We introduce semi-Euclidean analogues of circular Dini surfaces in $\mathbb{E}^4$ which generalize the classical Dini surfaces in $\mathbb{E}^3$ by inheriting their geometric features concerning the degeneration of the Bianchi-Bäcklund transformation. The construction is based on the use of semi-Euclidean analogues of circular tractrices in $\mathbb{E}^3$.

In this paper, we have analyzed the geometrical structure of the configurations of carbon atoms in nanocarbons, C$_{60}$, carbon nanotubes, C$_{60}$ dimer, and C$_{60}$-polymers (peanut-shaped fullerene polymers), determined by the first- principles calculations in terms of the discrete principal curvature based on the discrete differential geometry. While the C$_{60}$ and nanotubes have the constant discrete principal curvature (CDPC), we demonstrate that the C$_{60}$-polymers and C$_{60}$ dimer also have the almost constant (pre-constant) discrete principal curvature (pCDPC) with revolutionary symmetry due to discreteness, though constant principal curvature surfaces in a continuum picture are realized only by tubes and a sphere.

Beginning the study of non-Euclidean geometries, physical models or representations, such as crochet ones, provide a tangible portrayal of these advanced mathematical concepts. However, their connection to local Euclidean surfaces still needs further investigation. This work aims to explore how the characteristics of crochet models relate to non-Euclidean concepts by providing a parameterization of such surfaces.

Here we study the robustness of two modifications of quantum walk search algorithm on hypercube. In the first previously suggested modification, on each even iteration only quantum walk is applied. And in the second, the closest neighbors of the solution are measured classically. In this work we study the robustness of those two modifications when the traversing coin is constructed by both generalized Householder reflection and an additional phase multiplier. We investigate the stability of the algorithm to deviations in those phases.

Here we present an analytical proof assisted by computer calculations for the dynamical stability of the eight main planets and Pluto for the next 100 000 years. It means that the semi-major axes of the planets will not change significantly during this period. Also the eccentricities and the inclinations of the orbits will remain sufficiently small. A standard linear four-step numerical method is used to integrate approximately the orbits of Mercury, Venus, Earth+Moon, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto. Written in orbital elements, the dynamics of the nine planets manifests a system of 54 first-order ordinary differential equations. The step-size of the numerical method - about six days, has been performed 6 290 000 times. We estimate also the total accumulation of rounding-off errors, deviations related to possible uncertainty in the astronomical data and the accuracy of the computer calculations.