About some properties of algebras obtained by the Cayley-Dickson process
This paper is a short survey about some properties of algebras ob- tained by the Cayley-Dickson process and some of their applications.
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arXiv subjects
Cristina Flaut
Publications and source records attributed to Cristina Flaut.
Codes over a subset of Octonion Integers
In this paper we define codes over some Octonion integers. We prove that in some conditions these codes can correct up to two errors for a transmitted vector and the code rate of the codes is grater than the code rate of the codes defined on some subset of Quaternion integers.
cs.IT↗
Some properties of symbol algebras of degree three
In this paper we will study some properties of the matrix representations of symbol algebras of degree three, we study some equations with coefficients in these algebras, we find an octonion algebra in a symbol algebra of degree three, we define the Fibonacci symbol elements and we give some properties of them.
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Some examples of division symbol algebras of degree 3 and 5
In this paper we provide an algorithm to compute the product between two elements in a symbol algebra of degree n and we find an octonion non-division algebra in a symbol algebra of degree three. Starting from this last idea, we try to find an answer to the question if there are division symbol algebras of degree three. The answer is positive and we provide, using MAGMA software, some examples of division symbol algebras of degree 3 and of degree 5. Moreover, we will give some interesting applications of the symbol algebras in number theory.
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Cyclic codes over some special rings
In this paper we will study cyclic codes over some special rings: F_{q}[u]/(u^{i}), F_{q}[u_1,...u_{i}]/(u_1^2,u_2^2,...,u_{i}^2, u_1 u_2 - u_2 u_1,...,u_{i}u_{j} - u_{j}u_{i},...), F_{q}[u,v]/(u^{i},v^{j},uv-vu), q=p^{r}, where p is a prime number, r\in N-{0} and F_{q} is a field with q elements.
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On Generalized Fibonacci Quaternions and Fibonacci-Narayana Quaternions
In this paper, we investigate some properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions.
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Real matrix representations for the complex quaternions
Starting from known results, due to Y. Tian in [Ti; 00], referring to the real matrix representations of the real quaternions, in this paper we will investigate the left and right real matrix representations for the complex quaternions and we give some examples in the special case of the complex Fibonacci quaternions.
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Isotropy of some quadratic forms and its applications on levels and sublevels of algebras
In this paper, we give some properties of the levels and sublevels of algebras obtained by the Cayley-Dickson process. We will emphasize how isotropy of some quadratic forms can influence the levels and sublevels of algebras obtained by the Cayley-Dickson process.
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A note about algebras obtained by the Cayley-Dickson process
In this paper, we generalize the concepts of level and sublevels of a composition algebra to algebras obtained by the Cayley-Dickson process. In 1967, R. B. Brown constructed, for every $t\in \Bbb{N},$ a division algebra $A_{t}$ of dimension $2^{t}$ over the power-series field $K\{X_{1},X_{2},...,X_{t}\}.$ This gives us the possibility to construct a division algebra of dimension 2$^{t}$ and prescribed level 2$^{k}$ $ k, t\in \Bbb{N}^{*}.$
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A note about levels and sublevels of algebras obtained by the Cayley-Dickson process
In this paper, \thinspace \thinspace we generalize the concepts of \thinspace level \thinspace and \thinspace sublevel of a composition algebra to algebras obtained by the Cayley-Dickson process. In 1967, R. B. Brown constructed, for every}$t\in \Bbb{N},${\small % \ a division algebra}$A_{t}${\small \ of dimension}$2^{t}${\small \ over the power-series field}$K\{X_{1},X_{2},...,X_{t}\}.\,${\small \ This gives us the possibility to construct a division algebra of \thinspace dimension \thinspace 2}$^{t}\,${\small and prescribed \thinspace level and sublevel \thinspace \thinspace 2}$^{k}${\small ,\thinspace \thinspace}$k,\,t\in \Bbb{% N}^{*}${\small \ and a division algebra of \thinspace dimension \thinspace}$% 2^{t}+1,t\in \Bbb{N}\,${\small and prescribed \thinspace level and sublevel}% $\,2^{k}+1,k\in \Bbb{N}.\,\medskip $
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Levels and sublevels of division algebras obtained by the Cayley-Dickson process
{\small \ We generalize the concepts of \thinspace level \thinspace and \thinspace sublevel of a composition algebra to division algebras obtained by the Cayley-Dickson process. In 1967, R. B. Brown constructed, for every}$t\in \mathbb{N},${\small \ a nonassociative division algebra}$A_{t}${\small \ of dimension}$2^{t}${\small \ over the power-series field}$K\{X_{1},X_{2},...,X_{t}\}.\,${\small \ This gives us the possibility to construct a division algebra of \thinspace dimension}$% {\small \,2}^{t}\,${\small and prescribed \thinspace level and sublevel}$% {\small \,\,2}^{k}${\small ,\thinspace \thinspace}$k,\,t\in \mathbb{N}-\{0\} $ and dimension ${\small \,2}^{t}\,${\small and prescribed \thinspace level}${\small \,\,2}^{k}+1${\small ,\thinspace \thinspace}$k\in \mathbb{N}% -\{0\},t\in \mathbb{N},t\geq 2.$
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Values for the levels and sublevels of algebras obtained by the Cayley-Dickson process
In this paper we will prove that any number n\in N-{0} could be realised as level of an algebra obtained by the Cayley-Dickson process over a suitable field.
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Some identities in algebras obtained by the Cayley-Dickson process
In this paper we will prove that the Hall identity is true in all algebras obtained by the Cayley-Dickson process and, in some conditions, the converse is also true.
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Some properties of the symbol algebras
In this paper, we obtain some properties of the symbol algebras, starting from their connections with the quaternion and cyclic algebras over a field $K_{p},$% where $K$ is an algebraic number field, $p$ is a prime in $K$ and $K_{p}$ is the completion of $K$ with respect to $p-$ adic valuation, in the case when $K=\Bbb{Q}(\varepsilon), \varepsilon ^{3}=1,\varepsilon \neq 1.$
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