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
The complexity of semilinear problems in succinct representation
TL;DR: It is proved that the PNP[log]-completeness of deciding Zariski irreducibility exhibits for the first time a problem with a geometric nature complete in this class.
Journal ArticleDOI
Seven mutually touching infinite cylinders
TL;DR: The approach is algebraic and uses symbolic and numerical computational techniques to solve a problem of Littlewood: there exist seven infinite circular cylinders of unit radius which mutually touch each other.
Journal ArticleDOI
Optimal integration error on anisotropic classes for restricted Monte Carlo and quantum algorithms
Peixin Ye,Xiaofei Hu +1 more
TL;DR: It is shown that with clog"2n random bits the authors have the same optimal order for the nth minimal Monte Carlo integration error as with arbitrary random numbers.
Journal ArticleDOI
Control and Verification of the Safety-Factor Profile in Tokamaks Using Sum-of-Squares Polynomials.
TL;DR: In this article, the sum-of-squares methodology is used to synthesize controllers for plasma stabilization in Tokamak reactors, which are based on solving a dual version of the Lyapunov operator inequality.
Journal ArticleDOI
Condition Length and Complexity for the Solution of Polynomial Systems
TL;DR: This paper improves on both approaches to the main progress on Smale’s problem and builds on methods developed in Armentano et al. (2014) to prove an interesting intermediate result on the average value of the condition number of polynomial systems.
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.