|
Showing 1 - 5 of
5 matches in All Departments
This book presents recently obtained mathematical results on Gibbs
measures of the q-state Potts model on the integer lattice and on
Cayley trees. It also illustrates many applications of the Potts
model to real-world situations in biology, physics, financial
engineering, medicine, and sociology, as well as in some examples
of alloy behavior, cell sorting, flocking birds, flowing foams, and
image segmentation.Gibbs measure is one of the important measures
in various problems of probability theory and statistical
mechanics. It is a measure associated with the Hamiltonian of a
biological or physical system. Each Gibbs measure gives a state of
the system.The main problem for a given Hamiltonian on a countable
lattice is to describe all of its possible Gibbs measures. The
existence of some values of parameters at which the uniqueness of
Gibbs measure switches to non-uniqueness is interpreted as a phase
transition.This book informs the reader about what has been
(mathematically) done in the theory of Gibbs measures of the Potts
model and the numerous applications of the Potts model. The main
aim is to facilitate the readers (in mathematical biology,
statistical physics, applied mathematics, probability and measure
theory) to progress into an in-depth understanding by giving a
systematic review of the theory of Gibbs measures of the Potts
model and its applications.
'This book offers one of the few places where a collection of
results from the literature can be found ... The book has an
extensive bibliography ... It is very nice to have the compendium
of results that is presented here.'zbMATHA mathematical billiard is
a mechanical system consisting of a billiard ball on a table of any
form (which can be planar or even a multidimensional domain) but
without billiard pockets. The ball moves and its trajectory is
defined by the ball's initial position and its initial speed
vector. The ball's reflections from the boundary of the table are
assumed to have the property that the reflection and incidence
angles are the same. This book comprehensively presents known
results on the behavior of a trajectory of a billiard ball on a
planar table (having one of the following forms: circle, ellipse,
triangle, rectangle, polygon and some general convex domains). It
provides a systematic review of the theory of dynamical systems,
with a concise presentation of billiards in elementary mathematics
and simple billiards related to geometry and physics.The
description of these trajectories leads to the solution of various
questions in mathematics and mechanics: problems related to liquid
transfusion, lighting of mirror rooms, crushing of stones in a
kidney, collisions of gas particles, etc. The analysis of billiard
trajectories can involve methods of geometry, dynamical systems,
and ergodic theory, as well as methods of theoretical physics and
mechanics, which has applications in the fields of biology,
mathematics, medicine, and physics.
'This book offers one of the few places where a collection of
results from the literature can be found ... The book has an
extensive bibliography ... It is very nice to have the compendium
of results that is presented here.'zbMATHA mathematical billiard is
a mechanical system consisting of a billiard ball on a table of any
form (which can be planar or even a multidimensional domain) but
without billiard pockets. The ball moves and its trajectory is
defined by the ball's initial position and its initial speed
vector. The ball's reflections from the boundary of the table are
assumed to have the property that the reflection and incidence
angles are the same. This book comprehensively presents known
results on the behavior of a trajectory of a billiard ball on a
planar table (having one of the following forms: circle, ellipse,
triangle, rectangle, polygon and some general convex domains). It
provides a systematic review of the theory of dynamical systems,
with a concise presentation of billiards in elementary mathematics
and simple billiards related to geometry and physics.The
description of these trajectories leads to the solution of various
questions in mathematics and mechanics: problems related to liquid
transfusion, lighting of mirror rooms, crushing of stones in a
kidney, collisions of gas particles, etc. The analysis of billiard
trajectories can involve methods of geometry, dynamical systems,
and ergodic theory, as well as methods of theoretical physics and
mechanics, which has applications in the fields of biology,
mathematics, medicine, and physics.
The purpose of this book is to present systematically all known
mathematical results on Gibbs measures on Cayley trees (Bethe
lattices).The Gibbs measure is a probability measure, which has
been an important object in many problems of probability theory and
statistical mechanics. It is the measure associated with the
Hamiltonian of a physical system (a model) and generalizes the
notion of a canonical ensemble. More importantly, when the
Hamiltonian can be written as a sum of parts, the Gibbs measure has
the Markov property (a certain kind of statistical independence),
thus leading to its widespread appearance in many problems outside
of physics such as biology, Hopfield networks, Markov networks, and
Markov logic networks. Moreover, the Gibbs measure is the unique
measure that maximizes the entropy for a given expected energy.The
method used for the description of Gibbs measures on Cayley trees
is the method of Markov random field theory and recurrent equations
of this theory, but the modern theory of Gibbs measures on trees
uses new tools such as group theory, information flows on trees,
node-weighted random walks, contour methods on trees, and nonlinear
analysis. This book discusses all the mentioned methods, which were
developed recently.
A population is a summation of all the organisms of the same group
or species, which live in a particular geographical area, and have
the capability of interbreeding. The main mathematical problem for
a given population is to carefully examine the evolution (time
dependent dynamics) of the population. The mathematical methods
used in the study of this problem are based on probability theory,
stochastic processes, dynamical systems, nonlinear differential and
difference equations, and (non-)associative algebras.A state of a
population is a distribution of probabilities of the different
types of organisms in every generation. Type partition is called
differentiation (for example, sex differentiation which defines a
bisexual population). This book systematically describes the
recently developed theory of (bisexual) population, and mainly
contains results obtained since 2010.The book presents algebraic
and probabilistic approaches in the theory of population dynamics.
It also includes several dynamical systems of biological models
such as dynamics generated by Markov processes of cubic stochastic
matrices; dynamics of sex-linked population; dynamical systems
generated by a gonosomal evolution operator; dynamical system and
an evolution algebra of mosquito population; and ocean
ecosystems.The main aim of this book is to facilitate the reader's
in-depth understanding by giving a systematic review of the theory
of population dynamics which has wide applications in biology,
mathematics, medicine, and physics.
|
You may like...
Loot
Nadine Gordimer
Paperback
(2)
R398
R330
Discovery Miles 3 300
Loot
Nadine Gordimer
Paperback
(2)
R398
R330
Discovery Miles 3 300
Loot
Nadine Gordimer
Paperback
(2)
R398
R330
Discovery Miles 3 300
Gloria
Sam Smith
CD
R407
Discovery Miles 4 070
|