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Kuratowski Closure Operator
Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. He worked as a professor at the University of Warsaw and at the Mathematical Institute of the Polish Academy of Sciences (IM PAN). Between 1946 and 1953, he served as President of the Polish Mathematical Society. He is primarily known for his contributions to set theory, topology, measure theory and graph theory. Some of the notable mathematical concepts bearing Kuratowski's name include Kuratowski's theorem, Kuratowski closure axioms, Kuratowski-Zorn lemma and Kuratowski's intersection theorem. Life and career Early life Kazimierz Kuratowski was born in Warsaw, (then part of Congress Poland controlled by the Russian Empire), on 2 February 1896. He was a son of Marek Kuratow, a barrister, and Róża Karzewska. He completed a Warsaw secondary school, which was named after general Paweł Chrzanowski. In 191 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Warsaw
Warsaw, officially the Capital City of Warsaw, is the capital and List of cities and towns in Poland, largest city of Poland. The metropolis stands on the Vistula, River Vistula in east-central Poland. Its population is officially estimated at 1.86 million residents within a Warsaw metropolitan area, greater metropolitan area of 3.27 million residents, which makes Warsaw the List of cities in the European Union by population within city limits, 6th most-populous city in the European Union. The city area measures and comprises List of districts and neighbourhoods of Warsaw, 18 districts, while the metropolitan area covers . Warsaw is classified as an Globalization and World Cities Research Network#Alpha 2, alpha global city, a major political, economic and cultural hub, and the country's seat of government. It is also the capital of the Masovian Voivodeship. Warsaw traces its origins to a small fishing town in Masovia. The city rose to prominence in the late 16th cent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kuratowski Convergence
In mathematics, Kuratowski convergence or Painlevé-Kuratowski convergence is a notion of convergence for subsets of a topological space. First introduced by Paul Painlevé in lectures on mathematical analysis in 1902,This is reported in the Commentary section of Chapter 4 of Rockafellar and Wets' text. the concept was popularized in texts by Felix Hausdorff and Kazimierz Kuratowski. Intuitively, the Kuratowski limit of a sequence of sets is where the sets " accumulate". Definitions For a given sequence \_^ of points in a space X, a limit point of the sequence can be understood as any point x \in X where the sequence ''eventually'' becomes arbitrarily close to x. On the other hand, a cluster point of the sequence can be thought of as a point x \in X where the sequence ''frequently'' becomes arbitrarily close to x. The Kuratowski limits inferior and superior generalize this intuition of limit and cluster points to subsets of the given space X. Metric Spaces Let (X,d) be a metr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure Theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integral, integration theory, and can be generalized to assume signed measure, negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to Ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Torsion (mechanics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a Set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of List of continuity-related mathematical topics, continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and Homotopy, homotopies. A property that is invariant under such deformations is a to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of ''naive set theory''. After the discovery of Paradoxes of set theory, paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polish Mathematical Society
The Polish Mathematical Society () is the main professional society of Polish mathematicians and represents Polish mathematics within the European Mathematical Society (EMS) and the International Mathematical Union (IMU). History The society was established in Kraków, Poland on 2 April 1919 . It was originally called the Mathematical Society in Kraków, the name was changed to the Polish Mathematical Society on 21 April 1920. It was founded by 16 mathematicians, Stanisław Zaremba, Franciszek Leja, Alfred Rosenblatt, Stefan Banach and Otto Nikodym were among them. Ever since its foundation, the society's main activity was to bring mathematicians together by means of organizing conferences and lectures. The second main activity is the publication of its annals ''Annales Societatis Mathematicae Polonae'', consisting of: * Series 1''Commentationes Mathematicae'' * Series 2Wiadomości Matematyczne("Mathematical News"), in Polish * Series 3: '' Mathematica Applicanda'' (former ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polish Academy Of Sciences
The Polish Academy of Sciences (, PAN) is a Polish state-sponsored institution of higher learning. Headquartered in Warsaw, it is responsible for spearheading the development of science across the country by a society of distinguished scholars and a network of research institutes. It was established in 1951, during the early period of the Polish People's Republic following World War II. History The Polish Academy of Sciences is a Polish state-sponsored institution of higher learning, headquartered in Warsaw, that was established by the merger of earlier science societies, including the Polish Academy of Learning (''Polska Akademia Umiejętności'', abbreviated ''PAU''), with its seat in Kraków, and the Warsaw Society of Friends of Learning (Science), which had been founded in the late 18th century. The Polish Academy of Sciences functions as a learned society acting through an elected assembly of leading scholars and research institutions. The Academy has also, operating throug ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Warsaw School Of Mathematics
Warsaw School of Mathematics is the name given to a group of mathematicians who worked at Warsaw, Poland, in the two decades between the World Wars, especially in the fields of logic, set theory, point-set topology and real analysis. They published in the journal ''Fundamenta Mathematicae'', founded in 1920—one of the world's first specialist pure-mathematics journals. It was in this journal, in 1933, that Alfred Tarski—whose illustrious career would a few years later take him to the University of California, Berkeley—published his celebrated theorem on the undefinability of the notion of truth. Notable members of the Warsaw School of Mathematics have included: * Wacław Sierpiński * Kazimierz Kuratowski * Edward Marczewski * Bronisław Knaster * Zygmunt Janiszewski * Stefan Mazurkiewicz * Stanisław Saks * Karol Borsuk * Roman Sikorski * Nachman Aronszajn * Samuel Eilenberg Additionally, notable logicians of the Lwów–Warsaw School of Logic, working at War ... [...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] |
Tarski–Kuratowski Algorithm
In computability theory and mathematical logic the Tarski–Kuratowski algorithm is a non-deterministic algorithm that produces an upper bound for the complexity of a given formula in the arithmetical hierarchy and analytical hierarchy. The algorithm is named after Alfred Tarski and Kazimierz Kuratowski. Algorithm The Tarski–Kuratowski algorithm for the arithmetical hierarchy consists of the following steps: # Convert the formula to prenex normal form. (This is the non-deterministic part of the algorithm, as there may be more than one valid prenex normal form for the given formula.) # If the formula is quantifier-free, it is in \Sigma^0_0 and \Pi^0_0. # Otherwise, count the number of alternations of quantifiers; call this ''k''. # If the first quantifier is ∃, the formula is in \Sigma^0_. # If the first quantifier is ∀ A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knaster–Kuratowski–Mazurkiewicz Lemma
The Knaster–Kuratowski–Mazurkiewicz lemma is a basic result in mathematical fixed-point theory published in 1929 by Knaster, Kuratowski and Mazurkiewicz. The KKM lemma can be proved from Sperner's lemma and can be used to prove the Brouwer fixed-point theorem. Statement Let \Delta_ be an (n-1)-dimensional simplex with ''n'' vertices labeled as 1,\ldots,n. A KKM covering is defined as a set C_1,\ldots,C_n of closed sets such that for any I \subseteq \, the convex hull of the vertices corresponding to I is covered by \bigcup_C_i. The KKM lemma says that in every KKM covering, the common intersection of all ''n'' sets is nonempty, i.e.: :\bigcap_^n C_i \neq \emptyset. Example When n=3, the KKM lemma considers the simplex \Delta_2 which is a triangle, whose vertices can be labeled 1, 2 and 3. We are given three closed sets C_1,C_2,C_3 such that: * C_1 covers vertex 1, C_2 covers vertex 2, C_3 covers vertex 3. * The edge 12 (from vertex 1 to vertex 2) is covered by the set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kuratowski And Ryll-Nardzewski Measurable Selection Theorem
In mathematics, the Kuratowski–Ryll-Nardzewski measurable selection theorem is a result from measure theory that gives a sufficient condition for a set-valued function to have a measurable selection function. It is named after the Polish mathematicians Kazimierz Kuratowski and Czesław Ryll-Nardzewski. Many classical selection results follow from this theorem and it is widely used in mathematical economics and optimal control. Statement of the theorem Let X be a Polish space, \mathcal (X) the Borel σ-algebra of X , (\Omega, \mathcal) a measurable space and \psi a multifunction on \Omega taking values in the set of nonempty closed subsets of X . Suppose that \psi is \mathcal -weakly measurable, that is, for every open subset U of X , we have :\ \in \mathcal. Then \psi has a selection Selection may refer to: Science * Selection (biology), also called natural selection, selection in evolution ** Sex selection, in genetics ** Mate select ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
