Functions Of Matrices Research Articles

Applications of realizations (aka linearizations) to free probability

We show how the combination of new “linearization” ideas in free probability theory with the powerful “realization” machinery – developed over the last 50 years in fields including systems engineering and automata theory – allows solving the problem of determining the eigenvalue distribution (or even the Brown measure, in the non-selfadjoint case) of noncommutative rational functions of random matrices when their size tends to infinity. Along the way we extend evaluations of noncommutative rational expressions from matrices to stably finite algebras, e.g. type II1 von Neumann algebras, with a precise control of the domains of the rational expressions.The paper provides sufficient background information, with the intention that it should be accessible both to functional analysts and to algebraists.

Gramian-Based Reachability Metrics for Bilinear Networks

This paper studies Gramian-based reachability metrics for bilinear control systems. In the context of complex networks, bilinear systems capture scenarios where an actuator not only can affect the state of a node but also interconnections among nodes. Under the assumption that the input's infinity norm is bounded by some function of the network dynamic matrices, we derive a Gramian-based lower bound on the minimum input energy required to steer the state from the origin to any reachable target state. This result motivates our study of various objects associated with the reachability Gramian to quantify the ease of controllability of the bilinear network: the minimum eigenvalue (worst-case minimum input energy to reach a state), the trace (average minimum input energy to reach a state), and its determinant (volume of the ellipsoid containing the reachable states using control inputs with no more than unit energy). We establish an increasing returns property of the reachability Gramian as a function of the actuators, which, in turn, allows us to derive a general lower bound on the reachability metrics in terms of the aggregate contribution of the individual actuators. We conclude by examining the effect on the worst-case minimum input energy of the addition of bilinear inputs to difficult-to-control linear symmetric networks. We show that the bilinear networks resulting from the addition of either inputs at a finite number of interconnections or at all self loops with weight vanishing with the network scale remain difficult to control. Various examples illustrate our results.

Improved Explicit Integration Algorithms for Structural Dynamic Analysis with Unconditional Stability and Controllable Numerical Dissipation

ABSTRACTA computationally efficient one-parameter family of explicit (i.e., non iterative for nonlinear systems) direct integration algorithms featuring second-order accuracy, unconditional stability for linear systems, and controllable numerical dissipation is developed for solving inertial problems. Unconditional stability is attained within an explicit formulation using the concept of model-based algorithms in which the algorithmic parameters are functions of the model matrices. The proposed method improves the overshoot and nonlinear stability characteristics of the KR- method, an existing explicit-model-based algorithm. Numerical characteristics of the proposed method are compared with those of the KR- method and further demonstrated through representative numerical examples.

In vitro gastrointestinal digestion of pea protein isolate as a function of pH, food matrices, autoclaving, high-pressure and re-heat treatments

This study investigated the influence of pH and processing conditions (autoclave at 93 °C/13 min or high pressure processing (HPP) at 600 MPa/5 min without/with follow-up reheating at 80 °C/30 min) on the digestibility of pea protein isolate. Both aqueous solutions and real food matrices (apple and carrot purees) containing pea protein was examined at 37 °C. In vitro gastrointestinal digestion was followed using sodium dodecyl sulphate polyacrylamide gel electrophoresis, titrimetric techniques and theoretical calculations. Pea protein with HPP followed by re-heating showed the highest rate of proteolysis in gastric conditions. In case of sequential intestinal digestion of the gastric chyme, pea protein at pH 6.2 demonstrated higher degree and rate of digestibility as compared to that at pH 3.6, the latter being close to the isoelectric point of pea protein. However, autoclave treatments overshadowed such pH effects. Processing-induced enhancement in digestibility might be attributed to the unfolding of the globular pea protein subunits. Pea protein in the carrot puree was more digestible than in the apple puree, due to apple procyanidins binding to pea protein. These new findings might have important implications in designing the process parameters and selection of appropriate food matrices for delivering pea protein.

A reliable numerical algorithm for the fractional vibration equation

The key purpose of this article is to introduce a numerical algorithm for the solution of the fractional vibration equation (FVE). The numerical algorithm is based on the applications of the operational matrices of the Legendre scaling functions. The main advantage of the numerical algorithm is that it reduces the FVE into Sylvester form of algebraic equations which significantly simplify the problem. Error as well as convergence analysis of the proposed scheme are shown. Numerical results are discussed taking different initial conditions and wave velocities involved in the problem. Numerical results obtained by using suggested numerical algorithm are compared with the existing analytical methods for the different cases of FVE.

Large-sample approximations for variance-covariance matrices of high-dimensional time series

Distributional approximations of (bi--) linear functions of sample variance-covariance matrices play a critical role to analyze vector time series, as they are needed for various purposes, especially to draw inference on the dependence structure in terms of second moments and to analyze projections onto lower dimensional spaces as those generated by principal components. This particularly applies to the high-dimensional case, where the dimension $d$ is allowed to grow with the sample size $n$ and may even be larger than $n$. We establish large-sample approximations for such bilinear forms related to the sample variance-covariance matrix of a high-dimensional vector time series in terms of strong approximations by Brownian motions. The results cover weakly dependent as well as many long-range dependent linear processes and are valid for uniformly $ \ell_1 $-bounded projection vectors, which arise, either naturally or by construction, in many statistical problems extensively studied for high-dimensional series. Among those problems are sparse financial portfolio selection, sparse principal components, the LASSO, shrinkage estimation and change-point analysis for high--dimensional time series, which matter for the analysis of big data and are discussed in greater detail.

The nonlinear aeroelastic characteristics of a folding wing with cubic stiffness

This paper focuses on the nonlinear aeroelastic characteristics of a folding wing in the quasi-steady condition (namely at fixed folding angles) and during the morphing process. The structure model of the folding wing is formulated by the Lagrange equations, and the constraint equation is used to describe the morphing strategy. The aerodynamic influence coefficient matrices at several folding angles are calculated by the Doublet Lattice method, and described as rational functions in the Laplace domain by the rational function approximation, and then the Kriging agent model technique is adopted to interpolate the coefficient matrices of the rational functions, and the aerodynamics model of the folding wing during the morphing process is built. The aeroelastic responses of the folding wing with cubic stiffness are simulated, and the results show that the motion types of aeroelastic responses in the quasi-steady condition and during the morphing process are all sensitive to the initial condition and folding angle. During the morphing process, the transition of the motion types is observed. And apart from the period of transition, the aeroelastic response at some folding angles may exhibit different motion types, which can be found from the results in the quasi-steady condition.

On concavity of solutions of the Dirichlet problem for the equation $(-\Delta)^{1/2} \varphi = 1$ in convex planar regions

For a sufficiently regular open bounded set D \subset \mathbb R^2 let us consider the equation (-\Delta)^{1/2} \varphi(x) = 1 for x \in D with the Dirichlet exterior condition \varphi(x) = 0 for x \in D^c . Its solution \varphi(x) is the expected value of the first exit time from D of the Cauchy process in \mathbb R^2 . We prove that if D \subset \mathbb R^2 is a convex bounded domain then \varphi is concave on D . To do so we study the Hessian matrix of the harmonic extension of \varphi . The key idea of the proof is based on a deep result of Hans Lewy concerning the determinants of Hessian matrices of harmonic functions.

Direct numerical method for isoperimetric fractional variational problems based on operational matrix

In this paper, we applied a direct method for a solution of isoperimetric fractional variational problems. We use shifted Legendre orthonormal polynomials as basis function of operational matrices of fractional differentiation and fractional integration in combination with the Lagrange multipliers technique for converting such isoperimetric fractional variational problems into solving a system of algebraic equations. Also, we show the convergence analysis of the presented technique and introduce some test problems with comparisons between our numerical results with those introduced using different methods.

Starch-Derived Nanographene Oxide Paves the Way for Electrospinnable and Bioactive Starch Scaffolds for Bone Tissue Engineering.

A straightforward process that enabled electrospinning of bioactive starch-based nanofiber scaffolds was developed by utilizing starch derived nano graphene oxide (nGO) as a property enhancer and formic acid as a solvent and esterification reagent. The reaction mechanism and process were followed by detailed spectroscopic investigation. Furthermore, the incorporation of nGO as a "green bioactive additive" endorsed starch nanofibrous scaffolds several advantageous functionalities including improved electrospinnability and thermal stability, good cytocompatibility, osteo-bioactivity, and retained biodegradability. The biodegradable starch/nGO nanofibers underwent simultaneous degradation and mineralization process during 1 week of cell culture and mineralization test, thus, mimicking the structure and function of extracellular matrices (ECMs) and indicating promise for bone tissue engineering applications.

New spectral bounds on [formula omitted]-norm of linear dynamical networks

In this paper, we obtain new lower and upper bounds for the H2-norm of a class of linear time-invariant systems subject to exogenous noise inputs. We show that the H2-norm, as a performance measure, can be tightly bounded from below and above by some spectral functions of state and output matrices of the system. In order to show the usefulness of our results, we calculate bounds for the H2-norm of some network models with specific coupling or graph structures, e.g., systems with normal state matrices, linear consensus networks with directed graphs, and cyclic linear networks. As a specific example, the H2-norm of a linear consensus network over a directed cycle graph is computed and shown how its performance scales with the network size. Our proposed spectral bounds reveal the important role and contribution of fast and slow dynamic modes of a system in the best and worst achievable performance bounds under white noise excitation. Finally, we use several numerical simulations to show the superiority of our bounds over the existing bounds in the literature.

Covalent approach for in situ enhancement of interaction between pristine graphene and styrene‐butadiene‐p‐(2,2,2‐triphenylethyl)styrene rubber

ABSTRACTA facile method for enhancing the interaction between pristine graphene and nonpolar rubber matrices was developed by preparing a new solution‐polymerized styrene‐butadiene‐p‐(2,2,2‐triphenylethyl)styrene (TPES) rubber (SBTR). SBTR macroradicals were formed by the thermal decomposition of a 1,1,1,2‐tetraphenylethane pendant. The macroradicals were successfully trapped by graphene through the formation of covalent bonds. This was confirmed by rotorless rheometer and X‐ray photoelectron spectroscopy measurements. The dispersion of graphene and interfacial interaction between graphene and the SBTR was significantly increased by increasing the TPES content in SBTR composites, as demonstrated by differential scanning calorimetry, scanning electron microscopy, rubber process analysis, and dynamic mechanical thermal analysis. The mechanical properties of the SBTR/GNS composites were significantly improved by the improved dispersion and the increased surface affinity of SBTR for graphene. The developed approach can be generally applied to the functionalization of other polymer matrices by copolymerizing TPES with other vinyl monomers for pristine graphene‐based composite materials. © 2017 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2017, 134, 44923.

Biomineralization: From Material Tactics to Biological Strategy.

Biomineralization is an important tactic by which biological organisms produce hierarchically structured minerals with marvellous functions. Biomineralization studies typically focus on the mediation function of organic matrices on inorganic minerals, which helps scientists to design and synthesize bioinspired functional materials. However, the presence of inorganic minerals may also alter the native behaviours of organic matrices and even biological organisms. This progress report discusses the latest achievements relating to biomineralization mechanisms, the manufacturing of biomimetic materials and relevant applications in biological and biomedical fields. In particular, biomineralized vaccines and algae with improved thermostability and photosynthesis, respectively, demonstrate that biomineralization is a strategy for organism evolution via the rational design of organism-material complexes. The successful modification of biological systems using materials is based on the regulatory effect of inorganic materials on organic organisms, which is another aspect of biomineralization control. Unlike previous studies, this study integrates materials and biological science to achieve a more comprehensive view of the mechanisms and applications of biomineralization.

Identification of fractional-order systems with time delays using block pulse functions

In this paper, a novel method based on block pulse functions is proposed to identify continuous-time fractional-order systems with time delays. First, the operational matrices of block pulse functions for fractional integral operator and time delay operator are derived. Then, these operational matrices are applied to convert the continuous-time fractional-order systems with time delays to an algebraic equation. Finally, the system’s parameters along with the differentiation orders and the time delays are all simultaneously estimated through minimizing a quadric error function. The proposed method reduces the computation complexity of the identification process, and also it does not require the system’s differentiation orders to be commensurate. The effectiveness of the proposed method are demonstrated by several numerical examples.

Approximating Spectral Sums of Large-Scale Matrices using Stochastic Chebyshev Approximations

Computation of the trace of a matrix function plays an important role in many scientific computing applications, including applications in machine learning, computational physics (e.g., lattice quantum chromodynamics), network analysis, and computational biology (e.g., protein folding), just to name a few application areas. We propose a linear-time randomized algorithm for approximating the trace of matrix functions of large symmetric matrices. Our algorithm is based on coupling function approximation using Chebyshev interpolation with stochastic trace estimators (Hutchinson's method), and as such requires only implicit access to the matrix, in the form of a function that maps a vector to the product of the matrix and the vector. We provide rigorous approximation error in terms of the extremal eigenvalue of the input matrix, and the Bernstein ellipse that corresponds to the function at hand. Based on our general scheme, we provide algorithms with provable guarantees for important matrix computations, including log-determinant, trace of matrix inverse, Estrada index, Schatten $p$-norm, and testing positive definiteness. We experimentally evaluate our algorithm and demonstrate its effectiveness on matrices with tens of millions dimensions.

Inequalities of Hermite-Hadamard type for functions of selfadjoint operators and matrices

Inequalities of Hermite-Hadamard type for functions of selfadjoint operators and matrices

A Formula for the Fréchet Derivative of a Generalized Matrix Function

We state and prove an analogue of the Daleckii--Krein theorem, thus obtaining an explicit formula for the Fréchet derivative of generalized matrix functions. Moreover, we prove the differentiability of generalized matrix functions of real matrices under very mild assumptions. For complex matrices, we argue that, under the same assumptions, generalized matrix functions are real-differentiable but generally not complex-differentiable. Finally, we discuss the application of our results to the study of the condition number of generalized matrix functions. Along our way, we also derive generalized matrix functional analogues of a few classical theorems on polynomial interpolation of classical matrix functions and their derivatives.

Nanoconfined Ionic Liquids.

Ionic liquids (ILs) have been widely investigated as novel solvents, electrolytes, and soft functional materials. Nevertheless, the widespread applications of ILs in most cases have been hampered by their liquid state. The confinement of ILs into nanoporous hosts is a simple but versatile strategy to overcome this problem. Nanoconfined ILs constitute a new class of composites with the intrinsic chemistries of ILs and the original functions of solid matrices. The interplay between these two components, particularly the confinement effect and the interactions between ILs and pore walls, further endows ILs with significantly distinct physicochemical properties in the restricted space compared to the corresponding bulk systems. The aim of this article is to provide a comprehensive review of nanoconfined ILs. After a brief introduction of bulk ILs, the synthetic strategies and investigation methods for nanoconfined ILs are documented. The local structure and physicochemical properties of ILs in diverse porous hosts are summarized in the next sections. The final section highlights the potential applications of nanoconfined ILs in diverse fields, including catalysis, gas capture and separation, ionogels, supercapacitors, carbonization, and lubrication. Further research directions and perspectives on this topic are also provided in the conclusion.

A LASSO penalized regression approach for genome-wide association analyses using related individuals: application to the Genetic Analysis Workshop 19 simulated data.

We propose a novel LASSO (least absolute shrinkage and selection operator) penalized regression method used to analyze samples consisting of (potentially) related individuals. Developed in the context of linear mixed models, our method models the relatedness of individuals in the sample through a random effect whose covariance structure is a linear function of known matrices with elements combinations of the condensed coefficients of identity between the individuals in the sample. We implement our method to analyze the simulated family data provided by the 19th Genetic Analysis Workshop in an effort to identify loci regulating the simulated trait of systolic blood pressure. The analyses were performed with full knowledge of the simulation model. Our findings demonstrate that we can significantly reduce the rate of false positive signals by incorporating the relatedness of the study participants.

A single molecule assay to probe monovalent and multivalent bonds between hyaluronan and its key leukocyte receptor CD44 under force.

Glycosaminoglycans (GAGs), a category of linear, anionic polysaccharides, are ubiquitous in the extracellular space, and important extrinsic regulators of cell function. Despite the recognized significance of mechanical stimuli in cellular communication, however, only few single molecule methods are currently available to study how monovalent and multivalent GAG·protein bonds respond to directed mechanical forces. Here, we have devised such a method, by combining purpose-designed surfaces that afford immobilization of GAGs and receptors at controlled nanoscale organizations with single molecule force spectroscopy (SMFS). We apply the method to study the interaction of the GAG polymer hyaluronan (HA) with CD44, its receptor in vascular endothelium. Individual bonds between HA and CD44 are remarkably resistant to rupture under force in comparison to their low binding affinity. Multiple bonds along a single HA chain rupture sequentially and independently under load. We also demonstrate how strong non-covalent bonds, which are versatile for controlled protein and GAG immobilization, can be effectively used as molecular anchors in SMFS. We thus establish a versatile method for analyzing the nanomechanics of GAG·protein interactions at the level of single GAG chains, which provides new molecular-level insight into the role of mechanical forces in the assembly and function of GAG-rich extracellular matrices.