362 research outputs found
On Buffon Machines and Numbers
The well-know needle experiment of Buffon can be regarded as an analog (i.e.,
continuous) device that stochastically "computes" the number 2/pi ~ 0.63661,
which is the experiment's probability of success. Generalizing the experiment
and simplifying the computational framework, we consider probability
distributions, which can be produced perfectly, from a discrete source of
unbiased coin flips. We describe and analyse a few simple Buffon machines that
generate geometric, Poisson, and logarithmic-series distributions. We provide
human-accessible Buffon machines, which require a dozen coin flips or less, on
average, and produce experiments whose probabilities of success are expressible
in terms of numbers such as, exp(-1), log 2, sqrt(3), cos(1/4), aeta(5).
Generally, we develop a collection of constructions based on simple
probabilistic mechanisms that enable one to design Buffon experiments involving
compositions of exponentials and logarithms, polylogarithms, direct and inverse
trigonometric functions, algebraic and hypergeometric functions, as well as
functions defined by integrals, such as the Gaussian error function.Comment: Largely revised version with references and figures added. 12 pages.
In ACM-SIAM Symposium on Discrete Algorithms (SODA'2011
Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity
In each of the 10 cases with propagators of unit or zero mass, the finite
part of the scalar 3-loop tetrahedral vacuum diagram is reduced to 4-letter
words in the 7-letter alphabet of the 1-forms and , where is the sixth root of unity. Three diagrams
yield only . In two cases combines
with the Euler-Zagier sum ; in three cases it combines with the square of Clausen's
. The case
with 6 masses involves no further constant; with 5 masses a
Deligne-Euler-Zagier sum appears: . The previously unidentified term in the
3-loop rho-parameter of the standard model is merely . The remarkable simplicity of these results stems
from two shuffle algebras: one for nested sums; the other for iterated
integrals. Each diagram evaluates to 10 000 digits in seconds, because the
primitive words are transformable to exponentially convergent single sums, as
recently shown for and , familiar in QCD. Those are
SC constants, whose base of super-fast computation is 2. Mass involves
the novel base-3 set SC. All 10 diagrams reduce to SCSC constants and their products. Only the 6-mass case entails both bases.Comment: 41 pages, LaTe
Parallel integer relation detection: techniques and applications
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On a symmetric space attached to polyzeta values
Quickly convergent series are given to compute polyzeta numbers. The formula
involves an intricate combination of (generalized) polylogarithms at 1/2.
However, the combinatorics has a very simple geometric interpretation: it
corresponds with the square map on some symmetric space P.Comment: 18 page
Fault-Tolerant Quantum Computation with Local Gates
I discuss how to perform fault-tolerant quantum computation with concatenated
codes using local gates in small numbers of dimensions. I show that a threshold
result still exists in three, two, or one dimensions when
next-to-nearest-neighbor gates are available, and present explicit
constructions. In two or three dimensions, I also show how nearest-neighbor
gates can give a threshold result. In all cases, I simply demonstrate that a
threshold exists, and do not attempt to optimize the error correction circuit
or determine the exact value of the threshold. The additional overhead due to
the fault-tolerance in both space and time is polylogarithmic in the error rate
per logical gate.Comment: 14 pages, LaTe
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