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
Embracing Statistical Challenges in the Information Technology Age
TL;DR: It is concluded that for the healthy future of the field, computer technologies have to be integrated into statistics, and statistical thinking in turn must be Integrated into computer technologies.
Journal ArticleDOI
Discounted deterministic Markov decision processes and discounted all-pairs shortest paths
TL;DR: In this article, a randomized O(m 1/2n 2 )-time algorithm for discounted, infinite-horizon, determinitc Markov Decision Processes (DMDPs) was presented.
Journal ArticleDOI
Counting Complexity Classes for Numeric Computations. III: Complex Projective Sets
TL;DR: It is proved that the corresponding result is true over C, namely that the computation of the Euler characteristic of an affine or projective complex variety is complete in the class FPC#PC, and a corresponding completeness result for the Turing model.
Book ChapterDOI
In Praise of Numerical Computation
TL;DR: A pathway is suggested, by way of an exact numerical model of computation, that allows us to incorporate iteration and approximation into the authors' algorithms' design and suggests the name "numerical computational geometry" for such activities.
Journal ArticleDOI
Balancing the lifting values to improve the numerical stability of polyhedral homotopy continuation methods
TL;DR: This work proposes a procedure that operates as an intermediate stage between the mixed-volume computation and the tracing of solution curves and computes new lifting values of the support of a polynomial system to produce better-balanced powers of the continuation parameter in the polyhedral homotopies.
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.