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
A formula for separating small roots of a polynomial
Tateaki Sasaki,Akira Terui +1 more
TL;DR: This short article introduces the formula and checks its sharpness on many polynomials generated randomly, which is essentially the same as that derived independently by Yakoubsohn at almost the same time.
Journal ArticleDOI
Determining on-fault earthquake magnitude distributions from integer programming
Eric L. Geist,Tom Parsons +1 more
TL;DR: It is shown that the optimal magnitude distributions for each of the faults in the system span a rich diversity of shapes, ranging from characteristic to power-law distributions.
Book ChapterDOI
Constraint Databases, Data Structures and Efficient Query Evaluation
TL;DR: Constraint databases that can be described by boolean combinations of polynomial inequalities over the reals, as a constraint database query language, have received ample research attention but the difficulty of the effective evaluation of first-order queries has been largely neglected.
Posted Content
Crystalline Chebotarëv density theorems
Urs Hartl,Ambrus Pál +1 more
TL;DR: In this paper, the authors formulate a conjectural analogue of Chebotar\"ev's density theorem for convergent $F$-isocrystals over a smooth geometrically irreducible curve defined over a finite field using the Tannakian formalism.
Proceedings ArticleDOI
Approximating APSP without Scaling: Equivalence of Approximate Min-Plus and Exact Min-Max
TL;DR: In this paper, a strongly polynomial time algorithm for APSP on directed and undirected graphs was proposed. But this algorithm requires the scaling technique of Zwick's algorithm to be used.
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.