By examining asymptotic behavior of certain infinite basic ( -) hypergeometric sums at roots of unity (that is, at a ‘ -microscopic’ level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a -analogue of Ramanujan's formula of the two supercongruences valid for all primes , where denotes the truncation of the infinite sum at the -th place and stands for the quadratic character modulo 3.
We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the gradient estimate, the heat kernel estimate and the Hessian estimate to manifolds with integral Ricci lower bounds, without the non-collapsing conditions.
In this paper, we consider nonlinear equations involving the fractional p-Laplacian We prove a and obtain other key ingredients for carrying on the method of moving planes, such as a variant of the Hopf Lemma – which plays the role of the . Then we establish radial symmetry and monotonicity for positive solutions to semilinear equations involving the fractional p-Laplacian in a unit ball and in the whole space. We believe that the methods developed here can be applied to a variety of problems involving nonlinear nonlocal operators.
Recently, the second and third authors established the sharp Hardy–Adams inequalities of second order derivatives with the best constants on the hyperbolic space of dimension four ( , Adv. Math. (2017)). A key method used in dimension four in is that the following Hardy operator when and can be decomposed exactly as the product of fractional Laplacians. However, such a decomposition in dimension four is no longer possible in the case and and the situation here is thus considerably more difficult and complicated. Therefore, it remains open as whether a sharp Hardy–Adams inequality still holds on higher dimensional hyperbolic space for . The main purpose of this paper is to use a substantially new method of estimating the Hardy operator to establish the sharp Hardy–Adams inequalities on hyperbolic spaces for all even dimension and . As applications of such inequalities, we will improve substantially the known Adams inequalities on hyperbolic space in the literature and also strengthen the classical Adams' inequality and the Hardy inequality on Euclidean balls in any even dimension. The later inequality can be viewed as the borderline case of the sharp Hardy–Sobolev–Maz'ya inequalities for higher order derivatives in high dimensions obtained recently by the second and third authors .
Vindicating a sophisticated but non-rigorous physics approach called the cavity method, we establish a formula for the mutual information in statistical inference problems induced by random graphs and we show that the mutual information holds the key to understanding certain important phase transitions in random graph models. We work out several concrete applications of these general results. For instance, we pinpoint the exact condensation phase transition in the Potts antiferromagnet on the random graph, thereby improving prior approximate results (Contucci et al., 2013) . Further, we prove the conjecture from Krzakala et al. (2007) about the condensation phase transition in the random graph coloring problem for any number of colors. Moreover, we prove the conjecture on the information-theoretic threshold in the disassortative stochastic block model (Decelle et al., 2011) . Additionally, our general result implies the conjectured formula for the mutual information in Low-Density Generator Matrix codes (Montanari, 2005) .
The (p), dual curvature measure was introduced by Lutwak, Yang & Zhang in an attempt to unify the L-p Brunn Minkowski theory and the dual Brunn-Minkowski theory. The characterization problem for L-p dual curvature measure, called the L-p dual Minkowski problem, is a fundamental problem in this unifying theory. The L-p dual Minkowski problem contains the L-p Minkowski problem and the dual Minkowski problem, two major problems in modern convex geometry that remain open in general. In this paper, existence results on the L-p dual Minkowski problem in the weak sense will be provided. Moreover, existence and uniqueness of the solution in the smooth category will also be demonstrated. (C) 2018 Elsevier Inc. All rights reserved.
A new family of geometric Borel measures on the unit sphere is introduced. Special cases include the L-p surface area measures (which extend the classical surface area measure of Aleksandrov and Fenchel & Jessen) and L-p-integral curvature (which extends Alkesandrov's integral curvature) in the L-p, Brunn-Minkowski theory. It also includes the dual curvature measures (which are the duals of Federer's curvature measures) in the dual Brtum- Minkowski theory. This partially unifies the classical theory of mixed volumes and the newer theory of dual mixed volumes. (C) 2018 Elsevier Inc. All rights reserved.
In a recent paper Komornik et al. (2017) proved a long-conjectured formula for the Hausdorff dimension of the set of numbers having a unique expansion in the (non-integer) base , and showed that this Hausdorff dimension is continuous in . Unfortunately, their proof contained a gap which appears difficult to fix. This article gives a completely different proof of these results, using a more direct combinatorial approach.
We consider an asymptotic 1D (in space) rotation-Camassa–Holm (R-CH) model, which could be used to describe the propagation of long-crested shallow-water waves in the equatorial ocean regions with allowance for the weak Coriolis effect due to the Earth's rotation. This model equation has similar wave-breaking phenomena as the Camassa–Holm equation. It is analogous to the rotation-Green–Naghdi (R-GN) equations with the weak Earth's rotation effect, modeling the propagation of wave allowing large amplitude in shallow water. We provide here a rigorous justification showing that solutions of the R-GN equations tend to associated solution of the R-CH model equation in the Camassa–Holm regime with the small amplitude and the larger wavelength. Furthermore, we demonstrate that the R-GN model equations are locally well-posed in a Sobolev space by the refined energy estimates.
The dual curvature measure was introduced by Lutwak, Yang & Zhang in an attempt to unify the Brunn–Minkowski theory and the dual Brunn–Minkowski theory. The characterization problem for dual curvature measure, called the dual Minkowski problem, is a fundamental problem in this unifying theory. The dual Minkowski problem contains the Minkowski problem and the dual Minkowski problem, two major problems in modern convex geometry that remain open in general. In this paper, existence results on the dual Minkowski problem in the weak sense will be provided. Moreover, existence and uniqueness of the solution in the smooth category will also be demonstrated.
Given and , let Two conjectures in the coprime inhomogeneous Diophantine approximation state that for any irrational number and almost every , and that there exists , such that for all and , We prove the first conjecture and disprove the second one.
We describe how certain cyclotomic Nazarov–Wenzl algebras occur as endomorphism rings of projective modules in a parabolic version of BGG category of type D. Furthermore we study a family of subalgebras of these endomorphism rings which exhibit similar behaviour to the family of Brauer algebras even when they are not semisimple. The translation functors on this parabolic category are studied and proven to yield a categorification of a coideal subalgebra of the general linear Lie algebra. Finally this is put into the context of categorifying skew Howe duality for these subalgebras.
Let be a smooth, compact, oriented 4-manifold. Building upon work of Li-Liu, Ruberman, Nakamura and Konno, we consider a families version of Seiberg-Witten theory and obtain obstructions to the existence of certain group actions on by diffeomorphisms. The obstructions show that certain group actions on preserving the intersection form can not be lifted to an action of the same group on by diffeomorphisms. Using our obstructions, we construct numerous examples of group actions which can be realised continuously but can not be realised smoothly for any differentiable structure. For example, we construct compact simply-connected 4-manifolds and involutions such that can be realised by a continuous involution on or by a diffeomorphism, but not by an involutive diffeomorphism for any smooth structure on .
We use -tilting theory to give a description of the wall and chamber structure of a finite-dimensional algebra. We also study -generic paths in the wall and chamber structure of an algebra and show that every maximal green sequence in mod is induced by a -generic path.
We study bounded pseudoconvex domains in complex Euclidean space. We define an index associated to the boundary and show this new index is equivalent to the Diederich–Fornæss index defined in 1977. This connects the Diederich–Fornæss index to boundary conditions and refines the Levi pseudoconvexity. We also prove the -worm domain is of index . It is the first time that a precise non-trivial Diederich–Fornæss index in Euclidean spaces is obtained. This finding also indicates that the Diederich–Fornæss index is a continuum in , not a discrete set. The ideas of proof involve a new complex geometric analytic technique on the boundary and detailed estimates on differential equations.
We study quantum current algebra associated with the rational -matrix of and we give explicit formulae for the elements of its center at the critical level. Due to Etingof–Kazhdan's construction, the level vacuum module for the algebra possesses a quantum vertex algebra structure for any complex number . We prove that any module for the quantum vertex algebra is naturally equipped with a structure of restricted -module of level and vice versa.
An affine algebraic variety of dimension ≥2 is called if the subgroup generated by the one-parameter unipotent subgroups acts -transitively on for any . In we proved that any nondegenerate toric affine variety is flexible. In the present paper we show that one can find a subgroup of generated by a finite number of one-parameter unipotent subgroups which has the same transitivity property, provided the toric variety is smooth in codimension two. For with , three such subgroups suffice.
Inspired by Edward Witten's questions, we compute the mutual information associated with free fermions, and we deduce many results about entropies for chiral CFT's which are embedded into free fermions, and their extensions. Such relative entropies in CFT are here computed explicitly for the first time in a mathematical rigorous way, and Our results agree with previous computations by physicists based on heuristic arguments; in addition we uncover a surprising connection with the theory of subfactors, in particular by showing that a certain duality, which is argued to be true on physical grounds, is in fact violated if the global dimension of the conformal net is greater than 1.
The aims of this paper are to develop the LYZ ellipsoid and Petty projection body for log-concave functions, which correspond to the LYZ ellipsoid and Petty projection body for convex bodies when restricted to the subclass of characteristic functions. Moreover, the continuous, contravariant valuation on a subclass of log-concave functions is classified.
We define quasi-coherent parabolic sheaves with real weights on a fine saturated log analytic space, and explain how to interpret them as quasi-coherent sheaves of modules on its Kato–Nakayama space. This recovers the description as sheaves on root stacks of and for rational weights, but also includes the case of arbitrary real weights.