Lefschetz–Verdier Trace Formula and a Generalization of a Theorem of Fujiwara

The goal of this paper is to generalize a theorem of Fujiwara (Deligne’s conjecture) to the situation appearing in a joint work [KV] with David Kazhdan on the global Langlands correspondence over function fields. Moreover, our proof is more elementary than the original one and stays in the realm of ordinary algebraic geometry, that is, does not use rigid geometry. We also give a proof of the Lefschetz–Verdier trace formula and of the additivity of filtered trace maps, thus making the paper essentially self-contained. The work was supported by the Israel Science Foundation (Grant No. 555/04) Received: May 2005 Accepted: August 2005     

Determination of the electric field gradient asymmetry from Berry’s phase in NQR of powder samples

An effect of Berry’s phase on the NQR spectrum of the rotating powder sample is described and applied for the determination of the electric field gradient asymmetry. The proposed method involves the analysis of the frequency singularities in the NQR powder patterns of the rotating samples. The Berry’s phases for the eigenstates, associated with an adiabatically changing quadrupole hamiltonian, are calculated for nuclei with a spin I = 3/2 and I = 1 as a function of the asymmetry parameter.     

The Central Limit Problem for Random Vectors with Symmetries

Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. In particular, we obtain results for distributions which are coordinatewise symmetric, uniform in a regular simplex, or spherically symmetric. Our proofs are based on Stein’s method of exchangeable pairs; as far as we know, this approach has not previously been used in convex geometry. The spherically symmetric case is treated by a variation of Stein’s method which is adapted for continuous symmetries. This work was done while at Stanford University.     

Kinetics and mechanism of the formation of poly[(1,1,3,3-tetramethyldisiloxanyl)ethylene] and poly(methyldecylsiloxane) by hydrosilylation

The kinetics of the formation of poly(carbosiloxane), as well as of alkyl-substituted poly(siloxane), by Karstedt's catalyst catalyzed hydrosilylation were investigated. Linear poly(carbosiloxane), poly[(1,1,3,3-tetramethyldisiloxanyl)ethylene], (PTMDSE), was obtained by hydrosilylation of 1,3-divinyltetramethyldisiloxane (DVTMDS) and 1,1,3,3-tetramethyldisiloxane (TMDS), while alkyl-substituted poly(siloxane), poly(methyldecylsiloxane), (PMDS), was synthesized by hydrosilylation of poly(methylhydrosiloxane) (PMHS) and 1-decene. To investigate the kinetics of PTMDSE formation, two series of experiments were performed at reaction temperatures ranging from 25 to 56 °C and with catalyst concentrations ranging from 7.0 × 10⁻⁶ to 3.1 × 10⁻⁵ mol Pt/mol CHCH₂. A series of experiments was performed at reaction temperatures ranging from 28 to 48 °C, with catalyst concentrations of 7.0 ×10⁻⁶ mol of Pt per mol of CHCH₂, when kinetics of PMDS formation was investigated. All reactions were carried out in bulk, with equimolar amounts of the reacting Si H and CHCH₂ groups. The course of the reactions was monitored by following the disappearance of the Si H bands using quantitative infrared spectroscopy. The results obtained showed typical first order kinetics for the PTMDSE formation, consistent with the proposed reaction mechanism. In the case of PMDS an induction period occurred at lower reaction temperatures, but disappeared at 44 °C and the rate of Si H conversion also started to follow the first-order kinetics. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 2246–2258, 2007     

Ultrathin Fe film on Cu(0 0 1): Exchange splitting of image states from first principles

The theoretical investigation of the image states in front of an ultrathin iron film grown on copper has been performed by means of the embedding method and a recently developed procedure for the inclusion of the image potential tail in a first principle calculation. From the electronic response to an applied electric field, the image plane position has been evaluated. This also allows one to obtain useful information about the spin dependent screening properties of the system. Exchange splitting, effective mass, and lifetime of such surface states result in good agreement with recently performed two-photon photoemission experiments [see A.B. Schmidt, M. Pickel, M. Wiemhöfer, M. Donath, M. Weinelt, Phys. Rev. Lett. 95 (2005) 107402].     

Cauchy problem in the class of monotone increasing functions

The paper establishes some solvability conditions of the Cauchy problem for linear differential equation in the class of monotone increasing functions. The results are applied for clarifying the possibility of flight along a given trajectory under existence of braking forces.     

Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional

We study the generalization of the Willmore functional for surfaces in the three-dimensional Heisenberg group. Its construction is based on the spectral theory of the Dirac operator entering into theWeierstrass representation of surfaces in this group. Using the surfaces of revolution we demonstrate that the generalization resembles the Willmore functional for the surfaces in the Euclidean space in many geometrical aspects. We also observe the relation of these functionals to the isoperimetric problem.     

刘维尔与伽罗瓦数学手稿的发表

1846年,刘维尔在自己主办的杂志“纯粹与应用数学杂志”首次出版了伽罗瓦的数学研究,这对于伽罗瓦理论的传播与发展是具有决定意义的事件.伽罗瓦去世14年后,刘维尔发表伽罗瓦数学研究的原因是什么?采用数学历史文献分析法,得出四个重要原因:①伽罗瓦的朋友和弟弟的请求;②力图弥补科学院曾经造成的不公正;③刘维尔积极扶持年轻人的高贵品质使然;④刘维尔与利布里学术论战的促进.     

Irregular Behavior of Solutions for Fisher’s Equation

This paper is concerned with the irregular behavior of solutions for Fisher’s equation when initial data do not decay in a regular way at the spatial infinity. In the one-dimensional case, we show the existence of a solution whose profile and average speed are not convergent. In the higher-dimensional case, we show the existence of expanding fronts with arbitrarily prescribed profiles. We also show the existence of irregularly expanding fronts whose profile varies in time. Proofs are based on some estimate of the difference of two distinct solutions and a comparison technique. Dedicated to Professor Pavol Brunovsky on his 70th birthday.     

Sharp condition of global existence for second-order derivative nonlinear Schrödinger equations in two space dimensions

This paper discusses a class of second-order derivative nonlinear Schrödinger equations which are used to describe the upper-hybrid oscillation propagation. By establishing a variational problem, applying the potential well argument and the concavity method, we prove that there exists a sharp condition for global existence and blow-up of the solutions to the nonlinear Schrödinger equation. In addition, we also answer the question: how small are the initial data, the global solutions exist?