Questions tagged [examples]
For questions requesting examples of a certain structure or phenomenon
566 questions
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Is there a chordal graph whose toughness is exactly $\frac{3}{2}$ and which is non-Hamiltonian?
Kratsch once asked whether every $\frac{3}{2}$-tough chordal graph is Hamiltonian. This question was answered in the negative by Bauer et al., [1] who constructed a family of non-Hamiltonian chordal ...
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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 ...
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$L^2$-functions orthogonal to their own Fourier transform
It is well-known that, besides the standard Gaussian $e^{-|x|^2/2}$, 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 ...
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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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203
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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 ...
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187
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Interpreting $1/f$ 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 $\mathbb{R}^n \setminus \{0\}$, $f(0) = 0$, and $1/f$, viewed as a distribution on $\mathbb{R}^n \setminus \{0\}$, ...
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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 $0 \leq \mu(C) < 1$, which are perfect, nowhere dense, and totally ...
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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}{:=}
\...
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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 $x, y\in\mathcal{D}$, implies $\alpha x+(1-\alpha)y\in\mathcal{D}$ for all $\alpha\in[0,1]$, 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 ...
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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 ...
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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 ...
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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 $\alpha$, $D^\...
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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 ...
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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....
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What is the smallest and "best" 27 lines configuration? And what is its symmetry group?
I was this past year working with a bright high-schooler on algebraic geometry following Reid's book Undergraduate Algebraic Geometry, and we got all the way to proving that there is at least one line ...
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Proving that there are no solutions other than a few known ones
My question is mostly out of curiosity, with probably no other use, but here it is. I will need to provide a bit of background.
I heard from someone who works with elliptic curves that often proving ...
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Co-locating slowly increasing smooth functions in two different ways
This question is subsequent from my previous one.
I will write everything in detail for the sake of completeness.
Let $g_1$ and $g_2$ be smooth functions on $\mathbb{R}$, whose derivatives of all ...
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Subtlety of identifying $W^{k,p}\bigl([0,1] \bigr)$ and $W^{k,p}(S^1)$ - from ME
I apologize for repeating the same question from ME, but it seems more subtle than I expected.
Let me fix the notations here first:
\begin{equation}
C^\infty_c(0,1):= \{ f : (0,1) \to \mathbb{C} \mid ...
- 3,745
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348
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Existence of antiderivative w.r.t. any given multi-index for tempered distributions
I originally posted this question on ME, but I find it a lot more nontrivial than expected. So, I post it here.
Let $T$ be a tempered distribution on $\mathbb{R}^n$. Then, it is a well-known ...
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Elementary functions such that $\sum_{n=2}^{\infty} f(n) \left( \zeta(n)-1 \right)$ can be evaluated, but $\sum_{n=2}^{\infty} f(n)$ can't
Background
The general context for this question is the topic of rational zeta series. What I've found so far, is that it usually the case that sums of the form $$\zeta_{f} := \sum_{n=2}^{\infty} f(n) ...
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Twisting cochain intuition
I'm currently reading through Ed Brown's paper "Twisted tensor products, I", (MR105687, Zbl 0199.58201) and I couldn't find any simple examples of twisting cochains. I understand all ...
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244
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What is the most general notion of exactness for functors between triangulated categories?
For triangulated categories $T,T'$ I would like to define "weakly exact" functors as those that respect cones, that is, $F(Cone f)\cong Cone(F(f))$ for any $T$-morphism $f$, and I do not ...
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For any $p, q \in [1,\infty]$ and $s \in (0,\infty)$, can we find some $f \in L^q - W^{s,p}$?
Sobolev inequalities show us when we can embed a Sobolev space into another.
However, I wonder if these inclusions are always proper.
More specifically, let $\Omega \subset \mathbb{R}^n$ be a bounded ...
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Teaching suggestions for Kleene fixed point theorem
I will take over two lectures from a colleague in which we discuss fixed point theory in the context of complete partial orders, and culminates in showing the Kleene fixed point theorem (see f.e. ...
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568
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Concrete examples of derived categories
What examples of abelian categories $\mathcal{A}$ are there such that the derived category $\mathcal{D}(\mathcal{A})$ can be described concretely? For example, is there a concrete way of describing $\...
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Is the Schwartz space a tame Frechet space?
I ran into the following definition of tame Frechet spaces and Nash-Moser therem.
It says that the space of smooth functions on a compact manifold is tame Frechet.
However, I wonder if
The Schwartz ...
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320
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Hopf monads in categorical probability theory
1. Context. According to [1], probability monads are arguably the most important concept in categorical probability theory. In [2] Fritz and Perrone argue that "in order for a monad to really ...
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Nice, concrete example of pl-flipping contraction
In a course I'm giving on the MMP, I am discussing the importance of Shokurov's notion of a pl-flipping contraction for showing that flips exist for arbitrary flipping contractions. Does someone have ...
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Weakly contractible $X$, but none of the maps $*\to X$ are cofibrations
Let $\mathrm{Top}$ be the category of all topological spaces and continuous maps. The Quillen model structure on $\mathrm{Top}$ has weak equvalences $W = \{ \text{weak homotopy equivalences} \}$, ...
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An example of module which is square-free, CS, NOT C3, and NOT nonsingular
Let $M$ be a right $R$-module ($R$ has unity). Recall that $M$ is called square-free if $M$ does not contain two nonzero isomorphic submodules with zero intersection. $M$ is called CS if every ...
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Examples of non-polynomial comonads on Set?
Question: What are examples of comonads on $\mathbf{Set}$ that are not polynomial?
Background: polynomial functors and comonads on Set
A functor $F\colon\mathbf{Set}\to\mathbf{Set}$ is called ...
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Natural set-theoretic principles implying the Ground Axiom
The Ground Axiom states that the set-theoretic universe is not a set-forcing extension of an inner model. By
Reitz, it is first-order expressible and easy to force over any given ZFC model with class-...
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Is $L^2(I,\mathbb Z)$ homeomorphic to the Hilbert space?
I am somehow puzzled by the subset $G:=L^2(I,\mathbb Z)$ of $H:=L^2(I,\mathbb R)$ of all integer valued functions on $I=[0,1]$ (in fact I mentioned as an example in this old MO question).
Some simple ...
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Results that hold for the complex numbers but not for algebraically closed fields of characteristic zero
When a result is stated for the field of complex numbers it can usually be extended to a result for an algebraically closed field of characteristic zero. I would like to see a list of results that ...
- 157
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322
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Hardy space inclusion in the right-half plane
I'm looking for an example of a function $u \in H_2$ such that $u \notin H_\infty$, where $H_p$ is the Hardy space on the right-half plane. Since this notation is perhaps not standard, here is a ...
- 435
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Periodic tilings in finite type tiling spaces and substitution tiling spaces
I was reviewing the following statement from a survey by E. Arthur Robinson about tilings in $\mathbb{R}^d$ to better understand geometric tiling rather than tilings over symbols. I consider the ...
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Closed subset of unit ball with peculiar connected components
Let $n\geq 2$ and denote by $B\subset \mathbb{R}^n$ the closed unit ball.
Does there exist a closed subset $A\subset B$ containing $0\in \mathbb{R}^n$ with the following properties i,ii,iii?
i) $\{0\}$...
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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(...
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640
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Non-homeomorphic connected one-dimensional Hausdorff spaces that have continuous bijections between them in both sides
I need to construct an example of two non-homeomorphic connected one-dimensional Hausdorff spaces that have continuous bijections between them in both sides. Spaces should have induced ("good&...
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460
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Example of trickiness of finite lattice representation problem?
I'm trying to come up with a good explanation for my students of why the finite lattice representation problem is difficult. I've already shown that the "greedy approach" to representing the ...
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Torsion-free virtually free-by-cyclic groups
Is it known if there are any examples of a finitely generated group $G$ such that:
$G$ has a finite index subgroup $H$ which is free-by-cyclic
$G$ itself is not free-by-cyclic
$G$ is torsion-free
...
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Examples of convergence in distribution not implying convergence in moments
It is well know that the convergence in distributions does not necessarily imply convergence in expectation, but implies convergence in expectation of bounded continuous functions.
Let $\{X_n\}$ be a ...
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269
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Existence of the special entire Hardy space function with infinitely many zeros in the strip
Question. Does there exist an entire function $h$ satisfying three following assertions:
$h$ belongs to the $H^2$ Hardy space in every horizontal upper half-plane;
$zh - 1$ belongs to $H^2(\mathbb{C}...
- 607
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295
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Existence of nonzero entire function with restrictions of growth
Question. Is there an entire function $F$ satisfying first two or all three of the following assertions:
$F(z)\neq 0$ for all $z\in \mathbb{C}$;
$1/F - 1\in H^2(\mathbb{C}_+)$ -- the classical Hardy ...
- 607
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0
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245
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A zoo of derivations
Recall that given a $k$-algebra $A$, a derivation on $A$ is a $k$-linear morphism $d:A\to A$ such that $$d(ab)=d(a)b+ad(b).$$
The use of derivations is of paramount importance in mathematics. I think ...
9
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Is there a nonpolynomial $C^\infty$ function $f$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every $q >1$?
The question is as in the title:
Is there a nonpolynomial $C^\infty$ function $f$ on $\mathbb{R}$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every natural ...
- 3,745
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On the measure of nonconvexity (MNC)
I'm actually working on the measure of nonconvexity and its application. Especially, the Eisenfeld–Lakshmikantham MNC defined - in a Banach space - by:
$$\alpha(A)=\sup_{b\in\operatorname{conv}(A)} \...
- 301
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Natural cotransformations and "dual" co/limits
$\DeclareMathOperator{\id}{\mathrm{id}}\DeclareMathOperator{\Hom}{\mathrm{Hom}}\DeclareMathOperator{\UnCoNat}{\mathrm{UnCoNat}}\DeclareMathOperator{\UnNat}{\mathrm{UnNat}}\DeclareMathOperator{\CoNat}{\...
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