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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27 views

Prove or disprove for $ M_n (R ×R) \cong M_n(R) ×M_n(R) $

Prove or disprove for $ M_n (R ×R) \cong M_n(R) ×M_n(R) $ ($2≤n$, $R$ is a commutative ring with identity.)
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0answers
52 views

Prove or disprove $2abc(a+b+c)\ge 3(a^2b^2c^2+1)$

Let $a,b,c>0,ab+bc+ca=3$, prove or disprove $$2abc(a+b+c)\ge 3(a^2b^2c^2+1)$$ Now I can't find any counterexample
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1answer
45 views

Continuous or Differentiable but Nowhere Lipschitz Continuous Function

What is a real valued function that is continuous on a close interval but not Lipschitz continuous on any subinterval? What is a real valued function that is differentiable on a close interval but ...
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1answer
32 views

A graph with list chromatic number $4$ and chromatic number $3$

What is an example of a graph with chromatic number $\chi(G)=3$ and list-chromatic number $\chi_\ell(G)=4$? My first thought was to consider complete tripartite graphs since these will have chromatic ...
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1answer
24 views

Torsion elements and subgroups of nonabelian group

I am currently studying torsion groups and I am playing around with defintions to get used to them. An element $g \in G$ is a torsion element, if there exists $n \in \mathbb{N}$ so that $g^n = e$, ...
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1answer
16 views

Example of an uncountable metric space where every point is isolated

I was trying to come up with an example of an uncountable metric space all of whose points are isolated. I've had difficulty thinking of one, has anyone got any nice examples? Just in case: ...
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1answer
67 views

Examples of the use of $(p \vee q) \wedge (\neg p \vee r) \Rightarrow (q \vee r)$ in “real” mathematics?

Here, a proof is requested for the following tautology: $$(p \vee q) \wedge (\neg p \vee r) \Rightarrow (q \vee r)$$ Its pretty easy to prove; nonetheless, I don't find the formula at all intuitive, ...
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2answers
415 views

Does an abelian subgroup inject into the abelianisation of the whole group? [closed]

If $H <G $ are groups and H is abelian, do we get an injection from H into $G/[G,G] $?
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1answer
32 views

Is every such family induced by an associative operation?

Suppose we're given an associative operation $\star : X \times X \rightarrow X$. Then for each $n \in \mathbb{N}_{>0}$, there's a function $f_n : X^n \rightarrow X$ given as follows: ...
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1answer
36 views

Counterexample that a measurable function does not guarantee almost sure convergence

I try to find a counterexample that if $X_n\xrightarrow{\text{a.s.}}X$ then for a measurable function $g:\mathbb{R}\to \mathbb{R}$ this does not imply that $$g(X_n)\xrightarrow{\text{a.s.}}g(X)$$ ...
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2answers
25 views

Example for associative, commutative operations

I need examples of binary operations for real numbers that are associative and commutative associative but not commutative The examples are for a programming class and need to be rather simple. ...
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1answer
32 views

How many examples exist of Lie groups that are 2-dimesional surfaces?

It is relatively easy to show that $\mathbb{R}^2$ or $\mathbb{T}^2$ are 2-dimensional surfaces with a structure of Lie groups. I can not find other surface which are also a Lie group, there are more ...
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0answers
60 views

Counterexample for “Filtration of stopping time equals filtration generated by stopped process”

I am working in a discrete setting. Consider any stochastic process $(X_n)_{n\in\mathbb N}$ with its natural filtration $(\mathcal F_n)_{n\in\mathbb N}$ and a stopping time $\tau$. We know that ...
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1answer
35 views

Sum of closed spaces is not closed

I recently encountered the theorem that the sum of a closed (linear) subspace with a finite dimensional subspace is closed subspace of the Banach space in which it is contained. However, this came ...
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1answer
48 views

Is every set of measure zero countable?

I know it is true that every countable set has measure zero, but is the converse true. Is it true that every set of measure zero is countable?
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1answer
8 views

Counter example for The Composition Theorem for Riemann Integrability

Based on The Composition Theorem (found in the last Lemma here) we can say that if a function f is Riemann Integrable, then $f^n$ is Riemann Integrable as well. The converse is not true, but I can't ...
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0answers
59 views

Examples of provably${}^n$ unprovable statements

Given any statement $A$ and a classical theory $T$ which we assume is at least as strong as Peano Arithmetic ($\sf PA$), we have that $T\vdash A$ implies $T\vdash T\vdash A$ (that is, if a statement ...
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3answers
30 views

Continuous for each variables does not implies continuous

Prove or disprove the following statement: Statement. Continuous for each variables, when other variables are fixed, implies continuous? More clearly, prove or disprove the following problem: Let ...
6
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1answer
71 views

A smooth nowhere analytic function such that all derivatives are monotone

Related questions that might provide some context: (1) (2) (3) (4) Let's restrict our attention to real-values functions on an open unit interval $f:(0,1)\to\mathbb R$. There are examples ...
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0answers
34 views

Ergodic actions of orthogonal group $O(2)$

I am looking for explicit ergodic action of $O(2)$ on a von Neumann algebra $M$. ($O(2)$=orthogonal group of $2\times 2$ matrix)
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0answers
17 views

Counterexamples of Cumulative Distribution Function ( multidimensional )

For simplicity, we consider 2-dimensional. We consider the function $F:\mathbb{R}^2\to\mathbb{R}$. Let $F$ satisfies: $0\leq F(x_1,x_2)\leq1$ for $(x_1,x_2)\in\mathbb{R}^2$ ...
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0answers
14 views

Baire class one function

in here http://www.m-hikari.com/ijma/ijma-2013/ijma-5-8-2013/feneciosIJMA5-8-2013.pdf has been shown that if a function $f$ is a real valued function on $\mathbb{R}$ with a countable set of ...
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1answer
35 views

Is there a graph with $n$ vertices and $n^2/4$ edges that isn't bipartite? [closed]

Is there a graph with $n$ vertices and $\lfloor n^2/4\rfloor$ edges that isn't bipartite and contains no triangles ($K_3$)? Rather, what I am asking is whether Mantel's Theorem implies that every ...
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2answers
24 views

$A^\circ \cup B^\circ \subset (A \cup B )^\circ$ Counterexample for = instead of $\subset$

If $A$ and $B$ are sets of real numbers, then $(A \cup B)^{\circ} \supseteq A^ {\circ}\cup B^{\circ}$ But the same relation with a = isn't always true. Can someone find an example where the = ...
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1answer
55 views

counterexample for $\overline{A \cap B} = \overline{A} \cap \overline{B}$

Prove $(\overline{A \cap B}) \subseteq \overline{A} \cap \overline{B}$. But the same relation with a $=$ isn't always true. Can someone find an example where the $=$ doesn't hold, I can't seem to ...
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1answer
29 views

Is there an example where $ A \subseteq\mathcal {P}\bigcup A\ $ is no longer true?

I came up with the following: Let A = {x} Then $ \bigcup A = x $ $ \mathcal {P} \bigcup A =$ ? This is where I get stuck. The definition of power set is the set of all subsets of $A$ ...
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0answers
78 views

Algebra defined by $a^2=a,b^2=b,c^2=c,(a+b+c)^2=a+b+c$

Let $\cal A$ be the (noncommutative) unitary $\mathbb Z$-algebra defined by three generators $a,b,c$ and four relations $a^2=a,b^2=b,c^2=c,(a+b+c)^2=a+b+c$. Is it true that $ab\neq 0$ in $A$ ? This ...
7
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4answers
326 views

Ratio test for sequences, the other direction

Suppose I have a real sequence $x_n\to 0$. Is it true that: $$ \left|\frac{x_{n+1}}{x_n}\right|\to r<1 $$ for some $r\in\mathbb{R}$? If not, is it true that: $$\exists ...
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2answers
305 views

Distance from a compact subset need not be attained in a metric space?

Suppose we have a metric space $(X,d)$. Let $S$ be a compact subset of $X$. Provide me with an example of $X$, and $S$ (closed and bounded in $X$) such that $$\min \{d(p_0,p): p \in S\}$$ does not ...
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0answers
20 views

An example that does not satisfy the conditions of the Fourier inversion theorem?

Here is the Fourier inversion theorem page in Wikipedia. It states that for every function $f(x)$ that satisfy some conditions ($f(x)$ can be a function such as a Schwartz function, an integrable ...
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2answers
30 views

Series that converges on $[-1,1]$ [closed]

What is an example of a series that converges only on $[-1,1]$? I am unable to come up with one right now for some reason. Thanks
3
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1answer
46 views

Essential singularity of the resolvent operator of an unbounded operator

Is there an unbounded operator with isolated points in the spectrum, not all of which are eigenvalues? For unbounded operators it is known that isolated spectral points are either poles or essential ...
6
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2answers
87 views

Examples of Manifolds such that $\chi (X)=-3$

I am trying to come up with an example of a closed oriented manifold with euler characteristic equal to $-3$. I have tried to use $\chi (\underbrace{T^2\mathbin{\#}\cdots \mathbin{\#} T^2}_{\text{$g$ ...
2
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1answer
28 views

Integration obeys countable subadditivity?

Does Lebesgue integration have the property of countable subadditivity: 'if $f$ is integrable on $E$ and $E = \bigcup_{i=1}^{\infty} E_n$ then $\int_E f \le \sum_{i=1}^{\infty} \int_{E_n} f$'? You ...
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2answers
22 views

Example of a function $u \in W^{1,p}(\Omega)$ whose extension $\hat{u}$ to be $0$ outside $\Omega$ $\hat{u} \notin W^{1,p}(\mathbb{R}^n)$

Let's consider a function $u \in W^{1,p}(\Omega)$, where $W^{1,p}(\Omega)$ is the Sobolev Space and $\Omega$ is an open set. When we extend $u$ to $\hat{u}$ like this: ...
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1answer
19 views

Orthonormal basis for Hilbert space

Usually, an orthonormal basis for Hilbert space means an orthonormal subset which unconditionally spans the whole space. However, I'm curious whether there exists an orthonormal basis for every ...
7
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0answers
64 views

A List of Standard or “Cliche” Homeomorphisms [duplicate]

Learning topology has been hard. I just cannot see how some people can come up with complex functions that link one space to another, in a homeomorphic sense. The explanations are always "Well if you ...
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0answers
44 views

Hard counterexample to the fact that outer measure is additive

My question is very short: does there exist a couple of sets $A,B\subset[0;1]$ such that $A\cap B=\emptyset$, but $\mu(A)=\mu(B)=1$? Here $\mu(\cdot)$ is outer measure. It's easy to construct ...
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0answers
18 views

A illuminating theorem on general modules over a ring

I'm going to do a one hour presentation on modules, free modules and projective modules over a ring. While most of my motivation for studying them comes from my interest in starting homological ...
3
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1answer
45 views

A smooth function which is nowhere real analytic, and preserves rationality of its argument

There are examples $\!^{[1]}$$\!^{[2]}$ of continuous infinitely differentiable (class $C^\infty$) functions $\mathbb R\to\mathbb R$ that are nowhere real analytic. I wonder if it is possible to ...
16
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2answers
2k views

Does there exist a complex function which is differentiable at one point and nowhere else continuous?

Let $f\colon\mathbb{C}\to\mathbb{C}$. We know that if $f^{\prime}(a)$ exists for some $a\in\mathbb{C}$ then $f$ is continuous at $a$. This is because, from the definition of the derivative, ...
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1answer
37 views

two ways of counting

I'm reading Morris DeGroot's Probability and Statistics. In chapter 1.9 there's an example 1.9.3 says that suppose that 12 dice are to be rolled. We shall determine the probability $p$ that each ...
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2answers
37 views

Sequence of bounded Operators (Is this a counterexample?)

I've to proof the following statement Let $X,Y$ be to banach spaces and $(T_k)_{k \in \mathbb{N}} \subseteq L(X,Y)$ a bounded sequence of bounded linear operators. Further it exists a dense subset ...
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0answers
13 views

Proving not an equivalence relation -the basic case

For the basic case Let $X=Y= \mathbb{R}$ and $R(X,Y)= \{(x,y) \in X \times Y : y=x^{2} \}$. I know it's not symmetric, not reflexive, not transitive. How do I provide a counterexample that it's not ...
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1answer
38 views

Example of a field extension $K/F$ such that $K$ is the splitting field of some (non-separable) polynomial in $F$, but not Galois over $F$?

An extension $K/F$ is Galois if and only if $K$ is the splitting field of some separable polynomial over $F$. Is there an example of some non-Galois extension $K/F$ where a (necessarily) non-separable ...
2
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1answer
110 views

inflexion points of a composition of functions

Let's consider a smooth positive bounded and non-increasing function $h$ over $\mathbb{R}^{+}$ (for example some kind of decreasing sigmoid). A) Is it true that if $h$ has only one inflexion point, ...
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1answer
32 views

Results on “subtraction” of measures and outer measures?

Most results I have seen involves addition of measures For example, let $m^*$ and $m$ be Lebesgue outer measure and Lebesgue measure respectively, then given $A = \bigcup\limits_{n = 1}^\infty E_n, ...
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3answers
39 views

Proof intersection is finite and non-empty

Course: Analysis (1st year course). Question: If $A_3$ is a subset of $A_2$ and $A_2$ is a subset of $A_1$ and so on... are all finite, nonempty sets of real numbers, then the intersection ...
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0answers
22 views

Is the set of differentiable points of a monotonic function Borel-measurable?

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a monotonically increasing function. Then, it has a countable discontinuity and is differentiable almost everywhere with respect to the Lebesgue measure. ...
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1answer
81 views

need an example for an ode system with 3 limit cycles

I was trying to find an ode system in predator-pray model with at least 2 limit cycles and different foci but I had trouble finding any, does anybody have an example in mind? thanks in advance