We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas–Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas–Ni–Nirenberg type.
The classical Minkowski problem leads to the Minkowski problem and now to the Orlicz Minkowski problem. Existence is demonstrated for the even Orlicz Minkowski problem. A byproduct is a new approach to the solution of the classical Minkowski problem.
Minkowski's projection bodies have evolved into projection bodies and their asymmetric analogs. These all turn out to be part of a far larger class of Orlicz projection bodies. The analog of the classical Petty projection inequality is established for the new Orlicz projection bodies.
We introduce the idea of (and several variations) for a sequence of representations of groups . A central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability. This makes it possible to extend classical theorems of homological stability to a much broader variety of examples. Representation stability also provides a framework in which to find and to predict patterns, from classical representation theory (Littlewood–Richardson and Murnaghan rules, stability of Schur functors), to cohomology of groups (pure braid, Torelli and congruence groups), to Lie algebras and their homology, to the (equivariant) cohomology of flag and Schubert varieties, to combinatorics (the conjecture). The majority of this paper is devoted to exposing this phenomenon through examples. In doing this we obtain applications, theorems and conjectures. Beyond the discovery of new phenomena, the viewpoint of representation stability can be useful in solving problems outside the theory. In addition to the applications given in this paper, it is applied by Church–Ellenberg–Farb (in preparation) to counting problems in number theory and finite group theory. Representation stability is also used by Church (2012) to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds.
Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for affine surface areas are established.
We categorify Lusztig's — a version of the quantized enveloping algebra . Using a graphical calculus a 2-category is constructed whose split Grothendieck ring is isomorphic to the algebra . The indecomposable morphisms of this 2-category lift Lusztig's canonical basis, and the Homs between 1-morphisms are graded lifts of a semilinear form defined on . Graded lifts of various homomorphisms and antihomomorphisms of arise naturally in the context of our graphical calculus. For each positive integer a representation of is constructed using iterated flag varieties that categorifies the irreducible -dimensional representation of .
We give a general method based on dyadic Calderón–Zygmund theory to prove sharp one- and two-weight norm inequalities for some of the classical operators of harmonic analysis: the Hilbert and Riesz transforms, the Beurling–Ahlfors operator, the maximal singular integrals associated to these operators, the dyadic square function and the vector-valued maximal operator. In the one-weight case we prove the sharp dependence on the constant by finding the best value for the exponent such that For the Hilbert transform, the Riesz transforms and the Beurling–Ahlfors operator the sharp value of was found by Petermichl and Volberg (2007, 2008, 2002) ; their proofs used approximations by the dyadic Haar shift operators, Bellman function techniques, and two-weight norm inequalities. Our proofs again depend on dyadic approximation, but avoid Bellman functions and two-weight norm inequalities. We instead use a recent result due to A. Lerner (2010) to estimate the oscillation of dyadic operators. By applying this we get a straightforward proof of the sharp dependence on the constant for any operator that can be approximated by Haar shift operators. In particular, we provide a unified approach for the Hilbert and Riesz transforms, the Beurling–Ahlfors operator (and their corresponding maximal singular integrals), dyadic paraproducts and Haar multipliers. Furthermore, we completely solve the open problem of sharp dependence for the dyadic square functions and vector-valued Hardy–Littlewood maximal function. In the two-weight case we use the very same techniques to prove sharp results in the scale of bump conditions. For the singular integrals considered above, we show they map into , , if the pair satisfies where and are Orlicz functions. This condition is sharp. Furthermore, this condition characterizes (in the scale of these bump conditions) the corresponding two-weight norm inequality for the Hardy–Littlewood maximal operator and its dual: i.e., and . Muckenhoupt and Wheeden conjectured that these two inequalities for are sufficient for the Hilbert transform to be bounded from into . Thus, in the scale of bump conditions, we prove their conjecture. We prove similar, sharp two-weight results for the dyadic square function and the vector-valued maximal operator.
In this paper we develop a theory of slice regular functions on a real alternative algebra . Our approach is based on a well-known Fueter's construction. Two recent function theories can be included in our general theory: the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits to extend the range of these function theories and to obtain new results. In particular, we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in and we prove a Cauchy integral formula for slice functions of class .
This article gives a natural decomposition of the suspension of generalized moment-angle complexes or which arise as described below. The geometrical decomposition presented here provides structure for the stable homotopy type of these spaces including spaces appearing in work of Goresky–MacPherson concerning complements of certain subspace arrangements, as well as Davis–Januszkiewicz and Buchstaber–Panov concerning moment-angle complexes. Since the stable decompositions here are geometric, they provide corresponding homological decompositions for generalized moment-angle complexes for any homology theory.
The extremal index appears as a parameter in Extreme Value Laws for stochastic processes, characterising the clustering of extreme events. We apply this idea in a dynamical systems context to analyse the possible Extreme Value Laws for the stochastic process generated by observations taken along dynamical orbits with respect to various measures. We derive new, easily checkable, conditions which identify Extreme Value Laws with particular extremal indices. In the dynamical context we prove that the extremal index is associated with periodic behaviour. The analogy of these laws in the context of hitting time statistics, as studied in the authors’ previous works on this topic, is explained and exploited extensively allowing us to prove, for the first time, the existence of hitting time statistics for balls around periodic points. Moreover, for very well behaved systems (uniformly expanding) we completely characterise the extremal behaviour by proving that either we have an extremal index less than 1 at periodic points or equal to 1 at any other point. This theory then also applies directly to general stochastic processes, adding both useful tools to identify the extremal index and giving deeper insight into the periodic behaviour it suggests.
Let be a finite quiver without oriented cycles, let be the associated preprojective algebra, let be the associated Kac–Moody Lie algebra with Weyl group , and let be the positive part of . For each Weyl group element , a subcategory of was introduced by Buan, Iyama, Reiten and Scott. It is known that is a Frobenius category and that its stable category is a Calabi–Yau category of dimension two. We show that yields a cluster algebra structure on the coordinate ring of the unipotent group . Here is the pro-unipotent pro-group with Lie algebra the completion of . One can identify with a subalgebra of , the graded dual of the universal enveloping algebra of . Let be the dual of Lusztigʼs semicanonical basis of . We show that all cluster monomials of belong to , and that is a -basis of . Moreover, we show that the cluster algebra obtained from by formally inverting the generators of the coefficient ring is isomorphic to the algebra of regular functions on the unipotent cell of the Kac–Moody group with Lie algebra . We obtain a corresponding dual semicanonical basis of . As one application we obtain a basis for each acyclic cluster algebra, which contains all cluster monomials in a natural way.
We study the evolution of an -body weakly interacting system of Bosons. Our work forms an extension of our previous paper Grillakis, Machedon, and Margetis (2010) , in which we derived a second-order correction to a mean-field evolution law for coherent states in the presence of small interaction potential. Here, we remove the assumption of smallness of the interaction potential and prove global existence of solutions to the equation for the second-order correction. This implies an improved Fock-space estimate for our approximation of the -body state.
In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces and Triebel–Lizorkin spaces for all and , both in and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve on for all and . A metric measure space version of the above morphism property is also established.
We introduce the nuclear dimension of a -algebra; this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank. Our notion behaves well with respect to inductive limits, tensor products, hereditary subalgebras (hence ideals), quotients, and even extensions. It can be computed for many examples; in particular, it is finite for all UCT Kirchberg algebras. In fact, all classes of nuclear -algebras which have so far been successfully classified consist of examples with finite nuclear dimension, and it turns out that finite nuclear dimension implies many properties relevant for the classification program. Surprisingly, the concept is also linked to coarse geometry, since for a discrete metric space of bounded geometry the nuclear dimension of the associated uniform Roe algebra is dominated by the asymptotic dimension of the underlying space.
We prove the joints conjecture, showing that for any lines in , there are at most points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given lines in so that no lines lie in the same plane and so that each line intersects a set of points in at least points then the cardinality of the set of points is . Both our proofs are adaptations of Dvir's argument for the finite field Kakeya problem.
We introduce the algebraic entropy for endomorphisms of arbitrary abelian groups, appropriately modifying existing notions of entropy. The basic properties of the algebraic entropy are given, as well as various examples. The main result of this paper is the Addition Theorem showing that the algebraic entropy is additive in appropriate sense with respect to invariant subgroups. We give several applications of the Addition Theorem, among them the Uniqueness Theorem for the algebraic entropy in the category of all abelian groups and their endomorphisms. Furthermore, we point out the delicate connection of the algebraic entropy with the Mahler measure and Lehmer Problem in Number Theory.
We continue our study of tensor products in the operator system category. We define operator system quotients and exactness in this setting and refine the notion of nuclearity by studying operator systems that preserve various pairs of tensor products. One of our main goals is to relate these refinements of nuclearity to the Kirchberg conjecture. In particular, we prove that the Kirchberg conjecture is equivalent to the statement that every operator system that is (min,er)-nuclear is also (el,c)-nuclear. We show that operator system quotients are not always equal to the corresponding operator space quotients and then study exactness of various operator system tensor products for the operator system quotient. We prove that an operator system is exact for the min tensor product if and only if it is (min,el)-nuclear. We give many characterizations of operator systems that are (min,er)-nuclear, (el,c)-nuclear, (min,el)-nuclear and (el,max)-nuclear. These characterizations involve operator system analogues of various properties from the theory of C*-algebras and operator spaces, including the WEP and LLP.
bounds for solutions of a wide class of quasilinear degenerate elliptic inequalities are proved. As an outcome we deduce sharp Liouville theorems. Our investigation includes inequalities associated to -Laplacian and the mean curvature operators in Carnot groups setting. No hypotheses on the solutions at infinity are assumed. General results on the sign of solutions for quasilinear coercive/noncoercive inequalities are considered. Related applications to population biology and chemical reaction theory are also studied.
We develop the algebraic polynomial theory for “supertropical algebra,” as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of “ghost elements,” which also play the key role in our structure theory. Here, we work somewhat more generally over an ordered monoid, and develop a theory which contains the analogs of several basic theorems of classical commutative algebra. This structure enables one to develop a Zariski-type algebraic geometric approach to tropical geometry, viewing tropical varieties as sets of roots of (supertropical) polynomials, leading to an analog of the Hilbert Nullstellensatz. Particular attention is paid to factorization of polynomials. In one indeterminate, any polynomial can be factored into linear and quadratic factors, and although unique factorization may fail, there is a “preferred” factorization that is explained both geometrically and algebraically. The failure of unique factorization in several indeterminates is explained by geometric phenomena described in the paper.