64,187 research outputs found
Geometric Numerical Integration (hybrid meeting)
The topics of the workshop
included interactions between geometric numerical integration and numerical partial differential equations;
geometric aspects of stochastic differential equations;
interaction with optimisation and machine learning;
new applications of geometric integration in physics;
problems of discrete geometry, integrability, and algebraic aspects
Renewal theorems for a class of processes with dependent interarrival times and applications in geometry
Renewal theorems are developed for point processes with interarrival times
, where is a stochastic
process with finite state space and is
a H\"older continuous function on a subset .
The theorems developed here unify and generalise the key renewal theorem for
discrete measures and Lalley's renewal theorem for counting measures in
symbolic dynamics. Moreover, they capture aspects of Markov renewal theory. The
new renewal theorems allow for direct applications to problems in fractal and
hyperbolic geometry; for instance, results on the Minkowski measurability of
self-conformal sets are deduced. Indeed, these geometric problems motivated the
development of the renewal theorems.Comment: 2 figure
Dynamical Evolution in Noncommutative Discrete Phase Space and the Derivation of Classical Kinetic Equations
By considering a lattice model of extended phase space, and using techniques
of noncommutative differential geometry, we are led to: (a) the conception of
vector fields as generators of motion and transition probability distributions
on the lattice; (b) the emergence of the time direction on the basis of the
encoding of probabilities in the lattice structure; (c) the general
prescription for the observables' evolution in analogy with classical dynamics.
We show that, in the limit of a continuous description, these results lead to
the time evolution of observables in terms of (the adjoint of) generalized
Fokker-Planck equations having: (1) a diffusion coefficient given by the limit
of the correlation matrix of the lattice coordinates with respect to the
probability distribution associated with the generator of motion; (2) a drift
term given by the microscopic average of the dynamical equations in the present
context. These results are applied to 1D and 2D problems. Specifically, we
derive: (I) The equations of diffusion, Smoluchowski and Fokker-Planck in
velocity space, thus indicating the way random walk models are incorporated in
the present context; (II) Kramers' equation, by further assuming that, motion
is deterministic in coordinate spaceComment: LaTeX2e, 40 pages, 1 Postscript figure, uses package epsfi
Imprint of quantum gravity in the dimension and fabric of spacetime
We here conjecture that two much-studied aspects of quantum gravity,
dimensional flow and spacetime fuzziness, might be deeply connected. We
illustrate the mechanism, providing first evidence in support of our
conjecture, by working within the framework of multifractional theories, whose
key assumption is an anomalous scaling of the spacetime dimension in the
ultraviolet and a slow change of the dimension in the infrared. This sole
ingredient is enough to produce a scale-dependent deformation of the
integration measure with also a fuzzy spacetime structure. We also compare the
multifractional correction to lengths with the types of Planckian uncertainty
for distance and time measurements that was reported in studies combining
quantum mechanics and general relativity heuristically. This allows us to fix
two free parameters of the theory and leads, in one of the scenarios we
contemplate, to a value of the ultraviolet dimension which had already found
support in other quantum-gravity analyses. We also formalize a picture such
that fuzziness originates from a fundamental discrete scale invariance at short
scales and corresponds to a stochastic spacetime geometry.Comment: 6 pages; v2: phenomenology section adde
Discrete Dynamical Systems: A Brief Survey
Dynamical system is a mathematical formalization for any fixed rule that is described in time dependent fashion. The time can be measured by either of the number systems - integers, real numbers, complex numbers. A discrete dynamical system is a dynamical system whose state evolves over a state space in discrete time steps according to a fixed rule. This brief survey paper is concerned with the part of the work done by José Sousa Ramos [2] and some of his research students. We present the general theory of discrete dynamical systems and present results from applications to geometry, graph theory and synchronization
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