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In our paper [KR] we began a systematic study of representations of the universal central extension[InlineEquation not available: see fulltext.] of the Lie algebra of differential operators on the circle.
      
We express them in terms of generatorsEij ofU(gl(n)) and as differential operators on the space of matrices These expressions are a direct generalization of the classical Capelli identities.
      
Quantum integrable systems and differential Galois theory
      
This paper is devoted to a systematic study of quantum completely integrable systems (i.e., complete systems of commuting differential operators) from the point of view of algebraic geometry.
      
In particular, we show that the differential Galois group of this eigenvalue problem is reductive at generic eigenvalues.
      
This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues.
      
As an application, under the same assumption on the transitive group, we show that weakly symmetric spaces are precisely the homogeneous Riemannian manifolds with commutative algebra of invariant differential operators.
      
Structure of the Center of the Algebra of Invariant Differential Operators on Certain Riemannian Homogeneous Spaces
      
The homogeneous space X is called commutative or the pair (G, K) is called a Gelfand pair if the algebra of G-invariant differential operators on X is commutative.
      
Of special interest are the Mellin operators of differentiation and integration, more correctly of anti-differentiation, enabling one to establish the fundamental theorem of the differential and integral calculus in the Mellin frame.
      
They are of particular importance in solving differential and integral equations.
      
Calderón's formula associated with a differential operator on (0, ∞) and inversion of the generalized abel transform
      
We prove a Calderón reproducing formula for a continuous wavelet transform associated with a class of singular differential operators on the half line.
      
Fractal differential equations on the Sierpinski gasket
      
We study the analogs of some of the classical partial differential equations with Δ playing the role of the usual Laplacian.
      
It is shown that there is a similar identity when the inner product is replaced by an indefinite quadratic formq and h is a Л-harmonic distribution, where Л is the differential operator canonically associated toq.
      
Some applications are given to control theory for partial differential equations.
      
Their relationship with the heat equation and the newly introduced wavelet differential equation is established.
      
The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations
      
A partial differential equation from polymer science is shown to be solvable using the operational properties of the Euclidean-group Fourier transform.
      
 

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