The project method, which has enormous potential for the formation of non-versatile learning activities, is becoming more and more widespread in the school education system. But it is rather difficult to "fit" the project method into the classroom-lesson system. I include mini exploration in a regular lesson. This form of work opens up great opportunities for the formation cognitive activities and provides accounting individual characteristics students, sets the stage for skill development on large projects.
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"If a student at school has not learned to create anything himself, then in life he will only imitate, copy, since there are few who, having learned to copy, would be able to make an independent application of this information." Leo Tolstoy.
A characteristic feature of modern education is a sharp increase in the amount of information that students need to assimilate. A student's degree of development is measured and assessed by his ability to independently acquire new knowledge and use it in educational and practical activities. The modern pedagogical process requires the use of innovative technologies in teaching.
FSES of a new generation requires the use of activity-type technologies in the educational process, the methods of design and research activities are defined as one of the conditions for the implementation of the main educational program.
A special role is given to such activities in mathematics lessons, and this is not accidental. Mathematics is the key to understanding the world, the basis of scientific and technological progress and an important component of personality development. It is designed to educate in a person the ability to understand the meaning of the task assigned to him, the ability to reason logically, to learn the skills of algorithmic thinking.
It is difficult to fit the project method into the classroom system. I try to intelligently combine the traditional and the student-centered system by incorporating elements of research into a regular lesson. Here are some examples.
So when studying the topic "Circle" we conduct the following research with the students.
Mathematical research "Circle".
- Think about how to build a circle, what tools are needed for this. Circle designation.
- In order to define a circle, let's see what properties this geometric figure has. Connect the center of the circle to a point on the circle. Let's measure the length of this segment. Let's repeat the experiment three times. Let's make a conclusion.
- The segment connecting the center of the circle with any of its points is called the radius of the circle. This is the definition of the radius. Radius notation. Using this definition, construct a circle with a radius equal to 2cm5mm.
- Construct a circle of arbitrary radius. Plot a radius, measure it. Record your measurements. Build three more different radii. How many radii can you draw in a circle?
- Let us try, knowing the property of points on a circle, to define it.
- Construct a circle of arbitrary radius. Connect two points of the circle so that this segment passes through the center of the circle. This segment is called the diameter. Let us give a definition of the diameter. Diameter designation. Build three more diameters. How many diameters the circle has.
- Construct a circle of arbitrary radius. Measure the diameter and radius. Compare them. Repeat the experiment three more times with different circles. Make a conclusion.
- Connect any two points of the circle. The resulting segment is called a chord. Let us give a definition of a chord. Build three more chords. How many chords the circle has.
- Whether the radius is a chord. Prove it.
- Whether the diameter is chord. Prove it.
Research work may be of a propaedeutic nature. Having examined the circle, you can consider a number of interesting properties that students can formulate at the level of a hypothesis, and then prove this hypothesis. For example, the following study:
"Mathematical research"
- Construct a circle with a radius of 3 cm and draw its diameter. Connect the ends of the diameter to an arbitrary point on the circle and measure the angle formed by the chords. Carry out the same construction for two more circles. What do you notice.
- Repeat the experiment for a circle of arbitrary radius and formulate a hypothesis. Can it be considered proven with the help of the constructions and measurements carried out?
When studying the topic "Mutual arrangement of straight lines on a plane", mathematical research is carried out in groups.
Assignments for groups:
- group.
1.In one coordinate system, plot the graphs of the function
Y = 2x, y = 2x + 7, y = 2x + 3, y = 2x-4, y = 2x-6.
2. Answer the questions by filling out the table:
Mathematical methods are most widely used in systems research. In this case, the solution of practical problems by mathematical methods is sequentially carried out according to the following algorithm:
mathematical formulation of the problem (development of a mathematical model);
selection of a method for conducting research on the obtained mathematical model;
analysis of the obtained mathematical result.
Mathematical formulation of the problem usually represented in the form of numbers, geometric images, functions, systems of equations, etc. The description of an object (phenomenon) can be represented using continuous or discrete, deterministic or stochastic and other mathematical forms.
Mathematical model is a system of mathematical relationships (formulas, functions, equations, systems of equations) that describe certain aspects of the studied object, phenomenon, process or object (process) as a whole.
The first stage of mathematical modeling is the formulation of the problem, the definition of the object and objectives of the study, the setting of criteria (features) for the study of objects and their management. Incorrect or incomplete statement of the problem can negate the results of all subsequent stages.
The model is the result of a trade-off between two opposing goals:
the model should be detailed, take into account all real-life connections and factors and parameters involved in its work;
at the same time, the model should be simple enough so that acceptable solutions or results can be obtained within an acceptable time frame under certain resource constraints.
Modeling can be called approximate scientific research. And the degree of its accuracy depends on the researcher, his experience, goals, resources.
The assumptions made in developing the model are a consequence of the goals of the modeling and the capabilities (resources) of the researcher. They are determined by the requirements for the accuracy of the results and, like the model itself, are the result of a trade-off. After all, it is the assumptions that distinguish one model of the same process from another.
Usually, when developing a model, irrelevant factors are discarded (not taken into account). Constants in physical equations are considered constant. Sometimes some values are averaged that change in the process (for example, the air temperature can be considered constant over a certain period of time).
Model development process
This is a process of consistent (and possibly repeated) schematization or idealization of the phenomenon under study.
Adequacy of a model is its compliance with the real physical process (or object) that it represents.
To develop a model of a physical process, it is necessary to determine:
Sometimes an approach is used when a model of small completeness is used, which is of a probabilistic nature. Then, with the help of a computer, it is analyzed and refined.
Model check begins and takes place in the very process of its construction, when one or another relationship between its parameters is selected or established, the accepted assumptions are evaluated. However, after the formation of the model as a whole, it is necessary to analyze it from some general positions.
The mathematical basis of the model (ie, the mathematical description of physical relationships) must be consistent from the point of view of mathematics: functional dependencies must have the same tendencies of change as real processes; the equations must have a region of existence not less than the range in which the study is carried out; they should not have singular points or discontinuities if they are absent in the real process, etc. Equations should not distort the logic of the real process.
The model should adequately, that is, as accurately as possible, reflect reality. Adequacy is needed not in general, but in the considered range.
Discrepancies between the results of the analysis of the model and the real behavior of the object are inevitable, since the model is a reflection, not the object itself.
In fig. 3. a generalized representation is presented, which is used in the construction of mathematical models.
Rice. 3. Apparatus for constructing mathematical models
When using static methods, the most commonly used apparatus is algebra and differential equations with time-independent arguments.
In dynamic methods, differential equations are used in the same way; integral equations; partial differential equations; theory of automatic control; algebra.
The probabilistic methods use: probability theory; information theory; algebra; theory of random processes; theory of Markov processes; theory of automata; differential equations.
An important place in modeling is occupied by the question of the similarity of the model and the real object. Quantitative correspondences between individual parties to processes flowing in a real object and its model are characterized by scales.
In general, the similarity of processes in objects and the model is characterized by similarity criteria. The similarity criterion is a dimensionless set of parameters that characterize a given process. When conducting research, depending on the field of research, different criteria are applied. For example, in hydraulics, such a criterion is the Reynolds number (characterizes the fluidity of a fluid), in heat engineering, the Nussselt number (characterizes the heat transfer conditions), in mechanics, Newton's criterion, etc.
It is believed that if similar criteria for the model and the investigated object are equal, then the model is correct.
Another method of theoretical research adjoins the theory of similarity - dimensional analysis method, which is based on two provisions:
physical laws are expressed only by products of degrees of physical quantities, which can be positive, negative, whole and fractional; the dimensions of both parts of the equality expressing the physical dimension must be the same.
Mathematical Methods for Operations Research
regression analysis model software
Introduction
Description of the subject area and formulation of the research problem
Practical part
Conclusion
Bibliography
Introduction
In economics, almost any activity is based on a forecast. Already on the basis of the forecast, a plan of actions and measures is drawn up. Thus, we can say that the forecast of macroeconomic variables is a fundamental component of the plans of all subjects of economic activity. Forecasting can be carried out both on the basis of qualitative (expert) and using quantitative methods. The latter in themselves can do nothing without a qualitative analysis, just as expert assessments must be supported by sound calculations.
Now forecasts, even at the macroeconomic level, are of a scenario nature and are developed according to the principle: what if… , - and are often a preliminary stage and substantiation of large national economic programs. Macroeconomic forecasts are usually carried out with a lead time of one year. Modern practice the functioning of the economy requires short-term forecasts (six months, a month, a decade, a week). Designed for the tasks of providing advanced information to individual participants in the economy.
With the changes in the objects and tasks of forecasting, the list of forecasting methods has changed. Adaptive methods of short-term forecasting have been rapidly developed.
Modern economic forecasting requires a versatile specialization from developers, knowledge of various fields of science and practice. The tasks of the forecaster include possession of knowledge about the scientific (usually mathematical) forecasting apparatus, about theoretical foundations predictable process, information flows, software, interpretation of forecasting results.
The main function of the forecast is to substantiate the possible state of an object in the future or to determine alternative paths.
The importance of gasoline as the main type of fuel today is difficult to overestimate. And it is just as difficult to overestimate the impact of its price on the economy of any country. The nature of the development of the country's economy as a whole depends on the dynamics of fuel prices. An increase in gasoline prices causes an increase in prices for industrial goods, leads to an increase in inflationary costs in the economy and a decrease in the profitability of energy-intensive industries. The cost of petroleum products is one of the component parts prices of consumer goods, and transport costs affect the price structure of all consumer goods and services without exception.
Of particular importance is the question of the cost of gasoline in the developing Ukrainian economy, where any change in prices causes an immediate reaction in all its sectors. However, the influence of this factor is not limited only to the sphere of the economy; many political and social processes can also be attributed to the consequences of its fluctuations.
Thus, the study and forecasting of the dynamics of this indicator is of particular importance.
The purpose of this work is to predict fuel prices for the near future.
1. Description of the subject area and formulation of the research problem
The Ukrainian gasoline market can hardly be called constant or predictable. And there are many reasons for this, starting with the fact that the raw material for the production of fuel is oil, the prices and volume of production of which are determined not only by supply and demand in the domestic and foreign markets, but also by government policy, as well as by special agreements of manufacturing companies. Given the strong dependence of the Ukrainian economy, it is dependent on the export of steel and chemicals, and the prices for these products are constantly changing. And speaking of gasoline prices, one cannot fail to note their upward trend. Despite the restraining policy pursued by the state, it is their growth that is customary for the majority of consumers. Prices for petroleum products in Ukraine today change daily. They mainly depend on the price of oil on the world market ($ / barrel) and the level of the tax burden.
The study of gasoline prices is very relevant at the present time, since the prices of other goods and services depend on these prices.
This paper will consider the dependence of gasoline prices on time and factors such as:
ü oil prices, USD per barrel
ü official dollar exchange rate (NBU), hryvnia per US dollar
ü consumer price index
The price of petrol, which is a refined product, is directly related to the price of the specified natural resource and the volume of its production. The dollar exchange rate has a significant impact on the entire Ukrainian economy, in particular on the formation of prices in its domestic markets. The direct relationship of this parameter with gasoline prices directly depends on the US dollar exchange rate. The CPI reflects the general change in prices within the country, and since it is economically proven that the change in prices for some goods in the absolute majority of cases (in conditions of free competition) leads to an increase in the prices of other goods, it is reasonable to assume that the change in prices of goods across the country affects the studied indicator in operation.
Description of the mathematical apparatus used in the calculations
Regression analysis
Regression analysis is a method of modeling measured data and studying their properties. Data consists of pairs of values for the dependent variable (response variable) and the independent variable (explanatory variable). Regression model<#"19" src="doc_zip1.jpg" />... Regression analysis is the search for a function that describes this relationship. Regression can be represented as a sum of non-random and random components. where is the regression function, and is an additive random variable with zero expectation. The assumption about the nature of the distribution of this quantity is called the data generation hypothesis<#"8" src="doc_zip6.jpg" />has a Gaussian distribution<#"20" src="doc_zip7.jpg" />.
The problem of finding a regression model for several free variables is posed as follows. Sample set<#"24" src="doc_zip8.jpg" />values of free variables and the set of corresponding dependent variable values. These sets are designated as, a set of initial data.
A regression model is given - a parametric family of functions depending on parameters and free variables. It is required to find the most probable parameters:
The probability function depends on the hypothesis of data generation and is given by Bayesian inference<#"justify">Least square method
The least squares method is a method for finding the optimal parameters of linear regression, such that the sum of the squares of the errors (regression residuals) is minimal. The method consists in minimizing the Euclidean distance between two vectors - the vector of the reconstructed values of the dependent variable and the vector of the actual values of the dependent variable.
The task of the least squares method is to choose a vector to minimize the error. This error is the distance from vector to vector. The vector lies in the space of the columns of the matrix, since there is a linear combination of the columns of this matrix with the coefficients. Finding a solution by the least squares method is equivalent to finding a point that lies closest to and is in the column space of the matrix.
Thus, the vector must be a projection onto the column space and the residual vector must be orthogonal to this space. Orthogonality means that each vector in column space is a linear combination of columns with some coefficients, that is, it is a vector. For everyone in space, these vectors must be perpendicular to the residual:
Since this equality must be true for an arbitrary vector, then
The least squares solution to an inconsistent system consisting of equations with unknowns is the equation
which is called the normal equation. If the columns of a matrix are linearly independent, then the matrix is invertible and only decision
The projection of the vector onto the space of the columns of the matrix has the form
The matrix is called the matrix for projecting the vector onto the space of the columns of the matrix. This matrix has two main properties: it is idempotent, and it is symmetric,. The converse is also true: a matrix that has these two properties is a projection matrix onto its column space.
Let us have statistical data on the parameter y depending on x. We represent these data in the form
xx1 X2 …..Xi…..Xny *y 1*y 2*...... y i * … ..Y n *
The least squares method allows for a given type of dependence y = ?(x) choose its numerical parameters so that the curve y = ?(x) best displayed experimental data for a given criterion. Consider the rationale from the point of view of probability theory for the mathematical definition of the parameters included in? (x).
Suppose that the true dependence of y on x is exactly expressed by the formula y = ?(x). The experimental points presented in Table 2 deviate from this dependence due to measurement errors. Measurement errors obey the normal law by Lyapunov's theorem. Consider some value of the argument x i ... The result of the experiment is a random variable y i normally distributed with mathematical expectation ?(x i ) and with a mean square deviation ?i characterizing the measurement error. Let the measurement accuracy at all points x = (x 1, X 2, …, X n ) is the same, i.e. ?1=?2=…=?n =?. Then the normal distribution law Yi looks like:
As a result of a series of measurements, the following event occurred: random variables (y 1*, y 2*,…, Yn *).
Description of the selected software product
Mathcad - a computer algebra system from the class of systems computer-aided design <#"justify">4. Practical part
The research objective is to forecast gasoline prices. The initial information is a 36-week time series - from May 2012 to December 2012.
These statistics (36 weeks) are presented in the Y matrix. Next, we create the H matrix, which will be needed to find the vector A.
Let's present the initial data and values calculated using the model:
To assess the quality of the model, we use the coefficient of determination.
First, let's find the average value of Xs:
The part of the variance that is due to regression in the total variance of the indicator Y characterizes the coefficient of determination R2.
The coefficient of determination takes values from -1 to +1. The closer its value of the coefficient in modulus is to 1, the closer the relationship of the effective indicator Y with the investigated factors X.
The value of the coefficient of determination serves as an important criterion for assessing the quality of linear and nonlinear models. The greater the proportion of the explained variation, the less the role of other factors, which means that the regression model approximates the initial data well and such a regression model can be used to predict the values of the effective indicator. We received the coefficient of determination R2 = 0.78, therefore, the regression equation explains 78% of the variance of the effective trait, and other factors account for 22% of its variance (i.e., residual variance).
Therefore, we conclude that the model is adequate.
Based on the data obtained, it is possible to make a forecast of fuel prices for the 37th week of 2013. The formula for the calculation is as follows:
Calculated forecast using this model: the price of gasoline is equal to UAH 10.434.
Conclusion
This paper has shown the possibility of conducting regression analysis to predict gasoline prices for future periods. The purpose term paper were consolidating knowledge on the course "Mathematical Methods of Operations Research" and obtaining development skills software, allowing to automate the research of operations in a given subject area.
The forecast for the future gasoline price, of course, is not unambiguous, which is due to the peculiarities of the initial data and the developed models. However, based on the information received, it is reasonable to assume that in the near future gasoline prices, of course, will not decrease, but, most likely, they will remain at the same level or will grow slightly. Of course, factors related to consumer expectations, customs policy and many other factors are not taken into account here, but I would like to note that they are largely mutually redeemable ... And it would be quite reasonable to note that a sharp jump in gasoline prices at the moment is really extremely doubtful, which, first of all, is connected with the policy pursued by the government.
Bibliography
1.Bühl A., Zöfel P. SPSS: the art of information processing. Analysis of statistical data and restoration of hidden patterns. - SPb .: OOO "DiaSoftUP", 2001. - 608 p.
2. Internet resources http://www.ukrstat.gov.ua/
3. Internet resources http://index.minfin.com.ua/
Internet resources http://fx-commodities.ru/category/oil/
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FEDERAL EDUCATION AGENCY
State educational institution higher professional education "Ural State University named after "
History department
Department of Documentation and Information Support of Management
Mathematical Methods in Scientific Research
Course program
Standard 350800 "Document Management and Documentation Management"
Standard 020800 "Historical and Archival Science"
Yekaterinburg
I approve
Vice-rector
(signature)
The program of the discipline "Mathematical Methods in Scientific Research" is drawn up in accordance with the requirements university component to the mandatory minimum content and level of training:
certified specialist by specialty
Document management and documentation support of management (350800),
Historical and archival studies (020800),
on the cycle "General humanitarian and socio-economic disciplines" of the state educational standard of the highest vocational education.
Semester III
According to the curriculum of specialty No. 000 - Documentation and documentation support of management:
The total complexity of the discipline: 100 hours,
including lectures 36 hours
According to the curriculum of the specialty No. 000 - Historical and archival studies
The total complexity of the discipline: 50 hours,
including lectures 36 hours
Control activities:
Control works 2 people / hour
Compiled by: Cand. ist. Sciences, Associate Professor of the Department of Documentation and information support management of the Ural State University
Department of Documentation and Information Management
dated 01.01.01, No. 1.
Agreed:
Deputy chairman
Humanitarian Council
_________________
(signature)
(C) Ural State University
(WITH) , 2006
INTRODUCTION
The course "Mathematical Methods in Socio-Economic Research" is intended to familiarize students with the basic techniques and methods of processing quantitative information, developed by statistics. Its main task is to expand the methodological scientific apparatus of researchers, to teach how to apply in practical and research activities, in addition to traditional methods, basic on logical analysis, mathematical methods that help to quantitatively characterize historical phenomena and facts.
Currently, the mathematical apparatus and mathematical methods are used in almost all areas of science. This is a natural process, it is often called the mathematization of science. In philosophy, mathematization is usually understood as the application of mathematics in various sciences. Mathematical methods have long and firmly entered the arsenal of research methods for scientists; they are used to generalize data, identify trends and patterns in the development of social phenomena and processes, typology and modeling.
Knowledge of statistics is necessary in order to correctly characterize and analyze the processes taking place in the economy and society. To do this, you need to be familiar with the sampling method, summary and grouping of data, be able to calculate average and relative values, indicators of variation, correlation coefficients. An element of information culture is the skills of correct design of tables and construction of graphs, which are an important tool for systematizing primary socio-economic data and visual presentation of quantitative information. To assess temporal changes, it is necessary to have an idea of the system of dynamic indicators.
The use of the sampling methodology allows you to study large amounts of information provided by mass sources, save time and labor, while obtaining scientifically significant results.
Mathematical and statistical methods occupy auxiliary positions, complementing and enriching the traditional methods of socio-economic analysis, their development is a necessary component of qualifications modern specialist- document expert, historian-archivist.
Currently, mathematical and statistical methods are actively used in marketing, sociological research, in the collection of operational management information, reporting and analysis of document flows.
Skills quantitative analysis are necessary for the preparation of qualification papers, abstracts and other research projects.
Experience in using mathematical methods shows that their use should be carried out in compliance with the following principles in order to obtain reliable and representative results:
1) the decisive role is played by the general methodology and theory of scientific knowledge;
2) a clear and correct setting research task;
3) selection of quantitatively and qualitatively representative socio-economic data;
4) the correctness of the application of mathematical methods, that is, they must correspond to the research task and the nature of the data being processed;
5) meaningful interpretation and analysis of the results obtained, as well as mandatory additional verification of the information obtained as a result of mathematical processing, is necessary.
Mathematical methods help to improve the technology of scientific research: to increase its efficiency; they give a great saving of time, especially when processing large amounts of information, allow revealing hidden information stored in the source.
In addition, mathematical methods are closely related to such a direction of scientific and information activities as the creation of historical data banks and archives of machine-readable data. The achievements of the era cannot be ignored, and information technology is becoming one of the critical factors development of all spheres of society.
COURSE PROGRAM
Topic 1. INTRODUCTION. MATHEMATIZATION OF HISTORICAL SCIENCE
Purpose and objectives of the course. The objective need to improve historical methods by using the techniques of mathematics.
Mathematization of science, the main content. Prerequisites for mathematization: natural science prerequisites; socio-technical prerequisites. The boundaries of the mathematization of science. Mathematization levels for natural sciences, engineering, economics and humanities. The main laws of mathematization of science: impossibility to fully cover the research areas of other sciences by means of mathematics; correspondence of the applied mathematical methods to the content of the mathematized science. The emergence and development of new applied mathematical disciplines.
The mathematization of historical science. The main stages and their features. Preconditions for the mathematization of historical science. The importance of developing statistical methods for the development of historical knowledge.
Socio-economic research using mathematical methods in the pre-revolutionary and Soviet historiography of the 20s (, etc.)
Mathematical and statistical methods in the works of historians of the 60-90s. The computerization of science and the spread of mathematical methods. Creation of databases and prospects for the development of information support for historical research. The most important results of the application of the methods of mathematics in socio-economic and historical-cultural studies (, etc.).
Correlation of mathematical methods with other methods historical research: historical-comparative, historical-typological, structural, systemic, historical-genetic methods. The basic methodological principles of the application of mathematical and statistical methods in historical research.
Topic 2. STATISTICAL INDICATORS
Basic techniques and methods of statistical study of social phenomena: statistical observation, the reliability of statistical data. The main forms of statistical observation, the purpose of observation, object and unit of observation. Statistical document as a historical source.
Statistical indicators (indicators of volume, level and ratio), its main functions. The quantitative and qualitative side of the statistical indicator. Varieties of statistical indicators (volumetric and qualitative; individual and generalizing; interval and momentary).
Basic requirements for the calculation of statistical indicators, ensuring their reliability.
The relationship of statistical indicators. Scorecard. Generalizing indicators.
Absolute values, definition. Types of absolute statistical quantities, their meaning and methods of obtaining. Absolute values as a direct result of the summary of statistical observation data.
Units of measurement, their choice depending on the essence of the studied phenomenon. Natural, value and labor units of measurement.
Relative values. The main content of the relative indicator, the form of their expression (coefficient, percentage, ppm, decimille). Dependence of the form and content of the relative indicator.
Comparison base, choice of base when calculating relative values. Basic principles for calculating relative indicators, ensuring the comparability and reliability of absolute indicators (by territory, circle of objects, etc.).
The relative values of structure, dynamics, comparison, coordination and intensity. Methods for calculating them.
The relationship between absolute and relative values. The need for their complex application.
Topic 3. DATA GROUPING. TABLES.
Summary indicators and grouping in historical research. The tasks solved by these methods in scientific research: systematization, generalization, analysis, ease of perception. Statistical population, observation units.
Objectives and main content of the summary. Summary is the second stage of statistical research. Varieties of summary indicators (simple, auxiliary). The main stages of calculating summary indicators.
Grouping is the main method for processing quantitative data. The tasks of the grouping and their importance in scientific research. Types of groupings. The role of groupings in the analysis of social phenomena and processes.
The main stages of building a grouping: definition of the studied population; selection of a grouping attribute (quantitative and qualitative attributes; alternative and non-alternative; factorial and effective); distribution of the population into groups depending on the type of grouping (determination of the number of groups and the size of the intervals), the scale of measurement of signs (nominal, ordinal, interval); selection of the form of presentation of grouped data (text, table, graph).
Typological grouping, definition, main tasks, principles of construction. The role of typological grouping in the study of socio-economic types.
Structural grouping, definition, main tasks, principles of construction. The role of structural grouping in studying the structure of social phenomena
Analytical (factor) grouping, definition, main tasks, principles of construction, The role of analytical grouping in the analysis of the interrelationships of social phenomena. The need for complex use and study of groupings for the analysis of social phenomena.
General requirements for the construction and design of tables. Development of the layout of the table. Table details (numbering, heading, names of columns and lines, conventions, designation of numbers). The method of filling in the information in the table.
Topic 4. GRAPHIC METHODS FOR ANALYSIS OF SOCIO-ECONOMIC
INFORMATION
The role of graphs and graphic image in scientific research. Tasks of graphical methods: ensuring the visibility of the perception of quantitative data; analytical tasks; characteristic of the properties of signs.
Statistical graph, definition. The main elements of the graph: graph field, graphical image, spatial reference points, scale reference points, graph explication.
Types of statistical graphs: line diagram, features of its construction, graphic images; bar chart (histogram), defining the rule for constructing histograms in the case of equal and unequal intervals; pie chart, definition, construction methods.
Feature distribution polygon. Normal distribution of a feature and its graphical representation. Features of the distribution of signs that characterize social phenomena: skewed, asymmetric, moderately asymmetric distribution.
Linear dependence between signs, features of a graphic representation of a linear relationship. Features of the linear relationship with the characteristic social phenomena and processes.
Time series trend concept. Revealing the trend using graphical methods.
Topic 5. AVERAGE VALUES
Averages in scientific research and statistics, their essence and definition. Basic properties of average values as generalizing characteristics. Relationship between the method of means and groupings. General and group averages. Conditions for the typicality of averages. The main research tasks that averages solve.
Methods for calculating averages. The arithmetic mean is simple, weighted. Basic properties of the arithmetic mean. Features of calculating the average for discrete and interval distribution series. Dependence of the method for calculating the arithmetic mean, depending on the nature of the original data. Features of the interpretation of the arithmetic mean.
Median - the average indicator of the structure of the population, definition, basic properties. Determination of the median for a ranked quantitative series. Calculation of the median for the indicator represented by the interval grouping.
Fashion is an average indicator of the structure of the population, basic properties and content. Determination of the mode for discrete and interval series. Features of the historical interpretation of fashion.
The relationship of the arithmetic mean, median and mode, the need for their complex use, checking the typicality of the arithmetic mean.
Topic 6. INDICATORS OF VARIATION
The study of the variability (variability) of the values of the attribute. The main content of measures of scattering of a trait, and their use of research activities.
Absolute and average indicators of variation. Variational range, main content, methods of calculation. Average linear deviation. Mean square deviation, basic content, calculation methods for discrete and interval quantitative series. The concept of variance of a feature.
Relative indicators of variation. Oscillation coefficient, main content, calculation methods. Coefficient of variation, the main content of the calculation methods. The value and specificity of the application of each indicator of variation in the study of socio-economic characteristics and phenomena.
Topic 7.
The study of changes in social phenomena over time is one of the most important tasks of socio-economic analysis.
Dynamic series concept. Moment and interval time series. Requirements for constructing time series. Comparability in the series of dynamics.
Indicators of changes in the series of dynamics. The main content of the indicators of the series of dynamics. Row level. Basic and chain indicators. Absolute increase in the level of dynamics, basic and chain absolute increments, methods of calculation.
Growth rate indicators. Baseline and chain growth rates. Features of their interpretation. Growth rate indicators, main content, methods of calculating basic and chain growth rates.
The average level of a number of dynamics, the main content. Techniques for calculating the arithmetic mean for moment series with equal and unequal intervals and for interval series with equal intervals. Average absolute growth. Average growth rate. Average growth rate.
Comprehensive analysis of interconnected series of dynamics. Revealing the general tendency of development - the trend: the method of the moving average, the enlargement of intervals, analytical methods of processing the series of dynamics. The concept of interpolation and extrapolation of series of dynamics.
Topic 8.
The need to identify and explain the relationship for the study of socio-economic phenomena. Types and forms of relationships studied by statistical methods. The concept of functional and correlation communication. The main content of the correlation method and the tasks solved with its help in scientific research. The main stages of correlation analysis. Peculiarities of interpretation of correlation coefficients.
Linear correlation coefficient, feature properties for which the linear correlation coefficient can be calculated. Methods for calculating the linear correlation coefficient for grouped and ungrouped data. Regression coefficient, main content, calculation methods, interpretation features. The coefficient of determination and its meaningful interpretation.
The limits of application of the main varieties of correlation coefficients, depending on the content and form of presentation of the initial data. Correlation coefficient. Coefficient rank correlation... Association and contingency coefficients for alternative qualitative features. Approximate methods for determining the relationship between features: Fechner coefficient. Autocorrelation coefficient. Information coefficients.
Methods for ordering the correlation coefficients: correlation matrix, the method of the Pleiades.
Multivariate statistical analysis methods: factor analysis, component analysis, regression analysis, cluster analysis... Prospects for modeling historical processes for the study of social phenomena.
Topic 9. SELECTED STUDY
Reasons and conditions for conducting a sample study. The need for historians to use methods of partial study of social objects.
The main types of partial surveys: monographic, main body method, sample research.
Definition of the sampling method, the basic properties of the sampling. Sampling representativeness and sampling error.
Stages of sampling research. Determination of the sample size, basic techniques and methods for finding the sample size (mathematical methods, table large numbers). The practice of determining the sample size in statistics and sociology.
Methods of forming a sample population: proper random sampling, mechanical sampling, typical and nested sampling. Methodology for organizing sample censuses of the population, budget surveys of families of workers and peasants.
Methodology for proving the representativeness of the sample. Random, systematic sampling and observation errors. The role of traditional methods in determining the reliability of sample results. Mathematical methods for calculating sampling error. Dependence of the error on the size and type of sample.
Features of the interpretation of the results of the sample and the distribution of indicators of the sample population to the general population.
Natural selection, main content, formation features. The problem of representativeness of the natural sample. The main stages of proving the representativeness of a natural sample: the use of traditional and formal methods. The method of the criterion of signs, the method of series - as methods of proving the property of randomness of the sample.
Small sample concept. The basic principles of using it in scientific research
Topic 11. METHODS OF FORMALIZATION OF DATA OF MASS SOURCES
The need to formalize information from mass sources to obtain hidden information. The problem of measuring information. Quantitative and qualitative features. Scales for measuring quantitative and qualitative characteristics: nominal, ordinal, interval. The main stages of measuring source information.
Types of mass sources, features of their measurement. Methodology for constructing a unified questionnaire based on the materials of a structured, semi-structured historical source.
Features of information measurement of an unstructured narrative source. Content analysis, its content and prospects for use. Types of content analysis. Content analysis in sociological and historical research.
Interrelation of mathematical and statistical methods of information processing and methods of formalization of source information. Computerization of research. Databases and data banks. Database technology in socio-economic research.
Self-study assignments
To consolidate the lecture material, students are offered assignments for independent work on the following course topics:
Relative indicators Average indicators Grouping method Graphical methods Indicators of dynamics
The execution of assignments is supervised by the teacher and is a prerequisite admission to offset.
Indicative list of questions for offset
1. Mathematization of science, essence, prerequisites, levels of mathematization
2. The main stages and features of the mathematization of historical science
3. Prerequisites for the use of mathematical methods in historical research
4. Statistical indicator, essence, functions, varieties
3. Methodological principles of using statistical indicators in historical research
6. Absolute values
7. Relative values, content, forms of expression, basic principles of calculation.
8. Types of relative values
9. Objectives and main content of the data summary
10. Grouping, main content and objectives of the study
11. The main stages of building a grouping
12. The concept of a grouping attribute and its gradations
13. Types of grouping
14. Rules for the construction and design of tables
15. Time series, requirements for building a time series
16. Statistical graph, definition, structure, tasks to be solved
17. Types of statistical graphs
18. Polygon feature distribution. Normal distribution of the trait.
19. Linear relationship between features, methods for determining linearity.
20. The concept of the trend of the time series, how to determine it
21. Average values in scientific research, their essence and basic properties. Conditions for the typicality of averages.
22. Types of average indicators of the population. Interrelation of average indicators.
23. Statistical indicators of dynamics, general characteristics, types
24. Absolute indicators of changes in the series of dynamics
25. Relative indicators of changes in the series of dynamics (growth rates, growth rates)
26. Average indicators of the time series
27. Indicators of variation, main content and tasks to be solved, types
28. Types of discontinuous observation
29. Selective study, the main content and tasks to be solved
30. Custom and general population, the main properties of the sample
31. Stages of sampling research, general characteristics
32. Determination of the sample size
33. Methods of forming a sample
34. Sampling error and methods of its determination
35. Representativeness of the sample, factors affecting representativeness
36. Natural sampling, the problem of representativeness of natural sampling
37. The main stages of proving the representativeness of the natural sample
38. Correlation method, essence, main tasks. Peculiarities of Interpretation of Correlation Coefficients
39. Statistical observation as a method of collecting information, the main types of statistical observation.
40. Types of correlation coefficients, general characteristics
41. Coefficient of linear correlation
42. Coefficient of autocorrelation
43. Methods of formalizing historical sources: the method of a unified questionnaire
44. Methods of formalizing historical sources: the method of content analysis
III.Distribution of course hours by topic and type of work:
according to the curriculum of the specialty (No. 000 - documentation and management documentation)
Name
sections and topics
Auditory lessons
Independent work
including
Introduction. Mathematization of Science
Statistical indicators
Data grouping. Tables
Average values
Variation indicators
Statistical indicators of dynamics
Multivariate analysis methods. Correlation Coefficients
Selective study
Information formalization methods
Distribution of course hours by topic and type of work
according to the curriculum of the specialty No. 000 - historical and archival science
Name
sections and topics
Auditory lessons
Independent work
including
Practical (seminars, laboratory works)
Introduction. Mathematization of Science
Statistical indicators
Data grouping. Tables
Graphic methods for analyzing socio-economic information
Average values
Variation indicators
Statistical indicators of dynamics
Multivariate analysis methods. Correlation Coefficients
Selective study
Information formalization methods
IV. Final control form - offset
V. Educational and methodological support of the course
Slavko methods in historical research. Textbook. Yekaterinburg, 1995
Mazur Methods in Historical Research. Guidelines... Yekaterinburg, 1998
additional literature
Andersen T. Statistical analysis of time series. M., 1976.
Borodkin statistical analysis in historical research. M., 1986
Borodkin informatics: stages of development // New and recent history. 1996. № 1.
Tikhonov for the humanities. M., 1997
Garskova and data banks in historical research. Göttingen, 1994
Gerchuk methods in statistics. M., 1968
Druzhinin method and its application in socio-economic research. M., 1970
Jessen R. Methods of statistical surveys. M., 1985
Jeannie K. Average. M., 1970
Yuzbashev theory of statistics. M., 1995.
Rumyantsev theory of statistics. M., 1998
Shmoilova, the study of the main trend and relationship in the ranks of dynamics. Tomsk, 1985
Yates F. Selective method in censuses and surveys / per. from English ... M., 1976
Historical informatics. M., 1996.
Kovalchenko historical research. M., 1987
Computer in economic history... Barnaul, 1997
Circle of ideas: models and technologies of historical informatics. M., 1996
Circle of ideas: traditions and tendencies of historical informatics. M., 1997
Circle of ideas: macro - and micro approaches in historical informatics. M., 1998
Circle of Ideas: Historical Informatics on the Threshold of the 21st Century. Cheboksary, 1999
Circle of Ideas: Historical Informatics in information society... M., 2001
General theory of statistics: Textbook / ed. and. M., 1994.
Workshop on the theory of statistics: Textbook. manual. M., 2000
Eliseev statistics. M., 1990
Slavko-statistical methods in historical and research M., 1981
Slavko methods in the study of the history of the Soviet working class. M., 1991
Statistical Dictionary / ed. ... M., 1989
Theory of statistics: Textbook / ed. , M., 2000
Ursul Society. Introduction to Social Informatics. M., 1990
Schwartz G. Selective method / per. with him. ... M., 1978
In the history of mathematics, one can conditionally distinguish two main periods: elementary and modern mathematics. The borderline from which it is customary to count the era of new (sometimes they say - higher) mathematics was the 17th century - the century of the emergence of mathematical analysis. By the end of the 17th century. I. Newton, G. Leibniz and their predecessors created the apparatus of a new differential calculus and integral calculus, which forms the basis of mathematical analysis and even, perhaps, the mathematical basis of all modern natural science.
Mathematical analysis is a vast area of mathematics with a characteristic object of study (variable), a kind of research method (analysis by means of infinitesimal or by means of limit transitions), a defined system of basic concepts (function, limit, derivative, differential, integral, series) and constantly improving and a developing apparatus based on differential and integral calculus.
Let's try to give an idea of what kind of mathematical revolution took place in the 17th century, what characterizes the transition from elementary mathematics to the one that is now the subject of research in mathematical analysis and what explains its fundamental role in the entire modern system of theoretical and applied knowledge.
Imagine that before you is a beautifully executed color photograph of a stormy ocean wave running ashore: a mighty stooped back, a steep but slightly sunken chest, already tilted forward and ready to fall with a head torn by the wind with a gray mane. You stopped the moment, you managed to catch the wave, and now you can carefully study it in all its details without haste. The wave can be measured, and using the means of elementary mathematics, you will draw many important conclusions about this wave, and therefore all of its ocean sisters. But by stopping the wave, you have deprived it of movement and life. Its inception, development, run, the force with which it hits the shore - all this turned out to be out of your field of vision, because you do not yet have any language or mathematical apparatus suitable for describing and studying not static, but developing, dynamic processes, variables and their relationships.
"Mathematical analysis is no less comprehensive than nature itself: it defines all tangible relationships, measures times, spaces, forces, temperatures." J. Fourier
Movement, variables and their interconnections surround us everywhere. Various types of motion and their patterns constitute the main object of study of specific sciences: physics, geology, biology, sociology, etc. Therefore, an exact language and the corresponding mathematical methods for describing and studying variable quantities turned out to be necessary in all areas of knowledge to approximately the same extent as and arithmetic are necessary when describing quantitative relationships. So, mathematical analysis is the basis of the language and mathematical methods for describing variables and their relationships. Nowadays, without mathematical analysis, it is impossible not only to calculate space trajectories, work nuclear reactors, the running of the ocean wave and the laws of the development of the cyclone, but also economically manage production, the distribution of resources, the organization of technological processes, predict the course of chemical reactions or changes in the number of various species of animals and plants interconnected in nature, because all these are dynamic processes.
Elementary mathematics was mainly the mathematics of constants, it studied mainly the relationships between elements geometric shapes, arithmetic properties of numbers and algebraic equations... To some extent, her attitude to reality can be compared with a careful, even thorough and complete study of each fixed frame of a film that captures a changeable, developing living world in its movement, which, however, is not visible in a separate frame and which can be observed only by looking tape as a whole. But just as cinema is unthinkable without photography, so modern mathematics is impossible without that part of it, which we conventionally call elementary, without the ideas and achievements of many outstanding scientists, sometimes separated by tens of centuries.
Mathematics is one, and its "higher" part is connected with the "elementary" in about the same way as the next floor of a building under construction is connected with the previous one, and the width of the horizons that mathematics opens up to us in the world around us depends on which floor of this building we managed rise. Born in the 17th century. mathematical analysis opened up opportunities for scientific description, quantitative and qualitative study of variables and motion in the broad sense of the word.
What are the prerequisites for the emergence of mathematical analysis?
By the end of the 17th century. the following situation has developed. First, within the framework of mathematics itself, long years some important classes of similar problems have accumulated (for example, the problem of measuring the areas and volumes of non-standard figures, the problem of drawing tangents to curves) and methods for their solution have appeared in various special cases. Secondly, it turned out that these problems are closely related to the problems of describing arbitrary (not necessarily uniform) mechanical motion, and in particular with the calculation of its instantaneous characteristics (speed, acceleration at any time), as well as with finding the value of the distance traveled for movement occurring at a given variable speed. The solution of these problems was necessary for the development of physics, astronomy, and technology.
Finally, thirdly, to mid XVII v. the works of R. Descartes and P. Fermat laid the foundations analytical method coordinates (so-called analytical geometry), which made it possible to formulate geometrical and physical problems of heterogeneous origin in the general (analytical) language of numbers and numerical dependencies, or, as we now say, numerical functions.
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NIKOLAY NIKOLAEVICH LUZIN
NN Luzin - Soviet mathematician, founder of the Soviet school of function theory, academician (1929). Luzin was born in Tomsk, studied at the Tomsk gymnasium. The formalism of the gymnasium course in mathematics alienated the talented young man, and only a capable tutor was able to reveal to him the beauty and greatness of mathematical science. In 1901, Luzin entered the Mathematics Department of the Physics and Mathematics Faculty of Moscow University. From the first years of study, issues related to infinity fell into the circle of his interests. At the end of the XIX century. German scientist G. Cantor created the general theory of infinite sets, which has received numerous applications in the study of discontinuous functions. Luzin began to study this theory, but his studies were interrupted in 1905. revolutionary activity, I had to leave for France for a while. There he listened to lectures by the most prominent French mathematicians of the time. Upon his return to Russia, Luzin graduated from the university and was left to prepare for a professorship. Soon he again went to Paris, and then to Göttingen, where he became close with many scientists and wrote the first scientific works. The main problem that interested the scientist was the question of whether there can exist sets containing more elements than a set natural numbers, but less than the set of points of the segment (the problem of the continuum). For any infinite set that could be obtained from segments using the operations of joining and intersecting countable collections of sets, this hypothesis was fulfilled, and in order to solve the problem, it was necessary to find out what other ways of constructing the sets were. At the same time, Luzin studied the question of whether it is possible to represent any periodic function, even with infinitely many points of discontinuity, as a sum of a trigonometric series, i.e. the sum of an infinite set of harmonic vibrations. Luzin received a number of significant results on these issues and in 1915 defended his dissertation "Integral and Trigonometric Series", for which he was immediately awarded the degree of Doctor of Pure Mathematics, bypassing the intermediate master's degree that existed at that time. In 1917 Luzin became a professor at Moscow University. A talented teacher, he attracted the brightest students and young mathematicians. The Luzin school reached its peak in the first post-revolutionary years. Luzin's students formed a creative team, which they jokingly called "Lusitania". Many of them received first-class scientific results while still in college. For example, PS Aleksandrov and M. Ya. Suslin (1894-1919) discovered a new method for constructing sets, which served as the beginning of the development of a new direction - descriptive set theory. Research in this area, carried out by Luzin and his students, showed that the usual methods of set theory are not enough to solve many of the problems that have arisen in it. Luzin's scientific predictions were fully confirmed in the 60s. XX century Many of Luzin's students later became academicians and corresponding members of the USSR Academy of Sciences. Among them is P.S. Aleksandrov. A. N. Kolmogorov. M. A. Lavrent'ev, L. A. Lyusternik, D. E. Menshov, P. S. Novikov. L. G. Shnirelman and others. Contemporary Soviet and foreign mathematicians in their works develop the ideas of N. N. Luzin. |
The confluence of these circumstances and led to the fact that at the end of the XVII century. two scientists - I. Newton and G. Leibniz - independently of each other managed to create a mathematical apparatus for solving these problems, summarizing and generalizing individual results of their predecessors, including the ancient scientist Archimedes and contemporaries of Newton and Leibniz - B. Cavalieri, B. Pascal , D. Gregory, I. Barrow. This apparatus formed the basis of mathematical analysis - a new branch of mathematics that studies various developing processes, i.e. interconnections of variables, which in mathematics are called functional dependencies or, in other words, functions. By the way, the term "function" itself was required and naturally arose precisely in the 17th century, and by now it has acquired not only general mathematical, but also general scientific meaning.
Initial information about the basic concepts and the mathematical apparatus of analysis is given in the articles "Differential calculus" and "Integral calculus".
In conclusion, I would like to dwell on only one principle of mathematical abstraction, which is common for all mathematics and characteristic of analysis, and in this regard, explain in what form mathematical analysis studies variable quantities and what is the secret of such universality of its methods for studying all sorts of specific developing processes and their interrelationships. ...
Let's consider a few illustrative examples and analogies.
Sometimes we no longer realize that, for example, a mathematical relationship written not for apples, chairs or elephants, but in an abstract form abstracted from concrete objects, is an outstanding scientific achievement. This is a mathematical law that experience has shown is applicable to various specific objects. So, studying in mathematics general properties abstract, abstract numbers, we thereby study quantitative relationships the real world.
For example, it is known from a school mathematics course that, therefore, in a specific situation, you could say: “If two six-ton dump trucks are not allocated to me for transporting 12 tons of soil, then three four-ton dump trucks can be requested and the work will be done, and if only one four-ton is given, then she will have to make three flights. " So the abstract numbers and numerical patterns that are now familiar to us are associated with their specific manifestations and applications.
The laws of change of concrete variables and developing processes of nature are connected in approximately the same way with the abstract, abstract form-function in which they appear and are studied in mathematical analysis.
For example, an abstract relationship may reflect the dependence of a cinema's box office on the number of tickets sold, if 20 is 20 kopecks - the price of one ticket. But if we ride a bike on the highway, driving 20 km per hour, then the same ratio can be interpreted as the relationship between the time (hours) of our bike ride and the distance covered during this time (kilometers)., You can always argue that, for example, a change by several times leads to a proportional (i.e., by the same number of times) change in the value, and if, then the opposite conclusion is also true. This means, in particular, to double the box office of a cinema, you will have to attract twice as many viewers, and in order to ride a bicycle at the same speed twice the distance, you will have to ride twice as long.
Mathematics studies both the simplest dependence and other, much more complex dependences in a general, abstract form abstracted from a particular interpretation. The properties of a function or methods for studying these properties revealed in such a study will be in the nature of general mathematical techniques, conclusions, laws and conclusions applicable to each specific phenomenon in which the function studied in an abstract form occurs, regardless of which area of knowledge this phenomenon belongs to. ...
So, mathematical analysis as a branch of mathematics took shape at the end of the 17th century. The subject of study in mathematical analysis (as it is presented from modern positions) are functions, or, in other words, dependencies between variables.
With the emergence of mathematical analysis, mathematics became available to study and reflect the developing processes of the real world; variables and motion entered mathematics.