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- Mathematics In Europe - Mathematics in France

Mathématiques Appliquées et Industrielles SMAI smai emath fr lang fr Research institutions Centre International de Rencontres Mathématiques CIRM www cirm univ mrs fr Institut des Hautes Études Scientifiques IHÉS www ihes fr Institut Henri Poincaré Centre Émile Borel www ihp jussieu fr Exhibitions Salon Culture et Jeux Mathematiques http www cijm org Mathematiques d aujourd hui http smf emath fr content exposition mathematiques daujourdhui For a larger list of mathematical exhbitions see http smf emath fr content maths et travaux expositions sur le theme des mathematiques Popular web pages Images des Mathématiques http images math cnrs fr CultureMATH http www math ens fr culturemath Page Grand Public de la SMF http smf emath fr content grand public CIJM http www cijm org La bibliotheque des mathematiques http www bibmath net Maths et Travaux http smf emath fr content maths et travaux Recueil de problemes de momes net http www momes net education problemes problemes html Carre d as devinettes mathematiques http carredas free fr aMATHeur http www amatheur net maj html Raising public awareness journals Brochure Zoom sur les metiers des mathematiques http smf emath fr content zoom sur les metiers des mathematiques Brochure L Explosion des mathematiques http smf

Original URL path: http://mathematics-in-europe.eu/tr/halka-ulasma-faaliyetleri/ulusal-faaliyetler/46-information/math-in-europe/119-mathematics-in-france (2013-11-18)

Open archived version from archive - Mathematics In Europe - Mathematics in Italy

http specchi mat unimi it Matematica trasparente superfici minime e bolle di sapone Transparent mathematics minimal surfaces and soap bubbles http bolle science unitn it matemilano mathematical paths in the town http matemilano mat unimi it matetrentino mathematical paths in Trento and surrounding http matematita science unitn it matetrentino Un tuffo nella quarta dimensione Diving in 4 th dimension http www matematita it materiale index php p cat sc 919 Uguali Diversi Equal Different http www matematita it materiale index php p cat sc 1014 Besides these ones there have been in Italy other exhibitions about mathematics especially in the last few years mainly these were related to some local events and had just a temporary character Sometimes it happens these activities gave rise to a permanent structure Just to give one example we quote Giocare con le costruzioni la matematica che esiste Playing with constructions mathematics which exists an exhibition of mathematical objects in the maths department of the University of Perugia http www dmi unipg it iniziative mostre We don t give here a list which would be long and could never be complete rather we add two references not so recent but still valuable for anybody willing to know more about the development of mathematics exhibitions in Italy inside and outside science museums in particular you find here also informations about exhibitions like L occhio di Horus hystorically significant but not quoted here because no more available S Di Sieno Matematica al museo Mathematics in museums in Bollettino dell Unione Matematica Italiana A ser 8 vol 6 A 2003 pag 85 103 available online at the adress http www quadernoaquadretti it scuola riflessioni museo pdf S Di Sieno Mostre di matematica soltanto una nuova moda o una strategia interessante Mathematics exhibitions only a new fashion or an interesting strategy in Bollettino dell Unione Matematica Italiana A ser 8 vol 5 A 2002 pag 491 514 available online at the adress http www quadernoaquadretti it scuola riflessioni mostre pdf Popular web pages We prefer to give here a short list and to point out websites which maybe just for a specific reason highlighted in the comment can be particularly interesting either for mathematicians working in rpa activities or for laymen curious about maths The list is obviously and unavoidably incomplete http ulisse sissa it The website of SISSA InternationalSchool for Advanced Studies for general scientific information http ulisse sissa it controluce Inside the Ulisse website this is a page regarding scientific images divided in 3 main groups looking describing and thinking each one further divided in 3 subgroups looking near looking far looking inside describing objects describing places describing changes thinking elements thinking relations thinking spaces Mathematics is mainly in the subgroup thinking spaces http www matematita it materiale index php Inside the website of the Centre matematita this is a page with a large collection of Images for mathematics including some animations some further information for the layman some examples of paths using images for teaching and

Original URL path: http://mathematics-in-europe.eu/tr/halka-ulasma-faaliyetleri/ulusal-faaliyetler/46-information/math-in-europe/109-mathematics-in-italy (2013-11-18)

Open archived version from archive - Mathematics In Europe - Mathematics in Turkey

and bringing together mathematicians in Turkey It has 818 members from all over the country Turkish Mathematical Society has been a full member of the International Mathematical Union since 1960 Also Turkish Mathematical Society has been a member of European Mathematical Society since 2008 The Society is located in Istanbul and has also a branch in Ankara http www tmd org tr The Association of Mathematicians was founded in 1995 and is located in Ankara Those who are part as student or faculty at some university Mathematics Mathematics Engineering Mathematics Education Departments can apply for membership cf link in Turkish http www matder org tr Mathematics Foundation publishes mathematics teaching books organizes the yearly Cahit Arf Lectures and gives every year the Masatoshi Gündüz İkeda Research award cf link in Turkish http www matematikvakfi org tr Research institutions The Istanbul Center for Mathematical Sciences http www imbm org tr The Boğaziçi University TÜBİTAK Feza Gürsey Institute http www gursey gov tr http www3 iam metu edu tr iam index php Main Page Institute for Applied Mathematics METU http www3 iam metu edu tr Other regular research and education activities Antalya Algebra Days http www aad metu edu tr Gökova Geometry Topology Conference http gokovagt org Nesin Mathematics Village http matematikkoyu org en node Activities web pages http www turkmath org beta index php Exhibitions Popular web pages http www matematiktutkusu com Raising public awareness journals Matematik Dünyası the World of Mathematics MD is a quarterly journal aiming to convey abstract mathematics to young people Although its focus is high school and university students thanks to the universality of mathematics it has reached a larger scope of readers The journal is owned by the Turkish Mathematics Society and is run by professional mathematicians http www matematikdunyasi org Raising public awareness journalists

Original URL path: http://mathematics-in-europe.eu/tr/halka-ulasma-faaliyetleri/ulusal-faaliyetler/46-information/math-in-europe/148-mathematics-in-turkey (2013-11-18)

Open archived version from archive - Mathematics In Europe - Popular article of the day

languages Articles in Turkish can be found here Home Halka ulaşma faaliyetleri Ulusal faaliyetler Frontpage General Popular article of the day Anasayfa Haberler Bilgiler Halka ulaşma faaliyetleri Ulusal faaliyetler Halka açılma makaleler 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 Popular article of the day Ayrıntılar Kategori General The file phpfiles jumi popular

Original URL path: http://mathematics-in-europe.eu/tr/halka-ulasma-faaliyetleri/ulusal-faaliyetler?id=72 (2013-11-18)

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

set in such a way 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

Original URL path: http://mathematics-in-europe.eu/tr/halka-ulasma-faaliyetleri/ulusal-faaliyetler?id=77 (2013-11-18)

Open archived version from archive - Mathematics In Europe - How far?

to Ryczywól I have to cross the Vistula so I need to find a bridge The closest bridge is in Deblin and the road there is about 60 kilometres So by distance we need do not necessarily mean the shortest segment joining two points By the way we regularly think about distances in different ways sometimes not even realising what we are doing How do we answer the question about the distance between Berlin and Sydney Considering a straight line Not at all We would travel through the centre of the planet We investigate instead the length of an arc on the surface of the Earth These examples show that there are distances of various kinds No wonder then that we may decide consider a general definition of distance Such a procedure in which a number of examples provide the foundation for a general definition is quite frequent in mathematics We may introduce the notion of distance in any nonempty set We require that the distance between every two points contained in our set should be defined the distance between the point and itself should be equal to 0 the distance between two different points should be greater than 0 the distance is symmetric i e the distance between A and B is the same as the distance between B and A the triangle inequality holds i e the sum of distances between A and C and between C and B is greater or equal to the distance between A and B that is if we go from A to B and want visit C in our way then we cannot shorten the route by visiting C on the way Mathematicians call a distance with these properties a metric on the set and the set is called a metric s

Original URL path: http://mathematics-in-europe.eu/tr/halka-ulasma-faaliyetleri/ulusal-faaliyetler?id=78 (2013-11-18)

Open archived version from archive - Mathematics In Europe - Stefan Banach

surprising applications of mathematics He had very original ideas and knew a lot He was a man of great culture and deep knowledge of literature His aphorisms remarks and thoughts are famous to this day Here are a few that are unfortunately most of them including the best are not translatable It is easy to remove God from his place in the universe But such good positions don t remain vacant for long Strip tease should be strictly forbidden This is the only way of keeping this beautiful and useful custom alive It is easy to go from the house of reality to the forest of mathematics but only few know how to go back Once when somebody was decorated with a medal Steinhaus said Now I know what to do in order to be awarded a medal Nothing but for a very long time After the Second World War Steinhaus was elected a member of sone scientific committee Among the members of this committee there were many poor scientists who were the members of it because of political reasons Once Steinhaus did not come for the meeting of the committee and he was asked to explain the reason of his absence He answered I am not going to give any reason of my absence until some others will give reasons for their presence here He used to say that a computer is an extremely efficient idiot He was an accomplished populariser of mathematics His book Mathematical Snapshots first published in 1938 was translated into many languages Strangely enough it was not reissued in Poland between 1957 and 1990 Stanislaw Mazur 1905 1981 a mathematician who offered a goose as a reward for solving one problem Banach s pupil and friend was also an excellent mathematician Like Banach he did not publish many of his results but for another reason Banach had too many ideas and results whereas Mazur did not like publishing For Mazur only two things were interesting mathematics and communism Before the war it was not known that he was a member of the Communist Party Mazur was 13 years younger than Banach He started his study when Banach was already a professor Nevertheless Banach treated Mazur as a partner They worked together Mazur frequently judged Banach s ideas gave details of proofs Other Banach s excellent pupils working in Lvov were Juliusz Pawel Schauder and Wladyslaw Orlicz In the thirties Kazimierz Kuratowski a Warsaw mathematician came for some years to Lvov One of Kuratowski s pupils was Ulam Stanislaw Ulam known later throughout the world as Stan Ulam was 5 born in 1909 in Lvov where he studied and initially worked Already as a first year student he obtained original mathematical results which were soon published Following an invitation by John von Neumann one of the greatest mathematicians of the first half of the 20th century he went to the United States in 1935 and settled there Among other things Ulam is most famous for his research in nuclear physics performed at Los Alamos for 25 years 1943 1967 He was one of the discoverers of the theoretical foundations of the construction of the hydrogen bomb He had broke scientific interests and obtained important results in various areas of mathematics set theory topology measure theory group theory functional analysis ergodic theory probability and game theory as well as in a number of sciences technology computer science physics astronomy and biology He developed original methods of propulsion of vessels moving above the earth s atmosphere Ulam died in 1984 His life and work are described in the book From Cardinals to Chaos published posthumously at Los Alamos and in his autobiography Adventures of a Mathematician Another young mathematician who moved abroad from Lvov before the war was Mark Kac well known because of his achievements in probability theory and statistics The names of many mathematicians from the Lvov Mathematical School are until now very well known in the world but for sure the most frequently mentioned is Banach s name It turned out that Banach is the mathematician from the whole world who is mentioned most frequently in the titles of mathematical scientific papers in the 20th century Banach s name is associated with many important by now classical theorems the Hahn Banach Theorem the Banach Steinhaus Theorem the Banach Open Mapping Theorem the Banach Alaoglu Theorem the Banach Closed Graph Theorem and the Banach Fixed Point Theorem And above all there is the fundamental mathematical concept of a Banach space What is it We learn in school about straight lines planes and three dimensional space We can describe these geometric objects by means of numbers Specifically we can identify the points on a straight line with single real numbers the points on a plane with pairs of real numbers and the points in space with triples of real numbers This idea can be extended to the study of finite sequences of numbers These sequences can be added and multiplied by numbers very much like vectors In this way we create so called finitedimensional spaces We can and do go further We add and multiply by constants numerical valued functions by defining the sum of two such 6 functions at a point to be the sum of their values at the point in question In this we are no longer dealing with a finite dimensional spaces It turned out for a variety of reasons that function spaces are very useful in many investigations and applications To a large extent modern mathematics is concerned with the study of general structures specific models of which have been known for a long time One advantage of studying a general structure is the economy of thought a theorem proved for the general structure need not be re proved for its different models Moreover the general proof makes it easier to identify the properties utilized in the course of the proof and thus makes it more transparent It is paradoxical but true

Original URL path: http://mathematics-in-europe.eu/tr/halka-ulasma-faaliyetleri/ulusal-faaliyetler?id=79 (2013-11-18)

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

Bertrand Russell was the idea that mathematics should be given a secure foundation that would protect it from fallacies and misunderstandings of a logico mathematical or metaphysical nature This was to be accomplished by the definition of foundational mathematical concepts indeed those of arithmetic for the essential trouble spot seemed to lie in the supposed undefinability of such concepts Gottlob Frege Bertrand Russell Now to define a concept means to explain it in terms of an equivalent expression that is already understood In the case of arithmetic according to logicism logical symbols should play the role of the already known expressions Thus Frege and Russell gave definitions of the idea of number and the elementary calculational operations that contain exclusively logical constants such as and or there exists an x such that and allow for the proof of elementary arithmetic statements such as 2 2 4 by purely logical means What is remarkable here is that in spite of the collaborative nature of their program the philosophical positions of Russell and Frege exhibited clear differences Thus while the two were in agreement that logic should serve as the foundation of mathematics when it came to the nature of logical propositions

Original URL path: http://mathematics-in-europe.eu/tr/halka-ulasma-faaliyetleri/ulusal-faaliyetler?id=82 (2013-11-18)

Open archived version from archive