The evaluation of matrix elements for non-canonical Weyl tableau basis states adapted toU(n 1+n 2)⊃U(n 1)×U(n 2)

Summary This is the first paper of a series of two, which enables the evaluation ofU(n) generator matrix elements in the non-canonical Weyl tableau basis adapted to subgroupU(n ₁)×U(n ₂). In this paper the explicit closed formulae for subduction coefficients are presented. These formulae will become useful through an inductive method to be presented in the second paper.     

In situ mapping of high-pressure fluids using hydrothermal diamond anvil cells

We present new results combining high pressures and temperatures attainable in a diamond anvil cell with in situ synchrotron radiation induced micro-X-ray fluorescence measurements. Hydrothermal diamond anvil cells experiments have been performed by measuring the partitioning of Pb between aqueous fluids (pure water or NaCl-enriched water) and hydrous silicate melts of haplogranite composition using synchrotron X-ray fluorescence. The in situ measurements were performed in the range 0.3–1.2 GPa and 730–850 ^°C both in the aqueous fluid and in the silicate melts being in equilibrium. Pb is strongly partitioned into high-pressure–temperature hydrous melts when Cl is present in either the hydrous melt or the aqueous fluid. Moreover, our comparisons of in situ results with post-mortem results show that significant changes take place during rapid quenching especially when samples are small (few hundred of microns in diameter). Water exsolution is induced by the quench in the silicate melt showing the high mobility of Pb which immediately partitions into the water vapor phase during the quench. The current in situ approach offers thus a pertinent complementary method to the classical experimental petrology investigations.     

The subduction of vibrational states of molecules from the momentum space of simple harmonic motion

A single-particle model of molecular vibrational states is proposed in which the normal modes are projected out of the body vibrations of an infinite simple harmonic sphere. This model assigns the spurious change of mass or centre of mass and leads to removal of mass monopoles and dipoles from the system. These conservation conditions impose strict boundary conditions on the potential and basis functions. On incorporation into the model they result in a set of loop equations in which the potential is proportional to the fundamental vibration. The simplest solutions to these equations strongly resemble the Poschl-Teller generalization of the Morse potential. The solutions have been extended to incorporate the repulsive states and generate the set of net attractive states appropriate to the anharmonic potential.The basis functions of this potential display both angular and radial node structures. The degeneracies between radial and angular mode patterns can be studied by transformation into an angular coordinate space. In this way coupling to other phenomena described in similar angular momentum space can be performed directly before subduction to real displacement space.On leave from the Department of Chemistry, Catholic University of Leuven, Celestijnenlaan 200F, B-3030 Leuven-Belgium     

Subduction of dominant representations for combinatorial enumeration

Summary A new method for giving cycle indices is presented for combinatorial enumeration. Thus, cyclic groups are characterized by markaracter tables, the elements of which are determined by the orders of their subgroups. A set of such cyclic groups (defined as dominant subgroups) is used to characterize a group G of finite order, where the markaracter table for the group G is constructed with respect to dominant representations (DRs), which are defined as coset representations corresponding to the dominant subgroups. By starting from the markaracter table, we propose an essential set of subdominant markaracter tables and a magnification set for the group G; the latter concept clarifies the relationship between each subdominant markaracter table and the markaracter table of a dominant subgroup. The subduction of DRs is obtained by the markaracter table to produce a dominant subduction table and a dominant USCI (unit-subduced cycle index) table. The latter is used to evaluate a cycle index to be applied to combinatorial enumeration. The cycle index is shown to be equivalent to the couterpart of our previous approach concerning both cyclic and non-cyclic subgroups. The latter, in turn, has been proved to be equivaltent to the cycle index obtained by the Redfield-Pólya theorem.