Questions tagged [examples]
For questions requesting examples of a certain structure or phenomenon
554 questions
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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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Is there a finitely complete category with terminal object but NO subobject classifier?
This came up today while thinking about topoi in seminar, as the title suggests my question is;
Is there a finitely complete category with terminal object but NO subobject classifier?
Hopefully if ...
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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 are your favorite concrete examples of limits or colimits that you would compute during lunch?
(The title was initially "What are your favorite concrete examples that you would compute on the table during lunch to convince a working mathematician that the notions of limits and colimits are not ...
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are there natural examples of classical mechanics that happens on a symplectic manifold that isn't a cotangent bundle?
I'm curious about just how far the abstraction to a symplectic formalism can be justified by appeal to actual physical examples. There's good motivation, for example, for working over an arbitrary ...
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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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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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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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Examples of ubiquitous objects that are hard to find?
I've been wrestling with a certain research problem for a few years now, and I wonder if it's an instance of a more general problem with other important instances. I'll first describe a general ...
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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 $...
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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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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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Examples of seemingly elementary problems that are hard to solve?
I'm looking for a list of problems such that
a) any undergraduate student who took multivariable calculus and linear algebra can understand the statements, (Edit: the definition of understanding here ...
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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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books well-motivated with explicit examples
It is ultimately a matter of personal taste, but I prefer to see a long explicit example, before jumping into the usual definition-theorem path (hopefully I am not the only one here). My problem is ...
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Excellent uses of induction and recursion
Can you make an example of a great proof by induction or construction by recursion?
Given that you already have your own idea of what "great" means, here it can also be taken to mean that the chosen ...
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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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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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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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Algebraic varieties which are topological manifolds
Inspired by this thread, which concludes that a non-singular variety over the complex numbers is naturally a smooth manifold, does anyone know conditions that imply that the topological space ...
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What's the idea behind Carleman estimates?
A standard Carleman-type estimate is of the form
$$
\sum_{|\alpha|<m}{\tau^{2(m-|\alpha|-1)}\int{|D^{\alpha}u|^{2}e^{2\tau\phi}}dx}\leq K\int{|Pu|^{2}e^{2\tau\phi}dx},\quad u\in C_{0}^{\infty},
$$
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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 ...
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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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Regular spaces that are not completely regular
In the undergraduate toplogy course we were given examples of spaces that are $T_i$ but not $T_{i+1}$ for $i=0,\ldots,4$. However, no example of a space which is $T_3$ but not $T_{3.5}$ was given. ...
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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 ...
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Fundamental Examples
It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please)
I'd love to learn about ...
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Complete theory with exactly n countable models?
For $n$ an integer greater than $2$, Can one always get a complete theory over a finite language with exactly $n$ models (up to isomorphism)?
There’s a theorem that says that $2$ is impossible.
My ...
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A finitely generated, locally free module over a domain which is not projective?
This is a followup to a previous question
What is the right definition of the Picard group of a commutative ring?
where I was worried about the distinction between invertible modules and rank one ...
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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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Examples of Riemannian Submersions
Is there any example of a Riemannian submersion, which is no fibration?
As far as I know, a (any) submersion is locally, but not globally, given by a fibration. The converse holds globally. ...
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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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Difficult examples for Frankl's union-closed conjecture
Frankl's well-known union-closed conjecture states that if F is a finite family of sets that is closed under taking unions (that is, if A and B belong to the family then so does $A\cup B$), then there ...
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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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Exotic principal ideal domains
Recently I realized that the only PIDs I know how to write down that aren't fields are $\mathbb{Z}, F[x]$ for $F$ a field, integral closures of these in finite extensions of their fraction fields that ...
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Nonessential use of large cardinals
In Awfully sophisticated proof for simple facts, we are asked for examples of complex proofs of simple results. To quote from the questioner's post, we are asked for proofs that are akin to "nuking ...
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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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Elementary applications of Krein-Milman
This is a cross-post from MSE: Elementary applications of Krein-Milman. I'm starting to suspect that the question just doesn't really have a great answer, it's worth a try.
Recall that the Krein-...
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If $\Omega_{X/Y}$ is locally free of rank $\mathrm{dim}\left(X\right)-\mathrm{dim}\left(Y\right)$, is $X\rightarrow Y$ smooth?
Suppose I have a morphism $f:X\rightarrow Y$ such that the relative sheaf of differentials $\Omega_{X/Y}$ is locally free. Does it follow that $f$ is smooth?
The answer is no, but for a silly reason. ...
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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 ...
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