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
More filters
Proceedings ArticleDOI
Efficient construction of Drinfel'd doubles
TL;DR: The initial implemmtatiou of the Drinfel’tl tloublc construction aud report, which addresses performance issues and the pcrfi,rriiarwc impact implied by the represrIitat,ioIi used for Hopf algebras as well as the use of the Driufel‘tl dolllde.
Posted Content
The complexity and geometry of numerically solving polynomial systems
Carlos Beltrán,Michael Shub +1 more
TL;DR: An overview on the state of the art of efficient numerical analysis methods that solve systems of multivariate polynomial equations, culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo, Peter Buergisser and Felipe Cucker.
Journal ArticleDOI
The PCP Theorem for NP Over the Reals
Martijn Baartse,Klaus Meer +1 more
TL;DR: It is shown that the PCP theorem holds as well in the real number computational model introduced by Blum, Shub, and Smale as well as the classical Turing model class NP.
Proceedings Article
Transparent Long Proofs: A First PCP Theorem for NP R .
TL;DR: The first PCP theorem for NPR is stated, which states that each problem in NPR has transparent long proofs, i.e.,NPR \subseteq PCPR(poly,1), where poly denotes the class of univariate polynomial functions.
Posted Content
Output Feedback Control of Inhomogeneous Parabolic PDEs with Point Actuation and Point Measurement using SOS and Semi-Separable Kernels
Aditya Gahlawat,Matthew M. Peet +1 more
TL;DR: This paper uses Sum-of-Squares (SOS) and Semi-Definite Programming (SDP) to design output feedback controllers for a class of one-dimensional parabolic partial differential equations with point measurements and point actuation and develops an improved class of observer-based controllers.
References
More filters
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.