Remarks on Levy's reflection axiom

Adding higher types to set theory differs from adding inaccessible cardinals, in that higher type arguments apply to all sets rather than just ordinary ones. Levy's reflection axiom is justified, by considering the principle that we can pretend that the universe is a set, together with methods of Gaifman [8]. We reprove some results of Gaifman, and some facts about Levy's reflection axiom, including the fact that adding higher types yields no new theorems about sets. Some remarks on standard models are made. An obvious strengthening of Levy's axiom to higher types is considered, which implies the existence of indescribable cardinals. Other remarks about larger cardinals are made; some questions of Gloede [9] are settled. Finally we argue that the evidence for V = L is strong, and that CH is certainly true. MSC: 03E30, 03E55.     

Optimal control in nonlinear infinite-dimensional systems with nondifferentiability of two types

We consider the optimal control problem for systems described by nonlinear equations of elliptic type. If the nonlinear term in the equation is smooth and the nonlinearity increases at a comparatively low rate of growth, then necessary conditions for optimality can be obtained by well-known methods. For small values of the nonlinearity exponent in the smooth case, we propose to approximate the state operator by a certain differentiable operator. We show that the solution of the approximate problem obtained by standard methods ensures that the optimality criterion for the initial problem is close to its minimal value. For sufficiently large values of the nonlinearity exponent, the dependence of the state function on the control is nondifferentiable even under smoothness conditions for the operator. But this dependence becomes differentiable in a certain extended sense, which is sufficient for obtaining necessary conditions for optimality. Finally, if there is no smoothness and no restrictions are imposed on the nonlinearity exponent of the equation, then a smooth approximation of the state operator is possible. Next, we obtain necessary conditions for optimality of the approximate problem using the notion of extended differentiability of the solution of the equation approximated with respect to the control, and then we show that the optimal control of the approximated extremum problem minimizes the original functional with arbitrary accuracy.     

Measure of Non-Radon–Nikodym Property and Differentiability of Convex Functions on Banach Spaces

We introduce and discuss measure of non-Radon–Nikodym property to investigate differentiability of continuous convex functions on Banach spaces. Using this new concept, we establish some results about Asplund spaces and generic Fréchet differentiability of continuous convex functions on non-Asplund spaces.Mathematics Subject Classifications (2000) 46B22, 49J50, 49J53.     

Henstock引理,导函数的可积性,积分原函数的可导性

对经典实分析非绝对积分理论中的Henstock引理的本质特征进行了讨论,指出:Henstock引理的本质是刻划了导函数的可积性和积分原函数的可导性问题.     

Implicit Function Theorems for Nondifferentiable Mappings

Sufficient conditions are given for a mapping to be γ-G inverse differentiable. Constrained implicit function theorems for γ-G inverse differentiable mappings are obtained, where the constraint is taken to be either a closed convex cone or a closed subset. A theorem without assuming the γ-G inverse differentiability in a finite-dimensional space is also presented. Communicated by F. A. Potra The author thanks the referees for valuable suggestions concerning the presentation of this paper. He also thanks Professor J. R. L. Webb and Dr. M. French for help.     

高维非线性系统边值问题的奇摄动

利用渐近方法和对角化技巧研究了伴有边界摄动的高维非线性系统边值问题的奇摄动，在适当的假设下，证得摄动问题解的存在并导出其解关于ε的高阶近似．     

Antiproximinal convex bounded sets in the space c0(Γ) equipped with the day norm

We construct a convex smooth antiproximinal set in any infinite-dimensional space c ₀(Γ) equipped with the Day norm; moreover, the distance function to the set is Gâteaux differentiable at each point of the complement.     

Griffiths' singularities in diluted ising models on the Cayley tree

The Griffiths singularities are fully exhibited for a class of diluted ferromagnetic Ising models defined on the Cayley tree (Bethe lattice). For the deterministic model the Lee-Yang circle theorem is explicitly proven for the magnetization at the origin and it is shown that, in the thermodynamic limit, the Lee-Yang singularities become dense in the entire unit circle for the whole ferromagnetic phase. Smoothness (infinite differentiability) of the quenched magnetizationm at the origin with respect to the external magnetic field is also proven for convenient choices of temperature and disorder. From our analysis we also conclude that the existence of metastable states is impossible for the random models under consideration.     

Rigidity of higher elliptic genera

We prove some general rigidity theorems for both elliptic and higher elliptic genera under a natural condition on the first equivariant Pontrjagin classes. We also obtain the vanishing of some higher elliptic genera.Both authors are supported in part by NFS     

Investigation of the self association of higher fatty alcohols and acids on chromatographic sorbents

Summary Three different sorbents, one with high activity (silica gel), the second with medium activity (cellulose powder) and the third having low activity (chromatographic paper) were compared in respect of their usefulness for the evaluation of the self association of selected higher fatty alcohols and acids. The obtained results are correlated with IR spectroscopic data and theoretical considerations.