Dia 1

1
relations
A Core Course on Modeling
deterministic
Week 4-Dealing with mathematical relations
stochastic
functional
sets
non-functional
? ? ? ? ? Contents ? ? ? ? ?
1
logic
numeric
triples
tables
other
equation
graph shape
inequality
local behavior
optimality
algebraic
global behavior
difference
monotonous
differential
asymptotes
non-monotonous
integral
domain
integral
smooth
non-symmetric
non-smooth
symmetric
additive
equal-dim
rational
lower-dim
non-rational
min, max
mirror
rotational
abs
linear
periodic
translational
other
non-linear
other
other
modulo
log
trigonometry
exponential
positive power
affine
other
other
negative power
proportional
2
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Relations ? ? ? ? ?
2

• Intuition
• Mathematical notation
• Examples
• Remarks

Two (three, four, ...) quantities cannot
independently take arbitrary values there is
some mutual restriction between them.
does probability play an important role in the
relation? ? stochastic
R(x,y), where x and y can be arbitrary things.
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.

3
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Deterministic ? ? ? ? ?
3
relations

• Intuition
• Mathematical notation
• Examples
• Remarks

The relation between involved quantities does not
depend on chance.
R(x,y), where x and y can be arbitrary things,
not involving uncertainty.
is one quantity given in dependency of the
other(s)? ? functional
if not ? non-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?

4
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Stochastic ? ? ? ? ?
4
relations

• Intuition
• Mathematical notation
• Examples
• Remarks

The relation between involved quantities does
depend on chance.
R(x,y), where x and y can be arbitrary things,
characterized by some uncertainty distribution.

• the relation between the number of days of
  sunshine and the amount of kilograms of harvested
  tomatoes at the end of the season

5
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Functional ? ? ? ? ?
5
deterministic

• Intuition
• Mathematical notation
• Examples
• Remarks

There is a recipe to obtain output we need to
know (say, y) that is fully determined given some
known input (say, x)
should the recipe produce a number? ? numeric
yf(x) x and y can be any types (numbers,
vectors, objects, ...)
should the recipe produce true or false? ?
logic

• 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 IV/R

does the recipe involve sets (e.g. in the form
of tables) ?sets
should the recipe produce something else? ?
other
The collection of all xs is called domain of f
DOM(f) the collection of all ys is called range
of f, RANGE(f). 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.
6
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Equation ? ? ? ? ?
6
non-functional

• Intuition
• Mathematical notation
• Examples
• Remarks

There is enough information about some x we need
to know so that it is fully determined, but we
dont yet have x given in the form a recipe to
express x in other, known quantities
is the unknown x a number? ? algebraic
f(x)0 (equation in unknown x) Df(x)0 or
Df(x)h(x) or Df(x)h(x,f) (differential
equation D is differential operator and f is
unknown function other forms occur as well)
is the unknown a function and do we know
something about its derivative(s)? ?
differential equation

• 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.

is the unknown a function and do we know
something about differences? ? difference
equation
is the unknown a function and do we know
something about its integral?? integral
equation

• There is a relation between functions and
  equations for a function yf(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)

7
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Algebraic ? ? ? ? ?
7
equation

• Intuition
• Mathematical notation
• Examples
• Remarks

There is enough information about a number x we
need to know so that it is fully determined, but
we dont yet have a recipe to express x in other,
known quantities
f(x)0 (equation in unknown x). Often
x(x1,x2,..) and f(f1,f2,). Equations 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.

• Algebraic equations occur often in the form
  given a function yf(x), for what x does f attain
  yy0.
• How long do I need to put 100,- in the bank,
  such that 3 compound annual interest produces
  150,-?

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.
8
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Differential ? ? ? ? ?
8
equation

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
Df(x)0 (homogeneous) or Df(x)h(x) or
Df(x)h(x,f) (inhomogeneous differential
equation D is a differential operator such as
d/dx, and f is the unknown function other forms
occur as well)

• Dynamical systems model the temporal behavior of
  some signal sf(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.

Differential equations is a vast area of
mathematics that we dont even start to develop
here. Solving by far most differential equations
requires numerical approximation.
9
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Difference ? ? ? ? ?
9
equation

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
F(?f)H(x,f), where ?f f(xh)-f(x)

• 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.
10
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Integral ? ? ? ? ?
10
equation

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
Various forms see http//en.wikipedia.org/wiki/In
tegral_equation

• 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.

11
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Inequality ? ? ? ? ?
11
non-functional

• Intuition
• Mathematical notation
• Examples
• Remarks

There is enough information about a quantity x we
need to know, so that it is limited to one or
more ranges, but we dont yet have a recipe to
express this range in other, known quantities
Solve x from f(x)lt0 (often, there are multiple
xs and multiple f s)

• 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.

12
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Optimality ? ? ? ? ?
12
non-functional

• Intuition
• Mathematical notation
• Examples
• Remarks

There is a recipe to find y from x, yf(x) we
need to know x such that y is optimal (minimal or
maximal), often subject to additional conditions.
min x?DOM(f) f(x), subject to h(x)0 and/or
g(x)gt0 where there can be multiple xs, hs and
gs. There can be multiple xs and functions h
and g, there is only one f, though.

• Most design problems aim to get a situation where
  something (energy consumption, price, produced
  noise, comfort, ) is optimal either maximal or
  minimal

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, .
13
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Non-functional ? ? ? ? ?
13
deterministic

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
should we find a recipe to obtain one quantity
in terms of the other(s)? ? equation
For instance solve x from f(x,y)0 (equation),
from f(x,y)lt0 (inequality) or from min x?DOM(f)
f(x) (optimality)
should we find a value of x such that some y
is minimal or maximal? ? optimality

• 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)?

should we find a range of xs such that some
condition holds? ? inequality
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)?
14
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Sets ? ? ? ? ?
14
functional

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
do all concepts have a set of properties that is
known beforehand, concepts being represented as
tuples, i.e. rows in tables? ? tables
Concepts and their properties can be written e.g.
using the dot-notation selections are 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.

is the structure of concepts not known
beforehand, all info in a concept being written
as triples (concept, property, value)? ? triples
Most systems for interrogating data allow
conditions to use numerical expressions apart
from logical conditions and set-operations
15
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Tables ? ? ? ? ?
15
sets

• Intuition
• Mathematical notation
• Examples
• Remarks

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 the answer
to a question about the information stored in the
tables.
Languages such as MYSQL have constructs for
defining tables, inserting or deleting tuples,
and selecting tuples either existing tuples that
meet certain constraints, or combinations of
properties of existing tuples into new tuples.
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).
The vast majority of active websites (web shops
etc) use MYSQL or similar database architecture.
16
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Triples ? ? ? ? ?
16
functional

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
Languages such as SPARQL have constructs for
inserting, deleting, selecting and constructing
triples either existing triples that meet
certain constraints, or 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.
17
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Logic ? ? ? ? ?
17
functional

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
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 space of deducable propositions to see
if a given proposition is true.
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.
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.
18
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Numeric ? ? ? ? ?
18
functional

• Intuition
• Mathematical notation
• Examples
• Remarks

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, )
are we interested in all of DOM(f) ? ? global
y f(x), where x ?R and y?R. (Functions where x
and y are restricted to rationals or integers
also occur).
are we only interested in a limited region of
DOM(f)? ? local
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.
Arbitrary numeric relations typically not
correspond 1-to-1 to numeric functions. Example
Ohms law corresponds to three functions
Vf(I,R)IR If(V,R)V/R Rf(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 fbv/(bv). This could also be
fp/q, where pfp(b,v)bv and qfq(b,v)bv. 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.
19
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Other ? ? ? ? ?
19
functional

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
y f(x), where x and y are taken from arbitrary
(non-numerical) sets.

• 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.

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.
20
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Local behavior? ? ? ? ?
20
numerical

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
Is the behavior increasing (decreasing) over
the entire domain we are interested in? ?
monotonous
x0 ?DOM(f) ?x1

• World record times on 100 m sprint, W, descend as
  a function of time t. This behavior can be
  approximated as WfW(t)atb, with alt0. This only
  makes sense for t less than b/a.
• Following http//en.wikipedia.org/wiki/Growth_char
  t, an upper bound on the weight increase of 95
  of children can be approximated by
  wfw(t)4.02.0 ?t, w in kg and t in months. fW
  is meaningless (say) for tgt1200 and for tlt0.

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 dont get much older than 100
years moreover, the function is no longer
accurate for, say, tgt250) the lower bound t0,
however, comes from the used mathematical
expression (?t is undefined for tlt0).
21
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Global behavior? ? ? ? ?
21
numeric

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
Do we know something about the behavior in the
long range? ? asymptotes
DOM(f) R

• 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.

Do we (want to) know some- thing about a
restricted part of DOM(f) ? ? domain
Do we (want to) know some- thing about the area
under- neath the graph of the function? ?
integral
22
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Asymptotes? ? ? ? ?
22
global

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
? ? gt 0, ? xA xgtxA ? f(x)-fA(x) lt ?

• 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.

23
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Domain ? ? ? ? ?
23
global

• Intuition
• Mathematical notation
• Examples
• Remarks

The purpose of a model including a function f may
be, to assess for which part of the domain
something interesting happens.
Given yf(x), we are asked to give the set of xs
for which some condition P(y) holds.

• The income I of a company selling goods is
  IfI(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 QfQ(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 patients body.

24
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Integral ? ? ? ? ?
24
global

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
Q?ab f(x)dx

• 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.

25
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Non-monotonous ? ? ? ? ?
25
local

• Intuition
• Mathematical notation
• Examples
• Remarks

A function yf(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.
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
? x0 x0 ?D (? xltx0 desc(f(x)) ? ?
xgtx0asc(f(x))? (? xltx0 asc(f(x)) ? ?
xgtx0desc(f(x))), where desc(f(x)) means
??gt0?x1,x2 x-?ltx1ltx2ltx?f(x2)ltf(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.

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
non-smooth function that is monotonous (i.e.,
descends everywhere) is y1/x it is non-smooth
in x0 notice that 1/x has no local extrema.
26
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Monotonous? ? ? ? ?
26
local

• Intuition
• Mathematical notation
• Examples
• Remarks

A function yf(x), that is monotonous in some
domain D, either ascends (increases) or descends
(decreases) for all x in D.
is the behavior everywhere smooth (that is, if
we sufficiently zoom in in a part of the
function, does it resemble a straight line)? ?
smooth
(?x x ?D desc(f(x))) ? (?x x ?D asc(f(x)))

• 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.

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.
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Non-symmetric ? ? ? ? ?
27
non-monotonous

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
??MSR?R (?xf(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.
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Symmetric ? ? ? ? ?
28
non-monotonous

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
due to the symmetry in the behavior, can we
write the function with fewer arguments? ?
lower dimension
?MSR?R (?xf(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.

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.

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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Equal dimension ? ? ? ? ?
29
symmetric

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
does the symmetry behave like a mirror? ?
mirror
??g?x?DOM(f)?y?DOM(g)g(y)f(x)
?DIM(DOM(g))ltDIM(DOM(f)), where DIM(a) is the
dimension of a.
does the symmetry give rise to a repetitive
behavior? ? periodic

• 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,a
  2,s1,s2) symmetric in swapping some, but not all
  arguments.

is there any other form of symmetry? ? other
A function that is periodic, f(x)f(xp) 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.
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? Lower dimension ? ? ? ? ?
30
symmetric

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
does the symmetry cause the function to
invariant under rotation? ? rotational
?g?x?DOM(f)?y?DOM(g)g(y)f(x)
?DIM(DOM(g))ltDIM(DOM(f)), where DIM(a) is the
dimension of a.
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 instead of 2
• 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
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Mirror ? ? ? ? ?
31
equal dimension

• Intuition
• Mathematical notation
• Examples
• Remarks

In some functions yf(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.
f(x) f(-x), or, in general f(x)f(a-x) for some
a.

• 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.

• 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.

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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Periodic? ? ? ? ?
32
equal dimension

• Intuition
• Mathematical notation
• Examples
• Remarks

In some functions yf(x), replacing x by xp
gives the same result. We can repeat this
f(xp)f(x2p), and so on, so such functions
repeat themselves on the domain.
is the repetitive behavior like a smooth wave?
? trigonometric
f(x) f(xp) for some constant p.
does the repetitive behavior contain sharp bends
(e.g., like sawteeth)? ? modulo

• 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, ).

is there any other form of repetitive behavior?
? other

• 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)

33
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Other ? ? ? ? ?
33
equal dimension

• Intuition
• Mathematical notation
• Examples
• Remarks

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).
In each case, we have some mapping MS and
?x?DOM(f), f(x)f(MS(x)) .

• 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.

• 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.

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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Modulo? ? ? ? ?
34
periodic

• Intuition
• Mathematical notation
• Examples
• Remarks

A point in time is denoted as a number of hours,
minutes and seconds. All three repeatedly take a
sequence of values 0...23, 059, 059. This form
of periodicity is the result of integer division
the sequences are the possible remainders of
dividing, respectively, by 24, 60 and 60.
0? x mod p lt 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

35
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Trigonometric ? ? ? ? ?
35
periodic

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
sin(x)sin(x2?), cos(x)cos(x2?),
tan(x)tan(x?)

• 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 dont
  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.

• There is an intimate connection between
  trigonometry, exponents and complex numbers,
  expressed by Eulers 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.

36
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Other ? ? ? ? ?
36
periodic

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
f(x)f(xp) for constant p

• 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.

• 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.

37
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Rotational ? ? ? ? ?
37
lower dimension

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
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(?(x2y2))f(x,y). Example a
rotational paraboloid, f(x,y)x2y2, is identical
to g(r,?)r2, where r?(x2y2) the latter
function does not depend on ? g(r,?)g(r).

• 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

• 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,
  z1x1iy1, z2x2iy2, their product is
  x1x2-y1y2i(x1y2x2y1). 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.

38
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Translational ? ? ? ? ?
38
lower dimension

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
If ?(x,y)?DOM(f), ?p f(x1,x2)f(x1p,x2p),
then ?g g(x1-x2)f(x1,x2).

• 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.

39
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Other ? ? ? ? ?
39
lower dimension

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
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.

• 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
  efe(c1,c2). It turns out, however, that there is
  a simple function ege(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 (mltn) that
  each are a product of some of the pis (perhaps
  to some rational powers), and express q as a
  function of the fewer dimensionless quantities ri.

40
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Smooth ? ? ? ? ?
40
monotonous

• Intuition
• Mathematical notation
• Examples
• Remarks

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 dont want things in one place to be
too uncorrelated to things nearby. This is
expressed in the intuition of smoothness.
if we add some constant to x, is the difference
in f in- dependent of x? ? additive
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.
if we multiply x by some constant (for x
sufficiently large), is the change ratio of f
independent of x? ? rational

• 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.

none of the two above? ? non-rational
Differentiability (the existence of a derivative)
is loosely related to smoothness. The function
ysin(1/x) is differentiable but highly
non-smooth the function y10 for xlt0 and
y100.0001x for xgt0 is not differentiable in
x0, but it is very smooth.
41
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Non-smooth ? ? ? ? ?
41
monotonous

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
does the bahavior have (a) flat segment(s)
adjacent to a sharp bend? ? min,max
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.
does the behavior have local minimum or maximum
in a sharp bend? ? absolute

• 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.

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.
42
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Additive ? ? ? ? ?
42
smooth

• Intuition
• Mathematical notation
• Examples
• Remarks

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 0, corresponding to adding
nothing, or to not adding.
?p,q f(p)-f(q) p-q. Adding is commutative,
abba, and associative a(bc)(ab)c it
distributes over multiplication a(bc)abac

• 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 q1q2.

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.
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Rational ? ? ? ? ?
43
smooth

• Intuition
• Mathematical notation
• Examples
• Remarks

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.
if we multiply x with a constant, does f scale
with the same constant? ? linear
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.
if we multiply x with a constant, does f scale
differently? ? non-linear

• The focal distance of a lens so that a point at
  distance v is sharply projected onto a screen at
  distance b is bv/(bv) 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 ?.

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.
44
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Non-rational ? ? ? ? ?
44
smooth

• Intuition
• Mathematical notation
• Examples
• Remarks

If the evaluation of a function cannot be written
with finitely many addditions, subtractions,
multiplications or divisions, a function is
non-rational. An other word for non-rational is
transcendental.
if we multiply x with a constant, does f
increase (decrease) with a constant? ?
logarithmic
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.
if we add a constant to x does f scale with a
constant? ? exponential

• 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.

neither of the two? ? other
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.
45
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Linear ? ? ? ? ?
45
rational

• Intuition
• Mathematical notation
• Examples
• Remarks

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 yf(x).
if we scale x with a constant, does f scale
with the same constant? ? proportional
?p,q1,q2 (f(q1)-f(p))/(f(q2)-f(p))
(q1-p)/(q2-p). f(x)axb, so a can be found as
a(f(q)-f(p))/(q-p) and bf(0).
if not ? affine

• 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 ? 1x, sin(x) ? x and
  ?(1x) ? 1x/2.

• 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)axb.

46
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Proportional ? ? ? ? ?
46
linear

• Intuition
• Mathematical notation
• Examples
• Remarks

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 yf(x).
?p,q f(p)/f(q) p/q. Also ?p,q
f(pq)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, abba, and
associative a(,bc)(ab)c it does not distribute
over addition a(bc) ? (ab)(ac)

• Ohms law V? I and V? R, hence V ? IR, and the
  constant of proportionality is 1
• Gay-Lussacs 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.

The graph of proportional behavior is a straight
line through the origin..
47
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? Affine ? ? ? ? ?
47
linear

• Intuition
• Mathematical notation
• Examples
• Remarks

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)axb
the dimension of f(x) equals the offset x can
have a different dimension.
?p,q1,q2 (f(q1)-f(p))/(f(q2)-f(p))
(q1-p)/(q2-p). f(x)axb, so a can be found as
a(f(q)-f(p))/(q-p) and bf(0).

• All linear behavior that is not proportional, is
  affine.
• If we approximate some behavior yf(x) as linear
  behavior in the neighborhood of some xx0
  (sometimes called the equilibrium point, the
  starting position,