Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates

In this paper we will review the main results concerning the issue of stability for the determination unknown boundary portion of a thermic conducting body from Cauchy data for parabolic equations. We give detailed and selfcontained proofs. We prove that such problems are severely ill-posed in the sense that under a priori regularity assumptions on the unknown boundaries, up to any finite order of differentiability, the continuous dependence of unknown boundary from the measured data is, at best, of logarithmic type

Chemical composition of bottom sediments along the Transpacific section

Distribution of Fe, Mn, Ti, Cu, Ni, Co, V, Cr, Mo, As in bottom sediments of a section from the Hawaiian Islands to the coast of Mexico. In the surface layer and isochronic layers of sediments from biogenic-terrigenous sediments of the Mexico coast to pelagic red clays of the Northeast Basin contents of all studied elements increase, and more sharply for mobile ones - Mn, Mo, Cu, Ni, Co, As. In near Hawaii sediments rich in coarsely fragmented volcanic-terrigenous and pyroclastic material of basalt composition enriched in Ti, Fe, Cr, V, P contents of these elements in surface sediments and in sediment mass increase and contents of Mn, Mo, Ni, Co, Cu, As (for the same reason) decrease compared to red clays. An area of hemipelagic and transition sediments is identified; these sediments have much higher contents of Mn, Fe, Cu, Ni, Mo, As, (Ba) than red clays and similar sediments of the Northwest Pacific Ocean. This is due to hydrothermal activity in the tectonically active zone at the northern extension of the East Pacific Rise. Similar character of distribution of the elements in the surface layer and in the isochrone layers of bottom sediments along the most part of the section is shown. Similarity between distribution of the elements in sediments of the western and the eastern parts of the Transpacific section is established