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 Prospects for Mathematical Logic in the Twenty-First Century
TL;DR: In this paper, the authors present their speculations about the future developments of mathematical logic in the twenty-first century and discuss the areas of recursion theory, proof theory and logic for computer science, model theory, and set theory.
Journal Article
Continuous Amortization: A Non-Probabilistic Adaptive Analysis Technique.
TL;DR: This paper introduces a form of continuous amortization for adaptive complexity in subdivision algorithms based on purely numerical primitives such as function evaluation and provides an adaptive upper bound on the complexity of EVAL using an integral, analogous to integral bounds provided by Ruppert in a different context.
Journal ArticleDOI
On Defining Integers And Proving Arithmetic Circuit Lower Bounds
TL;DR: It is proved that if there are arithmetic circuits of size polynomial in n for computing the permanent of n by n matrices, then τ(n!) is polynomially bounded in log n.
Book ChapterDOI
Foundations of Computational Mathematics: Kronecker's smart, little black boxes
M. Giusti,J. Heintz +1 more
Journal ArticleDOI
Smale's point estimate theory for Newton's method on Lie groups
TL;DR: The notion of the (one-parameter subgroup) @c-condition for a map f from a Lie group to its Lie algebra is introduced and Smale's @a-theory and @c -theory are established.
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.