```A Core Course on Modeling
relations
deterministic
stochastic
Week 4-Dealing with mathematical relations
functional
non-functional
sets
logic
     Contents     
1
numeric
triples
tables
other
equation
graph shape
local behavior
global behavior
monotonous
non-monotonous
inequality
algebraic
difference
differential
asymptotes
optimality
integral
domain
integral
smooth
non-smooth
non-symmetric
symmetric
equal-dim
lower-dim
rational
non-rational
min, max
mirror
abs
linear
other
non-linear
positive power
negative power
proportional
periodic
translational
other
other
modulo
trigonometry
log
affine
rotational
exponential
other
other
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Relations     
Two (three, four, ...) quantities cannot
independently take arbitrary values: there is
some mutual restriction between them.
• Mathematical R(x,y), where x and y can be arbitrary things.
notation
• Examples
• Remarks
2
does probability play an
important role in the relation?
 stochastic
if not:
 deterministic
•correlatedTo(season,nrOfBookedHolidays) (stochastic)
•greaterThan(5,3) and darkerThan(night,day) (deterministic)
•Various types of relations include symmetric, reflexive and transitive relations.
•We give most examples for arity 2, but relations can have any arity.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Deterministic     
The relation between involved quantities does
not depend on chance.
• Mathematical R(x,y), where x and y can be arbitrary things,
notation
not involving uncertainty.
• Examples
• Remarks
relations
is one quantity given in
dependency of the other(s)?
 functional
•the relation between b and v, which are the
distances between an object and its image on a
screen in case of a sharp projection with a lens
with given focal distance;
•for a given distance b between a lens with
focal length f and a screen, what should the
distance to the object (v) be so that the image is
sharp?
3
if not:
 non-functional
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Stochastic     
The relation between involved quantities does
depend on chance.
4
relations
• Mathematical R(x,y), where x and y can be arbitrary things, characterized by some uncertainty
notation
distribution.
• Examples
• Remarks
•the relation between the number of days of sunshine and the
amount of kilograms of harvested tomatoes at the end of the
season
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Functional     
There is a recipe to obtain output we need to
know (say, y) that is fully determined given
some known input (say, x)
5
deterministic
should the recipe produce
• Mathematical y=f(x); x and y can be any types (numbers,
vectors, objects, ...)
notation
a number?  numeric
• Examples
true or false?  logic
• Remarks
•The value of a propery for a concept
•the sine of an angle
• the weekday of a date (e.g., 29 July 2012 is a Friday)
•square root
•the current I through a resistor R when
applying voltage V is given by I=V/R
The collection of all x’s is called domain of f; DOM(f);
the collection of all y’s is called range of f, RANGE(f).
should the recipe produce
does the recipe involve sets
(e.g. in the form of tables)
sets
should the recipe produce
something else?  other
In mathematics, the recipe need not to be computable. For instance, the
solutions of an equation of 5th degree are a function of its coefficients,
but this function is not computable in a finite number of steps. In
modeling, we assume that, for functional relations, the recipe can be
implemented on a computer in order to give a numerical approximation
of y.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Equation     
There is enough information about some x we
need to know so that it is fully determined, but
we don’t yet have x given in the form a recipe to
express x in other, known quantities
• Mathematical f(x)=0 (equation in unknown x); Df(x)=0 or
notation
Df(x)=h(x) or Df(x)=h(x,f) (differential equation;
• Examples
• Remarks
D is differential operator and f is unknown
function; other forms occur as well)
•Most physical and economical laws come in
the form of equations, relating quantities, but
not necessarily expressing the quantity you are
interested in as a function of known quantities.
To obtain these, equations need to be solved.
There is a relation between functions and
equations: for a function y=f(x) we may want to
know the x such that y equals some given y0.
Solving the equation is the same as finding an
inverse f-1 for the function f; the unknown x is
then f-1(y0)
6
non-functional
is the unknown x a
number?  algebraic
is the unknown a function
and do we know something
 differential equation
is the unknown a function
and do we know something
 difference equation
is the unknown a function
and do we know something
equation
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Algebraic     
There is enough information about a number x
we need to know so that it is fully determined,
but we don’t yet have a recipe to express x in
other, known quantities
7
equation
• Mathematical f(x)=0 (equation in unknown x). Often x=(x1,x2,..) and f=(f1,f2,…). Equations
notation
where the unknowns are numbers can be algebraic (involving +,-,x,/ only),
including linear, quadric and n-th degree equations, and rational equations; and
transcendental equations involving sin, cos, exp, log etc.
• Examples
•Algebraic equations occur often in the form: given a function y=f(x), for what x
does f attain y=y0.
•How long do I need to put €100,- in the bank, such that 3% compound annual
interest produces €150,-?
• Remarks
Usually, in modeling, we only need a numerical approximation to x. For most types of
equations (linear, quadratic, and some forms of triginometric equations form famous
exceptions), numerical solution is the only possible approach.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Differential     
We are interested in a function f(x), but we only
have information about its derivative(s), and
perhaps some boundary conditions such as
f(x0)=y0.We need to have f in a form such that
we can evaluate f(x) in arbitrary x.
8
equation
• Mathematical Df(x)=0 (homogeneous) or Df(x)=h(x) or Df(x)=h(x,f) (inhomogeneous
notation
differential equation; D is a differential operator such as d/dx, and f is the
unknown function; other forms occur as well)
• Examples
•Dynamical systems model the temporal behavior of some signal s=f(t) as a
function of time t,
•A vessel of water leaking: dh(t)/dt=-ch(t), h(t0)=h0, where h(t) is the water level
and c relates to the size of the opening.
• Remarks
Differential equations is a vast area of mathematics that we don’t even start to develop here.
Solving by far most differential equations requires numerical approximation.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Difference     
We are interested in a function f(x), but we only
have information about its increments or
decrements if x takes discrete steps, and
perhaps some boundary conditions such as
f(x0)=y0.We need to have f in a form such that
we can evaluate f(x) in arbitrary (discrete) x.
9
equation
• Mathematical F(f)=H(x,f), where f =f(x+h)-f(x)
notation
• Examples
• Remarks
•Phenomena where time can be treated discretely
•Financial systems with monthly or annual transactions (e.g., compound
interest) or systems that are sampled in time.
(Finite) difference equations occur when we try to approximate differential equations by
numerical procedures. We encountered difference equations in week 3 when dealing with
dynamical systems.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Integral     
We are interested in a function f(x), but we only
have information about an integral of f, and
perhaps some boundary conditions f(x0)=y0.We
need to have f in a form such that we can
evaluate f(x) in arbitrary x.
10
equation
• Mathematical Various forms; see
http://en.wikipedia.org/wiki/Integral_equation
notation
• Examples
• Remarks
•Light is distributed in a space, and reflected to the walls. To find the distribution
of illumination over the walls, we need to take the reflections into account. The
reflected light, incident in some point, is an integral over all wall area, visible from
that point, of the unknown light distribution.
•Google uses the so-called page-rank algorithm. This calculates the so-called
weight of a page, which is defined as the sum of the weights of the pages
referring to it. If we approximate an integral by a sum, finding the weight of pages
is an example of an integral equation.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Inequality     
There is enough information about a quantity x
we need to know, so that it is limited to one or
more ranges, but we don’t yet have a recipe to
express this range in other, known quantities
11
non-functional
• Mathematical Solve x from f(x)<0 (often, there are multiple x’s and multiple f ’s)
notation
• Examples
• Remarks
•Scheduling often means: finding an order for a set of tasks, some of which can
be executed simultaneously, such that a total passage time is not exceed,
whereas some tasks can only start after completion of others. This amounts to
solving a set of inequalities.
•Problems involving geometric tolerances (machine parts, manufacturing,
architecture) often give rise to sets of inequalities.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Optimality     
There is a recipe to find y from x, y=f(x); we
need to know x such that y is optimal (minimal
or maximal), often subject to additional
conditions.
12
non-functional
• Mathematical min xDOM(f) f(x), subject to h(x)=0 and/or g(x)>0 where there can be multiple x’s,
notation
h’s and g’s. There can be multiple x’s and functions h and g, there is only one f,
though.
• Examples
•Most design problems aim to get a situation where something (energy
consumption, price, produced noise, comfort, …) is optimal – either maximal or
minimal
• Remarks
Mathematical optimisation requires that there is only one function f(x) to be optimized. In case
we want several things f1, f2, … to be optimal (e.g. highest efficiency and lowest price), we
can form a penalty function, P(x)=1f12(x)+ 2f22(x)+ … and minimize P. The  determine
relative importances of the various criteria f1, f2, … .
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Non-functional     
Quantities x,y have a relation R, but R is not
given in the form of a recipe to immediately
obtain x from y, or y from x.
• Mathematical For instance: solve x from f(x,y)=0 (equation),
notation
from f(x,y)<0 (inequality) or from min xDOM(f) f(x)
• Examples
• Remarks
(optimality)
13
deterministic
should we find a recipe to
obtain one quantity in terms
of the other(s)?  equation
should we find a value of x
•A bottle of wine and a corkscrew together cost
20€; the corkscrew is 3x as expensive as the
wine; what does the wine cost (equation)?
•What is the maximum number of cars X on a
road with maximum velocity Y such that no
traffic jam occurs (optimality)?
Equations, inequalities and optimality often occur
together. For instance: what is the smallest amount of
fuel (optimality) such that a given car travels 100 km
(equation, a.k.a. constraint) in at most one hour
(inequality)?
such that some y is minimal
or maximal?  optimality
should we find a range of x’s
such that some condition
holds?  inequality
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Sets     
There is a discrete amount of information in the
form of elements (=concepts), grouped in one
or more sets (a database). We need the set of
concepts fulfilling certain conditions.
• Mathematical Concepts and their properties can be written
notation
e.g. using the dot-notation; selections are
• Examples
• Remarks
written using logic (AND, OR, NOT) and sets
are combined using set theory (, , , , \,
…)
•Given a table of employees in a firm, some
being salespersons, and a table of timestamped
sales transactions, find out which employee
sold most products during last month.
•Given a knowledge base (ontology) containing
related information about books, authors, and
countries, find books of some genre, written by
an author of some nationality.
Most systems for interrogating data allow conditions to
use numerical expressions apart from logical
conditions and set-operations
14
functional
do all concepts have a set of
properties that is known
beforehand, concepts being
represented as tuples, i.e.
rows in tables?  tables
is the structure of concepts
not known beforehand, all info
in a concept being written as
triples (concept, property,
value)?  triples
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Tables     
Tables are lists of tuples, a tuple being a list of
properties with values. Tuples in a table have
the same properties. We want one or more
tuples, perhaps combining tables, representing
stored in the tables.
15
sets
• Mathematical Languages such as MYSQL have constructs for defining tables, inserting or
deleting tuples, and selecting tuples: either existing tuples that meet certain
notation
constraints, or combinations of properties of existing tuples into new tuples.
• Examples
Given a table of patients in a hospital, and a table of medical staff, find out if two
patients were treated by the same doctor (e.g., as a possible cause fo the
occurrence of a contageous infection).
• Remarks
The vast majority of active websites (web shops etc) use MYSQL or similar database
architecture.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Triples     
Triple stores are lists of triples, a triple
consisting of (concept, property, value). We
want one or more triples, typically combining
existing triples, representing the answer to a
question about the information stored in the
triple store.
16
functional
• Mathematical Languages such as SPARQL have constructs for inserting, deleting, selecting
and constructing triples: either existing triples that meet certain constraints, or
notation
• Examples
• Remarks
combinations of existing triples into new triples.
If two knowledge bases (triple stores) agree on using some standardized sets
of properties (so called namespaces, typically targeted to an application
domain), the information in the two knowledge bases can be combined by
means of automated reasoning by a computer.
Information, stemming from different origins, is rarely organised into consistent table format.
MYSQL-type queries cannot handle such differences in structure. The triple-mechanism, being
the core technology of WEB 2.0, is a way to make inferences across various triple stores,
defined and maintained by different owners.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Logic     
Suppose we have a set of facts and a set of
rules. We might be interested in the truth or
falsehood of some new fact. To this end we use
functions f with RANGE(f)={true,false}: so called
predicates.
17
functional
• Mathematical Given a set of predicates and rules of the form P(x)Q(x), where P and Q are
predicates over dummy variable x, automated inference systems can search the
notation
space of deducable propositions to see if a given proposition is true.
• Examples
Suppose we have fact1: isFruit(appel) and rule1: isFruit(x)isEdible(x). Then we
can deduce (=assess the truth of ) isEdible(appel). With more extensive sets of
facts and rules, we can have an automated inference system to help us e.g.
drawing medical diagnoses or trouble shooting complex apparatus.
• Remarks
Reasoning on the basis of facts and implications is one form of (hard or classic) AI. Except for
limited knowledge domains, the strength of classic AI seems to be quite limited. More advanced
methods use statistics, fuzzy sets, neural networks and other means.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Numeric     
If we are interested in a numerical result, given
numerical values of known quantities, we use
algebraic computation (together with standard
functions such as sin, cos, exp, …)
18
functional
are we interested in all of
• Mathematical y = f(x), where x R and yR. (Functions where x
notation
and y are restricted to rationals or integers also occur).
DOM(f) ?  global
• Examples
limited region of DOM(f)?
• Remarks
Most of highschool physics and economy
formulas are functions. For instance, the
location of a falling object as function of time
(s(t)= -1/2 gt2), the volume of geometric objects
are functions of their size, etc.
are we only interested in a
 local
Arbitrary numeric relations typically not correspond 1-to-1 to numeric functions. Example:
Ohms law corresponds to three functions: V=f(I,R)=IR; I=f(V,R)=V/R; R=f(I,V)=V/I. Also,
numeric functions can often be decomposed into simpler functions. Example: the focal length f
of a lens to map an object at distance v to an image on distance b is f=bv/(b+v). This could
also be f=p/q, where p=fp(b,v)=bv and q=fq(b,v)=b+v. The latter functions are simpler, but
quantities p and q have no immediate meaning. Developing functions is often a trade-off
between simplicity and meaning.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Other     
If we regard a function as a ‘machine’ that
produces some y depending on specification x,
and we can give a precise format for y, we can
see the production of y as function application.
• Mathematical y = f(x), where x and y are taken from arbitrary
(non-numerical) sets.
notation
19
functional
• Examples
•A list is a function: x is an index in the list, and y is the object found on the x-th
location in the list.
•A tuple (=a concept, representing an object – as in conceptual modeling) is a
function: x is the name of the attribute and y is the value of that attribute.
•Types of objects that can be precisely formatted are, e.g., images (JPG, NPG,
…), sounds (WAV, MP3, …) geometries (VRML) and many others. An application
that takes input in the form of one of such formats and produces output in the
same or a different format can be viewed as a (computable) function.
• Remarks
Standardizing object formats such as JPG, MP3, … was a first step to interpret the execution
of software applications as function evaluation. The next step is, to have a standardized
language for defining object formats. This language is XML. Our earlier presentation of
conceptual modeling in terms of concepts, properties and values can be expressed
immediately in XML.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Local behavior    
Many quantities occurring as arguments for functions
can take an unbounded range of values. The practical
interpretation of these numbers, however, often
imposes natural bounds: it is meaningless to try to
evaluate the function beyond these bounds.
• Mathematical x DOM(f) x
0
1
notation
• Examples
• Remarks
•World record times on 100 m sprint, W, descend as a
function of time t. This behavior can be approximated
as W=fW(t)=at+b, with a<0. This only makes sense for t
less than –b/a.
•Following http://en.wikipedia.org/wiki/Growth_chart,
an upper bound on the weight increase of 95% of
children can be approximated by w=fw(t)=4.0+2.0 t, w
in kg and t in months. fW is meaningless (say) for
t>1200 and for t<0.
20
numerical
Is the behavior increasing
(decreasing) over the entire
domain we are interested
in?  monotonous
Is the behavior both
increasing in some places and
decreasing in other places of
the domain we are interested
in?  non-monotonous
Limiting the domain may be a consequence of the modeled system (the upper bound of 1200 in above
example: people don’t get much older than 100 years – moreover, the function is no longer accurate for, say,
t>250); the lower bound t=0, however, comes from the used mathematical expression (t is undefined for t<0).
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Global behavior    
Many quantities occurring as arguments for functions
can take an unbounded range of values. Sometimes,
we cannot a priori restrict the domain: we need to take
the global behavior of the function into account.
• Mathematical DOM(f) = R
notation
• Examples
• Remarks
•A model for illumination strength as a function of
distance to a lamp needs to give a decreasing behavior
as a function of distance for arbitrary large distance;
•A model for diagnosing tachycardia (a heart disease)
may use a 14-day ECG as input. It is not a priori known
which part of the data contains anomalous behavior;
•The probability density P(v), say, of finding value v for
some property as a function v, needs to fulfill the
condition that the area underneath the graph of P(v)
where v ranges from - to +  is euqal to 1.
21
numeric
the behavior in the long
range?  asymptotes
Do we (want to) know something about a restricted part
of DOM(f) ?  domain
Do we (want to) know something about the area ‘underneath’ the graph of the
function?  integral
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Asymptotes    
Some functions are such that, ‘in the long run’,
the function approximates some other function,
or even a constant value. It can be important to
know such ultimate behavior (asymptote or
asymptotic behavior); conversely, when we
know asymptotes, it can help constructing the
function.
22
global
• Mathematical   > 0,  xA : |x|>|xA |  |f(x)-fA(x)| < 
notation
• Examples
• Remarks
•Some dynamic processes show complex behavior, immediately after the occurrence of an
event, but ‘calm down’ after a while (e.g., a stone falling in a pond: the circular waves, after a
while, subside). The ‘calm’ state is an asymptote.
•The asymptotic state of a plucked guitar string is a decaying harmonic vibration, irrespective
of the initial shape of the string
•The asymptotic running time for a particular sorting algorithm for N numbers approaches the
function f(N)=cN log N for constant c, and N sufficiently large.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Domain     
The purpose of a model including a function f
may be, to assess for which part of the domain
something interesting happens.
23
global
• Mathematical Given y=f(x), we are asked to give the set of x’s
notation
for which some condition P(y) holds.
• Examples
• Remarks
•The income I of a company selling goods is I=fI(P,Q)=PQ where P is the price per sold item
and Q is the quantity of sold items. For larger P, however, Q will decrease (less people buy
expensive items), so Q=fQ(P). We may ask the range of prices such that I is at least some
minimum income I0.
•In physics: radiactive radiation is absorbed in lead. The intensity is a function of the led layer
thickness. What is the thickness of a layer of lead such that 95% of incoming radiation is
absorbed?
•MRI is a technique where medical images are formed, based on detecting radiation emitted
by Hydrogen atoms in a strong magnetic field. Algorithms for MRI imaging solve the problem
of finding the domain of the function that describes the radiation emission as a function of
location in the patient’s body.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Integral     
For a class of functions, called distributions,
meaningful (e.g., measurable) quantities only
correspond to segments of the area Q underneath the
graph of the function. We masy either be interested in
Q for a given function, or the function may have to be
constrained such that a given Q is obtained.
24
global
• Mathematical Q=a…b f(x)dx
notation
• Examples
• Remarks
•In statistics, a probability density or probability distribution P(x) is a function that tells, for
some quantity, how large the chance is that its value will be between x and x+ (for 
sufficiently small). So, the chance that x is larger than some x0 is x0… P(t)dt, and the fact
that it is certain that x must have some value is expressed by - …  P(t)dt = 1.
•Suppose we have some amount P of paint and we know that painting takes C kg/m2, and h
= f(x) is the height of some interestingly shaped wall (x and h in meters), for a segment x0 …
x1 of the wall we can paint we have that x0 … x1 f(x)dx = P/C. This can be used, e.g., to find
x1 for given x0 or vice versa.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
• Remarks
     Non-monotonous     
A function y=f(x), that is non monotonous in some
domain, both ascends and descends in that domain.
That is, there is at least one point x where y changes
its direction.
 x0: x0 D : ( x<x0: desc(f(x))   x>x0:asc(f(x)) (
x<x0: asc(f(x))   x>x0:desc(f(x))), where desc(f(x))
means: >0:x1,x2: x-<x1<x2<x+:f(x2)<f(x1). Similar
for asc(f(x)).
•A normal distribution has a local maximum (which is
also a global maximum) and therefore it is not
monotonous.
•A spectrum (e.g., in physics or chemistry) is a
distribution of something (say, energy) over something
else (say, frequency) which is often not monotonous.
25
local
is there some redundancy in
the behavior (i.e., if we know
the behavior for some x, we
also know it for other x)?
 symmetric
is there no redundancy in
the behavior?
 non-symmetric
If f is smooth, a non-monotonous function has at least one stationary point (a point where
f’(x)=0) which is a local extreme (a local maximum or a local minimum). An example of a nonsmooth function that is monotonous (i.e., descends everywhere) is y=1/x: it is non-smooth in
x=0; notice that 1/x has no local extrema.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
• Remarks
     Monotonous    
A function y=f(x), that is monotonous in some domain
D, either ascends (increases) or descends (decreases)
for all x in D.
(x: x D: desc(f(x)))  (x: x D: asc(f(x)))
26
local
is the behavior everywhere
smooth (that is, if we
sufficiently zoom in in a part
of the function, does
it resemble a straight line)?
•As a function of distance to a light source, the light
intensity monotonically decreases.
•As a function of time, the total amount of industrial
waste produced by human civilisation monotonically
increases.
 smooth
does the behavior have one
or more abrupt bends?
 non-smooth
Functions can monotonically increase or decrease yet never exceed some value. If they
increase or decrease on all of R with exceeding some value, they are said to have a
(horizontal) asymptote.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
• Remarks
     Non-symmetric     
Something is symmetric is it suffices to know only part
(say, the left half of the floorplan of a mirror symmetric
building) of it in order to know all of it. If there is no
(simple) way to fill in the missing part(s), the thing is
non symmetric.
27
non-monotonous
MS:RR : (x:f(x)=f(MS(x)), where MS is a symmetry mapping (such as rotation, translation,
…) (notice: there is no simple intensional definition of the collection of symmetry mappings)
•Macroscopic processes that develop in time are not reversible. If they are also non-periodic
(e.g., the growth of a population – perhaps represented by an exponential increase in time),
they are non-symmetric.
•Processes that are sufficiently stochastic typically loose any symmetry.
A sharp definition of non-symmetric is difficult, as the class of symmetry mappings cannot be formally
specified.
‘Symmetry’ also includes permutations. E.g., the outcome of a collision between two billiard balls is the same if
we swap the balls.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
• Remarks
     Symmetric     
Something is symmetric is it suffices to know only part
(say, the left half of the floorplan of a mirror symmetric
building) of it in order to know all of it.
MS:RR : (x:f(x)=f(MS(x)), where MS is a
symmetry mapping (such as rotation, translation, …)
(notice: there is no simple intensional definition of the
collection of symmetry mappings)
•Things that result from only non-oriented forces (e.g.,
electrostatic attraction by point-charge) are spherically
symmetric.
•Things that take place the same way everywhere
(say, the collision of billiard balls) are translationally
symmetric.
•Things that take place the same way always (say,
something cooling down) are time-shift symmetric.
28
non-monotonous
due to the symmetry in the
behavior, can we write the
function with fewer
arguments?
 lower dimension
despite the symmetry in the
behavior, do we still need the
same number of arguments
to evaluate the function?
 equal dimension
•A sharp definition of symmetry is difficult, as the class of symmetry mappings cannot be formally specified.
•‘Symmetry’ also includes permutations. E.g., the outcome of a collision between two billiard balls is the same
if we swap the balls.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
• Remarks
     Equal dimension     
Symmetry of a function f may allow to drop 1 or more
arguments. This lowers the dimension of DOM(f). If
not, the domain keeps the same dimension despite the
symmetry.
g:xDOM(f):yDOM(g):g(y)=f(x)
DIM(DOM(g))<DIM(DOM(f)), where DIM(a) is the
dimension of a.
29
symmetric
does the symmetry behave
like a mirror?
 mirror
does the symmetry give rise
•It requires a pressure P =fP(v) to move a fluid through
a pipe with speed v in the case of friction. If the fluid
should flow in the opposite direction, the needed
pressure has the same behavior: fP(v)=fP(-v).
•Sociology, among other things, studies the
distribution of people in a city in dependency of all
sorts of properties. The chance that two people with
salaries s1, s2 and ages a1, a2 are neighbours is
P(s1,s2,a1,a2)=P(s2,s1,a2,a1)P(a1,a2,s1,s2): symmetric
in swapping some, but not all arguments.
to a repetitive behavior?
 periodic
is there any other form of
symmetry?
 other
A function that is periodic, f(x)=f(x+p) for some p, only needs to be known on an interval with length p to be
known everywhere. But both the interval 0 … p and the entire set of real numbers R are 1-dimensional sets.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
• Remarks
    Lower dimension     
Symmetry of a function f may allow to drop 1 or more
arguments. This lowers the dimension of DOM(f). A
function with a lower dimensional domain is attractive:
it is usually simpler to compute. It is therefore
beneficient to exploit symmetry.
g:xDOM(f):yDOM(g):g(y)=f(x)
DIM(DOM(g))<DIM(DOM(f)), where DIM(a) is the
dimension of a.
30
symmetric
does the symmetry cause the
function to invariant under
rotation?  rotational
does the symmetry cause the
function to be invariant under
translation?  translational
•The distribution of light on a plane, originating from a
point source, is rotationally symmetric: 1 dimension
•The gravity attraction between two point masses at
locations r1 and r2 only depends on the difference r1-r2
(3 instead of 6 dimensions); the strength of this
interaction only depends on |r1-r2| (only 1 dimension).
is there any other symmetry?
 other
A Core Course on Modeling
Week 4-Dealing with mathematical relations
     Mirror     
31
• Intuition
In some functions y=f(x), replacing x by –x gives the
same result. It is as if we need only half the graph and
put it in front of a mirror to see the other half.
• Mathematical
notation
f(x)= f(-x), or, in general: f(x)=f(a-x) for some a.
• Examples
•Due to inaccuracy, repeatedly measured values for some quantity Q are not identical. They
form a distribution. Unless we make systematic errors, the distribution is often mirror
symmetric around the most probably value for Q.
• Remarks
•Functions f(x) for which f(x)=f(-x) are called even. Examples are f(x)=x2 and f(x)=cos(x).
Functions such as f(x)=x, f(x)=x3, f(x)=sin(x) have the property that f(-x)= -f(x). These are not
mirror symmetric (they are called odd), but they could be called symmetric in the sense that
knowledge of their behavior on part of the domain informs us about their behavior on the
entire domain.
equal dimension
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Periodic    
In some functions y=f(x), replacing x by x+p gives the
same result. We can repeat this: f(x+p)=f(x+2p), and so
on, so such functions repeat themselves on the
domain.
• Mathematical
notation
f(x)= f(x+p) for some constant p.
• Examples
•Many phenomena are periodic in time: all sorts of
oscillations (sound), rotations (planet orbits,
electrons), financial processes (monthly salaries),
biological processes (sleep-wake, reproductive
cycles), artefacts (traffic lights).
•Many phenomena are periodic in space: all sorts of
waves and ripples (sand dunes, some types of
clouds, radio waves), construction principles (cog
wheels, brick walls, …).
• Remarks
32
equal dimension
is the repetitive behavior like
a smooth wave?
 trigonometric
Processes that are periodic in time often occur in the
combination of damping or dissipation (energy leaking out of the
system): a vibrating string after a while stops making a sound.
Such behaviors are often the product of a periodic function and
a decreasing function (such as an exponential)
does the repetitive behavior
contain sharp bends
(e.g., like sawteeth)?
 modulo
is there any other form of
repetitive behavior?
 other
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
     Other     
There are many forms of symmetry, other than mirror,
translation or rotation. For instance: a spiral and a
screw are clearly symmetric, and so are various tilings
(2D) or crystal structures (3D).
33
equal dimension
In each case, we have some mapping MS and xDOM(f), f(x)=f(MS(x)) .
• Examples
•For a spiral (such as the shape of some shells), MS is a combination of a rotation and
applying a scale factor
•For a helix (such as the shape of a drill, or unfolded DNA), MS is a combination of a rotation
and a translation
•For the scrabble board, MS is a rotation over 0, /2, , or 3/2.
• Remarks
A symmetry map MS generates a set of points when repeatedly applied to some starting point.
For instance, a rotation generates circles, a translation generates lines, the combination of a
rotation and a scaling generates spirals. Combining multiple such mappings generates highly
complex, but sometimes very beautifull so-called iterated function sets.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
• Remarks
     Modulo    
A point in time is denoted as a number of hours,
minutes and seconds. All three repeatedly take a
sequence of values: 0...23, 0…59, 0…59. This form of
periodicity is the result of integer division: the
sequences are the possible remainders of dividing,
respectively, by 24, 60 and 60.
34
periodic
0 x mod p < p for integer x and p, where mod (from ‘modulo’) is the remainder by division.
•Processes involving time (e.g., energy consumption in an urban environment) shows
periodic behavior with several periods (24 hours; 7 days; 30 / 31 days; 365 / 366 days …);
•configurations in systems of cog wheels and other periodically re-used resources (shopping
carts, labour shifts, … ) can show complex periodic behavior
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Trigonometric     
Many periodic systems involve rotations, represented
by angles as function of time. When measuring an
angle, we encounter the periodicity of the circle, and
therefore all functions derived from angles (sin, cos,
tan, …) are perodic.
35
periodic
• Mathematical
notation
sin(x)=sin(x+2), cos(x)=cos(x+2), tan(x)=tan(x+)
• Examples
•Motions of the planets and the classical motion of electrons in magnetic fields (Lorentz force)
•In electric (resistor-capacity-induction), or mechanic (damper-spring-mass) systems we don’t
see anything rotating. Still, there is often periodic behavior. This is always caused by the
existence of two opposite causes (e.g., in a mass-spring system: the inertia of the mass, and
the elastic force in the spring) where alternatingly one and the other dominates. In a circular
motion (rotation), in hindsight, we also can identify such periodic competition between two
aspects: there, they are the vertical and horizontal deviation. If one is big, the other is small,
and vice versa. This is the reason that oscillations are well described with complex numbers:
the two competing aspects are the real and imaginary part of the complex number.
• Remarks
There is an intimate connection between trigonometry, exponents and complex numbers,
expressed by Euler’s theorem: e i=cos + i sin , which underlies all techniques for solving
linear 2nd order differential equations – such as mass-spring systems and electric networks.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Other     
Sometimes, periodicity results from a construction
principle. If many copies of the same thing are brought
close together there is little alternative for periodic
arrangement. – unless an external phenomenon
causes (stochastic) perturbation. Adding heat melts a
crystal structure, turning periodicity into chaos.
36
periodic
• Mathematical
notation
f(x)=f(x+p) for constant p
• Examples
•The arrangmenet of atoms in a crystal;
•The arrangement of leaves on the branch of a tree (Acacia!), the vertebrae in a spine, or the
optic cells in a retina;
•The repetitive arrangement of all the same houses in a suburb street, lamp posts near a
motorway, or rivets on a beam in a steel construction.
• Remarks
Although they are rare, there are some examples of non-periodic crystalline structures. A
famous example is the Penrose tiling, consisting of two types of elements (quadrilaterals with
angles that are multiples of 36 degrees). First constructed as a mathematical curiosity, it was
later discovered to occur in physical reality.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Rotational     
Most round things are round, either because they (need
to) rotate, or because their construction is isotropic
(meaning: no preferred direction). The properties of
something round are the same when being rotated. So
the representation of something round as a function of
location can safely ignore the angle-dependency.
37
lower dimension
• Mathematical
notation
If (x,y)DOM(f),  : f(x,y)=f((x,y)), where (x,y) is a rotation over angle  of the point
(x,y), then g: g((x2+y2))=f(x,y). Example: a rotational paraboloid, f(x,y)=x2+y2, is identical to
g(r,)=r2, where r=(x2+y2); the latter function does not depend on : g(r,)=g(r).
• Examples
•In 3D, spherical symmetry: planets are approximately spheres because they (presumably)
were formed under the infuence of gravity only, and gravity is isotropic;
•In 3D, axial symmetry: a ceramic vase has a round cross section because it results from a
process involving rotation;
•In 2D, a cog wheel has a round projection because it needs to rotate;
• Remarks
There is a close connection between rotation and complex multiplication. A complex number can be seen as a
vector in a 2D plane (the real-imaginary plane). For two complex numbers, z1=x1+iy1, z2=x2+iy2, their product is
x1x2-y1y2+i(x1y2+x2y1). The angle with the positive real axis of z1 is arctan (y1/x1); for z2 it is arctan (y2/x2); for
z1z2 it is the sum of the two (follows from summation formula for tan(x)). So rotating (in 2D) is the same as
multiplying with a complex number with length 1 and angle with the positive real axis equal to the desired
rotation angle.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Translational     
Most straight things are straight, either because they
(need to) translate, or because their construction is
translation-invariant (meaning: no preferred location).
The properties of something straight are the same
when being translated. The representation of
something translation-invariant as a function of location
does not have to depend on the individual locations,
only on the difference between locations.
38
lower dimension
• Mathematical
notation
If (x,y)DOM(f), p : f(x1,x2)=f(x1+p,x2+p), then g: g(x1-x2)=f(x1,x2).
• Examples
•The light intensity in a point r1, due to a lightsource in point r2 must not change if we displace
both r1 and r2 over the same vector. Therefore, the light intensity can only depend on the
distance r1-r2.
•The velocities of billiard balls after a collision cannot depend on the location of the collision.
Therefore, the formula for the new velocities can only contain the difference of the locations
of the balls.
• Remarks
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Other     
A function is simpler when it has fewer arguments. It is
therefore recommended to seek if, for some purpose,
multiple arguments can be replaced by a single
argument.
39
lower dimension
• Mathematical
notation
If g,h:(x,y)DOM(f), h(g(x,y))=f(x,y), then g(x,y) is the preferred variable to work with rather
than x and y separately.
• Examples
•It had long been assumed that cholesterol levels in humans relate to life expectancy, e.
There are two kinds of cholesterol, so two levels c1 and c2. It was very difficult to find a
function e=fe(c1,c2). It turns out, however, that there is a simple function e=ge(c1/c2).
Therefore, c1/c2 is a more meaningful quantity than c1 and c2 separately.
•In relation to dimensional analysis: if some quantity q, in principle, could depend on
quantities p1, p2, …pn it is recommended to seek dimensionless quantities r1, r2, … rm (m<n)
that each are a product of some of the pi’s (perhaps to some rational powers), and express q
as a function of the fewer dimensionless quantities ri.
• Remarks
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
• Remarks
     Smooth     
The world may be whimsical, but in models we often
want to ignore small irregular variations. We often first
want to capture the global behavior. We don’t want
things in one place to be too uncorrelated to things
nearby. This is expressed in the intuition of
smoothness.
One way to formalize smoothness is, to think of the
largest circle or sphere that can touch a function graph
or function surface on either side without intersecting it:
the larger its radius, the smoother the function.
40
monotonous
if we add some constant to x,
is the difference in f in-
dependent of x?
if we multiply x by some
constant (for x sufficiently
•We may be interested to know how smooth
something is: smoother behavior can be represented
with less information;
•We may want to make something smoother (e.g.,
remove noise introduced by measuring), typically
replacing values with averages between values and
their neighbors.
Differentiability (the existence of a derivative) is loosely related
to smoothness. The function y=sin(1/x) is differentiable but
highly non-smooth; the function y=10 for x<0 and y=10+0.0001x
for x>0 is not differentiable in x=0, but it is very smooth.
large), is the change ratio
of f independent of x?
 rational
none of the two above?
 non-rational
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
• Remarks
     Non-smooth     
The world may be whimsical, and some whimsicalities
may be the essential features of the modeled system.
In those cases our model must represent these
features. Often, they constitute jumps or abrupt
changes in slope.
41
monotonous
does the bahavior have (a)
a sharp bend?
One way to formalize smoothness is, to think of the
largest circle or sphere that can touch a function graph
or function surface on either side without intersecting it.
In a jump or abrupt slope change, the maximal radius is
zero.
•The path of a billiard ball is non-smooth at the
instance of a collision, as is a light ray when it passes
from one medium to another
•Non-smoothness is the characteristic of boundary
conditions, that is: the place or circumstance where
one condition abruptly changes to another condition.
 min,max
does the behavior have local
minimum or maximum in a
sharp bend?
 absolute
none of the two above?
 other
Usually,non smoothness occurs in isolated points, called singularities. The
behavior in between singularities is
smooth and can be represented with little or no information. Therefore, singularities in a phenomenon (say, an
image, a spectrum, a distribution) carry the bulk of the non-trivial information contents of the phenomenon.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Additive     
Adding corresponds to the intuition of combining sets
or quantities. The thing added has to be of the same
dimension as what it is added to. There is a notion of
42
smooth
• Mathematical
notation
p,q: f(p)-f(q) = p-q. Adding is commutative, a+b=b+a, and associative: a+(b+c)=(a+b)+c; it
distributes over multiplication: a*(b+c)=a*b+a*c
• Examples
•Suppose we are calculating the effect of thermal isolation of a house. The total energy loss
is the sum of the energy losses through the roof, through the walls and through the windows.
•The sum to be paid for a collection of goods is the sum of the amounts to pay for the
separate goods.
•Superposition in physics holds that if a quantity q1 corresponds to phenomenon p1, and q2
to p2, the quantity corresponding to the two phenomena working simultaneously is q1+q2.
• Remarks
Alternatives for additive behavior are for instance the root of the sum of squares, or the logarithm of the sum of
exponentials. An example of the first is the addition of the energy in two interfering waves; an example of the
second is the addition of the perceived loudness of two sources of sound.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Rational     
If evaluating f(x) only involves adddition, subtraction,
multiplication or division, f is rational. Plotting both the
output and the input on logarithmic scales, for
sufficiently large |x|, gives a straight line; the slope of
which is the power p of the asymptotic behavior,
f(x)=Cxp.
• Mathematical
notation
A rational function is the quotient of two polynomials; a polynomial
in variable x is the sum of integer powers of x, each with its own
coefficient.
• Examples
•The focal distance of a lens so that a point at distance v
is sharply projected onto a screen at distance b is
bv/(b+v): a rational function of b and v.
•The response of a linear dynamic system as a function
of the frequency of an input signal, i(t)=A0sin(t) is given
by the so-called transfer function H(). This is a rational
function of .
• Remarks
43
smooth
if we multiply x with a
constant, does f scale with
the same constant?
 linear
if we multiply x with a
constant, does f scale
differently?
 non-linear
Any function in x, only consisting of combinations of +, -, * and / can be re-written to contain only one division,
numerator and denominator being polynomials in x only. Let n and m be the degrees of numerator and
denominator (that is, the highest occurring power of x), respectively. For |x| sufficiently large, the entire function
approaches Cxn-m, C being the ratio of the coefficients for x in the numerator and the denominator: This is
called the asymptotic behavior of a function; it is extremely important in doing predictions about behavior of
processes. It may be hard to assess what ‘sufficiently large’ means, though.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
• Remarks
     Non-rational     
If the evaluation of a function cannot be written with
or divisions, a function is non-rational. An other word
for non-rational is transcendental.
A non-rational function of x is most often represented
as a Taylor series: a summation of infinitely many
terms of the form aixi. Transcendental functions such
as exp, log, sin etc. can all be defined as Taylor series
with appropriate coefficients ai.
•The Gaussian distribution from probability theory;
•The exponential increase or decay as a function of
time (e.g., unbounded growth or extinction) , or
exponential attenuation as a function of the thickness
of an absorption or filtering layer.
smooth
if we multiply x with a
constant, does f increase
(decrease) with a constant?
 logarithmic
if we add a constant to x
does f scale with a constant?
 exponential
neither of the two?
The value of non-rational functions, in general, cannot be
calculated in a finite amount of steps. Efficient numerical
procedures exist, however, to make accurate estimates with
arbitrary precision.
44
 other
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Linear     
Linear behavior corresponds to proportionality: if x is
scaled by a factor s, the function value f(x) also is
scaled by the same factor. It means that evaluation of
f(x) involves a multiplication: f(x)=px; the dimension of
x can differ from the dimension of y=f(x).
• Mathematical
notation
p,q1,q2: (f(q1)-f(p))/(f(q2)-f(p)) = (q1-p)/(q2-p).
f(x)=ax+b, so a can be found as a=(f(q)-f(p))/(q-p) and
b=f(0).
• Examples
•The temperature scales Centigrade, Fahrenheit and
Kelvin are linearly related: given one, the others are
found by applying linear functions.
•Many non-linear functions locally (i.e., in a small part
of the domain) can be approximated as linear
functions, e.g., ex  1+x, sin(x)  x and (1+x)  1+x/2.
• Remarks
45
rational
if we scale x with a
constant, does f scale
with the same constant?
 proportional
if not:
 affine
•The graph of linear behavior is a straight line. If it passes through the origin, the behavior is proportional;
otherwise it is affine, written as f(x)=ax+b.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Proportional     
Proportional means: if x is scaled by a factor s, the
function value f(x) also is scaled by the same factor. It
means that evaluation of f(x) involves a multiplication:
f(x)=Cx; the dimension of x can differ from the
dimension of y=f(x).
46
linear
• Mathematical
notation
p,q: f(p)/f(q) = p/q. Also: p,q: f(p+q)=f(p)+f(q) (although this equation, over R, has also other, albeit
highly pathological, solutions than f(x)=Cx) , and f(x)=Cx. Multiplying is commutative, ab=ba, and
associative: a(,bc)=(ab)c; it does not distribute over addition: a+(bc)  (a+b)(a+c)
• Examples
•Ohm’s law: V I and V R, hence V  IR, and the constant of proportionality is 1
•Gay-Lussac’s law: P T (pressure – temperature of an amount of gas with constant volume)
•Salary is proportional to time: if every month the same amount of money is earned, the
constant of proportionality is the monthly income.
• Remarks
The graph of proportional behavior is a straight line through the origin..
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
Remarks
     Affine     
Affine behavior means: proportional plus some offset.
The offset is the value that results if the input is 0.
Since the application of an affine function involves an
addition, f(x)=ax+b; the dimension of f(x) equals the
offset; x can have a different dimension.
47
linear
p,q1,q2: (f(q1)-f(p))/(f(q2)-f(p)) = (q1-p)/(q2-p). f(x)=ax+b, so a can be found as a=(f(q)-f(p))/(qp) and b=f(0).
•All linear behavior that is not proportional, is affine.
•If we approximate some behavior y=f(x) as linear behavior in the neighborhood of some
x=x0 (sometimes called the equilibrium point, the starting position, the rest position, te start
position, etc.), f(x0) is the offset. The coefficient a is found by studying the behavior for x near
x0: f(x)  f(x0)+(x-x0)df/dx(x=x0)
•When showing trends, the current situation is often arbitrarily set to some value (e.g., 1 or
100). The differences with respect to the current situation are said to be indexed. This is to
abstract from the precise value of the current situation. In linear approximation, this is to
eliminate an unimportant b.
Sometimes the distinction between proportional and affine is called homogenous (b=0) vs. inhomogenous
(b0)
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
• Remarks
     Positive power     
Rational behavior means that, for |x| sufficiently large,
the behavior approaches Cxp for either positive or
negative p. Such behavior is characterized in that
f(q1)/f(q2) equals (q1/q2)p, so log(f(q1)/f(q2)) = p log
(q1/q2). In other words, plotting the log of the ratio of
the function values agains the ratio of the arguments
gives a straight line through the origin with slope p.
48
non-linear
for |x| sufficiently large, f(x)=Cxp, p>0. For arbitrary x, f(x) is a ratio of two polynomials, where
p is the difference of the highest occurring powers in numerator and denominator.
•The area of a shape, or the volume of a body, increases as x2 or x3, respectively, with the
size x. So the amount of pixels in an image increases with the resolution squared.
•The distance for a vehicle with speed v to come to a standstill quadratically increases with v.
Not every super-linear increasing behavior is polynomial with an integer power. Two
examples: the volume of an object, given its surface area s is proportional to s(3/2) – which is
a rational function; the amount of processing to sort N numbers is proportional to N log N –
which increases faster than proportional with N, but slower than Np for any constant p (either
integer or non-integer). The latter is non-rational behavior.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
• Remarks
     Negative power     
Rational behavior means that, for |x| sufficiently large,
the behavior approaches Cxp for either positive or
negative p. Such behavior is characterized in that
f(q1)/f(q2) equals (q1/q2)p, so log(f(q1)/f(q2)) = p log
(q1/q2). In other words, plotting the log of the ratio of
the function values agains the ratio of the arguments
gives a straight line through the origin with slope p.
49
non-linear
for |x| sufficiently large, f(x)=Cxp, p<0. For arbitrary x, f(x) is a ratio of two polynomials, where
p is the difference of the highest occurring powers in numerator and denominator.
•The gravity force due to an object with mass decreases with the square of the distance to
that object. The same is true for electrostatic force; magnetic force between two magnets
decreases with distance to the power -4.
•Light intensity decreases with distance to the power -2 from a light source.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
• Remarks
     Non-linear     
Every smooth function, in a sufficiently small part of
the domain, can locally be approximated by a linear
function. So we never know if some perceived linear
behavior, if we extend the domain, could turn nonlinear in the long run. It requires at least three data
points to make sure behavior is non-linear.
If f is a non-linear ration function, there are two
polynomials, p1(x) and p2(x), at least one of them has
degree at least 2, such that f(x)=p1(x)/p2(x).
Consider a taperecorder playing. The tape goes from left reel to
right reel. The diameter of the left real decreases, the right one
increases. If the tape runs with constant speed, the diameters of
the reels change nearly, bot not exactly, linear with time. The
sums of the diameters is nearly, but not exactly, cosntant.
50
rational
does the behavior go to
infinity for some x?
 negative power
does f stay finite for all x?
 positive power
•From observing data only, it is impossible to assess if behavior is rational or non rational. Any non-rational
behavior (say, exponential) on a finite domain can be approximated with arbitrary accuracy by a rational
function.
•Assuming that linear behavior, on a limited domain, is globally linear is called linear extrapolation. Linear
extrapolation (and extrapolation in general) is dangerous; nevertheless, extrapolation underlies all of the
emperical laws that in turn underly quantitative science.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
• Remarks
     Logarithm     
51
Every constant increase in the output requires the
non-rational
multiplication with a constant, dimensionless factor of
the input. If some behavior, when plotted on an
exponential scale, gives a straight line, the behavior is
logarithmically.
x=Blog y is the solution of y=Bx. Usually, base B is 10 or e=2.71829… Let f(c1x)=c2+f(x) (c1 is
dimensionless). For logarithmic behavior, f(x)=a log bx, we have that f(c1x)=a log bx + a log
c1=f(x)+c2, so a=c2 /log c1. The value of b is given by ef(x)/a/x.
Quantities in chemistry are often defined as logarithms of physical quantities, e.q., pH= -log
[H+], where [H+] is the concentration of H+ ions in a diluted solution. (Notice that one can argue
whether or not [H+] is dimensionless).
Quantities related to perception are often defined as logarithms of physical quantities, e.g.,
dB=log(P1/P0) where P1 is some (audio) power to be measured and P0 is a reference power.
One property of the logarithm is, that a huge variation in the input (that needs to be strictly positive) gives
rise to a small variation in the output. The logarithmic function can be said to compress the input value. To
deal with a large dynamic range, in such a way that relative changes rather than absolute changes in the
input variable are reported, the logarithm is the ideal transformation, as it linearly depends on the ratio x1/x0
of input values x1 and x0. For this reason it is believed that many biological sensors (roughly) behave as
logarithms; also for artificial sensors, a logarithmic dependency is often desirable.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
• Remarks
     Exponential     
Every constant, dimensionless increase in the input
yields the multiplication with a constant factor in the
input. If some behavior, when plotted on a logarithmic
scale, gives a straight line, the behavior is exponential.
52
non-rational
Every exponential behavior, y=ax, can be written as y=e x ln a. Euler’s constant, e=2.71829… is
such that f(x)=ex=f ’(x). For exponential behavior, f(c1+x)=c2f(x) (c1 dimensionless). For f(x)=a
ebx, we have that f(c1+x)=a ebx ebc1=f(x)c2, so b=(log c2)/c1. The factor a follows from a=f(x)/ebx.
•eax with a>0 represents increase where a constant increment of the input causes
multiplication by a factor in the output. If x is time, this occurs in unbounded growth (bacteria,
capital in the case of accumulating interest).
•eax with a<0 represents decreas where a constant increment of the input causes reduction
with a factor in the output. If x is time, this occurs in (radioactive) decay. If x is thickness, it
occurrs in absorption.
Since the derivative of eax equals aeax, the exponential function plays a crucial role in solving differential
equations. Replacing the unknown function f(x) by F(s)=f(x)e-sxdx (this is the so-called Laplace transform)
means that f’(x) is to be replaced by sF(s)-f(0). Therefore the differential equation in f(x) becomes an
algebraic equation in F(s) – which is much easier to solve. The resulting F(s) can be transformed back to a
function in x (see http://en.wikipedia.org/wiki/Laplace_transform ). The Laplace transform replaces
differentiation to multiplication; this can be compared to the logaritmh chaning multiplication to addition.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
• Remarks
     Other     
Apart from logarithm and exponential, there are many
other forms of non-rational functions, that is: functions
that cannot be computed in finite amount of additions,
subtractions, multiplications, or divisions. The
trigonometric functions are one family of examples.
53
non-rational
Non-rational functions are called transcendental. Rational combinations of transcendental functions (that is,
quotients of polynomials in transcendental functions) are also transcendental.
In most cases, linear behavior does only apply to a limited domain. If argument x becomes
sufficiently large or sufficiently small, most behaviors saturate – that is: the output f(x) is
subject to some upper and lower bound. A simple example is in population dynamics: the
reduction of prey is only proportional to the amount of predators in a limited range; similarly,
the increase of predators is proportional to the amount of prey in a limited range only. A
function to formalise this behavior is the logistic function, f(x)=1/(1+e-x). Round x=0, this
behaves as f(x)  0.5+x, but 0  f(x)  1. Other applications of the logistic function include
economics (price elasticity), chemistry, and physics.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
     Min, max     
Often, increasing or decreasing behavior is limited by
some boundary. This merits the use of max or min
functions.
54
non-smooth
max(x,y)=x if x>y; max(x,y)=y if y>x. The max function is commutative and associative: max(a,b)=max(b,a);
max(max(a,b),c)=max(a,max(b,c)). Sometimes, max(a,b,c) is used as abbreviation for max(a,max(b,c)). The
min() function can be defined as min(x,y)=x+y-max(x,y). The functions max and min are continuous but not
differentiable in the situation where x=y; on left and right sides of this singularity, the derivatives are 1 and 0,
respectively.
• Examples
In the case of modeling (financial) transactions, chemical reactions or other producer –
consumer processes, an interaction can only occur if at least one of each type of occurring
ingredients is available. For instance, selling as good requires a buyer, a seller, at least one
good on the side of the seller, and at least a sufficient amount of money on the side of the
buyer. So the number of possible transactions of some sort is
• Remarks
Since min and max are not differentiable, they prevent methods for optimisation that involve
taking derivatives, such as steepest descent methods
(http://en.wikipedia.org/wiki/Steepest_descent ) .
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
• Mathematical
notation
• Examples
• Remarks
     Absolute value     
If a behavior is the same, irrespective of the sign of the
independent variable, we may encounter the absolute
value.
55
non-smooth
abs(x)=x if x>0; abs(x)= -x if x<0. The function abs(x) is continuous, but not differentiable. It is singular for
x=0; on both sides of the singularity its derivatives are -1 and +1 , respectively.
Sometimes, the difference between two quantities determines whether or not something will
happen. E.g., some demographic model may predict that the chance for moving depends on
the difference in income between two neighbors. This is the absolute difference: the chance
is symmetric on which of the two neighbors is richer.
The absolute value is not differentiable. Sometimes it can be replaced by an expression
involving squares and square roots, which is differentiable and which may have a similar
interpretation. For example: in some interpretation of ‘distance’ between points (x1,y1) and
(x2,y2) we may set d=abs(x1-x2)+abs(y1-y2); in another interpretation we may set d=((x1x2)2+(y1-y2)2). In the first case, the points (x2,y2) with distance at most 1 to the point (x1,y1)
are in a square with side 2; in the second case, they are in a circle round (x1,y1) with
diameter 2.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
• Intuition
     Other     
Behavior can be non-smooth for many different
reasons. Sometimes the non-smooth behavior is an
artefact that we may want to get rid of (byh smoothing);
sometimes it is essential. For instance, if we deal with
problems that essentially involve integers where
integer values depend on real-valued input quantities.
56
non-smooth
• Mathematical
notation
Non-smooth can mean discontontinuous. A function is discontinuous in x=x0 if p>0 :  >0 : |f(x0- )-f(x0+
|>p. A continuous function that is non-smooth is either (in some points) not differentiable, or the radius of the
largest touching circle (or ball) that stays on one side of the function graph (or surface) is zero.
• Examples
•Suppose we have a model that calculates the optimal number of counters n =fn(t) for a
super market in depedence of the average transaction time t per customer. There is no such
thing as a half counter, so the function fn(t) will be non-smooth.
•If a function is to be taken from a table, the index of the table is an integer. Even if the
values in the table are real numbers, the dependency of this function on its input will be nonsmooth, since the argument of the function takes discrete values only.
• Remarks
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