Minimal Matrix Representations for Six-Dimensional Nilpotent Lie Algebras

This paper is concerned with finding minimal dimension linear representations for six-dimensional real, indecomposable nilpotent Lie algebras. It is known that all such Lie algebras can be represented in gl(6, R). After discussing the classification of the 24 such Lie algebras, it is shown that only one algebra can be represented in gl(4, R). A Theorem is then presented that shows that 13 of the algebras can be represented in gl(5, R). The special case of filiform Lie algebras is considered, of which there are five, and it is shown that each of them can be represented in gl(6, R) and not gl(5, R). Of the remaining five algebras, four of them can be represented minimally in gl(5, R). That leaves one difficult case that is treated in detail in an Appendix

Lie Symmetries of the Canonical Geodesic Equations for Four-Dimensional Lie Groups

For each of the four-dimensional indecomposable Lie algebras the geodesic equations of the associated canonical Lie group connection are given. In each case a basis for the associated Lie algebra of symmetries is constructed and the corresponding Lie brackets are written down

Symmetry Algebras of the Canonical Lie Group Geodesic Equations in Dimension Three

For each of the two and three-dimensional indecomposable Lie algebras the geodesic equations of the associated canonical Lie group connection are given. In each case a basis for the associated Lie algebra of symmetries is constructed and the corresponding Lie brackets are written down

Classification of Symmetry Lie Algebras of the Canonical Geodesic Equations of Five-Dimensional Solvable Lie Algebras

In this investigation, we present symmetry algebras of the canonical geodesic equations of the indecomposable solvable Lie groups of dimension five, confined to algebras A_{5,7}^{abc} to A_{18}^a. For each algebra, the related system of geodesics is provided. Moreover, a basis for the associated Lie algebra of the symmetry vector fields, as well as the corresponding nonzero brackets, are constructed and categorized

Event-based Face Detection and Tracking in the Blink of an Eye

We present the first purely event-based method for face detection using the high temporal resolution of an event-based camera. We will rely on a new feature that has never been used for such a task that relies on detecting eye blinks. Eye blinks are a unique natural dynamic signature of human faces that is captured well by event-based sensors that rely on relative changes of luminance. Although an eye blink can be captured with conventional cameras, we will show that the dynamics of eye blinks combined with the fact that two eyes act simultaneously allows to derive a robust methodology for face detection at a low computational cost and high temporal resolution. We show that eye blinks have a unique temporal signature over time that can be easily detected by correlating the acquired local activity with a generic temporal model of eye blinks that has been generated from a wide population of users. We furthermore show that once the face is reliably detected it is possible to apply a probabilistic framework to track the spatial position of a face for each incoming event while updating the position of trackers. Results are shown for several indoor and outdoor experiments. We will also release an annotated data set that can be used for future work on the topic