A Characterization of Twin Buildings by Twin Apartments

A twin building is a pair of buildings along with a certain codistance' between chambers of one building, and chambers of the other. All spherical buildings can be regarded as twin buildings (such a building is twinned with itself in a natural way), and the aim of this paper is to give a definition of twin buildings along the lines that Tits originally gave for spherical buildings. This alternative approach via apartments has been used by the first author in studying some group actions on twin buildings.     

Valuations on PTR's induced by triangle buildings

A valuation v (in the sense of [10]) is defined on all coordinatizing planar ternary rings (R, T) of any projective plane which can be thought of as the spherical building at infinity of a triangle building. It is shown that (R, T) is complete with respect to v.The author's research was supported by the National Fund for Scientific Research (Belgium).     

Visibility metrics on the infinity boundary of the complex hyperbolic plane

Limiting spherical and horospherical metrics an the infinity boundary of the complex hyperbolic plane are constructed. It is proved that the limiting spherical metric, which automatically is the Carnot–Carathéodory metric, is also a visibility metric, i.e., it belongs to a canonical class of metrics on the infinity boundary. Bibliography: 6 titles.     

The direction of an automorphism of a totally disconnected locally compact group

We describe the asymptotic behavior of automorphisms of totally disconnected locally compact groups in terms of a set of `directions' which comes equipped with a natural pseudo-metric. The structure at infinity obtained by completing the induced metric quotient space of the set of directions recovers familiar objects such as: the set of ends of the tree for the group of inner automorphisms of the group of isometries of a regular locally finite tree; and the spherical Bruhat-Tits building for the group of inner automorphisms of the set of rational points of a semisimple group over a local field. Research supported by A.R.C. Grant DP0208137.     

THE UNIQUENESS OF LIMIT CYCLE AND THE STRUCTURE OF CRITICAL POINT AT INFINITY FOR A CLASS OF CUBIC SYSTEM

A class of cubic system, which is an accompany system of a quadratic differential one, is studied. It is proved that the system has at most one limit cycle, and the critical point at infinity is a higher order one. The structure and algebraic character of the critical point at infinity are obtained.     

Point source excitation in direct and inverse scattering: The soft and the hard small sphere

A spherical wave emanating from a point source is scatteredby either a soft or a hard body. The incident spherical wavehas a wavelength which is much larger than the characteristicdimension of the scatterer and it is modified in such a wayas to recover the plane wave incidence when the source pointrecedes to infinity. Using low frequency expansions the scatteringproblem is transformed to a sequence of exterior potential problemsin the presence of a monopole singularity located at the sourceof the incident wave field. Complete expansions for the scatteringamplitude are provided. The method is applied to the cases ofa soft and a hard sphere and the first three approximationsfor the near, as well as the far, field are evaluated. It isobserved that every one, after the first, low frequency approximationof the far field, involves one spherical multipole more thanthe corresponding approximation for the case of an incidentplane wave. As the point singularity tends to infinity, therelative results recover all the known expressions for planeincidence. It is shown that for point excitation the Rayleighapproximation of the scattering amplitude for a hard sphereis of the second order, in contrast to the case of plane excitationwhich is of the third order. Simple algorithms that specifythe radius and the position of a soft and a hard sphere areproposed, which are based on the additional dependence of thescattering amplitude represented by the distance from the pointsource to the centre of the scatterer. The inversion algorithmis shown to be stable whenever the source point is not too faraway from the target sphere. A simple way to decide whetherthe sphere is a soft or a hard body is also provided.     

On Green's function for the Helmholtz equation in a wedge

It is found that, in the spherical coordinate system, the fundamental solution of the Helmholtz equation in a wedge satisfies the Sommerfeld radiation conditions at infinity uniformly in angle coordinates.Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1312–1314, September, 1993.     

“Asymptotically non-minkowskian” bubbles with gravitational repulsion

It is usually assumed that spherical domain walls can be described by the same equation of state as planar domains. In this case, the spherical domain walls should also demonstrate gravitational repulsion, which would contradict the Birkhoff theorem. However, this theorem does not apply to solutions that are not described by Minkowskian geometry at infinity. In the thin-wall formalism, we consider the solution of the Einstein equations that describes a spherical domain wall with gravitational repulsion and which is “asymptotically non-Minkowskian.” Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 2, pp. 346–352, November, 1997.     

A new asymptotic bound of the minimum possible odd cardinality of spherical (2k−1)-designs

We apply polynomial techniques to investigate the structure of spherical designs in an asymptotic process with fixed odd strength while the dimension and odd cardinality tend to infinity in a certain relation. Our bounds for the extreme inner products of special points in such designs allow new lower bounds on the minimum possible odd cardinality.     

On Linear Programming Bounds for Spherical Codes and Designs

We investigate universal bounds on spherical codes and spherical designs that could be obtained using Delsartes linear programming methods. We give a lower estimate for the LP upper bound on codes, and an upper estimate for the LP lower bound on designs. Specifically, when the distance of the code is fixed and the dimension goes to infinity, the LP upper bound on codes is at least as large as the average of the best known upper and lower bounds. When the dimension n of the design is fixed, and the strength k goes to infinity, the LP bound on designs turns out, in conjunction with known lower bounds, to be proportional to k^n-1.     

Strongly transitive actions on euclidean buildings

We prove a decomposition result for a group G acting strongly transitively on the Tits boundary of a Euclidean building. As an application we provide a local to global result for discrete Euclidean buildings, which generalizes results in the locally compact case by Caprace–Ciobotaru and Burger–Mozes. Let X be a Euclidean building without cone factors. If a group G of automorphisms of X acts strongly transitively on the spherical building at infinity ?X, then the G-stabilizer of every affine apartment in X contains all reflections along thick walls. In particular G acts strongly transitively on X if X is simplicial and thick.     

Problem of magnetoelasticity for a solid with the spherical cavity

The solution of a static problem of magnetoelastisity for a soft ferromagnetic elastic solid with the spherical cavity is obtained on the base of the linear theory of Brown, Pao and Yeh. It is assumed that the solid has a multi-domain structure, so the hysteresis loss and remanent magnetization are neglected. The solid is affected by a magnetic field which is uniform at infinity and determined by the magnetic induction vector. The cavity causes some distortion of the field distribution near the interface. So the field induces magnetic moments and produces stresses and deformations in the body. The problem is solved for an unperturbed strain state. An approach is discussed to find the perturbed values on the base of the solution obtained. The Fourier variable separation method is used. The stresses are presented via harmonic functions. As a result magnetoelastic stresses are obtained in the closed form. Their distribution in the body is studied and some results of numerical calculations are shown. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Positive solutions and eigenvalue regions of two-delay singular systems with a twin parameter

The paper is concerned with a two-delay singular differential system with a twin parameter. Applying fixed-point index theory, we show the relationship between the asymptotic behaviors of nonlinearities (at zero and infinity) and the open regions (eigenvalue regions) of parameters, which are correlated with delays, such that the system has zero, one and two positive solution(s).     

Asymptotic behaviour and level-curve structure in plane subsonic potential flows

The problem of constructing asymptotic forms at infinity and the problem of determining the structure of the isoclines and isobars are considered for uniform plane subsonic potential flow, horizontal at infinity, around a large class of bodies. It is shown that these problems are intimately related. In fact, the construction of a solution in the neighbourhood of the point at infinity (PAI) reduces to (i) selecting a “correct” transformation of the physical plane (PP) onto an auxiliary plane (AP), under which the PAI of the PP goes into the origin of the AP and the gas dynamic equations at the origin of the AP reduce to a Cauchy-Riemann system; (ii) finding the number of isoclines that pass through the PAI and determining the inclinations of these isoclines. With this approach, the construction of asymptotic laws and the investigation of the level curve structure in the neighbourhood of the PAI have much in common with the analogous problem in the neighbourhood of an arbitrary point of the flow at a finite distance from the body. Asymptotic forms are constructed for two cases: lift-creating flow and symmetric flow around a body. The constant factors occurring in the asymptotic formulae are expressed in terms of the physical or geometrical parameters of the problems under consideration.     

On Opposition in Spherical Buildings and Twin Buildings

In this paper, we prove a combinatorial property of twin apartments and opposition of chambers in twin buildings. We then characterize adjacency of chambers in twin buildings by meansof opposition of chambers. As an application, we study maps which satisfy certain conditions related to opposition of chambers, e.g., maps that preserve opposition. Applied to the special case of spherical buildings, all our main results as well as their corollaries are new.     

Infinitesimal variations of Hodge structure at infinity

By analyzing the local and infinitesimal behavior of degenerating polarized variations of Hodge structure the notion of infinitesimal variation of Hodge structure at infinity is introduced. It is shown that all such structures can be integrated to polarized variations of Hodge structure and that, conversely, all are limits of infinitesimal variations of Hodge structure at finite points. As an illustration of the rich information encoded in this new structure, some instances of the maximal dimension problem for this type of infinitesimal variation are presented and contrasted with the “classical” case of IVHS at finite points.      

Scattering of elastic waves by nonaxisymmetric three-dimensional dipping layer

Using a boundary method, we investigated the scattering of elastic plane harmonic SH, SV, P, and Rayleigh waves by three-dimensional nonaxisymmetric dipping layers embedded in an elastic half-space. The valley was subjected to incident Rayleigh wave and oblique incident SH, SV, and P waves. The method utilized spherical wave functions to express the unknown scattered field. These functions satisfy the equation of motion and radiation conditions at infinity but they do not satisfy the stress-free boundary conditions at the surface of the half-space. The boundary and continuity conditions are imposed locally in the least-square-sense at several points on the layer interface and on the surface of the half-space. A comparative study was done to examine the validity and limitations of the two-dimensional approximations (antiplane and plane strain models) of three-dimensional models. It is demonstrated that the two-dimensional approximations may be inadequate to represent actual displacement field for three-dimensional irregularities.     

Structure of tensor fields of the second rank generated by an incompatibility operator

There is considered the structure of fields of symmetric tensors η generated by the incompatibility operator Ink acting according to the formula Inkη = × η × . This operator which is closely related to internal stresses in bodies was considered by Kroner /1/ in application to the continuum theory of dislocations. Allied to the study of the structure of tensor fields is the problem of their restoration according to a given incompatibility and divergence as well as the decomposition into incompatible and compatible strain. For a sufficiently smooth Kröner tensor field that vanishes at infinity, by starting from the analogy with the properties of a vector field, it is shown that such a decomposition exists and is unique. In the supplement to /2/, this problem is solved in practice, where an effective algorithm is developed for the decomposition into appropriate invariant components by using projection operators. Analogous questions are examined below in application to finite domains as well as in connection with the decomposition of a tensor into deviatoric and spherical parts.     

Coarse equivalences of Euclidean buildings

We prove the following rigidity results. Coarse equivalences between metrically complete Euclidean buildings preserve spherical buildings at infinity. If all irreducible factors have dimension at least two, then coarsely equivalent Euclidean buildings are isometric (up to scaling factors); if in addition none of the irreducible factors is a Euclidean cone, then the isometry is unique and has finite distance from the coarse equivalence.