Editing Mathematical model (section)

==Examples==
* One of the popular examples in [[computer science]] is the mathematical models of various machines, an example is the [[deterministic finite automaton]] (DFA) which is defined as an abstract mathematical concept, but due to the deterministic nature of a DFA, it is implementable in hardware and software for solving various specific problems. For example, the following is a DFA M with a binary alphabet, which requires that the input contains an even number of 0s:

[[File:DFAexample.svg|right|thumb|250px|The [[state diagram]] for <math>M</math>]]
:: <math>M = (Q, \Sigma, \delta, q_0, F)</math> where
::*<math>Q = \{S_1, S_2\},</math>
::*<math>\Sigma = \{0, 1\},</math>
::*<math>q_0 = S_1,</math>
::*<math>F = \{S_1\},</math> and
::*<math>\delta</math> is defined by the following [[state-transition table]]:
:::: {| border="1"
| || {{center|'''0'''}} || {{center|'''1'''}}
|-
|'''''S''<sub>1</sub>''' || <math>S_2</math> || <math>S_1</math>
|-
|'''''S''<sub>2</sub>''' || <math>S_1</math> || <math>S_2</math>
|}

:The state <math>S_1</math> represents that there has been an even number of 0s in the input so far, while <math>S_2</math> signifies an odd number. A 1 in the input does not change the state of the automaton. When the input ends, the state will show whether the input contained an even number of 0s or not. If the input did contain an even number of 0s, <math>M</math> will finish in state <math>S_1,</math> an accepting state, so the input string will be accepted.

:The language recognized by <math>M</math> is the [[regular language]] given by the [[regular expression]] 1*( 0 (1*) 0 (1*) )*, where "*" is the [[Kleene star]], e.g., 1* denotes any non-negative number (possibly zero) of symbols "1".
* Many everyday activities carried out without a thought are uses of mathematical models. A geographical [[map projection]] of a region of the earth onto a small, plane surface is a model which can be used for many purposes such as planning travel.<ref name="LAND INFO Worldwide Mapping">{{cite web | title=GIS Definitions of Terminology M-P | website=LAND INFO Worldwide Mapping | url=https://www.landinfo.com/resources_dictionaryMP.htm | access-date=January 27, 2020}}</ref>
* Another simple activity is predicting the position of a vehicle from its initial position, direction and speed of travel, using the equation that distance traveled is the product of time and speed. This is known as [[dead reckoning]] when used more formally. Mathematical modeling in this way does not necessarily require formal mathematics; animals have been shown to use dead reckoning.<ref>{{cite book |last=Gallistel |title=The Organization of Learning |location=Cambridge |publisher=The MIT Press |year=1990 |isbn=0-262-07113-4 }}</ref><ref>{{Cite journal | last1 = Whishaw | first1 = I. Q. | last2 = Hines | first2 = D. J. | last3 = Wallace | first3 = D. G. | doi = 10.1016/S0166-4328(01)00359-X | title = Dead reckoning (path integration) requires the hippocampal formation: Evidence from spontaneous exploration and spatial learning tasks in light (allothetic) and dark (idiothetic) tests | journal = Behavioural Brain Research | volume = 127 | issue = 1–2 | pages = 49–69 | year = 2001 | pmid =  11718884| s2cid = 7897256 }}</ref>
* ''[[Population]] Growth''. A simple (though approximate) model of population growth is the [[Malthusian growth model]]. A slightly more realistic and largely used population growth model is the [[logistic function]], and its extensions.
* ''Model of a particle in a potential-field''. In this model we consider a particle as being a point of mass which describes a trajectory in space which is modeled by a function giving its coordinates in space as a function of time. The potential field is given by a function <math>V\! : \Reals^3\! \to \Reals</math> and the trajectory, that is a function <math>\mathbf{r}\! : \Reals \to \Reals^3,</math> is the solution of the differential equation: <math display=block>-\frac{\mathrm{d}^2\mathbf{r}(t)}{\mathrm{d}t^2}m = \frac{\partial V[\mathbf{r}(t)]}{\partial x}\mathbf{\hat{x}} + \frac{\partial V[\mathbf{r}(t)]}{\partial y}\mathbf{\hat{y}} + \frac{\partial V[\mathbf{r}(t)]}{\partial z}\mathbf{\hat{z}},</math> that can be written also as <math display=block>m\frac{\mathrm{d}^2\mathbf{r}(t)}{\mathrm{d}t^2} = -\nabla V[\mathbf{r}(t)].</math>
:Note this model assumes the particle is a point mass, which is certainly known to be false in many cases in which we use this model; for example, as a model of planetary motion.
* ''Model of rational behavior for a consumer''.  In this model we assume a consumer faces a choice of <math>n</math> commodities labeled <math>1, 2, \dots, n</math> each with a market price <math>p_1, p_2, \dots, p_n.</math> The consumer is assumed to have an [[ordinal utility]] function <math>U</math> (ordinal in the sense that only the sign of the differences between two utilities, and not the level of each utility, is meaningful), depending on the amounts of commodities <math>x_1, x_2, \dots, x_n</math> consumed.  The model further assumes that the consumer has a budget <math>M</math> which is used to purchase a vector <math>x_1, x_2, \dots, x_n</math> in such a way as to maximize <math>U(x_1, x_2, \dots, x_n).</math>  The problem of rational behavior in this model then becomes a [[mathematical optimization]] problem, that is: <math display=block>\max \, U(x_1, x_2,\ldots, x_n)</math> subject to: <math display=block>\sum_{i=1}^n p_i x_i \leq M,</math><math display=block>x_i \geq 0  \; \; \; \text{ for all } i = 1, 2, \dots, n.</math> This model has been used in a wide variety of economic contexts, such as in [[general equilibrium theory]] to show existence and [[Pareto efficiency]] of economic equilibria.
* ''[[Neighbour-sensing model]]'' is a model that explains the [[mushroom]] formation from the initially chaotic [[fungus|fungal]] network.
* In [[computer science]], mathematical models may be used to simulate computer networks.
* In [[mechanics]], mathematical models may be used to analyze the movement of a rocket model.