The permanent of an $m$-by-$n$ matrix $A$ is the sum of all possible products of $m$ elements from $A$ with the property that the elements in each of the products lie on different lines of $A$. This scalar valued function of the matrix $A$ occurs throughout the combinatorial literature in connection with various enumeration and extremal problems. In this note, we construct a $(0,1)$-matrix with a prescribed permanent, $1$, $2$, $\ldots$, $2^{n-1}$.

In his paper [1], K. L. D. Gunawardena introduced two sequences of multiple integrals (see (1.1) and (1.2) below) encountered in one of his courses of mathematical statistics. Using a key probability result, he evaluated these integrals. At the end of [1], he asked the question whether these integrals could be computed without using results from probability theory. The aim of this paper is to answer that question and make direct computations of these integrals by elementary methods without using probabilistic tools.

A fundamental result of calculus states that a function $f$ with a positive derivative $Df$ on an interval is increasing on that interval. This result follows directly from the Mean Value Theorem. We explore the extent to which the hypotheses of the Mean Value Theorem can be weakened and $f$ still shown to be increasing. Let $f$ be a continuous function on an interval $[a,b]$. By constructing counterexamples using Cantor-Lebesgue functions, we show that the assumption $Df > 0 \ a.e.$ does not imply that $f$ is increasing. We can, however, show that if $Df > 0$ except on a countable subset of an interval, then $f$ is increasing. We call this the Countable Exceptional Set Theorem. This theorem is generalized by the Goldowsky-Tonelli Theorem, which tells us that if $Df$ exists except on a countable subset of an interval and $Df > 0 \ a.e.$, then $f$ is increasing. We then show that if the exceptional set $S$ is uncountable, then we can construct a continuous function $f$ for which $Df > 0$ on $[a,b] \backslash S$, but for which $f(a) > f(b)$. The key item is that the exceptional set contains a perfect set. Moreover, this occurs whenever the exceptional set is uncountable. Thus, in a very natural sense, Goldowsky-Tonelli is a vacuous extention of the Countable Exceptional Set Theorem.

We shall propose in this paper some practical divisibility tests as well as genuine expressions for the division's quotient. The latter is only based on successive operations using either addition or substraction and multiplication by some appropriate integer weights.