Some equivalent definitions of high order Sobolev spaces on stratified groups and generalizations to metric spaces

Recently, in the article [LW], the authors use the notion of polynomials in metric spaces of homogeneous type (in the sense of Coifman-Weiss) to prove a relationship between high order Poincaré inequalities and representation formulas involving fractional integrals of high order, assuming only that is a doubling measure and that geodesics exist. Motivated by this and by recent work in [H], [FHK], [KS] and [FLW] about first order Sobolev spaces in metric spaces, we define Sobolev spaces of high order in such metric spaces . We prove that several definitions are equivalent if functions of polynomial type exist. In the case of stratified groups, where polynomials do exist, we show that our spaces are equivalent to the Sobolev spaces defined by Folland and Stein in [FS]. Our results also give some alternate definitions of Sobolev spaces in the classical Euclidean case. Received: 10 February 1999 / Published online: 1 February 2002     

Hydraulic Theory and Particle-Driven Gravity Currents

Hydraulic theory, as it has been applied to compositionally driven gravity flows, involves the single simplifying assumption that the pressure in the fluid is hydrostatic [1]. This assumption provides, as a consequence, a depth independent horizontal velocity field. This approach has led to a greatly increased understanding of many of the phenomena associated with these complex flows, including issues surrounding internal hydraulic jumps and energy loss [2]. Recently, investigations into flow and deposition of particles from particle-driven gravity currents have been carried out using an approach that employs the hydraulic theory that had proved so successful in the case of homogeneous flows [3]–[7]. Unfortunately, as we show here, there is a fundamental contradiction in adopting this simplifying assumption when particles drive the flow. This contradiction is essentially that one cannot have a hydrostatic pressure that arises from the presence of particles while at the same time maintaining a depth-independent horizontal velocity field, as was assumed in references [3]–[7].     

Some algebraic properties of the hyperbolic systems

Abstract The technique of quasi-symmetrizer has been applied to the well-posedness of the Cauchy problem for scalar operators [10], [13] and linear systems [5], [15], [4], and to the propagation of analitycity for solutions to semi-linear systems [6]. In all these works, it is assumed that the principal symbol depends only on the time variable. In this note we illustrate, in some special cases, a new property of the quasisymmetrizer which allows us to generalize the result in [6] to semi-linear systems with coefficients depending also on the space variables [21]. Such a property is closely connected with some interesting inequalities on the eigenvalues of a hyperbolic matrix. We expect that this technique applies also to other problems. Keywords: First order hyperbolic systems, Quasi-symmetrizer, Glaeser inequality     

Exact sequences for the galois group

When F C→ C is a monoidal functor between suitable symmetric closed categories and H is a finite commutative and cocbmmuta-ive Hopf algebra in C, we obtain, generalizing results of S. Qaeneppel and M. Beattie, two exact sequences involving the Picard, Galois and Brauer groups defined in [21], [11]. Finally we describe the relation between this exact sequences.     

一类涉及两个单形的不等式及应用

本文给出了一类涉及两个单形的三角不等式，作为其应用；可以导出文［１］的主要结果，特别地可以证明文［５］提出的一个猜想．     

Dispersion relations for rotational gravity water flows having two jumps in the vorticity distribution

We derive the dispersion relation for periodic traveling waves propagating at the surface of water with a layer of constant non-zero vorticity situated between two layers of irrotational flow. Due to the complicated nature of the dispersion relation – a fourth order algebraic equation with intricate coefficients – we also give an estimate of a very simple form involving only the levels at which the vorticity has jumps. Our formula generalizes a corresponding one from [5].     

An H‐matrix Type Preconditioner For Frictional Contact Problems

The bilateral or unilateral contact problem with Coulomb friction between two elastic bodies is considered [1]. An algorithm is introduced to solve the resulting finite element system by a non‐overlapping domain decomposition method [2, 3]. The global problem is transformed to a independant local problems posed in each bodie and a problem posed on the contact surface (the interface problem). The solution is obtained by using a successive approximation method, in each step of this algorithm we solve two intermediate problems the first with prescribed tangential pressure and the second with prescribed normal pressure [8]. Our preconditioner construction is based on the application of the H‐matrix technique [6, 7] together with the representation of the H^1/2 seminorm by a sum of partial seminorms [4]. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

含间断项的非线性积分、微分方程极解的存在性

在不涉及紧型条件和耗散型条件的情形下，证明了Ｂａｎａｃｈ空间中含间断项的非线性积分、微分方程极解的存在性，推广了文［１］、［２］和［３］中的结果．     

Affine invariant local convergence theorems for inexact Newton-like methods

Affine invariant sufficient conditions are given for two local convergence theorems involving inexact Newton-like methods. The first uses conditions on the first Fréchet-derivative whereas the second theorem employs hypotheses on the second. Radius of convergence as well as rate of convergence results are derived. Results involving superlinear convergence and known to be true for inexact Newton methods are extended here. Moreover, we show that under hypotheses on the second Fréchet-derivative our radius of convergence is larger than the corresponding one in [10]. This allows a wider choice for the initial guess. A numerical example is also provided to show that our radius of convergence is larger than the one in [10].     

Representation of perfect and local MV-algebras

We describe representation theorems for local and perfect MV-algebras in terms of ultraproducts involving the unit interval [0, 1]. Furthermore, we give a representation of local Abelian ℓ-groups with strong unit as quasi-constant functions on an ultraproduct of the reals. All the above theorems are proved to have a uniform version, depending only on the cardinality of the algebra to be embedded, as well as a definable construction in ZFC.     

Solving a class of multiplicative programming problems via c-programming

In this note we show that many classes of global optimization problems can be treated most satisfactorily by classical optimization theory and conventional algorithms. We focus on the class of problems involving the minimization of the product of several convex functions on a convex set which was studied recently by Kunoet al. [3]. It is shown that these problems are typical composite concave programming problems and thus can be handled elegantly by c-programming [4]–[8] and its techniques.     

Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian

In this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional Laplacians and the critical exponents. This work can be seen as a nonlocal analog of the results of Han (1991) [24] and Rey (1990) [35].     

Validation and Limits of Finite Inflatable Beam Elements

Although nowadays inflatable tubular beams are often used in the field of civil engineering, by now there are only few publications dealing with finite deformation inflatable beam elements, see e.g. [1], [2] and [3]. All formulations of inflatable beams have several assumptions in common, as constant cross sections throughout the deformation, a constant internal gas pressure and the negligence of circumferential stresses. These assumptions have to be validated either by experiments or numerical analysis. In the current contribution beam–like structures are investigated using a finite element shell or membrane formulation and featuring a volume dependent gas loading, see e.g. [5] and [4]. In general the formulation substitutes the internal gas pressure by an energetically equivalent volume dependent loading and thus enables to check for potential gas pressure changes during the deformation process of the inflated beam as a consequence of volume changes. Further local deformations as occurring in the vicinity of supports or almost single loads can be considered. In this paper the focus will be only on the initial assumption of the beam theory that the biaxial stress state is neglected. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On τ–covers using ideals and some of its consequences

In this paper we primarily introduce an ideal version of τ -covers studied in [18, 19]. We establish the inter-relationships between ?-τ -covers and ?-γ, ?-large [3, 5] and κ-covers[2, 7]. We also make some investigations involving the splittability and preservation properties. Our results extend the earlier results proved in [18, 19, 2, 7] and present a more general version with respect to ideals.     

Local convergence theorems for Newton methods

Affine invariant sufficient conditions are given for two local convergence theorems involving inexact Newton-like methods. The first uses conditions on the first Fréchet-derivative whereas the second theorem employs hypotheses on themth (m ≥ 2 an integer). Radius of convergence as well as rate of convergence results are derived. Results involving superlinear convergence and known to be true for inexact Newton methods are extended here. Moreover, we show that under hypotheses on the mth Fréchet-derivative our radius of convergence can sometimes be larger than the corresponding one in [10]. This allows a wider choice for the initial guess. A numerical example is also provided to show that our radius of convergence is larger than the one in [10].     

亚纯函数微分多项式f~kQ[f]+P[f]的零点分布

文[4]对简单形式的微分多项式fkf’+a的零点分布进行了讨论,文[1]对一般形式的微分多项式fkQ[f]+P[f]的零点分布进行了讨论.但由于极点给证明带来的困难,这些工作主要是对整函数来做的.本文证明了任一满足δ(∞,f)>k+2ΓQ+3ΓP+2/2k+2ΓQ+1的超越亚纯函数f,微分多项式fkQ[f]+P[f]在不含f,Q[f]极点和P[f]零、极点的可数个圆盘并集之外有无穷多个零点,其中k≥3Γp+2,而ΓQ,ΓP分别是f的微分多项式Q[f],P[f]的权.文[1]和[2,4,6]中的结论是本文结论的特殊情况.     

Shape-preserving bivariate polynomial approximation inC([−1,1]×[−1,1])

In this paper we construct bivariate polynomials attached to a bivariate function, that approximate with Jackson-type rate involving a bivariate Ditzian-Totik ω2-modulus of smoothness and preserve some natural kinds of bivariate monotonicity and convexity of function.The result extends that in univariate case-of D. Leviatan in [5-6], improves that in bivariate case of the author in [3] and in some special cases, that in bivariate case of G. Anastassiou in [1].     

A characterization of the complex paraboloid

In this paper we classify in a global way the umbilical affine definite surfaces in R⁴ with respect to the Nomizu-Vrancken affine normalization introduced in [NV]. We prove that an affine complete affine definite surface in R⁴ is umbilical if and only if it is affine equivalent to the complex paraboloid.     

Shape-preserving bivariate polynomial approximation inC([?1,1]×[?1,1])

In this paper we construct bivariate polynomials attached to a bivariate function, that approximate with Jackson-type rate involving a bivariate Ditzian-Totik ω ₂ ^ρ of smoothness and preserve some natural kinds of bivariate monotonicity and convexity of function. The result extends that in univariate case-of D. Leviatan in [5–6], improves that in bivariate case of the author in [3] and in some special cases, that in bivariate case of G. Anastassiou in [1].     

The circular chromatic numbers of planar digraphs

The circular chromatic number is a refinement of the chromatic number of a graph. It has been established in [3,6,7] that there exists planar graphs with circular chromatic number r if and only if r is a rational in the set {1} ∪ [2,4]. Recently, Mohar, in [1,2] has extended the concept of the circular chromatic number to digraphs and it is interesting to ask what the corresponding result is for digraphs. In this article, we shall prove the new result that there exist planar digraphs with circular chromatic number r if and only if r is a rational in the interval [1,4]. © 2006 Wiley Periodicals, Inc. J Graph Theory 55: 14–26, 2007