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Advances in computer vision as well as increasingly widespread video-based behavioral monitoring have great potential for transforming how we study animal cognition and behavior. However, there is still a fairly large gap between the exciting prospects and what can actually be achieved in practice today, especially in videos from the wild. With this perspective paper, we want to contribute towards closing this gap, by guiding behavioral scientists in what can be expected from current methods and steering computer vision researchers towards problems that are relevant to advance research in animal behavior. We start with a survey of the state-of-the-art methods for computer vision problems that are directly relevant to the video-based study of animal behavior, including object detection, multi-individual tracking, individual identification, and (inter)action recognition. We then review methods for effort-efficient learning, which is one of the biggest challenges from a practical perspective. Finally, we close with an outlook into the future of the emerging field of computer vision for animal behavior, where we argue that the field should develop approaches to unify detection, tracking, identification and (inter)action recognition in a single, video-based framework.

We prove that the nonlinear Fourier transform of the Benjamin-Ono equation on $\mathbb{T}$, also referred to as Birkhoff map, is a real analytic diffeomorphism from the scale of Sobolev spaces $H^{s}_{0}(\mathbb{T},\mathbb{R})$, $s > -1/2$, to the scale of weighted $\ell^2-$sequence spaces, $\mathfrak{h}^{s +1/2}_{r,0}(\mathbb{N},\mathbb{C})$, $s >-1/2$. As an application we show that for any $-1/2

Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the Benjamin-Ono equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudo-differential operator of order 0 with principal part given by a modified Fourier transform (modification by a phase factor) and (2) the pullback of the Hamiltonian of the Benjamin-Ono is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of para-differential operators. Such coordinates are a key ingredient for studying the stability of finite gap solutions of the Benjamin-Ono equation under small, quasi-linear perturbations., Comment: arXiv admin note: substantial text overlap with arXiv:1812.05391

We prove smoothing properties of the solutions of the Benjamin-Ono equation in the Sobolev space $H^{s}(\mathbb{T},\mathbb{R})$ for any $s\ge 0$. To this end we show that Tao's gauge transform is a high frequency approximation of the nonlinear Fourier transform $\Phi$ for the Benjamin-Ono equation, constructed in our previous work. The results of this paper are manifestations of the quasi-linear character of the Benjamin-Ono equation.

The Thinking Wave is an ongoing development of visualization concepts showing the real-time effort and confidence of semi-autonomous vehicle (AV) systems. Offering drivers access to this information can inform their decision making, and enable them to handle the situation accordingly and takeover when necessary. Two different visualizations have been designed, Concept one, Tidal, demonstrates the AV systems effort through intensified activity of a simple graphic which fluctuates in speed and frequency. Concept two, Tandem, displays the effort of the AV system as well as the handling dynamic and shared responsibility between the driver and the vehicle system. Working collaboratively with mobility research teams at the University of Tokyo, we are prototyping and refining the Thinking Wave and its embodiments as we work towards building a testable version integrated into a driving simulator. The development of the thinking wave aims to calibrate trust by increasing the drivers knowledge and understanding of vehicle handling capacity. By enabling transparent communication of the AV systems capacity, we hope to empower AV-skeptic drivers and keep over-trusting drivers on alert in the case of an emergency takeover situation, in order to create a safer autonomous driving experience., Comment: 8 pages, 11 figures, 2021 IEEE Intelligent Vehicles Symposium (IV21)

In this paper, we survey our recent results on the Benjamin-Ono equation on the torus. As an application of the methods developed we construct large families of periodic or quasiperiodic solutions, which are not $C^\infty$-smooth.

In this paper we prove that the Benjamin-Ono equation admits an analytic Birkhoff normal form in an open neighborhood of zero in $H^{s}_{0}(\T, \R)$ for any $s>-1/2$ where $H^{s}_{0}(\T, \R)$ denotes the subspace of the Sobolev space $H^{s}(\T, \R)$ of elements with mean $0$. As an application we show that for any $-1/2

In this paper we obtain the following stability result for periodic multi-solitons of the KdV equation: We prove that under any given semilinear Hamiltonian perturbation of small size $\varepsilon > 0$, a large class of periodic multi-solitons of the KdV equation, including ones of large amplitude, are orbitally stable for a time interval of length at least $O(\varepsilon^{-2})$. To the best of our knowledge, this is the first stability result of such type for periodic multi-solitons of large size of an integrable PDE., Comment: The title has been changed. Other minor changes in the text

Two-sided bounds for the efficiency of the torsion function are obtained in terms of the square of the distance to the boundary function under the hypothesis that the Dirichlet Laplacian satisfies a strong Hardy inequality. Localisation properties of the torsion function are obtained under that hypothesis. An example is analysed in detail., Comment: 20 pages, 4 figures

We investigate the spectrum of the Lax operator $L_u$ of the Benjamin-Ono equation on the torus for complex valued potentials $u$ in the Sobolev space $H^{-s}(\mathbb{T},\mathbb{C})$, $0 \le s < 1/2$, with small imaginary part and prove analytic properties of the moment map, defined in terms of spectral data of $L_u$., Comment: In this revised version, we have corrected various misprints, updated and added some references, slightly simplified the exposition in Section 6, and made minor corrections to the estimate (50), the proof of Proposition 6, and the statement and proof of Lemma 18

We consider the torsion function for the Dirichlet Laplacian $-\Delta$, and for the Schr\"odinger operator $- \Delta + V$ on an open set $\Omega\subset \R^m$ of finite Lebesgue measure $0<|\Omega|<\infty$ with a real-valued, non-negative, measurable potential $V.$ We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction., Comment: 33 pages. The published version in Potential Analysis (2022) 57, 571--600 has some typos: Theorem 3(i): the first exponent should read $(m-2)/m$; Example 2 Line 2: .... $B(p_{n+1};cn^{-\beta})$.....; Formula (109): $\kappa^{-1}$ missing after the second inequality

We prove that the Benjamin--Ono equation on the torus is globally in time well-posed in the Sobolev space $H^{s}(\mathbb{T},\mathbb{R})$ for any $s > - 1/2$ and ill-posed for $s \le - 1/2$. Hence the critical Sobolev exponent $s_c=-1/2$ of the Benjamin--Ono equation is the threshold for well-posedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin-Ono equation on the torus are orbitally stable in $H^{s}(\mathbb{T},\mathbb{R})$ for any $ s > - 1/2$. Novel conservation laws and a nonlinear Fourier transform on $H^{s}(\mathbb{T},\mathbb{R})$ with $s > - 1/2$ are key ingredients into the proofs of these results., Comment: The paper is a greatly extended version of arXiv:1909.07314. In particular, we have included the result on ill-posedness of the Benjamin-Ono equation in $H^{-1/2}$ and added an appendix on the restriction of the Birkhoff map to the scale of Sobolev space $H^s$, $s>0$, and another one on a sharper form of ill-posedness of the Benjamin-Ono equation in $H^{s}_{r}$ with $s<-1/2$

The spectral curve associated with the sinh-Gordon equation on the torus is defined interms of the spectrum of the Lax operator appearing in the Lax pair formulation of the equation. If thespectrum is simple, it is an open Riemann surface of infinite genus. In this paper we construct normalizeddifferentials and derive estimates for the location of their zeroes., Comment: 58 pages, 7 figures

We prove an infinite dimensional version of the Arnold-Liouville theorem for integrable non-linear PDEs: In a case study we consider the {\em focusing} NLS equation with periodic boundary conditions.

We prove that for any $0 < s < 1/2$, the Benjamin--Ono equation on the torus is globally in time $C^0-$well-posed on the Sobolev space $H^{-s}(\T, \R)$,in the sense that the solution map, which is known to be defined for smooth data, continuously extends to $H^{-s}(\T,\R)$. The solution map does not extend continuously to $H^{-s}(\T, \R)$ with $s > 1/2$. Hence the critical Sobolev exponent $s_c=-1/2$ of the Benjamin--Ono equation is the threshold for well-posedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin--Ono equation on the torus are orbitally stable in $H^{-s}(\T,\R)$ for any $0\le s<1/2$., Comment: This is an extended version of the paper submitted on September 16, 2019. It contains additional new results

In this paper we prove the persistence of space periodic multi-solitons of arbitrary size under any quasi-linear Hamiltonian perturbation, which is smooth and sufficiently small. This answers positively a longstanding question whether KAM techniques can be further developed to prove the existence of quasi-periodic solutions of arbitrary size of strongly nonlinear perturbations of integrable PDEs., Comment: The new version contains a revised introduction

Image captioning models typically follow an encoder-decoder architecture which uses abstract image feature vectors as input to the encoder. One of the most successful algorithms uses feature vectors extracted from the region proposals obtained from an object detector. In this work we introduce the Object Relation Transformer, that builds upon this approach by explicitly incorporating information about the spatial relationship between input detected objects through geometric attention. Quantitative and qualitative results demonstrate the importance of such geometric attention for image captioning, leading to improvements on all common captioning metrics on the MS-COCO dataset., Comment: 10 pages

In this paper we prove that the Benjamin-Ono equation, when considered on the torus, is an integrable (pseudo)differential equation in the strongest possible sense: it admits global Birkhoff coordinates on the space $L^2(\T)$. These are coordinates which allow to integrate it by quadrature and hence are also referred to as nonlinear Fourier coefficients. As a consequence, all the $L^2(\T)$ solutions of the Benjamin--Ono equation are almost periodic functions of the time variable. The construction of such coordinates relies on the spectral study of the Lax operator in the Lax pair formulation of the Benjamin--Ono equation and on the use of a generating functional, which encodes the entire Benjamin--Ono hierarchy., Comment: 59 pages This new version corrected some typos and updated the references

Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the KdV equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudodifferential operator of order 0 with principal part given by the Fourier transform and (2) the pullback of the KdV Hamiltonian is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of a paradifferential operator. Such coordinates are a key ingredient for studying the stability of finite gap solutions of the KdV equation under small, quasi-linear perturbations.

We prove that there exist potentials so that near them the focusing non-linear Schr\"odinger equation does not admit local Birkhoff coordinates. The proof is based on the construction of a local normal form of the linearization of the equation at such potentials.

This preprint has been reviewed and recommended by Peer Community In Ecology (https://dx.doi.org/10.24072/pci.ecology.100001). In a recent article, Bicca-Marques and Calegaro-Marques [2016] discussed the putative assumptions related to an interpretation we provided regarding an observed positive relationship between weekly averaged parasite richness of a group of mandrills (Mandrillus sphinx) and their daily path lengths (DPL), published earlier in the same journal (Brockmeyer et al., 2015). In our article, we proposed, inter alia, that "the daily travels of mandrills could be seen as a way to escape contaminated habitats on a local scale". In their article, Bicca-Marques and Calegaro-Marques [2016] proposed an alternative mechanism that they considered to be more parsimonious. In their view, increased DPL also increases exposure to novel parasites from the environment. In other words, while we proposed that elevated DPL may be a consequence of elevated parasite richness, they viewed it as a cause. We are happy to see that our study attracted so much interest that it evoked a public comment. We are also grateful to Bicca-Marques and Calegaro-Marques [2016] for pointing out an obvious alternative scenario that we failed to discuss and for laying out several key factors and assumptions that should be addressed by future studies examining the links between parasite risk and group ranging. We use this opportunity to advance this discourse by responding to some of the criticisms raised in their discussion of our article. In this reply, we briefly contextualize the main object of criticism. We then discuss the putative parsimony of the two competing scenarios., Comment: 9 pages, no figure, no table

Bounds are obtained for the $L^p$ norm of the torsion function $v_{\Omega}$, i.e. the solution of $-\Delta v=1,\, v\in H_0^1(\Omega),$ in terms of the Lebesgue measure of $\Omega$ and the principal eigenvalue $\lambda_1(\Omega)$ of the Dirichlet Laplacian acting in $L^2(\Omega)$. We show that these bounds are sharp for $1\le p\le 2$., Comment: 14 pages

In this paper we prove a wellposedness result of the KdV equation on the space of periodic pseudo-measures, also referred to as the Fourier Lebesgue space $\mathscr{F}\ell^{\infty}(\mathbb{T},\mathbb{R})$, where $\mathscr{F}\ell^{\infty}(\mathbb{T},\mathbb{R})$ is endowed with the weak* topology. Actually, it holds on any weighted Fourier Lebesgue space $\mathscr{F}\ell^{s,\infty}(\mathbb{T},\mathbb{R})$ with $-1/2 < s \le 0$ and improves on a wellposedness result of Bourgain for small Borel measures as initial data. A key ingredient of the proof is a characterization for a distribution $q$ in the Sobolev space $H^{-1}(\mathbb{T},\mathbb{R})$ to be in $\mathscr{F}\ell^{\infty}(\mathbb{T},\mathbb{R})$ in terms of asymptotic behavior of spectral quantities of the Hill operator $-\partial_{x}^{2} + q$. In addition, wellposedness results for the KdV equation on the Wiener algebra are proved., Comment: 45 pages. arXiv admin note: text overlap with arXiv:1502.05857

In a case study for integrable PDEs, we construct real analytic, canonical coordinates for the defocusing NLS equation on the circle, specifically taylored towards the needs in perturbation theory. They are defined in neighbourhoods of families of finite dimensional invariant tori and are shown to satisfy together with their derivatives tame estimates. When expressed in these coordinates, the dNLS Hamiltonian is in normal form up to order three.

We prove that the renormalized defocusing mKdV equation on the circle is locally in time $C^{0}$-wellposed on the Fourier Lebesgue space ${\mathcal{F}\ell}^p$ for any $2 < p < \infty$. The result implies that the defocusing mKdV equation itself is illposed on these spaces since the renormalizing phase factor becomes infinite. The proof is based on the fact that the mKdV equation is an integrable PDE whose Hamiltonian is in the NLS hierarchy. A key ingredient is a novel way of representing the bi-infinite sequence of frequencies of the renormalized defocusing mKdV equation, allowing to analytically extend them to ${\mathcal{F}\ell}^p$ for any $2 \le p < \infty$ and to deduce asymptotics for $n \to \pm \infty$., Comment: 33 pages

In form of a case study for the KdV and the KdV2 equations, we present a novel approach of representing the frequencies of integrable PDEs which allows to extend them analytically to spaces of low regularity and to study their asymptotics. Applications include convexity properties of the Hamiltonians and wellposedness results in spaces of low regularity. In particular, it is proved that on $H^{s}$ the KdV2 equation is $C^{0}$-wellposed if $s\ge 0$ and illposed (in a strong sense) if $s < 0$., Comment: 76 pages

We prove that small, semi-linear Hamiltonian perturbations of the defocusing nonlinear Schr\"odinger (dNLS) equation on the circle have an abundance of invariant tori of any size and (finite) dimension which support quasi-periodic solutions. When compared with previous results the novelty consists in considering perturbations which do not satisfy any symmetry condition (they may depend on $x$ in an arbitrary way) and need not be analytic. The main difficulty is posed by pairs of almost resonant dNLS frequencies. The proof is based on the integrability of the dNLS equation, in particular the fact that the nonlinear part of the Birkhoff coordinates is one smoothing. We implement a Newton-Nash-Moser iteration scheme to construct the invariant tori. The key point is the reduction of linearized operators, coming up in the iteration scheme, to $ 2 \times 2 $ block diagonal ones with constant coefficients together with sharp asymptotic estimates of their eigenvalues.

The well-being of young adults in the United States today remains an area of key interest to the public and policy-makers alike. This age group faces the well-known challenges of achieving financial and social independence while forming their own households at a time of greater economic uncertainty than in the past. Better understanding of the achievements and needs of these young adults will inform approaches to best support this exciting and challenging transition to adulthood. Over the 20 years since it held its first organizational meetings, the Federal Interagency Forum on Child and Family Statistics (the Forum) has established a tradition of cooperation and commitment to understanding the challenges and opportunities facing children and families today. This year, in a Special Issue on America's Young Adults, the Forum extends that commitment to describing the well-being of youth as they transition into adulthood. This report focuses on key themes that are of particular relevance in describing the demographic status and well-being of today's young adults against a backdrop of the traditional milestones of adulthood--Education; Economic Circumstances; Family Formation; Civic, Social, and Personal Behavior; and Health and Safety. Hyperlinks to data sources referenced in the "America's Young Adults: Special Issue," are included in the report. This Special Issue provides data on young adults, focusing on their demographic characteristics, living situations, and overall well-being. The data come from several nationally representative, federally sponsored surveys. Together, the data from these surveys show a snapshot of the status and well-being of young adults, defined wherever possible as ages 18-24. [Additional assistance in writing and producing this report was provided by: Megan J. Benetsky, Thomas A. File, Tiffany A. Julian, Robin A. Cohen, Casey E. Copen, Catherine R. Duran, Charlotte A. Schoenborn, Joyce C. Abma, Ari Miniño, Stephanie J. Ventura, Cynthia A. Reuben, Cheryl D. Fryar, Holly B. Hedegaard, and Kenneth B. Quinto.]

In this paper we consider Schr{\"o}dinger equations with nonlinearities of odd order 2$\sigma$ + 1 on T^d. We prove that for $\sigma$d$\ge$2, they are strongly illposed in the Sobolev space H^s for any s \textless{} 0, exhibiting norm-inflation with infinite loss of regularity. In the case of the one-dimensional cubic nonlinear Schr{\"o}dinger equation and its renormalized version we prove such a result for H^s with s \textless{} --2/3., Comment: 18 pages

In this paper we prove approximation properties of the solutions of the defoucsing NLS equation on the circle by nearly linear flows. In addition we show that spatially periodic solutions of the defocusing NLS equation evolving in fractional Sobolev spaces $H^s$ with $s\geq 1$ remain bounded for all times.

In this paper we prove that the non-linear Fourier transform of the defocusing NLS equation on the circle is linear up to terms which are one order smoother.

In this paper we provide new asymptotic estimates of various spectral quantities of Zakharov-Shabat operators on the circle. These estimates are uniform on bounded subsets of potentials in Sobolev spaces.

We prove that the nonlinear part $H^{*}$ of the KdV Hamiltonian $H^{kdv}$, when expressed in action variables $I = (I_{n})_{n\ge 1}$, extends to a real analytic function on the positive quadrant $\ell^2_+(\mathbb N)$ of $\ell^{2}(\mathbb N)$ and is strictly concave near $0$. As a consequence, the differential of $H^{*}$ defines a local diffeomorphism near $0$ of $\ell_{\mathbb C}^{2}(\mathbb N)$., Comment: 48 pages

Background:The design of new nuclear reactors and transmutation devices requires to reduce the present neutron cross section uncertainties of minor actinides. Purpose: Reduce the $^{243}$Am(n,$\gamma$) cross section uncertainty. Method: The $^{243}$Am(n,$\gamma$) cross section has been measured at the n_TOF facility at CERN with a BaF$_{2}$ Total Absorption Calorimeter, in the energy range between 0.7 eV and 2.5 keV. Results: The $^{243}$Am(n,$\gamma$) cross section has been successfully measured in the mentioned energy range. The resolved resonance region has been extended from 250 eV up to 400 eV. In the unresolved resonance region our results are compatible with one of the two incompatible capture data sets available below 2.5 keV. The data available in EXFOR and in the literature has been used to perform a simple analysis above 2.5 keV. Conclusions: The results of this measurement contribute to reduce the $^{243}$Am(n,$\gamma$) cross section uncertainty and suggest that this cross section is underestimated up to 25% in the neutron energy range between 50 eV and a few keV in the present evaluated data libraries.

We develop a new approach for constructing normalized differentials on hyperelliptic curves of infinite genus and obtain uniform asymptotic estimates for the distribution of their zeros.

Let $(X_0, X_1)$ and $(Y_0, Y_1)$ be complex Banach couples and assume that $X_1\subseteq X_0$ with norms satisfying $\|x\|_{X_0} \le c\|x\|_{X_1}$ for some $c > 0$. For any $0<\theta <1$, denote by $X_\theta = [X_0, X_1]_\theta$ and $Y_\theta = [Y_0, Y_1]_\theta$ the complex interpolation spaces and by $B(r, X_\theta)$, $0 \le \theta \le 1,$ the open ball of radius $r>0$ in $X_\theta$, centered at zero. Then for any analytic map $\Phi: B(r, X_0) \to Y_0+ Y_1$ such that $\Phi: B(r, X_0)\to Y_0$ and $\Phi: B(c^{-1}r, X_1)\to Y_1$ are continuous and bounded by constants $M_0$ and $M_1$, respectively, the restriction of $\Phi$ to $B(c^{-\theta}r, X_\theta)$, $0 < \theta < 1,$ is shown to be a map with values in $Y_\theta$ which is analytic and bounded by $M_0^{1-\theta} M_1^\theta$.

For periodic Toda chains with a large number $N$ of particles we consider states which are $N^{-2}-$close to the equilibrium and constructed by discretizing any given $C^2-$functions with mesh size $N^{-1}$. For such states we derive asymptotic expansions of the Toda frequencies $(\omega^N_n)_{0 < n < N}$ and the actions $(I^N_n)_{0 < n < N},$ both listed in the standard way, in powers of $N^{-1}$ as $N \to \infty$. %listed in accordance with the ordering of the frequencies at the equilibrium, %$(2 \sin \frac{n\pi} {N})_{0 < n < N}$. At the two edges $n \sim 1$ and $N -n \sim 1$, the expansions of the frequencies are computed up to order $N^{-3}$ with an error term of higher order. Specifically, the coefficients of the expansions of $\omega^N_n$ and $\omega^N_{N-n}$ at order $N^{-3}$ are given by a constant multiple of the n'th KdV frequencies $\omega^-_n$ and $\omega^+_n$ of two periodic potentials, $q_{-}$ respectively $q_+$, constructed in terms of the states considered. The frequencies $\omega^N_n$ for $n$ away from the edges are shown to be asymptotically close to the frequencies of the equilibrium. For the actions $(I^N_n)_{0 < n < N},$ asymptotics of a similar nature are derived.

For periodic Toda chains with a large number $N$ of particles we consider states which are $N^{-2}$-close to the equilibrium and constructed by discretizing arbitrary given $C^2-$functions with mesh size $N^{-1}.$ Our aim is to describe the spectrum of the Jacobi matrices $L\_N$ appearing in the Lax pair formulation of the dynamics of these states as $N \to \infty$. To this end we construct two Hill operators $H\_\pm$ -- such operators come up in the Lax pair formulation of the Korteweg-de Vries equation -- and prove by methods of semiclassical analysis that the asymptotics as $N \rightarrow \infty $ of the eigenvalues at the edges of the spectrum of $L\_N$ are of the form $\pm (2-(2N)^{-2} \lambda ^\pm \_n + \cdots )$ where $(\lambda ^\pm \_n)\_{n \geq 0}$ are the eigenvalues of $H\_\pm $. In the bulk of the spectrum, the eigenvalues are $o(N^{-2})$-close to the ones of the equilibrium matrix. As an application we obtain asymptotics of a similar type of the discriminant, associated to $L\_N$.

In a case study on asymptotics of spectral quantities of Schr\"odinger operators we show how the Riesz-Thorin theorem on the interpolation of linear operators can be extended to nonlinear maps.

We present examples of isospectral operators that do not have the same heat content. Several of these examples are planar polygons that are isospectral for the Laplace operator with Dirichlet boundary conditions. These include examples with infinitely many components. Other planar examples have mixed Dirichlet and Neumann boundary conditions. We also consider Schr\"{o}dinger operators acting in $L^2[0,1]$ with Dirichlet boundary conditions, and show that an abundance of isospectral deformations do not preserve the heat content., Comment: 18 pages

In this paper we develop tools to study families of non-selfadjoint operators $L(\varphi), \varphi \in P$, characterized by the property that the spectrum of $L(\varphi)$ is (partially) simple. As a case study we consider the Zakharov-Shabat operators $L(\varphi)$ appearing in the Lax pair of the focusing NLS on the circle. The main result says that the set of potentials $\varphi $ of Sobolev class $H^N, N \geq 0$, so that all small eigenvalues of $L(\varphi)$ are simple, is path connected and dense.

In this paper we prove new qualitative features of solutions of KdV on the circle. The first result says that the Fourier coefficients of a solution of KdV in Sobolev space $H^N,\, N\geq 0$, admit a WKB type expansion up to first order with strongly oscillating phase factors defined in terms of the KdV frequencies. The second result provides estimates for the approximation of such a solution by trigonometric polynomials of sufficiently large degree.

In this paper we provide new asymptotic estimates of the Floquet exponents of Schr\"odinger operators on the circle. By the same techniques, known asymptotic estimates of various others spectral quantities are improved.

For $M$ being a closed manifold or the Euclidean space we present a detailed proof of regularity properties of the composition of $H^s$-regular diffeomorphisms of $M$ for $s > 1/2\dim M + 1$.

This paper describes the construction of a canonical compactification of the space of trajectories and of the unstable/stable sets of a generic gradient like vector field on a closed manifold as well as a canonical structure of a smooth manifold with corners of these spaces. As an application we discuss the geometric complex associated with a gradient like vector field and show how differential forms can be integrated on its unstable/stable sets. Integration leads to a morphism between the de Rham complex and the geometric complex., Comment: 65 pages, 19 figures

We study the Linial--Meshulam model of random two-dimensional simplicial complexes. One of our main results states that for $p\ll n^{-1}$ a random 2-complex $Y$ collapses simplicially to a graph and, in particular, the fundamental group $\pi_1(Y)$ is free and $H_2(Y)=0$, a.a.s. We also prove that, if the probability parameter $p$ satisfies $p\gg n^{-1/2+\epsilon}$, where $\epsilon>0$, then an arbitrary finite two-dimensional simplicial complex admits a topological embedding into a random 2-complex, with probability tending to one as $n\to \infty$. We also establish several related results, for example we show that for $p

In this paper we study the Linial-Meshulam model of random two-dimensional complexes. We prove that a random 2-complex is homotopically one dimensional, with probability tending to one as n tends to infitnity, assuming that the probability parameter p satisfies pn --> 0.

We construct normalized differentials on families of curves of infinite genus. Such curves are used to investigate integrable PDE's such as the focusing Nonlinear Schr{\"o}dinger equation.

We consider the nonlinear Schrodinger equation with a cubic nonlinearity on the circle, which is known to represent an integrable Hamiltonian system. We construct a global coordinate systems, which puts this Hamiltonian into standard normal form, so that it depends only on the actions of the system., Comment: March 2009 version

In this paper we present the evolution of a low mass model (initial mass M=1.5 Msun) with a very low metal content (Z=5x10^{-5}, equivalent to [Fe/H]=-2.44). We find that, at the beginning of the AGB phase, protons are ingested from the envelope in the underlying convective shell generated by the first fully developed thermal pulse. This peculiar phase is followed by a deep third dredge up episode, which carries to the surface the freshly synthesized 13C, 14N and 7Li. A standard TP-AGB evolution, then, follows. During the proton ingestion phase, a very high neutron density is attained and the s-process is efficiently activated. We therefore adopt a nuclear network of about 700 isotopes, linked by more than 1200 reactions, and we couple it with the physical evolution of the model. We discuss in detail the evolution of the surface chemical composition, starting from the proton ingestion up to the end of the TP-AGB phase., Comment: Accepted for Publication on PASA