On geometrical properties of the spaces defined by the Pfaff equations

The geometrical quantity in damped wave equations on a square

The energy in a square membrane $Ω$ subject to constant viscous damping on a subset $ω\subset Ω$ decays exponentially in time as soon as $ω$ satisfies a geometrical condition known as the "Bardos-Lebeau-Rauch" condition. The rate $τ(ω)$ of this decay satisfies $τ(ω)= 2 \min(-μ(ω), g(ω))$ (see Lebeau [Math. Phys. Stud. 19 (1996) 73-109]). Here $μ(ω)$ denotes the spectral abscissa of the damped wave equation operator and $g(ω)$ is a number called the geometrical quantity of $ω$ and defined as follows. A ray in $Ω$ is the trajectory generated by the free motion of a mass-point in $Ω$ subject to e…

对称空间上某些积分的行列式表达式

Determinantal expressions of certain integrals on symmetric spaces

本文提出对称空间上 $K$-不变且为单变量正函数张量幂的函数的积分可表示为行列式或普法夫行列式，并给出对称锥、格拉斯曼流形和经典域的例子，展望未来扩展。

The Riemann Surface of the Logarithm Constructed in a Geometrical Framework

The logarithmic Riemann surface Sigma_{log} is a classical holomorphic 1-manifold. It lives into R^4 and induces a covering space of C - 0 defined by exp. This paper suggests a geometric construction of it, derived as the limit of a sequence of vector fields extending exp suitably to embeddings of C into R^3, which turn to be helicoid surfaces living into C x R. In the limit we obtain a bijective complex exponential on the covering space in question, and thus a well-defined complex logarithm. In addition, the helicoids are diffeomorphic (not bi-holomorphic) copies of Sigma_{log} as smooth real…

Remarks on the statistical origin of the geometrical formulation of quantum mechanics

A quantum system can be entirely described by the Kähler structure of the projective space P(H) associated to the Hilbert space H of possible states; this is the so-called geometrical formulation of quantum mechanics. In this paper, we give an explicit link between the geometrical formulation (of finite dimensional quantum systems) and statistics through the natural geometry of the space Pn of non-vanishing probabilities p defined on a finite set En:={x1,...,xn}. More precisely, we use the Fisher metric and the exponential connection (both being natural statistical objects living on Pn) to con…