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Questions tagged [examples]

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4
votes
1answer
110 views

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 ...
0
votes
0answers
66 views

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^\...
1
vote
2answers
324 views

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 ...
7
votes
1answer
227 views

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 $r$ ...
9
votes
1answer
201 views

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 ...
0
votes
0answers
27 views

Example/Reference needed for Laplace equation coupled with another equation

I have been trying to solve a heat-exchanger problem where two fluids are separated by a conducting wall between them and the fluids flow perpendicular to each other. So i need to consider two ...
0
votes
0answers
66 views

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 ...
3
votes
2answers
132 views

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 ...
6
votes
0answers
193 views

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-...
6
votes
0answers
246 views

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 ...
1
vote
1answer
90 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^...
9
votes
2answers
905 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
97 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
135 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\...
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
188 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
443 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
308 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
110 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 ( \...
14
votes
5answers
804 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
166 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
134 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
113 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
904 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
165 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
138 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
127 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
364 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
68 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?
11
votes
2answers
597 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 ...
2
votes
0answers
125 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
150 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
46 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
396 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
122 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
381 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
308 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
239 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
179 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
319 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
177 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 ...