It is important to study asymptotic behavior of complete manifold without the assumption of pointwise Ricci curvature bound. A volume growth and curvature decay result was obtained in [4] for various classes of complete, noncompact, Bach-flat metrics in dimension 4. Some similar results were also claimed in [1]. In this note we consider a more general case, that is, the Bach tensor may not nece...

The goal of this paper is to express the Bach tensor of a four dimensional conformal geometry of an arbitrary signature by the Cartan normal conformal (CNC) connection. We show that the Bach tensor can be identified with the Yang–Mills current of the connection. It follows from that result that a conformal geometry whose CNC connection is reducible in an appropriate way has a degenerate Bach te...

The composition of polyphonic chorale music in the style of J.S Bach has represented a major challenge in automatic music composition over the last decades. The art of Bach chorales composition involves combining four-part harmony with characteristic rhythmic patterns and typical melodic movements to produce musical phrases which begin, evolve and end (cadences) in a harmonious way. To our know...

1 Sanders MH, Kern N. Obstructive sleep apnea treated by independently adjusted inspiratory and expiratory positive airway pressures via nasal mask: physiologic and clinical implications. Chest 1990; 98:317-24 2 Steljes DG, Kryger MH, Kirk BS, Millar TW Sleep in postpohio syndrome. Chest 1990; 98:133-40 3 Bach JR, Alba AS, Mosher R, Delaubier A. Intermittent positive pressure ventilation via na...

The Bach tensor and a vector which generates conformal symmetries allow a conserved 4-current to be defined. The Bach 4-current gives rise to a quasilocal 2-surface expression for power per luminosity distance in the Vaidya exterior of collapsing fluid interiors. This is interpreted in terms of entropy generation. PACS numbers: 0420J, 0440N