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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Proceedings ArticleDOI
Meeting in a Polygon by Anonymous Oblivious Robots
Giuseppe Antonio Di Luna,Paola Flocchini,Nicola Santoro,Giovanni Viglietta,Masafumi Yamashita +4 more
TL;DR: In this article, the meeting problem for k>=2 searchers in a polygon P (possibly with holes) was studied, and it was shown that k = sigma+1 searchers are sufficient to solve the problem, where sigma = 1 corresponds to no rotational symmetry.
Proceedings ArticleDOI
Descriptive complexity of real computation and probabilistic independence logic
TL;DR: In this paper, the authors introduce a variant of BSS machines called separate branching BSMs (S-BSS) and develop a Fagin-type logical characterisation for languages decidable in non-deterministic polynomial time by S-BSMs.
Homotopy Methods for Solving Polynomial Systems tutorial at ISSAC'05, Beijing, China, 24 July 2005
TL;DR: This tutorial is on linking recent algorithms in numerical algebraic geometry to the software package PHCpack, a collection of algorithms to solve polynomial systems.
Journal ArticleDOI
Wilkinson’s Bus: Weak Condition Numbers, with an Application to Singular Polynomial Eigenproblems
Martin Lotz,Vanni Noferini +1 more
TL;DR: In this paper, the authors propose a new approach to the theory of conditioning for numerical analysis problems for which both classical and stochastic perturbation theories fail to predict the observed accuracy of computed solutions.
Proceedings ArticleDOI
Monte Carlo Computability
TL;DR: It is proved that Monte Carlo computable functions are closed under composition and mutually separate the following classes of functions from each other: the class of multi-valued functions that are non-deterministically computable, that of Las Vegas computable function, and that of Monte Carlo Computable functions.
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.