| viXra.org > Functions and Analysis > 2006 |
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| viXra.org > Functions and Analysis > 2006 |
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[5] viXra:2006.0263 [pdf] submitted on 2020-06-29 15:53:50
Authors: Aryan Phadke
Comments: 9 Pages.
Background : Harmonic Series is the sum of Harmonic Progression. There have been multiple formulas to approximate the harmonic series, from Euler's formula to even a few in the 21st Century. Mathematicians have concluded that the sum cannot be calculated, however any approximation better than the previous others is always needed.
In this paper we will discuss the flaws in Euler's formula for approximation of harmonic series and provide a better formula. We will also use the infinite harmonic series to determine the approximations of finite harmonic series using the Euler-Mascheroni constant. We will also look at the Leibniz series for Pi and determine the correction factor that Leibniz discussed in his paper which he found using Euler numbers.
Each subsequent approximation we find in this paper is better than all previous ones. Different approximations for different types of harmonic series are calculated, best fit for the given type of harmonic series. The correction factor for Leibniz series might not provide any applied results but it is a great way to ponder some other infinite harmonic series.
Category: Functions and Analysis
[4] viXra:2006.0206 [pdf] replaced on 2020-07-04 13:01:43
Authors: George Precupescu
Comments: 30 Pages. This is an English version (the original v1 was in Romanian)
We define a 2-convex system by the restrictions $x_{1} + x_{2} + \ldots + x_{n} = ns$,
$e(x_{1}) + e(x_{2}) + \ldots + e(x_{n}) = nk$, $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ where $e:I \to \RR$ is a strictly convex function. We study the variation intervals for $x_k$ and give a more general version of the Boyd-Hawkins inequalities. Next we define a majorization relation on $A_S$ by $x\preccurlyeq_p y$ $\Leftrightarrow$ $T_k(x) \leq T_k(y) \ \ \forall 1 \leq k \leq p-1$ and $B_k(x) \leq B_k(y) \ \ \forall p+2 \leq k \leq n$ (for fixed $1 \leq p \leq n-1$) where $T_k(x) = x_1 + \ldots + x_k$, $B_k(x) = x_k + \ldots + x_n$. The following Karamata type theorem is given: if $x, y \in A_S$ and $x\preccurlyeq_p y$ then $f(x_1) + f(x_2) + \ldots + f(x_n) \leq f(y_1) + f(y_2) + \ldots + f(y_n)$ $\forall$$f:I \to \RR$ 3-convex with respect to $e$. As a consequence, we get an extended version of the equal variable method of V. Cîrtoaje
Category: Functions and Analysis
[3] viXra:2006.0105 [pdf] submitted on 2020-06-12 17:06:03
Authors: Eckhard Hitzer
Comments: 31 Pages. Proofs for all common formulas of vector differential calculus in an elementary step by step fashion.
This paper treats the fundamentals of the *vector differential calculus* part of *universal geometric calculus.* Geometric calculus simplifies and unifies the structure and notation of mathematics for all of science and engineering, and for technological applications. In order to make the treatment self-contained, I first compile all important *geometric algebra* relationships, which are necessary for vector differential calculus. Then *differentiation by vectors* is introduced and a host of major vector differential and vector derivative relationships is proven explicitly in a very elementary step by step approach. The paper is thus intended to serve as reference material, giving details, which are usually skipped in more advanced discussions of the subject matter.
Category: Functions and Analysis
[2] viXra:2006.0059 [pdf] submitted on 2020-06-07 19:31:47
Authors: Daniel Thomas Hayes
Comments: 1 Page.
A note on the instability of nothing.
Category: Functions and Analysis
[1] viXra:2006.0058 [pdf] submitted on 2020-06-07 19:38:45
Authors: Daniel Thomas Hayes
Comments: 1 Page.
A method for finding exact solutions of differential equations is proposed.
Category: Functions and Analysis
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