Cone Inequalities Research Articles

Perfect tensor hyperthreads

Bit threads, a dual description of the Ryu-Takyanagi formula for holographic entanglement entropy (EE), can be interpreted as a distillation of the quantum information to a collection of Bell pairs between different boundary regions. In this article we discuss a generalization to hyperthreads which can connect more than two boundary regions leading to a rich and diverse class of convex programs. By modeling the contributions of different species of hyperthreads to the EEs of perfect tensors we argue that this framework may be useful for helping us to begin to probe the multipartite entanglement of holographic systems. Furthermore, we demonstrate how this technology can potentially be used to understand holographic entropy cone inequalities and may provide an avenue to address issues of locking.

Redundancy in string cone inequalities and multiplicities in potential functions on cluster varieties

We study defining inequalities of string cones via a potential function on a reduced double Bruhat cell. We give a necessary criterion for the potential function to provide a minimal set of inequalities via tropicalization and conjecture an equivalence.

Holographic entropy cone in AdS-Vaidya spacetimes

We examine the five-region holographic entropy cone inequalities for the special case of the AdS$_3$-Vaidya metric for a variety of boundary configurations. This is done by numerically solving the geodesic equation in the bulk for various boundary configurations. In all the cases we examine, we find that all the inequalities are satisfied when the bulk satisfies the null energy condition, while the inequalities are all violated when the bulk spacetime violates the null energy condition. A proof of the five-region holographic entropy cone inequalities for the dynamical bulk case remains an open problem--our results provide evidence that these inequalities hold for dynamical bulk spacetimes.

Optimal Hardy inequalities in cones

Let Ω be an open connected cone in ℝn with vertex at the origin. Assume that the Operatoris subcritical in Ω, where δΩ is the distance function to the boundary of Ω and μ ⩽ 1/4. We show that under some smoothness assumption on Ω the improved Hardy-type inequalityholds true, and the Hardy-weight λ(μ)|x|–2 is optimal in a certain definite sense. The constant λ(μ) > 0 is given explicitly.

Hölder stability for Serrin’s overdetermined problem

In a bounded domain \(\varOmega \), we consider a positive solution of the problem \(\Delta u+f(u)=0\) in \(\varOmega \), \(u=0\) on \(\partial \varOmega \), where \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a locally Lipschitz continuous function. Under sufficient conditions on \(\varOmega \) (for instance, if \(\varOmega \) is convex), we show that \(\partial \varOmega \) is contained in a spherical annulus of radii \(r_i 0\) and \(\tau \in (0,1]\). Here, \([u_\nu ]_{\partial \varOmega }\) is the Lipschitz seminorm on \(\partial \varOmega \) of the normal derivative of u. This result improves to Holder stability the logarithmic estimate obtained in Aftalion et al. (Adv Differ Equ 4:907–932, 1999) for Serrin’s overdetermined problem. It also extends to a large class of semilinear equations the Holder estimate obtained in Brandolini et al. (J Differ Equ 245:1566–1583, 2008) for the case of torsional rigidity (\(f\equiv 1\)) by means of integral identities. The proof hinges on ideas contained in Aftalion et al. (1999) and uses Carleson-type estimates and improved Harnack inequalities in cones.

Optimal Hardy inequalities in cones

For a given subcritical Schrodinger operator in a cone in ℝ n with a given Hardy potential corresponding to the distance to the boundary of the cone, we present an explicit optimal Hardy-type improvement. In particular, we present an explicit expression for the associate best Hardy constant, and for the corresponding ground state.

Weighted isoperimetric inequalities in cones and applications

This paper deals with weighted isoperimetric inequalities relative to cones of RN. We study the structure of measures that admit as isoperimetric sets the intersection of a cone with balls centered at the vertex of the cone. For instance, in case that the cone is the half-space R+N={x∈RN:xN>0} and the measure is factorized, we prove that this phenomenon occurs if and only if the measure has the form dμ=axNkexp(c|x|2)dx, for somea>0, k,c≥0. Our results are then used to obtain isoperimetric estimates for Neumann eigenvalues of a weighted Laplace–Beltrami operator on the sphere, sharp Hardy-type inequalities for functions defined in a quarter space and, finally, via symmetrization arguments, a comparison result for a class of degenerate PDE’s.

Trace Hardy–Sobolev inequalities in cones

Sharp constants are found for the trace Hardy–Sobolev inequalities in cones. The question as to whether these constants are attained is discussed.

Poincaré inequalities in reflexive cones

We present an abstract result concerning Poincaré inequalities in cones. Some examples in Sobolev spaces are provided. We also discuss an application to a priori bounds of solutions for a general boundary value problem involving the vector p -Laplacian operator.

Cone inequalities and stability of differential systems

We investigate generalizations of classes of monotone dynamical systems in a partially ordered Banach space. We establish algebraic conditions for the stability of equilibrium states of differential systems on the basis of linearization and application of derivatives of nonlinear operators with respect to a cone. Conditions for the positivity and absolute stability of a certain class of differential systems with delay are proposed. Several illustrative examples are given.

Invariant sets and comparison of dynamical systems

We propose a method for the construction and investigation of invariant sets of differential systems described by cone inequalities with the use of the operator of differentiation along the trajectories of the system. Well-known conditions for the positivity of linear and nonlinear differential systems with respect to typical classes of cones are generalized. A method for comparison and ordering is developed for a family of dynamical systems.

An Interior Point Cutting Plane Method for the Convex Feasibility Problem with Second-Order Cone Inequalities

The convex feasibility problem in general is a problem of finding a point in a convex set that contains a full dimensional ball and is contained in a compact convex set. We assume that the outer set is described by second-order cone inequalities and propose an analytic center cutting plane technique to solve this problem. We discuss primal and dual settings simultaneously. Two complexity results are reported: the complexity of restoration procedure and complexity of the overall algorithm. We prove that an approximate analytic center is updated after adding a second-order cone cut (SOCC) in O(1) Newton step, and that the analytic center cutting plane method (ACCPM) with SOCC is a fully polynomial algorithm.

Non-existence of solutions of semilinear parabolic differential inequalities in cones

Problems of the non-existence of global non-trivial solutions of semilinear parabolic differential inequalities and second-order systems in cones are investigated. The proofs are based on the method of test functions and do not use comparison theorems or the maximum principle.

The absence of global positive solutions of systems of semilinear elliptic inequalities in cones

Let be a cone in , . We establish conditions for the absence of global non-trivial non-negative solutions of semilinear elliptic inequalities and systems of inequalities of the form We find the critical exponent that divides the domains of existence of these solutions from those of their absence. We prove that in the limiting case there are no solutions. The method is to multiply the system by a special factor and integrate the inequalities thus obtained.

Liouville theorems for elliptic inequalities and applications

In this paper we prove nonexistence of positiveC2solutions for systems of semilinear elliptic inequalities, for polyharmonic semilinear inequalities in cones and, under better conditions on the nonlinearity, for bounded positive solutions of elliptic semilinear equations in half spaces. Using a blow-up argument, these results allow us to provea-prioribounds for a class of semilinear elliptic systems of equations in bounded domains.

Parabolic differential inequalities in cones

Theory of parabolic differential inequalities, flow-invariance of solutions and comparison theorems are discussed relative to a cone. In this paper we investigate the theory of parabolic differential inequalities in arbitrary cones. After discussing the fundamental results concerning parabolic inequalities in cones, we prove a result on flow-invariance which is then used to obtain a comparison theorem. This comparison result is useful in deriving upper and lower bounds on solutions of parabolic differential equations in terms of the solutions of ordinary differential equations. We treat the Dirichlet problem in this paper since its theory follows the general pattern of ordinary differential equations and requires less restrictive assumptions. The treatment of Neumann problem, on the other hand, demands stronger smoothness assumptions and depends heavily on strong maximum principle. The study of the corresponding results relative to Neumann problem is discussed elsewhere.

Hölder type inequalities in cones

Hölder type inequalities in cones