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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12
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6answers
690 views

Induction - Examples where the induction step is correct but the base case is always wrong [duplicate]

I'd like to present to my students some induction examples that always satisfy the inductive step but never the base case. It could be for natural numbers, graphs or anything else.
2
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1answer
37 views

Is there a particularly simple example of geometric descent?

I'm looking for a particularly simple and familiar example of descent in geometry or topology in order to motivate the general definition. I'm not counting the definition of the arrow category ...
1
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2answers
55 views

Examples of Separable Spaces that are not Second-Countable

In this post I give an example of a separable space that is not second-countable. What are other good examples?
2
votes
1answer
248 views

Equicontinuity of a pointwise convergent sequence of monotone functions with continuous limit

I was looking at this question, and trying to come up with a counterexample. After thinking about it, I thought the following might be true: Claim: let $\{f_n\}$ be a sequence of continuous, ...
0
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5answers
825 views

Counter-examples of homeomorphism

Briefly speaking, we know that a map $f$ between 2 topological spaces is homeomorphic if $f$ is a bijection and the inverse of $f$ and itself are both continuous. So, can anyone give me 2 counter ...
0
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0answers
37 views

Practical examples/implementation details for Gauss-Seidel method

I'm having a presentation on Gauss-Seidel iterative method, and although it isn't mandatory , I would like to have some practical examples for this method (a system of linear equations with n>=1000, ...
1
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1answer
11 views

What is an example of a module over a division ring with two different ranks?

Let $R$ be a division ring and $M$ be an $R$-module. What is an example of $M$ and two bases $A,B$ of $M$ such that $|A|\neq |B|$?
1
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1answer
82 views

Parental Markov Condition Example

I'm currently reading a text on Bayesian networks and the text is giving some very crude interpretations of what appear to be some of the most important foundations of the subject. It states the ...
3
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1answer
132 views

Examples of 2-dimensional foliations of a 4-sphere.

This is a follow up to The 4-sphere does not admit dimension 2 foliations , where I asked about the existence of nonsingular foliations of a 4-sphere. Since that question determined there are no ...
3
votes
3answers
220 views

Examples of interesting / non-trivial manifolds that are direct products

What are interesting / non-trivial examples of smooth connected closed manifolds that are direct products or involve direct products? I am especially interested in orientable manifolds. Say, an ...
0
votes
2answers
38 views

Is a hausdorff perfect space which is not first countable neccessarily uncountable?

Counterexamples in Topology has a couple of countable spaces which aren't first countable, but none of them are perfect spaces. I'm looking either for a theorem that says that such a space can't exist ...
1
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1answer
64 views

Counterexample in optional stopping martingale

Problem: Give an example of submartingale $\{X_n\}$ with $\sup_nE |X_{n-1}-X_n|<\infty$ and stopping time $N$ with $E[N]<\infty$ such that $\{X_{n\wedge N}\}$ is not uniformly integrable. ...
5
votes
0answers
50 views

What is an importance of Gaussian and Eisenstein rings?

Among quadratic integer rings, $\mathbb{Z}[i]$ and $\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$ have their special names, namely Gaussian integers and Eisenstein integers respectively. I guess this is named so ...
0
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0answers
18 views

Finite fields and characteristics [duplicate]

I am just a little confused about finite fields and fields with characteristic $p$ (prime number). I want to know if the following proposition is true: A field $F$ is finite if and only if it is of ...
6
votes
5answers
538 views

Counterexamples for “every linear map on an infinite dimensional complex vector space has an eigenvalue”

Every linear map on a finite dimensional complex vector space has an eigenvalue. Not so in the infinite case. I'm interested in nice counterexamples anyone might have. Here's one: Consider the ...
1
vote
0answers
20 views

Is every subgroup of the quaternion group characteristic?

Let $Q$ be the quaternion group. I know that every subgroup of $Q$ is a normal subgroup of $Q$, but is every subgroup of $Q$ characteristic in $Q$? What is a counterexample?
1
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0answers
42 views

Find a function $f :\Bbb R^2\to \Bbb R$ which is not differentiable at $(0,0)$ even though all directional derivatives of $f$ exist at $(0,0).$ [duplicate]

This is a practice exam question and I have no idea how to start it. Find a function $f :\Bbb R^2\to \Bbb R$ which is not differentiable at $(0,0)$ even though all directional derivatives of $f$ ...
2
votes
0answers
39 views

What problems are related with the following type of FDE with delay?

Consider the following class of functional differential equations with delay: $$\begin{align} \frac{du}{dt} &= F(x,t,u(x,t),u_{t,x}), & (x,t) &\in [a,b] \times [0,T] \\ u(x,t) &= ...
9
votes
1answer
1k views

find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges

Find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges. To make it more challenging, produce examples where $a_n$ and $b_n$ are ...
0
votes
1answer
25 views

How do I answer questions that ask to construct examples with some given properties?

Give examples of sets and functions with the following properties: an open set $D_1 \subseteq \mathbb{R}$ and a continous function $f:D_1\rightarrow \mathbb{R}$ such that $f$ has both a maximum and ...
0
votes
0answers
73 views

If $f_{xy}$ , $f_{yx}$ are continuous at $(x_{0},y_{0})$,then $f_{x},f_{y}$ are continuous at $(x_{0},y_{0})$?

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ The second edition Let $f$ be a function of two variables,let$(x_{0},y_{0})$ be a point and let $U$ be an open disk with center $(x_{0},y_{0})$.Assume ...
2
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0answers
44 views

difficult example of a not differentiable function $f: \mathbb R^2 \to \mathbb R^2$ [closed]

Give an example of a function $f: \mathbb R^2 \to \mathbb R^2$ so that: 1) all its directional derivatives exist at $(0,0)$ ($D_{\vec u}f(0,0)$ exist for all $\vec u \in \mathbb R^2$ unitary), 2) ...
2
votes
2answers
44 views

Finite Space that is Not Normal

Is there any finite space that is not normal? By "normal", I refer to a space in which disjoint closed sets can be separated by disjoint open sets.
3
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4answers
91 views

Find a linear operator s.t. $A^{2}=A^{3}$ but $A^{2}\neq A$?

From Halmos's Finite-Dimensional Vector Spaces, question 6a section 43, the section after projections. Find a linear transformation A such that $A^{2}(1-A)=0$ but A is not idempotent (I remember A is ...
3
votes
5answers
265 views

Is it true that $2^n-1$ is prime whenever $n$ is prime?

In my discrete math book, I was tasked with finding a counterexample for this: If $n$ is prime, then $2^n-1$ is prime. Does there exist a counterexample for such a statement? Also, am I wrong in ...
1
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0answers
20 views

Double mapping cylinder- a point set question

We have the following set up: $X_0 \subset X_{\pm} \subset X$. Also interiors of $X_\pm$ cover $X$. Now let $Z$ be the double mapping cylinder of the maps $X_- \leftarrow X_0 \rightarrow X_+$. Define ...
0
votes
0answers
19 views

Nonlinear Programming examples

I'm an optimization newbie. I am looking for a simple nonlinear optimization problem that I can work through in Excel. For linear optimization I used the "Giappeto Inc" problem and I wonder if there ...
1
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2answers
50 views

Explaining roundoff error when row reducing matrices

In my linear algebra textbook (in the context of row reducing and obtaining a matrix in echelon or reduced echelon form), there is a numerical note that reads as follows: "A computer program usually ...
2
votes
0answers
73 views

Embedding vs embedding as a closed subset

A version of Whitney's Theorem state that any $n-$dimensional manifold can be embedded in $\mathbb{R}^{2n+1}$ as a closed subset. Another version states that that any $n-$dimensional manifold can be ...
1
vote
1answer
44 views

Example of non-commutative ring without unit such that…

Give an example of a non-commutative ring without unity such that $(xy)^2=x^2y^2$, for all $x,y\in R$.
34
votes
1answer
1k views

Counterexample Math Books

I have been able to find several counterexample books in some math areas. For example: $\bullet$ Counterexamples in Analysis, Bernard R. Gelbaum, John M. H. Olmsted $\bullet$ Counterexamples in ...
3
votes
3answers
68 views

What is an example of UFD such that a gcd of an infinite set does not exist?

Let $R$ be a unique factorization domain and $S$ be an infinite subset of $R$. What would be an example of $R$, $S$ such that a gcd of $S$ does not exist in $R$? That is, is there an infinite set ...
0
votes
1answer
46 views

Smooth function whose $(n+1)$th derivative is defined only on a proper subset of the domain of the $n$th, and the radius contract to $0$

I'm wondering if there exists such a function, whose $(n+1)$th derivative is defined only on a proper subset of the domain where the nth derivative is defined, and with the property that the diameter ...
5
votes
1answer
53 views

A compact infinite topological group with only two closed subgroups

It can be proved every compact infinite abelian topological group $(A,\tau )$, with $\tau$ nontrivial, has at least three distinct closed subgroups. Is there any compact infinite non-abelian ...
2
votes
1answer
58 views

Example of non-differentiable continuous function with all partial derivatives well defined

Give an example of a function $f : \mathbb{R}^3 \to \mathbb{R}$ such that the partial derivatives exist at $(0,0,0)$, and $f$ is continuous at $(0,0,0)$, but it is not differentiable at $(0,0,0)$. Any ...
75
votes
11answers
3k views

Problems that become easier in a more general form

When solving a problem, we often look at some special cases first, then try to work our way up to the general case. It would be interesting to see some counterexamples to this mental process, i.e. ...
4
votes
2answers
514 views

Does there exist a continuous and differentiable function which isn't smooth?

As I understand, a smooth function is continuously differentiable. But if I have a function which is continuous AND differentiable, I cannot automatically say that it is smooth. For it has to be so ...
1
vote
1answer
60 views

Caratheodory: Inner vs. Outer

Problem Given a plain space $\Omega$ and a ring $\mathcal{R}$. (In fact, a semiring would do the job, too.) Consider a premeasure $\mu:\mathcal{R}\to\overline{\mathbb{R}}_+$. For simplicity, ...
1
vote
1answer
98 views

Differentiable function with a set of critical points of second category.

I'm looking for an example of a nowhere constant differentiable function with a set of critical points of second category. In other words: Let $U \subset \mathbb{R}$ open. Is there a differentiable ...
5
votes
1answer
33 views

Find a sequence $(x_n)$ such that $(x_n)$ is monotonic, $\lim x_n=0$, $\sum_{n=1}^{\infty} ( 1-\frac{x_n}{x_{n+1}})$ converges

Find a sequence $(x_n)$ satisfying these conditions: $(x_n)$ is monotonic and $\lim x_n=0$ $\displaystyle \sum_{n=1}^{\infty} \left( 1-\frac{x_n}{x_{n+1}}\right)$ converges. This ...
1
vote
1answer
37 views

A compact Hausdorff space that is not Tychonoff

Is there an example of compact Haudorff that is not Tychonoff? As every continuous function on compact space is bounded, then I was thinking maybe every compact Haudorff is Tychonoff but I failed to ...
1
vote
1answer
42 views

A pushout of a homotopy equivalence along

Can anybody show me an example which prove that: A pushout of a homotopy equivalence along a arbitrary map (in Top) doesn't have to be a homotopy equivalence. I know that if we change "arbitrary ...
3
votes
0answers
33 views

Continuity of third derivative in extremum test

Consider the following standard real analysis textbook theorem: Let $I$ be a perfect interval, $f\colon I \to \mathbb{R}$ be $C^3$ (i.e. three times differentiable and $f'''$ continuous). If $x_0 ...
2
votes
1answer
44 views

Isomorphic relation between dihedral groups

Theorem Let $G,H$ be abelian groups such that $Dih(G)\cong Dih(H)$ If $G$ is finitely generated, then $G\cong H$. I'm curious whether "finitely generated" hypothesis can be removed. If ...
0
votes
1answer
21 views

Example of H-set

A subset $Z$ of a topological space $X$ is said to be an $\textbf{H-set}$ if there exists a transfinite decreasing sequence $\{ F_{\sigma}:\alpha < \kappa \}$ of closed subsets of $X$ such that ...
0
votes
1answer
39 views

$X$ is compact Hausdorff iff it is pseudocompact and realcompact

I just read this article http://en.wikipedia.org/wiki/Realcompact_space. I am interested with a property: $X$ is compact Hausdorff iff it is pseudocompact and realcompact. I don't know how to prove ...
0
votes
1answer
81 views

Finding a function ,satisfying the given properties

Finding a function of two variables , satisfying $$\lim_{\left(x,y \right)\rightarrow \left({x}_{0},{y}_{0} \right)}f\left(x,y \right)=+\infty ;$$ and for $\forall\delta>0, \exists y^{'},y^{''}\in ...
7
votes
1answer
357 views

What are the requirements for separability inheritance

Suppose we have an arbitrary separable topological space $X$. What are some (possibly nonequivalent) minimal requirements to put on $X$ to ensure that every subspace of $X$ is separable? This is not ...
2
votes
2answers
71 views

Relation between chief and compositions series of a group

Is there an example of a group with a composition series (of finite length) but without a chief series (of finite length)? Is there an example of a group with a chief series (of finite length) but ...
2
votes
1answer
94 views

Developing Examples for Basis $\mathcal{B}$ on Topology $\tau$

I put together my own example on the basis of topology. I wanted to know if it is a valid example displaying the properties of basis. Here is my example: Example: Since $X\subset\mathbb{R}$, let ...