Author: Kees van Overveld (tool support: Maikel Leemans)
Created with: Modeling Docgen
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 nonmathematical 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 followup 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 nonmathematical relation into mathematics.
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.





InformalA 'relation' between two or more quantities means that they cannot take independent values. There is some restriction or constraint among them. FormalA 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. 
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InformalThe relation between involved quantities does not depend on chance. Formal, where and can be arbitrary things, not involving uncertainty. 
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InformalThe relation between quantities does depend on chance. Formal, where and can be arbitrary things, characterized by some uncertainty distribution. 
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InformalThere 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, …) 
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InformalWe 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. 
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InformalThe 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 rewritten 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 nth degree equations, and rational equations. Transcendental functions lead to transcendental equations involving sin, cos, exp, log etc. 
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InformalWe want to know the value of an unknown quantitiy , and it occurs in an expression involving socalled 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 . 
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InformalWe 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 . FormalThe 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 nonlinear. 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 nonconstant coefficients. 
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InformalWe 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. 
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InformalWe 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 . FormalVarious forms occur. 
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InformalWe 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. FormalSolve from (often, there are multiple 's and multiple 's). The solution typically consists of one or more ranges of values. 
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InformalThere 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 . 
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InformalQuantities , have a relation , but is not given in the form of a recipe to immediately obtain from , or from . FormalFor instance: solve from (equation), from (inequality) or from (optimalization) 
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InformalThere 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. FormalConcepts and their properties can be written e.g. using the dotnotation; selections are written using logic () and sets are combined using set theory () 
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InformalTables 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. FormalLanguages 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. 
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InformalTriple 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. FormalLanguages 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. 
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InformalSuppose we have a set of facts and a set of rules. We might be interested in the truth or falsehood of some new fact. FormalGiven 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. 
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InformalIf 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). 
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InformalIf 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, nonnumerical sets. 
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InformalMany 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 
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InformalIn 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. FormalIf, for a function, , we can study features such as asymptotes (=the behavior of a function for the argument going to infinity). 
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InformalSome 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 
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InformalThe purpose of a model containing a function may be, to assess for which part of the domain something interesting happens. FormalGiven , we are asked to give the set of 's for which some condition holds. 
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InformalSome 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 2dimensional domain, represent a volume instead of an area. Formal 
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InformalA function , that is nonmonotonic in some domain, both ascends and descends in that domain. That is, there is at least one point where changes its direction. Formalwhere means: . Similar for . 
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InformalA function that is monotonic in some domain , either ascends (increases) or descends (decreases) for all in . Formal 
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InformalSomething 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). 
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InformalSomething 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). 
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InformalSymmetry 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. Formalwhere is the dimension of . 
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InformalSymmetry 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. Formalwhere is the dimension of . 
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InformalIn 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 . 
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InformalIn some functions , replacing by gives the same result. We can repeat this: , and so on, so such functions repeat themselves on the entire domain. Formalfor some constant . 
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InformalThere 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). FormalIn each case, we have some mapping and , . 
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InformalA 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. Formalfor integer and , where (from ‘modulo’) is the remainder by division. 
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InformalMany 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, , 
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InformalSometimes, 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. Formalfor constant 
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InformalMost 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 angledependency. FormalIf , , 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 : . 
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InformalMost straight things are straight, either because they (need to) translate, or because their construction is translationinvariant (that is: no preferred location along a line). The properties of something straight are the same when being translated. The representation of something translationinvariant as a function of location does not have to depend on the individual locations, only on the difference between locations. FormalIf , , then . 
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InformalA 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. FormalIf , , then is the preferred quantity to work with rather than and separately. 
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InformalThe 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. FormalOne 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. 
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InformalThe 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. FormalMost curves are smooth everywhere, except in finitely many points, where they have curvature radius 0. Some curves, however, are everywhere nonsmooth.Their curvature radius is 0 everywhere. Such curves have dimension larger than 1, a socalled 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. 
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InformalAdding 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’. FormalFor an additive function , . Adding is commutative, , and associative: ; it distributes over multiplication: 
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InformalIf 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, . FormalA rational function is the quotient of two polynomials; a polynomial in quantity is the sum of integer powers of , each with its own coefficient. 
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InformalIf the evaluation of a function cannot be written with finitely many addditions, subtractions, multiplications or divisions, a function is nonrational. An other word for nonrational is transcendental. FormalA nonrational 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 . 
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InformalLinear 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 . 
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InformalProportional 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: 
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InformalAffine 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 . 
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InformalRational 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 . Formalfor sufficiently large, , . For arbitrary , is a ratio of two polynomials, where is the difference of the highest occurring powers in numerator and denominator. 
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InformalRational 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 . Formalfor sufficiently large, , . For arbitrary , is a ratio of two polynomials, where is the difference of the highest occurring powers in numerator and denominator. 
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InformalEvery smooth function, in a sufficiently small part of the domain, can locally be approximated by a linear function. So we never know if some perceived linear behavior, if we extend the domain, could turn nonlinear in the long run. It requires at least three data points to make sure behavior is nonlinear. FormalAny function that cannot be written as for constant and is nonlinear. 
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InformalEvery 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. Formalis the solution of . Usually, base is 10 or Let ( is dimensionless). For logarithmic behavior, , we have that , so . The value of is given by . 
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InformalEvery 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. FormalEvery 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 . 
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InformalApart from logarithm and exponential, there are many other forms of nonrational 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. FormalNonrational functions are called transcendental. Rational combinations of transcendental functions (that is, quotients of polynomials in transcendental functions) are also transcendental. 
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InformalOften, increasing or decreasing behavior is limited by some boundary. This merits the use of max or min functions. Formalif ; 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. 
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InformalIf a behavior is the same, irrespective of the sign of the independent quantity, we may encounter the absolute value. Formalif ; 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. 
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InformalBehavior can be nonsmooth for many different reasons. Sometimes the nonsmooth 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 realvalued input quantities. FormalNonsmooth can mean discontontinuous. A function is discontinuous in if . A continuous function that is nonsmooth 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. 
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