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.

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23 views

Bound on matrix norms

Let $A,B$ be two $n\times n$ invertible matrices with complex entries. Assume that $B$ is very close to the identity matrix, i.e., $\|B-I\|\ll1$. Can there be a bound on $|\|ABA^{-1}\|-1|$ uniform ...
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40 views

Is this counterexample of $\mathbb{P}( | X | \ge c) \le \frac{\text{Var}[X]}{c^2}$ correct?

Let $X$ be a real valued random variable. Does $\mathbb{P}( | X | \ge c) \le \frac{\text{Var}[X]}{c^2}$ hold for $c > 0$? My counterexample: Let $X \sim \text{Bin}(n,0)$ for $n \in \mathbb{N}$. ...
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0answers
24 views

Counter-example wanted on a new formula for convex function

I'm confusing because in the inequality New bounds for convex function of 2 variables the RHS doesn't work (take $f(x)=\tan(x)$) so I propose a new formula that I have checked for all the elementary ...
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0answers
15 views

Examples of non submersions whose pullback does not exist

Let $f:M\rightarrow N$ is a submersion. Let $g:M’\rightarrow N$ be a smooth map, then, it is known that the pullback $M\times_NM’$ is an embedded submanifold of $M\times M’$ and that the map $M\times ...
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0answers
32 views

Nonisomorphic Groups with the Same Order and Exponent

I am trying to find two nonisomorphic finite abelian groups with the same order and exponent. I've tried solving this problem for a fews days, but I have had no luck. I tried looking for pairs of such ...
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1answer
54 views

If $\sum_{1}^{\infty} a_n^2$ converges and $a_n$ is complex valued, then how about $\sum_{1}^{\infty} a_n/n$

If $a_n$ is nonnegative-valued, then I can use the CS inequality to show that $\sum_{1}^{\infty} a_n/n$ converges. However, if $a_n$ is complex valued, then I think a sequence $a_n = 1/(\ln{n})^{0.5}$ ...
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3answers
57 views

Find $f:\mathbb{Q} \rightarrow \mathbb{R}$ where the mean value theorem fails

Find a differentiable function $f:\mathbb{Q} \rightarrow \mathbb{R}$ that is not monotonically increasing, but still fullfills $f'(x) \geq 0$ for all $x \in \mathbb{Q}$. The background of this ...
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0answers
36 views

A congruence involving generalized Lucas polynomials

This is a generalization of this claim . Can you provide a proof or a counterexample for the following claim? Let $n$ be a natural number greater than two . Let $L_n^{(a)}(x)=2^{-n}\cdot\left(\...
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2answers
38 views

Are there any concrete examples of $\sigma$-algebra generated by a random variable?

I have searched around the internet for any concrete example of $\sigma$ algebra generated by a random variable $X$ but failed to find any nontrivial, concrete examples. For example, https://stats....
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1answer
25 views

Function that grows as fast as sum of previous values

Does there exist a strictly positive, monotonically increasing function $f : \mathbb{N} \to \mathbb{R}$ such that $$ \lim_{n \to \infty} \sum_{k = 1}^n \frac{f(k)}{f(n)} = 1 $$ My guess is that there ...
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1answer
27 views

Is a continuous process with finite nonzero quadratic variation a semimartingale?

I know that every semimartingale has finite quadratic variation but the converse is not true. However, what if we assume continuity? If $X$ is a continuous process with finite nonzero quadratic ...
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1answer
44 views

When does path connectedness implies convexity?

It is easy to see that every convex set is path connected. What are some examples so that converse holds (not counting the (trivial) one dimensional case)? Is there a nice topology so that this holds? ...
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1answer
72 views

Is there any statistically convergent real sequence, which is not almost convergent?

I have read that almost convergence and statistical convergence are incompatible (i.e. not comparable). For this both of below must be satisfied : There exists a statistically convergent real ...
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2answers
75 views

$I \cap (J+K) =I \cap J + I\cap K $ [duplicate]

Let I, J, K be three ideals in a commutative ring R with unity. If R is ring of integers then above equation holds. I know the equation do not hold for arbitrary ring. Can you give me an ...
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1answer
66 views

Are there any interesting categories/objects whose products are isomorphic to themselves?

This was inspired from an exercise in Lawvere & Schanuel's Conceptual Mathematics. It asks what objects in $\mathbf{Set}$ (finite sets), $\mathbf{S^\circlearrowleft}$ (endomaps in $\mathbf{Set}$),...
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1answer
37 views

For ideals $\mathfrak{a,b}$ if a solution $\mathfrak{c}$ to $\mathfrak{cb}=\mathfrak{a}$ exists, is it unique and equal $(\mathfrak{a}:\mathfrak{b})$?

The motivation for this is to have a better intuition of how to think of the ideal quotient $(\mathfrak{a}:\mathfrak{b})$ as a quotient. Obviously for $a,b \in \mathbb{Q}$, the quotient $\frac{a}{b}$ ...
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1answer
33 views

Isomorph vector space such that X complete and Y not

Let $X,Y$ be some isomorph vector spaces and let $X $ be a Banach space. If this isomorphism is isometric $Y $ is complete, too. Could someone provide an example such that $Y $ is not complete?
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78 views

A counterexample in measure theory on $\sigma$-infinite spaces

Usually measure theory books include the following theorem (citing Proposition 5.1.3 in Cohn's measure theory book) Let $(X, \mathcal A , \mu )$ and $(Y, \mathcal B, \nu )$ be $\sigma$-finite ...
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5answers
58 views

Do we have for all $M \in SL_n(\Bbb K)$, $\lVert M \rVert \geq 1$ when $\lVert \cdot \rVert$ is a matrix norm?

Let be $\lVert \cdot \rVert$ a matrix norm (submultiplicative). Do we have for all matrices of determinant 1, the following lower bound: $$\lVert M \rVert \geq 1$$ I'm very confused and could not ...
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5answers
107 views

Is $A$ the $2 × 2$ identity matrix?

If $A$ is a $2 × 2$ complex matrix that is invertible and diagonalizable, and such that $A$ and $A^2$ have the same characteristic polynomial, then $A$ is the $2 × 2$ identity matrix. My claim: ...
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1answer
37 views

Two different versions of limit in topological spaces

Let $X,Y$ be topological spaces, and $f:X\to Y$. We say that $f(x)\to l$ as $x \to b$ iff for every open neighborhood $N$ of $l$, there exists an open neighborhood $M$ of $b$ such that $f(M)\subset N$...
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0answers
69 views

Compositeness tests for numbers of the form $\frac{k \cdot b^n \pm 1}{2}$

Can you provide proofs or counterexamples for the claims given below? First claim Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ . Let $M= \frac{k \cdot b^{n}-1}{2} $ where $k$ ...
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0answers
77 views

Constructions of Lebesgue non-measurable sets other than Vitali's

Let $f:\mathbb R\to\mathbb R$ a function such that $|f|$ is measurable. Is $f$ a measurable function? I know it is false. I have seen a few counterexamples where one takes the Vitali set $V\subseteq[...
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33 views

Examples of $C^*$-algebras without strict comparison

$\textbf{Definition: }$Let $\mathcal{A}$ be a unital $C^*$-algebra and $$\mathcal{T}(\mathcal{A})=Tracial~States~of~\mathcal{A}$$ $\mathcal{A}$ has strict comparison for projections if for all ...
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1answer
37 views

The image of a functor need not be a subcategory

Warning 1.2.19 gives an example when the image of a functor is not a subcategory: But I'm confused: the author defines a functor $F$ right away without saying what the codomain category is. This ...
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1answer
203 views

Is there a category where products don't exist because uniqueness fails?

I was looking at this question about categories without products, and the main examples are: fields manifolds with boundary posets But these all seem to fail for either structural reasons (fields/...
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1answer
45 views

Ricci tensor does not determine the curvature tensor

I've been asked to show that the Ricci tensor $$ \mathrm{Ric}(X,Y) = \mathrm{tr}(Z \mapsto \mathrm R(Z,X)Y),$$ where $\mathrm R$ is the curvature endomorphism, does not completely determine the ...
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0answers
26 views

Construct a 2 generator group $G$ that is an extension of a free abelian group $N$ of infinite rank by an infinite cyclic group

Construct a 2 generator group $G$ that is an extension of a free abelian group $N$ of infinite rank by an infinite cyclic group I'm looking for a quick example of this, I've been trying to figure ...
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109 views

Multiplication and division by a morphism under the “inner composition” in closed monoidal categories

Let ${\mathcal C}$ be a symmetric closed monoidal category, and let me denote the internal hom-functor by a fraction $$ (X,Y)\mapsto\frac{Y}{X}, $$ so that we have an isomorphism of functors $$ \...
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1answer
41 views

For non-diagonalizable matrices, the dimension of centralizer can be different from $\sum\limits_{j=1}^k d_j^2$

It is known that if a square matrix $A$ is diagonalizable the the subspace $$C(A)=\{X\in M_{n,n}; AX=XA\}$$ has the dimension $\sum\limits_{j=1}^k d_j^2$, where $d_j$ denotes the geometric ...
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2answers
28 views

an example related to derivatives

How can I construct an example of the following? $f$ is differentiable at $x_0$ but $\lim_{x\to x_0} f'(x)$ does not exist? A bit stupid but if a function that doesn't have a limit at a point does ...
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1answer
32 views

Counterexamples where $\sum_i (a_i : b) \subsetneq (\sum_i a_i :b)$ or where $\sum_i (a: b_i) \subsetneq (a : \bigcap_i b_i)$?

Better formatted title: Counterexamples where $\newcommand{\mf}[1]{\mathfrak{#1}}$$\sum_i (\mf{a}_i : \mf{b}) \subsetneq (\sum_i \mf{a}_i : \mf{b})$ and/or where $\sum_i (\mf{a}: \mf{b}_i) \subsetneq (...
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1answer
61 views

Understanding full and faithful functors

A functor $F:\mathscr A\to\mathscr B$ is faithful (resp., full) if for each $A,A'\in Ob(\mathscr A)$, the function $$\mathscr A(A,A')\to \mathscr B(F(A),F(A' ))\\ f\mapsto F(f)$$ is injective (resp., ...
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3answers
371 views

Eigenvalue and similar matrices

if $A$ and $B$ are two $n\times n$ matrices with same eigenvalues such that each eigenvalue has same algebraic and geometric multiplicity. Does $A$ and $B$ are similar? If $A$ is diagnalizable then ...
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2answers
65 views

Suppose that $f: \mathbb R^n \to \mathbb R^n$ is a bijection and $n\geq2$. Can $f$ send every open set onto non-open set?

Suppose that $f: \mathbb R^n \to \mathbb R^n$ is a bijection and $n\geq2$. Can $f$ send every open set onto non-open set? I do not know what exactly to write about this problem. Did I try anything? ...
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1answer
590 views

Counterexample to the primality test

This is a generalization of this claim . Can you provide a counterexample to the following claim? Let $n$ be a natural number greater than two . Let $r$ be the smallest odd prime number such that ...
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0answers
35 views

Can this set-theoretic vector space be endowed with this operation so that everything is fine?

Mike asked in this question to find a model of vector space with additional operation $\wedge$ so that we have $a \wedge (a+b) = (a+b) \wedge b = a \wedge b$ and $\wedge$ is non-distributive. I built ...
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1answer
69 views

Do these functions exist? [duplicate]

I created this question, but, I do not know the answer: Is there a function $f: \mathbb R \to \mathbb R$ such that for every interval $I \subseteq \mathbb R$ we have $f(I)=\mathbb R$? It seems to ...
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0answers
79 views

Compositeness test for numbers of the form $E_n(b)=\frac{b^{2^n}+1}{2}$

Can you provide a proof or a counterexample for the following claim? Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ . Let $E_n(b)= \frac{b^{2^n}+1}{2} $ where $b$ is an odd ...
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0answers
33 views

Defining Multiplication on Strongly Homogeneous Commutative Semigroups

Let us call any commutative semigroup $(S, +)$ strongly homogeneous if it satisfies the following three properties: P-1. Every endomorphism on $S$ is a bijection. P-2. Any two endomorphisms on $S$ ...
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0answers
24 views

Looking for an example of Lusin Theorem

Is there a particular short example and not hard to understand (could be basically trivial but not totally trivial) of the Lusin's Theorem that you can provide please? I have found a related question ...
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2answers
54 views

Counter example for Baire's Theorem

Theorem: Let $(X,d)$ be a complete metric space, and let $D_n, n\in \mathbb N$ be open, dense subsets of $X$. Then also $\bigcap_{n\in\mathbb N} D_n$ is dense in $X$. This statement is false if $X$ ...
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1answer
58 views

Hard inequality :$\sum_{cyc}\frac{a}{\sqrt[3]{a+b}}\leq a+b^{\frac{2}{3}}+c$

I'm interested by the following problem : Let $a,b,c$ be positive real numbers such that $a+b+c=1$ and $a\geq b \geq c$ then we have : $$\sum_{cyc}\frac{a}{\sqrt[3]{a+b}}\leq 1-b+b^{\frac{2}{3}}=...
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2answers
61 views

Counterexample to $ \frac{\mathrm{d}{y}}{\mathrm{d}{x}} = \frac{1}{\left( \frac{\mathrm{d}{x}}{\mathrm{d}{y}} \right)} $

I saw this question: Is $ \frac{\mathrm{d}{x}}{\mathrm{d}{y}} = \frac{1}{\left( \frac{\mathrm{d}{y}}{\mathrm{d}{x}} \right)} $? If this is true then see the following example: $y = sin(x)$, then $\...
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1answer
62 views

find 2×2 matrix such that $ A^2 =-I$

find 2×2 matrix such that $ A^2 =-I$ . I think that eigenvalue of $A^2$ is 0 or positive but eigenvalue of $-I$ is negative so we haven't matrix wih this condition.
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1answer
57 views

Set theory based on different logic

In HoTT book it is written that, We note that a set-theoretic foundation has two “layers”: the deductive system of first-order logic, and, formulated inside this system, the axioms of a particular ...
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1answer
62 views

If $f$ is absolutely continuous $\sqrt(f)$ may not be

The problem reads: Prove that if $ f:[0,1]\rightarrow(0,\infty) $ is absolutely continuous $ \sqrt{f} $ may not be. I am having trouble figuring out how to show this. I found that $x^2\sin\left(\...
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2answers
23 views

Questioning dense subset completeness (counterexample)

Let $X$ be a separable metric space and $A \subset$ X be countable and dense. Characterize the statements below as true or false (and why). If every Cauchy sequence in $A$ converges in $X$, $...
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1answer
39 views

Connection between composition and “inner composition” in closed monoidal categories

Let ${\mathcal C}$ be a symmetric closed monoidal category, $I$ its unit object, $\lambda_X:I\otimes X\to X$ the left unit morphism, and let me denote the internal hom-functor by a fraction $$ (X,Y)\...
1
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1answer
29 views

Uniformly Integrability and Non-Tightness

I want to costruct a measure space $(X,\mathcal{F},\mu)$ and a $\mathcal{C}\subset\mathrm{m}\mathcal{F}$, where $\mathrm{m}\mathcal{F}$ be the set of extended real-valued measurable functions on $X$, ...