Weak Order Research Articles

The \(s\)-Weak Order and \(s\)-Permutahedra I: Combinatorics and Lattice Structure

The \(s\)-Weak Order and \(s\)-Permutahedra I: Combinatorics and Lattice Structure

Electron series resonance excited in the 27.12 MHz magnetron sputtering discharge

Abstract The electron series resonance (ESR) excited in the 27.12 MHz magnetron sputtering discharge was investigated. By analyzing the discharge impedances, the imaginary part of impedance was found to undergo a transition from capacitive to inductive on varying RF power, and the conditions for ESR excited was satisfied at the 27.12 MHz magnetron sputtering. By analyzing the discharge current and its higher order harmonics, the near-sinusoidal current waveform and the weak 2nd order harmonic were obtained, showing a weak nonlinear effect of RF current. However, for the magnetron sputtering discharge, the non-uniform magnetic field has a significant effect on the sheath width and the transverse current, making the sheath thinner and the transverse current smaller. As a result, the small capacitive reactance was obtained and the inductive reactance was easily cancelled. Therefore, the electron series resonance excited in the 27.12 MHz magnetron sputtering was caused together by the strong effect of non-uniform magnetic field and the weak 2nd (H2) order current harmonic. By estimating the ESR frequency res,B, the 2nd order current harmonic (54.24 Hz) was found to be responsible for the ESR excitation.

A study on generalized balanced split drift stochastic Runge- Kutta methods for stochastic differential equations

To reduce computational complexity, the balanced numerical approximations of the general split drift stochastic Runge-Kutta methods are analyzed. The primary reasons for considering the numerical approximations of these balanced split stochastic Runge-Kutta methods are their improved stability characteristics and lower mean square error compared to other methods. By balancing the drift and diffusion components, the splitting techniques outperform the mean square error over longer time increments. For Ito multi-dimensional stochastic differential equations, we propose a novel family of balanced universal split stochastic Runge-Kutta procedures. The Kronecker product concept is utilized to derive the mean-square stability conditions. We conduct numerical tests to evaluate these methods against an existing weak order 2 split drift method. Ultimately, a specific numerical example validates the theoretical outcomes of the balanced general split stochastic Runge-Kutta procedures.

Green-synthesized BiFeO3 nanoparticles for efficient photocatalytic degradation of organic dyes, antibiotic and catalytic reduction of 4-nitrophenol

Green-synthesized nanoparticles have recently emerged as promising catalysts for the removal of toxic dyes from water bodies. In this article, Bismuth ferrite (BiFeO3) nanoparticles were synthesized by a simple and low-cost method using Couroupita guianensis plant leaf extract. The particles were used as photocatalysts for degrading RhB, MO dyes, and TC antibiotics, as well as for reducing toxic 4-NP to useful 4-AP. The synthesized nanoparticles were characterized using several state-of-the-art tools. The formation of perovskite structure and the presence of biomolecules on the surface of BiFeO3 nanoparticles was confirmed by FTIR analysis. Raman spectrum revealed a rhombohedral structure of BiFeO3 nanoparticles, corroborating the XRD analysis. FESEM micrograph demonstrated irregularly shaped agglomerated nanoparticles. The average hydrodynamic diameter was estimated to be 51 nm using the DLS technique. The optical energy bandgap of the bismuth ferrite was estimated using Tauc’s relation for direct bandgap semiconductors. SQUID measurements revealed that these nanoparticles exhibit a weak ferromagnetic ordering. Furthermore, these nanomaterials degraded the cationic and anionic dyes effectively because of the formation of hydroxyl radicals, confirmed by the scavenger test. The photocatalytic degradation pathway of RhB dye was investigated through LC-MS analysis, and the intermediate degradation products were identified. The prepared material also exhibited excellent catalytic efficiency in a short duration of time. This study proclaims that these nanoparticles can be used as potential catalysts for the purification of water.

Coarse-Grained Simulations of Columnar Ionic Liquid Crystals: Comparison with Experiments.

We simulate a homologous series of guanidinium-based columnar ionic liquid crystals (ILCs) using coarse-grained molecular dynamics (MD) simulations with the Martini force field. We systematically vary the length of alkyl side chains, ILC-n (n = 8, 12, 16), and compare our results with previous experimental findings. Experimentally, ILC-8 exhibits a narrow mesophase window and weak columnar order, while ILC-12 and ILC-16 display a broad mesophase window and high columnar order. The MD simulations show that ILC-8 forms a percolated structure, whereas the longer chain analogues self-assemble into columns, with columnar assembly becoming more prominent as the side chain length increases, in qualitative agreement with the experiments. Furthermore, the intercolumnar distance increases monotonically with increasing side chain length and decreases with increasing temperature. Finally, we find that the diffusion coefficient and ionic conductivity decrease substantially with increasing chain length, consistent with experimental observations. We attribute this decrease in mobility to the formation of hexagonally ordered columns, which restrict transport more than percolated networks.

The $s$-Weak Order and $s$-Permutahedra II: The Combinatorial Complex of Pure Intervals

This paper introduces the geometric foundations for the study of the $s$-permutahedron and the $s$-associahedron, two objects that encode the underlying geometric structure of the $s$-weak order and the $s$-Tamari lattice. We introduce the $s$-permutahedron as the complex of pure intervals of the $s$-weak order, present enumerative results about its number of faces, and prove that it is a combinatorial complex. This leads, in particular, to an explicit combinatorial description of the intersection of two faces. We also introduce the $s$-associahedron as the complex of pure $s$-Tamari intervals of the $s$-Tamari lattice, show some enumerative results, and prove that it is isomorphic to a well chosen $\nu$-associahedron. Finally, we present three polytopality conjectures, evidence supporting them, and some hints about potential generalizations to other finite Coxeter groups.

Understanding the nature of the magnetic coupling in transition metal doped Bi2Se3

In this paper, we employ electronic structure theory to investigate the nature of the exchange coupling in transition metal doped Bi2Se3. We focus on V, Cr, Mn, and Fe, which have been under scrutiny for the realization of the quantum anomalous Hall effect. For simplicity, we model the doping process as happening inside a single layer of Bi within a single quintuple layer inside a three-formula-unit conventional hexagonal cell and consider situations of full coverage or half coverage. Due to the covalent nature of the chemical bond between transition metal atoms and selenium, this simple model is capable of describing the fundamental features of the intraplane exchange coupling, while offering a clearer analysis of the response to structural, magnetic, and electronic perturbations. In agreement with recent literature, our results confirm that the van Vleck mechanism has a very marginal contribution to the exchange coupling. Depending on the filling of the 3d shell and on the details of the electronic structure, several other mechanisms compete and cooperate to induce the magnetic order. For V, double exchange and superexchange cooperate to have the strongest ferromagnetic coupling among the investigated elements, followed by Cr where only superexchange is active. For Mn, superexchange and double exchange compete to create an extremely weak ferromagnetic order. For Fe, a strong antiferromagnetic coupling caused by the double exchange is observed to dominate over the ferromagnetic Ruderman-Kittel-Kasuya-Yosida interaction, but the high sensitivity of this competition to the doping concentration suggests that our conclusion may not hold in the dilute limit. Overall, Fe doping seems to offer the most intriguing competition of exchange mechanisms that can be tuned by adjusting the doping concentration and the details of the host, as, e.g., by replacing Se with Te or Bi with Sb. Published by the American Physical Society 2024

Flat-band and diverse quasi-fermions in Pb10(PO4)6O4

Employing a combination of first-principles calculations and low-energy effective models, we present a comprehensive investigation on the electronic structure of Pb10(PO4)6O4, which exhibits remarkable quasi-one-dimensional topological flat-band around the Fermi level. These flat bands predominantly originate from the px/py orbitals of the oxygen molecules chain at the fully-occupied 4e Wyckoff positions and thus can be well-captured by a minimal four-band tight-binding model. Furthermore, the abundant crystal symmetry inherent in Pb10(PO4)6O4 provides an ideal platform for the emergence of various quasi-fermions characterized by different dispersion, degeneracy, and dimensionality. These include a 0D four-fold degenerate Dirac fermion exhibiting quadratic dispersion, a 1D quadratic/linear nodal-line fermion along symmetric k-paths, a 1D hourglass nodal-line (HNL) fermion associated with the Dirac fermion, and a 2D symmetry-enforced nodal surface located on the kz = π plane. Moreover, when considering the weak ferromagnetic order, Pb10(PO4)6O4 transforms into a rare semi-half-metal, which is characterized by the presence of Dirac fermion and HNL fermion at the Fermi level for a single spin channel exhibiting 100 % spin polarization. Our findings reveal the rarely coexistence of flat bands, diverse topological semimetal states and ferromagnetism in Pb10(PO4)6O4, which may provide valuable insights for further exploring the intriguing interplay between superconductivity and exotic electronic states.

Identifying the Origin of Thermal Modulation of Exchange Bias in MnPS3/Fe3GeTe2 van der Waals Heterostructures.

The exchange bias phenomenon, inherent in exchange-coupled ferromagnetic and antiferromagnetic systems, has intrigued researchers for decades. Van der Waals materials, with their layered structures, offer an ideal platform for exploring exchange bias. However, effectively manipulating exchange bias in van der Waals heterostructures remains challenging. This study investigates the origin of exchange bias in MnPS3/Fe3GeTe2 van der Waals heterostructures, demonstrating a method to modulate nearly 1000% variation in magnitude through simple thermal cycling. Despite the compensated interfacial spin configuration of MnPS3, a substantial 170 mT exchange bias is observed at 5 K, one of the largest observed in van der Waals heterostructures. This significant exchange bias is linked to anomalous weak ferromagnetic ordering in MnPS3 below 40 K. The tunability of exchange bias during thermal cycling is attributed to the amorphization and changes in the van der Waals gap during field cooling. The findings highlight a robust and adjustable exchange bias in van der Waals heterostructures, presenting a straightforward method to enhance other interface-related spintronic phenomena for practical applications. Detailed interface analysis reveals atom migration between layers, forming amorphous regions on either side of the van der Waals gap, emphasizing the importance of precise interface characterization in these heterostructures.

Combinatorial Results on Barcode Lattices

AbstractA barcode is a finite multiset of intervals on the real line. Jaramillo-Rodriguez (2023) previously defined a map from the space of barcodes with a fixed number of bars to a set of multipermutations, which presented new combinatorial invariants on the space of barcodes. A partial order can be defined on these multipermutations, resulting in a class of posets known as combinatorial barcode lattices. In this paper, we provide a number of equivalent definitions for the combinatorial barcode lattice, show that its Möbius function is a restriction of the Möbius function of the symmetric group under the weak Bruhat order, and show its ground set is the Jordan-Hölder set of a labeled poset. Furthermore, we obtain formulas for the number of join-irreducible elements, the rank-generating function, and the number of maximal chains of combinatorial barcode lattices. Lastly, we make connections between intervals in the combinatorial barcode lattice and certain classes of matchings.

Robust ordinal regression for subsets comparisons with interactions

This paper is devoted to a robust ordinal method for learning the preferences of a decision maker between subsets. The decision model, derived from Fishburn and LaValle (1996) and whose parameters we learn, is general enough to be compatible with any strict weak order on subsets, thanks to the consideration of possible interactions between elements. Moreover, we accept not to predict some preferences if the available preference data are not compatible with a reliable prediction. A predicted preference is considered reliable if all the simplest models (Occam’s razor) explaining the preference data agree on it. Following the robust ordinal regression methodology, our predictions are based on an uncertainty set encompassing the possible values of the model parameters. We define a new ordinal dominance relation between subsets and design a procedure to determine whether this dominance relation holds. Numerical tests are provided on synthetic and real-world data to evaluate the richness and reliability of the preference predictions made.

On preferences and reward policies over rankings

We study the rational preferences of agents participating in a mechanism whose outcome is a ranking (i.e., a weak order) among participants. We propose a set of self-interest axioms corresponding to different ways for participants to compare rankings. These axioms vary from minimal conditions that most participants can be expected to agree on, to more demanding requirements that apply to specific scenarios. Then, we analyze the theories that can be obtained by combining the previous axioms and characterize their mutual relationships, revealing a rich hierarchical structure. After this broad investigation on preferences over rankings, we consider the case where the mechanism can distribute a fixed monetary reward to the participants in a fair way (that is, depending only on the anonymized output ranking). We show that such mechanisms can induce specific classes of preferences by suitably choosing the assigned rewards, even in the absence of tie breaking.

Strong completeness of a class of 𝐿²(𝑇)-type Riesz spaces

The L p , 1 ≤ p ≤ ∞ L^{p}, 1\le p\le \infty , spaces have been generalized to the setting of Riesz spaces as L p ( T ) {L}^{p}(T) spaces, on which there are R ( T ) R(T) -valued norms. The strong sequential completeness of the space L 1 ( T ) {L}^{1}(T) and the strong completeness of L ∞ ( T ) {L}^{\infty }(T) with resepct to their respective R ( T ) R(T) -valued norms were established by Kuo, Rodda, and Watson. In the current work, the T T -strong completeness of L 2 ( T ) {L}^{2}(T) is established via the Riesz–Fischer type theorem given by Kalauch, Kuo, and Watson. It is also shown that the conditional expectation operator T T is a weak order unit for the T T -strong dual.

New line list for the ν4 bands of the trans (790.117 cm–1) and cis (851.943 cm–1) conformers of nitrous acid (HONO): Accurate positions and absolute intensities

The goal of this work was to update and significantly improve the line lists that have been generated recently for 11 µm bands of the trans- and cis- conformer forms of nitrous acid (HONO) [Armante R, Perrin A, Kwabia Tchana F, Manceron L. The ν4 bands at 11 μm: linelists for the trans- and cis- conformer forms of nitrous acid (HONO) in the 2019 version of the GEISA database. Molecular Physics 2021;120:e1951860]. That 2019 version of the 11 µm line list was generated using the spectroscopic parameters that were available, at that time, in the literature. During the present study, we used high-resolution Fourier transform spectra recorded at 11 µm to perform a large investigation of line positions and intensities for the ν4 bands of the trans- and cis-conformer of HONO. The resulting set of experimental ν4 (absolute) intensities and of 41 energy levels were used to determine, by least squares fit computations, improved position and intensity parameters for the ν4 bands of the trans- and cis-conformer of HONO. For trans-HONO, the ν4 band appeared not to be perturbed, while for cis-HONO a weak high order B-type Coriolis, coupling together the 41 and 61 energy levels was evidenced for the first time. This new list is of potential interest for the IASI-NG (Infrared Atmospheric Sounding Interferometer - New Generation) instrument which will be launched on board the METOP-SG satellite in 2025.

Zero energy bound states on nano atomic line defect in iron-based high temperature superconductors

Motivated by recent scanning tunneling microscopy experiments on Fe atomic line defect in iron-based high temperature superconductors, we explore the origin of the zero energy bound states near the endpoints of the line defect by employing the two-orbit four-band tight binding model. With increasing the strength of the Rashba spin–orbit coupling along the line defect, the zero energy resonance peaks move simultaneously forward to negative energy for s+− pairing symmetry, but split for s++ pairing symmetry. The superconducting order parameter correction due to As(Te, Se) atoms missing does not shift the zero energy resonance peaks. Such the zero energy bound states are induced by the weak magnetic order rather than the strong Rashba spin–orbit coupling on Fe atomic line defect.

Limited miscibility in hydrated DPPC – Lyso-PPC systems

Hydrated dipalmitoylphosphatidylcholine (DPPC) and lyso-palmitoylphosphatidylcholine (Lyso-PPC, abbreviated as LPC) mixtures form vesicles composed from periodically or randomly arranged layers and micelle-like nanoparticles, as revealed by a wide range of thermodynamical, structural, and morphological characterization methods. Based on five different experimental methods, we have reconstructed the phase diagram of the system. The micelle-like objects are strongly anisotropic and are identified as bicelles. Even a small amount of LPC induces drastic changes in the thermotropic phase behaviour of DPPC. Here, an interdigitated gel (LβI) structure, coexisting with the other gel phases, forms at moderate LPC concentration. Bicelles and LβI structured vesicles appear in the equimolar LPC ratio range (0.4 ≤ XLPC ≤ 0.6). Two complex phase transition regimes are detectable. Between 36–40 °C, bicelles are pre-melted while domains of LβI structure are “crystallized” through local endothermic and exothermic transitions, respectively. Then, around 42 °C the chains of both the LβI and bicelles melt completely. Above the equimolar LPC ratio range (XLPC > 0.6) the chain melting in the LβI structure vanishes, while this transition still exists in the chain region of bicelles. This wealth of structural forms, due to the weak first order transition characteristics, is highly sensitive for perturbations and interactions induced by e.g. the presence of membrane proteins.

Stochastic comparison results between two finite mixture models with generalized Weibull distributed components

In this paper, we establish sufficient conditions for stochastic comparisons of two finite mixture models (FMMs) with respect to the usual stochastic order, hazard rate order, and likelihood ratio order when the mixing components have generalized Weibull family of distributions. The established (sufficient) conditions are mainly based on the majorization order, weak supermajorization order, and weak submajorization order. The stochastic comparisons are studied when there is heterogeneity in one (model) parameter, and then in two parameters (model parameter and mixing proportion). Further, the concept of unordered majorization order is employed to establish the usual stochastic order between two FMMs. To illustrate the theoretical results established here, several numerical examples and counterexamples are presented. Finally, we have generalized some of the results to the case of τ-mixture models.

Anomalous Hall effect in 5d/5d SrTaO3/SrIrO3 superlattices driven by ferromagnetism and spin–orbit coupling

The observation of the anomalous Hall effect (AHE) in 5d perovskite oxides has been challenging due to their lack or weak ferromagnetic order, which is necessary for breaking time-reversal symmetry. Here, we present compelling evidence of ferromagnetism and consequent AHE in a series of carefully designed and fabricated 5d/5d SrTaO3/SrIrO3 (STO/SIO) superlattices. The coexistence of Ta5+ and Ta4+ chemical states induces ferromagnetism in the STO layer, while the interfacial magnetic proximity effect further enhances it in the SIO layer, resulting in both ferromagnetism and AHE within the STO/SIO superlattice. Additionally, the strong spin–orbit coupling between Ta and Ir elements positively contributes to enhancing the AHE. This work offers an alternative approach for designing artificial materials with AHE and holds potential for advancing spintronics.

A Facial Order for Torsion Classes

Abstract We generalize the “facial weak order” of a finite Coxeter group to a partial order on a set of intervals in a complete lattice. We apply our construction to the lattice of torsion classes of a finite-dimensional algebra and consider its restriction to intervals coming from stability conditions. We give two additional interpretations of the resulting “facial semistable order”: one using cover relations, and one using Bongartz completions of 2-term presilting objects. For $\tau $-tilting finite algebras, this allows us to prove that the facial semistable order is a semidistributive lattice. We then show that, in any abelian length category, our new partial order can be partitioned into a set of completely semidistributive lattices, one of which is the original lattice of torsion classes.

Signature of possible spin liquid state at 2K in spin-frustrated Cr1-xFexPSe3 alloy

Disordered spin systems typically exhibit glass-like characteristics below the glass transition temperature [1–4]. However, when the competition between ferromagnetic and antiferromagnetic couplings is balanced, the spin liquid state persists towards zero Kelvin rather than a glass state [5]. This unique state is known as the quantum spin liquid state. In this study, we examined a potential spin liquid state in the two-dimensional spin-frustrated Cr1-xFexPSe3 alloy by magnetic susceptibility measurements as a function of temperature and doping. Two end compounds, CrPSe3 and FePSe3, exhibit an antiferromagnetic order. With the Fe doping, the antiferromagnetic order of frustrated CrPSe3 gradually weakened and reached its minimum transition temperature at the critical concentration of x = 0.33. This weak order became indeterminate when a high magnetic field was applied, indicating the possibility of converting the magnetic order to the spin liquid state down to 2 K. With further doping, an antiferromagnetic order was observed in the Fe-dominated compound. Our results provide a new two-dimensional flat form for exploring quantum spin liquid states.