scholarly journals CONSTRUCTIVE GEOMETRY AND THE PARALLEL POSTULATE

2016 ◽  
Vol 22 (1) ◽  
pp. 1-104 ◽  
Author(s):  
MICHAEL BEESON
Keyword(s):  
Intuitionistic Logic ◽  
Euclidean Geometry ◽  
The Other ◽  
Field Theories ◽  
Kripke Models ◽  
Square Roots ◽  
The Third ◽  
Euclidean Field ◽  
Euclidean Field Theory ◽  
Independence Results

AbstractEuclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. This involves finding “uniform” constructions where normally a case distinction is used. For example, in finding a perpendicular to line L through point p, one usually uses two different constructions, “erecting” a perpendicular when p is on L, and “dropping” a perpendicular when p is not on L, but in constructive geometry, it must be done without a case distinction. Classically, the models of Euclidean (straightedge-and-compass) geometry are planes over Euclidean fields. We prove a similar theorem for constructive Euclidean geometry, by showing how to define addition and multiplication without a case distinction about the sign of the arguments. With intuitionistic logic, there are two possible definitions of Euclidean fields, which turn out to correspond to different versions of the parallel postulate.We consider three versions of Euclid’s parallel postulate. The two most important are Euclid’s own formulation in his Postulate 5, which says that under certain conditions two lines meet, and Playfair’s axiom (dating from 1795), which says there cannot be two distinct parallels to line L through the same point p. These differ in that Euclid 5 makes an existence assertion, while Playfair’s axiom does not. The third variant, which we call the strong parallel postulate, isolates the existence assertion from the geometry: it amounts to Playfair’s axiom plus the principle that two distinct lines that are not parallel do intersect. The first main result of this paper is that Euclid 5 suffices to define coordinates, addition, multiplication, and square roots geometrically.We completely settle the questions about implications between the three versions of the parallel postulate. The strong parallel postulate easily implies Euclid 5, and Euclid 5 also implies the strong parallel postulate, as a corollary of coordinatization and definability of arithmetic. We show that Playfair does not imply Euclid 5, and we also give some other independence results. Our independence proofs are given without discussing the exact choice of the other axioms of geometry; all we need is that one can interpret the geometric axioms in Euclidean field theory. The independence proofs use Kripke models of Euclidean field theories based on carefully constructed rings of real-valued functions. “Field elements” in these models are real-valued functions.

scholarly journals XIX. —Consideration of the objections raised against the geometrical representation of the square roots of negative quantities

1829 ◽  
Vol 119 ◽  
pp. 241-254 ◽  
Keyword(s):  
The Other ◽  
Square Root ◽  
Square Roots ◽  
The Third ◽  
Oblique Direction ◽  
Real Nature ◽  
Positive Quantity ◽  
Algebraic Operations ◽  
The One ◽  
Straight Lines

Some years ago my attention was drawn to those algebraic quantities, which are commonly called impossible roots or imaginary quantities: it appeared extraordinary, that mathematicians should be able by means of these quan­tities to pursue their investigations, both in pure and mixed mathematics, and to arrive at results which agree with the results obtained by other independent processes; and yet that the real nature of these quantities should be entirely unknown, and even their real existence denied. One thing was evident re­specting them; that they were quantities capable of undergoing algebraic operations analogous to the operations performed on what are called possible quantities, and of producing correct results: thus it was manifest, that the operations of algebra were more comprehensive than the definitions and funda­mental principles; that is, that they extended to a class of quantities, viz. those commonly called impossible roots, to which the definitions and funda­mental principles were inapplicable. It seemed probable, therefore, that there was a deficiency in the definitions and fundamental principles of algebra ; and that other definitions and fundamental principles might be discovered of a more comprehensive nature, which would extend to every class of quantities to which the operations of algebra were applicable; that is, both to possible and impossible quantities, as they are called. I was induced therefore to examine into the nature of algebraic operations, with a view, if possible, of arriving at these general definitions and fundamental principles: and I found, that, by considering algebra merely as applied to geometry, such principles and definitions might be obtained. The fundamental principles and definitions which I arrived at were these: that all straight lines drawn in a given plane from a given point, in any direction whatever, are capable of being algebra­ically represented, both in length and direction; that the addition of such lines (when estimated both in length and direction) must be performed in the same manner as composition of motion in dynamics; and that four such lines are proportionals, -both in length and direction, when they are proportionals in length, and the fourth is inclined to the third at the same angle that the second is to the first. From these principles I deduced, that, if a line drawn in any given direction be assumed as a positive quantity, and consequently its oppo­site, a negative quantity, a line drawn at right angles to the positive or nega­tive direction will be the square root of a negative quantity, and a line drawn in an oblique direction will be the sum of two quantities, the one either posi­tive or negative, and the other, the square root of a negative quantity.

Consideration of the objections raised against the geometrical repre­sentation of the square roots of negative quantities

1833 ◽  
Vol 2 ◽  
pp. 371-372
Keyword(s):  
A Priori ◽  
The Other ◽  
Square Root ◽  
Square Roots ◽  
Negative Direction ◽  
The Third ◽  
Oblique Direction ◽  
Positive Quantity ◽  
Algebraic Operations ◽  
The One

It has always appeared a paradox in mathematics, that by em­ploying what are called imaginary or impossible quantities, and sub­jecting them to the same algebraic operations as those which are performed on quantities that are real and possible, the results ob­tained should always prove perfectly correct. The author inferring from this fact, that the operations of algebra are of a more compre­hensive nature than its definitions and fundamental principles, was led to inquire what extension might be given to these definitions and principles, so as to render them strictly applicable to quantities of every description, whether real or imaginary. This deficiency, he conceives, may be supplied by having recourse to certain geometrical considerations. By taking into account the directions as well as the lengths of lines drawn in a given plane, from a given point, the ad­dition of such lines may admit of being performed in the same man­ner as the composition of motions in dynamics; and four such lines may be regarded as proportional, both in length and direction, when they are proportionals in length, and, when also the fourth is inclined to the third at the same angle that the second is to the first. From this principle he deduces, that if a line drawn in any given di­rection be assumed as a positive quantity, and consequently its op­posite a negative quantity, a line drawn at right angles to the posi­tive or negative direction will be represented by the square root of a negative quantity ; and a line drawn in an oblique direction will be represented by the sum of two quantities, the one either positive or negative, and the other the square root of a negative quantity. On this subject, the author published a treatise in April 1828; since which period several objections have been made to this hypothesis. The purpose of the present paper is to answer these objections. The first of these is, that impossible roots should be considered merely as the indications of some impossible condition, which the pro­position that has given rise to them involves; and that they have in fact no real or absolute existence. To this it is replied by the author, that although such a statement may be true in some cases, it is by no means necessarily so in all; and that these quantities re­semble in this respect fractional and negative roots, which, whenever they are excluded by the nature of the question, are indeed signs of impossibility, but yet in other cases are admitted to be real and significant quantities. We have therefore no stronger reasons, à priori , for denying the real existence of what are called impossible roots, because they are in some cases the signs of impossibility, than we should have for refusing that character to fractional or negative roots on similar grounds.

Efficacy of some insecticide modules against major insect pests and spider population of rice, Oryza sativa L.

ENTOMON ◽  
2018 ◽  
Vol 43 (4) ◽  
pp. 257-262
Author(s):  
Atanu Seni ◽  
Bhimasen Naik
Keyword(s):  
Oryza Sativa ◽  
Grain Yield ◽  
Untreated Control ◽  
Insect Pests ◽  
Oryza Sativa L ◽  
Gall Midge ◽  
Stem Borer ◽  
The Other ◽  
The Third ◽  
Treated Plot

Experiments were carried out to assess some insecticide modules against major insect pests of rice. Each module consists of a basal application of carbofuran 3G @ 1 kg a.i ha-1 at 20 DAT and Rynaxypyr 20 SC @ 30 g a.i ha-1 at 45 DAT except untreated control. All modules differ with each other only in third treatment which was applied in 65 DAT. The third treatment includes: Imidacloprid 17.8 SL @ 27 g a.i ha-1, Pymetrozine 50 WG @ 150 g a.i ha-1, Triflumezopyrim 106 SC @ 27 g a.i ha-1, Buprofezin 25 SC @ 250 g a.i ha-1; Glamore (Imidacloprid 40+Ethiprole 40% w/w) 80 WG @ 100 g a.i. ha-1, Thiacloprid 24 SC @ 60 g a.i ha-1, Azadirachtin 0.03 EC @ 8 g a.i ha-1, Dinotefuran 20 SG@ 40 g a.i ha-1 and untreated control. All the treated plots recorded significantly lower percent of dead heart, white ear- head caused by stem borer and silver shoot caused by gall midge. Module with Pymetrozine 50 WG @ 150 g a.i ha-1 treated plot recorded significantly higher per cent reduction of plant hoppers (>80% over untreated control) and produced higher grain yield (50.75 qha-1) than the other modules. Among the different treated modules the maximum number of spiders was found in Azadirachtin 0.03 EC @ 8 g a.i ha-1 treated module plot followed by other treatments.

Retroaktive Neuverhandlung

2018 ◽  
Vol 63 (1) ◽  
Author(s):  
Daniel Martin Feige
Keyword(s):  
Original Work ◽  
The Other ◽  
Final Part ◽  
Critical Reading ◽  
The Third ◽  
Works Of Art ◽  
The Relationship

Der Beitrag widmet sich der Frage historischer Folgeverhältnisse in der Kunst. Gegenüber dem Gedanken, dass es ein ursprüngliches Werk in der Reihe von Werken gibt, das späteren Werken seinen Sinn gibt, schlägt der Text vor, das Verhältnis umgekehrt zu denken: Im Lichte späterer Werke wird der Sinn früherer Werke neu ausgehandelt. Dazu geht der Text in drei Schritten vor. Im ersten Teil formuliert er unter der Überschrift ›Form‹ in kritischer Abgrenzung zu Danto und Eco mit Adorno den Gedanken, dass Kunstwerke eigensinnig konstituierte Gegenstände sind. Die im Gedanken der Neuverhandlung früherer Werke im Lichte späterer Werke vorausgesetzte Unbestimmtheit des Sinns von Kunstwerken wird im zweiten Teil unter dem Schlagwort ›Zeitlichkeit‹ anhand des Paradigmas der Improvisation erörtert. Der dritte und letzte Teil wendet diese improvisatorische Logik unter dem Label ›Neuaushandlung‹ dann dezidiert auf das Verhältnis von Vorbild und Nachbild an. The article proposes a new understanding of historical succession in the realm of art. In contrast to the idea that there is an original work in the series of works that gives meaning to the works that come later, the text proposes to think it exactly the other way round: in the light of later works, the meanings of earlier works are renegotiated. The text proceeds in three steps to develop this idea. Under the heading ›Form‹ it develops in the first part a critical reading of Danto’s and Eco’s notion of the constitution of the artworks and argues with Adorno that each powerful work develops its own language. In the second part, the vagueness of the meaning of works of art presupposed in the idea of renegotiating earlier works in the light of later works is discussed under the term ›Temporality‹ in terms of the logic of improvisation. The third and final part uses this improvisational logic under the label ›Renegotiation‹ to understand the relationship between model and afterimage in the realm of art.

Features of the course and treatment of glaucoma during pregnancy

2020 ◽  
Vol 98 (3) ◽  
pp. 178-184
Author(s):  
T. V. Chernyakova ◽  
A. Yu. Brezhnev ◽  
I. R. Gazizova ◽  
A. V. Kuroyedov ◽  
A. V. Seleznev
Keyword(s):  
Pregnant Women ◽  
Laser Treatment ◽  
Progressive Disease ◽  
Clinical Course ◽  
The Other ◽  
Third Trimester ◽  
The Third ◽  
Chronic Progressive Disease ◽  
Hypotensive Therapy ◽  
Eye Pressure

In the review we have integrated all up-to-date knowledge concerning clinical course and treatment of glaucoma among pregnant women to help specialists choose a proper policy of treatment for such a complicated group of patients. Glaucoma is a chronic progressive disease. It rarely occurs among childbearing aged women. Nevertheless the probability to manage pregnant patients having glaucoma has been recently increasing. The situation is complicated by the fact that there are no recommendations on how to treat glaucoma among pregnant women. As we know, eye pressure is progressively going down from the first to the third trimester, so we often have to correct hypotensive therapy. Besides, it is necessary to take into account the effect of applied medicines on mother health and evaluate possible teratogenic complications for a fetus. The only medicine against glaucoma which belongs to category B according to FDA classification is brimonidine. Medicines of the other groups should be prescribed with care. Laser treatment or surgery may also be a relevant decision when monitoring patients who are planning pregnancy or just bearing a child. Such treatment should be also accompanied by medicines.

The Nile Delta during the Early Dynastic and Old Kingdom periods. Preliminary remarks on the evolution of settlement landscape

2020 ◽  
pp. 13-61
Author(s):  
Natalia Małecka-Drozd
Keyword(s):  
Nile Delta ◽  
Ancient Egypt ◽  
The Other ◽  
Old Kingdom ◽  
The Third ◽  
Other Hand ◽  
Early Dynastic ◽  
Territorial Structure ◽  
Settlement Landscape ◽  
New Settlements

The 3rd millennium BC appears to be a key period of development of the historical settlement landscape in ancient Egypt. After the unification of the country, the process of disappearance of the predynastic socio-political structures and settlement patterns associated with them significantly accelerated. Old chiefdoms, along with their centres and elites, declined and vanished. On the other hand, new settlements emerging in various parts of the country were often strictly related to the central authorities and formation of the new territorial administration. Not negligible were climatic changes, which influenced the shifting of the ecumene. Although these changes were evolutionary in their nature, some important stages may be recognized. According to data obtained during surveys and excavations, there are a number of sites that were considerably impoverished and/or abandoned before and at the beginning of the Old Kingdom. On the other hand, during the Third and Fourth Dynasties some important Egyptian settlements have emerged in the sources and begun their prosperity. Architectural remains as well as written sources indicate the growing interest of the state in the hierarchy of landscape elements and territorial structure of the country.

Avoiding Shame and Guilt

2018 ◽  
Author(s):  
C. Daniel Batson
Keyword(s):  
Empathic Concern ◽  
The Other ◽  
The Third ◽  
Helping Relationship ◽  
Shame And Guilt

After rejecting the remove–empathy hypothesis (Chapter 4), attention turned to the other two classic egoistic explanations for the motivation to help produced by empathic concern: empathy-specific punishment and empathy-specific reward. This chapter considers two kinds of empathy-specific punishment, shame and guilt. The shame hypothesis claims that we help more when we feel empathic concern because we’re motivated to avoid disapproval and censure from others. The guilt hypothesis says that we’re motivated to avoid self-condemnation for failing to do what we should. Contradicting the shame hypothesis, experiments revealed that even when no one else will know if participants fail to help, empathy still increased helping. Contradicting the guilt hypothesis, participants induced to feel empathy helped more even when provided a good excuse for not helping. Given that neither form of empathy-specific punishment could account for the empathy–helping relationship, our search shifted to the third classic egoistic explanation: pursuing pride.

Introduction

2017 ◽  
Author(s):  
Patrick Colm Hogan
Keyword(s):  
Queer Theory ◽  
Social Constructionism ◽  
The Other ◽  
Sexual Identities ◽  
The Third ◽  
Sexuality And Gender ◽  
And Gender ◽  
Sexual Part ◽  
The One ◽  
And Behavior

The introduction first sets out some preliminary definitions of sex, sexuality, and gender. It then turns from the sexual part of Sexual Identities to the identity part. A great deal of confusion results from failing to distinguish between identity in the sense of a category with which one identifies (categorial identity) and identity in the sense of a set of patterns that characterize one’s cognition, emotion, and behavior (practical identity). The second section gives a brief summary of this difference. The third and fourth sections sketch the relation of the book to social constructionism and queer theory, on the one hand, and evolutionary-cognitive approaches to sex, sexuality, and gender, on the other. The fifth section outlines the value of literature in not only illustrating, but advancing a research program in sex, sexuality, and gender identity. Finally, the introduction provides an overview of the chapters in this volume.

Oxford Assess and Progress: Clinical Medicine

2019 ◽  
Keyword(s):  
Correct Answer ◽  
Clinical Medicine ◽  
The Other ◽  
The Third

Maximise your exam success with this essential revision guide. The third edition of Oxford Assess and Progress: Clinical Medicine features over 550 Single Best Answer questions. Packed with questions written by practicing clinicians and educators, this revision tool is an authoritative guide on core clinical topics and professional themes. Each question is accompanied by extensive feedback which explains not only the rationale of the correct answer, but why the other options are incorrect. Further reading resources and cross-references to the Oxford Handbook of Clinical Medicine have been fully updated to expand your revision further. Progess to exam success with the third edition of Oxford Assess and Progress: Clinical Medicine.

Coping with longitudinal data in research on developmental psychopathology

2006 ◽  
Vol 30 (1) ◽  
pp. 20-25 ◽  
Author(s):  
David A. Cole
Keyword(s):  
Data Collection ◽  
Longitudinal Data ◽  
Developmental Psychopathology ◽  
Careful Consideration ◽  
Two Dimensions ◽  
The Other ◽  
Outcome Variable ◽  
The Third ◽  
Data Analytic ◽  
Over Time

Many outcome variables in developmental psychopathology research are highly stable over time. In conventional longitudinal data analytic approaches such as multiple regression, controlling for prior levels of the outcome variable often yields little (if any) reliable variance in the dependent variable for putative predictors to explain. Three strategies for coping with this problem are described. One involves focusing on developmental periods of transition, in which the outcome of interest may be less stable. A second is to give careful consideration to the amount of time allowed to elapse between waves of data collection. The third is to consider trait-state-occasion models that partition the outcome variable into two dimensions: one entirely stable and trait-like, the other less stable and subject to occasion-specific fluctuations.

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