Open AccessBook
Complexity and Real Computation
TLDR
This chapter discusses decision problems and Complexity over a Ring and the Fundamental Theorem of Algebra: Complexity Aspects.Abstract:
1 Introduction.- 2 Definitions and First Properties of Computation.- 3 Computation over a Ring.- 4 Decision Problems and Complexity over a Ring.- 5 The Class NP and NP-Complete Problems.- 6 Integer Machines.- 7 Algebraic Settings for the Problem "P ? NP?".- 8 Newton's Method.- 9 Fundamental Theorem of Algebra: Complexity Aspects.- 10 Bezout's Theorem.- 11 Condition Numbers and the Loss of Precision of Linear Equations.- 12 The Condition Number for Nonlinear Problems.- 13 The Condition Number in ?(H(d).- 14 Complexity and the Condition Number.- 15 Linear Programming.- 16 Deterministic Lower Bounds.- 17 Probabilistic Machines.- 18 Parallel Computations.- 19 Some Separations of Complexity Classes.- 20 Weak Machines.- 21 Additive Machines.- 22 Nonuniform Complexity Classes.- 23 Descriptive Complexity.- References.read more
Citations
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Planning Algorithms: Introductory Material
TL;DR: This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms, into planning under differential constraints that arise when automating the motions of virtually any mechanical system.
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Stability of Time-Delay Systems
TL;DR: Preface, Notations 1.Introduction to Time-Delay Systems I.Robust Stability Analysis II.Input-output stability A.LMI and Quadratic Integral Inequalities Bibliography Index
Journal ArticleDOI
On the mathematical foundations of learning
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Computable Analysis : An Introduction
TL;DR: This book provides a solid fundament for studying various aspects of computability and complexity in analysis and is written in a style suitable for graduate-level and senior students in computer science and mathematics.
Journal ArticleDOI
Mathematical problems for the next century
TL;DR: Arnabels invitation is inspired in part by Hilbert's list of 1900 (see e.g. [Browder, 1976]) and I have used that list to help design this essay.
References
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Journal ArticleDOI
Recursive predicates and quantifiers
Proceedings ArticleDOI
On simultaneous resource bounds
TL;DR: It is shown here that simultaneous time and "reversal" bounds correspond to simultaneous size and depth bounds, and that simultaneousTime and space bounds Correspond to simultaneousSize and "width" bounds.
Book ChapterDOI
Complexity of Bezout’s Theorem II Volumes and Probabilities
Michael Shub,Steve Smale +1 more
TL;DR: In this paper, the volume of the subspace of badly conditioned problems is shown to be bounded by a small degree polynomial in n, N and D times the reciprocal of the condition number to the fourth power.
Journal ArticleDOI
Lower bounds for approximation by nonlinear manifolds
TL;DR: In this paper, it is known that there are positive constants A and B, independent of w and n, such that the modulus of continuity function w is dominated by continuity function n, and F is the family P, of polynomials of degree n-1.
Journal ArticleDOI
On the average number of steps of the simplex method of linear programming
TL;DR: The goal is to give some theoretical explanation for the efficiency of the simplex method of George Dantzig, and it is shown that the number of pivots required to solve a linear programming problem grows in proportion to thenumber of variables on the average.