Pneumocephalus and pneumorrhachis are related to transgression of the barriers to the central nervous system. We present a patient with a Pancoast tumor treated with palliative chemoradiation who developed symptomatic spinal and intracranial air caused by spontaneous bronchopleurodurosubarachnoid fistula secondary to direct tumor invasion into the thecal sac.

We establish several Witten type rigidity and vanishing theorems for twisted Toeplitz operators on odd dimensional manifolds. We obtain our results by combining the modular method, modular transgression and some careful analysis of odd Chern classes for cocycles in odd K-theory. Moreover we discover that in odd dimensions, the fundamental group of manifolds plays an important role in the rigidity.

Diatom assemblages were analysed to clarify the changes in sedimentary environments caused by the Holocene sea level changes, and the results were cross-checked with information from the field observations of sedimentary facies. Diatoms were grouped into three ecological categories; marine, brackish and freshwater. Based on the ecological spectrum of the diatoms, the sediments were divided into...