The period of periodic Young modules

The family of Young modules which are periodic has been determined in Hemmer and Nakano (J Algebra 254:422–440, 2002). We determine the period of all periodic Young modules in all characteristics. In particular, the period is dependent only on the characteristic. We calculate minimal projective resolutions of periodic Young modules in weight one blocks and in the principal block of S_2p{\mathcal {S}_{2p}} when p ≥ 3.     

Weakly almost periodic functionals on the measure algebra

It is shown that the collection of weakly almost periodic functionals on the convolution algebra of a commutative Hopf von Neumann algebra is a C*-algebra. This implies that the weakly almost periodic functionals on M(G), the measure algebra of a locally compact group G, is a C*-subalgebra of M(G)* = C ₀(G)**. The proof builds upon a factorisation result, due to Young and Kaiser, for weakly compact module maps. The main technique is to adapt some of the theory of corepresentations to the setting of general reflexive Banach spaces.     

Approximate computation of the permeability tensor of a periodic porous medium

The permeability tensor K of an infinite periodic porous medium, obtained using the homogenization theory, is considered. The solutions of an optimal control problem for the Dirichlet or Neumann equation are used to obtain optimal upper bounds for K. The test functions used for the estimations are simpler than those obtained by other authors. Some possibilities are given to obtain also lower bounds.     

On measure‐valued solutions to a two‐dimensional gravity‐driven avalanche flow model

This paper concerns measure‐valued solutions for the two‐dimensional granular avalanche flow model introduced by Savage and Hutter. The system is similar to the isentropic compressible Euler equations, except for a Coulomb–Mohr friction law in the source term. We will partially follow the study of measure‐valued solutions given by DiPerna and Majda. However, due to the multi‐valued nature of the friction law, new more sensitive measures must be introduced. The main idea is to consider the class of x‐dependent maximal monotone graphs of non‐single‐valued operators and their relation with 1‐Lipschitz, Carathéodory functions. This relation allows to introduce generalized Young measures for x‐dependent maximal monotone graph. Copyright © 2005 John Wiley & Sons, Ltd.     

The expectation and variance of the joint linear complexity of random periodic multisequences

The linear complexity of sequences is one of the important security measures for stream cipher systems. Recently, in the study of vectorized stream cipher systems, the joint linear complexity of multisequences has been investigated. By using the generalized discrete Fourier transform for multisequences, Meidl and Niederreiter determined the expectation of the joint linear complexity of random N-periodic multisequences explicitly. In this paper, we study the expectation and variance of the joint linear complexity of random periodic multisequences. Several new lower bounds on the expectation of the joint linear complexity of random periodic multisequences are given. These new lower bounds improve on the previously known lower bounds on the expectation of the joint linear complexity of random periodic multisequences. By further developing the method of Meidl and Niederreiter, we derive a general formula and a general upper bound for the variance of the joint linear complexity of random N-periodic multisequences. These results generalize the formula and upper bound of Dai and Yang for the variance of the linear complexity of random periodic sequences. Moreover, we determine the variance of the joint linear complexity of random periodic multisequences with certain periods.     

On the perturbation of the Q‐factor of the QR factorization

This paper gives normwise and componentwise perturbation analyses for the Q‐factor of the QR factorization of the matrix A with full column rank when A suffers from an additive perturbation. Rigorous perturbation bounds are derived on the projections of the perturbation of the Q‐factor in the range of A and its orthogonal complement. These bounds overcome a serious shortcoming of the first‐order perturbation bounds in the literature and can be used safely. From these bounds, identical or equivalent first‐order perturbation bounds in the literature can easily be derived. When A is square and nonsingular, tighter and simpler rigorous perturbation bounds on the perturbation of the Q‐factor are presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Non-uniform hyperbolicity and universal bounds for S-unimodal maps

An S-unimodal map f is said to satisfy the Collet-Eckmann condition if the lower Lyapunov exponent at the critical value is positive. If the infimum of the Lyapunov exponent over all periodic points is positive then f is said to have a uniform hyperbolic structure. We prove that an S-unimodal map satisfies the Collet-Eckmann condition if and only if it has a uniform hyperbolic structure. The equivalence of several non-uniform hyperbolicity conditions follows. One consequence is that some renormalization of an S-unimodal map has an absolutely continuous invariant probability measure with exponential decay of correlations if and only if the Collet-Eckmann condition is satisfied. The proof uses new universal bounds that hold for any S-unimodal map without periodic attractors. Oblatum 4-VII-1996 & 4-VII-1997     

Almost sure rates of mixing for i.i.d. unimodal maps

It has been known since the pioneering work of Jakobson and subsequent work by Benedicks and Carleson and others that a positive measure set of quadratic maps admit an absolutely continuous invariant measure. Young and Keller-Nowicki proved exponential decay of its correlation functions. Benedicks and Young [8], and Baladi and Viana [4] studied stability of the density and exponential rate of decay of the Markov chain associated to i.i.d. small perturbations. The almost sure statistical properties of the sample stationary measures of i.i.d. itineraries are more difficult to estimate than the “averaged statistics”. Adapting to random systems, on the one hand partitions associated to hyperbolic times due to Alves [1], and on the other a probabilistic coupling method introduced by Young [26] to study rates of mixing, we prove stretched exponential upper bounds for the almost sure rates of mixing.     

Obtaining upper bounds of heat kernels from lower bounds

We show that a near‐diagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an on‐diagonal upper bound. If in addition the Dirichlet form is local and regular, then we obtain a full off‐diagonal upper bound of the heat kernel provided the Dirichlet heat kernel on any ball satisfies a near‐diagonal lower estimate. This reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel. © 2007 Wiley Periodicals, Inc.     

Blister Patterns and Energy Minimization in Compressed Thin Films on Compliant Substrates

This paper is motivated by the complex blister patterns sometimes seen in thin elastic films on thick, compliant substrates. These patterns are often induced by an elastic misfit that compresses the film. Blistering permits the film to expand locally, reducing the elastic energy of the system. It is therefore natural to ask: what is the minimum elastic energy achievable by blistering on a fixed area fraction of the substrate? This is a variational problem involving both the elastic deformation of the film and substrate and the geometry of the blistered region. It involves three small parameters: the nondimensionalized thickness of the film, the compliance ratio of the film/substrate pair, and the mismatch strain. In formulating the problem, we use a small‐slope (Föppl–von Kármán) approximation for the elastic energy of the film, and a local approximation for the elastic energy of the substrate. For a one‐dimensional version of the problem, we obtain “matching” upper and lower bounds on the minimum energy, in the sense that both bounds have the same scaling behavior with respect to the small parameters. The upper bound is straightforward and familiar: it is achieved by periodic blistering on a specific length scale. The lower bound is more subtle, since it must be proved without any assumption on the geometry of the blistered region. For a two‐dimensional version of the problem, our results are less complete. Our upper and lower bounds only “match” in their scaling with respect to the nondimensionalized thickness, not in the dependence on the compliance ratio and the mismatch strain. The lower bound is an easy consequence of our one‐dimensional analysis. The upper bound considers a two‐dimensional lattice of blisters and uses ideas from the literature on the folding or “crumpling” of a confined elastic sheet. Our main two‐dimensional result is that in a certain parameter regime, the elastic energy of this lattice is significantly lower than that of a few large blisters. © 2015 Wiley Periodicals, Inc.     

Nonlinear integrodifferential equations anda priori bounds on periodic solutions

Summary This paper studies the existence of aperiodic solution of a nonlinear integrodifferential system of the form , for each continuous periodic function p and under suitable assumptions on f, k and g. A topological transversality method is employed to obtain the existence of periodic solutions. This method relies ona priori bounds on periodic solutions. Several examples are provided where a variant of Liapunov's direct method is employed to obtaina priori bounds on periodic solutions.     

Loss of polyconvexity by homogenization: a new example

This article is devoted to the study of the asymptotic behavior of the zero-energy deformations set of a periodic nonlinear composite material. We approach the problem using two-scale Young measures. We apply our analysis to show that polyconvex energies are not closed with respect to periodic homogenization. The counterexample is obtained through a rank-one laminated structure assembled by mixing two polyconvex functions with P-growth, where p ≥ 2 can be fixed arbitrarily.     

On the integral representation formula for a two‐component elastic composite

The aim of this paper is to derive, in the Hilbert space setting, an integral representation formula for the effective elasticity tensor for a two‐component composite of elastic materials, not necessarily well‐ordered. This integral representation formula implies a relation which links the effective elastic moduli to the N‐point correlation functions of the microstructure. Such relation not only facilitates a powerful scheme for systematic incorporation of microstructural information into bounds on the effective elastic moduli but also provides a theoretical foundation for inverse‐homogenization. The analysis presented in this paper can be generalized to an n‐component composite of elastic materials. The relations developed here can be applied to the inverse‐homogenization for a special class of linear viscoelastic composites. The results will be presented in another paper. Copyright © 2005 John Wiley & Sons, Ltd.     

Existence of μ−pseudo almost periodic solutions to some evolution equations

In this article, a new approach for pseudo almost periodic solution under the measure theory, under Acquistpace‐Terreni conditions. We make extensive use of interpolation spaces and exponential dichotomy techniques to obtain the existence of μ‐pseudo almost periodic solutions to some classes of nonautonomous partial evolution equations. For illustration, we propose some application to a nonautonomous heat equation. Copyright © 2017 John Wiley & Sons, Ltd.     

An application ofp-cyclic matrices,for solving periodic parabolic problems

Summary Finite-difference equations for the steady-state solution of a parabolic equation with periodic boundary conditions producep-cyclic matrices, wherep can be arbitrarily large. The periodic solution can be found by applying S.O.R. to the equations with thep-cyclic matrix, and this procedure is more economical than the standard step-by-step procedures for solving the parabolic equation. Bounds for the discretization error are found in terms of bounds for the known second derivatives of the boundary values.     

Class of tight bounds on the Q‐function with closed‐form upper bound on relative error

In this paper, we propose a novel class of parametric bounds on the Q‐function, which are lower bounds for 1 ≤ a < 3 and x > x_t = (a (a‐1) / (3‐a))^1/2, and upper bound for a = 3. We prove that the lower and upper bounds on the Q‐function can have the same analytical form that is asymptotically equal, which is a unique feature of our class of tight bounds. For the novel class of bounds and for each particular bound from this class, we derive the beneficial closed‐form expression for the upper bound on the relative error. By comparing the bound tightness for moderate and large argument values not only numerically, but also analytically, we demonstrate that our bounds are tighter compared with the previously reported bounds of similar analytical form complexity.     

Hyperbolic knots spanning accidental Seifert surfaces of arbitrarily high genus

A method for constructing hyperbolic knots each of which bounds accidental incompressible Seifert surfaces of arbitrarily high genus is given. Mathematics Subject Classification (2000):57N10, 57M25.The author was supported in part by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.     

Scattering by a highly oscillating surface

The boundary function method [A. B. Vasil'eva, V. F. Butuzov, and L. V. Kalachev, The boundary function method for singular perturbation problems, SIAM Studies in Applied Mathematics, Philadelphia, 1995] is used to build an asymptotic expansion at any order of accuracy of a scalar time‐harmonic wave scattered by a perfectly reflecting doubly periodic surface with oscillations at small and large scales. Error bounds are rigorously established, in particular in an optimal way on the relevant part of the field. It is also shown how the maximum principle can be used to design a homogenized surface whose reflected wave yields a first‐order approximation of the actual one. The theoretical derivations are illustrated by some numerical experiments, which in particular show that using the homogenized surface outperforms the usual approach consisting in setting an effective boundary condition on a flat boundary. Copyright © 2014 John Wiley & Sons, Ltd.     

Reiterated homogenization of the vector potential formulation of a magnetostatic problem in anisotropic composite media

We present an explicit characterization of the effective coefficients of a family of boundary value problems with multiscale periodic oscillatory coefficients, which correspond to the vector potential formulation of a magnetostatic problem in anisotropic composite media with periodic microstructures. Moreover, we study the Γ-convergence of sequences of multiscale periodic integral functionals depending on the curl of divergence-free fields applying the properties of multiscale Young measures associated with sequences of divergence-free fields.