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[4] viXra:2010.0234 [pdf] submitted on 2020-10-29 10:52:24
Authors: Quang Nguyen Van
Comments: 1 Page.
We give a new solvable sextic equation and its solution.
Category: Algebra
[3] viXra:2010.0227 [pdf] submitted on 2020-10-28 21:41:23
Authors: Theophilus Agama
Comments: 7 Pages.
This paper initializes the study of the Pierce-Birkhoff conjecture. We start by introducing the notion of the area and volume induced by a multivariate expansion and develop some inequalities for our next studies. In particular we obtain the inequality
\begin{align}
\sum \limits_{\substack{i,j\in [1,n]\\a_{i_{\sigma(s)}}<a_{j_{\sigma(s)}}\\s\in [1,l]\\v\neq i,j\\v\in [1,n] }}\bigg | \bigg |\vec{a}_{i} \diamond \vec{a}_{j}\diamond \cdots \diamond \vec{a}_v\bigg |\bigg |\sum \limits_{k=1}^{n}\int \limits_{a_{i_{\sigma(l)}}}^{a_{j_{\sigma(l)}}}\int \limits_{a_{i_{\sigma(l-1)}}}^{a_{j_{\sigma(l-1)}}}\cdots \int \limits_{a_{i_{\sigma(1)}}}^{a_{j_{\sigma(1)}}}g_kdx_{\sigma(1)}dx_{\sigma(2)}\cdots dx_{\sigma(l)}\nonumber\\ \leq 2C\times \binom{n}{2}\times \sqrt{n}\times \nonumber \\ \times \int \limits_{a_{i_{\sigma(l)}}}^{a_{j_{\sigma(l)}}}\int \limits_{a_{i_{\sigma(l-1)}}}^{a_{j_{\sigma(l-1)}}}\cdots \int \limits_{a_{i_{\sigma(1)}}}^{a_{j_{\sigma(1)}}}\sqrt{\bigg(\sum \limits_{k=1}^{n}(\mathrm{max}(g_k))^2\bigg)}dx_{\sigma(1)}dx_{\sigma(2)}\cdots dx_{\sigma(l)}\nonumber
\end{align}for some constant $C>0$, where $\sigma:\{1,2,\ldots,l\}\longrightarrow \{1,2,\ldots,l\}$ is a permutation for $g_k\in \mathbb{R}[x_1,x_2,\ldots,x_l]$ and $\vec{a}_{i} \diamond \vec{a}_{j}\diamond \cdots \diamond \vec{a}_k \diamond \vec{a}_{v}$ is the cross product of any of the $n-1$ fixed spots in $\mathbb{R}^{l}$ including the spots $\vec{a}_i,\vec{a}_j$.
Category: Algebra
[2] viXra:2010.0193 [pdf] submitted on 2020-10-23 19:52:23
Authors: Yaroslav Shitov
Comments: 23 Pages.
Let f be a homogeneous polynomial of degree d with coefficients in C. The Waring rank of f is the smallest integer r such that f is a sum of r powers of linear forms. We show that the Waring rank of the polynomial
x1 y2 z3 − x1 y3 z2 + x2 y3 z1 − x2 y1 z3 + x3 y1 z2 − x3 y2 z1
is at least 18, which matches the known upper bound.
Category: Algebra
[1] viXra:2010.0160 [pdf] submitted on 2020-10-20 19:27:31
Authors: Juan Elias Millas Vera
Comments: 3 Pages. Send your comment to: juanmillaszgz@gmail.com
In this paper I show how it is possible to find the value of an exponent in the square root of a negative number. Using four formulas whose I have develop.
Category: Algebra
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