Is the EMI model a QFT? An inquiry on the space of allowed entropy functions

The mutual information I(A, B) of pairs of spatially separated regions satisfies, for any d-dimensional CFT, a set of structural physical properties such as positivity, monotonicity, clustering, or Poincaré invariance, among others. If one imposes the extra requirement that I(A, B) is extensive as a function of its arguments (so that the tripartite information vanishes for any set of regions, I3(A, B, C ) ≡ 0), a closed geometric formula involving integrals over ∂A and ∂B can be obtained. We explore whether this “Extensive Mutual Information” model (EMI), which in fact describes a free fermion in d = 2, may similarly correspond to an actual CFT in general dimensions. Using the long-distance behavior of IEMI(A, B) we show that, if it did, it would necessarily include a free fermion, but also that additional operators would have to be present in the model. Remarkably, we find that IEMI(A, B) for two arbitrarily boosted spheres in general d exactly matches the result for the free fermion current conformal block $$ {G}_{\Delta =\left(d-1\right),J=1}^d $$ G ∆ = d − 1 , J = 1 d . On the other hand, a detailed analysis of the subleading contribution in the long-distance regime rules out the possibility that the EMI formula represents the mutual information of any actual CFT or even any limit of CFTs. These results make manifest the incompleteness of the set of known constraints required to describe the space of allowed entropy functions in QFT.

Stable s-minimal cones in ℝ3 are flat for s ~ 1

We prove that half spaces are the only stable nonlocal s-minimal cones in {\mathbb{R}^{3}}, for {s\in(0,1)} sufficiently close to 1. This is the first classification result of stable s-minimal cones in dimension higher than two. Its proof cannot rely on a compactness argument perturbing from {s=1}. In fact, our proof gives a quantifiable value for the required closeness of s to 1. We use the geometric formula for the second variation of the fractional s-perimeter, which involves a squared nonlocal second fundamental form, as well as the recent BV estimates for stable nonlocal minimal sets.

The geometric formulas of the Lewis’s law and Aboav-Weaire’s law in two dimensions based on ellipse packing

The two-dimensional (2D) Lewis’s law and Aboav-Weaire’s law are two simple formulas derived from empirical observations. Numerous attempts have been made to improve the empirical formulas. In this study, we simulated a series of Voronoi diagrams by randomly disordered the seed locations of a regular hexagonal 2D Voronoi diagram, and analyzed the cell topology based on ellipse packing. Then, we derived and verified the improved formulas for Lewis’s law and Aboav-Weaire’s law. Specifically, we found that the upper limit of the second moment of edge number is 3. In addition, we derived the geometric formula of the von Neumann-Mullins’s law based on the new formula of the Aboav-Weaire’s law. Our results suggested that the cell area, local neighbor relationship, and cell growth rate are closely linked to each other, and mainly shaped by the effect of deformation from circle to ellipse and less influenced by the global edge distribution.

The geometric formulas of the Lewis’s law and Aboav-Weaire’s law in two dimensions based on ellipse packing

The two-dimensional (2D) Lewis’s law and Aboav-Weaire’s law are two simple formulas derived from empirical observations. Numerous attempts have been made to improve the empirical formulas. In this study, we simulated a series of Voronoi diagrams by randomly disordered the seed locations of a regular hexagonal 2D Voronoi diagram, and analyzed the cell topology based on ellipse packing. Then, we derived and verified the improved formulas for Lewis’s law and Aboav-Weaire’s law. Specifically, we found that the upper limit of the second moment of edge number is 3. In addition, we derived the geometric formula of the von Neumann-Mullins’s law based on the new formula of the Aboav-Weaire’s law. Our results suggested that the cell area, local neighbor relationship, and cell growth rate are closely linked to each other, and mainly shaped by the effect of deformation from circle to ellipse and less influenced by the global edge distribution.

The geometric formulas of the Lewis’s law and Aboav-Weaire’s law in two dimensions based on ellipse packing

The two-dimensional (2D) Lewis’s law and Aboav-Weaire’s law are two simple formulas derived from empirical observations. Numerous attempts have been made to improve the empirical formulas. In this study, we simulated a series of Voronoi diagrams and analyzed the cell topology based on ellipse packing and then given the improved formulas. Specifically, we found that the upper limit of the second moment of edge number is 3. In addition, we derived the geometric formula of the von Neumann-Mullins’s law based on the improved formula of Aboav-Weaire’s law.

Classification of stable solutions for boundary value problems with nonlinear boundary conditions on Riemannian manifolds with nonnegative Ricci curvature

We present a geometric formula of Poincaré type, which is inspired by a classical work of Sternberg and Zumbrun, and we provide a classification result of stable solutions of linear elliptic problems with nonlinear Robin conditions on Riemannian manifolds with nonnegative Ricci curvature. The result obtained here is a refinement of a result recently established by Bandle, Mastrolia, Monticelli and Punzo.

Rigidity Results for Elliptic Boundary Value Problems: Stable Solutions for Quasilinear Equations with Neumann or Robin Boundary Conditions

We provide a general approach to the classification results of stable solutions of (possibly nonlinear) elliptic problems with Robin conditions. The method is based on a geometric formula of Poincaré type, which is inspired by a classical work of Sternberg and Zumbrun and which gives an accurate description of the curvatures of the level sets of the stable solutions. From this, we show that the stable solutions of a quasilinear problem with Neumann data are necessarily constant. As a byproduct of this, we obtain an alternative proof of a celebrated result of Casten and Holland, and Matano. In addition, we will obtain as a consequence a new proof of a result recently established by Bandle, Mastrolia, Monticelli, and Punzo.