Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [examples]

The tag has no usage guidance.

1
vote
0answers
58 views

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^...
0
votes
0answers
81 views

Example of a function satisfying certain conditions on its derivatives

I am searching for examples of a non-negative function $f \in C^1((0,1];C^\infty _b(\mathbb{R}^n))$ (where $C^\infty _b(\mathbb{R}^n)$ is set of all smooth functions with bounded derivatives, $f(t,x)$ ...
9
votes
2answers
903 views

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?
1
vote
2answers
92 views

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 ...
-1
votes
1answer
133 views

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\...
1
vote
0answers
238 views

Mathematical expressions involving weird constants [closed]

I hope this is a question that fits here and is not duplicated. Also that is clear since it can be a little ambiguous. I was wondering if you know deep expressions, theorems, isomorphisms or ...
2
votes
1answer
79 views

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 $n\in \mathbb Z$?...
5
votes
2answers
173 views

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 ...
8
votes
1answer
430 views

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 ...
16
votes
1answer
299 views

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 ...
4
votes
0answers
109 views

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 ( \...
13
votes
5answers
746 views

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 ...
7
votes
1answer
427 views

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 ...
9
votes
1answer
163 views

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 ...
1
vote
2answers
131 views

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 ...
3
votes
1answer
180 views

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 ...
1
vote
1answer
110 views

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.
17
votes
3answers
867 views

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 ...
3
votes
0answers
157 views

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 ...
1
vote
2answers
136 views

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 ?
2
votes
0answers
126 views

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 ...
8
votes
1answer
357 views

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 ...
0
votes
1answer
66 views

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?
10
votes
2answers
577 views

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 ...
0
votes
0answers
71 views

Amenability of Banach algebras

We know that $C_0(G)$ is Banach $M(G)$ bi-module for locally compact group $G$. I would like to know that is there derivation of $M(G)$ into $C_0(G)$? Or is it inner every derivation of $M(G)$ into $...
2
votes
0answers
123 views

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 "...
20
votes
7answers
2k views

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 $(M,d)$ ...
4
votes
0answers
147 views

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 ...
1
vote
0answers
45 views

Is it possible that a convex cone and its closure both induce vector lattices?

Given a convex cone $P\subset X$ where $X$ is a $K$-vector space, $K=\mathbb{R}\text{ or }\mathbb{C}$ is a field. Suppose that $P$ satisfies positive element stipulations. (1) $X=P-P$. (2) $P\cap-P=...
10
votes
1answer
387 views

Example of Banach spaces with non-unique uniform structures

While it is known that compact Hausdorff spaces admit unique uniform structures, it is further shown by Johson and Lindenstrauss's result that Banach spaces are characterized by their uniform ...
3
votes
1answer
118 views

Semi-metrizable spaces with countable chain condition

Note that $X$ is semi-metrisable iff $X$ is first countable and semi-stratifiable. Definition A topological space $(X,\tau)$ is called semi-metric if there exists a function $g:\omega\times X\to\tau$...
2
votes
3answers
358 views

For every monotonically-decreasing non-negative function $ f $, does there exist a function $ g $ so that $ f g $ is integrable? [closed]

Let $ f $ be a monotonically-decreasing non-negative function satisfying $ \displaystyle \lim_{x \to \infty} f(x) = 0 $. Is it true that the following claim holds? Claim: There exists a function $ ...
4
votes
2answers
173 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 \...
16
votes
2answers
2k views

Consequences of the Birch and Swinnerton-Dyer Conjecture?

Before asking my short question I had made some research. Unfortunately I did not find a good reference with some examples. My question is the following What are the consequences of the Birch and ...
6
votes
1answer
300 views

Non-trivial examples of Stably diffeomorphic 4-manifolds

I am looking for some non-trivial examples of (smooth) 4-mflds $M,N$ such that $M$ and $N$ are STABLY diffeomorphic. I.e. $$M\sharp_n (S^2\times S^2) \cong N \sharp_r (S^2\times S^2)$$ for $r,n$ not ...
27
votes
9answers
2k views

Phenomena of Gerbes

What is your favourite example of Gerbes? I would like to know Where do we find Gerbes in "nature"? The examples could vary from String theory to Galois theory. For example my favourite examples of ...
7
votes
0answers
231 views

Is there a list of examples of orthogonal spectra?

Schwede's symmetric spectra book project provides point-set models of many important spectra as symmetric spectra, including (in §I.1) the sphere spectrum, Eilenberg-Mac Lane spectra, several Thom ...
2
votes
2answers
178 views

Is there a known construction for heavy topologies of all sizes?

Given a set $A$ is there a known way to find a topological space $X$ such that $|A|=|X|<w(X)$? Here $w(X)$ is the weight of the topological space. This is clearly impossible for finite sets $A$. ...
22
votes
8answers
1k views

Applications of logic to group theory?

There seems to be an ever-growing literature on the first-order theory of groups. While I find this interaction between group theory and logic quite appealing, I was wondering the following: Are ...
2
votes
1answer
229 views

An example of Guillemin Sternberg Conjecture

Guillemin Sternberg Conjecture(proved) says that for symplectic manifold $(M,\omega)$ with $[Q,R]=0$ condiction, with compact group action $G$, such that $\mu:M\to \mathfrak g^*$ is regular at $0$, ...
11
votes
1answer
309 views

Torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z^2$

Is there a torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z \oplus \Bbb Z$ (as groups)? Notice that $\Bbb Z$ is not cancellable, so $A \oplus \Bbb Z \...
13
votes
4answers
1k views

Is the “Moebius Stairway” Graph Already Known?

It is a wellknown fact, that Moebius Ladder Graphs have $2n$ vertices, but nowhere could I find any hint of how to generalize them to Graphs with $2n+1$ vertices. Last week I had the idea of giving ...
5
votes
2answers
176 views

Confusion in some notations in Lie sub-algebras of exceptional Lie algebra

I was following Humphrey's Lie algebra for study, and came to study of Weyl groups of root systems. The book has stated orders of Weyl groups of exceptional Lie algebras, and there were no comments or ...
4
votes
2answers
148 views

more examples of non-weakly Lindelöf spaces

A space $X$ is called weakly-Lindelöf if every open cover $\mathcal{U}$ has a countable subcover $\mathcal{U'} \subseteq \mathcal{U}$ such that $\cup \mathcal{U}'$ is dense in $X$. This class seems ...
0
votes
3answers
163 views

Connected $T_2$-spaces with only constant maps between them

If $f:\mathbb{R}\to\mathbb{Q}$ is continuous, then it is constant. Are there infinite connected $T_2$-spaces $X,Y$ such that the only continuous maps $f:X\to Y$ are the constant maps?
2
votes
1answer
83 views

Worst Case Region for a Convex Hull Heuristic

I am currently implementing a heuristic algorithm for planar convex hulls hand would like to know, for which kind of strictly convex region it exhibits worst performance. I know that there are many ...
26
votes
0answers
816 views

Is there a field $F$ which is isomorphic to $F(X,Y)$ but not to $F(X)$?

Is there a field $F$ such that $F \cong F(X,Y)$ as fields, but $F \not \cong F(X)$ as fields? I know only an example of a field $F$ such that $F$ isomorphic to $F(x,y)$ : this is something like $F=k(...
59
votes
7answers
7k views

Theorems demoted back to conjectures

Many mathematicians know the Four Color Theorem and its history: there were two alleged proofs in 1879 and 1880 both of which stood unchallenged for 11 years before flaws were discovered. I am ...
6
votes
2answers
277 views

Applications of isotropic quadratic forms

I will soon be teaching an introductory course on bilinear algebra and quadratic forms. I will likely spend most of the time and effort on positive definite quadratic forms and euclidean spaces. These ...
7
votes
2answers
671 views

Second Stiefel-Whitney class is a square

I'm interested in examples of manifolds which are orientable and such that the second Stiefel-Whitney class is a square. (Of course the second Stiefel-Whitney class should be non-zero.) An easy ...