61,779 research outputs found
T-homotopy and refinement of observation (II) : Adding new T-homotopy equivalences
This paper is the second part of a series of papers about a new notion of
T-homotopy of flows. It is proved that the old definition of T-homotopy
equivalence does not allow the identification of the directed segment with the
3-dimensional cube. This contradicts a paradigm of dihomotopy theory. A new
definition of T-homotopy equivalence is proposed, following the intuition of
refinement of observation. And it is proved that up to weak S-homotopy, a old
T-homotopy equivalence is a new T-homotopy equivalence. The left-properness of
the weak S-homotopy model category of flows is also established in this second
part. The latter fact is used several times in the next papers of this series.Comment: 20 pages, 3 figure
A nonlinear Kolmogorov equation for stochastic functional delay differential equations with jumps
We consider a stochastic functional delay differential equation, namely an
equation whose evolution depends on its past history as well as on its present
state, driven by a pure diffusive component plus a pure jump Poisson
compensated measure. We lift the problem in the infinite dimensional space of
square integrable Lebesgue functions in order to show that its solution is an
valued Markov process whose uniqueness can be shown under standard
assumptions of locally Lipschitzianity and linear growth for the coefficients.
Coupling the aforementioned equation with a standard backward differential
equation, and deriving some ad hoc results concerning the Malliavin derivative
for systems with memory, we are able to derive a non--linear Feynman--Kac
representation theorem under mild assumptions of differentiability
Towards a Church-Turing-Thesis for Infinitary Computations
We consider the question whether there is an infinitary analogue of the
Church-Turing-thesis. To this end, we argue that there is an intuitive notion
of transfinite computability and build a canonical model, called Idealized
Agent Machines (s) of this which will turn out to be equivalent in
strength to the Ordinal Turing Machines defined by P. Koepke
Broadcasting Automata and Patterns on Z^2
The Broadcasting Automata model draws inspiration from a variety of sources
such as Ad-Hoc radio networks, cellular automata, neighbourhood se- quences and
nature, employing many of the same pattern forming methods that can be seen in
the superposition of waves and resonance. Algorithms for broad- casting
automata model are in the same vain as those encountered in distributed
algorithms using a simple notion of waves, messages passed from automata to au-
tomata throughout the topology, to construct computations. The waves generated
by activating processes in a digital environment can be used for designing a
vari- ety of wave algorithms. In this chapter we aim to study the geometrical
shapes of informational waves on integer grid generated in broadcasting
automata model as well as their potential use for metric approximation in a
discrete space. An explo- ration of the ability to vary the broadcasting radius
of each node leads to results of categorisations of digital discs, their form,
composition, encodings and gener- ation. Results pertaining to the nodal
patterns generated by arbitrary transmission radii on the plane are explored
with a connection to broadcasting sequences and ap- proximation of discrete
metrics of which results are given for the approximation of astroids, a
previously unachievable concave metric, through a novel application of the
aggregation of waves via a number of explored functions
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