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Neckpinch Dynamics for Asymmetric Surfaces Evolving by Mean Curvature Flow
The authors study noncompact surfaces evolving by mean curvature flow (mcf). For an open set of initial data that are $C^3$-close to round, but without assuming rotational symmetry or positive mean curvature, the authors show that mcf solutions become singular in finite time by forming neckpinches, and they obtain detailed asymptotics of that singularity formation. The results show in a precise way that mcf solutions become asymptotically rotationally symmetric near a neckpinch singularity.
Osnat and Her Dove
Osnat was born five hundred years ago – at a time when almost everyone believed in miracles. But very few believed that girls should learn to read. Yet Osnat's father was a great scholar whose house was filled with books. And she convinced him to teach her. Then she in turn grew up to teach others, becoming a wise scholar in her own right, the world's first female rabbi! Some say Osnat performed miracles – like healing a dove who had been shot by a hunter! Or saving a congregation from fire! But perhaps her greatest feat was to be a light of inspiration for other girls and boys; to show that any person who can learn might find a path that none have walked before.
Jimmy Carter
None
Public Papers of the Presidents of the United States
"Containing the public messages, speeches, and statements of the President", 1956-1992.
Framing the Early Middle Ages
The Roman empire tends to be seen as a whole whereas the early middle ages tends to be seen as a collection of regional histories, roughly corresponding to the land-areas of modern nation states. As a result, early medieval history is much more fragmented, and there have been few convincing syntheses of socio-economic change in the post-Roman world since the 1930s. In recent decades, the rise of early medieval archaeology has also transformed our source-base, but this has not been adequately integrated into analyses of documentary history in almost any country. In Framing the Early Middle Ages Chris Wickham combines documentary and archaeological evidence to create a comparative history of t...
Jimmy Carter
None
Public Papers of the Presidents of the United States: Jimmy Carter
Public Papers of the Presidents of the United States
