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Universidad Nacional Mayor de San MarcosFacultad de Ciencias
Matemticas
INTRODUCCIN A LOS MODELOS MATEMTICOS DE LABIOLOGA
Roxana Lpez Cruz, Ph.D.
Lima- PerJunio - Julio, 2006
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ndice general
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4 NDICE GENERAL
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ndice de cuadros
5
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6 NDICE DE CUADROS
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ndice de figuras
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8 NDICE DE FIGURAS
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Captulo 1
Como elaborar un Texto usandoLatex
1.1. Formulas
f : R2 R(x, y) x2 +y
1.2. Ecuaciones
1.2.1. Ecuaciones en una sola linea
Ecuaciones en Texto
la ecuacinb(b(a) +b(c)) = b(a+c).
Ecuaciones Centradas sin Numeracin
b(b(a) +b(c)) =b(a+c).
Ecuaciones Centradas con Numeracin
b(b(a) +b(c)) =b(a+c). (1.1)
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10 CAPTULO 1. COMO ELABORAR UN TEXTO USANDO LATEX
1.2.2. Sistema de Ecuaciones (Tipo Arreglo)
Sistema de Ecuaciones sin Numeracin
dS
dt = S I
dI
dt = S II
dR
dt = I
Sistema de Ecuaciones con Numeracin
dS
dt = S I
dIdt
= S I I
dR
dt = I
(1.2)
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12 CAPTULO 1. COMO ELABORAR UN TEXTO USANDO LATEX
Figura 1.1: Modelo SIR, S(0) = 1,I(0) = 1,27106,R(0) = 0
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Captulo 2
Basics of Extension and LiftingProblems
To boldly go where no map has gone before
2.1. Existence problems
We begin with some metamathematics. All problems about the
existence of
maps can be cast into one of the following two forms, which are
in a sense mu-
tually dual.
The Extension Problem Given an inclusionA i X, and a map A
f Y, does
there exist a mapf :XYsuch thatf agrees withf onA?
Here the appropriate source category for maps should be clear
from the context
and, moreover, commutativity through a candidatef is precisely
the restriction
requirement; that is,
f :f i = f|A= f .
If such an f exists1, then it is called anextensionoffand is
said toextendf.In any diagrams, the presence of a dotted arrow or
an arrow carrying a ? indicates
a pious hope, in no way begging the question of its existence.
Note that we shall
usually omitfrom composite maps.The Lifting Problem Given a pair
of mapsE
pB and X
fB , does there exist
a mapf :XE, withpf =f?
1 suggests striving for perfection, crusading
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14 CAPTULO 2. BASICS OF EXTENSION AND LIFTING PROBLEMS
N
N
Figura 2.1:The log-gamma family of densities with central mean
< N > = 12
as
a surface and as a contour plot.
Thatallexistence problems about maps are essentially of one type
or the other
from these two is seen as follows. Evidently, all existence
problems are represen-
table by triangular diagrams and it is easily seen that there
are only these six
possibilities:
? ? ? ? ? ?
1 2 3 4 5 6
