Eigenvalues Of Modulus Research Articles

Random and mean Lyapunov exponents for

AbstractWe consider orthogonally invariant probability measures on $\operatorname {\mathrm {GL}}_n(\mathbb {R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices with the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu and Shub [On random and mean exponents for unitarily invariant probability measures on $\operatorname {\mathrm {GL}}_n(\mathbb {C})$ . Astérisque287 (2003), xvii, 1–18]. A novel feature of our treatment is the use of the theory of spherical polynomials in the proof of our main result.

Vector Autoregression for Forecasting the Number of COVID-19 Cases and Analyzing Behavioral Indicators in the Philippines: Ecologic Time-Trend Study.

Traditional surveillance systems rely on routine collection of data. The inherent delay in retrieval and analysis of data leads to reactionary rather than preventive measures. Forecasting and analysis of behavior-related data can supplement the information from traditional surveillance systems. We assessed the use of behavioral indicators, such as the general public's interest in the risk of contracting SARS-CoV-2 and changes in their mobility, in building a vector autoregression model for forecasting and analysis of the relationships of these indicators with the number of COVID-19 cases in the National Capital Region. An etiologic, time-trend, ecologic study design was used to forecast the daily number of cases in 3 periods during the resurgence of COVID-19. We determined the lag length by combining knowledge on the epidemiology of SARS-CoV-2 and information criteria measures. We fitted 2 models to the training data set and computed their out-of-sample forecasts. Model 1 contains changes in mobility and number of cases with a dummy variable for the day of the week, while model 2 also includes the general public's interest. The forecast accuracy of the models was compared using mean absolute percentage error. Granger causality test was performed to determine whether changes in mobility and public's interest improved the prediction of cases. We tested the assumptions of the model through the Augmented Dickey-Fuller test, Lagrange multiplier test, and assessment of the moduli of eigenvalues. A vector autoregression (8) model was fitted to the training data as the information criteria measures suggest the appropriateness of 8. Both models generated forecasts with similar trends to the actual number of cases during the forecast period of August 11-18 and September 15-22. However, the difference in the performance of the 2 models became substantial from January 28 to February 4, as the accuracy of model 2 remained within reasonable limits (mean absolute percentage error [MAPE]=21.4%) while model 1 became inaccurate (MAPE=74.2%). The results of the Granger causality test suggest that the relationship of public interest with number of cases changed over time. During the forecast period of August 11-18, only change in mobility (P=.002) improved the forecasting of cases, while public interest was also found to Granger-cause the number of cases during September 15-22 (P=.001) and January 28 to February 4 (P=.003). To the best of our knowledge, this is the first study that forecasted the number of COVID-19 cases and explored the relationship of behavioral indicators with the number of COVID-19 cases in the Philippines. The resemblance of the forecasts from model 2 with the actual data suggests its potential in providing information about future contingencies. Granger causality also implies the importance of examining changes in mobility and public interest for surveillance purposes.

Application in Dynamic Stabilization System of Matrix Eigenvalue Estimation Based on Iterative Algorithm

The role of matrix eigenvalue in engineering technology and dynamic process and the necessity of eigenvalue estimation are described. Jury criterion and iterative recursion method are used to estimate the magnitude of matrix eigenvalue modulus, at last extending the application of eigenvalue to the discrete dynamic system.

A new empirical contribution to an old theoretical puzzle: what input–output matrix properties tells us about equilibrium prices and quantities

Eigenvalues of input-output matrices have significant implications for the structures of equilibrium prices and quantities. According to the Bródy Conjecture (BC), all subdominant eigenvalues of matrix would approach zero as matrix size approached infinity. Thus, any given initial quantity or price vector would converge to the corresponding equilibrium one in a single step. This paper adds significant empirical evidence to this theoretical discussion. We create a database of 307 different sizes matrices ranging over 30 years. Contrary to BC, we find that: the coefficient of variation and the subdominant eigenvalue moduli rise with matrix size; there’s a universal rank-size curve of eigenvalue moduli, but it is smooth and convex rather than L-shaped; the distribution of eigenvalue moduli is best fit by a Weibull probability distribution; the Weibull quantile function in turn yields a power law for eigenvalue moduli which is a better fit than a previously proposed exponential function.

An interval-valued linguistic Markov decision model with fast convergency

To rapidly generate stable solutions to the interval-valued linguistic decision-making problems, this paper proposes a novel interval-valued linguistic Markov decision model with fast convergency. A mapping framework is constructed to associate an interval-valued linguistic decision-making problem with a Markov chain. In consideration of two types of decision parameters including criterion weight and criterion reliability, two pairs of optimization problems are constructed under the framework to determine the optimized criterion weight interval and criterion reliability interval. A similarity measure between two rectangles formed by two pairs of optimized criterion weight and criterion reliability intervals is defined, and its relevant properties are theoretically proven. Based on the similarity measure, the transition probability of an alternative from one interval-valued linguistic term into another is constructed from the abstract combination of the developed similarity measure and an adaption parameter for fast convergency. The second largest eigenvalue modulus is then minimized to accelerate the generation of stable solutions, in which the corresponding adaption parameter is determined. The proposed decision model is used to help select suppliers of enterprise resource planning system for an enterprise that manufactures the key parts of high-speed trains, in which its applicability and validity are demonstrated. The necessity of the proposed decision model is further highlighted by comparative experiments.

Seeking Tracking Consensus for General Linear Multiagent Systems With Fixed and Switching Signed Networks.

The existing studies for tracking consensus of multiagent systems (MASs) are all restricted to networks with only cooperative relationships among agents. Tracking consensus, however, requires beyond these traditional models due to the ubiquitous competition in many real-world MASs, such as biological systems and social systems. Taking into account this fact, this article aims to extend the dynamics of tracking consensus to signed networks containing both cooperative and competitive relationships among agents. A group of agents with general linear dynamics is considered. The cases of the fixed network as well as switching networks are analyzed, respectively. In the end, some algebraic conditions related to the network structure and the positive/negative edge weight are established to ensure the implementation of tracking consensus. Moreover, the single decoupling system is allowed to be strictly unstable in theory, and the upper bound of the eigenvalue modulus of the system matrix related to the system instability is given.

Accelerating abelian random walks with hyperbolic dynamics

Given integers $$d \ge 2, n \ge 1$$ , we consider affine random walks on torii $$(\mathbb {Z}/ n \mathbb {Z})^{d}$$ defined as $$X_{t+1} = A X_{t} + B_{t} \mod n$$ , where $$A \in \mathrm {GL}_{d}(\mathbb {Z})$$ is a invertible matrix with integer entries and $$(B_{t})_{t \ge 0}$$ is a sequence of iid random increments on $$\mathbb {Z}^{d}$$ . We show that when A has no eigenvalues of modulus 1, this random walk mixes in $$O(\log n \log \log n)$$ steps as $$n \rightarrow \infty $$ , and mixes actually in $$O(\log n)$$ steps only for almost all n. These results are similar to those of Chung et al. (Ann Probab 15(3):1148–1165, 1987) on the so-called Chung–Diaconis–Graham process, which corresponds to the case $$d=1$$ . Our proof is based on the initial arguments of Chung, Diaconis and Graham, and relies extensively on the properties of the dynamical system $$x \mapsto A^{\top } x$$ on the continuous torus $$\mathbb {R}^{d} / \mathbb {Z}^{d}$$ . Having no eigenvalue of modulus one makes this dynamical system a hyperbolic toral automorphism, a typical example of a chaotic system known to have a rich behaviour. As such our proof sheds new light on the speed-up gained by applying a deterministic map to a Markov chain.

A governing equation for rotor and wavelet number in human clinical ventricular fibrillation: Implications for sudden cardiac death.

Ventricular fibrillation (VF) is characterized by multiple wavelets and rotors. No equation to predict the number of rotors and wavelets observed during fibrillation has been validated in human VF. The purpose of this study was to test the hypothesis that a single equation derived from a Markov M/M/∞ birth-death process could predict the number of rotors and wavelets occurring in human clinical VF. Epicardial induced VF (256-electrode) recordings obtained from patients undergoing cardiac surgery were studied (12 patients; 62 epochs). Rate constants for phase singularity (PS) (which occur at the pivot points of rotors) and wavefront (WF) formation and destruction were derived by fitting distributions to PS and WF interformation and lifetimes. These rate constants were combined in an M/M/∞ governing equation to predict the number of PS and WF in VF episodes. Observed distributions were compared to those predicted by the M/M/∞ equation. The M/M/∞ equation accurately predicted average PS and WF number and population distribution, demonstrated in all epochs. Self-terminating episodes of VF were distinguished from VF episodes requiring termination by a trend toward slower PS destruction, slower rates of PS formation, and a slower mixing rate of the VF process, indicated by larger values of the second largest eigenvalue modulus of the M/M/∞ birth-death matrix. The longest-lasting PS (associated with rotors) had shorter interactivation time intervals compared to shorter-lasting PS lasting <150 ms (∼1 PS rotation in human VF). The M/M/∞ equation explains the number of wavelets and rotors observed, supporting a paradigm of VF based on statistical fibrillatory dynamics.

Polynomial eigenvalue estimation: numerical radii versus norms

Bounds on the moduli of polynomial eigenvalues are typically expressed in terms of norms of the coefficient matrices of the polynomials. It is shown that using the numerical radii of these matrices instead of their norms can lead to improvements of those bounds and also, in special cases, to information on the angular distribution of the eigenvalues.

ON FRACTIONAL ORDER MAPS AND THEIR SYNCHRONIZATION

We study the stability of linear fractional order maps. We show that in the stable region, the evolution is described by Mittag-Leffler functions and a well-defined effective Lyapunov exponent can be obtained in these cases. For one-dimensional systems, this exponent can be related to the corresponding fractional differential equation. A fractional equivalent of map [Formula: see text] is stable for [Formula: see text] where [Formula: see text] is a fractional order parameter and [Formula: see text]. For coupled linear fractional maps, we can obtain ‘normal modes’ and reduce the evolution to an effective one-dimensional system. If the coefficient matrix has real eigenvalues, the stability of the coupled system is dictated by the stability of effective one-dimensional normal modes. If the coefficient matrix has complex eigenvalues, we obtain a much richer picture. However, in the stable region, evolution is dictated by a complex effective Lyapunov exponent. For larger [Formula: see text], the effective Lyapunov exponent is determined by modulus of eigenvalues. We extend these studies to fixed points of fractional nonlinear maps.

Eigenvalue contour lines of Kac–Murdock–Szegő matrices with a complex parameter

A previous paper studied the so-called borderline curves of the Kac–Murdock–Szegő matrix Kn(ρ)=[ρ|j−k|]j,k=1n, where ρ∈C. These are the level curves (contour lines) in the complex-ρ plane on which Kn(ρ) has a type-1 or type-2 eigenvalue of modulus n, where n is the matrix dimension. Those curves have cusps at all critical points ρ=ρc at which multiple (double) eigenvalues occur. The present paper determines corresponding curves pertaining to eigenvalues of modulus ν≠n. We find that these curves no longer present cusps; and that, when ν<n, the cusps have in a sense transformed into loops. We discuss the meaning of the winding numbers of our curves. Finally, we point out possible extensions to more general matrices.

Spectral analysis of dispersive shocks for quantum hydrodynamics with nonlinear viscosity

In this paper, we investigate spectral stability of traveling wave solutions to 1D quantum hydrodynamics system with nonlinear viscosity in the [Formula: see text], that is, density and velocity, variables. We derive a sufficient condition for the stability of the essential spectrum and we estimate the maximum modulus of eigenvalues with non-negative real part. In addition, we present numerical computations of the Evans function in sufficiently large domain of the unstable half-plane and show numerically that its winding number is (approximately) zero, thus giving a numerical evidence of point spectrum stability.

Real-World Networks Are Not Always Fast Mixing

Abstract The mixing time of random walks on a graph has found broad applications across both theoretical and practical aspects of computer science, with the application effects depending on the behavior of mixing time. It is extensively believed that real-world networks, especially social networks, are fast mixing with their mixing time at most $O(\log N)$ where $N$ is the number of vertices. However, the behavior of mixing time in the real-life networks has not been examined carefully, and exactly analytical research for mixing time in models mimicking real networks is still lacking. In this paper, we first experimentally evaluate the mixing time of various real-world networks with scale-free small-world properties and show that their mixing time is much higher than anticipated. To better understand the behavior of the mixing time for real-world networks, we then analytically study the mixing time of the Apollonian network, which is simultaneously scale-free and small-world. To this end, we derive the recursive relations for all eigenvalues, especially the second largest eigenvalue modulus of the transition matrix, based on which we deduce a lower bound for the mixing time of the Apollonian network, which approximately scales sublinearly with $N$. Our results indicate that real-world networks are not always fast mixing, which has potential implications in the design of algorithms related to mixing time.

Small C1 actions of semidirect products on compact manifolds

Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a compact, orientable surface $S$ with monodromy $\psi$, and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $\psi^*$ on $H^1(S,\mathbb{R})$ has no eigenvalues on the unit circle, then there exists a neighborhood $\mathcal U$ of the trivial action in the space of $C^1$ actions of $\pi_1(T)$ on $M$ such that any action in $\mathcal{U}$ is abelian. We will prove that the same result holds in the generality of an infinite cyclic extension of an arbitrary finitely generated group $H$, provided that the conjugation action of the cyclic group on $H^1(H,\mathbb{R})\neq 0$ has no eigenvalues of modulus one. We thus generalize a result of A. McCarthy, which addressed the case of abelian--by--cyclic groups acting on compact manifolds.

The central limit theorem for eigenvalues

We prove that the spectral radius of a strongly irreducible random walk on GLd(R) (or more generally the vector of moduli of eigenvalues of a Zariski-dense random walk on a reductive group) satisfies a central limit theorem under an order two moment assumption.

Bounds for the Right Spectral Radius of Quaternionic Matrices

UDC 517.5 In this paper we present bounds for the sum of the moduli of right eigenvalues of a quaternionic matrix. As a consequence, we obtain bounds for the right spectral radius of a quaternionic matrix. We also present a minimal ball in 4D spaces which contains all the Gersgorin balls of a quaternionic matrix. As an application, we introduce the estimation for the right ˇ eigenvalues of quaternionic matrices in the minimal ball. Finally, we suggest some numerical examples to illustrate of our results.

Edge Scaling Limit of the Spectral Radius for Random Normal Matrix Ensembles at Hard Edge

We investigate a random normal matrix model with eigenvalues forced to be in the droplet, the support of the equilibrium measure associated with an external field. For radially symmetric external fields, we show that the fluctuations of the spectral radius around a hard edge tend to follow an exponential distribution as the number of eigenvalues tends to infinity. As a corollary, we obtain the order statistics of the moduli of eigenvalues.

Upper Bounds for the Largest Singular Value of Certain Digraph Matrices

In this paper, we consider digraphs with possible loops and the particular case of oriented graphs, i.e. loopless digraphs with at most one oriented edge between every pair of vertices. We provide an upper bound for the largest singular value of the skew Laplacian matrix of an oriented graph, the largest singular value of the skew adjacency matrix of an oriented graph and the largest singular value of the adjacency matrix of a digraph. These bounds are expressed in terms of certain parameters related to vertex degrees. We also consider some bounds for the sums of squares of singular values. As an application, for the skew (Laplacian) adjacency matrix of an oriented graph and the adjacency matrix of a digraph, we derive some upper bounds for the spectral radius and the sums of squares of moduli of eigenvalues.

Bounds for the right spectral radius of quaternionicmatrices

UDC 517.5 In this paper we present bounds for the sum of the moduli of right eigenvalues of a quaternionic matrix. As a consequence, we obtain bounds for the right spectral radius of a quaternionic matrix. We also present a minimal ball in 4D spaces which contains all the Gersgorin balls of a quaternionic matrix. As an application, we introduce the estimation for the right ˇ eigenvalues of quaternionic matrices in the minimal ball. Finally, we suggest some numerical examples to illustrate of our results.

Thermodynamics of the Spin-1/2 Heisenberg–Ising Chain at High Temperatures: a Rigorous Approach

This work develops a rigorous setting allowing one to prove several features related to the behaviour of the Heisenberg–Ising (or XXZ) spin-1/2 chain at finite temperature T. Within the quantum inverse scattering method the physically pertinent observables at finite T, such as the per-site free energy or the correlation length, have been argued to admit integral representations whose integrands are expressed in terms of solutions to auxiliary non-linear integral equations. The derivation of such representations was based on numerous conjectures: the possibility to exchange the infinite volume and the infinite Trotter number limits, the existence of a real, non-degenerate, maximal in modulus Eigenvalue of the quantum transfer matrix, the existence and uniqueness of solutions to the auxiliary non-linear integral equations, as well as the possibility to take the infinite Trotter number limit on their level. We rigorously prove all these conjectures for temperatures large enough. As a by product of our analysis, we obtain the large-T asymptotic expansion for a subset of sub-dominant Eigenvalues of the quantum transfer matrix and thus of the associated correlation lengths. This result was never obtained previously, not even on heuristic grounds.