Iterative solving of variational inclusions under Wijsman perturbations

We prove that the metric regularity of set-valued mappings is stable under some Wijsman-type perturbations. Then, we solve a variational inclusion viewed as a limit-problem using assumptions on a sequence of associated problems. Finally, we apply our results to classical methods for solving variational inclusions.      

A Glivenko-Cantelli theorem for empirical measures of independent but non-identically distributed random variables

The bounded-dual-Lipschitz and Prohorov distances from the ‘empirical measure’ to the ‘average measure’ of independent random variables converges to zero almost surely if the sequence of average measures is tight. Three examples are also given.     

An assessment of two approaches to variable metric methods

Two recent suggestions in the field of variable metric methods for function minimization are reviewed: the self-scaling method, first introduced by Oren and Luenberger, and the method of Biggs. The two proposals are considered both from a theoretical and computational aspect. They are compared with methods which use correction formulae from the Broyden one-parameter family, in particular the BFGS formula and the Fletcher switching strategy.     

Variable metric methods for minimizing a class of nondifferentiable functions

We develop a class of methods for minimizing a nondifferentiable function which is the maximum of a finite number of smooth functions. The methods proceed by solving iteratively quadratic programming problems to generate search directions. For efficiency the matrices in the quadratic programming problems are suggested to be updated in a variable metric way. By doing so, the methods possess many attractive features of variable metric methods and can be viewed as their natural extension to the nondifferentiable case. To avoid the difficulties of an exact line search, a practical stepsize procedure is also introduced. Under mild assumptions the resulting method converge globally.Research supported by National Science Foundation under grant number ENG 7903881.     

On relations between weak approximation properties and their inheritances to subspaces

It is shown that for the separable dual X^∗ of a Banach space X, if X^∗ has the weak approximation property, then X^∗ has the metric weak approximation property. We introduce the properties W^∗D and MW^∗D for Banach spaces. Suppose that M is a closed subspace of a Banach space X such that M^⊥ is complemented in the dual space X^∗, where for all m∈M}. Then it is shown that if a Banach space X has the weak approximation property and W^∗D (respectively, metric weak approximation property and MW^∗D), then M has the weak approximation property (respectively, bounded weak approximation property).     

Pointwise Lipschitz functions on metric spaces

For a metric space X, we study the space D^∞(X) of bounded functions on X whose pointwise Lipschitz constant is uniformly bounded. D^∞(X) is compared with the space LIP^∞(X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare D^∞(X) with the Newtonian-Sobolev space N1,∞(X). In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D^∞(X)=N1,∞(X).     

New general convergence theory for iterative processes and its applications to Newton–Kantorovich type theorems

Let T:D⊂X→X$T:D\subset X\to X$ be an iteration function in a complete metric space X$X$. In this paper we present some new general complete convergence theorems for the Picard iteration _xn+1=T_xn$x_{n+1}=T{x}_n$ with order of convergence at least r≥1$r\ge 1$. Each of these theorems contains a priori and a posteriori error estimates as well as some other estimates. A central role in the new theory is played by the notions of a function of initial conditions   of T$T$ and a convergence function   of T$T$. We study the convergence of the Picard iteration associated to T$T$ with respect to a function of initial conditions E:D→X$E:D\to X$. The initial conditions in our convergence results utilize only information at the starting point _x0$x_0$. More precisely, the initial conditions are given in the form E(_x0)∈J$E\left(x_0\right)\in J$, where J$J$ is an interval on _R+${\mathbb{R}}_{+}$ containing 0. The new convergence theory is applied to the Newton iteration in Banach spaces. We establish three complete ω$\omega$-versions of the famous semilocal Newton–Kantorovich theorem as well as a complete version of the famous semilocal α$\alpha$-theorem of Smale for analytic functions.     

形式Laurent级数域上交错Oppenheim展开的研究

该文介绍了形式Laurent级数域上交错Oppenheim展开的算法,得到了该展开中数字的强(弱)大数定理、中心极限定理和重对数率,并且研究了这些级数部分和的逼近的度.     

On a generalization of distance sets

A subset X in the d-dimensional Euclidean space is called a k-distance set if there are exactly k distinct distances between two distinct points in X and a subset X is called a locally k-distance set if for any point x in X, there are at most k distinct distances between x and other points in X.Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the cardinalities of k-distance sets on a sphere in 1977. In the same way, we are able to give the same bound for locally k-distance sets on a sphere. In the first part of this paper, we prove that if X is a locally k-distance set attaining the Fisher type upper bound, then determining a weight function w, (X,w) is a tight weighted spherical 2k-design. This result implies that locally k-distance sets attaining the Fisher type upper bound are k-distance sets. In the second part, we give a new absolute bound for the cardinalities of k-distance sets on a sphere. This upper bound is useful for k-distance sets for which the linear programming bound is not applicable. In the third part, we discuss about locally two-distance sets in Euclidean spaces. We give an upper bound for the cardinalities of locally two-distance sets in Euclidean spaces. Moreover, we prove that the existence of a spherical two-distance set in (d−1)-space which attains the Fisher type upper bound is equivalent to the existence of a locally two-distance set but not a two-distance set in d-space with more than d(d+1)/2 points. We also classify optimal (largest possible) locally two-distance sets for dimensions less than eight. In addition, we determine the maximum cardinalities of locally two-distance sets on a sphere for dimensions less than forty.     

Metric and ultrametric spaces of resistances

Consider an electrical circuit, each edge e of which is an isotropic conductor with a monomial conductivity function . In this formula, y_e is the potential difference and current in e, while μ_e is the resistance of e; furthermore, r and s are two strictly positive real parameters common for all edges. In particular, the case r=s=1 corresponds to the standard Ohm’s law.In 1987, Gvishiani and Gurvich [A.D. Gvishiani, V.A. Gurvich, Metric and ultrametric spaces of resistances, in: Communications of the Moscow Mathematical Society, Russian Math. Surveys 42 (6 (258)) (1987) 235-236] proved that, for every two nodes a,b of the circuit, the effective resistance μ_a,b is well-defined and for every three nodes a,b,c the inequality holds. It obviously implies the standard triangle inequality μ_a,b≤μ_a,c+μ_c,b whenever s≥r. For the case s=r=1, these results were rediscovered in the 1990s. Now, after 23 years, I venture to reproduce the proof of the original result for the following reasons:

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It is more general than just the case r=s=1 and one can get several interesting metric and ultrametric spaces playing with parameters r and s. In particular, (i) the effective Ohm resistance, (ii) the length of a shortest path, (iii) the inverse width of a bottleneck path, and (iv) the inverse capacity (maximum flow per unit time) between any pair of terminals a and b provide four examples of the resistance distances μ_a,b that can be obtained from the above model by the following limit transitions: (i) r(t)=s(t)≡1, (ii) r(t)=s(t)→∞, (iii) r(t)≡1,s(t)→∞, and (iv) r(t)→0,s(t)≡1, as t→∞. In all four cases the limits μ_a,b=lim_t→∞μ_a,b(t) exist for all pairs a,b and the metric inequality μ_a,b≤μ_a,c+μ_c,b holds for all triplets a,b,c, since s(t)≥r(t) for any sufficiently large t. Moreover, the stronger ultrametric inequality μ_a,b≤max(μ_a,c,μ_c,b) holds for all triplets a,b,c in examples (iii) and (iv), since in these two cases s(t)/r(t)→∞, as t→∞.

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Communications of the Moscow Math. Soc. in Russ. Math. Surveys were (and still are) strictly limited to two pages; the present paper is much more detailed.Although a translation in English of the Russ. Math. Surveys is available, it is not free in the web and not that easy to find.

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The last but not least: priority.