Infinite Cone Research Articles

Detectability of gravity changes on the sea surface due to magma accumulation beneath submarine volcanoes

SUMMARY This study estimated the gravity change expected on the sea surface due to magma accumulation beneath submarine volcanoes. For calculation, a semi-analytical model describing the deformation of an infinite elastic cone was considered, in which a point spherical source was embedded on the axis. The expected gravity change exceeds 0.2 mGal in certain scenarios when $\delta V = 10^8\, \mathrm{m^3}$ of source volume change occurs 2 km below the summit, and the summit is shallower than 300 m below the sea surface. The steeper the slope of a submarine volcano, the greater the expected gravity change. 0.2 mGal of gravity variation can be detected with recent marine gravimeters such as cold atom interferometers. The computation method for the gravity change based on the integral transform (Mellin type) is elaborated in this paper.

Asymptotics of Robin eigenvalues on sharp infinite cones

Let \omega\subset\mathbb{R}^n be a bounded domain with Lipschitz boundary. For \varepsilon>0 and n\in\mathbb{N} , consider the infinite cone \Omega_{\varepsilon}:=\{(x_1,x')\in (0,\infty)\times\mathbb{R}^n\colon x'\in\varepsilon x_1\omega\}\subset\mathbb{R}^{n+1} and the operator Q_{\varepsilon}^{\alpha} acting as the Laplacian u\mapsto-\Delta u on \Omega_{\varepsilon} with the Robin boundary condition \partial_\nu u=\alpha u at \partial\Omega_\varepsilon , where \partial_\nu is the outward normal derivative and \alpha>0 . We look at the dependence of the eigenvalues of Q_\varepsilon^\alpha on the parameter \varepsilon : this problem was previously addressed for n=1 only (in that case, the only admissible \omega are finite intervals). In the present work, we consider arbitrary dimensions n\ge2 and arbitrarily shaped “cross-sections” \omega and look at the spectral asymptotics as \varepsilon becomes small, i.e., as the cone becomes “sharp” and collapses to a half-line. It turns out that the main term of the asymptotics of individual eigenvalues is determined by the single geometric quantity N_\omega:=\frac{\operatorname{Vol}_{n-1} \partial\omega}{\operatorname{Vol}_n \omega}. More precisely, for any fixed j\in \mathbb{N} and \alpha>0 , the j -th eigenvalue E_j(Q^\alpha_\varepsilon) of Q^\alpha_\varepsilon exists for all sufficiently small \varepsilon>0 and satisfies E_j(Q^\alpha_\varepsilon)=-\frac{N_\omega^2{}\alpha^2}{(2j+n-2)^2{}\varepsilon^2}+O\Bigl(\frac{1}{\varepsilon}\Bigr)\quad \text{as $\varepsilon\to 0^+$.} The paper also covers some aspects of Sobolev spaces on infinite cones, which can be of independent interest.

Electromagnetic field theory in superluminal spacetime

Recently, a number of experimental observations on the superluminal group velocities of pulses propagating in dispersive media have led to reconsidering electromagnetism theory in an unconventional framework. To consider faster-than-light phenomena, it is not necessary to replace the current relativistic theory, but it is sufficient to extend it to superluminal motions in a way that preserves the principle of causality. In the present paper, a new approach to study superluminal motions is proposed, which avoids introducing unphysical complex quantities and allows for the formulation of equations that are covariant according to a hyperbolic metric. In the framework of this formalism, Maxwell equations and the single-photon wave equation are obtained through superluminal transformations of ordinary equations. It is shown that the covariant and contravariant components of the superluminal electromagnetic field determine its magnitude and direction, respectively. Furthermore, the solutions of the transformed Maxwell equations are X-shaped ways propagating in the superluminal spacetime region that is delimited by the infinite light cone and the two-sheeted hyperboloid perpendicular to it. Instead, in quantum mechanics, the covariant and contravariant components of the electromagnetic field are proportional to the right- and left-handed helicities of the single photon, respectively.

Mutual visibility by fat robots with slim omnidirectional camera

In the existing literature of the Mutual Visibility problem for autonomous robot swarms, the adopted visibility models have some idealistic assumptions that are not consistent with practical sensing device implementations. This paper investigates the problem in the more realistic visibility model called opaque fat robots with slim omnidirectional camera. The robots are modeled as unit disks, each having an omnidirectional camera represented as a disk of smaller size. The region obstructed by a single robot in this model is a truncated infinite cone. This makes the visibility model significantly challenging compared to the previous ones. We assume that the robots have compasses that allow agreement in the direction and orientation of both axes of their local coordinate systems. The robots are equipped with visible lights which serve as a medium of communication and also as a form of memory. We present a distributed algorithm for the Mutual Visibility problem which is provably correct in the semi-synchronous setting. Our algorithm also provides a solution for Leader Election which we use as a subroutine in our main algorithm. Although Leader Election is trivial with two axis agreement in the full visibility model, it is challenging in our case and is of independent interest.

Deformation of an infinite elastic cone due to a point pressure source buried on the axis: implications to volcanic deformation

SUMMARY This paper presents a semi-analytical solution describing the deformation of an infinite elastic cone due to a point pressure source buried on the axis. The solution is based on the displacement potentials developed by Love in the spherical coordinate system and is solved as a boundary value problem given on the cone surface using the techniques of Mellin transform. Numerical integration of the inverse Mellin transform yields solutions for various cone angles. The displacements calculated for the cone angle α = 90° are confirmed to agree with the well-known Mogi’s solution for the infinite half-space. The present work will be useful for interpretation of the deformation of volcanoes with steep topography and for verification of other numerical methods.

Non-local tug-of-war with noise for the geometric

This paper concerns the fractional ${\mathrm{\bf p}}$-Laplace operator $\Delta_{\mathrm{\bf p}}^s$ in non-divergence form, which has been introduced in [2]. For any ${\mathrm{\bf p}}\in [2,\infty)$ and $s\in (\frac{1}{2},1)$, we first define two families of non-local, non-linear averaging operators, parametrized by $\epsilon$ and defined for all bounded, Borel functions $u:\mathbb R^N\to \mathbb R$. We prove that $\Delta_{\mathrm{\bf p}}^s u(x)$ emerges as the $\epsilon^{2s}$-order coefficient in the expansion of the deviation of each $\epsilon$-average from the value $u(x)$, in the limit of the domain of averaging exhausting an appropriate cone in $\mathbb R^N$ at the rate $\epsilon\to 0$. Second, we consider the $\epsilon$-dynamic programming principles modeled on the first average, and show that their solutions converge uniformly as $\epsilon\to 0$, to viscosity solutions of the homogeneous non-local Dirichlet problem for $\Delta_{\mathrm{\bf p}}^s$, posed in a domain ${\mathcal{D}}$ satisfying the external cone condition and subject to bounded, uniformly continuous data on $\mathbb R^N\setminus {\mathcal{D}}$. Finally, we interpret such $\epsilon$-approximating solutions as values to the non-local Tug-of-War game with noise. In this game, players choose directions while the game position is updated randomly within the infinite cone that aligns with the specified direction, whose aperture angle depends on ${\mathrm{\bf p}}$ and $N$, and whose $\epsilon$-tip has been removed.

Harmonic extension through conical surfaces

This paper establishes extension results for harmonic functions which vanish on a conical surface. These are based on a detailed analysis of expansions for the Green function of an infinite cone.

Numerical solution of compressible Euler and Magnetohydrodynamic flow past an infinite cone

A numerical scheme is developed for systems of conservation laws on manifolds which arise in high speed aerodynamics and magneto-aerodynamics. The systems are presented in an arbitrary coordinate system on the manifold and involve source terms which account for the curvature of the domain. In order for a numerical method to accurately capture the behavior of the system it is solving, the equations must be discretized in a way that is not only consistent in value, but also models the appropriate character of the system. Such a discretization is presented in this work which preserves the tensorial transformation relationships involved in formulating equations in a curved space. A numerical method is then developed and applied to the conical Euler and Ideal Magnetohydrodynamic equations. To the authors’ knowledge, this is the first demonstration of a numerical solver for the conical Ideal MHD equations. Along with the curvature of the domain, is the added challenge that both systems of equations are of mixed type. Both systems change between elliptic and hyperbolic type throughout the domain, which had to be accommodated by the numerical method.

The inhomogeneous wave equation with $L^p$ data

We prove existence and uniqueness of $L^2$ solutions to the inhomogeneous wave equation on $\mathbb{R}^{n-1}\times\mathbb{R}$ under the assumption that the inhomogeneous data lies in $L^p(\mathbb{R}^n)$ for $p=2n/(n+4)$. We also require the Fourier transform of the inhomogeneous data to vanish on an infinite cone where the solution could become singular. Subsequently, we show sharpness of the exponent $p$. This extends work of Michael Goldberg, in which similar Fourier-analytic techniques were used to study the inhomogeneous Helmholtz equation.

An index formula for the intersection Euler characteristic of an infinite cone

The aim of this article is to establish an index formula for the intersection Euler characteristic of a cone. The main actor is the model Witten Laplacian on the infinite cone. First, we study its spectral properties and establish a McKean-Singer type formula. We also give an explicit formula for the zeta function of the model Witten Laplacian. In a second step, we apply local index techniques to the model Witten Laplacian. By combining these two steps, we express the absolute and relative intersection Euler characteristic of the cone as a sum of two terms, a term which is local, and a second term which is the Cheeger invariant.

Homotopical complexity of a billiard flow on the 3D flat torus with two cylindrical obstacles

We study the homotopical rotation vectors and the homotopical rotation sets for the billiard flow on the unit flat torus with two disjoint and orthogonal toroidal (cylindrical) scatterers removed from it. The natural habitat for these objects is the infinite cone erected upon the Cantor set$\text{Ends}(G)$of all ‘ends’ of the hyperbolic group$G=\unicode[STIX]{x1D70B}_{1}(\mathbf{Q})$. An element of$\text{Ends}(G)$describes the direction in (the Cayley graph of) the group$G$in which the considered trajectory escapes to infinity, whereas the height function$s$($s\geq 0$) 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$\sqrt{3}$and, in any direction$e\in \text{Ends}(\unicode[STIX]{x1D70B}_{1}({\mathcal{Q}}))$, the escape is feasible with any prescribed speed$s$,$0\leq s\leq 1/(\sqrt{6}+2\sqrt{3})$. This means that the radial upper and lower bounds for the rotation set$R$are actually pretty close to each other. Furthermore, we prove the convexity of the set$\mathit{AR}$of constructible rotation vectors, and that the set of rotation vectors of periodic orbits is dense in$\mathit{AR}$. We also provide effective lower and upper bounds for the topological entropy of the studied billiard flow.

An index formula for the intersection Euler characteristic of an infinite cone

The aim of this note is to establish an index formula for the intersection Euler characteristic of a cone. The main actor of these notes is the model Witten Laplacian on the infinite cone. First, we study its spectral properties and establish a McKean–Singer-type formula. We also give an explicit formula for the zeta function of the model Witten Laplacian. In a second step, we apply local index techniques to the model Witten Laplacian. By combining these two steps, we express the absolute and relative intersection Euler characteristic of the cone as a sum of two terms, a term which is local, and a second term which is the Cheeger invariant.

On the logarithmic divergent part of entanglement entropy, smooth versus singular regions

The entanglement entropy for smooth regions $\cal A$ has a logarithmic divergent contribution with a shape dependent coefficient and that for regions with conical singularities an additional $\log ^2$ term. Comparing the coefficient of this extra term, obtained by direct holographic calculation for an infinite cone, with the corresponding limiting case for the shape dependent coefficient for a regularised cone, a mismatch by a factor two has been observed in the literature. We discuss several aspects of this issue. In particular a regularisation of $\cal A$, intrinsically delivered by the holographic picture, is proposed and applied to an example of a compact region with two conical singularities. Finally, the mismatch is removed in all studied regularisations of $\cal A$, if equal scale ratios are chosen for the limiting procedure.

The use of dimensional analysis to find fields sourced by distributions on self-scaled geometric shapes

Dimensional analysis (DA) is commonly used to solve problems in various fields in physics. In this work we concentrated on problems in electrostatics (and magneto-statics) that deal with finding the field (or potential) caused by a distribution of charges (or currents) on a family of scale-invariant geometrical shapes. An infinite cone is one example of such a shape; zooming-in or zooming-out of this shape will leave it unchanged. Once we choose the shape, a monomial length-dependence-of-charge distribution on such a shape is chosen. The interplay between the chosen geometry and the chosen distribution yields an added value to the DA method as shown in this paper. Examples, like finding the field of infinite cones, the field created by semi-infinite wires, and the distribution of current on a conducting spherical shell, are presented. The field of an infinite cone is calculated and found to be uniform in the region containing the axis of symmetry; moreover, for a specific opening angle the field vanishes. Another example of using DA is to show that the electric field caused by a moving charge is radial for any velocity which is constant without the need to use relativistic calculations.

Critical heights of destruction for a forest-fire model on the half-plane

Consider the following forest-fire model on the upper half-plane of the triangular lattice: Each site can be vacant or by a tree. At time 0 all sites are vacant. Then the process is governed by the following random dynamics: Trees grow at rate 1, independently for all sites. If an occupied cluster reaches the boundary of the half-plane or if it is about to become infinite, the cluster is instantaneously destroyed, i.e. all of its sites turn vacant. Let t(c) = log 2 denote the critical time after which an infinite cluster first appears in the corresponding pure growth process, where there is only the growth of trees but no destruction mechanism. Choose an arbitrary infinite cone in the half-plane whose apex lies on the boundary of the half-plane and whose boundary lines are non-horizontal. We prove that at time t(c) almost surely only finitely many sites inside the cone have been affected by destruction in the forest-fire process.

Asymptotics of solutions of second order parabolic equations near conical points and edges

The authors consider the first boundary value problem for a second order parabolic equation with variable coefficients in a domain with conical points or edges. In the first part of the paper, they study the Green function for this problem in the domain K× R n − m , where K is an infinite cone in R m , 2≤m≤n. They obtain the asymptotics of the Green function near the vertex (n=m) and edge (n>m), respectively. This result is applied in the second part of the paper, which deals with the initial-boundary value problem in this domain. Here, the right-hand side f of the differential equation belongs to a weighted L p space. At the end of the paper, the initial-boundary value problem in a bounded domain with conical points or edges is studied.

On the geometry of null cones to infinity under curvature flux bounds

The main objective of this paper is to control the geometry of a future outgoing truncated null cone extending smoothly toward infinity in an Einstein-vacuum spacetime. In particular, we wish to do this under minimal regularity assumptions, namely, at the (weighted) L2-curvature level. We show that if the curvature flux and the data on an initial sphere of the cone are sufficiently close to the corresponding values in a standard Minkowski or Schwarzschild null cone, then we can obtain quantitative bounds on the geometry of the entire infinite cone. The same bounds also imply the existence of limits at infinity, along the null cone, of naturally scaled geometric quantities. In Alexakis and Shao (2013 Bounds on the Bondi energy by a flux of curvature arXiv:1308.4170), we will apply these results in order to control various physical quantities—e.g., the Bondi energy and (linear and angular) momenta—associated with such infinite null cones in vacuum spacetimes.

Stokes flow driven by a Stokeslet in a cone

We consider an axisymmetric Stokes flow in an infinite right circular cone, which has a source of momentum (a Stokeslet) on its axis. It produces an infinite sequence of eddies in the conical flow region. A boundary problem for a stream function is solved. The picture of the streamlines is obtained. We investigate an eddy structure of the flow. The results can be used for constructing nanoreactors while carrying out chemical reactions in strictly localized nanosized spatial regions.

The torsion of the conical layered elastic cone

An infinite conic layered cone which represents the sequence of conical adjacent funnels, inserted one into another, is considered. Each of these funnels differs by their shear modulus. Conditions of an ideal contact are fulfilled between the adjacent conical surfaces. An external conical surface is loaded by tangent torsion stress. An exact solution of the corresponding one-dimensional boundary problem is constructed in the domain of Mellin’s transformation, which is applied directly to the torsion equation. A formulation of boundary conditions in matrix form allows to obtain recurrent-type relations for unknown constants of a general solution for each layer. The proposed solving method leads to the exact solution which is independent of the layers’ number. Mellin’s transformation is inversed to finish the construction of the formulas for displacements and stress. The detailed elaboration of the proposed approach is done for two particular cases: a cone without stratification and a two-layered cone.

Peak power in the 3D magnetic Schrödinger equation

This paper is devoted to the spectral analysis of the magnetic Neumann Laplacian on an infinite cone of aperture α. When the magnetic field is constant and parallel to the revolution axis and when the aperture goes to zero, we prove that the first n eigenvalues exist and admit asymptotic expansions in powers of α2.