Ground state at high density

Weak limits as the density tends to infinity of classical ground states of integrable pair potentials are shown to minimize the mean-field energy functional. By studying the latter we derive global properties of high-density ground state configurations in bounded domains and in infinite space. Our main result is a theorem stating that for interactions having a strictly positive Fourier transform the distribution of particles tends to be uniform as the density increases, while high-density ground states show some pattern if the Fourier transform is partially negative. The latter confirms the conclusion of earlier studies by Vlasov (1945), Kirzhnits and Nepomnyashchii (1971), and Likos et al. (2007). Other results include the proof that there is no Bravais lattice among high-density ground states of interactions whose Fourier transform has a negative part and the potential diverges or has a cusp at zero. We also show that in the ground state configurations of the penetrable sphere model particles are superposed on the sites of a close-packed lattice.Comment: Note adde

Fluorspar and fluorine chemicals. Part I, Economic aspects of the fluorspar industry

Bibliography: p. 39-40.Economic aspects of the fluorspar industry / N.T. Hamrick and W.H. Voskuil -- Fluorine chemicals in industry / G.C. Finger and F.H. Reed

Pines

Pinus is the most important genus within the Family Pinaceae and also within the gymnosperms by the number of species (109 species recognized by Farjon 2001) and by its contribution to forest ecosystems. All pine species are evergreen trees or shrubs. They are widely distributed in the northern hemisphere, from tropical areas to northern areas in America and Eurasia. Their natural range reaches the equator only in Southeast Asia. In Africa, natural occurrences are confined to the Mediterranean basin. Pines grow at various elevations from sea level (not usual in tropical areas) to highlands. Two main regions of diversity are recorded, the most important one in Central America (43 species found in Mexico) and a secondary one in China. Some species have a very wide natural range (e.g., P. ponderosa, P. sylvestris). Pines are adapted to a wide range of ecological conditions: from tropical (e.g., P. merkusii, P. kesiya, P. tropicalis), temperate (e.g., P. pungens, P. thunbergii), and subalpine (e.g., P. albicaulis, P. cembra) to boreal (e.g., P. pumila) climates (Richardson and Rundel 1998, Burdon 2002). They can grow in quite pure stands or in mixed forest with other conifers or broadleaved trees. Some species are especially adapted to forest fires, e.g., P. banksiana, in which fire is virtually essential for cone opening and seed dispersal. They can grow in arid conditions, on alluvial plain soils, on sandy soils, on rocky soils, or on marsh soils. Trees of some species can have a very long life as in P. longaeva (more than 3,000 years)