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The following statements For each $\epsilon>0$, there exists $K\in \mathbb{N}$ such that $n\geq K \Rightarrow |a_n-a|\leq \epsilon$ and For each $\epsilon>0$, there exists $K\in \mathbb{N}$ such that $n\geq K \Rightarrow |a_n-a|< \epsilon$ are equivalent. Proof of 1.$\Rightarrow$2. For any $\epsilon>0$, we have $\epsilon/2>0$. ...


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In a Dedekind domain every ideal is (in a unique way) the product of prime ideals. (The product of two ideal is the ideal generated by all products of elements.) The ring of integers of an algebraic number field is a Dedekind domain. The prime ideal decomposition of $p$ is the factorization of the principal ideal generated by $p$ into prime ideals of this ...


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In general if a topological group $G$ acts on a topological space $X$ then each $g\in G$ gives a homeomorphism $\theta_g:X\to X$ defined by $\theta_g(x)=g\cdot x$. Let $p:X'\to X$ be a covering map. We say that the action of $G$ lifts to $X'$ and is compatible with the action on $X$ if there exists a map $G\times X'\to X'$ such that the following diagram ...


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Generally speaking, the length $n$ of RS code $\mathcal{K}$ over $\mathbb{F}_q$ is not $q-1$. In the case, when $n=q-1$ we obtain cyclic RS-code. We can define RS-code also as a linear recursive code over $\mathbb{F}_q$. Let $\omega$ be a primitive element of $\mathbb{F}_q$ and $f(x) = (x-\omega^s)(x-\omega^{s+1})\ldots(x-\omega^{s+n-1})$, where $2\leq s\leq ...


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Given a point $x \in M$, its $\omega$-limit set with respect to a flow $\varphi_t$ is defined by $$ \omega(x)=\bigcap_{y \in \gamma(x)} \overline {\gamma^+(y)}, $$ where $$ \gamma(x)=\big\{\varphi_t(x): t \in \mathbb R\big\}\quad\text{and}\quad\gamma^+(y)=\big\{\varphi_t(y): t >0\big\}. $$ When $M$ is for example a smooth manifold (or simply $\mathbb ...


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It can be shown that rings of algebraic integers algebraic numbers which are roots of a monic polynomials are so-called Dedekind domains – a generalisation of both PIDs and UFDs. They have several characterisations, most notably that ideals have a decomposition as a product of prime ideals, and this decomposition is unique, up to the order of the factors. ...


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Perhaps a way to rewrite this definition is as follows (I am not checking correctness, just changing the language). The ordered list $\{P_0,P_1,\cdots,P_{n+1}\}\subseteq\mathbb{E}^n$ is called an Euclidean frame if the ordered list $\{Q_1,\cdots,Q_n\}$ where $Q_i=P_i-P_0$ has the property that for all $i\not=j$, $Q_i$ is orthogonal to $Q_j$ and $\|Q_i\|=1$. ...


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No, these formulations are not equivalent in general. As an example, consider $\mathbb N$ with the discrete topology, and consider the 'sheaf' of functions with values in $\mathbb C$ that are zero almost always. More precisely, for every $U \subseteq \mathbb N$ we define $$\mathcal F(U) = \{f: U \to \mathbb C: f(n) = 0\textrm{ for all but finitely many $n ...



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