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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Journal ArticleDOI
Approximate solutions of the hyperbolic Kepler equation
TL;DR: In this paper, the authors provided an approximate zero for the hyperbolic Kepler's equation and showed that Newton's method starting at this approximate zero converges to the actual solution at quadratic speed.
Journal ArticleDOI
Small Space Analogues of Valiant's Classes and the Limitations of Skew Formulas
TL;DR: In this paper, the width of an arithmetic circuit is defined as a measure of space, and the class VL is introduced as an algebraic variant of deterministic log-space L. The notion of read-once certificates for arithmetic circuits is introduced in the uniform setting, which coincides with that of VPSPACE at polynomial width.
BookDOI
Causality, meaningful complexity and embodied cognition
TL;DR: Carsetti as mentioned in this paper discusses the link between BRAIN LEARNING, ATTention, and CONSCIOUSNESS and discusses the role of selfishness in self-organizing networks.
Journal ArticleDOI
Elimination of parameters in the polynomial hierarchy
TL;DR: The present paper updates a technical report (LIP Research Report 97-37) with the same title, and in particular includes new results on interactive protocols and boolean parts.
Journal ArticleDOI
Statistics of stationary points of random finite polynomial potentials
TL;DR: In this article, a combination of the numerical polynomial homotopy continuation method and a certification method was used to obtain all the certified stationary points of the most general polynomials for each sample chosen from the Gaussian distribution with mean 0 and variance 1.
References
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Book
Principles of Algebraic Geometry
Phillip Griffiths,Joe Harris +1 more
TL;DR: In this paper, a comprehensive, self-contained treatment of complex manifold theory is presented, focusing on results applicable to projective varieties, and including discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex.
Book
The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
Proceedings ArticleDOI
The complexity of theorem-proving procedures
TL;DR: It is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be “reduced” to the problem of determining whether a given propositional formula is a tautology.