Questions tagged [examples]
For questions requesting examples of a certain structure or phenomenon
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Example of evaluation of $\int_0^1\left(\sum_{k=0}^n (f(x))^k\right)^{\alpha}dx$, for some choice of $f(x)$ satisfying certain requirements
Let $0<\alpha\leq\frac{1}{2}$ a fixed real number. I wondered if it is possible to evaluate the sequence of definite integrals $$\int_0^1\left(\sum_{k=0}^n (f(x))^k\right)^{\alpha}dx\tag{1}$$
for ...
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Which knot invariants have no known diagram-independent descriptions?
Many knot invariants in knot theory are discovered by finding a property of knot diagrams which is invariant under the three Reidemeister moves. Now in principle, any knot invariant can be described ...
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Examples of Steffensen's inequality at undergraduated level studies
I've known few days ago the known as Steffensen's inequality, see the article Steffensen's inequality from Wolfram MathWorld and the cited bibliography. It seems that there are applications (I don't ...
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Example of a sequence of logarithmically convex functions on $\mathbb{R}$ and for all $n\in\mathbb{N}$ in the spirit of one evoked in an article
To ask this question I was inspired in some words, if I understand well, from the authors of a preprint on arXiv in section 4.1, that I believe that is [1], to ask next question.
We consider the ...
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What can be an interesting problem of differential equations involving the definition of the Gudermannian function? [closed]
In this post I denote the Gudermannian function as $$\operatorname{gd}(x)=\int_0^x\frac{dt}{\cosh t}$$
and its inverse as $\operatorname{gd}^{-1}(x)$, please see if you need it the definitions, ...
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Examples of geometrical interpretations for sequences of particular values of Dirichlet series
The remark [1] (in Spanish) shows a geometric interpretation (linking two sequences) of particular values of a given Dirichlet series, that are $\zeta(k)$ and $\zeta(2k)$. I wondered about if it is ...
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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 ...
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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 ...
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Examples of hyperbolic groups
What are some other classes of word-hyperbolic groups other than the finite groups, fundamental groups of surfaces with Euler characteristics negative and virtually free groups?
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Lattices with no roots and spread out shells
I am looking for lattices with the following properties:
The lattice has no roots.
The norm (squared length) of the second shortest vectors should be at least twice as large as the norm of the ...
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Trivial fine Selmer group in the cyclotomic extension
In explicit examples that I have seen worked out, it appears that when the fine Selmer group is finite in the cyclotomic extension it is in fact trivial.
Is there any reason to expect that this ...
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Very canonical constructions
You have two categories $C_1$ and $C_2$. We call a map of the classes $\mathrm{Ob}(C_1)\rightarrow \mathrm{Ob}(C_2)$ a construction. Sometimes you can find a functor $C_1\rightarrow C_2$ inducing this ...
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A "concrete" example of a one-sided Hopf algebra
I came to know from the paper Left Hopf Algebras by Green, Nichols and Taft that one may consider a Hopf algebra whose antipode satisfies only the left (resp. right) antipode condition.
To be more ...
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Explicit computations with crystalline cohomology
I am currently studying crystalline cohomology and while all the talk about crystalline topoi is nice, I would like to see some explicit computations. What are some references on this subject which ...
user140761
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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
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Big list of comonads
The concept of a monad is very well established, and there are very many examples of monads pertaining almost all areas of mathematics.
The dual concept, a comonad, is less popular.
What are ...
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Example of a manifold with positive isotropic curvature but possibly negative Ricci curvature
Is there any example of a manifold with a positive isotropic curvature but it possibly obtains a negative Ricci curvature at some point and the direction? If we see the definition of the positive ...
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Equal volume and projections
Given three unit vectors $u_1,u_2,u_3$ in $\mathbb{R}^3$, can we find some body $K \subset \mathbb{R}^3$ (probably convex) such that the following three things hold
(1) $|P_{u_1^\perp}K|=|P_{u_2^\...
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When was the generalization easier to prove than the specific case? [duplicate]
I distinctly remember from my long-ago undergraduate math that there were some interesting cases where a solution (proof) was sought for some specific thing but it wasn't easy to find - and in a few ...
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Factoring $\frac{1}{1-rx}$ into an infinite products of polynomials
I am looking for examples of sequences of polynomials $(p_{k}(x))_{k=1}^{\infty}$ with positive integer coefficients where $p_{k}(0)=1$ for all $k\geq 1$ and where there is a positive integer $r$ ...
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Intuition behind orthogonality in category theory, and origin of name
In category theory, two morphisms $e:A\to B$ and $m:C\to D$ are said to be orthogonal if for any $f:A\to C$ and $g:B\to D$ with $m\circ f=g\circ e$, there exists a unique morphism $d:B\to C$ such that ...
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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 ...
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Example of convex functions fulfilling a (strange) lower bound
I am reading a preliminary version of a paper which focuses on some minimization problems connected to a class of integral functionals. Reading the assumptions of one of the theorems I cannot convince ...
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Does anyone use non-sober topological spaces?
Recall that a sober space is a topological space such that every irreducible closed subset is the closure of exactly one point.
Is there any area of mathematics outside of general topology where non-...
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Explicit examples of Azumaya algebras
I'm trying to understand the Brauer group of a scheme better. I know how to compute $\text{Br}(X)$ as an abstract group in some cases, but don't have a good idea of what the individual Azumaya ...
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An example of a measurable random process with non-measurable integral
Let $ \xi _t(\omega), t\in[0,\infty)$, be a random process and let $ \xi _t(\omega)\in \{\mathfrak F_t\}$ be some filtration. Even if $ \xi _t(\omega) $ is $ \mathfrak F_t $ measurable then $\int_0^...
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Examples of set theory problems which are solved using methods outside of logic
The question is essentially the one in the title.
Question. What are some examples of (major) problems in set theory which are solved using techniques outside of mathematical logic?
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Definition and examples of operator-stable distributions
I was trying to understand the basic ideas of the operator-stable distributions. I found the papers by Hudson and Sato. However, unfortunately, I am being unable to understand the mathematical ...
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How to choose compactly supported smooth $h$ so $h^2(x)+ h^2(x-1)=1$ for all $x\in [0,1],$ and $\int_{-3/4}^{3/4} |h(x)|^2 dx =3/2$? [closed]
It is known that we may choose smooth $f:\mathbb R \to [0,1]$ such that $f(x)=1$ if $x\geq \frac{3}{4} $ and $f(x)=0$ if $x\leq -\frac{3}{4}+1.$
Define $h(x)= \sin (\frac{\pi}{2} f(x+1))$ if $x\...
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How to choose function $\sum_{m\in \mathbb Z} (-1)^m f(x+m) f(x-m+n)=0$?
Can we expect to choose a function $f:\mathbb R \to \mathbb R$ (nonzero compactly supported) so that
$\sum_{m\in \mathbb Z} (-1)^m f(x+m) f(x-m+n)=0$ for all $x\in \mathbb R$ and $n\in \mathbb Z$?...
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Combinatorial proof that some model categories are monoidal/enriched?
I'm looking for examples of proofs that some Quillen model categories are monoidal, or enriched over an other model category, which are based on explicit computation of the "pushout product" of the ...
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Example of an abelian category with enough projectives and injectives which are not dual
For trying to understand how general a certain theorem is, I'm looking for an example of an essentially small abelian category which has enough projectives and enough injectives, but whose category of ...
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An orientable non-spin${}^c$ manifold with a spin${}^c$ covering space
Is there a closed, smooth, orientable manifold which is not spin${}^c$ but has a finite cover which is spin${}^c$?
Such examples exist when spin${}^c$ is replaced by spin: an Enriques surface is not ...
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What are some interesting examples of cooperative games that can be naturally generalised to a stochastic version of it?
In classical, deterministic cooperative game theory, there are $N$ players that can form $2^{N}$ coalitions. Each of these coalitions is assigned a value by means of the characteristic function $v ( \...
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Examples of residually-finite groups
One of the main reasons I only supervised one PhD student is that I find it hard to find an appropriate topic for a PhD project. A good approach, in my view, is to have on the one hand a list of ...
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Example of a smooth family of projective surfaces with non-vanishing integrals of Todd classes
Motivation:
Let $\pi\colon S \rightarrow B$ be smooth projective morphism of relative dimension 2 over a smooth projective scheme $B$. If the stucture sheaves of the fibres do not have higher ...
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Naturally occurring, non-amenable Zappa-Szep products of discrete amenable groups?
We say $G$ is the Zappa-Szep product of two subgroups $K$ and $P$ if $K\cap P = \{e\}$ and the function $K\times P \to G$, $(k,p)\mapsto kp$, is bijective.
The Iwasawa decomposition shows that we can ...
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Isolated periodic trajectories of Hamiltonian systems
Is there any example of an autonomous Hamiltonian system with a periodic trajectory isolated in the whole phase space? The Poincar\'e map of such a trajectory within its energy level should be very ...
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How could I see quickly that this space is not normal?
Recently, I read a paper in which the author construct a space $X$ which is dense in a $\sigma$-product $S$ of closed unit intervals. The space $X$ is CCC (denotes countable chain condition); it is ...
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Let $X$ be a Lindelof, perfectly normal, $\sigma$-space. Must $X$ be separable?
A space $X$ is a $\sigma$-space if $X$ has a $\sigma$-discrete network.
Let $X$ be a Lindelof, perfectly normal, $\sigma$-space.
Must $X$ be separable?
Thanks very much.
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For each $n$: show there is a genus $1$ curve over some field $k$ with no points of degree less than $n$, (simple argument / best reference)?
What is the simplest example (or perhaps best reference) for the fact that there are genus $1$ curves (over a field of your choice --- or if you wish, over $\mathbb{Q}$, to make it more exciting) with ...
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Applications of logic in theoretical and practical Computer Science [closed]
Can anyone suggest theoretical and/or practical applications of logic (modal, dynamic, Lukasiewici etc.) in Computer Science (like Markov Chains for linear algebra), as well as some open-source books ...
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Non-homogeneous space $X$ such that $X\cong X\setminus \{x\}$ for all $x\in X$
What is an example of a topological space $(X,\tau)$ with the properties that
$X\cong X\setminus \{x\}$ for all $x\in X$, and
$(X,\tau)$ is not topologically homogeneous
?
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An example of a finite group with some specific permutable subgroups
The following question is about finite groups.
Let $G$ be a finite group and let $H, K \leqslant G$. We say that $H$ permutes with $K$ if $HK = KH$ and in this case $HK \leqslant G$.
The Symbol $\pi ...
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Periodic function $f$ for which $f(x^2)$ is periodic too
There is the following question which was asked multiple times on Math.SE (e.g. here and here) without any final result:
Question: Is there a periodic function $f:\Bbb R \to\Bbb R$ of smallest ...
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Topology generated by complete and incomplete uniformities [closed]
Does there exist a topology which can be induced simultaneously by a complete and an incomplete uniformity?
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Examples of triality in mathematics
There are tons of interesting examples of duality in mathematics (Poincaré duality, Verdier duality, Stone duality, s-duality, Tannaka duality, Koszul duality, Spanier-Whitehead duality ... ). What ...
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Right adjoint completions
Forgive me if this question is not well thought out. I don't know how else to ask it.
The nlab page on completion gives some examples of completions which are left adjoints. These completions are "...
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When does a metric space have "infinite metric dimension"? (Definition of metric dimension)
Definition 1 A subset $B$ of a metric space $(M,d)$ is called a metric basis for $M$ if and only if $$[\forall b \in B,\,d(x,b)=d(y,b)] \implies x = y \,.$$
Definition 2 A metric space $(M,d)$ has &...
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In search for examples concerning pushforward of nef divisors and lc-trivial fibrations
My question is motivated by ideas around the moduli b-divisor of an lc-trivial fibration (see for instance the following paper by Ambro https://arxiv.org/pdf/math/0308143.pdf).
In such a setup, one ...
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