Free Boundary Regularity in the Parabolic Fractional Obstacle Problem

The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the asset prices are driven by pure‐jump Lévy processes. In this paper we study the regularity of the free boundary. Our main result establishes that, when , the free boundary is a C^1,α graph in x and t near any regular free boundary point . Furthermore, we also prove that solutions u are C^{1 + s} in x and t near such points, with a precise expansion of the form (1) with , and . © 2018 Wiley Periodicals, Inc.     

Absence of Infinite Cluster for Critical Bernoulli Percolation on Slabs

We prove that for Bernoulli percolation on a graph , there is no infinite cluster at criticality, almost surely. The proof extends to finite‐range Bernoulli percolation models on ?² that are invariant under ‐rotation and reflection.© 2016 Wiley Periodicals, Inc.     

On the Location of Maxima of Solutions of Schrödinger's Equation

We prove an inequality with applications to solutions of the Schrödinger equation. There is a universal constant c > 0 such that if is simply connected, vanishes on the boundary ∂Ω, and |u| assumes a maximum in , then (1) It was conjectured by Pólya and Szegő (and proven, independently, by Makai and Hayman) that a membrane vibrating at frequency λ contains a disk of size . Our inequality implies a refined result: the point on the membrane that achieves the maximal amplitude is at distance from the boundary. We also give an extension to higher dimensions (generalizing results of Lieb and of Georgiev and Mukherjee): if u solves on with Dirichlet boundary conditions, then the ball B with radius centered at the point in which |u| assumes a maximum is almost fully contained in Ω in the sense that © 2018 Wiley Periodicals, Inc.     

Regularity for Shape Optimizers: The Nondegenerate Case

We consider minimizers of (1) where F is a function strictly increasing in each parameter, and is the k^th Dirichlet eigenvalue of Ω. Our main result is that the reduced boundary of the minimizer is composed of C^1,α graphs and exhausts the topological boundary except for a set of Hausdorff dimension at most n – 3. We also obtain a new regularity result for vector‐valued Bernoulli‐type free boundary problems.© 2018 Wiley Periodicals, Inc.     

Some properties of infinite factorials

With the Axiom of Choice , for any infinite cardinal but, without , we cannot conclude any relationship between and for an arbitrary infinite cardinal . In this paper, we give some properties of in the absence of and compare them to those of for an infinite cardinal . Among our results, we show that “ for any infinite cardinal and any natural number n” is provable in although “ for any infinite cardinal ” is not.     

Scattering and Localization Properties of Highly Oscillatory Potentials

We investigate scattering, localization, and dispersive time decay properties for the one‐dimensional Schrödinger equation with a rapidly oscillating and spatially localized potential , where is periodic and mean zero with respect to y. Such potentials model a microstructured medium. Homogenization theory fails to capture the correct low‐energy (k small) behavior of scattering quantities, e.g., the transmission coefficient as ? tends to zero. We derive an effective potential well such that is small, uniformly for as well as in any bounded subset of a suitable complex strip. Within such a bounded subset, the scaled limit of the transmission coefficient has a universal form, depending on a single parameter, which is computable from the effective potential. A consequence is that if ?, the scale of oscillation of the microstructure potential, is sufficiently small, then there is a pole of the transmission coefficient (and hence of the resolvent) in the upper half‐plane on the imaginary axis at a distance of order from . It follows that the Schrödinger operator has an bound state with negative energy situated a distance from the edge of the continuous spectrum. Finally, we use this detailed information to prove the local energy time decay estimate: where denotes the projection onto the continuous spectral part of . © 2013 Wiley Periodicals, Inc.     

Ramsey Results for Cycle Spectra

Let denote the set of lengths of cycles of a graph G of order n and let denote the complement of G. We show that if , then contains all odd ? with and all even ? with , where and denote the maximum odd and the maximum even integer in , respectively. From this we deduce that the set contains at least integers, which is sharp.     

Littlewood–Paley characterizations of higher‐order Sobolev spaces via averages on balls

In this article, the authors characterize higher‐order Sobolev spaces , with , and , or with , and , via the Lusin area function and the Littlewood–Paley ‐function in terms of ball averages, where denotes the maximal integer not greater than . Moreover, the authors also complement the above results in the endpoint cases of p via establishing some weak type estimates. These improve and develop the corresponding known results for Sobolev spaces with smoothness order .     

Enclosings of decompositions of complete multigraphs in 2‐factorizations

Let k, m, n, λ, and μ be positive integers. A decomposition of into edge‐disjoint subgraphs is said to be enclosed by a decomposition of into edge‐disjoint subgraphs if and, after a suitable labeling of the vertices in both graphs, is a subgraph of and is a subgraph of for all . In this paper, we continue the study of enclosings of given decompositions by decompositions that consist of spanning subgraphs. A decomposition of a graph is a 2‐factorization if each subgraph is 2‐regular and spanning, and is Hamiltonian if each subgraph is a Hamiltonian cycle. We give necessary and sufficient conditions for the existence of a 2‐factorization of that encloses a given decomposition of whenever and . We also give necessary and sufficient conditions for the existence of a Hamiltonian decomposition of that encloses a given decomposition of whenever and either or and , or , , and .     

Eigenvalues of Robin Laplacians in infinite sectors

For , let denote the infinite planar sector of opening 2α, and be the Laplacian in , , with the Robin boundary condition , where stands for the outer normal derivative and . The essential spectrum of does not depend on the angle α and equals , and the discrete spectrum is non‐empty if and only if . In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle α. In particular, there is just one discrete eigenvalue for . As α approaches 0, the number of discrete eigenvalues becomes arbitrary large and is minorated by with a suitable , and the nth eigenvalue of behaves as and admits a full asymptotic expansion in powers of α². The eigenfunctions are exponentially localized near the origin. The results are also applied to δ‐interactions on star graphs.     

Boundary Value Problems for Second‐Order Elliptic Operators Satisfying a Carleson Condition

Let Ω be a Lipschitz domain in , and be a second‐order elliptic operator in divergence form. We establish the solvability of the Dirichlet regularity problem with boundary data in and of the Neumann problem with data for the operator L on Lipschitz domains with small Lipschitz constant. We allow the coefficients of the operator L to be rough, obeying a certain Carleson condition with small norm. These results complete the results of Dindo?, Petermichl, and Pipher (2007), where the Dirichlet problem was considered under the same assumptions, and Dindo? and Rule (2010), where the regularity and Neumann problems were considered on two‐dimensional domains.© 2016 Wiley Periodicals, Inc.     

Characterizations of the boundedness of generalized fractional maximal functions and related operators in Orlicz spaces

Given and a Young function η, we consider the generalized fractional maximal operator defined by where the supremum is taken over every ball B contained in . In this article, we give necessary and sufficient Dini type conditions on the functions , and η such that is bounded from the Orlicz space into the Orlicz space . We also present a version of this result for open subsets of with finite measure. Both results generalize those contained in 6 and 14 when , respectively. As a consequence, we obtain a characterization of the functions involved in the boundedness of the higher order commutators of the fractional integral operator with BMO symbols. Moreover, we give sufficient conditions that guarantee the continuity in Orlicz spaces of a large class of fractional integral operators of convolution type with less regular kernels and their commutators, which are controlled by .     

Two Questions of Erdős on Hypergraphs above the Turán Threshold

For ordinary graphs it is known that any graph G with more edges than the Turán number of must contain several copies of , and a copy of , the complete graph on vertices with one missing edge. Erd?s asked if the same result is true for , the complete 3‐uniform hypergraph on s vertices. In this note, we show that for small values of n, the number of vertices in G, the answer is negative for . For the second property, that of containing a , we show that for the answer is negative for all large n as well, by proving that the Turán density of is greater than that of .     

On the homology groups of the Brauer complex for a triquadratic field extension

The homology groups , , and of the Brauer complex for a triquadratic field extension are studied. In particular, given , we find equivalent conditions for the image of D in to be zero. We consider as well the second divided power operation , and show that there are nonstandard elements with respect to γ₂. Further, a natural transformation , which turns out to be nondegenerate on the left, is defined. As an application we construct a field extension such that the cohomology group of the Brauer complex contains the images of prescribed elements of , provided these elements satisfy a certain cohomological condition. At the final part of the paper examples of triquadratic extensions with nontrivial are given. As a consequence we show that the homology group can be arbitrarily big.     

Propagation of Reactions in Inhomogeneous Media

Consider reaction‐diffusion equation u _t =Δ u + f (x,u ) with and general inhomogeneous ignition reaction f ≥ 0 vanishing at u = 0,1. Typical solutions 0 ≤ u ≤ 1 transition from 0 to 1 as time progresses, and we study them in the region where this transition occurs. Under fairly general qualitative hypotheses on f we show that in dimensions d ≤ 3, the Hausdorff distance of the superlevel sets {u ≥ ε } and {u ≥ 1‐ε} remains uniformly bounded in time for each ε ? (0,1). Thus, u remains uniformly in time close to the characteristic function of in the sense of Hausdorff distance of superlevel sets. We also show that each {u ≥ ε} expands with average speed (over any long enough time interval) between the two spreading speeds corresponding to any x ‐independent lower and upper bounds on f . On the other hand, these results turn out to be false in dimensions d ≥ 4, at least without further quantitative hypotheses on f . The proof for d ≤ 3 is based on showing that as the solution propagates, small values of u cannot escape far ahead of values close to 1. The proof for d ≥ 4 is via construction of a counterexample for which this fails. Such results were before known for d =1 but are new for general non‐periodic media in dimensions d ≥ 2 (some are also new for homogeneous and periodic media). They extend in a somewhat weaker sense to monostable, bistable, and mixed reaction types, as well as to transitions between general equilibria of the PDE and to solutions not necessarily satisfying . © 2016 Wiley Periodicals, Inc.     

Many symmetrically indivisible structures

A structure in a first‐order language is indivisible if for every colouring of its universe M in two colours, there is a monochromatic such that . Additionally, we say that is symmetrically indivisible if can be chosen to be symmetrically embedded in (that is, every automorphism of can be extended to an automorphism of ). In the following paper we give a general method for constructing new symmetrically indivisible structures out of existing ones. Using this method, we construct many non‐isomorphic symmetrically indivisible countable structures in given (elementary) classes and answer negatively the following question from 6 : Let be a symmetrically indivisible structure in a language . Let . Is symmetrically indivisible?     

The Brezis–Nirenberg problem for fractional elliptic operators

Let be a uniformly elliptic operator in divergence form in a bounded open subset Ω of . We study the effect of the operator on the existence and nonexistence of positive solutions of the nonlocal Brezis–Nirenberg problem where denotes the fractional power of with zero Dirichlet boundary values on , , and λ is a real parameter. By assuming for all and near some point , we prove existence theorems for any , where denotes the first Dirichlet eigenvalue of . Our existence result holds true for and in the interior case () and for and in the boundary case (). Nonexistence for star‐shaped domains is obtained for any .     

Infinite Speed of Propagation and Regularity of Solutions to the Fractional Porous Medium Equation in General Domains

We study the positivity and regularity of solutions to the fractional porous medium equations in for m > 1 and s ∈ (0,1), with Dirichlet boundary data u = 0 in and nonnegative initial condition . Our first result is a quantitative lower bound for solutions that holds for all positive times t > 0. As a consequence, we find a global Harnack principle stating that for any t > 0 solutions are comparable to d^s/m , where d is the distance to ?Ω. This is in sharp contrast with the local case s = 1, where the equation has finite speed of propagation. After this, we study the regularity of solutions. We prove that solutions are classical in the interior (C ^∞ in x and C ^1,α in t ) and establish a sharp regularity estimate up to the boundary. Our methods are quite general and can be applied to wider classes of nonlocal parabolic equations of the form in Ω, both in bounded and unbounded domains.© 2016 Wiley Periodicals, Inc.     

Smallest domination number and largest independence number of graphs and forests with given degree sequence

For a sequence d of nonnegative integers, let and be the sets of all graphs and forests with degree sequence d, respectively. Let , , , and where is the domination number and is the independence number of a graph G. Adapting results of Havel and Hakimi, Rao showed in 1979 that can be determined in polynomial time. We establish the existence of realizations with , and with and that have strong structural properties. This leads to an efficient algorithm to determine for every given degree sequence d with bounded entries as well as closed formulas for and .