A periodic system of electrically permeable cracks at the interface between two piezoelectric materials

We constructed a closed solution for a bimaterial plane consisting of two dissimilar piezoelectric half-planes with a periodic system of electrically permeable cracks at the interface between these materials. The presence of zones of smooth contact of the crack lips near their tips was taken into account. By representing the characteristics of electromechanical fields via piecewise analytic functions, we reduced the problem to a Dirichlet–Riemann periodic problem, which was solved exactly. As a result of numerical analysis of the derived solution, we studied the dependence of the relative length of the contact zones and stress intensity factors on the ratio between the crack length and period for different combinations of piezoelectric materials.     

On the maximum size of an anti-chain of linearly separable sets and convex pseudo-discs

We answer a question raised by Walter Morris, and independently by Alon Efrat, about the maximum cardinality of an anti-chain composed of intersections of a given set of n points in the plane with half-planes. We approach this problem by establishing the equivalence with the problem of the maximum monotone path in an arrangement of n lines. A related problem on convex pseudo-discs is also discussed in the paper. This research was supported by a Grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

On one property of solutions of problems of the theory of elasticity for two half-planes or half-spaces

We have shown that the solution of any boundary-value problem for two conjugated half-planes with different elastic constants, in the case where the stresses are persistently continuable across the boundary between half-planes, can be expressed via one common elastic constant if the stresses and external force factors are nondimensionalized by the reduced modulus of elasticity. Owing to this, it is possible to obtain the solution of the problem for two conjugated half-planes directly from the solution of the corresponding problem for one elastic half-plane. This property also holds true for axially symmetric problems formulated for two conjugated half-spaces.     

Finding a minimum-weightk-link path in graphs with the concave Monge property and applications

LetG be a weighted, complete, directed acyclic graph (DAG) whose edge weights obey the concave Monge condition. We give an efficient algorithm for finding the minimum-weightk-link path between a given pair of vertices for any givenk. The time complexity of our algorithm is . Our algorithm uses some properties of DAGs with the concave Monge property together with the parametric search technique. We apply our algorithm to get efficient solutions for the following problems, improving on previous results: (1) Finding the largestk-gon contained in a given convex polygon. (2) Finding the smallestk-gon that is the intersection ofk half-planes out ofn half-planes defining a convexn-gon. (3) Computing maximumk-cliques of an interval graph. (4) Computing length-limited Huffman codes. (5) Computing optimal discrete quantization.     

On the solution of boundary value problems on an inhomogeneous plane with a crack and a barrier connected in series

We consider boundary value problems for an equation in divergence form on a plane divided into two inhomogeneous half-planes by a film inclusion in the form of a strongly permeable crack and a weakly permeable barrier connected in series; this models a contact of heterogeneous media under inhomogeneous external conditions. The desired potentials have prescribed singular points (sources, drains, etc.). The coefficients of the equation are nonconstant and may increase or decrease when moving away from the film inclusion along a family of parabolas. We obtain representations of solutions of the considered problems via harmonic functions with the corresponding singular points on the plane.     

Elastodynamical scattering by N parallel half-planes in R3: II. Explicit solutions forN=2 by explicit symbol factorization

In this paper, which is a continuation of [73] part I, explicit solutions of two mixed b.v.ps. for the vectorial Lamé equation with DDT/DDT data, TTD/TTD data, resp., given on a system ofN=2 parallel screen-crack half-planes, are derived by explicit calculation of the factors of the corresponding (residual)L ²-lifted nonrational 6×6 Wiener-Hopf-Fouriersymbol matrices, which were scalarized ton-part form (n=6) structures For a single screen two WHOs closely related to theRawlins problem and the impedance problem for the (scalar) Helmholtz equation are established to be Fredholm operators, the second when assuming the regularity higher thanH ¹ i.e. H ¹⁺, 0<<1/2. The WHO of theN-screen Dirichlet (and Neumann) problem for the Helmholtz equation is shown to be invertible by an operator Neumann series, even for small distances between the half-planes.Dedicated to Professor W. Wendland on the occasion of his 60 ^th birthday in September 1996Sponsored by the Deutsche Forschungsgemeinschaft under grant number KO 634/32-3     

A characterization of the zero-free region of the Riemann zeta function and its applications

We characterize the zero-free regions of a class of functions (including the Riemann zeta function) in half-planes in terms of closures of ranges of the corresponding multiplication operators on Hardy spaces. We give an explicit characterization of these closures. As applications, we obtain a weaker version of the Nyman–Beurling–Báez-Duarte criterion, and provide some investigations on a problem relating to the Riemann hypothesis proposed by Báez-Duarte et al. [Adv. Math. 149 (2000) 130-144].     

Some Isoperimetric Problems in Planes with Density

We study the isoperimetric problem in Euclidean space endowed with a density. We first consider piecewise constant densities and examine particular cases related to the characteristic functions of half-planes, strips and balls. We also consider continuous modification of Gauss density in ℝ². Finally, we give a list of related open questions.     

Elastic fields in two imperfectly bonded half-planes with a thermal inclusion of arbitrary shape

A general method is presented for the rigorous solution of Eshelby’s problem concerned with an arbitrary shaped inclusion embedded within one of two dissimilar elastic half-planes in plane elasticity. The bonding between the half-planes is considered to be imperfect with the assumption that the interface imperfections are uniform. Using analytic continuation, the basic boundary value problem is reduced to a set of two coupled nonhomogeneous first-order differential equations for two analytic functions defined in the lower half-plane which is free of the thermal inclusion. Using diagonalization, the two coupled differential equations are decoupled into two independent nonhomogeneous first-order differential equations for two newly defined analytic functions. The resulting closed-form solutions are given in terms of the constant imperfect interface parameters and the auxiliary function constructed from the conformal mapping which maps the exterior of the inclusion onto the exterior of the unit circle. The method is illustrated using several examples of an imperfect interface. In particular, when the same degree of imperfection is realized in both the normal and tangential directions between the two half-planes, a thermal inclusion of arbitrary shape in the upper half-plane does not cause any mean stress to develop in the lower half-plane. Alternatively, when the imperfect interface parameters are not equal, then a nonzero mean stress will be induced in the lower half-plane by the thermal inclusion of arbitrary shape in the upper half-plane. Detailed results are presented for the mean stress and the interfacial normal and shear stresses caused by a circular and elliptical thermal inclusion, respectively. Results from these calculations reveal that the imperfect bonding condition has a significant effect on the internal stress field induced within the inclusion as well as on the interfacial normal and shear stresses existing between the two half-planes especially when the inclusion is near the imperfect interface.     

Green's contact functions for heat conduction of two half-spaces and half-planes with application to several boundary contact problems

Green's contact functions are constructed for two half-spaces and two half-planes for materials with different thermal conductivities. With the aid of these contact functions some bimetal problems are reduced to boundary integral equations along the outer boundary where only the boundary conditions are to be satisfied. The boundary integral operators are investigated in the plane case. They are Fredholm operators with index zero. The asymptotics of the density of the potentials, which depends on the material parameters and on the angles between the contact line and the outer boundary, is determined by the Mellin transform technique.     

Thermoelastic contact interaction of bodies in the presence of surface thermophysical irregularities

We study the thermoelastic contact interaction (in the absence of friction) of half-spaces under conditions of planar deformation in the presence of thin surface thermophysical irregularities that are taken into account by means of generalized conditions of thermal contact with one another. The problem is reduced to solving a system of singular integrodifferential equations with respect to the jumps of temperature and heat flow on the boundary of a section. We analyze the influence of a nonuniform thermal resistance distributed periodically along the surface or localized in one region of it on the distribution of temperature and stresses in the bodies and on their boundary. Four figures.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 23–28.     

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.     

Determination of the temperatures and heat flows for a conical slide support taking account of frictional heating

We consider an axisymmetric problem of heat conduction taking account of frictional heating in a conetorus pair that models the functioning of a conical support. The bodies are pressed together and are rotating about a common axis. Heat is generated in the region of contact of the bodies due to frictional forces. Outside the region of contact there is heat exchange with the surrounding medium. The thermal contact between the two bodies is nonideal. The problem is reduced to a system of integral equations whose solution is constructed by the method of successive approximations. We give the results of numerical studies of the temperature distribution and heat flows from the geometric and thermophysical parameters of the body. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 19–27.     

Off-Line Maintenance of Planar Configurations

We achieve anO(log n) amortized time bound per operation for the off-line version of the dynamic convex hull problem in the plane. In this problem, a sequence ofninsert,delete, andqueryinstructions are to be processed, where each insert instruction adds a new point to the set, each delete instruction removes an existing point, and each query requests a standard convex hull search. We process the entire sequence in totalO(n log n) time andO(n) space. Alternatively, we can preprocess a sequence ofninsertions and deletions inO(n log n) time and space, then answer queries in history inO(log n) time apiece (a query in history means a query comes with a time parametert, and it must be answered with respect to the convex hull present at timet). The same bounds also hold for the off-line maintenance of several related structures, such as the maximal vectors, the intersection of half-planes, and the kernel of a polygon. Achieving anO(log n) per-operation time bound for theon-lineversions of these problems is a longstanding open problem in computational geometry.     

On the use of jump conditions at a thin inclusion in solving problems of elasticity theory for a half-space with protuberances

We propose a method of approximate solution of problems of elasticity theory for a half-space with protuberances based on the use of jump conditions in the stresses and displacements at a thin elastic element. The problem of determining the stresses reduces to a system of two-dimensional integral equations of Newtonian potential type for determining the contact stresses between the protuberances and the half-space. We consider the case when the elastic characteristics of the material of the protuberances are different from the material of the half-space.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 156–160.     

An inverse contact problem in the theory of elasticity

In this paper, we discuss an inverse problem in elasticity for determining a contact domain and stress on this domain. We show that this problem is an ill‐posed problem, and we establish the uniqueness and L²‐conditional stability estimation for the stress. Copyright © 1999 John Wiley & Sons, Ltd.     

The angular derivative of Fréchet-holomorphic maps in J*-algebras and complex Hilbert spaces

We study the problem of the existence of the angular derivative of Fréchet-holomorphic maps of the generalized right half-planes defined by non-zero partial isometries in infinite dimensional complex Banach spaces called J^*-algebras. These generalized right half-planes and open unit balls (which are bounded symmetric homogeneous domains) are holomorphically equivalent. The results obtained here also hold in C ^*-algebras, JC^*-algebras, B ^*-algebras and ternary algebras, containing non-zero partial isometries, and in complex Hilbert spaces. Some examples are given. The principal tool we use are general results of the Pick-Julia type in J^*-algebras.     

Nonlinear stability of viscous contact wave for the one‐dimensional compressible fluid models of Korteweg type

This paper is concerned with the large time behavior of solutions of the Cauchy problem to the one‐dimensional compressible fluid models of Korteweg type, which governs the motions of the compressible fluids with internal capillarity. When the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, it is shown that the viscous contact wave corresponding to the contact discontinuity is asymptotically stable provided that the strength of contact discontinuity and the initial perturbation are suitably small. The analysis is based on the elementary L²‐energy method together with continuation argument. Copyright © 2013 John Wiley & Sons, Ltd.     

Wiener-hopf equations for waves scattered by a system of parallel sommerfeld half-planes

A scalar time-harmonic wave (governed by Helmholtz's equation) impinges on N semi-infinite half-planes. The scattered field is sought when first, second, and third-kind boundary conditions or even general linear transmission conditions on the plates ∑_m and their complementary parts ∑ are prescribed. Making use of the Fourier transform a representation formula for H¹ (Ω) solutions is presented. The boundary/transmission problem is shown to be equivalent to a (2N × 2N)-Wiener–Hopf (WH) system for jumps of the Dirichlet–and Neumann–Cauchy data across the semi-infinite screens ∑_m. The (2N × 2N)-Fourier symbol matrix ??_?? contains N block matrices on the diagonal corresponding to Sommerfeld boundary/transmission problems for a single plate. These (2 × 2)-symbol matrices are factorizable and thus the full WH system is invertible by a perturbation argument for not too small spacings of neighbouring screens ∑_m.     

On the integrable case of a Riemann-Hilbert boundary value problem for two functions and the solutions of certain mixed problems for a composite elastic plane

A method for solving the Riemann-Hilbert boundary value problem with piecewise-constant coefficients is generalized /1/. It is shown that the following static problems of a composite elastic plane with three kinds of connection conditions allow of exact solutions: 1) the splicing line is weakened by a system of loaded slots and a transverse shear crack or the edges of one of the slots are partially contacting, or one of the slots is cleaved by a rigid insert; 2) the splicing line is reinforced by a system of thin rigid inclusions and there is one arbitrarily located delamination zone; 3) the elastic half-planes are contacting (with slip) on a certain section of their boundaries, and mixed boundary conditions in the displacements and stresses are given on the rest of the boundaries.

In the general case the Riemann-Hilbert boundary value problem for many functions reduces to the problem of a linear conjugation, and then to Fredholm integral Eqs./2/. Closed solutions are obtained in certain special cases /3–5/. For applications we mention the papers /6, 7/, where problems are considered concerning slits at the interface of two elastic media with two kinds of physical boundary conditions taken into account simultaneously.