47,206 research outputs found
Valadier-like formulas for the supremum function II: The compactly indexed case
We generalize and improve the original characterization given by Valadier
[20, Theorem 1] of the subdifferential of the pointwise supremum of convex
functions, involving the subdifferentials of the data functions at nearby
points. We remove the continuity assumption made in that work and obtain a
general formula for such a subdifferential. In particular, when the supremum is
continuous at some point of its domain, but not necessarily at the reference
point, we get a simpler version which gives rise to Valadier formula. Our
starting result is the characterization given in [10, Theorem 4], which uses
the epsilon-subdiferential at the reference point.Comment: 23 page
Valadier-like formulas for the supremum function I
We generalize and improve the original characterization given by Valadier
[18, Theorem 1] of the subdifferential of the pointwise supremum of convex
functions, involving the subdifferentials of the data functions at nearby
points. We remove the continuity assumption made in that work and obtain a
general formula for such a subdiferential. In particular, when the supremum is
continuous at some point of its domain, but not necessarily at the reference
point, we get a simpler version which gives rise to the Valadier formula. Our
starting result is the characterization given in [11, Theorem 4], which uses
the epsilon-subdifferential at the reference point.Comment: 27 page
Estimating a Signal In the Presence of an Unknown Background
We describe a method for fitting distributions to data which only requires
knowledge of the parametric form of either the signal or the background but not
both. The unknown distribution is fit using a non-parametric kernel density
estimator. The method returns parameter estimates as well as errors on those
estimates. Simulation studies show that these estimates are unbiased and that
the errors are correct
On algebraic classification of quasi-exactly solvable matrix models
We suggest a generalization of the Lie algebraic approach for constructing
quasi-exactly solvable one-dimensional Schroedinger equations which is due to
Shifman and Turbiner in order to include into consideration matrix models. This
generalization is based on representations of Lie algebras by first-order
matrix differential operators. We have classified inequivalent representations
of the Lie algebras of the dimension up to three by first-order matrix
differential operators in one variable. Next we describe invariant
finite-dimensional subspaces of the representation spaces of the one-,
two-dimensional Lie algebras and of the algebra sl(2,R). These results enable
constructing multi-parameter families of first- and second-order quasi-exactly
solvable models. In particular, we have obtained two classes of quasi-exactly
solvable matrix Schroedinger equations.Comment: LaTeX-file, 16 pages, submitted to J.Phys.A: Math.Ge
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