Foundations

21 pages, 3737 KiB

Open AccessArticle

Structural Analysis of Erbium-Doped Silica-Based Glass-Ceramics Using Anomalous and Small-Angle X-Ray Scattering

by Helena Cristina Vasconcelos, Maria Meirelles, Reşit Özmenteş and Luís Santos

Foundations 2025, 5(1), 5; https://doi.org/10.3390/foundations5010005 - 12 Feb 2025

This study employs advanced structural characterization techniques, including anomalous small-angle X-ray scattering (ASAXS), small-angle X-ray scattering (SAXS), and X-ray photoelectron spectroscopy (XPS), to investigate erbium (Er³⁺)-doped silica-based glass-ceramic thin films synthesized via the sol–gel method. This research examines the SiO₂-TiO₂ and SiO₂-TiO₂-PO_2.5 systems, focusing on the formation, dispersion, and structural integration of Er³⁺-containing nanocrystals within the amorphous matrix under different thermal treatments. Synchrotron radiation tuned to the L_III absorption edge of erbium enabled ASAXS measurements, providing element-specific details about the localization of Er³⁺ ions. The findings confirm their migration into crystalline phases, such as erbium phosphate (EPO) and erbium titanate (ETO). SAXS and Guinier analysis quantified nanocrystal sizes, revealing trends influenced by their composition and heat treatment. Complementary XPS analysis of the Er 5p core-level states provided detailed information on the chemical and electronic environment of the Er³⁺ ions, confirming their stabilization within the crystalline structure. Transmission electron microscopy (TEM) highlighted the nanoscale morphology, verifying the aggregation of Er³⁺ ions into well-defined nanocrystals. The results offer a deeper understanding of their size, distribution, and interaction with the surrounding matrix. Full article

(This article belongs to the Section Physical Sciences)

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14 pages, 483 KiB

Open AccessArticle

Linking Error Estimation in Fixed Item Parameter Calibration: Theory and Application in Large-Scale Assessment Studies

by Alexander Robitzsch

Foundations 2025, 5(1), 4; https://doi.org/10.3390/foundations5010004 - 11 Feb 2025

Abstract

In fixed item parameter calibration (FIPC), an item response theory (IRT) model is estimated with item parameters fixed at reference values to estimate the distribution parameters within a specific group. The presence of random differential item functioning (DIF) within this group introduces additional variability in the distribution parameter estimates, which is captured by the linking error (LE). Conventional LE estimates, based on item jackknife methods, are subject to positive bias due to sampling errors. To address this, this article introduces a bias-corrected LE estimate. Moreover, the use of statistical inference is examined using the newly proposed bias-corrected total error, which includes both the sampling error and LE. The proposed error estimates were evaluated through a simulation study, and their application is illustrated using PISA 2006 data for the reading domain. Full article

(This article belongs to the Section Mathematical Sciences)

19 pages, 640 KiB

Open AccessArticle

Exploring Order–Disorder Transitions Using a Two-State Master Equation

by Angelo Plastino and Diana Monteoliva

Foundations 2025, 5(1), 3; https://doi.org/10.3390/foundations5010003 - 14 Jan 2025

Viewed by 297

Abstract

In this study, we explore the order–disorder transition in the dynamics of a straightforward master equation that describes the evolution of a probability distribution between two states,

p_{1}

and

p_{2}

(with

p_{1} + p_{2} = 1

). We focus [...] Read more.

In this study, we explore the order–disorder transition in the dynamics of a straightforward master equation that describes the evolution of a probability distribution between two states,

p_{1}

and

p_{2}

(with

p_{1} + p_{2} = 1

). We focus on (1) the behavior of entropy S, (2) the distance D from the uniform distribution (

p_{1} = p_{2} = 1 / 2

), and (3) the free energy F. To facilitate understanding, we introduce two price-ratios:

η_{S} = \frac{d S / d t}{d F / d t}

and

η_{D} = \frac{d D / d t}{d F / d t}

. They respectively define the energetic costs of modifying (1) S and (2) D. Our findings indicate that both energy costs diverge to plus and minus infinity as the system approaches the uniform distribution, marking a critical transition point where the master equation temporarily loses its physical meaning. Following this divergence, the system stabilizes itself into a new well-behaved regime, reaching finite values that signify a new steady state. This two-regime behavior showcases the intricate dynamics of simple probabilistic systems and offers valuable insights into the relationships between entropy, distance in probability space, and free energy within the framework of statistical mechanics, making it a useful case study that highlights the underlying principles of the system’s evolution and equilibrium. Our discussion revolves about the order–disorder contrast that is important in various scientific disciplines, including physics, chemistry, and material science, and even in broader contexts like philosophy and social sciences. Full article

(This article belongs to the Section Mathematical Sciences)

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16 pages, 491 KiB

Open AccessArticle

Linking Error Estimation in Stocking–Lord Linking

by Alexander Robitzsch

Foundations 2025, 5(1), 2; https://doi.org/10.3390/foundations5010002 - 27 Dec 2024

Viewed by 440

Abstract

Stocking–Lord (SL) linking is a widely used linking method based on item response theory (IRT). This article examines the variability in SL linking parameter estimates within the two-parameter logistic (2PL) model. The uncertainty in SL linking arises from the sampling variability (standard error) and item selection (linking error), which can induce variability due to random differential item functioning (DIF). Three linking error estimation approaches are compared in this paper: the conventional jackknife linking error method, a newly developed approximate jackknife linking error method, and a Taylor approximation-based estimate. Simulation studies showed that the approximate jackknife method closely aligns with the traditional jackknife linking error method and outperforms the linking error estimation approach based on Taylor approximation. The adequacy of coverage rates for SL linking parameter estimates was also assessed using estimates of the total error. Results from a simulation study demonstrate that the bias-corrected total error provides superior coverage rates compared to both the conventional total error and the standard error, which does not account for item-related uncertainty due to random DIF. Full article

(This article belongs to the Section Mathematical Sciences)

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25 pages, 3400 KiB

Open AccessArticle

A Time-(Anti)symmetric Approach to the Double Solution Theory

by Pierre Jamet and Aurélien Drezet

Foundations 2025, 5(1), 1; https://doi.org/10.3390/foundations5010001 - 25 Dec 2024

Viewed by 381

Abstract

In this work, we present a new theoretical approach to interpreting and reproducing quantum mechanics using trajectory-guided wavelets. Inspired by the 1925 work of Louis de Broglie, we demonstrate that pulses composed of a difference between a delayed wave and an advanced wave (known as antisymmetric waves) are capable of following quantum trajectories predicted by the de Broglie–Bohm theory (also known as Bohmian mechanics). Our theory reproduces the main results of orthodox quantum mechanics and unlike Bohmian theory, is local in the Bell sense. We show that this is linked to the superdeterminism and past–future (anti)symmetry of our theory. Full article

(This article belongs to the Section Physical Sciences)

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Foundations, Volume 5, Issue 1 (March 2025) – 5 articles

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