Asymptotics of the reduced logarithmic capacity of a narrow cylinder

We consider the Dirichlet problem for the Laplace operator in the exterior of a narrow infinite cylinder with periodically varying directrix. The solution is sought in the class of functions logarithmically increasing as the distance from the cylinder is increased. The reduced logarithmic capacity is defined as a generalization of the logarithmic capacity (of the outer conformal radius).We construct and justify the asymptotics of the solution of the problem as the ratio of the diameter of the cross-section of the cylinder to its period tends to zero.     

Iterative solution of the secular equation

The secular equation with real symmetric positive definite n × n matrix is transformed into a system of n ² quadratic equations for which it is possible to construct a convergent procedure realizing an iterative solution. An example of the numerical realization of the method for solving the problem of determining the electronic energy spectrum is given.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 303–310.Original Russian Text Copyright © 2005 by L. S. Chkhartishvili.This revised version was published online in April 2005 with a corrected issue number.     

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.     

Bundles with multivalued automorphism groups

A category whose objects are principal bundles with fixed base (smooth manifold) B, structure group T ^k , and finite group of multivalued automorphisms is constructed; the morphisms are required to be equivariant with respect to . Invariants are found and used to calculate the group of equivalence classes of the category objects. Examples are given and applications to dynamical systems with gyroscopic forces are suggested.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 600–616.Original Russian Text Copyright © 2005 by A. V. Ryzhkova, E. I. Yakovlev.This revised version was published online in April 2005 with a corrected issue number.     

Existence of a Nonstationary Poiseuille Solution

The nonstationary Poiseuille solution describing the flow of a viscous incompressible fluid in an infinite cylinder is defined as a solution of the inverse problem for the heat equation. The existence and uniqueness of such nonstationary Poiseuille solution with the prescribed flux F(t) of the velocity field is studied. It is proved that under some compatibility conditions on the initial data and flux F(t) the corresponding inverse problem has a unique solution in Holder spaces.Original Russian Text Copyright © 2005 Pileckas K. and Keblikas V.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 649–662, May–June, 2005.     

On the distributions of the form ∑i (δpi−δni)

We present some properties of the distributions T of the form ∑_i (δ_pi−δ_ni), with ∑_i d(p_i,n_i)<∞, which arise in the study of the 3-d Ginzburg–Landau problem; see Bourgain et al. (C. R. Acad. Sci. Paris, Ser. I 331 (2000) 119–124). We show that there always exists an irreducible representation of T. We also extend a result of Smets (C. R. Acad. Sci. Paris, Ser. I 334 (2002) 371–374) which says that T is a measure iff T can be written as a finite sum of dipoles.     

An algorithm for matching point sets using the l_1 norm

The problem is considered of matching two sets of points in , by translation and rotation. There are many applications, for example in geodesy, computer vision and in the assessment of manufactured parts. When the matching criterion is least squares, there is a well known solution process based on the singular value decomposition of an matrix. Here we consider the use of the norm, which may be more appropriate than least squares in the context of wild points in the data. An algorithm is developed, and is illustrated by some examples for the case .     

Analytical solution of the quasi-static thermoelasticity problem in a pressurized thick-walled cylinder subjected to transient thermal loading

Thermoelasticity problem in a thick-walled cylinder is solved analytically using the finite Hankel transform. Time-dependent thermal boundary conditions are assumed to act on the inner surface of the cylinder. For the mechanical boundary conditions two different cases are assumed: Traction–displacement problem (traction is prescribed on the inner surface and the fixed displacement boundary condition on the outer one) and Traction–Traction problem (tractions are prescribed on both the inner and outer surfaces of the hollow cylinder). The quasi-static solution of the thermoelasticity problem is derived analytically, i.e., the transient thermal response of the cylinder is derived and then, quasi-static structural problem is solved and closed form relations are extracted for the thermal stresses in the two problems. The results show to be in accordance with that cited in the literature in the special cases.     

The Newton modified barrier method for QP problems

The Modified Barrier Functions (MBF) have elements of both Classical Lagrangians (CL) and Classical Barrier Functions (CBF). The MBF methods find an unconstrained minimizer of some smooth barrier function in primal space and then update the Lagrange multipliers, while the barrier parameter either remains fixed or can be updated at each step. The numerical realization of the MBF method leads to the Newton MBF method, where the primal minimizer is found by using Newton's method. This minimizer is then used to update the Lagrange multipliers. In this paper, we examine the Newton MBF method for the Quadratic Programming (QP) problem. It will be shown that under standard second-order optimality conditions, there is a ball around the primal solution and a cut cone in the dual space such that for a set of Lagrange multipliers in this cut cone, the method converges quadratically to the primal minimizer from any point in the aforementioned ball, and continues, to do so after each Lagrange multiplier update. The Lagrange multipliers remain within the cut cone and converge linearly to their optimal values. Any point in this ball will be called a hot start. Starting at such a hot start, at mostO(In In ^-1) Newton steps are sufficient to perform the primal minimization which is necessary for the Lagrange multiplier update. Here, >0 is the desired accuracy. Because of the linear convergence of the Lagrange multipliers, this means that onlyO(In ^-1)O(In In ^-1) Newton steps are required to reach an -approximation to the solution from any hot start. In order to reach the hot start, one has to perform Newton steps, wherem characterizes the size of the problem andC>0 is the condition number of the QP problem. This condition number will be characterized explicitly in terms of key parameters of the QP problem, which in turn depend on the input data and the size of the problem.Partially supported by NASA Grant NAG3-1397 and National Science Foundation Grant DMS-9403218.     

Monodromy-data parameterization of spaces of local solutions of integrable reductions of Einstein’s field equations

We show that for the fields depending on only two of the four space-time coordinates, the spaces of local solutions of various integrable reductions of Einsteins field equations are the subspaces of the spaces of local solutions of the null-curvature equations selected by universal (i.e., solution-independent conditions imposed on the canonical (Jordan) forms of the desired matrix variables. Each of these spaces of solutions can be parameterized by a finite set of holomorphic functions of the spectral parameter, which can be interpreted as a complete set of the monodromy data on the spectral plane of the fundamental solutions of associated linear systems. We show that both the direct and inverse problems of such a map, i.e., the problem of finding the monodromy data for any local solution of the null-curvature equations for the given Jordan forms and also of proving the existence and uniqueness of such a solution for arbitrary monodromy data, can be solved unambiguously (the monodromy transform). We derive the linear singular integral equations solving the inverse problem and determine the explicit forms of the monodromy data corresponding to the spaces of solutions of Einsteins field equations.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 2, pp. 278–304, May, 2005     

Existence and a priori bounds for steady stagnation flow toward a stretching cylinder

We investigate the nonlinear boundary value problem (BVP) that is derived from a similarity transformation of the Navier-Stokes equations governing fluid flow toward a stretching permeable cylinder. Existence of a solution is proven for all values of the Reynolds number and for both suction and injection, and uniqueness results are obtained in the case of a monotonic solution. A priori bounds on the skin friction coefficient are also obtained. These bounds achieve any desired order of accuracy as the injection parameter tends to negative infinity.     

Uniform and universal Glivenko-Cantelli classes

A class of measurable functions on a probability space is called a Glivenko-Cantelli class if the empirical measuresP _n converge to the trueP uniformly over almost surely. is a universal Glivenko-Cantelli class if it is a Glivenko-Cantelli Cantelli class for all lawsP on a measurable space, and a uniform Glivenko-Cantelli class if the convergence is also uniform inP. We give general sufficient conditions for the Glivenko-Cantelli and universal Glivenko-Cantelli properties and examples to show that some stronger conditions are not necessary. The uniform Glivenko-Cantelli property is characterized, under measurability assumptions, by an entropy condition.     

On complete-cocomplete subspaces of an inner product space

In this note we give a measure-theoretic criterion for the completeness of an inner product space. We show that an inner product space S is complete if and only if there exists a -additive state on C(S), the orthomodular poset of complete-cocomplete subspaces of S. We then consider the problem of whether every state on E(S), the class of splitting subspaces of S, can be extended to a Hilbertian state on E( ); we show that for the dense hyperplane S (of a separable Hilbert space) constructed by P. Pták and H. Weber in Proc. Am. Math. Soc. 129 (2001), 2111–2117, every state on E(S) is a restriction of a state on E( ).This revised version was published online in April 2005 with a corrected missing date string.     

Covering of nonlinear maps on a cone in neighborhoods of irregular points

Inverse function theorems for smooth nonlinear maps defined on convex cones in Banach spaces in a neighborhood of an irregular point are considered. The corresponding covering theorem is proved. The proofs are based on a Banach open mapping theorem for convex cones in Banach spaces, which is also proved in the paper. Sufficient conditions for tangency to the zero set of a nonlinear map without a priori regularity assumptions are obtained.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 483–497.Original Russian Text Copyright © 2005 by A. V. Arutyunov.This revised version was published online in April 2005 with a corrected issue number.     

On the singularities of the gradient of the solution to the Dirichlet-Neumann problem outside a plane cut

The problem considered in this paper deals with the Laplace equation outside a cut of sufficiently arbitrary form. The Dirichlet condition is given on one side of the cut and the Neumann condition on the other. Using the integral representation for the solution, we obtain explicit asymptotic formulas describing the singularity of the gradient of the solution at the edges of the cut. We discuss the effect of the disappearance of the singularity.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 364–377Original Russian Text Copyright © 2005 by P. A. Krutitskii, A. I. Sgibnev.This revised version was published online in April 2005 with a corrected issue number.     

Surface wave scattering by a rectangular impedance cylinder located on a reactive plane

The electromagnetic scattering of the surface wave by a rectangular impedance cylinder located on an infinite reactive plane is considered for the case that the impedances of the horizontal and vertical sides of the cylinder can have different values. Firstly, the diffraction problem is reduced into a modified Wiener–Hopf equation of the third kind and then solved approximately. The solution contains branch‐cut integrals and two infinite sets of constants satisfying two infinite systems of linear algebraic equations. The approximate analytical or numerical evaluations of corresponding integrals and numerical solution of the linear algebraic equation systems are obtained for various values of parameters such as the surface reactance of the plane, the vertical and horizontal wall impedances, the width and the height of the cylinder. Copyright © 2004 John Wiley & Sons, Ltd.     

Antiplane Problem of Electroelasticity for a Hollow Piezoceramic Cylinder Excited by a System of Surface Electrodes

An antiplane mixed boundary-value problem of electroelasticity is considered for a hollow piezoceramic cylinder with an arbitrary system of active surface electrodes generating its harmonic vibrations. The problem is solved using a method elaborated earlier for investigating vibrations of a solid piezoceramic cylinder with a system of active surface electrodes. The scheme of numerical solution of the obtained singular integro-differential equations of the boundary-value problem is based on the quadrature method. Calculation results are presented that describe the amplitude-frequency characteristics of a hollow cylinder and the behavior of some mechanical and electric quantities both within the cylinder and on its boundary.     

On asymptotics of solutions of parabolic equations with nonlocal high-order terms

In the paper, we study the Cauchy problem for second-order differential-difference parabolic equations containing translation operators acting to the high-order derivatives with respect to spatial variables. We construct the integral representation of the solution and investigate its long-term behavior. We prove theorems on asymptotic closeness of the constructed solution and the Cauchy problem solutions for classical parabolic equations; in particular, conditions of the stabilization of the solution are obtained. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 143–183, 2005.