V. E. Nazaikinskii

On the distribution of integer random variables related by a certain linear inequality: II

We continue our study of the problem on the allocation of indistinguishable particles to integer energy levels under the condition that the total energy of the system is bounded above. It is shown that the Bose condensation phenomenon can occur in this model. Systems of dimension d < 1 (including negative dimensions) are studied.

Methods of noncommutative analysis : theory and applications

The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics.While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.

Elliptic theory on singular manifolds

I Singular Manifolds and Differential Operators GEOMETRY OF SINGULARITIES Preliminaries Manifolds with conical singularities Manifolds with edges ELLIPTIC OPERATORS ON SINGULAR MANIFOLDS Operators on manifolds with conical singularities Operators on manifolds with edges Examples of elliptic edge operators II Analytical Tools PSEUDODIFFERENTIAL OPERATORS Preliminary remarks Classical theory Operators in sections of Hilbert bundles Operators on singular manifolds Ellipticity and finiteness theorems Index theorems on smooth closed manifolds LOCALIZATION (SURGERY) IN ELLIPTIC THEORY The index locality principle Localization in index theory on smooth manifolds Surgery for the index of elliptic operators on singular manifolds Relative index formulas on manifolds with isolated singularities III Topological Problems INDEX THEORY Statement of the problem Invariants of interior symbol and symmetries Invariants of the edge symbol Index theorems Index on manifolds with isolated singularities Supplement. Classification of elliptic symbols with symmetry and K-theory Supplement. Proof of Proposition 5.16 ELLIPTIC EDGE PROBLEMS Morphisms The obstruction to ellipticity A formula for the obstruction in topological terms Examples. Obstructions for geometric operators IV Applications and Related Topics FOURIER INTEGRAL OPERATORS ON SINGULAR MANIFOLDS Homogeneous canonical (contact) transformations Definition of Fourier integral operators Properties of Fourier integral operators The index of elliptic Fourier integral operators Application to quantized contact transformations Example RELATIVE ELLIPTIC THEORY Analytic aspects of relative elliptic theory Topological aspects of relative elliptic theory INDEX OF GEOMETRIC OPERATORS ON MANIFOLDS WITH CYLINDRICAL ENDS Operators on manifolds with cylindrical ends Index formulas HOMOTOPY CLASSIFICATION OF ELLIPTIC OPERATORS The homotopy classification problem Classification on smooth manifolds Atiyah-de Rham duality Abstract elliptic operators and analytic K-homology Classification on singular manifolds Some applications LEFSCHETZ FORMULAS Main result Proof of the theorem Contributions of conical points as sums of residues Supplement. The Lefschetz number Supplement. The Sternin-Shatalov method APPENDICES Spectral Flow Eta Invariants Index of Parameter-Dependent Elliptic Families Bibliographic Remarks Bibliography Index

On the distribution of integer random variables satisfying two linear relations

AbstractWe consider the multiplicative (in the sense of Vershik) probability measure corresponding to an arbitrary real dimension d on the set of all collections {Nj} of integer nonnegative numbers Nj, j = l0, l0 + 1, ..., satisfying the conditions

The Index Locality Principle in Elliptic Theory

\sum\limits_{j = l_0 }^\infty {jN_j \leqslant M,} \sum\limits_{j = l_0 }^\infty {N_j = N}

Pseudodifferential operators on stratified manifolds: II

, where l0, M, N are natural numbers. If M, N → ∞ and the rates of growth of these parameters satisfy a certain relation depending on d, and l0 depends on them in a special way (for d ≥ 2 we can take l0 = 1), then, in the limit, the “majority” of collections (with respect to the measure indicated above) concentrates near the limit distribution described by the Bose-Einstein formulas. We study the probabilities of the deviations of the sums Σj=l∞Nj from the corresponding cumulative integrals for the limit distribution. In an earlier paper (see [6]), we studied the case d = 3.

Conjugate variables in analytic number theory. Phase space and Lagrangian manifolds

For a given sequence λj of positive numbers whose counting function ρ(λ) has the asymptotics ρ(λ) = c0λ1+γ(1 + O(λ−ɛ)) with γ > −1 and ɛ > 0 as λ → ∞, we consider sequences {Nj} of nonnegative integers such that

On the distribution of integer random variables related by two linear inequalities: I

\sum\limits_{j = 1}^\infty {\lambda _j N_j \leqslant E} ,