On Fitting's lemma

Variations on Barbalat's Lemma

We first review recent versions of Barbălat's Lemma by comparing their proofs and by discussing a concrete example. Then we present a proof which allows for a quantitative interpretation. In 1959, Barbălat formalized the intuitive principle that a function whose integral up to infinity exists and whose oscillation is bounded needs to be small at infinity. Theorem 1. (Barbălat's Lemma [2, p. 269...

On the Hopf Lemma

The Hopf Lemma for second order elliptic operators is proved to hold in domains with C, and even less regular, boundaries. It need not hold for C boundaries. Corresponding results are proved for second order parabolic operators.

On Yao's XOR-Lemma

A fundamental lemma of Yao states that computational weakunpredictability of Boolean predicates is amplified when the results of several independent instances are XOR together. We survey two known proofs of Yao’s Lemma and present a third alternative proof. The third proof proceeds by first proving that a function constructed by concatenating the values of the original function on several indep...

Option of local anaesthetic for IUD fittings.

New light on Hensel’s lemma

The historical development of Hensel’s lemma is briefly discussed (section 1). Using Newton polygons, a simple proof of a general Hensel’s lemma for separable polynomials over Henselian fields is given (section 3). For polynomials over algebraically closed, valued fields, best possible results on continuity of roots (section 4) and continuity of factors (section 6) are demonstrated. Using this ...