Partial data inverse problems for the Hodge Laplacian

We prove uniqueness results for a Calderon type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth order potential. The method is based on Carleman estimates for the Hodge Laplacian with relative or absolute boundary conditions, and on the construction of complex geometric optics solutions which reduce the Calderon type problem to a tensor tomography problem for 2-tensors. The arguments in this paper allow to establish partial data results for elliptic systems that generalize the scalar results due to Kenig-Sjostrand-Uhlmann.Comment: 54 pages, updated versio

Resolvent estimates for the magnetic Schr\"odinger operator in dimension n≥2n \geq 2

It is well known that the resolvent of the free Schr\"odinger operator on weighted $L^2$ spaces has norm decaying like $\lambda^{-\frac{1}{2}}$ at energy $\lambda$. There are several works proving analogous high-frequency estimates for magnetic Schr\"odinger operators, with large long or short range potentials, in dimensions $n \geq 3$. We prove that the same estimates remain valid in all dimensions $n \geq 2$.Comment: 21 page

A monolithic collapse origin for the thin/thick disc structure of ESO 243-49

ESO 243-49 is a high-mass (circular velocity $v_{\rm c}\approx200\,{\rm km\,s^{-1}}$) edge-on S0 galaxy in the Abell 2877 cluster at a distance of $\sim95\,{\rm Mpc}$. To elucidate the origin of its thick disc, we use MUSE science verification data to study its kinematics and stellar populations. The thick disc emits $\sim80\%$ of the light at heights in excess of $3.5^{\prime\prime}$ ($1.6\,{\rm kpc}$). The rotation velocities of its stars lag by $30-40\,{\rm km\,s^{-1}}$ compared to those in the thin disc, which is compatible with the asymmetric drift. The thick disc is found to be more metal-poor than the thin disc, but both discs have old ages. We suggest an internal origin for the thick disc stars in high-mass galaxies. We propose that the thick disc formed either ${\rm a)}$ first in a turbulent phase with a high star formation rate and that a thin disc formed shortly afterwards, or ${\rm b)}$ because of the dynamical heating of a thin pre-existing component. Either way, the star formation in ESO 243-49 was quenched just a few Gyrs after the galaxy was born and the formation of a thin and a thick disc must have occurred before the galaxy stopped forming stars. The formation of the discs was so fast that it could be described as a monolithic collapse where several generations of stars formed in a rapid succession.Comment: Accepted for publication in A&A. The reduced data-cube as well as the data necessary to build the kinematic and stellar population maps are available at https://etsin.avointiede.fi/dataset/urn-nbn-fi-csc-kata2016092414291163237

THE USE OF MOTION ANALYSIS AS A COACHING AID TO IMPROVE THE INDIVIDUAL TECHNIQUE IN SPRINT HURDLES

Biomechanical data are oflen presented as a group average, which may not always help individual athletes to improve their own performance. The purpose of this study was to analyse techniques in sprint hurdles within the athlete and find critical individual aspects, which influence performance. The hurdle clearance of three athletes (eight trials each) were videotaped with four video camera recorders and analysed three-dimensionally. There were several statistically significant correlations between the critical overall horizontal velocity and other variables, especially for one athlete. Such trends in individual performance presented ideas to coaches, athletes and also to researchers, regarding what happened in less successful runs and which technical points were worth individual attention in training

Exponential instability in the fractional Calder\'on problem

In this note we prove the exponential instability of the fractional Calder\'on problem and thus prove the optimality of the logarithmic stability estimate from \cite{RS17}. In order to infer this result, we follow the strategy introduced by Mandache in \cite{M01} for the standard Calder\'on problem. Here we exploit a close relation between the fractional Calder\'on problem and the classical Poisson operator. Moreover, using the construction of a suitable orthonormal basis, we also prove (almost) optimality of the Runge approximation result for the fractional Laplacian, which was derived in \cite{RS17}. Finally, in one dimension, we show a close relation between the fractional Calder\'on problem and the truncated Hilbert transform.Comment: 17 page

Microgravity experiments on the collisional behavior of Saturnian ring particles

In this paper we present results of two novel experimental methods to investigate the collisional behavior of individual macroscopic icy bodies. The experiments reported here were conducted in the microgravity environments of parabolic flights and the Bremen drop tower facility. Using a cryogenic parabolic-flight setup, we were able to capture 41 near-central collisions of 1.5-cm-sized ice spheres at relative velocities between 6 and $22 \mathrm{cm s^{-1}}$. The analysis of the image sequences provides a uniform distribution of coefficients of restitution with a mean value of $\overline{\varepsilon} = 0.45$ and values ranging from $\varepsilon = 0.06$ to 0.84. Additionally, we designed a prototype drop tower experiment for collisions within an ensemble of up to one hundred cm-sized projectiles and performed the first experiments with solid glass beads. We were able to statistically analyze the development of the kinetic energy of the entire system, which can be well explained by assuming a granular `fluid' following Haff's law with a constant coefficient of restitution of $\varepsilon = 0.64$. We could also show that the setup is suitable for studying collisions at velocities of $< 5 \mathrm{mm s^{-1}}$ appropriate for collisions between particles in Saturn's dense main rings.Comment: Accepted for publication in the Icarus Special Issue "Cassini at Saturn

Spatial Structure and Coherent Motion in Dense Planetary Rings Induced by Self-Gravitational Instability

We investigate the formation of spatial structure in dense, self-gravitating particle systems such as Saturn's B-ring through local $N$-body simulations to clarify the intrinsic physics based on individual particle motion. In such a system, Salo (1995) showed that the formation of spatial structure such as wake-like structure and particle grouping (clump) arises spontaneously due to gravitational instability and the radial velocity dispersion increases as the formation of the wake structure. However, intrinsic physics of the phenomena has not been clarified. We performed local $N$-body simulations including mutual gravitational forces between ring particles as well as direct (inelastic) collisions with identical (up to $N\sim40000$) particles. In the wake structure particles no longer move randomly but coherently. We found that particle motion was similar to Keplerian motion even in the wake structure and that the coherent motion was produced since the particles in a clump had similar eccentricity and longitude of perihelion. This coherent motion causes the increase and oscillation in the radial velocity dispersion. The mean velocity dispersion is rather larger in a more dissipative case with a smaller restitution coefficient and/or a larger surface density since the coherence is stronger in the more dissipative case. Our simulations showed that the wavelength of the wake structure was approximately given by the longest wavelength \hs{\lambda}{cr} = 4\pi^2 G\Sigma/\kappa^2 in the linear theory of axisymmetric gravitational instability in a thin disk, where $G$, $\Sigma$, and $\kappa$ are the gravitational constant, surface density, and a epicyclic frequency.Comment: Accepted by Earth, Planets, and Space. 39 pages, 20 figures. PostScript files also available from http://www.geo.titech.ac.jp/nakazawalab/hdaisaka/works