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We refine a result concerning singular matrix pencils and the Wong sequences. In our recent paper [T. Berger and S. Trenn, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 336-368] we have shown that the Wong sequences are sufficient to obtain a quasi-Kronecker form. However, we applied the Wong sequences again on the regular part to decouple the regular matrix pencil corresponding to the finite and infinite eigenvalues. The current paper is an addition to [T. Berger and S. Trenn, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 336-368], which shows that the decoupling of the regular part can be done already with the help of the Wong sequences of the original matrix pencil. Furthermore, we show that the complete Kronecker canonical form can be obtained with the help of the Wong sequences. [ABSTRACT FROM AUTHOR]

In this paper we show that recently introduced quantics tensor train (QTT) decomposition call be considered as an algebraic wavelet transform with adaptively determined filters. The main algorithm for obtaining QTT decomposition can be reformulated as a method seeking "good subspaces" or "good bases" and considered as a parameterized transformation of an initial tensor into a sparse tensor. This interpretation allows us to introduce a modification of the tensor train-SVD (TT-SVD) algorithm to make it work in cases where the original algorithm does not work; it results in the new wavelet-like transforms called wavelet tensor train (WTT) transform. Properties of WTT transforms are studied numerically, and a theoretical conjecture on the number of vanishing moments is proposed. It is shown that WTT transforms are orthogonal by construction, and the efficiency of WTT is compared with and often outperforms Daubechies wavelet transforms on certain classes of function-related vectors and matrices. [ABSTRACT FROM AUTHOR]

This paper considers the Mosco convergence of Dirichiet forms υ(∫) = ∫ ∣∇∫∣² dμn, where the measures μn locally converge in variation and it is not necessary to have complete supports. [ABSTRACT FROM AUTHOR]

In this paper we introduce and analyze two augmented mixed finite element methods for a velocity-pressure-stress formulation of the stationary Stokes equations. Our approach, which extends analog results for linear elasticity problems, is based on the introduction of the Galerkin least-squares-type terms arising from the constitutive and equilibrium equations and the Dirichlet boundary condition for the velocity, all of them multiplied by suitable stabilization parameters. We show that these parameters can be chosen so that the resulting augmented variational formulations are defined by strongly coercive bilinear forms, whence the associated Galerkin schemes become well-posed for any choice of finite element subspaces. In particular, we can use continuous piecewise linear velocities, piecewise constant pressures, and Raviart-Thomas elements for the stresses, thus yielding a number of unknowns behaving asymptotically as 5 times the number of triangles of the triangulation. Alternatively, the above factor reduces to 4 when a second augmented variational formulation, involving only the velocity and the stress as unknowns, is employed. Next, we derive a reliable and efficient residual-based a posteriori error estimator for the augmented mixed finite element schemes. Finally, extensive numerical experiments illustrating the performance of the augmented mixed finite element methods, confirming the properties of the a posteriori estimators, and showing the behavior of the associated adaptive algorithms are reported. [ABSTRACT FROM AUTHOR]

Low-order Whitney elements are widely used for electromagnetic field problems. Higher-order approximations are receiving increasing interest, but their definition remains unduly complex. In this paper we propose a new simple construction for Whitney p-elements of polynomial degree higher than one that use only degrees of freedom associated to p-chains. We provide a basis for these elements on simplicial meshes and give a geometrical localization of all degrees of freedom. Properties of the higher-order Whitney complex are deeply investigated. [ABSTRACT FROM AUTHOR]

For given quasi-continuous (q.c.) functions g, h with g ≤ h and diffusion process M determined by stochastic differential equations or symmetric Dirichlet forms, characterizations of the value functions ẽg(s, x) = supσ J(s,x)(σ) and w¯(s, x) = infτ supσ J(s,x)(σ, τ) have been well studied. In this paper, by using the time-dependent Dirichlet forms, we generalize these results to time inhomogeneous diffusion processes. The difficulty of our case arises from the existence of essential semipolar sets [Y. Oshima, Tohoku Math. J. (2), 54 (2002), pp. 443–449]. In particular, excessive functions are not necessarily continuous along the sample paths. We get the result by showing such a continuity of the value functions. [ABSTRACT FROM AUTHOR]

In this paper we introduce three new classes of nonnegative forms (or equivalently, symmetric tensors) and their extensions. The newly identified nonnegative symmetric tensors constitute distinctive convex cones in the space of general symmetric tensors (order six or above). For the special case of quartic forms, they collapse into the set of convex quartic homogeneous polynomial functions. We discuss the properties and applications of the new classes of nonnegative symmetric tensors in the context of polynomial and tensor optimization. Numerical experiments for solving certain polynomial optimization models based on the new class es of nonnegative symmetric tensors are presented. [ABSTRACT FROM AUTHOR]

The family of Filippov systems constituted by planar discontinuous piecewise linear systems with two half-plane linearity zones is considered. Under generic conditions that amount to the boundedness of the sliding set, some changes of variables and parameters are used to obtain a Liénard-like canonical form with seven parameters. This canonical form is topologically equivalent to the original system if one restricts one's attention to orbits with no points in the sliding set. Under the assumption of focus-focus dynamics, a reduced canonical form with only five parameters is obtained. For the case without equilibria in both open half-planes we describe the qualitatively different phase portraits that can occur in the parameter space and the bifurcations connecting them. In particular, we show the possible existence of two limit cycles surrounding the sliding set. Such limit cycles bifurcate at certain parameter curves, organized around different codimension-two Hopf bifurcation points. The proposed canonical form will be a useful tool in the systematic study of planar discontinuous piecewise linear systems, in which this paper is a first step. [ABSTRACT FROM AUTHOR]

In this paper we study the shape and growth of structured pseudospectra for small matrix perturbations of the form A … AΔ = A + BΔC, Δ ∈ Δ, ||Δ|| ≤ δ. It is shown that the properly scaled pseudospectra components converge to nontrivial limit sets as δ tends to 0. We discuss the relationship of these limit sets with µ-values and structured eigenvalue condition numbers for multiple eigenvalues. [ABSTRACT FROM AUTHOR]

A square matrix can be reduced to simpler form via similarity transformations. Here \simpler form" may refer to diagonal (when possible), triangular (Schur), or Hessenberg form. Sim- ilar reductions exist for matrix pencils if we consider general equivalence transformations instead of similarity transformations. For both matrices and matrix pencils, well-established algorithms are available for each reduction, which are useful in various applications. For matrix polynomials, uni- modular transformations can be used to achieve the reduced forms but we do not have a practical way to compute them. In this work we introduce a practical means to reduce a matrix polynomial with nonsingular leading coefficient to a simpler (diagonal, triangular, Hessenberg) form while preserving the degree and the eigenstructure. The key to our approach is to work with structure preserving similarity transformations applied to a linearization of the matrix polynomial instead of unimodular transformations applied directly to the matrix polynomial. As an application, we illustrate how to use these reduced forms to solve parameterized linear systems. [ABSTRACT FROM AUTHOR]

We present a recursive function approximation technique that does not require the storage of the arrival data stream. Our work is motivated by algorithms in stochastic optimization which require approximating functions in a recursive setting such as a stochastic approximation algorithm. The unique collection of these features in this technique is essential for nonlinear modeling of large data sets where the storage of the data becomes prohibitively expensive and in circumstances where our knowledge about a given query point increases as new information arrives. The algorithm presented here employs radial basis functions (RBFs) to provide locally adaptive parametric models (such as linear models). The local models are updated using recursive least squares and only store the statistical representative of the local approximations. The resulting scheme is very fast and memory efficient without compromising accuracy in comparison to methods well accepted as the standard and some advanced techniques used for functional data analysis in the literature. We motivate the algorithm using synthetic data and illustrate the algorithm on several real data sets. [ABSTRACT FROM AUTHOR]

We analyze a family of degenerate parabolic equations with linear growth Lagrangian having the form ut = div(φ(u)ψ(∇u/u)). Here |ψ| ≤ 1 and saturates at infinity. We present a simple and natural set of assumptions on the functions ψ, φ, under which (1) these equations fall into the framework provided by [F. Andreu, V. Caselles, and J. M. Mazón, Nonlinear Anal., 61 (2005), pp. 637-669, and J. Eur. Math. Soc. (JEMS), 7 (2005), pp. 361-393] and hence they are well posed, (2) we can ensure finite propagation speed for these models, (3) a Rankine-Hugoniot analysis on traveling fronts is also performed. On the particular case of φ(u) = u we get more detailed information on the spreading rate of compactly supported solutions and some interesting connections with optimal mass transportation theory. [ABSTRACT FROM AUTHOR]

We present an efficient numerical method for computing Fourier-Taylor expansions of (un)stable manifolds associated with hyperbolic periodic orbits. Three features of the method are that (1) we obtain accurate representation of the invariant manifold as well as the dynamics on the manifold, (2) it admits natural a posteriori error analysis, and (3) it does not require numerically integrating the vector field. Our approach is based on the parameterization method for invariant manifolds, and studies a certain partial differential equation which characterizes a chart map of the manifold. The method requires only that some mild nonresonance conditions hold. The novelty of the present work is that we exploit the Floquet normal form in order to efficiently compute the Fourier-Taylor expansion. A number of example computations are given including manifolds in phase space dimension as high as ten and manifolds which are two and three dimensional. We also discuss computations of cycle-to-cycle connecting orbits which exploit these manifolds. [ABSTRACT FROM AUTHOR]

We study eigenfunctions Φj and eigenvalues Ej of the Dirichlet Laplacian on a bounded domain Ω ⊂ ℝn with piecewise smooth boundary. We bound the distance between all arbitrary parameter E > 0 and the spectrum {Ej} in terms of the boundary L²-norm of a normalized trial solution a of the Helmholtz equation (Δ + E)u = 0. We also bound the L²-norm of the error of this trial solution from an eigenfunction. Both of these results are sharp up to constants, hold for all E greater than a small coastant, and improve upon the best-known bounds of Moler-Payne by a factor of the wavenumber √E. One application is to the solution of cigenvalue problems at high frequency, via, for example, the method of particular solutions. In the case of planar, strictly star-shaped domaias we give an inclusion bound where the constant is also sharp. We give explicit constants in the theorems, and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy. The proof makes use of a new quasi-orthogonality property of the boundary normal derivatives of the eigenmodes (Theorem 1.3), of interest in its own right. Namely, the operator norm of the sum of rank 1 operators ∂nΦj<∂nΦj, ⋅> over all Ej in a spectral window of width √E-a sum with about E(n-l)/2 terms--is at most a constant factor (independent of E) larger than the operator norm of any one individual term. [ABSTRACT FROM AUTHOR]

We present a rigorous numerical analysis and computational tests for the Galerkin finite element discretization of the velocity-vorticity-helicity formulation of the equilibrium NavierStokes equations (NSEs). This formulation, recently derived by the authors, is the first NSE formulation that directly solves for helicity and the first velocity-vorticity formulation to naturally enforce incompressibility of the vorticity, and preliminary computations confirm its potential. We present a numerical scheme; prove stability, existence of solutions, uniqueness under a small data condition, and convergence; and provide numerical experiments to confirm the theory and illustrate the effectiveness of the scheme on a benchmark problem. [ABSTRACT FROM AUTHOR]

We consider the boundary controllability problem for a nonlinear Korteweg—de Vries equation with the Dirichlet boundary condition. We study this problem for a spatial domain with a critical length for which the linearized control system is not controllable. In order to deal with the nonlinearity, we use a power series expansion of second order. We prove that the nonlinear term gives the local exact controllability around the origin provided that the time of control is large enough. [ABSTRACT FROM AUTHOR]

Let W(T) and r(T) denote the numerical range and numerical radius of an n x n complex matrix T. Let H2n denote the space of pairs of n x n Hermitian matrices. Define a norm on H2n by ∥(X, Y )∥ = r(X + iY ). Take (A,B) ∊ H2n. It is shown that if 0 ∊ W(A + iB), then inf{∣μ∣ : μ ∉ W(A + iB)} is an upper bound on the distance to the nearest pair that is simultaneously diagonalizable by congruence. If 0 ∉ W(A + iB), then min{∣μ∣ : μ ∊ W(A + iB)}, which is the Crawford number of the pair (A,B), is equal to the distance to the nearest pair that is not simultaneously diagonalizable by congruence. The results are similar when the numerical radius is replaced by the spectral norm. [ABSTRACT FROM AUTHOR]

After the paper cited in the title [George and Ikramov, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 348–354] was sent to the printer, the authors succeeded in finding a way to show that the answer to the title question is no. This brief note contains a proof of the result. [ABSTRACT FROM AUTHOR]