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 On distinct unit fractions whose sum equals 1

2005 ◽  
Vol 300 (1-3) ◽  
pp. 213-217 ◽  
Author(s):  
Nechemia Burshtein
Keyword(s):  
Unit Fractions ◽  
Distinct Unit

scholarly journals On distinct unit fractions whose sum equals 1

1973 ◽  
Vol 5 (3) ◽  
pp. 201-206 ◽  
Author(s):  
Nechemia Burshtein
Keyword(s):  
Unit Fractions ◽  
Distinct Unit

Sums of Distinct Unit Fractions

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

Sums of Rational Numbers

1965 ◽  
Vol 17 ◽  
pp. 1019-1024
Author(s):  
W. A. Webb
Keyword(s):  
Arithmetic Progression ◽  
Rational Number ◽  
Rational Numbers ◽  
Unit Fractions ◽  
Finite Sums ◽  
Distinct Unit

Previously, the problem of expressing rational numbers as finite sums of rational numbers of a given type has been concerned with the Egyptian, or unit, fractions. It has long been known that any rational number is the sum of distinct unit fractions. In response to a problem proposed by E. P. Starke (4), R. Breusch (1) and B. M. Stewart (5) showed that every rational number with an odd denominator is a sum of distinct odd unit fractions. P. J. Van Albada and J. H. Van Lint (6) extended this result to show that any integer is a sum of unit fractions with denominators from an arithmetic progression.

Problem solving in mathematics, II

1965 ◽  
Vol 58 (7) ◽  
pp. 596-600
Author(s):  
Truman Botts
Keyword(s):  
Problem Solving ◽  
Rational Number ◽  
Positive Rational Number ◽  
Unit Fractions ◽  
Distinct Unit

Every positive rational number r is a sum of distinct 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 Has the fishery contributed to a major shift in the distribution of South African sardine?

2008 ◽  
Vol 65 (9) ◽  
pp. 1676-1688 ◽  
Author(s):  
Janet C. Coetzee ◽  
Carl D. van der Lingen ◽  
Laurence Hutchings ◽  
Tracey P. Fairweather
Keyword(s):  
South African ◽  
West Coast ◽  
Management Practices ◽  
Fishing Effort ◽  
South Coast ◽  
The West ◽  
Natal Homing ◽  
Distinct Unit ◽  
Major Shift ◽  
Sardine Population

Abstract Coetzee, J. C., van der Lingen, C. D., Hutchings, L., and Fairweather, T. P. 2008. Has the fishery contributed to a major shift in the distribution of South African sardine? – ICES Journal of Marine Science, 65: 1676–1688. A major shift in the distribution of South African sardine (Sardinops sagax) has resulted in a significant spatial mismatch in fishing effort vs. fish abundance in recent years. The sardine fishery started on the west coast during the 1940s, and processing capacity there increased rapidly. This trend together with increases in annual landings continued up to the early 1960s, but then the fishery collapsed as a consequence of overfishing. The population then recovered steadily during the 1980s and 1990s, coincident with, but perhaps not entirely attributable to, the inception of conservative management practices, to support catches similar to pre-collapse levels. Since 2001, however, most of the sardine population has been situated on South Africa’s south coast, far from processing facilities. Fishing effort has increased concomitantly on that coast, particularly during the past three years, reflecting the continued decline in the abundance of sardine on the west coast. Three hypotheses explaining the change in the distribution of sardine have been proposed: (i) intensely localized (i.e. west coast) fishing pressure depleted that part (or functionally distinct unit) of the population; (ii) the shift was environmentally induced; and (iii) successful spawning and recruit survival on the south coast contributed disproportionately more towards the bulk of recruitment, and progeny spawned there now dominate the population and exhibit natal homing. The first of these hypotheses is evaluated, and management implications of the shift discussed.

Sign in / Sign up

Export Citation Format

Share Document