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The high-pressure behaviour of inderborite [ideally CaMg[B3O3(OH)5]2(H2O)4⋅2H2O, space group C 2/ c with a ≈ 12.14, b ≈ 7.43, c ≈ 19.23 Å and β ≈ 90.3° at room conditions] has been studied by two in situ single-crystal synchrotron X-ray diffraction experiments up to ~10 GPa, using He as pressure-transmitting fluid. Between 8.11(5) and 8.80(5) GPa, inderborite undergoes a first-order phase transition to its high-pressure polymorph, inderborite-II (with a ≈ 11.37, b ≈ 6.96, c ≈ 17.67 Å, β ≈ 96.8° and ΔV ≈ 7.0%, space group unknown). The isothermal bulk modulus (KV0 = β−1P0,T0, where β P0,T0 is the volume compressibility coefficient) of inderborite was found to be KV0 = 41(1) GPa. The destructive nature of the phase transition prevented any structure resolution of inderborite-II or even the continuation of the experiments at pressures higher than 10.10(5) GPa. In the pressure range 0–8.11(5) GPa, the compressional anisotropy of inderborite, indicated by the ratio between the principal components of the Eulerian finite unit-strain ellipsoid, is ɛ1:ɛ2:ɛ3 = 1.4:1.05:1. The deformation mechanisms at the atomic scale in inderborite are here described. Our findings support the hypothesis of a quasi-linear correlation between the total H2O content and P -stability range in hydrated borates, as the pressure at which inderborite undergoes the phase transition falls in line with most of the hydrate borates studied at high-pressure so far. [ABSTRACT FROM AUTHOR]

Distinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton–Watson trees. For example, let $\mathcal{T}_1$ be the event that a Galton–Watson tree is infinite and let $\mathcal{T}_2$ be the event that it contains an infinite binary tree starting from its root. These events satisfy similar recursive properties: $\mathcal{T}_1$ holds if and only if $\mathcal{T}_1$ holds for at least one of the trees initiated by children of the root, and $\mathcal{T}_2$ holds if and only if $\mathcal{T}_2$ holds for at least two of these trees. The probability of $\mathcal{T}_1$ has a continuous phase transition, increasing from 0 when the mean of the child distribution increases above 1. On the other hand, the probability of $\mathcal{T}_2$ has a first-order phase transition, jumping discontinuously to a non-zero value at criticality. Given the recursive property satisfied by the event, we describe the critical child distributions where a continuous phase transition takes place. In many cases, we also characterise the event undergoing the phase transition. [ABSTRACT FROM AUTHOR]

Hydrothermally synthesised K2ZnSi5O12 has a polymerised framework structure with the same topology as leucite (KAlSi2O6, tetragonal I41/a), which has two tetrahedrally coordinated Al3+ cations replaced by Zn2+ and Si4+. At 293 K it has a cation-ordered framework P21/c monoclinic structure with lattice parameters a = 13.1773(2) Å, b = 13.6106(2) Å, c = 13.0248(2) Å and β = 91.6981(9)°. This structure is isostructural with K2MgSi5O12, the first cation-ordered leucite analogue characterised. With increasing temperature, the P21/c structure transforms reversibly to cation-ordered framework orthorhombic Pbca. This transition takes place over the temperature range 848−863 K where both phases coexist; there is an ~1.2% increase in unit cell volume between 843 K (P21/c) and 868 K (Pbca), characteristic of a first-order, displacive, ferroelastic phase transition. Spontaneous strain analysis defines the symmetry- and non-symmetry related changes and shows that the mechanism is weakly first order; the two-phase region is consistent with the mechanism being a strain-related martensitic transition. [ABSTRACT FROM AUTHOR]

I analyze the extent to which classical phase transitions, both first order and continuous, pose a challenge for intertheoretic reduction. My contention is that phase transitions are compatible with a notion of reduction that combines Nagelian reduction and what Thomas Nickles called Reduction2. I also argue that, even if the same approach to reduction applies to both types of phase transitions, there is a crucial difference in their physical treatment: in addition to the thermodynamic limit, in continuous phase transitions there is a second infinite limit involved, which marks an important difference in the reduction of first-order and continuous phase transitions. [ABSTRACT FROM AUTHOR]

Let A ≤ B be structures, and ${\cal K}$ a class of structures. An element b ∈ B is dominated by A relative to ${\cal K}$ if for all ${\bf{C}} \in {\cal K}$ and all homomorphisms g, g' : B → C such that g and g' agree on A, we have gb = g'b. Our main theorem states that if ${\cal K}$ is closed under ultraproducts, then A dominates b relative to ${\cal K}$ if and only if there is a partial function F definable by a primitive positive formula in ${\cal K}$ such that FB(a1,…,an) = b for some a1,…,an ∈ A. Applying this result we show that a quasivariety of algebras ${\cal Q}$ with an n-ary near-unanimity term has surjective epimorphisms if and only if $\mathbb{S}\mathbb{P}_n \mathbb{P}_u \left( {\mathcal{Q}_{{\text{RSI}}} } \right)$ has surjective epimorphisms. It follows that if ${\cal F}$ is a finite set of finite algebras with a common near-unanimity term, then it is decidable whether the (quasi)variety generated by ${\cal F}$ has surjective epimorphisms. [ABSTRACT FROM AUTHOR]

In this paper we investigate the reverse mathematics of higher-order analogues of the theory $$ATR_0$$ within the framework of higher order reverse mathematics developed by Kohlenbach [11]. We define a theory $$RCA_0^3$$, a close higher-type analogue of the classical base theory $$RCA_0$$ which is essentially a conservative subtheory of Kohlenbach’s base theory $$RCA_{\rm{0}}^\omega$$. Working over $$RCA_0^3$$, we study higher-type analogues of statements classically equivalent to $$ATR_0$$, including open and clopen determinacy, and examine the extent to which $$ATR_0$$ remains robust at higher types. Our main result is the separation of open and clopen determinacy for reals, using a variant of Steel’s tagged tree forcing; in the presentation of this result, we develop a new, more flexible framework for Steel-type forcing. [ABSTRACT FROM PUBLISHER]