Related papers: Spherical designs from norm-3 shell of integral la…

M.S. Rao recently investigated some sorts of special filters in distributive pseudocomplemented lattices. In our paper we extend this study to lattices which need neither be distributive nor pseudocomplemented. For this sake we define a…

We investigate several antipodal spherical designs on whether we can choose half of the points, one from each antipodal pair, such that they are balanced at the origin. In particular, root systems of type A, D and E, minimal points of Leech…

We consider the problem of finding the minimum of inhomogeneous Gaussian lattice sums: Given a lattice $L \subseteq \mathbb{R}^n$ and a positive constant $\alpha$, the goal is to find the minimizers of $\sum_{x \in L} e^{-\alpha \|x -…

The inhomogeneous six-vertex model is a 2$D$ multiparametric integrable statistical system. In the scaling limit it is expected to cover different classes of critical behaviour which, for the most part, have remained unexplored. For general…

Discrete fundamental and dipole solitons are constructed, in an exact analytical form, in an array of linear waveguides with an embedded $\mathcal{PT}$-symmetric dimer, which is composed of two nonlinear waveguides carrying equal gain and…

We prove the existence of static, spherically symmetric solutions of the stellar dynamic Vlasov-Poisson and Vlasov-Einstein systems, which have the property that their spatial support is a finite, spherically symmetric shell with a vacuum…

The dimension of a block design is the maximum positive integer $d$ such that any $d$ of its points are contained in a proper subdesign. Pairwise balanced designs PBD$(v,K)$ have dimension at least two as long as not all points are on the…

A lattice is called well-rounded, if its lattice vectors of minimal length span the ambient space. We show that there are interesting connections between the existence of well-rounded sublattices and coincidence site lattices (CSLs).…

We prove that the set of non-degenerate second order maximally superintegrable systems in the complex Euclidean plane carries a natural structure of a projective variety, equipped with a linear isometry group action. This is done by…

A spherical $L$-code, where $L \subseteq [-1,\infty)$, consists of unit vectors in $\mathbb{R}^d$ whose pairwise inner products are contained in $L$. Determining the maximum cardinality $N_L(d)$ of an $L$-code in $\mathbb{R}^d$ is a…

Let $L$ be an $n$-element finite lattice. We prove that if $L$ has strictly more than $2^{n-5}$ congruences, then $L$ is planar. This result is sharp, since for each natural number $n\geq 8$, there exists a non-planar lattice with exactly…

Motivated by the search for best lattice sphere packings in Euclidean spaces of large dimensions we study randomly generated perfect lattices in moderately large dimensions (up to d=19 included). Perfect lattices are relevant in the…

Optimal geometrical arrangements, such as the stacking of atoms, are of relevance in diverse disciplines. A classic problem is the determination of the optimal arrangement of spheres in three dimensions in order to achieve the highest…

In this study, we have identified $V_3$ slant helix ($2^{nd}$ type slant helix, $V_5$ slant helix ($3^{rd}$ type slant helix) and attained some characteristic properties in the Euclidean 5-Space $E^5$. In addition to this, we have proven…

A set ${X}_{N}=\{x_1,\ldots,x_N\}$ of $N$ points on the unit sphere $\mathbb{S}^d,\,d\geq 2$ is a spherical $t$-design if the average of any polynomial of degree at most $t$ over the sphere is equal to the average value of the polynomial…

We prove that the cosine law for spherical triangles and spherical tetrahedra defines integrable systems, both in the sense of multidimensional consistency and in the sense of dynamical systems.

We complete the intrinsic characterization of spherically symmetric solutions partially accomplished in a previous paper [Class.Quant.Grav. (2010) 27 205024]. In this approach we consider every compatible algebraic type of the Ricci tensor,…

Let $X$ be a finite set in a complex sphere of $d$ dimension. Let $D(X)$ be the set of usual inner products of two distinct vectors in $X$. A set $X$ is called a complex spherical $s$-code if the cardinality of $D(X)$ is $s$ and $D(X)$…

In 1947, Lehmer conjectured that the Ramanujan $\tau$-function $\tau (m)$ never vanishes for all positive integers $m$, where the $\tau (m)$ are the Fourier coefficients of the cusp form $\Delta_{24}$ of weight 12. Lehmer verified the…

In this paper, we count all non-isomorphic lattices on $n$ elements, containing four reducible elements and having nullity three. This work is in respect of Birkhoff's open problem (which is NP-complete) of counting all finite lattices on…