A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to certain equations. For example, the equation 2 = − 9 has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem; the idea is to extend the real numbers with an indeterminate i, taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: 2 = 2 = = 9 = − 9, 2 = 2 = 2 = 9 = − 9.
According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers; the 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i; this means that complex numbers can be added and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can be divided by nonzero complex numbers. Overall, the complex number system is a field. Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
The complex number a + bi can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary. A complex number whose imaginary part is zero can be viewed as a real number. Complex numbers can be represented in polar form, which associates each complex number with its distance from the origin and with a particular angle known as the argument of this complex number; the geometric identification of the complex numbers with the complex plane, a Euclidean plane, makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis; the addition of complex numbers is thus depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not available in a vector space. Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1.
For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials; the relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i, which hold for all integers k. The real number a is called the real part of the complex number a + bi. To emphasize, the imaginary part does not include a factor i and b, not bi, is the imaginary part. Formally, the complex numbers are defined as the quotient ring of the polynomia
Élie Joseph Cartan, ForMemRS was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems, differential geometry. He made significant contributions to general relativity and indirectly to quantum mechanics, he is regarded as one of the greatest mathematicians of the twentieth century. Cartan's recognition as a first–rate mathematician came to him only in his old age; this was due to his extreme modesty and to the fact that in France the main trend of mathematical research after 1900 was in the field of function theory, but chiefly to his extraordinary originality. It was only after 1930 that a younger generation started to explore the rich treasure of ideas and results that lay buried in his papers. Since his influence has been increasing, with the exception of Poincaré and Hilbert no one else has done so much to give the mathematics of 20th century its shape and viewpoints. Élie Cartan was born 9 April 1869 in the village of Dolomieu, Isère to Joseph Cartan and Anne Cottaz.
Joseph Cartan was the village blacksmith. Élie had an elder sister Jeanne-Marie. Élie Cartan was the best student in the school. One of his teachers, M. Dupuis, recalled "Élie Cartan was a shy student, but an unusual light of great intellect was shining in his eyes, this was combined with an excellent memory". Antonin Dubost the representative of Isère, visited the school and was impressed by Cartan's unusual abilities, he recommended Cartan to participate in a contest for a scholarship in a lycée. Cartan prepared for the contest under the supervision of M. Dupuis and passed at the age of ten years, he spent five years at the College of Vienne and two years at the Lycée of Grenoble. In 1887 he moved to the Lycée Janson de Sailly in Paris to study sciences for two years. Cartan enrolled in the École Normale Supérieure in 1888, he attended there lectures by Charles Hermite, Jules Tannery, Gaston Darboux, Paul Appell, Émile Picard, Edouard Goursat, Henri Poincaré whose lectures were what Cartan thought most of.
After graduation from the École Normale Superieure in 1891, Cartan was drafted into the French army, where he served one year and attained the rank of sergeant. For next two years Cartan returned to ENS and, following the advice of his classmate Arthur Tresse who studied under Sophus Lie in the years 1888–1889, worked on the subject of classification of simple Lie groups, started by Wilhelm Killing. In 1892 Lie came to Paris, at the invitation of Darboux and Tannery, met Cartan for the first time. Cartan defended his dissertation, The structure of finite continuous groups of transformations in 1894 in the Faculty of Sciences in the Sorbonne. Between 1894 and 1896 Cartan was a lecturer at the University of Montpellier. In 1903, while in Lyons, Cartan married Marie-Louise Bianconi. In 1904, Cartan's first son, Henri Cartan, who became an influential mathematician, was born. In 1909 Cartan moved his family to Paris and worked as a lecturer in the Faculty of Sciences in the Sorbonne. In 1912 Cartan became Professor there, based on the reference he received from Poincaré.
He remained in Sorbonne until his retirement in 1940 and spent the last years of his life teaching mathematics at the École Normale Supérieure for girls. In 1921 he became a foreign member of the Polish Academy of Learning and in 1937 a foreign member of the Royal Netherlands Academy of Arts and Sciences. In 1938 he participated in the International Committee composed to organise the International Congresses for the Unity of Science, he died in 1951 in Paris after a long illness. In the Travaux, Cartan breaks down his work into 15 areas. Using modern terminology, they are: Lie theory Representations of Lie groups Hypercomplex numbers, division algebras Systems of PDEs, Cartan–Kähler theorem Theory of equivalence Integrable systems, theory of prolongation and systems in involution Infinite-dimensional groups and pseudogroups Differential geometry and moving frames Generalised spaces with structure groups and connections, Cartan connection, Weyl tensor Geometry and topology of Lie groups Riemannian geometry Symmetric spaces Topology of compact groups and their homogeneous spaces Integral invariants and classical mechanics Relativity, spinorsCartan's mathematical work can be described as the development of analysis on differentiable manifolds, which many now conside
In mathematics, a heterogeneous relation is the same thing as a binary relation. The prefix hetero is from the Greek ἕτερος. A heterogeneous relation has been called a rectangular relation, suggesting that it does not have the square-symmetry of a homogeneous relation on a set where A = B. Developments in algebraic logic have facilitated usage of binary relations; the calculus of relations includes the algebra of sets, extended by composition of relations and the use of converse relations. The inclusion R ⊆ S, meaning that aRb implies aSb, sets the scene in a lattice of relations, but since P ⊆ Q ≡ ≡, the inclusion symbol is superfluous. Composition of relations and manipulation of the operators according to Schröder rules, provides a calculus to work in the power set of A × B. In contrast to homogeneous relations, the composition of relations operation is only a partial function; the necessity of matching range to domain of composed relations has led to the suggestion that the study of heterogeneous relations is a chapter of category theory as in the category of sets, except that the morphisms of this category are relations.
The objects of the category Rel are sets, the relation-morphisms compose as required in a category. 1) Let A =, the oceans of the globe, B =, the continents. Let aRb represent that ocean a borders continent b; the logical matrix for this relation is: R =. The connectivity of the planet Earth can be viewed through R RT and RT R, the former being a 4 × 4 relation on A, the universal relation; this universal relation reflects the fact that every ocean is separated from the others by at most one continent. On the other hand, RT R is a relation on B × B which fails to be universal because at least two oceans must be traversed to voyage from Europe to Oceania. 2) A geometric configuration can be considered a relation between its points and its lines. The relation is expressed as incidence. Finite and infinite projective and affine planes are included. Jakob Steiner pioneered the cataloguing of configurations with the Steiner systems S which have an n-element set S and a set of k-element subsets called blocks, such that a subset with t elements lies in just one block.
These incidence structures have been generalized with block designs. The incidence matrix used in these geometrical contexts corresponds to the logical matrix used with binary relations. An incidence structure is a triple D = where V and B are any two disjoint sets and I is a binary relation between V and B, i.e. I ⊆ V × B; the elements of V will be called points, those of B blocks and those of I flags.3) Visualization of relations leans on graph theory: For relations on a set, a directed graph illustrates a relation and a graph a symmetric relation. For heterogeneous relations a hypergraph has edges with more than two nodes, can be illustrated by a bipartite graph. Just as the clique is integral to relations on a set, so bicliques are used to describe heterogeneous relations. Heterogeneous relations have been described through their induced concept lattices: A concept C ⊂ R satisfies two properties: The logical matrix of C is the outer product of logical vectors C i j = u i v j, u, v logical vectors.
C is maximal, not contained in any other outer product. Thus C is described as a non-enlargeable rectangle. For a given relation R: X → Y, the set of concepts, enlarged by their joins and meets, forms an "induced lattice of concepts", with inclusion ⊑ forming a preorder; the MacNeille completion theorem is cited in a 2013 survey article "Decomposition of relations on concept lattices". The decomposition is R = f E g T, where f and g are functions, called mappings or left-total, univalent relations in this context; the "induced concept lattice is isomorphic to the cut completion of the partial order E that belongs to the minimal decomposition of the relation R."Particular cases are considered below: E total order corresponds to Ferrers type, E identity corresponds to difunctional, a generalization of equivalence relation on a set. Relations may be ranked by the Schein rank which cou
In set theory, a Cartesian product is a mathematical operation that returns a set from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs where a ∈ A and b ∈ B. Products can be specified using e.g.. A × B =. A table can be created by taking the Cartesian product of a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form. More a Cartesian product of n sets known as an n-fold Cartesian product, can be represented by an array of n dimensions, where each element is an n-tuple. An ordered pair is a couple; the Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, further generalized in terms of direct product. An illustrative example is the standard 52-card deck; the standard playing card ranks form a 13-element set. The card suits form a four-element set; the Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, which correspond to all 52 possible playing cards.
Ranks × Suits returns a set of the form. Suits × Ranks returns a set of the form. Both sets are distinct disjoint; the main historical example is the Cartesian plane in analytic geometry. In order to represent geometrical shapes in a numerical way and extract numerical information from shapes' numerical representations, René Descartes assigned to each point in the plane a pair of real numbers, called its coordinates; such a pair's first and second components are called its x and y coordinates, respectively. The set of all such pairs is thus assigned to the set of all points in the plane. A formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair; the most common definition of ordered pairs, the Kuratowski definition, is =. Under this definition, is an element of P, X × Y is a subset of that set, where P represents the power set operator. Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, power set, specification.
Since functions are defined as a special case of relations, relations are defined as subsets of the Cartesian product, the definition of the two-set Cartesian product is prior to most other definitions. Let A, B, C, D be sets; the Cartesian product A × B is not commutative, A × B ≠ B × A, because the ordered pairs are reversed unless at least one of the following conditions is satisfied: A is equal to B, or A or B is the empty set. For example: A =. × C ≠ A × If for example A = × A = ≠ = A ×. The Cartesian product behaves nicely with respect to intersections. × = ∩. × ≠ ∪ In fact, we have that: ∪ = ∪ ∪ [ ( B
Inner product space
In linear algebra, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors, they provide the means of defining orthogonality between vectors. Inner product spaces generalize Euclidean spaces to vector spaces of any dimension, are studied in functional analysis; the first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. An inner product induces an associated norm, thus an inner product space is a normed vector space. A complete space with an inner product is called a Hilbert space. An space with an inner product is called a pre-Hilbert space, since its completion with respect to the norm induced by the inner product is a Hilbert space.
Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. In this article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C. Formally, an inner product space is a vector space V over the field F together with an inner product, i.e. with a map ⟨ ⋅, ⋅ ⟩: V × V → F that satisfies the following three axioms for all vectors x, y, z ∈ V and all scalars a ∈ F: Conjugate symmetry: ⟨ x, y ⟩ = ⟨ y, x ⟩ ¯ Linearity in the first argument: ⟨ a x, y ⟩ = a ⟨ x, y ⟩ ⟨ x + y, z ⟩ = ⟨ x, z ⟩ + ⟨ y, z ⟩ Positive-definite: ⟨ x, x ⟩ > 0, x ∈ V ∖. Positive-definiteness and linearity ensure that: ⟨ x, x ⟩ = 0 ⇒ x = 0 ⟨ 0, 0 ⟩ = ⟨ 0 x, 0 x ⟩ = 0 ⟨ x, 0 x ⟩ = 0 Notice that conjugate symmetry implies that ⟨x, x⟩ is real for all x, since we have: ⟨ x, x ⟩ = ⟨ x, x ⟩ ¯. Conjugate symmetry and linearity in the first variable imply ⟨ x, a y ⟩ = ⟨ a y, x ⟩ ¯ = a ¯ ⟨ y, x ⟩ ¯ = a ¯ ⟨ x, y ⟩ ⟨ x, y + z ⟩ = ⟨ y + z, x ⟩ ¯ = ⟨ y, x ⟩ ¯ + ⟨ z, x ⟩ ¯ = ⟨ x, y ⟩ + ⟨ x, z ⟩.
Wolfgang Ernst Pauli was an Austrian-born Swiss and American theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or Pauli principle"; the discovery involved spin theory, the basis of a theory of the structure of matter. Pauli was born in Vienna to his wife Bertha Camilla Schütz. Pauli's middle name was given in honor of physicist Ernst Mach. Pauli's paternal grandparents were from prominent Jewish families of Prague. Pauli's father converted from Judaism to Roman Catholicism shortly before his marriage in 1899. Pauli's mother, Bertha Schütz, was raised in her own mother's Roman Catholic religion. Pauli was raised as a Roman Catholic, although he and his parents left the Church, he is considered to have been a mystic. Pauli attended the Döblinger-Gymnasium in Vienna, graduating with distinction in 1918.
Only two months after graduation, he published his first paper, on Albert Einstein's theory of general relativity. He attended the Ludwig-Maximilians University in Munich, working under Arnold Sommerfeld, where he received his PhD in July 1921 for his thesis on the quantum theory of ionized diatomic hydrogen. Sommerfeld asked Pauli to review the theory of relativity for the Encyklopädie der mathematischen Wissenschaften. Two months after receiving his doctorate, Pauli completed the article, it was praised by Einstein. Pauli spent a year at the University of Göttingen as the assistant to Max Born, the following year at the Institute for Theoretical Physics in Copenhagen, which became the Niels Bohr Institute in 1965. From 1923 to 1928, he was a lecturer at the University of Hamburg. During this period, Pauli was instrumental in the development of the modern theory of quantum mechanics. In particular, he formulated the theory of nonrelativistic spin. In 1928, he was appointed Professor of Theoretical Physics at ETH Zurich in Switzerland where he made significant scientific progress.
He held visiting professorships at the University of Michigan in 1931, the Institute for Advanced Study in Princeton in 1935. He was awarded the Lorentz Medal in 1931. At the end of 1930, shortly after his postulation of the neutrino and following his divorce and the suicide of his mother, Pauli experienced a personal crisis, he consulted psychotherapist Carl Jung who, like Pauli, lived near Zurich. Jung began interpreting Pauli's archetypal dreams, Pauli became one of the depth psychologist's best students, he soon began to criticize the epistemology of Jung's theory scientifically, this contributed to a certain clarification of the latter's thoughts about the concept of synchronicity. A great many of these discussions are documented in the Pauli/Jung letters, today published as Atom and Archetype. Jung's elaborate analysis of more than 400 of Pauli's dreams is documented in Psychology and Alchemy; the German annexation of Austria in 1938 made him a German citizen, which became a problem for him in 1939 after the outbreak of World War II.
In 1940, he tried in vain to obtain Swiss citizenship, which would have allowed him to remain at the ETH. Pauli moved to the United States in 1940, where he was employed as a professor of theoretical physics at the Institute for Advanced Study. In 1946, after the war, he became a naturalized citizen of the United States and subsequently returned to Zurich, where he remained for the rest of his life. In 1949, he was granted Swiss citizenship. In 1958, Pauli was awarded the Max Planck medal. In that same year, he fell ill with pancreatic cancer; when his last assistant, Charles Enz, visited him at the Rotkreuz hospital in Zurich, Pauli asked him: "Did you see the room number?" It was number 137. Throughout his life, Pauli had been preoccupied with the question of why the fine structure constant, a dimensionless fundamental constant, has a value nearly equal to 1/137. Pauli died in that room on 15 December 1958. Pauli made many important contributions as a physicist in the field of quantum mechanics.
He published papers, preferring lengthy correspondences with colleagues such as Niels Bohr and Werner Heisenberg, with whom he had close friendships. Many of his ideas and results were never published and appeared only in his letters, which were copied and circulated by their recipients. Pauli proposed in 1924 a new quantum degree of freedom with two possible values, in order to resolve inconsistencies between observed molecular spectra and the developing theory of quantum mechanics, he formulated the Pauli exclusion principle his most important work, which stated that no two electrons could exist in the same quantum state, identified by four quantum numbers including his new two-valued degree of freedom. The idea of spin originated with Ralph Kronig. George Uhlenbeck and Samuel Goudsmit one year identified Pauli's new degree of freedom as electron spin, a discovery in which Pauli for a long time wrongly refused to believe. In 1926, shortly after Heisenberg published the matrix theory of modern quantum mechanics, Pauli used it to derive the observed spectrum of the hydrogen atom.
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