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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Sets with one accumulation point

Are there any more examples of sets in $\mathbb R$ that have one accumulation point apart from convergent sequences? I can´t think of any
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41 views

Uniform convergence in series definitions of functions

Are there examples of well-known functions which are defined as the limit of a sequence of functions (for example, power series definitions) and are not uniformly convergent? Thanks!
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1answer
41 views

Examples when vector $(X,Y)$ is not normal 2D distribution, but X and Y are.

My question is: do you know any examples when $X$ and $Y$ are both normally distributed, but the two dimensional vector $(X,Y)$ is not? I found some example in book, but I don't understand it. The ...
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1answer
69 views

What is an example of $R\otimes_R M$ not isomorphic to $M$?

Let $R$ be a ring and $f:R\rightarrow R$ be a ring homomorphism. ($f(1)=1$) Let $R$ be given the left operation as the ring operation on $R$ and the right operation as $x•r=xf(r)$, so that $R$ is an ...
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2answers
75 views

Which optimization problem to use when I want to explain the concept of optimization to a layman?

I am looking some problem, which would be: Easy to understand Hard to solve intuitively Touch our everyday lives I am doing research in optimization using evolutionary computation. When people ...
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Example for $\dfrac{p_1p_2-1}{p_1+p_2}$ being odd natural number .

If $p_1,p_2$ are odd prime numbers , is it possible that $\dfrac{p_1p_2-1}{p_1+p_2}$ is odd natural number greater than 1.
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1answer
29 views

Integral on set sequence is not convergent to their countable intersection

Let $(\Omega,S,\mu)$ be a measure space. Suppose $f$ is integrable on $A_1\supset A_2 \supset A_3\dots$, a decreasing sequence of measurable sets $\{A_n\}_n\subset S$ and denote ...
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138 views

Example: satisfying $E(X_{n+1}\mid X_n)=X_n$ but not a martingale

I am wondering if there is such a sequence of random variables $(X_n)_{n=0}^\infty$ such that $\mathbb{E}(X_{n+1}\mid X_n)=X_n$ for all $n\geq0$ but which is not a martingale with respect to the ...
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76 views

understanding a theorem in c*algebra

I want to understand the following theorem: Theorem: Let A and B $C^*$-algebras with A unital, and let $\varphi:A\to B$ a bounded linear selfadjoint map such that: for every self-adjoint elements ...
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1answer
37 views

Examples for almost-semirings without absorbing zero

What is an instructive example of a set $X$ equipped with two monoid structures $(X,+,0)$, $(X,\cdot,1)$, such that $+$ is commutative, the distributive laws hold, but $0 \cdot x = 0$ or $x \cdot 0 = ...
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0answers
55 views

A metric space of which the geodesic is not a metric

The text book in my course has an exercise about finding a metric space whose (usual) length metric is not a metric. It wants me to find a metric space $(X,d)$ satisfying $d'(x,y)=0 \ \ $for some ...
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P = NP, NP example problems in our daily life

For a little presentation for school, i want to try to explain the P=NP? Problem. I'm searching for examples for daily life NP-problems. (example: is making the weather forecast a NP problem?) And if ...
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0answers
31 views

Similar function to Sinc function?

I am interested in a function which at $x=0$ starts from approximately $1$ and as you go on it decreases periodically to $0$ in a similar fashion to the Sinc function.
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1answer
37 views

Infinite sum of random variables is infinite

I am trying to better understand this statement and the assumptions made: If $X_1,X_2,\ldots$ are non-negative independently and identically distributed random variables with $P(X_i>0)>0$, ...
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1answer
55 views

Example of equivalence relation where reflexivity/symmetry is not trivial

For basically any equivalence relation that I've encountered in my mathematical studies, the only problematic part (if at all) was proving transitivity. Is there a "real-world" (i.e. appearing in some ...
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6answers
796 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.
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1answer
92 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, ...
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2answers
118 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?
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1answer
44 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 ...
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1answer
56 views

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
126 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
12 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
247 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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1answer
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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60 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
19 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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5answers
606 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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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 $f$ ...
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1answer
188 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
45 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
48 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
96 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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0answers
78 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
45 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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0answers
26 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 ...
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0answers
28 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
61 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 ...
3
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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 ...
1
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1answer
47 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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1answer
77 views

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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3answers
72 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
44 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
566 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
54 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
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1answer
99 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
40 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
43 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 ...