A Probabilistic Approach to Estimating Allowed SNR Values for Automotive LiDARs in “Smart Cities” under Various External Influences

The paper proposes an approach to assessing the allowed signal-to-noise ratio (SNR) for light detection and ranging (LiDAR) of unmanned autonomous vehicles based on the predetermined probability of false alarms under various intentional and unintentional influencing factors. The focus of this study is on the relevant issue of the safe use of LiDAR data and measurement systems within the “smart city” infrastructure. The research team analyzed and systematized various external impacts on the LiDAR systems, as well as the state-of-the-art approaches to improving their security and resilience. It has been established that the current works on the analysis of external influences on the LiDARs and methods for their mitigation focus mainly on physical (hardware) approaches (proposing most often other types of modulation and optical signal frequencies), and less often software approaches, through the use of additional anomaly detection techniques and data integrity verification systems, as well as improving the efficiency of data filtering in the cloud point. In addition, the sources analyzed in this paper do not offer methodological support for the design of the LiDAR in the very early stages of their creation, taking into account a priori assessment of the allowed SNR threshold and probability of detecting a reflected pulse and the requirements to minimize the probability of “missing” an object when scanning with no a priori assessments of the detection probability characteristics of the LiDAR. The authors propose a synthetic approach as a mathematical tool for designing a resilient LiDAR system. The approach is based on the physics of infrared radiation, the Bayesian theory, and the Neyman–Pearson criterion. It features the use of a predetermined threshold for false alarms, the probability of interference in the analytics, and the characteristics of the LiDAR’s receivers. The result is the analytical solution to the problem of calculating the allowed SNR while stabilizing the level of “false alarms” in terms of background noise caused by a given type of interference. The work presents modelling results for the “false alarm” probability values depending on the selected optimality criterion. The efficiency of the proposed approach has been proven by the simulation results of the received optical power of the LiDAR’s signal based on the calculated SNR threshold and noise values.

A Student's Guide to Laplace Transforms

The Laplace transform is a useful mathematical tool encountered by students of physics, engineering, and applied mathematics, within a wide variety of important applications in mechanics, electronics, thermodynamics and more. However, students often struggle with the rationale behind these transforms, and the physical meaning of the transform results. Using the same approach that has proven highly popular in his other Student's Guides, Professor Fleisch addresses the topics that his students have found most troublesome; providing a detailed and accessible description of Laplace transforms and how they relate to Fourier and Z-transforms. Written in plain language and including numerous, fully worked examples. The book is accompanied by a website containing a rich set of freely available supporting materials, including interactive solutions for every problem in the text, and a series of podcasts in which the author explains the important concepts, equations, and graphs of every section of the book.

Assessment of the Influence of Astronomical Cyclicity on Sedimentation Processes in the Eastern Paratethys Based on Paleomagnetic Measurements Using Discrete Mathematical Analysis

The introduction of modern methods for the mathematical processing of geological data is one of the promising areas of study and development in the field of geosciences. For example, today mathematical geology makes it possible to reliably identify astronomical cycles by measuring the scalar magnetic parameters of rocks (magnetic susceptibility). The main aim of this study is to develop a mathematical tool for identifying stable oscillation cycles (periods) in the dataset of the magnetic susceptibility of rocks in a geological section. The author’s method (algorithm) is based on the concept of discrete mathematical analysis—an innovative mathematical approach to the analysis of discrete geological and geophysical data. Its reliability is also demonstrated, by comparison with the results obtained by classical methods: Fourier analysis, Lomb periodogram, and REDFIT. The proposed algorithm was applied by the authors to analyze the material of field geological studies of the Zhelezny Rog section (Taman Peninsula). As a result, stable cycles were determined for the Pontian and Lower Maeotian sedimentary strata of the Black Sea Basin (Paratethys).

TriCCo v1.0.0 – a cubulation-based method for computing connected components on triangular grids

We present a new method to identify connected components on a triangular grid. Triangular grids are, for example, used in atmosphere and climate models to discretize the horizontal dimension. Because they are unstructured, neighbor relations are not self-evident and identifying connected components is challenging. Our method addresses this challenge by involving the mathematical tool of cubulation. We show that cubulation allows one to map the 2-d cells of the triangular grid onto the vertices of the 3-d cells of a cubic grid. The latter is structured and so connected components can be readily identified on the cubic grid by previously developed software packages. An advantage is that the cubulation, i.e., the mapping between the triangular and cubic grids, needs to be computed only once, which should be benifical for analysing many data fields for the same grid.We further implement our method in a python package that we name TriCCo and that is made available via pypi and gitlab. We document the package, demonstrate its application using cloud data from the ICON atmosphere model, and characterize its computational performance. This shows that TriCCo is ready for triangular grids with 100,000 cells, but that its speed and memory requirements need to be improved to analyse larger grids.

Sub-exact sequence of rough groups

Rough Set Theory (RST) is an essential mathematical tool to deal with imprecise, inconsistent, incomplete information and knowledge Rough Some algebra structures, such as groups, rings, and modules, have been presented on rough set theory. The sub-exact sequence is a generalization of the exact sequence. In this paper, we introduce the notion of a sub-exact sequence of groups. Furthermore, we give some properties of the rough group and rough sub-exact sequence of groups.

A New Formula for Calculating Uncertainty Distribution of Function of Uncertain Variables

As a mathematical tool to rationally handle degrees of belief in human beings, uncertainty theory has been widely applied in the research and development of various domains, including science and engineering. As a fundamental part of uncertainty theory, uncertainty distribution is the key approach in the characterization of an uncertain variable. This paper shows a new formula to calculate the uncertainty distribution of strictly monotone function of uncertain variables, which breaks the habitual thinking that only the former formula can be used. In particular, the new formula is symmetrical to the former formula, which shows that when it is too intricate to deal with a problem using the former formula, the problem can be observed from another perspective by using the new formula. New ideas may be obtained from the combination of uncertainty theory and symmetry.

Traveling wave solutions for the extended modified KORTEWEG-DE VRIES equation

In this paper, we study an extended modified Korteweg-de Vries equation, which contains the relevant higher-order nonlinear terms and fifth-order dispersion. This equation is the extension of the modified Korteweg-de Vries equation and described by the Ablowitz-Kaup-Newell-Segur hierarchy. The standard Korteweg-de Vries equation is the pioneer integrable model in solitary waves theory, which gives rise to multiple soliton solutions. The Korteweg-de Vries equation arises naturally from shallow water, plasma physics, and other fields of science. To obtain exact solutions the sine-cosine method is applied. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics. Traveling wave solutions are determined for extended modified Korteweg-de Vries equation. The study shows that the sine–cosine method is quite efficient and practically well suited for use in calculating traveling wave solutions for extended modified Korteweg-de Vries equation.

Mathematical Description of Energy Transition Scenarios Based on the Latest Technologies and Trends

This work, dedicated to a mathematical description of energy transition scenarios, consists of three main parts. The first part describes modern trends and problems of the energy sector. A large number of charts reflecting the latest updates in energy are provided. The COVID-2019 pandemic’s impacts on the energy sector are also included. The second part of the paper is dedicated to the analysis of energy consumption and the structure of the world fuel and energy balance. Furthermore, a detailed description of energy-efficient technologies is given. Being important and low-carbon, hydrogen is discussed, including its advantages and disadvantages. The last part of the work describes the mathematical tool developed by the authors. The high availability of statistical data made it possible to identify parameters used in the algorithm with the least squares method and verify the tool. Performing several not complicated steps of the algorithm, the tool allows calculating the deviation of the average global temperature of the surface atmosphere from preindustrial levels in the 21st century under different scenarios. Using the suggested mathematical description, the optimal scenario that makes it possible to keep global warming at a level below 1.7 °C was found.

Mathematical Tool Based on Breakthrough Curves to Evaluate the Economic Advantages of Chemical Regeneration of Activated Carbon in Power Plants: A Comparative Study

A mathematical tool has been developed to evaluate the economic advantages of in-situ chemical regeneration of fixed-bed industrial adsorbers of granular activated carbon for cooling water treatment systems in Cuban power plants. Two scenarios of activated carbon (AC) management in a power plant were compared by applying the proposed model. The economic profit by implementing the regeneration strategy as a function of the number of regeneration cycles was determined and optimized. Breakthrough curves were obtained to assess the adsorption performance of the AC after progressive saturation–chemical regeneration cycles using synthetic water and hydrochloric acid, respectively. For the first saturation cycle, the breakthrough time was 272 min and after 10 cycles, it was reduced to 58 min, indicating a decrease of the adsorption capacity of 21%. The AC adsorption performance in terms of saturation time as a function of the number of regeneration cycles was considered one of the tool parameters. The proposed tool allows to determine the optimal number of regeneration cycles for a maximum economic profit in the regeneration strategy. It was demonstrated, using the proposed tool, that after an optimum of seven regeneration cycles, the power plant expends only 26% of the total investment. The simplicity of the tool permits a rapid way to find the most profitable number of regeneration cycles by combining economic, technical and adsorption efficiency parameters in one function, thus improving the AC management strategy at an industrial scale with corresponding environmental and economic advantages, including sustainability.