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Generalized Catalan Numbers

Thomas Koshy

in Catalan Numbers with Applications

Published in print:
2008
Published Online:
January 2009
ISBN:
9780195334548
eISBN:
9780199868766
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195334548.003.0017
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter generalizes Catalan numbers and introduces Fuss numbers.


Divisibility Properties

Thomas Koshy

in Catalan Numbers with Applications

Published in print:
2008
Published Online:
January 2009
ISBN:
9780195334548
eISBN:
9780199868766
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195334548.003.0013
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter investigates several divisibility properties of shows Catalan numbers. They include their parity and primality. Mersenne numbers are also described.


A Catalan Triangle

Thomas Koshy

in Catalan Numbers with Applications

Published in print:
2008
Published Online:
January 2009
ISBN:
9780195334548
eISBN:
9780199868766
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195334548.003.0014
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter investigates an interesting triangular array that contains Catalan numbers and Cayley numbers.


Partitions and Catalan Numbers

Thomas Koshy

in Catalan Numbers with Applications

Published in print:
2008
Published Online:
January 2009
ISBN:
9780195334548
eISBN:
9780199868766
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195334548.003.0010
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter investigates several occurrences of Catalan numbers in the theory of partitioning. Noncrossing partition is explained in detail.


A Family of Binary Words

Thomas Koshy

in Catalan Numbers with Applications

Published in print:
2008
Published Online:
January 2009
ISBN:
9780195334548
eISBN:
9780199868766
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195334548.003.0015
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter generalizes the family of binary numbers studied in Chapter 5.


Notes on memory (1945)

Alan M. Turing

in Alan Turing's Automatic Computing Engine: The Master Codebreaker's Struggle to build the Modern Computer

Published in print:
2005
Published Online:
January 2008
ISBN:
9780198565932
eISBN:
9780191714016
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198565932.003.0022
Subject:
Mathematics, History of Mathematics

This chapter presents fragments from Turing's draft of ‘Proposed Electronic Calculator’. This material is of interest chiefly because of its remarks concerning the universal machine of Turing's 1936 ... More


Science in the Looking Glass: What Do Scientists Really Know?

E. Brian Davies

Published in print:
2007
Published Online:
September 2008
ISBN:
9780199219186
eISBN:
9780191711695
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199219186.001.0001
Subject:
Physics, History of Physics

How do scientific conjectures become laws? Why does proof mean different things in different sciences? Do numbers exist, or were they invented? Why do some laws turn out to be wrong? This book ... More


Catalan Numbers with Applications

Thomas Koshy

Published in print:
2008
Published Online:
January 2009
ISBN:
9780195334548
eISBN:
9780199868766
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195334548.001.0001
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

Fibonacci and Lucas sequences are “two shining stars in the vast array of integer sequences,” and because of their ubiquitousness, tendency to appear in quite unexpected and unrelated places, ... More


In the Wilderness: The Doctrine of Defilement in the Book of Numbers

Mary Douglas

Published in print:
2001
Published Online:
October 2011
ISBN:
9780199245413
eISBN:
9780191697463
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199245413.001.0001
Subject:
Religion, Biblical Studies

This book is a classic work on the anthropology of the Book of Numbers. Up to now Bible scholars have tended to dismiss Numbers — Wellhausen called it the junk room of the Bible, and most ... More


General Theorems From Specific Images

Marcus Giaquinto

in Visual Thinking in Mathematics

Published in print:
2007
Published Online:
September 2007
ISBN:
9780199285945
eISBN:
9780191713811
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199285945.003.0008
Subject:
Philosophy, Logic/Philosophy of Mathematics

This chapter shows that some general arithmetical theorems almost certainly can be discovered by visualizing. First, it describes two examples of what look like acceptable routes to general theorems ... More


Binomial Coefficients

Thomas Koshy

in Catalan Numbers with Applications

Published in print:
2008
Published Online:
January 2009
ISBN:
9780195334548
eISBN:
9780199868766
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195334548.003.0001
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter presents a brief introduction to binomial coefficients, the cornerstone of all the discussions in the book. Among the properties discussed are Hermite's divisibility properties. Catalan ... More


The Central Binomial Coefficient Revisited

Thomas Koshy

in Catalan Numbers with Applications

Published in print:
2008
Published Online:
January 2009
ISBN:
9780195334548
eISBN:
9780199868766
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195334548.003.0003
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter continues to investigate the central binomial coefficient in much greater detail. Topics explained include Wallis' product, Lucas numbers, Fermat's little theorem, Carlitz's formula, ... More


Letters-to-Numbers Problems

Patrick R. Laughlin

in Group Problem Solving

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691147918
eISBN:
9781400836673
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691147918.003.0006
Subject:
Psychology, Social Psychology

This chapter looks at letters-to-numbers problems, an interesting class of problems that entail many insightful strategies. Letters-to-numbers problems strongly fulfill the four conditions of ... More


On Measurement and Numbering

Rory Fox

in Time and Eternity in Mid-Thirteenth-Century Thought

Published in print:
2006
Published Online:
May 2006
ISBN:
9780199285754
eISBN:
9780191603563
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0199285756.003.0007
Subject:
Philosophy, Philosophy of Religion

This chapter examines 13th century views on issues of measurement, particularly temporal measurement. It shows that 13th century thinkers distinguished between a variety of ways in which the word ... More


The Concept of an Object in Formal Ontology

E. J. Lowe

in The Four-Category Ontology: A Metaphysical Foundation for Natural Science

Published in print:
2005
Published Online:
May 2006
ISBN:
9780199254392
eISBN:
9780191603600
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0199254397.003.0005
Subject:
Philosophy, Metaphysics/Epistemology

The formal ontological concept of an object is explicated and contrasted with that of a property. F. P. Ramsey’s objections to this distinction are challenged. The sense in which objects possess an ... More


The Mathematics of Various Entertaining Subjects: Research in Recreational Math

Jennifer Beineke and Jason Rosenhouse (eds)

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691164038
eISBN:
9781400881338
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691164038.001.0001
Subject:
Mathematics, History of Mathematics

The history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, ... More


Frege's Account of Classes

Michael Potter

in Reason's Nearest Kin: Philosophies of Arithmetic from Kant to Carnap

Published in print:
2002
Published Online:
May 2007
ISBN:
9780199252619
eISBN:
9780191712647
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199252619.003.0005
Subject:
Philosophy, Logic/Philosophy of Mathematics

In the Grundlagen, Frege sketched a proof of the basic properties of natural numbers from the numerical equivalence and showed how that equivalence could be proved if numbers were explicitly defined ... More


Mathematical Knowledge and the Interplay of Practices

José Ferreirós

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691167510
eISBN:
9781400874002
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691167510.001.0001
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction ... More


The Institutions of Democracy

André Béteille

in Democracy and Its Institutions

Published in print:
2012
Published Online:
September 2012
ISBN:
9780198080961
eISBN:
9780199082049
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198080961.003.0001
Subject:
Sociology, Politics, Social Movements and Social Change

This chapter examines the ideals of democracy and the actual operation of its institutions in India, focusing on those political institutions that began to emerge from the middle of the nineteenth ... More


Deductive Narrative and the Epistemological Function of Belief in Mathematics: On Bombelli and Imaginary Numbers

Federica La Nave

in Circles Disturbed: The Interplay of Mathematics and Narrative

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691149042
eISBN:
9781400842681
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691149042.003.0003
Subject:
Mathematics, History of Mathematics

This chapter explores the epistemological function of belief in mathematical thinking by considering Rafael Bombelli's contribution to the creation of imaginary numbers. The discussion focuses on ... More


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