A computational algorithm for the second term of the series of the ray method in an inhomogeneous isotropic elastic medium

An algorithm for computing the second term of the series of the ray method in the case of elastic inhomogeneous isotropic media is proposed. The main idea of the approach to the problem can be formulated as follows. Let a central (or support) ray of a ray tube be known. If we inroduce the ray-centered coordinates s, q₁, q₂ in the vicinity of the central ray, then the rays of the ray tube can be described by functions q_i=q_i(s, γ₁, γ₂), i=1,2, where s is the are length of the central ray and γ_j, j=1, 2, are the ray parameters. On the one hand, we show that the integrand of the second term of the series of the ray method can be expressed via the derivatives of the functions q_i with respect to γ_j of the first, second, and third orders. On the other hand, additional differential equations for the derivatives as functions of s can be obtained from Euler's equations for the rays. This paper also contains initial conditions for the derivatives in the case of a point source. Thus, we obtain an algorithm involving additional differential equations for the derivatives , and the initial conditions for them at the source. The algorithm for calculating the mixed components of a vector of displacement is elaborated in detail. Unfortunately, the Russian version of this paper contains some errors, which have been corrected in the English translation of the paper. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 73–93. Translated by M. M. Popov.     

Computation of the second term of the series of the ray method in the dimension two and a half

An algorithm for computing the second term of the series of the ray method suggested in this paper is specified for the case of the dimension two and a half. By this is meant that the elastic paramenters of a medium depend on x, z only and the source and observation points are placed in a plane orthogonal to the y axis. Thus, to compute the wave field we need to take into account the rays which are placed in the plane indicated. This enables us to simplify the algorithm. We discuss the algorithm in detail for the case of a point source. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp 94–108 Translated by N. Ya. Kirpichnikova.     

An Isoperimetric Theorem in Plane Geometry

Abstract. Let P be a simple polygon. Let the vertices of P be mapped, according to a counterclockwise traversal of the boundary, into a strictly increasing sequence of real numbers in [0, 2π) . Let a ray be drawn from each vertex so that the angle formed by the ray and a horizontal line pointing to the right equals, in measure, the number mapped to the vertex. Whenever the rays from two consecutive vertices intersect, let them induce the triangular region with extreme points comprising the vertices and the intersection point. It is shown that there is a fixed α such that if all of the assigned angles are increased by α , the triangular regions induced by the redirected rays cover the interior of P . This covering implies the standard isoperimetric inequalities in two dimensions, as well as several new inequalities, and resolves a question posed by Yaglom and Boltanskii.     

An Isoperimetric Theorem in Plane Geometry

   Abstract. Let P be a simple polygon. Let the vertices of P be mapped, according to a counterclockwise traversal of the boundary, into a strictly increasing sequence of real numbers in [0, 2π) . Let a ray be drawn from each vertex so that the angle formed by the ray and a horizontal line pointing to the right equals, in measure, the number mapped to the vertex. Whenever the rays from two consecutive vertices intersect, let them induce the triangular region with extreme points comprising the vertices and the intersection point. It is shown that there is a fixed α such that if all of the assigned angles are increased by α , the triangular regions induced by the redirected rays cover the interior of P . This covering implies the standard isoperimetric inequalities in two dimensions, as well as several new inequalities, and resolves a question posed by Yaglom and Boltanskii.     

Propagation of waves along a slightly bent piezoelectric rod

The waveguide modes of a curved piezoelectric rod are considered. The propagation of waves along the rod is studied by the ray method. The transport equations that describe the change in the intensity along the rays are solved. Bibliography:10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 191–202. Translated by V. V. Lukyanov     

Geometric heuristics for rural radio maps approximation

Given a terrain T and an antenna A located on it, we would like to approximate the Radio Map of A over T, namely, to associate a signal strength for each point p∈T as received from A. This work presents a new Radio Map approximation algorithm using an adaptive radial sweep-line technique. The suggested radar-like algorithm (RLA) uses a pipe-line method for computing the signal strength along points on a ray, and an adaptive method for interpolating the signal strength over regions between two consecutive rays. Whenever the difference between two consecutive rays is above a certain threshold, a middle ray is created. Thus, the density of the sampling rays is sensitive to the shape of the terrain. Finally, we report on an experiment which compares the new algorithm with other well-known methods. The main conclusion is that the new RLA is significantly faster than the others, i.e., its running time is 3–15 times faster for the same approximation accuracy.     

Paraxial ray theory for Maxwell’s equations

Paraxial ray theory for Maxwell’s equations in the case of an inhomogeneous isotropic medium with finite conductivity and smooth interfaces is developed. We show that ray centered coordinates are suitable for describing amplitudes and polarization of waves in their propagation and reflection/refraction on a smooth interface. Expressions for the geometric spreading and second order derivatives of the eikonal are obtained in terms of certain solutions of equations in variations, i.e., equations which describe rays close to a central ray in linear approximation. Bibliography: 18 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 190–212.     

The (2 n+1−2)-ray algorithm: A new simplicial algorithm to compute economic equilibria

A new variable dimension simplicial restart algorithm is introduced to compute economic equilibria. The number of rays along which the algorithm can leave the starting point differs from the thusfar known algorithms. More precisely, the new algorithm has one ray to each of the 2 ⁿ⁺¹−2 faces of then-dimensional price simplex, whereas the existing algorithms haven+1 rays either to each facet or to each vertex of the unit simplex. The path of points followed by the algorithm can be interpreted as a globally and universally convergent price adjustment process. The process is also economically meaningful and therefore it is a good alternative for the well-known Walras' tatonnement process. Computational results show that the algorithm is competitive with the most efficient simplicial algorithms developed thusfar. This work is part of theVF-program “Equilibrium and Disequilibrium in Demand and Supply”, which has been approved by the Netherlands Ministry of Education and Sciences.     

External rays to periodic points

We prove that for every polynomial-like holomorphic mapP, ifaεK (filled-in Julia set) and the componentK _aofK containinga is either a point ora is accessible along a continuous curve from the complement ofK andK _ais eventually periodic, thena is accessible along an external ray. Ifa is a repelling or parabolic periodic point, then the set of arguments of the external rays converging toa is a nonempty closed “rotation set”, finite (ifK _ais not a one point) or Cantor minimal containing a pair of arguments of external rays of a critical point in ℂ. In the Appendix we discuss constructions via cutting and glueing, fromP to its external map with a “hedgehog”, and backward. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis, sponsored by the Minerva Foundation (Germany). Supported by the Polish KBN Grants 210469101 “Iteracje i Fraktale” and 210909101 “Uklady Dynamiczne”.     

Viscoelastic Love waves in a layered structure with weak lateral inhomogeneity

High-frequency Love surface waves in a linear medium with Maxwell-Boltzmann-Volterra anelasticity are considered. Arbitrary vertical dependences of the material parameters are allowed. The weak lateral inhomqgeneity and anelasticity of the medium, assumed small in the high-frequency range, are treated as perturbations. The leading term of the ray expansion, which corresponds to the balance of energy along real surface rays, is provided. The additional components, i.e., the Rayleigh-type components of the displacement, described by a higher-order correction, are discussed.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 7–11.This work was supported in part by the Russian State Committee for Higher Education under grant 95-0-13.1-66.     

Extending external rays throughout the Julia sets of rational maps

For polynomial maps in the complex plane, the notion of external rays plays an important role in determining the structure of and the dynamics on the Julia set. In this paper we consider an extension of these rays in the case of rational maps of the form Fλ(z) = z ⁿ + λ/z ⁿ where n > 1. As in the case of polynomials, there is an immediate basin of ∞, so we have similar external rays. We show how to extend these rays throughout the Julia set in three specific examples. Our extended rays are simple closed curves in the Riemann sphere that meet the Julia set in a Cantor set of points and also pass through countably many Fatou components. Unlike the external rays, these extended rays cross infinitely many other extended rays in a manner that helps determine the topology of the Julia set.     

Rooms with a view: A novel approach to iterated multidimensional wave conversion

Ray tracing in the presence of linear mode conversion leads to a ‘splitting’ of an incoming ray into two outgoing rays. When the rays are confined to a cavity, the rays can re-enter the conversion region many times, leading to iterated conversion. In this paper, we present new methods for the analysis of this problem. These involve a shift from local to global methods of analysis, and a shift in emphasis from the study of ray evolution in the dispersion surface to the study of the iterated dynamics of rays crossing the conversion surface. The analytical methods are quite general and can be applied in phase spaces of arbitrary dimension. In two spatial dimensions, (x, y), i.e. with a four-dimensional ray space, (x, y, k_x, k_y), rays are confined to three-dimensional regions called rooms, with one room for each wave type. In these rooms the rays do not cross, but when they intersect the conversion surface a family of converted rays is produced in the other room. The use of rooms allows a full view of the phase space dynamics of the iterated conversion of ray families. A simple two-dimensional model, inspired by the Budden resonance model, is presented as an example of these ideas.     

Asymptotics of Weil–Petersson Geodesics I: Ending Laminations, Recurrence, and Flows

We define an ending lamination for a Weil–Petersson geodesic ray. Despite the lack of a natural visual boundary for the Weil–Petersson metric [Bro2], these ending laminations provide an effective boundary theory that encodes much of its asymptotic CAT(0) geometry. In particular, we prove an ending lamination theorem (Theorem 1.1) for the full-measure set of rays that recur to the thick part, and we show that the association of an ending lamination embeds asymptote classes of recurrent rays into the Gromov-boundary of the curve complex C(S){\mathcal{C}(S)}. As an application, we establish fundamentals of the topological dynamics of the Weil–Petersson geodesic flow, showing density of closed orbits and topological transitivity.     

Raising Roofs,Crashing Cycles,and Playing Pool: Applications of a Data Structure for Finding Pairwise Interactions

The straight skeleton of a polygon is a variant of the medial axis introduced by Aichholzer et al., defined by a shrinking process in which each edge of the polygon moves inward at a fixed rate. We construct the straight skeleton of an n -gon with r reflex vertices in time O(n ^1+ε + n ^8/11+ε r ^9/11+ε ) , for any fixed ε >0 , improving the previous best upper bound of O(nr log n) . Our algorithm simulates the sequence of collisions between edges and vertices during the shrinking process, using a technique of Eppstein for maintaining extrema of binary functions to reduce the problem of finding successive interactions to two dynamic range query problems: (1) maintain a changing set of triangles in R ³ and answer queries asking which triangle is first hit by a query ray, and (2) maintain a changing set of rays in R ³ and answer queries asking for the lowest intersection of any ray with a query triangle. We also exploit a novel characterization of the straight skeleton as a lower envelope of triangles in R ³ . The same time bounds apply to constructing non-self-intersecting offset curves with mitered or beveled corners, and similar methods extend to other problems of simulating collisions and other pairwise interactions among sets of moving objects. Received July 1, 1998, and in revised form March 29, 1999.     

On a theorem of bazilevic for areally meanp-valent functions attaining maximal growth onk rays

Letμ(p,k) denote the class of areally meanp-valent functions attaining maximal growth onk rays. The aim of this article is to get the sufficient and necessary condition for which Bazilevic’s theorem holds forf εμ(p,k). Project supported by the National Natural Science Foundation of China.     

On tree‐decompositions of one‐ended graphs

A graph is one‐ended if it contains a ray (a one way infinite path) and whenever we remove a finite number of vertices from the graph then what remains has only one component which contains rays. A vertex v dominates a ray in the end if there are infinitely many paths connecting v to the ray such that any two of these paths have only the vertex v in common. We prove that if a one‐ended graph contains no ray which is dominated by a vertex and no infinite family of pairwise disjoint rays, then it has a tree‐decomposition such that the decomposition tree is one‐ended and the tree‐decomposition is invariant under the group of automorphisms. This can be applied to prove a conjecture of Halin from 2000 that the automorphism group of such a graph cannot be countably infinite and solves a recent problem of Boutin and Imrich. Furthermore, it implies that every transitive one‐ended graph contains an infinite family of pairwise disjoint rays.     

The geodesic flow of a nonpositively curved graph manifold

We consider discrete cocompact isometric actions where X is a locally compact Hadamard space (following [B] we will refer to CAT(0) spaces — complete, simply connected length spaces with nonpositive curvature in the sense of Alexandrov — as Hadamard spaces) and G belongs to a class of groups (“admissible groups”) which includes fundamental groups of 3-dimensional graph manifolds. We identify invariants (“geometric data”) of the action which determine, and are determined by, the equivariant homeomorphism type of the action of G on the ideal boundary of X. Moreover, if are two actions with the same geometric data and is a G-equivariant quasi-isometry, then for every geodesic ray there is a geodesic ray (unique up to equivalence) so that . This work was inspired by (and answers) a question of Gromov in [Gr3, p. 136]. Submitted: May 2001.     

A new simplicial variable dimension algorithm to find equilibria on the product space of unit simplices

A new simplicial variable dimension restart algorithm is introduced to solve the nonlinear complementarity problem on the product spaceS of unit simplices. The triangulation which underlies the algorithm differs from the triangulations ofS used thus far. Moreover, the number of rays along which the algorithm can leave the arbitrarily chosen starting point is much larger. More precisely, there is a ray leading from the starting point to each vertex ofS. In caseS is the product ofn one-dimensional unit simplices the alogrithm is similar to the octahedral algorithm onR ⁿ having 2 ⁿ rays. Also, the accuracy of an approximate solution in the terminal simplex of the algorithm is in general better than for the other algorithms onS. Computational results will show that the number of iterations for the new algorithm is much less. The examples concern the computation of equilibria in noncooperative games, exchange economies and trade models. This author is financially supported by the Netherlands Organisation for the Advancement of Pure Research (ZWO), Grant 46-98. This research is part of the VF-program “Equilibrium and Disequilibrium in Demand and Supply” which has been approved by the Netherlands ministry of education and sciences.     

The Effects of Flexion and Torsion on a Fluid Flow Through a Curved Pipe

We consider an injection of incompressible viscous fluid in a curved pipe with a smooth central curve γ . The one-dimensional model is obtained via singular perturbation of the Navier—Stokes system as ɛ , the ratio between the cross-section area and the length of the pipe, tends to zero. An asymptotic expansion of the flow in powers of ɛ is computed. The first term in the expansion depends only on the tangential injection along the central curve γ of the pipe and the velocity as well as the pressure drop are in the tangential direction. The second term contains the effects of the curvature (flexion) of γ in the direction of the tangent while the effects of torsion appear in the direction of the normal and the binormal to γ . The boundary layers at the ends of the pipe are studied. The error estimate is proved. Accepted 21 March 2001. Online publication 9 August 2001.