A note on the convexity number for complementary prisms

In the geodetic convexity, a set of vertices $S$ of a graph $G$ is $\textit{convex}$ if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The cardinality $con(G)$ of a maximum proper convex set $S$ of $G$ is the $\textit{convexity number}$ of $G$. The $\textit{complementary prism}$ $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. In this work, we we prove that the decision problem related to the convexity number is NP-complete even restricted to complementary prisms, we determine $con(G\overline{G})$ when $G$ is disconnected or $G$ is a cograph, and we present a lower bound when $diam(G) \neq 3$.Comment: 10 pages, 2 figure

Mathematical Modeling Of Horizontal Displacement Of Above-ground Gas Pipelines

The modern geodetic equipment allows observations as soon as possible, providing high accuracy and productivity. Achieving high accuracy of measurement is impossible without taking into account external factors that create influence on an observation object. Therefore, in order to evaluate an influence of thermal displacement on the results of geodetic monitoring a mathematical model of horizontal displacement of above-ground pipelines was theoretically grounded and built. In this paper we used data of experimental studies on the existing pipelines "Soyuz" and "Urengoy - Pomary - Uzhgorod". Above-ground pipeline was considered as a dynamic system "building - environment". Based on the characteristics of dynamic systems the correlation between the factors of thermal influence and horizontal displacement of the pipeline axis was defined.Establishing patterns between input factors and output response of the object can be useful not only for geodetic control, but also for their consideration in the design of new objects. It was investigated that the greatest influence on the accuracy of geodetic observations can create dispersion of high-frequency oscillations caused by daily thermal displacement. The magnitude of displacement exceeds actual measurement error.The article presents the results of calculation of high-frequency oscillations of above-ground gas pipeline.The result made it possible to substantiate the accuracy and methodology of geodetic observations of the horizontal displacement of pipeline axes taking into account an influence of cyclical thermal displacement.Research results were recommended for use in practice for enterprises that serve the main gas pipelines and successfully tested by specialists of PJSC "Ukrtransgaz" (Kharkiv, Ukraine) during the technical state control of aerial pipeline crossing in Ukraine and also can be used to form the relevant regulations

On metric Ramsey-type phenomena

The main question studied in this article may be viewed as a nonlinear analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey theory in combinatorics. Given a finite metric space on n points, we seek its subspace of largest cardinality which can be embedded with a given distortion in Hilbert space. We provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion. Our main theorem states that for any epsilon>0, every n point metric space contains a subset of size at least n^{1-\epsilon} which is embeddable in Hilbert space with O(\frac{\log(1/\epsilon)}{\epsilon}) distortion. The bound on the distortion is tight up to the log(1/\epsilon) factor. We further include a comprehensive study of various other aspects of this problem.Comment: 67 pages, published versio

Precision surveying using very long baseline interferometry

Radio interferometry measurements were used to measure the vector baselines between large microwave radio antennas. A 1.24 km baseline in Massachusetts between the 36 meter Haystack Observatory antenna and the 18 meter Westford antenna of Lincoln Laboratory was measured with 5 mm repeatability in 12 separate experiments. Preliminary results from measurements of the 3,928 km baseline between the Haystack antenna and the 40 meter antenna at the Owens Valley Radio Observatory in California are presented