Asymptotic analysis of an elastic rod with rounded ends

We derive a one-dimensional model for an elastic shuttle, that is, a thin rod with rounded ends and small fixed terminals, by means of an asymptotic procedure of dimension reduction. In the model, deformation of the shuttle is described by a system of ordinary differential equations with variable degenerating coefficients, and the number of the required boundary conditions at the end points of the one-dimensional image of the rod depends on the roundness exponent m∈(0,1). Error estimates are obtained in the case m∈(0,1/4) by using an anisotropic weighted Korn inequality, which was derived in an earlier paper by the authors. We also briefly discuss boundary layer effects, which can be neglected in the case m∈(0,1/4) but play a crucial role in the formulation of the limit problem for m ≥ 1/4.     

Asymptotics of solutions to the spectral elasticity problem for a spatial body with a thin coupler

We construct asymptotics for the eigenvalues and vector eigenfunctions of the elasticity problem for an anisotropic body with a thin coupler (of diameter h) attached to its surface. In the spectrum we select two series of eigenvalues with stable asymptotics. The first series is formed by eigenvalues O(h ²) corresponding to the transverse oscillations of the rod with rigidly fixed ends, while the second is generated by the longitudinal oscillations and twisting of the rod, as well as eigenoscillations of the body without the coupler. We check the convergence theorem for the first series and derive the error estimates for both series.     

Asymptotically sharp weight Korn’s inequality for thin-walled elastic structures

An elastic junction of several thin plates is considered. All the plates, except for one, called the basic plate, are rigidly clamped along parts of lateral surfaces. We deduce an asymptotically sharp Korn inequality which is weighted and anisotropic. The constant in this Korn inequality is independent of two parameters, the thickness h ∈ (0, 1] and relative rigidity μ ∈ (0, +∞) of the supporting and basic plates. The weight factors in the Sobolev norm on the basic plate essentially depend on the parameters h, μ and on the mutual disposition of the supporting plates. Sufficient geometric and algebraic conditions for the validity of Korn inequalities with various groups of weight factors are given. We also describe special constructions that show the impossibility to improve the obtained inequalities and the necessity of the restrictions imposed on the junction structure. Bibliography: 29 titles. Illustrations: 12 figures. __________ Translated from Problemy Matematicheskogo Analiza, No. 36, 2007, pp. 29–64.     

Strict anisotropic norm bounded real lemma in terms of matrix inequalities

This paper is aimed at extending the H _∞ Bounded Real Lemma to stochastic systems under random disturbances with imprecisely known probability distributions. The statistical uncertainty is measured in the terms of information theory using the mean anisotropy functional. The disturbance attenuation capabilities of the system are quantified by the anisotropic norm which is a stochastic counterpart of the H _∞ norm. A state-space sufficient criterion for the anisotropic norm of a linear discrete time invariant system to be bounded by a given threshold value is derived. The resulting Strict Anisotropic Norm Bounded Real Lemma involves an inequality on the determinant of a positive definite matrix and a linear matrix inequality. These convex constraints can be approximated by two linear matrix inequalities.     

Analogs of Korn’s Inequality on Heisenberg Groups

We give two analogs of Korn’s inequality on Heisenberg groups. First, the norm of the horizontal differential is estimated in terms of the symmetric part of the differential. Second, Korn’s inequality is treated as a coercive estimate for a differential operator whose kernel coincides with the Lie algebra of the isometry group. For this purpose, we construct a differential operator whose kernel coincides with the Lie algebra of the isometry group on Heisenberg groups and prove a coercive estimate for the operator.

     

Justification of the asymptotic theory of thin rods. Integral and pointwise estimates

We present a general (without any condition on symmetry) and simplified procedure of obtaining onedimensional equations describing strains of thin rods that can be anisotropic, nonhomogeneous and have periodic structure as well. The presented asymptotics is justified with the help of the weighted Korn inequality, i.e., the difference of the exact solution and an asymptotic solution to the problem of elasticity theory is estimated in the energetic integral metric. Uniform (by the maximum of modulus) estimates for the error of approximation of 3-dimensional displacement fields and stresses are also obtained. As is shown, it is impossible to obtain the pointwise closeness with respect to stresses if the influence of the boundary layer near the end-walls of the rod is not taken into account. Bibliography: 44 titles. Translated fromProblemy Matematicheskogo Analiza, No. 17, 1997, pp. 101–152.     

On the coupling of regularization techniques and the boundary element method for a hemivariational inequality modelling a delamination problem

In this paper, we couple regularization techniques of nondifferentiable optimization with the h‐version of the boundary element method (h‐BEM) to solve nonsmooth variational problems arising in contact mechanics. As a model example, we consider the delamination problem. The variational formulation of this problem leads to a hemivariational inequality with a nonsmooth functional defined on the contact boundary. This problem is first regularized and then discretized by an h‐BEM. We prove convergence of the h‐BEM Galerkin solution of the regularized problem in the energy norm, provide an a priori error estimate and give a numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Notes to the proof of a weighted Korn inequality for an elastic body with peak-shaped cusps

We present the proof of the weighted anisotropic Korn inequality in a three-dimensional domain with peak-shaped cusps on the boundary. We verify the asymptotic accuracy of distribution of multipliers at the components of displacement vector and their derivatives in the corresponding weighted norm. We indicate conditions on a peak cusp under which the natural energy class is not embedded into a Sobolev or Lebesgue class. In the last case, the operator of elasticity problems possesses the continuous spectrum provoking wave processes in a finite volume (“black holes” for elastic waves). We also discuss possible generalizations of the result and open questions. Bibliography: 39 titles. Illustrations: 9 figures.     

Sequential stability of weak solutions in compressible self‐gravitating fluids and stationary problem

In this paper, we prove the sequential stability of weak solutions over time, in relation to the Navier–Stokes system of compressible self‐gravitating fluids in a three‐dimensional domain. As a byproduct, we show that there exists at least one non‐negative solution to the stationary problem in any bounded domain with a given mass for the adiabatic constant γ > 3 ∕ 2. In particular, for the spherically symmetric case, these conclusions still hold for γ > 4 ∕ 3 or γ = 4 ∕ 3 with a small mass. Copyright © 2012 John Wiley & Sons, Ltd.     

A Recursive Construction for New Symmetric Designs

We introduce a recursive construction of regular Handamard matrices with row sum 2h for h=±3ⁿ. Whenever q=(2h – 1)² is a prime power, we construct, for every positive integer m, a symmetric designs with parameters (4h²(q^m+1 – 1)/(q – 1), (2h² – h)q^m, (h² – h)q^m).     

Asymptotic Analysis and Modeling of the Jointing of a Massive Body with Thin Rods

Asymptotic formulas are obtained for solutions of the anisotropic elasticity problem for a body with cavities into which thin rods are inserted, the outer ends of the rods being rigidly fixed. The surface of the body and the lateral surface of the rods are assumed load-free, but the entire elastic junction is subject to mass forces. The elastic materials are inhomogeneous and the stiffness of the rods may differ greatly from that of the body, their ratio being of the order h with an arbitrary exponent ; for = 0, the junction is homogeneous. Together with the asymptotic formulas, we construct and justify an asymptotic model of the junction. This model is applicable for a wide range of the exponent and preserves the parameter h in the conjugation conditions but is represented by a regularly perturbed problem. Since the leading asymptotic term involves fields with strong singularities, we have to give correct statements of the limit problem for a body with one-dimensional rods. For this purpose, we use the theory of self-adjoint extensions of operators or the technique of weighted spaces with separated asymptotics. The justification of our asymptotic expansions utilizes weighted anisotropic Korn inequalities, which take into account the mutual position of the rods and provide the best possible a priori estimates of the solutions. In contrast to other investigations, we describe, in explicit terms, the dependence of the bounds in the error estimates on the right-hand sides of the original problem. We also discuss the relationship between the asymptotic ansatz formulas and the weighted norms in the asymptotically precise Korn inequality.__________Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 24, pp. 95–214, 2004.     

Graph norms and Sidorenko’s conjecture

Let H and G be two finite graphs. Define h _H (G) to be the number of homomorphisms from H to G. The function h _H (·) extends in a natural way to a function from the set of symmetric matrices to ℝ such that for A _G , the adjacency matrix of a graph G, we have h _H (A _G ) = h _H (G). Let m be the number of edges of H. It is easy to see that when H is the cycle of length 2n, then h _H (·)^1/m is the 2n-th Schatten-von Neumann norm. We investigate a question of Lovász that asks for a characterization of graphs H for which the function h _H (·)^1/m is a norm.     

A coefficient inequality for Bloch functions with applications to uniformly locally univalent functions

We give a Fekete-Szeg? type inequality for an analytic function on the unit disk with Bloch seminorm ≤1. As an application of it, we derive a sharp inequality for the third coefficient of a uniformly locally univalent function f(z) = z + a ₂ z ² + a ₃ z ³ + ⋯ on the unit disk with pre-Schwarzian norm ≤λ for a given λ > 0.     

Resolvable Mendelsohn triple systems with equal sized holes

An HMTS of type {n₁, n₂, ⋖, n_h} is a directed graph which can be decomposed into 3-circuits. If the 3-circuits can be partitioned into parallel classes, then the HMTS is called an RHMTS. In this article it is shown that the RHMTSs of type m^h exist when mh &equiv 0 (mod 3) and (m, h) &ne (1, 6), with the possible exception of h = 6 and , where M₁₇ = {m|m is divisible by a prime less than 17}. The existence of Mendelsohn frames, which is closely related to RHMTS, is also considered in this article. It is proved that a Mendelsohn frame of type t^u exists if and only if u ≥ 4 and t(u - 1) ≡ 0(mod 3) with 2 possible exceptions. © 1997 John Wiley & Sons, Inc. J Combin Designs 5:329–340, 1997     

Isoperimetric inequality from the poisson equation via curvature

In this paper, we establish an isoperimetric inequality in a metric measure space via the Poisson equation. Let (X,d,μ) be a complete, pathwise connected metric space with locally Ahlfors Q‐regular measure, where Q > 1, that supports a local L²‐Poincaré inequality. We show that, for the Poisson equation Δu = g, if the local L^∞‐norm of the gradient Du can be bounded by the Lorentz norm L^Q,1 of g, then we obtain an isoperimetric inequality and a Sobolev inequality in (X,d,μ) with optimal exponents. By assuming a suitable curvature lower bound, we establish such optimal bounds on $\||Du|\|_{L^\infty_{\rm loc}}$ . © 2011 Wiley Periodicals, Inc.     

Weighted korn inequalities in paraboloidal domains

A weighted Korn inequality in a domain Ω ⊂ ℝ ⁿ with paraboloidal exit II to infinity is obtained. Asymptotic sharpness of the inequality is achieved by using different weight factors for the longitudinal (with respect to the axis of II) and transversal displacement vector components and by making the weight factors of the derivatives depend on the direction of differentiation. The solvability of the elasticity problem in the energy class (the closure of in the norm generated by the elastic energy functional) is studied; the dimensions of the kernel and the cokerned of the corresponding operator depend on the exponents∈(−∞, 1) in the “rate of expansion” of the paraboloid II. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 751–765, November, 1997. Translated by V. N. Dubrovsky     

A problem in extremal quasiconformal extensions

A constantK ₀ ^(m) (h) is introduced for every quasisymmetric mappingh of the unit circle and every integerm≥4 which contains the constantK ₀(h) (indicated by the change in module of the quadrilaterals with vertices on the circle) as a special case. A necessary and sufficient condition is established forK ₀ ^(m) (h) =K ₁(h). It is shown that there are infinitely many quasisymmetric mappings of the unit circle having the property thatK ₀ ^(m) (h)<K ₁(h), wherek ₁(h) is the maximal dilatation ofh.     

Perfect Mendelsohn designs with equal-sized holes and block size four

Let M = {m₁, m₂, …, m_h} and X be a v-set (of points). A holey perfect Mendelsohn designs (briefly (v, k, λ) - HPMD), is a triple (X, H, B), where H is a collection of subsets of X (called holes) with sizes M and which partition X, and B is a collection of cyclic k-tuples of X (called blocks) such that no block meets a hole in more than one point and every ordered pair of points not contained in a hole appears t-apart in exactly λ blocks, for 1 ≤ t ≤ k − 1. The vector (m₁, m₂, …, m_h) is called the type of the HPMD. If m₁ = m₂ = … = m_h = m, we write briefly m^h for the type. In this article, it is shown that the necessary condition for the existence of a (v, 4, λ) - HPMD of type m^h, namely, is also sufficient with the exception of types 2⁴ and 1⁸ with λ = 1, and type m⁴ for odd m with odd λ. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 203–213, 1997     

Finding off‐diagonal entries of the inverse of a large symmetric sparse matrix

The method fast inverse using nested dissection (FIND) was proposed to calculate the diagonal entries of the inverse of a large sparse symmetric matrix. In this paper, we show how the FIND algorithm can be generalized to calculate off‐diagonal entries of the inverse that correspond to ‘short’ geometric distances within the computational mesh of the original matrix. The idea is to extend the downward pass in FIND that eliminates all nodes outside of each node cluster. In our advanced downwards pass, it eliminates all nodes outside of each ‘node cluster pair’ from a subset of all node cluster pairs. The complexity depends on how far (i,j) is from the main diagonal. In the extension of the algorithm, all entries of the inverse that correspond to vertex pairs that are geometrically closer than a predefined length limit l will be calculated. More precisely, let α be the total number of nodes in a two‐dimensional square mesh. We will show that our algorithm can compute O(α^{3 ∕ 2 + 2ε}) entries of the inverse in O(α^{3 ∕ 2 + 2ε}) time where l = O(α^{1 ∕ 4 + ε}) and 0 ≤ ε ≤1 ∕ 4. Numerical examples are given to illustrate the efficiency of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

A Concentration Theorem for the Heat Equation

 We compare the solution of to the solution of the same equation where f is replaced by a “concentrated” source . As a result we derive some estimates on the solution in spatial norm, locally uniformly in t, with respect to the norm of for any integer . In the case we obtain a critical inequality relating the norm of to an exponential norm of u. (Received 1 September 2000; in revised form 17 January 2001)