Questions tagged [examples-counterexamples]
To be used for questions whose central topic is a request for examples where a mathematical property holds, or counterexamples where it does not hold. This tag should be used in conjunction with another tag to clearly specify the subject.
5,484
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What is the importance of the density of $C_c^\infty(\mathbb R^n)$ and $L_c^\infty(\mathbb R^n)$ on $L^p(\mathbb R^n)$, for every $1 \leq p < \infty$?
Consider the usual Lebesgue spaces $L^p(\mathbb R^n)$, for $1 \leqslant p < \infty$.
It is well known that both the spaces $L_c^\infty(\mathbb R^n)$ of essentially bounded functions with compact ...
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What are other graphs of order $n$ than the star $K_{1, n-1}$ which are not packable?
We say, that a graph $G$ is packable, if it is isomorphic to a subgraph of its complement.
In more formal terms:
A graph $G$ is packable, if there is a permutation $\sigma : V(G) \to V(G)$ such that
$$...
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2
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60
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$\dim(U_1 \cap U_2 \cap U_3) = n − 3$, Give a proof or find a counterexample.
Suppose that $U_1, U_2, U_3$ are three distinct subspaces of $\dim = n-1$ from a vector space of $\dim = n$. where $n \gt 3$.
Give a proof or find a counterexample for $\dim(U_1 \cap U_2 \cap U_3) = n ...
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1
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27
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Attraction of events
I don't know if the next statement is true or false:
Let $(\Omega,\mathcal{F},P)$ be a probability space and let $A,B$ and $C$ be events in $\mathcal{F}$ such that $P(A)>0$. If $P(B|A)>P(B)$ and ...
4
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2
answers
87
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Example of Complex Pythagorean Triples
I am looking for example of a Pythagorean Triple with Gaussian Integers.
I followed the links and looked at followings :
Relation to Gaussian integers in
https://en.m.wikipedia.org/wiki/...
4
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3
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105
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If we remove the diagonal from $X\times X$, is it necessarily disconnected?
If $X$ is a compact, connected Hausdorff space, we know that the diagonal $\Delta_X=\{(x,x)\in X\times X\}$ is closed in $X\times X$ by Hausdorffness. But is $X\times X\setminus\Delta_X$ disconnected ...
2
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2
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391
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Coming up with a counter example - calculus
I have to come up with a counter example for the following statement:
Let $f$ be a function $f: [0,\infty)\longrightarrow R$, continuous and bounded. Prove that it receives either a minimum or a ...
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0
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94
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$\prod_{n=1}^{\infty}(1+e^{-n})=^{?}$ and a series .
Conjecture :
$$\prod_{n=1}^{\infty}(1+e^{-n})=^{?}1/1!+1/2!+1/3!+0/4!+1/5!+1/6!+7/7!+5/8!+9/9!+7/10!+\cdots+a_n/n!+\cdots$$
Where $a_n$ is an integer such that :
$$0\leq a_n\leq n$$
Some arguments :
...
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+200
Building a function $f$ such that $\| f - f_n \|_{L^p(B(x,r) \cap \Omega)} \to 0$ as $n \to \infty$ and $f \in L^p_{\text{loc}}(\Omega)$.
Consider an arbitrary open set $\Omega \subset \mathbb R^n$ and an arbitrary element $1 \leqslant p < \infty$. Moreover, let $(f_n)_{n \in \mathbb N} \subset L^p(B(x,r) \cap \Omega)$ denote a ...
2
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1
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45
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Must an infinite subset of a compact Hausdorff space contain a *sequential* accumulation point?
Def (Sequential Accumulation Point): Given a topological space $(X,\tau)$ and a subset $S\subseteq X$, we say that $a\in X$ is a sequential accumulation point if there exists $(s_n)_{n\geq1}\subseteq ...
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Confused regarding CMI question [closed]
I think there might be an error in the official solution to the following question:
CMI May 23, 2022 BSc entrance exam, Question A2(7):
You are asked to take three distinct points $1, \omega_1, \...
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0
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Counter example for the continuity for decreasing intersections property of measures
Let $\mathscr{R}$ be a $\sigma$-ring and $\mu$ be a (positive) measure on $\mathscr{R}$. Suppose that $\{A_n\}$ is in $\mathscr{R}$ with $A_1 \supseteq A_2 \supseteq A_3 \supseteq\cdots$ and $A = \...
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Given concave decreasing function, does $\exists c\in [0,1]$ s.t. $\frac{(f(c)-cf'(c))\left(c-\frac{f(c)}{f'(c)}\right)}{2}\leq2\int_{0}^{1}f(x)dx?$
For any curve $f:\mathbb{R}\to\mathbb{R},$ the gradient of $f(x)$ at the point $x=c$ is $f'(c).$ The curve passes through the point $(c,f(c)),$ and so the equation of the tangent to the curve at $x=c$ ...
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Analysis(convergence of series) If partial sum has a convergent subsequence with lim a_n=0, then series converges?
I have a question. I’m solving a problem;
Let <a_n>, <b_n> be two sequence in R. b_k=sum(a_n) from n=2^k-1 to 2(2^k-1) for each k in N. Show that if sigma(a_n)=1, then lim a_n=0 and sigma(...
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2
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78
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Real Polynomials on Compact sets of Complex numbers
Setting: $\mathbb{R}[x]$ is the set of polynomials with real coefficients. All $f\in \mathbb{R}[x]$ has domain $\mathbb{C}$. $K$ is a compact subset of $\mathbb{C}$. $\mathbb{R}[x]|_{K}$ is the set ...
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Second Derivative Test: Can we relax hypothesis that $f$ is twice differentiable on neighborhood? (Counterexample)
Second Derivative Test for Extrema:
Let $f:\mathbb R \to \mathbb R$ be a function that is twice-differentiable on $(c-\varepsilon,c+\varepsilon)$ for some $\varepsilon >0$. Suppose $f^{\prime}\...
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1
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98
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There is a metric space on $\mathbb{Q}$ such that this space be compact.
Prove or disprove:
There is a metric space on $\mathbb{Q}$ such that this space be compact.
I find some examples of non equivalent metric on rational numbers, euclidean metric, p-adic and discrete ...
2
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1
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187
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Let : $\frac{\Gamma(x+1)}{(x+x^2)}=f(x),x>0$ then the minimum over $(0,\infty)$ verify a particular power series
Problem :
Let :
$$\frac{\Gamma(x+1)}{(x+x^2)}=f(x),x>0$$
Then let $A=\sqrt{\frac{\pi}{4}}-1$ And $y$ be the global minimum over $x\in (0,\infty)$ of $f(x)$ then it seems we have :
$$2+A-A^4-30A^6-...
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1
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An ordinal $\nu$ is a natural iff there is no injection $f$ of $\nu$ into $X$ in $\mathscr P(\nu)\setminus\{\nu\}$.
Let's we prove the following theorem.
Theorem
An ordinal $\nu$ is a natural if and only if there is no injection of $\nu$ into $X$ in $\mathscr P(\nu)\setminus\{\nu\}$.
Proof.
Let's we assume there ...
0
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1
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59
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Finding a Lebesgue integrable function for every $1 \leqslant q < \infty$ that satisfies aditional requirement.
Consider the usual Lebesgue spaces. Amid one of my studies, I started wondering if it is possible to find an example that satisfies the following problem:
Problem. Consider arbitrary elements $1 \...
3
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1
answer
56
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Dirichlet's approximation theorem with even or odd denominators
It follows from Dirichlet's approximation theorem that for any irrational $\alpha,\ 0<\left\lvert \frac{p}{q} - \alpha \right\rvert < \frac{1}{q^2} $ for infinitely many pairs of integers $(p,q)....
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0
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49
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Proof that $P(A \cap B \mid C) \neq P(A \mid C) \, P(A \cap C)$
I want to find a counterexample to verify that the following is false $$
P(A \cap B \mid C) = P(A \mid C) \, P(A \cap C)
$$
So I thought I would take $A=C$ and then
$$
P(A \cap B \mid C) = \frac{P(C \...
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1
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86
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Necessity of Hausdorff-ness in "continuous function determined by its values on a dense subset"
It's well-known that if a continuous function taking values in a Hausdorff space is uniquely determined by its specification on a dense subset of the domain. Now, I contemplate on the necessity of ...
3
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1
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Exercise 1.1.9 from West
Currently I am reading the first edition "Introduction to Graph Theory" by Douglas B. West.
Exercise 1.1.9 states the following:
1.1.9. Suppose that $G$ is a simple graph having no vertex of ...
3
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Can an infinite sum of non-computable numbers be computable, such that all finite sums of subsets of terms are non-computable?
Background
In the following question, User1 asks whether an infinite sum of irrational numbers can be rational. Multiple answers1 indicate the answer to this question is 'yes'. For instance, Rasmus ...
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Using two copies of a decreasing $k$-tuple, can you form a convex decreasing $k$-tuple greater than the original sequence?
Suppose $(x_n)_{n=1}^{k}\ $ is a decreasing $k$-tuple of positive real numbers. Let $(y_n)_{n=1}^{2k}\ $ be two copies of $(x_n)_{n=1}^{k},\ $ that is,
$$
y_n=
\begin{cases}
x_n&\text{if}\ 1\leq ...
3
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1
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143
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Why isn't the third property in the definition of vector bundles redundant?
I am studying Manifold theory and it is essential for me to know vector bundles.The usual definition of vector bundles as given in the standard texts is a follows:
Suppose $M$ is a topological ...
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1
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41
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Example for a particular function
I am looking for an example of a nonnegative function (preferably continuous) that approaches $0$ near $0$, and for any positive number $B$, it is not monotonic on $(0,B)$ and $f(x/2) > f(x)$ for ...
2
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0
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Example of a functor that creates limits but not strictly (other than equivalences)
In Category Theory in Context, Riehl defines creating limits as follows, for a functor $F: \mathbb{C} \to \mathbb{D}$ and diagram $D: I \to \mathbb{C}$:
$F$ creates limits if whenever $FD: I \to \...
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0
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33
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Sequence in $\ell_p$ spaces
The sequence given by:
$$x_n=(1^{-1/q},2^{-1/q}-1^{-1/q}, 3^{-1/q}-2^{-1/q}, \dots)$$
That is,
$$\sum_{n=1}^{\infty}n^{-1/q}-(n-1)^{-1/q}$$
Is this sequence in the sequence space $\ell_p$ ? where for $...
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2
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150
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Find three non equivalent metrics on an infinite set.
Find three non equivalent metrics on infinite set.
For case infinite countable set we can use this answer for rational numbers.
Also for example in $ \mathbb {R}$ we can theree non equivalent metric :...
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Is there a monotone function that is not differentiable everywhere? [duplicate]
One of the most strange things that I learned in Real Analysis is a function that is continuous on $\mathbb{R}$ but that is not differentiable everywhere.
I wonder how to prove that no monotone ...
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75
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Does every positive real sequence whose series converges to $a$ have a "straight line" convex subsequence that also converges to $a?$
This question is possibly related to my previous question and it's answer.
Suppose $a_n>0\ \forall\ n\in\mathbb{N}\ $ and $\displaystyle\sum_{n=1}^{\infty} a_n = a.$
Does there exist a subsequence $...
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Can $x\sin(x)$ be algebraic when it is not $0$?
It's easy to show (using the Lindemann-Weierstrass theorem) that, for $x\ne 0$, at least one of $x$ and $\sin(x)$ must be transcendental.
But what about $x\sin(x)$?
After all, the product of two ...
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1
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56
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Example to show that the containment $\overline {f^{-1}(B)} \subset f^{-1}(\bar B) $ is proper where $f$ is continuous mapping
Let $f: X \to Y$ be a continuous function, where $B\subset Y$.
Then,
$\overline {f^{-1}(B)} \subset f^{-1}(\bar B) $ holds, here's the proof.
I am looking for an example to illustrate that the above ...
2
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1
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Is there a Hamiltonian cycle of $m$ x $n$ rectangular lattice points (these are the vertices) in $\mathbb{R}^2$ such that no two edges are parallel?
Let $m,n\geq 2$ and consider the rectangular lattice of $mn$ vertices in $\mathbb{R}^2,\ (i,j);\ i\in \{1,2,\ldots,m\},\ j\in \{1,2,\ldots,n\}.\ $ Call these vertices $X_1, X_2, \ldots, X_{mn}.$ Is ...
2
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1
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64
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Given a convergent sequence of complex numbers, is there a sequence of nested Jordan curves whose sum of members of $z_n$ in the regions converges?
Suppose $(z_n)_{n\in\mathbb{N}}\subset \mathbb{C}$ such that $\displaystyle\sum_{n=1}^{\infty} z_n$ converges.
Then, for example,
$$ \sum_{k=1}^{\infty}\left( \sum_{\frac{1}{k+1} < \left\lvert z_n \...
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An example of non simple group which is also a Lie group such that $G$ is a connected Lie group and has no non trivial normal Lie subgroup
I want to know if there exist a non simple group (as abstractl group) $G$ such that $G$ is a connected Lie group and has no non trivial normal Lie subgroup. I have tried some obvious examples like ...
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0
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48
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On definition of Gâteaux derivative
Definition. Let $X,Y$ be Banach spaces and $f:U\rightarrow Y$ a map on an open subset $U\subset X$.
The map $f$ is called Gâteaux differential at $x\in U$ if there exists a continous linear map $A:X\...
2
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0
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70
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Are there contiguous sequences of prime numbers of length $k$ which are convex (similarly, concave) for every $k\in\mathbb{N}?$
Does the sequence of prime numbers contain contiguous subsequences of length $k$ which are strictly convex (similarly, strictly concave), for every $k\in\mathbb{N}?$
For example,
$$ 17, 19, 23, 29 $$
...
6
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2
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499
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Looking for a counterexample for this statement regarding divergent series
I need counter for this statement :
If the series $ \sum a_n$ is divergent then the series $b_n$= $\sum \text{min}(a_n,\frac{1}{n})$ is also divergent.
I closest I reached to a counter is ,
Define $...
4
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1
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73
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Does $\exists\ n$ such that the first $2n$ digits of Thue Morse, $X_{2n},$ is the concatenated sequence $X_n X_n?$ If not then why not?
Background:
The Thue–Morse sequence is the binary sequence (an infinite
sequence of $0$s and $1$s) obtained by starting with $0$ and
successively appending the Boolean complement of the sequence ...
1
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1
answer
80
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Is there a function $f : \mathbb{R}-\{ 0\} \to \mathbb{R}$ such that $f(x)= - f(2x)$ for all $x \in \mathbb{R}-\{ 0\}$ , $f(x) \ne 0$?
In my problem book there is this question
Show by example that the condition $\lim\limits_{x \to 0} (f(x)+ f(2x) ) $ doesn't imply that $f$ has a limit at $0$.
Although I solved this question but I ...
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1
answer
68
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How to find solutions of PDEs or explain they don't exist
Consider the equation $u_x + 3x^2y^2u_y = 0$ :
Find a solution satisfying $u(x, 0) = 1/(1+x)$, or explain why no such solution exists.
My attempt:
I've managed to solve the equation and find the ...
2
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1
answer
47
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What are the compactness properties of $\mathbb R$, extended by a point with co-countable open neighborhoods?
In a remark in this answer regarding radiality and pseudoradiality in locally countable spaces, an example of a pseudoradial, countably tight space which failed to be sequential was constructed.
The ...
2
votes
1
answer
103
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Example of two group filtrations that induce different topologies but the same completion
This is a question about completion in the sense of commutative algebra, i.e. inverse limit of quotients. Let $G$ be a topological abelian group, $G_0 \supseteq G_1 \supseteq \cdots$ and $H_0 \...
2
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1
answer
74
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$m_i, n_j$ integers and $\{m_i\}_{i=1}^{k}\neq\{n_j\}_{j=1}^{k'}.$ Does $\sum\frac{1}{m_i}=\sum\frac{1}{n_j}\implies\sum m_i\neq\sum n_j?$
Suppose $\{m_i\}_{i=1}^{k}$ and $\{n_j\}_{j=1}^{k'}$ are each finite subsets of $\mathbb{N},$ $\{m_i\}_{i=1}^{k}\neq\{n_j\}_{j=1}^{k'},$ and $\displaystyle\sum_{i=1}^{i=k}\frac{1}{m_i} = \sum_{j=1}^{...
3
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0
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50
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Counterexamples to $\hat{A} \otimes_A M \cong \hat{M}$
I'm looking for counterexamples to Proposition 10.13 in Atiyah-Macdonald when the hypotheses are not satisfied (throughout, $A$ is a ring, and $\hat{\phantom{M}}$ denotes the $I$-adic completion for ...
2
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0
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43
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"Explicit" showing of non-flatness for a fractional ideal
I am not very familiar with flatness so I was trying to get a feel of what precisely is the crux of the notion, using objects I am used to. Sadly, most of the counter examples I could find where for ...
0
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0
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32
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Counter example: continuity assumption of mean value theorem [duplicate]
Let's consider $f:[a,b]\to\mathbb{R}$ which is differentiable in each $x\in ~\!]a,b[$. Consequently $f$ is also continuous in those points. But we don't require that $f$ is continuous in $a$ and $b$.
(...