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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Counterexamples to the Matrix norm AM-GM inequality?

I am new here and this my first question, I hope I am being as clear as possible and apologize in advance for any misunderstandings. I am researching the Arithmetic-Geometric Mean (AM-GM) inequality ...
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32 views

Simple examples of fractional ideals

Let $K$ be the quotient field of an integral domain $R$. A fractional ideal $I$ is a subset of $K$ not $\{0\}$, for which a $0 \neq r \in R$ exists so that $r I \subseteq R$ is an ideal in $R$. ...
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1answer
47 views

How differentiable is the function $g(x) = \sum_n 2^{-n} f(x-r_n)$ where $f(x)=x^2 \sin\frac1{x}$?

This is an auxiliary enquiry (something like it may well be already discussed on MSE, but I haven't found it) resulting from a feeling of unease provoked by the question of this post. Taking the ...
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2answers
54 views

Cancellative Abelian Monoids

Is there an example of cancellative Abelian monoid $M$ in which we may find two elements $x$ and $y$ such that they have a least common multiple but not a greatest common divisor?
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1answer
40 views

Prove by counterexample that $\gamma$ and $\delta$ are not necessarily unique

Assume $\mathbb Q[\sqrt{d}]$ is a Euclidean Field and $\alpha$, $\beta$ are two quadratic integers in $\mathbb Q[\sqrt{d}]$, so that there exists integers $\gamma$ and $\delta$ in $\mathbb ...
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3answers
311 views

Finding a pair of functions with properties

I need to find a pair of functions $f$, $g$ such that $f$ is not differentiable at $x = 0$ $g$ is not differentiable at $f(0)$ $g \circ f$ is differentiable at $x = 0$ I've tried a lot of ...
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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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1answer
57 views

Does an open map of the real line have to be one to one?

Is it true or not that open map $f:\mathbb{R} \rightarrow \mathbb{R}$ is a one one map? If it is one one explain how it is one one.
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2answers
40 views

An example of an ideal of order $12$

Provide an example of an ideal in $R=\mathbb{Z}_6\times\mathbb{Z}_{10}$ that has order $12$, and indicate whether the ideal is a principal ideal (if it is, then identify the generator for the ...
2
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1answer
63 views

What would be an example of free module such that cardinality of linearly independent set is greater than the rank?

Let $R$ be a commutative ring and $M$ be a free $R$-module. Since $R$ is commutative, $R$ has the IBN property, hence the rank of $M$ is uniquely well-defined. So set $n:=\mathrm{rank}(M)$. Let ...
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0answers
99 views

Request for counter examples in group theory

I am looking for books, papers, or even webpages, that have collected many counter examples in group theory (which, I guess, are just examples in group theory). I am particularly interested in ...
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4answers
340 views

Example of an associative binary operation, without identities or inverses.

In essence, I am looking for an example of a semigroup or a semicategory (closure is not that important, but it is useful) that is NOT a monoid or category. Hopefully, there is a neat and ...
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10answers
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Examples of finite nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
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0answers
37 views

Hahn-Banach Theorem for finite dimensional vector spaces

I am studying the Hahn-Banach Theorem but I'm having a hard time following some of the examples presented in my notes, so I would like to first study some small examples which help me to get some ...
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0answers
39 views

Prove that the Weierstrass-type function is nowhere differentiable

Given $0<\alpha\leq1$. Show that the function $$f(x)=\sum_{j=1}^\infty 2^{-j\alpha}\sin(2^jx)$$ is nowhere differentiable. I have solved the case $x=0$. Taking $t_l=2^{-l-1}\pi$, then ...
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2answers
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Is the axiom of choice really all that important?

According to this book: The Axiom of Choice is the most controversial axiom in the entire history of mathematics. Yet it remains a crucial assumption not only in set theory but equally in modern ...
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0answers
15 views

Non homeomorphic spaces with same homology groups [duplicate]

Is it possible for two spaces X and Y to have the same homology groups with X not homeomorhpic to Y.
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2answers
49 views

a discontinuous function the square of which is continuous

give an example of a discontinuous function the square of which is continuous. The domain is $[0,1]$. I tried to use the indicator function of rationals, but its square is not continuous. EDIT:I am ...
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3answers
3k 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?
10
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1answer
85 views

If all the numbers $(1^\alpha,\,2^\alpha,\,3^\alpha,\,\dotsc)$ are integer, then $\alpha$ is an integer.

A theorem of Siegel asserts that If $\beta>0$ and $2^\beta,\,3^\beta,\,5^\beta$ are integers, then $\beta$ is an integer. The following result is a beautiful consequence of this theorem ...
0
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1answer
15 views

Polynomial in $\mathbb{Z}_2[x]$ that is reducible but has no roots a prime $p$ for which $x+10$ divides $x^4+x^3+x+1$ in $\mathbb{Z}_p[x]$

First, I am suppose to find a prime $p\geq 4$ where $x+10$ divides $x^4+x^3+x+1$ in $\mathbb{Z}_p[x]$. Second, I am supposed to find a fifth degree polynomial in $\mathbb{Z}_2[x]$ that is reducible ...
2
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1answer
30 views

Integral domains examples

I am supposed to give an example of 1) an infinite integral domain of characteristic $5$, and 2) an integral domain which is not a field. Respectively, examples I chose were $\mathbb{Z}_5$ and ...
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1answer
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1answer
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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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1answer
301 views

An explicit example of a differentiable function taking rational values at rational points but whose derivative is irrational at rational points

Construct an example of a differentiable function such that $$ \forall r \in {\Bbb Q}\quad f(r) \in {\Bbb Q}\text{ but } f'(r) \notin {\Bbb Q} $$ this example is not trivial, in a paper they ...
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0answers
33 views

Difficult examples of invertible, differentiable functions

Give an example of: 1)$f:\mathbb R^2 \to \mathbb R^2$ such that $f$ is invertible in some neighbourhood of $x_0$ (that is $f$ is locally invertible) but $|Jf(x)|=0$ (jacobian determinant) $\forall x$ ...
2
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1answer
68 views

Blow-up toric varieties.

I have to take a talk of an hour and I have to talk about blow-up of toric varieties. Can you suggest me some interesting examples that I can present? How can I find a good reference for the theory ...
7
votes
1answer
112 views

Example of continuous function whose Fourier series doesn't converge on an uncountable dense set.

According to a well-known theorem (Theorem 5.12 in Rudin's Real and Complex Analysis), there is a dense $G_\delta$ set of continuous periodic functions $f:\mathbb{R}\to\mathbb{C}$ such that the ...
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2answers
42 views

What is the example of a not almost convergent sequence but whose Cesàro means converge?

It seems to me a sequence that is almost convergent implies that its Cesàro means converges but not vice versa. What is the example that a not almost convergent sequence whose Cesàro means converge. ...
0
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3answers
55 views

What is a counterexample to this one?

Let $R$ be a commutative ring and $A\in M_{n\times m}(R)$ where $n\neq m$. What is an example such that $\det(AA^t)\neq \det(A^tA)$? Indeed, I think it's true. If this is true, how do I prove this?
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0answers
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Find $f:C\to\mathbb{R}^2$ continuous and bijective but not open, $C\subset\mathbb{R}$ is closed and connected

Are there a closed connected subspace $C$ of $\mathbb{R}^2$ and a continuous, bijective function $f:C\to\mathbb{R}^2$ that is not open? If we remove the condition for $C$ to be connected, we have ...
4
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1answer
48 views

Zero-dimensional but not Hausdorff

Let's call a space zero-dimensional if it has a basis of clopen sets, and is $T_0$. Is there a zero-dimensional space that is not Hausdorff?
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4answers
554 views

How to come up with a counter example in linear algebra

This came up in a problem I was working on. Problem:Let $V$ be an $n$ dimensional vector space over a field $F$. Let $T:V\rightarrow V$ be a linear operator and let $W$ be a $T$ invariant subspace of ...
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3answers
98 views

Graph connected does not imply $f$ is continuous [closed]

Show an example of a function $\newcommand{\R}{\mathbb{R}} f: \R \times \R\to \R$ such that $f$ is not continuous, but its graph $$ \Gamma_f := \left\{\bigl((x, y), f(x, y)\bigr) \mid \text{$(x, y)$ ...
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0answers
47 views

Normal space is compact

I know that a compact Hausdorff space implies Normal, but does the converse holds? I.e. If a space is normal, it is compact and Haudorff. (Although $T_4$ imlicitly implies $T_2$)
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2answers
439 views

An example of a space which fails to be compactly generated

Does anyone know of an example of a topological space which is not compactly generated? I am using the definition in May's book "A Concise Course in Algebraic Topology." The definition is that a space ...
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1answer
56 views

Spaces in which “$A \cap K$ is closed for all compact $K$” implies “$A$ is closed.”

Let $X$ denote a topological space. For any $A \subseteq X$, consider two possible conditions on $A$. $A$ is closed $A \cap K$ is closed, for all compact $K \subseteq X$. If $X$ is Hausdorff, then ...
190
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28answers
18k views

Nice examples of groups which are not obviously groups

I am searching for some groups, where it is not so obvious that they are groups. In the lectures script there are only examples like $\mathbb{Z}$ under addition and other things like that. I ...
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1answer
52 views

Non-standard examples of Distributions

Can somebody point me to examples of distributions which are not sums of delta functions or derivatives thereof. Also of course not integrals with local integrable functions. Additional: is there a ...
0
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1answer
14 views

What is an example of $E/F,L/E$ are normal but $L/F$ is not. [duplicate]

Let $E/F,L/E$ be normal field extensions. What would be an example such that $L/F$ is not normal?
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2answers
15 views

What is an example of transcendental extension such that a monomorphism cannot be extended?

Let $E/F$ be a field extension and $\alpha\in E$ be transcendental over $F$. Let $\bar F$ be the algebraic closure of $F$ and $\sigma:F\rightarrow \bar{F}$ be a field monomorphism. What is an ...
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9answers
861 views

Advice on finding counterexamples

I am reaching out for specific advice on how one should go about finding counterexamples. It seems almost every time I've ever attempted a "find a counterexample" problem, I have to cheat by asking a ...
11
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4answers
211 views

Is $2^{\aleph_0} = \aleph_1$?

I was reading a thread on Examples of Common False Beliefs in Mathematics on MathOverflow, in which a user wrote: $$2^{\aleph_0} = \aleph_1$$ This is a pet peeve of mine, I'm always surprised ...
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1answer
56 views

Example of a white noise series that is not a martingale difference series with respect to its natural filtration

For a homework exercise, I am asked to find an example of a white noise series that is not a martingale difference series with respect to its natural filtration. Does anyone know an example? I read ...
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235 views

A strong Hausdorff condition

Is the following strong form of Hausdorff equivalent to usual Hausdorff? $X$ is strong Hausdorff if given distinct elements $x,y$ in $X$ there are open sets $U,V \subseteq X$ with $x \subseteq ...
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0answers
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How much regularity is needed, anyway?

When doing real analysis, the difference between functions which are continuous and functions which are not is intuitive. The graph of the later may exhibit shearing, or extreme distortion (in higher ...
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3answers
69 views

What are some illustrative examples that demonstrate how $\succ$ can differ in behavior from $>$ and/or $\geq$?

I really, really want to understand the generalization of metric spaces known as continuity spaces. Unfortunately, I always get tripped up right at the beginning. The problem is that I have little or ...
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1answer
22 views

Are there an completely regular, non-Lindelöf spaces with only constant real valued functions?

Is there an example for a topological space, which is not Lindelöf, but is completely regular, on which continuous real valued functions, are, constant for all $x \in X$, or, constant for all, except ...
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2answers
21 views

Example of $1-1$ Correspondence with Subgroups of Factor Group

I am working out an example to deomstrate the one-one correspondence between $\{\text{subgroups of}\ D_4/N\}$ and $\{\text{subgroups of $D_4$ that contain $N$}\}$ but I am short one in $D_4$. ...
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1answer
33 views

Finding a counter example for $ \left(A+A\right)'\subseteq\left(A'+A\right)\cup\left(A'+A'\right)$

let $A'$ be the set of limit points of $A\subseteq\mathbb{R}$ and $A+B=\{x+y:x\in A,y\in B\}$, I'm required to find a counter-example for: $$ ...