Volume 57, Issue 11, November 2016

We consider the 3body problem of celestial mechanics in Euclidean, elliptic, and hyperbolic spaces and study how the Lagrangian (equilateral) relative equilibria bifurcate when the Gaussian curvature varies. We thus prove the existence of new classes of orbits. In particular, we find some families of isosceles triangles, which occur in elliptic space.
 ARTICLES

 Partial Differential Equations

Well posedness and exponential stability in a wave equation with a strong damping and a strong delay
View Description Hide DescriptionIn this paper, we consider a wave equation with a strong damping and a strong constant (respectively, distributed) delay. We prove the wellposedness and establish an exponential decay result under a suitable assumption on the weight of the damping and the weight of the delay.

Multiplicity and concentration of positive solutions for a class of quasilinear problems through OrliczSobolev space
View Description Hide DescriptionIn this work, we study existence, multiplicity, and concentration of positive solutions for the following class of quasilinear problems where is a Nfunction, ΔΦ is the ΦLaplacian operator, ϵ is a positive parameter, V : ℝ^{N} → ℝ is a continuous function, and f : ℝ → ℝ is a C ^{1}function.

On effects of viscosity and magnetic fields on the largest growth rate of linear Rayleigh–Taylor instability
View Description Hide DescriptionIn this article, we investigate the effect of viscosity on the largest growth rate in the linear Rayleigh–Taylor (RT) instability of a threedimensional nonhomogeneous incompressible viscous flow in a bounded domain. By adapting a modified variational approach and careful analysis, we show that the largest growth rate in linear RT instability tends to zero as the viscosity coefficient goes to infinity. Moreover, the largest growth rate increasingly converges to one of the corresponding inviscid fluids as the viscosity coefficient goes to zero. Applying these analysis techniques to the corresponding viscous magnetohydrodynamic fluids, we can also show that the largest growth rate in linear magnetic RT instability tends to zero as the strength of horizontal (or vertical) magnetic field increasingly goes to a critical value.

Concentration in vanishing pressure limit of solutions to the modified Chaplygin gas equations
View Description Hide DescriptionTwo kinds of occurrence mechanism on the phenomenon of concentration and the formation of delta shock wave in vanishing pressure limit of solutions to the modified Chaplygin gas equations are analyzed and identified. The Riemann problem of the modified Chaplygin gas equations is first solved. Then it is shown that, as the pressure vanishes, any twoshock Riemann solution tends to a deltashock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted δmeasure which forms a delta shock wave; any tworarefactionwave Riemann solution tends to a twocontactdiscontinuity solution to the transport equations, and the nonvacuum intermediate state in between tends to a vacuum state. It is also shown that, as the pressure approaches the generalized Chaplygin gas pressure, any twoshock Riemann solution tends to a deltashock solution to the generalized Chaplygin gas equations. Some numerical results are presented to show the formation process of delta shock waves and vacuum states.

Existence and multiplicity of solutions for fourthorder elliptic equations of Kirchhoff type with critical growth in ℝ^{N}
View Description Hide DescriptionIn this paper, we deal with the existence and multiplicity of solutions for fourthorder elliptic equations of Kirchhoff type with critical nonlinearity: , (t, x) ∈ ℝ × ℝ^{N}. By using Lions’ second concentrationcompactness principle and concentrationcompactness principle at infinity to prove that (PS) condition holds locally and by variational method, we prove that it has at least one solution and for any m ∈ ℕ, it has at least m pairs of solutions.

Optimal decay rates for the compressible viscoelastic flows
View Description Hide DescriptionIn this paper, we study the compressible viscoelastic flows in threedimensional whole space. Under the assumption of small initial data, we establish the unique global solution by the energy method. Furthermore, we obtain the time decay rates of the higherorder spatial derivatives of the solution if the initial data belong to L ^{1}(ℝ^{3}) additionally.

Integrable multicomponent generalization of a modified short pulse equation
View Description Hide DescriptionWe propose a multicomponent generalization of the modified short pulse (SP) equation which was derived recently as a reduction of Feng’s twocomponent SP equation. Above all, we address the twocomponent system in depth. We obtain the Lax pair, an infinite number of conservation laws and multisoliton solutions for the system, demonstrating its integrability. Subsequently, we show that the twocomponent system exhibits cusp solitons and breathers for which the detailed analysis is performed. Specifically, we explore the interaction process of two cusp solitons and derive the formula for the phase shift. While cusp solitons are singular solutions, smooth breather solutions are shown to exist, provided that the parameters characterizing the solutions satisfy certain conditions. Last, we discuss the relation between the proposed system and existing twocomponent SP equations.
 Representation Theory and Algebraic Methods

Braid group representations from a deformation of the harmonic oscillator algebra
View Description Hide DescriptionWe describe a new technique to obtain representations of the braid group Bn from the ℜ–matrix of a quantum deformed algebra of the one dimensional harmonic oscillator. We consider the action of the ℜ–matrix not on the tensor product of representations of the algebra, that in the harmonic oscillator case are infinite dimensional, but on the subspace of the tensor product corresponding to the lowest weight vectors.

Oscillator versus prefundamental representations
View Description Hide DescriptionWe find the ℓweights and the corresponding ℓweight vectors for the finite and infinite dimensional representations of the quantum loop algebras Uq(ℒ(𝔰𝔩2)) and Uq(ℒ(𝔰𝔩3)) obtained from the Verma representations of the quantum groups Uq(𝔤𝔩2) and Uq(𝔤𝔩3) via the Jimbo’s homomorphism. Then we find the ℓweights and the ℓweight vectors for the qoscillator representations of the positive Borel subalgebras of the same quantum loop algebras. This allows, in particular, to relate the qoscillator and prefundamental representations.

On fixed point planar algebras
View Description Hide DescriptionTo a weighted graph can be associated a bipartite graph planar algebra . We construct and study the symmetric enveloping inclusion of . We show that this construction is equivariant with respect to the automorphism group of . We consider subgroups G of the automorphism of such that the Gfixed point space is a subfactor planar algebra. As an application we show that if G is amenable, then is amenable as a subfactor planar algebra. We define the notions of a cocycle action of a Hecke pair on a tracial von Neumann algebra and the corresponding crossed product. We show that a large class of symmetric enveloping inclusions of subfactor planar algebras can be described by such a crossed product.
 ManyBody and Condensed Matter Physics

LargeN expansion for the timedelay matrix of ballistic chaotic cavities
View Description Hide DescriptionWe consider the 1/Nexpansion of the moments of the proper delay times for a ballistic chaotic cavity supporting N scattering channels. In the random matrix approach, these moments correspond to traces of negative powers of Wishart matrices. For systems with and without broken time reversal symmetry (Dyson indices β = 1 and β = 2), we obtain a recursion relation, which efficiently generates the coefficients of the 1/Nexpansion of the moments. The integrality of these coefficients and their possible diagrammatic interpretation is discussed.
 Quantum Mechanics

Trace formulas for the matrix Schrödinger operator on the halfline with general boundary conditions
View Description Hide DescriptionWe prove BuslaevFaddeev trace formulas for the matrix Schrödinger operator on the halfline, with general boundary conditions at the origin and with selfadjoint matrix potentials.

Inverse eigenvalue problems
View Description Hide DescriptionIn this article we consider inverse eigenvalue problems for the Schrödinger operator on a finite interval. We extend and strengthen previously known uniqueness theorems. A partially known potential is identified by some sets of eigenvalues and norming constants.
 Quantum Information and Computation

There is no direct generalization of positive partial transpose criterion to the threebythree case
View Description Hide DescriptionWe show that there cannot exist a straightforward generalization of the famous positive partial transpose criterion to threebythree systems. We call straightforward generalizations that use a finite set of positive maps and arbitrary local rotations of the tested twopartite state. In particular, we show that a family of extreme positive maps discussed in a paper by Ha and Kye [Open Syst. Inf. Dyn. 18, 323–337 (2011)], cannot be replaced by a finite set of witnesses in the task of entanglement detection in threebythree systems. In a more mathematically elegant parlance, our result says that the convex cone of positive maps of the set of threedimensional matrices into itself is not finitely generated as a mapping cone.
 Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

The Moyal sphere
View Description Hide DescriptionWe construct a family of constant curvature metrics on the Moyal plane and compute the Gauss–Bonnet term for each of them. They arise from the conformal rescaling of the metric in the orthonormal frame approach. We find a particular solution, which corresponds to the Fubini–Study metric and which equips the Moyal algebra with the geometry of a noncommutative sphere.

On elliptic algebras and largen supersymmetric gauge theories
View Description Hide DescriptionIn this note we further develop the duality between supersymmetric gauge theories in various dimensions and elliptic integrable systems such as RuijsenaarsSchneider model and periodic intermediate long wave hydrodynamics. These models arise in instanton counting problems and are described by certain elliptic algebras. We discuss the correspondence between the two types of models by employing the largen limit of the dual gauge theory. In particular, we provide nonAbelian generalization of our previous result on the intermediate long wave model.

AdSCarroll branes
View Description Hide DescriptionCoset methods are used to determine the action of a codimension one brane (domain wall) embedded in (d + 1)dimensional AdS space in the Carroll limit in which the speed of light goes to zero. The action is invariant under the nonlinearly realized symmetries of the AdSCarroll spacetime. The NambuGoldstone field exhibits a static spatial distribution for the brane with a time varying momentum density related to the brane’s spatial shape as well as the AdSC geometry. The AdSC vector field dual theory is obtained.
 General Relativity and Gravitation

Regularization of Kepler problem in κspacetime
View Description Hide DescriptionIn this paper, we regularize the Kepler problem on κspacetime in several different ways. First, we perform a Mosertype regularization and then we proceed for the LigonSchaaf regularization to our problem. In particular, generalizing Heckman and de Laat [J. Symplectic Geom. 10, 463473 (2012)] in the noncommutative context, we show that the LigonSchaaf regularization map following from an adaptation of the Moser regularization can be generalized to the Kepler problem on κspacetime.

Cosmic anisotropic doomsday in Bianchi type I universes
View Description Hide DescriptionIn this paper we study finite time future singularities in anisotropic Bianchi type I models. It is shown that there exist future singularities similar to Big Rip ones (which appear in the framework of phantom FriedmannRobertsonWalker cosmologies). Specifically, in an ellipsoidal anisotropic scenario or in a fully anisotropic scenario, the three directional and average scale factors may diverge at a finite future time, together with energy densities and anisotropic pressures. We call these singularities “Anisotropic Big Rip Singularities.” We show that there also exist Bianchi type I models filled with matter, where one or two directional scale factors may diverge. Another type of future anisotropic singularities is shown to be present in vacuum cosmologies, i.e., Kasner spacetimes. These singularities are induced by the shear scalar, which also blows up at a finite time. We call such a singularity “Vacuum Rip.” In this case one directional scale factor blows up, while the other two and average scale factors tend to zero.

Extending the rigidity of general relativity
View Description Hide DescriptionWe give the most general conditions to date which lead to uniqueness of the general relativistic Hamiltonian. Namely, we show that all spatially covariant generalizations of the scalar constraint which extend the standard one while remaining quadratic in the momenta are second class. Unlike previous investigations along these lines, we do not require a specific Poisson bracket algebra, and the quadratic dependence on the momenta is completely general, with an arbitrary local operator as the kinetic term.