Normal Lattice Research Articles

Monotone-Cevian and Finitely Separable Lattices

AbstractA distributive lattice with zero is completely normal if its prime ideals form a root system under set inclusion. Every such lattice admits a binary operation $$(x,y)\mapsto x\mathbin {\smallsetminus }y$$ ( x , y ) ↦ x \ y satisfying the rules $$x\le y\vee (x\mathbin {\smallsetminus }y)$$ x ≤ y ∨ ( x \ y ) and $$(x\mathbin {\smallsetminus }y)\wedge (y\mathbin {\smallsetminus }x)=0$$ ( x \ y ) ∧ ( y \ x ) = 0 — in short a deviation. In this paper we study the following additional properties of deviations: monotone (i.e., isotone in x and antitone in y) and Cevian (i.e., $$x\mathbin {\smallsetminus }z\le (x\mathbin {\smallsetminus }y)\vee (y\mathbin {\smallsetminus }z)$$ x \ z ≤ ( x \ y ) ∨ ( y \ z ) ). We relate those matters to finite separability as defined by Freese and Nation. We prove that every finitely separable completely normal lattice has a monotone deviation. We pay special attention to lattices of principal $$\ell $$ ℓ -ideals of Abelian $$\ell $$ ℓ -groups (which are always completely normal). We prove that for free Abelian $$\ell $$ ℓ -groups (and also free $$\Bbbk $$ k -vector lattices) those lattices admit monotone Cevian deviations. On the other hand, we construct an Archimedean $$\ell $$ ℓ -group with strong unit, of cardinality $$\aleph _1$$ ℵ 1 , whose principal $$\ell $$ ℓ -ideal lattice does not have a monotone deviation.

Enhanced Energy Absorption with Bioinspired Composite Triply Periodic Minimal Surface Gyroid Lattices Fabricated via Fused Filament Fabrication (FFF)

Bio-inspired gyroid triply periodic minimum surface (TPMS) lattice structures have been the focus of research in automotive engineering because they can absorb a lot of energy and have wider plateau ranges. The main challenge is determining the optimal energy absorption capacity and accurately capturing plastic plateau areas using finite element analysis (FEA). Using nTop’s Boolean subtraction method, this study combined walled TPMS gyroid structures with a normal TPMS gyroid lattice. This made a composite TPMS gyroid lattice (CTG) with relative densities ranging from 14% to 54%. Using ideaMaker 4.2.3 (3DRaise Pro 2) software and the fused deposition modeling (FDM) Raise3D Pro 2 3D printer to print polylactic acid (PLA) bioplastics in 1.75 mm filament made it possible to slice computer-aided design (CAD) models and fabricate 36 lattice samples precisely using a layer-by-layer technique. Shimadzu 100 kN testing equipment was utilized for the mechanical compression experiments. The finite element approach validates the results of mechanical compression testing. Further, a composite CTG was examined using a field emission scanning electron microscope (FE-SEM) before and after compression testing. The composite TPMS gyroid lattice showed potential as shock absorbers for vehicles with relative densities of 33%, 38%, and 54%. The Gibson–Ashby model showed that the composite TPMS gyroid lattice deformed mainly by bending, and the size effect was seen when the relative densities were less than 15%. The lattice’s relative density had a significant impact on its ability to absorb energy. The research also explored the use of these innovative foam-like composite TPMS gyroid lattices in high-speed crash box scenarios to potentially enhance vehicle safety and performance. The structures have tremendous potential to improve vehicle safety by acting as advanced shock absorbers, which are particularly effective at higher relative densities.

A rationale for modelling hydrogen-induced softening in fcc single crystals

Several atomistic-scale mechanisms have been proposed to explain the phenomenon of hydrogen-enhanced localized plasticity (HELP). However, the relative contributions of each of these atomistic mechanisms in determining the constitutive behaviour of fcc single crystals undergoing HELP are yet to be fully understood. In this study, we address this issue by presenting a constitutive model of H-induced softening for fcc single crystals. Here, we first compute H concentrations in the trapped state close to the dislocation cores and in the normal interstitial lattice sites (NILS), which are then used to determine the efficacy of different HELP mechanisms contingent on them. The H in NILS forms atmospheres around dislocations and shields the long-range elastic forces acting on them, while the H in deep traps around dislocations reduces the activation barrier associated with short-range obstacles and enhances the ease of dislocation nucleation. These mechanisms are incorporated in a coarse-grained form into a dislocation density-based constitutive model of fcc single crystals, and the constitutive response is simulated for each one of the underlying mechanisms. Our simulations reveal that the mechanism of enhancement of dislocation nucleation in the presence of H is the most potent to cause softening of the constitutive response. In comparison, the flow softening caused by the mechanisms of reduction of the activation barrier of short-range obstacles and the shielding effect of H appears to be insignificant. The presented constitutive model can be extended to simulate H-induced softening in polycrystalline fcc materials as well.

Algebraic Proof Theory for LE-logics

In this article, we extend the research programme in algebraic proof theory from axiomatic extensions of the full Lambek calculus to logics algebraically captured by certain varieties of normal lattice expansions (normal LE-logics). Specifically, we generalize the residuated frames in Reference [ 34 ] to arbitrary signatures of normal lattice expansions (LE). Such a generalization provides a valuable tool for proving important properties of LE-logics in full uniformity. We prove semantic cut elimination for the display calculi \(\mathrm{D.LE}\) associated with the basic normal LE-logics and their axiomatic extensions with analytic inductive axioms. We also prove the finite model property (FMP) for each such calculus \(\mathrm{D.LE}\) , as well as for its extensions with analytic structural rules satisfying certain additional properties.

InCl3-Assisted Surface Defects Restoring to Enhance Lead-Free Cs2ZrCl6 Nanocrystals for X-Ray Imaging and Blue LED Applications.

As one type of recent emerging lead-free perovskites, Cs2ZrCl6 nanocrystals are widely concerned, benefiting from the eminent designability, high X-ray cutoff efficiency, and favorable stability. Improving the luminescence performance of Cs2ZrCl6 nanocrystals has great importance to cater for practical applications. In view of the surface defects frequently formed by the liquid phase method, the particle morphology and surface quality of this material are expected to be regulated if certain intervention is made in the synthesis process. In the work, differing from normal cell lattice modulation based on the ion doping, the grain size and surface morphology of Cs2ZrCl6 nanocrystals are optimized via adding a certain amount of InCl3 to the synthetic solution. The surface defects are restored to inhibit the defect-induced non-radiative transition, resulting in the improvement of the luminescence properties. Moreover, a flexible Cs2ZrCl6@polydimethylsiloxane film with excellent heat, water, and bending resistance and a light-emitting diode (LED) device are fabricated, exhibiting excellent application potential for X-ray imaging and blue LED.

A Novel 3D-Printed Negative-Stiffness Lattice Structure with Internal Resonance Characteristics and Tunable Bandgap Properties.

The bandgap tuning potential offered by negative-stiffness lattice structures, characterized by their unique mechanical properties, represents a promising and burgeoning field. The potential of large deformations in lattice structures to transition between stable configurations is explored in this study. This transformation offers a novel method for modifying the frequency range of elastic wave attenuation, simultaneously absorbing energy and effectively generating diverse bandgap ranges. In this paper, an enhanced lattice structure is introduced, building upon the foundation of the normal negative-stiffness lattice structures. The research examined the behavior of the suggested negative-stiffness lattice structures when subjected to uniaxial compression. This included analyzing the dispersion spectra and bandgaps across different states of deformation. It also delved into the effects of geometric parameter changes on bandgap properties. Furthermore, the findings highlight that the normal negative-stiffness lattice structure demonstrates restricted capabilities in attenuating vibrations. In contrast, notable performance improvements are displayed by the improved negative-stiffness lattice structure, featuring distinct energy band structures and variable bandgap ranges in response to differing deformation states. This highlights the feasibility of bandgap tuning through the deformation of negatively stiffened structures. Finally, the overall metamaterial structure is simulated using a unit cell finite element dynamic model, and its vibration transmission properties and frequency response patterns are analyzed. A fresh perspective on the research and design of negative-stiffness lattice structures, particularly focusing on their bandgap tuning capabilities, is offered in this study.

Interplanar Ferromagnetism Enhanced Ultrawide Zero Thermal Expansion in Kagome Cubic Intermetallic (Zr,Nb)Fe2.

A cubic metal exhibiting zero thermal expansion (ZTE) over a wide temperature window demonstrates significant applications in a broad range of advanced technologies but is extremely rare in nature. Here, enabled by high-temperature synthesis, we realize tunable thermal expansion via magnetic doping in the class of kagome cubic (Fd-3m) intermetallic (Zr,Nb)Fe2. A remarkably isotropic ZTE is achieved with a negligible coefficient of thermal expansion (+0.47 × 10-6 K-1) from 4 to 425 K, almost wider than most ZTE in metals available. A combined in situ magnetization, neutron powder diffraction, and hyperfine Mössbauer spectrum analysis reveals that interplanar ferromagnetic ordering contributes to a large magnetic compensation for normal lattice contraction upon cooling. Trace Fe-doping introduces extra magnetic exchange interactions that distinctly enhance the ferromagnetism and magnetic ordering temperature, thus engendering such an ultrawide ZTE. This work presents a promising ZTE in kagome metallic materials.

The design and implementation of folded adaptive lattice filter structures in FPGA for ECG signals

An adaptive filter is the utmost essential filter castoff in statistical signal dealing. The fine-tuning of the filter factor in relation to the response signal is the adaptive filter's key feature due to fewer calculations, Least Mean Square (LMS) adaptive filters are widely used to remove noise from Electrocardiograms (ECG). The adaptive filters are realized as signal processing algorithms in Digital Signal Processors (DSPs) or in VLSI Signal Processors (VSPs). The technique provides a way to create a folded adaptive lattice LMS filter, which requires less hardware than an adaptive lattice filter. Folding is an algorithm that uses a time scheduling technique that combines arithmetic operations into one operation which reduces Register and silicon chip areas. The design and implementation of a folded lattice adaptive filter remove Power Line Interference (PLI) noise from ECG signals. The MATLAB Xilinx System Generator tool is used to design the Adaptive Lattice LMS Filter and Folded Adaptive Lattice LMS Filter with Folding Order K = 2 and K = 4 and realized in the Virtex 5 FPGA KIT. The results of the folded architecture show that the area is reduced for K = 2 and K = 4 by 82.60% and 91.05%, respectively compared with a normal adaptive lattice filter.

Duality for normal lattice expansions and sorted residuated frames with relations

We revisit the problem of Stone duality for lattices with quasioperators, presenting a fresh duality result. The new result is an improvement over that of our previous work in two important respects. First, the axiomatization of frames is now simplified, partly by incorporating Gehrke’s proposal of section stability for relations. Second, morphisms are redefined so as to preserve Galois stable (and co-stable) sets and we rely for this, partly again, on Goldblatt’s recently proposed definition of bounded morphisms for polarities. In studying the dual algebraic structures associated to polarities with relations we demonstrate that stable/co-stable set operators result as the Galois closure of the restriction of classical (though sorted) image operators generated by the frame relations to Galois stable/co-stable sets. This provides a proof, at the representation level, that non-distributive logics can be regarded as fragments of sorted residuated (poly)modal logics, a research direction recently initiated by this author.

Phase-Transition Behaviors of Crystalline Hypophosphorous Acid and a Feasible Incommensurate Wave-like Array of the Proton Displacements from Each Center of Hydrogen Bonds along the 1-Dimensional Chain

Crystalline hypophosphorous acid, composed of strongly hydrogen-bonded chains, was investigated by adiabatic calorimetry in a temperature range between 20 and 306 K and by dielectrometry as a supplemental means. The heat-capacity curve showed subsequent phase transitions: a second-order one at 210.4 K followed by a first-order one at 159 K and the other two ones at 149 and 139.4 K with decreasing temperature. The total entropy of the transitions was assessed to be 0.35 J K–1 mol–1 by using the baseline heat capacities that were determined by means of a real-coded genetic algorithm. The small value was interpreted as the transitions were associated with displacements of the protons forming hydrogen bonds: namely, while they are located at the center between two oxygen atoms above 210.4 K, the protons are displaced close to one of the oxygen atoms with the bond length enlarged below it. Temperature dependence of dielectric constants displayed no sign, around 210 K, indicating the existence of (anti-)ferroelectricity. It was thus expected that the manner of off-center proton displacements along each hydrogen-bonded chain reveals a wave-like and incommensurate appearance below 210.4 K, referred to as the normal crystal lattice above it, and following locking to a wave-like but superlattice commensurate one at 159 K.

Syntactic Completeness of Proper Display Calculi

A recent strand of research in structural proof theory aims at exploring the notion of analytic calculi (i.e., those calculi that support general and modular proof-strategies for cut elimination) and at identifying classes of logics that can be captured in terms of these calculi. In this context, Wansing introduced the notion of proper display calculi as one possible design framework for proof calculi in which the analyticity desiderata are realized in a particularly transparent way. Recently, the theory of properly displayable logics (i.e., those logics that can be equivalently presented with some proper display calculus) has been developed in connection with generalized Sahlqvist theory (a.k.a. unified correspondence). Specifically, properly displayable logics have been syntactically characterized as those axiomatized by analytic inductive axioms , which can be equivalently and algorithmically transformed into analytic structural rules so the resulting proper display calculi enjoy a set of basic properties: soundness, completeness, conservativity, cut elimination, and the subformula property. In this context, the proof that the given calculus is complete w.r.t. the original logic is usually carried out syntactically , i.e., by showing that a (cut-free) derivation exists of each given axiom of the logic in the basic system to which the analytic structural rules algorithmically generated from the given axiom have been added. However, so far, this proof strategy for syntactic completeness has been implemented on a case-by-case base and not in general. In this article, we address this gap by proving syntactic completeness for properly displayable logics in any normal (distributive) lattice expansion signature. Specifically, we show that for every analytic inductive axiom a cut-free derivation can be effectively generated that has a specific shape, referred to as pre-normal form .

Abnormal trapping of hydrogen in the elastic stress field of dislocations in body-centered cubic iron

It has been widely accepted that the tetrahedral interstitial site is the normal lattice trapping site for H atoms in the body-centered cubic iron. In this work, we found H trapped in a tetrahedral site is unstable in the dislocation elastic stress field. By contrast, the octahedral site is energetically preferred. This abnormal trapping phenomenon was rooted in the H-dislocation elastic interaction and zero-point quantum vibration of the H atom. At the abnormal trapping state, the interaction between H-edge dislocation and H-screw dislocation was identified to be at a similar level. As a result, the diffusivity of H around dislocation is enhanced by the abnormal trapping, and the local H content is much higher than the prediction with the assumption of normal trapping. Our results suggest that the dislocation motion can shuffle the trapping position of H atoms and assist the immature failure in the hydrogen environment.

Sol—gel approach to low-temperature synthesis of single-phase metastable La2Ga3O7.5 melilite with enhanced grain-boundary oxide ionic conductivity via a kinetically favorable mechanism

Starting with the stoichiometric and highly homogeneous gel-precursor, single-phase metastable melilite La2Ga3O7.5, as the end-member of solid solution La1+xSr1−xGa3O7+x/2 (0≼x≼1), has been synthesized by solid-state reaction at 700 °C for 2 h via a kinetically favorable mechanism and characterized by X-ray diffraction (XRD), Raman, X-ray photoelectron spectroscopy (XPS), field emission scanning electron microscopy (FESEM), transmission electron microscopy (TEM), AC impedance spectroscopy, etc. It has been revealed that the as-synthesized melilite La2Ga3O7.5 shows an orthorhombic symmetry with crystal cell parameters a = 11.4690(1) Å, b = 11.2825(4) Å, and c = 10.3735(4) Å, while has more Raman active modes than LaSrGa3O7 with a tetragonal structure, which was also synthesized under the same conditions for comparison, but tends to slowly decompose into perovskite LaGaO3 and Ga2O3 when annealed at 700 °C for over 20 h driven by its meta-stability. Moreover, the metastable La2Ga3O7.5 shows a higher XPS binding energy for the excess oxide ions in the crystal structure than those at normal lattice sites. Its intrinsic grain oxide ion conductivity can reach as high as 0.04 and 0.51 mS·cm−1 at 550 and 700 °C, respectively, characterized by a simple Arrhenius relationship ln(σT)—1/T with invariable activation energy, Ea = 1.22 eV, over the temperature range from 300 to 700 °C, along with an apparent grain boundary conductivity that is about double that from the grains thanks to the clean grain boundaries. This paper provides a new strategic approach to the synthesis of complex oxides that may be of high performance but are difficultly achieved by the conventional ceramic method at high temperatures.

X-ray photoemission and absorption study of the pyrochlore iridates (EuBi x )2Ir2O7, 0 x 1

The pyrochlore iridates (EuBi x )2Ir2O7 (0 x 1) undergo an anomalous negative lattice expansion for small Bi-doping () (region I) and a normal lattice expansion for (region II); this is accompanied by a transition from an insulating (and magnetically ordered) to a metallic (and with no magnetic ordering) ground state. Here, we investigate (EuBi x )2Ir2O7 (0 x 1) using hard x-ray photoemission spectroscopy and x-ray absorption fine structure (XAFS) spectroscopy. By analyzing the Eu-L 3, Ir-L 3 and Bi-L 2 & L 3 edges x-ray absorption near edge structure spectra and Eu-3d core-level XPS spectra, we show that the metal cations retain their nominal valence, namely, Ir4+, Bi3+ and Eu3+, respectively, throughout the series. The Ir-4f and Bi-4f core-level XPS spectra consist of screened and unscreened doublets. The unscreened component is dominant In the insulating range (, and in the metallic region (), the screened component dominates the spectra. The Eu-3d core-level spectra remain invariant under Bi doping. The extended XAFS data show that the coordination around the Ir remains well preserved throughout the series. The evolution of the valence band spectra near the Fermi energy with increasing Bi doping indicates the presence of strong Ir(5d)–Bi(6p) hybridization which drives the metal-to-insulator transition.

Congruence Normality of Simplicial Hyperplane Arrangements via Oriented Matroids

A catalogue of simplicial hyperplane arrangements was first given by Grünbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and also the weak order through the poset of regions. The weak order is known to be a congruence normal lattice, and congruence normality of lattices of regions of simplicial arrangements can be determined using polyhedral cones called shards. In this article, we update Grünbaum’s catalogue by providing normals realizing all known simplicial arrangements with up to 37 lines and key invariants. Then we add structure to this catalogue by determining which arrangements always/sometimes/never lead to congruence normal lattices of regions. To this end, we use oriented matroids to recast shards as covectors to determine congruence normality of large hyperplane arrangements. We also show that lattices of regions coming from finite Weyl groupoids of any rank are always congruence normal.

Cevian properties in ideal lattices of Abelian ℓ-groups

Abstract We consider the problem of describing the lattices of compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. (Equivalently, describing the spectral spaces of Abelian lattice-ordered groups.) It is known that these lattices have countably based differences and admit a Cevian operation. Our first result says that these two properties are not sufficient: there are lattices having both countably based differences and Cevian operations, which are not representable by compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. As our second result, we prove that every completely normal distributive lattice of cardinality at most ℵ 1 {\aleph_{1}} admits a Cevian operation. This complements the recent result of F. Wehrung, who constructed a completely normal distributive lattice having countably based differences, of cardinality ℵ 2 {\aleph_{2}} , without a Cevian operation.

Elastic Properties of Taurine Single Crystals Studied by Brillouin Spectroscopy.

The inelastic interaction between the incident photons and acoustic phonons in the taurine single crystal was investigated by using Brillouin spectroscopy. Three acoustic phonons propagating along the crystallographic b-axis were investigated over a temperature range of −185 to 175 °C. The temperature dependences of the sound velocity, the acoustic absorption coefficient, and the elastic constants were determined for the first time. The elastic behaviors could be explained based on normal lattice anharmonicity. No evidence for the structural phase transition was observed, consistent with previous structural studies. The birefringence in the ac-plane indirectly estimated from the split longitudinal acoustic modes was consistent with one theoretical calculation by using the extrapolation of the measured dielectric functions in the infrared range.

Study on Helium Bubble Coalescence in the Slip Plane of Titanium

Helium bubble growth and coalescence in the slip plane as well as the influence on substrate were studied using the molecular dynamics method. In the slip plane, the helium bubbles grow first along the slip plane and then grow toward the side which is short one atomic layer in the form of a hexagonal structure at low temperature. The growth rates of helium bubbles are related to the addition rate of helium atoms and their surrounding environments. After coalescence, the coalesced helium bubble grows first toward the side that is short one atomic layer. Then it grows along the slip plane with a velocity less than the growth rate before coalescence. Helium bubble growth and coalescence in the slip plane have significant influence on the substrate. During the process, the preexisting slipping metal atoms are pushed back to the normal lattice sites, and the crystal structure of the metal is recovered around the helium bubbles. The recovered area changes with the number of helium atoms in the bubble and the temperature of the substrate. The simulation results indicate that the preexisting grain boundary is beneficial for enhancing the helium damage resistance of metal.

A note on the normal subgroup lattice of ultraproducts of finite quasisimple groups

Abel Stolz and Andreas Thom [Proc. Lond. Math. Soc. 108 (2014), pp. 73–102] stated that the lattice of normal subgroups of an ultraproduct of finite simple groups is always linearly ordered. This is false in this form in most cases for classical groups of Lie type. We correct the statement in this case and point out a version of \enquote*relative bounded normal generation for classical quasisimple groups and its implications on the structure of the lattice of normal subgroups of an ultraproduct of such groups.

Isotropic octet-truss lattice structure design and anisotropy control strategies for implant application

In the field of bone implantation, there is a large demand for isotropic lattice with appropriate mechanical properties and porosity. Two isotropic lattice structure design and anisotropy control strategies are proposed in this work. The first way to acquire isotropic lattice structures is to combine the cross-frustum with octet-truss structure. Then the anisotropy can be adjusted by changing the frustum parameters. The second strategy is implemented in the space of normal lattice to generate hierarchical structures to achieve lattice anisotropy control by changing the inner and outer diameters. In virtue of homogenization method and finite element analysis (FEA), the relationship between elastic modulus, anisotropy and lattice parameters can be established. Based on that, the lattice structures with required elastic modulus and isotropy can be obtained. In addition, the isotropic lattice structures acquired by these two methods are applied as acetabular cup implant in this work. Compared with normal octet-truss, experimental results validate that the elastic modulus of designed lattice implant fluctuates significantly smaller. The standard deviation of elastic modulus in different directions is reduced by about 80%.