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Value of cross section is a fundamental parameter to depict the transport of charged particles in matters. Due to masses of orders of magnitude higher than electrons and convenience of realistic calculation, the cross section of elastic nuclei-nuclei collision is usually treated via classical mechanics. The famous Bohr criterion was firstly proposed to judge whether the treatment via classical mechanics is reliable or not. Later, Lindhard generalized the results of Coulomb to screening potentials. Considering the increasing importance of detailed ion-ion interactions under modern simulation codes in inertial confinement fusion (ICF) researches, the validation of classical transport cross section for ion-ion interactions in a big range of parameter space is certainly required. In this work, the transport cross sections via classical mechanics under repulsive Yukawa potential are compared with those via quantum mechanics. Differences of differential cross sections are found with respect to scattering angles and velocities. Our results generally indicate that the classical picture fails at the cases of both low and high velocities, which represent a significant extension of the famous Bohr criterion and its generalized variations. Furthermore, the precise validation zones of classical picture is also analysed in this work. This work is of significant importance for benchmarking the modern ion-kinetic simulation codes in ICF researches, concerning the stopping power of $\alpha$ particles in DT fuels, ion-ion friction and viscous effects in the formation of kinetic shocks.

Although high energetic radiation from flares is a potential threat to exoplanet atmospheres and may lead to surface sterilization, it might also provide the extra energy for low-mass stars needed to trigger and sustain prebiotic chemistry. We investigate two flares on TRAPPIST-1, an ultra-cool dwarf star that hosts seven exoplanets of which three lie within its habitable zone. The flares are detected in all four passbands of the MuSCAT2 allowing a determination of their temperatures and bolometric energies. We analyzed the light curves of the MuSCAT1 and MuSCAT2 instruments obtained between 2016 and 2021 in $g,r,i,z_\mathrm{s}$-filters. We conducted an automated flare search and visually confirmed possible flare events. We studied the temperature evolution, the global temperature, and the peak temperature of both flares. For the first time we infer effective black body temperatures of flares that occurred on TRAPPIST-1. The black body temperatures for the two TRAPPIST-1 flares derived from the SED are consistent with $T_\mathrm{SED} = 7940_{-390}^{+430}$K and $T_\mathrm{SED} = 6030_{-270}^{+300}$K. The flare black body temperatures at the peak are also calculated from the peak SED yielding $T_\mathrm{SEDp} = 13620_{-1220}^{1520}$K and $T_\mathrm{SEDp} = 8290_{-550}^{+660}$K. We show that for the ultra-cool M-dwarf TRAPPIST-1 the flare black body temperatures associated with the total continuum emission are lower and not consistent with the usually adopted assumption of 9000-10000 K. This could imply different and faster cooling mechanisms. Further multi-color observations are needed to investigate whether or not our observations are a general characteristic of ultra-cool M-dwarfs. This would have significant implications for the habitability of exoplanets around these stars because the UV surface flux is likely to be overestimated by the models with higher flare temperatures., Comment: 17 pages, 11 figures, 2 tables, accepted for publication in A&A

In contrast with the 3D result, the Beth-Uhlenbeck (BU) formula in 1D contains an extra -1/2 term. The origin of this -1/2 term is explained using a spectral density approach. To be explicit, a delta-function potential is used to show that the correction term arises from a pole of the density of states at zero energy. The spectral density method shows that this term is actually an artifact of the non-normalizability of the scattering states and an infrared cutoff regularization scheme has to be used to get the correct result in 1D. The formal derivation of the BU formula would miss this term since it ignores the effects of the boundary terms. While the result is shown for the delta-function potential, the method and result are valid for more general potentials. Additionally, the 1D Levinson's theorem can be extracted from the spectral density method using the asymptotic form of general potentials. The importance of the result lies in the fact that all these correction terms in 1D have a universal source: a pole at zero energy. Similar calculations using quantum field theoretical approaches (without explicit infrared cutoff regularization schemes) also show the same subtleties with the correction term originating from the zero energy scattering states (appendix A)., Comment: 12 pages, 3 figures

In this paper, we would like to derive a quantitative uniqueness estimate, the three-region inequality, for the second order elliptic equation with jump discontinuous coefficients. The derivation of the inequality relies on the Carleman estimate proved in our previous work. We then apply the three-region inequality to study the size estimate problem with one boundary measurement., Comment: 16 pages, 1 figure

In this paper we prove a local Carleman estimate for second order elliptic equations with a general anisotropic Lipschitz coefficients having a jump at an interface. Our approach does not rely on the techniques of microlocal analysis. We make use of the elementary method so that we are able to impose almost optimal assumptions on the coefficients and, consequently, the interface.

This paper concerns about the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumption which called {\sl basic assumptions}, but also some technical assumptions which we called {\sl further assumptions}. It is shown as usual by first applying the Holmgren transform to this inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate given via a partition of unity and Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point. A little more details about this are as follows. Factorize this operator with constant coefficients into two first order differential operators. Conjugate each factor by a Carleman weight and derive an estimate which is uniform with respect to the point at which we froze the coefficients for each conjugated factor by constructing a parametrix for its adjoint operator.

In this paper we prove a quantitative form of the strong unique continuation property for the Lam\'e system when the Lam\'e coefficients $\mu$ is Lipschitz and $\lambda$ is essentially bounded in dimension $n\ge 2$. This result is an improvement of our earlier result \cite{lin5} in which both $\mu$ and $\lambda$ were assumed to be Lipschitz., Comment: 10 pages

Precision continuous-wave NMR measurements have been carried out over the entire magnetization curve of EuO and are presented in tabular form. Two very closely spaced resonances are observed and are attributed to domain and domain-wall signals. Both of the signals are useful for analysis in the spin-wave region. Only the domain signal is measurable above ~50K. The latter is used for fitting Tc and the critical exponent beta. The critical-region fits agree with previous measurements, within experimental error. The low-temperature data exhibit a clear-cut T^2 behavior, at variance with the expectations of conventional spin-wave theory. This result is discussed in relation to two semi-empirical spin-wave schemes, one formulated by N. Bykovetz, and one by U. Koebler. The NMR signal at 4.2K gives no indication of a quadrupole splitting, in contradiction to the interpretation of several previous spin-echo NMR spectra observed in EuO. This issue remains unresolved., Comment: 3 pages, 2 figures, 3 tables. in Proceedings of the 11TH Joint MMM-Intermag Conference, Washington, DC, 2010