On the fundamental group-scheme

We show that the fundamental group-scheme of a separably rationally connected variety defined over an algebraically closed field is trivial. Let X be a geometrically irreducible smooth projective variety defined over a finite field k admitting a k-rational point. Let {En,σn}_n?0 be a flat principal G-bundle over X, where G is a reductive linear algebraic group defined over k. We show that there is a positive integer a such that the principal G-bundle is isomorphic to E₀, where FX is the absolute Frobenius morphism of X. From this it follows that E₀ is given by a representation of the fundamental group-scheme of X in G.     

格林公式的教学探究

首先通过一个热点问题和两个计算题引出学习格林公式的目的和必要性,迅速抓住学生的兴趣点,从第二型曲线积分入手,通过逐步分析操作给出前提条件,并最终得到公式,打破了书上的传统推导模式.     

An isoperimetric result for the fundamental frequency via domain derivative

The Faber–Krahn deficit $\delta \lambda $ of an open bounded set $\Omega $ is the normalized gap between the values that the first Dirichlet Laplacian eigenvalue achieves on $\Omega $ and on the ball having same measure as $\Omega $ . For any given family of open bounded sets of $\mathbb R ^N$ ( $N\ge 2$ ) smoothly converging to a ball, it is well known that both $\delta \lambda $ and the isoperimetric deficit $\delta P$ are vanishing quantities. It is known as well that, at least for convex sets, the ratio $\frac{\delta P}{\delta \lambda }$ is bounded by below by some positive constant (Brandolini et al., Arch Math (Basel) 94(4): 391–400, 2010; Payne and Weinberger, J Math Anal Appl 2:210–216, 1961), and in this note, using the technique of the shape derivative, we provide the explicit optimal lower bound of such a ratio as $\delta P$ goes to zero.     

On the purely irregular fundamental group

Based on the (not yet fully understood analogy) between irregular connections and wild ramification, we define a purely irregular fundamental group for complex algebraic varieties and prove some results about this fundamental group which are analogous to the p‐adic étale fundamental group of algebraic varieties over fields of characteristic p (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On fundamental skew distributions

A new class of multivariate skew-normal distributions, fundamental skew-normal distributions and their canonical version, is developed. It contains the product of independent univariate skew-normal distributions as a special case. Stochastic representations and other main properties of the associated distribution theory of linear and quadratic forms are considered. A unified procedure for extending this class to other families of skew distributions such as the fundamental skew-symmetric, fundamental skew-elliptical, and fundamental skew-spherical class of distributions is also discussed.     

On the fundamental state for a Schrödinger operator with magnetic field in a domain with corners

We show that the Neumann realization for the Schrödinger operator with a constant magnetic field in a sector has at least one eigenvalue below the essential spectrum, when the angle is sufficiently small. We establish the complete asymptotics of the lowest eigenvalue as the angle tends to 0. This study is applied to the analysis of the bottom of the spectrum in the semi-classical case for domains with edges. To cite this article: V. Bonnaillie, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On fundamental Heronian triangles

Translated from Matematicheskie Zametki, Vol. 55, No. 2, pp. 72–79, February, 1994.     

On Noether's fundamental theorem

Ohne Zusammenfassung     

On arithmetic fundamental lemmas

We present a relative trace formula approach to the Gross–Zagier formula and its generalization to higher-dimensional unitary Shimura varieties. As a crucial ingredient, we formulate a conjectural arithmetic fundamental lemma for unitary Rapoport–Zink spaces. We prove the conjecture when the Rapoport–Zink space is associated to a unitary group in two or three variables.     

Upper bounds for the fundamental eigenvalue for a domain of unknown shape

For a particular eigenvalue problem in partial differential equations, upper bounds are established which do not depend on the shape of the domain but only on its size. The problem describes the free sloshing motions of an incompressible, inviscid fluid in a canal and furnishes upper bounds to the highest fundamental sloshing frequency which is attainable for a given cross-section area of the canal.This research was supported by the National Science Foundation under Grant No. GP-22587. The author wishes to thank Prof. J. Hersch for valuable discussions on the Stekloff problem in general and on the problem treated in this paper in particular.     

On the fundamental groups of one-dimensional spaces

We study here a number of questions raised by examining the fundamental groups of complicated one-dimensional spaces. The first half of the paper considers one-dimensional spaces as such. The second half proves related results for general spaces that are needed in the first half but have independent interest. Among the results we prove are the theorem that the fundamental group of a separable, connected, locally path connected, one-dimensional metric space is free if and only if it is countable if and only if the space has a universal cover and the theorem that the fundamental group of a compact, one-dimensional, connected metric space embeds in an inverse limit of finitely generated free groups and is shape injective.     

A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain

Barros-Neto and Gelfand (Duke Math. J. 98 (3) (1999) 465; Duke Math. J. 117 (2) (2003) 561) constructed for the Tricomi operator on the plane the fundamental solutions with the supports in the regions related to the geometry of the characteristics of the Tricomi operator. We give for the Tricomi-type operator a fundamental solution relative to an arbitrary point of Rⁿ⁺¹ with the support in the region t?0, where the operator is hyperbolic. Our key observation is that the fundamental solution for the Tricomi-type operator can be written like an integral of the distributions generated by the fundamental solution of the Cauchy problem for the wave equation. The application of that fundamental solution to the L^p-L^q estimate for the forced Tricomi-type equation is given as well.     

On Green's function for the Helmholtz equation in a wedge

It is found that, in the spherical coordinate system, the fundamental solution of the Helmholtz equation in a wedge satisfies the Sommerfeld radiation conditions at infinity uniformly in angle coordinates.Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1312–1314, September, 1993.