Generalized flatness constants, spanning lattice polytopes, and the Gromov width

In this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently large width contain a unimodular copy of a standard simplex. Following an argument of Eisenbrand and Shmonin, we prove that every lattice polytope contains a minimal generating set of the affine lattice spanned by its lattice points such that the number of generators (and the lattice width of their convex hull) is bounded by a constant which only depends on the dimension. We also discuss relations to recent results on spanning lattice polytopes and how our results could be viewed as the beginning of the study of generalized flatness constants. Regarding symplectic geometry, we point out how the lattice width of a Delzant polytope is related to upper and lower bounds on the Gromov width of its associated symplectic toric manifold. Throughout, we include several open questions.

Partial structure factors derived from coarse-grained phase-separation dynamics on a disordered lattice with density fluctuations

The simplest and most time-efficient phase-separation dynamics simulations are carried out on a disordered lattice to calculate the partial structure factors of coarse-grained A-B binary mixtures. The typical coarse-grained phase-separation models use regular lattices and can describe the local concentrations but cannot describe both local density and concentration fluctuations. To introduce fluctuation for local density in the model, the particle positions from a hard sphere fluid model are determined as disordered lattice points for the model. Then we place the local order parameter as the difference of the concentrations of A and B components on each lattice point. The concentration at each lattice point is time-evolved by discrete equations derived from the Cahn-Hilliard equation. From both fluctuations, Bhatia and Thornton’s structure factor can be accurately calculated. The structure factor for concentration fluctuations at the large wavenumber region gives us the correct mean concentrations of the components. Using the mean concentrations, partial structure factors can be converted from three of Bhatia and Thornton’s structure factors. The present model and procedures can provide a means of analysing the structural properties of many materials that exhibit complex morphological changes.

Proof the Skewes’ number is not an integer using lattice points and tangent line

Skewes’ number was discovered in 1933 by South African mathematician Stanley Skewes as upper bound for the first sign change of the difference π (x) − li(x). Whether a Skewes’ number is an integer is an open problem of Number Theory. Assuming Schanuel’s conjecture, it can be shown that Skewes’ number is transcendental. In our paper we have chosen a different approach to prove Skewes’ number is an integer, using lattice points and tangent line. In the paper we acquaint the reader also with prime numbers and their use in RSA coding, we present the primary algorithms Lehmann test and Rabin-Miller test for determining the prime numbers, we introduce the Prime Number Theorem and define the prime-counting function and logarithmic integral function and show their relation.

Efficient Covering of Thin Convex Domains Using Congruent Discs

We present efficient strategies for covering classes of thin domains in the plane using unit discs. We start with efficient covering of narrow domains using a single row of covering discs. We then move to efficient covering of general rectangles by discs centered at the lattice points of an irregular hexagonal lattice. This optimization uses a lattice that leads to a covering using a small number of discs. We compare the bounds on the covering using the presented strategies to the bounds obtained from the standard honeycomb covering, which is asymptotically optimal for fat domains, and show the improvement for thin domains.

Study on Icing Environment Judgment Based on Radar Data

As a major threat to aviation flight safety, it is particularly important to make accurate judgments and forecasts of the ice accumulation environment. Radar is widely used in civil aviation and meteorology, and has the advantages of high timeliness and resolution. In this paper, a variety of machine learning methods are used to establish the relationship between radar data and icing index (Ic) to determine the ice accumulation environment. The research shows the following. (1) A linear model was established, based on the scattering rate factor (Zh), radial velocity (v), spectral width (w), velocity standard deviation (σ) detected by 94 GHz millimeter wave radar, and backward attenuation coefficient (β) detected by 905 nm lidar, so linear regression was carried out. After principal component analysis (PCA), the correction determination coefficient of the linear equation was increased from 0.7127 to 0.7240. (2) Ice accumulation was unlikely for samples that were significantly off-center. By clustering the data into three or four categories, the proportion of icing lattice points could be increased from 18.81% to 33.03%. If the clustering number was further increased, the ice accumulation ratio will not be further increased, and the increased classification is reflected in the classification of pairs of noises and the possibility of omission is also increased. (3) Considering the classification and nonlinear factors of ice accumulation risk, the neural network method was used to judge the ice accumulation environment. Two kinds of neural network structures were established for quantitative calculation: Structure 1 first distinguished whether there was ice accumulation, and further calculated the icing index for the points where there was ice accumulation; Structure 2 directly calculated the temperature and relative humidity, and calculated the icing index according to definition. The accuracy of the above two structures could reach nearly 60%, but the quantitative judgment of the ice accumulation index was not ideal. The reasons for this dissatisfaction may be the small number of variables and samples, the interval between time and space, the difference in instrument detection principle, and the representativeness of the ice accumulation index. Further research can be improved from the above four points. This study can provide a theoretical basis for the diagnosis and analysis of the aircraft ice accumulation environment.

L2 Boundedness of Discrete Double Hilbert Transform Along polynomials

We will show $L^{2}$ boundedness of Discrete Double Hilbert Transform along polynomials satisfying some conditions. Double Hilbert exponential sum along polynomials:$\mu(\xi)$ is Fourier multiplier of discrete double Hilbert transform along polynomials. In chapter 1, we define the reverse Newton diagram. In chapter 2, We make approximation formula for the multiplier of one valuable discrete Hilbert transform by study circle method. In chapter 3, We obtain result that $\mu(\xi)$ is bounded by constants if $|D|\geq2$ or all $(m,n)$ are not on one line passing through the origin. We study property of $1/(qt^{n})$ and use circle method (Propsotion 2.1) to calculate sums. We also envision combinatoric thinking about $\mathbb{N}^{2}$ lattice points in j-k plane for some estimates. Finally, we use geometric property of some inequalities about $(m,n)\in\Lambda$ to prove Theorem 3.3. In chapter 4, We obtain the fact that $\mu(\xi)$ is bounded by sums which are related to $\log_{2}({\xi_{1}-a_{1}\slash {q}})$ and $\log_{2}({\xi_{2}-a_{2}\slash {q}})$ and the boundedness of double Hilbert exponential sum for even polynomials with torsion without conditions in Theorem 3.3. We also use $\mathbb{N}^{2}$ lattice points in j-k plane and Proposition 2.1 which are shown in chapter 2 and some estimates to show that Fourier multiplier of discrete double Hilbert transform is bounded by terms about $\log$ and integral this with torsion is bounded by constants.

Normal Polytopes and Ellipsoids

We show that: (1) unimodular simplices in a lattice 3-polytope cover a neighborhood of the boundary of the polytope if and only if the polytope is very ample, (2) the convex hull of lattice points in every ellipsoid in $\mathbb{R}^3$ has a unimodular cover, and (3) for every $d\geqslant 5$, there are ellipsoids in $\mathbb{R}^d$, such that the convex hulls of the lattice points in these ellipsoids are not even normal. Part (c) answers a question of Bruns, Michałek, and the author.

The Twinning Problem

In the field of crystallography, some crystals are not made of a single component but are instead twinned.In these cases, the observed intensities at some points in the lattice will be far larger than predictions. If we find the rotation associated to the twinned component, we can model this twin and improve our agreement with observations. In this report, we explore many routes to improve the process of identifying twins: Generation of fake data for better understanding and accurate testing. The representation of a rotation as defined by an axis and angle. The representation of a rotation as a quaternion. Using lattice points which must be equidistant from the origin to create our viable rotations. An algorithm focused on restricted possibilities. An exploration of 2D lattices for which twinning is mathematically impossible. We find that there is much to be investigated in the field of twinning.

Modified Born-Lande Equation to calculate Lattice Energy in a theoretical approach

Defects in ionic solid are very much common, which is increased with the rise in temperature. It causes the change in the value of many physical properties and varieties of physical parameters and the Lattice Energy is one such parameter to control the physical properties of the crystals. Considering the loss of ions from lattice points as random, the examination of each of the defects individually is going to be unpredictable, thus leading to almost nonattainment of the correct crystal structure with the theoretical calculations applying for available models. Here, in this present work, we have used some statistical methods and probabilistic approximation to introduce a novel idea of calculating the Madelung constant, and then Lattice Energy analytically. To make the understanding more lucid, we have taken one of the very common crystals, very popular in the crystallographic community, NaCl crystal having 6:6 co-ordination number, for which a significant number of Schottky defects are observed. During this study, we are bound to assume the random distribution of defects as Poisson distribution due to the fact that the number of defects is very less with respect to the total numbers of lattice points present in the crystal to calculate the Madelung Constant.