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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.

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Citations
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Journal ArticleDOI

Counterexamples to the uniformity conjecture

TL;DR: It is shown in this article that the uniformity conjecture is incorrect, and a technique is given for generating counterexamples which may be useful to check other conjectured constructive root bounds of this kind.
Journal ArticleDOI

Computing stationary probability distributions and large deviation rates for constrained random walks.: the undecidability results.

TL;DR: In this paper, it was shown that computing the stationary probability of a constrained homogeneous random walk in Z+d is an undecidable problem, and that computing large deviation rates for this model is also an unanswerable problem.
Book ChapterDOI

On the Complexity of Integer Programming in the Blum-Shub-Smale Computational Model

TL;DR: No algorithm can solve 0/1 KPR in o (n log n) f(a1, ..., an) time, even if f is an arbitrary continuous function of n variables, even as an alternative to the well-known Ben-Or's bound Ω(n2).
Posted Content

Semi-Algebraic Proofs, IPS Lower Bounds and the $\tau$-Conjecture: Can a Natural Number be Negative?

TL;DR: In this paper, the binary value principle is introduced, which is a simple subset-sum instance expressing that a natural number written in binary cannot be negative, relating it to central problems in proof and algebraic complexity.
Journal ArticleDOI

Computable general equilibrium with financial markets

TL;DR: In this paper, the authors give a general recipe for using homotopy algorithm to compute equilibria in general equilibrium models with incomplete security markets and describe how to implement the algorithm using a publicly available suite of subroutines.
References
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Book

Principles of Algebraic Geometry

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.
Book

Theory of Linear and Integer Programming

TL;DR: Introduction and Preliminaries.
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.
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