Tetyana Sviderska’s scientific contributions

It is difficult to align the measured elements of the triangle. A method of alignment using algebraic circles is proposed. If six elements are measured in a triangle: angles A, B, C, lengths of opposite sides a, b, c with mean square errors mA, mB, mC, ma, mb, mc, the values of the equalized elements must satisfy the requirements of the equations where D is the metric parameter of the triangle (the metric value through which the mathematical relationship between the sides and angles of the triangle is established), that is, the value of the diameter of the circle. If , which are found by formulas (1) and condition (2) is fulfilled, then this confirms the validity of the theorem of sines. In fact, it is a symmetrical set, because the number and composition of the upper indices corresponds to the number and composition of the lower indices, and this is a reflection of dimensions, since the triangle ABC has pairs of opposite elements: а- А; b- B; c- C. Let's construct the graph G Do of the diameter of the circle circumscribed around the aligned triangle, for which we will first construct the graphs G Di of the diameters of the circles circumscribed around the T-triangles, which are shown in fig. 1, and then combine them. Three vertices a, b, c form a "plane" - this is a metric field. The reflection of the metric field is the field of angular measurements - a "plane" defined by three points A, B, C. Note that for a measured triangle, the reflection of the metric field in the angular one is not unambiguous, so the first plane in fig. 1 is "inclined" to the second. After determining the most likely metric parameter Do of the triangle, it is possible to bring these planes into full correspondence. That is, Do is a "lens" [4],[8] with the help of which we obtain an unambiguous mapping of the metric field in the angular field and vice versa.

В наш час теорія надійності широко використовується в будівельному виробництві, а також використовується і в області геодезії. Шляхом абстрагування її положення можна з успіхом перенести і на системи, які, здавалося б, не знаходяться в динамічному стані. Візьмемо, наприклад, пункти полігонометричної (нівелірної) мережі в місті. Здавалося б, що така мережа знаходиться в статичному стані, однак з плином часу вона зазнає змін, тобто вона знаходиться в непомітній динаміці і надійність її поступово знижується. Під надійністю в широкому сенсі цього слова розуміють здатність технічного пристрою (системи, мережі) до безперебійної (безвідмовної) роботи упродовж заданого проміжку часу у певних умовах. Такий проміжок часу зазвичай зумовлено часом виконання деякої задачі, яка здійснюється приладом чи системою і є частиною загальної операційної задачі в вирішенні топографо-геодезичних робіт. В даний час проблема надійності стає однією з вузлових проблем техніки та організації управління. Забезпечення надійної роботи всіх елементів системи – є першорядною важливістю.

True errors can be found when we know the true value of the measured value. In geodetic measurements and in any other measurements, the true value of the measured quantity is unknown, therefore the most probable value of this quantity is taken, the sum of probable errors must be zero, the sum of true errors must be zero, therefore it is necessary to investigate the probable and true errors in an algebraic circle . A new method of equalizing and assessing the accuracy of measurements is considered. It differs from traditional methods in that, as a result of processing the measurements of one quantity, not only the total mean squared error is obtained, but also the mean squared errors of all measurements. The nature of true errors is manifested in closed systems, each (any) element of a closed system is neutral and opposite to all other elements and can be the beginning and end of the system. Therefore, a simple, closed system can be considered as an algebraic circle of elements, the dimensions of which are not measured precisely, but with some error. Definition 1. The set of all elements of a closed system is called a circle K, if the algebraic sum of the values of these elements is a true physical quantity. Definition 2. The set of all elements of a closed system is called a circle M if the algebraic sum of the values of these elements is zero. In circle K and circle M, we consider their own subsets, arcs D and which are opposite to each other and such that . Theorem. If the elements of the system create a circle K or a circle M, then the algebraic sum of their true errors is zero, and the algebraic sums of the errors of any two opposite arcs D and , circle К or circle М, will be the same in magnitude and have opposite signs. If three angles and three side lengths are measured in a triangle, then such a triangle has the properties of a closed circle M.

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All rights reserved. Printed in the United States of America. No part of this publication may be reproduced, distributed, or transmitted, in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. The content and reliability of the articles are the responsibility of the authors. When using and borrowing materials reference to the publication is required.

A formula for determining the correlation coefficients is proposed, according to which with a small number of measurements the values of the coefficients will be larger compared to the values calculated by known formulas, i.e., these values will be less offset relative to the true values of correlation coefficients. In geodesy in general are limited to a small number of measurements, in photogrammetry at mutual orientation of pictures of a stereo pair in an automatic mode at identification of points on the left and right pictures it is necessary to calculate precisely values of correlation coefficients.

When performing topographical surveying, when performing planning, construction - installation and other works, it is necessary to perform control - geodetic measurements of the quality of their execution. Until now, control and geodetic measurements were established for various reasons, which did not have a completely specific scientific and industrial approach. It will be most correct to perform control-geodetic measurements using the method of probabilistic-statistical sequential analysis, the foundations of which were developed by Wald. Sequential analysis has already been used to optimally determine the methods of angular measurements in triangulation, but it should be noted that the calculated number of measurements in triangulation (polygonometry, trilateration, GPS) is known in advance, it is given in the instructions. When using control-geodetic measurements, their number is unknown in advance, for this purpose, the theory of sequential analysis using Markov random processes was refined.