## The Diophantine Frobenius Problem

*Jorge L. Ramírez Alfonsín*

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198568209
- eISBN:
- 9780191718229
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568209.001.0001
- Subject:
- Mathematics, Algebra, Combinatorics / Graph Theory / Discrete Mathematics

During the early part of the last century, F. G. Frobenius raised, in his lectures, the following problem (called the Diophantine Frobenius Problem FP): given relatively prime positive integers a1, . ... More

## Hidden Markov Processes: Theory and Applications to Biology

*M. Vidyasagar*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691133157
- eISBN:
- 9781400850518
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691133157.001.0001
- Subject:
- Mathematics, Probability / Statistics

This book explores important aspects of Markov and hidden Markov processes and the applications of these ideas to various problems in computational biology. It starts from first principles, so that ... More

## Frobenius Conjugacy Classes

*Nicholas M. Katz*

### in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153308
- eISBN:
- 9781400842704
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153308.003.0006
- Subject:
- Mathematics, Number Theory

This chapter analyzes Frobenius conjugacy classes. It shows that in either the split or nonsplit case, when χ is good for N, the conjugacy class FrobE,X has unitary eigenvalues in every ... More

##
Frobenius Tori and Weights; Getting Elements of *G*_{arith}

_{arith}

*Nicholas M. Katz*

### in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153308
- eISBN:
- 9781400842704
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153308.003.0017
- Subject:
- Mathematics, Number Theory

This chapter works on Gₘ/k. It considers an arithmetically semisimple object N ɛ Garith which is pure of weight zero. It assumes it is of the form G[1], with G a middle extension sheaf. Thus, for ... More

## The Frobenius number for small n

*J. L. Ramírez Alfonsín*

### in The Diophantine Frobenius Problem

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198568209
- eISBN:
- 9780191718229
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568209.003.0002
- Subject:
- Mathematics, Algebra, Combinatorics / Graph Theory / Discrete Mathematics

It is easy to solve FP when n = 2. Indeed, g(a1, a2) = a1a2 - a1 - a2. However, the computation of a (simple) formula when n = 3 is much more difficult and has been the subject of numerous research ... More

## The general problem

*J. L. Ramírez Alfonsín*

### in The Diophantine Frobenius Problem

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198568209
- eISBN:
- 9780191718229
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568209.003.0003
- Subject:
- Mathematics, Algebra, Combinatorics / Graph Theory / Discrete Mathematics

This chapter provides a systematic exposition of the known formulas, including upper and lower bounds for g(a1, . . . , an) for general n, and for special sequences (for instance, when a1, . . . , an ... More

## Generalizations and related problems

*J. L. Ramírez Alfonsín*

### in The Diophantine Frobenius Problem

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198568209
- eISBN:
- 9780191718229
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568209.003.0006
- Subject:
- Mathematics, Algebra, Combinatorics / Graph Theory / Discrete Mathematics

Let g(n, t) and h(n, t) be the largest and smallest of three of the Frobenius numbers when a1 < · · · < an = t and t = a1 < · · · < an, respectively. This chapter reviews the results on these ... More

## INTRODUCTION TO MARKOV CHAINS

*D. A. Bini, G. Latouche, and B. Meini*

### in Numerical Methods for Structured Markov Chains

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198527688
- eISBN:
- 9780191713286
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527688.003.0001
- Subject:
- Mathematics, Numerical Analysis

This chapter contains a brief introduction to the fundamental concepts of Markov chains, including classification of states, irreducible classes, first passage times, stochastic complementation and ... More

## Consistent Orientation of Moduli Spaces

*Daniel S. Freed, Michael J. Hopkins, and Constantin Teleman*

### in The Many Facets of Geometry: A Tribute to Nigel Hitchin

- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199534920
- eISBN:
- 9780191716010
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199534920.003.0019
- Subject:
- Mathematics, Geometry / Topology

In a series of papers by Freed, Hopkins, and Teleman (2003, 2005, 2007a) the relationship between positive energy representations of the loop group of a compact Lie group G and the twisted ... More

##
Group-Theoretic Facts about *G*_{geom} and *G*_{arith}

_{geom}

_{arith}

*Nicholas M. Katz*

### in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153308
- eISBN:
- 9781400842704
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153308.003.0007
- Subject:
- Mathematics, Number Theory

This chapter presents proofs of following theorems. Theorem 6.1: Suppose N in Garith is geometrically semisimple. Then Ggeom,N is a normal subgroup of Garith,N. Theorem 6.2: Suppose that N in Garith ... More

## The Main Theorem

*Nicholas M. Katz*

### in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153308
- eISBN:
- 9781400842704
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153308.003.0008
- Subject:
- Mathematics, Number Theory

This chapter takes up the proof of the main theorem. Theorem 7.2: Suppose N in Garithι-pure of weight zero and arithmetically semisimple, such that the quotient group Garith,N/Ggeom,N is ℤ / nℤ. Fix ... More

## Nonnegative Matrices

*M. Vidyasagar*

### in Hidden Markov Processes: Theory and Applications to Biology

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691133157
- eISBN:
- 9781400850518
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691133157.003.0003
- Subject:
- Mathematics, Probability / Statistics

This chapter deals with nonnegative matrices, which are relevant in the study of Markov processes because the state transition matrix of such a process is a special kind of nonnegative matrix, known ... More

## Von Neumann's Economic Model

*S. N. Afriat*

### in Logic of Choice and Economic Theory

- Published in print:
- 1987
- Published Online:
- November 2003
- ISBN:
- 9780198284611
- eISBN:
- 9780191595844
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198284616.003.0023
- Subject:
- Economics and Finance, Microeconomics

This is the last of six chapters on the logic of price, and expands on the features of von Neumann's economic model, or activity system, which was first mentioned in Chapter IV.1. The fourteen ... More

## ‘The Benin Things are Fine, Fine’: The Art of Africa and the Pacific Islands in Pre-War London

*Rupert Richard Arrowsmith*

### in Modernism and the Museum: Asian, African, and Pacific Art and the London Avant-Garde

- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199593699
- eISBN:
- 9780191595684
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199593699.003.0009
- Subject:
- Literature, Film, Media, and Cultural Studies, 20th-century and Contemporary Literature

This chapter focuses on the shift in attention of the London avant-garde towards visual art from West Africa and the Pacific islands from late 1912 into the war years. New evidence is presented ... More

## Multi-parameter Singular Integrals. (AM-189)

*Brian Street*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691162515
- eISBN:
- 9781400852758
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691162515.001.0001
- Subject:
- Mathematics, Analysis

This book develops a new theory of multi-parameter singular integrals associated with Carnot–Carathéodory balls. The book first details the classical theory of Calderón–Zygmund singular integrals and ... More

## Second Order and Gaussian Fields

*Ulf Grenander and Michael I. Miller*

### in Pattern Theory: From representation to inference

- Published in print:
- 2006
- Published Online:
- November 2020
- ISBN:
- 9780198505709
- eISBN:
- 9780191916564
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198505709.003.0010
- Subject:
- Computer Science, Programming Languages

This chapter studies second order and Gaussian fields on the background spaces which are the continuum limits of the finite graphs. For this random processes in ... More

## Markov Processes and Random Sampling

*Ulf Grenander and Michael I. Miller*

### in Pattern Theory: From representation to inference

- Published in print:
- 2006
- Published Online:
- November 2020
- ISBN:
- 9780198505709
- eISBN:
- 9780191916564
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198505709.003.0019
- Subject:
- Computer Science, Programming Languages

The parameter spaces of natural patterns are so complex that inference must often proceed compositionally, successively building up more and more complex structures, ... More

## Categories for Quantum Theory: An Introduction

*Chris Heunen and Jamie Vicary*

- Published in print:
- 2019
- Published Online:
- January 2020
- ISBN:
- 9780198739623
- eISBN:
- 9780191802584
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198739623.001.0001
- Subject:
- Mathematics, Mathematical Physics, Applied Mathematics

Monoidal category theory serves as a powerful framework for describing logical aspects of quantum theory, giving an abstract language for parallel and sequential composition and a conceptual way to ... More

## Hamiltonian Fluid Dynamics

*Rick Salmon*

### in Lectures on Geophysical Fluid Dynamics

- Published in print:
- 1998
- Published Online:
- November 2020
- ISBN:
- 9780195108088
- eISBN:
- 9780197561034
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195108088.003.0010
- Subject:
- Earth Sciences and Geography, Geophysics: Earth Sciences

In this final chapter, we return to the subject of the first: the fundamental principles of fluid mechanics. In chapter 1, we derived the equations of fluid motion ... More

## Numerology

*Christophe Reutenauer*

### in From Christoffel Words to Markoff Numbers

- Published in print:
- 2018
- Published Online:
- January 2019
- ISBN:
- 9780198827542
- eISBN:
- 9780191866418
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198827542.003.0011
- Subject:
- Mathematics, Pure Mathematics

This chapter gives several examples, which may help the reader to work in concrete terms with Markoff numbers, Christoffel words, Markoff constants, and quadratic forms. In particular the thirteen ... More