To any finite collection of smooth real vector fields Xj in R we associate a metric in the phase space T ∗Rn. The relation between the asymptotic behavior of this metric and hypoellipticity of ∑ X j , in the smooth, real analytic, and Gevrey categories, is explored. To Professor P. Lelong, on the occasion of his 85th birthday.

We develop a new model of a spinning particle in Brans-Dicke space-time using a metric-compatible connection with torsion. The particle's spin vector is shown to be Fermi-parallel (by the Levi-Civita connection) along its worldline (an autoparallel of the metric-compatible connection) when neglecting spin-curvature coupling.

The sequential $p$-convergence in a fuzzy metric space, in the sense of George and Veeramani, was introduced by D. Mihet as a weaker concept than convergence. Here we introduce a stronger concept called $s$-convergence, and we characterize those fuzzy metric spaces in which convergent sequences are $s$-convergent. In such a case $M$ is called an $s$-fuzzy metric. If $(N_M,ast)$ is a fuzzy metri...

The Horrocks-Mumford bundle E is a famous stable complex vector bundle of rank 2 on 4-dimensional complex projective space. By construction, E has a natural Hermitian metric h1. On the other hand, stability implies the existence of a Hermitian-Einstein metric in E which is unique up to a positive scalar. Now the obvious question is if h1 is in fact the Hermitian-Einstein metric. In this note we...

In this paper, we show that in each nite dimensional Hilbert space, a frame of subspaces is an ultra Bessel sequence of subspaces. We also show that every frame of subspaces in a nite dimensional Hilbert space has frameness bound.

Let V be a vector space over R. A norm on V is a function || · || : V → R satisfying three properties: 1) ||v|| ≥ 0, with equality if and only if v = 0, 2) ||v + w|| ≤ ||v|| + ||w|| for v, w ∈ V , 3) ||αv|| = |α|||v|| for α ∈ R, v ∈ V. The same definition applies to a complex vector space. From a norm we get a metric on V by d(v, w) = ||v − w||.

Supervised and unsupervised prototype based vector quantization frequently are proceeded in the Euclidean space. In the last years, also non-standard metrics became popular. For classification by support vector machines, Hilbert or Banach space representations are very successful based on so-called kernel metrics. In this paper we give the mathematical justification that gradient based learning...

Abstract. A new supervised adaptive metric approach is introduced for mapping an input vector space to a plottable low-dimensional subspace in which the pairwise distances are in maximum correlation with distances of the associated target space. The formalism of multivariate subspace regression (MSR) is based on cost function optimization, and it allows assessing the relevance of input vector a...

in this paper, we introduce the cone normed spaces and cone bounded linear mappings. among other things, we prove the baire category theorem and the banach--steinhaus theorem in cone normed spaces.

In 1977, M. Matsumoto and R. Miron [9] constructed an orthonormal frame for an n-dimensional Finsler space, called ‘Miron frame’. The present authors [1, 2, 3, 10, 11] discussed four-dimensional Finsler spaces equipped with such frame. M. Matsumoto [7, 8] proved that in a three-dimensional Berwald space, all the main scalars are h-covariant constants and the h-connection vector vanishes. He als...