12 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Mathematical model | 5/5 | https://en.wikipedia.org/wiki/Mathematical_model | reference | science, encyclopedia | 2026-05-05T07:24:19.245253+00:00 | kb-cron |
The state
S
1
{\displaystyle S_{1}}
represents that there has been an even number of 0s in the input so far, while
S
2
{\displaystyle S_{2}}
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,
M
{\displaystyle M}
will finish in state
S
1
,
{\displaystyle S_{1},}
an accepting state, so the input string will be accepted. The language recognized by
M
{\displaystyle M}
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. 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. 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
V
:
R
3
→
R
{\displaystyle V\!:\mathbb {R} ^{3}\!\to \mathbb {R} }
and the trajectory, that is a function
r
:
R
→
R
3
,
{\displaystyle \mathbf {r} \!:\mathbb {R} \to \mathbb {R} ^{3},}
is the solution of the differential equation:
−
d
2
r
(
t
)
d
t
2
m
=
∂
V
[
r
(
t
)
]
∂
x
x
^
+
∂
V
[
r
(
t
)
]
∂
y
y
^
+
∂
V
[
r
(
t
)
]
∂
z
z
^
,
{\displaystyle -{\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}} ,}
that can be written also as
m
d
2
r
(
t
)
d
t
2
=
−
∇
V
[
r
(
t
)
]
.
{\displaystyle m{\frac {\mathrm {d} ^{2}\mathbf {r} (t)}{\mathrm {d} t^{2}}}=-\nabla V[\mathbf {r} (t)].}
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
n
{\displaystyle n}
commodities labeled
1
,
2
,
…
,
n
{\displaystyle 1,2,\dots ,n}
each with a market price
p
1
,
p
2
,
…
,
p
n
.
{\displaystyle p_{1},p_{2},\dots ,p_{n}.}
The consumer is assumed to have an ordinal utility function
U
{\displaystyle U}
(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
x
1
,
x
2
,
…
,
x
n
{\displaystyle x_{1},x_{2},\dots ,x_{n}}
consumed. The model further assumes that the consumer has a budget
M
{\displaystyle M}
which is used to purchase a vector
x
1
,
x
2
,
…
,
x
n
{\displaystyle x_{1},x_{2},\dots ,x_{n}}
in such a way as to maximize
U
(
x
1
,
x
2
,
…
,
x
n
)
.
{\displaystyle U(x_{1},x_{2},\dots ,x_{n}).}
The problem of rational behavior in this model then becomes a mathematical optimization problem, that is:
max
U
(
x
1
,
x
2
,
…
,
x
n
)
{\displaystyle \max \,U(x_{1},x_{2},\ldots ,x_{n})}
subject to:
∑
i
=
1
n
p
i
x
i
≤
M
,
{\displaystyle \sum _{i=1}^{n}p_{i}x_{i}\leq M,}
x
i
≥
0
for all
i
=
1
,
2
,
…
,
n
.
{\displaystyle x_{i}\geq 0\;\;\;{\text{ for all }}i=1,2,\dots ,n.}
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 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.
== See also ==
== References ==
== Further reading ==
=== Books === Aris, Rutherford [ 1978 ] ( 1994 ). Mathematical Modelling Techniques, New York: Dover. ISBN 0-486-68131-9 Bender, E.A. [ 1978 ] ( 2000 ). An Introduction to Mathematical Modeling, New York: Dover. ISBN 0-486-41180-X Gary Chartrand (1977) Graphs as Mathematical Models, Prindle, Webber & Schmidt ISBN 0871502364 Dubois, G. (2018) "Modeling and Simulation", Taylor & Francis, CRC Press. Gershenfeld, N. (1998) The Nature of Mathematical Modeling, Cambridge University Press ISBN 0-521-57095-6 . Lin, C.C. & Segel, L.A. ( 1988 ). Mathematics Applied to Deterministic Problems in the Natural Sciences, Philadelphia: SIAM. ISBN 0-89871-229-7 Models as Mediators: Perspectives on Natural and Social Science edited by Mary S. Morgan and Margaret Morrison, 1999. Mary S. Morgan The World in the Model: How Economists Work and Think, 2012.
=== Specific applications === Papadimitriou, Fivos. (2010). Mathematical Modelling of Spatial-Ecological Complex Systems: an Evaluation. Geography, Environment, Sustainability 1(3), 67–80. doi:10.24057/2071-9388-2010-3-1-67-80 Peierls, R. (1980). "Model-making in physics". Contemporary Physics. 21: 3–17. Bibcode:1980ConPh..21....3P. doi:10.1080/00107518008210938. An Introduction to Infectious Disease Modelling Archived February 22, 2016, at the Wayback Machine by Emilia Vynnycky and Richard G White.
== External links == General reference
Patrone, F. Introduction to modeling via differential equations, with critical remarks. Plus teacher and student package: Mathematical Modelling. Brings together all articles on mathematical modeling from Plus Magazine, the online mathematics magazine produced by the Millennium Mathematics Project at the University of Cambridge. Philosophical
Frigg, R. and S. Hartmann, Models in Science, in: The Stanford Encyclopedia of Philosophy, (Spring 2006 Edition) Griffiths, E. C. (2010) What is a model?