Entropy and Multivariable Interpolation

Entropy and Multivariable Interpolation
Author: Gelu Popescu
Publisher: American Mathematical Soc.
Total Pages: 98
Release: 2006
Genre: Mathematics
ISBN: 0821839128

We define a new notion of entropy for operators on Fock spaces and positive multi-Toeplitz kernels on free semigroups. This is studied in connection with factorization theorems for (e.g., multi-Toeplitz, multi-analytic, etc.) operators on Fock spaces. These results lead to entropy inequalities and entropy formulas for positive multi-Toeplitz kernels on free semigroups (resp. multi-analytic operators) and consequences concerning the extreme points of the unit ball of the noncommutative analytic Toeplitz algebra $F ninfty$. We obtain several geometric characterizations of the central intertwining lifting, a maximal principle, and a permanence principle for the noncommutative commutant lifting theorem. Under certain natural conditions, we find explicit forms for the maximal entropy solution of this multivariable commutant lifting theorem. All these results are used to solve maximal entropy interpolation problems in several variables. We obtain explicit forms for the maximal entropy solution (as well as its entropy) of the Sarason, Caratheodory-Schur, and Nevanlinna-Pick type interpolation problems for the noncommutative (resp. commutative) analytic Toeplitz algebra $F ninfty$ (resp. $W ninfty$) and their tensor products with $B({\mathcal H , {\mathcal K )$. In particular, we provide explicit forms for the maximal entropy solutions of several interpolation problems on the unit ball of $\mathbb{C n$.

Entropy and Multivariable Interpolation

Entropy and Multivariable Interpolation
Author: Gelu Popescu
Publisher: American Mathematical Soc.
Total Pages: 83
Release: 2006
Genre: Mathematics
ISBN: 9781470404727

We define a new notion of entropy for operators on Fock spaces and positive multi-Toeplitz kernels on free semigroups. This is studied in connection with factorization theorems for (e.g., multi-Toeplitz, multi-analytic, etc.) operators on Fock spaces. These results lead to entropy inequalities and entropy formulas for positive multi-Toeplitz kernels on free semigroups (resp. multi-analytic operators) and consequences concerning the extreme points of the unit ball of the noncommutative analytic Toeplitz algebra 'F ninfty'. We obtain several geometric characterizations of the central intertwining lifting, a maximal principle, and a permanence principle for the noncommutative commutant lifting theorem. Under certain natural conditions, we find explicit forms for the maximal entropy solution of this multivariable commutant lifting theorem. All these results are used to solve maximal entropy interpolation problems in several variables. We obtain explicit forms for the maximal entropy solution (as well as its entropy) of the Sarason, Caratheodory-Schur, and Nevanlinna-Pick type interpolation problems for the noncommutative (resp. commutative) analytic Toeplitz algebra 'F ninfty' (resp. In particular, we provide explicit forms for the maximal entropy solutions of several interpolation problems on the unit ball of '\mathbb{C n'.

Bernoulli Free-Boundary Problems

Bernoulli Free-Boundary Problems
Author: Eugene Shargorodsky
Publisher: American Mathematical Soc.
Total Pages: 86
Release: 2008
Genre: Mathematics
ISBN: 0821841890

Questions of existence, multiplicity, and regularity of free boundaries for prescribed data need to be addressed and their solutions lead to nonlinear problems. In this paper an equivalence is established between Bernoulli free-boundary problems and a class of equations for real-valued functions of one real variable.

Representations of Shifted Yangians and Finite $W$-algebras

Representations of Shifted Yangians and Finite $W$-algebras
Author: Jonathan Brundan
Publisher: American Mathematical Soc.
Total Pages: 122
Release: 2008
Genre: Mathematics
ISBN: 0821842161

The authors study highest weight representations of shifted Yangians over an algebraically closed field of characteristic $0$. In particular, they classify the finite dimensional irreducible representations and explain how to compute their Gelfand-Tsetlin characters in terms of known characters of standard modules and certain Kazhdan-Lusztig polynomials. The authors' approach exploits the relationship between shifted Yangians and the finite W-algebras associated to nilpotent orbits in general linear Lie algebras.

Noncommutative Maslov Index and Eta-Forms

Noncommutative Maslov Index and Eta-Forms
Author: Charlotte Wahl
Publisher: American Mathematical Soc.
Total Pages: 130
Release: 2007
Genre: Index theory
ISBN: 0821839977

The author defines and proves a noncommutative generalization of a formula relating the Maslov index of a triple of Lagrangian subspaces of a symplectic vector space to eta-invariants associated to a pair of Lagrangian subspaces. The noncommutative Maslov index, defined for modules over a $C *$-algebra $\mathcal{A}$, is an element in $K_0(\mathcal{A})$. The generalized formula calculates its Chern character in the de Rham homology of certain dense subalgebras of $\mathcal{A}$. The proof is a noncommutative Atiyah-Patodi-Singer index theorem for a particular Dirac operator twisted by an $\mathcal{A}$-vector bundle. The author develops an analytic framework for this type of index problem.

Multivariable Operator Theory

Multivariable Operator Theory
Author: Ernst Albrecht
Publisher: Springer Nature
Total Pages: 893
Release: 2024-01-22
Genre: Mathematics
ISBN: 3031505352

Over the course of his distinguished career, Jörg Eschmeier made a number of fundamental contributions to the development of operator theory and related topics. The chapters in this volume, compiled in his memory, are written by distinguished mathematicians and pay tribute to his many significant and lasting achievements.

Ramanujan's Forty Identities for the Rogers-Ramanujan Functions

Ramanujan's Forty Identities for the Rogers-Ramanujan Functions
Author: Bruce C. Berndt
Publisher: American Mathematical Soc.
Total Pages: 110
Release: 2007
Genre: Mathematics
ISBN: 082183973X

Sir Arthur Conan Doyle's famous fictional detective Sherlock Holmes and his sidekick Dr. Watson go camping and pitch their tent under the stars. During the night, Holmes wakes his companion and says, ``Watson, look up at the stars and tell me what you deduce.'' Watson says, ``I see millions of stars, and it is quite likely that a few of them are planets just like Earth. Therefore there may also be life on these planets.'' Holmes replies, ``Watson, you idiot. Somebody stole ourtent.'' When seeking proofs of Ramanujan's identities for the Rogers-Ramanujan functions, Watson, i.e., G. N. Watson, was not an ``idiot.'' He, L. J. Rogers, and D. M. Bressoud found proofs for several of the identities. A. J. F. Biagioli devised proofs for most (but not all) of the remaining identities.Although some of the proofs of Watson, Rogers, and Bressoud are likely in the spirit of those found by Ramanujan, those of Biagioli are not. in particular, Biagioli used the theory of modular forms. Haunted by the fact that little progress has been made into Ramanujan's insights on these identities in the past 85 years, the present authors sought ``more natural'' proofs. Thus, instead of a missing tent, we have had missing proofs, i.e., Ramanujan's missing proofs of his forty identities for theRogers-Ramanujan functions. in this paper, for 35 of the 40 identities, the authors offer proofs that are in the spirit of Ramanujan. Some of the proofs presented here are due to Watson, Rogers, and Bressoud, but most are new. Moreover, for several identities, the authors present two or threeproofs. For the five identities that they are unable to prove, they provide non-rigorous verifications based on an asymptotic analysis of the associated Rogers-Ramanujan functions. This method, which is related to the 5-dissection of the generating function for cranks found in Ramanujan's lost notebook, is what Ramanujan might have used to discover several of the more difficult identities. Some of the new methods in this paper can be employed to establish new identities for the Rogers-Ramanujanfunctions.

Maximum Entropy and Bayesian Methods

Maximum Entropy and Bayesian Methods
Author: John Skilling
Publisher: Springer
Total Pages: 344
Release: 1996-07-31
Genre: Mathematics
ISBN:

This volume records the proceedings of the Fourteenth International Workshop on Maximum Entropy and Bayesian Methods, held in Cambridge, England from August 1-5, 1994. Throughout applied science, Bayesian inference is giving high quality results augmented with reliabilities in the form of probability values and probabilistic error bars. Maximum Entropy, with its emphasis on optimally selected results, is an important part of this. Across wide areas of spectroscopy and imagery, it is now realistic to generate clear results with quantified reliability. This power is underpinned with a foundation of solid mathematics. The annual Maximum Entropy Workshops have become the principal focus of developments in the field, and which capture the imaginative research that defines the state of the art in the subject. The breadth of application is seen in the thirty-three papers reproduced here, which are classified into subsections on Basics, Applications, Physics and Neural Networks. Audience: This volume will be of interest to graduate students and researchers whose work involves probability theory, neural networks, spectroscopic methods, statistical thermodynamics and image processing.

Operator Algebras, Operator Theory and Applications

Operator Algebras, Operator Theory and Applications
Author: J. J. Grobler
Publisher: Springer Science & Business Media
Total Pages: 301
Release: 2009-12-24
Genre: Mathematics
ISBN: 3034601743

This volume contains the proceedings of the eighteenth International Workshop on Operator Theory and Applications (IWOTA), hosted by the Unit for Business Mathematics and Informatics of North-West University, Potchefstroom, South Africa from July 3 to 6, 2007. The conference (as well as these proceedings) was dedicated to Professors Joseph A. Ball and Marinus M. Kaashoek on the occasion of their 60th and 70th birthdays, respectively. This conference had a particular focus on Von Neumann algebras at the interface of operator theory with functional analysis and on applications of operator theory to differential equations.