35,525 research outputs found
A one line factoring algorithm
We describe a variant of Fermat’s factoring algorithm which is competitive with SQUFOF in practice but has heuristic run time complexity O(n1/3) as a general factoring algorithm. We also describe a sparse class of integers for which the algorithm is particularly effective. We provide speed comparisons between an optimised implementation of the algorithm described and the tuned assortment of factoring algorithms in the Pari/GP computer algebra package
Class invariants from a new kind of Weber-like modular equation
A technique is described for explicitly evaluating quotients of the Dedekind eta function at quadratic integers. These evaluations do not make use of complex approximations but are found by an entirely `algebraic' method. They are obtained by means of specialising certain modular equations related to Weber's modular equations of `irrational type'. The technique works for certain eta quotients evaluated at points in an imaginary quadratic field with discriminant d1 (mod 8)
Zero excess and minimal length in finite coxeter groups
Let \mathcal{W} be the set of strongly real elements of W, a Coxeter group. Then for ,
, the excess of w, is defined by
e(w) = \min min \{l(x)+l(y) - l(w)| w = xy; x^2 = y^2 =1}. When is finite we may also define E(w), the reflection excess of . The main result established here is that if is finite and is a -conjugacy class, then there
exists such that has minimal length in and
The locality of the square-root method for improved staggered quarks
We study the effects of improvement on the locality of square-rooted
staggered Dirac operators in lattice QCD simulations. We find the localisation
lengths of the improved operators (FAT7TAD and ASQTAD) to be very similar to
that of the one-link operator studied by Bunk et al., being at least the
Compton wavelength of the lightest particle in the theory, even in the
continuum limit. We conclude that improvement has no effect. We discuss the
implications of this result for the locality of the nth-rooted fermion
determinant used to reduce the number of sea quark flavours, and for possible
staggered valence quark formulations
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