A new mineral species of the pyrochlore supergroup, oxynatromicrolite (IMA2013-063), (Na,Ca,U)2Ta2O6O, was found in the No. 309 rare-metal granitic pegmatite vein, Guanpo, Lushi county, Henan Province, China, which is characterized by tantalum at theB site and oxygen at the Y site and is Na dominant at the A site. The mineral occurs as strongly metamict, and mostly euhedral octahedral crystals up to 0.05–0.20 mm across. The measured density of an unheated sample is 6.580(4) g cm–3, and the calculatedone is 6.506 g cm–3. Optically, the mineral is isotropic, with an index of refraction 1.999(5) and a reflectance of 11.88% (470 nm). When heated to 1000°C for 4 hours in N2, the mineral recrystallizes in the cubic system, with space group Fd3mand with unit-cell parameters similar those of other pyrochlore supergroup species: a = 10.420(6) Å, V = 1131.4(2) Å3. Electron microprobe analyses revealed the following composition of the mineral (in wt.%): Na2O 5.41, CaO 4.56, UO214.60, La2O3 0.16, Ce2O3 0.11, Nd2O3 0.13, PbO 0.62, Ta2O5 61.52, Nb2O5 8.21, Sb2O5 0.23, TiO2 0.05, SiO2 0.56, SnO2 0.29, F1.04, H2O 1.50 (calculated to correspond to 0.47 H2O pfu), F≡O –0.44, sum = 98.53%, which corresponds to the empirical formula (Na0.99Ca0.46U0.31Pb0.02La0.01H2O0.21)∑2.00(Ta1.58Nb0.35Si0.05Sn0.01Sb0.01)∑2.00O6 (O0.43F0.31H2O0.26)∑1.00, represented by the simplified formula (Na,Ca,U)2(Ta,Nb)2O6(O,F).Oxynatromicrolite crystallized during the late-stage of formation for the No. 309 pegmatite dyke and is associated with quartz, albite, potassium feldspar, muscovite, kaolinite, tantalite-Mn, stibiotantalite, pollucite, spodumene, montebrasite, Hf-rich zircon, a red tourmaline, polylithionite,trilithionite, luanshiweiite-2M 1 (IMA2011-102) and a hydrated derivative of oxynatromicrolite.

Tangdanite, ideally Ca2Cu9(AsO4)4(SO4)0.5(OH)9·9H2O and monoclinic, is a new mineral species (IMA No. 2011-096) occurring in the Tangdan and Nanniping mines, southeast Dongchuan copper mining district, Dongchuan County, Kunming City Prefecture, Yunnan Province, P. R. China (26°11’N 103°51’E). The mineral is found in the oxidized zone (gossan) of an As-bearing Cu sulfide deposit and is clearly of supergene origin. Associated minerals are chalcopyrite, bornite, chalcocite, covellite, tennantite, enargite, cuprite, malachite, azurite, copper and brochantite. Crystals form radiating or foliated aggregates of flaky crystals up to 3 mm, flattened parallel to (100) and elongated along [001]. It is emerald green with a light green streak, translucent and has a pearly to silky lustre. It is sectile having perfect cleavage on {100} although neither parting nor fracture was observed. No fluorescence in long- or short-wave ultraviolet radiation was observed. The hardness is VHN50 42.0−43.6, mean 42.8 kg mm−2 (2−2½ on the Mohs scale). The density measured by pycnometry is 3.22 g cm−3 (Ma et al., 1980). The calculated density from the empirical chemical formula is 3.32 g cm−3. The compatability index gives 1 − (Kp/Kc) = −0.041 (good). The empirical formula (based on 36 O a.p.f.u) of tangdanite is Ca2.05Cu9.08(As1.03O4)4(S0.63O4)0.5(OH)9·9H2.04O. The simplified formula is Ca2Cu9(AsO4)4(SO4)0.5(OH)9·9H2O. The strongest five reflections in the X-ray powder-diffraction pattern [d in Å(I) (hkl)] are: 4.782(100) ( 1 1), 4.333(71) (6 0 2), 5.263(54) ( 0 2), 3.949(47) (8 0 2) and 2.976(46) ( 1 1). The unit-cell parameters are a = 54.490(9), b = 5.5685(9), c = 10.4690(17) Å, β = 96.294(3)o, V = 3157.4(9) Å3, Z = 4. Its structure was solved and refined in space group C2/c, with R = 0.110.

Recently, the first author and Y. Wang proved that (n ≥ 2) is a Möbius transformation if and only if f is a non-degenerate circle-preserving map. In this paper, we will further the result to show that f is a Möbius transformation if and only if f is a non-degenerate r–dimensional sphere-preserving map. The versions for the Euclidean and hyperbolic cases are also obtained. These results make no surjectivity or injectivity or even continuity assumptions on f. Moreover, certain degenerate sphere-preserving maps are given, which completes the characterizations of sphere-preserving maps.