Nikulin involutions on <Emphasis Type="Italic">K</Emphasis>3 surfaces

We study the maps induced on cohomology by a Nikulin (i.e. a symplectic) involution on a K3 surface. We parametrize the 11-dimensional irreducible components of the moduli space of algebraic K3 surfaces with a Nikulin involution and we give examples of the general K3 surface in various components. We conclude with some remarks on Morrison–Nikulin involutions, these are Nikulin involutions which interchange two copies of E ₈(−1) in the Néron Severi group. The second author is supported by DFG Research Grant SA 1380/1-1.     

Multiphase double‐porosity homogenization for perimeter functionals

In this paper, we study a homogenization problem for perimeter energies in highly contrasted media; the analysis of the previous paper is carried out by removing the hypothesis that the perforated medium Rⁿ ? E is composed of disjoint compact components. Assuming E to be the union of a finite number N of connected components E¹, … ,E^N, the Γ‐limit F is a multiphase energy with a ‘decoupled’ surface part, obtained by homogenization from the surface tensions in each E ^j, a trivial bulk term obtained as a weak limit, and a further interacting term between the phases, involving an asymptotic formula for a family minimum problems on invading an asymptotic formula for a family of minimum problems on invading domains with prescribed boundary conditions. Copyright © 2012 John Wiley & Sons, Ltd.     

A cluster of many small holes with negative imaginary surface impedances may generate a negative refraction index

We deal with the scattering of an acoustic medium modeled by an index of refraction n varying in a bounded region Ω of and equal to unity outside Ω. This region is perforated with an extremely large number of small holes D_m's of maximum radius a, a << 1, modeled by surface impedance functions. Precisely, we are in the regime described by the number of holes of the order M:=O(a^{β ? 2}), the minimum distance between the holes is d ～ a^t, and the surface impedance functions of the form λ_m～λ_m,0a^?β with β > 0 and λ_m,0 being constants and eventually complex numbers. Under some natural conditions on the parameters β,t, and λ_m,0, we characterize the equivalent medium generating approximately the same scattered waves as the original perforated acoustic medium. We give an explicit error estimate between the scattered waves generated by the perforated medium and the equivalent one, respectively, as a→0. As applications of these results, we discuss the following findings:

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Exact Free Surface Flows for Shallow Water Equations II: The Compressible Case

Exact free surface flows with shear in a compressible barotropic medium are found, extending the authors’ earlier work for the incompressible medium. The barotropic medium is of finite extent in the vertical direction, while it is infinite in the horizontal direction. The “shallow water” equations for a compressible barotropic medium, subject to boundary conditions at the free surface and at the bottom, are solved in terms of double psi-series. Simple wave and time-dependent solutions are found; for the former the free surface is of arbitrary shape while for the latter it is a damping traveling wave in the horizontal direction. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed. In the case of an isothermal medium, when γ = 1, we again find simple wave and time-dependent solutions.     

Love waves of SH type in an inhomogeneous transversely isotropic elastic medium

The asymptotics of high-frequency Love waves, which are analogous to transverse surface SH waves, is considered for a special type of anisotropy (transverse isotropy) of elastic media. The wave field is represented as a sum of the space-time (ST) caustic expansion and two additional ST ray series for faster (relative to the transverse surface wave) body waves, decaying exponentially with depth. Near the surface, the coefficients of the ST caustic and ray series, as well as the eikonals of waves, are determined in the form of expansions in a small parameter, which characterizes the proximity of the caustic of the ray field to the surface. With regard for the specific structure of the elasticity tensor of a transversely isotropic medium, the surface is treated as a plane. Interrelations between the parameters of elasticity, which are consistent with the conditions of the positivity of the elastic deformation energy and provide for the origination of the surface waves considered, are obtained.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 243–262.This work was supported by the Russian Foundation for Basic Research under grant No. 96-01-00666.     

Automorphism groups of compact bordered Klein surfaces with invariant subsets

In this work we get upper bounds for the order of a group of automorphisms of a compact bordered Klein surface S of algebraic genus greater than 1. These bounds depend on the algebraic genus of S and on the cardinals of finite subsets of S which are invariant under the action of the group. We use our results to obtain upper bounds for the order of a group of automorphism whose action on the set of connected components of the boundary of S is not transitive. The bounds obtained this way depend only on the algebraic genus of S. The author is partially supported by the European Network RAAG HPRN-CT-2001-00271 and the Spanish GAAR DGICYT BFM2002-04797.     

Rayleigh waves from a point source on a boundary free of tensions

The solutions of equations of elasticity theory that have a discontinuity only on a boundary free of tensions (Rayleigh waves) are considered. Initial data for the complex intensity of the surface Rayleigh waves are found in two simple media. The first elastic medium fills a half-space with Lamé parameters and density dependent on depth. The second medium is bounded by a curve determined by a natural equation. The parameters of the second medium depend on the arc length along the curve. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 132–149.     

The space of surface group representations

In this note we prove that the number of irreducible components of Hom (π,G) is the same as π₁(G), where π is a surface group andG is complex semisimple. This is established by studying the flat bundles on Riemann surfaces. The present work is partially supported by NSF grant DMS89-04922     

Splitting (complicated) surfaces is hard

Let be an orientable combinatorial surface. A cycle on is splitting if it has no self-intersections and it partitions into two components, neither of which is homeomorphic to a disk. In other words, splitting cycles are simple, separating, and non-contractible. We prove that finding the shortest splitting cycle on a combinatorial surface is NP-hard but fixed-parameter tractable with respect to the surface genus g and the number of boundary components b of the surface. Specifically, we describe an algorithm to compute the shortest splitting cycle in (g+b)^O(g+b)nlogn time, where n is the complexity of the combinatorial surface.     

Flow in a porous medium induced by torsional oscillation of a disk near its surface

Torsional oscillation of an infinite disk in a viscous liquid bounded by a porous medium fully saturated with the liquid has been discussed. It is assumed that the flow between the disk and the porous medium is governed by Navier-Stokes equation and that in the porous medium by Brinkman equation. Flows in the two regions are matched at the interface by assuming that the velocity and stress components are continuous at it. It is found that the depth of penetration of the flow in the porous medium is proportional to the square root of the permeability of the medium. The oscillation of the disk induces a steady radial-axial flow in both the regions in such a way that there is a steady axial flow of the fluid from the porous medium to the free flow region i.e. the fluid is expelled out from the porous medium. The steady flow in the porous medium increases with the increase of the permeability of the medium and with the decrease of the distance between the oscillating disk and porous surface.     

Effective models of stratified media containing porous biot layers

Periodic stratified media in which either two porous Biot layer, or an elastic and a porous layers, or a fluid and a porous layer alternate are considered. The effective models of these media are constructed and investigated. In the case of alternating porous layers, the effective model is a generalized transversely isotropic Biot medium. In this medium, the density of the fluid phase and the mean density acquire tensor character. It is shown that the effective model of a porous-fluid medium is, on the one hand, a generalized transversely isotropic Biot medium of special type and, on the other hand, a generalization of the effective model of a stratified elastic-fluid medium.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 140–163.This work was supported by the Russian Foundation for Basic Research under grant Nos. 96-01-00666 and 96-05-66207.     

The mean curvature of the influence surface of wave equation with sources on a moving surface

The mean curvature of the influence surface of the space–time point ( x , t) appears in linear supersonic propeller noise theory and in the Kirchhoff formula for a supersonic surface. Both these problems are governed by the linear wave equation with sources on a moving surface. The influence surface is also called the Σ‐surface in the aeroacoustic literature. This surface is the locus, in a frame fixed to the quiescent medium, of all the points of a radiating surface f( x , t)=0 whose acoustic signals arrive simultaneously to an observer at position x and at the time t. Mathematically, the Σ‐surface is produced by the intersection of the characteristic conoid of the space–time point ( x , t) and the moving surface. In this paper, we derive the expression for the local mean curvature of the Σ‐surface of the space–time point ( x , t) for a moving rigid or deformable surface f( x , t)=0. This expression is a complicated function of the geometric and kinematic parameters of the surface f( x , t)=0. Using the results of this paper, the solution of the governing wave equation of high‐speed propeller noise radiation as well as the Kirchhoff formula for a supersonic surface can be written as very compact analytic expressions. Copyright © 1999 John Wiley & Sons, Ltd.     

Jumps of displacements and stresses as seismic sources in Biot's medium

Wave fields excited in a homogeneous isotropic Biot medium by point sources described in terms of discontinuities of displacements and stresses are determined. The results are represented in the form of relations involving Fourier-Bessel or Mellin integrals and in the form of Stokes-type formulas. The interrelations between these representations are established. Among all possible point sources exciting Biot's medium, the elementary sources, in terms of which any complicated linear source can be described, are selected. The special case where the wave fields in the two phases of the Biot medium are independent of one another is considered, and the corresponding sources in the Biot medium are compared with the known sources in elastic and fluid media.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 164–196.This work was supported by the Russian Foundation for Basic Research under grant No. 96-05-65904.     

Diffraction of Creeping Waves by a Line of Jump of Curvature (a Three-Dimensional Acoustic Medium)

The problem of diffraction of creeping waves by a line of jump of curvature in a three-dimensional acoustic medium is studied. Moreover, a sufficiently ``oblique' incidence is taken into account. Two cases of the curvature jump on the conjunction line of surfaces are considered: (i) the curvature does not change sign, but changes value, (ii) the surface of positive curvature is joined with a half-plane. Bibliography: 4 titles.     

On the problem of monitoring hydrocarbon reservoirs during production

The problem of monitoring the position of the boundary of a hydrocarbon collector as a source approaches, is discussed. Day surface seismograms are analyzed. Specific properties of the anomalous S* wave are used. The case of a porous medium inside a reservoir is also considered. Bibliography: 24 titles.     

On the problem of mapping deep cracked zones by using seismic observations

The problem of conversion of a slow wave propagating in a cracked collector into a tube wave in a borehole intersecting the collector is considered. The influences of the collector's porosity and of the parameters of the background medium on the dynamics of the tube wave are investigated. It is demonstrated that the elastic response of the medium to the wave incident on the borehole consists of a propagating and a diffusive field.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 123–128.This work was supported by the Russian Foundation for Basic Research under grant Nos. 96-05-66207 and 96-01-00666.     

Half- and finite-range completeness for the equation of transfer of polarized light

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Wärmewellen in einem leitenden und strahlenden Medium

Summary Plane thermal waves in a heat conducting and radiating (emitting and absorbing) medium that occupies the half-spacex>0 are investigated. The governing equations for a gray medium are linearized with regard to small perturbations of the radiative equilibrium. Solutions are given for the thermal wave that is due to harmonic oscillations of either the wall temperature or the radiative energy flux produced by an outer source. The behaviour of the thermal wave is then discussed for the asymptotic cases of weak, strong, optical thin and optical thick radiation, respectively, and also for the special case that the Bouguer numberBu and the radiation-conduction parameterK as defined in the text are equal to one. Then the equations and their solutions are generalized in order to apply to certain models of frequency-dependent absorption coefficients (non-gray media). Finally it is shown that nonlinear terms, although being small of higher order in the differential equations, cause the perturbation solution to be not uniformly valid as the distance from the boundary surface goes to infinity.     

Hamiltonian Dynamic Equations for Fluid Films

Two‐dimensional models for hydrodynamic systems, such as soap films, have been studied for over two centuries. Yet there has not existed a fully nonlinear system of dynamic equations analogous to the classical Euler equations. We propose the following exact system for the dynamics of a fluid film Here δ/δ t is the invariant time derivative, ρ is the two‐dimensional density of the film, C is the normal component of the velocity field, V^α are the tangential components, B_αβ is the curvature tensor, and ?_α is the covariant surface derivative. The surface energy density e(ρ) is a generalization of the common surface tension and e_ρ is its derivative. The Laplace model corresponds to e(ρ) =σ/ρ , where σ is the surface tension density. The proper choice of e(ρ) in paramount in capturing particular effects displayed by fluid films. The proposed system is exact in the sense that neither velocities nor deviation from the equilibrium are assumed small. The system is derived in the classical Hamiltonian framework. The assumption that e is a function of ρ alone can be relaxed in practical physical and biological applications. This leads to more complicated systems, briefly discussed in the text.     

One-dimensional drug release from finite Menger sponges: In silico simulation

The purpose of this work was to evaluate the consequences of the spatial distribution of components in pharmaceutical matrices type Menger sponge on the drug release kinetic from this kind of platforms by means of Monte Carlo computer simulation. First, six kinds of Menger sponges (porous fractal structures) with the same fractal dimension, d_f=2.727, but with different random walk dimension, d_w[2.149,3.183], were constructed as models of drug release device. Later, Monte Carlo simulation was used to describe drug release from these structures as a diffusion-controlled process. The obtained results show that drug release from Menger sponges is characterized by an anomalous behavior: there are important effects of the microstructure anisotropy, and porous structures with the same fractal dimension but with different topology produce different release profiles. Moreover, the drug release kinetic from heteromorphic structures depends on the axis used to transport the material to the external medium. Finally, it was shown that the number of releasing sites on the matrix surface has a significant impact on drug release behavior and it can be described quantitatively by the Weibull function.