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Semiholonomic jets and induced modules in Cartan geometry calculus

Jan Slovák, Vladimír Souček (2024)

Archivum Mathematicum

The famous Erlangen Programme was coined by Felix Klein in 1872 as an algebraic approach allowing to incorporate fixed symmetry groups as the core ingredient for geometric analysis, seeing the chosen symmetries as intrinsic invariance of all objects and tools. This idea was broadened essentially by Elie Cartan in the beginning of the last century, and we may consider (curved) geometries as modelled over certain (flat) Klein’s models. The aim of this short survey is to explain carefully the basic...

Sharp bounds for the intersection of nodal lines with certain curves

Junehyuk Jung (2014)

Journal of the European Mathematical Society

Let Y be a hyperbolic surface and let φ be a Laplacian eigenfunction having eigenvalue - 1 / 4 - τ 2 with τ > 0 . Let N ( φ ) be the set of nodal lines of φ . For a fixed analytic curve γ of finite length, we study the number of intersections between N ( φ ) and γ in terms of τ . When Y is compact and γ a geodesic circle, or when Y has finite volume and γ is a closed horocycle, we prove that γ is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between N ( φ ) and γ is O ( τ ) . This bound is sharp.

Sharp estimates of the Green function of hyperbolic Brownian motion

Kamil Bogus, Tomasz Byczkowski, Jacek Małecki (2015)

Studia Mathematica

The main objective of the work is to provide sharp two-sided estimates of the λ-Green function, λ ≥ 0, of the hyperbolic Brownian motion of a half-space. We rely on the recent results obtained by K. Bogus and J. Małecki (2015), regarding precise estimates of the Bessel heat kernel for half-lines. We also substantially use the results of H. Matsumoto and M. Yor (2005) on distributions of exponential functionals of Brownian motion.

Shrinkage strategies in some multiple multi-factor dynamical systems

Sévérien Nkurunziza (2012)

ESAIM: Probability and Statistics

In this paper, we are interested in estimation problem for the drift parameters matrices of m independent multivariate diffusion processes. More specifically, we consider the case where the m-parameters matrices are supposed to satisfy some uncertain constraints. Given such an uncertainty, we develop shrinkage estimators which improve over the performance of the maximum likelihood estimator (MLE). Under an asymptotic distributional quadratic risk criterion, we study the relative dominance of the...

Shrinkage strategies in some multiple multi-factor dynamical systems

Sévérien Nkurunziza (2012)

ESAIM: Probability and Statistics

In this paper, we are interested in estimation problem for the drift parameters matrices of m independent multivariate diffusion processes. More specifically, we consider the case where the m-parameters matrices are supposed to satisfy some uncertain constraints. Given such an uncertainty, we develop shrinkage estimators which improve over the performance of the maximum likelihood estimator (MLE). Under an asymptotic distributional quadratic risk criterion, we study the relative dominance of the...

Singular BGG sequences for the even orthogonal case

Lukáš Krump, Vladimír Souček (2006)

Archivum Mathematicum

Locally exact complexes of invariant differential operators are constructed on the homogeneous model for a parabolic geometry for the even orthogonal group. The tool used for the construction is the Penrose transform developed by R. Baston and M. Eastwood. Complexes constructed here belong to the singular infinitesimal character.

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