Viral Infection Spreading and Mutation in Cell Culture

A new model of viral infection spreading in cell cultures is proposed taking into account virus mutation. This model represents a reaction-diffusion system of equations with time delay for the concentrations of uninfected cells, infected cells and viral load. Infection progression is characterized by the virus replication number Rv, which determines the total viral load. Analytical formulas for the speed of propagation and for the viral load are obtained and confirmed by numerical simulations. It is shown that virus mutation leads to the emergence of a new virus variant. Conditions of the coexistence of the two variants or competitive exclusion of one of them are found, and different stages of infection progression are identified.

METHOD OF FINDING LOSSES IN THE THROTTLE WINDING AS A RESULT OF THE SKIN EFFECT AND THE PROXIMITY EFFECT

Представлен метод использования расчетных методик и моделирования магнитных полей в двухмерном пространстве для нахождения высокочастотных потерь в обмотке моточных изделий, таких как дроссель или трансформатор. Была проведена работа по анализу литературы по данной теме, а также поднят вопрос оптимизации и адаптации аналитических формул для случая использования проводников круглого сечения и намотки, имеющей неоднородное распределение слоев в окне сердечника. Был также поднят вопрос об аналитическом нахождении длины обмоточного провода намотки с различным количеством слоев и переменного количества используемых при этом витков. Для проведения автоматизации расчета с помощью формул был написан скрипт, строящий зависимость сопротивления переменному току относительно частоты, используя аналитические формулы. Была написана программа для автоматической постановки начальных условий и граничных значений параметров моделирования, процесса самого моделирования электромагнитных полей, анализа полученных данных, а также формирования массива для построения графика полученной при этом зависимости сопротивления от частоты. В данном методе используется свободно распространяемое программное обеспечение как для математических расчетов, так и моделирования электромагнитных полей. Итогом данной работы стало сравнение полученных результатов, которые показали хорошую сходимость и преемственность этапов данного метода Here we present a method for using computational methods and modeling magnetic fields in two-dimensional space to find high-frequency losses in the winding of winding products, such as a choke or transformer. We analyzed the literature on this topic, as well as the issue of optimization and adaptation of analytical formulas for the case of using round-section conductors and winding having a non-uniform distribution of layers in the core window. We discussed the analytical finding of the length of the winding wire of the winding with a different number of layers and a variable number of turns used in this case. To automate the calculation using formulas, we wrote a script that builds the dependence of the resistance to alternating current relative to the frequency using analytical formulas. In addition, we wrote a program for automatically setting the initial conditions and boundary values of the modeling parameters, the process of modeling electromagnetic fields itself, analyzing the data obtained, as well as forming an array for plotting the resulting dependence of resistance on frequency. This method uses freely distributed software for both mathematical calculations and modeling of electromagnetic fields. The result of this work was a comparison of the results obtained, which showed good convergence and continuity of the stages of this method

Time-Discrete Hedging of Down-and-Out Puts with Overnight Trading Gaps

Hedging down-and-out puts (and up-and-out calls), where the maximum payoff is reached just before a barrier is hit that would render the claim worthless afterwards, is challenging. All hedging methods potentially lead to large errors when the underlying is already close to the barrier and the hedge portfolio can only be adjusted in discrete time intervals. In this paper, we analyze this hedging situation, especially the case of overnight trading gaps. We show how a position in a short-term vanilla call option can be used for efficient hedging. Using a mean-variance hedging approach, we calculate optimal hedge ratios for both the underlying and call options as hedge instruments. We derive semi-analytical formulas for optimal hedge ratios in a Black–Scholes setting for continuous trading (as a benchmark) and in the case of trading gaps. For more complex models, we show in a numerical study that the semi-analytical formulas can be used as a sufficient approximation, even when stochastic volatility and jumps are present.

Implementation and use of the Levenberg-Marquard algorithm in the problems of calibration of robotic manipulators

The well-known problem of calibration of an arbitrary robotic manipulator, which is formulated in the most general form, is considered. To solve the direct problem of kinematics, an alternative to the Denavit-Hartenberg method, a universal analytical description of the kinematic scheme, taking into account possible errors in the manufacture and assembly of robot parts, is proposed. At the same time, a universal description of the errors in the orientation of the axes of the articulated joints of the links is proposed. On the basis of such a description, the direct and inverse problem of kinematics of robots as spatial mechanisms can be solved, taking into account the distortions of dimensions, the position of the axes of the joints and the positions of the zeros of the angles of their rotation. The problem of calibration of manipulators is formulated as a problem of the least squares method. Analytical formulas of the objective function of the least squares method for solving the problem are obtained. Expressions for the gradient vector and the Hessian of the objective function for the direct algorithm, Newton-Gauss and Levenberg-Marquardt algorithms are obtained by analytical differentiation using a special computer algebra system KiDyM. The procedures in the C ++ language for calculating the elements of the gradient and hessian are automatically generated. On the example of a projected angular 6-degree robot-manipulator, the results of modeling the solution to the problem of its calibration, that is, determination of 36 unknown angular and linear errors, are presented. A comparison is made of the solution of the calibration problem for simulated 64 and 729 experiments, in which the generalized coordinates - the angles in the joints took the values ±90° and -90°, 0, +90°.

Probabilistic Combinatorial Model of the Epidemic for a University

The aim of the research was to identify the main causes of infection of teachers and students in a university. Two probabilistic combinatorial problems are considered analytically to determine the probabilities and rates of infection of teachers and students in a university as a result of the appearance of infected persons among the contingent of students. The mathematical apparatus of probability theory and combinatorics is used to solve the problems. For the factorials of combinations arising in the structure, the asymptotic Stirling’s formula is used. Convergent series arise in the final formulas, reflecting the multiplicity of scenarios of the probabilistic approach. Analytical formulas for the sums of series, probabilities and rates of infection of teachers and students are obtained. It is shown that the infection of teachers and students occurs through «dangerous» spatially close contacts, when a teacher and a student talk at a distance of less than 0.5 meter. It is impossible to exclude such contacts in the students’ environment during full-time study. Among teachers, there is also a less probable classroom mechanism of infection through the volume of air infected with viruses.

Design Procedure Based on Maximum Efficiency for Wireless Power Transfer Battery Chargers with Lightweight Vehicle Assembly

This paper analyzes two different design procedures for a series-parallel compensated WPT battery charger, based on different definitions of the operating resonant frequency in the condition of maximum link efficiency. The behaviour of the voltage gain magnitude and the input impedance angle of the resulting WPT links is studied for different loads and coupling coefficients. The design algorithms are supported by analytical formulas derived from an exact circuit analysis of the WPT link, avoiding approximations as far as possible. To support the theoretical approach a case study is proposed, in which both design procedures are implemented considering specifications in line with the actual automotive standards. To conclude, a characterization of the efficiency in both cases is derived.

The Causal Interaction between Complex Subsystems

Information flow provides a natural measure for the causal interaction between dynamical events. This study extends our previous rigorous formalism of componentwise information flow to the bulk information flow between two complex subsystems of a large-dimensional parental system. Analytical formulas have been obtained in a closed form. Under a Gaussian assumption, their maximum likelihood estimators have also been obtained. These formulas have been validated using different subsystems with preset relations, and they yield causalities just as expected. On the contrary, the commonly used proxies for the characterization of subsystems, such as averages and principal components, generally do not work correctly. This study can help diagnose the emergence of patterns in complex systems and is expected to have applications in many real world problems in different disciplines such as climate science, fluid dynamics, neuroscience, financial economics, etc.

On Mass Spectra of Primordial Black Holes

Evidence for the primordial black holes (PBH) presence in the early Universe renews permanently. New limits on their mass spectrum challenge existing models of PBH formation. One of the known models is based on the closed walls collapse after the inflationary epoch. Its intrinsic feature is the multiple production of small mass PBH which might contradict observations in the nearest future. We show that the mechanism of walls collapse can be applied to produce substantially different PBH mass spectra if one takes into account the classical motion of scalar fields together with their quantum fluctuations at the inflationary stage. Analytical formulas have been developed that contain both quantum and classical contributions.

Some Rank Formulas for the Yang-Baxter Matrix Equation AXA = XAX

Let A be an arbitrary square matrix, then equation AXA =XAX with unknown X is called Yang-Baxter matrix equation. It is difficult to find all the solutions of this equation for any given A . In this paper, the relations between the matrices A and B are established via solving the associated rank optimization problem, where B = AXA = XAX , and some analytical formulas are derived, which may be useful for finding all the solutions and analyzing the structures of the solutions of the above Yang-Baxter matrix equation.

Distributions of Stress and Interlaminar Shear Stress in Laminates under Normal Load

The normal stress of each layer of the laminate composite material will undergo complex changes after normal compression, and shear stress will also appear between the layers. In order to explore the distribution laws of normal stress and shear stress, this paper uses Hooke's law and the equilibrium condition of force to carry out mathematical derivation, the analytical formulas for normal stress and shear stress are obtained, and their respective maximum values ​​are given. Studies have shown that the maximum normal stress occurs at the center of the laminate, and its value is proportional to the external load, and is also closely related to the length, width, thickness, elastic modulus of the cementing agent, elastic modulus and Poisson’s ratio of the laminate; The maximum shear stress occurs at the four corners of the laminate, and its value is proportional to the external load and the shear modulus of the cementing agent, inversely proportional to the thickness of the cementing agent layer, and its value is also closely related to the length, width, elastic modulus and Poisson's ratio of the laminate. The analytical formulas for normal stress and interlayer shear stress is helpful to deepen the understanding of the internal force distribution law of laminated plates, and the maximum value calculation formula can greatly facilitate the calculation of strength.