It is shown that the determinacy of $G_{\delta \sigma }$ games of length $\omega ^2$ is equivalent to the existence of a transitive model of ${\mathsf {KP}} + {\mathsf {AD}} + \Pi _1\textrm {-MI}_{\mathbb {R}}$ containing $\mathbb {R}$ . Here, $\Pi _1\textrm {-MI}_{\mathbb {R}}$ is the axiom asserting that every monotone $\Pi _1$ operator on the real numbers has an inductive fixpoint.

The crystal structure of tlapallite has been determined using single-crystal X-ray diffraction and supported by electron probe micro-analysis, powder diffraction and Raman spectroscopy. Tlapallite is trigonal, space group P321, with a = 9.1219(17) Å, c = 11.9320(9) Å and V = 859.8(3) Å3, and was refined to R1 = 0.0296 for 786 reflections with I > 2σ(I). This study resulted from the discovery of well-crystallised tlapallite at the Wildcat prospect, Utah, USA. The chemical formula of tlapallite has been revised to (Ca,Pb)3CaCu6[Te4+3Te6+O12]2(Te4+O3)2(SO4)2·3H2O, or more simply (Ca,Pb)3CaCu6Te4+8Te6+2O30(SO4)2·3H2O, from H6(Ca,Pb)2(Cu,Zn)3(TeO3)4(TeO6)(SO4). The tlapallite structure consists of layers containing distorted Cu2+O6 octahedra, Te6+O6 octahedra and Te4+O4 disphenoids (which together form the new mixed-valence phyllotellurate anion [Te4+3Te6+O12]12−), Te4+O3 trigonal pyramids and CaO8 polyhedra. SO4 tetrahedra, Ca(H2O)3O6 polyhedra and H2O groups fill the space between the layers. Tlapallite is only the second naturally occurring compound containing tellurium in both the 4+ and 6+ oxidation states with a known crystal structure, the other being carlfriesite, CaTe4+2Te6+O8. Carlfriesite is the predominant secondary tellurium mineral at the Wildcat prospect. We also present an updated structure for carlfriesite, which has been refined to R1 = 0.0230 for 874 reflections with I > 2σ(I). This updated structural refinement improves upon the one reported previously by refining all atoms anisotropically and presenting models of bond valence and Te4+ secondary bonding.

The effect of pressure on the naturally occurring hydroxide-perovskite stottite, FeGe(OH)6, has been studied in situ by micro-Raman spectroscopy to 21 GPa at 300 K. The ambient spectrum contains six OH-stretching bands in the range 3064 3352 cm–1. The presence of six non-equivalent OH groups is inconsistent with space group P42/n. In view of this inconsistency a new ambient structure determination of stottite from Tsumeb was carried out, but this did not allow the clear rejection of P42/n symmetry. However, a successful refinement was also carried out in space group P2/n, a subgroup of P42/n, which allows for six non-equivalent O atoms. The two refinements are of comparable quality and do not allow a choice to be made based purely on the X-ray data. However, taken with the ambient and 150 K Raman spectra, a good case can be made for stottite having P2/n symmetry at ambient conditions. On this basis, the pressure induced spectroscopic changes are interpreted in terms of a reversible phase transition P2/n ↔ P42/n.

The crystal structure of seeligerite, Pb3IO4Cl3, from the San Rafael mine, Sierra Gorda, Chile, was solved in the space group Cmm2, and refined to R = 3.07%. The unit-cell parameters are: a = 7.971(2), b = 7.976(2), c = 27.341(5) Å, V = 1738.3(6) Å3 and Z = 8. The crystal structure consists of a stacking sequence along [001] of square-net layers of O atoms and square-net layers of Cl atoms with Pb+ and I+ cations located in the voids of the packing. As is typical of cations with a stereoactive lone-pair of electrons, Pb2+ and I5+ adopt strongly-asymmetrical configurations. Pb2+ cations occur in a variety of coordination polyhedra, ranging from anticubes and monocapped anticubes to pyramidal ‘one-sided’ coordinations. I5+ is coordinated by a square of four oxygen atoms: I1 and I3 exhibit a ‘one-sided’ coordination, whereas I2 has square-planar coordination.

The TEM investigation has revealed additional superlattice reflections (which were not registered by X-ray diffraction (XRD)) in the hk0 diffraction pattern of seeligerite based upon a 0.158 Å-1 square net, which can be interpreted as arising from a 20-cation super-sheet motif (12.6 Å x 12.6 Å), likely related to a further level of Pb-I order superimposed upon the 8-site motif identified by XRD.

The commissioning and operation of apparatus for neutron diffraction at simultaneous high temperatures and pressures is reported. The basic design is based on the Paris-Edinburgh cell using opposed anvils, with internal heating. Temperature is measured using neutron radiography. The apparatus has been shown in both on-line and off-line tests to operate to a pressure of 7 GPa and temperature of 1700°C. The apparatus has been used in a neutron diffraction study of the crystal structure of deuterated brucite, and results for 520°C and 5.15 GPa are presented. The diffraction data that can be obtained from the apparatus are of comparable quality to previous high-pressure studies at ambient temperatures, and are clearly good enough for Rietveld refinement analysis to give structural data of reasonable quality.

Paratacamite-(Mg) (IMA 2013-014), Cu3(Mg, Cu)Cl2(OH)6, is the new Mg-analogue of paratacamite. It was found near the village of Cuya in the Camarones Valley, Arica Province, Chile. The mineral is a supergene secondary phase occurring in association with anhydrite, atacamite, chalcopyrite, copiapite, dolomite, epsomite, haydeeite, hematite, magnesite and quartz. Paratacamite-(Mg) crystals are rhombs and thick to thin prisms up to 0.3 mm in size exhibiting the forms {201} and {001}. Twinning by reflection on {10} is common. The mineral is transparent with a vitreous lustre, with medium to deep-green colour and light-green streak. Mohs hardness is 3–3½, the tenacity is brittle and the fracture is conchoidal. Paratacamite-(Mg) has one perfect cleavage on {201}. The measured and calculated densities are 3.50(2) and 3.551 g cm–3, respectively. The mineral is optically uniaxial (–) with ε = 1.785(5) and ω > 1.8 and slight pleochroism: O (bluish green) > E (green). Electron-microprobe analyses provided the empirical formula Cu3(Mg0.60Cu0.38Ni0.01Mn0.01)Cl2(OH)6. The mineral is easily soluble in dilute HCl. Paratacamite-(Mg) is trigonal, R, with cell parameters a = 13.689(1), c = 14.025(1) Å, V = 2275.8(3) Å3 and Z = 12. There is a pronounced sub-cell corresponding to a' ≈ ½a, c' ≈ c in space group Rm. The eight strongest lines in the X-ray powder diffraction pattern are [dobs Å(I)(hkl)]: 5.469(87)(021), 4.686(26)(003), 2.904(34)(401), 2.762(100)(22,042), 2.265(81)(404), 1.819(26)(603), 1.710 (34)(440) and 1.380(19)(446). The structure was refined to R1 = 0.039 for 480 Fo > 4σF reflections. Refinement using interlayer Mg-Cu site scattering factors indicated that Mg is distributed statistically between both interlayer octahedra M1O6 and M2O6. A comparison of the distortions associated with M1O6 and M2O6 octahedra suggest that the sample is near the upper compositional limit for stability of the R phase.

The crystal structure of eztlite has been determined using single-crystal synchrotron X-ray diffraction and supported using electron microprobe analysis and powder diffraction. Eztlite, a secondary tellurium mineral from the Moctezuma mine, Mexico, is monoclinic, space group Cm, with a = 11.466(2) Å, b = 19.775(4) Å, c = 10.497(2) Å, β = 102.62(3)° and V = 2322.6(9) Å3. The chemical formula of eztlite has been revised to ${\rm Pb}_{\rm 2}^{2 +} {\rm Fe}_3^{3 +} $(Te4+O3)3(SO4)O2Cl from that stated previously as ${\rm Fe}_6^{3 +} {\rm Pb}_{\rm 2}^{2 +} $(Te4+O3)3(Te6+O6)(OH)10·nH2O. This change has been accepted by the Commission on New Minerals, Nomenclature and Classification of the International Mineralogical Association, Proposal 18-A. Eztlite was reported originally to be a mixed-valence Te oxysalt; however the crystal structure, bond-valence analysis and charge balance considerations clearly show that all Te is tetravalent. Eztlite contains a unique combination of elements and is only the second Te oxysalt to contain both sulfate and chloride. The crystal structure of eztlite contains mitridatite-like layers, with a repeating triangular nonameric [${\rm Fe}_9^{3 +} $O36]45– arrangement formed by nine edge-sharing Fe3+O6 octahedra, decorated by four trigonal pyramidal Te4+O3 groups, compared to PO4 or AsO4 tetrahedra in mitridatite-type minerals. In eztlite, all four tellurite groups associated with one nonamer are orientated with the lone pair of the Te atoms pointing in the same direction, whereas in mitridatite the central tetrahedron is orientated in the opposite direction to the others. In mitridatite-type structures, interlayer connections are formed exclusively via Ca2+ and water molecules, whereas the eztlite interlayer contains Pb2+, sulfate tetrahedra and Cl–. Interlayer connectivity in eztlite is achieved primarily by connections via the long bonds of Pbφ8 and Pbφ9 groups to sulfate tetrahedra and to Cl–. Secondary connectivity is via Te–O and Te–Cl bonds.

Millsite, CuTeO3·2H2O, is a new mineral from Gråurdfjellet in Oppdal, Norway. It occurs as a minor secondary phase alongside teineite, other copper secondaries and relict primary tellurides in a boulder of quartz-rich granite, which is probably a glacial erratic. Millsite is bright cyan to royal blue in colour. The mineral is transparent to slightly translucent with a vitreous lustre and has a perfect (100) cleavage. It is brittle, has a conchoidal fracture and a pale green streak. Millsite is optically biaxial (+), α = 1.756(5), β = 1.794(5), γ = 1.925calc and 2Vmeas = 60(1)°. Millsite has monoclinic space group P21/c, with a = 7.4049(2) Å, b = 7.7873(2) Å, c = 8.5217(2) Å, β = 110.203(3)°, V = 461.17(2) Å3 and Z = 4. The empirical formula is Cu0.99(Te0.98Se0.02)O3(H2O)2. The five strongest reflections in the powder X-ray diffraction pattern are [dhkl in Å (hkl, Irel%)]: 6.954 (100, 100), 3.558 (012, 64), 2.838 (12$\bar 2$, 47), 2.675 (211, 43) and 3.175 (210, 39). The crystal structure has been determined to R1 = 0.016, wR2 = 0.036 and GooF = 1.049. The diagnostic structural unit of millsite consists of a Cu2O6(H2O)4 dimer that is decorated with four TeO3 groups connecting adjacent dimers and defining (100) heteropolyhedral sheets. These heteropolyhedral sheets are only connected by layers of structurally significant hydrogen bonds and correlate with the (100) cleavage. Millsite is a polymorph of teineite with a unique configuration of the M2O6(H2O)4 dimer that leads to a sheet topology. No isostructural selenium or tellurium analogue exists. The monoclinic polymorph (P21/c) of chalcomenite ‘monoclinic-CuSeO3·2H2O’ hereafter, ahlfeldite and MgSeO3·2H2O have M2O6(H2O)4 dimers, but their configuration differs significantly from that of millsite and leads to a framework topology rather than a sheet. Teineite does not have a dimeric structure and so is fundamentally different from millsite. The sheet topology of millsite appears to be unique among tellurites.

Objectives: The aim of this study was to identify whether the three main primary progressive aphasia (PPA) variants would show differential profiles on measures of visuospatial cognition. We hypothesized that the logopenic variant would have the most difficulty across tasks requiring visuospatial and visual memory abilities. Methods: PPA patients (n=156), diagnosed using current criteria, and controls were tested on a battery of tests tapping different aspects of visuospatial cognition. We compared the groups on an overall visuospatial factor; construction, immediate recall, delayed recall, and executive functioning composites; and on individual tests. Cross-sectional and longitudinal comparisons were made, adjusted for disease severity, age, and education. Results: The logopenic variant had significantly lower scores on the visuospatial factor and the most impaired scores on all composites. The nonfluent variant had significant difficulty on all visuospatial composites except the delayed recall, which differentiated them from the logopenic variant. In contrast, the semantic variants performed poorly only on delayed recall of visual information. The logopenic and nonfluent variants showed decline in figure copying performance over time, whereas in the semantic variant, this skill was remarkably preserved. Conclusions: This extensive examination of performance on visuospatial tasks in the PPA variants solidifies some previous findings, for example, delayed recall of visual stimuli adds value in differential diagnosis between logopenic variant PPA and nonfluent variant PPA variants, and illuminates the possibility of common mechanisms that underlie both linguistic and non-linguistic deficits in the variants. Furthermore, this is the first study that has investigated visuospatial functioning over time in the PPA variants. (JINS, 2018, 24, 259–268)

We show how in the hierarchies ${F_\alpha }$ of Fieldian truth sets, and Herzberger’s ${H_\alpha }$ revision sequence starting from any hypothesis for ${F_0}$ (or ${H_0}$) that essentially each ${H_\alpha }$ (or ${F_\alpha }$) carries within it a history of the whole prior revision process.

As applications (1) we provide a precise representation for, and a calculation of the length of, possible path independent determinateness hierarchies of Field’s (2003) construction with a binary conditional operator. (2) We demonstrate the existence of generalized liar sentences, that can be considered as diagonalizing past the determinateness hierarchies definable in Field’s recent models. The ‘defectiveness’ of such diagonal sentences necessarily cannot be classified by any of the determinateness predicates of the model. They are ‘ineffable liars’. We may consider them a response to the claim of Field (2003) that ‘the conditional can be used to show that the theory is not subject to “revenge problems”.’

We consider notions of truth and logical validity defined in various recent constructions of Hartry Field. We try to explicate his notion of determinate truth by clarifying the path-dependent hierarchies of his determinateness operator.

We give limits defined in terms of abstract pointclasses of the amount of determinacy available in certain canonical inner models involving strong cardinals. We show for example:

Theorem A. Det(-IND) ⇒ there exists an inner model with a strong cardinal.

Theorem B. Det(AQI) ⇒ there exist type-l mice and hence inner models with proper classes of strong cardinals.

where -IND(AQI) is the pointclass of boldface -inductive (respectively arithmetically quasi-inductive) sets of reals.

This paper continues the study of the Ramsey-like large cardinals introduced in [5] and [14]. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only α-many iterations for some countable ordinal α. Here we study such α-iterable cardinals. We show that the α-iterable cardinals form a strict hierarchy for α ≤ ω1, that they are downward absolute to L for , and that the consistency strength of Schindler's remarkable cardinals is strictly between 1-iterable and 2-iterable cardinals.

We show that the strongly Ramsey and super Ramsey cardinals from [5] are downward absolute to the core model K. Finally, we use a forcing argument from a strongly Ramsey cardinal to separate the notions of Ramsey and virtually Ramsey cardinals. These were introduced in [14] as an upper bound on the consistency strength of the Intermediate Chang's Conjecture.

The Infinite Time Turing Machine model [8] of Hamkins and Kidder is, in an essential sense, a “Σ2-machine” in that it uses a Σ2Liminf Rule to determine cell values at limit stages of time. We give a generalisation of these machines with an appropriate Σn rule. Such machines either halt or enter an infinite loop by stage , again generalising precisely the ITTM case.

The collection of such machines taken together computes precisely those reals of the least model of analysis.

We locate winning strategies for various -games in the L-hierarchy in order to prove the following:

Theorem 1. KP + Σ2-Comprehension -Determinacy.”

Alternatively: “there is a β-model of-Determinacy.” The implication is not reversible. (The antecedent here may be replaced with instances of Comprehension with only -lightface definable parameters—or even weaker theories.)

Theorem 2. KP + Δ2-Comprehension + Σ2-Replacement + -Determinacy.

(Here AQI is the assertion that every arithmetical quasi-inductive definition converges.) Alternatively: -Determinacy.

Hence the theories: , and are in strictly descending order of strength.