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
Spherical projective path tracking for homotopy continuation methods
Tianran Chen,Tien-Yien Li +1 more
TL;DR: Taking the Riemannian geometry of CPn into account, the basic algorithm for projective path tracking is derived using the sphere, S2n+1, as the model of computation.
Journal ArticleDOI
Space-efficient scheduling of stochastically generated tasks
TL;DR: Results on the random variable S^@s are presented modeling the maximal space needed by the processor to store the currently active tasks when acting under the scheduler @s, and tail bounds for the distribution are obtained.
Posted Content
Computing Spectral Measures and Spectral Types
TL;DR: In this paper, the first general algorithms for computing spectral measures and decompositions of a wide class of operators were presented. But these methods focus on specific examples where analytical formulae are available (or perturbations thereof) or on classes of operators with a lot of structure.
Book ChapterDOI
The Stability of Saturated Linear Dynamical Systems Is Undecidable
TL;DR: It is proved that several global properties (global convergence, global asymptotic stability, mortality, and nilpotence) of particular classes of discrete time dynamical systems are undecidable.
Proceedings ArticleDOI
Detectability of Denial-of-Service Attacks on Arbitrarily Varying Classical-Quantum Channels
TL;DR: In this article, the authors developed algorithms based on Turing machines and also Blum-Shub-Smale (BSS) machines, where the latter can process and store arbitrary real numbers.
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.