We introduce a new permutation statistic, which we call the k-window statistic, for permutations of $\{1,2,\ldots, n\}$. The aim is to add to the repertoire of potentially useful tools in the design of efficient algorithms. The k-window statistic counts the number of elements that are at a displacement at most k from their natural positions in the ascendingly sorted permutation. We find the exact mean and exact variance of the k-window statistic, revealing phases in the mean and variance in the different regimes of k. We find a weak (or even strong) law of large numbers in the regime where $k=k_n$ is growing but remains ${\mathrm{o}} (n)$. Via a representation of the statistic into dependent heterogeneous weakly correlated indicators, we establish the complete spectrum of distributions as $k_n$ spans its entire range using an appropriate method for each regime.

We consider general continuum percolation models obeying sparseness, translation invariance, and spatial decorrelation. In particular, this includes models constructed on general point sets other than the standard Poisson point process or the Bernoulli-percolated lattice. Moreover, in our setting the existence of an edge may depend not only on the two end vertices but also on a surrounding vertex set and models are included that are not monotone in some of their parameters. We study the critical annulus-crossing intensity $\widehat{\lambda}_\mathrm{c}$, which is smaller than or equal to the classical critical percolation intensity $\lambda_\mathrm{c}$ and derive a condition for $\widehat{\lambda}_\mathrm{c}\gt 0$ by relating the crossing of annuli to the occurrence of long edges. This condition is sharp for models that have a modicum of independence. In a nutshell, our result states that annuli are either not crossed for small intensities or crossed by a single edge. Our proof rests on a multiscale argument that further allows us to directly describe the decay of the annulus-crossing probability with the decay of long-edges probabilities. We apply our result to a number of examples from the literature. Most importantly, we extensively discuss the weight-dependent random connection model in a generalized version, for which we derive sufficient conditions for the presence or absence of long edges that are typically easy to check. These conditions are built on a decay coefficient $\zeta$ that has recently seen some attention due to its importance for various proofs of global graph properties.

We study the activated random walk model on the one-dimensional ring, in the high-density regime. We develop a toppling procedure that gradually builds an environment that can be used to show that activity will be sustained for a long time. This yields a self-contained and relatively short proof of existence of a slow phase for arbitrarily large sleep rates.

Expansive soils, characterized by large montmorillonite (Mnt) contents, exhibit significant swelling-shrinkage behavior, posing engineering challenges in regions with alternating wet and dry seasons, such as southern China. Recent studies have explored the possible use of microbes to control or modify the swelling-shrinkage behavior. This study has evaluated microbial influence on the swelling-shrinkage behaviors of expansive soils. A combination of aqueous phase characterization through inductively coupled plasma-optical emission spectroscopy (ICP-OES) and solid phase analyses including X-ray diffraction (XRD) and Fourier-transform infrared spectroscopy (FTIR) was adopted to elucidate the variations in system parameters. Results showed that Si, Al, K, and Mg concentrations in the supernatant increased with prolonged interaction, with K concentration peaking at 14 days and decreasing to 27.5 mg L−1 at 42 days, indicating K+ uptake by Mnt interlayers. The Si–O bonds in tetrahedral sheets, Al–O bonds in octahedral structures, and –OH bonds in water molecules were disrupted by Paenibacillus mucilaginosus, accelerating the montmorillonite to illite (Mnt-to-Ilt) transformation. An Ilt-Mnt interstratification peak appeared near 9.88°2θ after 14 days, and an Ilt peak emerged at 48.07°2θ after 28 days. The Mnt content decreased by 37.28%, while the Ilt content increased by 104.01% after 42 days, as calculated by the K-value method. Analysis of swelling-shrinkage properties revealed that reduced Mnt content led to a 69.01% decrease in the free swell index and a 24.04% decrease in the swell ratio under load. The linear and volumetric shrinkage ratios decreased by 32.90% and 30.98%, respectively. Microorganisms can promote the weathering and phase transformation of Mnt into non-expansive minerals, thereby reducing the amount of it and thus reducing the swelling-shrinkage potential in expansive soils.

Feldspars are among the most common phases in the Earth’s crust and are characterized by wide variations in chemical composition and crystal structure motifs. In this paper, we synthesized SrZn2P2O8 and studied it by means of scanning electron microscopy, electron microprobe analysis and powder X-ray diffraction at ambient and high-temperature (up to 900°C) conditions. We have structurally characterized for the first time layered triclinic β-modification of SrZn2P2O8 (space group P$\bar 1$, a = 5.0109(8), b = 8.6202(13), c = 9.7527(15) Å, α = 118.088(3), β = 74.622(6), γ = 87.525(6)° and V = 351.29(10) Å3), which is structurally related to hexacelsian. The high-temperature study demonstrated that β-SrZn2P2O8 transformed irreversibly into the framework monoclinic paracelsian-like α-SrZn2P2O8 (space group P21/c, a = 8.3163(2), b = 9.5023(2), c = 9.0248(2) Å, β = 92.285(1)°, V = 712.60(1) Å3) above 600°C, which, in turn, is stable up to 900°C, when it melts. Though the crystal structures of α- and β-modification of SrZn2P2O8 are different (framework and layered, respectively), their thermal expansion is similar: αV ≈ 26 ± 1 × 10–6 °C–1, αmax/αmin ≈ 5. The possible occurrence of the corresponding minerals in nature is discussed.

We study the bilateral preference graphs $\mathit{LK}(n, k)$ of La and Kabkab, obtained as follows. Put independent and uniform [0, 1] weights on the edges of the complete graph $K_n$. Then, each edge (i, j) is included in $\mathit{LK}(n,k)$ if it is bilaterally preferred, in the sense that it is among the k edges of lowest weight incident to vertex i, and among the k edges of lowest weight incident to vertex j. We show that $k = \log(n)$ is the connectivity threshold, solving a conjecture of La and Kabkab, and obtaining finer results about the window. We also investigate the asymptotic behavior of the average degree of vertices in $\mathit{LK}(n, k)$ as $n\rightarrow\infty$.

The leucite group structures are tetrahedrally coordinated silicate framework structures with some of the silicate framework cations partially replaced by divalent or trivalent cations. These structures have general formulae A2BSi5O12 and ACSi2O6, where A is a monovalent alkali metal cation, B is a divalent cation, and C is a trivalent cation. These leucites can have crystal structures in several different space groups, dependent on stoichiometry, synthesis conditions, and temperature. Phase transitions are known for temperature changes. This paper reports a high-temperature X-ray powder diffraction study on RbGaSi2O6, which shows a phase transition from I41/a tetragonal to Iad cubic on heating from room temperature to 733 K. On cooling to room temperature, the crystal structure reverts to I41/a tetragonal.

Chlorite plays an essential role in transporting water into the deep Earth. This study investigates the high-pressure behaviour of clinochlore up to ∼20 GPa at room temperature, using synchrotron X-ray diffraction and infrared spectroscopy. We identified a polytypic phase transition (from IIb-2 to IIa-1) at ∼9 GPa and a structural distortion at ∼12 GPa. The polytypic phase transition at ∼9 GPa is characterised by a significant reorganization of interlayer hydrogen bonding, transitioning to states of weakly bonded or non-hydrogen-bonded hydroxyl groups. The phase transition showed reversibility during the experiment. The structural distortion near 12 GPa indicates adjustments in the compression mechanism related to the hydroxyl groups across both the brucite-like and 2:1 sheets. The unit-cell volume of the low-pressure IIb-2 phase was fitted to the second-order Birch-Murnaghan equation of state, and the resulting parameters are V0 = 710.9(3) Å3 and K0 = 83.4(8) GPa. Our results exhibit a marked deviation from prior experimental findings, potentially attributable to the presence of Fe2+ cation incorporation in the natural chlorite investigated in this work. This study emphasises the complexity of the hydroxyl group in response to pressure in layered-structure silicates.

Feldspars are the most common minerals in the Earth’s crust, and commonly contain trace concentrations of gallium and germanium. However, their gallium/germanium synthetic analogues are widely used, both in industrial applications and geological reconstructions. Here KMeGe3O8 (Me = Al, Ga) compounds were synthesized using solid-state reactions and hydrothermal synthesis and studied by microprobe analysis and powder X-ray diffraction (under both ambient and high-temperature conditions). In both (Al and Ga) cases, paracelsian-like phases were the first to form, even though feldspar-like phases are predicted to be thermodynamically stable. Paracelsian-like phases are transformed irreversibly upon heating (930 ± 30 and 960 ± 30°C for KGaGe3O8 and KAlGe3O8, respectively) into feldspar-like modifications. According to the obtained data, the feldspar-like modifications exhibit a higher degree of anisotropy, whereas the volume thermal expansion coefficients are higher for paracelsian-like modifications. The influence of chemical composition and topology on the crystallization process, thermal transformation and thermal expansion of feldspar- and paracelsian-like compounds is discussed.

Groutite, α-Mn3⁺O(OH), is a mineral that belongs to the diaspore group. We used X-ray synchrotron radiation in the pressure range from 0.65(6) GPa to 39.85(8) GPa to collect diffraction data for three single crystals of groutite placed in a diamond anvil cell. Groutite is orthorhombic, space group Pbnm. Analysis of the reciprocal lattice planes and the intensities of the measured reflections revealed that above 30 GPa, a phase transformation begins in the structure. This transition is gradual, with observable changes in the intensity of specific reflections, particularly those that are forbidden in the Cmcm space group. Although we see significant decreases in the intensities of these reflections, many of them still have intensities significantly greater than zero after the beginning of transformation. This is accompanied by a smooth, continuous shrinkage in the unit cell volume, symmetrization of hydrogen positions, and distortion of the MnO6 octahedra. Analysis of the reflection intensities confirms that the observed changes are due to structural transformations and not experimental or sample effects. This means that the starting Pbnm symmetry is preserved although the mineral transforms as a function of pressure close to the Cmcm space-group symmetry. We propose a mechanism for this phase transition in the Pbnm space group.