A supersimple theory eliminates hyperimaginaries. In particular, in a supersimple theory Lascar strong type is the same as strong type, and every strong type has a canonical base in C . It follows that the Amalgamation Theorem (Independence Theorem) holds for types over algebraically closed sets.

An odd prime p is called a Wieferich prime if 2p−1 ≡ 1 (mod p); alternatively, a Wilson prime if (p− 1)! ≡ −1 (mod p). To date, the only known Wieferich primes are p = 1093 and 3511, while the only known Wilson primes are p = 5, 13, and 563. We report that there exist no new Wieferich primes p < 4×1012 , and no new Wilson primes p < 5×108. It is elementary that both defining congruences above h...

Let ψm be the smallest strong pseudoprime to the first m prime bases. This value is known for 1 ≤ m ≤ 11. We extend this by finding ψ12 and ψ13. We also present an algorithm to find all integers n ≤ B that are strong pseudoprimes to the first m prime bases; with a reasonable heuristic assumption we can show that it takes at most B2/3+o(1) time.

We prove a uniform functional law of the logarithm for the local empirical process. To accomplish this we combine techniques from classical and abstract empirical process theory, Gaussian distributional approximation and probability on Banach spaces. The body of techniques we develop should prove useful to the study of the strong consistency of d-variate kernel-type nonparametric function estim...

Given a domain $I\subset\Bbb{C}$ and an integer $N>0$, function $f: I\to\Bbb{C}$ is said to be {\it entrywise positivity preserving} on positive semidefinite $N\times N$ matrices $A=(a_{jk})\in I^{N\times N}$, if the application $f[A]=(f(a_{jk}))$ of $f$ $A$ for all such $A$. Such preservers in dimensions have been classified by Schoenberg Rudin as being absolutely monotonic. In fixed dimension...

Let p be a prime k|p−1, t = (p−1)/k and γ(k, p) be the minimal value of s such that every number is a sum of s k-th powers (mod p). We prove Heilbronn’s conjecture that γ(k, p) k1/2 for t > 2. More generally we show that for any positive integer q, γ(k, p) ≤ C(q)k1/q for φ(t) ≥ q. A comparable lower bound is also given. We also establish exact values for γ(k, p) when φ(t) = 2. For instance, whe...

Although the basic building blocks for working with strong encryption have become fairly widespread in the last few years, experience has shown that implementers frequently misuse them in a manner that voids their security properties. At least some of the blame lies with the tools themselves, which often make it unnecessarily easy to get things wrong. Just as no chainsaw manufacturer would thin...