Showing papers in "Advances in Mathematics in 2018"

TL;DR: A new family of geometric Borel measures on the unit sphere is introduced in this article, which partially unifies the classical theory of mixed volumes and the newer theory of dual mixed volumes.

TL;DR: In this paper, a higher Holder regularity was shown for non-homogeneous solutions of the fractional p-Laplacian of order s, when p ≥ 2 and 0 s 1.

TL;DR: In this article, the authors studied the set-theoretic solutions of the Yang-Baxter equation (YBE) and showed the intimate relation between the notions of "a braided involutive group" and "a left brace".

TL;DR: In this paper, the authors considered nonlinear equations involving the fractional p-Laplacian and established radial symmetry and monotonicity for positive solutions to positive solutions in a unit ball and in the whole space.

TL;DR: A formula for the mutual information in statistical inference problems induced by random graphs is established and it is shown that the Mutual information holds the key to understanding certain important phase transitions in random graph models.

TL;DR: In this article, a nonlocal discrete equation driven by fractional powers of the discrete Laplacian on a mesh of size h > 0 ( − Δ h ) s u = f, for u, f : Z h → R, 0 s 1, is analyzed.

TL;DR: In this article, the authors derived precise late-time asymptotics for solutions to the wave equation on spherically symmetric, stationary and spacetimes including as special cases the Schwarzschild and Reissner-Nordstrom families of black holes.

TL;DR: The L p dual curvature measure was introduced by Lutwak, Yang and Zhang as discussed by the authors in an attempt to unify the L p Brunn-Minkowski theory and the dual Minkowski theories.

TL;DR: The decomposition space is a simplicial ∞-groupoid satisfying an exactness condition, weaker than the Segal condition, expressed in terms of active and inert maps in this article.

TL;DR: In this paper, the authors introduce a dimension spectrum motivated by the Assouad dimension, a familiar notion of dimension which, for a given metric space, returns the minimal exponent α ⩾ 0 such that for any pair of scales 0 r R, any ball of radius R may be covered by a constant times (R / r ) α balls of radius r.

TL;DR: In this paper, it was shown that the local invariant cycle map is an isomorphism if and only if the special fiber has palindromic cohomology, which is independent of the Hessenberg variety context.

TL;DR: In this article, a generalization of the coinvariant algebra R n to R n, k has been introduced, which is called the graded Frobenius series of R n, k and is the t = 0 specialization of the combinatorial side of the Delta Conjecture.

TL;DR: In this article, a cyclotomic Nazarov-Wenzl algebras occur as endomorphism rings of projective modules in a parabolic version of BGG category O of type D. The translation functors on this parabolic category O are studied and proven to yield a categorification of a coideal subalgebra of the general linear Lie algebra.

TL;DR: In this paper, a new toolbox for the analysis of the global behavior of stochastic discrete particle systems has been developed, based on the notion of the Schur generating function of a random discrete configuration.

TL;DR: The Mirror Theorems for toric Deligne-Mumford stacks and toric complete intersections were proved in this article, and the Mellin-Barnes method for analytic continuation of hypergeometric functions.

TL;DR: In this article, the authors considered the third-order PDEs with critical or sub-critical nonlinearities and proved that nonnegative classical solutions u to (1.1) and 1.11) are radially symmetric about some point x 0 ∈ R n and derive the explicit forms for u in the critical case.

TL;DR: In this article, the authors studied the local structure of the extremal process associated with the Discrete Gaussian Free Field (DGFF) in scaled-up (square-lattice) versions of bounded open planar domains subject to mild regularity conditions on the boundary.

TL;DR: In this article, it was shown that an endomorphism f of a projective variety X is polarized (resp. quasi-polarized) if f ⁎ H ∼ q H (linear equivalence) for some ample Cartier divisor H and integer q > 1.

TL;DR: Theorem 1.2 can be thought of as the nonlocal counterpart of the result obtained in [30] for the case when s = 1, and the regularity theory of the extension problem for the equation (0.1) was developed in this paper.

TL;DR: The notion of decomposition spaces was introduced in this article as a general framework for coalgebra constructions, such as the Faa di Bruno and Connes-Kreimer bialgebras.

TL;DR: In this article, a coproduct on affine Kac-Moody algebras is constructed and shown to be an algebra homomorphism, and a minimalistic presentation of the Yangian is given when the kac-moody algebra is symmetrizable.

TL;DR: In this article, Kontsevich and Soibelman defined Donaldson-Thomas invariants of a 3d Calabi-Yau category equipped with a stability condition, which are encapsulated in a single formal automorphism of the cluster variety, called the DT-transformation.

TL;DR: A notion of an integral hyperring scheme is defined and it is proved that $\Gamma(X,\mathcal{O}_X)\simeq R$ for any integral affine hyperrings scheme $X=Spec R$.

TL;DR: In this paper, the mutual information associated with free fermions and their extensions is computed explicitly for the first time in a mathematical rigorous way, and the results agree with previous computations by physicists based on heuristic arguments.

TL;DR: In this paper, the authors proposed a new method of bounding the size of an s-distance set in compact two-point homogeneous spaces via zonal spherical functions, which allows them to prove that the maximum size of a spherical two-distance sets in R n, n ≥ 7, is n ( n + 1 ) 2 with possible exceptions for some n = ( 2 k + 1) 2 − 3, k ∈ N.

TL;DR: Recently, the sharp Hardy-Adams inequalities of second order derivatives with the best constants on the hyperbolic space B 4 of dimension four were shown to hold for all even dimensions n and n ≥ 4.

TL;DR: In this paper, the authors proposed a topological generator of the Platonic solids with optimal covering properties and efficient navigation. But this generator is not suitable for the design of universal quantum gates.

TL;DR: In this paper, necessary and sufficient conditions for stable-like estimates of the heat kernel for jump type Dirichlet forms on metric measure spaces are given in terms of the volume growth function, jump kernel and a generalized capacity.

TL;DR: In this article, the authors provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts, including stable homotopy theory and quasi-coherent sheaves.

TL;DR: In this paper, a parallel equivariant conjecture for toric Calabi-Yau 4-folds is proposed and proved for smooth toric divisors and verified for more general toric Divisors in many examples.