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Leray–Schauder Degree
In mathematics, the Leray–Schauder degree is an extension of the degree of a base point preserving continuous map between spheres (S^n, *) \to (S^n , *) or equivalently to a boundary sphere preserving continuous maps between balls (B^n, S^) \to (B^n, S^) to boundary sphere preserving maps between balls in a Banach space f: (B(V), S(V)) \to (B(V), S(V)), assuming that the map is of the form f = id - C where id is the identity map and C is some compact map (i.e. mapping bounded sets to sets whose closure is compact). The degree was invented by Jean Leray and Juliusz Schauder to prove existence results for partial differential equations.Mawhin, J. (2018). A tribute to Juliusz Schauder. ''Antiquitates Mathematicae The Polish Mathematical Society ( pl, Polskie Towarzystwo Matematyczne) is the main professional society of Polish mathematicians and represents Polish mathematics within the European Mathematical Society (EMS) and the International Mathematical Un ...'', ''12''. Referenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base Point
In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains unchanged during subsequent discussion, and is kept track of during all operations. Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e., a map f between a pointed space X with basepoint x_0 and a pointed space Y with basepoint y_0 is a based map if it is continuous with respect to the topologies of X and Y and if f\left(x_0\right) = y_0. This is usually denoted :f : \left(X, x_0\right) \to \left(Y, y_0\right). Pointed spaces are important in algebraic topology, particularly in homotopy theory, where many constructions, such as the fundamental group, depend on a choice of basepoint. The pointed set concept is less important; it is anyway the case of a pointed discrete space. Pointed spaces are often take ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre (geometry), centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the Greek mathematics, ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubble (physics), Bubbles such as soap bubbles take a spherical shape in equilibrium. spherical Earth, The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres rolling, roll smoothly in any direction, so mos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete norme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Map
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when is the identity function, the equality is true for all values of to which can be applied. Definition Formally, if is a set, the identity function on is defined to be a function with as its domain and codomain, satisfying In other words, the function value in the codomain is always the same as the input element in the domain . The identity function on is clearly an injective function as well as a surjective function, so it is bijective. The identity function on is often denoted by . In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or ''diagonal'' of . Algebraic properties If is any function, then we have (whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact (mathematics)
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topologi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Leray
Jean Leray (; 7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology. Life and career He was born in Chantenay-sur-Loire (today part of Nantes). He studied at École Normale Supérieure from 1926 to 1929. He received his Ph.D. in 1933. In 1934 Leray published an important paper that founded the study of weak solutions of the Navier–Stokes equations. In the same year, he and Juliusz Schauder discovered a topological invariant, now called the Leray–Schauder degree, which they applied to prove the existence of solutions for partial differential equations lacking uniqueness. From 1938 to 1939 he was professor at the University of Nancy. He did not join the Bourbaki group, although he was close with its founders. His main work in topology was carried out while he was in a prisoner of war camp in Edelbach, Austria from 1940 to 1945. He concealed his expertise on differential equations, fearing th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Juliusz Schauder
Juliusz Paweł Schauder (; 21 September 1899, Lwów, Austria-Hungary – September 1943, Lwów, Occupied Poland) was a Polish mathematician of Jewish origin, known for his work in functional analysis, partial differential equations and mathematical physics. Life and career Born on 21 September 1899 in Lwów, he was drafted into the Austro-Hungarian Army right after his graduation from school and saw action on the Italian front. He was captured and imprisoned in Italy. He entered the university in Lwów in 1919 and received his doctorate in 1923. He got no appointment at the university and continued his research while working as teacher at a secondary school. Due to his outstanding results, he obtained a scholarship in 1932 that allowed him to spend several years in Leipzig and, especially, Paris. In Paris he started a very successful collaboration with Jean Leray. Around 1935 Schauder obtained the position of a senior assistant in the University of Lwów. Schauder, along with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antiquitates Mathematicae
The Polish Mathematical Society ( pl, Polskie Towarzystwo Matematyczne) 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: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
