1,054,503 research outputs found
On Weighted Multivariate Sign Functions
Multivariate sign functions are often used for robust estimation and
inference. We propose using data dependent weights in association with such
functions. The proposed weighted sign functions retain desirable robustness
properties, while significantly improving efficiency in estimation and
inference compared to unweighted multivariate sign-based methods. Using
weighted signs, we demonstrate methods of robust location estimation and robust
principal component analysis. We extend the scope of using robust multivariate
methods to include robust sufficient dimension reduction and functional outlier
detection. Several numerical studies and real data applications demonstrate the
efficacy of the proposed methodology.Comment: Keywords: Multivariate sign, Principal component analysis, Data
depth, Sufficient dimension reductio
On almost Poisson commutativity in dimension two
Consider the following question: given two functions on a symplectic manifold
whose Poisson bracket is small, is it possible to approximate them in the
norm by commuting functions? We give a positive answer in dimension two, as a
particular case of a more general statement which applies to functions on a
manifold with a volume form. This result is based on a lemma in the spirit of
geometric measure theory. We give some immediate applications to function
theory and the theory of quasi-states on surfaces with area forms.Comment: 8 page
Optimal bounds for ancient caloric functions
For any manifold with polynomial volume growth, we show: The dimension of the
space of ancient caloric functions with polynomial growth is bounded by the
degree of growth times the dimension of harmonic functions with the same
growth. As a consequence, we get a sharp bound for the dimension of ancient
caloric functions on any space where Yau's 1974 conjecture about polynomial
growth harmonic functions holds.Comment: A stronger sharp dimension bound is added which is an equality on
Euclidean space. To appear in Duke Math. Journa
Generalized trace and modified dimension functions on ribbon categories
In this paper we use topological techniques to construct generalized trace
and modified dimension functions on ideals in certain ribbon categories.
Examples of such ribbon categories naturally arise in representation theory
where the usual trace and dimension functions are zero, but these generalized
trace and modified dimension functions are non-zero. Such examples include
categories of finite dimensional modules of certain Lie algebras and finite
groups over a field of positive characteristic and categories of finite
dimensional modules of basic Lie superalgebras over the complex numbers. These
modified dimensions can be interpreted categorically and are closely related to
some basic notions from representation theory.Comment: 44 page
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