Collapse transition and asymptotic scaling behavior of lattice animals: Low-temperature expansion

A low-temperature expansion for the free energy density of lattice animals is derived. Analysis of the series yields a collapse transition temperature ofT _c - 0.54, in close agreement with previous estimates. It is demonstrated that _p,k, the number ofp-particle,p-bond animals, obeys the asymptotic scaling law log _p,k pg(k/p) + o(p). The low-temperature series and numerical data are used to estimate the scaling function.     

Sandwich-type silicotungstates: structure and magnetic properties of the dimeric polyoxoanions[[SiM2W9O34(H2O)]2]12- (M = Mn2+, Cu2+, Zn2+)

The novel dimeric silicotungstates [[SiM2W9O34(H2O)]2]12- (M = Mn2+, Cu2+, Zn2+) have been synthesized and characterized by IR spectroscopy, elemental analysis, and magnetic measurements. X-ray single-crystal analyses were carried out on K4Na6Mn[[SiMn2W9O34(H2O)]2].33H2O (1), which crystallizes in the triclinic system, space group P1, with a = 12.2376(7) A, b = 13.6764(8) A, c = 15.6177(9) A, alpha = 70.2860(10) degrees, beta = 79.9150(10) degrees, gamma = 70.2760(10) degrees, and Z = 1; K3Na5[[SiCu2W9O34(H2O)]2].26H2O (2) crystallizes in the triclinic system, space group P1, with a = 11.4271(12) A, b = 12.5956(13) A, c = 15.3223(16) A, alpha = 80.456(2)degrees, beta = 76.383(2) degrees, gamma = 76.968(2) degrees, and Z = 1; K4Na6[[SiZn2W9O34(H2O)]2].34H2O (3) crystallizes also in the triclinic system, space group P1, with a = 12.2596(14) A, b = 13.2555(15) A, c = 16.2892(18) A, alpha = 96.431(2) degrees, beta = 100.944(2) degrees, gamma = 110.404(2) degrees, and Z = 1. The polyanions consist of two lacunary B-alpha-[SiW9O34]10- Keggin moieties linked via a rhomblike M4O16 (M = Mn, Cu, Zn) group leading to a sandwich-type structure. Magnetic measurements show that the central Mn4 unit in 1 exhibits antiferromagnetic (J = -1.77(5) cm(-1)) as well as weak ferromagnetic (J' = 0.08(2) cm(-1)) Mn-Mn exchange interactions. In 2 the Cu-Cu exchange interactions are antiferromagnetic (J = -0.10(2) cm(-1), J' = -0.29(2) cm(-1)).     

Chiral polyoxotungstates. 1. Stereoselective interaction of amino acids with enantiomers of [Ce(III)(alpha1-P2W17O61)(H2O)x]7-. The structure of DL-[Ce2(H2O)8(P2W17O61)2]14-

The ammonium salt of the 1:1complex (1) of Ce(III) with alpha(1)-[P(2)W(17)O(61)](10)(-) was prepared and characterized by elemental analysis, vibrational and NMR spectroscopy ((31)P, (183)W), cyclic voltammetry, and single-crystal X-ray analysis (P1; a = 15.8523(9) A, b = 17.4382(10) A, c = 29.3322(16) A, alpha = 99.617(1) degrees, beta = 105.450 (1) degrees, gamma = 101.132(1) degrees, V = 7460.9(7) A(3), Z = 2). The anion consists of a centrosymmetric head-to-head dimer, [[Ce(H(2)O)(4)(P(2)W(17)O(61))](2)],(14-) with each 9-coordinate Ce cation linked to four oxygens of one tungstophosphate anion and to one oxygen of the other anion. On the basis of P NMR spectroscopy, a monomer-dimer equilibrium exists in solution with K = 20 +/- 4 M(-1) at 22 degrees C. Addition of chiral amino acids to aqueous solutions of 1 results in splitting of the (31)P NMR signals as a result of diastereomer formation. No such splitting is observed with glycine or DL-proline, or when chiral amino acids are added to the corresponding complex of the achiral alpha(2)-isomer of [P(2)W(17)O(61)](10)(-). From analysis of the (31)P NMR spectra, formation constants of the two diastereomeric adducts of 1 with L-proline are 7.3 +/- 1.3 and 9.8 +/- 1.4 M(-1).     

Graphical Representations of Speed: Obstacles Preservice K‐8 Teachers Experience

This study examines the difficulties college students experience when creating and interpreting graphs in which speed is one of the variables. Nineteen students, all preservice elementary or middle school teachers, completed an upper‐level course exploring algebraic concepts. Although all of these preservice teachers had previously completed several mathematics courses, including calculus, they demonstrated widespread misconceptions about the variable speed. This study identifies four cognitive obstacles held by the students, provides excerpts of their graphical constructions and verbal interpretations, and discusses potential causes for the confusion. In particular, misconceptions arose when students interpreted the behavior and nature of speed within a graphical context, as well as in situations where they were required to construct a graph involving speed as a variable. The study concludes by offering implications for the teaching and learning of speed and its interpretation within a graphical setting.     

MR and CT imaging of ethanol-treated liver tumors in an animal model

To characterize the radiographic appearance of liver lesions over time following ethanol injection, seven New Zealand white rabbits underwent surgical implantation of small fragments of VX-2 carcinoma within the liver. Upon reaching 1 cm in diameter, a tumor nodule was directly injected with absolute ethanol. Another nodule in the same animal was injected with saline as a control. Imaging was performed 6-24 days after the injections by high resolution CT and MRI, and correlation obtained with the pathologic specimens. Long TR spin-echo MR sequences were found to characterize the ethanol-treated regions of liver most accurately. Liver tissue infarcted by alcohol could be differentiated from tumor and necrosis by virtue of its short T2 relaxation value. There were no distinguishing features by other imaging techniques between the ethanol-treated and control tumor nodules. Peripheral contrast enhancement was demonstrated in both, corresponding to fibrous tissue around the ethanol-injected regions, and to viable tumor in the case of controls.