164,960 research outputs found
Inverse zero-sum problems and algebraic invariants
In this article, we study the maximal cross number of long zero-sumfree
sequences in a finite Abelian group. Regarding this inverse-type problem, we
formulate a general conjecture and prove, among other results, that this
conjecture holds true for finite cyclic groups, finite Abelian p-groups and for
finite Abelian groups of rank two. Also, the results obtained here enable us to
improve, via the resolution of a linear integer program, a result of W. Gao and
A. Geroldinger concerning the minimal number of elements with maximal order in
a long zero-sumfree sequence of a finite Abelian group of rank two.Comment: 17 pages, to appear in Acta Arithmetic
Embedded minimal surfaces
The study of embedded minimal surfaces in \RR^3 is a classical problem,
dating to the mid 1700's, and many people have made key contributions. We will
survey a few recent advances, focusing on joint work with Tobias H. Colding of
MIT and Courant, and taking the opportunity to focus on results that have not
been highlighted elsewhere.Comment: To appear in proceedings of Madrid ICM200
Blind Construction of Optimal Nonlinear Recursive Predictors for Discrete Sequences
We present a new method for nonlinear prediction of discrete random sequences
under minimal structural assumptions. We give a mathematical construction for
optimal predictors of such processes, in the form of hidden Markov models. We
then describe an algorithm, CSSR (Causal-State Splitting Reconstruction), which
approximates the ideal predictor from data. We discuss the reliability of CSSR,
its data requirements, and its performance in simulations. Finally, we compare
our approach to existing methods using variable-length Markov models and
cross-validated hidden Markov models, and show theoretically and experimentally
that our method delivers results superior to the former and at least comparable
to the latter.Comment: 8 pages, 4 figure
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