On the contact geometry of nodal sets

In the 3-dimensional Riemannian geometry, contact structures equipped with an adapted Riemannian metric are divergence-free, nondegenerate eigenforms of the Laplace-Beltrami operator. We trace out a two-dimensional consequence of this fact: there is a close relationship between the topology of the contact structure on a convex surface in the 3-manifold (the dividing curves) and the nodal curves of Laplacian eigenfunctions on that surface. Motivated by this relationship, we consider a topological version of Payne's conjecture for the free membrane problem. We construct counterexamples to Payne's conjecture for closed Riemannian surfaces. In light of the correspondence between the nodal lines and dividing curves, we interpret the conjecture in terms of the tight versus overtwisted dichotomy for contact structures.

     

A Highly Sensitive Femtosecond Time-Resolved Sum Frequency Generation Vibrational Spectroscopy System with Simultaneous Measurement of Multiple Polarization Combinations

Characterization of real-time and ultrafast motions of the complex molecules at surface and interface is critical to understand how interfacial molecules function. It requires to develop surface-sensitive, fast-identification, and time-resolved techniques. In this study, we employ several key technical procedures and successfully develop a highly sensitive femtosecond time-resolved sum frequency generation vibrational spectroscopy (SFG-VS) system. This system is able to measure the spectra with two polarization combinations (ssp and ppp, or psp and ssp) simultaneously. It takes less than several seconds to collect one spectrum. To the best of our knowledge, it is the fastest speed of collecting SFG spectra reported by now. Using the time-resolved measurement, ultrafast vibrational dynamics of the N-H mode of α-helical peptide at water interface is determined. It is found that the membrane environment does not affect the N-H vibrational relaxation dynamics. It is expected that the time-resolved SFG system will play a vital role in the deep understanding of the dynamics and interaction of the complex molecules at surface and interface. Our method may also provide an important technical proposal for the people who plan to develop time-resolved SFG systems with simultaneous measurement of multiple polarization combinations.     

A reformulation of the nonsmooth approaches of M. Frémond and J.J. Moreau for rigid body dynamics

This paper deals with the nonsmooth dynamics of a rigid bodies system. The proposed theory is inspired by the formalism of J.J. Moreau and that of M. Frémond and relies on the notion of percussion which is the integral of the contact force during the duration of the collision. Contrary to classical discrete element models, it is here assumed that percussions can be expressed as a function of only the velocity before the impact. This assumption is checked for the usual mechanical constitutive laws for collisions derived from a pseudopotential of dissipation or the Coulomb friction law. Motion equations are then reformulated taking into account simultaneous collisions of solids. A mathematical study of the new model is presented: the existence and uniqueness of the solution are discussed according to the regularity of both the forces (Lebesgue‐density occurring during the regular evolution of the system) and the percussions (Dirac‐density describing the collision). In the light of the principles of thermodynamics, a condition on the internal percussion assuring that the collision is thermodynamically admissible, is established. Finally, an application of this new model to the motion of a system of rigid disks, including simultaneous collisions is presented.     

Effect of Optical Depth on Study of Chemical Properties of Massive Star Forming Clumps

Here we present the study on chemical properties of massive star forming clumps using N\begin{document}$ _2 $\end{document}H\begin{document}$ ^+ $\end{document}(1-0), H\begin{document}$ ^{13} $\end{document}CO\begin{document}$ ^+ $\end{document}(1-0), HCN(1-0) and HN\begin{document}$ ^{13} $\end{document}C(1-0) data from the literature [Astron. Astrophys. 563 , A97 (2014)]. We found that abundances of H\begin{document}$ ^{13} $\end{document}CO\begin{document}$ ^+ $\end{document} and HN\begin{document}$ ^{13} $\end{document}C are affected by H\begin{document}$ _2 $\end{document} column densities. As the median values of these two abundances increase by nearly 10 times from stages A to B, H\begin{document}$ ^{13} $\end{document}CO\begin{document}$ ^+ $\end{document} and HN\begin{document}$ ^{13} $\end{document}C are suitable for tracing the evolution of massive star forming clumps. The order of rapidity in growth of abundances of all the four studied molecules from stages A to B, is H\begin{document}$ ^{13} $\end{document}CO\begin{document}$ ^+ $\end{document}, HCN, HN\begin{document}$ ^{13} $\end{document}C, and N\begin{document}$ _2 $\end{document}H\begin{document}$ ^+ $\end{document}, from the highest to the lowest. Our results suggest that the observing optically thin molecular lines with high angular resolution are necessary to study the chemical evolution of massive star forming clumps.     

Theoretical studies of the effect of the dipolar field in multiple spin-echo sequences with refocusing pulses of finite duration

It has been observed recently that the finite duration of refocusing rf pulses in a multiecho acquisition of the signal formed under the influence of the dipolar field leads to significant signal attenuation [S. Kennedy, Z. Chen, C.K. Wong, E.W.-C. Kwok, J. Zhong, Investigation of multiple-echo spin-echo signal acquisition under distant dipole-dipole interactions, Proc. Int. Soc. Magn. Reson. Med. 13 (2005) 2288]. Hereto, we quantify the phenomenon by evaluating analytically the influences of both the distant dipolar field (DDF) and transverse relaxation T2 on the magnetization in a multiecho pulse sequence based on correlation spectroscopy revamped by asymmetric z-gradient echo detection (CRAZED). Analytic expressions for the magnetization were obtained, which demonstrate explicitly the origin of rephased signal in the presence of the finite pi pulses in the multiecho train. The expressions also explain the effects of the DDF and T2 during the refocusing pulses on the signal strength, and show the substantial signal dependence on the phase of the rf pulses. We show that when the DDF effect during the pulse is canceled, the signal rises primarily during the free evolution time in the acquisition period. This elucidates the signal attenuation when the rf pulses cover a significant proportion of time in the sequence. In addition, we performed an optimization on the number of refocusing pulses that maximizes the total acquired signal using parameters for water, brain white matter, and muscle. We found that maximal signal-to-noise ratio is obtained when the pulse duration approximately equals the free evolution time in the samples with a wide range of T2.     

A multiplicity theorem for problems with the p-Laplacian

We consider a nonlinear elliptic problem driven by the p-Laplacian, with a parameter λ∈R and a nonlinearity exhibiting a superlinear behavior both at zero and at infinity. We show that if the parameter λ is bigger than λ₂=the second eigenvalue of , then the problem has at least three nontrivial solutions. Our approach combines the method of upper-lower solutions with variational techniques involving the Second Deformation Theorem. The multiplicity result that we prove extends an earlier semilinear (i.e. p=2) result due to Struwe [M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990].     

多元模糊回归预测模型及其应用

论述多元模糊回归预测模糊的建模方法,探讨该预测模型在第二代玉米螟百株卵量各动态上的应用,研究结果表明,该预测模型为害害虫群动态的中长期预测预报提供了一种新的研究方法,是一种优良的模型。