Bispectral KP solutions and linearization of Calogero-Moser particle systems

Rational and soliton solutions of the KP hierarchy in the subgrassmannianGr ₁ are studied within the context of finite dimensional dual grassmannians. In the rational case, properties of the tau function, , which are equivalent to bispectrality of the associated wave function, , are identified. In particular, it is shown that there exists a bound on the degree of all time, variables in if and only if is a rank one bispectral wave function. The action of the bispectral involution, , in the generic rational case is determined explicitly in terms of dual grassmannian parameters. Using the correspondence between rational solutions, and particle systems, it is demonstrated that is a linearizing map of the Calogero-Moser particle system and is essentially the map introduced by Airault, McKean and Moser in 1977 [2].Research supported by NSA Grant MDA904-92-H-3032     

High‐energy‐resolution diced spherical quartz analyzers for resonant inelastic X‐ray scattering

A novel diced spherical quartz analyzer for use in resonant inelastic X‐ray scattering (RIXS) is introduced, achieving an unprecedented energy resolution of 10.53 meV at the Ir L₃ absorption edge (11.215 keV). In this work the fabrication process and the characterization of the analyzer are presented, and an example of a RIXS spectrum of magnetic excitations in a Sr₃Ir₂O₇ sample is shown.     

Darboux Transformations from n-KdV to KP

The iterated Darboux transformations of an ordinary differential operator are constructively parametrized by an infinite-dimensional Grassmannian of finitely supported distributions. In the case that the operator depends on time parameters so that it is a solution to the n-KdV hierarchy, it is shown that the transformation produces a solution of the KP hierarchy. The standard definitions of the theory of -functions are applied to this Grassmannian and it is shown that these new -functions are quotients of KP -functions. The application of this procedure for the construction of higher rank KP solutions is discussed.     

Factorization and Resultants of Partial Differential Operators

Comparatively little is known about commutative rings of partial differential operators, while in the ordinary case, concrete examples and an algebraic(-geometric) structure can be algorithmically determined for large classes. In this note, by the calculation of the partial μ-shifted differential resultant which we defined in a previous paper, we produce algebraic equations of spectral surfaces for commutative rings in two variables, and Darboux transformations of Airy-type operators that correspond to morphisms of surfaces. There are, however, many elementary differential-algebraic statements that we only observe experimentally, thus we offer open questions which seem to us quite significant in differential algebra, and access to Mathematica code to enable further experimentation.     

The impact of futures trading on volatility of the underlying asset in the Turkish stock market

This paper examines the impact of the introduction of stock index futures on the volatility of the Istanbul Stock Exchange (ISE), using asymmetric GARCH model, for the period July 2002-October 2007. The results from EGARCH model indicate that the introduction of futures trading reduced the conditional volatility of ISE-30 index. Results further indicate that there is a long-run relationship between spot and future prices. The results also suggest that the direction of both long- and short-run causality is from spot prices to future prices. These findings are consistent with those theories stating that futures markets enhance the efficiency of the corresponding spot markets.     

Universality of Rank 6 Plücker relations and Grassmann cone preserving maps

The Plücker relations define a projective embedding of the Grassmann variety . We give another finite set of quadratic equations which defines the same embedding, and whose elements all have rank 6. This is achieved by constructing a certain finite set of linear maps , and pulling back the unique Plücker relation on . We also give a quadratic equation depending on parameters having the same properties.

     

D-Modules and Darboux Transformations

A method of G. Wilson for generating commutative algebras of ordinary differential operators is extended to higher dimensions. Our construction, based on the theory of D-modules, leads to a new class of examples of commutative rings of partial differential operators with rational spectral varieties. As an application, we briefly discuss their link to the bispectral problem and to the theory of lacunas.     

Tau-functions,Grassmannians and rank one conditions

In integrable systems, specifically the KP hierarchy, there are functions known as “tau-functions”, closely related to the Schur polynomials in terms of which they are often written. Although they are generally viewed as the solutions to a collection of nonlinear PDEs, in this note they will equivalently be characterized by a quadratic difference equation. Sato's theorem associates tau-functions to the points of a Grassmann manifold. To make that amazing theorem clear to non-experts, we will first show an analogous (but easily understood) example of a linear ODE and its solution from a flow on the xy-plane. In each case the solution is created via a flow generated by a certain linear operator. The question we pose is this: “What other operators could have been used to generate solutions in the same way?” Although the answer is well known in the ODE case, the question in the nonlinear case is the main result of our new paper. We will state the result and discuss its relationship to the “trend” of writing tau-functions in terms of matrices satisfying certain rank one conditions. The elucidation of a geometric interpretation of the Hirota bilinear difference equation (HBDE) is a key feature of the proof and will be briefly described.     

On factoring an operator using elements of its kernel

A well-known theorem factors a scalar coefficient differential operator given a linearly independent set of functions in its kernel. The goal of this paper is to generalize this useful result to other types of operators. In place of the derivation ? acting on some ring of functions, this paper considers the more general situation of an endomorphism 𝔇 acting on a unital associative algebra. The operators considered, analogous to differential operators, are those which can be written as a finite sum of powers of 𝔇 followed by left multiplication by elements of the algebra. Assume that the set of such operators is closed under multiplication and that a Wronski-like matrix produced from some finite list of elements of the algebra is invertible (analogous to the linear independence condition). Then, it is shown that the set of operators whose kernels contain all of those elements is the left ideal generated by an explicitly given operator. In other words, an operator has those elements in its kernel if and only if it has that generator as a right factor. Three examples demonstrate the application of this result in different contexts, including one in which 𝔇 is an automorphism of finite order.