On a Definitizable Analog of the Trigonometric Moment Problem Generating an Indefinite Toeplitz Form

We prove the existence of an integro-polynomial representation for a sequence of numbers such that there exists a difference operator mapping this sequence to a sequence that generates the solvable trigonometric moment problem. A similar result related to the power moment problem was given in [12].     

Carousels,Zindler curves and the floating body problem

A carousel is a dynamical system that describes the movement of an equilateral linkage in which the midpoint of each rod travels parallel to it. They are closely related to the floating body problem. We prove, using the work of Auerbach, that any figure that floats in equilibrium in every position is drawn by a carousel. Of special interest are such figures with rational perimetral density of the floating chords, which are then drawn by carousels. In particular, we prove that for some perimetral densities the only such figure is the circle, as the problem suggests.     

Pseudorandom properties of prime factors

In this paper we study some pseudo-random properties related to the prime factors of integers. We prove some results concerning the well distribution of the greatest prime factor modulo 4, and give some computational insight on the correlations, which seem to be beyond reach of proof.     

Minimal expansions in redundant number systems: Fibonacci bases and Greedy algorithms

We study digit expansions with arbitrary integer digits in base q (q integer) and the Fibonacci base such that the sum of the absolute values of the digits is minimal. For the Fibonacci case, we describe a unique minimal expansion and give a greedy algorithm to compute it. Additionally, transducers to calculate minimal expansions from other expansions are given. For the case of even integer bases q, similar results are given which complement those given in [6].     

Planning District‐Wide Professional Development: Insights Gained From Teachers and Students Regarding Mathematics Teaching in a Large Urban District

This paper describes the mechanism used to gain insights into the state of the art of mathematics instruction in a large urban district in order to design meaningful professional development for the teachers in the district. Surveys of close to 2,000 elementary, middle school, and high school students were collected in order to assess the instructional practices used in mathematics classes across the district. Students were questioned about the frequency of use of various instructional practices that support the meaningful learning of mathematics. These included practices such as problem solving, use of calculators and computers, group work, homework, discussions, and projects, among others. Responses were analyzed and comparisons were drawn between elementary and middle school students' responses and between middle school and high school responses. Finally, fifth‐grade student responses were compared to those of their teachers. Student responses indicated that they had fewer inquiry‐based experiences, fewer student‐to‐student interactions, and fewer opportunities to defend their answers and justify their thinking as they moved from elementary to middle school to high school. In the elementary grades students reported an overemphasis on the use of memorization of facts and procedures and sparse use of calculators. Results were interpreted and specific directions for professional development, as reported in this paper, were drawn from these data. The paper illustrates how student surveys can inform the design of professional development experiences for the teachers in a district.     

The Effects of a Study‐Group Process on the Implementation of Reform in Mathematics Education

In each of three consecutive years, 16 teachers were admitted to a program in which they participated in a study group to reform their teaching practices in mathematics. The 48 (total) elementary school teachers in this program were selected to ensure diversity of setting— urban and suburban, as well as public and parochial. Teachers routinely met with the study group for 1 year, engaging in follow‐up activities after that year. Surveys, interviews, and site visits showed that teachers underwent significant changes in their knowledge, beliefs, and teaching practices and were attempting to implement reform consistent with current standards. Teachers with 11 to 25 years of experience demonstrated the greatest changes. Long‐term effects of the program and its impact on colleagues within the building and district have yet to be studied.     

Pure Skew Lattices in Rings

Given a ring $R$, let $S\subseteq R$ be a pure multiplicative band that is closed under the cubic join operation $x\nabla y = x+y+yx-xyx-yxy.$ We show that $\left( S,\cdot,\nabla\right) $ forms a pure skew lattice if and only if $S$ satisfies the polynomial identity $\left(xy-yx\right)^{2}z = z\left(xy-yx\right)^{2}$. We also examine properties of pure skew lattices in rings.