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Chelyabinskiy Fiziko-Matematicheskiy Zhurnal, 2017, Volume 2, Issue 1, Pages 30–45
(Mi chfmj43)
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This article is cited in 2 scientific papers (total in 2 papers)
Mathematics
Solving of functional equations associated with the scalar product
V. A. Kyrov Gorno-Altaisk State University, Gorno-Altaisk , Russia
Abstract:
The functional equations
$$\left[X\right]\frac{\partial \chi}{\partial \theta} +
X_{n+1}(x^{n+1})\frac{\partial \chi}{\partial x^{n+1}} +
X_{n+1}(y^{n+1})\frac{\partial \chi}{\partial y^{n+1}} = 0,$$ $$[X]\frac{\partial \sigma}{\partial \theta} +
(X_{n+1}(x) - X_{n+1}(y))\frac{\partial \sigma}{\partial w} = 0, [X]\frac{\partial \varkappa}{\partial \theta} +
(X_{n+1}(x) + X_{n+1}(y))\frac{\partial \varkappa}{\partial z} = 0,
$$ is solved in the paper. Here $[X] = \sum^{n}_{k=1}\bigl(\varepsilon_kx^kX_k(y) +
\varepsilon_ky^kX_k(x))$, $x = (x^1,\ldots,x^n,x^{n+1})$, $\varepsilon_k=\pm1$, the equations are arising in the embedding problem of the space $\mathbb R^n$ with the inner product of the form $\theta = \varepsilon_1x^1y^1 + \cdots + \varepsilon_nx^ny^n$.
In this problem, all kinds of functions $f = f(\theta,x^{n+ 1},y^{n+ 1}) $ are found that are two-point invariants of $n(n + 1)/2$-parametric group of transformations.
Keywords:
functional equation, functional-differential equation, differential equation, scalar product.
Received: 25.12.2016 Revised: 28.02.2017
Citation:
V. A. Kyrov, “Solving of functional equations associated with the scalar product”, Chelyab. Fiz.-Mat. Zh., 2:1 (2017), 30–45
Linking options:
https://www.mathnet.ru/eng/chfmj43 https://www.mathnet.ru/eng/chfmj/v2/i1/p30
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