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- Mathematics In Europe - The landscape: formulas

Society Our Sponsor Munich RE The landscape formulas Ayrıntılar Kategori Landscape Some important formulas from mathematics and physics Pythagoras formula pdf Newton s universal law of gravity pdf Euler s most beautiful formula of mathematics pdf The prime number theorem of Gauß pdf Maxwell s equations of electrodynamics pdf Clausius equation of thermodynamics pdf Einstein I How changes time for a moving object pdf Einstein II E mc 2 pdf

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Open archived version from archive - Mathematics In Europe - The hierarchy of numbers

integers is not necessarily an integer 2 Natural numbers are a special type of integer C Rational Numbers The rational numbers constitute the set of all quotients of the form m n where m is an integer and n is a natural number Thus 4 87 and 345 777 are examples of rational numbers Important information 1 Every integer m can be written as m 1 and therefore every integer is also an example of a rational number Of course there are rational numbers that are not integers such as 1 2 and 7 19 2 In the realm of natural numbers all four operations are allowed adding subtracting multiplying or dividing except by zero two rational numbers leads invariably to another rational number At the same time the rational numbers continue to enjoy the commutative associative and distributive properties D Real Numbers We have now reached a difficult juncture It took a long time in the history of mathematics in fact until the middle of the nineteenth century before mathematicians were able to work with real numbers in a sufficiently precise way For our purposes we are going to simplify our lives a bit and make the following definition real numbers are those that can be written in a possibly terminating decimal expansion for example 13 1212121212 or 4 5 or 896626 4142894110 The real numbers include practically all the numbers that one encounters in school and that are necessary in practical applications The number the fifth root of 323 the base of the natural logarithm e all are real numbers Important information 1 Every rational number has a finite or periodic decimal expansion Therefore all rational numbers are special examples of real numbers 2 It is not easy to see that there are indeed real numbers that are not rational The most famous example of such a number is surely the number whose square is 2 that is the square root of 2 Moreover this number is one that is encountered in elementary geometry according to the Pythagorean theorem the diagonal of the square with side length 1 is precisely equal to the square root of 2 3 Real numbers that are not rational are called irrational Thus as just noted the square root of two is an irrational number E Complex Numbers By definition the square root of a number a is a number r with the property that the square of r is equal to a Thus 10 is a square root of 100 and 11 is a square root of 121 Every positive real number has exactly two square roots one of which is positive and the other negative Therefore we may agree to call the positive of the two roots the square root In this sense 1 414213 is the square root of 2 and 7 is the square root of 49 Since the square of every real number is positive or 0 no negative real number can have a real square root However

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Open archived version from archive - Mathematics In Europe - Number theory

is 1 7 1 7 and the remainder is 0 and we conclude that the greatest common divisor of 247 and 16 is 1 Two integers are congruent to each other modulo an integer n if they give the same remainder when divided by n or equivalently if their difference is divisible by n For example congruences modulo 12 classify integers into 12 classes depending on their remainder modulo 12 Chinese remainder theorem known in 13th century China states that if you take a number of pairwise coprime integers n m and the same number of any integers a b there exist integers x that are at the same time congruent to a modulo n to b modulo m and so on all of these x are congruent to each other modulo the product n m Examples of more recent results in number theory are the Fermat s little theorem Lagrange s four square theorem and the prime number theorem Fermat s little theorem states that if a is an integer and p is a prime then a p a is divisible by p or equivalently if a is not divisible by a prime p then the remainder of the division of a p 1 by p is 1 For example as 127 is a prime number and 250 is not divisible by 127 without the actual calculation we know that 250 126 divided by 127 has a remainder 1 The Lagrange s four square theorem states that any positive integer can be written as the sum of four square numbers some can be zero For example 239 14 2 5 2 3 2 3 2 The prime number theorem gives an estimate for the number π x of primes up to a number x π x is approximately equal to x ln x and the approximation is the more correct the larger x is For example there are π 100 25 primes not greater than 100 and 100 ln 100 21 71 and there are π 1 000 000 78 498 primes not greater than 1 000 000 and 1 000 000 ln 1 000 000 72382 414 Many of the most famous open problems in mathematics are number theoretical The Goldbach s conjecture is one of the oldest stated by Goldbach in 1742 unsolved problems in mathematics it states that every even integer can be expressed as the sum of two primes It was checked by computers for all even integers up to the order of 10 18 but a general proof is still unknown The twin prime conjecture states that there is an infinite number of pairs of primes with difference 2 Diophantine equations are also a typical number theoretical topic These are indeterminate polynomial equations that are to be solved in integers only For example Pell s equation is x 2 n y 2 1 where n is a nonsquare integer e g 2 It was studied as early as 400 BC in India and Lagrange

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Open archived version from archive - Mathematics In Europe - Applied mathematics

The Turing machine is a mathematical model 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

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Open archived version from archive - Mathematics In Europe - Geometry

algebraic geometry studies the sets of solutions 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

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Open archived version from archive - Mathematics In Europe - Analysis

differential equations particularly often the ones that are good models of physical reality that can be solved only approximately using numerical analysis techniques The general theory of differential equations deals particularly with the problems of existence and uniqueness of solutions and although one can often prove that under given initial or boundary condition a differential equation has a unique solution the proof is usually nonconstructive i e does not give a method for finding a solution Figure 2 Ball trajectories with red and without blue air resistance as calculated from corresponding differential equations Consider now another physical problem Take a vertical plane and two points in it What is the shape of the curve such that a mass point travelling along the curve supposing the only force acting on it is gravitation comes from on point to the other in the shortest time This problem is known as the brachistochrone problem A similar problem is the ancient isoperimetric problem determine the shape of the curve of given length enclosing the largest area Other problems of the same kind include the tautuchrone problem find a curve such that if you place a bead anywhere on it it will fall to the bottom in the same amount of time the problem of minimal area given a borderline determine the shape of a surface with that border having minimal area and the problem of the geodesics given two points on a surface find the line on the surface of shortest connection between the points Such problems are typical for the calculus of variations Unlike classical calculus where one is often interested in determining the maximal and minimal values of a function depending on numbers in the calculus of variations one seeks for maximal or minimal values of functions of functions For example in the problem of the brachistochrone if we know the curve our point travels along the time required is described by an integral that includes the function f describing the curve But since we do not know the curve we have to find which function f makes this integral as small as possible i e we are comparing the values of a function in this case an integral depending on another function f The solution to the brachistochrone and the tautochrone problem problem is a cycloid and the solution to the isoperimetric problem is a circle The cycloid is shown in Figure 3 below Figure 3 The cycloid is a path of a point attached to a circular wheel of radius a as the wheel rolls along a straight line The calculus of variations is historically the first form of a later independent mathematical discipline known as functional analysis While in classical analysis one considers functions whose variables are numbers or n tuples of them functional analysis studies more general functions particularly those whose variables are also functions functions of functions More precisely it studies various kinds of vector spaces usually those supplied with a norm and operators acting upon them Functional analysis has many interconnections with topology and algebra Typical terms from functional analysis are Banach and Hilbert spaces and bounded operators A vector space is a set such that addition between its elements and their multiplication by real or complex numbers is defined Elements of vector spaces are called vectors A norm is a function assigning a nonnegative number to each vector it is a generalisation of the usual notion of the length of a vector In a normed space there is a natural way to define convergence of sequences that generalises the idea of convergence of sequences of real numbers If such a space has the property that every convergent sequence converges to an element of the same space and not some element outside the normed space is called a Banach space An example of a Banach space is the space C 0 a b of all continuous functions from a b to a b with the norm of a function defined to be equal to its maximal value on a b Sometimes the norm arises from an inner product product of vectors which results in a number and if such a normed space has the just mentioned property of closedness upon convergence it is called a Hilbert space The space C 0 a b is not a Hilbert space Simple examples of Hilbert spaces are the Euclidean line plane and space R R 2 and R 3 A linear map is a mapping between two vector spaces behaving nicely with respect to the two vector space operators If a linear map maps from one to another normed space then it is bounded if for every vector in the domain the norm of the image of the vector after the operator is applied is not larger than a positive constant independent of the vector times the norm of the vector One of the most important theorems in functional analysis is the Banach fixed point theorem if a linear map is a contraction the constant in the above definition of boundedness is at most 1 i e the map decreases the norm of every vector acting from a closed subset of a Banach space into the same subset then it has an unique fixed point a vector that is mapped into itself This theorem implies for example the Picard Lindelöf theorem ordinary differential equations of the form y f x y with a initial condition y x 0 y 0 have a unique solution if f satisfies a certain condition called Lipschitz continuity There are other kinds of fixed point theorems guranteeing that a function satisfying certain conditions has at least one fixed point For example a continuous mapping f defined on 0 1 having values also in 0 1 has at least one fixed point as illustrated by Figure 4 below This has many applications For example imagine somebody climbs a mountain for six hours say from 8AM until 2PM camps overnight and then at 8AM the next

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Open archived version from archive - Mathematics In Europe - Topology

to enter and leave each vertex via 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

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Open archived version from archive - Mathematics In Europe - historical_reminder_09-13

Manzara Araştırma Tarih Felsefe Müzik sanat Avrupa da matematik Halka ulaşma faaliyetleri Yarışmalar Matematik Yardım Meslek olarak matematik Karışık Misyon Hoşgeldin Mesajları EMS Destekçiler Diller Iletişim Yasal Bilgi arama The European Mathematical Society Our Sponsor Munich RE historical reminder 09 13 Ayrıntılar Kategori Historical reminder On September 13 1873 Constantin Carathéodory was born in Berlin Germany He obtained fundamental results in many areas of mathematics in particular in the theory

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