P. L. PROPER KNOT EQUIVALENCE CLASSES ARE GENERATED BY LOCALLY FLAT ISOTOPIES

1994 ◽  
Vol 03 (04) ◽  
pp. 497-509
Author(s):  
OLLIE NANYES
Keyword(s):  
Real Line ◽  
Knot Theory ◽  
A Priori ◽  
Equivalence Classes ◽  
The Real ◽  
Smooth Category

A p. 1. proper knot is a proper p. 1. embedding of the real line into a p. 1. open 3-manifold. Two p. 1. proper knots are said to be equivalent if there is a p. 1. proper isotopy (possibly non-ambient) connecting them. A priori such an isotopy need not be locally flat: for example a local knot in the range could be shrunk to a point within a 3-cell. In this paper we show that, in essence, this is the only type of non-locally flat phenomena. We show that such a local knot can always be "combed to infinity". Thus if two p. 1. proper knots are connected by a p. 1. proper isotopy then the connecting isotopy can be modified to be a locally flat p. 1. proper isotopy. Hence, in proper knot theory, the smooth category and the p. 1. category are similar.

A Revised Version of P. L. PROPER KNOT EQUIVALENCE CLASSES ARE GENERATED BY LOCALLY FLAT ISOTOPIES (Vol. 3, No. 4 (December 94), pp. 497–509)

1995 ◽  
Vol 04 (02) ◽  
pp. 329-342 ◽  
Author(s):  
OLLIE NANYES
Keyword(s):  
Real Line ◽  
Knot Theory ◽  
A Priori ◽  
Equivalence Classes ◽  
The Real ◽  
Smooth Category

A p. 1. proper knot is a proper p. 1. embedding of the real line into a p. 1. open 3-manifold. Two p. 1. proper knots are said to be equivalent if there is a p. 1. proper isotopy (possibly non-ambient) connecting them. A priori such an isotopy need not be locally flat: for example a local knot in the range could be shrunk to a point within a 3-cell. In this paper we show that, in essence, this is the only type of non-locally flat phenomenon. We show that such a local knot can always be “combed to infinity”. Thus if two p. 1. proper knots are connected by a p. 1. proper isotopy then the connecting isotopy can be modified to be a locally flat p. 1. proper isotopy. Hence, in proper knot theory, the smooth category and the p. 1. category are similar.

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

AN OSCILLATION FUNCTION ON THE REAL LINE

2000 ◽  
Vol 26 (1) ◽  
pp. 237
Author(s):  
Duszyński
Keyword(s):  
Real Line ◽  
The Real

DERIVATION BASES ON THE REAL LINE (I)

1982 ◽  
Vol 8 (1) ◽  
pp. 67 ◽  
Author(s):  
Thomson
Keyword(s):  
Real Line ◽  
The Real

On finite sums of periodic functions

2020 ◽  
Vol 27 (2) ◽  
pp. 265-269
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Analytic Function ◽  
Real Line ◽  
Real Analytic Function ◽  
Periodic Functions ◽  
Uniform Limit ◽  
The Real ◽  
Pointwise Limit ◽  
Finite Sums ◽  
Real Analytic

AbstractIt is shown that any function acting from the real line {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

scholarly journals Heisenberg–Weyl Groups and Generalized Hermite Functions

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1060
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano A. del del Olmo
Keyword(s):  
Hilbert Space ◽  
Fourier Transform ◽  
Heisenberg Group ◽  
Real Line ◽  
Hermite Functions ◽  
Unitary Representations ◽  
Weyl Groups ◽  
The Real ◽  
Symmetry Properties ◽  
The Fourier Transform

We introduce a multi-parameter family of bases in the Hilbert space L2(R) that are associated to a set of Hermite functions, which also serve as a basis for L2(R). The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these “generalized Hermite functions”. The construction of these new bases is grounded on some symmetry properties of the real line under translations, dilations and reflexions as well as certain properties of the Fourier transform. We show how these generalized Hermite functions are transformed under the unitary representations of a series of groups, including the Weyl–Heisenberg group and some of their extensions.

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