Higher order bending and membrane responses of thin linearly elastic plates

The limit behaviors of three-dimensional displacements in thin linearly elastic plates, as the half-thickness ε tends to zero, is now known for various lateral boundary conditions (see [1], [5]). In the generic case one obtains that the leading term of the asymptotic series u⁰ + ge¹+ε²u² +… of the scaled displacement is a Kirchhoff-Love field. In this Note we investigate the case where this leading term vanishes, giving the structure of the first non-vanishing term u^k and an error estimate for its deviation from the scaled solution u(ε) multiplied by ε^−k. There are essentially only three new cases (uncoupling in membrane and bending). Finally, in these situations a boundary layer term of the same order as the actual leading term appears in a generic way.     

A nonlinear elliptic problem with terms concentrating in the boundary

In this paper, we investigate the behavior of a family of steady‐state solutions of a nonlinear reaction diffusion equation when some reaction and potential terms are concentrated in a ε‐neighborhood of a portion Γ of the boundary. We assume that this ε‐neighborhood shrinks to Γ as the small parameter ε goes to zero. Also, we suppose the upper boundary of this ε‐strip presents a highly oscillatory behavior. Our main goal here was to show that this family of solutions converges to the solutions of a limit problem, a nonlinear elliptic equation that captures the oscillatory behavior. Indeed, the reaction term and concentrating potential are transformed into a flux condition and a potential on Γ, which depends on the oscillating neighborhood. Copyright © 2012 John Wiley & Sons, Ltd.     

Some properties of asymptotic energy of a class of functionals of Ginzburg-Landau type endowed with generalized lower-order oscillatory term in one dimension

We consider a generalization of the functional of Ginzburg-Landau type studied in the paper A. Raguž: A note on calculation of asymptotic energy for Ginzburg-Landau functional with externally imposed lower-order oscillatory term in one dimension, Boll. Un. Mat. Ital. (8)10-B , 1125-1142 (2007), whereby the oscillatory term a(ε^−βs) (where a ∈ L¹_per(0, 1) and β > 0) is replaced by a(ρ_εs) (where lim_ε−→0 ρ_ε = +∞). We describe how the expression for the rescaled asymptotic energy of such class of functionals depends on the properties of the sequence (ρ_ε). (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Averaging of a 3D primitive equations with oscillating external forces

In this article, we consider a non-autonomous three-dimensional primitive equations of the ocean with a singularly oscillating external force g ^ε?=?g ₀(t)?+?ε^?ρ g ₁(t/ε) depending on a small parameter ε?>?0 and ρ?∈?[0,?1) together with the averaged system with the external force g ₀(t), formally corresponding to the case ε?=?0. Under suitable assumptions on the external force, we prove as in [V.V. Chepyzhov, V. Pata, and M.M.I. Vishik, Averaging of 2D Navier–Stokes equations with singularly oscillating forces, Nonlinearity, 22 (2009), pp. 351–370] the boundness of the uniform global attractor 𝒜^ε as well as the convergence of the attractors 𝒜^ε of the singular systems to the attractor 𝒜⁰ of the averaged system as ε?→?0⁺. When the external force is small enough and the viscosity is large enough, the convergence rate is controlled by Kε^(1?ρ). Let us note that the main difference between this work and that of Chepyzhov et al. (2009) is that the non-linearity involved in the three-dimensional primitive equation is stronger than the one in the two-dimensional Navier–Stokes equations considered in Chepyzhov et al. (2009), which makes the analysis of the problem studied in this article more involved.     

Universality at the edge of the spectrum for unitary,orthogonal, and symplectic ensembles of random matrices

We prove universality at the edge of the spectrum for unitary (β = 2), orthogonal (β = 1), and symplectic (β = 4) ensembles of random matrices in the scaling limit for a class of weights w(x) = e^?V(x) where V is a polynomial, V(x) = κ_2mx^2m + · · ·, κ_2m > 0. The precise statement of our results is given in Theorem 1.1 and Corollaries 1.2 and 1.4 below. For the same class of weights, a proof of universality in the bulk of the spectrum is given in [12] for the unitary ensembles and in [9] for the orthogonal and symplectic ensembles. Our starting point in the unitary case is [12], and for the orthogonal and symplectic cases we rely on our recent work [9], which in turn depends on the earlier work of Widom [46] and Tracy and Widom [42]. As in [9], the uniform Plancherel‐Rotach‐type asymptotics for the orthogonal polynomials found in [12] plays a central role. The formulae in [46] express the correlation kernels for β = 1, 4 as a sum of a Christoffel‐Darboux (CD) term, as in the case β = 2, together with a correction term. In the bulk scaling limit [9], the correction term is of lower order and does not contribute to the limiting form of the correlation kernel. By contrast, in the edge scaling limit considered here, the CD term and the correction term contribute to the same order: this leads to additional technical difficulties over and above [49]. © 2006 Wiley Periodicals, Inc.     

Some metric properties on the GCF fraction expansion

In Schweiger (2003) [1], Fritz Schweiger introduced the algorithm of the generalized continued fraction (GCF), and in Zhong (2008) [2], T. Zhong studied some basic metric properties of the GCF. In this paper, under the restriction of −1<ε(k)?1, the “0-1” law and the central limit theorem of quotients in the GCF expansions are studied.     

A note on periodic solutions of a differential equation with two parameters

In [1] (p. 215), the authors Andronov, Leontovich-Andronova, Gordon, and Maier, consider the following equation: $$\left\{ \begin{gathered} \tfrac{{dx}}{{dt}} = y, \hfill \\ \tfrac{{dy}}{{dt}} = x + x^2 - \left( {\varepsilon _1 + \varepsilon _2 x} \right)y, \hfill \\ \end{gathered} \right.$$ whereε ₁ andε ₂ are real constants andε ₁ andε ₂ are not both zero. They proved that there are no non-trivial periodic solutions except possibly for the case $0< \tfrac{{\varepsilon _1 }}{{\varepsilon _2 }}< \tfrac{3}{2}$ . They left that case as an open problem. In this note we prove that there are indeed no non-trivial periodic solutions in the case $0< \tfrac{{\varepsilon _1 }}{{\varepsilon _2 }}< \tfrac{3}{2}$ either. Our method of proof consists essentially of constructing a Dulac function (see [6] and [9]) and using the conception of Duff's rotated vector field (see [4], [7], [8], [10], and [11]).     

Retrieving random media

Benjamini asked whether the scenery reconstruction methods of Matzinger (see e.g. [21], [22], [20]) can be done in polynomial time. In this article, we give the following answer for a 2-color scenery and simple random walk with holding: We prove that a piece of the scenery of length of the order 3 ⁿ around the origin can be reconstructed – up to a reflection and a small translation – with high probability from the first 2 · 3^{10 αn} observations with a constant α > 0 independent of n. Thus, the number of observations needed is polynomial in the length of the piece of scenery which we reconstruct. The probability that the reconstruction fails tends to 0 as n→∞. In contrast to [21], [22], and [20], the proofs in this article are all constructive. Our reconstruction algorithm is an algorithm in the sense of computer science. This is the first article which shows that the scenery reconstruction is also possible in the 2-color case with holding. The case with holding is much more difficult than [22] and requires completely different methods.     

A singularly perturbed nonlinear traction problem in a periodically perforated domain: a functional analytic approach

We consider a periodically perforated domain obtained by making in a periodic set of holes, each of them of size proportional to ε. Then, we introduce a nonlinear boundary value problem for the Lamé equations in such a periodically perforated domain. The unknown of the problem is a vector‐valued function u, which represents the displacement attained in the equilibrium configuration by the points of a periodic linearly elastic matrix with a hole of size ε contained in each periodic cell. We assume that the traction exerted by the matrix on the boundary of each hole depends (nonlinearly) on the displacement attained by the points of the boundary of the hole. Then, our aim is to describe what happens to the displacement vector function u when ε tends to 0. Under suitable assumptions, we prove the existence of a family of solutions {u(ε, ? )}_{ε ∈ ]0,ε ′ [} with a prescribed limiting behavior when ε approaches 0. Moreover, the family {u(ε, ? )}_{ε ∈ ]0,ε ′ [} is in a sense locally unique and can be continued real analytically for negative values of ε. Copyright © 2013 John Wiley & Sons, Ltd.     

Values of rational functions on non-hilbertian fields and a question of Weissauer

We answer in the negative a question raised by Fried and Jarden, asking whether the quotient field of a unique factorization domain with infinitely many primes is necessarily hilbertian. This implies a negative answer to a related question of Weissauer. Our constructions are simple and take place inside the field of algebraic numbers. Simultaneously we investigate the relation of hilbertianity of a fieldK with the structure of the value sets of rational functions onK: we construct a non-hilbertian subfieldK of such that, given anyf ₁ ,…,f _h ∈K(x), each of degree ≥2, the union ∪ _z=1 ^h f _z(K) does not containK. See e.g. [FrJ], [L1], [L2], [Sch], [Se1], or [Se2] for the classical theory of hilbertian fields.     

On the global existence of solutions to a singular semilinear parabolic equation arising from the study of autocatalytic chemical kinetics

In this paper we consider the question of the existence of solutions to an initial-boundary value problem for a singular, semilinear, parabolic equation arising from the study of autocatalytic chemical kinetics of the typeA B at ratek[A][B] ^P , whereA is a reactant,B is the autocatalyst,k>0 the rate constant and 0<p<1 the order of the reaction, King and Needham [1]. A monotone iteration method is adopted in the spirit of Sattinger [2], However the general theory due to Sattinger [2] cannot be applied directly because of the singular nature of the source term in the equation.     

A Two-Dimensional Bisection Envelope Algorithm for Fixed Points

In this paper we present a new algorithm for the two-dimensional fixed point problem f(x)=x on the domain [0, 1]×[0, 1], where f is a Lipschitz continuous function with respect to the infinity norm, with constant 1. The computed approximation x satisfies 6f(x)−x6_∞⩽ε for a specified tolerance ε<0.5. The upper bound on the number of required function evaluations is given by 2⌈log₂(1/ε)⌉+1. Similar bounds were derived for the case of the 2-norm by Z. Huang et al. (1999, J. Complexity15, 200–213), our bound is the first for the infinity norm case.     

A pair of forbidden subgraphs and perfect matchings in graphs of high connectivity

Sumner [7] proved that every connected K _1,3-free graph of even order has a perfect matching. He also considered graphs of higher connectivity and proved that if m ≥ 2, every m-connected K _1,m+1-free graph of even order has a perfect matching. In [6], two of the present authors obtained a converse of sorts to Sumner’s result by asking what single graph one can forbid to force the existence of a perfect matching in an m-connected graph of even order and proved that a star is the only possibility. In [2], Fujita et al. extended this work by considering pairs of forbidden subgraphs which force the existence of a perfect matching in a connected graph of even order. But they did not settle the same problem for graphs of higher connectivity. In this paper, we give an answer to this problem. Together with the result in [2], a complete characterization of the pairs is given.     

Quasidiagonality and the hyperinvariant subspace problem

In a sequence of recent papers, [11], [13], [9] and [5], the authors (together with H. Bercovici and C. Foias) reduced the hyperinvariant subspace problem for operators on Hilbert space to the question whether every C ₀₀-(BCP)-contraction that is quasidiagonal and has spectrum the unit disc has a nontrivial hyperinvariant subspace (n.h.s.). An essential ingredient in this reduction was the introduction of two new equivalence relations, ampliation quasisimilarity and hyperquasisimilarity, defined below. This note discusses the question whether, by use of these relations, a further reduction of the hyperinvariant subspace problem to the much-studied class (N + K) (defined below) might be possible.     

Nonuniqueness for a bounded, differentiable complete orthonormal system inL 2[0, 1]

The argument of Müsegian and Ovsepjan is adapted to produce a complete orthonormal system on [0, 1] of uniformly bounded functions, differentiable on [0, 1], andC ^∞ on [0, 1], for which the analogue of Cantor's uniqueness theorem is false. We also construct a complete orthonormal system ofC ^∞ functions which vanish to infinite order at both endpoints.     

Engouffrement symplectique et intersections lagrangiennes

Let }L _t{,t ∈ [0, 1], be a path of exact Lagrangian submanifolds in an exact symplectic manifold that is convex at infinity and of dimension ≥6. Under some homotopy conditions, an engulfing problem is solved: the given path }L _t{ is conjugate to a path of exact submanifolds inT ^*L_o. This impliesL _t must intersectL _o at as many points as known by the generating function theory. Our Engulfing theorem depends deeply on a new flexibility property of symplectic structures which is stated in the first part of this work.

     

Feebly projectable ℓ-groups

In the article [17], we introduced and investigated feebly and flatly projectable frames. In this article, we apply these two properties to lattice-ordered groups. An example is constructed to illustrate that the two properties are distinct, which solves a question from [17]. We also investigate these properties with respect to archimedean ℓ-groups with weak order unit, as well as commutative semiprime f-rings.     

A Banach space with,up to equivalence,precisely two symmetric bases

It is well know that the classical sequence spaces c_o andl _p (1≦p<∞) have, up to equivalence, just one symmetric basis. On the other hand, there are examples of Orlicz sequence spaces which have uncountably many mutually non-equivalent symmetric bases. Thus in [4], p. 130, the question is asked whether there is a Banach space with, up to equivalence, more than one symmetric basis, but not uncountably many. In this paper we answer the question positively, by exhibiting a Banach space with, up to equivalence, precisely two symmetric bases.     

Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3:2:1

We consider a boundary-value problem for the Poisson equation in a thick junction Ω_ε, which is the union of a domain Ω₀ and a large number of ε-periodically situated thin curvilinear cylinders. The following nonlinear Robin boundary condition ∂_νu_ε + εκ(u_ε)=0 is given on the lateral surfaces of the thin cylinders. The asymptotic analysis of this problem is performed as ε → 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove the convergence theorem and show that the nonlinear Robin boundary condition is transformed (as ε → 0) in the blow-up term of the corresponding ordinary differential equation in the region that is filled up by the thin cylinders in the limit passage. The convergence of the energy integral is proved as well. Using the method of matched asymptotic expansions, the approximation for the solution is constructed and the corresponding asymptotic error estimate in the Sobolev space H¹(Ω_ε) is proved. Copyright © 2007 John Wiley & Sons, Ltd.