Dynamic Windows Scheduling with Reallocation

We consider the Windows Scheduling (WS) problem, which is a restricted version of Unit-Fractions Bin Packing, and it is also called Inventory Replenishment in the context of Supply Chain. In brief, WS problem is to schedule the use of communication channels to clients. Each client c i is characterized by an active cycle and a window w i . During the period of time that any given client c i is active, there must be at least one transmission from c i scheduled in any w i consecutive time slots, but at most one transmission can be carried out in each channel per time slot. The goal is to minimize the number of channels used. We extend previous online models, where decisions are permanent, assuming that clients may be reallocated at some cost. We assume that such cost is a constant amount paid per reallocation. That is, we aim to minimize also the number of reallocations. We present three online reallocation algorithms for Windows Scheduling. We evaluate experimentally multiple variants of these protocols showing that, in practice, all three achieve constant amortized reallocations with close to optimal channel usage. Our simulations also expose interesting tradeoffs between reallocations and channel usage. We introduce a new objective function for WS with reallocations that can be also applied to models where reallocations are not possible. We analyze this metric for one of the algorithms that, to the best of our knowledge, is the first online WS protocol with theoretical guarantees that applies to scenarios where clients may leave and the analysis is against current load rather than peak load. Using previous results, we also observe bounds on channel usage for one of the algorithms.

Continuous facility location on graphs

We study a continuous facility location problem on undirected graphs where all edges have unit length and where the facilities may be positioned on the vertices as well as on interior points of the edges. The goal is to cover the entire graph with a minimum number of facilities with covering range $$\delta >0$$ δ > 0 . In other words, we want to position as few facilities as possible subject to the condition that every point on every edge is at distance at most $$\delta $$ δ from one of these facilities. We investigate this covering problem in terms of the rational parameter $$\delta $$ δ . We prove that the problem is polynomially solvable whenever $$\delta $$ δ is a unit fraction, and that the problem is NP-hard for all non unit fractions $$\delta $$ δ . We also analyze the parametrized complexity with the solution size as parameter: The resulting problem is fixed parameter tractable for $$\delta <3/2$$ δ < 3 / 2 , and it is W[2]-hard for $$\delta \ge 3/2$$ δ ≥ 3 / 2 .

Adding value in barley malt rootlets as a source of 5’-phosphodiesterase

A thermostable 5’-phosphodiesterase (5’-PDE, EC 3.1.4.1) was extracted from barley (Hordeum distichum var. Rex) malt rootlets. The purification procedure comprised acetone precipitation, S-Sepharose cation-exchange and DEAE-Sepharose anion-exchange chromatography. The enzyme was purified 101-fold with a recovery of 22% and a specific activity of 81.9 U mg-1 protein, Optimum enzyme activity was obtained at 70 °C, and pH 8.9. The SDS-PAGE profiling of the purified protein exhibited molecular weight of 116 kDa and revealed three sub-unit fractions of 26, 43, and 56 kDa making up its active configuration. The kinetic constants Km and Vmax were determined as 0.25 mM and 0.816 mmol min-1, respectively. Thermodynamic studies showed that the thermal inactivation of purified barley malt rootlets 5’-PDE followed the first-order kinetics, indicating inactivation energy (Ed) of 134 kJ mol-1. The half-life (t1/2) at 70 °C was estimated as 169 min. Thermodynamic parameters ΔH*, ΔS* and ΔG* were determined as a function of temperature and were 131.15 kJ mol-1, 37.01 kJ mol-1 K-1 and 118.4 kJ mol-1, respectively. The purified enzyme has long half-life with 11 days at 0 °C, 37 hours at 4 °C and 11 hours at room temperature. These results provide useful information about the factors that affects the activity of barley malt rootlets 5’-PDE and suggests a good indication for application of this enzyme in pharmaceutical and food industry.

Unit Fractions as Superheroes for Instruction

The complexity of understanding unit fractions is often underappreciated in instruction. We introduce a continuum of children's understanding of unit fractions to explore this complexity and to help teachers make sense of children's strategies and recognize milestones in the development of unit-fraction understanding. Suggestions for developing this understanding are provided.

How does a fraction get its name?

Philosophical and cultural perspectives shape how a fraction is named and defined. In turn, these perspectives have consequences for learners' conceptualization of fractions. We examine historical foundations of two perspectives of what are fractions—partitioning and measuring—and how these views influence fraction knowledge. For the dominant perspective, partitioning, we indicate how its approach to what is a fraction that discretizes objects and its well-meaning visual correlates cause learners a host of perceptual difficulties. Based on the human cultural and social practice of measuring continuous quantities, we then offer an alternative understanding of what is a fraction and illustrate the promise of this view for fraction knowledge. We introduce pedagogical tools, Cuisenaire rods, and illustrate how they can be used to implement a measuring perspective to comprehending properties and a definition of fractions. We end by sketching how to initiate a measuring perspective in a mathematics classroom.Keywors: Fractions; Gattegno; Measuring; Partitioning; Unit fractions. Como uma fração recebe seu nome?Resumo: Perspectivas filosóficas e culturais moldam como uma fração é nomeada e definida. Por sua vez, essas perspectivas têm consequências para a conceitualização de frações dos estudantes. Examinamos os fundamentos históricos de duas perspectivas do que são frações—particionamento e medição—e como essas visões influenciam o conhecimento das frações. Para a perspectiva dominante, partição, indicamos como sua abordagem ao que é uma fração que discretiza objetos e seu correlato visual bem-intencionado causa aos alunos uma série de dificuldades perceptivas. Com base na prática cultural e social humana de medir quantidades contínuas, oferecemos um entendimento alternativo do que é uma fração e ilustramos a promessa dessa visão para o conhecimento da fração. Introduzimos ferramentas pedagógicas, varas Cuisenaire e ilustramos como elas podem ser usadas para implementar uma perspectiva de medição para compreender propriedades e uma definição de frações. Terminamos esboçando como iniciar uma perspectiva de medição em uma sala de aula de matemática.Palavras-chave: Frações; Gattegno; Medição; Partição; Frações unitárias.

Egyptian Fractions Re-Revisited

Ancient Egyptians represented each fraction as a sum of unit fractions, i.e., fractions of the type 1/n. In our previous papers, we explained that this representation makes perfect sense: e.g., it leads to an efficient way of dividing loaves of bread between people. However, one thing remained unclear: why, when representing fractions of the type 2/(2k+1), Egyptians did not use a natural representation 1/(2k+1)+1/(2k+1), but used a much more complicated representation instead. In this paper, we show that the need for such a complicated representation can be explained if we take into account that instead of cutting a rectangular-shaped loaf in one direction – as we considered earlier – we can simultaneously cut it in two orthogonal directions. For example, to cut a loaf into 6 pieces, we can cut in 2 pieces in one direction and in 3 pieces in another direction. Together, these cuts will divide the original loaf into 2 * 3 = 6 pieces. It is known that Egyptian fractions are an exciting topics for kids, helping them better understand fractions. In view of this fact, we plan to use our new explanation to further enhance this understanding.

The number of solutions of the Erdős-Straus Equation and sums of k unit fractions

We prove new upper bounds for the number of representations of an arbitrary rational number as a sum of three unit fractions. In particular, for fixed m there are at most ${\cal O}_{\epsilon }(n^{{3}/{5}+\epsilon })$ solutions of ${m}/{n} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$. This improves upon a result of Browning and Elsholtz (2011) and extends a result of Elsholtz and Tao (2013) who proved this when m=4 and n is a prime. Moreover, there exists an algorithm finding all solutions in expected running time ${\cal O}_{\epsilon }(n^{\epsilon }({n^3}/{m^2})^{{1}/{5}})$, for any $\epsilon \gt 0$. We also improve a bound on the maximum number of representations of a rational number as a sum of k unit fractions. Furthermore, we also improve lower bounds. In particular, we prove that for given $m \in {\open N}$ in every reduced residue class e mod f there exist infinitely many primes p such that the number of solutions of the equation ${m}/{p} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$ is $\gg _{f,m} \exp (({5\log 2}/({12\,{\rm lcm} (m,f)}) + o_{f,m}(1)) {\log p}/{\log \log p})$. Previously, the best known lower bound of this type was of order $(\log p)^{0.549}$.

Utilizing Analytics to show Representations used in Comparing and Ordering Unit Fractions

Video Analytics bring together the world of educational research and classroom teaching with technology and the internet. Through use of more than 4500 hours of video data, an open source analytic creation tool, this study creates a video analytic that supports a research paper. In addition to supporting research, analytics can be a reflective tool for teachers, as well as support professional development as all levels. This report illustrates the video analytic, Using Meredith’s models to reason about comparing and ordering unit fractions, (Horwitz, 2015, available at http://dx.doi.org/doi:10.7282/T33J3FQG), as well as the methods used in the creation of the analytic used to support research in student use of representations to make sense of fractions.