scholarly journals Decoding Book II of the Elements

2020 ◽  
Vol 12 ◽  
pp. 39-88
Author(s):  
Piotr Błaszczyk ◽  
Kazimierz Mrówka ◽  
Anna Petiurenko
Keyword(s):  
Geometric Mean ◽  
Mathematical Problem ◽  
Pythagorean Theorem ◽  
The Other ◽  
Foundations Of Mathematics ◽  
Modern Solution

The paper is a commentary to the Polish translation of the Elements Book II, included in this volume.We focus on relations between figures represented and not represented on diagrams and identify rules which enableEuclid to bridge these two kinds of objects. Also, we argue that the main mathematical problem addressed in Book II is constructing a leg of a rightangled triangle, given its hypotenuse and the other leg. In proposition II.14,Euclid solves it through the construction called the geometric mean. We trace the problem in Book III and beyond the Elements: in Heron’s Metrica, Descartes’ La Géométrie, and modern foundations of mathematics. We showthat Descartes, by novel interpretation of the Pythagorean theorem, provides a modern solution to this problem.

Assessment of the exposure of swimmers to microbiological contaminants in fresh waters

1997 ◽  
Vol 35 (11-12) ◽  
pp. 157-163 ◽  
Author(s):  
G. J. Medema ◽  
I. A. van Asperen ◽  
A. H. Havelaar
Keyword(s):  
Geometric Mean ◽  
Microbiological Quality ◽  
The Other ◽  
Pollution Level ◽  
Intestinal Infection ◽  
E Coli ◽  
Faecal Pollution ◽  
Run Off ◽  
Thermotolerant Coliforms ◽  
Mean Concentration

As part of a prospective cohort study among triathletes to determine a relationship between the microbiological quality of fresh bathing water and the risk of acquiring an intestinal infection, the exposure of the triathletes to microbiological contaminants was assessed. Waters were collected at seven triathlons (swimming course 1–1.5km) held in the summer of 1993 and 1994 to have a range of water qualities. All were influenced by sewage effluents, most also by agricultural run-off. Samples were collected several weeks before the event to establish a sampling programme (1993) and during the actual exposure of the triathletes (1993 and 1994) and examined for thermotolerant coliforms alone (samples preceding the event) and for E. coli, faecal enterococci, Staphylococcus aureus, F-specific RNAphages, enteroviruses (1993 and 1994) and for thermophilic Campylobacter, Salmonella, Aeromonas, Plesiomonas shigelloides and Pseudomonas aeruginosa (1993). The samples taken in the weeks before the exposure showed significant differences in thermotolerant coliform concentration between locations, depths and times. Also during swimmer exposure, significant differences occurred in microorganism levels at the different sampling points over the swimming course. As the triathletes swam as a group, they were exposed to approximately the same water at the same time. The geometric mean concentration was used to characterise each site. In the epidemiological study, the risk of an intestinal infection correlated with the concentration of thermotolerant coliforms and E. coli but not with the other parameters. The geometric mean concentration of thermotolerant coliforms at the triathlons ranged from 11–330/100mL and 54–1,200/100mL E. coli. Ranking of the seven sites by faecal pollution level, based on the geometric mean concentration of a faecal indicator, resulted in a different ranking for each indicator. At the fresh water sites studied, only the ratio between the geometric mean density of E. coli and thermotolerant coliforms was constant. The ratio between the other parameters related to faecal pollution (faecal enterococci, F-specific RNA phages, enteroviruses) varied considerably. Water quality standards relating to faecal pollution can only be based on parameters that show a significant correlation with risk of intestinal illness.

scholarly journals Biases Associated with Four Stream Substrate Samplers

1991 ◽  
Vol 48 (10) ◽  
pp. 1882-1886 ◽  
Author(s):  
Michael K. Young ◽  
Wayne A. Hubert ◽  
Thomas A. Wesche
Keyword(s):  
Particle Size ◽  
Geometric Mean ◽  
The Other ◽  
Substrate Composition ◽  
Mean Particle Size ◽  
Core Samples ◽  
Freeze Core ◽  
Single Probe ◽  
Core Sampler ◽  
Stream Substrate

We compared samples collected from 10 substrates of various compositions with a single-probe freeze-core sampler, a triple-probe freeze-core sampler, a McNeil sampler, and a shovel. The accuracy with which these devices sampled particles larger than 50 mm in diameter varied; they were oversampled by the freeze-core devices, sampled in proportion to their availability by a shovel, and sampled inconsistently by the McNeil sampler. The geometric mean particle size and variance of single-probe freeze-core samples consistently exceeded those of samples collected with the other devices. Most sample means also exceeded the test substrate means. By excluding the proportions of particles larger than 50 mm in diameter in our analyses, we found that proportions of several particles sizes in samples collected by different methods differed significantly from the actual proportions in test substrates. There were few differences between the single- and triple-probe freeze-core samples or between McNeil and shovel samples. All four samplers were biased, but the McNeil sampler most frequently produced samples that approximated the true substrate composition.

scholarly journals Haemagglutination-inhibiting (HI) antibodies against four prototype strains of influenza A virus in different age groups

1979 ◽  
Vol 82 (2) ◽  
pp. 225-230 ◽  
Author(s):  
A. L. Terzin ◽  
S. Djurišić ◽  
B. Vuković ◽  
V. Vujkov
Keyword(s):  
Influenza Virus ◽  
Hong Kong ◽  
New Jersey ◽  
Antibody Response ◽  
Influenza A ◽  
Geometric Mean ◽  
Age Groups ◽  
The Other ◽  
Younger Age ◽  
Lower Antibody Response

SUMMARYSera of 197 apparently well persons were tested for residual haemagglutination-inhibiting antibodies against live Hong Kong/68, A/FM/47 and A/PR/34 strains. Sera of 62 well persons, regularly exposed to contacts with swine, were tested against an inactivated A/New Jersey/76 antigen.Those born some time before and during a certain influenza era showed a significantly greater proportion of homologous residual titres against the subtype prevailing in that influenza era, than those born after the termination of the same era.In each of the seven age groups tested both the percentage of positives and the geometric mean titres were usually highest against the Hong Kong strain (representing the most recent era); the next highest were those against the FM1 strain and the lowest were those against the PR8 strain (representing the most distant of these three influenza eras).The serological involvement of donors exposed to regular contacts with swine was relatively stronger against the New Jersey antigen than the response of other serum donors shown against the other three, more recent, prototypes of influenza virus A. The oldest age groups showed significantly lower antibody response against the PR8, FM1 and Hong Kong strains (but not against the New Jersey antigen) than the next one or two of the younger age groups.

Aging Population, Retirement, and Risk Taking: Reply

2020 ◽  
Vol 66 (6) ◽  
pp. 2796-2799
Author(s):  
Haim Levy
Keyword(s):  
Empirical Evidence ◽  
Risk Taking ◽  
Geometric Mean ◽  
Aging Population ◽  
The Other ◽  
Investment Horizon ◽  
Technical Error ◽  
Long Run ◽  
Original Theorem

In theorem 1 given in my paper, “Aging Population, Retirement, and Risk Taking” [Levy H (2016a) Aging population, retirement, and risk taking. Management Sci. 62(5):1415–1430.], there is indeed a technical error. Yet, adding one condition to the theorem (which can be added in two alternate ways) is sufficient to ensure the dominance of stocks over bonds in the very long run. For the commonly employed preferences, the empirical evidence conforms with the claim given in my original theorem 1, asserting that the portfolio with the higher geometric mean (stocks) dominates the other portfolio under consideration (bonds) as the investment horizon increases indefinitely. Thus, as advocated in my paper, stocks dominate bonds for investors with typical preferences who save for retirement. This paper was accepted by Karl Deither, finance.

Enlightenment Infinitesimals and Tolstoy’s War and Peace

2020 ◽  
Vol 24 (2) ◽  
pp. 433-451
Author(s):  
Russell Winslow ◽  
Keyword(s):  
Lived Experiences ◽  
Mathematical Problem ◽  
Human Science ◽  
The Other ◽  
The Novel ◽  
War And Peace ◽  
Human Reason ◽  
Extreme Form ◽  
The Enlightenment ◽  
The One

During the Enlightenment period the concept of the infinitesimal was developed as a means to solve the mathematical problem of the incommensurability between human reason and the movements of physical beings. In this essay, the author analyzes the metaphysical prejudices subtending Enlightenment Humanism through the lens of the infinitesimal calculus. One of the consequences of this analysis is the perception of a two-fold possibility occasioned by the infinitesimal. On the one hand, it occasions an extreme form of humanism, “transhumanism,” which exhibits limitless confidence in the possibility of human science. On the other hand, the concept of the infinitesimal also contains within itself a source for a critical “posthumanism,” that is to say, a source which initiates the dissolution of the presuppositions of humanism while simultaneously announcing a different ontological organization. In , Tostoy’s novel takes up the problem of the relation between reason and motion and makes the two-fold possibility visible by presenting a contrast between its theoretical presentations and the lived experiences of the characters in the novel. Thus, is the setting in which the author has chosen to conduct this analysis.

scholarly journals Overnight variations in cortisol, interleukin 6, tumour necrosis factor α and other cytokines in people with rheumatoid arthritis

2008 ◽  
Vol 68 (1) ◽  
pp. 63-68 ◽  
Author(s):  
M G Perry ◽  
J R Kirwan ◽  
D S Jessop ◽  
L P Hunt
Keyword(s):  
Rheumatoid Arthritis ◽  
Interleukin 6 ◽  
Repeated Measures ◽  
Tumour Necrosis ◽  
Geometric Mean ◽  
The Other ◽  
Tumour Necrosis Factor Α ◽  
Random Coefficient Model ◽  
Factor Α ◽  
Necrosis Factor

Objective:To investigate overnight variations in absolute values and patterns of cytokines including interleukin 6 (IL6) and tumour necrosis factor α (TNFα) in rheumatoid arthritis (RA), and to relate any changes to those occurring in blood cortisol.Methods:A total of 16 people (8 female) with active RA and who had received no recent glucocorticoids were admitted overnight. Blood samples were obtained at 13 time points between 21.00 and 10.00.Results:The geometric mean IL6 concentration rose significantly from 35 pg/ml at 22:00 to 64 pg/ml at 07:15 (repeated measures analysis of variance (ANOVA), p<0.001). The geometric mean cortisol concentration rose significantly overnight from 57 ng/ml at 01:00 to 229 ng/ml at 07:15 (repeated measures ANOVA, p<0.001). Neither TNFα nor the other cytokines measured changed significantly. Using cubic regression modelling IL6 began to rise before cortisol (range 0.01 to 4.83 h) in eight participants and after cortisol (range 1.11 to 5.14 h) in three participants. In a random coefficient model including data from all participants, the estimated mean IL6 value began to rise 3.05 h before the estimated mean cortisol value, with the IL6 peak occurring 0.70 h before the cortisol peak.Conclusion:The mean IL6 and cortisol concentrations showed a significant overnight variation. Neither TNFα nor the other cytokines measured changed significantly. In a random coefficient model IL6 began to rise approximately 3 h, and reached a peak about 40 min, before cortisol. These studies confirm that there are abnormalities in plasma cortisol and IL6 concentrations and dynamics. The data also link the overnight rise in IL6 to the circadian variation in symptoms.

scholarly journals ARITHMETIC-GEOMETRIC MEANS FOR HYPERELLIPTIC CURVES AND CALABI–YAU VARIETIES

2010 ◽  
Vol 21 (07) ◽  
pp. 939-949 ◽  
Author(s):  
KEIJI MATSUMOTO ◽  
TOMOHIDE TERASOMA
Keyword(s):  
Projective Space ◽  
Hyperelliptic Curve ◽  
Geometric Mean ◽  
Hyperelliptic Curves ◽  
Double Cover ◽  
The Other ◽  
Geometric Means ◽  
Dimensional Projective Space ◽  
Period Integral ◽  
Generalized Arithmetic

In this paper, we define a generalized arithmetic-geometric mean μg among 2g terms motivated by 2τ-formulas of theta constants. By using Thomae's formula, we give two expressions of μg when initial terms satisfy some conditions. One is given in terms of period integrals of a hyperelliptic curve C of genus g. The other is by a period integral of a certain Calabi–Yau g-fold given as a double cover of the g-dimensional projective space Pg.

scholarly journals Dirichlet Polynomials and Entropy

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 1085
Author(s):  
David I. Spivak ◽  
Timothy Hosgood
Keyword(s):  
Probability Distribution ◽  
Shannon Entropy ◽  
Empirical Distribution ◽  
Geometric Mean ◽  
The Other ◽  
Cross Entropy ◽  
Dirichlet Polynomial ◽  
Area Formula ◽  
Additive Structure ◽  
Dirichlet Polynomials

A Dirichlet polynomial d in one variable y is a function of the form d(y)=anny+⋯+a22y+a11y+a00y for some n,a0,…,an∈N. We will show how to think of a Dirichlet polynomial as a set-theoretic bundle, and thus as an empirical distribution. We can then consider the Shannon entropy H(d) of the corresponding probability distribution, and we define its length (or, classically, its perplexity) by L(d)=2H(d). On the other hand, we will define a rig homomorphism h:Dir→Rect from the rig of Dirichlet polynomials to the so-called rectangle rig, whose underlying set is R⩾0×R⩾0 and whose additive structure involves the weighted geometric mean; we write h(d)=(A(d),W(d)), and call the two components area and width (respectively). The main result of this paper is the following: the rectangle-area formula A(d)=L(d)W(d) holds for any Dirichlet polynomial d. In other words, the entropy of an empirical distribution can be calculated entirely in terms of the homomorphism h applied to its corresponding Dirichlet polynomial. We also show that similar results hold for the cross entropy.

Interplay between reproduction and age selective harvesting: A case study of Hilsa (Tenualosa ilisha) fish at Sundarban estuary of northern Bay of Bengal, India

2019 ◽  
Vol 12 (02) ◽  
pp. 1950023 ◽  
Author(s):  
Debaldev Jana ◽  
Sachinandan Dutta ◽  
G. P. Samanta
Keyword(s):  
Bay Of Bengal ◽  
Sexual Maturation ◽  
Mathematical Problem ◽  
Fish Population ◽  
The Other ◽  
Selective Harvesting ◽  
Tenualosa Ilisha ◽  
Northern Bay Of Bengal ◽  
Sexually Mature

New offspring due to sexual reproduction is not an instantaneous process from its birth, it takes time to be sexually mature. On the other hand, harvesting of commercially profitable fish population before the perfect size or weight is reached is not only a commercial loss but also risks the extinction of the population. Now, we discuss the issue of Ganges-Brahmaputra-Meghna basin, northern Bay of Bengal for the age-selective harvesting of Hilsa shad (Tenualosa ilisha) which lays eggs after its sexual maturation. Harvesting of hilsa before its sexual maturation risks its extinction and due to lamer body weight, it is not a commercially profitable policy. This is a reality of Sundarban estuary for hilsa fish harvesting, therefore, biologically and economically both India and Bangladesh are facing several problems. Empirical data of Frasergunje Fishing Harbor shows a clear picture as the supporting document of this mathematical problem.

Relations between MaxRotD50 and Some Horizontal Components of Ground-Motion Intensity Used in Practice

2021 ◽  
Author(s):  
Alan Poulos ◽  
Eduardo Miranda
Keyword(s):  
Ground Motion ◽  
Earthquake Engineering ◽  
Ground Motions ◽  
Geometric Mean ◽  
The Other ◽  
Horizontal Projection ◽  
Site Class ◽  
Engineering Research ◽  
Natural Logarithm ◽  
Horizontal Ground Motion

ABSTRACT The most commonly used intensity measure of ground motion in earthquake engineering is the 5% damped spectral ordinate, which varies in different directions. Several different measures have been proposed over the years to combine the intensity of the two horizontal recorded ground motions to derive ground-motion models as well as for design purposes. This study provides the relation to seven previously used measures of horizontal ground motion with respect to a recently proposed orientation-independent measure of horizontal ground-motion intensity referred to as MaxRotD50. This new measure of horizontal intensity is defined as the median value of the maximum spectral ordinate of two orthogonal directions computed for all possible nonredundant orientations. The relations are computed using 5065 pairs of horizontal ground motions taken from the database of ground motions recorded in shallow crustal earthquakes in active tectonic regions developed as part of the Pacific Earthquake Engineering Research Center’s Next Generation Attenuation-West2 project. Empirically derived period-dependent relations are presented for three quantities that permit transforming any of the seven other definitions of horizontal ground-motion intensity to MaxRotD50, namely, (1) geometric mean of the ratio of MaxRotD50 to any of the seven other measures of intensities, (2) standard deviation of the natural logarithm of the ratio of MaxRotD50 to any of the seven other measures of intensities, and (3) the correlation between the natural logarithm of the ratio of MaxRotD50 to the other measures of intensities and the natural logarithm of the other measure of intensity. In addition, the influence of site class at the recording station, earthquake magnitude, and distance to the horizontal projection of the rupture is examined on the geometric mean of the ratio of MaxRotD50 to the median intensity of all nonredundant orientations (i.e., RotD50), showing negligible influence of site class and only a relatively small influence of magnitude and distance.

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