The multicomponent 2D Toda hierarchy: Discrete flows and string equations

The multicomponent 2D Toda hierarchy is analyzed through a factorization problem associated to an infinite-dimensional group. A new set of discrete flows is considered and the corresponding Lax and Zakharov--Shabat equations are characterized. Reductions of block Toeplitz and Hankel bi-infinite matrix types are proposed and studied. Orlov--Schulman operators, string equations and additional symmetries (discrete and continuous) are considered. The continuous-discrete Lax equations are shown to be equivalent to a factorization problem as well as to a set of string equations. A congruence method to derive site independent equations is presented and used to derive equations in the discrete multicomponent KP sector (and also for its modification) of the theory as well as dispersive Whitham equations.Comment: 27 pages. In the revised paper we improved the presentatio

Ladders operators for general discrete Sobolev orthogonal polynomials

We consider a general discrete Sobolev inner product involving the Hahn difference operator, so this includes the well--known difference operators $\mathscr{D}_{q}$ and $\Delta$ and, as a limit case, the derivative operator. The objective is twofold. On the one hand, we construct the ladder operators for the corresponding nonstandard orthogonal polynomials and we obtain the second--order difference equation satisfied by these polynomials. On the other hand, we generalise some related results appeared in the literature as we are working in a more general framework. Moreover, we will show that all the functions involved in these constructions can be computed explicitly

The multicomponent 2D Toda hierarchy: dispersionless limit

The factorization problem of the multi-component 2D Toda hierarchy is used to analyze the dispersionless limit of this hierarchy. A dispersive version of the Whitham hierarchy defined in terms of scalar Lax and Orlov--Schulman operators is introduced and the corresponding additional symmetries and string equations are discussed. Then, it is shown how KP and Toda pictures of the dispersionless Whitham hierarchy emerge in the dispersionless limit. Moreover, the additional symmetries and string equations for the dispersive Whitham hierarchy are studied in this limit.Comment: Revised version with an overall improved presentatio

Associations between sedentary time, physical activity and bone health among older people using compositional data analysis

Introduction : Aging is associated with a progressive decrease in bone mass (BM), and being physical active is one of the main strategies to combat this continuous loss. Nonetheless, because daily time is limited, time spent on each movement behavior is co-dependent. The aim of this study was to determine the relationship between BM and movement behaviors in elderly people using compositional data analysis. Methods : We analyzed 871 older people [395 men (76.9 +/- 5.3y) and 476 women (76.7 +/- 4.7y)]. Time spent in sedentary behavior (SB), light physical activity (LPA) and moderate-to-vigorous physical activity (MVPA), was assessed using accelerometry. BM was determined by bone densitometry (DXA). The sample was divided according to sex and bone health indicators. Results : The combined effect of all movement behaviors (PA and SB) was significantly associated with whole body, leg and femoral region BM in the whole sample (p<0.05), with leg and pelvic BM (p<0.05) in men and, with whole body, arm and leg BM (p<0.05) in women. In men, arm and pelvic BM were negatively associated with SB and whole body, pelvic and leg BM were positively associated with MVPA (p<0.05). In women, whole body and leg BM were positively associated with SB. Arm and whole body BM were positively associated and leg BM was negatively associated with LPA and arm BM was negatively associated with MVPA (p<0.05). Women without bone fractures spent less time in SB and more in LPA and MVPA than the subgroup with bone fractures. Conclusion : We identified that the positive effect of MVPA relative to the other behaviors on bone mass is the strongest overall effect in men. Furthermore, women might decrease bone fracture risk through PA increase and SB reduction, despite the fact that no clear benefits of PA for bone mass were found

On the Whitham hierarchy: dressing scheme, string equations and additional symmetrie

A new description of the universal Whitham hierarchy in terms of a factorization problem in the Lie group of canonical transformations is provided. This scheme allows us to give a natural description of dressing transformations, string equations and additional symmetries for the Whitham hierarchy. We show how to dress any given solution and prove that any solution of the hierarchy may be undressed, and therefore comes from a factorization of a canonical transformation. A particulary important function, related to the $\tau$-function, appears as a potential of the hierarchy. We introduce a class of string equations which extends and contains previous classes of string equations considered by Krichever and by Takasaki and Takebe. The scheme is also applied for an convenient derivation of additional symmetries. Moreover, new functional symmetries of the Zakharov extension of the Benney gas equations are given and the action of additional symmetries over the potential in terms of linear PDEs is characterized

On the Whitham hierarchy: dressing scheme, string equations and additional symmetrie

A new description of the universal Whitham hierarchy in terms of a factorization problem in the Lie group of canonical transformations is provided. This scheme allows us to give a natural description of dressing transformations, string equations and additional symmetries for the Whitham hierarchy. We show how to dress any given solution and prove that any solution of the hierarchy may be undressed, and therefore comes from a factorization of a canonical transformation. A particulary important function, related to the $\tau$-function, appears as a potential of the hierarchy. We introduce a class of string equations which extends and contains previous classes of string equations considered by Krichever and by Takasaki and Takebe. The scheme is also applied for an convenient derivation of additional symmetries. Moreover, new functional symmetries of the Zakharov extension of the Benney gas equations are given and the action of additional symmetries over the potential in terms of linear PDEs is characterized

S-functions, reductions and hodograph solutions of the r-th dispersionless modified KP and Dym hierarchies

We introduce an S-function formulation for the recently found r-th dispersionless modified KP and r-th dispersionless Dym hierarchies, giving also a connection of these $S$-functions with the Orlov functions of the hierarchies. Then, we discuss a reduction scheme for the hierarchies that together with the $S$-function formulation leads to hodograph systems for the associated solutions. We consider also the connection of these reductions with those of the dispersionless KP hierarchy and with hydrodynamic type systems. In particular, for the 1-component and 2-component reduction we derive, for both hierarchies, ample sets of examples of explicit solutions.Comment: 35 pages, uses AMS-Latex, Hyperref, Geometry, Array and Babel package

Additional symmetries and solutions of the dispersionless KP hierarchy

The dispersionless KP hierarchy is considered from the point of view of the twistor formalism. A set of explicit additional symmetries is characterized and its action on the solutions of the twistor equations is studied. A method for dealing with the twistor equations by taking advantage of hodograph type equations is proposed. This method is applied for determining the orbits of solutions satisfying reduction constraints of Gelfand--Dikii type under the action of additional symmetries.Comment: 21 page