Infinite Cone Research Articles

Exact solution of a mixed boundary-value problem of elasticity theory for an infinite cone truncated along an axis

Exact solution of a mixed boundary-value problem of elasticity theory for an infinite cone truncated along an axis

The Mellin-Edge Quantisation for Corner Operators

We establish a quantisation of corner-degenerate symbols, here called Mellin-edge quantisation, on a manifold $M$ with second order singularities. The typical ingredients come from the "most singular" stratum of $M$ which is a second order edge where the infinite transversal cone has a base $B$ that is itself a manifold with smooth edge. The resulting operator-valued amplitude functions on the second order edge are formulated purely in terms of Mellin symbols taking values in the edge algebra over $B.$ In this respect our result is formally analogous to a quantisation rule of a joint paper with J. Gil and J. Seiler for the simpler case of edge-degenerate symbols that corresponds to the singularity order 1. However, from the singularity order 2 on there appear new substantial difficulties for the first time, partly caused by the edge singularities of the cone over $B$ that tend to infinity.

Universal Polynomial Expansions of Harmonic Functions

Let Ω be a domain in ℝN such that \(\left(\mathbb{R}^{N}\cup\lbrace\infty\rbrace\right)\setminus\Omega\) is connected. It is known that, for each w ∈ Ω, there exist harmonic functions on Ω that are universal at w, in the sense that partial sums of their homogeneous polynomial expansion about w approximate all plausibly approximable functions in the complement of Ω. Under the assumption that Ω omits an infinite cone, it is shown that the connectedness hypothesis on \(\left(\mathbb{R}^{N}\cup\lbrace\infty\rbrace\right)\setminus\Omega\) is essential, and that a harmonic function which is universal at one point is actually universal at all points of Ω.

Homotopical complexity of 2D billiard orbits

Traditionally, rotation numbers for toroidal billiard flows are defined as the limiting vectors of average displacements per time on trajectory segments. Naturally, these creatures live in the (commutative) vector space ℝn, if the toroidal billiard is given on the flatn-torus. The billiard trajectories, being curves, often getting very close to closed loops, quite naturally define elements of the fundamental group of the billiard table. The simplest non-trivial fundamental group obtained this way belongs to the classical Sinai billiard, i.e. the billiard flow on the 2-torus with a single, strictly convex obstacle (with smooth boundary) removed. This fundamental group is known to be the groupF2freely generated by two elements, which is a heavily noncommutative, hyperbolic group in Gromov’s sense. We define the homotopical rotation number and the homotopical rotation set for this model, and provide lower and upper estimates for the latter one, along with checking the validity of classically expected properties, like the density (in the homotopical rotation set) of the homotopical rotation numbers of periodic orbits.The natural habitat for these objects is the infinite cone erected upon the Cantor set Ends (F2) of all ŋds" of the hyperbolic groupF2. An element of Ends (F2) describes the direction in (the Cayley graph of) the groupF2in which the considered trajectory escapes to infinity, whereas the height functiont(t≧ 0) of the cone gives us the average speed at which this escape takes place.The main results of this paper claim that the orbits can only escape to infinity at a speed not exceeding\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sqrt 2 $ \end{document}, and any directione∈ Ends (F2) for the escape is feasible with any prescribed speeds, 0 ≦s≦\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sqrt 2 $ \end{document}/2. This means that the radial upper and lower bounds for the rotation setRare actually pretty close to each other.

The steady-state oscillations of the elastic infinite cone loaded at a vertex by a concentrated force

In the proposed paper, a new problem on the steady-state oscillations of an elastic infinite cone is solved for the point oscillating force enclosed to a cone’s vertex. On a cone’s conic surface, the stresses are given. The auxiliary problem for a cone truncated by a spherical surface is formulated with the aim to obtain the solution of the initial problem. The solution is constructed with the help of the integral transformation method and the method of Green’s function. The limiting transition in the solution’s formulas is carried out to obtain the exact solution of the initial problem. The calculation of the displacements and the stress wave field of the cone is performed.

CYCLIC SUBGROUPS ARE QUASI-ISOMETRICALLY EMBEDDED IN THE THOMPSON–STEIN GROUPS

We give criteria for determining the approximate length of elements in any given cyclic subgroup of the Thompson–Stein groups F(n1,…,nk) such that n1 - 1|ni - 1 ∀i ∈ {1,…,k} in terms of the number of leaves in the minimal tree-pair diagram representative. This leads directly to the result that cyclic subgroups are quasi-isometrically embedded in the Thompson–Stein groups. This result also leads to the corollaries that ℤn is also quasi-isometrically embedded in the Thompson–Stein groups for all n ∈ ℕ and that the Thompson–Stein groups have infinite dimensional asymptotic cone.

On optimal estimates for the Laplace-Leray commutator in planar domains with corners

For smooth domains, Liu et al. (Comm. Pure Appl. Math. 60: 1443-1487, 2007) used optimal estimates for the commutator of the Laplacian and the Leray projection operator to establish well-posedness of an extended Navier-Stokes dynamics. In their work, the pressure is not determined by incompressibility, but rather by a certain formula involving the Laplace-Leray commutator. A key estimate of Liu et al. controls the commutator strictly by the Laplacian in L 2 norm at leading order. In this paper we show that this strict control fails in a large family of bounded planar domains with corners. However, when the domain is an infinite cone, we find that strict control may be recovered in certain power-law weighted norms.

Parameter-dependent edge operators

We study parameter-dependent operators on a manifold with edge and construct new classes of elliptic elements in the corner calculus on an infinite cone with a singular base.

The Solution of Mitchell's Problem for the Elastic Infinite Cone with a Spherical Crack

The new problem about the stress concentration around a spherical crack inside of an elastic cone is solved for the point tensile force enclosed to a cone's edge. The constructed discontinuous solutions of the equilibrium equations have allowed to express the displacements and stress in a cone through their jumps and the jumps of their normal derivatives across the crack's surface. The application of the integral transformation method under the generalized scheme has reduced the problem solving to the solving of the integrodifferential equation system with regard to the displacements' jumps. This system was solved approximately by the orthogonal polynomial method. The use of this method has allowed to take into consideration the order of the solution's singularities at the ends of an integral interval. The correlation between the crack's geometrical parameters, its distance from an edge, and the SIF values is established after the numerical analysis. The limit of the proposed method applicability is specified.

Edge quantisation of elliptic operators

The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy $${\sigma=(\sigma_\psi,\sigma_\wedge)}$$ , where the second component takes values in operators on the infinite model cone of the local wedges. In the general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the ellipticity of the principal edge symbol $${\sigma_\wedge}$$ which includes the (in general not explicitly known) number of additional conditions of trace and potential type on the edge. We focus here on these questions and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet–Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich–Dynin formula for edge boundary value problems.

GLOBAL MULTIDIMENSIONAL SHOCK WAVE FOR THE STEADY SUPERSONIC FLOW PAST A THREE-DIMENSIONAL CURVED CONE

In this paper, we establish the global existence and stability of a multidimensional conic shock wave for three-dimensional steady supersonic flow past an infinite cone. The flow is assumed to be hypersonic and described by a steady potential flow equation. Under an appropriate boundary condition on the curved cone, we show that a pointed shock attached at the vertex of the cone will exist globally in the whole space.

Power sums of roots of a nonlinear system

For a system of meromorphic functions f = (f 1, …, fn ) in n , an explicit formula is given for evaluating negative power sums of the roots of the nonlinear system f (z) = 0. Under reasonable assumptions on the system, we introduce the concept of a zeta-function associated with the system. Using residue theory we get an integral representation for the zeta-function which allows us to construct an analytic extension of the zeta-function to an infinite cone in n .

Solutions globales de systèmes hyperboliques non linéaires sur un cône caractéristique

We solve globally, in the unbounded domain Y interior to an infinite characteristic cone C , the Cauchy problem for a class of systems of non linear hyperbolic equations of the second order. The initial data, with small norm and verifying some decay property at infinity, are given on C . To cite this article: M. Dossa, F. Touadera, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Acoustic diffraction by a material cone

The boundary-value problem for the Helmholtz equation connected with an infinite circular cone has been analyzed. The proposed scheme of solution includes application of the Kontorovich–Lebedev (KL) transform. The fields excited by a rotationally invariant ring source have been considered. Uncoupled singular integral equations (SIEs) satisfied by the spectral functions were derived. The singularities of the spectral functions were deduced. Some asymptotic approximation for the field with source or observation point near the tip of the cone was obtained. Alternative representations for the far fields were derived. The conditions of validity of the derived field representations for a given set of problem parameters were studied.

A geometric approach to portfolio optimization in models with transaction costs

We consider a continuous-time stochastic optimization problem with infinite horizon, linear dynamics, and cone constraints which includes as a particular case portfolio selection problems under transaction costs for models of stock and currency markets. Using an appropriate geometric formalism we show that the Bellman function is the unique viscosity solution of a HJB equation.

Symbolic calculus for boundary value problems on manifolds with edges

Boundary value problems for (pseudo-) differential operators on a manifold with edges can be characterised by a hierarchy of symbols. The symbolic structure is responsible for ellipticity and for the nature of parametrices within an algebra of “edge-degenerate” pseudo-differential operators. The edge symbolic component of that hierarchy takes values in boundary value problems on an infinite model cone, with edge variables and covariables as parameters. Edge symbols play a crucial role in this theory, in particular, the contribution with holomorphic operator-valued Mellin symbols. We establish a calculus in a framework of “twisted homogeneity” that refers to strongly continuous groups of isomorphisms on weighted cone Sobolev spaces. We then derive an equivalent representation with a particularly transparent composition behaviour.

The thermoelastic equilibrium of conical bodies

An effective solution of various boundary-value problems of thermoelasticity for a hollow infinite cone and an infinite conical panel, when conditions of symmetry or antisymmetry are specified on the plane boundaries of the panel, is constructed in a spherical system of coordinates by the method of separation of variables. A solution of the boundary-contact problems of thermoelasticity is constructed in the case when such bodies are multilayer bodies. The contact surfaces are conical surfaces. A steady temperature field and surface perturbations act on the body. Moreover, certain boundary-value problems of the theory of elasticity are solved by this method for bodies bounded by coordinate surfaces of a spherical system of coordinates, when inhomogeneous boundary conditions are specified on the conical surfaces of the body, and conditions of symmetry or antisymmetry are specified on the plane boundaries, while special homogeneous boundary conditions are specified on the spherical surfaces.

Invariance relations for laminar forced convection in ducts with slowly varying cross-section

Laminar flow forced convective heat transfer in ducts with slowly varying cross-section is analysed. The analysis is based on a simplifying assumption concerning the velocity field, called the similarity assumption. It is assumed that the axial velocity profile is locally fully developed, irrespective of the variation of the cross-sectional area with axial position. It is shown that in the axisymmetric case (circular tube with varying diameter) for boundary conditions of the first kind the diameter variation has no influence at all on the heat transferred. As in the case of a tube with constant diameter, the degree of temperature equilibration depends only on the Fourier number Fo= πaL Q that is on the ratio of length of the tube, L, to flow rate Q. In the case of a plane channel, however, variations of the channel height 2 H have a pronounced influence on the transfer coefficient. It is shown that the degree of equilibration depends only on the effective Fourier number Fo eff = aB Q ·∫ 0 L dz H(z) . The heat transfer is enhanced by a narrow design of the duct. These results hold for any shape of the axial velocity distribution over the duct cross-section, provided the above-mentioned similarity assumption is justified. By analogy, all foregoing results are also applicable to the convective–diffusive mass transfer in flow through ducts with slowly varying cross-section. The range of validity of the similarity assumption is discussed in a hydrodynamic section. It can be shown that for creeping flow of a Newtonian liquid in an infinite cone or wedge this assumption is a very good approximation to the exact solution of Stokes' equations for cone or wedge angles up to 30°. Apart from Newtonian creeping flow, the assumption might well be applicable in the duct flow of a power-law fluid and in pure elongational flow as encountered in stretching filaments or sheets of liquid, where the axial velocity is constant over the cross-section.

Le Schéma de carte et ses applications

In this article, we define a new combinatorial concept, the sketch of map. Intuitively, the sketch of map of a rooted map is a map (in a generalized way) which describes how the map “rolls” itself up round the holes of the surface. Thus the sketch of map is associated with an “infinite cone” of maps, obtained from the given sketch by “enriching” itself in all the possible ways. The first part presents the topological operation of extraction of a sketch of map, its properties and the translation of this operation in terms of operators (derivation, composition) on generating series. In the second part, we apply this new topological operation to obtain new classifications and associated functional equations for rooted maps, respectively, on the projective plane, the Klein bottle and the two-hole torus.

Biconical antennas with elliptical cross-sections

Co-ordinate transformation and conformal mapping methods have been applied to study the problem of radiation from a biconical antenna with elliptical cross-section. The infinite elliptical cone geometry can be transformed into the two-dimensional plane under the assumption of pure TEM spherical waves. Closed-form solutions for the characteristic parameters are presented.