Remark on the global existence and large time asymptotics of solutions for the quadratic NLS

We study the global in time existence of small solutions to the nonlinear Schrödinger equation with quadratic interactions (0.1) We prove that if the initial data u₀ satisfy smallness conditions in the weighted Sobolev norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore, we prove the existence of the usual scattering states and find the large time asymptotics of the solutions.     

Canonical and monophonic convexities in hypergraphs

Known properties of “canonical connections” from database theory and of “closed sets” from statistics implicitly define a hypergraph convexity, here called canonical convexity (c-convexity), and provide an efficient algorithm to compute c-convex hulls. We characterize the class of hypergraphs in which c-convexity enjoys the Minkowski-Krein-Milman property. Moreover, we compare c-convexity with the natural extension to hypergraphs of monophonic convexity (or m-convexity), and prove that: (1) m-convexity is coarser than c-convexity, (2) m-convexity and c-convexity are equivalent in conformal hypergraphs, and (3) m-convex hulls can be computed in the same efficient way as c-convex hulls.     

A note on K-theory and triangulated derivators

In this paper we show an example of two differential graded algebras that have the same derivator K-theory but non-isomorphic Waldhausen K-theory. We also prove that Maltsiniotis?s comparison and localization conjectures for derivator K-theory cannot be simultaneously true.     

The topology of moduli spaces of group representations: The case of compact surface

Let G be a connected complex semisimple affine algebraic group, and let K be a maximal compact subgroup of G. Let X be a noncompact oriented surface. The main theorem of Florentino and Lawton (2009) [3] says that the moduli space of flat K-connections on X is a strong deformation retraction of the moduli space of flat G-connections on X. We prove that this statement fails whenever X is compact of genus at least two.     

Oscillation in the initial segment complexity of random reals

We study oscillation in the prefix-free complexity of initial segments of 1-random reals. For upward oscillations, we prove that _∑n∈ω²−g(n) diverges iff (^∃∞n)K(X?n)>n+g(n) for every 1-random X∈ω². For downward oscillations, we characterize the functions g such that (^∃∞n)K(X?n)<n+g(n) for almost every X∈ω². The proof of this result uses an improvement of Chaitin's counting theorem—we give a tight upper bound on the number of strings σ∈n² such that K(σ)<n+K(n)−m.The work on upward oscillations has applications to the K-degrees. Write XK_?Y to mean that K(X?n)?K(Y?n)+O(1). The induced structure is called the K-degrees. We prove that there are comparable () 1-random K-degrees. We also prove that every lower cone and some upper cones in the 1-random K-degrees have size continuum.Finally, we show that it is independent of ZFC, even assuming that the Continuum Hypothesis fails, whether all chains of 1-random K-degrees of size less than ℵ₀² have a lower bound in the 1-random K-degrees.     

Geodesic convexity of the relative entropy in reversible Markov chains

We consider finite-dimensional, time-continuous Markov chains satisfying the detailed balance condition as gradient systems with the relative entropy E as driving functional. The Riemannian metric is defined via its inverse matrix called the Onsager matrix K. We provide methods for establishing geodesic λ-convexity of the entropy and treat several examples including some discretizations of one-dimensional Fokker–Planck equations.     

The hyperbolic mean curvature flow

We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established.     

Convexity conditions and normal structure of Banach spaces

We prove that F-convexity, the property dual to P-convexity of Kottman, implies uniform normal structure. Moreover, in the presence of the WORTH property, normal structure follows from a weaker convexity condition than F-convexity. The latter result improves the known fact that every uniformly nonsquare space with the WORTH property has normal structure.     

?-adic properties of smallest parts functions

We prove explicit congruences modulo powers of arbitrary primes for three smallest parts functions: one for partitions, one for overpartitions, and one for partitions without repeated odd parts. The proofs depend on ?-adic properties of certain modular forms and mock modular forms of weight 3/2 with respect to the Hecke operators T(?2m).     

The upper bound of the $${{\rm L}_{\mu}^2}$$ spectrum

Let M be an n-dimensional complete non-compact Riemannian manifold, dμ = e ^h (x)dV(x) be the weighted measure and \triangle_m{\triangle_{\mu}} be the weighted Laplacian. In this article, we prove that when the m-dimensional Bakry–émery curvature is bounded from below by Ric _m ≥ −(m − 1)K, K ≥ 0, then the bottom of the L_m²{{\rm L}_{\mu}^2} spectrum λ₁(M) is bounded by

The Baum–Connes conjecture for free orthogonal quantum groups

We prove an analogue of the Baum–Connes conjecture for free orthogonal quantum groups. More precisely, we show that these quantum groups have a γ-element and that γ=1. It follows that free orthogonal quantum groups are K-amenable. We compute explicitly their K-theory and deduce in the unimodular case that the corresponding reduced C^?-algebras do not contain nontrivial idempotents.Our approach is based on the reformulation of the Baum–Connes conjecture by Meyer and Nest using the language of triangulated categories. An important ingredient is the theory of monoidal equivalence of compact quantum groups developed by Bichon, De Rijdt and Vaes. This allows us to study the problem in terms of the quantum group SUq(2). The crucial part of the argument is a detailed analysis of the equivariant Kasparov theory of the standard Podle? sphere.     

Conformal metrics with prescribed nonpositive Gaussian curvature on {\mathbb R}^2

In this paper, we consider the equation where is a nonpositive function in . A solution u is said to be complete if the conformal metric is complete in . Let Assuming only that , we prove that equation (0.1) possesses infinitely many complete solutions. If in addition, K is assumed to satisfy for some positive constant m, then is also necessary for equation (0.1) to have a complete solution with finite total curvature. We are also able to classify the solution set of equation (0.1) for a wider class of the curvature function K than those considered in [5, 6]. Received October 1, 1997 / Revised version August 10, 1999 / Published online April 6, 2000     

On the stability of Jensen’s functional equation on groups

In this paper we establish the stability of Jensen’s functional equation on some classes of groups. We prove that Jensen equation is stable on noncommutative groups such as metabelian groups and T (2, K), where K is an arbitrary commutative field with characteristic different from two. We also prove that any group A can be embedded into some group G such that the Jensen functional equation is stable on G.     

Convexities of metric spaces

We introduce two kinds of the notion of convexity of a metric space, called k-convexity and L-convexity, as generalizations of the CAT(0)-property and of the nonpositively curved property in the sense of Busemann, respectively. 2-uniformly convex Banach spaces as well as CAT(1)-spaces with small diameters satisfy both these convexities. Among several geometric and analytic results, we prove the solvability of the Dirichlet problem for maps into a wide class of metric spaces.      

Dissimilarity maps on trees and the representation theory of SL<Subscript><Emphasis Type="Italic">m</Emphasis></Subscript>(ℂ)

We prove that m-dissimilarity vectors of weighted trees are points on the tropical Grassmannian, as conjecture by Cools in response to a question of Sturmfels and Pachter. We accomplish this by relating m-dissimilarity vectors to the representation theory of SL _m (ℂ).     

Convergence of the Dirichlet solutions of the very fast diffusion equation

For any −1<m<0, positive functions f, g and u₀≥0, we prove that under some mild conditions on f, g and u₀ as R→∞ the solution u^R of the Dirichlet problem u_t=(u^m/m)_xx in (−R,R)×(0,∞), u(R,t)=(f(t)|m|R)^1/m, u(−R,t)=(g(t)|m|R)^1/m for all t>0, u(x,0)=u₀(x) in (−R,R), converges uniformly on every compact subset of R×(0,T) to the solution of the equation u_t=(u^m/m)_xx in R×(0,T), u(x,0)=u₀(x) in R, which satisfies some mass loss formula on (0,T) where T is the maximal time such that the solution u is positive. We also prove that the solution constructed is equal to the solution constructed in Hui (2007) [15] using approximation by solutions of the corresponding Neumann problem in bounded cylindrical domains.     

Extensions of regular mappings and the ?ojasiewicz exponent at infinity

Let F:V→Cm be a regular mapping, where V⊂Cn is an algebraic set of positive dimension and m?n?2, and let L_∞(F) be the ?ojasiewicz exponent at infinity of F. We prove that F has a polynomial extension G:Cn→Cm such L_∞(G)=L_∞(F). Moreover, we give an estimate of the degree of the extension G. Additionally, we prove that if then for any β∈Q, β?L_∞(F), the mapping F has a polynomial extension G with L_∞(G)=β. We also give an estimate of the degree of this extension.     

An extrapolation theorem for nonlinear approximation and its applications

We prove an extrapolation theorem for the nonlinear m-term approximation with respect to a system of functions satisfying very mild conditions. This theorem allows us to prove endpoint L^p-L^q estimates in nonlinear approximation. As a consequence, some known endpoint estimates can be deduced directly and some new estimates are also obtained. Finally, applications of these new estimates are given to spherical m-widths and m-term approximation of the weighted Besov classes.     

$${\cal K}$$-Convexity1 in $$\Re^{n}$$

We generalize the concept of K-convexity to an n-dimensional Euclidean space. The resulting concept of -convexity is useful in addressing production and inventory problems where there are individual product setup costs and/or joint setup costs. We derive some basic properties of -convex functions. We conclude the paper with some suggestions for future research. Support from Columbia University and University of Texas at Dallas is gratefully acknowledged. Helpful comments from Qi Feng are appreciated.     

The implicit function theorem in a non-Archimedean setting

In this paper, the inverse function theorem and the implicit function theorem in a non-Archimedean setting will be discussed. We denote by N any non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order; and we study the properties of locally uniformly differentiable functions from Nⁿ to N^m. Then we use that concept of local uniform differentiability to formulate and prove the inverse function theorem for functions from Nⁿ to Nⁿ and the implicit function theorem for functions from Nⁿ to N^m with m<n.