Minimum Eigenvalue Research Articles

Fast Algorithms for Maximizing the Minimum Eigenvalue in Fixed Dimension

In the minimum eigenvalue problem, we are given a collection of vectors in Rd, and the goal is to pick a subset B to maximize the minimum eigenvalue of the matrix ∑i∈Bvivi⊤. We give a O(nO(dlog⁡(d)/ϵ2))-time randomized algorithm that finds an assignment subject to a partition constraint whose minimum eigenvalue is at least (1−ϵ) times the optimum, with high probability. As a byproduct, we also get a simple algorithm for an algorithmic version of Kadison-Singer problem.

New inequalities on the Fan product of M-matrices

This paper focuses on the minimum eigenvalue involving the Fan product. By utilizing the Hölder inequality and the classic eigenvalue inclusion theorem, we introduce two novel lower bounds for τ(A1⋆A2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\ au \\left ( A_{1}\\star A_{2} \\right )$\\end{document}, representing the minimum eigenvalue involving the Fan product of two M-matrices A1,A2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$A_{1},A_{2}$\\end{document}. The newly derived lower bounds are then compared with the traditional findings. Numerical tests are presented to illustrate that the new lower bound formulas significantly enhance Johnson and Horn’s results in certain scenarios and are more precise than other existing findings.

Heat transfer inspection and entropy optimization in water/MoS2-SiO2 Casson hybrid nanofluid flow with quadratic radiation ray over an exponentially contracting permeable Riga surface: Duality and stability analysis

In this study, we have examined the primary behaviour of a hybridized electro-magneto-non-Newtonian MoS2-SiO2/H2O nanofluid over a porous Riga surface that is exponentially contracting, in particular its flow dynamics and heat transfer characteristics. The key factors of this research are the presence of quadratic thermal radiation over the Riga surface under velocity slip and convective boundary. We have solved the highly circuitous coupled partial differential equations system by utilizing the bvp4c numerical method with MATLAB programming. A dual solution is determined based on a certain amount of mass suction parameter and shrinking parameter under the critical value range; beyond that value, no solution exists. We investigated the stability of the dual solutions by determining the minimal eigenvalue. An evaluation of the temporal stability of these solutions reveals that only the first solution remains stable. This is supported by the positivity of the smallest eigenvalues (β1> 0), indicating stability. Conversely, for the second solution, the smallest eigenvalues (β1< 0) are negative, signifying its instability. It can be seen that the nanofluid temperature is substantially enhanced for both branches of solution for the quadratic thermal radiating parameter, Eckert number, and Biot number. It was found that, in the first solution branch, the velocity layout was boosted with the velocity slip factor while reduced by enhancing the Casson parameter, whereas, in the second solution branch, the opposite behavior was observed. As the suction parameter increases, the heat transfer coefficient for the first solution is decreased by about 0.5 % and increases by about 0.75 % for the second solution branch. Quadratic thermal radiation parameter Nr significantly impacts the Nusselt number, as it is reduced by approximately 27.44 % and 28.72 % for the first and second solution branches, respectively. The shear stress coefficient is decreased by about 29.32 % for the first solution branch with contracting parameter λ. Because the Riga plate, paired with Casson nanofluid flows, has industrial and civil engineering applications, this study's thermal relevance makes it applicable to a wide range of industrial and engineering disciplines.

Spectral Pseudorandomness and the Road to Improved Clique Number Bounds for Paley Graphs

We study subgraphs of Paley graphs of prime order p induced on the sets of vertices extending a given independent set of size a to a larger independent set. Using a sufficient condition due to Kunisky (2023), we show that estimates on a new family of necklace character sums would imply that, as p → ∞ , the empirical spectral distributions of the adjacency matrices of any sequence of such subgraphs have the same weak limit (after rescaling) as those of subgraphs induced on a random set including each vertex independently with probability 2 − a , namely, a Kesten-McKay law with parameter 2 a . We prove the necessary estimates for a = 1, obtaining in the process an alternate proof of a character sum equidistribution result of Xi (2022), and provide numerical evidence for this weak convergence for a ≥ 2 . We also conjecture that the minimum eigenvalue of any such sequence converges (after rescaling) to the left edge of the corresponding Kesten-McKay law, and provide numerical evidence for this convergence. Finally, we show that, once a ≥ 3 , this (conjectural) convergence of the minimum eigenvalue would imply bounds on the clique number of the Paley graph improving on the current state of the art due to Hanson and Petridis (2021), and that this convergence for all a ≥ 1 would imply that the clique number is o ( p ) .

Smoluchowski temperature and Maxwell velocity slip impacts on magneto-radiative Fe 3 O 4-Al 2 O 3-Cu/H 2 O tri-hybrid nanofluid flow over a shrinking surface with entropy generation: a stability performance

ABSTRACT This investigation aims to find entropy production and to determine the existence of a dual solution in magnetohydrodynamic (MHD) thermal radiative tri-hybrid nanofluid flow over an exponentially shrinking surface by considering suction effects at the surface. Furthermore, the effects of Smoluchowski temperature and Maxwell velocity slip are considered. The products on various flow fields are examined for three different types of fluid: mono nanofluid ( F e 3 O 4 / H 2 O ), hybrid nanofluid ( F e 3 O 4 - A l 2 O 3 / H 2 O ), and tri-hybrid nanofluid ( F e 3 O 4 - A l 2 O 3 - Cu / H 2 O ). The similarity transformation translates the nonlinear partial differential equations (PDEs) into a set of ordinary differential equations (ODEs). The numerical MATLAB function bvp4c is utilized to construct and solve governing higher-order nonlinear ODEs. A tabular and graphical analysis is conducted on the impact of emergent factors on velocity, temperature, skin friction coefficient, Nusselt number, and entropy generation. The acceptable parameter ranges for the computation are as follows: 0.01 < ϕ 1 < 0.1 , 0.01 < ϕ 2 < 0.1 , 0.01 < ϕ 3 < 0.1 , − 0.7 < ξ < 1.6 , 2.0 < S < 3.0 , 0.0 < M < 0.2 , 0.5 < Nr < 1.5 , 0.1 < ω < 0.5 , and 0.1 < ε < 0.5 . Due to the contraction of the surface, dual solutions are found, but dual solutions cannot be found beyond the critical values. The critical values ξ λ are − 1.44285 , − 1.44804 , and 1.51550 for mono nanofluid, hybrid nanofluid, and tri-hybrid nanofluid, respectively. Including A l 2 O 3 and Cu nanoparticles inside F e 3 O 4 nanoparticles containing nanofluid depicted an accelerating effect on skin friction and Nusselt number rates. The skin friction coefficient and heat transference rate in both the solution branches improved as the suction parameter S was increased. It has been demonstrated that the fluid’s temperature is enlarged by the thermal radiation parameter Nr and increasing nanoparticle volume fractions ϕ 1 , ϕ 2 , ϕ 3 . However, the temperature is reduced greatly by the suction parameter S and Smoluchowski temperature slip parameter ε . Since dual solutions are present, stability analysis is performed by finding the minimum eigenvalue. A positive minimum eigenvalue ( β 1 ) denotes the upper stable solution branch, whereas a minimal eigenvalue that is negative indicates the bottom branch is unstable. The entropy of the flow was found to increase with thermal radiation parameter Nr and magnetic field parameter M in a physically stable solution. Still, it decreased with a higher nanoparticle volume fraction ( ϕ 1 , ϕ 2 , ϕ 3 ). The current optimization technique offers a new and beneficial insight through numerous applications in various fields, i.e. polymer engineering, extrusion of polymer, plastic sheet compression, glass production, fiber, metallic furnace, medicine, and the field of biomedical technology.

The Structure of Field‐Aligned Current Systems as Inferred From the Multiscale Minimum Variance Analysis

AbstractAuroral field‐aligned currents (FACs) have an intrinsic complexity caused by the superposition of contributions from a broad spectrum of scales and diversity of locations. The complex FAC systems are investigated by using the multiscale minimum variance analysis. This technique provides a scale based decomposition of the FAC systems by identifying the constituting FAC elements as well as their structure. At the basis, the analysis exploits the scale dependence of the eigenvalues of the magnetic field variance matrix. The scale decomposition along the transversal (latitudinal) direction results from the scale derivative of the maximum eigenvalue. The complementary information from the scale derivative of the minimum eigenvalue helps to infer the full structure of each FAC element by providing estimates of the FAC length (longitudinal) scale. The scale derivative of minimum and maximum eigenvalues are used to identify FAC signatures associated to different types of aurora (e.g., highly extended, finite arcs, gradient regions) as well as to characterize the influence of the crossing location with respect to the FAC structures (e.g., near edge crossings). The multiscale analysis is applied to simulated FACs and to spacecraft observations made by the Swarm mission. The use with real world data illustrates the power of this analysis, whose full benefits for magnetosphere‐ionosphere coupling investigations are yet to be explored.

A topological network connectivity design problem based on spectral analysis

How to improve network connectivity and which parts of the network are vulnerable are critical issues. We begin by defining an equal distribution problem, in which supplies are distributed equally to all nodes in the network. We then derive a topological network connectivity measure from the convergence speed, which is the second minimum eigenvalue of a Laplacian network matrix. Based on the equal distribution problem, we propose a method for identifying critical links for network connectivity using the derivative of the second minimum eigenvalue. Furthermore, we develop a network design problem that maximizes topological connectivity within a budget creating strengthening network links. The problem is convex programming, and the solution is global. Furthermore, it can be converted into an identical semidefinite programming problem, which requires less computational effort. Finally, we test the developed problems on road networks in the Japanese prefectures of Ishikawa and Toyama to determine their applicability and validity.

Comparative analysis of features and classification techniques in breast cancer detection for Biglycan biomarker images.

Breast cancer (BC) is considered the world's most prevalent cancer. Early diagnosis of BC enables patients to receive better care and treatment, hence lowering patient mortality rates. Breast lesion identification and classification are challenging even for experienced radiologists due to the complexity of breast tissue and variations in lesion presentations. This work aims to investigate appropriate features and classification techniques for accurate breast cancer detection in 336 Biglycan biomarker images. The Biglycan biomarker images were retrieved from the Mendeley Data website (Repository name: Biglycan breast cancer dataset). Five features were extracted and compared based on shape characteristics (i.e., Harris Points and Minimum Eigenvalue (MinEigen) Points), frequency domain characteristics (i.e., The Two-dimensional Fourier Transform and the Wavelet Transform), and statistical characteristics (i.e., histogram). Six different commonly used classification algorithms were used; i.e., K-nearest neighbours (k-NN), Naïve Bayes (NB), Pseudo-Linear Discriminate Analysis (pl-DA), Support Vector Machine (SVM), Decision Tree (DT), and Random Forest (RF). The histogram of greyscale images showed the best performance for the k-NN (97.6%), SVM (95.8%), and RF (95.3%) classifiers. Additionally, among the five features, the greyscale histogram feature achieved the best accuracy in all classifiers with a maximum accuracy of 97.6%, while the wavelet feature provided a promising accuracy in most classifiers (up to 94.6%). Machine learning demonstrates high accuracy in estimating cancer and such technology can assist doctors in the analysis of routine medical images and biopsy samples to improve early diagnosis and risk stratification.

Short-term voltage stability analysis in power systems: A voltage solvability indicator

This paper presents an analytical study for short-term voltage stability based on a novel indicator. First, it is shown that the reduced Jacobian matrix of network equations can be transformed to obtain an approximately symmetric matrix which is usually positive definite when the system under study is stable. The minimum eigenvalue of the matrix is suggested as a short-term Voltage Solvability Indicator (VSI). Then, properties of the reduced conductive matrix and susceptance matrix which have significant impacts on VSI are examined. The focus of the study is power systems with multiple constant power injections, while VSI can also be applied to analyze the impact of generic power system models. It is shown that as the penetration level of constant power injections increases, VSI decreases and the system becomes more vulnerable to voltage collapse. Finally, the relationship between the eigenvalues of the reduced Jacobian matrix and the structure-preserving Jacobian matrix is established. As a result, the suggested VSI can be efficiently computed by exploring the sparsity of structure-preserving Jacobian matrix of network equations. Case studies of several test systems including a simplified 575-machine 8117-bus East China power system are reported, demonstrating the effectiveness and practicability of the proposed method.

Specific target pinning control of complex dynamical networks based on semidefinite programming strategy

Pinning control of complex networks aims to force the states of all nodes to reach the desired goal by controlling a few nodes in the network. The easiness of a pinning strategy can be measured by the minimum eigenvalue λ1 of the specific matrix L+Γ, where L is the Laplacian matrix of the underlying network and Γ is a diagonal matrix with Γi,i=1 if node i is pinned. A larger value λ1 indicates a smaller coupling strength. We here are interested in two key questions on this issue. The first one is, what is the minimum number of pinned nodes to achieve system synchronization when individual node dynamics and the coupling strength of the network are given? We show an upper bound of the minimum eigenvalue λ1, by which a lower bound of the number of pinned nodes is presented. The second one is, which nodes should be pinned such that the system achieves synchronization much easier if a potential pinning target node set S0 (instead of the whole node set) and the number of pinned nodes l (l<|S0| ) are both restricted beforehand? This specific target pinning control problem is raised for the first time. By transforming it into a semi-definite programming problem, we then use a convex optimization tool to solve this problem. Experimental results show that the strategy is more effective than other classical node selection indices.

Evaluating non-consumptive household water uses in a growing urban centre in Nigeria

Efficient use of water could be partly achieved with sound management strategies of the non-consumptive uses (N-CUs) of water in homes being put in place. This research evaluated the non-consumptive water use component in Iwo, Osun State, Nigeria. Data required for the investigation was generated from the administration of 325 questionnaires across the five Quarters into which the town is divided, out of which 269 were completed and retrieved. Both descriptive and inferential analysis of the data were carried out. Descriptive analysis showed that households engage absolutely in different non-consumptive uses such as bathing, clothe washing, drainage cleaning and dish washing while households’ engagement in other N-CUs were in varying proportions. The results of Factor Analysis (FA) revealed that five out of the 13 variables identified and analyzed with a minimum eigen value of 1.000 were strong explanatory variables of 73.674% when engaging in issues relating to N-CUs at household level. These are water use for the following (i) drainage cleaning (16.153%); (ii) Dish washing (15.922%); (iii) Toilet cleaning (14.547%); (iv) Auto-wash (14.238%); and Bathing (12.814%). Regression analysis (RA) of the data revealed that three variables namely clothe washing, Incidental washing and auto-washing were significant (p < 0.001) in generating predictive model of N-CUs of water in homes. The combined results of FA and RA implied that the set of variables in both analysis need to be considered in any issue involving the management and control of N-CUs of water in homes for a result-oriented water use efficiency at household level.

On convergence properties of the brain-state-in-a-convex-domain

Convergence in the presence of multiple equilibrium points is one of the most fundamental dynamical properties of a neural network (NN). Goal of the paper is to investigate convergence for the classic Brain-State-in-a-Box (BSB) NN model and some of its relevant generalizations named Brain-State-in-a-Convex-Body (BSCB). In particular, BSCB is a class of discrete-time NNs obtained by projecting a linear system onto a convex body of Rn. The main result in the paper is that the BSCB is convergent when the matrix of the linear system is symmetric and positive semidefinite or, otherwise, it is symmetric and the step size does not exceed a given bound depending only on the minimum eigenvalue of the matrix. This result generalizes previous results in the literature for BSB and BSCB and it gives a solid foundation for the use of BSCB as a content addressable memory (CAM). The result is proved via Lyapunov method and LaSalle’s Invariance Principle for discrete-time systems and by using some fundamental inequalities enjoyed by the projection operator onto convex sets as Bourbaki–Cheney–Goldstein inequality.

Robust design against frequency variation for amplitude death in delay-coupled oscillators.

Amplitude death has the potential to suppress unwanted oscillations in various engineering applications. However, in some engineering applications, such as dc microgrids, airfoil systems, and thermoacoustic systems, oscillation frequency is highly susceptible to external influences, leading to considerable variations. To maintain amplitude death amidst these frequency variations, we propose a design procedure that is robust against frequency variation for inducing amplitude death in delay-coupled oscillators. We first analytically derive the oscillator frequency band in which amplitude death can occur. The frequency bandwidth is maximized when the coupling strength is inversely proportional to the connection delay. Furthermore, our analysis reveals that the oscillator frequency band is influenced by the minimum eigenvalue of the normalized adjacency matrix (i.e., network topology) and that bipartite networks exhibit limited robustness to frequency variations. Our design procedure maintains the stability of amplitude death even under substantial frequency variations and is applicable to various network topologies. Numerical simulations confirm the validity of the proposed design.

Note on the product of the largest and the smallest eigenvalue of a graph

Abstract In this note, we use eigenvalue interlacing to derive an inequality between a graph’s maximum degree and its maximum and minimum adjacency eigenvalues. The equality case is fully characterized.

The Eigenvalue Properties of a Kind of Singular Differential Equations in 3-Dimensional Space

In this paper, we consider the eigenvalue problem of the singular differential equation [see formula in PDF] in a bounded open ball with Dirichlet boundary condition in 3-dimensional space, where, [see formula in PDF][see formula in PDF], [see formula in PDF] is a given constant[see formula in PDF]. And we have made a detailed characterization of the weak solution space. Furthermore, the existence of the minimum eigenvalue and the fundamental gap are provided.

Lp maximal regularity for vector-valued Schrödinger operators

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

Factorial structure of the Comprehensive Assessment of At-Risk Mental States in help-seeking individuals: mapping the structure and the prediction of subsequent transition to psychosis

ObjectivesThe aim of the current study was 3-fold: 1) to examine the factorial structure of the Comprehensive Assessment of At-Risk Mental States (CAARMS) in help-seeking individuals undergoing an assessment on suspicion of psychosis risk; 2) to investigate the association of CAARMS factors with functioning; 3) and to test the association of any derived factors with the longitudinal outcome of transition to psychosis.MethodsThe study included 101 patients. First, a principal component analysis (PCA) was conducted using the Varimax rotation method. A minimum initial eigenvalues of greater than or equal to 1.0, analysis of Scree plots, percentage of variance explained by each component, reliability (Cronbach’s alpha) of factors above 0.7 and Parallel Analysis were the criteria used to determine the appropriate number of factors Second, Spearman correlations were run to analyze the relationship between CAARMS factors and sociodemographic and functional variables (i.e. age, schooling, Social and Occupational Functioning Assessment Scale-SOFAS- and Health of the Nation Outcome Scales-HoNOS- scores). Third, we performed a Logistic regression analysis to evaluate the association between baseline CAARMS factors and the risk of transition to psychosis at the 6-month follow-up.ResultsA total of 101 consecutive patiens were recruited. We found that: 1) a 6 factor model solution as the most appropriate, jointly accounting for 65% of the variance; 2) factors 1 (“negative-interpersonal”), 2 (“cognitive-disorganization”), 3 (“positive”), and 4 (“motor-physical changes”) were negatively correlated with SOFAS total score; factors 1, 2, and 3 showed positive correlations with HoNOS total score; factors 2 and 3 present similar patterns of correlations, factor 3 manifesting the strongest association with HoNOS symptoms, HONOS and SOFAS total score. Both factors 5 and 6 show significant associations with HoNOS behavioral impairment; 3) after 6 months 28 participants (30.1%) converted to psychosis. Factors 2 and 3 were positively associated with the risk of transition to psychosis; whereas, the factor 5 (“affective factor”) was negatively associated with the outcome variable.ConclusionsIt is thus crucial to recognize the type and severity of psychopathology in help-seeking individuals in order to intensive clinical monitoring of subclinical psychopathology risk profiles, and design specific care pathways.

Mathematical modeling of entropy generation in MHD mixed convective Cu–Ag-Al2O3/H2O tri-hybrid nanofluid over an exponential permeable shrinking surface with radiation and slip impacts: Multiple solutions with stability analysis

The present investigation aims to determine the existence of a dual solution in magnetohydrodynamic (MHD) mixed convective radiative flow of Cu-Ag- A l 2 O 3 / H 2 O tri-hybrid nanofluid in a permeable Darcy–Forchheimer porous medium over an exponentially shrinking surface by considering velocity, thermal slips, and suction effects at the surface. This study focuses on assessing entropy production in novel coolant applications. To streamline the analysis, the complex nonlinear partial differential equations (PDEs) are simplified by converting them into a set of ordinary differential equations (ODEs) through the utilization of the similarity transformation technique. These ODEs are then solved utilizing the bvp4c numerical MATLAB function. Graphical and tabular analyses are conducted to investigate the impact of emerging variables on entropy generation, as well as velocity, temperature, skin friction coefficient, and Nusselt number. Due to the contraction of the surface, dual solutions are found, but dual solutions cannot be found beyond the critical values. The critical values of S C are 1.58928, 1.48700, and 1.66058, but ξ C are − 1.239499 , − 1.416999 , and − 1.130030 for Cu / H 2 O nanofluid, Cu − Ag / H 2 O hybrid nanofluid, and Cu − Ag − A l 2 O 3 / H 2 O tri-hybrid nanofluid, respectively. A positive minimum eigenvalue γ 1 signifies the existence of the upper stable solution branch, while a negative minimal eigenvalue indicates the presence of the lower unstable solution branch. The tri-hybrid nanofluid exhibits superior thermal properties compared to nanofluid and hybrid fluids. Adding nanoparticles to traditional fluids is perceived as improving their ability to transmit heat. The analysis has demonstrated that the thermal radiation and temperature slip parameters significantly influence the heat transfer rate. Thermal radiation is found to speed up the creation of entropy. Additionally, in the case of a stable solution, an increase in the Forchheimer number results in a deceleration of the liquid flow, an effect which is further amplified by the presence of the magnetic field and velocity slip parameter.

Lower bounds on the minimum eigenvalue of the Fan product of several M-matrices

The concept of the Fan product of several M-matrices is introduced. Furthermore, two new lower bounds of the minimum eigenvalue of the Fan product of several M-matrices are proposed. These obtained new lower bounds generalize and improve some earlier findings. One example is presented to illustrate the precision of the given lower bounds.

Graph Laplace Regularization-based pressure sensor placement strategy for leak localization in the water distribution networks under joint hydraulic and topological feature spaces

Urban water distribution networks (WDNs) have wide range and intricate topology, which include leakage, pipe burst and other abnormal states during production and operation. With the continuous development of the Internet of Things (IoT) technology in recent years, the means of monitoring the WDNs by using wireless sensor network technology has gradually received attention and extensive research. Most of the existing researches select the deployment location of sensors according to the hydraulic state of the WDNs, but the connectivity and topology between the nodes of the WDNs are not fully considered and analyzed. In this study, a new method that can integrate the topological features and hydraulic model information of the WDN is proposed to solve the problem of optimal sensor placement. First, the method preprocesses the covariance matrix of the pressure sensitivity matrix of the water distribution network by a diffusion kernel-based data prefiltering method and obtains the new network topology weights and its Laplacian matrix under the constraints of the network topology through a data-based graphical Laplacian learning method. Then, the sensor placement problem is transformed into a matrix minimum eigenvalue constraint problem by the Graph Laplace Regularization (GLR)-based method, and finally the selection of sensor nodes is accomplished by the method based on Gershgorin Disc Alignment (GDA). The proposed strategy is tested on a passive Hanoi network, an active Net 3 network, and a larger network, PA2, and is compared with some existing methods. The results show that the proposed solution achieves good performance in three different leak localization methods.