Infinity In Space Research Articles

Maximal surfaces in Anti-de Sitter space, width of convex hulls and quasiconformal extensions of quasisymmetric homeomorphisms

We give upper bounds on the principal curvatures of a maximal surface of nonpositive curvature in three-dimensional Anti-de Sitter space, which only depend on the width of the convex hull of the surface. Moreover, given a quasisymmetric homeomorphism \phi , we study the relation between the width of the convex hull of the graph of \phi , as a curve in the boundary of infinity of Anti-de Sitter space, and the cross-ratio norm of \phi . As an application, we prove that if \phi is a quasisymmetric homeomorphism of \mathbb{R}\mathrm P^1 with cross-ratio norm ||\phi|| , then ln K\leq C||\phi|| , where K is the maximal dilatation of the minimal Lagrangian extension of \phi to the hyperbolic plane.

Coarse proximity and proximity at infinity

We define coarse proximity structures, which are an analog of small-scale proximity spaces in the large-scale context. We show that metric spaces induce coarse proximity structures, and we construct a natural small-scale proximity structure, called the proximity at infinity, on the set of equivalence classes of unbounded subsets of an unbounded metric space given by the relation of having finite Hausdorff distance. We show that this construction is functorial. Consequently, the proximity isomorphism type of the proximity at infinity of an unbounded metric space X is a coarse invariant of X.

Phases of mathcal{N}=1 theories in 2 + 1 dimensions

We study the dynamics of 2+1 dimensional theories with mathcal{N}=1 supersymmetry. In these theories the supersymmetric ground states behave discontinuously at codimension one walls in the space of couplings, with new vacua coming in from infinity in field space. We show that the dynamics near these walls is calculable: the two-loop effective potential yields exact results about the ground states near the walls. Far away from the walls the ground states can be inferred by decoupling arguments. In this way, we are able to follow the ground states of mathcal{N}=1 theories in 2+1 dimensions and construct the infrared phases of these theories. We study two examples in detail: Adjoint SQCD and SQCD with one fundamental quark. In Adjoint QCD we show that for sufficiently small Chern-Simons level the theory has a non-perturbative metastable supersymmetry-breaking ground state. We also briefly discuss the critical points of this theory. For SQCD with one quark we establish an infrared duality between a U(N) gauge theory and an SU(N) gauge theory. The duality crucially involves the vacua that appear from infinity near the walls.

Existence of solutions for a fractional semilinear parabolic equation with singular initial data

In this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the Cauchy problem ∂tu+(−Δ)θ2u=up,x∈RN,t>0,u(0)=μ≥0 in RN,where N≥1, 0<θ≤2, p>1 and μ is a Radon measure or a measurable function in RN. Our conditions lead optimal estimates of the life span of the solution with μ behaving like λ|x|−A (A>0) at the space infinity, as λ→+0.

Weakly coupled systems of fully nonlinear parabolic equations in the Heisenberg group

This paper is devoted to viscosity solutions to weakly coupled systems of fully nonlinear parabolic equations in the first Heisenberg group. We extend well-posedness results in the Euclidean space to the Heisenberg group, including the uniqueness and existence of solutions with exponential growth at space infinity under monotonicity and other regularity assumptions on the parabolic operators. In addition, Lipschitz preserving properties of the system are also studied.

Global existence and blow-up of solutions for the heat equation with exponential nonlinearity

We are concerned with the existence of global in time solution for a semilinear heat equation with exponential nonlinearity(P){∂tu=Δu+eu,x∈RN,t>0,u(x,0)=u0(x),x∈RN, where u0 is a continuous initial function. In this paper, we consider the case where u0 decays to −∞ at space infinity, and study the optimal decay bound classifying the existence of global in time solutions and blowing up solutions for (P). In particular, we point out that the optimal decay bound for u0 is related to the decay rate of forward self-similar solutions of ∂tu=Δu+eu.

Second critical exponent for a nonlinear nonlocal diffusion equation

This paper studies the Cauchy problem for a nonlocal diffusion equation u t = J ∗ u − u + u p . We determine the second critical exponent, namely the optimal decay order for initial data at space infinity to distinguish global and blow-up solutions in the “super Fujita” range. Similar to the critical Fujita exponent obtained very recently by Alfaro (2017), we find that the second critical exponent also relies heavily on the behavior of the Fourier transform of the kernel function J .

Multidimensional stability of traveling fronts in combustion and non-KPP monostable equations

This paper is concerned with the multidimensional stability of traveling fronts for the combustion and non-KPP monostable equations. Our study contains two parts: in the first part, we first show that the two-dimensional V-shaped traveling fronts are asymptotically stable in $$\mathbb {R}^{n+2}$$ with $$n\ge 1$$ under any (possibly large) initial perturbations that decay at space infinity, and then, we prove that there exists a solution that oscillates permanently between two V-shaped traveling fronts, which implies that even very small perturbations to the V-shaped traveling front can lead to permanent oscillation. In the second part, we establish the multidimensional stability of planar traveling front in $$\mathbb {R}^{n+1}$$ with $$n\ge 1$$ .

Towards causal patch physics in dS/CFT

This contribution is a status report on a research program aimed at obtaining quantum-gravitational physics inside a cosmological horizon through dS/CFT, i.e. through a holographic description at past/future infinity of de Sitter space. The program aims to bring together two main elements. The first is the observation by Anninos, Hartman and Strominger that Vasiliev’s higher-spin gravity provides a working model for dS/CFT in 3+1 dimensions. The second is the proposal by Parikh, Savonije and Verlinde that dS/CFT may prove more tractable if one works in so-called “elliptic” de Sitter space – a folded-in-half version of global de Sitter where antipodal points have been identified. We review some relevant progress concerning quantum field theory on elliptic de Sitter space, higher-spin gravity and its holographic duality with a free vector model. We present our reasons for optimism that the approach outlined here will lead to a full holographic description of quantum (higher-spin) gravity in the causal patch of a de Sitter observer.

Prescribed conditions at infinity for fractional parabolic and elliptic equations with unbounded coefficients

We investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed point-wise conditions at infinity (in space), which can be time-dependent. Moreover, we study the asymptotic behavior of such solutions. We also consider solutions of elliptic equations satisfying appropriate conditions at infinity.

On the Asymptotic Behavior of Eigenfunctions of the Continuous Spectrum at Infinity in Configuration Space for the System of Three Three-Dimensional Like-Charged Quantum Particles

To our knowledge there are no complete results, even not rigorously mathematically justified, related to a system of three and more quantum particles, interacting by Coulomb pair potentials, and expressed in terms of eigenfunctions. For the system of three such identical particles, asymptotic formulas describing the behavior of eigenfunctions at infinity in configuration space are suggested.

To the Question on the Resolvent Kernel Asymptotics in the Three-Body Scattering Problem

The present paper aims at announcing a new approach to the construction of the asymptotics (at infinity in configuration space) of the resolvent kernel of the Schrodinger operator in the scattering problem of three one-dimensional quantum particles interacting by compactly supported pair repulsive potentials. Within the framework of this approach, the asymptotics of eigenfunctions of the absolutely continuous spectrum of the Schrodinger operator can be constructed explicitly. It should be emphasized that the restriction of the consideration to the case of compactly supported pair potentials does not lead to a simplification of the problem in its essence, since the potential of the interaction of all three particles remains nondecreasing at infinity, but allows one to put aside a certain number of technical details.

Linking past and future null infinity in three dimensions

We provide a mapping between past null and future null infinity in three-dimensional flat space, using symmetry considerations. From this we derive a mapping between the corresponding asymptotic symmetry groups. By studying the metric at asymptotic regions, we find that the mapping is energy preserving and yields an infinite number of conservation laws.

Kant's Conceptualism: a New Reading of the Transcendental Deduction

AbstractI defend a novel interpretation of Kant's conceptualism regarding the contents of our perceptual experiences. Conceptualist interpreters agree that Kant's Deduction aims to prove that intuitions require the categories for their spatiality and temporality. But conceptualists disagree as to which features of space and time make intuitions require the categories. Interpreters have cited the singularity, unity, infinity, and homogeneity of space and time. But this is incompatible with Kant's Aesthetic, which aims to prove that these same features qualify space and time as intuitions, not concepts. On my interpretation, the feature is objectivity. Space and time are objective, in that they ground our judgments in geometry and mechanics.

Sztuka i rytuały. Koji Kamoji i jego wizje nieskończoności

Sztuka i rytuały. Koji Kamoji i jego wizje nieskończoności

Searching for Transcendence: Drives, Emotions, and Infinity

There seems to be a need to theorize about transcendence in the actual relational literature. This may be related to the abandonment of drive theory. Emotions conceived by Matte-Blanco as infinite sets seem to recover that transcendence. In the clinical situation references to God seem to point many times to this infinity of emotionality. In these cases the emergence of art in the clinical dialogue—as long it can provide symbols of the infinite—seem to be able to unfold in our finiteness the infinite. These symbols can fill the infinity of potential space, so as not to experience an infinite and silent void that terrifies.

The Hardy inequality and the heat equation with magnetic field in any dimension

ABSTRACTIn the Euclidean space of any dimension d, we consider the heat semigroup generated by the magnetic Schrödinger operator from which an inverse-square potential is subtracted to make the operator critical in the magnetic-free case. Assuming that the magnetic field is compactly supported, we show that the polynomial large-time behavior of the heat semigroup is determined by the eigenvalue problem for a magnetic Schrödinger operator on the (d − 1)-dimensional sphere whose vector potential reflects the behavior of the magnetic field at the space infinity. From the spectral problem on the sphere, we deduce that in d = 2 there is an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta, while there is no extra polynomial decay rate in higher dimensions. To prove the results, we establish new magnetic Hardy-type inequalities for the Schrödinger operator and develop the method of self-similar variables and weighted Sobolev spaces for the associated heat equation.

Lipschitz continuity and convexity preserving for solutions of semilinear evolution equations in the Heisenberg group

In this paper we study viscosity solutions of semilinear parabolic equations in the Heisenberg group. We show uniqueness of viscosity solutions with exponential growth at space infinity. We also study Lipschitz and horizontal convexity preserving properties under appropriate assumptions. Counterexamples show that in general such properties that are well-known for semilinear and fully nonlinear parabolic equations in the Euclidean spaces do not hold in the Heisenberg group.

Pure Strategy Nash Implementation with Finite Mechanisms

Canonical mechanisms have unattractive features because they have to cover every instances of the implementation problem. This is unavoidable but at the same time it leaves open whether these features can be avoided in all specific cases or even most of them. We focus on one such feature: The infinity of the message space. It has been know since Jackson [A crash course in implementation theory, Soc Choice Welfare 18 (2001) 655-708.] that allowing only finite message spaces is a constraint. What is not known, however, is how bad this constraint is. Not surprisingly it all depends on what is known about the utility representations. What is surprising, though, is that allowing only finite message spaces does not severely limit implementability. In economic environments the constraints are particularly weak.

English

In many theories of gravity, the cause of gravity is neither mentioned nor clarified yet. Dark energy, which makes the universe accelerate and push all matters in the universe apart, has repulsive power. Every matter in space is equally pressured by this repulsive force from all directions. It is assumed that this force comes from infinity in space. However, if matters exist nearby, the matters shield each other from the repulsive force; and thus, the repulsive force acting on the matter is not isotropic anymore. When two nearby bodies are applied with this shielding effect, they are pushed toward each other; and the theoretically derived net pushing force on each body is mathematically equivalent to the Newton’s attractive gravitational force. This result shows that the gravity may come from dark energy. Key words: Dark energy, gravity, cosmic repulsive force, shielding effect, expanding universe.