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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New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems
TL;DR: This paper presents some new general complete convergence theorems for the Picard iteration x"n"+"1=Tx"n with order of convergence at least r>=1 and establishes three complete @w-versions of the famous semilocal Newton-Kantorovich theorem as well as a complete version of the Famous Semilocal @a-theorem of Smale for analytic functions.
Proceedings ArticleDOI
Dynamic planar convex hull
Gerth Stølting Brodal,Riko Jacob +1 more
TL;DR: In this article, the authors present a data structure that maintains a finite set of n points in the plane under insertion and deletion of points in amortized O(log n) time per operation.
BookDOI
Advances in convex optimization: conic programming
TL;DR: The major components of the resulting theory (conic duality and primal-dual interior point polynomial time algorithms) are overview, the extremely rich �expressive abilities� of conic quadratic and semidefinite programming are outlined and a number of instructive applications are discussed.
Journal Article
NP-complete Problems and Physical Reality
TL;DR: Can NP-complete problems be solved efficiently in the physical universe? as mentioned in this paper survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and anthropic computing.
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A counterexample to the Hirsch conjecture
TL;DR: This paper presents the rst counterexample to the Hirsch Conjecture, obtained from a 5-dimensional polytope with 48 facets that violates a certain generalization of the d-step conjecture of Klee and Walkup.
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.