An official website of the United States government
Green used an arithmetic analogue of Szemerédi's celebrated regularity lemma to prove the following strengthening of Roth's theorem in vector spaces. For every α>0, β<α3, and prime number p, there is a least positive integer n_p(α,β) such that if n≥n_p(α,β), then for every subset of 𝔽np of density at least α there is a nonzero d for which the density of three-term arithmetic progressions with common difference d is at least β. We determine for p≥19 the tower height of n_p(α,β) up to an absolute constant factor and an additive term depending only on p. In particular, if we want half the random bound (so β=α^{3}/2), then the dimension n required is a tower of twos of height Θ((log p)loglog(1/α)). It turns out that the tower height in general takes on a different form in several different regions of α and β, and different arguments are used both in the upper and lower bounds to handle these cases.
more »
« less
PhaseSplit-FH3: A dataset of ternary separation per Flory-Huggins theory.
This dataset holds 1036 ternary phase diagrams and how points on the diagram phase separate if they do. The data is provided as a serialized object using the `pickle' Python module. The data was compiled using Python version 3.8. ReferencesThe specific applications and analyses of the data are described in 1. Dhamankar, S.; Jiang, S.; Webb, M.A. "Accelerating Multicomponent Phase-Coexistence Calculations with Physics-informed Neural Networks" UsageTo access the data in the .pickle file, users can execute the following: # LOAD SIMULATION DATADATA_DIR = "your/custom/dir/" filename = os.path.join(DATA_DIR, f"data_clean.pickle")with open(filename, "rb") as handle: (x, y_c, y_r, phase_idx, num_phase, max_phase) = pickle.load(handle) x: Input x = (χ_AB, χ_BC, χ_AC, v_A, v_B, v_C, φ_A, φ_B) ∈ ℝ^8. y_c: Output one-hot encoded classification vector y_c ∈ ℝ^3. y_r: Output equilibrium composition and abundance vector y_r = (φ_A^α, φ_B^α, φ_A^β, φ_B^β, φ_A^γ, φ_B^γ, w^α, w^β, w^γ) ∈ ℝ^9. phase_idx: A single integer indicating which unique phase system it belongs to. num_phase: A single integer indicates the number of equilibrium phases the input splits into. max_phase: A single integer indicates the maximum number of equilibrium phases the system splits into. Help, Suggestions, Corrections?If you need help, have suggestions, identify issues, or have corrections, please send your comments to Shengli Jiang at sj0161@princeton.edu GitHubAdditional data and code relevant for this study is additionally accessible at hthttps://github.com/webbtheosim/ml-ternary-phase
more »
« less
- PAR ID:
- 10544580
- Publisher / Repository:
- Zenodo
- Date Published:
- Format(s):
- Medium: X
- Right(s):
- Creative Commons Attribution 4.0 International
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Phase transitions in metastable α-, κ(ε)-, and γ-Ga2O3 films to thermodynamically stable β-Ga2O3 during annealing in air, N2, and vacuum have been systematically investigated via in situ high-temperature x-ray diffraction (HT-XRD) and scanning electron microscopy (SEM). These respective polymorphs exhibited thermal stability to ∼471–525 °C, ∼773–825 °C, and ∼490–575 °C before transforming into β-Ga2O3, across all tested ambient conditions. Particular crystallographic orientation relationships were observed before and after the phase transitions, i.e., (0001) α-Ga2O3 → (2¯01) β-Ga2O3, (001) κ(ε)-Ga2O3 → (310) and (2¯01) β-Ga2O3, and (100) γ-Ga2O3 → (100) β-Ga2O3. The phase transition of α-Ga2O3 to β-Ga2O3 resulted in catastrophic damage to the film and upheaval of the surface. The respective primary and possibly secondary causes of this damage are the +8.6% volume expansion and the dual displacive and reconstructive transformations that occur during this transition. The κ(ε)- and γ-Ga2O3 films converted to β-Ga2O3 via singular reconstructive transformations with small changes in volume and unchanged surface microstructures.more » « less
-
Abstract (Mg, Fe)SiO3orthopyroxene is an abundant mineral of oceanic subducting slabs. In‐situ high‐pressure and high‐temperature single‐crystal X‐ray diffraction has been used to investigate the phase transition of orthopyroxene across the enstatite‐ferrosilite (En‐Fs) join (En70Fs30, En55Fs45, En44Fs56and Fs100) up to 24.3 GPa and 800 K, simulating conditions within the coldest part of a subduction zone consisting of an old and rapidly subducting slab. Instead of the orthopyroxene → high‐pressure clinopyroxene transition, the α‐opx → β‐opx and β‐opx → γ‐opx phase transition are observed at 7.2–15.3 and 11.6–21.1 GPa (depending on the Fs content), respectively. This study indicates that the pressure‐induced phase transition of (Mg, Fe)SiO3orthopyroxene under relatively low temperature (<800 K) could be different than those occurring under relatively high temperature (>800 K). Additionally, the α‐opx → β‐opx → γ‐opx phase transition could exist within the center of the extremely cold slabs (like Tonga), where such low temperature persists to ~600‐km depth.more » « less
-
For a finite point set P⊂R^d, denote by diam(P) the ratio of the largest to the smallest distances between pairs of points in P. Let c_{d,α}(n) be the largest integer c such that any n-point set P⊂R^d in general position, satisfying diam(P)<αn^{1/d}, contains an c-point convex independent subset. We determine the asymptotics of c_{d,α}(n) as n→∞ by showing the existence of positive constants β=β(d,α) and γ=γ(d) such that βn^{(d−1)/(d+1)}≤c_{d,α}(n)≤γn^{(d−1)/(d+1)} for α≥2.more » « less
-
β-Ga2O3 is a promising ultra-wide bandgap semiconductor whose properties can be further enhanced by alloying with Al. Here, using atomic-resolution scanning transmission electron microscopy, we find the thermodynamically unstable γ-phase is a ubiquitous structural defect in both β-(AlxGa1−x)2O3 films and doped β-Ga2O3 films grown by molecular beam epitaxy. For undoped β-(AlxGa1−x)2O3 films, we observe γ-phase inclusions between nucleating islands of the β-phase at lower growth temperatures (∼500–600 °C). In doped β-Ga2O3, a thin layer of the γ-phase is observed on the surfaces of films grown with a wide range of n-type dopants and dopant concentrations. The thickness of the γ-phase layer was most strongly correlated with the growth temperature, peaking at about 600 °C. Ga interstitials are observed in the β-phase, especially near the interface with the γ-phase. By imaging the same region of the surface of a Sn-doped β-(AlxGa1−x)2O3 after ex situ heating up to 400 °C, a γ-phase region is observed to grow above the initial surface, accompanied by a decrease in Ga interstitials in the β-phase. This suggests that the diffusion of Ga interstitials toward the surface is likely the mechanism for growth of the surface γ-phase and more generally that the more-open γ-phase may offer diffusion pathways to be a kinetically favored and early forming phase in the growth of Ga2O3. However, more modeling and simulation of the γ-phase and the interstitials are needed to understand the energetics and kinetics, the impact on electronic properties, and how to control them.more » « less
