A polynomial algorithm for minimal interval representation

The following problem arises in the context of object representation: given two endpoints of an interval in a Gray code table, find a Boolean function in DNF that represents this interval, with as few prime implicants as possible. This paper shows that there is a unique minimal representation and presents a polynomial algorithm that finds it.     

Current challenges in compound-specific stable isotope analysis of environmental organic contaminants

Compound-specific stable-isotope analysis (CSIA) has greatly facilitated assessment of sources and transformation processes of organic pollutants. Multielement isotope analysis is one of the most promising applications of CSIA because it even enables distinction of different transformation pathways. This review introduces the essential features of continuous-flow isotope-ratio mass spectrometry (IRMS) and highlights current challenges in environmental analysis as exemplified for the isotopes of nitrogen, hydrogen, chlorine, and oxygen. Strategies and recent advances to enable isotopic measurements of polar contaminants, for example pesticides or pharmaceuticals, are discussed with special emphasis on possible solutions for analysis of low concentrations of contaminants in environmental matrices. Finally, we discuss different levels of calibration and referencing and point out the urgent need for compound-specific isotope standards for gas chromatography-isotope-ratio mass spectrometry (GC-IRMS) of organic pollutants.     

Optimizing buffer size for the retrial queue: two state space collapse results in heavy traffic

We study a single server queueing model with admission control and retrials. In the heavy traffic limit, the main queue and retrial queue lengths jointly converge to a degenerate two-dimensional diffusion process. When this model is considered with holding and rejection costs, formal limits lead to a free boundary curve that determines a threshold on the main queue length as a function of the retrial queue length, above which arrivals must be rejected. However, it is known to be a notoriously difficult problem to characterize this curve. We aim instead at optimizing the threshold on the main queue length independently of the retrial queue length. Our main result shows that in the small and large retrial rate limits, this problem is governed by the Harrison–Taksar free boundary problem, which is a Bellman equation in which the free boundary consists of a single point. We derive the asymptotically optimal buffer size in these two extreme cases, as the scaling parameter and the retrial rate approach their limits.     

Scan Statistic Tail Probability Assessment Based on Process Covariance and Window Size

A scan statistic is examined for the purpose of testing the existence of a global peak in a random process with dependent variables of any distribution. The scan statistic tail probability is obtained based on the covariance of the moving sums process, thereby accounting for the spatial nature of the data as well as the size of the searching window. Exact formulas linking this covariance to the window size and the correlation coefficient are developed under general, common and auto covariance structures of the variables in the original process. The implementation and applicability of the formulas are demonstrated on multiple processes of t-statistics, treating also the case of unknown covariance. A sensitivity analysis provides further insight into the variant interaction of the tail probability with the influence parameters. An R code for the tail probability computation and the data analysis is offered within the supplementary material.     

Quasi-interpolation and outliers removal

In this work, we present a method that will allow us to remove outliers from a data set. Given the measurements of a function f = g + e on a set of sample points \(X \subset \mathbb {R}^{d}\), where \(g \in C^{M+1}(\mathbb {R}^{d})\) is the function of interest and e is the deviation from the function g. We will say that a sample point x ∈ X is an outlier if the difference e(x) = f(x) ? g(x) is large. We show that by analyzing the approximation errors on our sample set X, we may predict which of the sample points are outliers. Furthermore, we can identify outliers of very small deviations, as well as ones with large deviations.     

The missing link in teachers’ knowledge about common fractions division

The current study explored the difficulties teachers encounter when teaching common fractions division, focusing on teachers’ knowledge concerning this issue. Nine teachers who study towards a M.Ed. degree in mathematics education demonstrated the algorithms they apply in order to solve fractions division problems, described how they teach the subject, and attempted to explain a student's mistake, in understanding a word problem involving dividing by fraction. The findings indicate there is a missing link in the teachers’ pedagogical capability, stemming from insufficient content knowledge. They presented different solution algorithms and reported using constructivist teaching methods, yet the methods they described couldn't lead a student to understand the logic behind the algorithm they teach (invert-and-multiply – multiplication by an inverse number, in accordance with the requirements of the curriculum). Furthermore, the participating teachers did not possess specialized mathematics content knowledge (SCK) and knowledge of content and students (KCS), enabling them to identify the source of a student's misconception.