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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0answers
24 views

Verification: Closed Set Expands to Fill Space, but Contains No Open Ball $B_\epsilon(0) $?

I have the proof that $C$ closed, convex, symmetric in Banach space $X$ and $\cup_{n \in N \setminus 0} n.C= X $ then $B_\epsilon(0) \subset C$ for some $\epsilon > 0$. I also have the proof for $...
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2answers
106 views

Concrete examples and computations in differential geometry

I've been studying differential geometry by myself for some time now. I studied a fair amount of the basic general theory and gone through a lot of the exercises from several textbooks. Lately I ...
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3answers
5k views

Examples for proof of geometric vs. algebraic multiplicity

Here you see a supposedly easy proof of a well-known theorem in linear algebra: Although I know I should understand this, I don't :-( Obviously there are too many indices and stuff, so I don't see ...
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3answers
1k views

Convergence in measure of products

Let $\mu$ be a measure on $(X,\mathcal A)$ and let $f, f_1, f_2,\dots$ and $g, g_1, g_2,\dots $be real valued $\mathcal A$- measureable functions on $X$. Show that if $\mu$ is finite, $(f_n)$ ...
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0answers
69 views

Is there a field extension $K / \Bbb Q$ such that $\text{Aut}_{\Bbb Q}(K) \cong \Bbb Z$?

I'm not requiring this extension to be Galois, that's why I wrote $\text{Aut}$ instead of $\text{Gal}$. I'm not very familiar with infinite extensions nor with profinite groups. I don't know if my ...
2
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2answers
48 views

Vertex Coloring Optimal Sum vs Chromatic Number

I am having trouble coming up with an example of when the number of colors used in the optimal solution of the sum coloring problem of a graph is strictly greater than the chromatic number of that ...
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1answer
230 views

Does $A^2 \cong B^2$ imply $A \cong B$ for rings?

If $A$ and $B$ are two unital rings such that $A \times A \cong B \times B$, as rings, does it follows that $A$ and $B$ are isomorphic (as rings)? I believe that the answer is no, but I can't come ...
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2answers
42 views

Continuous functions with orbit of period $3$

I would like to build some continuous functions $f : E \to \Bbb R$ (where $E \subset \Bbb R$ is an interval), such that $$\exists x \in E,\;\; [f(x)≠x≠f(f(x)),\;\; f^3(x):= f(f(f(x)))=x]$$ I tried ...
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1answer
5k views

Bounded Function Which is Not Riemann Integrable

This problem is taken from Problem 2.4.31 (page 84) from Problems in Mathematical Analysis: Integration by W. J. Kaczor, Wiesława J. Kaczor and Maria T. Nowak. Give an example of a bounded function ...
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3answers
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Example for non-Riemann integrable functions

According to Rudin (Principles of Mathematical Analysis) Riemann integrable functions are defined for bounded functions.For every bounded function defined on a closed interval $[a,b]$ Lower Riemann ...
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1answer
117 views

Example of non-homeomorphic compact spaces $K_1$ and $K_2$ such that $K_1\oplus K_1$ is homeomorphic to $K_2\oplus K_2$

Once I heard that there exists two compact spaces $K_1$ and $K_2$ which are non-homeomorphic, but with $K_1\oplus K_1$ homeomorphic to $K_2\oplus K_2$ (where $\oplus$ denotes the topological sum). Is ...
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2answers
42 views

constructing a specific (real-) analytic function

Im searching for an example of a special-behaving analytic function. Maybe you can beat me to constructing such one. The criterias are $g :\mathbb{R}\rightarrow \mathbb{R^+}$ is analytic $g$ is $\...
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3answers
52 views

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 ...
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2answers
350 views

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, ...
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1answer
55 views

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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1answer
50 views

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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30answers
18k views

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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1answer
37 views

If 2 loops with equal base points are homotopic, must they be homotopic relative to the base point?

Let $X $ be a topological space and $\mathbb {S}^1$ be the set of complex numbers with magnitude 1 equipped with the inherited topology from $\mathbb {C} $. If we have 2 loops $f,g:\mathbb {S}^1\...
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6answers
5k views

Simplest Example of a Poset that is not a Lattice

A partially ordered set $(X, \leq)$ is called a lattice if for every pair of elements $x,y \in X$ both the infimum and suprememum of the set $\{x,y\}$ exists. I'm trying to get an intuition for how a ...
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2answers
38 views

Useful analogy to interpret the notion of evolutionary stable strategy (ESS)

I am seeking a good analogy to understand the concept of evolutionary stable strategy (state) Let $\pi$ denote the fitness of a population, $\pi_{ij}$ is the fitness of strategy $i$ against strategy $...
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1answer
42 views

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"? ...
5
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1answer
47 views

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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0answers
64 views

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 ...
6
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5answers
912 views

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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0answers
31 views

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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0answers
78 views

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 ...
0
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2answers
44 views

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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14answers
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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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2answers
80 views

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 (...
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1answer
49 views

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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32 views

Example of function

Consider $O(2)$, the orthogonal group in dimension 2, $C_{k}$ the cyclic group of order $k$ and $D_{k}$ the dihedral group of order $2k$. Is it possible to have an example of function $f\in L^{\infty}(...
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2answers
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Lebesgue measurable but not Borel measurable

I'm trying to find a set which is Lebesgue measurable but not Borel measurable. So I was thinking of taking a Lebesgue set of measure zero and intersecting it with something so that the result is not ...
2
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1answer
28 views

If $f:X\to Y$ is continuous and $X$ is a totally bounded metric space, is $f(X)$ also bounded?

If $X$ and $Y$ are metric spaces and $X$ is totally bounded and $f:X\to Y$ is continuous (not necessarily uniform), is it true that $f(X)$ is also bounded? How can we prove it or is there any counter ...
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1answer
73 views

Counterexamples in Analysis

I want to (dis)prove the following statement: A sequence of functions which converges almost uniformly implies uniform convergence for that sequence of functions. I'm sure I've read up on a ...
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3answers
825 views

M compact $p\in M$ , there exist $f:M-p\to M-p$ continuous bijection but not homeomorphism?

Let M be a compact metric space. We know that if $ g:M\to M$ is a continuous bijection then it's a homeomorphism. But I want to know, if I have a continuous bijection $ f:M - \left\{ p \right\} \to M -...
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5answers
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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 ...
2
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1answer
24 views

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 ...
2
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1answer
37 views

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 ...
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1answer
25 views

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)\...
3
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0answers
35 views

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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2answers
64 views

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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2answers
51 views

An example of subset $A$ such that $A \cap K$ is open in $K$ for each compact set $K$, but $A$ is not open. [duplicate]

Let $X$ be a topological space. For any $A \subseteq X$, consider two possible conditions on $A$: 1) $A$ is open in $X$; 2) $A \cap K$ is open in $K$, for each compact set $K \subseteq X$. Then $(...
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2answers
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Examples of compact sets that are infinite dimensional and not bounded

In an infinite dimensional Banach space, does a compact subset have to be finite dimensional? I know it cannot contain any infinite dimensional balls, if this mean it has to be finite dimensional, ...
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0answers
56 views

Collection is uniformly integrable, but individual is not integrable

Could you give me an example about: "a collection of functions that is uniformly integrable but each (or some) function in the collection is not integrable." This sounds counterintuitive? However ...
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Example of Non-Measurable Sets in Product Space

If $\mu$ and $\nu$ are measures on $X$ and $Y$, is there an example of a set $E\subset X\times Y$ such that $E_x,E^y$ are measurable for all $(x,y)$ but $E$ is not measurable with respect to $\mu\...
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1answer
21 views

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 ...