61 research outputs found
Symmetries of Nonlinear PDEs on Metric Graphs and Branched Networks
This Special Issue focuses on recent progress in a new area of mathematical physics and applied analysis, namely, on nonlinear partial differential equations on metric graphs and branched networks. Graphs represent a system of edges connected at one or more branching points (vertices). The connection rule determines the graph topology. When the edges can be assigned a length and the wave functions on the edges are defined in metric spaces, the graph is called a metric graph. Evolution equations on metric graphs have attracted much attention as effective tools for the modeling of particle and wave dynamics in branched structures and networks. Since branched structures and networks appear in different areas of contemporary physics with many applications in electronics, biology, material science, and nanotechnology, the development of effective modeling tools is important for the many practical problems arising in these areas. The list of important problems includes searches for standing waves, exploring of their properties (e.g., stability and asymptotic behavior), and scattering dynamics. This Special Issue is a representative sample of the works devoted to the solutions of these and other problems
Foundations of Stochastic Thermodynamics
Small systems in a thermodynamic medium --- like colloids in a suspension or
the molecular machinery in living cells --- are strongly affected by the
thermal fluctuations of their environment. Physicists model such systems by
means of stochastic processes. Stochastic Thermodynamics (ST) defines entropy
changes and other thermodynamic notions for individual realizations of such
processes. It applies to situations far from equilibrium and provides a unified
approach to stochastic fluctuation relations. Its predictions have been studied
and verified experimentally.
This thesis addresses the theoretical foundations of ST. Its focus is on the
following two aspects: (i) The stochastic nature of mesoscopic observations has
its origin in the molecular chaos on the microscopic level. Can one derive ST
from an underlying reversible deterministic dynamics? Can we interpret ST's
notions of entropy and entropy changes in a well-defined
information-theoretical framework? (ii) Markovian jump processes on finite
state spaces are common models for bio-chemical pathways. How does one quantify
and calculate fluctuations of physical observables in such models? What role
does the topology of the network of states play? How can we apply our abstract
results to the design of models for molecular motors?
The thesis concludes with an outlook on dissipation as information written to
unobserved degrees of freedom --- a perspective that yields a consistency
criterion between dynamical models formulated on various levels of description.Comment: Ph.D. Thesis, G\"ottingen 2014, Keywords: Stochastic Thermodynamics,
Entropy, Dissipation, Markov processes, Large Deviation Theory, Molecular
Motors, Kinesi
New Directions for Contact Integrators
Contact integrators are a family of geometric numerical schemes which
guarantee the conservation of the contact structure. In this work we review the
construction of both the variational and Hamiltonian versions of these methods.
We illustrate some of the advantages of geometric integration in the
dissipative setting by focusing on models inspired by recent studies in
celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282
Sublinear Computation Paradigm
This open access book gives an overview of cutting-edge work on a new paradigm called the âsublinear computation paradigm,â which was proposed in the large multiyear academic research project âFoundations of Innovative Algorithms for Big Data.â That project ran from October 2014 to March 2020, in Japan. To handle the unprecedented explosion of big data sets in research, industry, and other areas of society, there is an urgent need to develop novel methods and approaches for big data analysis. To meet this need, innovative changes in algorithm theory for big data are being pursued. For example, polynomial-time algorithms have thus far been regarded as âfast,â but if a quadratic-time algorithm is applied to a petabyte-scale or larger big data set, problems are encountered in terms of computational resources or running time. To deal with this critical computational and algorithmic bottleneck, linear, sublinear, and constant time algorithms are required. The sublinear computation paradigm is proposed here in order to support innovation in the big data era. A foundation of innovative algorithms has been created by developing computational procedures, data structures, and modelling techniques for big data. The project is organized into three teams that focus on sublinear algorithms, sublinear data structures, and sublinear modelling. The work has provided high-level academic research results of strong computational and algorithmic interest, which are presented in this book. The book consists of five parts: Part I, which consists of a single chapter on the concept of the sublinear computation paradigm; Parts II, III, and IV review results on sublinear algorithms, sublinear data structures, and sublinear modelling, respectively; Part V presents application results. The information presented here will inspire the researchers who work in the field of modern algorithms
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Superprobability on Graphs
The classical random walk isomorphism theorems relate the local times of a continuous-time random walk to the square of a Gaussian free field. The Gaussian free field is a spin system (or sigma model) that takes values in Euclidean space; in this work, we generalise the classical isomorphism theorems to spin systems taking values in hyperbolic and spherical geometries. The corresponding random walks are no longer Markovian: they are the vertex-reinforced and vertex-diminished jump processes. We also investigate supersymmetric versions of these formulas, which give exact random walk representations.
The proofs are based on exploiting the continuous symmetries of the corresponding spin systems. The classical isomorphism theorems use the translation symmetry of Euclidean space, while in hyperbolic and spherical geometries the relevant symmetries are Lorentz boosts and rotations, respectively. These very short proofs are new even in the Euclidean case.
To illustrate the utility of these new isomorphism theorems, we present several applications. These include simple proofs of exponential decay for spin system correlations, exact formulas for the resolvents of the joint processes of random walks together with their local times, and a new derivation of the SabotâTarrĂšs magic formula for the limiting local time of the vertex-reinforced jump process.
The second ingredient is a new MerminâWagner theorem for hyperbolic sigma models. This result is of intrinsic interest for the sigma models, and together with the aforementioned isomorphism theorems, implies our main theorem on the VRJP, namely, that it is recurrent in two dimensions for any translation invariant finite-range initial jump rates.
We also use supersymmetric hyperbolic sigma models to study the arboreal gas. This is a model of unrooted random forests on a graph, where the probability of a forest with edges is multiplicatively weighted by a parameter . In simple terms, it can be defined to be Bernoulli bond percolation with parameter conditioned to be acyclic, or as the limit with of the random cluster model.
It is known that on the complete graph with there is a phase transition similar to that of the ErdĆs--RĂ©nyi random graph: a giant tree percolates for and all trees have bounded size for . This result is again a consequence of our hyperbolic MerminâWagner theorem, and is used in conjunction with a version of the principle of dimensional reduction. To further illustrate our methods, we also give a spin-theoretic proof of the phase transition on the complete graph.University of Sydney, Cambridge Trus
PASSIVE THERMAL CONTROL SYSTEMS FOR SPACE INSTRUMENTS MAKING â SCIENTIFIC BACKGROUND, QUALIFICATION, EXPLOITATION IN SPACE
Passive thermal control systems (TCS) are one of obligatory system of any space
mission, used as on large spacecraft and microsatellites Supporting of required temperature
range for space instruments is supported by rational design of TCS with optimal choice of
main thermal control components such as multilayer insulation, optical coatings, heat
conductive elements, heat insulation supports, thermal conductive gaskets, radiating surfaces
and other elements. New ideology in TCS design has come after appearance of new element
â heat pipe(s) which is a super heat conductive thermal conductor with constant or variable
thermal properties
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
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