Series: Grundlehren der mathematischen Wissenschaften, Vol. 344
2011, XII, 252 p.
Hardcover, ISBN 978-3-642-21146-1
Due: July 2011
Manifolds over complete nonarchimedean fields together with notions like tangent spaces and vector fields form a convenient geometric language to express the basic formalism of p-adic analysis. The volume starts with a self-contained and detailed introduction to this language. This includes the discussion of spaces of locally analytic functions as topological vector spaces, important for applications in representation theory. The author then sets up the analytic foundations of the theory of p-adic Lie groups and develops the relation between p-adic Lie groups and their Lie algebras. The second part of the book contains, for the first time in a textbook, a detailed exposition of Lazard's algebraic approach to compact p-adic Lie groups, via his notion of a p-valuation, together with its application to the structure of completed group rings.
Introduction.- Part A: p-Adic Analysis and Lie Groups.- I.Foundations.- I.1.Ultrametric Spaces.- I.2.Nonarchimedean Fields.- I.3.Convergent Series.- I.4.Differentiability.- I.5.Power Series.- I.6.Locally Analytic Functions.- II.Manifolds.- II.7.Charts and Atlases.- II.8.Manifolds.- II.9.The Tangent Space.- II.10.The Topological Vector Space C^an(M,E), part 1.- II.11 Locally Convex K-Vector Spaces.- II.12 The Topological Vector Space C^an(M,E), part 2.- III.Lie Groups.- III.13.Definitions and Foundations.- III.14.The Universal Enveloping Algebra.- III.15.The Concept of Free Algebras.- III.16.The Campbell-Hausdorff Formula.- III.17.The Convergence of the Hausdorff Series.- III.18.Formal Group Laws.- Part B:The Algebraic Theory of p-Adic Lie Groups.- IV.Preliminaries.- IV.19.Completed Group Rings.- IV.20.The Example of the Group Z^d_p.- IV.21.Continuous Distributions.- IV.22.Appendix: Pseudocompact Rings.- V.p-Valued Pro-p-Groups.- V.23.p-Valuations.- V.24.The free Group on two Generators.- V.25.The Operator P.- V.26.Finite Rank Pro-p-Groups.- V.27.Compact p-Adic Lie Groups.- VI.Completed Group Rings of p-Valued Groups.- VI.28.The Ring Filtration.- VI.29.Analyticity.- VI.30.Saturation.- VII.The Lie Algebra.- VII.31.A Normed Lie Algebra.- VII.32.The Hausdorff Series.- VII.33.Rational p-Valuations and Applications.- VII.34.Coordinates of the First and of the Second Kind.- References.- Index.
Series: Lecture Notes in Mathematics, Vol. 2023
2011, 200 p.
Softcover, ISBN 978-3-642-21334-2
Due: July 2011
The theory of linear damped oscillations was originally developed more than hundred years ago and is still of vital research interest to engineers, mathematicians and physicists alike. This theory plays a central role in explaining the stability of mechanical structures in civil engineering, but it also has applications in other fields such as electrical network systems and quantum mechanics.
This volume gives an introduction to linear finite dimensional damped systems as they are viewed by an applied mathematician. After a short overview of the physical principles leading to the linear system model, a largely self-contained mathematical theory for this model is presented. This includes the geometry of the underlying indefinite metric space, spectral theory of J-symmetric matrices and the associated quadratic eigenvalue problem. Particular attention is paid to the sensitivity issues which influence numerical computations. Finally, several recent research developments are included, e.g. Lyapunov stability and the perturbation of the time evolution.
1 The model.- 2 Simultaneous diagonalisation (Modal damping).- 3 Phase space.- 4 The singular mass case.- 5 "Indefinite metric".- 6 Matrices and indefinite scalar products.- 7 Oblique projections.- 8 J-orthogonal projections.- 9 Spectral properties and reduction of J-Hermitian matrices.- 10 Definite spectra.- 11 General Hermitian matrix pairs.- 12 Spectral decomposition of a general J-Hermitian matrix.- 13 The matrix exponential.- 14 The quadratic eigenvalue problem.- 15 Simple eigenvalue inclusions.- 16 Spectral shift.- 17 Resonances and resolvents.- 18 Well-posedness .- 19 Modal approximation.- 20 Modal approximation and overdampedness.- 21 Passive control.- 22 Perturbing matrix exponential.- 23 Notes and remarks.
Series: Lecture Notes in Mathematics, Vol. 2026
Subseries: Ecole d'Ete de Probabilites de Saint-Flour
2012, VIII, 134 p. 11 illus.
Softcover, ISBN 978-3-642-21215-4
Due: July 2011
The purpose of these notes is to explore some simple relations between Markovian path and loop measures, the Poissonian ensembles of loops they determine, their occupation fields, uniform spanning trees, determinants, and Gaussian Markov fields such as the free field. These relations are first studied in complete generality for the finite discrete setting, then partly generalized to specific examples in infinite and continuous spaces.
1 Symmetric Markov processes on finite spaces.- 2 Loop measures.- 3 Geodesic loops.- 4 Poisson process of loops.- 5 The Gaussian free field.- 6 Energy variation and representations.- 7 Decompositions.- 8 Loop erasure and spanning trees.- 9 Reflection positivity.- 10 The case of general symmetric Markov processes.
Series: SpringerBriefs in Mathematics, Vol. 1
2011, XV, 102 p. 6 illus.
Softcover, ISBN 978-1-4419-9907-8
Due: September 29, 2011
Content Level ≫ Research
Keywords ≫ foliation of codimension one - geometric flow - integral formula - mean curvatures - variation formula
Related subjects ≫ Dynamical Systems & Differential Equations - Geometry & Topology
1. Integral Formulae (Introduction, Preliminaries, Integral Formulae for Codimension-one foliations).-2.Variation Formulae (Introduction, Auxiliary results, Variations of extrinsic geometric quantities, Variations of general functional, Variations of particular functional, Applications and examples).-3. Extrinsic Geometric Flows (Introduction, The systems of PDE’s related to EGF, Auxiliary results, Existence and uniqueness results, A solution to general case, Global existence of EGF, Variation formulae for EGF, Extrinsic geometric solitons, Applications and examples).- References.
Series: Pseudo-Differential Operators, Vol. 8
2011, 325 p.
Softcover, ISBN 978-3-0348-0165-2
Due: August 2011
Pseudodifferential analysis, introduced in this book in a way adapted to the needs of number theorists, relates automorphic function theory in the hyperbolic half-plane Π to automorphic distribution theory in the plane. Spectral-theoretic questions are discussed in one or the other environment: in the latter one, the problem of decomposing automorphic functions in Π according to the spectral decomposition of the modular Laplacian gives way to the simpler one of decomposing automorphic distributions in R2 into homogeneous components. The Poincare summation process, which consists in building automorphic distributions as series of g-transforms, for g E SL(2;Z), of some initial function, say in S(R2), is analyzed in detail. On Π, a large class of new automorphic functions or measures is built in the same way: one of its features lies in an interpretation, as a spectral density, of the restriction of the zeta function to any line within the critical strip. Several new ideas are far from being pushed to the end, and call for many possible generalizations, only hinted at.
Introduction.- The Weyl calculus.- The Radon transformation and applications.- Automorphic functions and automorphic distributions.- A class of Poincare series.- Spectral decomposition of the Poincare summation process.- The totally radial Weyl calculus and arithmetic.- Should one generalize the Weyl calculus to an adelic setting?.- Index of notation.- Subject Index.- Bibliography.
Series: Progress in Mathematics, Vol. 292
2011, XXXVI, 372 p. 5 illus.
Hardcover, ISBN 978-0-8176-8243-9
Due: August 15, 2011
Hans Duistermaat, an influential geometer-analyst, made substantial contributions to the theory of ordinary and partial differential equations, symplectic, differential, and algebraic geometry, minimal surfaces, semisimple Lie groups, mechanics, mathematical physics, and related fields. Written in his honor, the invited and refereed articles in this volume contain important new results as well as surveys in some of these areas, clearly demonstrating the impact of Duistermaat's research and, in addition, exhibiting interrelationships among many of the topics.
The well-known contributors to this text cover a wide range of topics: semi-classical inverse problems; eigenvalue distributions; symplectic inverse spectral theory for pseudodifferential operators; solvability for systems of pseudodifferential operators; the Darboux process and a noncommutative bispectral problem; a proof of the Atiyah-Weinstein conjecture on the index of Fourier integral operators and the relative index of CR structures; relations between index theory and localization formulas of Duistermaat?Heckman; non-Abelian localization; symplectic implosion and nonreductive quotients; conjugation spaces; and Hamiltonian geometry. Also included are several articles in memory of Hans Duistermaat.
Preface.- About J.J. Duistermaat.- Hans Duistermaat (1942-2010).- Recollections of Hans Duistermaat.- Recollections of Hans Duistermaat.- Recollections of Hans Duistermaat.- Classical Mechanics and Hans Duistermaat.- Duistermaat-Heckman formulas and index theory.- Asymptotic equivariant index of Toeplitz operators and relative index of CR structures.- A semi-classical inverse problem I: Taylor expansions.- A semi-classical inverse problem II: reconstruction of the potential.- On the solvability of systems of pseudodifferential operators.- The Darboux process and a noncommutative bispectral problem: some explorations and challenges.- Conjugation spaces and edges of compatible torus actions.- Non-Abelian localization for U(1) Chern-Simons theory.- Symplectic implosion and non-reductive quotients.- Quantization of q-Hamiltonian SU(2)-spaces.- Wall-crossing formulas in Hamiltonian geometry.- Eigenvalue distributions and Weyl laws for semi-classical non-self-adjoint operators in 2 dimensions.- Symplectic inverse spectral theory for pseudodifferential operators.