quotient

Biproportional apportionment methods provide two-way proportionality in electoral systems where the electoral region is subdivided into electoral districts. The problem is to assign integral values to the elements of a matrix that are proportional to a given input matrix, and such that a set of row- and column-sum requirements are fulfilled. In a divisor-based method for biproportional apportionment, the problem

Let X be a Hausdorff topological group and G a locally compact subgroup of X. We show that the natural action of G on X is proper in the sense of R. Palais. This is applied to prove that there exists a closed set F ⊂ X such that FG = X and the restriction of the quotient projection X → X/G to F is a perfect map F → X/G. This is a key result to prove that many topological properties (among them, paracompactness and normality) are transferred from X to X/G, and some others are transferred from X/G to X. Yet another application leads to the inequality dimX ≤ dimX/G+dimG for every paracompact topological group X and a locally compact subgroup G of X having a compact group of connected components.

The development of two middle school students' reasoning about quotients in arithmetic and algebraic contexts was studied through a series of individual teaching experiments. Initially, when children were confronted with algebraic contexts, they viewed algebraic fractions as a new case. As they progressed, it appeared that understanding of quotients of algebraic expressions depended heavily on the students' understanding of arithmetical quotients. They used analogy to arithmetic fractional reasoning as a tool for initial understanding, followed similar developmental routes when constructing more advanced knowledge, and folded back to arithmetic analogies when new information was presented. In particular, similar to students' development of understanding in numeric fractions, their thinking began with an understanding of Algebraic Fractions as Part Whole, moved to a notion of Algebraic Fractions as Division, then to seeing Division of Polynomials as Algebraic Fractions. At the end of the teaching experiments, they approached situations involving algebraic fractions or division of polynomials flexibly and used multiple-constructed methods.

Abstract. In this note we prove a new result about (finite) multiplier sequences, i.e. linear operators acting diagonally in the standard monomial basis of R[x] and sending polynomials with all real roots to polynomials with all real roots. Namely, we show that any such operator does not decrease the logarithmic mesh when acting on an arbitrary polynomial having all roots real and of the same sign. The logarithmic mesh of such a polynomial is defined as the minimal quotient of its consecutive roots taken in the nondecreasing order of their absolute values. Multiplicateurs et mailles logarithmiques Résumé. Les multiplicateurs considérés dans cette note sont les opérateurs linéaires qui agissent diagonalement sur R[x] muni de sa base standard (les monômes) et qui transforment les polynômes à racines réelles en polynômes à racines réelles. Nous montrons qu’un tel opérateur, appliqué à un polynôme dont toutes les racines sont réelles et de même signe, ne diminue pas la maille logarithmi...