The Minkowski content of a compact set is fine measure its geometric scaling. For Lebesgue null sets it measures the decay epsilon neighbourhoods set. It well known that self-similar sets, satisfying reasonable separation conditions and non-log commensurable contraction ratios, have well-defined content. When dropping conditions, more general notion average still exists. random recursive also e...

In this article, we give explicit examples of infinitely many non-commensurable (non-arithmetic) hyperbolic $3$-manifolds admitting exactly $k$ totally geodesic surfaces for any positive integer $k$, answering a question Bader, Fisher, Miller and Stover. The construction comes from family twist knot complements their dihedral covers. case $k=1$ arises the uniqueness an immersed thrice-punctured...

In this paper, a constructive description of Hölder classes functions on certain compact sets in $${{\mathbb{R}}^{m}}$$ (m $$ \geqslant 3) is given terms the rate approximation by harmonic shrinking neighborhoods these sets. The under consideration are higher-dimensional generalization subsets curves $${{\mathbb{R}}^{3}}$$ whose arc commensurable with chord. neighborhood size directly connected...

The geodesic length spectrum of a complete, finite volume, hyperbolic 3-orbifold M is fundamental invariant the topology via Mostow-Prasad Rigidity. Motivated by this, second author and Reid defined two-dimensional analogue given multiset isometry types totally geodesic, immersed, finite-area surfaces called geometric genus spectrum. They showed that if $M$ arithmetic contains surface, then det...

The process of diffusion admixture particles in a multiphase randomly nonhomogeneous body with spherical inclusions different materials commensurable volume fractions phases is investigated. According to the theory binary systems, mathematical model disposed radii constructed. dense packing spheres used modeling skeleton body. contact initial-boundary value problem reduced mass transfer equatio...

In this paper we show that many projective Anosov representations act convex cocompactly on some properly domain in real space. particular, if a non-elementary word hyperbolic group is not commensurable to non-trivial free product or the fundamental of closed surface, then any representation acts We also preserves domain, it (possibly different) domain. give three applications. First, into gene...

In this paper we consider a class of the 2D integrable models. These models are higher spin XXZ chains with an extra condition of the commensurability between spin and anisotropy. The mathematics underlying this commensurability is provided by the quantum groups with deformation parameter being an Nth root of unity. Our discussion covers a range of topics including new integrable deformations, ...

for multi-objective optimal reactive power dispatch (morpd), a new approach is proposed where simultaneous minimization of the active power transmission loss, the bus voltage deviation and the voltage stability index of a power system are achieved. optimal settings of continuous and discrete control variables (e.g. generator voltages, tap positions of tap changing transformers and the number of...

Abstract Using the CO molecule as target, we investigate high-order harmonic generation by a bichromatic elliptically polarized laser field. This field consists of two components with commensurable frequencies and mutually orthogonal semi-major axes. Both odd even harmonics are emitted their ellipticity can be large depending on values laser-field parameters. It is often case that subsequent su...

In this study, we perform a theoretical study of light propagation properties in two-dimensional square photonic crystals (PCs) following Bravais–Moiré (BM) patterns composed copper oxide high-temperature superconductors (HTSCs). The BM PCs are made cylindrical cores formed from the combination two Bravais lattices. Moiré pattern forms due to commensurable rotation one these lattices with respe...