An Approximate Vertex-Isoperimetric Inequality for $r$-sets

We prove a vertex-isoperimetric inequality for \([n]^{(r)}\), the set of all \(r\)-element subsets of \(\{1,2,\ldots,n\}\), where \(x,y \in [n]^{(r)}\) are adjacent if \(|x \Delta y|=2\). Namely, if \(\mathcal{A} \subset [n]^{(r)}\) with \(|\mathcal{A}|=\alpha {n \choose r}\), then its vertex-boundary \(b(\mathcal{A})\) satisfies\[|b(\mathcal{A})| \geq c\sqrt{\frac{n}{r(n-r)}} \alpha(1-\alpha) {n \choose r},\]where \(c\) is a positive absolute constant. For \(\alpha\) bounded away from 0 and 1, this is sharp up to a constant factor (independent of \(n\) and \(r\)).

The growth of univalent functions with an initial gap II

We consider the class Sp of functions univalent in the unit disk Δ.In [1] it was shown that if f∈Sp and p is large, (0.1) Here we show that there exists f in Sp for p=1,2,. . . such that where C0 is a positive absolute constant.

THE SECOND DERIVATIVE OF A MEROMORPHIC FUNCTION

Let $f$ be meromorphic of finite order in the plane, such that $f^{(k)}$ has finitely many zeros, for some $k\geq2$. The author has conjectured that $f$ then has finitely many poles. In this paper, we strengthen a previous estimate for the frequency of distinct poles of $f$. Further, we show that the conjecture is true if either $f$ has order less than $1+\varepsilon$, for some positive absolute constant $\varepsilon$, or$f^{(m)}$, for some $0\leq m lt k$, has few zeros away from the real axis.AMS 2000 Mathematics subject classification: Primary 30D35

Probabilities of ruin when the safety loading tends to zero

When the premium rate is a positive absolute constant throughout the time period of observation and the safety loading of the insurance business is positive, a classical result of collective risk theory claims that probabilities of ultimate ruin ψ(u) and of ruin within finite time ψ(t,u) decrease as eϰu with a constant ϰ>0, as the initial risk reserve u increases. This paper establishes uniform approximations to ψ(t,u) with slower rates of decrease when the premium rate depends on u in such a way that the safety loading decreases to zero as u→∞.

Probabilities of ruin when the safety loading tends to zero

When the premium rate is a positive absolute constant throughout the time period of observation and the safety loading of the insurance business is positive, a classical result of collective risk theory claims that probabilities of ultimate ruin ψ(u) and of ruin within finite time ψ(t,u) decrease as eϰu with a constant ϰ>0, as the initial risk reserve u increases. This paper establishes uniform approximations to ψ(t,u) with slower rates of decrease when the premium rate depends on u in such a way that the safety loading decreases to zero as u→∞.

Small Solutions of the Congruence ax2 + by2 ≡ c(mod k)

In 1957, Mordell [3] provedTheorem. If p is an odd prime there exist non-negative integers x, y ≤ A p3/4 log p, where A is a positive absolute constant, such that(1.1)provided (abc, p) = 1.Recently Smith [5] has obtained a sharp asymptotic formula for the sum where r(n) denotes the number of representations of n as the sum of two squares.