pr.probability – Is there a name for a random variable that is the absolute value of the difference between two iid discrete uniform variables?

I’m working on a project and I needed to calculate the distribution of the difference between two iid discrete uniform variables (sorry for the long title).
That is, let $I, J$ be two iid discrete uniform variables with support ${1, ldots, k}$. I calculated the distribution of $Z = |I – J|$:
$P(Z = 0) = frac{1}{k}$, $P(Z = y) = frac{2(k-y)}{k^2} forall y in {1, ldots, k-1}$.
I was wondering if such a distribution has a name, so that I can get more info and maybe some results about it.

Thanks a lot.

Packing in uniform domains

We consider the square lattice $mathbb{Z}^2$ then we can define $r(N)$ to be the number of coordinates $(x,y)in mathbb{Z}^2$ such that $$x^2+y^2=N.$$

It is a theorem by Gauß (Theorem 9.4) that
$$sum_{i=1}^{N} r(i) = pi N+ mathcal O(sqrt{N}).$$

This shows that it is possible to fit $N$ points in a domain of diameter $mathcal O(sqrt{N})$ in a regular way.

Given $N$ points $X:=(x_i)_{i in {1,..,N}}$, we now define a score function $S:X rightarrow mathbb{N}$ that is $S(X)= sum_{i=1}^N S(x_i)$ where the score of $S(x_i)$ is

$$S(x_i) = 2* vert {x_j; vert x_i-x_j vert in (1,2)} vert+ vert {x_j; vert x_i-x_j vert in (2,3)} vert$$
where $vert bullet vert$ denotes the cardinality of the set.

Question: Is it true that any configuration of $N$ points with maximal possible score is in a domain of diameter $csqrt{N}$ for some fixed c?

complex analysis – Real roots of a uniform compact limit funtion, if the sequence of functions only have real roots

Let $(f_n)_n$ be a sequence of entire functions that converge uniformly on compact subsets of $mathbb{C}$ to some function $f$. Then $f$ is entire, by Morera’s theorem. The question is,

If $f_n$ has $n$ simple real roots (just say all of $f_n$‘s all have only real roots), does $f$ have only real roots as well?

What comes in my mind has something with the following theorem, what other call it as the Hurwitz Theorem,

Let $G$ be a region and suppose the sequence $(f_n)$ of holomorphic functions on $G$ converging to a limit function $f$ on $G$. If $fnotequiv 0$, $overline{B}(a;R)subseteq G$, and $f(z)neq 0$ for $zin partial B(a,R)$, then there is an integer $N$ such that for $ngeq N$, $f$ and $f_n$ have the same number of zeros in $B(a,R)$.

Correct me, if I am wrong. Assume contrary that $f$ has a simple non-real root $z_0in mathbb{C}$. Let $r>0$ be such that the closed disc $overline{B}(z_0,r)$ does not intersect the real axis at all. Then, for sufficiently large $n$, both $f$ and $f_n$ have the same number of roots in $B(z_0,r)$. This result indicates that $f_n$ has roots in $B(z_0,r)$, which is impossible, because it does not contain real roots.

My wordings may sound a bit confusing. What do you think? Are there some details I should mention or not mention?

Thank you

co.combinatorics – making a random uniform hypergraph linear

Let $mathcal{H}_{n,p,h}=(V,E)$ be a random $h$-uniform hypergraph on $(n)$, sampled according to the usual binomial distribution. We known that with high probability, the number of edges in $mathcal{H}_{n,p,h}$ is
$$m = (1+o(1))binom{n}{h}p$$

Let $ell$ be given. I would like to delete some edges in order to

  • have a linear hypergraph (any two edges share at most one vertex)
  • remove all cycles of length at most $ell$

I expect that we should be able to do so by deleting with high probabilities $o(m)$ edges, however simple first moment method are failing me… I try to count the number of Berge-cycle of length of length at most $ell$, but simply looking at potential cycles for each pair of vertices I over-count way too much.

Is there any known upper bound for the number of cycles ? I found some literature on the probability threshold for the appearance of cycles, but not much on counting the cycles.

Is the conditional probability uniform if the joint is uniform?

I mean this questions in the general sense. Suppose, we have two random variables $X$ and $Y$ that are jointly uniformly distributed, i.e.
$$f_{X,Y}(x,y) = k space forall x,y$$

This should mean that all (x,y) points are equally likely. Now if we find out one of the X’s or Y’s, we’re left with just finding the other one using the conditional distribution. So,
$$f_{X|Y}(x|y) = frac{f_{X,Y}(x,y)}{f_Y(y)} = frac{k}{f_Y(y)}$$

So given we know $Y = y$, the joint uniform density then becomes a conditional uniform density? Is this correct? It should also hold for the discrete case. Can someone point me to a more mathematical proof?

probability theory – Sampling from the uniform distribution

Is there an efficient classical algorithm that generates samples from the uniform distribution (or a distribution that is close to the uniform distribution in total variation distance), over the set ${0, 1}^{n}$, for a fixed $n$?

We can easily express the uniform distribution in terms of a formula:

$Pr(X = x) = frac{1}{N}$, for each $x in {0, 1}^{n}$ and $N = 2^{n}$.

My guess is that any distribution that can be expressed in terms of an explicit formula can be efficiently sampled from, but I can’t find a concrete proof.

Uniform continuity of complex valued function $f(z)=frac{1}{z}$ where $0

Complex Analysis Experts

I have discussed same question for Real valued function, now my questions is

Is complex valued function $f(z)=frac{1}{z}$ uniformly continuous on

  1. 0leq |z| leq 1

  2. 0< |z| leq 1

  3. 0leq |z|= 1

  4. 0< |z| < 1

1, 3 are not possible because f is not continuous there.

also, relate it with real valued function.

fa.functional analysis – does the Skorokhod space with the uniform topology admit a smooth partition of unity?

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