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- Mathematics In Europe - Applied mathematics

of a computer It consists of an infinite tape divided into successive cells a head a finite set of states and a set of rules for transitions from one state to another The cells of the tape can hold one symbol each from a given set of symbols say and as symbols used to represent data and a blank to represent the end of the data The cells are numbered with integers The head can read or write a symbol from to one cell and move one place forward or backward At any moment the machine is in one state special states are the initial one and one or more final states Each transition rule is of the form given a state and a symbol read by the head change the state write a symbol under the head and move the head in a specific direction For example the Turing machine that recognizes if the tape holds data of the form i e first a number at least two of then a number at least one of has to have two final states one when the answer is yes and another for a no answer Denote them by Y and N and by 0 the initial state start of the program Suppose that in the initial state the head is above the leftmost symbol on the tape Obviously at no moment the head needs to write anything The first rule is that if in the initial state the head reads a or a blank denoted by the state should switch to N the sequence is not of the requested form and head should stop Otherwise move the head one place to the right and switch to a state we shall denote by 1 In state 1 if the head reads it is possible that the sequence is of the required form so the machine should switch to a state 2 and move the head one place to the right otherwise it should stop the head and switch to N By further analysis of the problem one gets all the needed rules and states one can represent them graphically as in figure 3 Figure 3 Turing machine recognizing a sequence of the form The circles represent states the arrows represent transition rules and a label a n should be read if head reads a move the head by n and switch to the state indicated by arrow where n 1 means move the head one place to the right and n 0 means stop the head Mathematical physics studies interconnections between mathematics and physics It is concerned with applications of mathematics to physics It also develops mathematical models and methods suitable for describing physical phenomena and theories Although physical problems were solved by mathematical methods since ancient times mathematical physics in the modern sense of the word was created when sir Isaac Newton developed calculus to solve problems related to motion The power of calculus for modeling physical laws was recognized by Newton and his contemporaries and differential equations as models were used since the beginnings of calculus Up to this day ordinary and more often partial differential equations are a typical ingredient of mathematical physics and many results abut them were discovered and proven because of physics A typical partial differential equation from mathematical physics is the Poisson equation 2 φ f The symbol 2 denotes an operator known as the Laplacian it is a way to generalize taking second derivatives to multivariable functions It arises in physical problems about finding potentials φ for a known density function f The best known example comes from electrostatics where f ρ ε 0 and the equation describes the relationship of the electric potential φ and the charge density ρ Many other areas of analysis potential theory variational calculus Fourier thery are also used in all areas of physics and developed for physical reasons Functional analysis and operator theory were developed partly as mathematical foundations of quantum theory But other areas of mathematics also partly belong to mathematical physics Algebra particularly group theory and topology play a fundamental role in the theory of relativity and quantum field theory Geometry interestingly not only differential and classical geometry but also abstract geometry e g noneuclidean geometries are also substantial for modern physics relativity theory string theory Combinatorics and probability are foundations of statistical mechanics this is the study of thermodynamics of systems consisting of a large number of particles Because there is no sharp distinction where e g analysis ends and mathematical physic begins many of the results cannot be clearly classified Thus Lagrangian mechanics a re formulation of classical Newtonian mechanics in terms of differential equations about potentials is considered mathematical physics although the general study of differential equations where some results applicable to Lagrangian mechanics are also proven is pure mathematics It may also be interesting to know that 10 of the 64 official mathematical areas as listed in the Mathematical Subject Classification are mathematical physics subdisciplines mechanics of particles and systems mechanics of deformable solids fluid mechanics optics and electromagnetic theory classical thermodynamics and heat transfer quantum theory statistical mechanics and structure of matter relativity and gravitational theory astronomy and astrophysics geophysics Although probably the oldest field of applications of mathematical results physics is by far not the only one As already said computer science is nowadays a separate field in applied mathematics Mathematical biology and chemistry also exist Also in the 20th century so many applications of mathematics to economy emerged that now we have two mathematical disciplines operations research and game theory that were developed for applications in economy and social and behavioral sciences Operations research is the mathematical discipline studying various mathematical methods coming from various fields probability and statistics graph theory optimization that help make better decisions It originated in military problems before World War II and after the war the techniques were applied to problems in economics and society A typical class of problems studyed in operations

Original URL path: http://mathematics-in-europe.eu/tr/45-information/landscape-areas/92-applied-mathematics (2013-11-18)

Open archived version from archive - Mathematics In Europe - Combinatorics

at the same time there are m n ways to do one of the things the product rule if there are m ways to do one thing and n ways to do another there are mn ways to do both things and the pigeonhole principle if n things are to be put into m boxes and m n then at least one box has to contain more than one thing Figure 2 a The sum and product rules the objects to choose from If you are given three cylinders four balls and five cubes and want to choose two objects of different kinds in how many ways can you do it You can choose one cylinder and one ball in 3 4 ways one cylinder and one cube in 3 5 ways and one ball and one cube in 4 5 ways product rule Consequently you can choose two objects of different kinds in 3 4 3 5 4 5 47 ways sum rule The 47 choices are shown below Figure 2 b The 47 possible choices of two objects of different kinds There are also other more advanced counting techniques For example the cardinalities of sequences of sets are often arranged into generating functions Generating functions are formal power series with coefficients that contain information about a sequence They are analysed with techniques of analysis For example the famous Fibonacci numbers are defined as the sequence of numbers formed from the starting numbers 0 and 1 in such a way that each next Fibonacci number is the sum of the previous two Thus one obtains the Fibonacci sequence 0 1 1 2 3 5 8 13 21 34 The easiest way to obtain a formula for the n th Fibonacci number f n is by using a generating function We define it as the sum F x of all terms of the form f n x n for n 0 1 2 This sum does not converge for any x this means that no matter what value one inserts for x there is no reasonable way to assign a number to the infinite sum Still but by performing formal computations one can still obtain many insights in the sequence of coefficients Since by definition f 0 0 f 1 1 f n f n 1 f n 2 the sum of all f n x n can be written as x plus the sum of all terms of the form f n 1 f n 2 x n for n 1 After rearrangements one gets that F x x xF x x 2 F x so F x x 1 x x 2 The last fraction can be written as 1 1 Φ x 1 1 φ x 5 where Φ 1 5 2 and φ 1 5 2 As 1 1 ax can be written in analysis only if x a because otherwise the series does not converge but here one performs the expansion formally without considering the

Original URL path: http://mathematics-in-europe.eu/tr/45-information/landscape-areas/90-combinatorics (2013-11-18)

Open archived version from archive - Mathematics In Europe - Set theory and logic

out that they are equivalent by means of propositional logic In predicate logic statements P x containing a variable x like the mentioned x is even become propositions if either a value is assigned to the variable e g put x 5 to get 5 is even and now one can determine that it is a false proposition or by using one of the two quantifiers for all universal quantifier and exists existential quantifier where x P x means for all x in the universe P x and x P x means there exists a x in the universe P x In both cases the truth depends on what one takes as the universe For example x such that x is even is true if the universe is that of integers but false if the universe is the set of all odd integers Proof theory is a branch of mathematical logic in which proofs are considered as mathematical objects It is of a syntactic nature meaning that it deals with formal rules without consideration of their interpretation In contrast model theory the study of mathematical structures using the tools of mathematical logic is of a semantic nature it is interested in interpreations Other subdisciplines of mathematical logic include type theory computability and recursion theory algebraic logic Consider two sets of points in the coordinate plane the yellow rectangle A and the red dots B as shown above the third picture in the row above shows the relative position of the two sets The orange region on the left is the intersection and the blue on the right the union of A and B Figure 1 Illustration of two basic operations from algebra of sets intersection and union Many school maths curricula include the description of some topics from naive set thery sets and a few basic operations with sets union intersection difference But rare nonprofessionals know that set theory is not about finding an intersection of a set of groceries with a set of fruits from a market stand or the intersection union and diference of sets shown in figure 1 above Set theory provides an exact way to deal with the infinite as well as answering fundamental questions like what a positive integer is Cantor s proof from 1873 showing that there are various levels of infinity not all infinities are equally infinite is usually taken as the birthmark of set theory Cantor found a way to compare the sizes of infinite sets The idea is simple if you want to know if two finite sets have the same number of elements the cardinality all one has to do is to check if each of the elements of one set can be paired with one element of the other set For example if you want to check if the number of seats in the classroom is enough for all the students you can instead of counting both the seats and the students let each student sit on a chair

Original URL path: http://mathematics-in-europe.eu/tr/45-information/landscape-areas/91-set-theory-and-logic (2013-11-18)

Open archived version from archive - Mathematics In Europe - Geometry

of systems of polynomial equations with coefficients from a specified field e g with rational coefficients Such sets are known as algebraic varieties and in a very simplified way one can say that algebraic geometry is the study of algebraic varieties As already Descartes noted a polynomial equation in two unknowns represents a curve in a coordinate plane a two dimensional geometric space For example the set of the solutions of the polynomial equation x 3 y 3 3 xy can be represented in the Cartesian coordinate plane as a curve called the folium of Descartes Figure 2 below This idea was generalized to more unknowns and higher dimensional geometric spaces For example the set of the solutions of the equation x 3 3 xy 2 z can be represented in the Cartesian coordinate space as a surface called the monkey saddle Figure 2 below The probably most famous application of algebraic geometry was its use in the proof of Fermat s last theorem by Andrew Wiles but this geometric discipline has many other applications both in other mathematical fields and in sciences Figure 2 The folium of Descartes left and the monkey saddle right Projective geometry is nowadays considered as a part of algebraic geometry Typical ideas in projective geometry include the following two all parallel lines intersect in a point at infinity and the set of all these points representing all possible directions of lines form a line at infinity Projective geometry developed from the theory of perspective where it was noted that parallel lines seem to meet at infinity Altough Desargues is considered the founder or one of the founders of this discipline there are much older results in projective geometry The most famous of them is the Pappus theorem if A B and C are three points on a line and A B and C are three points on another line then the intersection points 1 of AB with BA 2 of AC with CA and 3 of BC with CB all lie on the same line see Figure 3 below A generalization of this theorem is the Pascal s mystic hexagram theorem if 1 2 3 1 2 and 3 are six points on a conic section circle ellipse hyperbola parabola then the intersection points of lines 12 with 21 13 with 31 and of 23 with 32 are collinear see Figure 3 below This theorem has to be proved only for the case of a circle because all other conic sections are projective transformations of a circle projective transformations can be described as mappings that map lines to lines and one can define projective geometry as the study of properties that do not change when projective transformations are applied Figure 3 Pappus theorem left and Pascal s mystic hexagram theorem right Differential geometry uses calculus techniques in geometric studies It was first developed at the turn of the 18 19th century Monge and in its beginnings it studied curves and surfaces in the three

Original URL path: http://mathematics-in-europe.eu/tr/45-information/landscape-areas/86-geometry (2013-11-18)

Open archived version from archive - Mathematics In Europe - Probability and statistics

to heads and another say 1 to tails In this way we define a real valued function X on Ω Such a function is called a random variable a function X defined on the sample space is a random variable if it assigns unique numerical values to elementary events the function must also meet some other requirements and there are also random variable taking values in other sets besides R If one is interested in how many people come to concerts in a city and is therefore recording the numbers for as many concerts as possible then each counting of the number of visitors at a particular concert can be considered as an experiment As the results will be integers but there is no explicit upper bound you cannot identify the largest possible number of visitors the range of the corresponding random variable is best taken to be the set of all possible nonnegative integers Another example of a random variable would be the following if one measures the pH of samples of various solutions the corresponding random variable would assign to each experiment i e sample solution the value of its pH Compared to the previous two examples one can note while in the first three the possible values are clearly separated in the last experiment the possible values cannot be separated pH can take any real value usually between 0 and 14 The first three cases are examples of discrete random variables while the last one is an example of a continuous one Consider now the following sequence of experiments A person is repeatedly throwing a die and keeping track how many times a 6 rolls This is an example of a stochastic process a sequence of random variables their values are usually called states Even more it is an example of a Markov chain a discrete stochastic process with the property that the next state does not depend on previous ones except possibly on the current one For a given number n of rolls we consider the random variable X that describes the number of sixes rolled The probability of rolling a 6 in one roll can now be denoted by p 1 6 and the probability of not rolling a six is q 1 p 5 6 As each roll is independent on the previous one it is not hard to see that the probability of rolling exactly k sixes in n rolls is B n k p k q n k where B n k is the binomial coefficient equal to n n k k One can now define a function F from R into 0 1 by setting F x equal to the probability of of X having value at most x i e of having rolled at most x sixes in n rolls Such functions F R 0 1 with the property that F x equal to the probability of of X having value at most x are known as probability distributions and

Original URL path: http://mathematics-in-europe.eu/tr/45-information/landscape-areas/89-probability-and-statistics (2013-11-18)

Open archived version from archive - Mathematics In Europe - Topology

different edges This solution as well as the discovery of the Euler polyhedron formula also by Euler in 1750 mark the beginnings of topology more precisely of a topological subdiscipline nowadays called graph theory Figure 2 The Koenigsberg bridges problem and the corresponding graph In the 19th century topology developed into the modern theory of properties preserved under continuous transformations Mathematicians like the Germans August Ferdinand Moebius and Johann Benedikt Listing discovered surfaces with interesting properties like the one sided surface known now as the Moebius strip You can make a Moebius strip from a strip of paper if you make a half turn in the paper before glueing the ends If you draw a line continuously halfway between the edges after drawing arroung you will end up with your endpoint of the line meeting the starting point the Moebius strip is as said one sided If you cut the Moebius strip along the middle line you will end up with only one band The images shown on Figure 3 below show three models of a Moebius strip the Figure on the right shows the slow burning of potassium nitrate along the middle line of the band Listing is also credited with the first use of the word topology German Topologie in a a letter written in 1836 and then in a book published in 1847 Still until the beginnings of the 20th century the old Euler s term analysis situs was more frequently used than topology Figure 3 The Moebius strip End of the 19th century topology evolved into its modern form with several subdisciplines The word topology now has two meanings one is the mathematical discipline the other a particular kind of structure the topologists are interested in In this second sense a topology is a family O of possibly but usually not all subsets of a given set X that are called its open subsets There are three requirements for a family O of sets to be a topology 1 the empty set and the whole of X must be contained in O 2 if you take any number even infinite of sets in O their union must also be in O and 3 if you take any two sets in O their intersection must also be in O If you take a set X and define a topology O on it you call the pair a topological space A familiar example is the set R of all real numbers with the topology consisting of all open intervals and all possible unions of open intervals A complement of an open set is called a closed set In the example of R with unions of open intervals as open sets any segment closed interval is closed because its complement in R is a union of two open intervals an that is open e g the complement of 0 1 in R is the union of 0 and 1 Besides openness and closedness of sets typical topological characteristics of

Original URL path: http://mathematics-in-europe.eu/tr/45-information/landscape-areas/88-topology (2013-11-18)

Open archived version from archive - Mathematics In Europe - Countable vs. uncountable

that no two elements of the first set are mapped to the same element of the second set and such that every element of the second set is the image of some element of the first set For example a function that maps every integer n to its additive inverse n is a bijection of the set of integers to itself For example 3 is mapped to 3 and 14 is mapped to 14 No matter which integer you consider it has precisely one negative and so the conditions of a bijection are satisfied We note that 0 is mapped to itself since 0 0 Now let us attempt to prove that the set of natural numbers 0 1 2 is infinite To do so we must find a proper subset of the natural numbers that we can map bijectively onto the full set of natural numbers We choose as our proper subset the even natural numbers 0 2 4 We now choose the function that maps every natural number to its double This is easily seen to be a bijection between the two sets and we have shown that the set of natural numbers is infinite Now that we know what it means for a set to be infinite we come to the next step 2 Comparing the Sizes of Different Sets We begin here as well with a definition in order to clarify what it means for two sets to be of the same size Two sets are said to be of the same size if there exists a bijection between them Since we have just defined the notion of bijection above this definition should give us no trouble We have also shown above that the set of even natural numbers is of the same size as the complete set of natural numbers even though intuitively one might suppose that there are many more natural numbers than just the even ones A further interesting result is that the set of natural numbers is the same size as the set of integers all positive and negative whole numbers plus zero To prove this we define a mapping in the following way zero is mapped to itself 1 to 1 2 to 1 3 to 2 4 to 2 and so on It should be clear that this mapping is a bijection Now we can approach the definitions of countable and uncountable 3 Countable Sets Definition A set M is said to be countable if there exists a bijection between M and the natural numbers Thus the countable sets are precisely those that are of the same size as the natural numbers What about the set of rational numbers all the fractions Is it countable We must look for a bijection from the natural numbers to the rational numbers Such a bijection indeed exists and it most easily demonstrated as follows We map the natural numbers to the rational numbers by following the arrow in the figure that is 0

Original URL path: http://mathematics-in-europe.eu/tr/80-information/landscapes-concepts/77-countable-vs-uncountable (2013-11-18)

Open archived version from archive - Mathematics In Europe - Associative, commutative and distributive law

creature that eats purple people purple people eater In another newspaper article one could read that girls and boys from homes with educated parents are given more encouragement by their teachers It became clear in the course of the article that what was meant is that girls and boys from homes with educated parents are given more encouragement by their teachers The point here is that girls and boys from homes with educated parents is not the same as girls and boys from homes with educated parents 2 When the associative law holds our work is simplified No matter how many objects are to be combined once the left to right order of the objects has been set no matter how they are paired off and operated on the result will be the same Therefore it is not necessary to use parentheses to give the operation an unambiguous meaning and that makes for easier reading The Commutative Law a b b a The commutative law states that the order in which two objects appear does not affect the result of the operation operating on a and b always leads to the same result as operating on b and a Well known elementary examples of commutative operations are addition and multiplication of numbers In other domains we have the examples of logical AND and OR for propositions as well as the union and intersection operations in set theory But not all operations have this nice property If the operation is the quotient of two natural numbers m and n it is certainly not the case that in general n m m n so this operation is not commutative Some Important Points 1 In contrast to the associative law which holds for almost all operations that one is likely to encounter the commutative law is frequently violated It is often the case that theories in which the commutative law holds are much simpler than generalizations in which one is forced to abandon commutativity One such example encountered already by high school students is the composition of functions For example if function 1 has the rule multiply by 3 and function 2 has the rule take the square then depending on the order in which the functions are applied we have either 3x 2 or 3x 2 Since these two rules are different we conclude that the composition of functions does not obey the commutative law 2 Many well known rules for operations depend heavily on the commutative law For example it plays an important role in proving the law of exponents a b n a n b n 3 In applications to physics one often treats processes as mathematical objects and interprets the successive applications of processes as an operation And then it is quite important whether the commutative law holds In quantum mechanics for example measurements play the role of these processes The violation of the commutative law reveals itself as a deep basis of the Heisenberg uncertainty principle which states

Original URL path: http://mathematics-in-europe.eu/tr/80-information/landscapes-concepts/136-associative-commutative-and-distributive-law (2013-11-18)

Open archived version from archive