Coupling coefficients of SO(n) and integrals over triplets of Jacobi and Gegenbauer polynomials

The expressions of the coupling coefficients (3j-symbols) for the most degenerate (symmetric) representations of the orthogonal groups SO(n) in a canonical basis (with SO(n) restricted to SO(n-1)) and different semicanonical or tree bases [with SO(n) restricted to SO(n'})\times SO(n''), n'+n''=n] are considered, respectively, in context of the integrals involving triplets of the Gegenbauer and the Jacobi polynomials. Since the directly derived triple-hypergeometric series do not reveal the apparent triangle conditions of the 3j-symbols, they are rearranged, using their relation with the semistretched isofactors of the second kind for the complementary chain Sp(4)\supset SU(2)\times SU(2) and analogy with the stretched 9j coefficients of SU(2), into formulae with more rich limits for summation intervals and obvious triangle conditions. The isofactors of class-one representations of the orthogonal groups or class-two representations of the unitary groups (and, of course, the related integrals involving triplets of the Gegenbauer and the Jacobi polynomials) turn into the double sums in the cases of the canonical SO(n)\supset SO(n-1) or U(n)\supset U(n-1) and semicanonical SO(n)\supset SO(n-2)\times SO(2) chains, as well as into the_4F_3(1) series under more specific conditions. Some ambiguities of the phase choice of the complementary group approach are adjusted, as well as the problems with alternating sign parameter of SO(2) representations in the SO(3)\supset SO(2) and SO(n)\supset SO(n-2)\times SO(2) chains.Comment: 26 pages, corrections of (3.6c) and (3.12); elementary proof of (3.2e) is adde

Art History and Geo-Politics in an (Un-)Divided Europe

The Desire for Freedom: Art in Europe Since 1945, an ambitious project organized as the thirtieth edition of the Council for Europe’s exhibition series, takes the novel approach of trying to fight European integration via neo-liberalism with an equally integrationist Enlightenment model of freedom posed as “critique.” Beginning at the German Historical Museum in Berlin in 2012, the core project included one hundred and eighty works produced in the post-war and contemporary periods from forty ..

Nutrient Reserves Of Lesser Snow Geese During Winter And Spring Migration

I studied body composition, diet, and behavior of midcontinent lesser snow geese (Chen caerulescens caerulescens) on their major winter areas and during spring migration from January to May 1983-84. Members of this population regularly overwinter in three habitats--marshes along the Gulf of Mexico, rice fields in Texas and Louisiana contiguous to coastal marshes, and corn fields in Iowa, Missouri, and Kansas. Lesser snow geese did not store nutrient reserves that could be considered to have been for the purpose of reproduction the following spring. Differences in accessibility of foods in different winter habitats and the feeding mode required to obtain them was related to differences in bill morphology. Body size was also different among different winter habitats. Accessibility and nutrient composition of food in each of the habitats interacted with local climate to affect the body composition of geese differently in different winter habitats. Snow geese in marsh habitats appeared to be able to regulate fat reserves in response to prevailing weather. An important preadaptation of lesser snow geese was the highly labile gut morphology, which can rapidly accommodate changes in diet quality.;During spring, lesser snow geese switched to a diet of green vegetation and seeds (mostly by-product corn), and granivory eventually replaced herbivory the farther north that geese had migrated on the prairies. Both males and females stored fat and protein, but only females stored mineral reserves. There were consistent patterns between the sexes and between years of study in the timing of nutrient accumulation. Rates of protein and mineral storage were highest during early migration. Fat storage was related to vernal equinox, and most fat reserves were stored in the northern prairies. As fat was accumulated, it became increasingly related to body size, attaining a maximum in southern Manitoba. After snow geese left the prairies, a second episode of protein storage, but no fat storage, occurred on southern Hudson Bay. Drought on the prairies was related to reduced body protein when geese were in southern Manitoba, but snow geese were able to store enough protein on coastal Hudson Bay to erase any between-year differences in body protein on the prairies