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E. E. Levi
Eugenio Elia Levi (18 October 1883 – 28 October 1917) was an Italian mathematician, known for his fundamental contributions in group theory, in the theory of partial differential operators and in the theory of functions of several complex variables. He was a younger brother of Beppo Levi and was killed in action during First World War. Work Research activity He wrote 33 papers, classified by his colleague and friend Mauro Picone according to the scheme reproduced in this section. Differential geometry Group theory He wrote only three papers in group theory: in the first one, discovered what is now called Levi decomposition, which was conjectured by Wilhelm Killing and proved by Élie Cartan in a special case. Function theory In the theory of functions of several complex variables he introduced the concept of pseudoconvexity during his investigations on the domain of existence of such functions: it turned out to be one of the key concepts of the theory. Cauchy and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torino
Turin ( , Piedmontese: ; it, Torino ) is a city and an important business and cultural centre in Northern Italy. It is the capital city of Piedmont and of the Metropolitan City of Turin, and was the first Italian capital from 1861 to 1865. The city is mainly on the western bank of the Po River, below its Susa Valley, and is surrounded by the western Alpine arch and Superga Hill. The population of the city proper is 847,287 (31 January 2022) while the population of the urban area is estimated by Eurostat to be 1.7 million inhabitants. The Turin metropolitan area is estimated by the OECD to have a population of 2.2 million. The city used to be a major European political centre. From 1563, it was the capital of the Duchy of Savoy, then of the Kingdom of Sardinia ruled by the House of Savoy, and the first capital of the Kingdom of Italy from 1861 to 1865. Turin is sometimes called "the cradle of Italian liberty" for having been the political and intellectual cent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First World War
World War I (28 July 1914 11 November 1918), often abbreviated as WWI, was one of the deadliest global conflicts in history. Belligerents included much of Europe, the Russian Empire, the United States, and the Ottoman Empire, with fighting occurring throughout Europe, the Middle East, Africa, the Pacific, and parts of Asia. An estimated 9 million soldiers were killed in combat, plus another 23 million wounded, while 5 million civilians died as a result of military action, hunger, and disease. Millions more died in genocides within the Ottoman Empire and in the 1918 influenza pandemic, which was exacerbated by the movement of combatants during the war. Prior to 1914, the European great powers were divided between the Triple Entente (comprising France, Russia, and Britain) and the Triple Alliance (containing Germany, Austria-Hungary, and Italy). Tensions in the Balkans came to a head on 28 June 1914, following the assassination of Archduke Franz Ferdina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Differential Equation
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, where , , , , , , and are functions of and and where u_x=\frac, u_=\frac and similarly for u_,u_y,u_. A PDE written in this form is elliptic if :B^2-AC, applying the chain rule once gives :u_=u_\xi \xi_x+u_\eta \eta_x and u_=u_\xi \xi_y+u_\eta \eta_y, a second application gives :u_=u_ _x+u_ _x+2u_\xi_x\eta_x+u_\xi_+u_\eta_, :u_=u_ _y+u_ _y+2u_\xi_y\eta_y+u_\xi_+u_\eta_, and :u_=u_ \xi_x\xi_y+u_ \eta_x\eta_y+u_(\xi_x\eta_y+\xi_y\eta_x)+u_\xi_+u_\eta_. We can replace our PDE in x and y with an equivalent equation in \xi and \eta :au_ + 2bu_ + cu_ \text= 0,\, where :a=A^2+2B\xi_x\xi_y+C^2, :b=2A\xi_x\eta_x+2B(\xi_x\eta_y+\xi_y\eta_x) +2C\xi_y\eta_y , and :c=A^2+2B\eta_x\eta_y+C^2. To transform our PDE into the desired canonical fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not address boundary conditions). In terms of the Dirac delta "function" , a fundamental solution is a solution of the inhomogeneous equation Here is ''a priori'' only assumed to be a distribution. This concept has long been utilized for the Laplacian in two and three dimensions. It was investigated for all dimensions for the Laplacian by Marcel Riesz. The existence of a fundamental solution for any operator with constant coefficients — the most important case, directly linked to the possibility of using convolution to solve an arbitrary right hand side — was shown by Bernard Malgrange and Leon Ehrenpreis. In the context of functional analysis, fundamental solutions are usually developed via the Fredholm alternative and explored in F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parametrix
In mathematics, and specifically the field of partial differential equations (PDEs), a parametrix is an approximation to a fundamental solution of a PDE, and is essentially an approximate inverse to a differential operator. A parametrix for a differential operator is often easier to construct than a fundamental solution, and for many purposes is almost as good. It is sometimes possible to construct a fundamental solution from a parametrix by iteratively improving it. Overview and informal definition It is useful to review what a fundamental solution for a differential operator with constant coefficients is: it is a distribution on ℝ''n'' such that :P(D) = \delta(x)~, in the weak sense, where is the Dirac delta distribution. In a similar way, a parametrix for a variable coefficient differential operator is a distribution such that :P(x,D) = \delta(x) + \omega(x) ~, where is some function with compact support. The parametrix is a useful concept in the study of e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wirtinger Derivatives
In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of several complex variables, are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives with respect to one real variable, when applied to holomorphic functions, antiholomorphic functions or simply differentiable functions on complex domains. These operators permit the construction of a differential calculus for such functions that is entirely analogous to the ordinary differential calculus for functions of real variables. Historical notes Early days (1899–1911): the work of Henri Poincaré Wirtinger derivatives were used in complex analysis at least as early as in the paper , as briefly noted by and by . As a matter of fact, in the third paragraph of his 1899 paper, Henri Poi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variable (mathematics)
In mathematics, a variable (from Latin '' variabilis'', "changeable") is a symbol that represents a mathematical object. A variable may represent a number, a vector, a matrix, a function, the argument of a function, a set, or an element of a set. Algebraic computations with variables as if they were explicit numbers solve a range of problems in a single computation. For example, the quadratic formula solves any quadratic equation by substituting the numeric values of the coefficients of that equation for the variables that represent them in the quadratic formula. In mathematical logic, a ''variable'' is either a symbol representing an unspecified term of the theory (a meta-variable), or a basic object of the theory that is manipulated without referring to its possible intuitive interpretation. History In ancient works such as Euclid's ''Elements'', single letters refer to geometric points and shapes. In the 7th century, Brahmagupta used different colours to represent the u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudoconvexity
In mathematics, more precisely in the theory of functions of several complex variables, a pseudoconvex set is a special type of open set in the ''n''-dimensional complex space C''n''. Pseudoconvex sets are important, as they allow for classification of domains of holomorphy. Let :G\subset ^n be a domain, that is, an open connected subset. One says that G is ''pseudoconvex'' (or '' Hartogs pseudoconvex'') if there exists a continuous plurisubharmonic function \varphi on G such that the set :\ is a relatively compact subset of G for all real numbers x. In other words, a domain is pseudoconvex if G has a continuous plurisubharmonic exhaustion function. Every (geometrically) convex set is pseudoconvex. However, there are pseudoconvex domains which are not geometrically convex. When G has a C^2 (twice continuously differentiable) boundary, this notion is the same as Levi pseudoconvexity, which is easier to work with. More specifically, with a C^2 boundary, it can be shown t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He also made significant contributions to general relativity and indirectly to quantum mechanics. He is widely regarded as one of the greatest mathematicians of the twentieth century. His son Henri Cartan was an influential mathematician working in algebraic topology. Life Élie Cartan was born 9 April 1869 in the village of Dolomieu, Isère to Joseph Cartan (1837–1917) and Anne Cottaz (1841–1927). Joseph Cartan was the village blacksmith; Élie Cartan recalled that his childhood had passed under "blows of the anvil, which started every morning from dawn", and that "his mother, during those rare minutes when she was free from taking care of the children and the house, was working with a spinning-wheel". Élie had an elder sister Je ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilhelm Killing
Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry. Life Killing studied at the University of Münster and later wrote his dissertation under Karl Weierstrass and Ernst Kummer at Berlin in 1872. He taught in gymnasia (secondary schools) from 1868 to 1872. He became a professor at the seminary college Collegium Hosianum in Braunsberg (now Braniewo). He took holy orders in order to take his teaching position. He became rector of the college and chair of the town council. As a professor and administrator Killing was widely liked and respected. Finally, in 1892 he became professor at the University of Münster. In 1886, Killing and his spouse entered the Third Order of Franciscans. Work In 1878 Killing wrote on space forms in terms of non-Euclidean geometry in Crelle's Journal, which he further developed in 1880 as well as in 1885. Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
