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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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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Do you know a useful application of this type of differential equation?

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

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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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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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
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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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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
36 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
255 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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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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24 views

Group and subgroup question/example

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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3answers
57 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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2answers
459 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 ...
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1answer
50 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
39 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
29 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 ...
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1answer
33 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
39 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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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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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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78 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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34 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
96 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
62 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
25 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
73 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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2answers
31 views

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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3answers
59 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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2answers
43 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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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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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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2answers
51 views

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
43 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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60 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 ...
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1answer
39 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 ...
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0answers
37 views

Both $F$ and $C$ are closed sets but their sum $F+C$ is not closed. [duplicate]

In context to the question what will be an counter example such that both $F$ and $C$ are closed sets in $ \Bbb R^n$ but their sum $F+C$ is not closed in $ \Bbb R^n$?
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1answer
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Strongly mixing uniquely ergodic dynamical system

I'm looking for an example of a dynamical system which is both (measure-theoretically) strongly mixing and uniquely ergodic. I've looked around and found lots of discussion of systems which are ...
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1answer
20 views

Metric spaces - Limit points and Isolated points

I apologise for this, but I have numerous potential misunderstandings. $\Bbb R^2$ is a metric space, since it is a subspace of the Euclidean space, $\Bbb R^n$ I can look at a set $E$ with elements ...
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1answer
24 views

Difference between finite and discrete set of discontinuities

I am reading this paper. In page 395 and 396, theorems $1$ and $2$ use the terms 'finite' and 'discrete' to refer to sets, in this case sets of discontinuities. What I don't understand is: what is the ...
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2answers
50 views

Lattice of integers $\mathbf{Z}$ in $\mathbb{R^2}$

Lattice of integers $\mathbf{Z}$ in $\mathbb{R^2}$ The questions: Give an example of a nonempty subset of $\mathbb{R^2}$ (noted $M$) which is closed under addition and for all $m\in M$ we have $-m\in ...
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0answers
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Vanishing cech cohomology of a concratible space.

Let $C$ be a concratible space. Is it possible to find a sheaf $F$ such that two Cech cohomology groups don't vanish? E.g. Can I find a sheaf (or a local system) and two integers $i,j$ such that ...
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1answer
22 views

Relation between convergence in distribution and in probability

Does convergence in distribution imply convergence in probability ? I suppose no, but I need a counterexample. Does anyone know any counterexamples ?
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2answers
273 views

Example of non-Abelian group with 4, 5, or 6 elements of order 2

Is there any example for non-Abelian group in which has more than $3$ elements of order $ 2$, but has less than 7 elements of order $2$?
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1answer
665 views

Difficulty in finding a counterexample

I am finding difficulties in finding a counterexample that if $f\colon (0,\infty) \to(0,\infty) $ is uniformly continuous, this implies that $$\lim_{x\to \infty} \frac{f(x+\frac{1}{x})}{f(x)} =1.$$
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13answers
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An easy example of a non-constructive proof without an obvious “fix”?

I wanted to give an easy example of a non-constructive proof, or, more precisely, of a proof which states that an object exists, but gives no obvious recipe to create/find it. Euclid's proof of the ...
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1answer
38 views

Properties of Jacobson radical

I'm looking for an example of ring epimorphism $\varphi:R\rightarrow S$ such that the natural homomorphism $\tilde\varphi:J(R)\rightarrow J(S)$ is not surjective, where $J(R)$ is a Jacobson radical.
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Does every ball of boundary point contain both interior and exterir points?

My question is If $x$ is a boundary point of $S$ ($S$ is subset of $R$), does every ball of $x$ contain both interior points and exterior points of $S$? I think this is false. Since $R$ is union of ...
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59 views

Nyquist–Shannon Sampling Theorem Counter Example?

I was learning about the Nyquist theorem regards signal processing the area of interest which I will rephrase below: Given a signal lasting infinitely long with a maximum frequency of f, then you can ...