Trace classIn mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are compact operators. In quantum mechanics, mixed states are described by density matrices, which are certain trace class operators.
Charlotte, North CarolinaCharlotte (ˈʃɑrlət ) is the most populous city in the U.S. state of North Carolina. Located in the Piedmont region, it is the county seat of Mecklenburg County. The population was 874,579 at the 2020 census, making Charlotte the 15th-most populous city in the U.S., the seventh-most populous city in the South, and the second-most populous city in the Southeast behind Jacksonville, Florida. The city is the cultural, economic, and transportation center of the Charlotte metropolitan area, whose 2020 population of 2,660,329 ranked 22nd in the U.
Defective matrixIn linear algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable. In particular, an n × n matrix is defective if and only if it does not have n linearly independent eigenvectors. A complete basis is formed by augmenting the eigenvectors with generalized eigenvectors, which are necessary for solving defective systems of ordinary differential equations and other problems.
Trace (linear algebra)In linear algebra, the trace of a square matrix A, denoted tr(A), is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of A. The trace is only defined for a square matrix (n × n). It can be proven that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proven that tr(AB) = tr(BA) for any two matrices A and B. This implies that similar matrices have the same trace.
Linear algebraLinear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.
World Championship of Online PokerThe World Championship of Online Poker (WCOOP) is an online poker tournament series sponsored by PokerStars. It is played on the PokerStars website in September. Established in 2002, WCOOP is PokerStars' attempt to establish the online equivalent of the World Series of Poker. The WCOOP tournament series is the largest of its kind on the Internet. The fifteen WCOOP events in 2005 generated $12,783,900 in prize money, making it not only the biggest ever online poker event, but the third biggest poker series (live or online) in all of 2005.
Minimal polynomial (linear algebra)In linear algebra, the minimal polynomial μA of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P(A) = 0. Any other polynomial Q with Q(A) = 0 is a (polynomial) multiple of μA. The following three statements are equivalent: λ is a root of μA, λ is a root of the characteristic polynomial χA of A, λ is an eigenvalue of matrix A. The multiplicity of a root λ of μA is the largest power m such that ker((A − λIn)m) strictly contains ker((A − λIn)m−1).
Kivu Ebola epidemicThe Kivu Ebola epidemic was an outbreak of Ebola virus disease (EVD) mainly in eastern Democratic Republic of the Congo (DRC), and in other parts of Central Africa, from 2018 to 2020. Between 1 August 2018 and 25 June 2020 it resulted in 3,470 reported cases. The Kivu outbreak also affected Ituri Province, whose first case was confirmed on 13 August 2018. In November 2018, the outbreak became the biggest Ebola outbreak in the DRC's history, and had become the second-largest Ebola outbreak in recorded history worldwide, behind only the 2013–2016 Western Africa epidemic.
General linear groupIn mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix as the identity element of the group. The group is so named because the columns (and also the rows) of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.
PolynomialIn mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1. Polynomials appear in many areas of mathematics and science.
Jacobian matrix and determinantIn vector calculus, the Jacobian matrix (dʒəˈkəʊbiən, dʒᵻ-,_jᵻ-) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature.
.mz.mz is the Internet country code top-level domain (ccTLD) for Mozambique. Registrations are at the third level beneath the second-level names co.mz, net.mz, adv.mz, mil.mz, org.mz, ac.mz, gov.mz and edu.mz.
