Self-adjoint operatorIn mathematics, a self-adjoint operator on an infinite-dimensional complex vector space V with inner product (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map A (from V to itself) that is its own adjoint. If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A^∗. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers.
Unbounded operatorIn mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The term "unbounded operator" can be misleading, since "unbounded" should sometimes be understood as "not necessarily bounded"; "operator" should be understood as "linear operator" (as in the case of "bounded operator"); the domain of the operator is a linear subspace, not necessarily the whole space; this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense; in the special case of a bounded operator, still, the domain is usually assumed to be the whole space.
Hermitian adjointIn mathematics, specifically in operator theory, each linear operator on an inner product space defines a Hermitian adjoint (or adjoint) operator on that space according to the rule where is the inner product on the vector space. The adjoint may also be called the Hermitian conjugate or simply the Hermitian after Charles Hermite. It is often denoted by A† in fields like physics, especially when used in conjunction with bra–ket notation in quantum mechanics.
Self-adjointIn mathematics, and more specifically in abstract algebra, an element x of a *-algebra is self-adjoint if . A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if then since in a star-algebra, the set {x,y} is a self-adjoint set even though x and y need not be self-adjoint elements. In functional analysis, a linear operator on a Hilbert space is called self-adjoint if it is equal to its own adjoint A^∗.
Differential operatorIn mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative.
Normal operatorIn mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N : H → H that commutes with its hermitian adjoint N*, that is: NN* = NN. Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood. Examples of normal operators are unitary operators: N = N−1 Hermitian operators (i.e., self-adjoint operators): N* = N Skew-Hermitian operators: N* = −N positive operators: N = MM* for some M (so N is self-adjoint).
X-rayX-ray radiation, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 nanometers to 10 picometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz (3e16Hz to 3e19Hz) and energies in the range 124 keV to 145 eV, respectively. X-ray wavelengths are shorter than those of UV rays and typically longer than those of gamma rays.
Extensions of symmetric operatorsIn functional analysis, one is interested in extensions of symmetric operators acting on a Hilbert space. Of particular importance is the existence, and sometimes explicit constructions, of self-adjoint extensions. This problem arises, for example, when one needs to specify domains of self-adjointness for formal expressions of observables in quantum mechanics. Other applications of solutions to this problem can be seen in various moment problems. This article discusses a few related problems of this type.
Σ-algebraIn mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair is called a measurable space. The σ-algebras are a subset of the set algebras; elements of the latter only need to be closed under the union or intersection of finitely many subsets, which is a weaker condition.
X-ray tubeAn X-ray tube is a vacuum tube that converts electrical input power into X-rays. The availability of this controllable source of X-rays created the field of radiography, the imaging of partly opaque objects with penetrating radiation. In contrast to other sources of ionizing radiation, X-rays are only produced as long as the X-ray tube is energized. X-ray tubes are also used in CT scanners, airport luggage scanners, X-ray crystallography, material and structure analysis, and for industrial inspection.
History of AlagoasThe history of Alagoas begins before the discovery of Brazil by the Portuguese, when the territory was inhabited by the Caeté people. The coast of the current state of Alagoas, recognized since the first Portuguese expeditions, was also visited early on by vessels of other nationalities for the barter of brazilwood (Caesalpinia echinata). Within the Captaincies of Brazil system (1534), the territory was part of the captaincy of Pernambuco, and its occupation dates back to the foundation of the village of Penedo (1545), on the banks of the São Francisco River, by the donee Duarte Coelho Pereira, who encouraged the construction of sugar cane mills in the region.
Compact operator on Hilbert spaceIn the mathematical discipline of functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators (representable by finite-dimensional matrices) in the topology induced by the operator norm. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments.
Mafra, PortugalMafra (ˈmafɾɐ) is a city and a municipality in the district of Lisbon, on the west coast of Portugal, and part of the urban agglomeration of the Greater Lisbon subregion. The population in 2011 was 76,685, in an area of 291.66 km2. It is mostly known for the sumptuous Mafra National Palace inscribed as a UNESCO World Heritage Site. Built in the baroque style, the Mafra National Palace also inspired Portuguese Nobel Prize laureate José Saramago to write his novel Baltasar and Blimunda (Memorial do Convento).
Rio de Janeiro (state)Rio de Janeiro (ˈʁi.u dʒi ʒɐˈne(j)ɾu, ˈʁi.u dʒɐˈ-) is one of the 27 federative units of Brazil. It has the second largest economy of Brazil, with the largest being that of the state of São Paulo. The state, which has 8.2% of the Brazilian population, is responsible for 9.2% of the Brazilian GDP. The state of Rio de Janeiro is located within the Brazilian geopolitical region classified as the Southeast (assigned by IBGE). Rio de Janeiro shares borders with all the other states in the same Southeast macroregion: Minas Gerais (N and NW), Espírito Santo (NE) and São Paulo (SW).
Tourism in BrazilTourism is a growing sector and key to the economy of several regions of Brazil. The country had 6.589 million visitors in 2018, ranking in terms of the international tourist arrivals as the second main destination in South America after Argentina and third in Latin America after Mexico and Argentina. Revenues from international tourists reached in 2015, continuing a recovery trend from the 2008–2009 economic crisis.
CoimbraCoimbra (koʊˈɪmbrə, also USkuˈ-,_ˈkwɪmbrə, UKˈkɔɪmbrə, kuˈĩbɾɐ or ˈkwĩbɾɐ) is a city and a municipality in Portugal. The population of the municipality at the 2011 census was 143,397, in an area of . The fourth-largest agglomerated urban area in Portugal after Lisbon, Porto, and Braga, it is the largest city of the district of Coimbra and the Centro Region. About 460,000 people live in the Região de Coimbra, comprising 19 municipalities and extending into an area of .
Σ-finite measureIn mathematics, a positive (or signed) measure μ defined on a σ-algebra Σ of subsets of a set X is called a finite measure if μ(X) is a finite real number (rather than ∞), and a set A in Σ is of finite measure if μ(A) < ∞. The measure μ is called σ-finite if X is a countable union of measurable sets each with finite measure. A set in a measure space is said to have σ-finite measure if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.
X-ray astronomyX-ray astronomy is an observational branch of astronomy which deals with the study of X-ray observation and detection from astronomical objects. X-radiation is absorbed by the Earth's atmosphere, so instruments to detect X-rays must be taken to high altitude by balloons, sounding rockets, and satellites. X-ray astronomy uses a type of space telescope that can see x-ray radiation which standard optical telescopes, such as the Mauna Kea Observatories, cannot.
Roman Catholic Diocese of FunchalThe Diocese of Funchal (Dioecesis Funchalensis) is a Latin Church ecclesiastical territory or patriarchal archdiocese of the Catholic Church in Portugal. It was originally created on 12 June 1514 by the papal bull Pro excellenti præeminentia from Pope Leo X, following the elevation of Funchal from a village to the status of city, by King Manuel I of Portugal (Royal Decree of 21 August 1508). The diocese was a suffragan of the Archdiocese of Lisbon.
