72 research outputs found
Topological Complexity with Continuous Operations
AbstractThe topological complexity of algorithms is studied in a general context in the first part and for zero-finding in the second part. In the first part thelevel of discontinuityof a functionfis introduced and it is proved that it is a lower bound for the total number of comparisons plus 1 in any algorithm computingfthat uses only continuous operations and comparisons. This lower bound is proved to be sharp if arbitrary continuous operations are allowed. Then there exists even a balanced optimal computation tree forf. In the second part we use these results in order to determine the topological complexity of zero-finding for continuous functionsfon the unit interval withf(0) ·f(1) < 0. It is proved that roughly log2log2ϵ−1comparisons are optimal during a computation in order to approximate a zero up to ϵ. This is true regardless of whether one allows arbitrary continuous operations or just function evaluations, the arithmetic operations {+, −, *, /}, and the absolute value. It is true also for the subclass of nondecreasing functions. But for the subclass of increasing functions the topological complexity drops to zero even for the smaller class of operations
08271 Abstracts Collection -- Topological and Game-Theoretic Aspects of Infinite Computations
From June 29, 2008, to July 4, 2008, the Dagstuhl Seminar 08271 ``Topological and Game-Theoretic Aspects of Infinite Computations\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, many participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available
Computation with Advice
Computation with advice is suggested as generalization of both computation
with discrete advice and Type-2 Nondeterminism. Several embodiments of the
generic concept are discussed, and the close connection to Weihrauch
reducibility is pointed out. As a novel concept, computability with random
advice is studied; which corresponds to correct solutions being guessable with
positive probability. In the framework of computation with advice, it is
possible to define computational complexity for certain concepts of
hypercomputation. Finally, some examples are given which illuminate the
interplay of uniform and non-uniform techniques in order to investigate both
computability with advice and the Weihrauch lattice
Donagi-Markman cubic for the generalised Hitchin system
Donagi and Markman (1993) have shown that the infinitesimal period map for an algebraic completely integrable Hamiltonian system (ACIHS) is encoded in a section of the third symmetric power of the cotangent bundle to the base of the system. For the ordinary Hitchin system the cubic is given by a formula of Balduzzi and Pantev. We show that the Balduzzi\u2013Pantev formula holds on maximal rank symplectic leaves of the G-generalised Hitchin system
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