then L=H and consequently Lie(L)≠V.

EFFECT OF JOB REWARDS AND LEARNING INNOVATION ON TEACHER PROFESSIONAL COMMITMENT IN THE ERA OF THE FOURTH INDUSTRIAL REVOLUTION

This study aims to analyze the effects of employee benefits and learning innovation on teacher professional commitment. After calculating the correlation of all research variables in the form of matrices, the data were examined using path analysis. The total sample comprised 251 teachers at Banda Aceh State Senior High School. Based on the study findings, job rewards had a 0.466-point positive direct effect on teachers' professional commitment to their careers. The direct positive influence of learning innovation on teachers' professional commitment was 0.589 points. New research shows that the positive and significant impact of job rewards and learning innovations on the teachers' professional commitment can be enhanced through higher wages, high dedication, effective training programs, appropriate curriculum requirements, and appropriate technology support. Keywords: Job Rewards, Learning Innovation, Professional Commitment DOI： https://doi.org/10.35741/issn.0258-2724.58.3.7

Mixed Chebyshev and Legendre polynomials differentiation matrices for solving initial-boundary value problems

<abstract><p>A new form of basis functions structures has been constructed. These basis functions constitute a mix of Chebyshev polynomials and Legendre polynomials. The main purpose of these structures is to present several forms of differentiation matrices. These matrices were built from the perspective of pseudospectral approximation. Also, an investigation of the error analysis for the proposed expansion has been done. Then, we showed the presented matrices' efficiency and accuracy with several test functions. Consequently, the correctness of our matrices is demonstrated by solving ordinary differential equations and some initial boundary value problems. Finally, some comparisons between the presented approximations, exact solutions, and other methods ensured the efficiency and accuracy of the proposed matrices.</p></abstract>

Intelligent multicast routing method based on multi-agent deep reinforcement learning in SDWN.

Multicast communication technology is widely applied in wireless environments with a high device density. Traditional wireless network architectures have difficulty flexibly obtaining and maintaining global network state information and cannot quickly respond to network state changes, thus affecting the throughput, delay, and other QoS requirements of existing multicasting solutions. Therefore, this paper proposes a new multicast routing method based on multiagent deep reinforcement learning (MADRL-MR) in a software-defined wireless networking (SDWN) environment. First, SDWN technology is adopted to flexibly configure the network and obtain network state information in the form of traffic matrices representing global network links information, such as link bandwidth, delay, and packet loss rate. Second, the multicast routing problem is divided into multiple subproblems, which are solved through multiagent cooperation. To enable each agent to accurately understand the current network state and the status of multicast tree construction, the state space of each agent is designed based on the traffic and multicast tree status matrices, and the set of AP nodes in the network is used as the action space. A novel single-hop action strategy is designed, along with a reward function based on the four states that may occur during tree construction: progress, invalid, loop, and termination. Finally, a decentralized training approach is combined with transfer learning to enable each agent to quickly adapt to the dynamic changes of network link information and accelerate convergence. Simulation experiments show that MADRL-MR outperforms existing algorithms in terms of throughput, delay, packet loss rate, etc., and can establish more intelligent multicast routes. Code and model are available at https://github.com/GuetYe/MADRL-MR_code.

Quasi self-dual codes over non-unital rings from three-class association schemes

<abstract><p>Let $ E $ and $ I $ denote the two non-unital rings of order 4 in the notation of (Fine, 93) defined by generators and relations as $ E = \langle a, b \mid 2a = 2b = 0, a^2 = a, b^2 = b, ab = a, ba = b\rangle $ and $ I = \langle a, b \mid 2a = 2b = 0, a^2 = b, ab = 0\rangle $. Recently, Alahmadi et al classified quasi self-dual (QSD) codes over the rings $ E $ and $ I $ for lengths up to 12 and 6, respectively. The codes had minimum distance at most 2 in the case of $ I $, and 4 in the case of $ E $. In this paper, we present two methods for constructing linear codes over these two rings using the adjacency matrices of three-class association schemes. We show that under certain conditions the constructions yield QSD or Type Ⅳ codes. Many codes with minimum distance exceeding 4 are presented. The form of the generator matrices of the codes with these constructions prompted some new results on free codes over $ E $ and $ I $.</p></abstract>

Infrared nanospectroscopy depth-dependent study of modern materials: morpho-chemical analysis of polyurethane/fibroin binary meshes.

Infrared scattering-type scanning near-field optical microscopy (IR s-SNOM) and imaging is here exploited together with attenuated total reflection (ATR) IR imaging and scanning electron microscopy (SEM) to depict the chemical composition of fibers in hybrid electrospun meshes. The focus is on a recently developed bio-hybrid material for vascular tissue engineering applications, named Silkothane®, obtained in the form of nanofibrous matrices from the processing of a silk fibroin-polyurethane (SFPU) blend via electrospinning. Morphology and chemistry of single fibers, at both surface and subsurface level, have been successfully characterized with nanoscale resolution, taking advantage of the IR s-SNOM capability to portray the nanoscale depth profile of this modern material working at diverse harmonics of the signal. The applied methodology allowed to describe the superficial characteristics of the mesh up to a depth of about 100 nm, showing that SF and PU do not tend to co-aggregate to form hybrid fibers, at least at the length scale of hundreds of nanometers, and that subdomains other than the fibrillar ones can be present. More generally, in the present contribution, the depth profiling capabilities of IR s-SNOM, so far theoretically predicted and experimentally proven only on model systems, have been corroborated on a real material in its natural conditions with respect to production, opening the room for the exploitation of IR s-SNOM as valuable technique to support the production and the engineering of nanostructured materials by the precise understanding of their chemistry at the interface with the environment.

Bayesian hierarchical modeling on covariance valued data

Analysis of structural and functional connectivity (FC) of human brains is of pivotal importance for diagnosis of cognitive ability. The Human Connectome Project (HCP) provides an excellent source of neural data across different regions of interest (ROIs) of the living human brain. Individual specific data were available in the form of time varying covariance matrices representing the brain activity as the subjects perform a specific task. As a preliminary objective of studying the heterogeneity of brain connectomics across the population, we develop a probabilistic model for a sample of covariance matrices using a scaled Wishart distribution. We stress here that our data units are available in the form of covariance matrices, and we use the Wishart distribution to create our likelihood function rather than its more common usage as a prior on covariance matrices. Based on empirical explorations suggesting the data matrices to have a low effective rank, we further model the center of the Wishart distribution using an orthogonal factor model type decomposition. We encourage shrinkage toward a low rank structure through a novel shrinkage prior and discuss strategies to sample from the posterior distribution using a combination of Gibbs and slice sampling. The efficacy of the approach is explored in various simulation settings and exemplified on several case studies including our motivating HCP data. We extend our modeling framework to a dynamic setting to detect change points.

Засоби керування кроковим двигуном та вимірювання зусиль під час дослідження процесу згинання аркушевого матеріалу

The improvement of the equipment for the forming of the volumetric shape of packages and containers requires a detailed study of the phenomena in the folding process of sheet material with a complex structure (corrugated cardboard). A stand has been created to study process parameters, successful research on which should be provided by programmed control of tool movement and synchronous transfer of measured data to a computer for further processing. Moving tools (punches) are driven by a stepper motor. The movement of the stepper motor shaft is programmed to obtain both a uniform movement and movement according to a certain law, which can be different for the working and reverse moves. Forces are measured by a strain gauge clamp, the signal from which is also transmitted to external devices for visualization and processing. The basis of the motor control and measurement unit is the Arduino Uno microcontroller, which, together with the Arduino Nano controller and the analog-to-digital converter, significantly expands the functionality of the stand. The diagram and operation of the created control and measurement unit is described. A special program has been created to study the geometric, kinematic and power parameters of the process, the possibility of quickly changing the law of motion, providing various ratios and laws of motion to the working and reverse motion of punches. The laws of motion for the motor shaft are presented in it in the form of matrices of values, which are accessed before starting the measurement from the stand control panel. The parameters of the process of simultaneous folding of corrugated cardboard, which were investigated using a stand, are described. The developed measurement technique and means of controlling the movement of the stepper motor of the tool drive allowed to estimate the maximum effort, the nature of the change during the work cycle, to compare different types of material and folding modes. The result of determining the optimal parameters of the process of operational packaging & container formation is the creation of recommendations necessary for the design of modern energy-saving equipment.

Eigenvalue bounds for double saddle-point systems

Abstract We derive bounds on the eigenvalues of a generic form of double saddle-point matrices. The bounds are expressed in terms of extremal eigenvalues and singular values of the associated block matrices. Inertia and algebraic multiplicity of eigenvalues are considered as well. The analysis includes bounds for preconditioned matrices based on block diagonal preconditioners using Schur complements, and it is shown that in this case the eigenvalues are clustered within a few intervals bounded away from zero. Analysis for approximations of Schur complements is included. Some numerical experiments validate our analytical findings.

Interval linear algebra – A new perspective

The purpose of this article is to put the concept of interval linear algebra on a sound algebraic setting, by judiciously defining a Field and a Vector Space over a field involving equivalence classes. Classifying mathematical objects by bringing them into canonical forms, is an important subject of study in every branch of Mathematics. The concepts of eigenvalues and eigenvectors are used in several fields including machine learning, quantum computing, communication system design, construction design, electrical and mechanical engineering etc. The study of interval matrices leads to canonical forms of interval matrices, which will help us classify and study interval matrices more effectively and more easily.

O(n)-invariant Riemannian metrics on SPD matrices

Symmetric Positive Definite (SPD) matrices are ubiquitous in data analysis under the form of covariance matrices or correlation matrices. Several O(n)-invariant Riemannian metrics were defined on the SPD cone, in particular the kernel metrics introduced by Hiai and Petz. The class of kernel metrics interpolates between many classical O(n)-invariant metrics and it satisfies key results of stability and completeness. However, it does not contain all the classical O(n)-invariant metrics. Therefore in this work, we investigate super-classes of kernel metrics and we study which key results remain true. We also introduce an additional key result called cometric-stability, a crucial property to implement geodesics with a Hamiltonian formulation. Our method to build intermediate embedded classes between O(n)-invariant metrics and kernel metrics is to give a characterization of the whole class of O(n)-invariant metrics on SPD matrices and to specify requirements on metrics one by one until we reach kernel metrics. As a secondary contribution, we synthesize the literature on the main O(n)-invariant metrics, we provide the complete formula of the sectional curvature of the affine-invariant metric and the formula of the geodesic parallel transport between commuting matrices for the Bures-Wasserstein metric.

Universality of certain diagonal quadratic forms for matrices over a ring of integers

Universality of certain diagonal quadratic forms for matrices over a ring of integers

An Optimized Flower Categorization Using Customized Deep Learning

Categorizing flowers is quite a challenging task as there is so much diversity in the species, and the images of the different flower species could be pretty similar. Flower categorization involves many issues like low resolution and noisy images, occluded images with the leaves and the stems of the plants and sometimes even with the insects. The traditional handcrafted features were used for extraction of the features and the machine learning algorithms were applied but with the advent of the deep neural networks. The focus of the researchers has inclined towards the use of the non-handcrafted features for the image categorization tasks because of their fast computation and efficiency. In this study, the images are pre-processed to enhance the key features and suppress the undesired information’s and the objects are localized in the image through the segmentation to extract the Region of Interest, detect the objects and perform feature extraction and the supervised classification of flowers into five categories: daisy, sunflower, dandelion, tulip and rose. First step involves the pre-processing of the images and the second step involves the feature extraction using the pre-trained models ResNet50, MobileNet, DenseNet169, InceptionV3 and VGG16 and finally the classification is done into five different categories of flowers. Ultimately, the results obtained from these proposed architectures are then analyzed and presented in the form of confusion matrices. In this study, the CNN model has been proposed to evaluate the performance of categorization of flower images, and then data augmentation is applied to the images to address the problem of overfitting. The pre-trained models ResNet50, MobileNet, DenseNet169, InceptionV3 and VGG16 are implemented on the flower dataset to perform categorization tasks. The pre-trained models are empirically implemented and assessed on the various flower datasets. Performance analysis has been done in terms of the training, validation accuracy, validation loss and training loss. The empirical assessment of these pre-trained models demonstrate that these models are quite effective for the categorization tasks. According to the performance analysis, the VGG16 outperforms all the other models and provides a training accuracy of 99.01%. Densenet169 and MobileNet also give comparable validation accuracy. ResNet50 gives the lowest training accuracy of 60.46% as compared with the rest of the pre-trained replica or models.

Fast calculation of gravitational effects using tesseroids with a polynomial density of arbitrary degree in depth

Fast and accurate calculation of gravitational effects on a regional or global scale with complex density environment is a critical issue in gravitational forward modelling. Most existing significant developments with tessroid-based modelling are limited to homogeneous density models or polynomial ones of a limited order. Moreover, the total gravitational effects of tesseroids are often calculated by pure summation in these methods, which makes the calculation extremely time-consuming. A new efficient and accurate method based on tesseroids with a polynomial density up to an arbitrary order in depth is developed for 3D large-scale gravitational forward modelling. The method divides the source region into a number of tesseroids, and the density in each tesseroid is assumed to be a polynomial function of arbitrary degree. To guarantee the computational accuracy and efficiency, two key points are involved: (1) the volume Newton’s integral is decomposed into a one-dimensional integral with a polynomial density in the radial direction, for which a simple analytical recursive formula is derived for efficient calculation, and a surface integral over the horizontal directions evaluated by the Gauss–Legendre quadrature (GLQ) combined with a 2D adaptive discretization strategy; (2) a fast and flexible discrete convolution algorithm based on 1D fast Fourier transform (FFT) and a general Toepritz form of weight coefficient matrices is adopted in the longitudinal dimension to speed up the computation of the cumulative contributions from all tesseroids. Numerical examples show that the gravitational fields predicted by the new method have a good agreement with the corresponding analytical solutions for spherical shell models with both polynomial and non-polynomial density variations in depth. Compared with the 3D GLQ methods, the new algorithm is computationally more accurate and efficient. The calculation time is significantly reduced by 3 orders of magnitude as compared with the traditional 3D GLQ methods. Application of the new algorithm in the global crustal CRUST1.0 model further verifies its reliability and practicability in real cases. The proposed method will provide a powerful numerical tool for large-scale gravity modelling and also an efficient forward engine for inversion and continuation problems.

Permanents of Hexagonal and Armchair Chains

The permanent is important invariants of a graph with some applications in physics. If G is a graph with adjacency matrix A = a i j , then the permanent of A is defined as perm A = ∑ σ ∈ S n ∏ i = 1 n a i σ i , where S n denotes the symmetric group on n symbols. In this paper, the general form of the adjacency matrices of hexagonal and armchair chains will be computed. As a consequence of our work, it is proved that if G k and H k denote the hexagonal and armchair chains, respectively, then perm A G 1 = 4 , perm A G k = k + 1 2 , k ≥ 2 , and perm A H k = 4 k with k ≥ 1 . One question about the permanent of a hexagonal zig-zag chain is also presented.

Dmscatter: A fast program for WIMP-nucleus scattering

Recent work [1,2], using an effective field theory framework, have shown the number of possible couplings between nucleons and the dark-matter-candidate Weakly Interacting Massive Particles (WIMPs) is larger than previously thought. Inspired by an existing Mathematica script that computes the target response [2], we have developed a fast, modern Fortran code, including optional OpenMP parallelization, along with a user-friendly Python wrapper, to swiftly and efficiently explore many scenarios, with output aligned with practices of current dark matter searches. A library of most of the important target nuclides is included; users may also import their own nuclear structure data, in the form of reduced one-body density matrices. The main output is the differential event rate as a function of recoil energy, needed for modeling detector response rates, but intermediate results such as nuclear form factors can be readily accessed. Program summaryProgram Title: dmscatterCPC Library link to program files:https://doi.org/10.17632/6fm7d4m2py.1Developer's repository link:https://github.com/ogorton/dmscatterLicensing provisions: MITProgramming languages: Fortran, PythonNature of problem: Simulating the event rate of nuclear recoils from collisions with dark matter for a variety of different nuclear targets and different nucleon-dark matter couplings.Solution method: The event rate is an integral over the product of the nuclear and dark matter response functions, weighted by the expected dark matter flux. To compute the nuclear response, reduced one-body nuclear density matrices are coupled to multipole expansion of operators computed analytically in a harmonic oscillator basis. The nuclear structure input, in the form of one-body densities matrices, are supplied by outside code; we provide a library of the most prominent targets.

A Jordan-Schur Algorithm for Solving Sylvester and Lyapunov Matrix Equations

This paper presents a version of the Bartels-Stewart algorithm for solving the Sylvester and Lyapunov equations that utilizes the Jordan-Schur form of the equation matrices. The Jordan-Schur form is a type of Schur form which contains additional information about the Jordan structure of the corresponding matrix. This information can be used to solve more efficiently the Sylvester and Lyapunov equations in some cases. A two-level algorithm is implemented which allows us to find directly non-scalar blocks of the solution matrix. These blocks have sizes that are determined by the Weyr characteristics associated with the eigenvalues of the equation matrices. In the case of large elements of the Weyr characteristics associated with multiple eigenvalues, the determination of the solution blocks can be done more efficiency. Also, the blocks equations can be more appropriate in solving the Sylvester and Lyapunov equations in the case of parallel computations. Results obtained from numerical experiments confirm that the accuracy of the new algorithm is comparable with the accuracy of the Bartels-Stewart algorithm.

Approaches to the Creation of a Comprehensive Typology for the Monitoring System of Socio-Economic Development of Municipalities of the Russian Federation

The article proposes approaches to the typologization of municipalities of the Russian Federation on the basis of objective indicators in the framework of one of the urgent tasks of the state policy of spatial development on the introduction of a system for monitoring socio-economic development at the municipal level. The analysis of many parameters of socio-economic development of municipalities allowed us to identify 9 most acceptable typologies for the development of this kind: specialization of the economy, socio-economic risks, geo-economic situation, agglomeration situation, population, urbanization, status and settlement role, climatic conditions, area. In the article in the case of regions located in different parts of the country: Kursk, Kaliningrad and Irkutsk regions. In the article shows that the selected indicators reflect various aspects of the economic, social and geographical location of the municipality and allow us to develop a stable typology of municipalities taking into account differences in their level of economic development, demographic potential and geographical location features and can be presented in the form of matrices with letter encodings. The obtained typological matrices in the future, depending on the combination of letter encodings, will become the basis for a comprehensive typology of municipalities.