Distribution with two (or more) medians

Is there any example with a distribution with two or more medians? I was reading about the median on wikipedia: https://en.wikipedia.org/wiki/Median and here it says that there may be more than one median, but I haven´t been able to give such an example. I would really appreciate if you can help me Answer Let random … Read more

Counterexample to G2/G1G_2/G_1 abelian implies (A∩G2)/(A∩G1)(A\cap G_2)/(A\cap G_1) abelian.

My officemate and I are currently procrastinating on research/grading (by doing some math, obviously), and he came up with this statement, which we are both convinced is false, but are having trouble coming up with a good counterexample: Suppose G1, G2, A are subgroups of a group G, G1⊴, A\unlhd G, and G_2/G_1 is abelian. … Read more

If G is open and dense subset in R\mathbb R then show that G∖{x}G\setminus\{x\} is also open and dense in R. is it true in general metric space?

If G is open and dense subset in R then show that G∖{x} is also open and dense in R. is it true in general metric space? I know as G is open and singleton set {x} is closed so that G∖{x} has to be open, but how to show it is dense in R. … Read more

What are some interesting cases where the two obvious definitions of “discrete object” diverge?

The nLab page defines “discrete object” as follows: Definition. [nLab] Let C denote a concrete category whose forgetful functor U has a left adjoint F. Call the counit of this adjunction ε. Then X∈C is discrete iff the morphism εX:X←FUX is an isomorphism. Another possible definition, that seems to give the correct answer in most … Read more

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, and I’d be impressed if this has been used to prove things in “real” mathematics. Question. Are there any theorems … Read more

Vertex Coloring Optimal Sum vs Chromatic Number

I am having trouble coming up with an example of when the number of colors used in the optimal solution of the sum coloring problem of a graph is strictly greater than the chromatic number of that graph. Answer The tree that is Figure 3.2 in this paper is an example. The chromatic number is … Read more

Counterexample to the Gronwall lemma

Gronwall Lemma. Let $I := [x_0,x_1]$, $x_1, > x_0$, $a,b \in \mathbb{R}$ where $b \geqslant 0$, $y \in C(I)$, such that $$y(x) \leqslant a + b\int_{x_0}^x y(t) dt$$ holds for all $x \in I$. Then $$y(x) \leqslant a\exp\left( b(x – x_0)\right)$$ for all $x \in I$. Now I am asked to provide a counterexample if … Read more

Composition of multiplicative functions which is not multiplicative

A function $f\colon\mathbb N\to\mathbb C$ is called multiplicative if $f(1)=1$ and $$\gcd(a,b)=1 \implies f(ab)=f(a)f(b).$$ It is called completely multiplicative if the equality $f(ab)=f(a)f(b)$ holds for any pair of positive integers $a$, $b$. (In the definition of multiplicative function we have this condition only for $a$, $b$ coprime.) If is not difficult to show that if … Read more

Is knowing the sum , the product and the sum of squares of $k$ different positive integers enough to find them?

This post posed an interesting question; its answer is negative. Now, we take the sum of squares into consideration. Let $a_1,a_2,a_3…a_k$ be $k$ positive integers such that any two of them are different. Caculate: $$A=\displaystyle \sum_{i=1}^{k} a_i$$ $$B=\displaystyle \prod_{i=1}^{k} a_i$$ $$C=\displaystyle \sum_{i=1}^{k} a_i^{2}$$ $A,B$, $C$ and $k$ are known to us. Can we uniquely identify … Read more

a compact subset of uncountable ordinal space ω1\omega_1

It is known that the topological space ω1 is not compact. Further, ω1 has a least upper bound. Hence, by Theorem 27.1 of Munkres, every closed subset of ω1 is compact. My question is: Is true that every compact subset of ω1 closed? I do not know how to prove it and I also do … Read more