# Algebra

## 1912 Submissions

[4] **viXra:1912.0529 [pdf]**
*submitted on 2019-12-31 05:39:24*

### On a Local Spectra Inequality

**Authors:** Theophilus Agama

**Comments:** 5 Pages.

In this note we show that under certain conditions the inequality holds \begin{align}\sum \limits_{\lambda_i\in \mathrm{Spec}(ab^{T})}\mathrm{min}\{\log |t-\lambda_i|\}_{[||a||,||b||]}&\leq \# \mathrm{Spec}(ab^T)\log\bigg(\frac{||b||+||a||}{2}\bigg)\nonumber \\&+\frac{1}{||b||-||a||}\sum \limits_{\lambda_i\in \mathrm{Spec}(ab^T)}\log \bigg(1-\frac{2\lambda_i}{||b||+||a||}\bigg).\nonumber
\end{align}Also under the same condition, the inequality also holds\begin{align}\int \limits_{||a||}^{||b||}\log|\mathrm{det}(ab^{T}-tI)|dt&\leq \# \mathrm{Spec}(ab^T)(||b||-||a||)\log\bigg(\frac{||b||+||a||}{2}\bigg)\nonumber \\&+\sum \limits_{\lambda_i\in \mathrm{Spec}(ab^T)}\log \bigg(1-\frac{2\lambda_i}{||b||+||a||}\bigg).\nonumber
\end{align}

**Category:** Algebra

[3] **viXra:1912.0435 [pdf]**
*submitted on 2019-12-24 03:27:19*

### On Linear Algebra

**Authors:** Henry Wong

**Comments:** 1 Page.

An addendum to linear algebra.

**Category:** Algebra

[2] **viXra:1912.0073 [pdf]**
*submitted on 2019-12-04 07:10:56*

### V={e, (1 2)(3 4), (1 3)(2 4),(1 4)(2 3)} is a Characteristic Subgroup of the Alternating Group of Degree 4

**Authors:** Henry Wong

**Comments:** 1 Page.

An addendum to group theory.

**Category:** Algebra

[1] **viXra:1912.0002 [pdf]**
*submitted on 2019-12-01 07:44:01*

### Refutation of the Quaternion Based on Its Implication Truth Table

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the standard definition of the quaternion as four equations to produce its multiplication table. We then derive the implication table for the quaternion which is not tautologous, and hence refutes the quaternion. This result forms a non tautologous fragment of the universal logic VŁ4.

**Category:** Algebra