2,212 research outputs found
Linear operators with compact supports, probability measures and Milyutin maps
The notion of a regular operator with compact supports between function
spaces is introduced. On that base we obtain a characterization of absolute
extensors for zero-dimensional spaces in terms of regular extension operators
having compact supports. Milyutin maps are also considered and it is
established that some topological properties, like paracompactness,
metrizability and k-metrizability, are preserved under Milyutin maps.Comment: 26 page
Metrizability and pattern recognition
Metrizability and pattern recognitio
Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations
We use Fr\"olicher-Nijenhuis theory to obtain global Helmholtz conditions,
expressed in terms of a semi-basic 1-form, that characterize when a semispray
is locally Lagrangian. We also discuss the relation between these Helmholtz
conditions and their classic formulation written using a multiplier matrix.
When the semi-basic 1-form is 1-homogeneous (0-homogeneous) we show that two
(one) of the Helmholtz conditions are consequences of the other ones. These two
special cases correspond to two inverse problems in the calculus of variation:
Finsler metrizability for a spray, and projective metrizability for a spray
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