Israel M. Gelfand

Webs, Veronese curves, and bihamiltonian systems

Abstract We define a special kind of multidimensional webs, connected with the Veronese curve. For these webs the foliations in question depend not on a discrete parameter, but on the point on a projective line. For each bihamiltonian system of odd dimension in general position we construct such a web and show how to reconstruct the original bihamiltonian system based on these data.

Changes in the force-sharing pattern induced by modifications of visual feedback during force production by a set of fingers

Abstract We investigated force-sharing among three fingers which acted in parallel and produced ramp profiles of total force from zero to the maximal voluntary force. The feedback to the subject was provided by a visual signal on the monitor and could correspond to the sum of forces of all the fingers or to the sum of forces of two fingers, while the force of the third finger was added with a coefficient 2 or 0.5. If the subjects did not know about the distorted feedback, they showed a template-sharing pattern within the whole range of total force values. This pattern did not depend on which finger force was multiplied and by which coefficient. If the subjects knew in advance how the feedback signal would be calculated, they tried to perform the task using either only the finger whose force was multiplied by 2 or two fingers when the force of the third one was multiplied by 0.5. Further into the trial, however, they were unable to track the ramp pattern using only one or two fingers and demonstrated a search activity that could continue until the end of the trial or lead eventually to a three-finger sharing pattern similar to the template pattern used in cases of undistorted feedback. We conclude that the limited number of preferred sharing pattern within the studied task reflects an organization of the fingers into a structural unit (involving one, two, or all three fingers) by the central nervous system. The availability of structural units defines the presence of stable solutions available for the system.

Combinatorics of hypergeometric functions associated with positive roots

In this paper we study the hypergeometric system on unipotent matrices. This system gives a holonomic D-module. We find the number of independent solutions of this system at a generic point. This number is equal to the famous Catalan number. An explicit basis of Γ-series in solution space of this system is constructed in the paper. We also consider restriction of this system to certain strata. We introduce several combinatorial constructions with trees, polyhedra, and triangulations related to this subject.

Integrability of linear and nonlinear evolution equations and the associated nonlinear fourier transforms

The inverse spectral method is a nonlinear Fourier transform method for solving certain equations. Here, we emphasize that such transforms should be considered in their own right. We also elucidate further the connection between the Fourier transform and inverse spectral methods by establishing that linear equations can also be solved through the inverse spectral method.

Duality and minors of secondary polyhedra

Abstract Using Minkowski integration, we define the secondary polyhedron of a vector configuration A and study its behavior under the matroidal operations of duality, deletion, and contraction. A main tool is the identification of the regular polyhedral subdivisions of A with the cells in the dual chamber complex. As an application we construct a non-regular triangulation of a cyclic polytope.

Common features in structures and sequences of sandwich-like proteins

The goal of this work is to define the structural and sequence features common to sandwich-like proteins (SPs), a group of very different proteins now comprising 69 superfamilies in 38 protein folds. Analysis of the arrangements of strands within main sandwich sheets revealed a rigorously defined constraint on the supersecondary substructure that holds true for 94% of known SP structures. The invariant substructure consists of two interlocked pairs of neighboring β-strands. It is even more typical for centers of SP than the well-known “Greek key” strands arrangement for their edges. As homology among these proteins is not usually detectable even with the most powerful sequence-comparing algorithms, we employed a structure-based approach to sequence alignment. Within the interlocked strands we found 12 positions with fixed structural roles in SP. A residue at any of these positions possesses similar structural properties with residues in the same position of other SPs. The 12 positions lie at the center of the interface between the β-sheets and form the common geometrical core of SPs. Of the 12 positions, 8 are occupied by only four hydrophobic residues in 80% of all SPs.

A formula for constructing infinitely many surfaces on Lie algebras and integrable equations

Abstract. Surfaces immersed in Lie algebras can be characterized by the so called fundamental forms. The coefficients of these forms satisfy a system of nonlinear partial differential equations (PDEs), the Gauss–Mainardi–Codazzi–Ricci equations. For particular surfaces, this system of PDEs belongs to a distinguished class of equations known as integrable equations. Such an example in

On the Local Geometry of a Bihamiltonian Structure

{\Bbb R}^3

Inversion method for magnetoencephalography

is the class of surfaces of constant mean curvature which is associated with the sinh-Gordon equation. Here an explicit formula is presented which associates with a given system of integrable nonlinear PDEs infinitely many surfaces immersed in Lie algebras. This formula is based on the general construction of surfaces on Lie algebras introduced recently by the first two authors, and on the fact that integrable equations possess infinitely many symmetries. Several examples of surfaces immersed in the 3-dimensional Euclidean space are discussed, including the list of integrable surfaces recently presented by Bobenko and certain deformations thereof.

New classification of supersecondary structures of sandwich-like proteins uncovers strict patterns of strand assemblage.

We give several examples of bihamiltonian manifolds and show that under very mild assumptions a bihamiltonian structure in “general position” is locally of one of these types. This shows, in particular, that a bihamiltonian manifold in general position is always a moduli space of some kind. In the even-dimensional case it is a Hubert scheme of a surface, in the odd-dimensional case it is a sub- cotangent bundle of a moduli space of rational curves on a surface.