# Questions tagged [examples]

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### Real-world example of a Banach *-algebra with a nonzero *-radical

Is there a real-world example of a Banach *-algebra with a nonzero *-radical (intersection of kernels of all *-representations)? Textbooks give examples of finite-dimensional algebras with degenerate ...
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### Proofs by Schubert calculus and combinatorics

Do you know some examples proved by two different methods: 1. Schubert calculus, 2. combinatorial method.
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### Contrasting theorems in classical logic and constructivism

Is it possible there are examples of where classical logic proves a theorem that provably is false within constructivism? Is so what are some examples? What are some examples of most contrasting ...
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### Examples of faithful functors not injective on objects

As is well-known, a faithful functor need not be injective on objects. What are some good examples to illustrate this point?
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### Examples of complicated parametric Jordan curves

For test purposes I need parametric Jordan curves that are complicated in the sense of having many inflection points and ideally no symmetries. When doing online search I always land at complex ...
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### What's an illustrative example of a tame algebra?

A finite-dimensional associative $\mathbf{k}$-algebra $\mathbf{k}Q/I$ is of tame representation type if for each dimension vector $d\geq 0$, with the exception of maybe finitely many dimension vectors ...
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### Horizontal categorification: Two questions

According to the nlab, horizontal categorification is a process in which a concept is realized to be equivalent to a certain type of category with a single object, and then this concept is generalized ...
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### Infinitely generated non-free group with all proper subgroups free

Is there any example of group $G$ satisfying the following properties? $G$ is non-abelian, infinitely generated (i.e. it is not finitely generated) and not a free group. $H< G$ implies that $H$ is ...
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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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### Examples of proofs using induction or recursion on a big recursive ordinal

There are many proofs use induction or recursion on $\omega$, or on an arbitary (may be uncountable) ordinal. Are there some good examples of proofs which use a big but computable ordinal? The ...
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### What are examples of (collections of) papers which “close” a field?

There is sometimes talk of fields of mathematics being "closed", "ended", or "completed" by a paper or collection of papers. It seems as though this could happen in two ways: A total characterisation,...
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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 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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### 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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### 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 , 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  (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 , 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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### Eventually non vanishing tors

Let $A$ be a commutative $k$-algebra, for $k$ a field of characteristic $0$. Let $Perf_{A}$ denote the dg category of cohomologically graded $A$-modules and let $M\in Perf_{A}$ be a classical perfect ...
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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 ...
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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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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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