Geometric estimates of variations in space geodetic data

Well-known statistical parameters have some disadvantages when analyzing space geodetic data. Geometric parameters are proposed here for estimating the variation properties of samples for various discrete datasets. The proposed parameters are logically related to each other and are based on the simplest well-known statistical parameters; they do not depend on the type of distribution of the sample under study. “Variation asymmetry” shows the shift of the arithmetic mean relative to the center of the variation interval in the units of the studied sample. “Density of variation” characterizes the level of average variability in sample units. This parameter has several times greater discriminatory sensitivity to extremely different types of variations than linear and standard deviations. The relative parameter “proportion of maximum density” shows the closeness of variation to a uniform distribution in the ranked sample and complements the indicator of variation density. An algorithm for separating different structural levels of the useful signal from emissions (noise) is proposed here based on the calculation of geometric characteristics. The iterations of dividing the sample into structurally homogeneous segments can be stopped at the level of the proportion of maximum density ≥0.9 when analyzing real GPS coordinates.

Random inscribed polytopes in projective geometries

We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are obtained. We deduce these results by proving a general central limit theorem for the weighted volume of the convex hull of random points chosen from the boundary of a smooth convex body according to a positive and continuous density in Euclidean space. In the background are geometric estimates for weighted surface bodies and a Berry–Esseen bound for functionals of independent random variables.

Geometric issues of transforming a topographic surface into a design project

Natural relief adjustment for functional purpose - the task of vertical planning - is one of the main complexes of engineering preparation problems of urban areas, industrial sites, reclamation of irrigated land. Numerous methods of designing vertical levelling are aimed at building algorithms that would allow us to obtain an optimal solution in an automated mode. In the set of problems of engineering preparation of urban areas and industrial sites, vertical planning is defined as: a set of engineering and aesthetic measures aimed at adapting the natural relief for the needs of development and subsequent operation, taking into account the functional characteristics of the site a part of engineering preparation, which consists in providing a height arrangement of buildings and structures necessary for the best technological connection between individual objects, as well as a quick collection of atmospheric waters Reclamation of irrigated lands has specific requirements to design surface - maximum preservation of fertile layer, the satisfaction of cultivation technology conditions of different crops and irrigation technique. The main disadvantage of all known methods of designing vertical levelling is a cumbersome solution - a very labor-intensive way of successive approximation to an acceptable solution requires a large expenditure of computer time and subsequent manual revision. The article proposes one of the approaches to the design of levelling using geometric estimates.

Geometric estimates for complex Monge–Ampère equations

We prove uniform gradient and diameter estimates for a family of geometric complex Monge–Ampère equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge–Ampère equations. We also prove a uniform diameter estimate for collapsing families of twisted Kähler–Einstein metrics on Kähler manifolds of nonnegative Kodaira dimensions.

Geometric Estimates in Interpolation on an n-Dimensional Ball

Suppose \(n\in {\mathbb N}\). Let \(B_n\) be a Euclidean unit ball in \({\mathbb R}^n\) given by the inequality \(\|x\|\leq 1\), \(\|x\|:=\left(\sum\limits_{i=1}^n x_i^2\right)^{\frac{1}{2}}\). By \(C(B_n)\) we mean a set of continuous functions \(f:B_n\to{\mathbb R}\) with the norm \(\|f\|_{C(B_n)}:=\max\limits_{x\in B_n}|f(x)|\). The symbol \(\Pi_1\left({\mathbb R}^n\right)\) denotes a set of polynomials in \(n\) variables of degree \(\leq 1\), i.e. linear functions upon \({\mathbb R}^n\). Assume that \(x^{(1)}, \ldots, x^{(n+1)}\) are vertices of an \(n\)-dimensional nondegenerate simplex \(S\subset B_n\). The interpolation projector \(P:C(B_n)\to \Pi_1({\mathbb R}^n)\) corresponding to \(S\) is defined by the equalities \(Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).\) Denote by \(\|P\|_{B_n}\) the norm of \(P\) as an operator from \(C(B_n)\) on to \(C(B_n)\). Let us define \(\theta_n(B_n)\) as the minimal value of \(\|P\|_{B_n}\) under the condition \(x^{(j)}\in B_n\). We describe the approach in which the norm of the projector can be estimated from the bottom through the volume of the simplex. Let \(\chi_n(t):=\frac{1}{2^nn!}\left[ (t^2-1)^n \right] ^{(n)}\) be the standardized Legendre polynomial of degree \(n\). We prove that \(\|P\|_{B_n}\geq\chi_n^{-1}\left(\frac{vol(B_n)}{vol(S)}\right).\) From this, we obtain the equivalence \(\theta_n(B_n)\) \(\asymp\) \(\sqrt{n}\). Also we estimate the constants from such inequalities and give the comparison with the similar relations for linear interpolation upon the \(n\)-dimensional unit cube. These results have applications in polynomial interpolation and computational geometry.