Functions Of Matrices Research Articles

Adaptive Exponential Synchronization of Multislave Time-Delayed Recurrent Neural Networks With Lévy Noise and Regime Switching

This paper discusses the problem of adaptive exponential synchronization in mean square for a new neural network model with the following features: 1) the noise is characterized by the Lévy process and the parameters of the model change in line with the Markovian process; 2) the master system is also disturbed by the same Lévy noise; and 3) there are multiple slave systems, and the state matrix of each slave system is an affine function of the state matrices of all slave systems. Based on the Lyapunov functional theory, the generalized Itô's formula, -matrix method, and the adaptive control technique, some criteria are established to ensure the adaptive exponential synchronization in the mean square of the master system and each slave system. Moreover, the update law of the control gain and the dynamic variation of the parameters of the slave systems are provided. Finally, the effectiveness of the synchronization criteria proposed in this paper is verified by a practical example.

Automatic Multivector Differentiation and Optimization

In this work, we present a novel approach to nonlinear optimization of multivectors in the Euclidean and conformal model of geometric algebra by introducing automatic differentiation. This is used to compute gradients and Jacobian matrices of multivector valued functions for use in nonlinear optimization where the emphasis is on the estimation of rigid body motions.

A numerical method to compute derivatives of functions of large complex matrices and its application to the overlap Dirac operator at finite chemical potential

We present a method for the numerical calculation of derivatives of functions of general complex matrices. The method can be used in combination with any algorithm that evaluates or approximates the desired matrix function, in particular with implicit Krylov–Ritz-type approximations. An important use case for the method is the evaluation of the overlap Dirac operator in lattice Quantum Chromodynamics (QCD) at finite chemical potential, which requires the application of the sign function of a non-Hermitian matrix to some source vector. While the sign function of non-Hermitian matrices in practice cannot be efficiently approximated with source-independent polynomials or rational functions, sufficiently good approximating polynomials can still be constructed for each particular source vector. Our method allows for an efficient calculation of the derivatives of such implicit approximations with respect to the gauge field or other external parameters, which is necessary for the calculation of conserved lattice currents or the fermionic force in Hybrid Monte-Carlo or Langevin simulations. We also give an explicit deflation prescription for the case when one knows several eigenvalues and eigenvectors of the matrix being the argument of the differentiated function. We test the method for the two-sided Lanczos approximation of the finite-density overlap Dirac operator on realistic SU(3) gauge field configurations on lattices with sizes as large as 14×143 and 6×183.

Electrolocation of objects in fluids by means of active sensor movements based on discrete EEVs

Weakly electric fish use self-generated electric fields for communication and for active electrolocation. The sensor part of the biological system consists of a vast amount of electroreceptors which are distributed across the skin of the electric fish. Fish utilise changes of their position and body geometry to aid in the extraction of sensory information. Inspired by the biological model, this study looks for a fixed, minimal scanning strategy compiled of active receptor-system movements that allows unique identification of the positions of objects in the vicinity. The localisation method is based on the superposition of numerical extracted contour-rings of rotated and/or linearly shifted EEVs (Solberg et al 2008 Int. J. Rob. Res. 27 529–48), simulated by means of FEM. For the evaluation of a movement sequence, matrices of unique intersection points and respective contrast functions are introduced. The resultant optimal scanning strategy consists of a combination of a linear shift and a rotation of the original EEV.

The Schrödinger Equation in the Mean-Field and Semiclassical Regime

In this paper, we establish (1) the classical limit of the Hartree equation leading to the Vlasov equation, (2) the classical limit of the $N$-body linear Schr\"{o}dinger equation uniformly in N leading to the N-body Liouville equation of classical mechanics and (3) the simultaneous mean-field and classical limit of the N-body linear Schr\"{o}dinger equation leading to the Vlasov equation. In all these limits, we assume that the gradient of the interaction potential is Lipschitz continuous. All our results are formulated as estimates involving a quantum analogue of the Monge-Kantorovich distance of exponent 2 adapted to the classical limit, reminiscent of, but different from the one defined in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016), 165-205]. As a by-product, we also provide bounds on the quadratic Monge-Kantorovich distances between the classical densities and the Husimi functions of the quantum density matrices.

A New Approach based on Triangular Functions for Solving N-dimensional Stochastic Differential Equations

In this article, we prepare a new numerical method based on triangular functions for solving n-dimensional stochastic differential equations. At first stochastic operational matrices of triangular functions are derived then n-dimensional stochastic differential equations are solved recently. Convergence analysis and numerical examples are prepared to illustrate accuracy and efficiency of this approach.

An inequality for the matrix pressure function and applications

We prove an a priori lower bound for the pressure, or p-norm joint spectral radius, of a measure on the set of d×d real matrices which parallels a result of J. Bochi for the joint spectral radius. We apply this lower bound to give new proofs of the continuity of the affinity dimension of a self-affine set and of the continuity of the singular-value pressure for invertible matrices, both of which had been previously established by D.-J. Feng and P. Shmerkin using multiplicative ergodic theory and the subadditive variational principle. Unlike the previous proof, our lower bound yields algorithms to rigorously compute the pressure, singular value pressure and affinity dimension of a finite set of matrices to within an a priori prescribed accuracy in finitely many computational steps. We additionally deduce a related inequality for the singular value pressure for measures on the set of 2×2 real matrices, give a precise characterisation of the discontinuities of the singular value pressure function for two-dimensional matrices, and prove a general theorem relating the zero-temperature limit of the matrix pressure to the joint spectral radius.

Bandit online optimization over the permutahedron

The permutahedron is the convex polytope with vertex set consisting of the vectors (π(1),…,π(n)) for all permutations (bijections) π over {1,…,n}. We study a bandit game in which, at each step t, an adversary chooses a hidden weight vector st, a player chooses a vertex πt of the permutahedron and suffers an observed instantaneous loss of ∑i=1nπt(i)st(i).We study the problem in two different approaches. In the two approaches, we assume that st is a point in the polytope dual to the permutahedron. Algorithm CombBand of Cesa-Bianchi et al. (2012) guarantees a regret of O(nTlog⁡n) after T steps. Unfortunately, CombBand requires at each step an n-by-n matrix permanent computation, a #P-hard problem. Approximating the permanent is possible in the impractical running time of O(n10), with an additional heavy inverse-polynomial dependence on the sought accuracy. In the first approach, we provide an algorithm of slightly worse regret O(n3/2T) but with more realistic time complexity O(n3) per step. The technical contribution is a bound on the variance of the Plackett–Luce noisy sorting process's ‘pseudo loss’, obtained by establishing positive semi-definiteness of a family of 3-by-3 matrices of rational functions in exponents of 3 parameters.In the second approach, we present and analyze an algorithm based on Bubeck et al.'s (2012) OSMD approach with a novel projection and decomposition technique for the permutahedron. The second algorithm's running time and regret guarantees are similar to our first algorithm, modulo a numerical line search procedure the running time of which we have not been able to analyze. It is interesting that the two approaches are totally different.The main open problem from this work is whether there exists a bandit algorithm for this problem with both optimal regret of O(nT) and running time of O(n3) for either regime, or there is an inherent tradeoff between the two performance measures.

Chitosan Nanofibers for Specific Capture and Nondestructive Release of CTCs Assisted by pCBMA Brushes.

Over the last decade, significant progress has been made to develop sensitive devices for the capture of circulating tumor cells (CTCs) from blood of cancer patients. However, simple capture and counting of CTCs cannot provide effective information for understanding the biology of them. In this work, a functional biointerface is fabricated for specific capture and nondestructive release of CTCs from blood samples. A nanostrucure of porous network based on chitosan nanofibers is fabricated by electrospinning, to mimic the function of extracellular matrices, and then the poly(carboxybetaine methacrylate) (pCBMA) brushes integrating onto nanofiber interface provide the effect of interfacial properties to control nonspecific cell adhesion and the multivalent immobilization of aptamers to induce high efficient and specific CTC capture. Furthermore, a complementary sequence is used to efficiently hybridize with the aptamer to achieve nondestructive release of the captured target cells, assisted by the flexible space provided by pCBMA brushes. This work also shows how nanostructure and the interface molecules regulate the morphology of the captured CTCs, and reveals the importance of the controllable cell morphology on biointerface for an effective nondestructive release of the captured CTCs.

Monotone convergence of the extended Krylov subspace method for Laplace–Stieltjes functions of Hermitian positive definite matrices

The extended Krylov subspace method is known to be very efficient in many cases in which one wants to approximate the action of a matrix function f(A) on a vector b, in particular when f belongs to the class of Laplace–Stieltjes functions. We prove that the Euclidean norm of the error decreases strictly monotonically in this situation when A is Hermitian positive definite. Similar results are known for the (polynomial) Lanczos method for f(A)b, and we demonstrate how the techniques of proof used in the polynomial Krylov case can be transferred to the extended Krylov case.

Approximation of functions of large matrices with Kronecker structure

We consider the numerical approximation of $$f(\mathcal{A})b$$f(A)b where $$b\in {\mathbb {R}}^{N}$$bźRN and $$\mathcal A$$A is the sum of Kronecker products, that is $$\mathcal{A}=M_2 \otimes I + I \otimes M_1\in {\mathbb {R}}^{N\times N}$$A=M2źI+IźM1źRN×N. Here f is a regular function such that $$f(\mathcal{A})$$f(A) is well defined. We derive a computational strategy that significantly lowers the memory requirements and computational efforts of the standard approximations, with special emphasis on the exponential and on completely monotonic functions, for which the new procedure becomes particularly advantageous. Our findings are illustrated by numerical experiments with typical functions used in applications.

Zero-term rank and zero-star cover number of symmetric matrices and their linear preservers

This paper concerns two notions of matrix functions over semirings; zero-term rank and zero-star cover number. We study linear operators that preserve these two matrix functions of symmetric matrices. Consequently, we determine the form of linear operators T on symmetric matrices that preserves zero-term rank (zero-star cover number). We also show that a linear operator T on symmetric matrices preserves zero-term rank (zero-star cover number) if and only if T preserves zero-term ranks (zero-star cover numbers, respectively) 1 and 2. Other characterizations of zero-term rank (zero-star cover number, respectively) preservers are also given.

Matrix metalloproteinase-13 mediated degradation of hyaluronic acid-based matrices orchestrates stem cell engraftment through vascular integration

A critical design parameter for the function of synthetic extracellular matrices is to synchronize the gradual cell-mediated degradation of the matrix with the endogenous secretion of natural extracellular matrix (ECM) (e.g., creeping substitution). In hyaluronic acid (HyA)-based hydrogel matrices, we have investigated the effects of peptide crosslinkers with different matrix metalloproteinases (MMP) sensitivities on network degradation and neovascularization in vivo. The HyA hydrogel matrices consisted of cell adhesive peptides, heparin for both the presentation of exogenous and sequestration of endogenously synthesized growth factors, and MMP cleavable peptide linkages (i.e., QPQGLAK, GPLGMHGK, and GPLGLSLGK). Sca1+/CD45−/CD34+/CD44+ cardiac progenitor cells (CPCs) cultured in the matrices with the slowly degradable QPQGLAK hydrogels supported the highest production of MMP-2, MMP-9, MMP-13, VEGF165, and a range of angiogenesis related proteins. Hydrogels with QPQGLAK crosslinks supported prolonged retention of these proteins via heparin within the matrix, stimulating rapid vascular development, and anastomosis with the host vasculature when implanted in the murine hindlimb.

Biochemical Composition and Assembly of Biosilica-associated Insoluble Organic Matrices from the Diatom Thalassiosira pseudonana

The nano- and micropatterned biosilica cell walls of diatoms are remarkable examples of biological morphogenesis and possess highly interesting material properties. Only recently has it been demonstrated that biosilica-associated organic structures with specific nanopatterns (termed insoluble organic matrices) are general components of diatom biosilica. The model diatom Thalassiosira pseudonana contains three types of insoluble organic matrices: chitin meshworks, organic microrings, and organic microplates, the latter being described in the present study for the first time. To date, little is known about the molecular composition, intracellular assembly, and biological functions of organic matrices. Here we have performed structural and functional analyses of the organic microrings and organic microplates from T. pseudonana. Proteomics analysis yielded seven proteins of unknown function (termed SiMat proteins) together with five known silica biomineralization proteins (four cingulins and one silaffin). The location of SiMat1-GFP in the insoluble organic microrings and the similarity of tyrosine- and lysine-rich functional domains identifies this protein as a new member of the cingulin protein family. Mass spectrometric analysis indicates that most of the lysine residues of cingulins and the other insoluble organic matrix proteins are post-translationally modified by short polyamine groups, which are known to enhance the silica formation activity of proteins. Studies with recombinant cingulins (rCinY2 and rCinW2) demonstrate that acidic conditions (pH 5.5) trigger the assembly of mixed cingulin aggregates that have silica formation activity. Our results suggest an important role for cingulins in the biogenesis of organic microrings and support the hypothesis that this type of insoluble organic matrix functions in biosilica morphogenesis.

Mixed, Componentwise Condition Numbers and Small Sample Statistical Condition Estimation for Generalized Spectral Projections and Matrix Sign Functions

The explicit expressions of mixed and componentwise condition numbers for the spectral projections, generalized spectral projections and sign functions for matrices and regular matrix pairs are derived. The condition numbers improve some known results of the normwise type and reveal the structured perturbations. Statistical condition estimation is applied to these problems which can be calculated efficiently.

Robust estimation in structural equation models using Bregman and other divergences with t-centre approach to estimate the covariance matrix

Structural equation models seek to find causal relationships between latent variables by analysing the mean and the covariance matrix of some observable indicators of the latent variables. Under a multivariate normality assumption on the distribution of the latent variables and of the errors, maximum likelihood estimators are asymptotically efficient. The estimators are significantly influenced by violation of the normality assumption and hence there is a need to robustify the inference procedures. Previous work minimized the von Neuman divergence or its variant the total von Neumann divergence to estimate the parameters, with the minimum covariance determinant used as a robust estimator of the covariance matrix. We extend this approach by considering other divergences and by developing a robust estimate of the covariance matrix. The robust estimator of the covariance matrix developed is a t-centre like estimator based on several minimum covariance determinant estimators ranging from 0% contamination to 50% contamination. The simulation results are promising. The results can be used for robustifying the fit of structural equation models. References K. A. Bollen. Structural equations with latent variables . Wiley, New York, 1989. http://au.wiley.com/WileyCDA/WileyTitle/productCd-0471011711.html R. W. Butler, P. L. Davies and M. Jhun. Asymptotics for the minimum covariance determinant estimator. Ann. Stat. , 21(3):1385–1400, 1993. http://www.jstor.org/stable/2242201 N. J. Higham. Functions of matrices: Theory and computation. SIAM, Philadelphia, 2008. doi:10.1137/1.9780898717778 S. Jayasumana, R. Hartley, M. Salzmann, H. Li and M. Harandi. Kernel methods on the Riemannian manifold of symmetric positive definite matrices. Proc. CVPR IEEE 2013 , 73–80, 2013. doi:10.1109/CVPR.2013.17 S. Penev and T. Prvan. Robust estimation in structural equation models using Bregman divergences. CTAC2012, ANZIAM J. , 54:C574–C589, 2013. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6306 B. C. Vemuri, M. Liu, S.-I. Amari, and F. Nielsen. Total Bregman divergence and its applications to DTI analysis. IEEE T. Med. Imaging. , 30(2):475–483, 2011. doi:10.1109/TMI.2010.2086464 S. Verboven and M. Hubert. LIBRA: a MATLAB library for robust analysis, Chemometr. Intell. Lab. , 75(2):127–136, 2005. doi:10.1016/j.chemolab.2004.06.003

Fluctuations of functions of Wigner matrices

We show that matrix elements of functions of $N\times N$ Wigner matrices fluctuate on a scale of order $N^{-1/2}$ and we identify the limiting fluctuation. Our result holds for any function $f$ of the matrix that has bounded variation and thus considerably relaxes the regularity requirement imposed in [7,11].

Aeroelastic study for folding wing during the morphing process

This paper focuses on the aeroelastic characteristics of a folding wing during the morphing process. The folding wing structure is modeled by using the flexible multi-body dynamics approach, and an efficient method is proposed to calculate the aerodynamic force of the folding wing during the morphing process. The aerodynamic influence coefficient (AIC) matrices at different folding angles are obtained by the Doublet Lattice based aerodynamics theory, and then the orders of these AIC matrices are reduced by the spline interpolation technique. Through the minimum state approximation, the reduced AIC matrices are described as rational functions in the Laplace domain. Then the Kriging agent model technique is used to interpolate the coefficient matrices of the rational functions obtained from several different folding angles and to build the aerodynamics model in the time domain. At some different folding angles, the element values of the coefficient matrices before and after the interpolation are compared to verify the accuracy of the aerodynamics model, and then the aeroelastic responses of the folding wing during its morphing processes are simulated. The results demonstrate that the folding and unfolding processes have opposite influences on the dynamic aeroelastic stability of the folding wing, and the influences become much more significant with the increasing of the folding and unfolding rates. When the folding wing is morphing with a very slow rate, the dynamic aeroelastic stability will be similar to that obtained by the quasi-steady aeroelastic analysis.

On the use of functional calculus for phase-type and related distributions

The area of phase-type distributions is renowned for its ability to obtain closed form formulas or algorithmically exact solutions to many complex stochastic models. The method of functional calculus will provide an additional tool along these lines for establishing results in terms of functions of matrices. Functional calculus, which is a branch of operator theory frequently associated with complex analysis, can be applied to phase-type and matrix-exponential distributions in a rather straightforward way. In this article we provide a number of examples of how to execute the formal arguments.

Stochastic estimates for the trace of functions of matrices via Hadamard matrices

ABSTRACTLet be a symmetric matrix. Stochastic estimates have been introduced in the literature for estimating the trace of a function of the matrix A by computing the mean value of the quantities xTif(A)xi, where xi are appropriate sample vectors forming the columns of a sample matrix X. In this work, the selection of the vectors xi from the columns of a Hadamard matrix is discussed and the employment of the Hadamard matrices for the construction of the sample matrix X is studied and tested throughout several experiments. In addition, the combination of Hadamard matrices with a central-composite design (CCD) for the construction of the sample matrix X is introduced.