Realization Of Matrix Research Articles

Equilibrium and surviving species in a large Lotka-Volterra system of differential equations.

Lotka-Volterra (LV) equations play a key role in the mathematical modeling of various ecological, biological and chemical systems. When the number of species (or, depending on the viewpoint, chemical components) becomes large, basic but fundamental questions such as computing the number of surviving species still lack theoretical answers. In this paper, we consider a large system of LV equations where the interactions between the various species are a realization of a random matrix. We provide conditions to have a unique equilibrium and present a heuristics to compute the number of surviving species. This heuristics combines arguments from Random Matrix Theory, mathematical optimization (LCP), and standard extreme value theory. Numerical simulations, together with an empirical study where the strength of interactions evolves with time, illustrate the accuracy and scope of the results.

Root vectors of polynomial and rational matrices: Theory and computation

The notion of root polynomials of a polynomial matrix P(λ) was thoroughly studied in Dopico and Noferini (2020) [6]. In this paper, we extend such a systematic approach to general rational matrices R(λ), possibly singular and possibly with coalescent pole/zero pairs. We discuss the related theory for rational matrices with coefficients in an arbitrary field. As a byproduct, we obtain sensible definitions of eigenvalues and eigenvectors of a rational matrix R(λ), without any need to assume that R(λ) has full column rank or that the eigenvalue is not also a pole. Then, we specialize to the complex field and provide a practical algorithm to compute them, based on the construction of a minimal state space realization of the rational matrix R(λ) and then using the staircase algorithm on the linearized pencil to compute the null space as well as the root polynomials in a given point λ0. If λ0 is also a pole, then it is necessary to apply a preprocessing step that removes the pole while making it possible to recover the root vectors of the original matrix: in this case, we study both the relevant theory (over a general field) and an algorithmic implementation (over the complex field), still based on minimal state space realizations.

A digraph approach to the state-space model realization of MIMO non-commensurate fractional order systems

This paper proposes a novel approach that can generate a state-space model with low inner dimension for an MIMO non-commensurate fractional order (NCFO) system. Specifically, the notion of an admissible digraph is firstly introduced associated with a fractional order transfer (function column) vector. Then, new state-space model realization conditions and corresponding procedures based on this admissible digraph are proposed for the state-space model realization of an NCFO polynomial transfer matrix. Finally, a new necessary and sufficient state-space model realization condition is proposed for the rational transfer matrix of an MIMO NCFO system, and it is shown, based on a matrix fractional description (MFD) of the given rational transfer matrix, a state-space model realization can be obtained by firstly converting it to the polynomial case and then utilizing the digraph approach for polynomial case. Symbolic and numerical examples are provided to demonstrate the main ideas and effectiveness of the proposed digraph approach.

Planar Compliance Realization With Two 3-Joint Serial Mechanisms

Abstract In this paper, the realization of any specified planar compliance with two 3R serial elastic mechanisms is addressed. Using the concept of dual elastic mechanisms, it is shown that the realization of a compliant behavior with two serial mechanisms connected in parallel is equivalent to its realization with a 6-spring fully parallel mechanism. Since the spring axes of a 6-spring parallel mechanism indicate the geometry of a dual 3R serial mechanism, a new synthesis procedure for the realization of a stiffness matrix with a 6-spring parallel mechanism is first developed. Then, this result is extended to a geometric construction-based synthesis procedure for two 3-joint serial mechanisms.

Noise sensitivity for the top eigenvector of a sparse random matrix

We investigate the noise sensitivity of the top eigenvector of a sparse random symmetric matrix. Let v be the top eigenvector of an N×N sparse random symmetric matrix with an average of d non-zero centered entries per row. We resample k randomly chosen entries of the matrix and obtain another realization of the random matrix with top eigenvector v[k]. Building on recent results on sparse random matrices and a noise sensitivity analysis previously developed for Wigner matrices, we prove that, if d≥N2∕9, with high probability, when k≪N5∕3, the vectors v and v[k] are almost collinear and, on the contrary, when k≫N5∕3, the vectors v and v[k] are almost orthogonal. A similar result holds for the eigenvector associated to the second largest eigenvalue of the adjacency matrix of an Erdős-Rényi random graph with average degree d≥N2∕9.

A new approach for finite element based reliability evaluation of offshore corroded pipelines

Internal corrosion is recognized as a serious treat in offshore pipeline services which are supposed to transport vast quantities of oil and natural gases over many long and complicated subsea routes. This paper aims to present a new approach for reliability evaluation of pipelines against corrosion using finite element simulation. In this process the effective random variables are selected by an efficient sensitivity analysis named Tornado Diagram Analysis. Afterwards, a combination of Latin Hypercube Sampling method along with Simulated Annealing technique is used to generate the random simulations which represent the existent epistemic uncertainties. Each realization of this simulation matrix is introduced into the numerical model as the necessary inputs to evaluate the probabilistic responses of the corroded pipeline from linear behavior to failure limit state. This methodology is implemented on a sample offshore pipeline and its results indicate that the maximum Von-Mises stress at all the internal pressure levels follows lognormal distribution. Additionally, as per the findings of this study, the deterministic analysis appears to predict the internal pressure corresponding to the failure limit state conservatively and, therefore, may lead to less remaining life and capacity for corroded pipelines. According to the estimated failure fragility curve, the failure probability of the aforementioned internal pressure level is around 65%.

Signaling Equilibria for Dynamic LQG Games With Asymmetric Information

In this article, we consider a finite-horizon dynamic game with two players who observe their types privately and take actions that are publicly observed. Players' types evolve as conditionally independent, controlled linear Gaussian processes, and players incur quadratic instantaneous costs. This forms a dynamic linear quadratic Gaussian game with asymmetric information. We show that under certain conditions, players' strategies that are linear in their private types, together with Gaussian beliefs, form a perfect Bayesian equilibrium (PBE) of the game. This is a signaling equilibrium due to the fact that future beliefs on players' types are affected by the equilibrium strategies. We provide a backward-forward algorithm to find the PBE. Each step of the backward algorithm reduces to solving an algebraic matrix equation for every possible realization of the state estimate covariance matrix. The forward algorithm consists of Kalman filter recursions, where state estimate covariance matrices depend on equilibrium strategies. We provide necessary and sufficient conditions for the existence of the solution of the algebraic matrix equation for scalar actions. Finally, we present numerical examples.

FSID - A Frequency Weighted MIMO Frequency Domain Identification and Rational Matrix Approximation Method for Python, Julia and Matlab

An open source toolbox, FSID, implemented in the Python, Julia and MATLAB programming languages is described. The toolbox provides scripts which estimates linear multiinput multi-output state-space models from sample data using frequency-domain subspace algorithms. Algorithms which estimate models based on samples of the transfer function matrix as well as frequency domain input and output vectors are provided. The algorithms can be used for discrete-time models, continuous-time models as well as for approximation of rational matrices from samples corresponding to arbitrary points in the complex plane. The algorithms can handle frequency dependent weighting which enable to obtain approximate BLUE estimates. To reduce the computational complexity for the estimation algorithms, an accelerated algorithm is provided which evaluate the state-space realization of the transfer function matrix at arbitrary points.

Noise sensitivity of the top eigenvector of a Wigner matrix

We investigate the noise sensitivity of the top eigenvector of a Wigner matrix in the following sense. Let v be the top eigenvector of an \(N\times N\) Wigner matrix. Suppose that k randomly chosen entries of the matrix are resampled, resulting in another realization of the Wigner matrix with top eigenvector \(v^{[k]}\). We prove that, with high probability, when \(k \ll N^{5/3-o(1)}\), then v and \(v^{[k]}\) are almost collinear and when \(k\gg N^{5/3}\), then \(v^{[k]}\) is almost orthogonal to v.

A direct method to obtain a realization of a polynomial matrix and its applications

In this paper we present a Silverman-Ho algorithm-based method to obtain a realization of a polynomial matrix. This method provides the final formulation of a minimal realization directly from a full rank factorization of a specific given matrix. Also, some classical problems in control theory such as model reduction in singular systems or the positive realization problem in standard systems are solved with this method.

Realization of Photo-curing Gelatin Hydrogel using a Commercial Projector for Culturing Mesenchymal Cells

This paper investigates the realization of synthetic extracellular matrix with visible light photo-patterning gelatin in a simpler manner. A synthetic extracellular matrix provides an initial attachment for the seeded cells on the experimental substrate such as glass plate or multi-well until they realized their the natural extracellular matrix. Here, a commercial Digital Light Projector (DLP) was used to induce gelatin with Rose Bengal as a crosslinking agent to form the thin layer on the experimental substrate. Various gelatin concentrations from 2%-10% were exposed at different times in order to optimize the patterning process. A geometrical characterization on the patterned gelatin, such as contour measurement and resolution, were taken place. Results showed that the thickness of patterned gelatin was in the range of 10 µm - 60 µm depends on the exposure time of the DLP projector. Moreover, a visual method aided by the Fiji toolbox from NIH ImageJ image processing was used to observe the density and spatial arrangement of the cultured cells on the substrate. Ultimately, biocompatibility using MTT assay was also employed to confirm the viability of the cells on the gelatin substrate. The results show that we are able to control the physical and spatial arrangement of the gelatin substrate, and they with cell viability depend on 6 days of observation. It was found that the gelatin substrate provides faster growth on cultured cells compared to the control study. This finding leads to the possibility to realize the automation system in cell culture technology with an affordable investment.

Packing tree degree sequences

A degree sequence $D=(d_1,\dots, d_n)$ is a series of non-ne\-ga\-tive integers. A degree sequence is graphical if there exists a vertex labeled graph $G$ in which the degree of vertex $v_i$ is exactly $d_i$ for $i=1,\dots,n$. The graph $G$ is called a realization of $D$. The \emph{edge-disjoint realization problem} is the following: given a $c \times n$ degree matrix $D=\{(d_{1,1},\dots, d_{1,n} ),$ $(d_{2,1},\dots,d_{2,n} ),\dots, (d_{c,1},\dots, d_{c,n})\}$, in which each row of the matrix is a graphical degree sequence, decide if there exists pairwise edge-disjoint realizations of these degree sequences on the same vertex set $V=\{v_1,v_2,\dots,v_n\}$. Such set of edge-disjoint graphs is called a realization of the degree matrix. A realization can also be presented as an edge colored simple graph, in which the edges with a given color form a realization of the degree sequence in a given row of the degree matrix. It is known that the edge-disjoint realization problem is $\NP$-complete even if the number of rows is three and the degrees on each vertex sum up to $n-1$, that is, when a decomposition of the complete graph is required into three subgraphs with prescribed degrees. Moreover, the problem is also $\NP$-complete when the number of rows is two and the sum of the degrees on some of the vertices is less than $n-1$. However, special cases that are computationally tractable are also of interest. A classical result of Kundu \cite{kundutree} shows that deciding if two \emph{tree degree sequences} have edge-disjoint realizations is in $\P$. Motivated by the aforementioned result, we consider special cases of the problem for two tree degree sequences. We show that if there is no node which have degree one in both degree sequences (that is, the requested trees do not share a common leaf node) then they always have edge-disjoint caterpillar realizations. By using a probabilistic method, we prove that two tree degree sequences always have edge-disjoint realizations if each vertex is a leaf in at least one of the trees. This theorem can be extended to more trees: we show that the edge-disjoint realization problem is in $\P$ for an arbitrary number of tree sequences with the property that each vertex is a non-leaf in at most one of the trees, i.e., in every column of the degree matrix there is at most one entry which is greater than one. We also consider the following variant of the edge-disjoint realization problem: given two graphical degree sequences $D_1$ and $D_2$ such that $D_2$ is a tree degree sequence, decide if there exists edge-disjoint realizations of $D_1$ and $D_2$ where the realization of $D_2$ does not need to be a tree. We show that this problem is already $\NP$-complete. Counting, or just estimating the number of distinct realizations of degree sequences is challenging in general. We show that efficient approximations for the number of solutions as well as an almost uniform sampler exist for two tree degree sequences if each vertex is a leaf in at least one of the trees.

All Passive Realization of Lossy Coupling Matrices Using Resistive Decomposition Technique

A complex coupling matrix has been extensively used in lossy filters and negative group delay devices. For the realization, conventional technique decomposes the complex coupling matrix into lossy resonators and complex inverters. Since the complex inverter does not follow the passivity in some cases, the resultant realization may be globally passive but locally active. This paper proposes a new decomposition technique to ensure the passivity everywhere. It decomposes the complex coupling matrix into a resistive connection matrix and a conventional real coupling matrix, which are both passively realizable. This technique provides a passive realization of the complex coupling matrix. Furthermore, a loss equalization technique is also proposed, to further achieve a uniform quality factor (Q) distribution among all the lossy resonators. Several illustrative examples and an experimental validation are finally provided.

Coherent chaos in a recurrent neural network with structured connectivity.

We present a simple model for coherent, spatially correlated chaos in a recurrent neural network. Networks of randomly connected neurons exhibit chaotic fluctuations and have been studied as a model for capturing the temporal variability of cortical activity. The dynamics generated by such networks, however, are spatially uncorrelated and do not generate coherent fluctuations, which are commonly observed across spatial scales of the neocortex. In our model we introduce a structured component of connectivity, in addition to random connections, which effectively embeds a feedforward structure via unidirectional coupling between a pair of orthogonal modes. Local fluctuations driven by the random connectivity are summed by an output mode and drive coherent activity along an input mode. The orthogonality between input and output mode preserves chaotic fluctuations by preventing feedback loops. In the regime of weak structured connectivity we apply a perturbative approach to solve the dynamic mean-field equations, showing that in this regime coherent fluctuations are driven passively by the chaos of local residual fluctuations. When we introduce a row balance constraint on the random connectivity, stronger structured connectivity puts the network in a distinct dynamical regime of self-tuned coherent chaos. In this regime the coherent component of the dynamics self-adjusts intermittently to yield periods of slow, highly coherent chaos. The dynamics display longer time-scales and switching-like activity. We show how in this regime the dynamics depend qualitatively on the particular realization of the connectivity matrix: a complex leading eigenvalue can yield coherent oscillatory chaos while a real leading eigenvalue can yield chaos with broken symmetry. The level of coherence grows with increasing strength of structured connectivity until the dynamics are almost entirely constrained to a single spatial mode. We examine the effects of network-size scaling and show that these results are not finite-size effects. Finally, we show that in the regime of weak structured connectivity, coherent chaos emerges also for a generalized structured connectivity with multiple input-output modes.

Third family quark–lepton unification at the TeV scale

We construct a model of quark–lepton unification at the TeV scale based on an SU(4) gauge symmetry, while still having acceptable neutrino masses and enough suppression in flavor changing neutral currents. An approximate U(2) flavor symmetry is an artifact of family-dependent gauge charges leading to a natural realization of the CKM mixing matrix. The model predicts sizeable violation of PMNS unitarity as well as a gauge vector leptoquark U1μ=(3,1,2/3) which can be produced at the LHC – both effects within the reach of future measurements. In addition, recently reported experimental anomalies in semi-leptonic B-meson decays, both in charged b→cτν and neutral b→sμμ currents, can be accommodated.

Unlabeled Sensing With Random Linear Measurements

We study the problem of solving a linear sensing system when the observations are unlabeled. Specifically we seek a solution to a linear system of equations ${\mathbf{y}}= {\mathbf{A}} {\mathbf{x}} $ when the order of the observations in the vector ${\mathbf{y}}$ is unknown. Focusing on the setting in which ${\mathbf{A}}$ is a random matrix with i.i.d. entries, we show that if the sensing matrix ${\mathbf{A}}$ admits an oversampling ratio of 2 or higher, then, with probability 1, it is possible to recover ${\mathbf{x}}$ exactly without the knowledge of the order of the observations in ${\mathbf{y}}$ . Furthermore, if ${\mathbf{x}}$ is of dimension $K$ , then any $2K$ entries of ${\mathbf{y}}$ are sufficient to recover ${\mathbf{x}}$ . This result implies the existence of deterministic unlabeled sensing matrices with an oversampling factor of 2 that admit perfect reconstruction. The result is universal in that conditioned on the realization of matrix ${\mathbf{A}}$ , recovery is guaranteed for all possible choices of ${\mathbf{x}}$ . While the proof is constructive, it uses a combinatorial algorithm which is not practical, leaving the question of complexity open. We also analyze a noisy version of the problem and show that local stability is guaranteed by the solution. In particular, for every ${\mathbf{x}}$ , the recovery error tends to zero as the signal-to-noise ratio tends to infinity. The question of universal stability is unclear. In addition, we obtain a converse of the result in the noiseless case: If the number of observations in ${\mathbf{y}}$ is less than $2K$ , then with probability 1, universal recovery fails, i.e., with probability 1, there exist distinct choices of ${\mathbf{x}}$ which lead to the same unordered list of observations in ${\mathbf{y}}$ . We also present extensions of the result of the noiseless case to special cases with non-i.i.d. entries in ${\mathbf{A}}$ , and to a different setting in which the labels of a portion of the observations ${\mathbf{y}}$ are known. In terms of applications, the unlabeled sensing problem is related to data association problems encountered in different domains including robotics where it is appears in a method called “simultaneous localization and mapping”, multi-target tracking applications, and in sampling signals in the presence of jitter.

Development of a Connector Element for Single-Bolted, Single-Lap Joint Under Flexion–Torsion

In this research, analytical, numerical and experimental methods were used to develop and investigate a connector element for a bolted joint under a tensile load. In the experimental part of the research, various mechanical solicitations (tensile, torsion and shear tests) were applied to steel bolted connections in different configurations having multiple columns and either a single or multiple rows. In the numerical method, to determine the stress and strain distributions due to tensile load forces in specimens, ABAQUS software was used to generate 3D finite element models to compare with our results. However, our analytical solution, which leads to realization of the stiffness matrix under the Rayleigh–Ritz hypothesis, permits us to replace a bolted connection with a simpler connective element and thus to deduce the most intensely loaded zones. Numerical simulation, analytical studies and experimental studies showed coherent results for different configurations, emphasizing the importance of our developed connector for bolted joints. This allows reduction of the time required for calculations.

The Generalized Conditional Autoregressive Wishart Model for Multivariate Realized Volatility

It is well known that in finance variances and covariances of asset returns move together over time. Recently, much interest has been aroused by an approach involving the use of the realized covariance (RCOV) matrix constructed from high-frequency returns as the ex-post realization of the covariance matrix of low-frequency returns. For the analysis of dynamics of RCOV matrices, we propose the generalized conditional autoregressive Wishart (GCAW) model. Both the noncentrality matrix and scale matrix of the Wishart distribution are driven by the lagged values of RCOV matrices, and represent two different sources of dynamics, respectively. The GCAW is a generalization of the existing models, and accounts for symmetry and positive definiteness of RCOV matrices without imposing any parametric restriction. Some important properties such as conditional moments, unconditional moments, and stationarity are discussed. Empirical examples including sequences of daily RCOV matrices from the New York Stock Exchange illustrate that our model outperforms the existing models in terms of model fitting and forecasting.

On the Limitation of Spectral Methods: From the Gaussian Hidden Clique Problem to Rank One Perturbations of Gaussian Tensors

We consider the following detection problem: given a realization of a symmetric matrix X of dimension $n$ , distinguish between the hypothesis that all upper triangular variables are independent and identically distributed (i.i.d). Gaussians variables with mean 0 and variance 1 and the hypothesis, where X is the sum of such matrix and an independent rank-one perturbation. This setup applies to the situation, where under the alternative, there is a planted principal submatrix B of size $L$ for which all upper triangular variables are i.i.d. Gaussians with mean 1 and variance 1, whereas all other upper triangular elements of X not in B are i.i.d. Gaussians variables with mean 0 and variance 1. We refer to this as the “Gaussian hidden clique problem.” When $L=(1+\epsilon )\sqrt {n}$ ( $\epsilon >0$ ), it is possible to solve this detection problem with probability $1-o_{n}(1)$ by computing the spectrum of X and considering the largest eigenvalue of X. We prove that this condition is tight in the following sense: when $L no algorithm that examines only the eigenvalues of X can detect the existence of a hidden Gaussian clique, with error probability vanishing as $n\to \infty $ . We prove this result as an immediate consequence of a more general result on rank-one perturbations of $k$ -dimensional Gaussian tensors. In this context, we establish a lower bound on the critical signal-to-noise ratio below which a rank-one signal cannot be detected.

The best-approximate realization of a spatial stiffness matrix with simple springs connected in parallel

This paper presents an approximation algorithm to simplify the physical implementation of the non-isotropic stiffness matrices. The algorithm is inspired by the best approximation theorem in Matrix theory, by which the original stiffness matrix is projected onto a special linear space, thus an approximate isotropic matrix is derived. Accordingly, the desired force–deflection characteristics can be realized by purely simple springs (linear/torsional springs) connected in parallel. In mathematics, the approximation only changes the diagonal entries of the off-diagonal blocks of the original stiffness matrix. Moreover, the physical properties of the approximate isotropic matrix are identical to that of the original stiffness matrix in many aspects, including the center of stiffness, the wrench compliance axes as well as the translational and rotational stiffness value. The approximation is unique in both mathematics and physics. In order to illustrate the effectiveness of the proposed method, a numerical example is studied with comparison to some traditional algorithms of stiffness decomposition. The advantages of the approximation are well verified through the example.