Questions tagged [examples]
For questions requesting examples of a certain structure or phenomenon
565 questions
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Example(s) of presheaves on a category C failing to discriminate between objects of a category D into which C maps
I’ve been trying to refine my intuition of the Yoneda Lemma, and in the process of doing so, I’ve thought a lot about the following situation. Suppose $F:C \to D$ is a functor between locally small ...
3
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$L^2$-functions orthogonal to their own Fourier transform
It is well-known that, besides the standard Gaussian $e^{-|x|^2/2},ドル there are many interesting functions which are eigenfunctions of the Fourier transform, for example the Hermite functions.
Mainly ...
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Examples for the use of AI and especially LLMs in notable mathematical developments
The purpose of this question is to collect examples where large language models (LLMs) like ChatGPT have led to notable mathematical developments.
The emphasis in this question is on LLMs, but ...
3
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Is there a countably infinite pre-closure with no circuits and no co-circuits?
For any $X$ call $f:2^X\to 2^X$ a pre-closure on $X$ when $\small\forall S,Q\subseteq X[S\subseteq Q\implies S\subseteq f(S)\subseteq f(Q)]$ while the complement of $T\subseteq X$ is $T^{\complement}=...
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Cumulants and { concentration / deviation } inequalities
In some recent reading, I was reminded of the following (trimmed) quote from Terry Speed (from Cumulants and partition lattices, Australian Journal of Statistics 25(2) (1983),
378–388.)
In a sense ...
4
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0
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183
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Interpreting 1ドル/f$ as a distribution when $f$ is only smooth
My first question is: does there exist a smooth function $f$ such that $f \neq 0$ on $\mathbb{R}^n \setminus \{0\},ドル $f(0) = 0,ドル and 1ドル/f,ドル viewed as a distribution on $\mathbb{R}^n \setminus \{0\},ドル ...
3
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How close can a meager set of full measure be to a perfect nowhere dense set?
Motivation:
On any interval of the real line (say, $[0,1]$ without loss of generality), we can construct Cantor-type sets $C$ with 0ドル \leq \mu(C) < 1,ドル which are perfect, nowhere dense, and totally ...
1
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Convexity of 2-Wasserstein metric
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\sP}{\mathcal{P}}
\newcommand{\sW}{W}
\newcommand{\coloneq}{:=}
\...
2
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3
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Examples of functions that vanish on a closed convex region and are positive outside
Question:
given a convex region $\mathcal{D}\subset\mathbb{R}^n$ i.e. a region for which $x, y\in\mathcal{D},ドル implies $\alpha x+(1-\alpha)y\in\mathcal{D}$ for all $\alpha\in[0,1],ドル what are examples ...
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Injections from the symmetric square of $P^3$ and other symmetric powers of projective spaces into projective spaces of small dimension
I am interested in "simple" projective varieties that are of "small" codimension in some $\mathbb{P}^N$ and are not set-theoretic complete intersections there. In particular, I am ...
1
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0
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Epimorphisms with kernel pairs
I am a bit lost understanding some subtleties in various form of epimorphy.
The nLab reports that an effective epimorphism is one that coequalizes its kernel pair. A regular epimorphism is simply one ...
1
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0
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On the Schaffer constant of finite dimensional complex normed spaces
The Schaffer constant of a normed space $X$ is given by
$$S(X)= \hbox{inf}\{\hbox{max} \{\|x+y\|, \|x-y\| \}: \|x\|=\|y\|=1\}.$$
I am interested in knowing whether there exists a finite-dimensional ...
1
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1
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405
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The space of bounded smooth functions with rapidly decaying derivatives
Let $f : \mathbb{R}^n \to \mathbb{C}$ be a bounded smooth function such that all of its partial derivatives are rapidly decaying. That is, for any nonzero $n$-dimensional multi-index $\alpha,ドル $D^\...
4
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3
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Non metrizable uniform spaces
Bourbaki's book on general topology states that a uniform space is metrizable iff it is Hausdorff and the filter of entourages of the uniformity has a countable basis. However, he doesn't provide an ...
3
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How irregular can the set of points of non-differentiability for an L1 function's primitive F get, before the FTC fails?
A Fundamental Theorem of Calculus for Lebesgue Integration, J. J. Koliha begins with the passage
Lebesgue proved a number of remarkable results on the relation between integration and differentiation....