The document provides an overview of functions including definitions, examples, and properties. It defines a function as a relation that assigns each element in the domain to a single element in the range. Examples of functions expressed by formulas, numerically, graphically, and verbally are given. Properties like monotonicity, symmetry, evenness, and oddness are defined and illustrated with examples. The document aims to introduce the fundamental concepts of functions to readers.
This document is from a Calculus I class at New York University. It provides an overview of functions, including the definition of a function, different ways functions can be represented (formulas, tables, graphs, verbal descriptions), properties of functions like monotonicity and symmetry, and examples of determining domains and ranges of functions. It aims to help students understand functions and their representations as a foundation for calculus.
This document contains exercises related to limits and continuity. It provides examples of functions and asks the reader to (a) evaluate the functions at given x-values to determine apparent behavior, and (b) find the indicated limit. It also contains exercises asking the reader to find vertical asymptotes and sketch graphs of given functions.
The document discusses inverse functions and logarithms. It begins by introducing the concept of an inverse function using an example of bacteria population growth over time. It then defines inverse functions formally and discusses their key properties. The document explains that a function must be one-to-one to have an inverse function. It introduces the natural logarithm as the inverse of the exponential function with base e and discusses properties of logarithmic functions like logarithmic laws. Graphs of exponential, logarithmic and natural logarithmic functions are presented.
This document provides an overview of functions and function notation that will be used in Calculus. It defines a function as an equation where each input yields a single output. Examples demonstrate determining if equations are functions and evaluating functions using function notation. The key concepts of domain and range of a function are explained. The document concludes by finding the domains of various functions involving fractions, radicals, and inequalities.
- A function is a rule that maps each input to a unique output. Not every rule defines a valid function.
- For a rule to be a valid function, it must map each input to only one output. The domain is the set of valid inputs, and the range is the set of corresponding outputs.
- Functions can be represented graphically by plotting the input-output pairs. The graph of a valid function should only intersect the vertical line above each input once.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
The document provides an overview of functions including definitions, examples, and properties. It defines a function as a relation that assigns each element in the domain to a single element in the range. Examples of functions expressed by formulas, numerically, graphically, and verbally are given. Properties like monotonicity, symmetry, evenness, and oddness are defined and illustrated with examples. The document aims to introduce the fundamental concepts of functions to readers.
This document is from a Calculus I class at New York University. It provides an overview of functions, including the definition of a function, different ways functions can be represented (formulas, tables, graphs, verbal descriptions), properties of functions like monotonicity and symmetry, and examples of determining domains and ranges of functions. It aims to help students understand functions and their representations as a foundation for calculus.
This document contains exercises related to limits and continuity. It provides examples of functions and asks the reader to (a) evaluate the functions at given x-values to determine apparent behavior, and (b) find the indicated limit. It also contains exercises asking the reader to find vertical asymptotes and sketch graphs of given functions.
The document discusses inverse functions and logarithms. It begins by introducing the concept of an inverse function using an example of bacteria population growth over time. It then defines inverse functions formally and discusses their key properties. The document explains that a function must be one-to-one to have an inverse function. It introduces the natural logarithm as the inverse of the exponential function with base e and discusses properties of logarithmic functions like logarithmic laws. Graphs of exponential, logarithmic and natural logarithmic functions are presented.
This document provides an overview of functions and function notation that will be used in Calculus. It defines a function as an equation where each input yields a single output. Examples demonstrate determining if equations are functions and evaluating functions using function notation. The key concepts of domain and range of a function are explained. The document concludes by finding the domains of various functions involving fractions, radicals, and inequalities.
- A function is a rule that maps each input to a unique output. Not every rule defines a valid function.
- For a rule to be a valid function, it must map each input to only one output. The domain is the set of valid inputs, and the range is the set of corresponding outputs.
- Functions can be represented graphically by plotting the input-output pairs. The graph of a valid function should only intersect the vertical line above each input once.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
The document provides information about Lagrangian interpolation, including:
1. It introduces Lagrangian interpolation as a method to find the value of a function at a discrete point using a polynomial that passes through known data points.
2. It gives the formula for the Lagrangian interpolating polynomial and provides an example of using it to find the velocity of a rocket at a certain time.
3. It discusses using higher order polynomials for interpolation, providing another example that calculates velocity using quadratic and cubic polynomials.
The document contains exercises, hints, and solutions for analyzing algorithms from a textbook. It includes problems related to brute force algorithms, sorting algorithms like selection sort and bubble sort, and evaluating polynomials. The solutions analyze the time complexity of different algorithms, such as proving that a brute force polynomial evaluation algorithm is O(n^2) while a modified version is linear time. It also discusses whether sorting algorithms like selection sort and bubble sort preserve the original order of equal elements (i.e. whether they are stable).
X01 Supervised learning problem linear regression one feature theorieMarco Moldenhauer
1. The document describes supervised learning problems, specifically linear regression with one feature. It defines key concepts like the hypothesis function, cost function, and gradient descent algorithm.
2. A data set with one input feature and one output is defined. The goal is to learn a linear function that maps the input to the output to best fit the training data.
3. The hypothesis function is defined as h(x) = θ0 + θ1x, where θ0 and θ1 are parameters to be estimated. Gradient descent is used to minimize the cost function and find the optimal θ values.
Linear and discrete systems analysis Jntu Model Paper{Www.Studentyogi.Com}guest3f9c6b
This document contains information about a linear and discrete systems analysis exam, including 8 questions on topics such as:
- Writing state equations and finding responses for electrical circuits
- Determining Fourier series expansions and transforms
- Analyzing RLC circuits and finding voltages/currents over time
- Checking if polynomials are Hurwitz and determining positive real functions
- Synthesizing impedance functions in Foster and Cauer forms
- Analyzing sampling of bandpass signals and power density spectra
- Finding fundamental periods and energies of sequences
Students were asked to answer 5 out of the 8 questions.
This document contains a final exam with 7 multiple part questions testing concepts in statistics, probability, and signal processing. The exam allows standard calculators and requires showing all work. Questions cover topics like distributions, confidence intervals, Monte Carlo integration, random processes, and sampling. Students are advised that reasonable assumptions can be made if they have difficulties with a part of a question. The exam is worth a total of 80 marks plus 4 bonus marks.
The document discusses combinational logic circuits and Boolean algebra. It contains 17 questions covering topics such as:
1. The three basic logic operations - AND, OR, NOT - and their truth tables and logic diagrams.
2. Boolean algebra, functions, truth tables, and logic diagrams.
3. Identities of Boolean algebra, dual expressions, and the consensus theorem.
4. Minimizing logic functions using Boolean algebra identities and Karnaugh maps.
5. Minterms, maxterms, and representing logic functions with sums of products and products of sums.
6. Demorgan's theorem and complementing logic functions.
7. Simplifying logic functions through
Integral Calculus. - Differential Calculus - Integration as an Inverse Process of Differentiation - Methods of Integration - Integration using trigonometric identities - Integrals of Some Particular Functions - rational function - partial fraction - Integration by partial fractions - standard integrals - First and second fundamental theorem of integral calculus
The document summarizes key concepts about functions and lines. It defines a function as a rule that assigns unique output values to inputs. Functions can be represented graphically by plotting the points (x, f(x)). Lines on a plane can be defined by two points and have a slope that represents the rise over run. The slope formula is used to find the equation of a line in y=mx+b form, where m is the slope and b is the y-intercept. Perpendicular and parallel lines have specific relationships between their slopes.
This document discusses composite functions and the order of operations when combining functions.
It provides an example of a mother converting the temperature of her baby's bath water from Celsius to Fahrenheit using two separate functions. The first function converts the Celsius reading to Fahrenheit, and the second maps the Fahrenheit reading to whether the water is too cold, alright, or too hot. Together these functions form a composite function.
Algebraically, a composite function f∘g(x) is defined as applying the inner function f first to the input x, and then applying the outer function g to the output of f. The domain of the inner function must be contained within the range of the outer function. The order of
This document discusses deep generative models including variational autoencoders (VAEs) and generational adversarial networks (GANs). It explains that generative models learn the distribution of input data and can generate new samples from that distribution. VAEs use variational inference to learn a latent space and generate new data by varying the latent variables. The document outlines the key concepts of VAEs including the evidence lower bound objective used for training and how it maximizes the likelihood of the data.
The document discusses functions and their graphical representations. It defines key terms like domain, range, and one-to-one and many-to-one mappings. It then focuses on quadratic functions, showing that their graphs take characteristic U-shaped or inverted U-shaped forms. The document also examines inequalities involving quadratic expressions and how to determine the range of values satisfying such inequalities by analyzing the graph of the quadratic function.
This document provides an overview of reduced-order models and emulators. It discusses two main approaches: polynomial chaos expansions (PCE) and Gaussian process (GP) emulators. PCE approximates quantities of interest using orthogonal polynomials and provides error bounds, while GP emulators approximate computer model outputs as realizations of random processes. Both rely on designs of experiments, with Latin hypercube designs commonly used. The document compares the approaches and discusses pros and cons, noting they are complementary rather than competing methods. It concludes by emphasizing the importance of accounting for model discrepancy in surrogate modeling.
This document discusses different methods for interpolation and approximation, including polynomial interpolation, spline interpolation, and parametric interpolation. Polynomial interpolation finds an interpolating polynomial that passes through discrete data points. It can be done using Lagrange polynomials or by solving a Vandermonde system of equations. Spline interpolation fits piecewise polynomials over intervals defined by interpolation nodes, ensuring smoothness at interval boundaries. Parametric interpolation treats variables equally by interpolating as functions of a parameter.
Handling missing data with expectation maximization algorithmLoc Nguyen
Expectation maximization (EM) algorithm is a powerful mathematical tool for estimating parameter of statistical models in case of incomplete data or hidden data. EM assumes that there is a relationship between hidden data and observed data, which can be a joint distribution or a mapping function. Therefore, this implies another implicit relationship between parameter estimation and data imputation. If missing data which contains missing values is considered as hidden data, it is very natural to handle missing data by EM algorithm. Handling missing data is not a new research but this report focuses on the theoretical base with detailed mathematical proofs for fulfilling missing values with EM. Besides, multinormal distribution and multinomial distribution are the two sample statistical models which are concerned to hold missing values.
This document discusses solving problems related to quantum mechanics and waves. It provides solutions to several problems involving waves on drum membranes, classical wave equations, particles in infinite and finite boxes, and the time evolution of waves. The document solves these problems through separation of variables, normal mode expansions, computing expectation values, and discussing qualitative features like dephasing and rephasing of waves. It also briefly discusses parameters for a two-slit light experiment.
The document contains 16 multiple choice questions about algorithms, data structures, and graph theory. Each question has 4 possible answers and the correct answer is provided. The maximum number of comparisons needed to merge sorted sequences is 358, and depth first search on a graph represented with an adjacency matrix has a worst case time complexity of O(n^2).
This document provides an overview of linear and quadratic functions and modeling. It defines polynomial functions and discusses linear functions, their graphs, and applications in business and economics. Quadratic functions and their graphs are also introduced, including how to find the vertex and axis of a quadratic function. Examples are provided to illustrate related concepts like finding equations of linear functions, average rate of change, and describing quadratic graphs algebraically.
Neuron Synchronization and Representation of Space and Time in Neural NetworksSSA KPI
AACIMP 2010 Summer School lecture by Witali Dunin-Barkowski. "Physics, Chemistry and Living Systems" stream. "Problems of Synchronization and Representation of Time and Space in Neural Networks" course.
More info at http://summerschool.ssa.org.ua
Partitioning procedures for solving mixed-variables programming problemsSSA KPI
This document presents two procedures for solving mixed-variables programming problems. The key steps are:
1. The problem is partitioned into two subproblems - a programming problem defined on the set S, and a linear programming problem defined in Rp.
2. A partitioning theorem shows that solving the programming problem on S determines whether the original problem is feasible, feasible but not optimal, or determines an optimal solution and reduces the original problem to a linear programming problem.
3. To efficiently solve the programming problem on S without computing the entire constraint set, the problem is decomposed into subproblems where only constraints determining optimality need to be identified.
Search and Structure Design of Physiologically Active CompoundsSSA KPI
This document discusses the history of applying mathematics to chemistry. It notes that in 1830, Augusto Compte argued that using mathematics in chemistry was "profoundly irrational." However, in 1808, Louis Joseph Gay-Lussac stated that the application of calculation to chemistry was not far off. The document provides these two quotes contrasting early views on quantitative chemistry.
AACIMP 2010 Summer School lecture by Anton Chizhov. "Physics, Chemistry and Living Systems" stream. "Neuron-Computer Interface in Dynamic-Clamp Experiments. Models of Neuronal Populations and Visual Cortex" course. Part 2.
More info at http://summerschool.ssa.org.ua
The document provides information about Lagrangian interpolation, including:
1. It introduces Lagrangian interpolation as a method to find the value of a function at a discrete point using a polynomial that passes through known data points.
2. It gives the formula for the Lagrangian interpolating polynomial and provides an example of using it to find the velocity of a rocket at a certain time.
3. It discusses using higher order polynomials for interpolation, providing another example that calculates velocity using quadratic and cubic polynomials.
The document contains exercises, hints, and solutions for analyzing algorithms from a textbook. It includes problems related to brute force algorithms, sorting algorithms like selection sort and bubble sort, and evaluating polynomials. The solutions analyze the time complexity of different algorithms, such as proving that a brute force polynomial evaluation algorithm is O(n^2) while a modified version is linear time. It also discusses whether sorting algorithms like selection sort and bubble sort preserve the original order of equal elements (i.e. whether they are stable).
X01 Supervised learning problem linear regression one feature theorieMarco Moldenhauer
1. The document describes supervised learning problems, specifically linear regression with one feature. It defines key concepts like the hypothesis function, cost function, and gradient descent algorithm.
2. A data set with one input feature and one output is defined. The goal is to learn a linear function that maps the input to the output to best fit the training data.
3. The hypothesis function is defined as h(x) = θ0 + θ1x, where θ0 and θ1 are parameters to be estimated. Gradient descent is used to minimize the cost function and find the optimal θ values.
Linear and discrete systems analysis Jntu Model Paper{Www.Studentyogi.Com}guest3f9c6b
This document contains information about a linear and discrete systems analysis exam, including 8 questions on topics such as:
- Writing state equations and finding responses for electrical circuits
- Determining Fourier series expansions and transforms
- Analyzing RLC circuits and finding voltages/currents over time
- Checking if polynomials are Hurwitz and determining positive real functions
- Synthesizing impedance functions in Foster and Cauer forms
- Analyzing sampling of bandpass signals and power density spectra
- Finding fundamental periods and energies of sequences
Students were asked to answer 5 out of the 8 questions.
This document contains a final exam with 7 multiple part questions testing concepts in statistics, probability, and signal processing. The exam allows standard calculators and requires showing all work. Questions cover topics like distributions, confidence intervals, Monte Carlo integration, random processes, and sampling. Students are advised that reasonable assumptions can be made if they have difficulties with a part of a question. The exam is worth a total of 80 marks plus 4 bonus marks.
The document discusses combinational logic circuits and Boolean algebra. It contains 17 questions covering topics such as:
1. The three basic logic operations - AND, OR, NOT - and their truth tables and logic diagrams.
2. Boolean algebra, functions, truth tables, and logic diagrams.
3. Identities of Boolean algebra, dual expressions, and the consensus theorem.
4. Minimizing logic functions using Boolean algebra identities and Karnaugh maps.
5. Minterms, maxterms, and representing logic functions with sums of products and products of sums.
6. Demorgan's theorem and complementing logic functions.
7. Simplifying logic functions through
Integral Calculus. - Differential Calculus - Integration as an Inverse Process of Differentiation - Methods of Integration - Integration using trigonometric identities - Integrals of Some Particular Functions - rational function - partial fraction - Integration by partial fractions - standard integrals - First and second fundamental theorem of integral calculus
The document summarizes key concepts about functions and lines. It defines a function as a rule that assigns unique output values to inputs. Functions can be represented graphically by plotting the points (x, f(x)). Lines on a plane can be defined by two points and have a slope that represents the rise over run. The slope formula is used to find the equation of a line in y=mx+b form, where m is the slope and b is the y-intercept. Perpendicular and parallel lines have specific relationships between their slopes.
This document discusses composite functions and the order of operations when combining functions.
It provides an example of a mother converting the temperature of her baby's bath water from Celsius to Fahrenheit using two separate functions. The first function converts the Celsius reading to Fahrenheit, and the second maps the Fahrenheit reading to whether the water is too cold, alright, or too hot. Together these functions form a composite function.
Algebraically, a composite function f∘g(x) is defined as applying the inner function f first to the input x, and then applying the outer function g to the output of f. The domain of the inner function must be contained within the range of the outer function. The order of
This document discusses deep generative models including variational autoencoders (VAEs) and generational adversarial networks (GANs). It explains that generative models learn the distribution of input data and can generate new samples from that distribution. VAEs use variational inference to learn a latent space and generate new data by varying the latent variables. The document outlines the key concepts of VAEs including the evidence lower bound objective used for training and how it maximizes the likelihood of the data.
The document discusses functions and their graphical representations. It defines key terms like domain, range, and one-to-one and many-to-one mappings. It then focuses on quadratic functions, showing that their graphs take characteristic U-shaped or inverted U-shaped forms. The document also examines inequalities involving quadratic expressions and how to determine the range of values satisfying such inequalities by analyzing the graph of the quadratic function.
This document provides an overview of reduced-order models and emulators. It discusses two main approaches: polynomial chaos expansions (PCE) and Gaussian process (GP) emulators. PCE approximates quantities of interest using orthogonal polynomials and provides error bounds, while GP emulators approximate computer model outputs as realizations of random processes. Both rely on designs of experiments, with Latin hypercube designs commonly used. The document compares the approaches and discusses pros and cons, noting they are complementary rather than competing methods. It concludes by emphasizing the importance of accounting for model discrepancy in surrogate modeling.
This document discusses different methods for interpolation and approximation, including polynomial interpolation, spline interpolation, and parametric interpolation. Polynomial interpolation finds an interpolating polynomial that passes through discrete data points. It can be done using Lagrange polynomials or by solving a Vandermonde system of equations. Spline interpolation fits piecewise polynomials over intervals defined by interpolation nodes, ensuring smoothness at interval boundaries. Parametric interpolation treats variables equally by interpolating as functions of a parameter.
Handling missing data with expectation maximization algorithmLoc Nguyen
Expectation maximization (EM) algorithm is a powerful mathematical tool for estimating parameter of statistical models in case of incomplete data or hidden data. EM assumes that there is a relationship between hidden data and observed data, which can be a joint distribution or a mapping function. Therefore, this implies another implicit relationship between parameter estimation and data imputation. If missing data which contains missing values is considered as hidden data, it is very natural to handle missing data by EM algorithm. Handling missing data is not a new research but this report focuses on the theoretical base with detailed mathematical proofs for fulfilling missing values with EM. Besides, multinormal distribution and multinomial distribution are the two sample statistical models which are concerned to hold missing values.
This document discusses solving problems related to quantum mechanics and waves. It provides solutions to several problems involving waves on drum membranes, classical wave equations, particles in infinite and finite boxes, and the time evolution of waves. The document solves these problems through separation of variables, normal mode expansions, computing expectation values, and discussing qualitative features like dephasing and rephasing of waves. It also briefly discusses parameters for a two-slit light experiment.
The document contains 16 multiple choice questions about algorithms, data structures, and graph theory. Each question has 4 possible answers and the correct answer is provided. The maximum number of comparisons needed to merge sorted sequences is 358, and depth first search on a graph represented with an adjacency matrix has a worst case time complexity of O(n^2).
This document provides an overview of linear and quadratic functions and modeling. It defines polynomial functions and discusses linear functions, their graphs, and applications in business and economics. Quadratic functions and their graphs are also introduced, including how to find the vertex and axis of a quadratic function. Examples are provided to illustrate related concepts like finding equations of linear functions, average rate of change, and describing quadratic graphs algebraically.
Neuron Synchronization and Representation of Space and Time in Neural NetworksSSA KPI
AACIMP 2010 Summer School lecture by Witali Dunin-Barkowski. "Physics, Chemistry and Living Systems" stream. "Problems of Synchronization and Representation of Time and Space in Neural Networks" course.
More info at http://summerschool.ssa.org.ua
Partitioning procedures for solving mixed-variables programming problemsSSA KPI
This document presents two procedures for solving mixed-variables programming problems. The key steps are:
1. The problem is partitioned into two subproblems - a programming problem defined on the set S, and a linear programming problem defined in Rp.
2. A partitioning theorem shows that solving the programming problem on S determines whether the original problem is feasible, feasible but not optimal, or determines an optimal solution and reduces the original problem to a linear programming problem.
3. To efficiently solve the programming problem on S without computing the entire constraint set, the problem is decomposed into subproblems where only constraints determining optimality need to be identified.
Search and Structure Design of Physiologically Active CompoundsSSA KPI
This document discusses the history of applying mathematics to chemistry. It notes that in 1830, Augusto Compte argued that using mathematics in chemistry was "profoundly irrational." However, in 1808, Louis Joseph Gay-Lussac stated that the application of calculation to chemistry was not far off. The document provides these two quotes contrasting early views on quantitative chemistry.
AACIMP 2010 Summer School lecture by Anton Chizhov. "Physics, Chemistry and Living Systems" stream. "Neuron-Computer Interface in Dynamic-Clamp Experiments. Models of Neuronal Populations and Visual Cortex" course. Part 2.
More info at http://summerschool.ssa.org.ua
This document summarizes models of synaptic transmission, including:
- Models of the NMDA receptor and its voltage-dependent magnesium block.
- Models showing activity-dependent recovery from depression at synapses increases their range of effective information transmission frequencies.
- Models examining how short-term synaptic dynamics like depression, facilitation, and activity-dependent recovery affect information transmission and processing at the network level.
Methods from Mathematical Data Mining (Supported by Optimization)SSA KPI
This document summarizes a presentation on cluster stability estimation and determining the optimal number of clusters in a dataset. The presentation proposes a method that draws random samples from the dataset and compares the partitions obtained from each sample to estimate cluster stability. It quantifies the consistency between partitions using minimal spanning trees and the Friedman-Rafsky test statistic. Experiments on synthetic and real-world datasets show that the method can accurately determine the true number of clusters by finding the partition that maximizes cluster stability.
International Industrial Marketing course AI212V. Day 4SSA KPI
This document discusses various topics related to marketing including managing personal interaction, non-interactive selling methods, word-of-mouth marketing, managing the sales force, new market offerings, going global, holistic marketing trends, corporate social responsibility, and profitability control and analysis. The chapters cover direct marketing, ethics of non-interactive selling, locating opinion leaders, motivating the sales force, cross-functional new product development, and benchmarking marketing performance on a global scale.
Neuron-computer interface in Dynamic-Clamp experimentsSSA KPI
AACIMP 2010 Summer School lecture by Anton Chizhov. "Physics, Chemistry and Living Systems" stream. "Neuron-Computer Interface in Dynamic-Clamp Experiments. Models of Neuronal Populations and Visual Cortex" course. Part 1.
More info at http://summerschool.ssa.org.ua
Investigation of Heliocollectors Work EfficiencySSA KPI
AACIMP 2010 Summer School lecture by Gennady Varlamov. "Sustainable Development" stream. "Clean Energy Production Technologies" course.
More info at http://summerschool.ssa.org.ua
International Industrial Marketing course AI212V. Day 3SSA KPI
The document outlines chapters from an industrial marketing course, covering topics like product strategy, pricing, marketing channels, promotion, communication, and personal selling. It provides an overview of key concepts in each area, such as the 4 P's of marketing, product differentiation, pricing models, channel structures, the AIDA communication model, and roles of a sales force. Examples and frameworks are presented to illustrate different approaches in industrial marketing management.
Uncertainty Problem in Control & Decision TheorySSA KPI
AACIMP 2010 Summer School lecture by Viktor Ivanenko. "Applied Mathematics" stream. "On the Models of Uncertainty in Decision and Control Problems" course. Part 1.
More info at http://summerschool.ssa.org.ua
Derivative Free Optimization and Robust OptimizationSSA KPI
The document summarizes presentations on nonsmooth optimization, robust optimization, and derivative-free optimization given at the 4th International Summer School in Kiev, Ukraine. Gerhard Weber and Basak Akteke-Ozturk presented on nonsmooth optimization and its applications in clustering problems. Robust optimization approaches were discussed for problems with uncertain parameters. Methods for minimizing functions without computing derivatives, known as derivative-free optimization, were also covered. Examples included trust-region algorithms and using models based on interpolation. Semidefinite programming relaxations for support vector clustering were mentioned.
Some Engg. Applications of Matrices and Partial DerivativesSanjaySingh011996
This document contains a submission by three students to Dr. Sona Raj Mam regarding partial differentiation, matrices and determinants, and eigenvectors and eigenvalues. It provides examples of how these mathematical concepts are applied in fields like engineering. Partial differentiation is used in economics to analyze demand and in image processing for edge detection. Matrices and determinants allow representing linear transformations in graphics software. Eigenvalues and eigenvectors have applications in areas like computer science, smartphone apps, and modeling structures in civil engineering. The document also provides real-world examples and references textbooks and websites for further information.
The document discusses the Fundamental Theorem of Calculus, which has two parts. Part 1 establishes the relationship between differentiation and integration, showing that the derivative of an antiderivative is the integrand. Part 2 allows evaluation of a definite integral by evaluating the antiderivative at the bounds. Examples are given of using both parts to evaluate definite integrals. The theorem unified differentiation and integration and was fundamental to the development of calculus.
1) The document outlines a course curriculum that covers functions, rational functions, one-to-one functions, exponential functions, and logarithmic functions over 32 hours spread across 8 weeks.
2) It provides the chapter titles and learning objectives for each chapter, along with the topics and hours allocated to each lesson.
3) Key concepts covered include functions as models of real-life situations, representing functions as sets of ordered pairs, tables, graphs, and piecewise functions.
This document discusses four ways to represent functions: verbally, numerically, visually through graphs, and algebraically through explicit formulas. It provides examples of each type of representation and discusses key properties of functions like domain, range, and the vertical line test. The document also covers piecewise-defined functions and functions with symmetry properties like even functions whose graphs are symmetric about the y-axis.
This document summarizes a paper that presents new algorithms for solving the cyclic order-preserving assignment problem (COPAP) and related sub-problem, the linear order-preserving assignment problem (LOPAP). It introduces a new point-assignment cost function called the Procrustean local shape distance (PLSD) and explores heuristics for using the A* search algorithm to more efficiently solve COPAP and LOPAP. Experimental results on the MPEG-7 shape dataset are presented and recommendations are made for solving COPAP/LOPAP in practice.
This document discusses functions and their representations. It begins by defining relations and functions, and providing examples of each. It then discusses different types of functions including linear, quadratic, constant, identity, absolute value, and piecewise functions. Examples are provided for each type. The document ends with exercises asking the reader to determine if given relations are functions, and to identify what type of function is being described.
- The document provides an overview of mathematical concepts for control systems including complex variables, complex functions, differential equations, Laplace transforms, and their applications.
- It introduces complex numbers and variables, complex functions including poles and zeros. Basic concepts of differential equations and Laplace transforms are reviewed.
- Methods for solving differential equations using Laplace transforms are described, including taking the inverse Laplace transform using partial fraction expansion.
- A function is a rule that maps an input number (independent variable) to a unique output number (dependent variable).
- To determine if a rule describes a valid function, you can plot points from the rule on a graph and check that each input only maps to one output using a vertical ruler.
- For a rule to describe a valid function, its domain must be restricted if multiple outputs are possible for any single input. The domain is the set of possible inputs, and the range is the set of corresponding outputs.
- A function is a rule that maps an input number (independent variable) to a unique output number (dependent variable).
- To determine if a rule describes a valid function, you can plot points from the rule on a graph and check that each input only maps to one output using a vertical ruler.
- For a rule to describe a valid function, its domain must be restricted if multiple outputs are possible for any single input. The domain is the set of possible inputs, and the range is the set of corresponding outputs.
The document discusses functions and algorithms. It defines what a function is and provides examples. It also discusses different types of functions such as one-to-one, onto, and inverse functions. The document then discusses algorithms and complexity analysis. It provides examples of linear search algorithms and analyzes their worst case and average case time complexities.
The document provides an introduction to integral calculus. It discusses how integral calculus is motivated by the problem of defining and calculating the area under a function's graph. The key points are:
1) Integration is the inverse process of differentiation, where we find the original function given its derivative. This results in families of functions that differ by an arbitrary constant.
2) Indefinite integrals represent families of functions, while definite integrals have practical uses in science, engineering, economics and other fields.
3) Standard formulae for integrals are provided that correspond to common derivative formulae, which can be used to evaluate more complex integrals.
Numerical method for pricing american options under regime Alexander Decker
This document presents a numerical method for pricing American options under regime-switching jump-diffusion models. It begins with an abstract that describes using a cubic spline collocation method to solve a set of coupled partial integro-differential equations (PIDEs) with the free boundary feature. The document then provides background on regime-switching Lévy processes and derives the PIDEs that describe the American option price under different regimes. It presents the time and spatial discretization methods, using Crank-Nicolson for time stepping and cubic spline collocation for the spatial variable. The method is shown to exhibit second order convergence in space and time.
The document defines key terms related to functions including univariate and bivariate data, independent and dependent variables, domain and range, and linear, exponential, quadratic, and step functions. It provides examples of evaluating various functions and finding linear and quadratic models to describe relationships between variables from sets of data points. The overall content describes different types of mathematical functions and how to analyze and model real-world data using functions.
This chapter discusses state feedback and output feedback control of linear systems. It introduces the concepts of reachability and observability, which describe whether a system's states can be influenced by inputs or estimated from outputs respectively. The chapter shows that under certain conditions, feedback can be used to assign a system's eigenvalues, allowing its dynamics to be designed. State feedback is developed by placing closed-loop eigenvalues in desired locations. If states are not directly measurable, an observer can estimate them from input and output measurements.
MAT-121 COLLEGE ALGEBRAWritten Assignment 32 points eac.docxjessiehampson
MAT-121: COLLEGE ALGEBRA
Written Assignment 3
2 points each except for 5, 6, 9, 15, 16, which are 4 points each as indicated.
SECTION 3.1
Algebraic
For the following exercise, determine whether the relationship represents y as a function of x. If the relationship represents a function then write the relationship as a function of
x
using
f
as the function.
x+y2=5
Consider the relationship 7n-5m=4.
Write the relationship as a function
n
=
k
(
m
).
Evaluate
k
(
5
).
Solve for
k
(
m
) = 7.
Graphical
Given the following graph
Evaluate
f
(4)
Solve for
f
(x) = 4
Numeric
For the following exercise, determine whether the relationship represents a function.
{(0, 5), (-5, 8), (0, -8)}
For the following exercise, use the function
f
represented in table below. (4 points)
x
-18
-12
-6
0
6
12
18
f(x)
24
17
10
3
-4
-11
-18
Answer the following:
Evaluate
f
(-6).
Solve
f
(
x
) = -11
Evaluate
f
(12)
Solve
f
(
x
) = -18
For the following exercise, evaluate the expressions, given functions
f
,
g
, and
h
:
f(x)=4x+2
; g(x)=7-6x; h(x)=7x2-3x+6
f(-1)g(1)h(0) (4 points)
Real-world applications
The number of cubic yards of compost,
C
, needed to cover a garden with an area of
A
square feet is given by
C
=
h
(
A
).
A garden with an area of 5,000 ft2 requires 25 yd3 of compost. Express this information in terms of the function
h
.
Explain the meaning of the statement
h
(2500) = 12.5.
SECTION 3.2
Algebraic
For the following exercise, find the domain and range of each function and state it using interval notation.
f(x)=9-2x5x+13
Numeric
For the following exercise, given each function
f
, evaluate
f
(3),
f
(-2),
f
(1), and f (0). (4 points)
Real-World Applications
The height,
h,
of a projectile is a function of the time,
t,
it is in the air. The height in meters for
t
seconds is given by the function h(t)= -9.8t2+19.6t. What is the domain of the function? What does the domain mean in the context of the problem?
SECTION 3.3
Algebraic
For the following exercise, find the average rate of change of each function on the interval specified in simplest form.
k(x)=23x+1
on [2, 2+h]
Graphical
For the following exercise, use the graph of each function to
estimate
the intervals on which the function is increasing or decreasing.
For the following exercise, find the average rate of change of each function on the interval specified.
g(x)=3x2-23x3 on [1, 3]
Real-World Applications
Near the surface of the moon, the distance that an object falls is a function of time. It is given by d(t)=1.6t2, where
t
is in seconds and d(t) is in meters. If an object is dropped from a certain height, find the average velocity of the object from t = 2 to t = 5.
SECTION 3.4
Algebraic
For the following exercise, determine the domain for each function in interval notation. (4 points)
f(x)=2x+5 and g(x)=4x+9, find f-g, f+g, fg, and fg
For.
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Some Two-Steps Discrete-Time Anticipatory Models with ‘Boiling’ Multivaluedness
1. Some Two- Steps Discrete- Time Anticipatory Models
with ‘Boiling’ Multivaluedness.
Alexander.S.Makarenko, Alexander S. Stashenko
Institute of Applied System Analysis at
National Technical University of Ukraine (KPI),
37 Pobedy Avenue, Kiev, 03056, Ukraine,
E-mail: makalex@i.com.ua ; makalex@mmsa.ntu-kpi.kiev.ua.
Abstract
In this paper it is described and investigated some class of models and the
concept which can make a universal methodological background for difficult
social, economic and public systems concerning different spatial and time
scales and hierarchical levels. These are nonlinear models of difficult processes
with foresight expectations. In the review some existing models with foresight
expectations is presented, and the new nonlinear model with the behavior
similar to models of neural network is offered. The offered model has next
differences from existing. At first the model is anticipatory, that is passing on
two steps ahead, and secondly the function f (?) has a piecewise - linear
character, and looks like the activation function of neurons.
The condition for multivaludness had been found. Such multivaluedness is of
special type of 'boiling tank' when the multiplicity had created at the restricted
region of space. Suggested concept and principles allow developing some
practical applications of models.
Keywords: anticipatory element; multivaluedness; society models
1. Introduction
It is known well, that the systems with anticipating have large prospects both
in theoretical, and in the applied aspects [1, 2]. But for subsequent development of the
theory of anticipatory systems a large number of concrete examples of such systems
should be investigated. As it was indicated in the previous works of one of the authors
(A.Makarenko) very prospect and interesting are the neuronets with anticipating
elements, which corresponds to the models of society with accounting the mentality
of individuals [3 - 5].
The mathematical investigation of complex hierarchical models with
anticipatory property is the further research task. At present work we consider the
example of the system from one basic element of anticipatory network. We had
considered as the prototype the investigations of some economical models [6 - 8].
Remark that it allows considering some presumable economical applications at the
end of our paper.
2. In the given work the offered two-step discrete model in time with anticipating
had been proposed and investigated. The main result of analysis of the maps is the
possibility of multy - valued transitions. In the paper all solutions of model are
explored. Some explanation of choice between these branches of solutions is also
proposed.
Remark that earlier a one – dimensional model with anticipation had been
used for money-economic processes [7, 8]. But the offered model has some principle
differences. The first is two- steps in transitions, that is passing on two steps ahead,
and second is that the function f(?) has a piece-vise linear character and belongs to the
class of neuron responses function which is usual for neural networks [9]. Piece -vise
linear character of function allowed making the thorough numerical - analytical
analysis. A new type of significant behavior had been found, when the region of
multy – valuedness in solutions is localized in space. At the end of work the example
of multy – step model is proposed is of which is more complicated by the dimension
of map. Because of presumable importance of results we also pose some discussion of
its possible applications to economical problems.
2. Model description
2.1 Prototype from economics
First of all we shortly describe the prototype model from the field of
economics. Here we pose the description of economical terms for illustration the
origin of prototype models. The nonlinear model of money - credit dynamics in
continuous time consists of equating of the price adjusting [6- 8]:
&
p = α [m − p − f (π )], (1)
&
where p - logarithm of price level, m - logarithm of amount (here it is foreseen to be
constant) of money, and p - the expected norm of inflation, that is:
&
π (t ) = Et [ p (t )], (2)
In accordance with equality (1), the norm of change of price at the market of
commodities relies on the surplus of demand for the real balances. The function f is
logarithm of demand for the real balances of money.
The discrete version of evolution equation of price (1) has such form:
pt +1 = α ⋅ m + (1 − α ) pt − αf (π t ,t +1 ), (3)
where now pt means logarithm of level of price in the moment of time of t. In
equality (3) the function of demand of money in the moment of time of t relies on the
norm of inflation of expected in a next period.
π t , t +1 = Et ( pt +1 − pt ), (4)
3. That to complete a definite model in (3) it is needed to take into account the
condition of passing ahead expectation. The hypothesis of passing ahead expectation
results in the one-dimensional map in a form for pt+1:
pt +1 = α ⋅ m + (1 − α ) pt − α ⋅ f ( pt +1 − pt ), (5)
Remark that the equation (5) had been derived as economical model but it is
(and other equations with anticipatory property) interesting mathematical object itself.
2.3 Two – step model of anticipatory element
Here we introduce a new nonlinear model which substantially extends the one
– dimensional equation (5). New features in proposed model are the next. The first is
two- steps nature (that is passing on two steps ahead). The second is that the function
f(?) has a piecewise - linear character, and looks like the transition function of neurons
in neuronets [9]. Remark that piecewise character of nonlinearity usually allows
developed mathematical investigations (see for example [10]).
We will write down the offered model as follows:
pt +1 = α ⋅ m + (1 − α ) pt − α ⋅ f ( pt + 2 − pt +1 ) ( 6)
pt + 2 = α ⋅ m + (1 − α ) pt +1 − α ⋅ f ( pt + 2 − pt +1 ) (7 )
The function f(x) depends on to the parameter a and has the following
expression:
f ( x) = 0, x≤0
f ( x) = α ⋅ x, x ∈ (0, 1α ] (8)
f ( x) = 1,
x > 1α
3 Model investigations
3.1 Inverse function representation
We will rewrite equation (7) thus, that pt + 2 were found for the right side of
equation.
( pt + 2 − pt +1 ) + α ⋅ f ( pt + 2 − pt +1 ) = α ⋅ m − α ⋅ pt +1 (9)
We will put this in right part of equation (6). Converting into a comfortable
form we will get the following equality:
( pt + 2 − pt +1 ) + α (1 − α ) f ( pt + 2 − pt +1 ) = α (1 − α )(m − pt ) (10)
We will enter the following function:
V ( x) = x + α (1 − α ) f ( x) (11)
4. Then we can write a next correlation
V ( pt + 2 − pt +1 ) = α (1 − α )(m − pt ) , from which we can receive
pt + 2 = pt +1 + V −1[α (1 − α )(m − pt )] ≡ F ( pt , pt +1 ) (12)
V −1 in equation (12) means the inverse function V, when V is invertible or
the proper function is definite by means one of inverting of function V, when V is not
uniquely invertible.
We will write down first derivative to the function V:
V ′( x) = 1 + α 2 f ′( x) − α 3 f ′( x) (13)
First derivative (13) to the function V at x < 0 that x > 1α has independent
from a value, according to properties of function f, equal to 1 accordant (11). We are
interested at the value of derivative function V at x ∈ (0,1/ α ] , so at exactly in this
interval the function of f matters dependency upon x accordant (8):
V ′( x) = 1 + α 2 − α 3 (14)
From equation (14) we have next results
α = 1/3 + 1/3{29/2 − 3(√ 93/2)}1/ 3 + 1/3{1/2{29 + 3 √ 93}}1/ 3
Approximate value is the next: a = 1.465...
2 3
Consequently subject to the condition 1 + α − α < 0 derivative to the
function V has the negative value, that can mean that the points of maximum appear,
and minimum of function V. And accordingly the ambiguousness of invertability of
function V appears which is represented on Picture 1.
5. Picture 1. The graph of function V, subjecting to the condition of ambiguous
invertability. Intervals (−∞,ν m ) and (VM , +∞) have one solution, interval (ν m , VM ) -
three solutions. Dotted line represents the reverse transformation of function V.
On Picture 1. we see a form of function V under a definite condition on a
parameter. We have the maximum of function V - VM in the point π 2 ( VM = V( π 2 ))
and minimum ν m in a point π 1 (ν m = V( π 1 )). There on graphic we see prototypes
?
V( π 2 ) and V( π 1 ) in other points π 2 and π1 accordingly. It is visible therefore, that
?
V −1 ( y ) has three separate points when vm < y <VM and one point otherwise. There
is a question how to represent a reverse function because the three special cases.
It is needed to remark that there is the fourth case, subject to the condition
1 + α 2 − α 3 = 0 for inverse function of V when there are not two points, but the
whole interval of points, which causes the endless quantity of possible variants of the
map F ( pt , pt +1 ) .
−1
Consequently, a function V determines the behavior of the map
F ( pt , pt +1 ) as follows. When V simply reversible, F ( pt , pt +1 ) is continuous, and non-
continuous, when V is not uniquely reversible.
3.2 Map F ( pt , pt +1 )
We will write out equation for the map F ( pt , pt +1 ) as follows:
F ( pt , pt +1 ) = pt +1 + V −1[α (1 − α )(m − pt )] (15)
The map F ( pt , pt +1 ) is continuous, when V reversible and is discontinuous
otherwise.
The fixed point:
V −1[α (1 − α )(m − pt )] = 0
⇒ α (1 − α )(m − pt ) = α (1 − α ) f (0)
V (0) = 0 + α (1 − α ) f (0)
*
pt = m − f (0) = m (16)
As soon as F ( pt , pt +1 ) = pt +1 is achieved a condition
V −1[α (1 − α )(m − pt )] = 0 is executed, that is α (1 − α )(m − pt ) = V (0) , or
specifying the last expression α (1 − α )(m − pt ) = α (1 − α ) f (0) . When
6. F ( pt , pt +1 ) has the points of break, there it can be nonexisting of the fixed point,
however if exists, has a form in obedience to equality (16).
We will consider the cut of the map F ( pt , pt +1 ) =0. How visible with
Picture 2, right overhead part has the positive value, and the left lower part of cut
accordingly negative value. That is getting in a positive region, value of the map
F ( pt , pt +1 ) to be increased and increased to endlessness. Like there is the
reduction of value of map, at the hit in the negative region of values. If to set the
initial value from the region of ambiguousness close to the fixed point, on a next step,
the map will get a few values, one of which, will translate the map in a positive
region, the second value vice versa will translate the map in a negative region, that
will result in rejection of the state in endlessness, one in positive, second in negative.
And another solution will arise up, which will leave the state of map in the region of
ambiguousness. Thus there will be the permanent troop landing from the region of
ambiguousness in a plus and minus of endlessness. We can mark the region of
multivaluedness origin as the ‘boiling’ multy - valuedness. The term ‘boiling’ had
been introduced following visible analogy with boiling water tank when the
molecules of water leave tank through the free surface of water. Some solutions cross
the boundary of ‘rhomb’ and then tend to infinity. But some branches of solutions
stay within the ‘rhomb’ to undergo to further multiplication.
Picture 2. Cut of the map F ( pt , p t +1 ) = 0 at a parameter b>1.465.... Considered
equations (6) – (8) have three solutions in the rhomb on the plane (with marked three
vertexes m, 1* and 2*) and one solution in the rest of the plane.
Consequently, we are interested in the case, when the state of map is
constantly found in the region of ambiguousness. The most adjusted for such
investigation is computer calculation.
3.3 Behavior of the map F ( pt , pt +1 ) in state space
The software had been developed so that it is possible to visualize some
branches of the states of map, but only those, that are not thrown out to infinity, that
much facilitated understanding of processes, which take place in the region of
7. ambiguousness. Namely, on Picture 3 we can see two cycles of period 6, and the
quantity of cycles is multiplied in course of time, that is we can see a tendency to
phenomena which we may called „chaos”.
Picture 3. The graph of forks of the map F ( pt , p t +1 ) . Parameters: a = 3.2, m = 0, p0
=m-0.15, p1 =m-0.2, limitation is maximal: m+2, minimum: m-2, without the forks,
that are thrown out to infinity. The cycles are within the ‘rhomb’ with the multy -
valuedness above.
Picture 4. Map in state space. The sequence of values of ( pt , pt +1 ) is represented.
With reduction of parameter a the period of cycles diminishes. Diminishing
takes place until then, while a model can generate new cycles. As soon as a model
loses such power, a process goes out slowly. That it is possible to see on Picture 5
(cycles of period 4).
8. Picture 5. The graph of forks of map F ( pt , p t +1 ) . Parameters: a = 2.5, m = 5, p0=m-
0.15, p1=m-0.2, limitation is maximal: m+1, minimum: m-1, without the forks, that
are thrown out on endlessness.
3.4 Generalization to the N-step model with anticipating
Further possible extension of the model consist in considering the possibilities
of increasing the number of steps in the model For subsequent investigations it is
possible to consider the model of such form:
pt +1 = α ⋅ m + (1 − α ) pt − α ⋅ f ( pt + N − pt + N −1 )
p
t + 2 = α ⋅ m + (1 − α ) pt +1 − α ⋅ f ( pt + N − pt + N −1 )
pt + 3 = α ⋅ m + (1 − α ) pt + 2 − α ⋅ f ( pt + N − pt + N −1 )
pt + 4 = α ⋅ m + (1 − α ) pt + 3 − α ⋅ f ( pt + N − pt + N −1 )
...........................................
pt + N −1 = α ⋅ m + (1 − α ) pt + N − 2 − α ⋅ f ( pt + N − pt + N −1 )
pt + N = α ⋅ m + (1 − α ) pt + N −1 − α ⋅ f ( pt + N − pt + N −1 )
Research of nonlinear anticipating model with piece-vise linear functions
confirmed basic conformities to the law in the offered nonlinear model. At first
−1
accordance of behavior is confirmed by nature of reverse functions V ( pt ) which
remember the two – dimensional case. A parameter a would turn out very influential
on the behavior and nature of the map F ( pt , pt +1 ) , that was expected at
construction of the given nonlinear two-step model with anticipating and with piece-
vise linear functions. Critical value to the parameter which follows to bifurcation
points appear at:
α = 1/3 + 1/3{29/2 − 3(√ 93/2)}1/ 3 + 1/3{1/2{29 + 3 √ 93}}1/ 3
9. It is needed to remark that there is the fourth case, subject to the condition at
the inverse to function V when there are not two points, but a whole interval of points,
which causes the infinite number of possible variants of the map F ( pt , pt +1 ) .
−1
A function V determines the conduct of the map F ( pt , pt +1 ) as follows.
When V simply reversible, F ( pt , pt +1 ) is continuous, and otherwise when a
function V is not uniquely reversible. But just in the preliminary investigations some
new possibilities had been found. For examples for some parameters value we had
found the possibilities of increasing the number of branches during time increasing.
Summary
So in proposed paper we have considered some examples of anticipatory
models – namely discrete – time models of single element with two – step anticipation
in time. Chosen form of nonlinearity (piecewise - linear) allowed considering in
details the dynamical behavior of solutions, branching of solutions and possible ways
for some type of complex behavior related with possible multy - valuedness. These
results are interesting and new per se. But it may be supposed that such type of
models may constitute one of the interesting fields of mathematical investigations of
anticipatory system. Just many – step in time equations from Paragraph 3.4 are
interesting objects. But much more interesting may be investigations of coupled
systems of anticipated elements. One of the most important classes of such systems
constitutes the multy – valued neuronal networks [5]. In case of the artificial neuronal
networks usually some of the research problems are the architecture of networks,
leading principles and investigations of their behavior. Remark that now we make
some investigations on such networks.
Other wide new class of research problem is the investigation of self –
organization processes in the anticipating media, in particular in discrete chains,
lattices, networks from anticipating elements. In such case the main problems are self
– organization, emergent structures including dissipative, bifurcations,
synchronization and chaotic behavior [11]. As it is seen from previous paragraphs
such problems take new forms of presumable possible multy – valuedness in
anticipatory systems. For example just definition of ‘chaos’ in such case should be
reconsidered. Remark that such problems are new for recent theory. But currently
already understanding of such phenomena possibilities may help in investigation and
managing real systems. Especially important may be applications to social,
economical etc. systems. Some outlines of possibilities were discussed in [3, 4]. The
realizations of such research programs are the goals for further investigations. Here
we pose only some discussion on possible applications of proposed models in
economics.
Recently the ideas of anticipatory nature of ‘homo economicus’ (participant of
economical relation) and organizations explicitly (but sometimes only verbally)
penetrate into the community of theoretic and practitioners in economy. Currently
some explicit investigations of macro economical models with anticipation had been
proposed [12, 13]. But these investigations are concentrated mainly on the stability
problems.
Described in present paper results extended to the new society models open
the new possibilities for exploring economical behavior. The key is possible multy –
valuedness in such systems and new understanding on decision – making role. As one
10. of possible topics for considerations we may foresee the investigation if uncertainty in
such systems. Now one of the leading ideas is the the uncertainty in economical
systems origins from dynamical chaos in it [14]. But as it follows from our
investigations anticipation and multy – valuedness also may serve as the source of
uncertainty in economical systems. Then presumable new tools for managing such
uncertainty may follows from mathematical modeling of anticipatory economical
systems.
Thus in proposed paper we have discussed strict results on some mathematical
models with anticipation and possible related issues, especially for economical
systems. We hope that further investigation will follow to next new and interesting
results.
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