Posted by: holdenlee | May 9, 2010

## B7s: April 24, 2010; May 8, 2010

4/24 Problem: Let $X$ be a closed subset of $\mathbb{R}^n$. Prove that there exists a function $f:\mathbb{R}^n\to\mathbb{R}$ that is zero on $X$, positive outside $X$, and whose partial derivatives of all orders all exist.

5/8 Problem 1: [MS 1965/1] Let $p$ be a prime, $n\in \mathbb{N}$, and $S$ a set with $p^n$ elements. Let $P$ be a family of partitions into nonempty parts of sizes divisible by $p$ such that the intersection of any two parts that occur in any of the partitions has at most 1 element in common. Find $\max(P)$. (Scroll down for solution)

5/8 Problem 2: [MS 1965/2] Let $R$ be a finite commutative ring. Prove that $R$ has an identity element iff the annihilator of $R$ is 0. (The annihilator of $R$ is the set $\{a\in R\mid ar=0 \text{ for all } r\in R\}$.)

Note: Here a commutative ring does not by definition need to have an identity. Multiplication is associative and commutative (i.e. $R$ is a commutative semigroup), distributive over addition, and $R$ is an additive group.

$\displaystyle \frac{p^n-1}{p-1}.$

(A) The bound is attainable: Think of $S$ as the vector space $\mathbb{F}_p^n$. For a line $W$ through the origin (i.e. a one-dimensional subspace of $S$), let the partition $P_W$ consist of the cosets of $W$. The family of such partitions satisfies the given conditions: each part has $p$ elements and every two lines can intersect at only one point. The number of such partitions is just the number of lines passing through the origin. There are $p^n-1$ choices for a point other than the origin for the line to pass through, which determines the line. However, each of the $p-1$ points beside the origin on the line give the same line, so we must divide by $p-1$.

(B) The bound is optimal. Let $m$ be the number of partitions in $P$. Take any point $O$. Each partition must have exactly one set containing $O$; call them $S_1,\ldots,S_m$. Since two sets in the partition cannot intersect in more than one point, $S_i-\{O\}$ are all disjoint. Each has at least $p-1$ elements since each set in a partition has to have at least $p$ elements. The union $\bigcup (S_i-\{O\})$ must have at most $|S-\{O\}|=p^n-1$ elements, so

$(p-1)m\leq p^n-1,$

giving the desired bound.