Two-loop n -point anomalous amplitudes in N=4 supergravity

We compute the anomalous two-loop four-point amplitudes in N = 4 pure supergravity, using unitarity and the double-copy construction. We also present all terms determined by four-dimensional cuts in two all-multiplicity two-loop anomalous superamplitudes. This result provides the first two-loop n -point gravity amplitude, up to a class of undetermined terms, which we determine completely at four points. We show that a recently proposed finite counterterm cancels these amplitudes to this order. We argue that the counterterm does not spoil the three-loop finiteness of anomalous amplitudes in the N = 4 theory.

Mechanisms for pattern formation leading to axis formation and lineage allocation in mammals: a marsupial perspective

Developing patterns in early embryogenesis are analysed in conceptuses from several families, including Dasyuridae, Phalangeridae, Macropodidae and Didelphidae, in which cleavage has been examined in some detail. Features common to cleavage and blastocyst formation, and in some cases to hypoblast formation, are used to develop an outline of possible mechanisms leading to axis formation and lineage allocation. Relevant features that have been described only in some species are also included. It is suggested that certain features of marsupial cleavage establish patterns in the developing blastocyst epithelia, pluriblast, trophoblast and hypoblast that contribute to axis formation and lineage allocation. All marsupials examined had a polarized oocyte or conceptus, the polarity of which was related to the conceptus embryonic-abembryonic axis and, eventually, the conceptus dorsal-ventral axis and the formation of the pluriblast (future embryo) and trophoblast. The embryonic dorsal-ventral and anterior-posterior axes were established after the allocation of hypoblast and epiblast. Mechanisms that appear to result in patterning of the developing epithelia leading to axis formation and lineage allocation are discussed, and include sperm entry point, gravity, conceptus polarity, differentials in cell-zona, cell-cell and cell-type (boundary effects) contacts, cell division order during cleavage and signals external to the conceptus. A model of the patterning effects is included. The applicability of these mechanisms to other amniotes, including eutherian mammals, is also examined.

Mechanics: Rectilinear Motion of a Point. Gravity.

In the last article we were concerned with finding the final velocity, being given the initial velocity, acceleration and the time. We did not consider the distance passed over during any interval of time.