scholarly journals Trade-offs of certified fixed-point code synthesis for linear algebra basic blocks

2017 ◽  
Vol 76 ◽  
pp. 133-148
Author(s):  
Matthieu Martel ◽  
Amine Najahi ◽  
Guillaume Revy
Keyword(s):  
Fixed Point ◽  
Linear Algebra ◽  
Trade Offs ◽  
Point Code ◽  
Basic Blocks

Trade-Offs Between Synchronization, Communication, and Computation in Parallel Linear Algebra Computations

2016 ◽  
Vol 3 (1) ◽  
pp. 1-47 ◽  
Author(s):  
Edgar Solomonik ◽  
Erin Carson ◽  
Nicholas Knight ◽  
James Demmel
Keyword(s):  
Linear Algebra ◽  
Trade Offs

Thurston's Proof

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
General Theory ◽  
Linear Algebra ◽  
Point Theorem ◽  
Original Proof ◽  
Closed Curves ◽  
Measured Foliation ◽  
Geodesic Laminations ◽  
Train Tracks

This chapter describes Thurston's original path of discovery to the Nielsen–Thurston classification theorem. It first provides an example that illustrates much of the general theory, focusing on Thurston's iteration of homeomorphisms on simple closed curves as well as the linear algebra of train tracks. It then explains how the general theory works and presents Thurston's original proof of the Nielsen–Thurston classification. In particular, it considers the Teichmüller space and the measured foliation space. The chapter also discusses measured foliations on a pair of pants, global coordinates for measured foliation space, the Brouwer fixed point theorem, the Thurston compactification for the torus, and Markov partitions. Finally, it evaluates other approaches to proving the Nielsen–Thurston classification, including the use of geodesic laminations.

scholarly journals A new method for solving nonlinear systems of equations that is based on functional iterations

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 605-630 ◽  
Author(s):  
Joaquín Moreno ◽  
Miguel A. López ◽  
Raquel Martínez
Keyword(s):  
Fixed Point ◽  
Nonlinear Systems ◽  
Linear Algebra ◽  
Linear Equations ◽  
Continuous Functions ◽  
Compact Set ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Banach Theorem ◽  
Non Linear Equations

Abstract Regarding solving nonlinear equations systems, there is a main problem that is the number and complexity of the linear algebra operations, and the functional evaluations of the applied algorithm. In this paper, an alternative solution will be proposed by means of constructing a converse of the Banach Theorem fixed-point, only to ℝ2 and ℝ3, in the following sense, this being: each root of a non-linear equations system has been considered as a fixed-point. Besides, the compact set and the continuous functions that fulfil the Banach Theorem are built under certain conditions, those that must satisfy the systemfunctions. Thus each iteration only requires the evaluation of two or three functions.

ON THE DIMENSIONS OF CARTESIAN SYMMETRY CLASSES

2012 ◽  
Vol 05 (03) ◽  
pp. 1250046
Author(s):  
Yousef Zamani ◽  
Mohammad Shahryari
Keyword(s):  
Fixed Point ◽  
Linear Algebra ◽  
Symmetry Class ◽  
Class V ◽  
Idempotent Matrix ◽  
Symmetry Classes

The notion of Cartesian symmetry classes is introduced in [T. G. Lei, Notes on Cartesian symmetry classes and generalized trace functions, Linear Algebra Appl.292 (1999) 281–288]. In this paper, we discuss these classes and compute the dimensions of these classes in terms of the fixed point character of Sm. Also, we give a formula for the dimension of Cartesian symmetry class Vχ(G) in terms of the rank of an idempotent matrix M(χ). Some properties of generalized trace functions of a matrix are concluded.

scholarly journals Product-Normed Linear Spaces

2018 ◽  
Vol 11 (3) ◽  
pp. 740-750
Author(s):  
Benedict Barnes ◽  
I. A. Adjei ◽  
S. K. Amponsah ◽  
E. Harris
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Differential Equations ◽  
Linear Space ◽  
Linear Algebra ◽  
Functional Properties ◽  
Normed Linear Space ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Space Product

In this paper, both the product-normed linear space $P-NLS$ (product-Banach space) and product-semi-normed linear space (product-semi-Banch space) are introduced. These normed linear spaces are endowed with the first and second product inequalities, which have a lot of applications in linear algebra and differential equations. In addition, we showed that $P-NLS$ admits functional properties such as completeness, continuity and the fixed point.

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