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 real dimension problem is NP R -complete
TL;DR: It is shown that computing the dimension of a semi-algebraic set of R n is a NP R -complete problem in the Blum–Shub–Smale model of computation over the reals.
Proceedings ArticleDOI
An almost optimal algorithm for computing nonnegative rank
TL;DR: Arora, Ge, Kannan, and Moitra as discussed by the authors presented a nonnegative matrix factorization algorithm that runs in time (nm)o(r) under the Exponential Time Hypothesis.
Book ChapterDOI
Verification of Hybrid Systems
TL;DR: Hybrid systems are models which combine discrete and continuous behavior which occur frequently in safety-critical applications in various domains such as health care, transportation, and robotics, as a result of interactions between a digital controller and a physical environment.
Journal ArticleDOI
Uncomputability and undecidability in economic theory
TL;DR: Interpreting the core of mathematical economic theory to be defined by General Equilibrium Theory and Game Theory, a general - but concise - analysis of the computable and decidable content of the implications of these two areas are discussed.
Posted Content
Certified numerical homotopy tracking
Carlos Beltrán,Anton Leykin +1 more
TL;DR: In this article, a regular homotopy path connecting an approximate zero of the start system to an approximate 0 of the target system is tracked using the alpha theory of Smale.
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.