Terms Of Lattice Research Articles

Instrumented Dynamic Penetration of Sandwich Panels with Auxetic and Non‐Auxetic Core using Direct Impact Hopkinson Bar

This article deals with the dynamic penetration of 3D‐printed panels with auxetic and conventional honeycomb unit cell‐based cores. The geometry of the unit cells and their periodic assembly in the resulting lattices are selected to ensure the same relative density and overall weight of the individual sample types. Such a similarity of both specimen types allows for the evaluation of differences between conventional and auxetic lattices in terms of penetration characteristics and deformation energy mitigation properties. Dynamic penetration of the samples is performed using a fully strain‐gauge instrumented open Hopkinson pressure bar at three impact velocities resulting in three loading scenarios. All the performed experiments are captured by two optical cameras for detailed observation and to track the impactor movement using digital image correlation. The force‐penetration depth relation is used to evaluate the elastic and postyield compression characteristics of the lattices together with their deformation energy mitigation capabilities. The results show that the main differences in the deformation response of the lattices consist of lower overall stiffness and effective yielding of the auxetic lattices at a higher penetration depth. Numerical simulation using an explicit solver is performed to analyze the deformation mechanism of the individual core types.

A unified way to studies of t-seminorms, t-semiconorms and semi-uninorms on a complete lattice in terms of behaviour operations

We present a unified framework involved to the extensions of a t-seminorm, a t-semiconorm and a semi-uninorm in the lattice-valued setting through the use of the particular uplift and downlift of a binary operation linked by a behaviour operation. We also investigate the preservations of the left- and right-conjunctive of semi-uninorms as well as the left- and right-disjunctive of semi-uninorms under the extension approach studied in this paper. Moreover, for the extensions of t-seminorms, t-semiconorms and semi-uninorms, we provide their corresponding characterizations that depend on behaviour operations in terms of connected points. Finally, we summarize the dual results with respect to a unified way to investigations of t-seminorms, t-semiconorms and semi-uninorms.

Brillouin propagation modes of cold atoms undergoing Sisyphus cooling.

An exact expression for the average velocity of cold atoms in a driven, dissipative optical lattice in terms of the amplitudes of atomic density waves is derived from semiclassical equationsfor the phase space densities of the Zeeman ground-state sublevels. The calculations are for a J_{g}=1/2→J_{e}=3/2 transition, as it is customary in theoretical studies of Sisyphus cooling. While the driver, an additional beam of small amplitude, sets the atoms into directed motion, the new expression permits the quantification of the contribution to the atomic motion of a specific atomic wave, revealing unexpected counterpropagating contributions from many modes. Additionally, the method is shown to provide the generic threshold for the transition into the regime of infinite density, regardless of the details, or even the presence, of driving.

Lattice distortion and re-distortion affecting irradiation tolerance in high entropy alloys.

High entropy alloys (HEAs) are promising nuclear structural materials due to their excellent irradiation resistance. However, the essential mechanisms of irradiation tolerance in HEAs remain largely inferential and imperfectly understood. This study investigates the evolution of irradiation-induced nano-scale microstructures in Ni, FeNiCr, FeNiCrCoCu and FeNiCrCuAl HEA models by molecular dynamics simulations to elucidate the conundrums. As fewer irradiation-induced Frenkel pair (FP) residuals were found in the FeNiCrCuAl HEA model in comparison with other models, a high resistance of the HEAs to the generation of permanent defects was indicated, while also the associated relatively long thermal spike and slow recrystallization stimulated a high efficiency for the recombination/annihilation of FPs to underscore a superior structural recovery of the HEAs. Under the influence of compositional increases of constituent elements, the effect of severe lattice distortion by energetics modifications was found to stimulate decreased atomic mobility accompanied by inhibited dislocation formation. The evolution of the models' lattices in terms of their capacity to restrict interstitials and repair defects revealed that the self-healing/recovery mechanism that confirmed the highest initial lattice distortion value accompanied by the least lattice re-distortion value in the FeNiCrCuAl HEA model is key to the observed superior irradiation tolerance of the HEA models. Thus, by feasibly enhancing lattice distortion in crystalline materials, notably in HEAs, promising and potentially high irradiation-resistant structural materials can be developed.

Discrete Integrable Systems and Random Lax Matrices

We study properties of Hamiltonian integrable systems with random initial data by considering their Lax representation. Specifically, we investigate the spectral behaviour of the corresponding Lax matrices when the number N of degrees of freedom of the system goes to infinity and the initial data is sampled according to a properly chosen Gibbs measure. We give an exact description of the limit density of states for the exponential Toda lattice and the Volterra lattice in terms of the Laguerre and antisymmetric Gaussian beta -ensemble in the high temperature regime. For generalizations of the Volterra lattice to short range interactions, called INB additive and multiplicative lattices, the focusing Ablowitz–Ladik lattice and the focusing Schur flow, we derive numerically the density of states. For all these systems, we obtain explicitly the density of states in the ground states.

English

Meet-distributive lattices form an intriguing class of lattices, because they are precisely the lattices obtainable from a closure operator with the so-called anti-exchange property. Moreover, meet-distributive lattices are join semidistributive. Therefore, they admit two natural, secondary structures: the core label order is an alternative order on the lattice elements and the canonical join complex is the flag-simplicial complex on canonical join representations. In this article we present a characterization of finite meet-distributive lattices in terms of the core label order and the canonical join complex, and we show that the core label order of a finite meet-distributive lattice is always a meet-semilattice.

The First Syzygy of Hibi Rings Associated with Planar Distributive Lattices

In this article, we give explicit minimal generators of the first syzygy of the Hibi ring for a planar distributive lattice in terms of sublattices. We also give a characterization when it is linearly related and derive an exact formula for the first Betti number of a planar distributive lattice.

Effective stiffness, strength, buckling and anisotropy of foams based on nine unique triple periodic minimal surfaces

Recent advancements in the field of additive manufacturing has led to an increased interest in the study of cellular architectures especially triply periodic minimal surfaces (TPMS) and their properties. In this study, 9 sheet-based TPMS architectures are studied by comparing their effective stiffness, strength, buckling and anisotropic properties. The TPMS lattices are generated using level set equations where the relative density of each lattice is varied to study the effective properties as a function of relative density. Finite element models of these lattices are built to study the elastic and plastic deformation behaviour under uniaxial, shear, hydrostatic and buckling loading conditions. Effective properties such as Young’s modulus, shear modulus, bulk modulus, Poisson’s ratio and strengths under uniaxial, shear, hydrostatic and buckling loading are computed against each other and other common lattice structures. In addition, anisotropic behaviour as well as the orientation dependence of the properties of each lattice are also studied. The geometric efficiency criterion is predominantly used to evaluate the different structures and the results obtained are consolidated in both quantitative and qualitative manner. SC lattice outperforms all the other lattices in terms of strength and stiffness efficiencies at lower and higher relative densities. On the other hand, I2 − Y** lattice shows the weakest stiffness and strength efficiency especially at higher relative densities. The current study focuses on the properties of some unique lattice architectures that have not been studied previously and shows promising efficiencies compared to the common architectures studied such as the Gyroid lattice. This study opens the door for employing these lattices in various applications and the results are very useful to multi-scale topology-optimization studies.

(F,G)-Derivations on a Lattice

In the present paper, we introduce the notion of (F,G)-derivation on a lattice as a generalization of the notion of (∧,∨)-derivation. This newly notion is based on two arbitrary binary operations F and G instead of the meet (∧) and the join (∨) operations. Also, we investigate properties of (F,G)-derivation on a lattice in details. Furthermore, we define and study the notion of principal (F,G)-derivations as a particular class of (F,G)-derivations. As applications, we provide two representations of a given lattice in terms of its principal (F,G)-derivations.

A characterization of idempotent nullnorms on bounded lattices

We characterize the class of idempotent nullnorms on a bounded lattice in terms of particular common solutions to two equations related to the underlying meet and join operations. When this common solution is unique, it is an idempotent nullnorm if and only if it is increasing on a particular set. As an application of this characterization, we present several construction methods for idempotent nullnorms on a bounded lattice. These construction methods unify and generalize several known ones in the literature.

Extremely Overdoped Superconducting Cuprates via High Pressure Oxygenation Methods

Within the cuprate constellation, one fixed star has been the superconducting dome in the quantum phase diagram of transition temperature vs. the excess charge on the Cu in the CuO2-planes, p, resulting from O-doping or cation substitution. However, a more extensive search of the literature shows that the loss of the superconductivity in favor of a normal Fermi liquid on the overdoped side should not be assumed. Many experimental results from cuprates prepared by high-pressure oxygenation show Tc converging to a fixed value or continuing to slowly increase past the upper limit of the dome of p = 0.26–0.27, up to the maximum amounts of excess oxygen corresponding to p values of 0.3 to > 0.6. These reports have been met with disinterest or disregard. Our review shows that dome-breaking trends for Tc are, in fact, the result of careful, accurate experimental work on a large number of compounds. This behavior most likely mandates a revision of the theoretical basis for high-temperature superconductivity. That excess O atoms located in specific, metastable sites in the crystal, attainable only with extreme O chemical activity under HPO conditions, cause such a radical extension of the superconductivity points to a much more substantial role for the lattice in terms of internal chemistry and bonding.

Expansion for the critical point of site percolation: the first three terms

AbstractWe expand the critical point for site percolation on the d-dimensional hypercubic lattice in terms of inverse powers of 2d, and we obtain the first three terms rigorously. This is achieved using the lace expansion.

Unbounded continuous operators and unbounded Banach–Saks property in Banach lattices

Motivated by the equivalent definition of a continuous operator between Banach spaces in terms of weakly null nets, we introduce unbounded continuous operators by replacing weak convergence with the unbounded absolutely weak convergence (uaw-convergence) in the definition of a continuous operator between Banach lattices. We characterize order continuous Banach lattices and reflexive Banach lattices in terms of these spaces of operators. Moreover, motivated by characterizing of a reflexive Banach lattice in terms of unbounded absolutely weakly Cauchy sequences, we consider pre-unbounded continuous operators between Banach lattices which maps uaw-Cauchy sequences to weakly (uaw- or norm) convergent sequences. This allows us to characterize KB-spaces and reflexive spaces in terms of these operators, too. Furthermore, we consider the unbounded Banach–Saks property as an unbounded version of the weak Banach–Saks property. There are many relations between spaces possessing the unbounded Banach–Saks property with spaces fulfilled by different types of the known Banach–Saks property. In particular, we characterize order continuous Banach lattices in terms of these relations, as well.

The double hypergeometric series for the partition function of the 2D anisotropic Ising model

In 1944 Lars Onsager published the exact partition function of the ferromagnetic Ising model on the infinite square lattice in terms of a definite integral. Only in the literature of the last decade, however, has it been recast in terms of special functions. Until now all known formulas for the partition function in terms of special functions have been restricted to the important special case of the isotropic Ising model with symmetric couplings. Indeed, the anisotropic model is more challenging because there are two couplings and hence two reduced temperatures, one for each of the two axes of the square lattice. Hence, standard special functions of one variable are inadequate to the task. Here, we reformulate the partition function of the anisotropic Ising model in terms of the Kampé de Fériet function, which is a double hypergeometric function in two variables that is more general than the Appell hypergeometric functions. Finally, we present hypergeometric formulas for the generating function of multipolygons of given length on the infinite square lattice, for isotropic as well as anisotropic edge weights. For the isotropic case, the results allow easy calculation, to arbitrary order, of the celebrated series found by Cyril Domb.

T-norms and t-conorms on a family of lattices

This paper provides a complete classification of all t-norms on a family of lattices in terms of t-norms on discrete chains. Moreover, the cardinal of some classes on discrete chains is computed. Therefore, the number of t-norms on the family of lattices is obtained. Also, new results involving Archimedean and divisible t-norms are presented. Finally, we bring out dual results for t-conorms.

A modular characterization of supersolvable lattices

We characterize supersolvable lattices in terms of a certain modular type relation. McNamara and Thomas earlier characterized this class of lattices as those graded lattices having a maximal chain that consists of left-modular elements. Our characterization replaces the condition of gradedness with a second modularity condition on the maximal chain of left-modular elements.

Random crystal field effects on antiferromagnetic spin-1 Blume–Capel model

The outcome of the random crystal field effects on the antiferromagnetic spin-1 Blume–Capel model and external magnetic field are examined on the Bethe Lattice in terms of exact recursion relations. It is assumed that the crystal field is either turned on or off randomly with probability [Formula: see text] and [Formula: see text], respectively. The phase diagrams are constructed from the thermal analysis of the order parameters with the coordination number [Formula: see text] which corresponds to honeycomb lattice. It is explored that the system goes both second- and first-order phase transitions, along with the reentrant behavior and a few critical points. The reentrant behavior is stronger for lower values of [Formula: see text] and disappears as [Formula: see text] gets closer to 1.0. The first-order lines are observed to be either linked to the tricritical points or decomposed. The critical end points and double critical points are also observed.

Trimodal-random field Blume-Capel model

The external random magnetic field [Formula: see text] with three nodes, i.e. acting up and down along the [Formula: see text]-axis and zero, effective on the spins in the Blume-Capel model is analyzed on the Bethe lattice in terms of the exact recursion relations. All the nodes are assumed to have the same probability, [Formula: see text], so that the model could give various kinds of phase transitions. As a mapping of the phase transitions, the phase diagrams are constructed on two different planes which present very rich and interesting phase diagrams. In addition to the second- and first-order phase transitions, a few critical points, reentrant and double reentrant behaviors are also observed.

On sparse geometry of numbers

Let L be a lattice of full rank in n-dimensional real space. A vector in L is called i-sparse if it has no more than i nonzero coordinates. We define the ith successive sparsity level of L, $$s_i(L)$$ , to be the minimal s so that L has s linearly independent i-sparse vectors, then $$s_i(L) \le n$$ for each $$1 \le i \le n$$ . We investigate sufficient conditions for $$s_i(L)$$ to be smaller than n and obtain explicit bounds on the sup-norms of the corresponding linearly independent sparse vectors in L. These results can be viewed as a partial sparse analogue of Minkowski’s successive minima theorem. We then use this result to study virtually rectangular lattices, establishing conditions for the lattice to be virtually rectangular and determining the index of a rectangular sublattice. We further investigate the 2-dimensional situation, showing that virtually rectangular lattices in the plane correspond to elliptic curves isogenous to those with real j-invariant. We also identify planar virtually rectangular lattices in terms of a natural rationality condition of the geodesics on the modular curve carrying the corresponding points.

Inverse spectral problem for Jacobi operators and Miura transformation

Abstract We study a Miura-type transformation between Kac - van Moerbeke (Volterra) and Toda lattices in terms of the inverse spectral problem for Jacobi operators, which appear in the Lax representation for such systems. This inverse problem method, which amounts to reconstruction of the operator from the moments of its Weyl function, can be used in solving initial-boundary value problem for both systems. It is shown that the Miura transformation can be easily described in terms of these moments. Using this description we establish a bijection between the Volterra lattices and the class of Toda lattices which is characterized by positivity of Jacobi operators in their Lax representation. Also, we discuss an implication of the latter result to the spectral theory.