Functions and Analysis

1901 Submissions

[3] viXra:1901.0341 [pdf] replaced on 2019-01-24 13:38:55

Necessary and Sufficient Conditions for a Factorable Matrix to be Hyponormal

Authors: H. C. Rhaly Jr.
Comments: 4 Pages.

Necessary and sufficient conditions are given for a special subclass of the factorable matrices to be hyponormal operators on $\ell^2$. Three examples are then given of polynomials that generate hyponormal weighted mean operators, and one example that does not. Paired with the main result presented here, various computer software programs may then be used as an aid for classifying operators in that subclass as hyponormal or not.
Category: Functions and Analysis

[2] viXra:1901.0294 [pdf] submitted on 2019-01-20 04:27:04

Ramanujan Summation of the Ln(n) Series

Authors: Jesús Sánchez
Comments: 9 Pages.

In this paper it will calculated that the Ramanujan summation of the Ln(n) series is: lim┬█(n→∞)⁡(Ln(1)+Ln(2)+Ln(3)+⋯Ln(n))=Ln(-γ)=Ln(γ)+(2k+1)πi Being γ the Euler-Mascheroni constant 0.577215... The solution is valid for every integer number k (it has infinite solutions). The series are divergent because Ln(n) tends to infinity as n tends to infinity. But, as in other divergent series, a summation value can be associated to it, using different methods (Cesàro, Abel or Ramanujan). If we take the logarithm of the absolute value (this is, we take only the real part of the solution), the value corresponds to the smooth continuation to the y axis of the curve that calculates the partial sums at every point, as we will see in the paper. lim┬█(n→∞)⁡(Ln|1|+Ln|2|+Ln|3|+⋯Ln|n|)=Ln|-γ|=Ln|γ|
Category: Functions and Analysis

[1] viXra:1901.0134 [pdf] submitted on 2019-01-10 21:01:16

Next to Nothing - a Single Paradigm

Authors: Mark C Marson
Comments: 18 Pages.

To gain true understanding of a subject it can help to study its origins and how its theory and practice changed over the years – and the mathematical field of calculus is no exception. But calculus students who do read accounts of its history encounter something strange – the claim that the theory which underpinned the subject for long after its creation was wrong and that it was corrected several hundred years later, in spite of the fact that the original theory never produced erroneous results. I argue here that both this characterization of the original theory and this interpretation of the paradigm shift to its successor are false. Infinitesimals, used properly, were never unrigorous and the supposed rigor of limit theory does not imply greater correctness, but rather the (usually unnecessary) exposition of hidden deductive steps. Furthermore those steps can, if set out, constitute a proof that original infinitesimals work in accordance with limit theory – contrary to the common opinion that the two approaches represent irreconcilable philosophical positions. This proof, demonstrating that we can adopt a unified paradigm for calculus, is to my knowledge novel although its logic may have been employed in another context. I also claim that non-standard analysis (the most famous previous attempt at unification) only partially clarified the situation because the type of infinitesimals it uses are critically different from original infinitesimals.
Category: Functions and Analysis