Lattice Points Research Articles

Local structural disorder introduced by Cr in fcc high-/medium-entropy alloys consisting of 3d transition metal elements

High-/medium-entropy alloys (H/MEAs) have attracted attention for their excellent mechanical properties. According to first-principles calculations, the atomic displacements from their lattice points are element-dependent and correlated with the yield strength of H/MEAs. To investigate experimentally the element dependence of the local structure, we measured the extended x-ray absorption fine structure in seven kinds of face-centered cubic H/MEAs consisting of 3d transition metal elements. Our experimental results indicate that the local disorder around chromium (Cr) is larger than that around other elements, which is a common feature in all the measured fcc H/MEAs, aligning with prior theoretical studies indicating larger displacements of Cr in comparison to other elements. We discuss the mechanism underlying the local structural disorder induced by the constituent element Cr.

Research on match fluctuation prediction and strategy based on OFS algorithm

The aim of this paper is to construct a model for predicting on-field fluctuations in matches by using OFS algorithm, combined with random forest and lattice point search algorithms, and to analyse the key factors in order to improve the coach's decision-making ability in matches. Cross-validation and hyper-parameter tuning techniques were used during the model building process to find the best hyper-parameter combinations through lattice point search to optimise the model performance and generalisation ability. The study provides a detailed description of the model implementation process, including data processing, feature selection, parameter setting, model building, training and evaluation. Meanwhile, a series of targeted recommendations are proposed to address the variability of match fluctuations, including the analysis of past matches, the focus on recent performance, the assessment of psychological quality, the study of opponent's weaknesses, the flexible response to changes in matches and the maintenance of focus, with a view to helping athletes and coaches to formulate a more effective match strategy and to achieve better match results.

Some Improvements on Good Lattice Point Sets

Good lattice point (GLP) sets are a type of number-theoretic method widely utilized across various fields. Their space-filling property can be further improved, especially with large numbers of runs and factors. In this paper, Kullback-Leibler (KL) divergence is used to measure GLP sets. The generalized good lattice point (GGLP) sets obtained from linear-level permutations of GLP sets have demonstrated that the permutation does not reduce the criterion maximin distance. This paper confirms that linear-level permutation may lead to greater mixture discrepancy. Nevertheless, GGLP sets can still enhance the space-filling property of GLP sets under various criteria. For small-sized cases, the KL divergence from the uniform distribution of GGLP sets is lower than that of the initial GLP sets, and there is nearly no difference for large-sized points, indicating the similarity of their distributions. This paper incorporates a threshold-accepting algorithm in the construction of GGLP sets and adopts Frobenius distance as the space-filling criterion for large-sized cases. The initial GLP sets have been included in many monographs and are widely utilized. The corresponding GGLP sets are partially included in this paper and will be further calculated and posted online in the future. The performance of GGLP sets is evaluated in two applications: computer experiments and representative points, compared to the initial GLP sets. It shows that GGLP sets perform better in many cases.

Photonic crystal fibers based on Dirac point-guiding

—In this work photonic crystal fibers of different cross-sectional patterns are studied. Dirac points of these various lattices are explored, propagation diagrams showing positions of the Dirac spectrums are obtained, and their application in fiber guiding is discussed. A quasi-3D FDTD simulation method, which is simpler and more efficient than a commercial 3D FDTD simulation software, is developed for photonic crystal fiber. This method is then applied to the new photonic crystal fiber proposed in [Fiber guiding at the Dirac frequency beyond photonic bandgaps, Light Sci. & App. 4 (2015) e304.]. Dirac-point guidance, which is based on Dirac localization of photons, is verified for these fibers.

Analysis of differential scanning calorimetry data for aged plutonium.

Differential scanning calorimetry data for samples of a 52 year old plutonium alloy with 3.3 at. % Ga that were heated beyond the melting point is analyzed using transition state theory to find activation energies for the δ to ɛ and ɛ to liquid phase transitions. A Bayesian statistical method involving a Gaussian process model is used to find mean values and confidence intervals for the activation energies. The activation energy for the δ to ɛ phase transition increases by 3.3 ± 3.8% per decade, relative to the case when all age related plutonium lattice point defects have been removed through annealing. The corresponding increase in activation energy for the ɛ to liquid transition is shown to be 7.1 ± 1.8% per decade. It is postulated that the change in activation energy with age for both phase transitions is caused, in part, by the accumulation of the same type of lattice point defects associated with the observed increase in elastic bulk modulus over time.

Semistable torsion classes and canonical decompositions in Grothendieck groups

AbstractWe study two classes of torsion classes that generalize functorially finite torsion classes, that is, semistable torsion classes and morphism torsion classes. Semistable torsion classes are parametrized by the elements in the real Grothendieck group up to TF equivalence. We give a close connection between TF equivalence classes and the cones given by canonical decompositions of the spaces of projective presentations due to Derksen–Fei. More strongly, for ‐tame algebras and hereditary algebras, we prove that TF equivalence classes containing lattice points are exactly the cones given by canonical decompositions. One of the key steps in our proof is a general description of semistable torsion classes in terms of morphism torsion classes. We also answer a question by Derksen–Fei negatively by giving examples of algebras that do not satisfy the ray condition. As an application of our results, we give an explicit description of TF equivalence classes of preprojective algebras of type .

Lattice points on polyominoes of inversion sequences

Inversion sequences of length n are positive integer sequences e 1 e 2 …e n such that 1 ≤ ei ≤ i for all 1 ≤ i ≤ n. These sequences are in bijection with the permutations of [n]. This paper focuses on the polyomino or barpgraph representation of the inversion sequences. More precisely, we study the distribution of lattice points on these polyominoes. We find the generating functions respect to the length, the number of interior vertices, corners, and vertices of a given degree. We also give simple explicit formulas for the total values of these parameters over all polyominoes of inversion sequences of length n. Throughout this work, we use symbolic computer algebra to facilitate the computations.

A novel directional simulation method for estimating failure possibility

Failure possibility plays a crucial role in assessing the safety level of structures under fuzzy uncertainty. However, the traditional fuzzy simulation method suffers from computational inefficiency as it requires a large number of samples for accurate estimation. To address this issue, a directional simulation method is proposed to improve the efficiency of estimating failure possibility. The directional simulation method reformulates the failure possibility estimation into two key steps: the generation of direction samples and the estimation of conditional failure possibility under each direction sample in the polar coordinate system of the standard fuzzy space. To ensure direction uniformity, these direction samples are generated by adopting a good lattice point set based on stratified sampling on the unit hypercube. The conditional failure possibility under each direction sample is estimated by solving the minimum root of a nonlinear equation. The proposed method effectively reduces the dimensionality of the fuzzy input and greatly improves the computational efficiency. To further enhance efficiency, an adaptive Kriging model is embedded into the directional simulation method to reduce the number of performance function evaluations. Four examples are performed to illustrate the accuracy and efficiency of the proposed method. The results highlight the superiority of the directional simulation method over the traditional fuzzy simulation method, offering substantial improvements in computational efficiency while maintaining high estimation accuracy.

Large-Area Perovskite Nanocrystal Metasurfaces for Direction-Tunable Lasing.

Perovskite nanocrystals (PNCs) are attractive emissive materials for developing compact lasers. However, manipulation of PNC laser directionality has been difficult, which limits their usage in photonic devices that require on-demand tunability. Here we demonstrate PNC metasurface lasers with engineered emission angles. We fabricated millimeter-scale CsPbBr3 PNC metasurfaces using an all-solution-processing technique based on soft nanoimprinting lithography. By designing band-edge photonic modes at the high-symmetry X point of the reciprocal lattice, we achieved four linearly polarized lasing beams along a polar angle of ∼30° under optical pumping. The device architecture further allows tuning of the lasing emission angles to 0° and ∼50°, respectively, by adjusting the PNC thickness to shift other high-symmetry points (Γ and M) to the PNC emission wavelength range. Our laser design strategies offer prospects for applications in directional optical antennas and detectors, 3D laser projection displays, and multichannel visible light communication.

Shape-Memory Effect and the Topology of Minimal Surfaces

Martensitic transformations, viewed as continuous mappings between triply periodic minimal surfaces (TPMSs), as suggested by Hyde and Andersson (Z. Kristallogr. 1986, 174, 225–236), are extended to include paths between the initial and final phases. Reversible transformations, which correspond to shape-memory materials, occur only if lattice points remain at flat points on a TPMS throughout a continuous transformation. For the shape-memory material NiTi, the density functional calculations by Hatcher et al. [Phys. Rev. B2009, 80, 144203] yield irreversible and reversible paths with and without energy barriers, respectively, in agreement with our theory. Although there are TPMSs for face-centered and body-centered cubic crystals for iron, the deformation between them is not reversible, in agreement with the non-vanishing energy barriers obtained from the density functional calculations of Zhang et al. (RSC Advances2021, 11, 3043–3048).

Around the Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circles

AbstractHardy conjectured that the error term arising from approximating the number of lattice points lying in a radius‐ disc by its area is . One source of support for this conjecture is a folklore heuristic that uses i.i.d. random variables to model the lattice points lying near the boundary and square root cancellation of sums of these random variables. We examine this heuristic by studying how these lattice points interact with one another and prove that their autocorrelation is determined in terms of a random model. Additionally, it is shown that lattice points near the boundary which are “well separated” behave independently. We also formulate a conjecture concerning the distribution of pairs of these lattice points.

Symmetry change in LaNiO3 films caused by epitaxial strain from LaAlO3, SrTiO3, and DyScO3 pseudocubic (001) surfaces

To elucidate the epitaxial strain effect over a wide range of lattice mismatch, we investigated the structures of ∼25 nm thick LaNiO3 films grown on the pseudocubic (001) surfaces of three different substrates, namely, LaAlO3 (LAO), SrTiO3 (STO), and DyScO3 (DSO). Such structural information had been inferred from the intensities of a small number of Bragg reflections that relate to the NiO6 octahedral tilting in previous studies. Here, we measured more than 100 reciprocal lattice points to derive reliable structural information. The procedure of ordinary crystal structure analysis is hampered by the multidomain structure and limited volume of measurable reciprocal space, both caused by a huge, highly symmetric substrate. To overcome this difficulty, we employed the Bayesian inference to obtain the detailed atomic positions in film samples. Octahedral tilting about the c axis was dominant for the compressively strained film grown on LAO, whereas tilting about the a and b axes was dominant for the tensile strained films grown on STO and DSO. The film lattice parameters of the samples grown on STO and DSO were nearly identical, whereas additional twofold lattice modulation, including cation displacement, was only observed in the latter.

К теории рентгеновской Лауэ дифракции в термомиграционном кристаллическом канале с легирующей примесью

X-ray Laue diffraction in a silicon crystal with Si(Al) thermomigration channels has been theoretically considered. Based on the model of elastic fields of atomic displacements in the channel, expressions for the distribution of strains have been obtained to describe diffraction in the Laue geometry. A numerical calculation of the X-ray scattering intensity distribution near a reciprocal lattice point has been performed. The difference between diffraction in a perfect and strained crystal has been shown.

High-Speed Rail Multiple-Input–Multiple-Output Channel Model Based on Reconfigurable Intelligent Surface System

This paper presents a geometry-based stochastic model (GBSM) for a reconfigurable intelligent surface (RIS)-assisted multiple-input–multiple-output (MIMO) system tailored to high-speed rail environments, addressing energy leakage at space lattice points caused by the time-dependent reception direction and differing antenna array resolutions. Initially, this study explores the spatial domain feature map and spreading antenna lobe characteristics from various RIS array configurations to reshape the polarization distribution and facilitate a virtual antenna array. Subsequently, the time-dependent reception direction and position are derived by integrating the dynamics of train operations. Finally, the received signal is aligned with the polarization direction to assess the signal reception efficiency and finalize the model output. Research findings indicate that the number of RIS elements, spacing between RIS elements, and mobile relay (MR) movement characteristics significantly influence the performance of the system. Compared to existing models, the proposed model proficiently captures the effects of time-dependent receiver angle properties and RIS array configuration.

Monocular Metasurface for Structured Light Generation and 3D Imaging with a Large Field-of-View.

Structured light three-dimensional (3D) imaging technology captures the geometric information on 3D objects by recording waves reflected from the objects' surface. The projection angle and point number of the laser dots directly determine the field-of-view (FOV) and the resolution of the reconstructed image. Conventionally, diffractive optical elements with micrometer-scale pixel size have been used to generate laser dot arrays, leading to limited FOV and point number within the projection optical path. Here, we theoretically put forward and experimentally demonstrate a monocular geometric phase metasurface composed of deep subwavelength meta-atoms to generate a 10 798 dot array within an FOV of 163°. Attributed to the vast number and high-density point cloud generated by the metasurface, the 3D reconstructed results showcase a maximum relative error in depth of 5.3 mm and a reconstruction error of 6.07%. Additionally, we propose a spin-multiplexed metasurface design method capable of doubling the number of lattice points. We demonstrate its application in the field of 3D imaging through experiments, where the 3D reconstructed results show a maximum relative depth error of 0.44 cm and a reconstruction error of 2.78%. Our proposed metasurface featuring advanced point cloud generation holds substantial potential for various applications such as facial recognition, autonomous driving, virtual reality, and beyond.

Geography of symplectic Lefschetz fibrations and rational blowdowns

We produce simply-connected, minimal, symplectic Lefschetz fibrations realizing all the lattice points in the symplectic geography plane below the Noether line. This provides a symplectic extension of the classical works populating the complex geography plane with holomorphic Lefschetz fibrations. Our examples are obtained by rationally blowing down Lefschetz fibrations with clustered nodal fibers, the total spaces of which are potentially new homotopy elliptic surfaces. Similarly, clustering nodal fibers on higher genera Lefschetz fibrations on standard rational surfaces, we get rational blowdown configurations that yield new constructions of small symplectic exotic 4 4 –manifolds. We present an example of a construction of a minimal symplectic exotic C P 2 # 5 C P ¯ 2 {\mathbb {CP}}{}^{2}\# \, 5 \overline {\mathbb {CP}}{}^{2} through this procedure applied to a genus– 3 3 fibration.

Crystal Symmetry-Inspired Algorithm for Optimal Design of Contemporary Mono Passivated Emitter and Rear Cell Solar Photovoltaic Modules

A metaheuristic algorithm named the Crystal Structure Algorithm (CrSA), which is inspired by the symmetric arrangement of atoms, molecules, or ions in crystalline minerals, has been used for the accurate modeling of Mono Passivated Emitter and Rear Cell (PERC) WSMD-545 and CS7L-590 MS solar photovoltaic (PV) modules. The suggested algorithm is a concise and parameter-free approach that does not need the identification of any intrinsic parameter during the optimization stage. It is based on crystal structure generation by combining the basis and lattice point. The proposed algorithm is adopted to minimize the sum of the squares of the errors at the maximum power point, as well as the short circuit and open circuit points. Several runs are carried out to examine the V-I characteristics of the PV panels under consideration and the nature of the derived parameters. The parameters generated by the proposed technique offer the lowest error over several executions, indicating that it should be implemented in the present scenario. To validate the performance of the proposed approach, convergence curves of Mono PERC WSMD-545 and CS7L-590 MS PV modules obtained using the CrSA are compared with the convergence curves obtained using the recent optimization algorithms (OAs) in the literature. It has been observed that the proposed approach exhibited the fastest rate of convergence on each of the PV panels.

On the distribution of powerful and r-free lattice points

On the distribution of powerful and r-free lattice points

A facile route to synthesis NixFe1−xFe2O4 ferrofluids with optimal rheological and magneto‐optical properties

AbstractThis study presents an easy method for synthesizing ultrafine NixFe1–xFe2O4 nanoparticles with adjustable composition (x = 0–.8), followed by their stabilization into ferrofluids. Structural identification of the crystalline structure, lattice points, and grain boundaries from the broadened diffraction peaks reveal an average crystalline size of the nanoparticles as 10–16.5 nm. Transmission electron microscopy images reveal spherical magnetite nanoparticles with a particle size ranging from 6 to 13 nm, consistent with diffraction studies. In ferrofluids, the NixFe1–xFe2O4 nanoparticles are stabilized in kerosene with oleic acid, a surfactant. Absorbance data of the ferrofluids is seen in the 200–400 nm wavelength region of UV–vis spectra. The magnetic properties of the samples are probed using a Superconducting Quantum Interference Device. The synthesized samples exhibit superparamagnetic behavior at room temperature (300 K). The saturation magnetization of the samples decreases with an increase in Ni composition (x = 0–.8), ranging from 54 to 28 emu/g. This study explores the magnetic and magneto‐optical properties of NixFe1–xFe2O4 ferrofluids. Magneto‐viscosity of ferrofluids is also studied, and the final application of such ferrofluids in data storage, catalysis, and biomedical applications is discussed.

Sparse distribution of lattice points in annular regions

This paper is inspired by Richards' work on large gaps between sums of two squares [10]. It is shown in [10] that there exist arbitrarily large values of λ and μ, where μ≥Clog⁡λ, such that intervals [λ,λ+μ] do not contain any sums of two squares. Geometrically, these gaps between sums of two squares correspond to annuli in R2 that do not contain any integer lattice points. A major objective of this paper is to investigate the sparse distribution of integer lattice points within annular regions in R2. Specifically, we establish the existence of annuli {x∈R2:λ≤|x|2≤λ+κ} with arbitrarily large λ and κ≥Cλs for 0<s<14, satisfying that any two integer lattice points within any one of these annuli must be sufficiently far apart. This result is sharp, as such a property ceases to hold at and beyond the threshold s=14. Furthermore, we extend our analysis to include the sparse distribution of lattice points in spherical shells in R3.