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  • Mathematics In Europe - 300th birthday of Ruđer Josip Bošković (Roger Joseph Boscovich)
    quantities to arbitrary exactness Boscovich also comments that even in the case of need for great exactness his geometric method can serve as a means to check the corectness of calculations He published several other texts on spheric trigonometry and its applications to astronomy and the best known of them is De formules différentielles de trigonométrie published as part of Opera pertinentia ad opticam et astronomiam etc 1785 where he introduced the four fundamental equations of differential plane and spheric trigonometry out of which all the others can be deduced His theory of conic sections presented in the third volume of Elementa universae matheseos 1754 was the first complete and systematic theory of conic sections with several new ideas developed using purely geometric arguments In particular he chose a unified definition of all the conics as geometrical loci of points in the plane having constant ratio of distances to a given point focus and a given line directrix and deduced all their other properties from this definition This defining ratio he called ratio determinans of the conic section and it is nowadays called numerical eccentricity Although this property was known to ancient Greek mathematicians in particular to Pappus of Alexandria it was never before used as a definition before Boscovich conics were defined as conic intersections while Boscovich reduced the theory of conics to a purely planimetric theory He also introduced a powerful instrument for proving various properties of conics This is the famous eccentric circle of Boscovich the circle with arbitrary centre with the property that the ratio of its radius to the distance of its centre to the directrix is equal to the ratio determinans Using the eccentric circle Boscovich described many constructions and proved many properties of conics in particular he could deduce their properties from the properties of a circle by using the corresponence between the points of the eccentric circle and the points of the conic As an appendix De transformatione locorom geometricorum to the third volume of Elementa universae matheseos Boscovich presented a purely new theory of geometric transformations which are basically collineations and can thus be considered a forerunner of the development of projective and synthetic geometry in the 19th century Boscovich was also the first to present a systematic and general method for curve fitting He developed it to balance the results of geodetic measurements the data Boscovich fitted by his method were collected on an expedition from Rome to Rimini he undertook with the English jesuit Cristopher Maire to collect data for measurements of two meridian degrees The collected data as well as his method were described in De Litteraria expeditione per pontificam ditionem ad dimetiendos duos meridiani gradus a P P Maire et Boscovich 1755 and in more detail in some later texts Boscovich s idea was to obtain the best fit curve by minimizing the sum of absolute values of all the deviations of measured data from the quantities that would be obtained using the curve Boscovich s method

    Original URL path: http://mathematics-in-europe.eu/tr/47-information/math-calendar/600-300th-birthday-of-ruder-josip-boskovic-roger-joseph-boscovich (2013-11-18)
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  • Mathematics In Europe - 2011: 75 Years of Finiteness Theorems in Mathematical Logic
    what is termed an elementary language see below A statement H is the logical consequence of a set X of hypotheses a system of axioms if every interpretation of the language that makes X true also makes H true Therefore logical consequence is a semantic notion It refers to the unavoidable particularly in mathematics possibility that symbols or words can have various meanings associated with them In contrast proof is a syntactic concept It refers to the formal derivation of statements from hypotheses based on a defined system of rules Gödel s completeness theorem after a small generalization states that there exist proof calculi that can completely replace logical consequence in finite axiom systems and thereby automate them The Gödel Mal cev theorem states that one may do without the restriction to finite axiom systems Thus the equivalence of logical consequence and proof via a specifiable calculus holds for all theories that can be formulated in an elementary language Here elementary means essentially that in determining logical consequences all interpretations that are compatible with the structure of the language must be considered As Hilbert is supposed to have said in connection with the foundations of geometry Instead of points lines and planes one should be able to speak of tables benches and beer mugs Mal cev s original theorem goes as follows If every finite subset of an infinite axiom system possesses a model then the entire axiom system possesses a model This theorem admits even without its relation to Gödel s theorem a variety of equivalent formulations and has some weighty consequences one of which is that every consequence of an axiom system always follows from a certain finite portion of that system From the viewpoint consequence equals proof this is clear In any given proof one has always

    Original URL path: http://mathematics-in-europe.eu/tr/47-information/math-calendar/599-2011-75-years-of-finiteness-theorems-in-mathematical-logic (2013-11-18)
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  • Mathematics In Europe - 1911: The Mathematical Concept of Dimension
    points in an apparently two dimensional surface could be described by a single real number In 1890 Giuseppe Peano 1858 1932 modified Cantor s idea to obtain a continuous but now no longer one to one mapping of a parametric interval to a two dimensional triangle In the meantime there evolved the concept of a topological mapping as a combination of the properties of bijectivity and continuity in both directions Mathematicians were convinced that such mappings in the apparently accessible two and three dimensional cases would achieve precisely those results that could be described by stretching contracting folding and deforming without tearing or gluing The open problems were now the following To give a definition of dimension that would be independent of the parameterization by n real numbers and invariant under topological mappings To prove that the dimension of a space is unchanged under a topological mapping to itself or to another space In 1911 partial solutions to both of these problems were obtained In that year the Dutch mathematician Luitzen Egbertus Jan Brouwer 1881 1966 found a proof of the impossibility of mapping an n dimensional simplex topologically onto an m dimensional simplex where m n By an m dimensional simplex is meant the simplest so to speak m dimensional set in the real m dimensional space R m namely the convex hull of its m 1 vertices P 0 P 1 P m where the vectors P 0 P 1 P 0 P m are linearly independent Subsets of the space R m of greater complexity can be approximated as the union of simplicial sets while on the other hand finite dimensional metric spaces can be isometrically embedded in R m so that Brouwer s result solved problem B at least for finite dimensional metric spaces which however

    Original URL path: http://mathematics-in-europe.eu/tr/47-information/math-calendar/589-1911-the-mathematical-concept-of-dimension (2013-11-18)
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  • Mathematics In Europe - 1611: Kepler’s Essay on the Six-Cornered Snowflake
    the enormous variety of snow crystals which always display sixfold symmetry Nevertheless along the way to his unachievable goal he developed a methodology illustrated by a variety of examples for which he might today be considered not without justification a pioneer in the field of biomathematics Unfortunately it was a methodology that was ineffective in elucidating the shape of snowflakes because there was much about the crystalline structure of a variety of substances about which he could have known nothing Kepler begins his series of examples with the architecture of honeycombs and discovers along the way two Archimedean dual or Catalan solids whose faces are rhombuses namely the rhombic dodecahedron which is dual to the cuboctahedron and the rhombic triacontahedron dual to the icosidodecahedron This leads to cuboctahedron its dual the rhombic dodecahedron with inscribed cube icosidodecahedron its dual the thirty faced rhombic triacontahedron The not very user friendly names of the Archimedean polyhedra were invented by Kepler in 1619 on the occasion of their first complete listing after Archimedes Some of the dual Catalan polyhedra received names only in the twentieth century and a few of the solids offer two alternatives see also http math world wolfram com CatalanSolid html Since instead of a circumscribed sphere they have only an inscribed sphere which is not so eyecatching the Catalan polyhedra have not received the same degree of attention as the Archimedean polyhedra which have excited the imagination of artists from the Renaissance to today ideas about filling all of three dimensional space without gaps using polyhedra Kepler discovered that space can be filled gaplessly with rhombidodecahedra and optimal sphere packings which he again relates to form for example that assumed by pomegranate seeds or peas as they grow inside a bounded hull under pressure from all sides From there

    Original URL path: http://mathematics-in-europe.eu/tr/47-information/math-calendar/572-1611-kepler-s-essay-on-the-six-cornered-snowflake (2013-11-18)
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  • Mathematics In Europe - 1910: The first volume of Russell and Whitehead’s “Principia Mathematica” is published
    extensive introduction the second edition 1925 1927 was distinguished by little more than the correction of typographical errors Both authors had attended the consequential 1900 International Congress of Philosophy in Paris and were particularly impressed with the presentation given by Giuseppe Peano As Russell later wrote The Congress was the turning point of my intellectual life because there I met Peano It became clear to me that his notation afforded an instrument of logical analysis such as I had been seeking for years Russell s contradiction free set theory rests on an a posteriori simplification of a hierarchical ranking of sets Beginning with primordial elements ur elements that is objects of the real world that are themselves not sets one forms sets of the first level from such ur elements Then sets of the second level families of sets are formed from sets of the first level and so on As auspicious as the foundation of this world of concepts based on the ur elements may be from a pedagogical point of view it nonetheless has its pitfalls To get even as far as the set of all natural numbers it is necessary to postulate the existence of infinitely many ur elements and from the viewpoint of modern physics and cosmology that is extremely problematic Later axiomatic systems for set theory that were free of such a hierarchy such as the Zermelo Fraenkel and the von Neumann Bernays systems overcome this difficulty elegantly for example by beginning with the empty set Ø and forming the smallest set that contains the empty set Ø as well as x υ x whenever it contains x more precisely the intersection of all sets satisfying the preceding property Such a countably infinite set fits in no fixed level In this way the entire cosmos of mathematical concepts is created independently of material reality Among the many pearls of the Principia which today remains by and large only of historical interest is Russell s definition of the finiteness of a set A subset of a set E is finite if it appears as an element in every family of sets that contains the empty set Ø as well as M υ e for every e in E for each set M that it contains Today there is a multitude of ways of defining finiteness set theoretically yet we know that thanks to a number of nonstandard models these definitions are in no way equivalent under all circumstances If we have given the impression that the Principia was nothing more than some kind of textbook on set theory we should add here that with the Principia Russell and Whitehead created the first consistent and comprehensive treatise of mathematical logic together with the required formulaic language It is essentially the proof of its thesis that mathematics is set theory which in turn is nothing other than another language for logic Bertrand Russell Alfred North Whitehead About the authors Bertrand Russell 1872 1970 was a member of an

    Original URL path: http://mathematics-in-europe.eu/tr/47-information/math-calendar/291-1910-the-first-volume-of-russell-and-whitehead-s-principia-mathematica-is-published (2013-11-18)
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  • Mathematics In Europe - 1810: Gergonne founds one of the first mathematical journals
    The first independent mathematical journal in France and one of the world s first was the Annales de mathématiques pures et appliquées which was published from1810 until 1832 and played a vital role above all in the development of projective synthetic and algebraic geometry An issue appeared every month Many famous French mathematicians published there including J V Poncelet M Chasles Ch J Brianchon P Dupin G Lamé and É Galois and also foreigners such as J Steiner J Plücker and S l Huilier Indeed the Swiss l Huilier was an associate editor during the journal s first four years of existence The Annales was founded by Joseph Gergonne 1771 1859 and later the journal was often referred to simply as Gergonne s Annals Gergonne who was born in Nancy the son of a painter and architect was essentially self taught as a mathematician As was customary for one of his class he spent the years 1791 1796 as a soldier and officer and then accepted a position as a teacher of mathematics at the newly founded École Centrale in Nîmes Later he distinguished himself as a disciple of Gaspard Monge whose strong influence is reflected in his research although he had never attended the École Polytechnique in Paris of which Monge was the director In 1816 he accepted a chair of astronomy at the University of Montpellier whose rector he became in 1830 and where he died in 1859 His founding of the Annales was due as was also the case with Hindenburg Crelle and others to the lack of a suitable venues in which to publish the results of his enormous productivity Over a period of 22 years he wrote more than 200 contributions to his journal often published anonymously These include such momentous works as the formulation

    Original URL path: http://mathematics-in-europe.eu/tr/47-information/math-calendar/282-1810-gergonne-founds-one-of-the-first-mathematical-journals (2013-11-18)
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  • Mathematics In Europe - 1810: Gaussian elimination
    Liber Abaci 1202 of Leonardo of Pisa Fibonacci the method is explained by the example of a few merchants who are able to make a purchase only if they join forces Much later in 1770 Leonhard Euler in his Complete Introduction to Algebra first treated the cases of one two and three variables in great detail Then he wrote If more than three unknown quantities and the same number of equations should occur then it would be possible to solve the problem in a similar manner as in the above cases which however in general would lead to tiresome calculations But in all concrete cases you can in general find means by which the solution will become much simpler and this happens by introducing additional unknown quantities e g the sum of all given quantities This will be managed quite simply by those have already gained some experience in such calculations Part 2 Chapter 1 2 4 There is nothing in this passage that suggests systematic elimination In 1810 Gauss published his method in his astronomical paper on the disturbances of the orbit of the asteroid Pallas which had been discovered in 1802 In 1809 Gauss for the first time published his method of least squares in his Theoria motus corporum coelestium in sectionibus coniis solem ambientum and used it for the calculation of the orbit of the small planet Ceres which had vanished after a small number of rough observations The fact that his calculation made possible the rediscovery of Ceres caused Gauss to become famous in all of Europe which led to his lifelong appointment as professor of astronomy and director of the newly founded observatory in Göttingen The method of least squares leads to what are called normal equations and is historically apparently the first serious problem that requires the solution of a system of linear equation systems in many unknowns The following year 1810 Gauss revealed almost as an aside his method for solving such systems of equations Gauss in 1803 painted by J C A Schwartz Among extant portraits it is the nearest to 1810 Again one may wonder why two such important pieces of pure mathematics as the least squares method and Gaussian elimination appeared as by products in astronomical works instead of as separate publications However at the beginning of the nineteenth century pure mathematics was regarded in the eyes of the public and even of the mathematicians themselves as something of a hobby without serious importance in contrast to astronomy and the other natural sciences As Gauss himself wrote in 1801 to his former teacher and patron E A W Zimmermann after the Duke of Brunswick had increased his stipend following the publication of his great number theoretic work Disquisitiones arithmeticae But I have not earned it I have not yet done anything for the nation The main use of Gaussian elimination today is in the insights it reveals into the conditions of solvability and the structure of the solutions Gauss was

    Original URL path: http://mathematics-in-europe.eu/tr/47-information/math-calendar/124-1810-gaussian-elimination (2013-11-18)
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  • Mathematics In Europe - Mathematics and music
    Website Skip to main content Press Enter Here are some articles concerning the interesting connections between mathematics and music Ehrhard Behrends Freie Universität Berlin Chance as composer random compositions Ehrhard Behrends Freie Universität Berlin Mathematics that you can hear Fourier analysis Ehrhard Behrends Freie Universität Berlin On semitones and twelfth roots The mathematical background of our musical scales JavaScript is currently disabled Please enable it for a better experience of

    Original URL path: http://mathematics-in-europe.eu/tr/44-information/math-and/715-mathematics-and-music (2013-11-18)
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