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Stieltjes Measure
Thomas Joannes Stieltjes ( , ; 29 December 1856 – 31 December 1894) was a Dutch mathematician. He was a pioneer in the field of moment problems and contributed to the study of continued fractions. The Thomas Stieltjes Institute for Mathematics at Leiden University, dissolved in 2011, was named after him, as is the Riemann–Stieltjes integral. Biography Stieltjes was born in Zwolle on 29 December 1856. His father (who had the same first names) was a civil engineer and politician. Stieltjes Sr. was responsible for the construction of various harbours around Rotterdam, and also seated in the Dutch parliament. Stieltjes Jr. went to university at the Polytechnical School in Delft in 1873. Instead of attending lectures, he spent his student years reading the works of Gauss and Jacobi — the consequence of this being he failed his examinations. There were two further failures (in 1875 and 1876), and his father despaired. His father was friends with H. G. van de Sande Bakhuyzen (wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zwolle
Zwolle () is a List of cities in the Netherlands by province, city and Municipalities of the Netherlands, municipality in the Northeastern Netherlands. It is the Capital city, capital of the Provinces of the Netherlands, province of Overijssel and the province's second-largest municipality, after Enschede, and has a population of 132,441 as of December 2023. Zwolle borders the province of Gelderland and lies on the eastern side of the River IJssel. History Archeology, Archaeological findings indicate that the area surrounding Zwolle has been inhabited for a long time. A Henge, woodhenge that was found in the Zwolle-Zuid suburb in 1993 was dated to the Bronze Age period. During the Roman era, the area was inhabited by Salian Franks. The modern city was founded around 800 CE by Frisians, Frisian merchants and troops of Charlemagne. Previous spellings of its name include the identically pronounced ''Suolle'', which means "hill" (cf. the English cognate verb "to swell"). This re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev–Markov–Stieltjes Inequalities
In mathematical analysis, the Chebyshev–Markov–Stieltjes inequalities are inequalities related to the problem of moments that were formulated in the 1880s by Pafnuty Chebyshev and proved independently by Andrey Markov and (somewhat later) by Thomas Jan Stieltjes. Informally, they provide sharp bounds on a measure from above and from below in terms of its first moments. Formulation Given ''m''0,...,''m''2''m''-1 ∈ R, consider the collection C of measures ''μ'' on R such that : \int x^k d\mu(x) = m_k for ''k'' = 0,1,...,2''m'' − 1 (and in particular the integral is defined and finite). Let ''P''0,''P''1, ...,''P''''m'' be the first ''m'' + 1 orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geom ... with respect to ''μ'' ∈ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dutch Parliament
The States General of the Netherlands ( ) is the supreme bicameral legislature of the Netherlands consisting of the Senate () and the House of Representatives (). Both chambers meet at the Binnenhof in The Hague. The States General originated in the 15th century as an assembly of all the provincial states of the Burgundian Netherlands. In 1579, during the Dutch Revolt, the States General split as the northern provinces openly rebelled against Philip II, and the northern States General replaced Philip II as the supreme authority of the Dutch Republic in 1581. The States General were replaced by the National Assembly after the Batavian Revolution of 1795, only to be restored in 1814, when the country had regained its sovereignty. The States General was divided into a Senate and a House of Representatives in 1815, with the establishment of the United Kingdom of the Netherlands. After the constitutional amendment of 1848, members of the House of Representatives were directly elected, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotterdam
Rotterdam ( , ; ; ) is the second-largest List of cities in the Netherlands by province, city in the Netherlands after the national capital of Amsterdam. It is in the Provinces of the Netherlands, province of South Holland, part of the North Sea mouth of the Rhine–Meuse–Scheldt delta, via the Nieuwe Maas, New Meuse inland shipping channel, dug to connect to the Meuse at first and now to the Rhine. Rotterdam's history goes back to 1270, when a dam was constructed in the Rotte (river), Rotte. In 1340, Rotterdam was granted city rights by William II, Count of Hainaut, William IV, Count of Holland. The Rotterdam–The Hague metropolitan area, with a population of approximately 2.7 million, is the List of urban areas in the European Union, 10th-largest in the European Union and the most populous in the country. A major logistic and economic centre, Rotterdam is Port of Rotterdam, Europe's largest seaport. In 2022, Rotterdam had a population of 655,468 and is home to over 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harbour
A harbor (American English), or harbour (Commonwealth English; see American and British English spelling differences#-our, -or, spelling differences), is a sheltered body of water where ships, boats, and barges can be Mooring, moored. The term ''harbor'' is often used interchangeably with ''port'', which is a man-made facility built for loading and unloading Watercraft, vessels and dropping off and picking up passengers. Harbors usually include one or more ports. Alexandria Port in Egypt, meanwhile, is an example of a port with two harbors. Harbors may be natural or artificial. An artificial harbor can have deliberately constructed breakwater (structure), breakwaters, sea walls, or jetties or they can be constructed by dredging, which requires maintenance by further periodic dredging. An example of an artificial harbor is Long Beach Harbor, California, United States, which was an array of salt marshes and tidal flats too shallow for modern merchant ships before it was first ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leiden University
Leiden University (abbreviated as ''LEI''; ) is a Public university, public research university in Leiden, Netherlands. Established in 1575 by William the Silent, William, Prince of Orange as a Protestantism, Protestant institution, it holds the distinction of being the oldest university in the Netherlands of today. During the Dutch Golden Age scholars from around Europe were attracted to the Dutch Republic for its climate of intellectual tolerance. Individuals such as René Descartes, Rembrandt, Christiaan Huygens, Hugo Grotius, Benedictus Spinoza, and later Baron d'Holbach were active in Leiden and environs. The university has seven academic faculties and over fifty subject departments, housing more than forty national and international research institutes. Its historical primary campus consists of several buildings spread over Leiden, while a second campus located in The Hague houses a liberal arts college (Leiden University College The Hague) and several of its faculties. It i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continued Fraction
A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, the continued fraction is finite or infinite. Different fields of mathematics have different terminology and notation for continued fraction. In number theory the standard unqualified use of the term continued fraction refers to the special case where all numerators are 1, and is treated in the article simple continued fraction. The present article treats the case where numerators and denominators are sequences \,\ of constants or functions. From the perspective of number theory, these are called generalized continued fraction. From the perspective of complex analysis or numerical analysis, however, they are just standard, and in the present article they will simply be called "continued fraction". Formulation A continued fraction is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment Problem
In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure \mu to the sequence of moments :m_n = \int_^\infty x^n \,d\mu(x)\,. More generally, one may consider :m_n = \int_^\infty M_n(x) \,d\mu(x)\,. for an arbitrary sequence of functions M_n. Introduction In the classical setting, \mu is a measure on the real line, and M is the sequence \. In this form the question appears in probability theory, asking whether there is a probability measure having specified mean, variance and so on, and whether it is unique. There are three named classical moment problems: the Hamburger moment problem in which the support of \mu is allowed to be the whole real line; the Stieltjes moment problem, for ,\infty); and the Hausdorff moment problem for a bounded interval, which without loss of generality may be taken as ,1/math>. The moment problem also extends to complex analysis as the trigonometric moment problem in which the Hankel matric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematical model, models, and mathematics#Calculus and analysis, change. History One of the earliest known mathematicians was Thales of Miletus (); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem. The number of known mathematicians grew when Pythagoras of Samos () established the Pythagorean school, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Montel's Theorem
In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after French mathematician Paul Montel, and give conditions under which a family of holomorphic functions is normal. Locally uniformly bounded families are normal The first, and simpler, version of the theorem states that a family of holomorphic functions defined on an open subset of the complex numbers is normal if and only if it is locally uniformly bounded. This theorem has the following formally stronger corollary. Suppose that \mathcal is a family of meromorphic functions on an open set D. If z_0\in D is such that \mathcal is not normal at z_0, and U\subset D is a neighborhood of z_0, then \bigcup_f(U) is dense in the complex plane. Functions omitting two values The stronger version of Montel's theorem (occasionally referred to as the Fundamental Normality Test) states that a family of holomorphic functions, all of which omi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mertens Conjecture
In mathematics, the Mertens conjecture is the statement that the Mertens function M(n) is bounded by \pm\sqrt. Although now disproven, it had been shown to imply the Riemann hypothesis. It was conjectured by Thomas Joannes Stieltjes, in an 1885 letter to Charles Hermite (reprinted in ), and again in print by , and disproved by . It is a striking example of a mathematical conjecture proven false despite a large amount of computational evidence in its favor. Definition In number theory, the Mertens function is defined as : M(n) = \sum_ \mu(k), where μ(k) is the Möbius function; the Mertens conjecture is that for all ''n'' > 1, : , M(n), < \sqrt. Disproof of the conjecture Stieltjes claimed in 1885 to have proven a weaker result, namely that was[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
