Questions tagged [nonstandard-analysis]

Non-standard ordered fields are fields which have infinitesimals, that is, positive numbers which are smaller than any positive *real* number. Non-standard analysis is analysis done over such fields (e.g. hyperreal fields).

373 questions
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Using hyperreals to express the interval of uniform convergence of $x^n$ in a closed form

The sequence of functions $f_n: E \rightarrow \mathbb{R}$ where $f(x) = x^n$ converges uniformly to $g(x) = 0$ for $E = [0,1-\varepsilon]$, $\forall \varepsilon > 0$. Yet, it doesn't converge ...
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It would be possible to define an uniform distribution on $\Bbb N$ using infinitesimals?

In standard analysis it is clear that it is impossible to define an uniform probability distribution on $\Bbb N$ because there is no constant $c\in\Bbb R$ such that $\sum_{k=1}^\infty c=1$. Using ...
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Can the real numbers be embedded into all non-Archimedean real closed fields?

Every Archimedean real closed field is isomorphic to a subfield of $\mathbb{R}$. But I’m wondering if something in the opposite direction is true. Suppose that $F$ is a non-Archimedean real closed ...
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Finding functions $f:\Bbb{R}\to\Bbb{R}$ for which $\sum_{x\in\Bbb{R}}f(x)$ converges

A while ago, a friend asked me "for which functions $f:\Bbb{R}\to\Bbb{R}$ does the sum $\sum_{x\in\Bbb{R}}f(x)$ converge?", and a few days ago I completed a sketch of a proof that the sum converges ...
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Prove the nonstandard part is in a closed and bounded subset of $\mathbb{R^2}$

problem : Let K be a closed and bounded subset of $\mathbb{R^2}$ and $^*K \subset ^* \mathbb{R^2}$ it's * - extension. Prove that for any $x \in ^* K$ , it's standard part $st(x) \in K$ My Attempt ...
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i want reference about the method(below) of describing Infinity… [closed]

let think x is real number, then \begin{align} x&=x\\ x&=(x/2)+(x/2), \\ x&=(x/3)+(x/3)+(x/3), \\ x&=(x/4)+(x/4)+(x/4)+(x/4), \\ &\vdots \\ x&=(x/n)+(x/n)+(x/n)+ \cdots +(x/n), ...