Tannaka duality for proper Lie groupoids

By replacing the category of smooth vector bundles of finite rank over a manifold with the category of what we call smooth Euclidean fields, which is a proper enlargement of the former, and by considering smooth actions of Lie groupoids on smooth Euclidean fields, we are able to prove a Tannaka duality theorem for proper Lie groupoids. The notion of smooth Euclidean field we introduce here is the smooth, finite dimensional analogue of the usual notion of continuous Hilbert field.     

Very smooth points of spaces of operators

In this paper we study very smooth points of Banach spaces with special emphasis on spaces of operators. We show that when the space of compact operators is anM-ideal in the space of bounded operators, a very smooth operatorT attains its norm at a unique vectorx (up to a constant multiple) andT(x) is a very smooth point of the range space. We show that if for every equivalent norm on a Banach space, the dual unit ball has a very smooth point then the space has the Radon-Nikodym property. We give an example of a smooth Banach space without any very smooth points.     

Biholomorphic mappings between bounded domains in C n

In this paper we prove a global boundary behavior result on biholomorphic mappings between smooth bounded domains in complex space of any dimension. As a consquence, we have that the inverse of a biholomorphic mapping between smooth bounded domains is smooth up to the boundary if the mapping is smooth up to the boundary.     

一类光滑凸规划的牛顿法

给出了一个求解一类光滑凸规划的算法,利用光滑精确乘子罚函数把一个光滑凸规划的极小化问题化为一个紧集上强凸函数的极小化问题,然后在给定的紧集上用牛顿法对这个强凸函数进行极小化．     

Geometricity for derived categories of algebraic stacks

We prove that the dg category of perfect complexes on a smooth, proper Deligne–Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of triangulated categories, this means that the derived category of perfect complexes embeds as an admissible subcategory into the bounded derived category of coherent sheaves on a smooth, projective variety. The same holds for a smooth, projective, tame Artin stack over an arbitrary field.     

The Volume Blow-Up and Characteristic Classes for Transverse, Type-Changing, Pseudo-Riemannian Metrics

Consider a smooth manifold with smooth (0, 2)-tensor which changes bilinear type on a hypersurface. We show that there are natural tensors on this hypersurface which control the smooth extension of sectional, Ricci, and scalar curvature. This enables us to adapt the classical characteristic class construction to a large collection of manifolds with such singular pseudo-Riemannian metrics.     

Uniform Convergence of Discrete Curvatures from Nets of Curvature Lines

We study discrete curvatures computed from nets of curvature lines on a given smooth surface and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.     

The Maximum Number of Zeros of Functions with Parameters and Application to Differential Equations

In this paper, we first study the problem of finding the maximum number of zeros of functions with parameters and then apply the results obtained to smooth or piecewise smooth planar autonomous systems and scalar periodic equations to study the number of limit cycles or periodic solutions, improving some fundamental results both on the maximum number of limit cycles bifurcating from an elementary focus of order $k$ or a limit cycle of multiplicity $k$, or from a period annulus, and on the maximum number of periodic solutions for scalar periodic smooth or piecewise smooth equations as well.     

Higher Intersection Theory on Algebraic Stacks: II

This is the second part of our work on the intersection theory of algebraic stacks. The main results here are the following. We provide an intersection pairing for all smooth Artin stacks (locally of finite type over a field) which we show reduces to the known intersection pairing on the Chow groups of smooth Deligne–Mumford stacks of finite type over a field as well as on the Chow groups of quotient stacks associated to actions of linear algebraic groups on smooth quasi-projective schemes modulo torsion. The former involves also showing the existence of Adams operations on the rational étale K-theory of all smooth Deligne–Mumford stacks of finite type over a field. In addition, we show that our definition of the higher Chow groups is intrinsic to the stack for all smooth stacks and also stacks of finite type over the given field. Next we establish the existence of Chern classes and Chern character for Artin stacks with values in our Chow groups and extend these to higher Chern classes and a higher Chern character for perfect complexes on an algebraic stack, taking values in cohomology theories of algebraic stacks that are defined with respect to complexes of sheaves on a big smooth site. As a by-product of our techniques we also provide an extension of higher intersection theory to all schemes locally of finite type over a field. As the higher cycle complex, by itself, is a bit difficult to handle, the stronger results like contravariance for arbitrary maps between smooth stacks and the intersection pairing for smooth stacks are established by comparison with motivic cohomology.     

光滑区域上二维无黏性无热传导Boussinesq方程组与三维轴对称不可压Euler方程组的指数增长全局光滑解

研究二维无黏性无热传导Boussinesq方程组和三维轴对称不可压Euler方程组光滑解的增长情况,找各种区域使其上的方程组有快增长的解。对Boussinesq方程组,通过选取初始温度和速度的一个分量,可以把方程去耦为两部分。从关于涡量的部分求出涡量、速度场和使结论成立的区域,从关于温度的部分,可见温度的高阶导的增长仅依赖于速度场的一个分量。通过适当选取该分量,得到温度高阶导有指数增长的全局光滑解。对轴对称Euler方程组做类似的处理,适当选取速度场的径向分量,可把方程组去耦,最终得到一类光滑区域,在其上方程组有指数增长全局光滑解。该研究把Chae、Constantin、Wu对一个二维锥形区域上无黏性无热传导Boussinesq方程的结果,推广到一类光滑区域上, 并把他们的方法应用到三维轴对称不可压Euler方程组, 得到了类似的结果。     

Hochschild Homology of Singular Algebras

One of the main properties of Hochschild homology of the algebra of smooth functions on a smooth manifold is its local character. In this paper, we consider subalgebras of smooth functions which are significant for singular spaces such that simplicial complexes or cones over smooth manifolds. We compute their Hochschild homology and investigate the local character. Our computations show that, in opposition with the smooth case, the (local part of the) Hochschild homology is not always isomorphic to the corresponding de Rham complex of differential forms. The method we use is a slight modification of the localization procedure introduced in by Sullivan.     

一个模拟趋化现象的广义双曲-抛物系统的光滑解的全局分析

考虑一个模拟趋化现象的广义双曲-抛物系统的Cauchy问题,当动能函数为非线性函数且初始值具有小的L~2能量但其H~2能量可能任意大时,得到了全局光滑解的存在性和渐近行为.这些结果推广了以前的关于动能函数为线性函数或初始值具有小的H~2能量情形下的相关结果,首次获得了关于全局光滑大解方面的结果.这些结果的证明基于构造一个新的非负凸熵和做精细的能量估计.     

Generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on domains

In this paper, we consider a non‐smooth atomic decomposition by using a smooth atomic decomposition. Applying the non‐smooth atomic decomposition, a local means characterization and a quarkonical decomposition, we obtain a pointwise multiplier and a trace operator for generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on the whole space. We also develop the theory of those spaces on domains. We consider an extension operator and a trace operator on the upper half space and on compact oriented Riemannian manifolds.     

Surface in R^3 with Prescribed Gauss Curvature

Assume that a smooth function K(x, y) vanishes on a simple closed smooth curve \Gamma and dK|_\Gamma \ne 0. Then the result of the present paper shows that there is a sufficiently smooth surface z=z(x, y) defined in a neighbourhood of \Gamma with curvature K.     

Fourier analysis on trapezoids with curved sides

It is well known that smooth periodic functions can be expanded into Fourier series and can be approximated by trigonometric polynomials. The purpose of this paper is to do Fourier analysis for smooth functions on planar domains. A planar domain can often be divided into some trapezoids with curved sides, so first we do the Fourier analysis for smooth functions on trapezoids with curved sides. We will show that any smooth function on a trapezoid with curved sides can be expanded into Fourier sine series with simple polynomial factors, and so it can be well approximated by a combination of sine polynomials and simple polynomials. Then we consider the Fourier analysis on the global domain. Finally, we extend these results to the three-dimensional case.     

On homomorphisms between generalized polygons

We consider homomorphisms between abstract, topological, and smooth generalized polygons. It is shown that a continuous homomorphism is either injective or locally constant. A continuous homomorphism between smooth generalized polygons is always a smooth embedding. We apply this result to isoparametric submanifolds.Dedicated to Prof. Dr. H. R. Salzmann on the occasion of his 65th anniversary     

Smooth exact penalty functions: a general approach

In the article, we present a new perspective on the method of smooth exact penalty functions that is becoming more and more popular tool for solving constrained optimization problems. In particular, our approach to smooth exact penalty functions allows one to apply previously unused tools (namely, parametric optimization) to the study of these functions. We give a new simple proof of local exactness of smooth penalty functions that significantly generalizes all similar results existing in the literature. We also provide new necessary and sufficient conditions for a smooth penalty function to be globally exact.     

Discontinuous symplectic capacities

We show that the spherical capacity is discontinuous on a smooth family of ellipsoidal shells. Moreover, we prove that the shell capacity is discontinuous on a family of open sets with smooth connected boundaries.     

Solutions of singular elliptic equations via the oblique derivative problem

We study a class of second elliptic equations whose highest order coefficients vanish everywhere on the boundary. Under suitable conditions on the lower order coefficients, Langlais proved in 1985 that such equations have unique smooth solutions up to the boundary provided the data are smooth enough. Our goal here is to prove some Schauder estimates for these equations and to obtain results even in Lipschitz domains. In addition, we show that bounded solutions of such problems are as smooth as the data allow. A key step is to observe that smooth solutions must satisfy an oblique derivative boundary condition.     

Fenchel Problem of Level Sets

The Fenchel problem of level sets in the smooth case is solved by deducing a new and nice geometric necessary and sufficient condition for the existence of a smooth convex function with the level sets of a given smooth pseudoconvex function. The main theorem is based on a general differential geometric tool, the space of paths defined on smooth manifolds. This approach provides a complete geometric characterization of a new subclass of pseudoconvex functions originating from analytical mechanics and a new view on convexlike and generalized convexlike mappings in image analysis. This paper is dedicated to the memory of Guido Stampacchia. The author thanks K. Balla for assistance in solving the system of differential equations of Example 2.1 and for helpful remarks. This research was supported in part by the Hungarian Scientific Research Fund, Grants OTKA-T043276 and OTKA-T043241, and by CNR, Rome, Italy. An erratum to this article is available at .