Non-degenerate solutions of universal Whitham hierarchy

The notion of non-degenerate solutions for the dispersionless Toda hierarchy is generalized to the universal Whitham hierarchy of genus zero with $M+1$ marked points. These solutions are characterized by a Riemann-Hilbert problem (generalized string equations) with respect to two-dimensional canonical transformations, and may be thought of as a kind of general solutions of the hierarchy. The Riemann-Hilbert problem contains $M$ arbitrary functions $H_a(z_0,z_a)$, $a = 1,...,M$, which play the role of generating functions of two-dimensional canonical transformations. The solution of the Riemann-Hilbert problem is described by period maps on the space of $(M+1)$-tuples $(z_\alpha(p) : \alpha = 0,1,...,M)$ of conformal maps from $M$ disks of the Riemann sphere and their complements to the Riemann sphere. The period maps are defined by an infinite number of contour integrals that generalize the notion of harmonic moments. The $F$-function (free energy) of these solutions is also shown to have a contour integral representation.Comment: latex2e, using amsmath, amssym and amsthm packages, 32 pages, no figur

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

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

Large N expansions and Painlev\'e hierarchies in the Hermitian matrix model

We present a method to characterize and compute the large N formal asymptotics of regular and critical Hermitian matrix models with general even potentials in the one-cut and two-cut cases. Our analysis is based on a method to solve continuum limits of the discrete string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. This method also leads to an explicit formulation, in terms of coupling constants and critical parameters, of the members of the Painlev\'e I and Painlev\'e II hierarchies associated with one-cut and two-cut critical models respectively

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

Gender bias in ecosystem restoration: from science to practice

The declaration of the United Nations Decade on Ecosystem Restoration 2020–2030 has established the need to focus on human rights in restoration initiatives, including gender equality. Although this goal raises a need to monitor gender biases on ecosystem restoration, we still lack basic gender information and evaluations on the current situation. The main purpose of this study is to analyze gender bias in ecosystem restoration covering three dimensions: research, outreach, and practice. We used scientific publications from the Restoration Ecology journal, mentions of these articles in Altmetric Explorer and Twitter, and projects from the Society for Ecological Restoration's database. First, we study gender bias among people leading ecosystem restoration initiatives in the three dimensions. Second, we assessed factors that could influence gender bias, including year, target ecosystem, and socioeconomic country development. Third, we analyzed whether the impact of scientific knowledge in society depends on the gender of the scientific team. Our results indicate that men were primary leaders in research, outreach, and practice initiatives in ecosystem restoration. There seems to be a trend over time toward equality in research, but gender inequality is still present in most types of ecosystems, with women leading more projects in more developed countries. The impact of scientific knowledge is independent of the author's gender, but research of male senior authors seems to reach society more easily. This broad perspective of inequality in the three dimensions can evolve toward gender equality, by applying gender approaches in restoration policies and initiatives. © 2022 Society for Ecological Restoration.Raw data is publicly available thanks to Web of Science, Altmetrics, Twitter and SER. Data and scripts used for the analysis are available via Figshare (Cruz‐Alonso et al. 2022 ). Funding: V.C.‐A.—Real Colegio Complutense postdoc fellowship; A.R.‐U.—Spanish State Research Agency through María de Maeztu Excellence Unit accreditation 2018–2022 (MDM‐2017‐0714); L.M.—Swiss National Science Foundation (PCEFP2_181115) and a Margarita Salas Postdoctoral Fellowship from Universidad de Alcalá; L.M.‐B.—Ministerio de Ciencia e Innovación (PID2019‐106806GB‐I00) and a Margarita Salas Postdoctoral Fellowship from Universidad de Alcalá; N.M.—predoctoral grant from Universidad de Alcalá; E.V.‐A.—European Commission (project SHOWCASE, H2020: 862480). We appreciate the support of the FIRE Foundation and the comments of M. Almaraz, M. Pajares, A. S. Moya, and D. Rohrer to improve the manuscript.Raw data is publicly available thanks to Web of Science, Altmetrics, Twitter and SER. Data and scripts used for the analysis are available via Figshare (Cruz‐Alonso et al. 2022 ). Funding: V.C.‐A.—Real Colegio Complutense postdoc fellowship; A.R.‐U.—Spanish State Research Agency through María de Maeztu Excellence Unit accreditation 2018–2022 (MDM‐2017‐0714); L.M.—Swiss National Science Foundation (PCEFP2_181115) and a Margarita Salas Postdoctoral Fellowship from Universidad de Alcalá; L.M.‐B.—Ministerio de Ciencia e Innovación (PID2019‐106806GB‐I00) and a Margarita Salas Postdoctoral Fellowship from Universidad de Alcalá; N.M.—predoctoral grant from Universidad de Alcalá; E.V.‐A.—European Commission (project SHOWCASE, H2020: 862480). We appreciate the support of the FIRE Foundation and the comments of M. Almaraz, M. Pajares, A. S. Moya, and D. Rohrer to improve the manuscript

The time traveler's guide to the quantization of zero modes

We study the relationship between the quantization of a massless scalar field on the two-dimensional Einstein cylinder and in a spacetime with a time machine. We find that the latter picks out a unique prescription for the state of the zero mode in the Einstein cylinder. We show how this choice arises from the computation of the vacuum Wightman function and the vacuum renormalized stress-energy tensor in the time-machine geometry. Finally, we relate the previously proposed regularization of the zero mode state as a squeezed state with the time-machine warp parameter, thus demonstrating that the quantization in the latter regularizes the quantization in an Einstein cylinder

Charge control in laterally coupled double quantum dots

We investigate the electronic and optical properties of InAs double quantum dots grown on GaAs (001) and laterally aligned along the [110] crystal direction. The emission spectrum has been investigated as a function of a lateral electric field applied along the quantum dot pair mutual axis. The number of confined electrons can be controlled with the external bias leading to sharp energy shifts which we use to identify the emission from neutral and charged exciton complexes. Quantum tunnelling of these electrons is proposed to explain the reversed ordering of the trion emission lines as compared to that of excitons in our system.Comment: 4 pages, 4 figures submitted to PRB Rapid Com