33 research outputs found
Sampling-based proofs of almost-periodicity results and algorithmic applications
We give new combinatorial proofs of known almost-periodicity results for
sumsets of sets with small doubling in the spirit of Croot and Sisask, whose
almost-periodicity lemma has had far-reaching implications in additive
combinatorics. We provide an alternative (and L^p-norm free) point of view,
which allows for proofs to easily be converted to probabilistic algorithms that
decide membership in almost-periodic sumsets of dense subsets of F_2^n.
As an application, we give a new algorithmic version of the quasipolynomial
Bogolyubov-Ruzsa lemma recently proved by Sanders. Together with the results by
the last two authors, this implies an algorithmic version of the quadratic
Goldreich-Levin theorem in which the number of terms in the quadratic Fourier
decomposition of a given function is quasipolynomial in the error parameter,
compared with an exponential dependence previously proved by the authors. It
also improves the running time of the algorithm to have quasipolynomial
dependence instead of an exponential one.
We also give an application to the problem of finding large subspaces in
sumsets of dense sets. Green showed that the sumset of a dense subset of F_2^n
contains a large subspace. Using Fourier analytic methods, Sanders proved that
such a subspace must have dimension bounded below by a constant times the
density times n. We provide an alternative (and L^p norm-free) proof of a
comparable bound, which is analogous to a recent result of Croot, Laba and
Sisask in the integers.Comment: 28 page
The Frobenius problem, rational polytopes, and Fourier-Dedekind Sums
We study the number of lattice points in integer dilates of the rational
polytope ,
where are positive integers. This polytope is closely related to
the linear Diophantine problem of Frobenius: given relatively prime positive
integers , find the largest value of t (the Frobenius number) such
that has no solution in positive integers
. This is equivalent to the problem of finding the largest dilate
tP such that the facet contains no lattice point. We
present two methods for computing the Ehrhart quasipolynomials of P which count
the integer points in the dilated polytope and its interior. Within the
computations a Dedekind-like finite Fourier sum appears. We obtain a
reciprocity law for these sums, generalizing a theorem of Gessel. As a
corollary of our formulas, we rederive the reciprocity law for Zagier's
higher-dimensional Dedekind sums. Finally, we find bounds for the
Fourier-Dedekind sums and use them to give new bounds for the Frobenius number.Comment: Added journal referenc
Implementation of a New Quasi-Optimal Controller Tuning Algorithm for Time-Delay Systems
The aim of this chapter is to describe, demonstrate and implement a new quasi-optimal pole placement algorithm for SISO LTI-TDS based on the quasi-continuous pole shifting to the prescribed positions. The desired positions are obtained by overshoot analysis of the step response for a dominant pair of complex conjugate poles. A controller structure is initially obtained by algebraic controller design in RMS. Note that the maximum number of prescribed poles (including their multiplicities) equals the number of unknown parameters. If the prescribed roots locations can not be reached, the optimizing of an objective function involving the distance of shifting poles to the prescribed ones and the roots dominancy is utilized. The optimization is made via Self-Organizing Migration Algorithm (SOMA). Matlab m-file environment is utilized for the algorithm implementation and, consequently, results are tested in Simulink on an attractive example of unstable SISO LTI-TDS.P(ED2.1.00/03.0089), Z(MSM7088352102
A friendly introduction to Fourier analysis on polytopes
This book is an introduction to the nascent field of Fourier analysis on
polytopes, and cones. There is a rapidly growing number of applications of
these methods, so it is appropriate to invite students, as well as
professionals, to the field. We assume a familiarity with Linear Algebra, and
some Calculus. Of the many applications, we have chosen to focus on: (a)
formulations for the Fourier transform of a polytope, (b) Minkowski and
Siegel's theorems in the geometry of numbers, (c) tilings and multi-tilings of
Euclidean space by translations of a polytope, (d) Computing discrete volumes
of polytopes, which are combinatorial approximations to the continuous volume,
(e) Optimizing sphere packings and their densities, and (f) use iterations of
the divergence theorem to give new formulations for the Fourier transform of a
polytope, with an application. Throughout, we give many examples and exercises,
so that this book is also appropriate for a course, or for self-study.Comment: 204 pages, 46 figure