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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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Book ChapterDOI
Condition number analysis for sparse polynomial systems
TL;DR: A new invariant, the sparse condition number, is introduced and it is shown that a sparse polynomial system analysis in terms of this invariant is easier to solve than a non sparse one.
Journal ArticleDOI
Counting Complexity Classes for Numeric Computations I: Semilinear Sets
Peter Bürgisser,Felipe Cucker +1 more
TL;DR: This work defines a counting class in the Blum--Shub--Smale setting of additive computations over the reals and characterize in terms of completeness results the complexity of computing basic topological invariants of semilinear sets given by additive circuits.
Proceedings ArticleDOI
Computational and computer complexity of analogic cellular wave computers
TL;DR: It is shown that algorithms using spatial-temporal continuous elementary instructions (a-recursive functions) represent not only a new world in computing, but also a more general type of logic inferencing.
Journal ArticleDOI
Elimination of Constants from Machines over Algebraically Closed Fields
TL;DR: It is shown that constants can be removed efficiently from any machine over K solving a problem which is definable without constants, which gives a new proof of the Blum?Cucker?Shub?Smale transfer theorem for the problem P=?NP.
Proceedings ArticleDOI
Structure from Motion with Missing Data is NP-Hard
TL;DR: It is shown that structure from motion is NP-hard for most sensible cost functions when missing data is allowed, and that the NP-complete problem 3SAT could be solved in polynomial time, which would imply that P=NP.
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.