Enumeration of substitutional isomers with restrictive mutual positions of ligands: I. Overall counts

We address systematics for the enumeration of substitutional isomers when there is constrained positioning of ligands on a molecular skeleton. One constraint involves ‘restrictive ligands’ where two of the same kind are forbidden to occupy adjacent sites in a molecular skeleton. This may arise because of steric hindrance, or because of groups which in neighbor proximity react to eliminate one. For instance, no pair of –OH groups attach to the same C atom in a molecular skeleton. In another case, malonic acid residues decarboxylate leaving no more than one decarboxylation in each residue. The enumeration with such restrictive ligands may be addressed via a Polya-theoretic cycle index hybridized with the graph-theoretic independence polynomial (when there is just a single such neighbor-excluding ligand and another which is not), while more generally a hybridization with the chromatic polynomial is needed. Another substitional-isomer constraint involves bidentate ligands, with each ligand-part occupying adjacent sites, and possibly also with additional separate unidentate ligands. Here, the set of all pure & mixed such ligand placements is analytically represented by a ‘symmetry-reduced’ matching polynomial (which is a hybrid now of the matching polynomial and Polya’s cycle index). This result gives the generating function for isomer enumeration, taking into account every possible so-restricted assortment of the employed ligands. Here we make such novel hybridizations (for these and other graphtheoretic polynomials) to deal with such oft-encountered chemical problems, which nevertheless transcend typical earlier unconstrained formulizations. Further subsymmetry classification & enumerations, along with examples are considered in a further article.     

Comparison of phantom materials for use in quality assurance of microbeam radiation therapy

Microbeam radiation therapy (MRT) is a promising radiotherapy modality that uses arrays of spatially fractionated micrometre‐sized beams of synchrotron radiation to irradiate tumours. Routine dosimetry quality assurance (QA) prior to treatment is necessary to identify any changes in beam condition from the treatment plan, and is undertaken using solid homogeneous phantoms. Solid phantoms are designed for, and routinely used in, megavoltage X‐ray beam radiation therapy. These solid phantoms are not necessarily designed to be water‐equivalent at low X‐ray energies, and therefore may not be suitable for MRT QA. This work quantitatively determines the most appropriate solid phantom to use in dosimetric MRT QA. Simulated dose profiles of various phantom materials were compared with those calculated in water under the same conditions. The phantoms under consideration were RMI457 Solid Water (Gammex‐RMI, Middleton, WI, USA), Plastic Water (CIRS, Norfolk, VA, USA), Plastic Water DT (CIRS, Norfolk, VA, USA), PAGAT (CIRS, Norfolk, VA, USA), RW3 Solid Phantom (PTW Freiburg, Freiburg, Germany), PMMA, Virtual Water (Med‐Cal, Verona, WI, USA) and Perspex. RMI457 Solid Water and Virtual Water were found to be the best approximations for water in MRT dosimetry (within ±3% deviation in peak and 6% in valley). RW3 and Plastic Water DT approximate the relative dose distribution in water (within ±3% deviation in the peak and 5% in the valley). PAGAT, PMMA, Perspex and Plastic Water are not recommended to be used as phantoms for MRT QA, due to dosimetric discrepancies greater than 5%.     

The diameter of a fuzzy set

In pattern recognition one often wants to measure geometric properties of imprecisely defined subsets of an image. This paper proposes definitions of intrinsic and extrinsic diameter for fuzzy subsets which reduce to the ordinary definitions when the subsets are crisp. We also define height and width for a fuzzy subset and show how they relate to the area (i.e., integral of membership). For convex fuzzy subsets the intrinsic diameter cannot exceed the extrinsic diameter, but it can be smaller. Finally, for piecewise constant convex fuzzy subsets the intrinsic diameter cannot exceed half the fuzzy perimeter, but this need not be true in the nonconvex case.     

Physical reasons for the emergence of jumps during rotation of the magnetization of a two-sublattice ferrimagnet at low temperatures

It is shown that the contribution of the intersublattice exchange interaction to the magnetic anisotropy energy of a two-sublattice ferrimagnet can come only from higher order constants satisfying the condition 2K₂+3K₃+?>0. For this reason, for different signs of the first anisotropy constants of the sublattices, this contribution may cause a spontaneous spin-reorientation second-order transition, but not first-order transitions and jumps during magnetization rotation, which are associated with such a transition. Such jumps can appear only when the opposite inequality is satisfied, and the corresponding contribution to anisotropy can be ensured only by a fairly strong magnetoelastic interaction.     

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