Functions and Analysis

2308 Submissions

[4] viXra:2308.0157 [pdf] submitted on 2023-08-24 01:14:01

Calculation of the Mathematical Expectancy of Cauchy's Law and Extension to Other Improper Integrals

Authors: André Pérennec, Gilles Burel
Comments: 2 Pages.

It is well known that the calculation of the mathematical expectancy of Cauchy'slaw in probability generates an indeterminate form. We show here that this indeterminacy canbe lifted and the calculation leads to a fixed value. Moreover, we show that other improperintegrals with an indeterminate result can be computed.
Category: Functions and Analysis

[3] viXra:2308.0124 [pdf] submitted on 2023-08-18 05:57:57

Embedding of Octonion Fourier Transform in Geometric Algebra of R^3 and Polar Representations of Octonion Analytic Signals in Detail

Authors: Eckhard Hitzer
Comments: 17 Pages. 3 tables, 1 figure. Published in Mathematical Methods in the Applied Sciences, 2023. DOI: 10.1002/mma.9639.

We show how the octonion Fourier transform can be embedded and studied in Clifford geometric algebra of three-dimensional Euclidean space Cl(3,0). We apply a new form of dimensionally minimal embedding of octonions in geometric algebra, that expresses octonion multiplication non-associativity with a sum of up to four (individually associative) geometric algebra product terms. This approach leads to new polar representations of octonion analytic signals and signal reconstruction formulas.
Category: Functions and Analysis

[2] viXra:2308.0089 [pdf] submitted on 2023-08-13 14:11:14

Correcting Rudin's Perfect Sets Are Uncountable Proof

Authors: Timothy W. Jones
Comments: 4 Pages.

It is fascinating fact that the reals are uncountably infinite. Usually Cantor's diagonal method is used to show this. Rudin gives a second proof that promises to be more rigorous than this method. But his proof is a little confusing, if not incorrect. His proof does not stipulate that the perfect set be bounded, but its proof hinges on a local, bounded phenomenon. We duplicate Rudin's proof and argue using two examples that assuming any indexing scheme for the presumed countable set can't work. We then give two proofs: one re-indexes points and the other indexes in the course of the proof.
Category: Functions and Analysis

[1] viXra:2308.0054 [pdf] submitted on 2023-08-10 16:22:03

New Series and Concise Algorithm for "Lambert W Function"

Authors: Warren D. Smith
Comments: 2 Pages.

We give a new series expression, and code up a concise algorithm,for the "Lambert W function" W(X) such that WeW=X with W≥-1.
Category: Functions and Analysis