6,452 research outputs found
General Statistical Design of an Experimental Problem for Harmonics
Four years ago, the Michelin Tire Corporation proposed a problem on experimental design, to improve the manufacturing process for their tires. The idea is basically to determine the effects of placements for various layers built up in the construction of a tire, to allow the design of a smooth tire with a smooth ride. A highly success solution was developed, and it has been reported that this method introduced savings of over half a million dollars in their test processes. This year, Michelin returned to the workshop with an extension to the original problem, to address specific refinements in the testing method. This report summarizes the work completed in course of the five day workshop.
It was clear early in the workshop that this problem could be handled quickly by reviewing the analysis which was done in 2000, and extending those ideas to the new problems at hand. We reviewed the required Fourier techniques to describe the harmonic problem, and statistical techniques to deal with the linear model that described how to accurately measure quantities that come from real experimental measurements. The âprime methodâ and âgood lattice points methodâ were reviewed and re-analysed so we could understand (and prove) why they work so well. We then looked at extending these methods and successfully found solutions to problem 1) and 2) posed by Michelin. Matlab code was written to test and verify the algorithms developed. We have some ideas on problems 3) and 4), which are also described
Cyclotomic and simplicial matroids
Two naturally occurring matroids representable over Q are shown to be dual:
the {\it cyclotomic matroid} represented by the roots of unity
inside the cyclotomic extension ,
and a direct sum of copies of a certain simplicial matroid, considered
originally by Bolker in the context of transportation polytopes. A result of
Adin leads to an upper bound for the number of -bases for among
the roots of unity, which is tight if and only if has at most two
odd prime factors. In addition, we study the Tutte polynomial of in the
case that has two prime factors.Comment: 9 pages, 1 figur
Computing Bits of Algebraic Numbers
We initiate the complexity theoretic study of the problem of computing the
bits of (real) algebraic numbers. This extends the work of Yap on computing the
bits of transcendental numbers like \pi, in Logspace.
Our main result is that computing a bit of a fixed real algebraic number is
in C=NC1\subseteq Logspace when the bit position has a verbose (unary)
representation and in the counting hierarchy when it has a succinct (binary)
representation.
Our tools are drawn from elementary analysis and numerical analysis, and
include the Newton-Raphson method. The proof of our main result is entirely
elementary, preferring to use the elementary Liouville's theorem over the much
deeper Roth's theorem for algebraic numbers.
We leave the possibility of proving non-trivial lower bounds for the problem
of computing the bits of an algebraic number given the bit position in binary,
as our main open question. In this direction we show very limited progress by
proving a lower bound for rationals
Crystal constructions in Number Theory
Weyl group multiple Dirichlet series and metaplectic Whittaker functions can
be described in terms of crystal graphs. We present crystals as parameterized
by Littelmann patterns and we give a survey of purely combinatorial
constructions of prime power coefficients of Weyl group multiple Dirichlet
series and metaplectic Whittaker functions using the language of crystal
graphs. We explore how the branching structure of crystals manifests in these
constructions, and how it allows access to some intricate objects in number
theory and related open questions using tools of algebraic combinatorics
On numbers dividing the th term of a linear recurrence
Here, we give upper and lower bounds on the count of positive integers dividing the th term of a nondegenerate linearly recurrent sequence with
simple roots
- âŠ