104 research outputs found
Towards a theory of arithmetic degrees
The aim of this paper is to start a systematic investigation of the
arithmetic degree of projective schemes as introduced by D. Bayer and D.
Mumford. One main theme concerns itself with the behaviour of this arithmetic
degree under hypersurface sections. The notion of arithmetic degree involves
the new concept of length-multiplicity of embedded primary ideals. Therefore it
is much harder to control the arithmetic degree under a hypersurface section
than in the case for the classical degree theory. Nevertheless it has important
and interesting applications. We describe such applications to the
Castelnuovo-Mumford regularity and to Bezout-type theorems.Comment: LaTeX, 14 page
Stochastic Perturbations of Periodic Orbits with Sliding
Vector fields that are discontinuous on codimension-one surfaces are known as
Filippov systems and can have attracting periodic orbits involving segments
that are contained on a discontinuity surface of the vector field. In this
paper we consider the addition of small noise to a general Filippov system and
study the resulting stochastic dynamics near such a periodic orbit. Since a
straight-forward asymptotic expansion in terms of the noise amplitude is not
possible due to the presence of discontinuity surfaces, in order to
quantitatively determine the basic statistical properties of the dynamics, we
treat different parts of the periodic orbit separately. Dynamics distant from
discontinuity surfaces is analyzed by the use of a series expansion of the
transitional probability density function. Stochastically perturbed sliding
motion is analyzed through stochastic averaging methods. The influence of noise
on points at which the periodic orbit escapes a discontinuity surface is
determined by zooming into the transition point. We combine the results to
quantitatively determine the effect of noise on the oscillation time for a
three-dimensional canonical model of relay control. For some parameter values
of this model, small noise induces a significantly large reduction in the
average oscillation time. By interpreting our results geometrically, we are
able to identify four features of the relay control system that contribute to
this phenomenon.Comment: 44 pages, 9 figures, submitted to: J Nonlin. Sc
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