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
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
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Friedrich Moritz "Fritz" Hartogs was a German-Jewish mathematician, known for his work on set theory and foundational results on several complex variables. Hartogs was the son of the merchant Gustav Hartogs and his wife Elise Feist and grew up in Frankfurt am Main, he studied at the Königliche Technische Hochschule Hannover, at the Technische Hochschule Charlottenburg, at the University of Berlin, at the Ludwig Maximilian University of Munich, graduating with a doctorate in 1903 (supervised by Alfred Pringsheim. He was Privatdozent and Professor in Munich; as a Jew, he suffered under the Nazi regime: he was fired in 1935, was mistreated and interned in KZ Dachau in 1938, committed suicide in 1943. Hartogs main work was in several complex variables where he is known for Hartogs's theorem, Hartogs's lemma and the concepts of holomorphic hull and domain of holomorphy. In set theory, he contributed to the theory of wellorders and proved what is known as Hartogs's theorem: for every set x there is a wellordered set that cannot be injectively embedded in x.
The smallest such set is known as the Hartogs number or Hartogs Aleph of x. Hartogs domain Hartogs–Laurent expansion Hartogs's extension theorem Hartogs's lemma Hartogs number Hartogs's theorem Hartogs–Rosenthal theorem Hartogs, Fritz, "Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veränderlichen.", Sitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften zu München, Mathematisch-Physikalische Klasse, 36: 223–242, JFM 37.0443.01. Hartogs, Fritz, "Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselber durch Reihen welche nach Potentzen einer Veränderlichen fortschreiten", Mathematische Annalen, 62: 1–88, doi:10.1007/BF01448415, JFM 37.0444.01. Available at the DigiZeitschriften. Hartogs, Fritz, "Über das Problem der Wohlordnung", Mathematische Annalen, 76: 438–443, doi:10.1007/BF01458215, JFM 45.0125.01. Available at the DigiZeitschriften. O'Connor, John J.. Biography
In mathematics, a theorem is a statement, proven on the basis of established statements, such as other theorems, accepted statements, such as axioms. A theorem is a logical consequence of the axioms; the proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, experimental. Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called premises. In light of the interpretation of proof as justification of truth, the conclusion is viewed as a necessary consequence of the hypotheses, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.
Although they can be written in a symbolic form, for example, within the propositional calculus, theorems are expressed in a natural language such as English. The same is true of proofs, which are expressed as logically organized and worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, from which a formal symbolic proof can in principle be constructed; such arguments are easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but explains in some way why it is true. In some cases, a picture alone may be sufficient to prove a theorem; because theorems lie at the core of mathematics, they are central to its aesthetics. Theorems are described as being "trivial", or "difficult", or "deep", or "beautiful"; these subjective judgments vary not only from person to person, but with time: for example, as a proof is simplified or better understood, a theorem, once difficult may become trivial.
On the other hand, a deep theorem may be stated but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a well-known example of such a theorem. Logically, many theorems are of the form of an indicative conditional: if A B; such a theorem does not assert B, only that B is a necessary consequence of A. In this case A is called B the conclusion; the theorem "If n is an natural number n/2 is a natural number" is a typical example in which the hypothesis is "n is an natural number" and the conclusion is "n/2 is a natural number". To be proved, a theorem must be expressible as a formal statement. Theorems are expressed in natural language rather than in a symbolic form, with the intention that the reader can produce a formal statement from the informal one, it is common in mathematics to choose a number of hypotheses within a given language and declare that the theory consists of all statements provable from these hypotheses. These hypotheses are called axioms or postulates.
The field of mathematics known as proof theory studies formal languages and the structure of proofs. Some theorems are "trivial", in the sense that they follow from definitions and other theorems in obvious ways and do not contain any surprising insights. Some, on the other hand, may be called "deep", because their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics. A theorem might be simple to state and yet be deep. An excellent example is Fermat's Last Theorem, there are many other examples of simple yet deep theorems in number theory and combinatorics, among other areas. Other theorems have a known proof that cannot be written down; the most prominent examples are the Kepler conjecture. Both of these theorems are only known to be true by reducing them to a computational search, verified by a computer program. Many mathematicians did not accept this form of proof, but it has become more accepted.
The mathematician Doron Zeilberger has gone so far as to claim that these are the only nontrivial results that mathematicians have proved. Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities and hypergeometric identities. To establish a mathematical statement as a theorem, a proof is required, that is, a line of reasoning from axioms in the system to the given statement must be demonstrated. However, the proof is considered as separate from the theorem statement. Although more than one proof may be known for a single theorem, only one proof is required to establish the status of a statement as a theorem; the Pythagorean theorem and the law of quadratic reciprocity are contenders for the title of theorem with the greatest number of distinct proofs. Theorems in mathematics and theories in science are fundamentally different in their epistemology. A scientific theory cannot be proved.
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f to every input x. We say the function has a limit L at an input p: this means f gets closer and closer to L as x moves closer and closer to p. More when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs close to p are taken to outputs that stay a fixed distance apart, we say the limit does not exist; the notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the limit: a function is continuous if all of its limits agree with the values of the function, it appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.
Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique to define continuous functions. However, his work was not known during his lifetime. Cauchy discussed variable quantities and limits and defined continuity of y = f by saying that an infinitesimal change in x produces an infinitesimal change in y in his 1821 book Cours d'analyse, while claims that he only gave a verbal definition. Weierstrass first introduced the epsilon-delta definition of limit in the form it is written today, he introduced the notations lim and limx→x0. The modern notation of placing the arrow below the limit symbol is due to Hardy in his book A Course of Pure Mathematics in 1908. Imagine a person walking over a landscape represented by the graph of y = f, her horizontal position is measured by the value of x, much like the position given by a map of the land or by a global positioning system.
Her altitude is given by the coordinate y. She is walking towards the horizontal position given by x; as she gets closer and closer to it, she notices that her altitude approaches L. If asked about the altitude of x = p, she would answer L. What does it mean to say that her altitude approaches L? It means that her altitude gets nearer and nearer to L except for a possible small error in accuracy. For example, suppose we set a particular accuracy goal for our traveler: she must get within ten meters of L, she reports back that indeed she can get within ten meters of L, since she notes that when she is within fifty horizontal meters of p, her altitude is always ten meters or less from L. The accuracy goal is changed: can she get within one vertical meter? Yes. If she is anywhere within seven horizontal meters of p her altitude always remains within one meter from the target L. In summary, to say that the traveler's altitude approaches L as her horizontal position approaches p means that for every target accuracy goal, however small it may be, there is some neighborhood of p whose altitude fulfills that accuracy goal.
The initial informal statement can now be explicated: The limit of a function f as x approaches p is a number L with the following property: given any target distance from L, there is a distance from p within which the values of f remain within the target distance. This explicit statement is quite close to the formal definition of the limit of a function with values in a topological space. To say that lim x → p f = L, means that ƒ can be made as close as desired to L by making x close enough, but not equal, to p; the following definitions are the accepted ones for the limit of a function in various contexts. Suppose f: R → R is defined on the real line and p,L ∈ R, it is said the limit of f, as x approaches p, is L and written lim x → p f = L, if the following property holds: For every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < | x − p | < δ implies | f − L | < ε. The value of the limit does not depend on the value of f, nor that p be in the domain of f. A more general definition applies for functions defined on subsets of the real line.
Let be an open interval in R, p a point of. Let f be a real-valued function defined on all of except at p itself, it is said that the limit of f, as x approaches p, is L if, for every real ε > 0, there exists a real δ > 0 such that 0 < | x − p | < δ and x ∈ implies | f − L | < ε. Here again the limit does not depend on f being well-defined; the letters ε and δ can be understood as "error" and "distance", in fact Cauchy used ε as an abbreviation for "error" in some of his work, though in his definition of continuity he used an infinitesimal α rather than either ε or δ. In these terms, the error in the measurement of the value at the limit can be made as small as desired by reducing the distance to the limit point; as discussed below this definition works for functions in a more general context. The idea that δ and ε represent distances helps suggest these generalizations. Alternatively x may approach p from