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.
5,176
questions
2
votes
2
answers
125
views
Directional derivative zero, minimum and convexity
Let $f:\mathbb R^n \to \mathbb R$ be convex, let $u\in \mathbb R^n$, $v\in \mathbb R^n\setminus \{0\}$ and assume that the directional derivative of $f$ at $u$ in direction $v$ is $0$.
I'm wondering ...
- 689
1
vote
1
answer
40
views
Bound of the norm of $f^{-1}$ in Wiener algebra
Setup. Wiener algebra $W$ is a set of all functions $f(\zeta) = \sum_{n\in \mathbb{Z}}c_n\zeta^n$ on the unit circle with $\|f\|_W = \sum_{n\in \mathbb{Z}}|c_n| < \infty$. The famous Wiener $1/f$ ...
- 740
0
votes
0
answers
139
views
+50
Prove or disprove that $\gcd\left(n!+1,f\left(n\right)\right)=1$
Problem/Conjecture :
Let :
$$f(x)=\sum_{i=1}^{\infty}\left\lfloor\frac{x^{i}}{i!}\right\rfloor$$
Then it seems we have $n\geq 1$ an integer :
$$\gcd\left(n!+1,f\left(n\right)\right)=1$$
This question ...
- 3,243
0
votes
0
answers
26
views
If $C$ is closed, is $\mathbb{R}^+C=\{kc:k\geq0,c\in C\}$ also closed?
If $C$ is closed, is $\mathbb{R}^+C=\{kc:k\geq0,c\in C\}$ also closed? Here for simplicity, assume $C\subset\mathbb{E}$ which is a finite dimensional Euclidean space.
It is clear that $kC$ is closed. $...
- 619
1
vote
1
answer
39
views
If $N$ is a subset of a monoid $(M,\bot ,e)$ with identity element $\epsilon$ with respect $\bot$ then does the equality $\epsilon=e$ holds?
If $(M,\bot,e)$ is a monoid then it is usually to say that $N$ in $\mathcal P(M)$ is a submonoid of $(M,\bot, e)$ if it is closed under $\bot$ and it contains $e$. So by this definition I suspect that ...
- 4,382
0
votes
1
answer
40
views
Regarding a false claim about extrema
Consider the following false claim:
Let $f:[a,b]\to\mathbb{R}$ be a continuous function. A point $x_0\in(a,b)$ is called a local maximum point iff there is a neighborhood $N$ of $x_0$ such that $\...
- 99
2
votes
1
answer
51
views
Algebraic structures with opposite subalgebra lattices and congruence lattices that are not groups with operators
Are there any algebraic structures where the congruence lattice and subalgebra lattice are opposites that do not look like groups with operators?
My motivation for this question is noticing that ...
- 9,298
14
votes
5
answers
1k
views
What are examples of Halmos's claim that a single small concrete special case can capture every instance of a concept of great generality?
Paul Halmos states:
It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.
What are examples of ...
- 2,161
0
votes
1
answer
45
views
Axler Linear Algebra Done Right 3.f.26
Here's the problem: Suppose $V$ is finite-dimensional and $\Gamma$ is a subspace of $V'$. Show that
$$\Gamma=\{v \in V:\varphi(v)=0 \forall \varphi \in \Gamma\}^0.$$
Note that $V'$ denotes the dual ...
1
vote
0
answers
34
views
Example of little $\alpha$ Hölder function that is not $\beta$ Hölder, for any $\beta>\alpha$?
What functions are in the set $$ c^\alpha \setminus \bigcup_{\beta \in (\alpha,1)} C^\beta?$$
A single example will do, but the more the merrier.
This question was natural to me after writing this ...
- 32.9k
1
vote
0
answers
165
views
How to explain it ? Difference between two continued fraction very small
Well I find it as a coincidence but how to explain :
$$\frac{1}{1+\frac{a}{1+\frac{a^{2}}{1+\frac{a^{3}}{1+\frac{\cdot\cdot\cdot}{a^{2n}}}}}}-\frac{1}{1+\frac{b}{1+\frac{b^{2}}{1+\frac{b^{3}}{1+\frac{\...
- 3,243
0
votes
2
answers
75
views
Conjecture about the minimum of the Gamma function
Problem/Conjecture:
Let the function :
$$f(x)=\frac{((x+x_{\min})!-(x_{\min})!)^{\frac{1}{x}}}{x^{\frac{1}{x^2}}}$$
Where $x_\min$ denotes the minimum abscissa of the Gamma function near by $0.4616$
...
- 3,243
4
votes
1
answer
49
views
Can every subsequence of a non-computable sequence be itself non-computable?
So I was playing around with a small conjecture (not in computability theory), trying to find a counterexample, and I realized that if a function $f$ with very specific properties exists, then I could ...
1
vote
0
answers
45
views
Real analytic periodic function $f$ such that $\nabla f(x)=0 \Rightarrow \nabla^2f(x)=0$.
Is there any non-constant real analytic periodic function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that, $$\{x\in\mathbb{R}^n\mid\nabla f(x)=0 \}\subset\{x\in\mathbb{R}^n\mid\nabla^2 f(x)=0 \}? $$
...
- 87
1
vote
0
answers
73
views
Prove that if $(X,⊥)$ is abelian then for any permutation $σ$ the equality $x_{σ(1)}⊥...⊥x_{σ(n)}=x_1⊥...⊥x_n$ holds.
If $M$ and $N$ are a set then for pratical reason let's we put
$$
\mathcal F(N,M):=M^N
$$
Now if $(X,\bot)$ a semigroup then by the recursion theorem it is not hard to prove (see here for details) the ...
- 4,382
1
vote
1
answer
68
views
Prove or disprove the equality $\biggl\langle\bigcap_{Y\in\mathcal Y}Y\biggl\rangle = \bigcap_{Y\in\mathcal Y}\langle Y\rangle$
A magma is a pair given $(X,\bot)$ given by a set $X$ and a function from $X\times X$ to $X$ so that we say that $Y\in\mathcal P(X)$ is closed under $\bot$ if the inclusion
$$
\bot[Y\times Y]\subseteq ...
- 4,382
0
votes
1
answer
22
views
Any complex lattice is equivalent to a lattice of the form …
Follow-up question to this one.
A complex lattice consists of a pair of $\mathbb{R}$-linearly independent vectors of the real-vector space $\mathbb{C}$. We call two lattices $(\lambda_1,\lambda_2)$ ...
- 525
0
votes
1
answer
15
views
Any complex lattice admits a certain lattice basis
A complex lattice consists of a pair of $\mathbb{R}$-linearly independent vectors of the real-vector space $\mathbb{C}$. We call two lattices $(\lambda_1,\lambda_2)$ and $(\mu_1,\mu_2)$ equivalent if $...
- 525
5
votes
1
answer
100
views
Examples of rings where every left ideal is two-sided but not every right ideal
Is there an example of a ring where every left-ideal is two-sided but not every right ideal?
WHAT FOLLOWS IS A FAILED EXAMPLE
As an argument but not a proof that the example fails, consider:
$ f(a) \;...
- 9,298
5
votes
1
answer
80
views
If $G_1$, $G_2$ and $G_3$ are subgroup of $G$ then does the equality $\big\langle⟨ G_1\cup G_2⟩\cup G_3\big\rangle=⟨ G_1\cup G_2\cup G_3⟩$ holds?
So it is a well knew result that any intersection of subgroup is a subgroup and even it is a well knew result that the union of subgroup is not generally a subgroup. However, if $\mathcal S(G)$ is the ...
- 4,382
1
vote
0
answers
52
views
$(X,Z) \sim (Y,Z) \implies X \sim Y$? [duplicate]
Let $X,Y,Z$ be random variables such that the random vectors $(X,Z)$ and $(Y,Z)$ have the same law. Is it true that $X$ and $Y$ have the same law ?
I can't find a counterexample to it but I can't see ...
- 1,470
2
votes
1
answer
29
views
Non-trivial (regular) open semigroups of the open unit interval $(0,1)$?
This is a follow-up question to the one I asked here.
Namely, are there examples of open (or even regular open) subsets of $(0,1)$ which are multiplicative subsemigroups of $(0,1)$ but are not of the ...
- 525
2
votes
3
answers
75
views
Non-trivial semisubgroups of the unit interval? [closed]
The open unit interval $(0,1)\subset \mathbb{R}$ forms a semigroup under multiplication. What are examples of subsemigroups of this semigroup, which are not intervals of the form $(0,a)$ for $a\in (0,...
- 525
0
votes
1
answer
24
views
If $f$ is a epimorphism from the monoid $(X,⊕,x_0)$ to the monoid $(Y,⊗, y_1)$ then $f(x_0)=y_1$ and $f(x^{-1})=f(x)^{-1}$ provided $x^{-1}$ exists.
Well, I know that if $f$ is a homomorphism from the group $(X,\oplus, x_0)$ to the group $(Y,\otimes, y_1)$ then $f(x_0)$ is $y_1$ and moreover the image of an invertible element of $X$ is an ...
- 4,382
2
votes
0
answers
139
views
Combinatorial game greater than its left options and smaller than its right options.
A number in combinatorial game theory is a game $x=\{x^L\mid x^R\}$ such that all its options are numbers and there are no $x^L,x^R$ such that $x^L\geq x^R$.
It turns out, after some work, that if $x$ ...
- 1,393
3
votes
1
answer
82
views
A problem from Applied Asymptotic Analysis by Peter D. Miller (Exercise 1.8)
transcribed exercise
Exercise 1.8. Suppose that $\mu$ is a continuous parameter and that for each $\mu \in[0,1]$, we have $f(z, \mu)=O(g(z, \mu))$ as $z \rightarrow z_0$ from $D$. The above proof ...
- 147
3
votes
2
answers
147
views
How can I think of counterexamples of abstract mathematical objects?
I noticed that I have an intuition for theory (ie making connections between theorems, lemmas, and proving related statements).
However, when a problem gives me the choice between proving or finding ...
- 73
1
vote
1
answer
39
views
Relation between transtive, minimal and uniquely ergodic systems
I was wondering whether about several dynamical notions, and their relations.
I am considering a amenable second countable Hausdorff group $G$, with Haar measure which acts topologically on a compact ...
- 5,915
0
votes
2
answers
141
views
Why does the equality $\langle x\rangle=\{x^{-1},1,x\}$ not generally hold?
So if $X$ is a not empty subset of a group $(G,\cdot,1)$ then the collection
$$
\mathcal X:=\{H\in\mathcal P(X):H\,\text{subgroup containing X}\}
$$
is not empty since it contains $G$ and moreover its ...
- 4,382
1
vote
1
answer
48
views
Pullback of coproduct inclusions
Let $X$, $Y$ and $Z$ be objects of a category.
Consider the coproduct inclusions $X+Y\to X+Y+Z$ and $Y+Z\to X+Y+Z$.
Is it always true that their pullback is $Y$?
I cannot find a counterexample.
- 7,567
0
votes
1
answer
41
views
Error in proof of $X:=\{0,1\}^\mathbb{R}$ not sequentially compact?
Set $X:=\{0,1\}^\mathbb{R}$ equipped with the product topology, where $\{0,1\}$ is equipped with the discrete topology. By Tychonoff's Theorem, $X$ is compact. I want to show that $X$ is not ...
- 3
0
votes
1
answer
78
views
Are there any non-trivial mathematical hypotheses that cannot be disproved by counterexample?
Proving a hypothesis often requires the development of new and powerful techniques and maybe even new branches of mathematics.
However disproving such a hypothesis could result from a single ...
2
votes
0
answers
67
views
Is Lagrange group theorem still valid for infinite groups?
So it is a well know result that the quotient space $G/\cal R$ corresponding to right congruent relation $\cal R$ is equipotent to the quotient space $G/\cal L$ corresponding to the left congruent ...
- 4,382
1
vote
2
answers
60
views
True or false: For continuous random variable $X$, if $E(X)$ has a closed form, then $P(X<E(X))$ has a closed form.
For this question, let us define "closed form" as an expression restricted to addition, subtraction, multiplication, and division; exponents and logarithms, including $e^x$ and $\ln{x}$; ...
- 8,500
8
votes
2
answers
116
views
Example of a function that is continuous at $c$ whose inverse is discontinuous at $f(c)$
I'd like an example of a function $f:(a,b)\to\mathbb R$ and a point $c\in(a,b)$ such that:
$f$ is invertible.
$f$ is continuous at $c$.
$f^{-1}$ is discontinuous at $f(c)$.
Motivation: There is a ...
- 17.6k
1
vote
1
answer
47
views
Can the modulus of an entire function of the form $ze^{g(z)}$ tend to $1$ in the upper half-plane?
Does there exist an entire function $g$ such that $|ze^{g(z)}| = 1 + o(1)$ as $\text{Im}(z)\to+\infty$?
Question can be reformulated in terms of $u(z) = \text{Re}(g(z))$. Namely, does there exist a ...
- 740
0
votes
2
answers
89
views
Solve $x^n=1$ in a monoid.
Let $(X, *)$ a monoid with identity $e$. So can the equality
$$
x^n=e
$$
hold for some $n\ge 1$ when $x$ is not equal to $e$? If this can be true what is an example? If this is not true how prove it?
- 4,382
0
votes
2
answers
72
views
Examples of unital non-associative algebras [closed]
What is an example of a unital non-associative algebra?
- 129
1
vote
3
answers
74
views
Smallest group that is not a complex reflection group
What is the smallest group which is not a complex reflection group?
Many well known families of finite groups are complex reflection groups https://en.wikipedia.org/wiki/Complex_reflection_group For ...
- 5,860
0
votes
0
answers
11
views
Examples of non-convex Nash equilibrium problems with only inequality constraints?
I am trying to find real life application examples for the Nash equilibrium problem of finding $(\overline{x},\overline{y})$ with
$$\overline{x} \in \begin{array}{cc}argmin_x & f_1(x,\overline{y}) ...
- 999
1
vote
1
answer
25
views
Example of a differentiable function such that its partial derivatives are not continues at some point
I am currently reviewing James Stewart's Multivariable Calculus. In it, we are given the following theorem regarding a function $f: \mathbb{R}^2 \to \mathbb{R}$.
If the partial derivatives $f_x$ and $...
- 598
4
votes
0
answers
74
views
Examples of Fano varieties having $h^i(X, O_X) \neq 0$ for some $i >0$ in positive characteristic
A Fano variety $X$ is a (smooth) projective algebraic variety whose anticanonical bundle $-K_X$ is ample.
In characteristic $0$, by Kodaira vanishing, $h^i(X, O_X) = 0$ for all $i>0$.
In his paper &...
- 73
1
vote
2
answers
72
views
Example of $g, h:\mathbb{R}^3\setminus\{0\}\to\mathbb{R}$ with this behavior in the origin and this condition on second partial derivatives
Last week I had a calculus test consisting in some T/F questions and some open questions. I solved correctly all the TF questions but I got some troubles about two open questions.
The first one was: ...
- 3,128
0
votes
1
answer
47
views
Let $a\in \mathbb Z$ be such that $a=b^2+c^2,$ where $b,c\in \mathbb Z-\{0\}.$ Then $a$ can be written as ...
Let $a\in \mathbb Z$ be such that $a=b^2+c^2,$ where $b,c\in \mathbb
Z-\{0\}.$ Then $a$ can be written as
$(1)pd^2,$ where $d\in \mathbb Z$ and $p$ is a prime with
$p\equiv1\pmod 4$
$(2)pd^2,$ where ...
- 49
0
votes
0
answers
49
views
We know that every normal topology is a regular topology
We know that every normal topology is a regular topology, but I'm trying to find an example to show that the opposite is not true.
- 11
1
vote
1
answer
51
views
Is there everywhere large real-part analytic function on the upper half-plane?
Let $\mathbb{H}$ be the upper half-plane of the complex plane. Does there exist an analytic function $A$ on $\mathbb{H}$ satisfying the inequality $\text{Re}(A(z))\ge \text{Im} (z)^{\alpha}$ for some $...
- 740
0
votes
1
answer
64
views
What is an $E-$ invariant set in this context?
I'm having some trouble understanding the following definition from Marker's Model Theory, Definition 1.3.9:
We say that an $\cal L_0$-structure $\cal N$ is interpretable in an $\cal L$-structure $\...
- 3,959
3
votes
1
answer
153
views
Constructing a counter-example of an increasing function such that $|\alpha(x)/x|, |\alpha(x)/\alpha(x/2)| \to \infty$ as $x\to 0^+$
Consider a continuous function $\alpha:[0,\infty) \to [0,\infty)$ satisfying the following properties:
$\alpha(0)=0;$
$\alpha$ is a strictly increasing function;
$\alpha$ is smooth in $(0,\infty);$ ...
- 3,469
3
votes
2
answers
94
views
Upper Bounding the Discrete Entropy by the Expectation
Let $\mathcal P$ be the set of probability mass functions (pmfs) on $\mathbb Z_{>0}$, i.e. for $p=(p(x))_{x\in\mathbb Z_{>0}}\in\mathcal P$ we have $p\ge 0$ and $\sum_{x=1}^\infty p(x)=1$.
Let $...
- 2,862
1
vote
0
answers
45
views
Simplest application of Picard-Lindelöf in the sciences
I am teaching single-variable real analysis and I want to give the students a concrete example of application of the Picard--Lindelöf theorem for a first-order ODE
$$
\frac{dx}{dt}=f(t,x),$$
where $t$ ...
