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- Mathematics In Europe - Recreational Mathematics

many candies do you have By Franka Brückler The Three Ducks Trick by Franka Brückler The clairvoyant mathe magician by Franka Brückler Gergonne s trick by Franka Brückler Two rows of pebbles by Franka Brückler The Four Ducks Trick by Franka Brückler Inductive Magic by Franka Brückler Magical mathematics The Integers by Ehrhard Behrends Order amidst Chaos by Ehrhard Behrends Magical invariants by Ehrhard Behrends A Beer Trick by Franka

Original URL path: http://mathematics-in-europe.eu/nb/78-enjoy-maths/recreational-mathematics/611-recreational-mathematics (2013-11-18)

Open archived version from archive - Mathematics In Europe - The European launch of MPE2013

Audouze the president of International Mathematical Union Ingrid Daubechies the Fields medalist Wendelin Werner and the president of the European Mathematical Society Marta Sanz Solé Then Christiane Rousseau from Montreal the coordinator of MPE2013 gave a survey on the initiatives that are planned in this year The next item of the agenda was the presentation of the open source platform IMAGINARY by Martin Greuel see the next picture and Andreas Matt Finally at the end of the morning session the winners of the MPE2013 competition were announced in a ceremony moderated by Ehrhard Behrends chair of the jury the pdf of his talk can be found here Third prize 2000 How to predict the future of glaciers by the team of Guilleaume Jouvet France Switzerland Germany In an entertaining way this video illustrates the collaboration between a mathematician and a glacier expert as they develop a dynamic model for the evolution of glaciers At the end of the video the user can choose among alternative scenarios to see possible futures for the Aletsch glacier in the Alps From the minutes of the meeting of the jury In the picture E Behrends G Jouvet Ch Rousseau Second prize 3000 Dune Ash by the team of Tobias Malkmus Germany This interactive computer program graphically simulates the dispersion of a volcanic ash cloud using a mathematical model The user chooses the location of the volcano sketches the direction and strength of the winds and sets the rate of dispersion An original interface allows the user to specify complex wind patterns and invites repeated exploration From the minutes of the meeting of the jury In the picture E Behrends two members of the Malkmus team Ch Rousseau First prize 5000 Sphere of the Earth by the team of Daniel Ramos Spain This exhibit shows

Original URL path: http://mathematics-in-europe.eu/nb/49-popularization/general/809-the-european-launch-of-mpe2013 (2013-11-18)

Open archived version from archive - Mathematics In Europe - Mathematics of Planet Earth 2013 in Europe

the WELT Luftballons im Handy Akku can be found here and here is the English version Mathematical modelling for energy research April 18 2013 Jürgen Sprekels Produktion von Halbleiterkristallen ein Fall für Mathematik the article in the WELT was called Simulierte Produktion The English version Production of Semiconductor Crystals a Case for Mathematics May 23 2013 Rupert Klein Mathematik und Klima The online version of the article in the WELT can be found here and here is pdf of the paper version An English translation is also available Mathematics and Climate June 20 2013 Peter Deuflhard Erdbeben Vorhersage in Italien führt ins Gefängnis The version of the article in the WELT can be found here and here is the English translation July 25 2013 Volker Mehrmann Mathe spart Energie The online version of this article in the WELT can be found here An English version is also available August 15 2013 Ralf Kornhuber Virtuelles Kniegelenk in German the slightly longer version Immer auf die Knochen of this article can be found here The English version has the title Close to the Bone Mathematics in Orthopaedic Surgery September 26 2013 Christof Schütte Moderne Schamanen in German the slightly longer version Die Medizin braucht die Mathematik of this article can be found here The English version has the title Medicine needs Mathematics October 24 2013 Peter Gritzmann Fisch sucht Fahrrad in German a slightly longer version can be found here The mpe2013 article of Günter Ziegler appeared on October 11 2013 Die alltägliche Vermessung der Welt On May 23 one of the lectures that are sponsored by the Simons foundation at the occasion of MPE2013 took place at Freie Uiversität in Berlin The lecture was presented by Rupert Klein details can be cound here in German In total there will be nine such lectures in 2013 six of them in North America The open source platform IMAGINARY is administrated in the research center Oberwolfach Back to the survey ITALY Mathematical Models and Methods for Planet Earth http www altamatematica it mmmpe2013 earth html this is a workshop organized by INDAM Istituto nazionale di Alta Matematica which is a MPE partner in Rome 27 29 May Conference on the Mathematical and Computational Issues in the Geosciences GS13 a conference in Padova by SIAM 17 23 June 2013 http www siam org meetings gs13 As regards rpa activities the journal XlaTangente has scheduled one article in each issue since December 2012 to December 2013 for seven issues and the website associated to the journal is building a section http www xlatangente it page php id 1571 where it will be possible to find besides these articles a list of mpe activities and other news Back to the survey PORTUGAL There are five MPE2013 international partners in Portugal CIM International Centre of Mathematics SPM Portuguese Mathematics Society APM Mathematics Teachers Association MNHNCUL National Museum of Natural History and Science of Lisbon University and MCUC Science Museum of the Coimbra University Launch of MPE2013 in Portugal

Original URL path: http://mathematics-in-europe.eu/nb/49-popularization/general/790-mathematics-of-planet-earth-2013-in-europe (2013-11-18)

Open archived version from archive - Mathematics In Europe - Heidelberg laureate forum

Our Sponsor Munich RE Heidelberg laureate forum Detaljer Kategori News The Klaus Tschira Stiftung will establish the Heidelberg Laureate Forum where Abel Fields and Turing laureates will meet the next generation of scientists The Klaus Tschira Stiftung will establish the Heidelberg Laureate Forum as an annual meeting bringing together winners of the most prestigious scientific awards in Mathematics Abel Prize and Fields Medal and Computer Science Turing Award with a selected group of highly talented young researchers The Forum has been initiated by the Heidelberg Institute for Theoretical Studies HITS the research institute of the Klaus Tschira Stiftung KTS a German foundation which promotes Natural Sciences Mathematics and Computer Science The Heidelberg Laureate Forum is modeled after the annual Lindau Nobel Laureate Meetings established more than 60 years ago to bring forward new ideas Klaus Tschira founder and managing partner of the foundation states Meeting with the scientific leaders of Mathematics and Computer Science will be extremely inspiring and encouraging for the young scientists The agreement on collaborating in the Heidelberg Laureate Forum between the organizers and the award granting institutions Norwegian Academy of Science and Letters International Mathematical Union and Association for Computing Machinery has been signed in Oslo

Original URL path: http://mathematics-in-europe.eu/nb/nyheter/10-frontpage/news/695-heidelberg-laureate-forum (2013-11-18)

Open archived version from archive - Mathematics In Europe - Abel prize ceremony

The European Mathematical Society Our Sponsor Munich RE Abel prize ceremony Detaljer Kategori News This year s Abel prize ceremony took place in Oslo on May 22 The first event was a meeting on Monday morning May 21 in Abel s school Katedralskole There the winners of the Holmboe prizes for Norwegian school children and mathematics teachers were announced Holmboe was Abel s teacher In the afternoon there took place

Original URL path: http://mathematics-in-europe.eu/nb/nyheter/10-frontpage/news/694-abel-prize-ceremony (2013-11-18)

Open archived version from archive - Mathematics In Europe - Philosophy of mathematics: intuitionism

of a mathematical theorem then is to express the implementation of such a mental construction And while formalism emphasized the importance of mathematical symbols intuitionism assigns essential significance to verbal expression in any formal system The verbal formulation of mathematical results serves exclusively communicative purposes and represents the essentials of mathematics the mental constructions only in imperfect approximation According to intuitionism mathematical axioms need not be reduced to logic or proven to be consistent rather it is their intuitive rightness that somehow determines their validity To such an extent intuitionism is a purely philosophical position It becomes a foundational program for mathematics only through its attack on traditional logic led above all by Brouwer In a number of papers and lectures Brouwer advocated a restriction on the application of the law of the excluded middle and the method of indirect proof based on that law The law of the excluded middle asserts that every assertion is either true or false and a proof is said to be indirect if it proves the validity of an assertion by showing that its negation leads to an absurdity that is leads to some sort of contradiction In the case of universally quantified statements that is statements of the form all x are or among all x there exists one such that then according to Brouwer the two principles are valid only if applied to finite sets represented by the variable x In that case universally quantified propositions can always be verified in principle by checking each individual case and according to Brouwer it is precisely this condition the ability to verify the truth of every statement in principle that expresses the law of the excluded middle In contrast when we are dealing with infinite sets in which the possibility of a direct verification

Original URL path: http://mathematics-in-europe.eu/tr/43-information/history-philosophy/83-philosophy-of-mathematics-intuitionism (2013-11-18)

Open archived version from archive - Mathematics In Europe - Algebra

in which we perform the operation when calculating 2 3 4 it is unimportant if we calculate this as 2 3 4 5 4 9 or as 2 3 4 2 7 9 In both cases there is a special element called the neutral element in the set such that the operation performed on him and any other element doesn t change the other element if you add 0 to any integer you don t change it and if you multiply 1 by any positive real you don t change it Finally for every element of the set we can find another such that the performance of the operation on these two elements results in the neutral element if you add n to any integer n you get 0 if you multiply any positive real number x by 1 x you get 1 When a set and the operation on it have all the above mentioned properties we call it a commutative group the sets of integers with addition and positive reals with multiplications are examples of commutative groups The group D 2 described above is also a commutative group but contrary to the group of natural numbers with addition and positive real numbers with multiplication it is finite it has only four elements Not all groups are commutative For example if you consider the set S 3 the possible rearrangements of three objects A B and C and take the composition of rearrangements as the operation of the group you will notice that the order of rearrangements matters see Figure 31 below the three squares start in order shown in the first row We consider the following two rearrangements switching the first two squares on the left rearrangement 1 and switching the first and the last one rearrangement 2 The second row of Figure 3 shows what happens if we first perform rearrangement 1 and then rearrangement 2 and the third row shows what happens if we first perform rearrangement 2 and then rearrangement 1 You can see that we end up with two different arrangements if we change the order of the rearrangements Thus S 3 with the operation of composition of rearrangements is not commutative but it is a group i e all of the other properties mentioned for integers with addition hold Figure 3 Rearrangements of three squares illustrating the noncommutativity of the group S 3 Besides groups there are many other kinds of algebraic structures rings fields vector spaces modules The study of algebraic structures has its origins in the study of solutions to algebraic equations linear quadratic cubic quartic After the formulas for solutions of cubic and quartic equations were discovered in the reneissance mathematicians worked on equations of higher order In the beginning of the 19th century N H Abel and E Galois proved that it is not possible to describe the solutions of higher degree equations by radicals i e by formulas giving the rule how to calculate the solutions from the

Original URL path: http://mathematics-in-europe.eu/tr/45-information/landscape-areas/85-algebra (2013-11-18)

Open archived version from archive - Mathematics In Europe - Analysis

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 day starts his descent Provided that the

Original URL path: http://mathematics-in-europe.eu/tr/45-information/landscape-areas/87-analysis (2013-11-18)

Open archived version from archive