Bernard Bolzano was a Bohemian mathematician, philosopher and Catholic priest of Italian extraction known for his antimilitarist views. Bolzano wrote in his native language. For the most part, his work came to prominence posthumously. Bolzano was the son of two pious Catholics, his father, Bernard Pompeius Bolzano, was an Italian who had moved to Prague, where he married Maria Cecilia Maurer who came from Prague's German-speaking family Maurer. Only two of their twelve children lived to adulthood. Bolzano entered the University of Prague in 1796 and studied mathematics and physics. Starting in 1800, he began studying theology, becoming a Catholic priest in 1804, he was appointed to the new chair of philosophy of religion at Prague University in 1805. He proved to be a popular lecturer not only in religion but in philosophy, he was elected Dean of the Philosophical Faculty in 1818. Bolzano alienated many faculty and church leaders with his teachings of the social waste of militarism and the needlessness of war.
He urged a total reform of the educational and economic systems that would direct the nation's interests toward peace rather than toward armed conflict between nations. Upon his refusal to recant his beliefs, Bolzano was dismissed from the university in 1819, his political convictions, which he was inclined to share with others with some frequency proved to be too liberal for the Austrian authorities. He was exiled to the countryside and devoted his energies to his writings on social, religious and mathematical matters. Although forbidden to publish in mainstream journals as a condition of his exile, Bolzano continued to develop his ideas and publish them either on his own or in obscure Eastern European journals. In 1842 he moved back to Prague, where he died in 1848. Bolzano made several original contributions to mathematics, his overall philosophical stance was that, contrary to much of the prevailing mathematics of the era, it was better not to introduce intuitive ideas such as time and motion into mathematics.
To this end, he was one of the earliest mathematicians to begin instilling rigor into mathematical analysis with his three chief mathematical works Beyträge zu einer begründeteren Darstellung der Mathematik, Der binomische Lehrsatz and Rein analytischer Beweis. These works presented "...a sample of a new way of developing analysis", whose ultimate goal would not be realized until some fifty years when they came to the attention of Karl Weierstrass. To the foundations of mathematical analysis he contributed the introduction of a rigorous ε–δ definition of a mathematical limit. Bolzano was the first to recognize the greatest lower bound property of the real numbers. Like several others of his day, he was skeptical of the possibility of Gottfried Leibniz's infinitesimals, the earliest putative foundation for differential calculus. Bolzano's notion of a limit was similar to the modern one: that a limit, rather than being a relation among infinitesimals, must instead be cast in terms of how the dependent variable approaches a definite quantity as the independent variable approaches some other definite quantity.
Bolzano gave the first purely analytic proof of the fundamental theorem of algebra, proven by Gauss from geometrical considerations. He gave the first purely analytic proof of the intermediate value theorem. Today he is remembered for the Bolzano–Weierstrass theorem, which Karl Weierstrass developed independently and published years after Bolzano's first proof and, called the Weierstrass theorem until Bolzano's earlier work was rediscovered. Bolzano's posthumously published work Paradoxien des Unendlichen was admired by many of the eminent logicians who came after him, including Charles Sanders Peirce, Georg Cantor, Richard Dedekind. Bolzano's main claim to fame, however, is his 1837 Wissenschaftslehre, a work in four volumes that covered not only philosophy of science in the modern sense but logic and scientific pedagogy; the logical theory that Bolzano developed in this work has come to be acknowledged as ground-breaking. Other works are a four-volume Lehrbuch der Religionswissenschaft and the metaphysical work Athanasia, a defense of the immortality of the soul.
Bolzano did valuable work in mathematics, which remained unknown until Otto Stolz rediscovered many of his lost journal articles and republished them in 1881. In his 1837 Wissenschaftslehre Bolzano attempted to provide logical foundations for all sciences, building on abstractions like part-relation, abstract objects, sentence-shapes and propositions in themselves and sets, substances, subjective ideas and sentence-occurrences; these attempts were an extension of his earlier thoughts in the philosophy of mathematics, for example his 1810 Beiträge where he emphasized the distinction between the objective relationship between logical consequences and our subjective recognition of these connections. For Bolzano, it was not enough that we have confirmation of natural or mathematical truths, but rather it was the proper role of the sciences to seek out justification in terms of the fundamental truths that may or may not appear to be obvious to our intuitions. Bolzano begins his work by explainin
Michiel Hazewinkel is a Dutch mathematician, Emeritus Professor of Mathematics at the Centre for Mathematics and Computer and the University of Amsterdam known for his 1978 book Formal groups and applications and as editor of the Encyclopedia of Mathematics. Born in Amsterdam to Jan Hazewinkel and Geertrude Hendrika Werner, Hazewinkel studied at the University of Amsterdam, he received his BA in Mathematics and Physics in 1963, his MA in Mathematics with a minor in Philosophy in 1965 and his PhD in 1969 under supervision of Frans Oort and Albert Menalda for the thesis "Maximal Abelian Extensions of Local Fields". After graduation Hazewinkel started his academic career as Assistant Professor at the University of Amsterdam in 1969. In 1970 he became Associate Professor at the Erasmus University Rotterdam, where in 1972 he was appointed Professor of Mathematics at the Econometric Institute. Here he was thesis advisor of Roelof Stroeker, M. van de Vel, Jo Ritzen, Gerard van der Hoek. From 1973 to 1975 he was Professor at the Universitaire Instelling Antwerpen, where Marcel van de Vel was his PhD student.
From 1982 to 1985 he was appointed part-time Professor Extraordinarius in Mathematics at the Erasmus Universiteit Rotterdam, part-time Head of the Department of Pure Mathematics at the Centre for Mathematics and Computer in Amsterdam. In 1985 he was appointed Professor Extraordinarius in Mathematics at the University of Utrecht, where he supervised the promotion of Frank Kouwenhoven, Huib-Jan Imbens, J. Scholma and F. Wainschtein. At the Centre for Mathematics and Computer CWI in Amsterdam in 1988 he became Professor of Mathematics and head of the Department of Algebra and Geometry until his retirement in 2008. Hazewinkel has been managing editor for journals as Nieuw Archief voor Wiskunde since 1977, he was managing editor for the book series Mathematics and Its Applications for Kluwer Academic Publishers in 1977. Hazewinkel was member of 15 professional societies in the field of Mathematics, participated in numerous administrative tasks in institutes, Program Committee, Steering Committee, Consortiums and Boards.
In 1994 Hazewinkel was elected member of the International Academy of Computer Sciences and Systems. Hazewinkel has authored and edited several books, numerous articles. Books, selection: 1970. Géométrie algébrique-généralités-groupes commutatifs. With Michel Demazure and Pierre Gabriel. Masson & Cie. 1976. On invariants, canonical forms and moduli for linear, finite dimensional, dynamical systems. With Rudolf E. Kalman. Springer Berlin Heidelberg. 1978. Formal groups and applications. Vol. 78. Elsevier. 1993. Encyclopaedia of Mathematics. Ed. Vol. 9. Springer. Articles, a selection: Hazewinkel, Michiel. "Moduli and canonical forms for linear dynamical systems II: The topological case". Mathematical Systems Theory. 10: 363–385. Doi:10.1007/BF01683285. Archived from the original on 12 December 2013. Hazewinkel, Michiel. "On Lie algebras and finite dimensional filtering". Stochastics. 7: 29–62. Doi:10.1080/17442508208833212. Archived from the original on 12 December 2013. Hazewinkel, M.. J.. "Nonexistence of finite-dimensional filters for conditional statistics of the cubic sensor problem".
Systems & Control Letters. 3: 331–340. Doi:10.1016/0167-691190074-9. Hazewinkel, Michiel. "The algebra of quasi-symmetric functions is free over the integers". Advances in Mathematics. 164: 283–300. Doi:10.1006/aima.2001.2017. Homepage
In mathematics, a monotonic function is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, was generalized to the more abstract setting of order theory. In calculus, a function f defined on a subset of the real numbers with real values is called monotonic if and only if it is either non-increasing, or non-decreasing; that is, as per Fig. 1, a function that increases monotonically does not have to increase, it must not decrease. A function is called monotonically increasing, if for all x and y such that x ≤ y one has f ≤ f, so f preserves the order. A function is called monotonically decreasing if, whenever x ≤ y f ≥ f, so it reverses the order. If the order ≤ in the definition of monotonicity is replaced by the strict order < one obtains a stronger requirement. A function with this property is called increasing. Again, by inverting the order symbol, one finds a corresponding concept called decreasing. Functions that are increasing or decreasing are one-to-one If it is not clear that "increasing" and "decreasing" are taken to include the possibility of repeating the same value at successive arguments, one may use the terms weakly increasing and weakly decreasing to stress this possibility.
The terms "non-decreasing" and "non-increasing" should not be confused with the negative qualifications "not decreasing" and "not increasing". For example, the function of figure 3 first falls rises falls again, it is therefore not decreasing and not increasing, but it is neither non-decreasing nor non-increasing. A function f is said to be monotonic over an interval if the derivatives of all orders of f are nonnegative or all nonpositive at all points on the interval; the term monotonic transformation can possibly cause some confusion because it refers to a transformation by a increasing function. This is the case in economics with respect to the ordinal properties of a utility function being preserved across a monotonic transform. In this context, what we are calling a "monotonic transformation" is, more called a "positive monotonic transformation", in order to distinguish it from a “negative monotonic transformation,” which reverses the order of the numbers; the following properties are true for a monotonic function f: R → R: f has limits from the right and from the left at every point of its domain.
F can only have jump discontinuities. The discontinuities, however, do not consist of isolated points and may be dense in an interval; these properties are the reason. Two facts about these functions are: if f is a monotonic function defined on an interval I f is differentiable everywhere on I, i.e. the set of numbers x in I such that f is not differentiable in x has Lebesgue measure zero. In addition, this result cannot be improved to countable: see Cantor function. If f is a m
Encyclopedia of Mathematics
The Encyclopedia of Mathematics is a large reference work in mathematics. It is available in book form and on CD-ROM; the 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduate level, the presentation is technical in nature. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer; the CD-ROM contains three-dimensional objects. The encyclopedia has been translated from the Soviet Matematicheskaya entsiklopediya edited by Ivan Matveevich Vinogradov and extended with comments and three supplements adding several thousand articles; until November 29, 2011, a static version of the encyclopedia could be browsed online free of charge online. This URL now redirects to the new wiki incarnation of the EOM. A new dynamic version of the encyclopedia is now available as a public wiki online; this new wiki is a collaboration between the European Mathematical Society. This new version of the encyclopedia includes the entire contents of the previous online version, but all entries can now be publicly updated to include the newest advancements in mathematics.
All entries will be monitored for content accuracy by members of an editorial board selected by the European Mathematical Society. Vinogradov, I. M. Matematicheskaya entsiklopediya, Sov. Entsiklopediya, 1977. Hazewinkel, M. Encyclopaedia of Mathematics, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 1, Kluwer, 1987. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 2, Kluwer, 1988. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 3, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 4, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 5, Kluwer, 1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 6, Kluwer, 1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 7, Kluwer, 1991. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 8, Kluwer, 1992. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 9, Kluwer, 1993. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 10, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement I, Kluwer, 1997. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement II, Kluwer, 2000.
Hazewinkel, M. Encyclopaedia of Mathematics, Supplement III, Kluwer, 2002. Hazewinkel, M. Encyclopaedia of Mathematics on CD-ROM, Kluwer, 1998. Encyclopedia of Mathematics, public wiki monitored by an editorial board under the management of the European Mathematical Society. List of online encyclopedias Official website Publications by M. Hazewinkel, at ResearchGate
Karl Theodor Wilhelm Weierstrass was a German mathematician cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a teacher teaching mathematics, physics and gymnastics. Weierstrass formalized the definition of the continuity of a function, proved the intermediate value theorem and the Bolzano–Weierstrass theorem, used the latter to study the properties of continuous functions on closed bounded intervals. Weierstrass was born in part of Ennigerloh, Province of Westphalia. Weierstrass was the son of Wilhelm Weierstrass, a government official, Theodora Vonderforst, his interest in mathematics began. He was sent to the University of Bonn upon graduation to prepare for a government position; because his studies were to be in the fields of law and finance, he was in conflict with his hopes to study mathematics. He resolved the conflict by paying little heed to his planned course of study, but continued private study in mathematics.
The outcome was to leave the university without a degree. After that he studied mathematics at the Münster Academy and his father was able to obtain a place for him in a teacher training school in Münster, he was certified as a teacher in that city. During this period of study, Weierstrass attended the lectures of Christoph Gudermann and became interested in elliptic functions. In 1843 he taught in Deutsch Krone in West Prussia and since 1848 he taught at the Lyceum Hosianum in Braunsberg. Besides mathematics he taught physics and gymnastics. Weierstrass may have had an illegitimate child named Franz with the widow of his friend Carl Wilhelm Borchardt. After 1850 Weierstrass suffered from a long period of illness, but was able to publish papers that brought him fame and distinction; the University of Königsberg conferred an honorary doctor's degree on him on 31 March 1854. In 1856 he took a chair at the Gewerbeinstitut, which became the Technical University of Berlin. In 1864 he became professor at the Friedrich-Wilhelms-Universität Berlin, which became the Humboldt Universität zu Berlin.
At the age of fifty-five, Weierstrass met Sonya Kovalevsky whom he tutored after failing to secure her admission to the University. They had a fruitful intellectual, but troubled personal relationship that "far transcended the usual teacher-student relationship"; the misinterpretation of this relationship and Kovalevsky's early death in 1891 was said to have contributed to Weierstrass' ill-health. He was immobile for the last three years of his life, died in Berlin from pneumonia. Weierstrass was interested in the soundness of calculus, at the time, there were somewhat ambiguous definitions regarding the foundations of calculus, hence important theorems could not be proven with sufficient rigour. While Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 his work remained unknown to most of the mathematical community until years and many mathematicians had only vague definitions of limits and continuity of functions. Delta-epsilon proofs are first found in the works of Cauchy in the 1820s.
Cauchy did not distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 Cours d'analyse, Cauchy argued that the limit of continuous functions was itself continuous, a statement interpreted as being incorrect by many scholars; the correct statement is. This required the concept of uniform convergence, first observed by Weierstrass's advisor, Christoph Gudermann, in an 1838 paper, where Gudermann noted the phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, both formalized it and applied it throughout the foundations of calculus; the formal definition of continuity of a function, as formulated by Weierstrass, is as follows: f is continuous at x = x 0 if ∀ ε > 0 ∃ δ > 0 such that for every x in the domain of f, | x − x 0 | < δ ⇒ | f − f | < ε. In simple English, f is continuous at a point x = x 0 if for each ε > 0 there exists a δ > 0 such that the function f lies between f − ε and f