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
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Computable structures on topological manifolds.
Marcelo A. Aguilar,Rodolfo Conde +1 more
TL;DR: Using the framework of computable topology and Type-2 theory of effectivity, computable versions of all the basic concepts needed to define manifolds are developed, like computable atlases and (computably) compatible computableAtlases.
Posted Content
A Measure of Space for Computing over the Reals
TL;DR: In this article, the authors proposed a new complexity measure of space for the BSS model of computation, which they called LOGSPACE, and proved that it is P\_R-complete.
Book ChapterDOI
A Continuation Method for Visualizing Planar Real Algebraic Curves with Singularities.
Changbo Chen,Wenyuan Wu +1 more
TL;DR: A new method for visualizing planar real algebraic curves inside a bounding box based on numerical continuation and critical point methods that is more stable for defining the numerical connectedness of the complement of the curve, which is important for applications such as solving bi-parametric polynomial systems.
Journal ArticleDOI
Digital hyperplane recognition in arbitrary fixed dimension within an algebraic computation model
TL;DR: An algorithm for the integer linear programming (ILP) problem within an algebraic model of computation is presented and used to solve the following digital plane segment recognition problem: Given a set of points M, decide whether M is a portion of a digital hyperplane and, if so, determine its analytical representation.
Book ChapterDOI
Few Product Gates But Many Zeros
TL;DR: In this paper, it was shown that 2 n -gems for n ≤ 4 are skew, that is, each { +,? }-gate adds an integer, and that these for n ≥ 5 would imply new solutions to the Prouhet-Tarry-Escott problem.
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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.