Some Results on Integer Solutions of Quadratic Polynomials in Two Variables

Maguiña, B. M. Cerna; Ramos, Janet Mamani

doi:10.13189/ms.2021.090609

2021

DOI: 10.13189/ms.2021.090609

|View full text |Cite

Some Results on Integer Solutions of Quadratic Polynomials in Two Variables

B. M. Cerna Maguiña¹,

Janet Mamani Ramos²

Abstract: In this work we prove some theorems that allow us to find integer solutions of quadratic equations in two variables that represent a natural number.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2024

Publication Types

Select...

Preprint1

Relationship

Self Cite0

Independent1

Authors

Journals

Cited by 1 publication

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Some Results on Maxima and Minima of Real Functions of Vector Variables: A New Perspective

Maguiña,

Lujerio Garcia,

Pocoy Yauri

et al. 2024

Preprint

View full text Add to dashboard Cite

In this work, the extreme points of real vector variable functions are obtained without the use of the classical theory that involves the use of partial derivatives. We illustrate with several theorems and examples a new method that consists of establishing an appropriate link between the function to be optimized, its restrictions and the result, stating that: given \(n\) non-zero real numbers \(a_{1},a_{2},\cdots,a_{n}\in\mathbb{R}\), then there exists a unique \(\lambda\in\mathbb{R}\) such that: This relation is obtained by decomposing the Hilbert space \(\mathbb{R}^{n}\) as the direct sum of a closed subspace and its orthogonal complement. Since the dimension of the space \(\mathbb{R}^{n}\) is finite, this guarantees that any linear functional defined on the space \(\mathbb{R}^{n}\) is continuous, and this guarantees that the kernel of said linear functional is closed in the space \(\mathbb{R}^{n}\), therefore we have that the space \(\mathbb{R}^{n}\) breaks down, as the direct sum of the kernel of the continuous linear functional \(f\) and its orthogonal complement, that is: \(\mathbb{R}^{n}\,-\,\ker f\,\bigoplus\,\left[\,\ker f\,\right]^{\perp}\), where the dimension of \(\ker f\,-\,n\,-\,1\) and the dimension of \(\left[\,\ker f\,\right]^{\perp}\,-\,1\). Adding to the link found new definitions about the hierarchy of one variable in relation to another and the fact that if \(x_{1}^{2}+x_{2}^{2}+...+x_{n}^{2}\,-\,r^{2}\) then the \(\max\{x_{1}+x_{2}+\cdots+x_{n}\}\,-\,r\sqrt{n}\) and the \(\min\{x_{1}+x_{2}+\cdots+x_{n}\}\,-\,-r\sqrt{n}\) we solve the optimization problem without using classical theory.

show abstract

Some Results on Maxima and Minima of Real Functions of Vector Variables: A New Perspective

Maguiña,

Lujerio Garcia,

Pocoy Yauri

et al. 2024

Preprint

View full text Add to dashboard Cite

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Some Results on Integer Solutions of Quadratic Polynomials in Two Variables

Abstract: In this work we prove some theorems that allow us to find integer solutions of quadratic equations in two variables that represent a natural number.

Cited by 1 publication

References 3 publications

Some Results on Maxima and Minima of Real Functions of Vector Variables: A New Perspective

Some Results on Maxima and Minima of Real Functions of Vector Variables: A New Perspective

Contact Info

Product

Resources

About