All Questions
Tagged with examples counterexamples
22
questions
2
votes
0
answers
94
views
Real analytic periodic function whose critical points are fully denegerated
I have asked this question on MathStackExchange. My question: is there any non-constant real analytic function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that, $$\nabla f(x_0)=0 \Rightarrow \nabla^2 f(...
- 21
4
votes
1
answer
145
views
When does the refinement of a paracompact topology remain paracompact?
Let $(X,\tau)$ be a Hausdorff paracompact space. Let $\tau'$ be the smallest $P$-topology refining $(X,\tau)$, i.e. the topology which has for base the $G_\delta$-subsets of $(X,\tau)$.
Is it true ...
- 665
13
votes
0
answers
195
views
Examples and counterexamples to Lack's coherence observation
In Lack's A 2-categories companion, he states
There are general results asserting that any bicategory is biequivalent to
a 2-category, but in fact naturally occurring bicategories tend to be ...
- 7,889
8
votes
2
answers
613
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Existence of nontrivial categories in which every object is atomic
An object $X$ of a cartesian closed category $\mathbf C$ is atomic if $({-})^X \colon \mathbf C \to \mathbf C$ has a right adjoint (hence is also internally tiny). Intuitively, atomic objects are &...
- 7,889
5
votes
1
answer
151
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Finding non-inner derivations of simple $\mathbb Q$-algebras
What's a good example of a simple algebra over a field of characteristic $0$ which has a non-inner derivation but also has the invariant basis number property (IBN)?
I'm under the impression that when ...
- 1,593
6
votes
1
answer
151
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Mañé's example of an attractor with no natural measure
I'm reading Milnor's notes on dynamical systems and in Lecture 3 he gives an example of an attractor with no natural measure, which he attributes to Mañé. I can find no other reference in which this ...
- 63
7
votes
2
answers
201
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Examples of 2-categories with multiple interesting proarrow equipment structures
Proarrow equipments (also known as framed bicategories) are identity-on-objects locally fully faithful pseudofunctors $({-})_* \colon \mathcal K \to \mathcal M$ for which every 1-cell $f_*$ in the ...
- 7,889
1
vote
1
answer
216
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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:
$$
\|...
- 5,001
0
votes
1
answer
306
views
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 ...
1
vote
2
answers
206
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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&...
- 203
0
votes
0
answers
22
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A linear map satisfying the given property
Let $A$ and $B$ be two Banach algebras such that $B$ is a Banach $A$-bimodue and $T:A\rightarrow B$ a linear map satisfying
$T(aa')=aT(a')+T(a)a'+T(a)T(a')$ for all $a,a'\in A$.
If the algerba ...
- 167
0
votes
0
answers
56
views
Examples for a Golomb's result, and rationals as $\sum_{n\geq 1}\frac{|G_n|}{P(n)}$, where $G_n$ are Gregory coefficients and $P(x)$ a polynomial
After I was stuying the first pages of a chapter of the book [1], in particular the statement of Corollary 10.3 and its proof, I wondered what can be interesting examples of irrational numbers that ...
10
votes
0
answers
196
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Examples of automorphic representations to keep in mind
I have recently started studying the automorphic science and find it somewhat hard to form intuition. Can we have a list of examples of automorphic representations that you usually use to test a new ...
user140615
1
vote
0
answers
87
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An example of a Banach algebra with a specified property
I asked this question
(https://math.stackexchange.com/questions/3076735/an-example-of-a-banach-algebra-satisfying-given-conditions)
but unfortunately no one answered it. Please help me to find an ...
- 167
4
votes
2
answers
191
views
A result on spaces with countable pseudocharacter and countable tightness
There is a statement as follows:
If a Hausdorff (regular, Tychonoff) space $X$ has countable pseudocharacter and countable tightness, then the closure of any set $Y\subset X$ of cardinality $\le \...
- 601
3
votes
1
answer
168
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p-Group satisfying the minimal condition on abelian subgroups
Are there examples of $p$-groups satisfying the minimal condition on abelian subgroups but do not satisfying the minimal condition on subgroups?
Obviously such a group cannot be locally finite.
I've ...
- 137
4
votes
0
answers
55
views
Looking for a Collection of Examples and Counter Examples for Assumptions about the Properties of Planar Euclidean TSP Instances?
Where can I find example and counter examples to seemingly plausible assumption about the properties of optimal solutions of planar euclidean TSP instances?
The reason for asking is that the ...
- 12.5k
18
votes
1
answer
580
views
Bialgebras with Hopf restricted (or Sweedler) duals
It is known from the general theory that, given a bialgebra (over a field $k$)
\begin{equation}
\mathcal{B}=(B,\mu,1_B,\Delta,\epsilon)
\end{equation}
the Sweedler's dual $\mathcal{B}^0$ (called also ...
- 3,666
14
votes
3
answers
1k
views
Are all vector-space valued functors on sets free?
Let $\mathbf{Set}$ be the category of finite sets and functions between them, and let $\mathbf{Vect}$ be the category of finite-dimensional complex vector spaces and linear transformations between ...
- 3,909
0
votes
1
answer
351
views
Examples of groups such that order isomorphism of the subgroups of $G\times G$ and $H\times H$ does not imply isomorphism of $G$ and $H$
Let $G$ and $H$ be groups, $\operatorname{Sub}(G\times G)$ be the set of all subgroups of $G\times G$ and $\operatorname{Sub}(H\times H)$ be the set of all subgroups of $H\times H$. Assume there ...
- 1,145
5
votes
2
answers
1k
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Example for the Sobolev embedding theorem when p=n.
Let $\Omega$ be a bounded domain in $\mathbb R^2$. By the Sobolev embedding theorem, if $k>\frac np$ (in this case $k>\frac 2p$) then
$u\in W^{k,p}(U) \implies u\in C^{k-[\frac 2 p]-1,\gamma}(...
- 51
0
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0
answers
220
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Example of function with a certain behavior.
Let $f: R \rightarrow R$. Consider the following properties:
$(1)$ - There are positive constants $a$ and $r$ such that $\forall x, y$
$$|f(x)-f(y)|\leq a(1 + |x|^r+|y|^r)|x-y|.$$
$(2)$ - There is a ...
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