Eigenvalues Λ1 Research Articles

Extreme values of the Fiedler vector on trees

Let G be a tree on n vertices and let L=D−A denote the Laplacian matrix on G. The second-smallest eigenvalue λ2(G)>0, also known as the algebraic connectivity, as well as the associated eigenvector have been of substantial interest. We investigate the question of when the maxima and minima of an associated eigenvector are assumed at the endpoints of the longest path in G. Our results also apply to more general graphs that ‘behave globally’ like a tree but can exhibit more complicated local structure. The crucial new ingredient is a reproducing formula for eigenvectors of graphs.

Relationships between quantitative magnetic resonance imaging measures at the time of return to sport and clinical outcomes following acute hamstring strain injury

Hamstring strain injuries (HSI) are a common occurrence in athletics and complicated by high rates of reinjury. Evidence of remaining injury observed on magnetic resonance imaging (MRI) at the time of return to sport (RTS) may be associated with strength deficits and prognostic for reinjury, however, conventional imaging has failed to establish a relationship. Quantitative measure of muscle microstructure using diffusion tensor imaging (DTI) may hold potential for assessing a possible association between injury-related structural changes and clinical outcomes. The purpose of this study was to determine the association of RTS MRI-based quantitative measures, such as edema volume, muscle volume, and DTI metrics, with clinical outcomes (i.e., strength and reinjury) following HSI. Spearman’s correlations and Firth logistic regressions were used to determine relationships in between-limb imaging measures and between-limb eccentric strength and reinjury status, respectively. Twenty injuries were observed, with four reinjuries. At the time of RTS, between-limb differences in eccentric hamstring strength were significantly associated with principal effective diffusivity eigenvalue λ1 (r = -0.64, p = 0.003) and marginally associated with mean diffusivity (r = -0.46, p = 0.056). Significant relationships between other MRI-based measures of morphology and eccentric strength were not detected, as well as between any MRI-based measure and reinjury status. In conclusion, this preliminary evidence indicates DTI may track differences in hamstring muscle microstructure, not captured by conventional imaging at the whole muscle level, that relate to eccentric strength.

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.

The Existence and Uniqueness of Radial Solutions for Biharmonic Elliptic Equations in an Annulus

This paper concerns with the existence of radial solutions of the biharmonic elliptic equation ▵2u=f(|x|,u,|∇u|,▵u) in an annular domain Ω={x∈RN:r1&lt;|x|&lt;r2}(N≥2) with the boundary conditions u|∂Ω=0 and ▵u|∂Ω=0, where f:[r1,r2]×R×R+×R→R is continuous. Under certain inequality conditions on f involving the principal eigenvalue λ1 of the Laplace operator −▵ with boundary condition u|∂Ω=0, an existence result and a uniqueness result are obtained. The inequality conditions allow for f(r,ξ,ζ,η) to be a superlinear growth on ξ,ζ,η as |(ξ,ζ,η)|→∞. Our discussion is based on the Leray–Schauder fixed point theorem, spectral theory of linear operators and technique of prior estimates.

Effect of temperature and magnetic field induced hysteresis on reversibility of magnetocaloric effect and its minimization by optimizing the geometrical compatibility condition in Mn–Ni–Fe–Si alloy

The advancement of magnetic materials with coupled magneto-structural phase transition (MST) to fulfill the ultimate objectives of practical solid-state cooling applications requires a better understanding of the hysteresis phenomenon linked across the phase transition region along with the large magnetocaloric parameters. For the present sample Mn0.65Ni0.65Fe0.70Si, the MST is associated with a sharp jump in magnetization along with a small thermal hysteresis of ∼13 K. A giant isothermal magnetic entropy change (|ΔSMmax|) of ∼37.6 J kg−1 K−1 at 299 K and effective refrigerant capacity (RCeffe) of ∼214.3 J kg−1 under ΔH = 30 kOe is obtained with excellent compatibility between the martensite and austenite phases. The geometrical compatibility condition, i.e., very small (∼0.55%) deviation of the middle eigenvalue (λ2) from unity justifies the observation of small hysteresis in the present material. The investigation of hysteresis behavior under different extents of the driving forces (temperature or magnetic field) reveals that both the driving forces trigger equally the phase transition and are responsible equivalently for the hysteresis phenomenon. The present study provides a pathway to understand the complexity of the hysteresis behavior, its impact on the reversibility of magnetocaloric effect, and its minimization by optimizing the geometrical compatibility condition between the austenite and martensite phases.

Analysis of Epidemic Models in Complex Networks and Node Isolation Strategie Proposal for Reducing Virus Propagation

Many models of virus propagation in Computer Networks inspired by epidemic disease propagation mathematical models that can be found in the epidemiology field (SIS, SIR, SIRS, etc.) have been proposed in the last two decades. The purpose of these models has been to determine the conditions under which a virus becomes rapidly extinct in a network. The most common models of virus propagation in networks are SIS-type models or their variants. In such models, the conditions that lead to a rapid extinction of the spread of a computer virus have been calculated and its dependence on some parameters inherent to the mathematical model has been observed. In this article, we will try to analyze a particular SIS-type model proposed in the past by Chakrabarti as well as an SIRS-type variation of this model proposed in the past by myself. I will show through simulations the influence that the topology of a network has on the dynamics of the spread of a virus in different network types. In the recent past, there have been interesting articles that demonstrate the relationship between the eigenvalue λ1 of the adjacency matrix and the reduction in the spread of a virus in a network. From this, the minimization of the spectral radius strategies by edge suppression has been proposed. This problem is NP-complete in its general case and for this reason, heuristic algorithms have been proposed. In this article, I will perform simulations of an SIS-type model in topologies with the same number of nodes but with different structures to compare their epidemic behavior. The simulations will show that regular topologies with small node degrees, i.e., of degree 4, as is the case of the topology that I call Lattice4, have favorable behavior in terms of the fast extinction property, with respect to other denser and less regular topologies such as the binomial topologies as well as Power law topologies. Based on the results of the simulations, my contribution will consist of proposing, as a node isolation strategy, a transformation of the original topology into an approximately regular topology by edge elimination. Although such a transformed topology is not optimal in terms of reducing the propagation of a virus, it induces the rapid extinction of the virus in the network.

Changes in lumbar muscle diffusion tensor indices with age.

To investigate differences in diffusion tensor imaging (DTI) parameters and proton density fat fraction (PDFF) in the spinal muscles of younger and older adult males. Twelve younger (19-30 years) and 12 older (61-81years) healthy, physically active male participants underwent T1W, T2W, Dixon and DTI of the lumbar spine. The eigenvalues (λ1, λ2, and λ3), fractional anisotropy (FA), and mean diffusivity (MD) from the DTI together with the PDFF were determined in the multifidus, medial and lateral erector spinae (ESmed, ESlat), and quadratus lumborum (QL) muscles. A two-way ANOVA was used to investigate differences with age and muscle and t-tests for differences in individual muscles with age. The ANOVA gave significant differences with age for all DTI parameters and the PDFF (P < .01) and with muscle (P < .01) for all DTI parameters except for λ1 and for the PDFF. The mean of the eigenvalues and MD were lower and the FA higher in the older age group with differences reaching statistical significance for all DTI measures for ESlat and QL (P < .01) but only in ESmed for λ3 and MD (P < .05). Differences in DTI parameters of muscle with age result from changes in both in the intra- and extra-cellular space and cannot be uniquely explained in terms of fibre length and diameter. Previous studies looking at age have used small groups with uneven age spacing. Our study uses two well defined and separated age groups.

Evolution of the first eigenvalue along the inverse mean curvature flow in space forms

In this paper, we show that the first nonzero eigenvalues λ1 of the Laplacian and the p-Laplacian are decreasing along the inverse mean curvature flow in the hyperbolic space and in the sphere. For a convex closed surface M⊂S3, we show that the geometric quantity λ1⋅Area(M) is monotonically decreasing along the inverse mean curvature flow in S3.

Quantitative bounds for unconditional pairs of frames

We formulate a quantitative finite-dimensional conjecture about frame multipliers and prove that it is equivalent to Conjecture 1 in [22]. We then present solutions to the conjecture for certain classes of frame multipliers. In particular, we prove that for all C>0 and N∈N the following is true: Let (xj)j=1N and (fj)j=1N be sequences in a finite dimensional Hilbert space which satisfy ‖xj‖=‖fj‖ for all 1≤j≤N and‖∑j=1Nεj〈x,fj〉xj‖≤C‖x‖,for all x∈ℓ2M and |εj|=1. If the frame operator for (fj)j=1N has eigenvalues λ1≥...≥λM and β>0 is such that λ1≤βM−1∑j=1Mλj then (fj)j=1N has Bessel bound 27β2C. The same holds for (xj)j=1N.

Connected [formula omitted]-free graphs whose second largest eigenvalue does not exceed 1

Connected graphs whose second largest eigenvalue λ2 does not exceed 1 have been investigated in the last four decades. Over the years only few particular classes with this spectral property are completely determined. For example, they include bipartite graphs, line graphs and threshold graphs. In this paper we determine all connected (K4−e)-free graphs, i.e., the graphs that do not contain an induced subgraph isomorphic to the graph obtained by removing an edge from the complete graph of order 4; such graphs are also called diamond-free graphs. Since every triangle-free graph is automatically diamond-free, our results includes all triangle-free graphs with λ2≤1. Moreover, it includes all connected bipartite graphs with the same spectral property, and therefore strengthens the result of Petrović (J. Combin. Theory B, 1991).

Time-dependent diffusion tensor imaging and diffusion modeling of age-related differences in the medial gastrocnemius and feasibility study of correlations to histopathology.

Implement STEAM-DTI to model time-dependent diffusion eigenvalues using the random permeable barrier model (RPBM) to study age-related differences in the medial gastrocnemius (MG) muscle. Validate diffusion model-extracted fiber diameter for histological assessment. Diffusion imaging at different diffusion times (Δ) was performed on seven young and six senior participants. Time-dependent diffusion eigenvalues (λ2 (t), λ3 (t), and D⊥ (t); average of λ2 (t) and λ3 (t)) were fit to the RPBM to extract tissue microstructure parameters. Biopsy of the MG tissue for histological assessment was performed on a subset of participants (four young, six senior). λ3 (t) was significantly higher in the senior cohort for the range of diffusion times. RPBM fits to λ2 (t) yielded fiber diameters in agreement to those from histology for both cohorts. The senior cohort had lower values of volume fraction of membranes, ζ, in fits to λ2 (t), λ3 (t), and D⊥ (t) (significant for fit to λ3 (t)). Fits of fiber diameter from RPBM to that from histology had the highest correlation for the fit to λ2 (t). The age-related patterns in λ2 (t) and λ3 (t) could tentatively be explained from RPBM fits; these patterns may potentially arise from a decrease in fiber asymmetry and an increase in permeability with age.

A universal description of stochastic oscillators

Many systems in physics, chemistry, and biology exhibit oscillations with a pronounced random component. Such stochastic oscillations can emerge via different mechanisms, for example, linear dynamics of a stable focus with fluctuations, limit-cycle systems perturbed by noise, or excitable systems in which random inputs lead to a train of pulses. Despite their diverse origins, the phenomenology of random oscillations can be strikingly similar. Here, we introduce a nonlinear transformation of stochastic oscillators to a complex-valued function [Formula: see text](x) that greatly simplifies and unifies the mathematical description of the oscillator's spontaneous activity, its response to an external time-dependent perturbation, and the correlation statistics of different oscillators that are weakly coupled. The function [Formula: see text] (x) is the eigenfunction of the Kolmogorov backward operator with the least negative (but nonvanishing) eigenvalue λ1 = μ1 + iω1. The resulting power spectrum of the complex-valued function is exactly given by a Lorentz spectrum with peak frequency ω1 and half-width μ1; its susceptibility with respect to a weak external forcing is given by a simple one-pole filter, centered around ω1; and the cross-spectrum between two coupled oscillators can be easily expressed by a combination of the spontaneous power spectra of the uncoupled systems and their susceptibilities. Our approach makes qualitatively different stochastic oscillators comparable, provides simple characteristics for the coherence of the random oscillation, and gives a framework for the description of weakly coupled oscillators.

Quantitative muscle MRI displays clinically relevant myostructural abnormalities in long-term ICU-survivors: a case–control study

BackgroundLong-term data on ICU-survivors reveal persisting sequalae and a reduced quality-of-life even after years. Major complaints are neuromuscular dysfunction due to Intensive care unit acquired weakness (ICUAW). Quantitative MRI (qMRI) protocols can quantify muscle alterations in contrast to standard qualitative MRI-protocols.MethodsUsing qMRI, the aim of this study was to analyse persisting myostructural abnormalities in former ICU patients compared to controls and relate them to clinical assessments. The study was conducted as a cohort/case–control study. Nine former ICU-patients and matched controls were recruited (7 males; 54.8y ± 16.9; controls: 54.3y ± 11.1). MRI scans were performed on a 3T-MRI including a mDTI, T2 mapping and a mDixonquant sequence. Water T2 times, fat-fraction and mean values of the eigenvalue (λ1), mean diffusivity (MD), radial diffusivity (RD) and fractional anisotropy (FA) were obtained for six thigh and seven calf muscles bilaterally. Clinical assessment included strength testing, electrophysiologic studies and a questionnaire on quality-of-life (QoL). Study groups were compared using a multivariate general linear model. qMRI parameters were correlated to clinical assessments and QoL questionnaire using Pearson´s correlation.ResultsqMRI parameters were significantly higher in the patients for fat-fraction (p < 0.001), water T2 time (p < 0.001), FA (p = 0.047), MD (p < 0.001) and RD (p < 0.001). Thighs and calves showed a different pattern with significantly higher water T2 times only in the calves. Correlation analysis showed a significant negative correlation of muscle strength (MRC sum score) with FA and T2-time. The results were related to impairment seen in QoL-questionnaires, clinical testing and electrophysiologic studies.ConclusionqMRI parameters show chronic next to active muscle degeneration in ICU survivors even years after ICU therapy with ongoing clinical relevance. Therefore, qMRI opens new doors to characterize and monitor muscle changes of patients with ICUAW. Further, better understanding on the underlying mechanisms of the persisting complaints could contribute the development of personalized rehabilitation programs.

More limiting distributions for eigenvalues of Wigner matrices

The Tracy–Widom distributions are among the most famous laws in probability theory, partly due to their connection with Wigner matrices. In particular, for A=1n(aij)1≤i,j≤n∈Rn×n symmetric with (aij)1≤i≤j≤n i.i.d. standard normal, the fluctuations of its largest eigenvalue λ1(A) are asymptotically described by a real-valued Tracy–Widom distribution TW1:n2/3(λ1(A)−2)⇒TW1. As it often happens, Gaussianity can be relaxed, and this results holds when E[a11]=0, E[a112]=1 and the tail of a11 decays sufficiently fast: limx→∞x4P(|a11|>x)=0, whereas when the law of a11 is regularly varying with index α∈(0,4), ca(n)n1/2−2/αλ1(A) converges to a Fréchet distribution for ca:(0,∞)→(0,∞), slowly varying and depending solely on the law of a11. This paper considers a family of edge cases, limx→∞x4P(|a11|>x)=c∈(0,∞), and unveils a new type of limiting behavior for λ1(A): a continuous function of a Fréchet distribution in which 2, the almost sure limit of λ1(A) in the light-tailed case, plays a pivotal role: f(x)=2,0<x<1,x+1 x,x≥1.

The Mechanism of Slip System Activation With Grain Rotation During Superplastic Forming

Abstract The activated slip system of Ti-6Al-4V alloy during the superplastic forming (SPF) was investigated by the in-grain misorientation axes analysis (IGMA), and the mechanisms of slip system activation have been discussed. Depending on the distribution of IGMA, one significant discovery from this study is that all the basal, prismatic, and pyramidal slip systems would be activated. Considering the effective slip systems, Schmid factors, and the Euler angles together, it is suggested that the dominant slip systems not only desired the largest Schmid factors but strongly demand continuous Schmid factors among the adjacent grains. Meanwhile, the estimated critical resolved shear stress (CRSS) on basal &amp;lt;a&amp;gt; and prismatic &amp;lt;a&amp;gt; at the temperature of 920 °C with the strain rate of 10−3 s is given. An original method of roughly estimating dominant slip systems with Euler angles has been introduced, which predicts that grain rotation may change the slip system. Furthermore, the crystal plasticity finite element method (CPFEM) is employed to simulate the evolution of Euler angles, and the grain orientation presents the largest set of significant clusters around the (1¯100) after deformation. Besides, the continuity of the Schmid factor assumption for the activated slip system has also been verified by CPFEM. In addition, the eigenvector corresponding to the eigenvalue λ1 = 1 of Euler angle rotation matrix is calculated to be aligned with the grain rotation axis, which can be applied to describe the grain rotation.

Eigenvalues of the Dirac operator on compact spin manifolds under Ricci flow

In this paper, we first derive the evolution formula for the square of the first eigenvalue λ2 of the Dirac operator under the metric flow ∂∂tg=h on compact spin manifolds. We then prove that λ2 is nondecreasing under the Ricci flow on compact spin surfaces with non-negative Gauss curvature.

On computing root polynomials and minimal bases of matrix pencils

We revisit the notion of root polynomials, thoroughly studied in (Dopico and Noferini, 2020 [9]) for general polynomial matrices, and show how they can efficiently be computed in the case of a matrix pencil λE+A. The method we propose makes extensive use of the staircase algorithm, which is known to compute the left and right minimal indices of the Kronecker structure of the pencil. In addition, we show here that the staircase algorithm, applied to the expansion (λ−λ0)E+(A−λ0E), constructs a block triangular pencil from which a minimal basis and a maximal set of root polynomials at the eigenvalue λ0, can be computed in an efficient manner.

On the Sum and Spread of Reciprocal Distance Laplacian Eigenvalues of Graphs in Terms of Harary Index

The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G)=RT(G)−RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Clearly, RDL(G) is a real symmetric matrix, and we denote its eigenvalues as λ1(RDL(G))≥λ2(RDL(G))≥…≥λn(RDL(G)). The largest eigenvalue λ1(RDL(G)) of RDL(G), denoted by λ(G), is called the reciprocal distance Laplacian spectral radius. In this paper, we obtain several upper bounds for the sum of k largest reciprocal distance Laplacian eigenvalues of G in terms of various graph parameters, such as order n, maximum reciprocal distance degree RTmax, minimum reciprocal distance degree RTmin, and Harary index H(G) of G. We determine the extremal cases corresponding to these bounds. As a consequence, we obtain the upper bounds for reciprocal distance Laplacian spectral radius λ(G) in terms of the parameters as mentioned above and characterize the extremal cases. Moreover, we attain several upper and lower bounds for reciprocal distance Laplacian spread RDLS(G)=λ1(RDL(G))−λn−1(RDL(G)) in terms of various graph parameters. We determine the extremal graphs in many cases.

Exact eigenvalue order statistics for the reduced density matrix of a bipartite system

We consider the reduced density matrix ρa(m) of a bipartite system AB of dimensionality mn (n⩾m without loss of generality) in a Gaussian ensemble of random, complex pure states of the composite system. For a given dimensionality m of the subsystem A, the eigenvalues λ1(m),…,λm(m) of ρa(m) are correlated random variables because their sum equals unity. The following quantities are known, among others: The joint probability density function (PDF) of the eigenvalues λ1(m),…,λm(m) of ρa(m), the PDFs of the smallest eigenvalue λmin(m) and the largest eigenvalue λmax(m), and the family of average values 〈Tr(ρa(m))q〉 parametrized by q. Using these as inputs, we find the exact eigenvalue order statistics for any arbitrary value of m and n, i.e., explicit analytic expressions for the PDFs of each of the m eigenvalues arranged in ascending order from the smallest to the largest one. For the sake of clarity, we first present the eigenvalue order statistics for values of m running from 2 to 6, before going on to the general expressions. When m=n (respectively, m<n) these PDFs are polynomials of order m2−2 (respectively, mn−2) with support in specific sub-intervals of the unit interval, demarcated by appropriate unit step functions. Our exact results are fully corroborated by numerically generated histograms of the ordered set of eigenvalues corresponding to ensembles of over 105 random complex pure states of the bipartite system.

Screening for shape memory alloys with narrow thermal hysteresis using combined XGBoost and DFT calculation

Shape memory alloys (SMAs) are desirable candidates for elastocaloric effect materials, but they all suffer from large thermal hysteresis (Thys). This study analyzes multicomponent TiNi-based SMAs dataset by machine learning (ML) to explore new SMAs with narrow Thys. The second-largest eigenvalue λ2 of the stretch transformation matrix U is added to the original dataset to guide the ML process as a feature. Firstly, λ2 is obtained by first-principles calculations combined with ML. XGBoost Regressor (XGBR) combined with Leave-One-Out Cross-Validation (LOO-CV) is selected from four algorithms for modeling with the highest coefficient of determination R2 of 0.87. The introduction of λ2 improves the performance of the model. The dataset is divided into 15 groups based on different doping elements (such as Hf, Cu, Zr, etc.), among which TiNiCu is the most predictive component with the R2 of 0.89. Over 500 TiNiCu components are randomly generated and predicted Thys. Based on the contour maps created from the prediction results, it is found that Thys is likely to decrease with the increase of Cu doping in general, and minimum Thys occurs when the Cu is about 15 at. %, which is consistent with the existing experimental results. Eventually, a potential Thys minimum (1.2 K) region of TixNiyCuz (58.3%≤x ≤ 58.5%, 26.5%≤y ≤ 27%, 14.8%≤z ≤ 15.3%, x + y + z = 100%) SMA composition is predicted. Our study not only provides a potential selection of narrow Thys TiNi-based SMAs but also indicates combining of XGBoost and DFT calculation is an effective strategy for materials design.