Relation Wizard

Author: Kees van Overveld (tool support: Maikel Leemans)
Created with: Modeling Docgen

Introduction

The Relation Wizard is a tool to help finding useful notions from mathematics while translating an intuitive relationship into a formal representation (a formula, an expression or otherwise). It assumes that the user studies a non-mathematical problem or situation involving a relation between quantities, and tries to find a mathematical representation for this relation.

Starting from the notion of 'relation', the Relation Wizard poses a sequence of multiple choice questions. Each subsequent question aims to narrow down the scope of mathematical tools or notions that could be helpful for the translation. Each of the options in a multiple choice question corresponds to a mathematical notion. These are the clickable terms after '--->' in a 'Children'-section. So a 'Children'-section as a whole represents one multiple choice question.

Every mathematical notion in a 'Children'-section corresponds to one lemma (=little article in the Relation Wizard). So each multiple choice question leads to some lemmas; most lemmas contain follow-up multiple choice questions. In this way, a sequence of questions can be followed that help guiding the user through a collection of (hopefully) meaningful notions in the pursuit of translating his/her non-mathematical relation into mathematics.

Structure of the Relation Wizard

Every lemma in the Relation Wizard starts with a pink bar containing the title of the lemma. Next, lemmas can have the following sections:

Furthermore, most lemmas are graphically illustrated.

Prior to the list of lemmas, there is an aphabetically sorted overview of all lemmas in the Relation Wizard by means of clickable index. By means of this index, the Relation Wizard doubles as a compact dictionary of mathematical terminology as it is frequently used in modeling.

Lemma overview

Relation

Informal

A 'relation' between two or more quantities means that they cannot take independent values. There is some restriction or constraint among them.

Formal

A relation between quantities and is denoted as , where and can be arbitrary quantities. Sometimes, a relation is denoted by if we consider only one relation, or if the relation should be distinguished from other relations.

Children (=follow up lemmas)

Examples

  • correlatedTo(season, nrOfBookedHolidays) (stochastic: there are numerous other factors that relate the number of booked holidays to the season, but on average there may be more Summer holdiays than Winter holidays).
  • greaterThan(5, 3) and darkerThan(night, day) (deterministic: there is no uncertainty about 5 being greater than 3, and night being darker than day).

Remarks

  • Various types of relations include symmetric, reflexive and transitive relations. A symmetric relation means that ; a reflecive relation means than holds for any ; a transitive relation means that and implies that .
  • We give most examples for arity 2, but relations can have any arity.The 'arity' of a relation tells how many quantities are involved in the relation. Examples of arity=2 are , or . Both are symmetric. Examples of arity=3 are , or . Definitions of symmetry, reflexiveness and transitivity don't straightforwardly generalize to arities larger than 2.

Deterministic relation

Informal

The relation between involved quantities does not depend on chance.

Formal

, where and can be arbitrary things, not involving uncertainty.

Parent (previous lemma)

  • Relation: does probability play no important role in the relation?

Children (=follow up lemmas)

  • is one quantity given in dependency of the other(s)?
    ---> Function
  • is it not the case that one quantity given in dependency of the other(s)?
    ---> Non-functional relation

Examples

  • for a given distance between a lens with focal length and a screen, what should the distance between the object and the lens be so that the image is sharp? The theoretical relationship between , and is deterministic. In the case of practical experiments, we should repeat the measurement of , and/or a number of times. This gives collections of values for each of the quantities, and these collections have some stochastical distribution. We often take the average of the values in such a distributions as the best estimate for the theoretical values.

Stochastic relation

Informal

The relation between quantities does depend on chance.

Formal

, where and can be arbitrary things, characterized by some uncertainty distribution.

Parent (previous lemma)

  • Relation: does probability play an important role in the relation?

Examples

  • The relation between the number of days of sunshine in a season and the amount of harvested tomatoes at the end of that season is a stochastic relation. It is likely that more sunshine means more tomatoes, but due to numerous other factors, the amount of tomatoes is not fully determined by the amount of sunshine.

Function

Informal

There is a recipe to obtain output we need to know that is fully determined given some known input . This recipe is a function.

Formal

; and can be any types (numbers, vectors, objects, …)

Parent (previous lemma)

Children (=follow up lemmas)

Examples

  • The value of a propery for a concept is a function of that concept: , for instance the population (=a number) of a country is a function of that country: any other country might have another population. Similar, the dominant export product (although not a number) of a country is also a function of that country.
  • The sine of an angle is a number between -1 and 1. So the domain of function is the set of reals, the range is the set of reals between -1 and 1.
  • The weekday of a date (e.g., 29 July 2012 is a Friday). So 'weekday' is a function, the domain is a date and the range is the set Monday, Tuesday, ..., Sunday .
  • The square root of a number is the number that, multiplied with itself, yields . It can be approximated by starting with an arbitrary, non-negative and repeatedly replace by . In this case, the recipe to compute the function value does not terminate, but if we need only the first few decimals of the answer, this is an efficient way to do so.
  • The current through a resistor when applying voltage is given by , which is a function of and .
  • The voltage through a resistor when applying current is given by , which is a function of and .

Remarks

  • The collection of all 's is called domain of , ;
  • the collection of all 's is called range of , .
  • In mathematics, the recipe need not to be computable in a finite number of steps. 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. Even familiar functions such as , and can only be numerically approximated. Still, we commonly work with such functions. First, because analytic results exist that allow to rewrite functions into other expressions, even if we cannot evaluate such functions. For instance, . Further, if the answer to a problem is given in terms of functions of known quantities, where these functions can be numerically approximated to arbitrary precision, we usually say that the problem is solved. In modeling, we assume that the recipe to (approximately) evaluate a function can be implemented on a computer in order to give a numerical approximation to y.

Equation

Informal

We need to know the value of some quantity . There is enough information available to find this value, but we don't have an expression that can be straightforwardly evaluated to produce as a function of some other, known quantities. Solving an equation means: finding an expression that states as a function of the other quantities.

Formal

: an equation with unknown quantity . The unknown 'quantity' can also be a function, say . If the equation involves and its derivative(s), the equations is a differential equation. If it involves an integral of it is called an integral equation. If it involves differences such as it is called a diffrence equation.

Parent (previous lemma)

Children (=follow up lemmas)

  • is the unknown quantity a number, and does the expression in which occurs only involve addition, subtraction, multiplication or divisions?
    ---> Algebraic equation
  • is the unknown quantity a number, and does the expression in which occurs involve functions that cannot be expressed in finitely many additions, subtractions, multiplications or divisions? In other words, do expressions such as or occur in ?
    ---> Non-algebraic equation
  • is the unknown 'quantity' a function and do we know something about its derivative(s)?
    ---> Differential equation
  • is the unknown 'quantity' a function and do we know something about differences between and for some ?
    ---> Difference equation
  • is the unknown 'quantity' a function and do we know something about its integral?
    ---> Integral equation

Examples

  • Many riddles are equations in disguise: 'when John was five years younger than Suzy, Suzy was twice as old as John' can be expressed as and ; it follows that and which can be interpreted as the ages of John and Suzy at the time the riddle refers to. Strictly speaking, this is a case of two equations, namely and , with two unknowns, namely and . The unknown, in this case, is therefore a set of two numbers, and . Both equations have to hold for one pair of values for and , hence the name 'simultaneous equations'.
  • Most physical and economical laws come in the form of equations. These relate quantities, but they not necessarily express the quantity you are interested in as a function of known quantities. To obtain functional dependencies, equations need to be solved. For instance, Ohms law is usually stated . Suppose that you need to know to ensure that, given a voltage , a current results. Then you need to 'solve' , treating as an unknown. The solution is the function . In this case the solution could be obtained in closed form. That is: there is a function, involving only additions, multiplications, divisions and subtractions, that expresses in terms of the known quantities. Most often, equations cannot be solved in closed form: numerical approximtion is needed in stead.

Remarks

  • There is a relation between functions and equations: for a function we may want to know the such that equals some given . Solving the equation is the same as finding an inverse for the function ; the unknown is then . This inverse, however, not always exists in the form of a single function. The function , for instance, occurring in the equation , has two corresponding inverse functions: and , leading to the two solutions of the equation, 2 and -2.

Algebraic equation

Informal

The unknown occurs in an expression with only additions, subtractions, multiplications and divissions, such as . Also cases where more than one unknown occurs classify as algebraic equations, such as . The equation is solved, in case of one unknown , when we express as a function of known quantities only. In case of multiple unknowns, say and , the equation is equivalent to a relation between and .

Formal

(equation in unknown ). Often and . If only requires (finitely many) additions, subtractions and multiplications, it is called a polynomial function. If it also requires a division, it is called a rational function. Both polynomials and rational functions are called algebraic functions. Functions with only additions, subtractions, multiplications and multiple divisions can always be re-written to functions with merely a single division (making equal denominators). Functions that are not algebraic are called transcendental. Algebraic functions lead to algebraic equations, including linear, quadric and n-th degree equations, and rational equations. Transcendental functions lead to transcendental equations involving sin, cos, exp, log etc.

Parent (previous lemma)

  • Equation: is the unknown a number, and does the expression involving contain no other operations than additions, subtractions, multiplications and divisions?

Examples

  • An important class of algebraic equations is linear equations, where is a linear function. Solving a single linear equation amounts to solving , for with solution . In case is a vector , and is a set of linear functions, the problem is written as where is a matrix, and and are -dimensional vectors. The formal solution is , where is the inverse matrix of . The inverse of a matrix can be written in closed form by the so-called Cramer's rule, but this is not practical unless is very small. In all other cases numerical methods are used. Some are found here: http://en.wikipedia.org/wiki/System_of_linear_equations. Numerically solving sets of linear equations lies at the core of many numerical methods.
  • An algebraic equation of degree 2, , is solved in closed form by the abc-formula: .
  • Rational equations (=equations involving a division) of degree 1 lead to polynomial equations (=involving no divisions) of degree 2, and therefore can be solved with the abc formula as well. An example is the famous golden ratio, the aspect ratio of a rectangle such that the rectangle with the largest inscribing square cut off has the same aspect ratio, , defined as with solution .

Remarks

  • Usually, in modeling, we only need a numerical approximation to . For most types of equations (linear, quadratic, and some forms of trigonometric equations form important exceptions), numerical solution is the only possible approach.
  • Rational equations, such as can always be re-written to polynomial equations, as in this case: by left and right multiplying with the product of the denominators.

Non-algebraic equation

Informal

We want to know the value of an unknown quantitiy , and it occurs in an expression involving so-called transcendental functions. Transcendental functions are functions such as or that cannot be evaluated with finitely many additions, multiplications, divisions or subtractions.

Formal

(equation in unknown ). Often and .

Parent (previous lemma)

  • Equation: is the unknown quantity a number, and does the expression involving contain transcendental functions?

Examples

  • In most cases, there is no closed-form solution for non-algebraic equations. A simple heuristic approach is the following: write . Then the original equation, becomes . Start with a guess for , and repeatedly replace by . If the sequence of -values converges, it converge to a solution of the initial equation. There is no guarantee that the sequence converges; furthermore, in case there are multiple solutions, there is no straightforward way to obtain other solutions. More advanced methods for numerically solving non-algebraic are found in http://en.wikibooks.org/wiki/Numerical_Methods/Equation_Solving
  • Some non-algebraic solutions can be solved in closed form. For instance: How long do I need to put € 100,- in the bank, such that 3% compound annual interest produces € 150,-? Every year, the amount increases with a factor 1.03, so in years, the growth rate is . So , and therefore . Conversely, I could ask 'how much interest rate is required if I want this growth to occur in 10 years?'; the answer then is .

Remarks

  • Usually, in modeling, we only need a numerical approximation to . For most types of transcendental equations, numerical solution is the only possible approach.
  • An overview of two important classes of numerical methods (so-called bracketing methods and so-called open methods is found here: http://mech.utah.edu/~pardyjak/me2040/Lect4_RootsofEquations.pdf).

Differential equation

Informal

We are interested in a function , but we only have information about its derivative(s), and perhaps some boundary conditions such as . We need to have in a form such that we can evaluate in arbitrary .

Formal

The order of a differential equation is the highest derivative that occurs. A differential equation is called ordinary differential equation (abbreviated as ODE) if is a function of a single quantity. Differential equations where the unknown function is a function of multiple quantitys, say or is called a partial differential equation or PDE for short. An ODE or PDE is linear if and its derivatives occur as separate terms each with exponent 1; a term such as or makes the differential equation non-linear. A homogeneous differential equation contains no terms that don't contain . An inhomogenous differential equation is for example . Coefficients in a differential equation can be constant, as in ; a differential equation of the form has non-constant coefficients.

Parent (previous lemma)

  • Equation: is the unknown 'quantity' a function and do we know something about its derivative(s)?

Examples

  • Dynamical systems model the temporal behavior of some quantity as a function of time . Often, we don't know immediately, but we know something about derivatives of . An example is Newton's law of motion, .
  • A vessel of water leaking: , , where is the water level and 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.

Difference equation

Informal

We are interested in a function , but we only have information about its increments or decrements if takes discrete steps, and perhaps some boundary conditions such as . We need to have in a form such that we can evaluate in arbitrary , where can be either discrete or continuous.

Formal

, where . Just as with differential equations, where second order derivatives, of the unknown function occur, we have second order differences: Another name for difference equations is recurrence relations.

Parent (previous lemma)

  • Equation: is the unknown a function and do we know something about differences?

Examples

  • Phenomena where time can be treated discretely, for instance sampling sound on a CD. Most physical measurements nowadays are recorded by means of electronic equipment which takes samples at regular time intervals.
  • Financial systems with monthly or annual transactions (e.g., compound interest).

Remarks

  • (Finite) difference equations occur when we try to approximate differential equations by numerical procedures. We encounter difference equations in the lecture notes, chapter 3, when dealing with dynamical systems.
  • If we take the limit for to 0, difference equations become differential equations. It is therefore not surprising that many of the rules for calculating differences are very similar (although not totally equal) to the rules for calculating derivatives. For instance, the product rule: (notice the additional term ). The quotient rule reads: .

Integral equation

Informal

We are interested in a function , but the expression we have for involves an integral of , and perhaps some boundary conditions . We need to have in a form such that we can evaluate in arbitrary .

Formal

Various forms occur.

Parent (previous lemma)

  • Equation: is the unknown a function and do we know something about its integral?

Examples

  • 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.

Inequality

Informal

We need to know some quantity , occurring in a function , and we seek those such that the function is larger than or less than some constant.

Formal

Solve from (often, there are multiple 's and multiple 's). The solution typically consists of one or more ranges of -values.

Parent (previous lemma)

Examples

  • Scheduling tasks or processes in time often means: finding an order for these tasks or processes, some of which can be executed simultaneously, such that a total passage time is not exceeded, whereas some tasks can only start after completion of others. This amounts to solving a set of inequalities.
  • Suppose that we have a number of resources, e.q. raw materials, a certain amount of each, and a number of products we could make from these, each giving a certain market price. We can ask which choice of products we should manufacture such that the income is maximal. This is an optimization problem, but the solution should be such that none of the raw materials is used more than the available amount. This makes it a so-called linear programming problem, which amounts to dealing with a number of simultaneous inequalities (one inequality for each raw material, stating that the amount of used raw material should not be more than the available amount). See http://en.wikipedia.org/wiki/Linear_programming for a treatment of this important class of problems.
  • Problems involving geometric tolerances (machine parts, manufacturing, architecture) often give rise to sets of inequalities.

Remarks

  • An inequality often occurs as boundary condition. That is, the goal is not so much to solve the inequality, but to find solutions of an other problem such that the inequalities are met. This other problem is often an problem where something should be optimized. For example: what is the volume of the largest rectangular box such that the area is less a given amount . For dimensions width , height , and depth , this problem reads: what is the maximum of , where the inequality < should hold? A good heuristic to deal with such bounded optimization problems is to use the inequality to eliminate as much as possible of the unknown quantities: in general, an optimum is attained at the border of the feasibile domain. See http://www.math.dartmouth.edu/archive/m8f02/public_html/pauls_mws/boxeg.pdf for a further elaboration of this problem.

Optimality relation

Informal

There is a function to find from , ; we need to know such that is optimal (minimal or maximal), often subject to additional conditions.

Formal

, subject to and/or where there can be multiple 's, 's and 's. There is only one numeric function .

Parent (previous lemma)

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 to be optimized. In case we want several things , , … to be optimal (e.g. highest efficiency and lowest price), we can form a penalty function, and minimize . The determine relative importances of the various criteria , , … .
  • For optimization of subject to boundary conditions of the form , we can apply the so-called Lagrange multiplier method, see http://en.wikipedia.org/wiki/Lagrange_multiplier. As follows: form , and find an extreme of . Call this extreme ; in general, depends on . In order to establish the value of , substitute into and demand . Solve this equation for .

Non-functional relation

Informal

Quantities , have a relation , but is not given in the form of a recipe to immediately obtain from , or from .

Formal

For instance: solve from (equation), from (inequality) or from (optimalization)

Links

Parent (previous lemma)

Children (=follow up lemmas)

  • should we find a recipe to obtain one quantity in terms of the other(s)?
    ---> Equation
  • should we find a value of x such that some y is minimal or maximal?
    ---> Optimality relation
  • should we find a range of x's such that some condition holds?
    ---> Inequality

Examples

  • A bottle of wine and a corkscrew together cost € 20; the corkscrew is 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)?

Remarks

  • 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)?

Set-valued function

Informal

There is an amount of information in the form of elements (=concepts), grouped in one or more sets (typically: tables in a database). We need the set of concepts fulfilling certain conditions.

Formal

Concepts and their properties can be written e.g. using the dot-notation; selections are written using logic () and sets are combined using set theory ()

Parent (previous lemma)

  • Function: does the recipe involve sets (e.g. in the form of tables)

Children (=follow up lemmas)

  • 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

Examples

  • 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.

Remarks

  • Most systems for interrogating data allow conditions to use not only logical conditions and set-operations but also numerical expressions.

Tables

Informal

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.

Formal

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.

Parent (previous lemma)

  • Set-valued function: do all concepts have a set of properties that is known beforehand, concepts being represented as tuples, i.e. rows in tables?

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.

Triples

Informal

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.

Formal

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.

Parent (previous lemma)

  • Set-valued function: is the structure of concepts not known beforehand, all info in a concept being written as triples (concept, property, value)?

Examples

  • 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.

Remarks

  • 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.

Logic function

Informal

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.

Formal

Given a set of predicates and rules of the form , where and are predicates over dummy quantity , automated inference systems can search the space of deducable propositions to see if a given proposition is true. To this end we use functions f with : so called predicates.

Parent (previous lemma)

  • Function: should the recipe produce true or false?

Examples

  • Suppose we have fact1: and rule1: . Then we can deduce (= assess the truth of) . 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.

Numeric function

Informal

If we are interested in a numerical result of evaluating a function, given numerical values of known quantities, we use algebraic operations (addition, multiplication, subtraction and division), together with algebraic approximations for transcendental functions such as sin, cos, exp, …

Formal

, where and (Functions where and are restricted to rationals or integers also occur).

Parent (previous lemma)

  • Function: should the recipe produce a number?

Children (=follow up lemmas)

Examples

  • Most of highschool physics and economy formulas are functions. For instance, the location of a falling object as function of time (), the volume of geometric objects are functions of their size, etc.

Remarks

  • Arbitrary numeric relations typically not correspond 1-to-1 to numeric functions. Example: Ohms law corresponds to three functions: ; ; .
  • Numeric functions can often be decomposed into simpler functions. Example: the focal length of a lens to map an object at distance to an image on distance is . This could also be , where and . The latter functions are simpler, but quantities and have no immediate meaning. Developing functions is often a trade-off between simplicity and meaning.

Other function

Informal

If we regard a function as a 'machine' that produces some depending on specification , and we can give a precise format for , we can see the production of as function application.

Formal

, where and are taken from arbitrary, non-numerical sets.

Parent (previous lemma)

  • Function: should the recipe produce something else than a number?

Examples

  • A list is a function: is an index in the list, and is the object found on the -th location in the list.
  • A tuple (=a concept, representing an object as in conceptual modeling) is a function: is the name of the attribute and 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 for arbitrary types of objects, e.i.: one format both for images, for sounds, for texts, for software code, ... . This language is XML. The presentation of conceptual modeling in terms of concepts, properties and values from the lecture notes can be expressed immediately in XML.

Local features

Informal

Many functions that occur in modeling can take an unbounded range of values. Such functions have an unbounded domain: their domain is the set of all real numbers, or all points in the plane or in space. In practice, however, only a limited region, say of the domain is interesting for a modeling purpose. For this purpose, it is meaningless to try to evaluate the function beyond such a region.

Formal

Links

Parent (previous lemma)

Children (=follow up lemmas)

  • Is the behavior increasing (decreasing) over the entire region we are interested in?
    ---> Monotonic
  • Is the behavior both increasing in some places and decreasing in other places of the region we are interested in?
    ---> Non-monotonic

Examples

  • Yearly world record times on 100 m sprint, , descend as a function of the year . This behavior can be approximated as , with . This only makes sense for less than .
  • Following http://en.wikipedia.org/wiki/Growth\_chart, an upper bound on the weight increase of 95% of children can be approximated by , in kg and in months. is meaningless (say) for and for .

Remarks

  • Limiting the domain of a function to an interesting region, given the modeling purpose, 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, ). The lower bound , however, comes from the used mathematical expression ( is undefined for ); this lower bound has nothing to do with the modeled system. It is possible that a modeled system asks for a wider range of values than contained by the domain of any occurring function in the model: this is a fundamentel limitation to the applicability of the model in such case.

Global features

Informal

In some cases the entire domain of a function, occurring in a model, is relevant for the modeling purpose. In such cases we may learn something from the modeled system by studying the behavior of the function over its entire domain.

Formal

If, for a function, , we can study features such as asymptotes (=the behavior of a function for the argument going to infinity).

Links

Parent (previous lemma)

Children (=follow up lemmas)

  • Do we know something about the behavior in the long range?
    ---> Asymptotes
  • Do we (want to) know something about a restricted part of ?
    ---> Domain
  • Do we (want to) know something about the area ‘underneath’ the graph of the function?
    ---> Integral

Examples

  • 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 , say, of finding value for some property, as a function of , needs to fulfill the condition that the area underneath the graph of where ranges from to is equal to 1.

Asymptotes

Informal

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 or asymptotic behavior; conversely, when we know asymptotes, it can help constructing the function.

Formal

Parent (previous lemma)

  • Global features: Do we know something about the behavior in the long run?

Examples

  • Some dynamic processes show complex behavior, immediately after the occurrence of an event, but 'calm down' after a while. For instance, for a stone falling in a pond: the chaotic splash, after a short while, gives rise to quiet circular waves that are well-described by simple mathematical functions. This 'calm' state is an asymptote.
  • The asymptotic state of a metal rod with inhomogeneous temperature distribution is one with uniform temperature.
  • The asymptotic running time for a particular sorting algorithm for numbers approaches the function for constant , and sufficiently large (see also http://en.wikipedia.org/wiki/Computational_complexity_theory for the relation between the complexity of algorithms and asymptotic behavior).

Domain

Informal

The purpose of a model containing a function may be, to assess for which part of the domain something interesting happens.

Formal

Given , we are asked to give the set of 's for which some condition holds.

Links

Parent (previous lemma)

  • Global features: Do we (want to) know something about a restricted part of ?

Examples

  • The income of a company selling goods is where is the price per sold item and is the quantity of sold items. For larger , however, will decrease (less people buy expensive items), so . We may ask the range of prices such that is at least some minimum income .
  • In physics: radiactive radiation is absorbed in lead. The radiation 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.

Integral

Informal

Some meaningful quantities correspond the area , or segments thereof, underneath the graph of a function. We may either be interested in for a given function, or the function may have to be constrained such that a given is obtained. For a function on a 2-dimensional domain, represent a volume instead of an area.

Formal

Parent (previous lemma)

  • Global features: Do we (want to) know something about the area underneath the graph of the function?

Examples

  • In statistics, a probability density or probability distribution is a function that tells, for some quantity, how large the chance is that its value will be between and (for sufficiently small). So, the chance that is larger than some is , and the fact that it is certain that must have some value is expressed by .
  • Suppose we have some amount of paint and we know that painting takes kg/m, and is the height of some interestingly shaped wall ( and in meters), for a segment of the wall we can paint, we have that . This can be used, e.g., to find for given or vice versa.

Non-monotonic

Informal

A function , that is non-monotonic in some domain, both ascends and descends in that domain. That is, there is at least one point where changes its direction.

Formal

where means: . Similar for .

Parent (previous lemma)

  • Local features: Is the behavior both increasing in some places and decreasing in other places of the domain we are interested in?

Children (=follow up lemmas)

  • is there some redundancy in the behavior (i.e., if we know the behavior for some , we also know it for other )?
    ---> Symmetric
  • is there no redundancy in the behavior?
    ---> Non-symmetric

Examples

  • A normal distribution has a local maximum (which is also a global maximum) and therefore it is not monotonic.
  • A spectrum (e.g., in physics or chemistry) is a distribution of something (say, energy) over something else (say, frequency) which is often not monotonic.

Remarks

  • If is smooth, a non-monotonic function has at least one stationary point (a point where ) which is a local extreme (a local maximum or a local minimum). An example of a non-smooth function that is monotonic (i.e., descends everywhere) is : it is non-smooth in ; notice that has no local extrema.

Monotonic

Informal

A function that is monotonic in some domain , either ascends (increases) or descends (decreases) for all in .

Formal

Parent (previous lemma)

  • Local features: Is the behavior increasing (decreasing) over the entire domain we are interested in?

Children (=follow up lemmas)

  • 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
  • does the behavior have one or more abrupt bends?
    ---> Non-smooth

Examples

  • As a function of distance to a light source, the light intensity monotonically decreases.
  • Assuming that industrial waste never vanishes, as a function of time, the total amount of industrial waste produced by human civilisation (in kg) monotonically increases. The production of industrial waste (in kg/year), in good approximation, thus far also has been a monotonically increasing function.

Remarks

  • Functions can monotonically increase or decrease yet never exceed some value. If they increase or decrease on all of without exceeding some value, they are said to have a (horizontal) asymptote.

Non-symmetric

Informal

Something is symmetric if it suffices to know only part of it in order to know all of it. The left half of the floorplan of a mirror symmetric building is enough to know the entire floorplan. If there is no (simple) way to fill in the missing part(s), the thing is non symmetric.

Formal

, where is a symmetry mapping (such as rotation, translation, …) (notice: there is no simple intensional definition of the collection of symmetry mappings).

Parent (previous lemma)

Examples

  • A macroscopic processes that develop in time is not reversible. If it is also non-periodic (e.g., the growth of a population perhaps represented by an exponential increase in time), it is non-symmetric.
  • Processes that are sufficiently stochastic typically loose any symmetry.

Remarks

  • A sharp definition of non-symmetric is difficult, as the class of symmetry mappings cannot be formally specified.
  • 'Symmetry' also includes permutations. For instance, there is a symmetry relation between the string 'abcd' and 'dcba'. An example of permutation symmetry occurs in billiard: the outcome of a collision between two billiard balls is the same if we swap the balls prior to the collision.

Symmetric

Informal

Something is symmetric is it suffices to know only part of it in order to know all of it. The left half of the floorplan of a mirror symmetric building is enough to know the entire floorplan.

Formal

, where is a symmetry mapping (such as rotation, translation, …) (notice: there is no simple intensional definition of the collection of symmetry mappings).

Parent (previous lemma)

  • Non-monotonic: is there some redundancy in the behavior (i.e., if we know the behavior for some , we also know it for other )?

Children (=follow up lemmas)

  • 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

Examples

  • Things that result from centrally directed forces (e.g., electrostatic attraction by a point-charge, or gravity) are spherically symmetric. Since gravity is a central force, planets as well as liquid drops in zero-gravity are 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.

Remarks

  • A sharp definition of symmetry is difficult, as the class of symmetry mappings cannot be formally specified.
  • 'Symmetry' also includes permutations. For instance, there is a symmetry relation between the string 'abcd' and 'dcba'. An example of permutation symmetry occurs in billiard: the outcome of a collision between two billiard balls is the same if we swap the balls prior to the collision.

Equal dimension

Informal

Symmetry of a function sometimes allows the function to be described with fewer arguments. This lowers the dimension of . If not, the domain keeps the same dimension despite the symmetry.

Formal

where is the dimension of .

Parent (previous lemma)

  • Symmetric: despite symmetry, do we still need the same number of arguments to evaluate the function?

Children (=follow up lemmas)

  • does the symmetry behave like a mirror?
    ---> Mirror
  • does the symmetry give rise to a repetitive behavior?
    ---> Periodic
  • is there any other form of symmetry?
    ---> Other (equal dimension)

Examples

  • It requires a pressure to move a fluid through a pipe with speed in the case of friction. If the fluid should flow in the opposite direction, the needed pressure is known immediately: .
  • 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 , and ages , are neighbours is the function is symmetric in swapping some, but not all arguments.

Remarks

  • A function that is periodic, for some , only needs to be known on an interval with length to be known everywhere. But both the interval and the entire set of real numbers are 1-dimensional sets. So, periodicity does not reduce the dimension of the domain.

Lower dimension

Informal

Symmetry of a function f may allow to drop 1 or more arguments. This lowers the dimension of . A function with a lower dimensional domain is attractive: it is usually simpler to compute. It is therefore beneficient to exploit symmetry.

Formal

where is the dimension of .

Parent (previous lemma)

  • Symmetric: due to symmetry, can we write the function with fewer arguments?

Children (=follow up lemmas)

  • does the symmetry cause the function to be invariant under rotation?
    ---> Rotational
  • does the symmetry cause the function to be invariant under translation?
    ---> Translational
  • is there any other symmetry?
    ---> Other (lower dimension)

Examples

  • 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 and only depends on the difference (3 instead of 6 dimensions); the strength of this interaction only depends on (1 instead of 6 dimensions).

Mirror

Informal

In some functions , replacing by - 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. Another term for mirror symmetry is reflection symmetry.

Formal

, or, in general: for some .

Parent (previous lemma)

Examples

  • Due to inaccuracy, repeatedly measured values for some quantity are not identical. They form a distribution. Unless we make systematic errors, the distribution is often mirror symmetric around the most probable value for : the chance of finding a value larger than is equal to the chance of finding a value less than .

Remarks

  • Functions for which are called even. Examples are and . Functions such as , , have the property that . 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.
  • For even functions, the derivative for the value can be arbitrary.
  • For odd functions, the derivative for can be arbitrary; the value of is 0.

Periodic

Informal

In some functions , replacing by gives the same result. We can repeat this: , and so on, so such functions repeat themselves on the entire domain.

Formal

for some constant .

Parent (previous lemma)

Children (=follow up lemmas)

  • is the repetitive behavior like a smooth wave?
    ---> Trigonometric
  • does the repetitive behavior contain sharp bends (e.g., like sawteeth)?
    ---> Modulo
  • is there any other form of repetitive behavior?
    ---> Other (periodic)

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

  • 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 a negative exponential).

Other (equal dimension)

Informal

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).

Formal

In each case, we have some mapping and , .

Parent (previous lemma)

Examples

  • For a spiral (such as the shape of some shells), is a combination of a rotation and applying a scale factor.
  • For a helix (such as the shape of a drill, or unfolded DNA), is a combination of a rotation and a translation.
  • For the scrabble board, is a rotation over , , , or .

Remarks

  • A symmetry map 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.

Modulo

Informal

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.

Formal

for integer and , where (from ‘modulo’) is the remainder by division.

Parent (previous lemma)

  • Periodic: does the repetitive behavior contain sharp bends (e.g., like sawteeth)?

Examples

  • 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.

Trigonometric

Informal

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.

Formal

, ,

Parent (previous lemma)

  • Periodic: is the repetitive behavior like a smooth wave?

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 forms of energy. E.g., in a mass-spring system: the inertia of the mass relates to a kinetic energy , for mass and speed , and the elastic force in the spring relates to potential energy for spring constant and deviation , where alternatingly one and the other dominates. In a circular motion (rotation), in hindsight, we also can identify 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: , which underlies all techniques for solving linear 2nd order differential equations such as mass-spring systems and electric networks.

Other (periodic)

Informal

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, such as in crystals, unless an external phenomenon disturbs this regular structure. Adding heat melts a crystal structure, turning periodicity into chaos.

Formal

for constant

Parent (previous lemma)

  • Periodic: is there any other form of repetitive behavior?

Examples

  • Periodicity in non-living physical systems: the arrangement of atoms in a crystal, or the repetitive pattern in sand dunes, caused by wind and friction.
  • Periodicity in living systems: the arrangement of leaves on the branch of a tree (Acacia), the vertebrae in a spine, or the optic cells in a retina.
  • Periodicity in artifical systems: 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.

Rotational

Informal

Most round things are round, either because they (need to) rotate, or because their construction is isotropic (that is: 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 ignore the angle-dependency.

Formal

If , , where is a rotation over angle of the point , then . Example: a rotational paraboloid, , is identical to , where ; the latter function does not depend on : .

Parent (previous lemma)

  • Lower dimension: does the symmetry cause the function to invariant under rotation?

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, , , their product is . The angle with the positive real axis of is ; for it is ; for it is the sum of the two (follows from summation formula for ). 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.

Translational

Informal

Most straight things are straight, either because they (need to) translate, or because their construction is translation-invariant (that is: no preferred location along a line). 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.

Formal

If , , then .

Parent (previous lemma)

  • Lower dimension: does the symmetry cause the function to be invariant under translation?

Examples

  • The light intensity in a point , due to a lightsource in point must not change if we displace both and over the same vector. Therefore, the light intensity can only depend on the distance .
  • 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.

Other (lower dimension)

Informal

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.

Formal

If , , then is the preferred quantity to work with rather than and separately.

Links

Parent (previous lemma)

Examples

  • It had long been assumed that cholesterol levels in humans relate to life expectancy, . There are two kinds of cholesterol, so two levels and . It was very difficult to find a function . It turns out, however, that there is a simple function . Therefore, is a more meaningful quantity than and separately.
  • In relation to dimensional analysis: if some quantity , in principle, could depend on quantities it is recommended to seek dimensionless quantities that each are a product of some of the 's (perhaps to some rational powers), and express as a function of the fewer dimensionless quantities .

Smooth

Informal

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.

Formal

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.

Parent (previous lemma)

  • Monotonic: is the behavior everywhere smooth (that is, if we sufficiently zoom in in a part of the function, does it resemble a straight line)?

Children (=follow up lemmas)

  • if we add some constant to , is the difference in independent of ?
    ---> Additive
  • if we multiply by some constant (for sufficiently large), is the change ratio of independent of ?
    ---> Rational
  • none of the two above?
    ---> Non-rational

Examples

  • 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.

Remarks

  • Differentiability (the existence of a derivative) is loosely related to smoothness. The function is differentiable but highly non-smooth; the function for and for is not differentiable in , but it is very smooth.

Non-smooth

Informal

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.

Formal

Most curves are smooth everywhere, except in finitely many points, where they have curvature radius 0. Some curves, however, are everywhere non-smooth.Their curvature radius is 0 everywhere. Such curves have dimension larger than 1, a so-called fractal dimension. The intuition of fractal dimension is as follows. Consider a shape . Estimate the perimeter of by picking a point every meters apart on the circumference; suppose that we encounter points, then the coarsest estimate of the perimeter is . Nest we halve , and repeat the experiment for a more accurate estimate, giving . This gives a sequence of estimates for =0, 1, 2, ... For smooth shapes , this sequence converges to a constant. In other words, the ratio approaches 2. For very rough shapes, however, the ratio converges to a number larger than 2. The fractal dimension of the shape is defined as . For rough shapes, the fractal dimension is between 1 and 2; the larger the fractal dimension, the rougher the shape.

Parent (previous lemma)

  • Monotonic: does the behavior have one or more abrupt bends?

Children (=follow up lemmas)

  • does the bahavior have (a) flat segment(s) adjacent to a sharp bend?
    ---> Min or max
  • does the behavior have local minimum or maximum in a sharp bend?
    ---> Absolute value
  • none of the two above?
    ---> Other (non-smooth)

Examples

  • 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.

Remarks

  • Usually,non smoothness occurs in finitely many 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 information contents of the phenomenon.

Additive

Informal

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’.

Formal

For an additive function , . Adding is commutative, , and associative: ; it distributes over multiplication:

Parent (previous lemma)

  • Smooth: if we add some constant to , is the difference in independent of ?

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 corresponds to phenomenon , and to , the quantity corresponding to the two phenomena working simultaneously is .

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.

Rational

Informal

If evaluating only involves adddition, subtraction, multiplication or division, is rational. Plotting both the output and the input on logarithmic scales, for sufficiently large , gives a straight line; the slope of which is the power of the asymptotic behavior, .

Formal

A rational function is the quotient of two polynomials; a polynomial in quantity is the sum of integer powers of , each with its own coefficient.

Parent (previous lemma)

  • Smooth: if we multiply by some constant (for sufficiently large), is the change ratio of independent of ?

Children (=follow up lemmas)

  • if we multiply with a constant, does scale with the same constant?
    ---> Linear
  • if we multiply with a constant, does scale differently?
    ---> Non-linear

Examples

  • The focal distance of a lens so that a point at distance is sharply projected onto a screen at distance is : a rational function of and .
  • The response of a linear dynamic system as a function of the frequency of an input signal, is given by the so-called transfer function . This is a rational function of .

Remarks

  • Any function in , only consisting of combinations of , , and can be re-written to contain only one division, numerator and denominator being polynomials in only. Let and be the degrees of numerator and denominator (that is, the highest occurring power of ), respectively. For sufficiently large, the entire function approaches , being the ratio of the coefficients for 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.

Non-rational

Informal

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.

Formal

A non-rational function of x is often approximated by a power series, such as a Taylor series: a summation of infinitely many terms of the form . Transcendental functions such as exp, log, sin etc. can all be defined as Taylor series with appropriate coefficients .

Parent (previous lemma)

  • Smooth: none of the two above?

Children (=follow up lemmas)

Examples

  • The Gaussian distribution from probability theory is a non-rational function.
  • 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 are non-rational.

Remarks

  • The value of non-rational functions, in general, cannot be calculated in a finite amount of steps. Efficient numerical procedures are used to make accurate estimates with arbitrary precision.

Linear

Informal

Linear behavior means that the increment of a function value for a constant increment of its argument is the same for all . This is equivalent to saying that the graph of a linear function is a straight line in the -plane.

Formal

. , so a can be found as and .

Parent (previous lemma)

  • Rational: if we multiply with a constant, does scale with the same constant?

Children (=follow up lemmas)

  • if we scale with a constant, does scale with the same constant?
    ---> Proportional
  • if we scale with a constant, does not scale with the same constant?
    ---> Affine

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., , and .

Remarks

  • 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 .

Proportional

Informal

Proportional means: if is scaled by a factor , the function value also is scaled by . It means that evaluation of involves a multiplication: ; the dimension of can differ from the dimension of .

Formal

. Also: (although this equation, over , has also other, albeit highly pathological, solutions than ) , and . Multiplying is commutative, , and associative: ; it does not distribute over addition:

Parent (previous lemma)

  • Linear: if we scale with a constant, does scale with the same constant?

Examples

  • Ohm's law: and , hence , and the constant of proportionality is 1
  • Gay-Lussac's law: (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.

Affine

Informal

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, ; the dimension of equals the offset; can have a different dimension.

Formal

. , so a can be found as and .

Parent (previous lemma)

  • Linear: if we scale with a constant, does not scale with the same constant?

Examples

  • If we approximate some behavior as linear behavior in the neighborhood of some (sometimes called the equilibrium point, the starting position, the rest position, the start position, etc.), is the offset. The coefficient is found by studying the behavior for near : , the derivative estimates the value of .
  • When showing trends, say in economy (the value of a company, the average spending in a population), 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 .

Remarks

  • All linear behavior that is not proportional, is affine.
  • Sometimes the distinction between proportional and affine is called homogenous vs. inhomogenous .

Positive power

Informal

Rational behavior means that, for sufficiently large, the behavior approaches for either positive or negative . Such behavior is characterized in that equals , so . 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 .

Formal

for sufficiently large, , . For arbitrary , is a ratio of two polynomials, where is the difference of the highest occurring powers in numerator and denominator.

Parent (previous lemma)

Examples

  • The area of a shape, or the volume of a body, increases as or , respectively, with the size . So the amount of pixels in an image, and hence the size of the file in uncompressed ('raw') formats, increases with the resolution squared.
  • The distance for a vehicle with speed to come to a standstill quadratically increases with .

Remarks

  • The behavior of a function to increase faster than linear is called super linear. Not every super-linear increasing behavior is polynomial with an integer power. Two examples: the volume of an object, given its surface area is proportional to which is a non-rational function of , but a rational function of ; the amount of processing to sort numbers is proportional to which increases faster than proportional with , but slower than for any constant (either integer or non-integer). The latter is also non-rational behavior.

Negative power

Informal

Rational behavior means that, for sufficiently large, the behavior approaches for either positive or negative . Such behavior is characterized in that equals , so . 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 .

Formal

for sufficiently large, , . For arbitrary , is a ratio of two polynomials, where is the difference of the highest occurring powers in numerator and denominator.

Parent (previous lemma)

  • Non-linear: does the behavior go to infinity for some ?

Examples

  • 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.

Non-linear

Informal

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 non-linear in the long run. It requires at least three data points to make sure behavior is non-linear.

Formal

Any function that cannot be written as for constant and is non-linear.

Parent (previous lemma)

  • Rational: if we multiply with a constant, does scale differently?

Children (=follow up lemmas)

Examples

  • 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, linearly with time. The sum of the diameters is nearly, but not exactly, constant.

Remarks

  • 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 most of the emperical laws that in turn underly quantitative science. For instance, our knowledge on the universe (age, size, mass) are based on extreme extrapolations from measurments done on laboratory scale.

Logarithmic

Informal

Every constant increase in the output requires the 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.

Formal

is the solution of . Usually, base is 10 or Let ( is dimensionless). For logarithmic behavior, , we have that , so . The value of is given by .

Parent (previous lemma)

  • Non-rational: if we multiply with a constant, does increase (decrease) with a constant?

Examples

  • Quantities in chemistry are often defined as logarithms of physical quantities, e.q., , where is the concentration of ions in a diluted solution. (Notice that one can argue whether or not is dimensionless).
  • Quantities related to perception are often defined as logarithms of physical quantities, e.g., where is some (audio) power to be measured and is a reference power.

Remarks

  • 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 quantity are reported, the logarithm is the ideal transformation, as it linearly depends on the ratio of input values and . For this reason it is believed that many biological sensors (roughly) behave as logarithms; also for artificial sensors, a logarithmic dependency is often desirable.

Exponential

Informal

Every constant, dimensionless increase in the input yields the multiplication with a constant factor in the ouput. If some behavior, when plotted on a logarithmic scale, gives a straight line, the behavior is exponential.

Formal

Every exponential behavior, , can be written as . Euler's constant, is such that . For exponential behavior, ( dimensionless). For , we have that , so . The factor follows from .

Parent (previous lemma)

  • Non-rational: if we add a constant to does scale with a constant?

Examples

  • with represents increase where a constant increment of the input causes multiplication by a factor in the output. If is time, this occurs in unbounded growth (bacteria, capital in the case of accumulating interest).
  • with represents decrease where a constant increment of the input causes reduction with a factor in the output. If is time, this occurs in (radioactive) decay. If is thickness, it occurrs in absorption.

Remarks

  • Since the derivative of equals , the exponential function plays a crucial role in solving differential equations. Replacing the unknown function by (this is the so-called Laplace transform) means that is to be replaced by . Therefore the differential equation in becomes an algebraic equation in which is much easier to solve. The resulting can be transformed back to a function in (see http://en.wikipedia.org/wiki/Laplace\_transform). The Laplace transform replaces differentiation to multiplication; this can be compared to the logarithm changing multiplication to addition.

Other (non-rational)

Informal

Apart from logarithm and exponential, there are many other forms of non-rational functions, that is: functions that cannot be computed using finite amount of additions, subtractions, multiplications, or divisions. The trigonometric functions are one family of examples.

Formal

Non-rational functions are called transcendental. Rational combinations of transcendental functions (that is, quotients of polynomials in transcendental functions) are also transcendental.

Links

Parent (previous lemma)

Examples

  • In most cases, linear behavior does only apply to a limited domain. If argument becomes sufficiently large or sufficiently small, most behaviors saturate that is: the output 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, . Round , this behaves as , but . Other applications of the logistic function include economics (price elasticity), chemistry, and physics.

Min or max

Informal

Often, increasing or decreasing behavior is limited by some boundary. This merits the use of max or min functions.

Formal

if ; if . The max function is commutative and associative: ; . Sometimes, is used as abbreviation for . The function can be defined as . The functions max and min are continuous but not differentiable in the situation where on left and right sides of this singularity, the derivatives are 1 and 0, respectively.

Parent (previous lemma)

  • Non-smooth: does the bahavior have (a) flat segment(s) adjacent to a sharp bend?

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

Absolute value

Informal

If a behavior is the same, irrespective of the sign of the independent quantity, we may encounter the absolute value.

Formal

if ; if . The function is continuous, but not differentiable. It is singular for ; on both sides of the singularity its derivatives are -1 and +1, respectively.

Parent (previous lemma)

  • Non-smooth: does the behavior have local minimum or maximum in a sharp bend?

Examples

  • 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.

Remarks

  • 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 and we may set ; in another interpretation we may set . In the first case, the points with distance at most 1 to the point are in a square with side 2; in the second case, they are in a circle round with diameter 2.

Other (non-smooth)

Informal

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 (by 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.

Formal

Non-smooth can mean discontontinuous. A function is discontinuous in if . 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.

Parent (previous lemma)

Examples

  • Suppose we have a model that calculates the optimal number of counters for a super market in depedence of the average transaction time per customer. There is no such thing as a half counter, so the function 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 non-smooth, since the argument of the function takes discrete values only.