scholarly journals USING A FUNCTION DERIVATIVE IN ECONOMIC TASKS

2021 ◽  
Vol 16 (1(21)) ◽  
pp. 93-99
Author(s):  
Maka Lomtadze
Keyword(s):  
Average Cost ◽  
Initial Period ◽  
Rate Of Change ◽  
Order Derivative ◽  
Mathematical Methods ◽  
Instantaneous Rate ◽  
Economic Problems ◽  
Per Capita ◽  
Volume Output ◽  
The Given

The article focuses on the application of mathematical methods in economics, in particular discussing economic problems that are easily solved using derivatives. The purpose of the article is to show students the way and opportunity to use mathematical methods to solve economic problems. To this end, the article discusses and analyzes several economic tasks in detail, which will be interesting and easy for students to master. I considered the derivative of a function as the rate of change and introduced the definition: The instantaneous rate of change of the function f with respect to x at a point is called the derivative if it exists. With the help of this definition I have discussed and explained Task 1: Suppose that the increase in production of a certain product over a period of time is described by a function And population growth is described by the following function: Where is the number of years from the initial period, then the production of these products per capita is given by the function: Find the growth rate of product production. By solving this task I came to the conclusion that after a year the production of products per capita increases. In the following tasks I used the method of finding the extremum values of a function using a derivative, namely I equated the first-order derivative to 0, found the critical points, and with the help of the second-order derivative I determined the extremes of the function. I discussed task 2: For the production of X volume of products, the firm plans a cost that is calculated by the formulan . For what volume of production will the average cost be the smallest? Find the numerical value of this small expense. Solving this problem, I came to the conclusion that the given function of the average cost takes on the least value when the output volume is a unit , and this value is equal to: which is the marginal cost when producing the volume output. I discussed Task 3: How many products should be sold in order for a firm to profit maximally if the derivative cost function is known: And return function: Here I came to the conclusion: if 600 units of the product are sold, then the firm's profit will be maximum and it will be numerically equal At the end of the article I discussed such a task 4 of applied optimization. What is the minimum amount of material needed to make a 2 liter cylindrical jar? Where I came to the conclusion: the smallest amount of material will be spent to get a cylindrical vessel if we take the height 2 times the radius of the base.

ACHIEVEMENT OF SINGAPORE'S INDEPENDENCE AND DIFFICULTIES ON THE INITIAL PERIOD OF DEVELOPMENT

2019 ◽  
Vol 17 (2) ◽  
pp. 75-84
Author(s):  
Ilyas Shakirov ◽  
Keyword(s):  
Initial Period ◽  
Historical Sources ◽  
History Of ◽  
The Given

In the article considered events between 1945-1965 years in Singapore. On the ground of historical sources author of the given article learned the history of gaining independence by Singapore, as well, difficulties country carried out over 20 years

APPLICATION OF MATHEMATICAL METHODS IN THE STUDY OF ECONOMIC MODELS

2020 ◽  
Author(s):  
Tanzila Muhtarovna Dakasheva ◽  
Keyword(s):  
Economic Models ◽  
Mathematical Methods ◽  
Economic Problems

This article discusses the ways of applying mathematical methods in the study of economic processes. The economic meaning of such a mathematical category as a derivative is considered, its meaning and ways of application in solving economic problems are studied. Several economic models are used as examples.

scholarly journals The Position of Heterodox Economics in Economic Science

2021 ◽  
Vol 66 (2) ◽  
pp. 275-290
Author(s):  
Eleonóra Matoušková
Keyword(s):  
Point Of View ◽  
Austrian School ◽  
Economic Science ◽  
Left Wing ◽  
Mathematical Methods ◽  
Heterodox Economics ◽  
Homo Economicus ◽  
Common Features ◽  
Economic Problems ◽  
Theoretical Approaches

In economic science dominate orthodox economics (mainstream economics respectively neoclassical economics). Despite its numerous intellectual failures, orthodox economics continue to prevail in teaching at universities. A certain alternative to orthodox economics is heterodox economics, which consists of three groups of theoretical approaches, represented by the Left-wing heterodoxy and Neo-Austrian school (we include them together in the Old heterodoxy) and the New heterodoxy. The objective of this article is to define the differences between orthodox economics and heterodox economics, to find common features of individual heterodox approaches and identify substantial differences between them and also highlight the relevance of these heterodox approaches from the point of view of the challenges we are facing today. A common characteristic of heterodoxy is the rejection of orthodoxy, especially its research methods. Heterodox economists reject the axiom that individuals are always rational, the concept of ‘homo economicus’, the application of a formal-deductive approach, the use of mathematical methods in cases that are not appropriate for this, and access from a closed system position. Heterodoxy is a very diverse theoretical tradition, and there are differences not only between the Left-wing heterodoxy, Neo-Austrian school and New heterodoxy, but also within these heterodox groups. They differ on specific topics they deal with and proposed solutions to socio-economic problems.

scholarly journals SOLVING INVERSE PROBLEM OF CHEMICAL KINETICS WITH USE OF CUBIC SPLINES

2020 ◽  
Vol 63 (7) ◽  
pp. 61-66
Author(s):  
Nikolay I. Kol'tsov
Keyword(s):  
Experimental Data ◽  
Inverse Problem ◽  
Chemical Kinetics ◽  
Good Accuracy ◽  
Cubic Splines ◽  
Rate Of Change ◽  
Reference Points ◽  
Additional Advantage ◽  
Instantaneous Rate ◽  
Relaxation Curves

A simple effective method for solving the inverse problem of chemical kinetics based on non-stationary experiments for multistage reactions occurring in an isothermal reactor of ideal mixing is described. The idea of the method is based on taking into account the distinctive features (informativeness) of different fragments of relaxation curves for chemical reactions with arbitrary (non-monotonic) kinetics and their as accurate approximation as possible. For this purpose, non-linear (cubic) splines are used to describe different informative fragments of relaxation curves, which allow to approximate and interpolate experimental data as accurately as possible. An additional advantage of cubic splines, from the point of view of the implementation of the described method, is their continuity at all given points up to and including second-order derivatives (smoothness). This allows us to calculate with good accuracy not only the concentration of reagents, but also the instantaneous rate of change at any time. The consequence of this is the possibility of a sufficiently accurate solution of the inverse problem based on the data of non-stationary experiments. The correctness of the mathematical model used and the stability of the method were tested using variations of the original data. An example of using the method for determining the intervals of physical values of the rate constants of the stages of a two-stage reaction is given. The influence of the method of selecting the reference points (structure) of the spline and measurement errors (noise) of experimental data on the error of determining the speed constants of the stages is estimated. The efficiency of application and good accuracy of the method for solving the inverse problem of chemical kinetics of multistage reactions occurring in non-gradient systems with taking into account of noise is shown.

Probability and the Single Neuron: Decisions, Uncertainty and the Brain: The Science of Neuroeconomics, by P.-W. Glimcher. 2003. Cambridge, MA: MIT Press. 375 pp., $40.00.

2004 ◽  
Vol 10 (7) ◽  
pp. 1027-1029
Author(s):  
M. Kinsbourne
Keyword(s):  
Decision Making ◽  
Single Neuron ◽  
Prior Probability ◽  
Inclusive Fitness ◽  
Neural Circuitry ◽  
Inferior Parietal Lobule ◽  
Mathematical Methods ◽  
Economic Problems ◽  
Neural Substrate ◽  
The Brain

In an uncertain world, people and other animals make their living by predicting which of alternative courses of action is likely to yield the best return. For humans the return might take many forms, such as material, financial, social, or esthetic, but the underlying currency involved for any species is “inclusive fitness,” the rate at which an animal's genes are propagated. Professor Glimcher demonstrates that Economics methods are applicable to decision-making under conditions of uncertainty, both at the behavioral and the neuronal level. This approach has been called neuro-economics, although “econometrics” characterizes it more precisely. Econometrics is the application of statistical and mathematical methods in the field of economics to test and quantify economic theories and the solution to economic problems. Specifically, individuals' decision-making benefits from knowing how likely a response is to be reinforced, and knowing the reinforcement's value. Even single neurons are sensitive to these variables. Glimcher reaches beyond the heavily studied neural substrate for sensation and response to predictive neural circuitry that factors in the prior probability of reward, and its expected value. Indeed, he and his colleagues have identified neurons in monkey's inferior parietal lobule whose firing rates reflect both probability and value.

scholarly journals The role of big eddies in homogeneous turbulence

1949 ◽  
Vol 195 (1043) ◽  
pp. 513-532 ◽  
Keyword(s):  
Fourier Transforms ◽  
Stokes Equations ◽  
Initial Period ◽  
Rate Of Change ◽  
Homogeneous Turbulence ◽  
Wave Number ◽  
Small Range ◽  
Large Wave ◽  
Correlation Tensor ◽  
Spectrum Function

Recent experimental work on the decay of isotropic turbulence has shown that big eddies play an important part in the motion. There is a range of eddy sizes which, during the initial period of decay, contains a negligible proportion of the total energy and is excluded from the similarity possessed by the smaller eddies. This paper examines the motion associated with this small range of large wave-lengths in the more general case of homogeneous turbulence. For this purpose it is convenient to introduce a spectrum tensor, defined as the three-dimensional Fourier transform of the double-velocity correlation tensor. This spectrum function is also suitable for the application of similarity hypotheses, unlike the conventional one-dimensional spectrum function. The properties of the spectrum as a function of the wave-number vector k, are discussed with particular reference to small values of the magnitude k . When k is small the energy per unit interval of wave-number magnitude varies as k 4 . The rate of change of the spectrum function is obtained from the Navier-Stokes equations in terms of Fourier transforms of the triple-velocity and pressure-velocity mean values. After taking into account the continuity condition it is found that the terms of the first and second degree in the expansion of the spectrum function in powers of components of k are constant throughout the decay. The biggest eddies of the turbulence are therefore permanent, being determined wholly by the initial conditions, and are dominant in the final period when the smaller eddies have decayed. The action of smaller eddies on the invariant big eddies is equivalent to that of a turbulent viscosity, the value of which may vary with direction. The implications of the analysis for similarity hypotheses are discussed briefly.

Fractional Derivative Consideration on Nonlinear Viscoelastic Dynamical Behavior Under Statical Pre-Displacement

2005 ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu ◽  
Hiroshi Nasuno
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Dynamical Behavior ◽  
Variable Coefficients ◽  
Rate Of Change ◽  
Order Derivative ◽  
Nonlinear Viscoelastic ◽  
Higher Order Derivative ◽  
The Difference ◽  
Different Order

Nonlinear fractional calculus model for the viscoelastic material is examined for oscillation around the off-equilibrium point. The model equation consists of two terms of different order fractional derivatives. The lower order derivative characterizes the slow process, and the higher order derivative characterizes the process of rapid oscillation. The measured difference in the order of the fractional derivative of the material, that the order is higher when the material is rapidly oscillated than when it is slowly compressed, is partly attributed to the difference in the frequency dependence between the two fractional derivatives. However, it is found that there could be possibility for the variable coefficients of the two terms with the rate of change of displacement.

The Engine or the Caboose? Resource Industries and Twentieth-Century Canadian Economic Performance

2007 ◽  
Vol 67 (1) ◽  
pp. 1-32 ◽  
Author(s):  
Ian Keay
Keyword(s):  
Twentieth Century ◽  
Empirical Evidence ◽  
Natural Resource ◽  
Economic Performance ◽  
Positive Impact ◽  
Rate Of Change ◽  
Domestic Economy ◽  
Intensive Production ◽  
Resource Endowment ◽  
Per Capita

The Canadian economy, already wealthy, diverse, and relatively industrial at the dawn of the twentieth century, had not yet outgrown its reliance on resource-intensive production. Empirical evidence indicates that the exploitation of Canada's natural resource endowment made direct and indirect contributions to the size and efficiency of the twentieth-century domestic economy. I conclude that the concentration of capital and labor in resource industries did not constrain the rate of change of Canadian real GNP per capita between 1900 and 1999, and it appears to have had a substantial positive impact on the level of real GNP per capita.

scholarly journals Decay of isotropic turbulence in the initial period

1948 ◽  
Vol 193 (1035) ◽  
pp. 539-558 ◽  
Keyword(s):  
Correlation Function ◽  
Isotropic Turbulence ◽  
Initial Period ◽  
Rate Of Change ◽  
Velocity Correlation ◽  
Integral Scale ◽  
Direct Measurements ◽  
Theoretical Predictions ◽  
History Of ◽  
Velocity Correlation Functions

In a previous paper the authors described direct measurements of all the terms in the equation for the rate of change of mean square vorticity in isotropic turbulence. The present paper is concerned with developments arising from the earlier work and with the experimental verification of some recent theoretical investigations. The results of measurements of the turbulent intensity u ' and of λ are presented; these establish that u' -2 and λ 2 are each proportional to the time of decay provided that the time is not too large. Within this initial period of the decay, the double and triple velocity correlation functions are found to maintain their form, i.e. to be self-preserving, for small values of the distance r between the two points at which the correlations are taken. For larger separations the double velocity correlation function changes its form slightly during decay and direct measurements of λ and of the integral scale L show that λ/ L increases during the decay. Theoretical predictions about the shape of the correlation function, for limited ranges of r , at high and at low Reynolds numbers are compared with measurements. Theory has shown that the above decay law cannot persist indefinitely, and the present experiments confirm that the decay law changes in the expected direction when the time is large. A division of the life-history of the turbulence into initial, transition and final periods is suggested; within the initial period, a classification based on the Reynblds number is also possible. Some speculations on the interpretation of the initial period are presented.

scholarly journals A dynamical time operator in Dirac's relativistic quantum mechanics

2014 ◽  
Vol 29 (06) ◽  
pp. 1450036 ◽  
Author(s):  
M. Bauer
Keyword(s):  
Quantum Mechanics ◽  
Wave Packet ◽  
Phase Velocity ◽  
Relativistic Quantum ◽  
Rate Of Change ◽  
Time Operator ◽  
Relativistic Quantum Mechanics ◽  
Instantaneous Rate ◽  
Expectation Value ◽  
Dynamical Time

A self-adjoint dynamical time operator is introduced in Dirac's relativistic formulation of quantum mechanics and shown to satisfy a commutation relation with the Hamiltonian analogous to that of the position and momentum operators. The ensuing time-energy uncertainty relation involves the uncertainty in the instant of time when the wave packet passes a particular spatial position and the energy uncertainty associated with the wave packet at the same time, as envisaged originally by Bohr. The instantaneous rate of change of the position expectation value with respect to the simultaneous expectation value of the dynamical time operator is shown to be the phase velocity, in agreement with de Broglie's hypothesis of a particle associated wave whose phase velocity is larger than c. Thus, these two elements of the original basis and interpretation of quantum mechanics are integrated into its formal mathematical structure. Pauli's objection is shown to be resolved or circumvented. Possible relevance to current developments in electron channeling, in interference in time, in Zitterbewegung-like effects in spintronics, graphene and superconducting systems and in cosmology is noted.

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