Necessary Conditions for Differentiability of Distance Functions

Necessary conditions for the Gâteaux differentiability of the distance function to a set are considered. A series of characterizing results is obtained.     

Simulation of the nonlinear vibration of a string using the Cellular Automata based on the reflection rule

In this study, the nonlinear dynamic responses of a string are simulated using the Cellular Automata method based on the reflection rule. In the case of nonlinear systems, the velocity of wave propagation is not constant and depends on the amplitude. A new treatment of the dynamic time step is proposed for the Cellular Automata method considering the effect of the propagation velocity. As numerical examples, first, the dynamic responses of a string with linear characteristic are simulated using the Cellular Automata method. A typical resonance curve can be obtained. Second, the dynamic responses of a string with nonlinear characteristic are simulated using the proposed method. Some characteristic types of vibration can be obtained. It is concluded that the linear and nonlinear dynamic responses of a string may be obtained by simulation using the Cellular Automata method.     

ON THE PRODUCT OF GATEAUX DIFFERENTIABILITY LOCALLY CONVEX SPACES

A locally convex space is said to be a Gateaux differentiability space (GDS)provided every continuous convex function defined on a nonempty convex open subset D of the space is densely Gateaux differentiable in D.This paper shows that the product of a GDS and a family of separable Frechet spaces is a GDS,and that the product of a GDS and an arbitrary locally convex space endowed with the weak topology is a GDS.     

Integral Representations,Differentiability Properties and Limits at Infinity for Beppo Levi Functions

The first aim in the present paper is to give an integral representation for Beppo Levi functions on R ⁿ. Our integral representation is an extension of Sobolev's integral representation given for infinitely differentiable functions with compact support. As applications, continuity and differentiability properties of Beppo Levi functions are studied.Our second aim in this paper is to study the existence of limits at infinity for Beppo Levi functions. We also consider the existence of fine-type limits at infinity with respect to Bessel capacities, which yields the radial limit result at infinity.     

f(R) theories of gravity in the Palatini approach matched with observations

We investigate the viability of f(R) theories in the framework of the Palatini approach as solutions to the problem of the observed accelerated expansion of the universe. Two physically motivated popular choices for f(R) are considered,: power law, f(R) = β R ⁿ , and logarithmic, f(R) = α ln R. Under the Palatini approach, both Lagrangians give rise to cosmological models comprising only standard matter and undergoing a present phase of accelerated expansion. We use the Hubble diagram of type Ia Supernovae and the data on the gas mass fraction in relaxed galaxy clusters to see whether these models are able to reproduce what is observed and to constrain their parameters. It turns out that they are indeed able to fit the data with values of the Hubble constant and of the matter density parameter in agreement with some model independent estimates, but the today deceleration parameter is higher than what is measured in the concordance ΛCDM model.     

Elliptic systems with nonlinearity q greater or equal than two with controlled growth. Global Hölder continuity of solutions to the Dirichlet problem

Hölder regularity up to the boundary of the solutions to the Dirichlet problem for second order elliptic systems with nonlinearity q>2 and with controlled growth is proved when n?q+2.     

Shape Hessian for generalized Oseen flow by differentiability of a minimax: A Lagrangian approach

The goal of this paper is to compute the shape Hessian for a generalized Oseen problem with nonhomogeneous Dirichlet boundary condition by the velocity method. The incompressibility will be treated by penalty approach. The structure of the shape gradient and shape Hessian with respect to the shape of the variable domain for a given cost functional are established by an application of the Lagrangian method with function space embedding technique. This work was supported by the National Natural Science Fund of China (No. 10371096) for ZM Gao and YC Ma.     

On a problem of Zygmund

It is proved that there exists an integrable function on [0, 1]² whose integral is nondifferentiable in each direction belonging to a set everywhere dense in [0, 2π] but is strongly differentiable. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 749–762, November, 1998.     

Continuity properties of the normal cone to the level sets of a quasiconvex function

Two main properties of the subgradient mapping of convex functions are transposed for quasiconvex ones. The continuity of the functionxf(x)^–1f(x) on the domain where it is defined is deduced from some continuity properties of the normal coneN to the level sets of the quasiconvex functionf. We also prove that, under a pseudoconvexity-type condition, the normal coneN(x) to the set {x:f(x)f(x)} can be expressed as the convex hull of the limits of type {N(x _n)}, where {x _n} is a sequence converging tox and contained in a dense subsetD. In particular, whenf is pseudoconvex,D can be taken equal to the set of points wheref is differentiable.This research was completed while the second author was on a sabbatical leave at the University of Montreal and was supported by a NSERC grant. It has its origin in the doctoral thesis of the first author (Ref. 1), prepared under the direction of the second author.The authors are grateful to an anonymous referee and C. Zalinescu for their helpful remarks on a previous version of this paper.     

低碳醇合成Raney Cu－Co催化剂的表征

本文采用ＸＲＤ、ＸＰＳ、及Ｈ_２—ＴＰＤ技术对不同Ｃｕ／Ｃｏ比的几个ＲａｎｅｙＣｕ－Ｃｏ催化剂进行了表征．并与反应醇选择性进行了关联．结果表明：ＲａｎｅｙＣｕ－Ｃｏ催化剂均由不同Ｃｕ／Ｃｏ比的两个Ｃｕ－Ｃｏ固溶体相组成，还原（Ｈ_２，５６３Ｋ，２ｈ）前，表面除有Ｃｕ￣０和Ｃｏ￣０外，尚有少量Ｃｕ￣（＋１）和显著量的Ｃｏ￣（＋２）存在，还原后，Ｃｕ￣（＋１）几乎消失，Ｃｏ￣（＋２）仍有一定量存在，在反应温度下（５６３Ｋ），ＣＯ和ＣＯ_２均能使表面Ｃｏ￣０部分氧化，且ＣＯ_２氧化能力大于ＣＯ，而合成气（Ｈ_２／ＣＯ＝２）表现为还原的性质。三种气氛对Ｃｕ￣０无明显影响。催化剂表面Ｃｕ／Ｃｏ比高于体相Ｃｕ／Ｃｏ比，表面Ｃｕ富集显著．ＲａｎｅｙＣｕ－Ｃｏ催化剂表面有四种吸氢中心：Ｃｕ（弱吸氢中心），高配位Ｃｏ（弱吸氢中心），低配位Ｃｏ（活化吸氢中心），及强吸氢中心；对不同催化剂，表面低配位Ｃｏ中心的比例与醇选择性有一致的变化规律。基于上述结果，就ＣＯ播入中心进行了讨论。