Format: Hardback, 220 pages, height x width: 235x155 mm, XX, 220 p., 1 Hardback
Series: Mathematical Physics Studies
Pub. Date: 03-Jul-2024
ISBN-13: 9789819726233
This book provides a systematic study of spectral and scattering theory for many-body Schrodinger operators at
two-cluster thresholds. While the two-body problem (reduced after separation of the centre of mass motion to
a one-body problem at zero energy) is a well-studied subject, the literature on many-body threshold problems
is sparse. However, the authors analysis covers for example the system of three particles interacting by Coulomb
potentials and restricted to a small energy region to the right of a fixed nonzero two-body eigenvalue. In general,
the authors address the question: How do scattering quantities for the many-body atomic and molecular models
behave within the limit when the total energy approaches a fixed two-cluster threshold? This includes mapping
properties and singularities of the limiting scattering matrix, asymptotics of the total scattering cross section,
and absence of transmission from one channel to another in the small inter-cluster kinetic energy region.
The authors principal tools are the FeshbachGrushin dimension reduction method and spectral analysis based
on a certain Mourre estimate. Additional topics of independent interest are the limiting absorption principle,
micro-local resolvent estimates, Rellich- and Sommerfeld-type theorems and asymptotics of the limiting
resolvents at thresholds. The mathematical physics field under study is very rich, and there are many open
problems, several of them stated explicitly in the book for the interested reader.
Introduction.- Many-Body Schrodinger Operators, Conditions and
Notation.- Reduction to a One-Body Problem.- Spectral Analysis of H0 near 0.-
Rellich-Type Theorems.- Resolvent Asymptotics near a Two-Cluster Threshold.-
Elastic Scattering at a Threshold.- Non-Transmission at a Threshold for
Physical Models.- Threshold Behaviour of Cross-Sections in AtomIon
Scattering.
Format: Hardback, 800 pages, height x width: 235x155 mm, 157 Illustrations,
color; Approx. 800 p. 157 illus. in color., 1 Hardback
Pub. Date: 25-Jun-2024
ISBN-13: 9783031595776
This updated/augmented second edition retains it class-tested content and pedagogy as a core text for graduate
courses in advanced fluid mechanics and applied science. The new edition adds revised sections, clarification,
problems, and chapter extensions including a rewritten section on Schauder bases for turbulent pipe flow, coverage
of Cantwell’s mixing length closure for turbulent pipe flow, and a section on the variational Hessian.
Consisting of two parts, the first provides an introduction and general theory of fully developed turbulence,
where treatment of turbulence is based on the linear functional equation derived by E. Hopf governing the
characteristic functional that determines the statistical properties of a turbulent flow. In this section,
Professor Kollmann explains how the theory is built on divergence free Schauder bases for the phase space of
the turbulent flow and the space of argument vector fields for the characteristic functional. The second segment,
presented over subsequent chapters, is devoted to mapping methods, homogeneous turbulence based upon the
hypotheses of Kolmogorov and Onsager, intermittency, structural features of turbulent shear flows and their recognition.
Introduction.- Navier-Stokes equations.- Basic properties of turbulent flows.- Flow domains and bases.
- Phase and test function spaces.- Probability measure and characteristic functional.- Functional differential equations.
- Characteristic functionals for incompressible turbulent flows.- Fdes for the characteristic functionals.
- Solution of Hopf type equations in the spatial description.- The role of the pressure .
- Properties and construction of Mappings.- M(): Single scalar in homogeneous turbulence.- M(N): Mappings for velocity
-scalar and position-scalar Pdfs.- Integral transforms and spectra.- Intermittency.
- Equilibrium theory of Kolmogorov and Onsager.- Homogeneous turbulence.- Length and time scales.
- The structure of turbulent ows.- Wall-bounded turbulent ows.
- The limit of in_nite Reynolds number for incompressible uids.- Appendix A: Mathematical tools.
- Appendix B: Example for a measure on a ball in Hilbert space.- Appendix C: Scalar and vector bases for periodic pipe ow.
- Modi_ed Jacobi polynomials Pa;b.- n (r).- Orthonormalisation of the modi_ed polynomials Pa;b.- n (r).
- Test function space Np: Scalar _elds.- (i) Bases for the test function space Np.- Function spaces: Vector _elds.
- (i) Construction of a general vector basis.- (ii) Construction of a solenoidal vector basis.- Gram
-Schmidt orthonormalisation.- Appendix D: Green's function for periodic pipe ow.- .- Leray version of the Navier
-Stokes pdes.- Appendix E: Semi-empirical treatment of simple wall-bounded ows.- Appendix F: Solutions to problems.
- Bibliography.
Format: Hardback, 200 pages, height x width: 235x155 mm, 5 Illustrations, black and white;
Approx. 200 p. 5 illus., 1 Hardback
Series: Probability Theory and Stochastic Modelling 106
Pub. Date: 19-Sep-2024
ISBN-13: 9783031579226
This book presents fundamental relations between random walks on graphs and field theories of mathematical physics.
Such relations have been explored for several decades and remain a rapidly developing research area in probability
theory.
The main objects of study include Markov loops, spanning forests, random holonomies, and covers, and the purpose
of the book is to investigate their relations to Bose fields, Fermi fields, and gauge fields. The book starts with
a review of some basic notions of Markovian potential theory in the simple context of a finite or countable graph,
followed by several chapters dedicated to the study of loop ensembles and related statistical physical models.
Then, spanning trees and Fermi fields are introduced and related to loop ensembles. Next, the focus turns to
topological properties of loops and graphs, with the introduction of connections on a graph, loop holonomies,
and YangMills measure. Among the main results presented is an intertwining relation between merge-and-split
generators on loop ensembles and Casimir operators on connections, and the key reflection positivity property
for the fields under consideration.
Aimed at researchers and graduate students in probability and mathematical physics, this concise monograph is
essentially self-contained. Familiarity with basic notions of probability, Poisson point processes, and discrete
Markov chains are assumed of the reader.
1 Markov Chains and Potential Theory on Graphs.- 2 Loop Measures.- 3 Decompositions, Traces and Excursions.-
4 Occupation Fields.- 5 Primitive Loops, Loop Clusters, and Loop Percolation.- 6 The Gaussian Free Field.-
7 Networks, Ising Model, Flows, and Configurations.- 8 Loop Erasure, Spanning Trees and Combinatorial Maps.-
9 Fock Spaces, Fermi Fields, and Applications.- 10 Groups and Covers.- 11 Holonomies and Gauge Fields.-
12 Reflection Positivity and Physical Space.
Format: Paperback / softback, 347 pages, height x width: 240x168 mm, 1 Illustrations,
color; 1 Illustrations, black and white; VIII, 347 p. 2 illus., 1 illus. in color., 1 Paperback / softback
Series: Frontiers in Elliptic and Parabolic Problems
Pub. Date: 07-Jul-2024
ISBN-13: 9783031583551
This monograph proposes a unified theory of the calculus of fractional and standard derivatives by means of an abstract
operator-theoretic approach. By highlighting the axiomatic properties shared by standard derivatives, Riemann-Liouville
and Caputo derivatives, the author introduces two new classes of objects. The first class concerns differential triplets
and differential quadruplets; the second concerns boundary restriction operators. Instances of boundary restriction
operators can be generalized fractional differential operators supplemented with homogeneous boundary conditions.
The analysis of these operators comprises:
The computation of adjoint operators; The definition of abstract boundary values; The solvability of equations
supplemented with inhomogeneous abstract linear boundary conditions; The analysis of fractional inhomogeneous
Dirichlet Problems.
As a result of this approach, two striking consequences are highlighted: Riemann-Liouville and Caputo operators appear
to differ only by their boundary conditions; and the boundary values of functions in the domain of fractional operators
are closely related to their kernel.
Unified Theory for Fractional and Entire Differential Operators will appeal to researchers in analysis and those who work
with fractional derivatives. It is mostly self-contained, covering the necessary background in functional analysis and
fractional calculus.
Introduction.- Background on Functional Analysis.- Background on Fractional Calculus.
- Differential Triplets on Hilbert Spaces.- Differential Quadruplets on Banach Spaces.
- Fractional Differential Triplets and Quadruplets on Lebesgue Spaces.- Endogenous Boundary Value Problems.
- Abstract and Fractional Laplace Operators.
Format: Hardback, 260 pages, height x width: 235x155 mm, 40 Illustrations,
black and white; X, 260 p. 50 illus., 10 illus. in color., 1 Hardback
Series: Mathematical Physics Studies
Pub. Date: 06-Jul-2024
ISBN-13: 9783031591198
This book provides an introduction to noncommutative geometry and presents a number of its recent applications
to particle physics. In the first part, we introduce the main concepts and techniques by studying finite noncommutative
spaces, providing a light approach to noncommutative geometry. We then proceed with the general framework by
defining and analyzing noncommutative spin manifolds and deriving some main results on them, such as the local
index formula. In the second part, we show how noncommutative spin manifolds naturally give rise to gauge theories,
applying this principle to specific examples. We subsequently geometrically derive abelian and non-abelian
Yang-Mills gauge theories, and eventually the full Standard Model of particle physics, and conclude by explaining
how noncommutative geometry might indicate how to proceed beyond the Standard Model.
The second edition of the book contains numerous additional sections and updates. More examples of noncommutative
manifolds have been added to the first part to better illustrate the concept of a noncommutative spin manifold and to
showcase some of the key results in the field, such as the local index formula. The second part now includes
the complete noncommutative geometric description of particle physics models beyond the Standard Model.
This addition is particularly significant given the developments and discoveries at the Large Hadron Collider at CERN
over the last few years. Additionally, a chapter on the recent progress in formulating noncommutative quantum theory
has been included.
The book is intended for graduate students in mathematics/theoretical physics who are new to the field of
noncommutative geometry, as well as for researchers in mathematics/theoretical physics with an interest in
the physical applications of noncommutative geometry.
Finite noncommutative spaces.- Finite real noncommutative spaces.- Noncommutative Riemannian spin manifolds.
- The local index formula in noncommutative geometry.- Gauge theories from noncommutative manifolds.
- Spectral invariants.- Almost-commutative manifolds and gauge theories.
- The noncommutative geometry of electrodynamics.- The noncommutative geometry of Yang-Mills fields.
- The noncommutative geometry of the Standard Model.- Phenomenology of the noncommutative Standard Model.
- Towards a quantum theory.
Format: Paperback / softback, 340 pages, height x width: 235x155 mm, Approx. 500 p., 1 Paperback
Series: UNITEXT 161
Pub. Date: 05-Sep-2024
ISBN-13: 9783031546877
This book offers accessible probabilistic modelling of relevant financial problems. It is divided into two parts.
The first part (cookbook) is written by emphasizing the key definitions and theorems without burden too much
the reader with unnecessary technical details. Here a first kind of target audience are graduate students in
Economics with no prior exposition to probability theory (except undergraduate courses in Applied Statistics)
which are provided by a self-contained account of probabilistic modelling mainly applied to finance.
The fundamental concepts of random variable/vector and probability distributions are introduced beforehand with
respect to the usual treatment of this subject in standard probability textbook, trying to strike a balance between
precise mathematical definitions and their applied knowledge. All the analytic tools developed are illustrated
through examples of probability distributions of future stock prices, returns and profit and loss, together with their
main characteristics such as moments, moment generating and characteristic functions, location-scale families,
quantiles. The extension to the multivariate case for fixed time horizons is presented, together with the fundamentals
of stochastic processes both in discrete and continuous time as candidate models for asset prices and return dynamics.
Convergence concepts are presented as applied to the problem of point estimation of means, variances, correlation
coefficients and risk measures. Short sections on risk and copula functions, credit risk and extreme value theory further
illustrate the potential application of probability models to financial problems. The second part of the book can be
accessed by those students with more mathematical preparation. It presents all the relevant proofs of results which
are only stated in the first part and some advanced exercises with complete solutions.
Memoire 180 - 2024
148 pages
For a simple real Lie group G with Heisenberg parabolic subgroup P, we study the corresponding degenerate
principal series representations. For a certain induction parameter the kernel of the conformally invariant
system of second order differential operators constructed by Barchini, Kable and Zierau is a subrepresentation
which turns out to be the minimal representation. To study this subrepresentation, we take the Heisenberg group
Fourier transform in the non-compact picture and show that it yields a new realization of the minimal representation
on a space of L2-functions. The Lie algebra action is given by differential operators of order ?3 and we find explicit
formulas for the functions constituting the lowest K-type.
These L2-models were previously known for the groups SO(n,n), E6(6), E7(7) and E8(8) by Kazhdan and Savin,
for the group G2(2) by Gelfand, and for the group SL?(3,?) by Torasso, using different methods. Our new approach
provides a uniform and systematic treatment of these cases and also constructs new L2-models for E6(2), E7(?5)
and E8(?24) for which the minimal representation is a continuation of the quaternionic discrete series, and for
the groups SO?(p,q) with either p?q=3 or p,q?4 and p+q
even.
As a byproduct of our construction, we find an explicit formula for the group action of a non-trivial
Weyl group element that, together with the simple action of a parabolic subgroup, generates G.