scholarly journals Sums of four and more unit fractions and approximate parametrizations

2021 ◽  
Author(s):  
Christian Elsholtz ◽  
Stefan Planitzer
Keyword(s):  
Unit Fractions

scholarly journals Dynamic Windows Scheduling with Reallocation

2021 ◽  
Vol 26 ◽  
pp. 1-19
Author(s):  
Martín Farach-Colton ◽  
Katia Leal ◽  
Miguel A. Mosteiro ◽  
Christopher Thraves Caro
Keyword(s):  
Supply Chain ◽  
Bin Packing ◽  
Time Slot ◽  
Peak Load ◽  
Communication Channels ◽  
Current Load ◽  
Constant Amount ◽  
Unit Fractions ◽  
Consecutive Time ◽  
Multiple Variants

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.

scholarly journals Monochromatic solutions to equations with unit fractions

1991 ◽  
Vol 43 (3) ◽  
pp. 387-392 ◽  
Author(s):  
Tom C. Brown ◽  
Voijtech Rödl
Keyword(s):  
Solutions To Equations ◽  
Unit Fractions ◽  
Image Position ◽  
Positive Integers

Our main result is that if G(x1, …, xn) = 0 is a system of homogeneous equations such that for every partition of the positive integers into finitely many classes there are distinct y1,…, yn in one class such that G(y1, …, yn) = 0, then, for every partition of the positive integers into finitely many classes there are distinct Z1, …, Zn in one class such thatIn particular, we show that if the positive integers are split into r classes, then for every n ≥ 2 there are distinct positive integers x1, x1, …, xn in one class such thatWe also show that if [1, n6 − (n2 − n)2] is partitioned into two classes, then some class contains x0, x1, …, xn such that(Here, x0, x2, …, xn are not necessarily distinct.)

scholarly journals The Number of Huffman Codes, Compact Trees, and Sums of Unit Fractions

2013 ◽  
Vol 59 (2) ◽  
pp. 1065-1075 ◽  
Author(s):  
Christian Elsholtz ◽  
Clemens Heuberger ◽  
Helmut Prodinger
Keyword(s):  
Huffman Codes ◽  
Unit Fractions

Of Camels, Inheritance, and Unit Fractions

Math Horizons ◽  
2013 ◽  
Vol 21 (1) ◽  
pp. 8-11
Author(s):  
Paul K. Stockmeyer
Keyword(s):  
Unit Fractions

scholarly journals Sums of distinct unit fractions

1963 ◽  
Vol 14 (1) ◽  
pp. 126-126 ◽  
Author(s):  
Paul Erdős ◽  
Sherman Stein
Keyword(s):  
Unit Fractions ◽  
Distinct Unit

scholarly journals Canonical Trees, Compact Prefix-Free Codes, and Sums of Unit Fractions: A Probabilistic Analysis

2015 ◽  
Vol 29 (3) ◽  
pp. 1600-1653
Author(s):  
Clemens Heuberger ◽  
Daniel Krenn ◽  
Stephan Wagner
Keyword(s):  
Probabilistic Analysis ◽  
Unit Fractions

Activities for Students: Unit Fractions and Their Basimal Representations: Exploring Patterns

2004 ◽  
Vol 98 (4) ◽  
pp. 274-284
Author(s):  
Marlena Herman ◽  
Eric Milou ◽  
Jay Schiffman
Keyword(s):  
Preservice Teachers ◽  
Secondary Mathematics ◽  
Rational Numbers ◽  
College Level ◽  
Unit Fractions ◽  
Flexible Thinking ◽  
Mathematics Course

Major foci of secondary mathematics include understanding numbers, ways of representing numbers, and relationships among numbers (NCTM 2000). This article considers different representations of rational numbers and leads students through activities that explore patterns in base ten, as well as in other bases. These activities encourage students to solve problems and investigate situations designed to foster flexible thinking about rational numbers. Preservice teachers in a college-level mathematics course carried out these activities. Their conjectures and ideas are incorporated throughout this article.

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