5,692 research outputs found
Semi-Markov approach to continuous time random walk limit processes
Continuous time random walks (CTRWs) are versatile models for anomalous
diffusion processes that have found widespread application in the quantitative
sciences. Their scaling limits are typically non-Markovian, and the computation
of their finite-dimensional distributions is an important open problem. This
paper develops a general semi-Markov theory for CTRW limit processes in
with infinitely many particle jumps (renewals) in finite time
intervals. The particle jumps and waiting times can be coupled and vary with
space and time. By augmenting the state space to include the scaling limits of
renewal times, a CTRW limit process can be embedded in a Markov process.
Explicit analytic expressions for the transition kernels of these Markov
processes are then derived, which allow the computation of all finite
dimensional distributions for CTRW limits. Two examples illustrate the proposed
method.Comment: Published in at http://dx.doi.org/10.1214/13-AOP905 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
-Schur functions and affine Schubert calculus
This book is an exposition of the current state of research of affine
Schubert calculus and -Schur functions. This text is based on a series of
lectures given at a workshop titled "Affine Schubert Calculus" that took place
in July 2010 at the Fields Institute in Toronto, Ontario. The story of this
research is told in three parts: 1. Primer on -Schur Functions 2. Stanley
symmetric functions and Peterson algebras 3. Affine Schubert calculusComment: 213 pages; conference website:
http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/, updates
and corrections since v1. This material is based upon work supported by the
National Science Foundation under Grant No. DMS-065264
Reasoning about goal-directed real-time teleo-reactive programs
The teleo-reactive programming model is a high-level approach to developing real-time systems that supports hierarchical composition and durative actions. The model is different from frameworks such as action systems, timed automata and TLA+, and allows programs to be more compact and descriptive of their intended behaviour. Teleo-reactive programs are particularly useful for implementing controllers for autonomous agents that must react robustly to their dynamically changing environments. In this paper, we develop a real-time logic that is based on Duration Calculus and use this logic to formalise the semantics of teleo-reactive programs. We develop rely/guarantee rules that facilitate reasoning about a program and its environment in a compositional manner. We present several theorems for simplifying proofs of teleo-reactive programs and present a partially mechanised method for proving progress properties of goal-directed agents. © 2013 British Computer Society
A Meshfree Generalized Finite Difference Method for Surface PDEs
In this paper, we propose a novel meshfree Generalized Finite Difference
Method (GFDM) approach to discretize PDEs defined on manifolds. Derivative
approximations for the same are done directly on the tangent space, in a manner
that mimics the procedure followed in volume-based meshfree GFDMs. As a result,
the proposed method not only does not require a mesh, it also does not require
an explicit reconstruction of the manifold. In contrast to existing methods, it
avoids the complexities of dealing with a manifold metric, while also avoiding
the need to solve a PDE in the embedding space. A major advantage of this
method is that all developments in usual volume-based numerical methods can be
directly ported over to surfaces using this framework. We propose
discretizations of the surface gradient operator, the surface Laplacian and
surface Diffusion operators. Possibilities to deal with anisotropic and
discontinous surface properties (with large jumps) are also introduced, and a
few practical applications are presented
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