scholarly journals Three Proofs that the Square Root of 2 Is Irrational

2021 ◽  
Author(s):  
Shalin Shah
Keyword(s):  
Field Theory ◽  
Simple Proof ◽  
Abstract Algebra ◽  
Background Information ◽  
Square Root ◽  
Clemson University ◽  
Short Article ◽  
Proof By Contradiction

This short article gives three proofs that the square root of 2 is irrational. The article is written in an expository tutorial format and the background information is provided in brief. The first proof is a simple proof by contradiction and the second and third proofs use field theory from abstract algebra. All three topics are developed and explained. For more details, please see this excellent course at Clemson University.

scholarly journals A Simple Proof of the Polar Decomposition Theorem

2017 ◽  
Vol 31 (1) ◽  
pp. 165-171
Author(s):  
Paweł Wójcik
Keyword(s):  
Simple Proof ◽  
Polar Decomposition ◽  
Decomposition Theorem ◽  
Square Root ◽  
Positive Matrix ◽  
Broad Audience ◽  
Expository Paper

Abstract In this expository paper, we present a new and easier proof of the Polar Decomposition Theorem. Unlike in classical proofs, we do not use the square root of a positive matrix. The presented proof is accessible to a broad audience.

The Living Tradition of "Yaren Talks" as an Indigenous Community of Practice in Today's Knowledge Society

2011 ◽  
pp. 353-356 ◽  
Author(s):  
Tunç Medeni
Keyword(s):  
Communities Of Practice ◽  
Community Of Practice ◽  
Ottoman Empire ◽  
Middle Ages ◽  
Knowledge Society ◽  
Indigenous Community ◽  
Background Information ◽  
Short Article ◽  
Living Tradition ◽  
Ancient Times

The “corporations” of craftsmen and guilds of artisans can be considered as the communities of practice of ancient times and the Middle Ages. The “Ahi (Brotherhood) Organization” in the Ottoman Empire is an example of these. Today, although the Ahi Organization itself has ceased to exist, the “Yaren (Friend) Talks” still continue in certain parts of Turkey as a living tradition of the Ahi principles and practices. Viewing the continuing custom of the Yaren Talks as a precious practice of society and a unique community of practice, in this short article, we will first provide some background information about this tradition and then address some related discussions, mainly about how to make this tradition continue. We believe that the Yaren Talks will provide a rich source that can be fruitful for improving the understanding and use of communities of practice in a knowledge society.

scholarly journals A minimal model for household effects in epidemics

2020 ◽  
Author(s):  
Greg Huber ◽  
Mason Kamb ◽  
Kyle Kawagoe ◽  
Lucy Li ◽  
Boris Veytsman ◽  
...  
Keyword(s):  
Field Theory ◽  
Numerical Simulations ◽  
Minimal Model ◽  
Reproduction Number ◽  
Mean Field Theory ◽  
Mean Field ◽  
Household Size ◽  
Transmission Dynamics ◽  
Square Root ◽  
Close Contacts

Shelter-in-place and other confinement strategies implemented in the current COVID-19 pandemic have created stratified patterns of contacts between people: close contacts within households and more distant contacts between the households. The epidemic transmission dynamics is significantly modified as a consequence. We introduce a minimal model that incorporates these household effects in the framework of mean-field theory and numerical simulations. We show that the reproduction number R0 depends on the household size in a surprising way: linearly for relatively small households, and as a square root of size for larger households. We discuss the implications of the findings for the lockdown, test, tracing, and isolation policies.

scholarly journals Special methods of analytic completion in field theory

1960 ◽  
Vol 256 (1284) ◽  
pp. 39-52 ◽  
Keyword(s):  
Field Theory ◽  
Integral Representation ◽  
Simple Proof ◽  
Vertex Function ◽  
Distribution Theory ◽  
Complex Variables ◽  
Scattering Function ◽  
Envelope Of Holomorphy

By combining known theorems in the theory of functions of many complex variables and distribution theory a technique of analytic completion is developed; this provides a simple proof of the ‘edge of the wedge* theorem. An integral representation for the double commutator is derived, and it is used to simplify some of the work involved in computing the envelope of holomorphy for the vertex function. The Jacobi identities have not been incorporated, with the consequence that the threefold case cannot be completely solved by this method. The technique is applied to the fourfold (scattering) function, again without the Jacobi identities being included. By this technique analytic completion can be performed for some but not all the domains encountered in the fourfold problem.

Encodability of Kleene's O

1973 ◽  
Vol 38 (3) ◽  
pp. 437-440 ◽  
Author(s):  
Carl G. Jockusch ◽  
Robert I. Soare
Keyword(s):  
Simple Proof ◽  
Order Type ◽  
Background Information ◽  
Suitable Sense ◽  
Image Position ◽  
Infinite Set

Let ω be the nonnegative integers. G. E. Sacks once asked whether there exists an infinite X ⊆ ω such that, for all Y ⊆ X, ω1Yω1 where ω1 is the first nonrecursive ordinal. In this note we negatively answer this question by giving a simple proof that for every infinite set X ⊆ ω there exists Y ⊆ X such that the first recursively inaccessible ordinal. This is accomplished by proving that Hα is hyper-arithmetic in Y whereHα is the αth hyperjump of the empty set ∅, defined in a suitable sense for all ordinals Background information and undefined notation can be found in Rogers [11]. In particular, we write A ≤hB(A ≤T B) if A is hyperarithmetical (recursive) in B, and A ≡hB if A ≤hB and B ≤hA. We will say that a set A is hyperarithmetically (recursively) encodable if, for every infinite set X ⊆ ω, there exists Y ⊆ X such that A ≤hY (A ≤TY). For any set A (hyperdegree a) let A′ (a′) denote the hyperjump of A (a). Let 0 denote the hyperdegree of ∅. A function f majorizes a function g if f(n) ≥ g(n) for every n. E1 is the representing (type-2) functional ofintroduced by Tugué [13] (also Kleene [6]). Let be the smallest ordinal which is not the order type of any well-ordering recursive in E1. Information on can be found in Richter [9] and [10].

From Equations to Structures: Linking Abstract Algebra and High-School Algebra for Secondary School Teachers

2019 ◽  
pp. 517-522
Author(s):  
Josephine Shamash
Keyword(s):  
Field Theory ◽  
High School ◽  
Group Theory ◽  
Secondary School ◽  
Secondary School Teachers ◽  
Abstract Algebra ◽  
School Curriculum ◽  
Algebraic Structures ◽  
High School Curriculum ◽  
Fundamental Theorems

The high-school curriculum in algebra deals mainly with the solution of different types of equations. Modern algebra has a completely different viewpoint and is concerned with algebraic structures and operations. The course Algebra: From Equations to Structures is part of an M.Sc. programme for Israeli secondary school mathematics teachers. It provides an introduction to algebraic structures and modern abstract algebra, and links abstract algebra to the high-school curriculum in algebra. It follows the historical attempts of mathematicians to solve polynomial equations of higher degrees, attempts which resulted in the development of group theory and field theory by Galois and Abel. This approach leads naturally to examining topics and fundamental theorems in both group theory and field theory. Along the historical “journey”, many other major results in algebra in the past 150 years are introduced, and current research in algebra is highlighted. We examine the relevance of the course to the teachers' work.

Passy-Muir Speaking Valve Use in a Children's Hospital: An Interdisciplinary Approach

2008 ◽  
Vol 18 (2) ◽  
pp. 76-86 ◽  
Author(s):  
Lauren Hofmann ◽  
Joseph Bolton ◽  
Susan Ferry
Keyword(s):  
Tracheostomy Tube ◽  
Pediatric Population ◽  
Interdisciplinary Approach ◽  
Extended Period ◽  
Speech Development ◽  
Tube Placement ◽  
Background Information ◽  
Speaking Valve ◽  
Children's Hospital ◽  
Children’S Hospital

Abstract At The Children's Hospital of Philadelphia (CHOP) we treat many children requiring tracheostomy tube placement. With potential for a tracheostomy tube to be in place for an extended period of time, these children may be at risk for long-term disruption to normal speech development. As such, speaking valves that restore more normal phonation are often key tools in the effort to restore speech and promote more typical language development in this population. However, successful use of speaking valves is frequently more challenging with infant and pediatric patients than with adult patients. The purpose of this article is to review background information related to speaking valves, the indications for one-way valve use, criteria for candidacy, and the benefits of using speaking valves in the pediatric population. This review will emphasize the importance of interdisciplinary collaboration from the perspectives of speech-language pathology and respiratory therapy. Along with the background information, we will present current practices and a case study to illustrate a safe and systematic approach to speaking valve implementation based upon our experiences.

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