We develop a theory of quasi-Newton and least-change update methods for solving systems of nonlinear equations F (x) = 0. In this theory, no diierentiability conditions are necessary. Instead, we assume that F can be approximated, in a weak sense, by an aane function in a neighborhood of a solution. Using this assumption, we prove local and ideal convergence. Our theory can be applied to B-diie...

We investigate properties of measures in infinite dimensional spaces in terms of Poincaré inequalities. A Poincaré inequality states that the L2 variance of an admissible function is controlled by the homogeneous H1 norm. In the case of Loop spaces, it was observed by L. Gross [17] that the homogeneous H1 norm alone may not control the L2 norm and a potential term involving the end value of the...

In this paper we obtain almost sure convergence theorems for vectorvalued uniform amarts and C-sequences without assuming the Radon-Nikodym Property. Specifically, it is shown that if a limit exists in a weak sense for these martingale generalizations, then a.s. scalar and strong convergence follow. These results lead to some new versions of the Ito-Nisio theorem. Convergence results for random...

In this paper, we derive a necessary and sufficient condition on the parameters of the Hypergeometric distribution for weak convergence to a Normal limit. We establish a Berry-Esseen theorem for the Hypergeometric distribution solely under this necessary and sufficient condition. We further derive a nonuniform Berry-Esseen bound where the tails of the difference between the Hypergeometric and t...

GP studied preferences over menus (sets) of alternatives. This implicitly means that the decision maker chooses a menu and then selects an alternative from the menu. Consider a compact metric space Z of prizes. Let ∆ be the set of all Borel probability measures over Z. An element in ∆ represents an alternative for (implicit) choice. Assume that ∆ is endowed with the topology of weak convergence...

In this paper, we establish the relationships between spaces with a compact-countable weak-base and spaces with various compact-countable networks, and give two mapping theorems on spaces with compact-countable weakbases. Weak-bases and g-first countable spaces were introduced by A.V.Arhangel’skii [1]. Spaces with a point-countable weak-base were discussed in [5,6], and spaces with a locally co...

The convergence of multilocus systems under viability selection with constant fitnesses is investigated. Generations are discrete and nonoverlapping; the monoecious population mates at random. The number of multiallelic loci, the linkage map, dominance, and epistasis are arbitrary. It is proved that if epistasis or selection is sufficiently weak (and satisfies a certain nondegeneracy assumption...

begin{abstract} in this paper, we prove some strong and weak convergence of three step random iterative scheme with errors to common random fixed points of three asymptotically nonexpansive nonself random mappings in a real uniformly convex separable banach space. end{abstract}

We discuss three forms of convergence in distribution which are stronger than the normal weak convergence. They have the advantage that they are non-topological in nature and are inherited by discontinuous functions of the original random variables—clearly an improvement on ‘normal’ weak convergence. We give necessary and sufficient conditions for the three types of convergence and go on to giv...

‎In this paper‎, ‎first we study the weak and strong convergence of solutions to the‎ ‎following first order nonhomogeneous gradient system‎ ‎$$begin{cases}-x'(t)=nablaphi(x(t))+f(t), text{a.e. on} (0,infty)\‎‎x(0)=x_0in Hend{cases}$$ to a critical point of $phi$‎, ‎where‎ ‎$phi$ is a $C^1$ quasi-convex function on a real Hilbert space‎ ‎$H$ with ${rm Argmin}phineqvarnothing$ and $fin L^1(0...