A Note on Fractional KdV Hierarchies

We introduce a hierarchy of mutually commuting dynamical systems on a finite number of Laurent series. This hierarchy can be seen as a prolongation of the KP hierarchy, or a ``reduction'' in which the space coordinate is identified with an arbitrarily chosen time of a bigger dynamical system. Fractional KdV hierarchies are gotten by means of further reductions, obtained by constraining the Laurent series. The case of sl(3)^2 and its bihamiltonian structure are discussed in detail.Comment: Final version to appear in J. Math. Phys. Some changes in the order of presentation, with more emphasis on the geometrical picture. One figure added (using epsf.sty). 30 pages, Late

Quasi-BiHamiltonian Systems and Separability

Two quasi--biHamiltonian systems with three and four degrees of freedom are presented. These systems are shown to be separable in terms of Nijenhuis coordinates. Moreover the most general Pfaffian quasi-biHamiltonian system with an arbitrary number of degrees of freedom is constructed (in terms of Nijenhuis coordinates) and its separability is proved.Comment: 10 pages, AMS-LaTeX 1.1, to appear in J. Phys. A: Math. Gen. (May 1997

On third Poisson structure of KdV equation

The third Poisson structure of KdV equation in terms of canonical ``free fields'' and reduced WZNW model is discussed. We prove that it is ``diagonalized'' in the Lagrange variables which were used before in formulation of 2D gravity. We propose a quantum path integral for KdV equation based on this representation.Comment: 6pp, Latex. to appear in ``Proceedings of V conference on Mathematical Physics, String Theory and Quantum Gravity, Alushta, June 1994'' Teor.Mat.Fiz. 199

Influence of infection on the distribution patterns of NIH-Chronic Prostatitis Symptom Index scores in patients with chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS)

Chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS) is a complex condition for which the etiological determinants are still poorly defined. To better characterize the diagnostic and therapeutic profile of patients, an algorithm known as UPOINT was created, addressing six major phenotypic domains of CP/CPPS, specifically the urinary (U), psycho-social (P), organ-specific (O), infection (I), neurological/systemic (N) and muscular tenderness (T) domains. An additional sexual dysfunction domain may be included in the UPOINT(S) system. The impact of the infection domain on the severity of CP/CPPS symptoms is a controversial issue, due to the contradictory results of different trials. The aim of the present retrospective study was to further analyze the extent to which a positive infection domain of UPOINTS may modify the pattern of CP/CPPS symptom scores, assessed with the National Institutes of Health-Chronic Prostatitis Symptom Index (NIH-CPSI). In a cohort of 935 patients that was divided on the basis of the presence or absence of prostatic infection, more severe clinical symptoms were shown by the patients with infection (median NIH total score: 24 versus 20 points in uninfected patients; P<0.001). Moreover, NIH-CPSI score distribution curves were shifted towards more severe symptoms in patients with a positive infection domain. Division of the patients into the six most prominent phenotypic clusters of UPOINTS revealed that the 'prostate infection-related sexual dysfunction' cluster, including the highest proportion of patients with evidence of infection (80%), scored the highest number of NIH-CPSI points among all the clusters. To assess the influence of the infection domain on the severity of patients' symptoms, all subjects with evidence of infection were withdrawn from the 'prostate infection-related sexual dysfunction' cluster. This modified cluster showed symptom scores significantly less severe than the original cluster, and the CPSI values became comparable to the scores of the five other clusters, which were virtually devoid of patients with evidence of infection. These results suggest that the presence of pathogens in the prostate gland may significantly affect the clinical presentation of patients affected by CP/CPPS, and that the infection domain may be a determinant of the severity of CP/CPPS symptoms in clusters of patients phenotyped with the UPOINTS system. This evidence may convey considerable therapeutic implications

Compatible Poisson-Lie structures on the loop group of SL2SL_{2}

We define a 1-parameter family of $r$-matrices on the loop algebra of $sl_{2}$, defining compatible Poisson structures on the associated loop group, which degenerate into the rational and trigonometric structures, and study the Manin triples associated to them.Comment: 5 pages, amstex, no figure

On the integrability of stationary and restricted flows of the KdV hierarchy.

A bi--Hamiltonian formulation for stationary flows of the KdV hierarchy is derived in an extended phase space. A map between stationary flows and restricted flows is constructed: in a case it connects an integrable Henon--Heiles system and the Garnier system. Moreover a new integrability scheme for Hamiltonian systems is proposed, holding in the standard phase space.Comment: 25 pages, AMS-LATEX 2.09, no figures, to be published in J. Phys. A: Math. Gen.

A Novel Hierarchy of Integrable Lattices

In the framework of the reduction technique for Poisson-Nijenhuis structures, we derive a new hierarchy of integrable lattice, whose continuum limit is the AKNS hierarchy. In contrast with other differential-difference versions of the AKNS system, our hierarchy is endowed with a canonical Poisson structure and, moreover, it admits a vector generalisation. We also solve the associated spectral problem and explicity contruct action-angle variables through the r-matrix approach.Comment: Latex fil

Vertex Operator Superalgebras and Odd Trace Functions

We begin by reviewing Zhu's theorem on modular invariance of trace functions associated to a vertex operator algebra, as well as a generalisation by the author to vertex operator superalgebras. This generalisation involves objects that we call `odd trace functions'. We examine the case of the N=1 superconformal algebra. In particular we compute an odd trace function in two different ways, and thereby obtain a new representation theoretic interpretation of a well known classical identity due to Jacobi concerning the Dedekind eta function.Comment: 13 pages, 0 figures. To appear in Conference Proceedings `Advances in Lie Superalgebras

Extension of Hereditary Symmetry Operators

Two models of candidates for hereditary symmetry operators are proposed and thus many nonlinear systems of evolution equations possessing infinitely many commutative symmetries may be generated. Some concrete structures of hereditary symmetry operators are carefully analyzed on the base of the resulting general conditions and several corresponding nonlinear systems are explicitly given out as illustrative examples.Comment: 13 pages, LaTe