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 ...

learn more… | top users | synonyms (2)

0
votes
1answer
27 views

Martingale $X_n \to \infty$ a.s.

Construct a martingale $X_n$ such that $X_n \to \infty $ a.s. I have trouble coming up with such an example and prove it. Can someone provide an example?
1
vote
1answer
37 views

Counterexample for the infinitely many primes between two primes in a Noetherian ring

Consider the following Proposition: Proposition: Let $R$ be a noetherian ring. If $p_0 \subsetneq p_1 \subsetneq p_2$ is a chain of distinct prime ideals in $R$, then there exist infinitely many ...
0
votes
1answer
30 views

A set of real numbers whose limit points from a countable set

Construct a set of real numbers whose limit points from a countable set. Is the set you constructed closed? Is it compact? My example is $$G=\{1/n+1/m: n, m \in \mathbb N\}\cup \{0\}$$ and as ...
0
votes
1answer
40 views

Example of a Differential equation whose solution is not defined for all time $t$ [on hold]

Give an example of a differential equation with its domain $R$ and an initial condition for this equation such the solution is not defined for all time $t$.
3
votes
1answer
20 views

Examples spectral sequence

I have to make a talk about spectral sequences, so I'd like to present some concrete examples of computation, after the general definition. I'd like to present three examples of spectral sequences: ...
1
vote
1answer
23 views

Premeasures: Inner Measure vs. Outer Measure

Problem Given a plain space $\Omega$ and a ring $\mathcal{R}$. (In fact, a semiring would do the job, too.) Consider a premeasure $\mu:\mathcal{R}\to\overline{\mathbb{R}}_+$. For simplicity, ...
1
vote
2answers
54 views

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$ I'm pretty sure it's not necessarily true, but can't think of a counter example. Can you help me think of ...
0
votes
1answer
49 views

Monoidal categories in which $\mathrm{Aut}(X \otimes Y) \cong \mathrm{Aut}(X) \sqcup \mathrm{Aut}(Y).$

I'm looking for examples of monoidal categories $\mathbf{C}$ such that one of the following two statements holds. For all objects $X$ and $Y$ of $\mathbf{C},$ $\mathrm{Aut}(X \otimes Y) \cong ...
2
votes
1answer
33 views

Example of topological space where there is a point and a subset $A $: $x \in \overline A $, but no sequence in $A $ converging to $x $?

It is known that if a space $X $ is metricable then for any subset $A $ of $X $ and a point $x \in \overline A $ there is a sequence of points in $A $ converging to $x $. I wonder if there is an ...
4
votes
0answers
21 views

Function in Lipschitz space

I'm looking for a function that is in $W^{1,1}(0,1)$ but only in the Lipschitz space $\mathrm{Lip} (\alpha, L_2(0,1))$ for $0<\alpha < 1$. $\mathrm{Lip}(\alpha, L_2(0,1))$ is defined as the set ...
2
votes
2answers
28 views

Dimension Field True/False.

I'm having trouble approaching how to determine truthfulness and falsehood of the following type of problems. $F$ and $K$ are fields. 1) Suppose that $F\subseteq K$ and $r\in K$. If $[F(r):F]=4$ ...
2
votes
1answer
61 views

Proof or counterexample: If $F\subseteq K$ and $r\in K$. If $[F(r):F]=4$ then $F(r)=F(r^3)$.

$F$ and $K$ are fields. Proof or counterexample: If $F\subseteq K$ and $r\in K$. If $[F(r):F]=4$ then $F(r)=F(r^3)$. I think I need to find a polynomial in $F(r^3)[x]$ that has $r$ as a root. I ...
0
votes
1answer
18 views

The set of differentiability of an extension from half-plane to the plane

This question is related to: Differentiability: Partially Defined Functions Consider a real-valued function $f:\mathbb{H}^2\to\mathbb{R}$. Are there some that admit no extension differentiable in ...
1
vote
1answer
32 views

Example of metric space completion

I'm looking for examples of noncomplete metric spaces and their completions. I know of some basic examples like completion of open intervals and rational numbers(both with the reals and p-adic ...
2
votes
2answers
41 views

Counterexample for “if every continuous function on $E$ is bounded, then $E$ is compact” [on hold]

Give me counter example for this false statement: "Every continuous function on the set $E$ is bounded this implies $E$ is compact".
7
votes
1answer
63 views

If two elements a,b have a gcd, do then a*t,b*t also have a gcd?

Let $M$ be a commutative cancellative monoid. For elements $a,b \in M$ a gcd of $a,b$ is an element $\mathrm{gcd}(a,b)$ with the (universal) property $\forall c \in M (c |\mathrm{gcd}(a,b) ...
2
votes
0answers
37 views

Example of a series parallel graph with toughness greater than $\frac{4}{7}$

Can anyone lead me to an example of a "more than $\frac{4}{7}$ tough series parallel graph"? Graph toughness is defined as $T = \min \left\{\frac{|a|}{\omega{(G\backslash A)}}\right\}$ over all ...
0
votes
1answer
34 views

How do I prove that this map is a homeomorphism?

Let $X$ be a topological space. Let $\{X_i\}$ be a family of mutually disjoint open subsets of $X$ such that $\bigcup X_i = X$. Let $a_i$ be a point of $X_i$ for each $i$. Consider a quotient map ...
1
vote
1answer
50 views

Zero sets in Mrówka spaces

For a maximal almost disjoint family $\mathcal A$ of subsets of $\omega$ we choose a set $\{x_A:A\in\mathcal A\}$ of distinct points not in $\omega$ and define $\Psi (\mathcal A)=\omega\cup \{x_A:A\in ...
1
vote
1answer
26 views

What are some examples of these kinds of commutative semirings?

What are some examples of commutative semirings such that the following hold? Multiplication is idempotent i.e. we have $xx=x$ for all elements $x$. Addition is not idempotent i.e. there is at least ...
1
vote
2answers
23 views

Examples: Non-Step & Non-Measurable

Reference This question is related to: Banach Spaces: Uniform Integral vs. Riemann Integral Problem What are examples of real-valued functions: Bounded & Non-Step & Non-Measurable ...
3
votes
3answers
85 views

Example of Hausdorff and Second Countable Space that is Not Metrizable

Does there exist topological space that is Hausdorff and second countable but not metrizable?
5
votes
2answers
84 views

Example (?) of a Banach space containing an uncomplemented copy of itself

I was wondering the following: Background Question: Does there exist a Banach space $X$ which contains a copy $X_0 \subset X$ of itself that is not complemented? By "$X_0$ is a copy of $X$", I ...
1
vote
3answers
98 views

First Countable Spaces are Hausdorff or Not?

Does first countable imply Hausdorff? If not, what is an example of first countable space that is not Hausdorff?
0
votes
0answers
24 views

Analytic function with inconsistent asymptotic behaviour on rays

Consider an function $f$, defined continuously on the closed upper half plane, and analytic on the upper half plane. Going along any ray from the origin that go strictly up (ie. not along the real ...
0
votes
1answer
28 views

Is this question wrong? Sequence of polynomial approximates $\sin$

Prove that there cannot be a sequence of polynomial $p_n$ converging uniformly to $\cos$ or $\sin$ on $\Bbb R$. Doesn't the Taylor series completely contradict this question?
0
votes
1answer
32 views

What do the operator semi-groups have to do with PDE's?

Can anybody please help me to understand what does the semi-group do with partial differential equations? We started this subject very recently and we are now in the proof of Hille-Yosida Theorem, ...
1
vote
0answers
16 views

Find ideal defining $Gr_2(\mathbb{C}^5)$ in Pluker embedding

Let $Gr_k(\mathbb{C}^n)$ the Grassmannian variety of $k$-planes in the complex space $\mathbb{C}^n$. We can consider the Pluker embedding $$ \mathcal{P}: Gr_k(\mathbb{C}^n) \to \mathbb{P}(\Lambda^k ...
2
votes
1answer
72 views

Just How Strong is Associativity?

A friend of mine is using a lot of Non-associative Algebra for an advanced Chemistry project. We were discussing it recently and I found it rather amusing how often she said things like "brackets ...
0
votes
1answer
25 views

Is a ring $R$ factorial $\iff$ $R[X]$ factorial?

Let $R$ be a factorial ring. Then, the polynomial ring $R[X]$ is factorial. I was wondering if the other direction also works (i.e. $R[X]$ factorial $\implies$ $R$ factorial)? If not, please give ...
3
votes
2answers
67 views

Does existence of non-trivial solution of $S(x,y,z) = 0, \; S(y,z,x) = 0, \; S(z,x,y) = 0$ implies existence of trivial solution at $x=y=z$ axis?

My question is following. Suppose that you have an implicit surface given by equation $S(x,y,z) = 0$ (if it matters, now $S(x,y,z)$ is a polynomial function). I'm interested only in $\mathbb{R}^3_{+}$ ...
2
votes
1answer
66 views

What would be an example such that $\langle a\rangle\langle b\rangle \neq \langle ab\rangle$?

Let $R$ be an rng (no unity). Define $IJ$ as the ideal generated by $\{ab:a\in I, b\in J\}$ for every ideals $I,J$ of $R$. Let $I=\langle a\rangle , J=\langle b\rangle $ be principal ideals. What ...
0
votes
2answers
24 views

Existence Adjoint Operator: Boundedness?

Context This would make the check on the GNS construction much more simple. Problem Given a Hilbert space $\mathcal{H}$. Consider a merely linear operator $A:\mathcal{H}\to\mathcal{H}$. Suppose ...
3
votes
1answer
95 views

Generalized Riemann Integral: Nonexample?

Reference For a comparison of integrals see: Uniform Integral vs. Riemann Integral Definition Given a finite measure space $\mu(\Omega)<\infty$ and a Banach space $E$. (In fact, a Hausdorff TVS ...
2
votes
1answer
37 views

Examples of Free Lie Algebra

In wikipedia, free Lie algebras are defined using the universal property. Can anyone give some concrete examples of free Lie algebras? "In mathematics, a free Lie algebra, over a given field $K$, is ...
0
votes
1answer
21 views

Finding an specific counterexample of the interchanging of the limit and integral.

Hi I found the following exercise where I'm stuck, I'd appreciate any help. Thanks in advances Let $0\le f_n\to f$ and $\int f_n\to c>0$ pointwise. Show that $\int \lim f_n d\mu\in [0,c]$ and ...
0
votes
1answer
56 views

Prove or disprove regarding sequences

Question: Find a sequence $\displaystyle \{a_n\}_{n=1}^{\infty}$ such that $a_n\rightarrow 0$ and $n\left|a_{n+1}-a_n\right|\rightarrow \infty$. If no such sequence exists, prove it. My try: At ...
0
votes
1answer
62 views

Not Quite Metrization

Let's say I have a space $X$ with a function $d\colon X \times X \to \mathbb R$ that has the following 2 properties: $d(x,y)\ge 0$ for all $x$, $y \in X$ and $d(x,x) = 0$ for all $x$, $y \in X$. ...
0
votes
2answers
50 views

Fun results from modular arithmetic

I'm trying to go through various bits of neat, fun math with some junior-high-school students in my local area, and am thinking of doing a short unit on modular arithmetic/finite groups. I'm looking ...
-1
votes
0answers
18 views

theorem applicable in real function and not in complex function apart from MVT

please quote a theorem which is applicable to real valued functions and not to complex valued functions apart from MVT and ROLLES Theorem along with example
5
votes
0answers
80 views

Connected space such that (almost) no subspace is connected [duplicate]

Is there a connected space $(X,\tau)$ such that $X$ has more than $2$ points and the only proper connected subsets of $X$ are the singletons?
0
votes
1answer
23 views

What is an example of ideals such that $IJ\neq JI$?

Let $R$ be an rng. Let $I,J$ be ideals of $R$. What is an example of $IJ\neq JI$?
5
votes
1answer
76 views

Counter-example for abelian category that is not concrete

I am trying to figure out a counter-example for abelian category that is not concrete category. Does the category of representations over a quiver work? Thanks,
5
votes
1answer
180 views

Can the composite of two projections really fail to be a projection?

Let $H$ denote a Hilbert space. For any closed subspace $C \subseteq H$, write $P_C$ for the orthogonal projection onto $C$. Then according to wikipedia, the composite $P_U \circ P_V$ needn't be a ...
0
votes
3answers
39 views

Interior of a set?

I'm trying to think if their is any topology for which this is false: If G is an open set, then G = interior(G) Can anybody think of anything? I'm pretty sure it's straight forward.
6
votes
1answer
136 views

Topological spaces with unknown fundamental group

Are there any well known topological spaces for which the fundamental group is not known yet?
1
vote
1answer
28 views

Examples of the Geometric Realization of a Semi-Simplicial Complex

I am reading The Geometric Realization of a Semi-Simplicial Complex by J. Milnor and here are the definitions: I find it difficult to visualize without specific examples. Can anyone help to provide ...
3
votes
3answers
52 views

An example of a Banach space whose evaluation map is not surjective?

I have been giving the following corollary while studying functional analysis Let $X$ be a normed vector space. Then the evaluation map $$ev : X \to X'' , x \mapsto (f \mapsto fx) $$ is an ...
0
votes
2answers
36 views

A function with a finite limit at (both) 0 and infinity

I ask myself if there exists a $f(x)$ function that limit goes to a finite number for both when x goes to 0 and to infinity. Is it possible in some way ?
11
votes
5answers
1k views

Humorous integration example?

I was just reading though an introductory calculus book and it has the note: NOTE When integrating quotients, do not integrate the numerator and denominator separately. This is no more valid in ...