The Metric Topology

2021 ◽  
pp. 103-190
Author(s):  
Adel N. Boules
Keyword(s):  
Continuous Functions ◽  
Function Spaces ◽  
Space Filling ◽  
Space Filling Curves ◽  
Local Compactness ◽  
Metric Properties ◽  
Differentiable Functions ◽  
Metric Topology ◽  
Nowhere Differentiable Functions ◽  
And Function

The chapter is an extensive account of the metric topology and is a prerequisite for all the subsequent chapters. The leading sections develop the basic metric properties such as closure and interior, continuity and equivalent metrics, separation properties, product spaces, and countability axioms. This is followed by a detailed study of completeness, compactness, local compactness, and function spaces. Chapter applications include contraction mappings, continuous nowhere differentiable functions, space-filling curves, closed convex subsets of ?n, and a number of approximation results. The chapter concludes with a detailed section on orthogonal polynomials and Fourier series of continuous functions, which, together with section 3.7, provides an excellent background for Hilbert spaces. The study of sequence and function spaces in this chapter leads up gradually into Banach spaces.

Local Compactness in Set Valued Function Spaces

1976 ◽  
Vol 19 (2) ◽  
pp. 193-198 ◽  
Author(s):  
Saroop K. Kaul
Keyword(s):  
Uniform Convergence ◽  
Uniform Space ◽  
Continuous Functions ◽  
Function Spaces ◽  
Range Space ◽  
Locally Compact ◽  
Local Compactness

Recently Hunsaker and Naimpally [2] have proved: The pointwise closure of an equicontinuous family of point compact relations from a compact T2-space to a locally compact uniform space is locally compact in the topology of uniform convergence. This is a generalization of the same result of Fuller [1] for single valued continuous functions.For a range space which is locally compact normal and uniform theorem B below is an improvement on the result of Hunsaker and Naimpally quoted above [see Remark 3 at the end of this paper].

scholarly journals Multiplicativity of the uniform norm and independent functions

1990 ◽  
Vol 42 (1) ◽  
pp. 153-155
Author(s):  
A. Guyan Robertson
Keyword(s):  
Recent Work ◽  
Lebesgue Measure ◽  
Uniform Norm ◽  
Continuous Functions ◽  
Space Filling ◽  
Stochastic Independence ◽  
Space Filling Curve ◽  
Space Filling Curves ◽  
Independent Functions ◽  
Filling Curve

It has long been known that there is a close connection between stochastic independence of continuous functions on an interval and space-filling curves [9]. In fact any two nonconstant continuous functions on [0, 1] which are independent relative to Lebesgue measure are the coordinate functions of a space filling curve. (The results of Steinhaus [9] have apparently been overlooked in more recent work in this area [3, 5, 6].)

Kolmogorov Superpositions

2013 ◽  
pp. 219-245 ◽  
Author(s):  
David Sprecher
Keyword(s):  
Continuous Function ◽  
Computational Algorithm ◽  
Continuous Functions ◽  
Computational Algorithms ◽  
Space Filling ◽  
Space Filling Curves ◽  
Suitable Space

Kolmogorov’s superpositions enable the representation of every real-valued continuous function f defined on the Euclidean n-cube in the form , with continuous functions that compute f, and fixed continuous functions dependent only on n. The functions specify space-filling curves that determine characteristics that are not suitable for efficient computational algorithms. Reversing the process, we specify suitable space-filling curves that enable new functions that give a computational algorithm better adaptable to applications. Detailed numerical constructions are worked out for the case n = 2.

scholarly journals On one approach to the approximation of solutions to the direct kinematics problem of parallel manipulators

2020 ◽  
Vol 10 (1) ◽  
pp. 65-70
Author(s):  
Andrei Gorchakov ◽  
Vyacheslav Mozolenko
Keyword(s):  
Neural Network ◽  
Bounded Function ◽  
Parallel Manipulators ◽  
Feedforward Neural Network ◽  
Numerical Implementation ◽  
Continuous Bounded Function ◽  
Space Filling ◽  
Space Filling Curves ◽  
Direct Kinematics

AbstractAny real continuous bounded function of many variables is representable as a superposition of functions of one variable and addition. Depending on the type of superposition, the requirements for the functions of one variable differ. The article investigated one of the options for the numerical implementation of such a superposition proposed by Sprecher. The superposition was presented as a three-layer Feedforward neural network, while the functions of the first’s layer were considered as a generator of space-filling curves (Peano curves). The resulting neural network was applied to the problems of direct kinematics of parallel manipulators.

Computing Space-Filling Curves

2010 ◽  
Vol 50 (2) ◽  
pp. 370-386 ◽  
Author(s):  
P. J. Couch ◽  
B. D. Daniel ◽  
Timothy H. McNicholl
Keyword(s):  
Space Filling ◽  
Space Filling Curves
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