Related papers: From the discrete to the continuous - towards a cy…
The emergence of macroscopic variables can be effected through {\it coarse graining}. Despite practical and fundamental benefits conveyed by this partitioning of state space, the apparently subjective nature of the selection of coarse…
Lumping a Markov process introduces a coarser level of description that is useful in many contexts and applications. The dynamics on the coarse grained states is often approximated by its Markovian component. In this letter we derive…
We first discuss a framework for discrete quantum processes (DQP). It is shown that the set of q-probability operators is convex and its set of extreme elements is found. The property of consistency for a DQP is studied and the quadratic…
The interaction between discrete and continuous mathematics lies at the heart of many fundamental problems in applied mathematics and computational sciences. In this paper we discuss the problem of discretizing vector-valued functions…
We consider fully discrete numerical approximations for axisymmetric Willmore flow that are unconditionally stable and work reliably without remeshing. We restrict our attention to surfaces without boundary, but allow for spontaneous…
The purpose of physics is to describe nature from elementary particles all the way up to cosmological objects like cluster of galaxies and black holes. Although a unified description for all this spectrum of events is desirable, this would…
The renormalization group has proven to be a very powerful tool in physics for treating systems with many length scales. Here we show how it can be adapted to provide a new class of algorithms for discrete optimization. The heart of our…
We propose and illustrate an approach to coarse-graining the dynamics of evolving networks (networks whose connectivity changes dynamically). The approach is based on the equation-free framework: short bursts of detailed network evolution…
As a complementary tool to laboratory experiments, discrete numerical simulation, applied to granular materials, provides valuable information on the grain and contact scale microstructure, thereby enabling one to better understand the…
We discuss the nonlinear dynamics and fluctuations of interfaces with bending rigidity under the competing attractions of two walls with arbitrary permeabilities. This system mimics the dynamics of confined membranes. We use a two-dimension…
I review the decoherent (or consistent) histories approach to quantum mechanics, due to Griffiths, to Gell-Mann and Hartle, and to Omnes. This is an approach to standard quantum theory specifically designed to apply to genuinely closed…
Discontinuous dynamical systems with grazing solutions are discussed. The group property, continuation of solutions, continuity and smoothness of motions are thoroughly analyzed. A variational system around a grazing solution which depends…
In the last 20 years, loop quantum gravity, a background independent approach to unify general relativity and quantum mechanics, has been widely investigated. The aim of loop quantum gravity is to construct a mathematically rigorous,…
Simplicity constraints play a crucial role in the construction of spin foam models, yet their effective behaviour on larger scales is scarcely explored. In this article we introduce intertwiner and spin net models for the quantum group…
The loss of coherence is one of the main obstacles for the implementation of quantum information processing. The efficiency of dynamical decoupling schemes, which have been introduced to address this problem, is limited itself by the…
A variational coarse-graining framework for heterogeneous media is developed that allows for a seamless transition from the traditional static scenario to a arbitrary loading conditions, including inertia effects and body forces. The…
How does one generalize differential geometric constructs such as curvature of a manifold to the discrete world of graphs and other combinatorial structures? This problem carries significant importance for analyzing models of discrete…
Thermodynamic irreversibility is a fundamental concept in statistical physics, yet its experimental measurement remains challenging, especially for complex systems. We introduce a novel random coarse-graining framework to identify…
The Marginally Rigid State is a candidate paradigm for what makes granular material a state of matter distinct from both liquid and solid. Coordination number is identified as a discriminating characteristic, and for rough-surfaced…
In loop quantum gravity (LQG), states of the gravitational field are represented by labeled graphs called spin networks. Their dynamics can be described by a Hamiltonian constraint, { which acts on the spin network states modifying both…