Questions tagged [examples]
For questions requesting examples of a certain structure or phenomenon
538
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Simple example of nontrivial simplicial localization
Does anyone has a simple example of a 1-category $\mathcal{C}$ and a collection of morphisms W such that the infinity-categorical / simplicial localization $\mathcal{C}\left[W^{-1}\right]$ is not a 1-...
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Does a quaternion algebra exist over a number field that is split over some infinite real places, but not others?
Let $F$ be a totally real number field having at least two different real embeddings $\sigma_1 : F \hookrightarrow \mathbb{R}$ and $\sigma_2 : F \hookrightarrow \mathbb{R}$.
Does a quaternion algebra $...
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1
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An example of a $T_1$ space where all closed $G_\delta$ sets are zero-sets, but it isn't normal
In Engelking's General topology, in the exercises section, there is Ju. M. Smirnov's characterization of normal spaces:
A $T_1$ space is normal iff the following properties hold (both):
Every closed $...
1
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1
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Examples of $C^{k,1}$ functions which are not $C^{k+1}$?
I'm currently reading this paper and the authors define the set $C^{k,1}(\mathbb{R}^n)$ as consisting of all functions $f:\mathbb{R}^n\rightarrow \mathbb{R}$ having $k$ derivatives and for which:
$$
\|...
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Example of a non-cocomplete model category of a realized limit sketch
Let $(\mathcal{E},\mathcal{S})$ be a realized limit sketch, i.e. a locally small category $\mathcal{E}$ with a class $\mathcal{S}$ of limit cones in it. It is not assumed that $\mathcal{E}$ is small, ...
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Mechanical systems with their configuration space being a Lie group
Cross-posted from Physics.SE
In Marsden, Ratiu - Introduction To Mechanics And Symmetry there is a certain focus on reducing cotangent bundles of Lie groups. More precisely, if $G$ is a Lie group, ...
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What are examples of mathematical objects that are 'constructed out of' a range of other objects but fall out of them? [closed]
What are examples of mathematical objects that are somehow 'constructed out of' a whole range of other objects but fall out of them? One example that comes to my mind is that of ordinal numbers: $\...
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Examples of multiplier Hopf algebras
A multiplier Hopf-algebra (introduced by Van Daele) is a pair $(A, \Delta)$ where $A$ is a non-degenerate algebra $A$ together with a non-degenerate algebra morphism $\Delta: A \to M(A \otimes A)$ ...
user167952
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Making coherent sheaves with nonvanishing higher Chern classes
Let $\mathcal{F}$ be a coherent sheaf on a variety $X$, and assume $\mathcal{F}$ has generic rank $n$. I expect (see e.g. here) that this actually puts no conditions on its Chern classes $c_1(\mathcal{...
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description of very ample bundle of Hirzebruch surface
I learned some basic properties of Hirzebruch surface mainly from Vakil's notes "the rising sea", section 20.2.9. the Hirzebruch surface is defined as $\mathbb{F}_n:=\operatorname{Proj} (\...
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On cospectral graphs
Is there examples of non-isomorphic cospectral graphs having
Non-isomorphic automorphism groups?
Isomorphic automorphism groups?
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answer
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Find strictly subharmonic function that vanishes at infinity
I am not sure about the term "strictly" subharmonic.
What I want is a function $\psi\in C^{\infty}(\mathbb{R}^3)$ with $\Delta\psi>0$ and $\lim\limits_{|x|\rightarrow\infty}\psi(x)=0$.
I ...
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Uniform approximation of indicator function of a point
Fix $x \in \mathbb{R}$ and let $I_{[x]}$ be its indicator function. Does anyone know of a sequence of (obviously) discontinuous approximations $g_n$ to $I_{[x]}$ such that
$g_n$ converge uniformly ...
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Push-out in the category of coherent sheaves over the complex projective plane
I'm trying to deal with an example of a rank two vector bundle over the complex projective plane which is non slope-stable (because the associated sheaf of sections has a coherent subsheaf of equal ...
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"Non-categorical" examples of $(\infty, \infty)$-categories
This title probably seems strange, so let me explain.
Out of the several different ways of modeling $(\infty, n)$-categories, complicial
sets and comical sets allow $n = \infty$,
providing ...
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Examples for certain class of projective varieties
I am looking for specific varieties that satisfy certain property. I call them symmetric varieties, I want to know what varieties are in it. It contains the projective spaces $\mathbb{P}^n$ for all $n$...
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Example of a PID with a residue field of finite characteristic and a residue field of characteristic 0?
I understand that for any nonempty set $S$ of characteristics, there exists a PID $R$ such that the set of characteristics of residue fields of $R$ (i.e. quotients by of $R$ by maximal ideals -- I'm ...
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Multivarate "RKHS" Examples
I've been reading about RKHSs and Hilbert spaces of functions these days a bit these days and I haven't yet come across an example of a hilbert space $H$ whose elements are all functions $f:\mathbb{R}^...
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Interesting "epimorphisms" of $E_\infty$-ring spectra
$\newcommand{\Mod}{\mathbf{Mod}} \newcommand{\map}{\mathrm{map}_{E_\infty-A}}$ Suppose $i:A\to B$ is a map of $E_\infty$-ring spectra. It induces a functor of $\infty$-categories $\Mod_B\to\Mod_A$ by ...
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Examples of conjectures whose direct falsity implies different consequences than indirect falsity
Mathematics several times has statements of form
$$\mathsf{Statement A}\implies\mathsf{Statement B}$$
where $\mathsf{Statement A}$ and $\mathsf{Statement B}$ are conjectures while the implication is ...
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Lagrange’s interpolation formula: Theoreme and Example [closed]
I would like to know where they come up with the formula of Lagrange interpolation (Lagrange’s interpolation formula),Lagrange_polynomial because I did some research, but I find a different definition ...
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Known dense subset of Schwartz-like space and $C_c^{\infty}$?
After reading this question, which asked for some examples of commonly used (proper) dense subsets of $C_0^{\infty}(\mathbb{R}^n)$ with the $L^p$-norm I wonder. What are some "well-known" ...
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Folding a non-rectangular shape into a rectangle of uniform thickness
I think the following might be an interesting subproblem of this question:
Question: For an odd number $n\ge 3$, is there a non-rectangular but still convex shape of area $A=1$, that can be folded (...
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What are some examples of proving that a thing exists by proving that the set of such things has positive measure?
Suppose we want to prove that among some collection of things, at least one
of them has some desirable property. Sometimes the easiest strategy is to
equip the collection of all things with a measure, ...
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Is there an orientable prime manifold covered by a non-prime manifold?
A manifold is called prime if whenever it is homeomorphic to a connected sum, one of the two factors is homeomorphic to a sphere.
Is there an example of a finite covering $\pi : N \to M$ of closed ...
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Examples of additive categories [closed]
I already this question here but I didn't get any satisfactory answer, so I will try in MO now.
There are a lot of interesting and creative examples of categories, such as for example, the category ...
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A diameter 2 arc-transitive graph whose complement is not arc-transitive?
A graph $G=(V,E)$ is arc-transitive if its symmetry group acts transitively on ordered pairs of adjacent vertices.
In general, the complement of an arc-transitive graph is not arc-transitive.
But I ...
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A connected topological space whose points cannot be connected by irreducible components
Does there exist a topological space $X$ with the following properties?
$X$ is connected.
The set of irreducible components of $X$ is locally finite.
Not every pair of points in $X$ can be "connected ...
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Request for examples: verifying vs understanding proofs
My colleague and I are researchers in philosophy of mathematical practice and are working on developing an account of mathematical understanding. We have often seen it remarked that there is an ...
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Real-world example of a Banach *-algebra with a nonzero *-radical
Is there a real-world example of a Banach *-algebra with a nonzero *-radical (intersection of kernels of all *-representations)? Textbooks give examples of finite-dimensional algebras with degenerate ...
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Proofs by Schubert calculus and combinatorics
Do you know some examples proved by two different methods: 1. Schubert calculus, 2. combinatorial method.
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Counterexample for absolute summability of autocovariances of strictly stationary strongly mixing sequence
Suppose $(X_i)_{i\in\mathbb{Z}}$ is a strictly stationary, strongly (i.e. $\alpha-$)mixing sequence of real random variables. If we have $\mathbb{E}[|X_1|^{2+\epsilon}]<\infty$ for some $\epsilon&...
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about simple non-abelian 2-generated group [closed]
Does there exist a simple non-abelian 2-generated group $G$ and two elements $a, b \in G$, such that $\langle \{a, b\} \rangle = G$, $a^2 =1$ and $\forall c, d \in G$ $\langle \{c^{-1}bc, d^{-1}bd \} \...
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Contrasting theorems in classical logic and constructivism
Is it possible there are examples of where classical logic proves a theorem that provably is false within constructivism? Is so what are some examples?
What are some examples of most contrasting ...
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Examples of faithful functors not injective on objects
As is well-known, a faithful functor need not be injective on objects. What are some good examples to illustrate this point?
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Examples of complicated parametric Jordan curves
For test purposes I need parametric Jordan curves that are complicated in the sense of having many inflection points and ideally no symmetries.
When doing online search I always land at complex ...
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What's an illustrative example of a tame algebra?
A finite-dimensional associative $\mathbf{k}$-algebra $\mathbf{k}Q/I$ is of tame representation type if for each dimension vector $d\geq 0$, with the exception of maybe finitely many dimension vectors ...
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Horizontal categorification: Two questions
According to the nlab, horizontal categorification is a process in which a concept is realized to be equivalent to a certain type of category with a single object, and then this concept is generalized ...
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Infinitely generated non-free group with all proper subgroups free
Is there any example of group $G$ satisfying the following properties?
$G$ is non-abelian, infinitely generated (i.e. it is not finitely generated) and not a free group.
$H< G$ implies that $H$ is ...
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On a continuous function as a substitute of the prime-counting function in the second Hardy–Littlewood conjecture satisfying certain asymptotics
It it well-known that the prime-counting function $\pi(x)$ satisfies the prime number theorem and that were in the literature two related conjectures to this arithmetic function, these are: the ...
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Examples of "irregularities" in mathematics, other than prime numbers [closed]
Prime numbers are the prime example (no pun intended) for something that arises apparently without describable patters; we know that infinitely many exist, that gaps between them can be arbitrarily ...
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Examples of proofs using induction or recursion on a big recursive ordinal
There are many proofs use induction or recursion on $\omega$, or on an arbitary (may be uncountable) ordinal. Are there some good examples of proofs which use a big but computable ordinal?
The ...
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What are examples of (collections of) papers which "close" a field?
There is sometimes talk of fields of mathematics being "closed", "ended", or "completed" by a paper or collection of papers. It seems as though this could happen in two ways:
A total characterisation,...
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Is there a Borel-measurable function which maps every interval onto $\mathbb R$?
Using AC, one easily defines a function $F:\mathbb R\to \mathbb R$ such that the $F$-image of any real interval $(a,b)$ ($a<b$) is equal to $\mathbb R$.
(Equivalently, the $F$-preimage of any real ...
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What is a non-trivial example of an unbounded subdifferential?
Let $f: X \to [ -\infty, \infty]$ be some function,
Can someone provide a non-trivial example where the subdifferential evaluated at a point $x$,
$$\partial f(x)$$ is "unbounded"? (trivial examples ...
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Mathematical Proofs [closed]
Create an example of a function $f: \mathbb{R} \to \mathbb{R}$ such that $f(f(f(\mathbb{R}))) = f(f(\mathbb{R})) \neq f(\mathbb{R})$
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Manifolds with nonwhere vanishing closed one forms
I am trying to find examples of closed manifolds $M$ admitting a nowhere vanishing closed one form. I am wondering if there are any examples beyond $N\times S^1$.
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Original examples of functions of slow increase in the spirit of Jakimczuk
I believe that it is possible to prove that $$f(x)=e^{\operatorname{Ai}(x)}\log x$$ is a function of slow increase in the spirit of the definition given by the author of [1], where $\operatorname{Ai}(...
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Symmetric products of varieties and projective bundles
Given a smooth projective geometrically connected curve $C$, a symmetric product of $C$ has the structure of a projective bundle over the Jacobian of $C$ (e.g. see Symmetric powers of a curve = ...
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Example of an integer $n_0$ such that $1+\sum_{k=2}^{n_0} \zeta(k)^s=0$ has repeated roots
After I was studying the exercise Problem 4.20 from [1] I was inspired to ask about next problem, where $\zeta(k)$ denotes, for integers $k>1$, particular values of the Riemann zeta function. And $...