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## ALTERNATIVE APPROACHES

*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/9780198527688.003.0008
- Subject:
- Mathematics, Numerical Analysis

Alternative numerical approaches for solving matrix equations associated with M/G/1-type Markov chains are considered in this chapter. A general shift technique for accelerating the convergence of ... More

## Smoothness and Asymptotic Smoothness

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

### in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0008
- Subject:
- Mathematics, Analysis

This chapter describes the modulus of smoothness of a function in the direction of a family of subspaces and the much simpler notion of upper Fréchet differentiability. It also considers the notion ... More

##
Porosity, Γ*N*- and Γ-Null Sets

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

### in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0010
- Subject:
- Mathematics, Analysis

This chapter introduces the notion of porosity “at infinity” (formally defined as porosity with respect to a family of subspaces) and discusses the main result, which shows that sets porous with ... More

## Efficient Computations

*Lars Peter Hansen and Thomas J. Sargent*

### in Recursive Models of Dynamic Linear Economies

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691042770
- eISBN:
- 9781400848188
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691042770.003.0003
- Subject:
- Economics and Finance, History of Economic Thought

This chapter describes fast algorithms for computing the value function and optimal decision rule for the type of social planning problem to be described in Chapter 5. The chapter is organized as ... More

## Krylov Subspace Methods: Principles and Analysis

*Jörg Liesen and Zdenek Strakos*

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199655410
- eISBN:
- 9780191744174
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199655410.001.0001
- Subject:
- Mathematics, Applied Mathematics, Algebra

This book offers a detailed treatment of the mathematical theory of Krylov subspace methods with focus on solving systems of linear algebraic equations. Starting from the idea of projections, Krylov ... More

## Introduction

*Jörg Liesen and Zdenek Strakos*

### in Krylov Subspace Methods: Principles and Analysis

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199655410
- eISBN:
- 9780191744174
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199655410.003.0001
- Subject:
- Mathematics, Applied Mathematics, Algebra

The Introduction consists of three sections that establish the general setting and give background information that is useful to know before starting to read the other chapters. In particular, it ... More

## Direct Density-Ratio Estimation with Dimensionality Reduction

*Masashi Sugiyama and Motoaki Kawanabe*

### in Machine Learning in Non-Stationary Environments: Introduction to Covariate Shift Adaptation

- Published in print:
- 2012
- Published Online:
- September 2013
- ISBN:
- 9780262017091
- eISBN:
- 9780262301220
- Item type:
- chapter

- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262017091.003.0005
- Subject:
- Computer Science, Machine Learning

This chapter discusses a dimensionality reduction scheme for density-ratio estimation, called direct density-ratio estimation with dimensionality reduction (D3; pronounced as “D-cube”). The basic ... More

## The Standard Metric

*Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss*

### in Descent in Buildings (AM-190)

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691166902
- eISBN:
- 9781400874019
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691166902.003.0026
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine ... More

## Solution of discrete convection–diffusion problems

*A. M. Stuart and E. Söli*

### in Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics

- Published in print:
- 2014
- Published Online:
- September 2014
- ISBN:
- 9780199678792
- eISBN:
- 9780191780745
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199678792.003.0008
- Subject:
- Mathematics, Numerical Analysis, Computational Mathematics / Optimization

This chapter concerns iterative methods for solution of discrete convection–diffusion equations. It discusses Krylov subspace methods, principally the generalized minimum residual method, together ... More

## Linear symplectic geometry

*Dusa McDuff and Dietmar Salamon*

### in Introduction to Symplectic Topology

- Published in print:
- 2017
- Published Online:
- June 2017
- ISBN:
- 9780198794899
- eISBN:
- 9780191836411
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198794899.003.0003
- Subject:
- Mathematics, Analysis, Geometry / Topology

The second chapter introduces the basic concepts of symplectic topology in the linear algebra setting, such as symplectic vector spaces, the linear symplectic group, Lagrangian subspaces, and the ... More

## Subspaces and Substructures; Straightness and Differentiability

*Tim Maudlin*

### in New Foundations for Physical Geometry: The Theory of Linear Structures

- Published in print:
- 2014
- Published Online:
- April 2014
- ISBN:
- 9780198701309
- eISBN:
- 9780191771613
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198701309.003.0007
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

Parts of spaces are themselves spaces with their own geometrical structure. The way that the part of a space inherits its geometry from an embedding space in the Theory of Linear Structures is ... More

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