Methods Of Differential Calculus Research Articles

ПРОДОВОЛЬСТВЕННАЯ БЕЗОПАСНОСТЬ РОССИИ: АНАЛИЗ СОВРЕМЕННОГО СОСТОЯНИЯ

The article analyzes the current state of Russian food security. During the study, methods of comparison, statistical analysis and grouping, and the method of differential calculus were used. The article analyzes the indicators of food independence for basic food products to determine their compliance with the Food Security Doctrine. Evaluation of the current situation in the field of food consumption in Russia is given. Indicators of economic accessibility are calculated, and the dynamics of consumption levels for the main types of products are presented. An assessment was made of the compliance of actual food consumption with established rational consumption standards, what made it possible to identify high differentiation in consumption by decile groups of the population and a worse situation among its low-income part.

On the yield to maturity of a coupon bond

The article is devoted to one of the main characteristics of the bond-the yield to maturity. The subject of the study is the type of yield to maturity indicator. It is known, that there are two approaches to determining the yield to maturity of a bond: the nominal interest rate and the effective interest rate method. The relevance of the study is due to the fact that, as preliminary comparison has shown, these two approaches to determining the yield to maturity may be unequal in research. The purpose of this paper is to conduct a study of the dependence of the research results on the type of yield to maturity indicator. For this purpose, the problem of the dependence of the interest rate risk of a bond on the number of coupon payments per year was chosen. The literature contains reports on the dependence on the frequency of coupon payments over the term a bond that evaluates interest rate risk. The problem of the dependence directly of the interest rate risk of a bond on the number of coupon payments per year has not been considered in the literature. The task was set to determine which of the two approaches to determining the yield to maturity allows us to obtain results for interest rate risk that are consistent with the dependence of the duration of the bond on the number of coupon payments per year. Methods of differential calculus are used to solve the problem. As a result, it was proved that the use of the yield to maturity determined by the effective interest rate method allows us to obtain results consistent with the dependence of the duration of the bond on the number of coupon payments per year. The results obtained by using the yield to maturity determined by the nominal interest rate method do not agree with the dependence of the duration of the bond on the number of coupon payments per year. It is concluded that the yield to maturity determined by the nominal interest rate method in research may lead to incorrect results, in contrast to the yield to maturity in the form of an effective interest rate. Results of the work can be useful to both the bond issuer and the investor, as well as in theoretical studies of investments in bonds.

ПРИМЕНЕНИЕ ПРОИЗВОДНОЙ ФУНКЦИИ ПРИ РЕШЕНИИ ОПТИМИЗАЦИОННЫХ ЗАДАЧ

The paper considers the application of methods of differential calculus in solving optimization problems

Mathematical substantiation of mine locomotive braking efficiency under pulsiting braking torque

Purpose. Determination of braking torque parameters that provide high braking performance of a mine locomotive by mathematical modeling of the braking of a mine locomotive by a disc brake that creates a pulsating braking torque. The methods. Forced vibrations of the elements of the wheel-motor unit (WMU) of a mine locomotive in the process of braking with a disc brake are studied by the methods of differential calculus and mathematical modeling. Integration of the system of differential equations is performed by the Runge-Kutt method. Findings. A comparative analysis of a mine locomotive braking by a disc brake, which creates a constant and pulsating sinusoidal braking torque with a different number of sinusoid periods per revolution of the wheel pair, is carried out. It has been proved that the pulsating sinusoidal braking moment generated on the axle of the wheel pair equal to the sum of the constant component and the amplitude of the oscillations of the variable component multiplied by the sine of the product of the number of sinusoid periods per revolution of the wheel pair by its angular coordinate provides higher braking characteristics than the constant braking moment. It is shown that pulsating braking torque reduces the braking time and braking distance of the mine locomotive. The values of the amplitude of oscillations, which depend on the average value of the braking torque, and the number of periods of the sinusoid per one revolution of the wheelset, at which the greatest effect is achieved, are determined. The originality. For the first time, a mathematical model has been developed for braking a mine locomotive by a disc brake, which creates a pulsating braking torque on the axle of the wheel pair, depending on its angular coordinate, taking into account the nonlinear dependence of the adhesion coefficient on relative slip, on the basis of which the efficiency of a disc brake with a multi-sector disc is shown in comparison with a disc brake with a homogeneous disk under various conditions of the rail track. Practical implementation. The research results make it possible to determine the parameters of a disc brake with a multi-sector disc located on the wheelset axle and on the engine shaft, which provide the highest braking efficiency.

The dynamical problem on acting distributed load on the elastic layer

The wave field of an elastic half-layer is constructed, when a dynamic normal load distributed over a rectangular area acts on upper face at the initial moment of time. The lower face of the half-layer is rigidly fixed to the foundation, and the side border is in the conditions of a smooth contact. The method of decomposing the system of motion equations into a system of equations and an independently solvable equation is used, this approach was proposed by Popov~G.~Ya. Laplace and Fourier integral transformations are applied directly to the motion equations and boundary conditions, which reduces the problem to a vector one-dimensional boundary value problem, which is solved by the matrix differential calculus method. The output displacements are obtained using inverse integral transformations. The case of steady oscillations was considered and the amplitude of vertical displacement occurring in the layer was analyzed depending on the shape of the distributed load section, the material of the layer medium and the values of the natural frequency of the layer oscillations.

Interest Rate Risk of Bonds in the Condition of a Changing Key Rate

The article is devoted to the analysis of the behavior of the interest rate risk of bonds in the conditions of a changing key interest rate. As known, the key rate is an instrument of monetary regulation of the Central Bank of the Russian Federation. During periods of instability, the key rate may change, which leads to changes in yields in the bond market. The latter, in turn, means that bonds on the Russian market are exposed to interest rate risk. When investing in federal loan bonds (OFZ), interest rate risk becomes the main type of risk, since there is no credit risk in such bonds. The aim of the paper is to obtain proof of the dependence of the interest rate risk of bonds on the term to maturity. The author applies methods of differential calculus to obtain the proof. The novelty of the research is that there is no similar proof in the literature. The instability of interest rates in the market persists at the present time, which allows us to speak about the relevance of this work. Results: it is established that with fixed values of the coupon rate, initial yield and the amount of interest rate change, the interest risk of bonds increases with an increase in the term to maturity. For longterm bonds sold at a discount, there is a term of a maximum interest rate risk. The formula for the approximate value of the term of maximum is obtained. Proven statements are confirmed by calculations, are consistent with previously performed studies, and are in line with market observations. The author comes to the conclusion that the proof obtained in the article of the dependence of the interest rate risk of bonds on the term to maturity can be used to analyze the behavior of the interest rate risk of bonds in the conditions of a changing key interest rate. The practical significance: the results of the research can be useful to the issuer and investor, as well as in theory when studying the investment properties of bonds.

Mathematical modeling of heat and mass transfer in the process of drying seeds of agricultural crops in vacuum conditions

Purpose. To analyze the effect of vacuum on the process of drying seeds of agricultural crops using mathematical modeling. Methods. Methods of differential calculus were used, in particular first-order differential equations, as well as differential geometry. The analysis of the mathematical model was carried out using graphs that were built by computer programs for solving differential equations. Results. On the basis of balance equations of heat and humidity flows of the seed drying process in the drying chamber, a system of differential equations was compiled and solved. Analytical dependencies describing the kinetics of drying and heating crop seeds when using vacuum in the drying chamber are obtained. On the example of soybean seeds, graphic dependencies of the kinetics of drying and heating at different pressures and the initial absolute humidity of the air in the drying chamber are constructed. These graphic dependences show that an increase in the vacuum in the drying chamber leads to an increase in the rate of seed drying, and a decrease in the initial absolute air humidity also increases the drying rate and decreases the equilibrium humidity. Analysis of the graphical dependencies of seed heating kinetics shows that drying in a vacuum reduces some of the thermal energy used directly to heat the seeds and reduces their heating temperature. Conclusions. To describe the process of heat and mass transfer in the process of drying seeds of agricultural crops under vacuum conditions, a mathematical model has been developed using a system of differential equations. The solution of which, on the example of soybean seeds, shows that when using a vacuum in the process of drying seeds, the exposure time and the proportion of thermal energy used directly to heat the seeds are reduced. Reducing the initial absolute humidity of the air inside the drying chamber also reduces the exposure time and equilibrium humidity. Therefore, drying in a vacuum reduces the negative impact of the heater on the seeds. Keywords: seeds, seed moisture, seed temperature, vacuum, absolute humidity, drying kinetics, drying chamber, equilibrium humidity.

MODEL OF THEMATIC INTERPRETOTION OF THE VIEW IMAGES

The subject matter of the article is thematic interpretation of species images. The goal of the work is to develop a model for thematic interpretation of species images, which will be based on the model for the formation of species images and additionally take into account geographic zoning. The following task was solved in the article: development of a model of thematic interpretation of view images, which takes into account the shortcomings of existing models and is based on the model of formation of view images, carries out the reverse transformation of the image coordinates into geospatial coordinates and clustering the image into separate classes according to their color and texture, additionally taking into account geographic zoning, which provides the possibility of advanced analysis and forecasting the temporal dynamics of data in geospatial information processing systems. The following methods used are – mathematical apparatus of the theory of matrices, methods of mathematical modelling, methods of data clustering, methods of differential calculus, methods of digital image processing. The following results were obtained – groups of existing models for interpreting the results of remote sensing of the Earth, their advantages and disadvantages have been analyzed; a mathematical model of the formation of a species image for a section of the earth's surface has been formulated in general form; a mathematical model of thematic interpretation of species images was formulated in general form; the operator of transformation of coordinates, operators of clustering, operators of zoning and their explicit form are considered; a model of thematic interpretation of species images in operator form is obtained. Conclusions: for the first time, a model of thematic interpretation of view images was developed, which, based on the model of formation of view images, carries out the inverse transformation of coordinates into geospatial coordinates and clustering by their color and texture, additionally taking into account geographical zoning, which provides the possibility of advanced analysis and forecasting of the temporal dynamics of data in processing systems geospatial information.

Petrol engine workflow model for researching unconventional engines

The work lies with the study of petrol engines of an unconventional design with improved fuel and economic characteristics developed at the Automobile and Road Institute of Donetsk National Technical University for several decades. It describes the obtained mathematical model of the working process of a petrol piston engine to study the operation of engines. The authors used analytical methods of differential calculus to obtain a mathematical model. They solved the obtained equations using numerical methods and Simulink algorithms. Implementation required the development of a model that has sufficient speed to complete the tasks. The paper presents some modelling results, which makes it possible to judge the correctness of the obtained model. The implemented approach makes it possible to use this model in multi-domain modelling of the engine as a whole. The use of such a model opens up broad prospects for optimizing engine parameters, both its mechanical elements and the parameters of the working process. In addition, it becomes possible to develop and debug a control system for a petrol internal combustion engine using simulation in a Simulink environment.

MODELS FOR DETERMINING MAXIMUM DEGREE FILLING CHANNELS OF CIRCULAR SECTION SHAPE

Objectives The task was to obtain a model for determining the maximum possible degree of filling the circular section channels in the earthen channel based on the condition of ensuring the stability of slopes and the minimum volume of excavation during their construction, as well as finding the optimum degree of filling fortified channels of a closed profile corresponding to maximum throughput.Method In work analytical methods of differential calculus and the solution of implicit equations are used.Result The experience of domestic and foreign researchers was taken into account to solve the set tasks, select research methods and criteria for optimizing channel parameters. Two cases of circular-shaped channels are considered: 1) in the earth channel, 2) reinforced with a closed transverse profile. For the case of a hydraulically most advantageous circular channel in the earthchannel channel, equating the first derivative of the equation of a circle with the reciprocal of the allowable embedding coefficient of the slopes obtained an analytical solution for determining the maximum degree of filling from the slope stability condition, which was not dependent on hydraulic flow elements. In the case of non-cohesive soils that form the channel, the circular channel can be filled to a depth not exceeding 20 percent of the channel radius. At the same time, the average flow rate should remain in the range from non-venting to non-blurring. In order to be able to determine the flow rate, analytical expressions are given for finding hydraulic flow elements in a circular channel. When determining the volume of excavation for the construction of the channel, the excess of the channel edge above the maximum water level in the channel was taken into account. For the case of a fortified channel of a closed transverse profile, by taking the derivatives from the Chezy formula, we obtained the optimal values of flow rate and average velocity.Conclusion A fortified closed circular profile channel has a maximum capacity with a relative degree of filling of 0.938, and the maximum average velocity of a fluid in a pressureless channel is achieved with a degree of filling of 0.815. To determine the maximum permissible relative degree of filling in the case of a circular section channel in the earthen channel, analytical dependences were obtained, before using which, according to the reference literature, it is necessary to take the value of the slope coefficient for this type of channel bed soil.

Controllable generation of nanofibers through a magnetic-field-assisted electrospinning design

Electrospinning technology is taken as the most versatile process to generate continuous nanofibers. However, effects should be taken to achieve better understanding and precisely control the actual mechanics in the formation of nanofibers through better system design. In this article, the diameter and the deposition area of the as-prepared nanofibers can be controlled by added helix tube which can provide external electric-magnetic field. The spinning process can be controlled employing a magnetic flux (B) analytical model which is based on the magnetic field calculation modelof the solution and the electrostatic charge repulsion in the jet. This method includes the thermo-mechanical properties based on as-prepared electrospun nanofiber films and the capability of tribology-mechanical and mathematical physics. A novel investigation of the whole output increase at per unit spinning area is presented in this article to describe the whipping instability which is an important feature of the Taylor Cone phenomenon. The present design and process for electrospinning are based on a solution differential calculus, which a single stainless steel spinning nozzle can generate multi-squirt flow. With the purpose of providing useful jet initial behaviour design for the further technology applications, we attempt to set up a differential calculus method with evenly distributed circumferential electric field, yet giving more design guidance to produce controllable nanofibers through magnetic-field-assisted electrospinning technology.

Застосування системи Maple при викладанні «диференціального числення» студентам-іноземцям у ЗВТО

The article shows the relevance of the use of the Maple system during the study of mathematics with students of a foreign branch. On the example of practical lesson «Application and methods of differential calculus», the experience of overcoming the problem of activating the independent work of foreign students in the technical ZVO is highlighted. The method of adaptation of the Maple system to the study of the derivative with students of the foreign branch is described. The content and organizational form of such classes can be implemented in the process of training students of technical field. The current tendency of education is characterized by an increase in the share of independent work. Therefore, one of the most important tasks of training is the formation of skills and abilities of independent work, activating it in various accessible ways. The practice of teaching mathematics to foreign students in the preparatory department shows that the success of mastering the educational material in the process of independent work is not ideal. The organization of independent work in traditional education is complicated by the lack of educational textbooks for foreign students. The authors developed their approach to organizing independent work of the student in its various forms. Since higher mathematics is compulsory for all faculties of FTI, in the process of training foreign-language students in the preparatory department it is necessary to solve the problem of equalization of knowledge in mathematics to the school curriculum of Ukraine. Solving the problems of overcoming the academic differences in mathematical preparation and the formation of the readiness of foreign students for the preparatory department to study higher mathematics should take place on the basis of an individual approach during the educational process, which is based on the application of ICT.

A Study of the Extremum of the Total Energy of the Selective Signals Constructed by Quadratic Splines

The development of mobile communication networks in the direction of 5G networks implies the use of the radio interface that is based on new signal-code structures, the choice of which will determine their further development and the ability of operators to provide innovative services. The use of quadratic splines for the synthesis of selective signals with a finite spectrum, free from intersymbol interference, is proposed. An analytical expression for the synthesized signal in the time and frequency domains is obtained. A study was made of the dependence of the total energy of a selective signal on the parameters of a quadratic spline, which is used to interpolate the spectral density in the transition region using the methods of differential calculus of functions of several variables. To study the extremum of the total energy of a selective signal, the parameters of the width of the transition area α was used and the coefficient of rounding the spectrum ρ, the variation of whose spectral density allowed us to establish the limits of the change in the total energy of the signal in question. Conducted studies allow us to synthesize the signal and formulate recommendations on how by changing the parameters of the signal, you can get a signal with the desired properties. This will allow to obtain the optimal waveform in accordance with the selected criteria, providing for the required energy performance of the signal in the radio interface of 5G networks.

Optimal pricing and lot-sizing policy for supply chain system with deteriorating items under limited storage capacity

This research work considers a two-echelon supply chain model for deteriorating items in which the retailer's warehouse capacity of display area is limited. Therefore, the retailer stores remaining units in the back room which has unlimited capacity. The demand rate is assumed to be dependent on the retailer's selling price and displayed stock level. The proposed approach explicitly models the interdependence among price of a product, demand for the product and the integration among the retailer and the supplier under four different policies: non-integrated, integrated, supplier-led Stackelberg policy and retailer-led Stackelberg policy. Furthermore, this paper also establishes several theoretical results to show that the solution provides the maximum total relevant profit. By using differential calculus method, it is proved that the optimal solution not only exists but also is unique. In this study, the impact of price, inventory level and the integration among retailer and the supplier is explicitly analyzed in the emerging retail scenario with limited shelf space.

Застосування системи Maple при викладанні «диференціального числення» студентам-іноземцям у ЗВТО

The article shows the relevance of the use of the Maple system during the study of mathematics with students of a foreign branch. On the example of practical lesson «Application and methods of differential calculus», the experience of overcoming the problem of activating the independent work of foreign students in the technical ZVO is highlighted. The method of adaptation of the Maple system to the study of the derivative with students of the foreign branch is described. The content and organizational form of such classes can be implemented in the process of training students of technical field. The current tendency of education is characterized by an increase in the share of independent work. Therefore, one of the most important tasks of training is the formation of skills and abilities of independent work, activating it in various accessible ways. The practice of teaching mathematics to foreign students in the preparatory department shows that the success of mastering the educational material in the process of independent work is not ideal. The organization of independent work in traditional education is complicated by the lack of educational textbooks for foreign students. The authors developed their approach to organizing independent work of the student in its various forms. Since higher mathematics is compulsory for all faculties of FTI, in the process of training foreign-language students in the preparatory department it is necessary to solve the problem of equalization of knowledge in mathematics to the school curriculum of Ukraine. Solving the problems of overcoming the academic differences in mathematical preparation and the formation of the readiness of foreign students for the preparatory department to study higher mathematics should take place on the basis of an individual approach during the educational process, which is based on the application of ICT.

CEO discretion, CEO tenure and executive pay-performance sensitivity: A dynamic model based on game theory

Following the latest research trend which argues that the effects of CEO discretion on executive pay-performance sensitivity (EPPS) are some “contingent effects” instead of the “fixed effects”, the paper innovatively addresses the moderating role of CEO tenure in the relationship between CEO discretion and EPPS. For this purpose, based on the insights of game theory, the study creates a comprehensive theoretical algorithm model of EPPS with the introduction of two new critical variables, namely CEO discretion and CEO tenure into the classical principal-agent models of EPPS. By adopting the method of differential calculus, the model solution is calculated and further discussed. According to the model results, CEO tenure can moderate the relationship between CEO discretion and EPPS. To be specific, CEO discretion of fresh CEOs is positively related to EPPS; while for senior CEOs, CEO discretion is negatively related to EPPS. This finding is of great significance to reconcile the conflicting views on the relationship between CEO discretion and EPPS by proving fresh CEOs’ firm-serving motivations and senior CEOs’ self-serving motivations.

El cálculo diferencial y las cotas de errores en las mediciones indirectas, en el trabajo experimental de la Física General

The article deals with the problem of the use of error theory in the propagation of indirect measurements carried out in the teaching laboratories of the institution from the realization of physical experiments, experimental problems and laboratory practices, among them, elements and methods of differential calculus most used in the propagation of error values of indirect measurements. Some experiences in the use of methodological actions designed with the objective of developing skills in students, with the fundamental contents of the theory of errors that they receive in the careers in subjects of the discipline General Physics, to facilitate and propitiate the development of experimental activities of the students of the careers of Mathematics-Physics, Industrial Engineering, Agronomy and Computer Science, are also exposed. Translated with www.DeepL.com/Translator (free version)

Детерминированный факторный анализ: методы интегро-дифференцирования нецелого порядка

Objective: to summarize the methods of deterministic factor economic analysis, namely the differential calculus and the integral method. Methods: mathematical methods for integro-differentiation of non-integral order, the theory of derivatives and integrals of fractional (non-integral) order.Results: the basic concepts are formulated and the new methods are developed that take into account the memory and non-locality effects in the quantitative description of the influence of individual factors on the change in the effective economic indicator. Two methods are proposed for integro-differentiation of non-integral order for the deterministic factor analysis of economic processes with memory and non-locality. It is shown that the method of integro-differentiation of non-integral order can give more accurate results compared with standard methods (method of differentiation using the first order derivatives and the integral method using the integration of the first order) for a wide class of functions describing effective economic indicators.Scientific novelty: the new methods of deterministic factor analysis are proposed: the method of differential calculus of non-integral order and the integral method of non-integral order.Practical significance: the basic concepts and formulas of the article can be used in scientific and analytical activity for factor analysis of economic processes. The proposed method for integro-differentiation of non-integral order extends the capabilities of the determined factorial economic analysis. The new quantitative method of deterministic factor analysis may become the beginning of quantitative studies of economic agents behavior with memory, hereditarity and spatial non-locality. The proposed methods of deterministic factor analysis can be used in the study of economic processes which follow the exponential law, in which the indicators (endogenous variables) are power functions of the factors (exogenous variables), including the processes described by the Cobb - Douglas production function, since these methods allow to more accurately describe the total influence of the factors in comparison with the standard method. The proposed methods can be used in the study of economic processes described by equations with a power-law non-locality in factor space and in state space.

Differential Calculus Based on the Double Contradiction

The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. Next, the geometrical meaning of the double contradiction is considered as follows. A tangent at a point on a convex curve is iterated. Then, the slope of the tangent at the point is sandwiched by two kinds of lines. The first kind of line crosses the curve at the original point and a point to the right of it. The second kind of line crosses the curve at the original point and a point to the left of it. Then, the double contradiction can be applied, and the slope of the tangent is determined as a single value. Finally, the meaning of this method for the foundation of mathematics is considered. We reflect on Dehaene’s notion that the foundation of mathematics is based on the intuitions, which evolve independently. Hence, there may be gaps between intuitions. In fact, the Ancient Greeks identified inconsistency between arithmetic and geometry. However, Eudoxus developed the theory of proportion, which is equivalent to the Dedekind Cut. This allows the iteration of an irrational number by rational numbers as precisely as desired. Simultaneously, we can define the irrational number by the double contradiction, although its existence is not guaranteed. Further, an area of a curved figure is iterated and defined by rectilinear figures using the double contradiction.

Maximum and minimum problems in secondary school education

The aim of this paper is to offer some possible ways of solving extreme value problems by elementary methods with which the generally available method of differential calculus can be avoided. We line up some problems which can be solved by the usage of these elementary methods in secondary school education. The importance of the extremum problems is ignored in the regular curriculum; however they are in the main stream of competition problems – therefore they are useful tools in the selection and development of talented students. The extremum problem-solving by elementary methods means the replacement of the methods of differential calculus (which are quite stereotyped) by the elementary methods collected from different fields of Mathematics, such as elementary inequalities between geometric, arithmetic and square means, the codomain of the quadratic and trigonometric functions, etc. In the first part we show some patterns that students can imitate in solving similar problems. These patterns could also provide some ideas for Hungarian teachers on how to introduce this topic in their practice. In the second part we discuss the results of a survey carried out in two secondary schools and we formulate our conclusion concerning the improvement of students' performance in solving these kind of problems.