Lipschitz-Volume rigidity on limit spaces with Ricci curvature bounded from below

We prove a Lipschitz-Volume rigidity theorem for the non-collapsed Gromov-Hausdorff limits of manifolds with Ricci curvature bounded from below. This is a counterpart of the Lipschitz-Volume rigidity in Alexandrov geometry

On the choice of the exchange-rate regimes

This paper utilizes recent research developments in portfolio balance theory and in real exchange-rate instability to synthesize, update, and test the optimum currency area (OCA) theory. Four hypotheses, capturing the central features of the OCA theory, are advanced and tested in a multinomial-logit setup. the empirical results establish the linkage between a fixed rate and financial integration, trade integration, plus inflation convergence. the Mundell-Fleming ranking of regime is refuted in a fundamental way. these findings are applied to a discussion of European monetary integration, in relation to both its final objective and its intermediate procedure.Foreign exchange rates

Tensor Hierarchy and Generalized Cartan Calculus in SL(3)×\timesSL(2) Exceptional Field Theory

We construct exceptional field theory for the duality group SL(3)$\times$SL(2). The theory is defined on a space with 8 `external' coordinates and 6 `internal' coordinates in the $(3,2)$ fundamental representation, leading to a 14-dimensional generalized spacetime. The bosonic theory is uniquely determined by gauge invariance under generalized external and internal diffeomorphisms. The latter invariance can be made manifest by introducing higher form gauge fields and a so-called tensor hierarchy, which we systematically develop to much higher degree than in previous studies. To this end we introduce a novel Cartan-like tensor calculus based on a covariant nil-potent differential, generalizing the exterior derivative of conventional differential geometry. The theory encodes the full $D=11$ or type IIB supergravity, respectively.Comment: 49 page

A Monte Carlo exploration of threefold base geometries for 4d F-theory vacua

We use Monte Carlo methods to explore the set of toric threefold bases that support elliptic Calabi-Yau fourfolds for F-theory compactifications to four dimensions, and study the distribution of geometrically non-Higgsable gauge groups, matter, and quiver structure. We estimate the number of distinct threefold bases in the connected set studied to be $\sim { 10^{48}}$. The distribution of bases peaks around $h^{1, 1}\sim 82$. All bases encountered after "thermalization" have some geometric non-Higgsable structure. We find that the number of non-Higgsable gauge group factors grows roughly linearly in $h^{1,1}$ of the threefold base. Typical bases have $\sim 6$ isolated gauge factors as well as several larger connected clusters of gauge factors with jointly charged matter. Approximately 76% of the bases sampled contain connected two-factor gauge group products of the form SU(3)$\times$SU(2), which may act as the non-Abelian part of the standard model gauge group. SU(3)$\times$SU(2) is the third most common connected two-factor product group, following SU(2)$\times$SU(2) and $G_2\times$SU(2), which arise more frequently.Comment: 38 pages, 22 figure

On the Fractional Landis Conjecture

In this paper we study a Landis-type conjecture for fractional Schr\"odinger equations of fractional power $s\in(0,1)$ with potentials. We discuss both the cases of differentiable and non-differentiable potentials. On the one hand, it turns out for \emph{differentiable} potentials with some a priori bounds, if a solution decays at a rate $e^{-|x|^{1+}}$, then this solution is trivial. On the other hand, for $s\in(1/4,1)$ and merely bounded \emph{non-differentiable} potentials, if a solution decays at a rate $e^{-|x|^\alpha}$ with $\alpha>4s/(4s-1)$, then this solution must again be trivial. Remark that when $s\to 1$, $4s/(4s-1)\to 4/3$ which is the optimal exponent for the standard Laplacian. For the case of non-differential potentials and $s\in(1/4,1)$, we also derive a quantitative estimate mimicking the classical result by Bourgain and Kenig.Comment: comments are welcom