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
Journal ArticleDOI
What is numerical algebraic geometry
TL;DR: This article provides a short introduction to numerical algebraic geometry with the subsequent articles in this special issue considering three current research topics: solving structured systems, certifying the results of numerical computations, and performing algebraic computations numerically via Macaulay dual spaces.
Journal ArticleDOI
Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares
Reza Kamyar,Matthew M. Peet +1 more
TL;DR: In this article, the authors explore the merits of various algorithms for solving polynomial optimization and optimization of polynomials, focusing on alternatives to sum-of-squares programming.
Journal ArticleDOI
A Dichotomy for Real Weighted Holant Problems
Sangxia Huang,Pinyan Lu +1 more
TL;DR: This paper proves a dichotomy for the general Holant framework where all the constraints are real symmetric functions on Boolean inputs, and introduces a new reduction technique, namely realizing a constraint function by approximating it.
Posted Content
Computability of Julia sets
Mark Braverman,Michael Yampolsky +1 more
TL;DR: Computational hardness of Julia sets is the main subject of this book, which says that while a dynamical system can be described numerically with an arbitrary precision, the picture of the dynamics cannot be visualized.
Journal ArticleDOI
Computer-Sensors: Spatial-Temporal Computers for Analog Array Signals, Dynamically Integrated with Sensors
TL;DR: The systematic description of the Non-equilibrium Spatial-temporal (NEST) algorithms is given, as a new way of array signal processing, and some practical aspects of NEST algorithms are discussed.
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.