For origin-symmetric convex bodies (i.e., the unit balls of finite dimensional Banach spaces) it is conjectured that there exist a family of inequalities each of which is stronger than the classical Brunn–Minkowski inequality and a family of inequalities each of which is stronger than the classical Minkowski mixed-volume inequality. It is shown that these two families of inequalities are “equivalent” in that once either of these inequalities is established, the other must follow as a consequence. All of the conjectured inequalities are established for plane convex bodies.
Consider Hermitian or symmetric random matrices with independent entries, where the distribution of the matrix element is given by the probability measure with zero expectation and with variance . We assume that the variances satisfy the normalization condition for all and that there is a positive constant such that . We further assume that the probability distributions have a uniform subexponential decay. We prove that the Stieltjes transform of the empirical eigenvalue distribution of is given by the Wigner semicircle law uniformly up to the edges of the spectrum with an error of order where is the imaginary part of the spectral parameter in the Stieltjes transform. There are three corollaries to this strong local semicircle law: (1) Rigidity of eigenvalues: If denotes the of the -th eigenvalue under the semicircle law ordered in increasing order, then the -th eigenvalue is close to in the sense that for some positive constants , for large enough. (2) The proof of (Dyson, 1962 ) which states that the time scale of the Dyson Brownian motion to reach local equilibrium is of order up to logarithmic corrections. (3) The edge universality holds in the sense that the probability distributions of the largest (and the smallest) eigenvalues of two generalized Wigner ensembles are the same in the large limit provided that the second moments of the two ensembles are identical.
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.
The present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable Kac–Moody algebras. The resulting quantum symmetric pairs are right coideal subalgebras of quantized enveloping algebras. They give rise to triangular decompositions, including a quantum analog of the Iwasawa decomposition, and they can be written explicitly in terms of generators and relations. Moreover, their centers and their specializations are determined. The constructions follow G. Letzter's theory of quantum symmetric pairs for semisimple Lie algebras. The main additional ingredient is the classification of involutive automorphisms of the second kind of symmetrizable Kac–Moody algebras due to Kac and Wang. The resulting theory comprises various classes of examples which have previously appeared in the literature, such as -Onsager algebras and the twisted -Yangians introduced by Molev, Ragoucy, and Sorba.
In this paper, we study the birational geometry of the Hilbert scheme P-2[n] of n-points on P-2. We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the models in terms of moduli spaces of Bridgeland semi-stable objects. We construct these moduli spaces as moduli spaces of quiver representations using G.I.T. and thus show that they are projective. There is a precise correspondence between wall-crossings in the Bridgeland stability manifold and wall-crossings between Mori cones. For n <= 9, we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions. (C). 2012 Elsevier Inc. All rights reserved.
In this paper, we consider the following Dirichlet problem for poly-harmonic operators on a half space : First, under some very mild growth conditions, we show that problem is equivalent to the integral equation where is the Greenʼs function on the half space. Then, by combining the method of moving planes in integral forms with some new ideas, we prove that there is no positive solution for integral equation in both subcritical and critical cases. This partially solves an open problem posed by Reichel and Weth (2009) . We also prove non-existence of weak solutions for problem .
In this paper, we solve a long-standing problem on Bernoulli convolutions. In particular, we show that the Bernoulli convolution with contraction rate admits a spectrum if and only if is the reciprocal of an even integer.
In the first part of this paper, we establish the global existence of solutions of the liquid crystal (gradient) flow for the well-known Oseen–Frank model. The liquid crystal flow is a prototype of equations from the Ericksen–Leslie system in the hydrodynamic theory and generalizes the heat flow for harmonic maps into the 2-sphere. The Ericksen–Leslie system is a system of the Navier–Stokes equations coupled with the liquid crystal flow. In the second part of this paper, we also prove the global existence of solutions of the Ericksen–Leslie system for a general Oseen–Frank model in .
Let be the -dimensional Heisenberg group, be the homogeneous dimension of , , and be the homogeneous norm of . Then we prove the following sharp Moser–Trudinger inequality on (Theorem 1.6): there exists a positive constant such that for any pair satisfying , there holds The constant is best possible in the sense that the supremum is infinite if . Here is any positive number, and . Our result extends the sharp Moser–Trudinger inequality by Cohn and Lu (2001) on domains of finite measure on and sharpens the recent result of Cohn et al. (2012) where such an inequality was studied for the subcritical case . We carry out a completely different and much simpler argument than that in Cohn et al. (2012) to conclude the critical case. Our method avoids using the rearrangement argument which is not available in an optimal way on the Heisenberg group and can be used in more general settings such as Riemanian manifolds, appropriate metric spaces, etc. As applications, we establish the existence and multiplicity of nontrivial nonnegative solutions to certain nonuniformly subelliptic equations of -Laplacian type on the Heisenberg group (Theorems 1.8, 1.9, 1.10 and 1.11): with nonlinear terms of maximal exponential growth as . In particular, when , the existence of a nontrivial solution is also given.
A Steiner type formula for continuous translation invariant Minkowski valuations is established. In combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations, this new Steiner type formula is used to obtain a family of Brunn–Minkowski type inequalities for rigid motion intertwining Minkowski valuations.
We produce Brill–Noether general graphs in every genus, confirming a conjecture of Baker and giving a new proof of the Brill–Noether Theorem, due to Griffiths and Harris, over any algebraically closed field.
Let be a Poisson manifold with Poisson bivector field . We say that is -Poisson if the map intersects the zero section transversally on a codimension one submanifold . This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of in the neighborhood of and using symplectic techniques define topological invariants which determine the structure up to isomorphism. We also investigate a variant of de Rham theory for these manifolds and its connection with Poisson cohomology.
In the first part of this paper, we establish the global existence of solutions of the liquid crystal (gradient) flow for the well-known Oseen-Frank model. The liquid crystal flow is a prototype of equations from the Ericksen-Leslie system in the hydrodynamic theory and generalizes the heat flow for harmonic maps into the 2-sphere. The Ericksen-Leslie system is a system of the Navier-Stokes equations coupled with the liquid crystal flow. In the second part of this paper, we also prove the global existence of solutions of the Ericksen-Leslie system for a general Oseen-Frank model in R-2. Crown Copyright (C) 2012 Published by Elsevier Inc. All rights reserved.
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.
Inspired by a previous work of Nakajima, we consider perverse sheaves over acyclic graded quiver varieties and study the Fourier–Sato–Deligne transform from a representation theoretic point of view. We obtain deformed monoidal categorifications of acyclic quantum cluster algebras with specific coefficients. In particular, the (quantum) positivity conjecture is verified whenever there is an acyclic seed in the (quantum) cluster algebra. In the second part of the paper, we introduce new quantizations and show that all quantum cluster monomials in our setting belong to the dual canonical basis of the corresponding quantum unipotent subgroup. This result generalizes previous work by Lampe and by Hernandez–Leclerc from the Kronecker and Dynkin quiver case to the acyclic case. The Fourier transform part of this paper provides crucial input for the second author's paper where he constructs bases of acyclic quantum cluster algebras with arbitrary coefficients and quantization.
In this paper we study a general class of “quasilinear non-local equations” depending on the gradient which arises from tug-of-war games. We establish a regularity theory for these equations (the kind of regularity depending on the assumptions on the kernel), and we construct different non-local approximations of the -Laplacian.
It is proven here that the diameter of the -dimensional associahedron is when is greater than 9. Two maximally distant vertices of this polytope are explicitly described as triangulations of a convex polygon, and their distance is obtained using combinatorial arguments. This settles two problems posed about twenty-five years ago by Daniel Sleator, Robert Tarjan, and William Thurston.
We consider the most general Dunkl shift operator with the following properties: (i) is of first order in the shift operator and involves reflections; (ii) preserves the space of polynomials of a given degree; (iii) is potentially self-adjoint. We show that under these conditions, the operator has eigenfunctions which coincide with the Bannai–Ito polynomials. We construct a polynomial basis which is lower-triangular and two-diagonal with respect to the action of the operator . This allows to express the BI polynomials explicitly. We also present an anti-commutator AW(3) algebra corresponding to this operator. From the representations of this algebra, we derive the structure and recurrence relations of the BI polynomials. We introduce new orthogonal polynomials – referred to as the complementary BI polynomials – as an alternative limit of the Askey–Wilson polynomials. These complementary BI polynomials lead to a new explicit expression for the BI polynomials in terms of the ordinary Wilson polynomials.