Bispectral algebras of commuting ordinary differential operators

We develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank $N$. It combines and unifies the ideas of Duistermaat-Gr\"unbaum and Wilson. Our construction is completely algorithmic and enables us to obtain all previously known classes or individual examples of bispectral operators. The method also provides new broad families of bispectral algebras which may help to penetrate deeper into the problem.Comment: 46 pages, LaTeX2e, uses amsfonts.sty and latexsym.sty, no figures, rearrangement of the introduction, skipping Conjecture 0.2 of the first version, to appear in Communications in Mathematical Physic

B\"acklund--Darboux transformations in Sato's Grassmannian

We define B\"acklund--Darboux transformations in Sato's Grassmannian. They can be regarded as Darboux transformations on maximal algebras of commuting ordinary differential operators. We describe the action of these transformations on related objects: wave functions, tau-functions and spectral algebras. This paper is the second of a series of papers (hep-th/9510211, q-alg/9602011, q-alg/9602012) on the bispectral problem.Comment: 13 pages, LaTeX2e, uses amsfonts.sty and latexsym.sty, no figure

Lie conformal algebra cohomology and the variational complex

We find an interpretation of the complex of variational calculus in terms of the Lie conformal algebra cohomology theory. This leads to a better understanding of both theories. In particular, we give an explicit construction of the Lie conformal algebra cohomology complex, and endow it with a structure of a g-complex. On the other hand, we give an explicit construction of the complex of variational calculus in terms of skew-symmetric poly-differential operators.Comment: 56 page

Geometric construction of modular functors from Conformal Field Theory

This is the second paper in a series of papers aimed at providing a geometric construction of modular functors and topological quantum field theories from conformal field theory building on the constructions in [TUY] and [KNTY]. We give a geometric construct of a modular functor for any simple Lie-algebra and any level by twisting the constructions in [TUY] by a certain fractional power of the abelian theory first considered in [KNTY] and further studied in our first paper [AU1].Comment: Paper considerably expanded so as to make it self containe

Jacobi Identity for Vertex Algebras in Higher Dimensions

Vertex algebras in higher dimensions provide an algebraic framework for investigating axiomatic quantum field theory with global conformal invariance. We develop further the theory of such vertex algebras by introducing formal calculus techniques and investigating the notion of polylocal fields. We derive a Jacobi identity which together with the vacuum axiom can be taken as an equivalent definition of vertex algebra.Comment: 35 pages, references adde

Progressor: Personalized visual access to programming problems

This paper presents Progressor, a visualization of open student models intended to increase the student's motivation to progress on educational content. The system visualizes not only the user's own model, but also the peers' models. It allows sorting the peers' models using a number of criteria, including the overall progress and the progress on a specific topic. Also, in this paper we present results of a classroom study confirming our hypothesis that by showing a student the peers' models and ranking them by progress it is possible to increase the student's motivation to compete and progress in e-learning systems. © 2011 IEEE

Finiteness and orbifold Vertex Operator Algebras

In this paper, I investigate the ascending chain condition of right ideals in the case of vertex operator algebras satisfying a finiteness and/or a simplicity condition. Possible applications to the study of finiteness of orbifold VOAs is discussed.Comment: 12 pages, comments are welcom

Precision spectroscopy of the molecular ion HD+: control of Zeeman shifts

Precision spectroscopy on cold molecules can potentially enable novel tests of fundamental laws of physics and alternative determination of some fundamental constants. Realizing this potential requires a thorough understanding of the systematic effects that shift the energy levels of molecules. We have performed a complete ab initio calculation of the magnetic field effects for a particular system, the heteronuclear molecular hydrogen ion HD+. Different spectroscopic schemes have been considered, and numerous transitions, all accessible by modern radiation sources and exhibiting well controllable or negligible Zeeman shift, have been found to exist. Thus, HD+ is a perspective candidate for determination of the ratio of electron-to-nuclear reduced mass, and for tests of its time-independence.Comment: A Table added, references and figures update

General methods for constructing bispectral operators

We present methods for obtaining new solutions to the bispectral problem. We achieve this by giving its abstract algebraic version suitable for generalizations. All methods are illustrated by new classes of bispectral operators.Comment: 11 pages, LaTeX2e, uses amsfonts.sty and latexsym.sty, no figure