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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97 views

Handwaving gone wrong

My motivation for this question is twofold: On one hand, I'm studying algebraic topology, where - at least in the book written by Hatcher - there is quite a lot of handwaving (e.g. maps are continous ...
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0answers
43 views

Examples of reduced associative algebras

An associative $K$-algebra A is called reduced if $A/rad(A)$ has no nilpotent elements. It can be shown that this is equivalent to that $A/rad(A)$ is a isomorphic to a direct sum of division algebras. ...
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0answers
40 views

How to find intelligently counterexamples for (dis)proofs about matrices?

Let's say I'm asked to give a counterexample for a claim about matrices, such as The elementwise product of two positive semi-definite matrices is positive semi-definite. It's easy enough to do ...
2
votes
3answers
113 views

What are some examples of coolrings that cannot be expressed in the form $R[X]$?

Let $K$ denote a field. Then the polynomial ring $K[x]$ has the property that the sum of two units is either a unit, or zero. I'll bet there's heaps of other examples, though. So let a coolring be a ...
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2answers
41 views

Compact Set: Cover by Merely Neighborhoods

Disclaimer: This thread is just a record of thoughts and written in Q&A style. A subset is compact if every open cover admits a finite subcover. What if one replaces open covers with covers by ...
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1answer
104 views

How to convert a problem to a stars and bars problem?

Continued question from here. With certain questions I have $x_i$ being constrained by various different inequalities, I want to know how to remove these from the problem, to bring me back to a ...
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1answer
59 views

Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}\;$

I am unsatisfied with the answers here. (Half of which used algebraic methods despite being advised not to!) Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}$ ...
3
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1answer
59 views

Examples of division algebras

Together with the Grunwald–Wang theorem, the Albert–Brauer–Hasse–Noether theorem implies that every central simple algebra over an algebraic number field is cyclic, i.e. can be obtained by an explicit ...
0
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2answers
39 views

Is the function $x \mapsto \max\{f(x),g(x)\}$ always an automorphism if $f$ and $g$ are?

Consider a totally ordered set $X$ and a pair of automorphisms (i.e., order-preserving bijections) $f,g$ of $X.$ Then the function $$h:x \in X \mapsto \max\{f(x),g(x)\} \in X$$ is an order-preserving ...
6
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1answer
58 views

A locally metrizable, Lindelöf Hausdorff space that is not metrizable

I am looking for an example of locally metrizable, Lindelöf Hausdorff space that is not metrizable. I've proved that if such space is regular, then it is metrizable. The proof relies on the ability ...
0
votes
3answers
93 views

How to use stars and bars(combinatorics)

How to use the stars and bars method? Say I want to find number of combinations I can get with $x_1+x_2+x_3+x_4=22$ Where $x_i\in\mathbb{N}$ Is this the correct time to apply the method?
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0answers
18 views

Examples of inner forms

Let $G$ and $G′$ be two linear algebraic groups over a field $F$. From what I understand, $G$ is called an inner form of $G′$ if $G$ and $G′$ are isomorphic over a the (or an?) algebraic closure of ...
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2answers
35 views

Connectedness and non-local-connectedness of a subspace of $\mathbb R^2$

Let $(X,\tau)$ be the subspace of $\mathbb R^2$ consisting of the points in the line segments joining $(0,1)$ to $(0,0)$ and to all the points $(1/n,0)$, $n=1,2,3,\ldots$. Show that $(X,\tau)$ ...
53
votes
4answers
2k views

Is it possible for a function to be in $L^p$ for only one $p$?

I'm wondering if it's possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain). One can use interpolation to show that ...
3
votes
3answers
51 views

Metrizability of quotient spaces of metric spaces

Suppose $X$ a metric space and $\sim$ an equivalence relation on $X$. Is the space $X/\mathord{\sim}$ metrizable? I think that the answer is no, but I could not arrive at a counterexample.
261
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31answers
16k views

Examples of apparent patterns that eventually fail

Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of 'proof'. I receive responses like: "surely if the Collatz ...
7
votes
3answers
729 views

Examples of uncountable sets with zero Lebesgue measure

I would like examples of uncountable subsets of $\mathbb{R}$ that have zero Lebesgue measure and are not obtained by the Cantor set. Thanks.
3
votes
4answers
686 views

A set that it is uncountable, has measure zero, and is not compact

I want a example of a set that it is uncountable and has measure zero and not compact? Cantor set has these properties except not compactness.
64
votes
31answers
7k views

What are some conceptualizations that work in mathematics but are not strictly true?

I am having an argument with someone who thinks that it's never justified to teach something that is not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is ...
0
votes
0answers
15 views

An example of $k$-independent distributions.

I'm trying to better understand the idea of $k$-independence in distributions. The idea is that a distribution $\mu$ over $\{0,1\}^n$ is $k$-independent if any restriction of $\mu$ to $k$ variables ...
53
votes
8answers
2k views

Problems that become easier in a more general form.

When solving a problem, we often look at some special cases first, and then try to work our way up to the general case. It would be interesting to see some counterexamples to this mental process, ...
5
votes
2answers
66 views

A Hausdorff space which is not completely regular

My example is, $f : \mathbb{R}^+ \to \mathbb{R}$ defined by: $$f(x) = \begin{cases} x, &\text{if }0 \leq x < 1 \\ \tfrac{1}{x}, &\text{if }x \geq 1. \end{cases}$$ Even though $f(0)=0$ but ...
1
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0answers
13 views

Inclusion of commutators on classical pseudodifferential operators

We denote by $Cl^\mu$ the class of classical pseudo-differential operators of order $\mu$. Consider the notation $$[Cl^{a},Cl^{b}]\hookrightarrow [Cl^{a'},Cl^{b'}]$$ which means that a commutator on ...
3
votes
2answers
290 views

Interesting Problems for NonMath Majors

Sometime in the upcoming future, I will be doing a presentation as a college alumni to a bunch of undergrads from an organization I was in college. I did a dual major in mathematics and computer ...
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1answer
35 views

One-sided and two-sided cancellability

Is there a semigroup $S$ with right- and left-cancellable elements but no elements cancellable from both sides? $s\in S$ is left-cancellable if for any $a,b\in S$ we have $$sa=sb\implies a=b$$ and ...
4
votes
2answers
84 views

A non-UFD such that $a^2 \mid b^2$ does not lead to $a\mid b$

Is there any non-UFD that is a commutative ring such that $a^2 \mid b^2$ does not always lead to $a\mid b$? It would be preferable if examples are something that does not involve ...
1
vote
1answer
23 views

Converse to Exactness of localization of modules

It is a standard fact that if $R$ is a ring, and $$\tag{1} 0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0 $$ a short exact sequence of $R$-modules, if $S$ is a multiplicative ...
0
votes
1answer
8 views

closedness of a sublattice under complements

Let $A$ be a bounded sublattice of the bounded lattice $(X,\le)$ with $$\max A=\max X, ~~\min A=\min X$$ Let $a,b\in X$ be complements and $a\in A$. Is $b\in A$?
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votes
2answers
1k views

The direct sum of two closed subspace is closed? (Hilbert space)

I know that if $X$ is a Banach space, then, the direct sum of two closed subspace $X_1$ and $X_2$ is not necessarily closed. But what if $X$ is Hilbert? I assume there is something to do with the ...
0
votes
1answer
33 views

Nowhere dense set - coarser vs. finer topology

Let $X$ be a set and let $\tau_1\subseteq\tau_2$ be topologies on $X$. Suppose that $A\subseteq X$ is nowhere dense in $\left(X, \tau_2\right)$. I was wondering if it follows that $A$ is nowhere dense ...
4
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1answer
48 views

Does $S^\bot+T^\bot = (S\cap T)^\bot$ hold in infinite-dimensional spaces?

If $S$ and $T$ are subspaces of some finite-dimensional inner product space then $$S^\bot+T^\bot = (S\cap T)^\bot.$$ See, for example, this post or this post Does it hold in infinite-dimensional ...
0
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1answer
47 views

Are there standard parameters for the Weierstrass nowhere differentiable function?

On Wikipedia, the Weierstass non-differentiable function is defined as: $$f(x)=\sum^{\infty}_{n=0}a^n\cos(b^n\pi x)$$ where $0<a<1$, $0<b$, and $ab>1+\frac 32 \pi$ Since it seems like, ...
0
votes
1answer
46 views

Need an example/counterexample of continuous and increasing function.

If $\mu$ is a finite measure on the measurable space $\big( X, \mathscr{F} \big)$, $f : X\to [ 0, +\infty)$ is measurable. Then $\textbf{does it exist a continuous function $g : [ 0, +\infty)\to [ ...
0
votes
2answers
48 views

Significance of low order terms in base expansion of integer square root

My head is turning into a uniform gel of random thoughts! I cannot see a proof or find a counterexample to the following: Conjecture: Let integer $x$ be expressed as $a_3 \, b^3 + a_2 \, b^2 + a_1 \, ...
12
votes
6answers
285 views

Function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times

Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times? If not, how could I prove that such a function does not exist?
1
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1answer
25 views

Diff. Eq. Example with Matrices

I'm currently working on a side project of mine that deals with $\sin(A)$ and $\cos(B)$, where $A,B\in\mathbb{C}^{nxn}$. I'm trying to find some interesting (or non-interesting) examples where one ...
4
votes
3answers
102 views

Is every such function convex or concave?

I was pondering convex functions today, and the following questions naturally posed themselves. Call a function $f : \mathbb{R} \rightarrow \mathbb{R}$ $2$-limited iff for all $m,c \in \mathbb{R},$ ...
1
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1answer
31 views

Counterexample: Convergence in finite dimensional distributions does not imply weak convergence

I'm working at the following exercise: Give an example of a sequence of stochastic processes $(\mathbb{X}^n)_{n\geq 1}$ such that the finite dimensional distributions converge to the finite ...
1
vote
1answer
44 views

Series converges but term by term integration not valid?

Give an example of a series $\sum g_n$ of Lebesgue integrable functions on $\mathbb{R}$ that converges but for which term by term integration is not valid. This is last minute exam revision so I do ...
9
votes
3answers
288 views

Examples of non-Riemann integrable functions that appear “in nature”?

I am teaching an honours calculus class, and am looking for examples on non-integrable functions that occur somewhere real in mathematics. (The standard example of 1 on $\mathbb{Q}$ and 0 elsewhere ...
3
votes
1answer
49 views

Closed surjection that does not preserve regularity

Def Map $p\colon X\rightarrow Y$ is perfect if it is a closed surjection and $p^{-1}\left(\left\{y\right\}\right)$ is compact for each $y\in Y$ It is well known that perfect maps preserve regularity, ...
16
votes
10answers
2k views

Polygons with equal area and perimeter but different number of sides?

Let's say we have two polygons with different numbers of sides. They can be any sort of shape, but they have to have the same area, and perimeter. There could be such possibilities, but can someone ...
2
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1answer
24 views

Absolutely continuous probability measures example

I was given the following definition: Then this example: It is said that $\mathbb P_1\ll\mathbb P_2$ , but I don't really see it.Please help.
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0answers
27 views

Kullback-Leibler $KL(p,q)\neq KL(q,p)$

I'm doing a course of Artificial Intelligence and in my homework I must to provide a counter example to show that the Kullback-Leibler distance is not a symmetric function of its arguments: $$ ...
3
votes
2answers
41 views

Example of a function $F(x,y)$

I'm trying to find a non trivial function $F(x,y)$ such that $div F(x,y)=0$ everywhere and $F(x,y)=0$ on the unit square. I know that there are some books that provide such example but I didn't find ...
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2answers
35 views

A non-algebraic complete lattice

Do you have an example of a complete lattice which is not algebraic‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌?
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2answers
31 views

Example: Algebraic Multiplicity vs Geometric Multiplicity

Is there a simple example of a matrix having an eigenvalue whose geometric multiplicity is strictly smaller than its algebraic multiplicity?
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2answers
75 views

Example of a function between boolean lattices that preserves $(\top,\bot,\wedge,\vee)$ but not complements.

Its easy to find boolean lattices $A$ and $B$ together with a function $f : A \rightarrow B$ such that $f$ preserves both top and bottom elements, as well as binary meets, but not complements. ...
9
votes
7answers
417 views

Constructing a family of distinct curves with identical area and perimeter

Two recent questions were posed by Arjuba [1] [2] asking for counter-examples regarding whether two different figures could have the same perimeter and area. Responders quickly raised a number of such ...
4
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0answers
37 views

Self-duality in a lattice

Is there any finite self-dual lattice $(X,\le)$ such that there is not any self-duality $f:X\to X$ such that $f\circ f = 1_X$? Let $f,g:X\to X$ be a self-dualities. Then $f^{-1}\circ g$ is an ...