Dimensional Matrices Research Articles

Random matrices and the free energy of Ising-like models with disorder

We consider an Ising model with quenched surface disorder, the disorder average of the free energy is the main object of interest. Explicit expressions for the free energy distribution are difficult to obtain if the quenched surface spins take values of ± 1±1. Thus, we choose a different approach and model the surface disorder by Gaussian random matrices. The distribution of the free energy is calculated. We chose skew-circulant random matrices and analytically compute the characteristic function of the free energy distribution. From the characteristic function we numerically calculate the distribution and show that it becomes log-normal for sufficiently large dimensions of the disorder matrices, and in the limit of infinitely large matrices tends to a Gaussian. Furthermore, we establish a connection to the central limit theorem.

Clusters of African countries based on the social contacts and associated socioeconomic indicators relevant to the spread of the epidemic

IntroductionIt is well known that social contact patterns differ from country to country. This variation coincides with significant socioeconomic heterogeneity that complicates the design of effective nonpharmaceutical interventions. This study examined how socioeconomic heterogeneity in selected African countries might be factored in to explain better patterns of social contact mix between countries.MethodsWe used a standardized contact matrix for 32 African countries, estimated in (Prem et al. in PLoS Comput Biol 17(7):e1009098, 2021). We scaled the matrices using an epidemic model from (Röst et al. in Viruses 12(7):708, 2020). We also analyzed aggregated data from the World Bank country website. The data includes 28 variables; social, economic, environmental, institutional, governance, health and well-being, education, gender inequality, and other development-related indicators that describe countries. Principal component analysis was used to visualize socioeconomic similarities between countries and identify the indicators for maximum variation. The (2D)2PCA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(2D)^{2} PCA$\\end{document} approach was used to reduce the dimension of synthetic contact matrices for each country to avoid the dimensionality curse. Then, hierarchical aggregate clustering was used to identify groups of countries with similar social patterns, taking into account the country’s socioeconomic performance.ResultsOur model yielded four meaningful clusters, each with a few distinguishing features. Social contacts varied between groups, but were generally similar within each set. The socioeconomic performance of the country influenced the clusters.ConclusionsOur results suggest that integrating socioeconomic factors into social contacts can better explain infectious disease transmission dynamics and that similar interventions can be implemented in countries within the cluster

Three–dimensional matrices for enhanced coral settlement through design for additive manufacturing

Purpose The effects of climate change have been contributing to coral reef degradation. Artificial reefs are one method being used to counteract this destruction. However, the most common artificial approaches, such as sunken vehicles and prefabricated cement reefs, do not allow adequate coral development. This paper aims to demonstrate how designers, using additive manufacturing and computational design techniques, can create artificial reefs that better mimic natural reef structures. Design/methodology/approach This research focuses on developing three-dimensional matrices through computational design using additive manufacturing to achieve better coral settlement. A “Nature Centered Design” approach was followed, with the corals at the center of the design project. Samples with different geometries and roughness, produced using paste-based extrusion with porcelain and porcelain with oyster shell, were tested in a controlled environment to investigate the settlement preference of soft corals. Findings The rapid prototyping of samples confirmed the preference of corals to settle to complex surfaces compared to smooth surfaces. Porcelain showed comparable results to Portland cement, suggesting further testing potential. A closer resemblance to the natural and intricate forms found in coral reefs was achieved through computational design. Originality/value This paper proposes a new approach combining rapid prototyping with coral’s biological responses to enhance the understanding of their surface settlement preference. The Nature Centered Design approach, with additive manufacturing and computational design, made it possible to create an innovative working model that could be customized depending on the implementation area or intended coral species, validating the design approach as a method to support environmental conservation.

Interest HD: An Interest Frame Model for Recommendation Based on HD Image Generation.

This work is inspired by high-definition (HD) image generation techniques. When the user's interests are viewed as different frames of varying clarity, the unclear parts of one interest frame can be clarified by other interest frames. The user's overall HD interest portrait can be viewed as a fusion of multiple interest frames through detail compensation. Based on this inspiration, we propose a model for generating HD interest portrait called interest frame for recommendation (IF4Rec). First, we present a fine-grained pixel-level user interest mining method, Pixel embedding (PE) uses positional coding techniques to mine atomic-level interest pixel matrices in multiple dimensions, such as time, space, and frequency. Then, using an atomic-level interest pixel matrix, we propose Item2Frame to generate several interest frames for a user. The similarity score of each item is calculated to fill the multi-interest pixel clusters, through an improved self-attention mechanism. Finally, stimulated by HD image generation techniques, we initially present an interest frame noise compensation method. By utilizing the multihead attention mechanism, pixel-level optimization and noise complementation are performed between multi-interest frames, and an HD interest portrait is achieved. Experiments show that our model mines users' interests well. On five publicly available datasets, our model outperforms the baselines.

On arbitrary matrix coefficient assignment for the characteristic matrix polynomial of block matrix linear control systems

For block matrix linear control systems, we study the property of arbitrary matrix coefficient assignability for the characteristic matrix polynomial. This property is a generalization of the property of eigenvalue spectrum assignability or arbitrary coefficient assignability for the characteristic polynomial from system with scalar $(s=1)$ block matrices to systems with block matrices of higher dimensions $(s>1)$. Compared to the scalar case $(s=1)$, new features appear in the block cases of higher dimensions $(s>1)$ that are absent in the scalar case. New properties of arbitrary (upper triangular, lower triangular, diagonal) matrix coefficient assignability for the characteristic matrix polynomial are introduced. In the scalar case, all the described properties are equivalent to each other, but in block matrix cases of higher dimensions this is not the case. Implications between these properties are established.

Sensitivity analysis of frequency response functions with imaginary parts decoupling based on multicomplex-step perturbation

Imaginary perturbation is used in the complex step differentiation method to compute first-order derivatives, widely known as an effective approach for sensitivity analysis in structural dynamics. However, coupling of imaginary parts occurs in the damped frequency response functions when employing this method. To mitigate this coupling, a novel approach for sensitivity analysis based on multicomplex-step perturbation is proposed in this paper, for sensitivity analysis of Frequency Response Functions in structural dynamics. The structural parameters are perturbed in multicomplex domain, the dimensions of structural matrices are expanded using the Cauchy Riemann matrix representation, the equation of motion for sensitivity analysis in frequency domain is transformed to matrix operation in field of real numbers, imaginary term will not exist in the equation of motion for sensitivity analysis, the imaginary part of the frequency response function and the imaginary part of the perturbation are decoupled, the structural frequency response functions and the corresponding sensitivities are obtained from the dimension-expanded equation of motion. A truss structure and a solar wing are adopted to verify the accuracy of the proposed method. Results show that the sensitivity of FRFs can be effectively calculated using the proposed method. Compare to the finite difference method, the proposed method is not depended on the step-size selection procedure. The multi-order and mixed-order sensitivity matrices, especially Hessian matrix can also be obtained using the proposed method.

Using nested tensor train contracted basis functions with group theoretical techniques to compute (ro)-vibrational spectra of molecules with non-Abelian groups.

In this paper, we use nested tensor-train contractions to compute vibrational and ro-vibrational energy levels of molecules with five and six atoms. At each step, we fully exploit symmetry by using symmetry adapted basis functions obtained from an irreducible tensor method. Contracted basis functions are determined by diagonalizing reduced dimensional Hamiltonian matrices. The size of matrices of eigenvectors, used to account for coupling between groups of coordinates, is reduced by discarding rows and columns. The size of the matrices that must be diagonalized is thus substantially reduced, making it possible to use direct eigensolvers, even for molecules with five and six atoms. The symmetry-adapted contracted vibrational basis functions have been used to compute J = 0 energy levels of the CH3CN (C3v) and J > 0 levels of CH4.

Variational Quantum Algorithms for Semidefinite Programming

A semidefinite program (SDP) is a particular kind of convex optimization problem with applications in operations research, combinatorial optimization, quantum information science, and beyond. In this work, we propose variational quantum algorithms for approximately solving SDPs. For one class of SDPs, we provide a rigorous analysis of their convergence to approximate locally optimal solutions, under the assumption that they are weakly constrained (i.e., N≫M, where N is the dimension of the input matrices and M is the number of constraints). We also provide algorithms for a more general class of SDPs that requires fewer assumptions. Finally, we numerically simulate our quantum algorithms for applications such as MaxCut, and the results of these simulations provide evidence that convergence still occurs in noisy settings.

Algebraic curves associated with centrosymmetric matrices of orders up to 6

In this paper the boundary generating curves and numerical ranges of centrosymmetric matrices of orders up to 6 are characterized in terms of the matrices entries. These results extend previous ones concerning Kac-Sylvester matrices. The classification of all the possible boundary generating curves for centrosymmetric matrices of higher dimensions remains open.

COMPARISON MRCD AND ORACLE FOR ESTIMATING THE DETERMINANT OF HIGH DIMENSIONAL COVARIANCE MATRIX

Estimating the variance matrix has an important role in statistical applications and conclusions, in high–dimensional matrices if the number of variables is greater than the number of observations P > n, the traditional statistical methods are not reliable because they give uncontrolled estimates. Shrinkage methods are used to estimate the high–dimensional variance matrix. In this research, the high–dimensional variance matrix was estimated using the robust Nonparametric method Minimum Regularized Covariance Determinant (MRCD), which is based on Mahalanobis distance, and compared with the variance matrix estimated by the Oracle method, which is based on the Frobenius criterion

Jacobian-free Locally Linearized Runge–Kutta method of Dormand and Prince for large systems of differential equations

This paper introduces Jacobian-free Locally Linearized Runge–Kutta formulas of Dormand and Prince for integrating large systems of initial value problems. With these new Jacobian-free formulas, an adaptive time-stepping variable-order scheme with regulated Krylov dimension is constructed with new developments in the computation of the phi-function action over vectors through Jacobian-free Krylov–Padé approximations, and in the procedures for controlling the approximation errors and for estimating the dimension of the Krylov subspaces. At each integration step, the implemented scheme performs only one Krylov subspace decomposition and a few computations of low dimensional exponential matrices, which contrasts with the implementation of other exponential-type integrators of high order. Regarding to existing schemes, the performed numerical study demonstrates a higher effectiveness of the constructed scheme in the integration of a variety of test equations.

Bulk universality and quantum unique ergodicity for random band matrices in high dimensions

Bulk universality and quantum unique ergodicity for random band matrices in high dimensions

DIGITAL TOOLS FOR MATCHING QUALIFICATIONS TO THE LEVELS OF THE NATIONAL QUALIFICATIONS FRAMEWORK

Mutual compatibility of different national qualifications frameworks (NQFs) based on their compatibility with the European Qualifications Framework (or another international one) is crucially important for the effective recognition of qualifications between states. In turn, it depends on the quality of “filling” NQF levels with qualifications. Right comparing professional (occupational) qualifications with NQF level is a non-trivial problem for standard developers. The quality of any national system of qualifications depends on the comparability of qualifications with the level of the NQF so the comparison process should be strongly argued to secure the comparability. For qualification standard developers (especially for occupational standard developers) there were no strictly justified recommendations on how to compare qualification with the NQF level. Th. Saaty’s Analytic Hierarchy Process (AHP) is developed to resolve the problem of evidence-based comparing educational or professional (occupational) qualifications with the level of the National Qualification Framework (NQF). Research results give standard developers software tools based on a strong mathematical background to determine NQF level for developed standards. It is shown that Th. Saaty’s Analytic Hierarchy Process (AHP) is close to optimal for solving the problem of qualifications comparing and therefore looks like the best option for such methods. However, AHP demands non-trivial qualifications in mathematics and computing. The key problem resolved by this research is simplifying procedures to ensure effective access to the tool for qualification standard developers with minimal qualification in mathematics and computing. It is proven that each problem of qualification comparison with NQF level may be reduced to three options of decision. At the lower level of the decision-making process, there are 3-4 descriptors of qualification. Therefore, a user should be capable of forming at most four matrices of judgments and computing the main eigenvectors with some level of accuracy. The maximal dimension of matrices is four (for example it's true for the Ukrainian case). But for some national qualification frameworks that use only three descriptors maximal dimension of matrices equals three. Therefore, some simple approximation methods for eigenvector computing may be applied using only minimal means of Microsoft Excel or analogous applications. For the most general case of four NQF descriptors, Microsoft Excel macro is developed to secure achieving any level of accuracy. Corresponding API is developed by PHP programming language. Both Excel and API are accessible for users at the website of the Institute of Educational Analytics in Kyiv. The novelty of the article is that for the first time in national and international practice, it proposes an alternative/supplementary algorithmic method for determining the level of certain full and/or partial professional qualifications by the National Qualifications Framework, thus creating prerequisites for further automation of the activities of professional standards developers.

Random-matrix model for thermalization

We show that for a system governed by a random-matrix Hamiltonian (a member of the time-reversal invariant Gaussian Orthogonal Ensemble (GOE) of random matrices of dimension N), all functions Tr(Aρ(t)) in the ensemble thermalize: For N→∞ every such function tends to the value Tr(Aρeq(∞))+Tr(Aρ(0))g2(t) . Here ρ(t) is the time-dependent density matrix of the system, A is a Hermitean operator standing for an observable, and ρeq(∞) is the equilibrium density matrix at infinite temperature. The oscillatory function g(t) is the Fourier transform of the average GOE level density and falls off as 1/|t|3/2 for large t. With g(t)=g(−t) , thermalization is symmetric in time. Analogous results, including the symmetry in time of thermalization, are derived for the time-reversal non-invariant Gaussian Unitary Ensemble of random matrices. Comparison with the ‘eigenstate thermalization hypothesis’ of (Srednicki 1999 J. Phys. A: Math. Gen. 32 1163) shows overall agreement but raises significant questions.

Matrix Representation of Magnetic Pseudo-Differential Operators via Tight Gabor Frames

In this paper we use some ideas from [12, 13] and consider the description of Hörmander type pseudo-differential operators on Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^d$$\\end{document} (d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 1$$\\end{document}), including the case of the magnetic pseudo-differential operators introduced in [15, 16], with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calderón-Vaillancourt theorem and Beals’ commutator criterion, and also establish local trace-class criteria.

Abstract 4907: RanBALL: Identifying B-cell acute lymphoblastic leukemia subtypes based on an ensemble random projection model

Abstract As the most common pediatric malignancy, B-cell acute lymphoblastic leukemia (B-ALL) has multiple distinct subtypes characterized by recurrent and sporadic somatic and germline genetic alterations like chromosomal alteration, transcription factor rearrangement or kinase inhibition. The treatment of B-ALL patients is personalized based on specific subtypes, as the treatment responses for different B-ALL subtypes may vary considerably. Identification of B-ALL subtypes can facilitate risk stratification and enable tailored therapeutic approaches. Existing methods for B-ALL subtyping primarily depend on immunophenotypic, cytogenetic and genomic analyses, which would be costly, complicated, and laborious in clinical practice applications. To overcome these challenges, we present RanBALL (an Ensemble Random Projection-Based Model for Identifying B-Cell Acute Lymphoblastic Leukemia Subtypes), an accurate and cost-effective model for B-ALL subtype identification based on transcriptomic profiling only. RanBALL leverages random projection (RP) to construct an ensemble of dimension-reduced multi-class classifiers for B-ALL subtyping. Specifically, the transcriptomic profiling features were projected onto low-dimensional spaces by random projection matrices whose elements conform to a distribution characterized by zero mean and unit variance. To ensure reliable and robust performance, we selected 20 subspace dimensions ranging from 600 to 2500, with intervals of 100. The transformed low dimensional data matrix was used for training an ensemble of multi-class support vector machine (SVM) classifiers, each corresponding to one of the RP matrices of various dimensions. The predicted probabilistic scores of each B-ALL subtype were integrated for determining the final decision. Results based on 10 times 10-fold cross validation tests for >1700 B-ALL patients demonstrated that the proposed model achieved an accuracy of 93.7%, indicating promising prediction capabilities of RanBALL for B-ALL subtyping. Furthermore, the 30% held-out tests suggested that the model was robust and consistent to maintain high confidence levels for accurate predictions. The high accuracies of RanBALL suggested that our model could effectively capture underlying patterns of transcriptomic profiling for accurate B-ALL subtype identification. To extend the impact of RanBALL, we have established a free and publicly available python package for RanBALL available at https://github.com/wan-mlab/RanBALL. We believe RanBALL will facilitate the discovery of B-ALL subtype-specific marker genes and therapeutic targets, and eventually have consequential positive impacts on downstream risk stratification and tailored treatment design. Citation Format: Lusheng Li, Hanyu Xiao, Joseph D. Khoury, Jieqiong Wang, Shibiao Wan. RanBALL: Identifying B-cell acute lymphoblastic leukemia subtypes based on an ensemble random projection model [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2024; Part 1 (Regular Abstracts); 2024 Apr 5-10; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2024;84(6_Suppl):Abstract nr 4907.

A symmetric substructuring method for analyzing the natural frequencies of conical origami structures

Conical origami structures are characterized by their substantial out-of-plane stiffness and energy-absorption capacity. Previous investigations have commonly focused on the static characteristics of these lightweight structures. However, the efficient analysis of the natural vibrations of these structures is pivotal for designing conical origami structures with programmable stiffness and mass. In this paper, we propose a novel method to analyze the natural vibrations of such structures by combining a symmetric substructuring method (SSM) and a generalized eigenvalue analysis. SSM exploits the inherent symmetry of the structure to decompose it into a finite set of repetitive substructures. In doing so, we reduce the dimensions of matrices and improve computational efficiency by adopting the stiffness and mass matrices of the substructures in the generalized eigenvalue analysis. Finite element simulations of pin-jointed models are used to validate the computational results of the proposed approach. Moreover, the parametric analysis of the structures demonstrates the influences of the number of segments along the circumference and the radius of the cone on the structural mass and natural frequencies of the structures. Furthermore, we present a comparison between six-fold and four-fold conical origami structures and discuss the influence of various geometric parameters on their natural frequencies. This study provides a strategy for efficiently analyzing the natural vibration of symmetric origami structures and has the potential to contribute to the efficient design and customization of origami metastructures with programmable stiffness.

Optimal modeling of nonlinear systems Method of variable injections

Our work addresses a development and justification of the new approach to the modeling of nonlinear systems. Let $\f$ be an unknown input-output map of the system with a random input and output $\y$ and $\x$, respectively. It is assumed that $\y$ and $\x$ are available and covariance matrices formed from $\y$ and $\x$ are known. We determine a model of $\f$ so that an associated error is minimized. To this end, the model $\ttt_p$ is constructed as a sum of $p+1$ particular parts, in the form $\ttt_p (\y) = \sum_{j=0}^{p}G_j H_j Q_j(\vv_j)$ where $G_j$ and $ H_j$, for $j=0,\ldots, p$, are matrices to be determined, and $\vv_j$, for $j=1,\ldots,p$, is a special random vector called the injection. We denote $\vv_0=\y$. Injections $\vv_1,\ldots, \vv_p$ are aimed to diminish the error associated with the proposed model $\ttt_p$. Further, $Q_j$ is a special transform aimed to facilitate the numerical realization of model $\ttt_p$. It is determined in the way allowing us to optimally determine $G_j$ and $ H_j$ as a solution of $p+1$ separate error minimization problems which are simpler than the original minimization problem. The empirical determination of injections $\vv_1,\ldots, \vv_p$ is considered. The proposed method has several degrees of freedom to diminish the associated error. They are `degree' $p$ of $\ttt_p$, choice of matrices $G_0, H_0, \ldots,$ $G_p, H_p$, dimensions of matrices $G_0, H_0, \ldots,$ $G_p, H_p$ and injections $\vv_1,\ldots, \vv_p$, respectively. Four numerical examples are provided. At the end, the open problem is formulated.

Bayesian estimation of cluster covariance matrices of unknown form

We develop a flexible Bayesian model for cluster covariance matrices in large dimensions where the number of clusters and the assignment of cross-sectional units to a cluster are a-priori unknown and estimated from the data. In a cluster covariance matrix, the variances and covariances are equal within each diagonal block, while the covariances are equal in each off-diagonal block. This reduces the number of parameters by pooling those parameters together that are in the same cluster. In order to treat the number of clusters and the cluster assignments as unknowns, we build a random partition model which assigns a prior distribution over the space of partitions of the data into clusters. Sampling from the posterior over the space of partitions creates a flexible estimator because it averages across a wide set of cluster covariance matrices. We illustrate our methods on linear factor models and large vector autoregressions.

Avoiding matrix exponentials for large transition rate matrices.

Exact methods for the exponentiation of matrices of dimension N can be computationally expensive in terms of execution time (N3) and memory requirements (N2), not to mention numerical precision issues. A matrix often exponentiated in the natural sciences is the rate matrix. Here, we explore five methods to exponentiate rate matrices, some of which apply more broadly to other matrix types. Three of the methods leverage a mathematical analogy between computing matrix elements of a matrix exponential process and computing transition probabilities of a dynamical process (technically a Markov jump process, MJP, typically simulated using Gillespie). In doing so, we identify a novel MJP-based method relying on restricting the number of "trajectory" jumps that incurs improved computational scaling. We then discuss this method's downstream implications on mixing properties of Monte Carlo posterior samplers. We also benchmark two other methods of matrix exponentiation valid for any matrix (beyond rate matrices and, more generally, positive definite matrices) related to solving differential equations: Runge-Kutta integrators and Krylov subspace methods. Under conditions where both the largest matrix element and the number of non-vanishing elements scale linearly with N-reasonable conditions for rate matrices often exponentiated-computational time scaling with the most competitive methods (Krylov and one of the MJP-based methods) reduces to N2 with total memory requirements of N.