906,350 research outputs found
Algorithms in algebraic number theory
In this paper we discuss the basic problems of algorithmic algebraic number
theory. The emphasis is on aspects that are of interest from a purely
mathematical point of view, and practical issues are largely disregarded. We
describe what has been done and, more importantly, what remains to be done in
the area. We hope to show that the study of algorithms not only increases our
understanding of algebraic number fields but also stimulates our curiosity
about them. The discussion is concentrated of three topics: the determination
of Galois groups, the determination of the ring of integers of an algebraic
number field, and the computation of the group of units and the class group of
that ring of integers.Comment: 34 page
Quantum algorithms for problems in number theory, algebraic geometry, and group theory
Quantum computers can execute algorithms that sometimes dramatically
outperform classical computation. Undoubtedly the best-known example of this is
Shor's discovery of an efficient quantum algorithm for factoring integers,
whereas the same problem appears to be intractable on classical computers.
Understanding what other computational problems can be solved significantly
faster using quantum algorithms is one of the major challenges in the theory of
quantum computation, and such algorithms motivate the formidable task of
building a large-scale quantum computer. This article will review the current
state of quantum algorithms, focusing on algorithms for problems with an
algebraic flavor that achieve an apparent superpolynomial speedup over
classical computation.Comment: 20 pages, lecture notes for 2010 Summer School on Diversities in
Quantum Computation/Information at Kinki Universit
Algorithms in number theory
Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe
Group theory factors for Feynman diagrams
We present algorithms for the group independent reduction of group theory
factors of Feynman diagrams. We also give formulas and values for a large
number of group invariants in which the group theory factors are expressed.
This includes formulas for various contractions of symmetric invariant tensors,
formulas and algorithms for the computation of characters and generalized
Dynkin indices and trace identities. Tables of all Dynkin indices for all
exceptional algebras are presented, as well as all trace identities to order
equal to the dual Coxeter number. Further results are available through
efficient computer algorithms (see http://norma.nikhef.nl/~t58/ and
http://norma.nikhef.nl/~t68/ ).Comment: Latex (using axodraw.sty), 47 page
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