# Tagged Questions

Examples and counterexamples are great ways to learn about the intricacies of definitions in mathematics. Counterexamples are especially useful in topology and analysis where most things are fairly intuitive, but every now and then one may run into borderline cases where the naive intuition may ...

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### Using symmetry to solve Non-Linear Ordinary Differential Equations [on hold]

I know that general rules or general guidance for solving nonlinear differential equations do not exist, but im curious about the various ingenious ways that are being used to solve some of them. I ...
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### Intermediate value property with no continuity

Definition: A real function f has the intermediate value property on an interval I containing [a,b] if f(a) < v < f(b) or f(b) < v < f(a); that is, if v is between f(a) and f(b), there is ...
2answers
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### My problem in understanding the minimal counterexample technique

If minimal counterexample method of proof is to assume to opposite of an argument is true and then finding a counterexample for the opposite and then concluding the validity of the original argument, ...
1answer
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### In which topologies do open sets maintain open under countable or arbitrary intersection?

We know that in the usual topology, countable or arbitrary intersection of open sets can zoom into a singleton, hence is not in the topology. I am curious if there is well known classes of ...
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### Not Abelian group G with Z(G) that contains only two elements? [closed]

Is there an example of which is not Abelian group G, and Z(G) contains only two elements?
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### A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language

The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to ...
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### Formal definition of “proexample”. [closed]

Where in the literature do we find the preferred formal definition of “proexample” as in: the number zero is a proexample for the existential sentence "some integer is neither positive nor negative"? ...
1answer
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### What is a measurable set $E \subset [0, 1]$ such that the closure of $E$ is $[0, 1]$ and $m(E) = \epsilon$?

Suppose $\epsilon \in (0, 1)$ and $m$ is Lebesgue measure. What is a measurable set $E \subset [0, 1]$ such that the closure of $E$ is $[0, 1]$ and $m(E) = \epsilon$?
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### If $f$ is defined on $\mathbb{R}$ and $f$ is unbounded, is it necessarily true that $\lim_{|x|\to\infty} |f(x)| = \infty$?

This question comes from the following problem: A real-valued function $f$ defined on $\mathbb{R}$ has the following property: For every positive $\epsilon$, there exists positive $\delta$ such ...
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### Formal definition of “counterexample”.

What is the preferred formal definition of “counterexample” as in: zero is a counterexample for "every integer is either positive or negative". Where in the literature is the notion of “counterexample”...
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### What is an example of lower semicontinuous functions not satisfying this?

Let $X$ be a locally compact Hausdorff space and $\mu$ be a Radon measure on $X$. Let $u:X\rightarrow [0,\infty]$ be a lower semicontinuous function such that $\int_X u d\mu <\infty$. Then, does ...
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### Counterexample 3n + 1 problem (Collatz) Exponential and linear Diophantine equation

So, I have found a sufficient condition (not necessary) for finding a counterexample to the 3n + 1 problem, namely the existence of solutions for the following two-parameter family of Diophantine ...
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### Examples of $R$-modules $X$ such that $(X \setminus TX) \cup \{0\}$ isn't a submodule.

Work over an ambient commutative ring with unity. Given a module $X$, write $TX$ for its submodule of torsion elements. Suppose we want to find the "submodule" of torsion-free elements of $X$. So ...
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### How are proofs formatted when the answer is a counterexample?

Suppose it is asked: Prove or find a counterexample: the sum of two integers is odd The fact that 1 + 1 = 2 is a counterexample that disproves that statement. What is the proper format in which ...
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### Noteworthy examples of finite categories

So far all the finite categories I have encountered fall into one of these c̶a̶t̶e̶g̶o̶r̶i̶e̶s̶ sets: finite monoids finite preorders just formal devices to explain, what a "diagram" in another (...
1answer
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### In what space is a closed set is not or not necessarily $G_\delta$

We know that A closed set in a metric space is $G_\delta$ Is there any topological space where a closed set is not necessarily $G_\delta$? I am thinking a space where singletons are well known to be ...
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### Examples and further results about the order of the product of two elements in a group

Let $G$ be a group and let $a,b$ be two elements of $G$. What can we say about the order of their product $ab$? Wikipedia says "not much": There is no general formula relating the order of a ...
1answer
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### Discontinuous (no continuous representative) function $u \in W^{1,p}(\Omega)$?

For an open set $\Omega \subset \mathbb{R}^N$ and $p \leq N$ we know that there are functions in $W^{1,p}(\Omega)$ which don't belong to $L^{\infty}(\Omega)$. We also know that for $p>N$ there is a ...
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### Counterexamples about locally compact sets on the real line

Is there a counterexample in the space $\mathbb{R}$ with it's usual metric to the statements: The union of two locally compact subsets of $\mathbb{R}$ is locally compact The complement of a locally ...
1answer
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### Example of nonuniqueness of asymptotes of a ray

Let $(M, g)$ be a complete Riemannian manifold and let $\gamma : [0, \infty) \to \mathbb{R}$ be a ray, i.e. a unit speed geodesic such that for every $s, t \ge 0$ :  dist\big(\gamma(s), \gamma(t)\...
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### On the right adjoints of inverse image functors ($f^* \dashv \forall_f$)

Given is an ambient category $\mathcal{A}$ with finite limits. For the remainder of this post, a subobject of an object $A$ is a mono $m : M\to A$ and $\operatorname{Sub} A$ is the preordered set (/...
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### Basis Function Algorithm, In The NURBS book

On page 74, Peigl explained an algorithm about computing a single basis function. first lines of this algorithm are handling some special cases. ...
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### Suppose $ab\equiv 0 \pmod{n}$, and that $a$ and $b$ are positive integers both less than $n$. Does it follow that either $a | n$ or $b | n$?

Suppose $ab\equiv 0 \pmod{n}$, and that $a$ and $b$ are positive integers both less than $n$. Does it follow that either $a | n$ or $b | n$? If it does follow, give a proof. If it doesn’t, then give ...
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### Obvious application of internal category theory outside from topoi

The nLab lists a bunch of examples for internal categories in various categories. If we think of a topos as a "universe" for mathematics the need for internal categories in a topos becomes obvious. ...
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### Simple example of a function which is in $W^{1,p}(\Omega)$ but not in $L^{\infty}(\Omega)$?

I am looking for a simple (intuitive) example of a function $u \in W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is an open set and, obviously, $p \leq N$. Sobolev embedding theorem asserts ...