Lattice Points Research Articles

Castelnuovo Polytopes

It is known that the sectional genus of a polarized variety has an upper bound, which is an extension of the Castelnuovo bound on the genus of a projective curve. Polarized varieties whose sectional genus achieve this bound are called Castelnuovo. On the other hand, a lattice polytope is called Castelnuovo if the associated polarized toric variety is Castelnuovo. Kawaguchi characterized Castelnuovo polytopes having interior lattice points in terms of their h∗-vectors. In this paper, as a generalization of this result, we present a characterization of all Castelnuovo polytopes. Finally, as an application of our characterization, we give a sufficient criterion for a lattice polytope to be IDP.

An Evans function for the linearised 2D Euler equations using Hill’s determinant

We study the point spectrum of the linearisation of Euler’s equation for the ideal fluid on the torus about a shear flow. By separation of variables the problem is reduced to the spectral theory of a complex Hill’s equation. Using Hill’s determinant an Evans function of the original Euler equation is constructed. The Evans function allows us to completely characterise the point spectrum of the linearisation, and to count the isolated eigenvalues with non-zero real part. We prove that the number of discrete eigenvalues of the linearised operator for a specific shear flow is exactly twice the number of non-zero integer lattice points inside the so-called unstable disc.

Prime representations in the Hernandez–Leclerc category: classical decompositions

Abstract We use the dual functional realization of loop algebras to study the prime irreducible objects in the Hernandez–Leclerc (HL) category for the quantum affine algebra associated with $\mathfrak {sl}_{n+1}$ . When the HL category is realized as a monoidal categorification of a cluster algebra (Hernandez and Leclerc (2010, Duke Mathematical Journal 154, 265–341); Hernandez and Leclerc (2013, Symmetries, integrable systems and representations, 175–193)), these representations correspond precisely to the cluster variables and the frozen variables are minimal affinizations. For any height function, we determine the classical decomposition of these representations with respect to the Hopf subalgebra $\mathbf {U}_q(\mathfrak {sl}_{n+1})$ and describe the graded multiplicities of their graded limits in terms of lattice points of convex polytopes. Combined with Brito, Chari, and Moura (2018, Journal of the Institute of Mathematics of Jussieu 17, 75–105), we obtain the graded decomposition of stable prime Demazure modules in level two integrable highest weight representations of the corresponding affine Lie algebra.

Deep Lattice Points in Zonotopes, Lonely Runners, and Lonely Rabbits

Abstract Let $P \subseteq {\mathbb {R}}^{d}$ be a polytope and let $\textbf {w}$ be an interior point of $P$. The coefficient of asymmetry$\operatorname {ca}(P,\textbf {w}):= \min \{ \lambda \geq 1: \textbf {w} - P \subseteq \lambda (P - \textbf {w}) \}$ of $P$ about $\textbf {w}$ has been studied extensively in the realm of Hensley’s conjecture on the maximal volume of a $d$-dimensional lattice polytope that contains a fixed positive number of interior lattice points. We zero in on the coefficient of asymmetry for lattice zonotopes, that is, Minkowski sums of line segments with integer endpoints. Our main result gives the existence of an interior lattice point for which the coefficient of asymmetry is bounded above by an explicit constant in $\Theta (d \log \log d)$, for any lattice zonotope that has an interior lattice point. Our work is both inspired by and feeds on Wills’ lonely runner conjecture from Diophantine approximation: we make intensive use of a discrete version of this conjecture (which, in fact, has been proved), and reciprocally, we reformulate the lonely runner conjecture in terms of the coefficient of asymmetry for certain lattice zonotopes.

The Effect of Exchange Interaction on Phase Transition Behavior of Magnetic Nanotubes

In this paper, the phase diagram of the Blume-Capel model under exchange interaction between lattice points in different positions in magnetic nanotube lattice is studied with effective field theory. The research shows that the interaction between nearest neighbors and the lattice field strength strongly influences the system’s three critical points and phase transition temperature. Under certain conditions, there are three critical points in the system. That is, the relationship between the crucial points is linear from the second-order phase transition to the first-order phase transition. First-order phase transition replaces second-order phase transition in the system’s phase transition. The findings demonstrate that exchange interaction significantly affects the system’s phase transition, and they offer a theoretical framework for both industrial production and experimental study.

The Topological Phase Transitions near the Exceptional Point of a Non-Hermitian Electromagnetic Lattice Model

In this paper, we take the double-layer three-component non-Hermitian Lieb lattice system as an example to discuss the topological properties of the non-Hermitian lattice point electromagnetic field system. The topological phase transitions of the Lieb lattice system near the exceptional point are analyzed by calculating the topological invariants. We find that there are abundant topological phase transitions near the exceptional point. This study will help us explore more topological physical phenomena of electromagnetic fields of lattice systems in condensed-matter physics.

Partition the total energy of Al3Ti to characterize the Al3Ti/Al interface properties: a first-principles study

The geometric model of coplanar reciprocal lattice points (CRLPs) was used to predict the preferred orientation relationships (ORs) between the Al3Ti intermetallic particle and the Al matrix. The calculation result indicates that there exist two preferred ORs, i.e. (001)Al3Ti//(001)Al, [100]Al3Ti//[100]Al and (010)Al3Ti//(010)Al, [100]Al3Ti//[101¯]Al. The periodic models with a specific interface structure and surface slab models on a particular crystallographic plane were proposed to partition the total energy of the Al3Ti compound into its constituent atoms’ energies. The Ti and two kinds of Al atoms (Wyckoff notation 2b and 4d) possess the energy of -119.80325739 Ry, -5.09778518 Ry, and -5.03677683 Ry, respectively. Compared with the Al3Ti total energy of -269.94919247 Ry, the proposed energy partition scheme presents a satisfactory accuracy better than ∼10−7 Ry. Owing to the data of the atom's energy in the solid state, the definite value of surface energy can be obtained whether or not it is a stoichiometric or non-stoichiometric surface. The interface energy calculation, which was carried out on all the preferred ORs with all the possible atomic configurations, indicates that the energy-preferred interface possesses the orientation relationship of (001)Al3Ti//(001)Al, [100]Al3Ti//[100]Al. The atomic configuration is that the atoms on the surface of the Al slab face directly to atoms on the second layer of the Al3Ti slab with the Al-Ti terminated surface on account of its lowest interface energy of 0.005 J/m2. The predicted OR agrees well with the previously published experimental results, and the significantly lower interface energy explains the potency of the Al3Ti particle as a heterogeneous nucleation site in the aluminum solidification process.

Effective Results on the Size and Structure of Sumsets

Let A subset {mathbb {Z}}^d be a finite set. It is known that NA has a particular size (vert NAvert = P_A(N) for some P_A(X) in {mathbb {Q}}[X]) and structure (all of the lattice points in a cone other than certain exceptional sets), once N is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary A. Such explicit results were only previously known in the special cases when d=1, when the convex hull of A is a simplex or when vert Avert = d+2 Curran and Goldmakher (Discrete Anal. Paper No. 27, 2021), results which we improve.

Trianguloids and triangulations of root polytopes

Triangulations of a product of two simplices and, more generally, of root polytopes are closely related to Gelfand-Kapranov-Zelevinsky's theory of discriminants, to tropical geometry, tropical oriented matroids, and to generalized permutohedra. We introduce a new approach to these objects, identifying a triangulation of a root polytope with a certain bijection between lattice points of two generalized permutohedra. In order to study such bijections, we define trianguloids as edge-colored graphs satisfying simple local axioms. We prove that trianguloids are in bijection with triangulations of root polytopes.

Visibility phenomena in hypercubes

We study the set of visible lattice points in multidimensional hypercubes. The problems we investigate mix together geometric, probabilistic and number theoretic themes. For example, we prove that almost all self-visible triangles with vertices in the lattice of points with integer coordinates in W=([0,N]∩Z)d are almost equilateral having all sides almost equal to dN/6, and the sine of the typical angle between rays from the visual spectra from the origin of W is, in the limit, equal to 7/4, as d and N/d tend to infinity. We also show that there exists an interesting number theoretic constant Λd,K, which is the limit probability of the chance that a K-polytope with vertices in the lattice W has all vertices visible from each other.

The role of the substrate on the structure of reactive sputtered Co3O4: From polycrystalline to highly oriented films

We report a systematic study of the substrate's influence on the structure of Co3O4 thin films grown by direct current reactive magnetron sputtering. Three different substrates have been simultaneously loaded to the deposition chamber and held at 620 K during growth: amorphous fused silica (a-SiO2), LaAlO3 (001), and α-Al2O3 (0001). Samples were characterized using atomic force microscopy, Raman spectroscopy, and high-resolution X-ray using synchrotron radiation (HR-XRD). Raman spectra showed bands corresponding to modes of Co3O4. The θ-2θ (HR-XRD) scans confirmed only the presence of the cubic spinel Co3O4 phase with its texture reliant on the sample substrate used. Rocking curves indicate that α-Al2O3 favors (111) Co3O4 film orientation (smaller mosaic spread). High-resolution 3-D Reciprocal Space Map analysis covering both (0006) α-Al2O3 and the (222) Co3O4 reflections have shown out-of-plane strain (0.1905%) and presented an in-plane isotropic intensity distribution of the film reciprocal lattice point, indicating no preferred mosaic direction. The substrate epitaxy governs the structure and morphology of the three samples, going from a rather flat, but polycrystalline textured Co3O4 film on a-SiO2, to a flat highly oriented strained film on α-Al2O3 and a rough highly textured and strained film on LaAlO3.

The extensible No-Three-In-Line problem

The classical No-Three-In-Line problem seeks the maximum number of points that may be selected from an n×n grid while avoiding a collinear triple. The maximum is well known to be linear in n. Following a question of Erde, we seek to select sets of large density from the infinite grid Z2 while avoiding a collinear triple. We show the existence of such a set which contains Θ(n/log1+ɛn) points in [1,n]2 for all n, where ɛ>0 is an arbitrarily small real number. We also give computational evidence suggesting that a set of lattice points may exist that has at least n/2 points on every large enough n×n grid.

On the reconstruction of functions from values at subsampled quadrature points

This paper is concerned with function reconstruction from samples. The sampling points used in several approaches are (1) structured points connected with fast algorithms or (2) unstructured points coming from, e.g., an initial random draw to achieve an improved information complexity. We connect both approaches and propose a subsampling of structured points in an offline step. In particular, we start with quasi-Monte Carlo (QMC) points with inherent structure and stable L 2 L_2 reconstruction properties. The subsampling procedure consists of a computationally inexpensive random step followed by a deterministic procedure to further reduce the number of points while keeping its information. In these points functions (belonging to a reproducing kernel Hilbert space of bounded functions) will be sampled and reconstructed from whilst achieving state of the art error decay. Our method is dimension-independent and is applicable as soon as we know some initial quadrature points. We apply our general findings on the d d -dimensional torus to subsample rank-1 lattices, where it is known that full rank-1 lattices lose half the optimal order of convergence (expressed in terms of the size of the lattice). In contrast to that, our subsampled version regains the optimal rate since many of the lattice points are not needed. Moreover, we utilize fast and memory efficient Fourier algorithms in order to compute the approximation. Numerical experiments in several dimensions support our findings.

Study of impact dynamics of porous brittle materials based on a three-dimensional lattice point-spring model

Due to manufacturing, processing, etc., pores are flaws in a variety of brittle materials that can affect the material's impact dynamics properties. Studying the influence of pores and other defects on impact mesoscopic deformation characteristics can further utilize the beneficial functions of pores for engineering applications. Through the use of a 3D lattice point spring model with rotational degrees of freedom (3D-LSMR) that can precisely computationally simulate the fracture evolution mechanism of brittle materials, this paper reveals the mesoscopic deformation properties of spherical pores in brittle materials under impact loading. Under impact loading, pore collapse and shear cracking due to spherical pores can generate stress relaxation. The stress relaxation can evolve the shock-wave into a dual-wave structure of elastic wave (EW) and deformation wave (DW) in samples containing many spherical pores. The calculation results show that the porosity in the samples is an important factor influencing the Hugoniot elastic limit (HEL) of brittle materials. The impact velocity, porosity and material parameter variation factors is the key in influencing the propagation velocity of DWs. The results of the impact mesoscopic deformation characteristics obtained from the study are useful for the topology optimization of porous and brittle materials.

Horospherical coordinates of lattice points in hyperbolic spaces: effective counting and equidistribution

Horospherical coordinates of lattice points in hyperbolic spaces: effective counting and equidistribution

Optimization of the photoelectric performances of Mg and Al co-doped ZnO films by a two-step heat treatment process

Mg and Al co-doped ZnO (AZMO) films were prepared by magnetron co-sputtering to improve the optical and electrical performances of the ZnO-based films simultaneously. A two-step heat treatment process was introduced to obtain high-quality AZMO films. AZMO films were first deposited with substrates heating of 300 °C and then annealed at 500 °C in vacuum. The individual and coupling effects of substrate heating during sputtering deposition and post-annealing treatment on AZMO films were investigated. The results show that substrate heating during sputtering deposition is mainly on increasing the grain size and crystallinity of AZMO films. However, the effects of post-annealing treatment are mainly on making Mg and Al atoms move to the lattice points and ionize to from MgZn and AlZn. Besides, post-annealing treatment can effectively desorb adsorbed oxygen from the surface or grain boundaries of AZMO films and also make lattice oxygen break away from the lattice and change into oxygen vacancy. Compared with one-step heat treatment alone, it is with the two-step heat treatment process that the coupling effects can be exerted by combining the different effects of the two heat treatments, so as to achieve more performances optimization of AZMO films.

Growth of pseudo-Anosov conjugacy classes in Teichmüller space

Athreya, Bufetov, Eskin and Mirzakhani (2012) have shown that the number of mapping class group lattice points intersecting a closed ball of radius R in Teichmüller space is asymptotic to e^{hR} , where h is the dimension of the Teichmüller space. We show for any pseudo-Anosov mapping class f , there exists a power n , such that the number of lattice points of the f^n conjugacy class intersecting a closed ball of radius R is coarsely asymptotic to e^{\frac{h}{2}R} .

Equitable design of material processing parameters by utilizing probability‐based method for multi‐objective optimization

AbstractIn this paper, the equitable design is regulated, two steps are involved in the main processes. Preliminarily, the newly proposed probability‐based method for multi‐objective optimization is used to transfer the multi‐objective problem into a single objective one in the viewpoint of probability theory; subsequently the “uniform design method” and “good lattice point” are employed to complete the discretization in the sequential optimization. Moreover, the detailed parameters design of heat treatment for steel 30CrMnSiA and the proper cutting of titanium alloy are given as examples. The experimental results show the rationality of the proposed approach reasonably.

On a Reduced Component-by-Component Digit-by-Digit Construction of Lattice Point Sets

Abstract In this paper, we study an efficient algorithm for constructing point sets underlying quasi-Monte Carlo integration rules for weighted Korobov classes. The algorithm presented is a reduced fast component-by-component digit-by-digit (CBC-DBD) algorithm, which is useful for situations where the weights in the function space show a sufficiently fast decay. The advantage of the algorithm presented here is that the computational effort can be independent of the dimension of the integration problem to be treated if suitable assumptions on the integrand are met. By considering a reduced digit-by-digit construction, we allow an integration algorithm to be less precise with respect to the number of bits in those components of the problem that are considered less important. The new reduced CBC-DBD algorithm is designed to work for the construction of lattice point sets, and the corresponding integration rules (so-called lattice rules) can be used to treat functions in different kinds of function spaces. We show that the integration rules constructed by our algorithm satisfy error bounds of almost optimal convergence order. Furthermore, we give details on an efficient implementation such that we obtain a considerable speed-up of a previously known CBC-DBD algorithm that has been studied in the paper Digit-by-digit and component-by-component constructions of lattice rules for periodic functions with unknown smoothness by Ebert, Kritzer, Nuyens, and Osisiogu, published in the Journal of Complexity in 2021. This improvement is illustrated by numerical results.

Experimental Revelation of Surface and Bulk Lattices in Faceted Cu2 O Crystals.

Semiconductor crystals have generally shown facet-dependent electrical, photocatalytic, and optical properties. These phenomena have been proposed to result from the presence of a surface layer with bond-level deviations. To provide experimental evidence of this structural feature, synchrotron X-ray sources areused to obtain X-ray diffraction (XRD) patterns of polyhedral cuprous oxide crystals. Cu2 O rhombic dodecahedra display two distinct cell constants from peak splitting. Peak disappearance during slow Cu2 O reduction to Cu with ammonia borane differentiates bulk and surface layer lattices. Cubes and octahedra also show two peak components, while diffraction peaks of cuboctahedra are comprised of three components. Temperature-varying lattice changes in the bulk and surface regions also show shape dependence. From transmission electron microscopy (TEM) images, slight plane spacing deviations in surface and inner crystal regions are measured. Image processing provides visualization of the surface layer with depths of about 1.5-4nm giving dashed lattice points instead of dots from atomic position deviations. Close TEM examination reveals considerable variation in lattice spot size and shape for different particle morphologies, explaining why facet-dependent properties are emerged. Raman spectrum reflects the large bulk and surface lattice difference in rhombic dodecahedra. Surface lattice difference can change the particle bandgap.