New structure of norms on $\mathbb{R}^n$ and their relations with the curvature of the plane curves
Let $f_1, f_2, \ldots, f_n$ be fixed nonzero real-valued functions on $\mathbb{R}$, the real numbers. Let $\varphi_n(X_n)= \big(x_1^2f_1^2+x_2^2f_2^2+ \ldots + x_n^2f_n^2 \big)^{\frac{1}{2}}$, where $X_n=(x_1, x_2, \ldots, x_n) \in \mathbb{R}^n$. We show that $\varphi_n$ has properties similar to a...
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| Format: | Article |
| Language: | English |
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Accademia Piceno Aprutina dei Velati
2020-12-01
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| Series: | Ratio Mathematica |
| Online Access: | http://eiris.it/ojs/index.php/ratiomathematica/article/view/552 |
| _version_ | 1819057556854669312 |
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| author | Amir Veisi Ali Delbaznasab |
| author_facet | Amir Veisi Ali Delbaznasab |
| author_sort | Amir Veisi |
| collection | DOAJ |
| description | Let $f_1, f_2, \ldots, f_n$ be fixed nonzero real-valued functions on $\mathbb{R}$, the real numbers. Let $\varphi_n(X_n)= \big(x_1^2f_1^2+x_2^2f_2^2+ \ldots + x_n^2f_n^2 \big)^{\frac{1}{2}}$, where $X_n=(x_1, x_2, \ldots, x_n) \in \mathbb{R}^n$. We show that $\varphi_n$ has properties similar to a norm on the normed linear space. Although $\varphi_n$ is not a norm on $\mathbb{R}^n$ in general, it induces a norm on $\mathbb{R}^n$. For the nonzero function $F : \mathbb{R}^2 \to \mathbb{R}$, a curvature formula for the implicit curve $G(x, y)=F^2(x, y)=0$ at any regular point is given. A similar result is presented when $F$ is a nonzero function from $\mathbb{R}^3$ to $\mathbb{R}$. In continued, we concentrate on $F(x, y)=\int_a^b \varphi_2(x, y)dt$. It is shown that the curvature of $F(x, y)=c$ when $c>0$ is a positive multiple of $c^2$. Particularly, we observe that $F(x, y)=\int_0^{\frac{\pi}{2}} \sqrt{x^2 \cos^2 t + y^2 \sin^2 t} dt$ is an elliptic integral of the second kind. |
| first_indexed | 2024-12-21T13:41:12Z |
| format | Article |
| id | doaj.art-44b1a1872dd541119a21b285f2c9edd0 |
| institution | Directory Open Access Journal |
| issn | 1592-7415 2282-8214 |
| language | English |
| last_indexed | 2024-12-21T13:41:12Z |
| publishDate | 2020-12-01 |
| publisher | Accademia Piceno Aprutina dei Velati |
| record_format | Article |
| series | Ratio Mathematica |
| spelling | doaj.art-44b1a1872dd541119a21b285f2c9edd02022-12-21T19:02:01ZengAccademia Piceno Aprutina dei VelatiRatio Mathematica1592-74152282-82142020-12-01390556710.23755/rm.v39i0.552502New structure of norms on $\mathbb{R}^n$ and their relations with the curvature of the plane curvesAmir Veisi0Ali Delbaznasab1Faculty of Petrolem and Gas, Yasouj University, Gachsaran, IranDepartment of Mathematics, Bahonar University, Kerman, IranLet $f_1, f_2, \ldots, f_n$ be fixed nonzero real-valued functions on $\mathbb{R}$, the real numbers. Let $\varphi_n(X_n)= \big(x_1^2f_1^2+x_2^2f_2^2+ \ldots + x_n^2f_n^2 \big)^{\frac{1}{2}}$, where $X_n=(x_1, x_2, \ldots, x_n) \in \mathbb{R}^n$. We show that $\varphi_n$ has properties similar to a norm on the normed linear space. Although $\varphi_n$ is not a norm on $\mathbb{R}^n$ in general, it induces a norm on $\mathbb{R}^n$. For the nonzero function $F : \mathbb{R}^2 \to \mathbb{R}$, a curvature formula for the implicit curve $G(x, y)=F^2(x, y)=0$ at any regular point is given. A similar result is presented when $F$ is a nonzero function from $\mathbb{R}^3$ to $\mathbb{R}$. In continued, we concentrate on $F(x, y)=\int_a^b \varphi_2(x, y)dt$. It is shown that the curvature of $F(x, y)=c$ when $c>0$ is a positive multiple of $c^2$. Particularly, we observe that $F(x, y)=\int_0^{\frac{\pi}{2}} \sqrt{x^2 \cos^2 t + y^2 \sin^2 t} dt$ is an elliptic integral of the second kind.http://eiris.it/ojs/index.php/ratiomathematica/article/view/552 |
| spellingShingle | Amir Veisi Ali Delbaznasab New structure of norms on $\mathbb{R}^n$ and their relations with the curvature of the plane curves Ratio Mathematica |
| title | New structure of norms on $\mathbb{R}^n$ and their relations with the curvature of the plane curves |
| title_full | New structure of norms on $\mathbb{R}^n$ and their relations with the curvature of the plane curves |
| title_fullStr | New structure of norms on $\mathbb{R}^n$ and their relations with the curvature of the plane curves |
| title_full_unstemmed | New structure of norms on $\mathbb{R}^n$ and their relations with the curvature of the plane curves |
| title_short | New structure of norms on $\mathbb{R}^n$ and their relations with the curvature of the plane curves |
| title_sort | new structure of norms on mathbb r n and their relations with the curvature of the plane curves |
| url | http://eiris.it/ojs/index.php/ratiomathematica/article/view/552 |
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