Text 8. Classification of Mathematical Models

Many mathematical models can be classified in the following ways:

Linear vs nonlinear. Mathematical models are usually composed of variables, which are abstractions of quantities of interest in the described systems, and operators that act on these variables, which can be algebraic operators, functions, differential operators, etc. If all the operators in a mathematical model present linearity, the resulting mathematical model is defined as linear. A model is considered to be nonlinear otherwise.

The question of linearity and nonlinearity is dependent on the context, and linear models may have nonlinear expressions in them. For example, in a static linear model, it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables. Similarly, a differential equation is said to be linear if it can be written with linear differential operators, but it can still have nonlinear expressions in it. In a mathematical programming model, if the objective functions and constraints are represented entirely by linear equations, then the model is regarded as a linear one. If one or more of the objective functions or constraints are represented with a nonlinear equation, then the model is known as a nonlinear one.

Nonlinearity, even in fairly simple systems, is often associated with phenomena such as chaos and irreversibility. Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. A common approach to nonlinear problems is linearization, but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity.

Deterministic vs probabilistic (stochastic). A deterministic model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables. Therefore, deterministic models perform the same way for a given set of initial conditions. Conversely, in a stochastic model, randomness is present, and variable states are not described by unique values, but rather by probability distributions.

Static vs dynamic. A static model does not account for the element of time, while a dynamic model does. Dynamic models typically are represented with differential equations.

Lumped vs distributed parameters. If the model is homogeneous (consistent state throughout the entire system) the parameters are distributed. If the model is heterogeneous (varying state within the system) then the parameters are lumped. Distributed parameters are typically represented with partial differential equations.

 

Exercise 3. Give Ukrainian equivalents for the following word combinations.

Linear vs nonlinear; predictor variables, differential equation, objective functions and constraints; in fairly simple systems; to be often associated with phenomena such as chaos and irreversibility; a common approach to nonlinear problems; aspects such as irreversibility, which are strongly tied to nonlinearity; deterministic vs probabilistic (stochastic) model; every set of variable states; to be uniquely determined by parameters; unique value; probability distribution; lumped vs distributed parameters; consistent state throughout the entire system; varying state within the system.

 

Exercise 4. Find in the text the English for:

величини, що нас цікавлять; оператори, що діють на змінні; бути лінійним; питання лінійності та нелінійності; бути залежним від контексту; припускати, що; прогностичний параметр; диференційне рівняння; цільові функції та обмеження; бути представленим виключно лінійними рівняннями; досить прості системи; бути пов’язаним з такими явищами, як хаос та незворотність; загальний підхід; набір (сукупність) змінних; однозначно визначатися параметрами; зосереджені та розподілені параметри; однорідна модель; неоднорідна модель; стабільний стан; мінливий стан.

 

Exercise 5. Write out all verbs from the text. Identify their tense and voice.

 

Exercise 6. Write out all Non-Finite forms of the verb (Infinitives, Participles and Gerunds) from the text. Identify their forms and functions.

Exercise 7. Put questions to the underlined words.

1. Mathematical models are usually composed of variables.

2. The question of linearity and nonlinearity is dependent on the context.

3. A differential equation is said to be linear if it can be written with linear differential operators.

4. Nonlinearity is often associated with phenomena such as chaos and irreversibility.

5. A common approach to nonlinear problems is linearization.

6. Deterministic models perform the same way for a given set of initial conditions.

7. Dynamic models typically are represented with differential equations.

 

Exercise 8. Answer the questions on the text.

1. What ways can many mathematical models be classified?

2. What are variables?

3. What mathematical model is defined as linear?

4. What model is known as nonlinear?

5. What is deterministic model?

6. What is present in a stochastic model?

7. What is the difference between static and dynamic models?

8. When are the parameters distributed?

9. When are the parameters lumped?

 

Варіант 9

Exercise 1. Memorize the following words and word combinations.

Collage – колаж, комбінація різних елементів

general-purpose – універсальний, загального призначення

specify – визначати, задавати

visualize – наглядно представляти, візуалізовувати

software intensive system – система з громіздким ПЗ

blueprint – ескіз, креслення, проект

actor – актор (в мові UML – людина або пристрій, що взаємодіє з системою; зображується у вигляді фігурки людини)

database schema – схема даних, логічна структура даних (зовнішній опис або діаграма заданої в СКБД структури запису; термін було запроваджено у 1971 р. для дворівневого підходу до опису структури БД)

entity relationship diagram – діаграма відношень логічних об’єктів-сутностей, ER-діаграма

business modelling – бізнес-моделювання

workflow – послідовність операцій

throughout – впродовж

software development life cycle – життєвий цикл розробки ПЗ

synthesize – синтезувати

notation – система позначень

Object-modelling technique (OMT) – метод об’єктного моделювання

Object-oriented software engineering (OOSE) – об’єктно-орієнтована програмотехніка

fuse – комбінувати, поєднувати

aim – 1) мета, намір, прагнення 2) прагнути

distributed system – розподілена система

de facto – фактично, де-факто

evolve – розвиватися

auspice – сприяння

customization – пристосування під вимоги замовника

profile – профіль, сукупність параметрів користувача

transformation – перетворення, трансформація

stereotype – стереотип (в мові UML – створення нових елементів моделі шляхом розширення функціональності базових елементів)

compatible – сумісний

recast – змінювати, переробляти

take advantage of – скористатися, використовувати

Rational Unified Process (RUP) – раціональний уніфікований процес (розробки)

 

Exercise 2. Read and translate the text in writing.