The rate of convergence of GMRES on a tridiagonal Toeplitz linear system

The Generalized Minimal Residual method (GMRES) is often used to solve a nonsymmetric linear system Ax = b. But its convergence analysis is a rather difficult task in general. A commonly used approach is to diagonalize A = XΛ X ⁻¹ and then separate the study of GMRES convergence behavior into optimizing the condition number of X and a polynomial minimization problem over A’s spectrum. This artificial separation could greatly overestimate GMRES residuals and likely yields error bounds that are too far from the actual ones. On the other hand, considering the effects of both A’s spectrum and the conditioning of X at the same time poses a difficult challenge, perhaps impossible to deal with in general but only possible for certain particular linear systems. This paper will do so for a (nonsymmetric) tridiagonal Toeplitz system. Sharp error bounds on and sometimes exact expressions for residuals are obtained. These expressions and/or bounds are in terms of the three parameters that define A and Chebyshev polynomials of the first kind.     

A rigorous construction of maximum recoverable work

A simple and rigorous derivation of the maximum recoverable work is presented. In contrast to previous derivations it is based on simple and rigorous projectional methods. The principle holds if the stress has a non-trivial Newtonian component.      

Agmon’s condition for incompressible elasticity: A formulation of Mielke–Sprenger type

Agmon’s Condition for incompressible elasticity is recast in a tractable variational form analogous to that developed by Mielke and Sprenger for the compressible case.

“A teacher affects eternity; he can never tell where his influence will stop.” – Henry Adams, The Education of Henry Adams, 1918.

To Professor Michael Hayes on the Occasion of his Seventieth Birthday     

A Class of Integral Operators on the Unit Ball of \mathbb{C}^{n}

For real parameters a, b, c, and t, where c is not a nonpositive integer, we determine exactly when the integral operator

Derivation of the material momentum equation from the energy balance

Based on a recent paper of Yavari et al, we show how the equation of material momentum can be obtained from invariance of the energy balance under translations in the material space. This approach further highlights the duality between the physical and material space.      

Smoluchowski-Kramers approximation for a general class of SPDEs

We prove that the so-called Smoluchowski-Kramers approximation holds for a class of partial differential equations perturbed by a non-Gaussian noisy term. Namely, we show that the solution of the one-dimensional semi-linear stochastic damped wave equations , u(0) = u₀, u_t (0) = v₀, endowed with Dirichlet boundary conditions, converges as the parameter μ goes to zero to the solution of the semi-linear stochastic heat equation , u(0) = u₀, endowed with Dirichlet boundary conditions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Asymptotic Behaviour of Iterates of Volterra Operators on L p (0, 1)

Given k ∈ L¹ (0,1) satisfying certain smoothness and growth conditions at 0, we consider the Volterra convolution operator V_k defined on L^p (0,1) by

Small amplitude waves and stability for a pre-stressed viscoelastic solid

We study the propagation of small amplitude waves superimposed on a large static deformation in a nonlinear viscoelastic material of differential type. We use bulk waves and surface waves to address the questions of dissipation and of material and geometric stability. In particular, the analysis provides bounds on the constitutive parameters and on the predeformation that ensure linearized stability in the neighbourhood of a large pre-stretch. This type of result is relevant to the imaging of biological soft tissues using acoustical techniques, where pre-deformation is known to increase contrast and reduce de-correlation noise. This work was supported by GNFM (Italy) and by an International Joint Project grant awarded jointly by the Royal Society of London (UK) and the CNRS (France).     

Blow-up and critical exponents for parabolic equations with non-divergent operators: dual porous medium and thin film operators

Our first basic model is the fully nonlinear dual porous medium equation with source

Strain gradient plasticity solution for an internally pressurized thick-walled cylinder of an elastic linear-hardening material

An elastic-plastic solution is presented for an internally pressurized thick-walled plane strain cylinder of an elastic linear-hardening plastic material. The solution is derived in a closed form using a strain gradient plasticity theory. The inner radius of the cylinder enters the solution not only in non-dimensional forms but also with its own dimensional identity, which differs from that in classical plasticity based solutions and makes it possible to capture the size effect at the micron scale. The classical plasticity solution of the same problem is recovered as a special case of the current solution. To further illustrate the newly derived solution, formulas and numerical results for the plastic limit pressure are provided. These results reveal that the load-carrying capacity of the cylinder increases with decreasing inner radius at the micron scale. It is also seen that the macroscopic behavior of the pressurized cylinder can be well described by using classical plasticity based solutions.     

Thermomechanics of twinning from a parent to a twin with mirror image symmetry

In the paper we develop a modeling with multiple configurations and mirror image of parent crystal in the twinned structure, to describe the behavior of partially twinned structure. In the constitutive framework we take into account that: (1) the untwinned and twinned material have distinct natural configurations by virtue of their miscrostructure being different, (2) the material symmetry groups of the untwinned and twinned structures characterize the peculiar feature that the presence of the mirror image structure is related to the untwinned structure, but it can exist only as a counterpart of the previous one. The partially twinned structure is described by the evolution equations for the growth of twins, characterized by a pair of a deformation like tensorial variable and a scalar field with meaning of the volume fraction for the twins. The capability of the material to twin and untwin at a constant rate of strain in uniaxial compression has been analyzed and the oscillatory behavior predicted by the model reveals qualitative agreement with experimental evidences.      

Simple estimation on effective transport properties of a random composite material with cylindrical fibres

Improved bounds for effective transport properties of a random non-percolated composite with cylindrical fibres are developed by means of the security-spheres approach. The key point of the method is to obtain a solution for a regular composite that can be valid for all values of the volume fractions and properties of the components. For this aim we use the asymptotic homogenization method; a cell problem is solved by a modified version of the boundary shape perturbation technique.      

On the socles of fully invariant subgroups of Abelian p-groups

The classification of the fully invariant subgroups of a reduced Abelian p-group is a difficult long-standing problem when one moves outside of the class of fully transitive groups. In this work we restrict attention to the socles of fully invariant subgroups and introduce a new class of groups which we term socle-regular groups; this class is shown to be large and strictly contains the class of fully transitive groups. The basic properties of such groups are investigated but it is shown that the classification of even this simplified class of groups, seems extremely difficult. Received: 4 September 2008     

On Vector Integral Inequalities

The first author introduced an integration theory of vector functions with respect to an operator-valued measure in complete bornological locally convex vector spaces. In this paper some important results behind this Dobrakov-type integration technique in non-metrizable spaces are given. Received: December 10, 2007., Accepted: May 6, 2008.     

Gradient Estimates for Solutions of Ultraparabolic Equations

We establish a priori estimates for solutions to ultraparabolic equations which play a crucial role in the solvability of the initial value problem. A class of these equations came from population dynamics, namely from a fish larvae model.      

Skew representations of twisted Yangians

Analogs of the classical Sylvester theorem have been known for matrices with entries in noncommutative algebras including the quantized algebra of functions on GL _N and the Yangian for $$ \mathfrak{g}\mathfrak{l}_{{N}} $$ . We prove a version of this theorem for the twisted Yangians $$ {\text{Y(}}\mathfrak{g}_{N} {\text{)}} $$associated with the orthogonal and symplectic Lie algebras $$ \mathfrak{g}_{N} = \mathfrak{o}_{N} {\text{ or }}\mathfrak{s}\mathfrak{p}_{N} $$. This gives rise to representations of the twisted Yangian $$ {\text{Y}}{\left( {\mathfrak{g}_{{N - M}} } \right)} $$ on the space of homomorphisms $$ {\text{Hom}}_{{\mathfrak{g}_{M} }} {\left( {W,V} \right)} $$, where W and V are finite-dimensional irreducible modules over $$ \mathfrak{g}_{{M}} {\text{ and }}\mathfrak{g}_{{N}} $$, respectively. In the symplectic case these representations turn out to be irreducible and we identify them by calculating the corresponding Drinfeld polynomials.We also apply the quantum Sylvester theorem to realize the twisted Yangian as a projective limit of certain centralizers in universal enveloping algebras.     

Conditions for the spectrum associated with an asymptotically straight leaky wire to contain an interval [− α2/4, ∞)

The method of singular sequences is used to provide a simple and, in some respects, a more general proof of a known spectral result for leaky wires. The method introduces a new concept of asymptotic straightness. Received: 4 October 2007     

On the Orbit of an m-Isometry

A bounded linear operator T on a Hilbert space H is called an m-isometry for a positive integer m if . We prove some properties concerning the behaviour of the orbit of an m-isometry. For example, every orbit of an m-isometry is eventually norm increasing and some m-isometries can not be N-supercyclic, that is, there does not exist an N-dimensional subspace E_N such that the orbit of T at E_N is dense in H.

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To Professor José Rodríguez Expósito on his 60th birthday