Akns Hierarchy, Self-Similarity, String Equations and the Grassmannian

In this paper the Galilean, scaling and translational self--similarity conditions for the AKNS hierarchy are analysed geometrically in terms of the infinite dimensional Grassmannian. The string equations found recently by non--scaling limit analysis of the one--matrix model are shown to correspond to the Galilean self--similarity condition for this hierarchy. We describe, in terms of the initial data for the zero--curvature 1--form of the AKNS hierarchy, the moduli space of these self--similar solutions in the Sato Grassmannian. As a byproduct we characterize the points in the Segal--Wilson Grassmannian corresponding to the Sachs rational solutions of the AKNS equation and to the Nakamura--Hirota rational solutions of the NLS equation. An explicit 1--parameter family of Galilean self--similar solutions of the AKNS equation and the associated solution to the NLS equation is determined.Comment: 25 pages in AMS-LaTe

Evaluation of CNN architectures for gait recognition based on optical flow maps

This work targets people identification in video based on the way they walk (\ie gait) by using deep learning architectures. We explore the use of convolutional neural networks (CNN) for learning high-level descriptors from low-level motion features (\ie optical flow components). The low number of training samples for each subject and the use of a test set containing subjects different from the training ones makes the search of a good CNN architecture a challenging task.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tec

String Equations for the Unitary Matrix Model and the Periodic Flag Manifold

The periodic flag manifold (in the Sato Grassmannian context) description of the modified Korteweg--de Vries hierarchy is used to analyse the translational and scaling self--similar solutions of this hierarchy. These solutions are characterized by the string equations appearing in the double scaling limit of the symmetric unitary matrix model with boundary terms. The moduli space is a double covering of the moduli space in the Sato Grassmannian for the corresponding self--similar solutions of the Korteweg--de Vries hierarchy, i.e. of stable 2D quantum gravity. The potential modified Korteweg--de Vries hierarchy, which can be described in terms of a line bundle over the periodic flag manifold, and its self--similar solutions corresponds to the symmetric unitary matrix model. Now, the moduli space is in one--to--one correspondence with a subset of codimension one of the moduli space in the Sato Grassmannian corresponding to self--similar solutions of the Korteweg--de Vries hierarchy.Comment: 21 pages in LaTeX-AMSTe

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

The transitional gap transient AT 2018hso: new insights into the luminous red nova phenomenon

Context. The absolute magnitudes of luminous red novae (LRNe) are intermediate between those of novae and supernovae (SNe), and show a relatively homogeneous spectro-photometric evolution. Although they were thought to derive from core instabilities in single stars, there is growing support for the idea that they are triggered by binary interaction that possibly ends with the merging of the two stars. Aims. AT 2018hso is a new transient showing transitional properties between those of LRNe and the class of intermediate-luminosity red transients (ILRTs) similar to SN 2008S. Through the detailed analysis of the observed parameters, our study supports that it actually belongs to the LRN class and was likely produced by the coalescence of two massive stars. Methods. We obtained ten months of optical and near-infrared photometric monitoring, and 11 epochs of low-resolution optical spectroscopy of AT 2018hso. We compared its observed properties with those of other ILRTs and LRNe. We also inspected the archival Hubble Space Telescope (HST) images obtained about 15 years ago to constrain the progenitor properties. Results. The light curves of AT 2018hso show a first sharp peak (reddening-corrected M-r = -13.93 mag), followed by a broader and shallower second peak that resembles a plateau in the optical bands. The spectra dramatically change with time. Early-time spectra show prominent Balmer emission lines and a weak [Ca II] doublet, which is usually observed in ILRTs. However, the strong decrease in the continuum temperature, the appearance of narrow metal absorption lines, the great change in the H alpha strength and profile, and the emergence of molecular bands support an LRN classification. The possible detection of a M-I similar to -8 mag source at the position of AT 2018hso in HST archive images is consistent with expectations for a pre-merger massive binary, similar to the precursor of the 2015 LRN in M101. Conclusions. We provide reasonable arguments to support an LRN classification for AT 2018hso. This study reveals growing heterogeneity in the observables of LRNe than has been thought previously, which is a challenge for distinguishing between LRNe and ILRTs. This suggests that the entire evolution of gap transients needs to be monitored to avoid misclassifications

Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking

We have recently solved the inverse spectral problem for integrable PDEs in arbitrary dimensions arising as commutation of multidimensional vector fields depending on a spectral parameter $\lambda$. The associated inverse problem, in particular, can be formulated as a non linear Riemann Hilbert (NRH) problem on a given contour of the complex $\lambda$ plane. The most distinguished examples of integrable PDEs of this type, like the dispersionless Kadomtsev-Petviashivili (dKP), the heavenly and the 2 dimensional dispersionless Toda equations, are real PDEs associated with Hamiltonian vector fields. The corresponding NRH data satisfy suitable reality and symplectic constraints. In this paper, generalizing the examples of solvable NRH problems illustrated in \cite{MS4,MS5,MS6}, we present a general procedure to construct solvable NRH problems for integrable real PDEs associated with Hamiltonian vector fields, allowing one to construct implicit solutions of such PDEs parametrized by an arbitrary number of real functions of a single variable. Then we illustrate this theory on few distinguished examples for the dKP and heavenly equations. For the dKP case, we characterize a class of similarity solutions, a class of solutions constant on their parabolic wave front and breaking simultaneously on it, and a class of localized solutions breaking in a point of the $(x,y)$ plane. For the heavenly equation, we characterize two classes of symmetry reductions.Comment: 29 page

Use of camera trapping in determining Iberian lynx population parameters: The use area and its limitations

Below are the results of the survey of the Iberian lynx obtained with camera-trapping between 2000 and 2007 in Sierra Morena. Two very important aspects of camera-trapping concerning its efficiency are also analyzed. The first is the evolution along years according to the camera-trapping type used of two efficiency indicators. The results obtained demonstrate that the most efficient lure is rabbit, though it is the less proven (92 trap-nights), followed by camera-trapping in the most frequent marking places (latrines). And, we propose as a novel the concept of use area as a spatial reference unit for the camera-trapping monitoring of non radio-marked animals is proposed, and its validity discussed