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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Counterexample for an isometric homomorphism of algebras which is not involutive.

I am finding difficulties in finding a counterexample that if $f:A\to B$ is a homomorphism of $C^*$algebras A and B (which means: f is linear and multiplicative) and let f be isometric, this implies ...
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2answers
132 views

If $\frac{a+b}2$ is rational, can we say that $a,b$ are rational?

The question is if it's given that $$ {a+b\over 2} \in \Bbb Q $$ prove or disprove $a,b \in \Bbb Q$. Since it is to disprove, I tried the following method by using examples. Take $$a = 1 + \sqrt{2} ...
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1answer
14 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 $|A|\neq |B|$?
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3answers
257 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 ...
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2answers
42 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 ...
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79 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. ...
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62 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 ...
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0answers
20 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 ...
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621 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 ...
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0answers
22 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?
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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 $f$ ...
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222 views

Convergence types in probability theory : Counterexamples

I know that the following implications are true: $$\text{Almost sure convergence} \Rightarrow \text{ Convergence in probability } \Leftarrow \text{ Convergence in }L^p $$ $$\Downarrow$$ ...
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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 ...
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0answers
46 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) &= ...
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0answers
51 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) ...
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4answers
97 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 ...
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79 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 ...
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2answers
46 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.
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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 ...
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0answers
34 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 ...
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2answers
64 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 ...
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5answers
270 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 ...
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1answer
49 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 $x,y\in R$.
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Normal subgroup question/example [duplicate]

Give an example(s) of a group $G$ and $H\leq G$ (i.e., $H$ is a subgroup of $G$) where $H$ is not normal in $G$. What about $S_3$ and $\langle(1\;2)\rangle=\{(1),(1\;2)\}$? [Since ...
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74 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 ...
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1answer
46 views

What are other examples of complex associative operators besides, x + y +rxy, rxy, and x + y + 1/r?

I have been having fun (and frustration) in finding complex associative operators over the complex numbers. So far, I have found the 3 listed in the title (r is a constant), and also know about ...
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2answers
587 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 ...
5
votes
1answer
56 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 ...
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1answer
121 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 ...
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1answer
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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 ...
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1answer
44 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 ...
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1answer
45 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 ...
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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 ...
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1answer
41 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 ...
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2answers
93 views

Examples of a cayley table that represents a structure that satisfies all group axioms except associativity

I'm curious if there are any cayley tables on a finite amount of elements that satisfy the axioms of a) closure, b) identity, and c) inverse, but that for at least one triple of elements do not ...
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1answer
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McLain's characteristically simple group

Given a field $\mathbb{F}$, one can construct the free $\mathbb{F}$-module over the set of rational numbers. Assume that $(v_x)_{x \in \mathbb{Q}}$ is a basis of this linear space (indexed by the ...
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1answer
85 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 ...
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1answer
43 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 ...
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1answer
101 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 ...
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2answers
91 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 ...
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0answers
33 views

Example of infinite sequences that produces different norms

So I have an infinite sequence: $u = (u_1, u_2, u_3, . . .)$, where $u_j$ are real numbers. $l_1, l_2, l_\infty$ is defined as $$||u||_1=\sum_{j\in \mathbb{N}} |u_j|$$ ...
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1answer
152 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 ...
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Why does $f(U)$ is open for every open $U\subset M$ not imply $f$ is continuous?

Let $f:M \to N$ be a map from a metric space $M$ to a metric space $N$. Does "$f(U)$ is open for every open $U\subset M$" imply $f$ is continuous? I think it's wrong but I can't find a counter ...
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102 views

Examples (trivial and non-trivial) of computable functions whose inverse is not computable

Can you give some examples (some trivial and some non-trivial) of computable functions whose inverse is not computable?
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116 views

Why must a locally compact second countable Hausdorff space be second countable to imply paracompactness?

The textbook version of the result I've seen states: A locally compact second countable Hausdorff space is paracompact. Is the property of being second countable needed, or have I missed something? ...
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1answer
49 views

Locally Hamiltonian vector fields

Consider the following definitions (taken from [1]) Definition. Let $E$ be a Banach space and $B: E \times E \to \mathbb R$ a continuous bilinear mapping. Then $B$ induces a map $B^\natural: E \to ...
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53 views

Counterexample: Nonvanishing derivative (constant) on $\Bbb R$ implies injectivity?

I know the case is false if $S \subset \Bbb R$ by Rolle's Theorem, what is a counter example if $S = \Bbb R$? I think the "analog" is false on $\Bbb C$, but that's yet to be proven? Addedeum: ...
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give an example of a cyclic group with 6 generators.

Give an example of a cyclic group with 6 generators. Give the generators, explain how you know that these are generators and that they are the only generators. I don't even know how to begin this ...
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1answer
54 views

A nonmetrizable image of a metrizable space

It is well known that a hausdorff continuous image of a compact metric space is metrizable. What is a counterexample for noncompact case?
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66 views

An example of a nilpotent group

Is there an example of a nilpotent group such that $G/G'$ is (non-trivial) torsion-free while $G$ is not? I cannot think of any example of this kind and I think that it is not proved any result like ...