GEODESY, CARTOGRAPHY AND AERIAL PHOTOGRAPHY

The aim. The study of formulas determination of the point coordinates by the inverse linear-angular intersection method. Previously, we investigated the possibility of using electronic total stations to control the geometric parameters of industrial buildings. The applied application of electronic total stations for high-precision measurements has been investigated as well. [Vivat, 2018]. The formula for optimal use of the device with certain accuracy characteristics relative to the measured basis is analytically proved and derived [Litynskyi, 2014]. Measurements on the basis of the II category are performed and theoretical calculations are confirmed. The possibility of achieving high accuracy in determining the segment by the method of linear-angular measurements is shown [Litynsky, 2015]. The influence of the angle value on the accuracy of determining the coordinates by the sine theorem is investigated and the possibility of optimizing the determination of coordinates by the method of inverse linear-angular serif by the formulas of cosines and sines is investigated [Litynskyi, 2019]. Method. Establishing a mathematical interconnection between measured values (distances and angles) with the required (flat coordinates of a point), differentiation and finding the minima of functions. Results.There were five formulas selected, of which six combinations had been created to calculate the increments of coordinates and to estimate their accuracy. Numerical experiments show that neither method has a significant advantage, which is supported by the results presented in the graphs and tables. It is worth noting one feature of the second method - in which it is possible to determine the increments of coordinates with an accuracy that exceeds the accuracy of measuring the sides. The possibility of optimizing the coordinate increments determination due to the choice of calculation formulas is considered. The possibility of increasing the accuracy of determination of the coordinates increments using different calculation formulas is researched. Consequently, it is suggested to optimize the choice of calculation formulas depending on the position of the desired point. The results of these studies can be used to create electronic total station or laser tracker application software in order to improve the accuracy of coordinate determination.

$\vee$ -Systems, Holonomy Lie Algebras, and Logarithmic Vector Fields

It is shown that the description of certain class of representations of the holonomy Lie algebra $\mathfrak g_{\Delta}$ associated with hyperplane arrangement $\Delta$ is essentially equivalent to the classification of $\vee$-systems associated with $\Delta.$ The flat sections of the corresponding $\vee$-connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane arrangement of any $\vee$-system is free in Saito's sense and show this for all known $\vee$-systems and for a special class of $\vee$-systems called harmonic, which includes all Coxeter systems. In the irreducible Coxeter case the potentials of the corresponding gradient vector fields turn out to be Saito flat coordinates, or their one-parameter deformations. We give formulas for these deformations as well as for the potentials of the classical families of harmonic $\vee$-systems.

Minimal string theory and the Douglas equation

We use the connection between the Frobenius manifold and the Douglas string equation to further investigate Minimal Liouville gravity. We search for a solution of the Douglas string equation and simultaneously a proper transformation from the KdV to the Liouville frame which ensures the fulfilment of the conformal and fusion selection rules. We find that the desired solution of the string equation has an explicit and simple form in the flat coordinates on the Frobenius manifold in the general case of (p,q) Minimal Liouville gravity.

Singular polynomials from orbit spaces

We consider the polynomial representation S(V*) of the rational Cherednik algebra Hc(W) associated to a finite Coxeter group W at constant parameter c. We show that for any degree d of W and m∈ℕ the space S(V*) contains a single copy of the reflection representation V of W spanned by the homogeneous singular polynomials of degree d−1+hm, where h is the Coxeter number of W; these polynomials generate an Hc (W) submodule with the parameter c=(d−1)/h+m. We express these singular polynomials through the Saito polynomials which are flat coordinates of the Saito metric on the orbit space V/W. We also show that this exhausts all the singular polynomials in the isotypic component of the reflection representation V for any constant parameter c.

GRAVITATIONAL THEORY WITH A DYNAMICAL SPACE–TIME

A gravitational theory involving a vector field χμ, which has the properties of a dynamical space–time, is studied. The variation of the action with respect to χμ gives the covariant conservation of an energy–momentum tensor [Formula: see text]. Studying the theory in a background that has Killing vectors and Killing tensors, we find appropriate shift symmetries of the field χμ which lead to conservation laws. The energy–momentum that is the source of gravity [Formula: see text] is different but related to [Formula: see text] and the covariant conservation of [Formula: see text] determines in general the vector field χμ. When [Formula: see text] is chosen to be proportional to the metric, the theory coincides with the Two Measures Theory, which has been studied before in relation to the Cosmological Constant Problem. When the matter model consists of point particles, or strings, the form of [Formula: see text], solutions for χμ are found. In a locally inertial frame, the vector field corresponds to the locally flat coordinates. For the case of a string gas cosmology, we find that the Milne universe can be a solution, where the gas of strings does not curve the space–time since although [Formula: see text], [Formula: see text], as a model for the early universe, this solution is also free of the horizon problem. There may also be an application to the "time problem" of quantum cosmology.

Flat Coordinates and Hidden Symmetry for Superintegrable Benenti Systems

In this talk I present the results from my paper Exact solvability of superintegrable Benenti systems, J. Math. Phys. 48 (2007), 052114.