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Aspects of topology: On dimension theory

Recently, Azam et al. introduced new spaces called the complex valued metric spaces and established the existence of fixed point theorems under the contraction condition. In this article, we extend and improve the condition of contraction of the results of Azam et al. and also apply the main result to the unique common solution of system of Urysohn integral equation. Mathematics Subject Classification (2000): 47H09; 47H10.

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Generalized common fixed point theorems in complex valued metric spaces and applications

Motivated by the work of Musiela and Zariphopoulou [Backward and forward utilities and the associated pricing systems: The case study of the binomial model, in Indifference Pricing, Princeton Unive...

http://www.cmap.polytechnique.fr/~mrad/SPDECPDUEVnek

An Exact Connection between Two Solvable SDEs and a Nonlinear Utility Stochastic PDE

Recently, the complex-valued metric spaces which are more general than the metric spaces were first introduced by Azam et al. (Numer. Funct. Anal. Optim. 32:243-253, 2011). They also established the existence of fixed point theorems under the contraction condition in these spaces. The aim of this paper is to introduce the concepts of a C-Cauchy sequence and C-complete in complex-valued metric spaces and establish the existence of common fixed point theorems in C-complete complex-valued metric spaces. Furthermore, we apply our result to obtain the existence theorem for a common solution of the Urysohn integral equations

/pdf/urysohn-integral-equations-approach-by-common-fixed-points-2kl6mafbn4.pdf

Urysohn integral equations approach by common fixed points in complex-valued metric spaces

In standard reinforcement learning (RL), a learning agent seeks to optimize the overall reward. However, many key aspects of a desired behavior are more naturally expressed as constraints. For instance, the designer may want to limit the use of unsafe actions, increase the diversity of trajectories to enable exploration, or approximate expert trajectories when rewards are sparse. In this paper, we propose an algorithmic scheme that can handle a wide class of constraints in RL tasks: specifically, any constraints that require expected values of some vector measurements (such as the use of an action) to lie in a convex set. This captures previously studied constraints (such as safety and proximity to an expert), but also enables new classes of constraints (such as diversity). Our approach comes with rigorous theoretical guarantees and only relies on the ability to approximately solve standard RL tasks. As a result, it can be easily adapted to work with any model-free or model-based RL. In our experiments, we show that it matches previous algorithms that enforce safety via constraints, but can also enforce new properties that these algorithms do not incorporate, such as diversity.

/pdf/reinforcement-learning-with-convex-constraints-3f7lxwi4sp.pdf

Reinforcement learning with convex constraints

Projections onto convex cones in Hilbert space

Fixed point theorems for partially outward mappings

$u$-mappings on trees.

In [1] David Bellamy gave an example of a tree-like continuum without the fixed point property. Another example of a tree-like continuum which admits a fixed point free mapping was given in [5] by L. G. Oversteegen and J. T. Rogers, Jr. We prove in this paper that a certain inverse limit of fans has the fixed point property. Certain possible methods for simplifying Bellamy's and Oversteegen and Roger's examples are eliminated by our theorem. A continuum is a nondegenerate compact connected metric space. A continuous function will be referred to as a map or mapping. A continuum X has the fixed point property provided that whenever f is a mapping of X into X, there is a point x in X such that f (x) = x. For each i > 1, let B1 be a 0-dimensional compact subset of El x (-1). Let z be the point of E2 with first coordinate 0 and second coordinate 1. For each a in B1 U {z), let La = {taIO 1, b' is a surjection, b1 I (0) = 0, and if a EcB , there is a c E B1 such that b +'(La) = Lc. Then Xhas the fixed point property. PROOF. Let d denote the metric on X and, for each n > 1, let rn be the projection mapping of X onto Tn. Let p be the point of X such that 7?n( p) = 0 for each n > 1. We assume that f: X -X is a fixed point free mapping and that E is a positive number for which d(x, f (x)) > E for each x E X. Let F = lim (F, b+ 'IF ) First we point out that if x E Xand there is a positive integer k such that 7Tk(x) e Fk, then, for each i > k, 7i(x) e Fi. Suppose that ( * ) is not the case. Then there is a point x in X and positive integers k and i with k < i such that 7Tk(x) e Fk but 7i(x) is in Fi. However, this implies that 7Tk(x) = b 7, (x) E Fk, which is a contradiction. Received by the editors September 10, 1982 and, in revised form, March 15, 1983. Presented to the 795th meeting of the Society, June 19, 1982 at Western Washington University, under the title Certain n-od like continua and the fixed point property. 1980 Mathematics Subject Classification. Primary 54H25; Secondary 54F20.

/pdf/a-fixed-point-theorem-for-inverse-limits-of-fans-5ajlxglzjj.pdf

A fixed point theorem for inverse limits of fans

We prove that products of surjective mappings from continua onto span zero continua are universal. It follows that products of span zero continua have the fixed point property.

/pdf/products-of-span-zero-continua-and-the-fixed-point-property-1pgm84e38h.pdf

M. M. Marsh

Papers

Projections onto convex cones in Hilbert space

Fixed point theorems for partially outward mappings

$u$-mappings on trees.

A fixed point theorem for inverse limits of fans

Products of span zero continua and the fixed point property