Three basic properties are associated with any quantity:
- Name,
- meaning,
- type.
Quantity name May be semantic and symbolic . An example of a semantic name is “gas pressure”; a symbolic name for the same quantity is R.
If value of quantity does not change, then it is called constant value or constant . An example of a constant is the Pythagorean number ¶=3.14259... . A quantity whose value can change is called variable . For example, in the description of the process of falling of a body, the variable quantities are the height H and the falling time t.
Type defines the set of values that a quantity can take. Basic types of quantities : numeric, symbolic, logical. Dimensions determine the units in which the values of quantities are represented. For example, t (s) is the time of fall; N (m) - fall height.
Mathematical models
If the relationship between quantities can be represented in mathematical form, that is mathematical model .
A mathematical model is a set of quantitative characteristics of a certain object (process) and the connections between them, presented in the language of mathematics.
This is an example of the dependency presented in functional form. This dependence is called root (time is proportional to square root height).
In more complex tasks mathematical models are represented in the form of equations or systems of equations.
Tabular and graphical models
These are other, non-formular, ways of representing dependencies between quantities. For example, we decided to test the law free fall bodies experimentally.
| We will organize an experiment in the following way: we will throw a steel ball from a 6-meter height, 9-meter height, etc. (after 3 meters), measuring the height initial position ball and falling time. Based on the results of the experiment, we will create a table and draw a graph.If each pair of values H and t from this table is substituted into the previously given formula for the dependence of height on time, then the formula will turn into equality (within accuracy up to measurement error). This means the model works well. However, if you throw not a steel ball, but big light ball, then equality will not be achieved, and if it is an inflatable ball, then the values of the left and right sides of the formula will differ very much. Why do you think? |  |  |
In the same way, you can display the dependence of any phenomenon physical nature, described by well-known formulas.
Information models that describe the development of systems over time have a special name: dynamic models . In physics, dynamic information models describe the movement of bodies, in biology - the development of organisms or animal populations, in chemistry - the flow chemical reactions etc.
Statistical Forecasting Models
Statistics- the science of collecting, measuring and analyzing mass quantitative data.
There are medical statistics, economic statistics, social statistics and others. The mathematical apparatus of statistics is developed by a science called math statistics
.
| For example, most strong influence has an effect on bronchial-pulmonary diseases carbon monoxide- . With the goal of determining this relationship, medical statisticians collect data. The obtained data can be summarized in a table and also presented in the form of a scatter plot. How to build a mathematical model of this phenomenon? Obviously, you need to obtain a formula reflecting the dependence of the number of chronic patients P on the concentration of carbon monoxide C. In the language of mathematics, this is called the function of the dependence of P on C: P(C). The type of such a function is unknown; it should be sought by a selection method based on experimental data. |  |  |
This implies the basic requirements for the required function:
it should be simple enough to be used in further calculations;
the graph of this function should pass near the experimental points so that the deviations of these points from the graph are minimal and uniform. In statistics, the resulting function is usually called regression model.
Method least squares
Obtaining a regression model occurs in two stages:
1) selection of the type of function;
2) calculation of function parameters.
The first task does not have strict decision.
Most often, the choice is made among the following functions:
y = ax + b - linear function (1st degree polynomial);
y = ax 2 + bx + c - quadratic function
y =a n x n + a (n-1) x n-1 +...+ a 2 x 2 + a 1 x + a 0 -nth degree polynomial;
y = a ln(x) + b - logarithmic function;
y = ae bx - exponential function;
y = ax b - power function.
After choosing one of the proposed functions, you need to select the parameters (a, b, c, etc.) so that the function is located as close as possible to the experimental points, using the parameter calculation method. This method was proposed in the 18th century German mathematician K. Gauss. It is called the method of least squares (OLS) and is very widely used in statistical data processing and is built into many mathematical software packages. It is important to understand the following: using the least squares method, any function can be constructed from a given set of experimental points. But whether it will satisfy us is a question of compliance criterion. For our example, consider three functions constructed by the least squares method.
These figures were obtained using a spreadsheet processor Microsoft Excel. The regression model graph is called trend.
English word"trend" can be translated as " general direction", or "trend".
Schedule linear function- this is a straight line. From this graph it is difficult to say anything about the nature of this growth. But quadratic and exponential trends plausible.
The graphs contain the value obtained as a result of building trends. It is designated as R2. In statistics this quantity is called coefficient of determinism. It is this that determines how successful the resulting regression model is. Coefficient of determinism always ranges from 0 to 1. The closer R2 is to 1, the better the regression model.
Of the three selected models, the R2 value is the smallest for the linear one. This means she is the most unfortunate. The R2 values of the other two models are quite close (the difference is less than 0.01). They are equally successful.
Forecasting using a regression model
Having obtained a regression mathematical model, it is possible to predict the process by calculations, i.e. estimate the level of asthma incidence not only for those values that were obtained by measurements, but also for other values.
If the forecast is made within experimental values, then it's called restoration of value
.
Prediction beyond experimental data is called extrapolation.
Having a regression model makes it easy to make predictions using spreadsheets.
In some cases, you need to be careful with extrapolation. The applicability of any regression model is limited, especially outside
experimental area. In our example, when extrapolating, one should not go far from the value of 5 mg/m 3. We don't know what will happen away from this area. Any extrapolation rests on a hypothesis: “let us assume that the pattern persists outside the experimental area.” What if it doesn’t save?
For example, the quadratic model in our example at a concentration close to 0 will produce 150 sick people, i.e. more than at 5 mg/m 3 . Obviously this is nonsense. In the region of small values of C, the exponential model works better. By the way, that's pretty typical situation: different areas different models may fit the data better.
Modeling correlation dependencies
Let an important characteristic of some complex system is factor A. It can be influenced simultaneously by many other factors: B, C, D, etc.
Dependencies between quantities, each of which is subject to completely uncontrolled scatter, are called correlation dependencies.
Chapter mathematical statistics, which explores such dependencies is called correlation analysis. Correlation analysis studies the average law of behavior of each quantity depending on the values of another quantity, as well as the measure of such dependence.
The assessment of the correlation of values begins with a hypothesis about the possible nature of the relationship between their values. Most often it is assumed that there is linear dependence. In this case, the measure of correlation dependence is a quantity called correlation coefficient.
correlation coefficient (usually denoted Greek letter ρ ) is a number from the range from -1 to +1;
Ifρ modulus is close to 1, then there is a strong correlation, if close to 0, then it is weak;
closenessρ to +1 means that an increase in the values of one set corresponds to an increase in the values of another set, close to -1 means that an increase in the values of one set corresponds to a decrease in the values of another set;
meaningρ easy to find with using Excel, since the corresponding formulas are built into this program.
| As an example of a complex system, consider a school. Let the school's business expenses be expressed by the number of rubles per number of students in the school (rub/person) spent over a certain period of time (for example, over the last 5 years). Let academic performance be assessed by the average score of school students based on the results of their last school year. The results of data collection for 20 schools, entered into a spreadsheet andscatter plotare presented in the figures. Values for both quantities: financial costs and student performance - have a significant scatter and, at first glance, the relationship between them is not visible. However, it may well exist. IN Excel function calculating the correlation coefficient is called CORREL and is included in the group of statistical functions. We'll show you how to use it. On the same Excel sheet where the table is located, you need to place the cursor on any free cell and run the CORREL function. It will ask for two ranges of values. We indicate, respectively, B2:B21 and C2:C21. After entering them, the answer will be displayed: p = 0.500273843. This value indicates an average level of correlation. Now let’s consider which of the 2 parameters: the availability of textbooks or computers is correlated in to a greater extent, i.e. has a greater impact on academic performance BelowThe figure shows the results of measuring both factors in 11 different schools. For both dependencies the coefficients were obtained linear correlation. As can be seen from the table, the correlation between the provision of textbooks and academic performance is stronger than the correlation between computer provision and academic performance (although both correlation coefficients are not very large). From this we can conclude that the book still remains a more significant source of knowledge than the computer. |  |
 |  |
MODELING DEPENDENCIES BETWEEN VARIABLES
INFORMATION MODELING TECHNOLOGIES
- Magnitude
- Characteristics of the quantity: name, type, value
- Functional and other types of dependencies
- Mathematical models
- Dynamic models
Key Concepts
Application mathematical modeling
The use of mathematical modeling constantly requires taking into account the dependencies of some quantities on others.
Examples of dependencies:
- the time a body falls to the ground depends on its initial height;
- the gas pressure in the cylinder depends on its temperature;
- morbidity rate of city residents bronchial asthma depends on concentration harmful impurities in the city air.
Implementation mathematical model requires knowledge of techniques for representing dependencies between quantities.
Dependency representation methods
Magnitude – quantitative characteristic object under study
Quantity characteristics
reflects the meaning of the quantity
defines possible values quantities
Meaning
constant
variable
Main types of quantities:
An example of a constant is the Pythagorean number
The value name can be
semantic
semantic
numerical
"gas pressure"
In describing the process of a body falling variable quantities are height H and time of fall t
symbolic
symbolic
logical
Types of dependencies
Functional dependence is a relationship between two quantities in which a change in one of them causes a change in the other.
Example 1: t(c) – fall time; H(m) – fall height. We will represent the dependence, neglecting air resistance; the acceleration of free fall g (m/s 2) will be considered a constant.
Example 2: P(n/m 2) – gas pressure (in SI units, pressure is measured in newtons per square meter); t°C – gas temperature. Pressure at zero degrees P We will consider 0 a constant for a given gas.
certain .
Types of dependencies
Other addiction wears more complex nature, the same value can take different meanings, since it may be influenced by other indicators.
Example 3: Air pollution is characterized by the concentration of impurities – C (mg/m3). The unit of measurement is the mass of impurities contained in 1 cubic meter air, expressed in milligrams. The incidence rate will be characterized by the number of chronic asthma patients per 1000 inhabitants of this city P(bol/thousand)
The relationship between quantities is completely certain .
Mathematical models
Mathematical models - this is a set of quantitative characteristics of some object (process) and connections between them, presented in the language of mathematics.
Mathematical models reflect physical laws and are presented in the form of formulas:
Linear dependence
Root dependence (time is proportional to the square root of the height)
In complex problems, mathematical models are represented as equations or systems of equations.
Tabular and graphical models
Let's experimentally check the law of free fall of a body
Experiment: a steel ball is dropped from a 6-meter, 9-meter height, etc. (after 3 meters), measuring the height of the initial position of the ball and the time of fall
The result of the experiment is presented in the table and graph
N , m
t , c
Tabular and graphical representation dependence of the time of falling of a body on height
Dynamic models
Information models that describe the development of systems over time have a special name: dynamic models .
In physics this is the movement of bodies, in biology - the development of organisms or animal populations,
in chemistry – the occurrence of chemical reactions.
The most basic
- Quantity is a quantitative characteristic of the object under study.
- Size characteristics:
Name – reflects the meaning of the quantity
Type – defines possible values of quantities
Value: constant value (constant) or variable
- Name – reflects the meaning of the quantity Type – defines the possible values of the quantities Meaning: constant value (constant) or variable
- A functional dependence is a relationship between two quantities in which a change in one of them causes a change in the other.
- There are three ways to model quantities: functional (formula), tabular and graphical
- The formula is more versatile; Having a formula, you can easily create a table and plot a graph.
- Description of the development of systems over time - a dynamic model.
Questions and tasks
- What forms of representation of dependencies between quantities do you know?
- What is a mathematical model?
- Can a mathematical model include only constants?
- Give an example of something you know functional dependence(formulas) between the characteristics of an object or process.
- Justify the advantages and disadvantages of each of the three forms of dependency representation.
- Present a mathematical model of the dependence of gas pressure on temperature in the form of a tabular and graphical model, if it is known that at a temperature of 27 °C the gas pressure in a closed vessel was 75 kPa.
- Computer Science and ICT. A basic level of: textbook for grades 10-11 / I.G. Semakin, E.K. Henner. – 7th ed. – M.: Binom. Laboratory of Knowledge, 2011. – 246.: ill.
Illustrations:
Sources
- http://1.bp.blogspot.com/-u7m70qcqIdw/Ukh9R4Ga-9I/AAAAAAAAEkk/wIqkfCqOgGo/s1600/%25D0%2593%25D0%25B0%25D0%25BB%25D0%25B8%25D0%25BB%25D0%25B5% 25D0%25BE.gif
- http://ehsdailyadvisor.blr.com/wpcontent/uploads/2015/11/EHSDA_110615.jpg
- http://himki.blizhe.ru/userfiles/Image/MIL-GRAFIK/dop-photo/PRIMESI.JPG
- http://f.10-bal.ru/pars_docs/refs/12/11350/11350_html_mbb50c21.jpg
Preliminary preparation. Questions and tasks
When solving what information problems are they used?
spreadsheets?
a) How is data addressed in spreadsheet?
b) What types of data can be stored in ET cells?
c) What is the principle of relative addressing?
d) How can you undo the effect of relative addressing?
What is the purpose of diagrams?
How is the area for selecting data from a table to construct a chart and the order of selection determined? What quantities are plotted along the horizontal (OX) axis and vertical (OY) axis?
In what situations is it preferable to use: histograms; graphics; pie charts?
Information Modeling in production planning and management
Questions studied
The most common types of planning and control problems
Representation of dependencies between quantities
Statistics and statistical data
Least square method
Building regression models using a spreadsheet processor
Forecasting using a regression model
The concept of correlation dependencies. Calculation of correlation dependencies in a spreadsheet
Optimal planning. Using MS Excel to solve the optimal planning problem
The most common types of planning and control problems
In management and planning there is whole line typical tasks that can be delegated to a computer. The user of such software may not even know deeply the mathematics behind the apparatus used. He only needs to understand the essence of the problem being solved, prepare and enter initial data into the computer, and interpret the results obtained.
In this topic, we will consider three types of problems that specialists in the field of planning and management often have to solve:
1) forecasting- searching for answers to the questions “What will happen after some time?”, or “What will happen if...?”;
2) determining the influence of some factors on others- searching for an answer to the question “How strongly does factor B influence factor A?”, or “Which factor - B or C - influences factor A more strongly?”;
3) search for optimal solutions- searching for an answer to the question “How to plan production in order to achieve the optimal value of a certain indicator (for example, maximum profit, or minimum energy consumption)? "
Tool information technologies The spreadsheet we will use is MS Excel.
Representation of dependencies between quantities
Solving planning and management problems constantly requires taking into account the dependencies of some factors on others. Examples of dependencies:
- the time a body falls to the ground depends on the initial height;
- pressure depends on the temperature of the gas in the cylinder;
‒ the incidence of bronchial asthma among residents depends on the quality of urban air.
Let's look at various dependency representation methods.
Any research must begin by identifying the quantitative characteristics of the object (process, phenomenon) being studied. Such characteristics are called quantities.
Associated with any quantity three main properties: name, value, type.
The name of a quantity can be complete (emphasizing its meaning), or it can be symbolic. An example of a full name is "Gas Pressure"; and the symbolic name for the same value is P. In databases, values are record fields. For them, as a rule, they are used full names, for example: “Last name”, “Weight”, “Rating”, etc. In physics and other sciences that use mathematical apparatus, symbolic names are used to denote quantities.
If s meaning quantity does not change, it is called a constant quantity or constant. Example constants- Pythagorean number π=3.14159... A quantity that changes its value is called variable. For example, in describing the process of a body falling, the variable quantities are height (H) and falling time (t).
The third property of a quantity is its type. A type defines the set of values that a value can take. Basic types of values: numeric, symbolic, logical.
Now let’s return to examples 1-3 and denote (name) all the variable quantities, the dependencies between which will interest us. In addition to the names, we indicate the dimensions of the quantities. Dimensions define the units in which the values of quantities are represented.
1. t (sec) - fall time; N (m) - fall height. We will represent the dependence, neglecting air resistance. Gravity acceleration g (m/sec 2) - constant.
2. P (kg/m2) - gas pressure; t (C) - gas temperature. Pressure at zero degrees P o is considered a constant for a given gas.
3. We will characterize air pollution by the concentration of impurities - C (mg/cubic m). The unit of measurement is the mass of impurities contained in 1 cubic meter of air, expressed in milligrams. The incidence rate will be characterized by the number of chronic asthma patients per 1000 residents of a given city - P (patients/thousand).
If the relationship between quantities can be represented in mathematical form, then we have a mathematical model.
Mathematical model is a set of quantitative characteristics of some object (process) and connections between them, presented in the language of mathematics.
Mathematical models for the first two examples listed above are well known. They reflect physical laws, and are presented in the form of formulas:
These are examples of dependencies represented in functional form. The first dependence is called root (time is proportional to the square root of height), the second is linear (pressure is directly proportional to temperature).
In more complex problems, mathematical models are represented as equations or systems of equations. In this case, to extract the functional dependence of quantities, you need to be able to solve these equations. At the end of this chapter, we will consider an example of a mathematical model that is expressed by a system of inequalities.
Let's look at examples of two other ways to represent dependencies between quantities: tabular and graphical. Imagine that we decided to test the law of free fall of a body experimentally. The experiment was organized as follows: we throw a steel ball from the balcony of the 2nd floor, 3rd floor (and so on) of a ten-story building, measuring the height of the initial position of the ball and the time of fall. Based on the results of the experiment, we compiled a table and drew a graph.
Planned results of teaching mathematics in grades 5-6
Arithmetic
Understand the features decimal system calculus;
Use concepts related to the divisibility of natural numbers;
Express numbers in equivalent forms, choosing the most appropriate depending on the specific situation;
Compare and order rational numbers;
Perform calculations with rational numbers, combining oral and written calculation methods, use a calculator;
Use concepts and skills related to the proportionality of quantities and percentages when solving mathematical problems and tasks from related subjects, perform simple practical calculations;
Analyze graphs of relationships between quantities (distance, time, temperature, etc.).
Become familiar with positional number systems with bases other than 10;
Deepen and develop ideas about natural numbers and divisibility properties;
Learn to use techniques that rationalize calculations, acquire the skill of controlling calculations, choosing the method appropriate for the situation.
Upon completion of the course, the student will learn:
· Perform operations with numerical expressions;
· perform transformations of literal expressions (expanding parentheses, casting similar terms);
· decide linear equations, solve word problems using the algebraic method.
The student will have the opportunity to:
· develop ideas about literal expressions and their transformations;
· master special techniques for solving equations, apply the apparatus of equations to solve both textual and practical problems.
Geometric figures. Measurement of geometric quantities
Upon completion of the course, the student will learn:
Recognize flat and spatial geometric figures and their elements in drawings, drawings, models and in the surrounding world;
Construct angles, determine their degree measure;
Recognize and depict the development of a cube, rectangular parallelepiped, regular pyramid, cylinder and cone;
Determine the linear dimensions of the figure itself from the linear dimensions of the figure’s development and vice versa;
Calculate the volume of a rectangular parallelepiped and a cube.
The student will have the opportunity to:
Learn to calculate the volume of spatial geometric figures composed of rectangular parallelepipeds;
Deepen and develop ideas about spatial geometric shapes;
Learn to apply the concept of sweep to perform practical calculations.
Upon completion of the course, the student will learn:
Use the simplest methods of presenting and analyzing statistical data;
Solve combinatorial problems to find the number of objects or combinations.
The student will have the opportunity to:
Gain initial experience in organizing data collection during a survey. public opinion, carry out their analysis, present the survey results in the form of a table, diagram;
Learn some special techniques for solving combinatorial problems.
Arithmetic
Integers
A series of natural numbers. Decimal notation natural numbers. Rounding natural numbers.
Coordinate beam.
Comparison of natural numbers. Addition and subtraction of natural numbers. Properties of addition.
Multiplication and division of natural numbers. Properties of multiplication. Division with remainder. Power of a number with a natural exponent.
Divisors and multiples natural number. Largest common divisor. Least common multiple. Signs of divisibility by 2, by 3, by 5, by 9, by 10.
Simple and composite numbers. Factoring numbers into prime factors. „
Ordinary fractions. The main property of a fraction. Finding a fraction from a number. Finding a number by the value of its fraction. Correct and improper fractions. Mixed numbers.
Comparison of common fractions and mixed numbers. Arithmetic operations with ordinary fractions and mixed numbers.
Decimal fractions. Comparing and rounding decimals. Arithmetic operations with decimals. Estimates of the calculation results. Representation of a decimal fraction as common fraction and ordinary in the form of a decimal. Infinite periodic decimals. Decimal approximation of a common fraction.
Attitude. Percentage two numbers. Dividing a number in this ratio. Scale.
Proportion. The main property of proportion. Direct and inverse proportional relationships. Interest. Finding percentages of a number. Finding a number by its percentage.
Solution word problems arithmetic ways.
Rational numbers
Positive, negative numbers and the number 0.
Opposite numbers. The absolute value of a number.
Whole numbers. Rational numbers. Comparison of rational numbers. Arithmetic operations with rational numbers. Properties of addition and multiplication of rational numbers.
Coordinate line. Coordinate plane.
Quantities. Dependencies between quantities
Units of length, area, volume, mass, time, speed.
Examples of dependencies between quantities. Representation of dependencies in the form of formulas. Calculations using formulas.
Numeric and letter expressions. Equations
Numeric expressions. The value of a numeric expression. Procedure in numerical expressions. Literal expressions. Expanding parentheses. Similar terms, reduction of similar terms. Formulas.
Equations. Root of the equation. Basic properties of equations. Solving word problems using equations.
Elements of statistics, probability. Combinatorial problems
Presentation of data in the form of tables, circular and bar charts, graphs.
Average. Average value of the quantity.
Random event. Reliable and impossible events. Probability random event. Solving combinatorial problems.
Geometric figures. Measurements of geometric quantities
Line segment. Construction of a segment. Length of the segment, broken line. Measuring the length of a segment, constructing a segment given length. Perimeter of a polygon. Plane. Straight. Ray.
Corner. Types of angles. Degree measure corner. Measuring and constructing angles using a protractor.
Rectangle. Square. Triangle. Types of triangles. Circle and circle. Circumference. Number.
Equality of figures. The concept and properties of area. Area of a rectangle and square. Area of a circle. Axis of symmetry of a figure.
Visual representations O spatial figures: rectangular parallelepiped, cube, pyramid, cylinder, cone, ball, sphere. Examples of developments of polyhedra, cylinder, cone. The concept and properties of volume. Volume of a rectangular parallelepiped and a cube.
Mutual arrangement two straight lines. Perpendicular lines. Parallel lines.
Axial and central symmetries.
>>Informatics: Representation of dependencies between quantities
Representation of dependencies between quantities
Solving planning and management problems constantly requires taking into account the dependencies of some factors on others.
Examples of dependencies:
1) the time a body falls to the ground depends on the initial height;
2) the pressure depends on the temperature of the gas in the cylinder;
Mathematical model is a set of quantitative characteristics of some object (process) and connections between them, presented in the language of mathematics.
Mathematical models for the first two examples listed above are well known. They reflect physical laws and are presented in the form of formulas:
These are examples of dependencies represented in a sawtooth function. The first dependence is called root (time is proportional to the square root of height), the second is linear (pressure is directly proportional to temperature).
In more complex problems, mathematical models are represented as equations or systems of equations. In this case, to extract the functional dependence of quantities, you need to be able to solve these equations. At the end of this chapter, we will consider an example of a mathematical model that is expressed by a system of inequalities.
Let's look at examples of two other ways to present dependencies between quantities: tabular and graphical.
Imagine that we decided to test the law of free fall of a body experimentally. The experiment was organized as follows; we throw a steel ball from the balcony of the 2nd floor, 3rd floor (and so on) of a ten-story building, measuring the height of the initial position of the ball and the time of fall. Based on the results of the experiment, we compiled a table and drew a graph.
"
Rice. 2.11. Tabular and graphical representation of the dependence of the time of falling of a body on height
If each pair of values of H and t from this table is substituted into the above formula for the dependence of height on time, then it will turn into an equality (to within the measurement error). This means the model works well. (However, if you throw not a steel ball, but a large light ball, then this model will correspond less to the formula, and if it is an inflatable ball, it will not correspond at all - why do you think?)
In this example, we looked at three ways to display the dependence of quantities: functional (formula), tabular and graphical. However, only a formula can be called a mathematical model of the process of a body falling to the ground. Why? Because the formula is universal. It allows you to determine the time of a body falling from any height, and not just for the experimental set of H values shown in Fig. 2.11.
In addition, the table and diagram(graph) state the facts, and the mathematical model makes it possible to forecast, predict through calculations.
In the same way, you can display the dependence of pressure on temperature in three ways. Both examples are related to known physical laws - the laws of nature. Knowledge physical laws allow to produce accurate calculations, they form the basis of modern technology.
Briefly about the main thing
Magnitude is some quantitative characteristic of an object.
Dependencies between quantities can be presented in the form of a mathematical model, in tabular and graphical forms.
The relationship, presented in the form of a formula, is a mathematical model.
Questions and tasks
1. a) What forms of representation of dependencies between quantities do you know?
b) What is a mathematical model?
c) Can a mathematical model include only constants?
2. Give an example of a functional relationship (formula) known to you between the characteristics of a certain system.
3. Justify the advantages and disadvantages of each of the three forms of representing dependencies.
Semakin I.G., Henner E.K., Computer Science and ICT, 11
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