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
Implicit Complexity over an Arbitrary Structure: Sequential and Parallel Polynomial Time
TL;DR: This work provides several machine-independent characterizations of deterministic complexity classes in the model of computation proposed by L. Blum, M. Shub and S. Smale and shows that polynomial time over an arbitrary structure can be characterized in terms of safe recursion.
Proceedings ArticleDOI
On the complexity of diophantine geometry in low dimensions
TL;DR: It is shown that the following two problems can be solved within PSPACE, and the truth of the Generalized Riemann Hypothesis implies that detecting roots in Q/sup n/ for the polynomial systems in problem (I) can be done via a two-round Arthur-Merlin protocol.
Proceedings ArticleDOI
Experimentation in the Schubert Calculus
TL;DR: A well-known web of conjectures and results in the real Schubert calculus has been inspired by this continuing experimentation as mentioned in this paper, which enables large-scale experimentation to investigate subtle and ill-understood phenomena in the SchUbert calculus.
Posted Content
Using SOS for Optimal Semialgebraic Representation of Sets: Finding Minimal Representations of Limit Cycles, Chaotic Attractors and Unions
Morgan Jones,Matthew M. Peet +1 more
TL;DR: It is shown that Sum-of-Squares optimization can be used to find optimal semialgebraic representations of sets, and defines optimality in the sense of minimum volume, while satisfying constraints that can include set containment, convexity, or Lyapunov stability conditions.
Journal ArticleDOI
Discreteness is undecidable
TL;DR: It is proved that the discreteness problem for two-generated nonelementary subgroups of SL(2, ℂ) is undecidable in the Blum–Shub–Smale (BSS) computability model.
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.