Skew-symmetric matrixIn mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if denotes the entry in the -th row and -th column, then the skew-symmetric condition is equivalent to The matrix is skew-symmetric because Throughout, we assume that all matrix entries belong to a field whose characteristic is not equal to 2.
Frobenius theorem (differential topology)In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields.
Fubini's theoremIn mathematical analysis, Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by its absolute value. Fubini's theorem implies that two iterated integrals are equal to the corresponding double integral across its integrands.
Leibniz's notationIn calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small (or infinitesimal) increments of x and y, respectively, just as Δx and Δy represent finite increments of x and y, respectively. Consider y as a function of a variable x, or y = f(x).
Old FrenchOld French (franceis, françois, romanz; Modern French: ancien français) was the language spoken in most of the northern half of France approximately between the 8th and the 14th century. Rather than a unified language, Old French was a linkage of Romance dialects, mutually intelligible yet diverse. These dialects came to be collectively known as the langue d'oïl, contrasting with the langue d'oc in the south of France. The mid-14th century witnessed the emergence of Middle French, the language of the French Renaissance in the Île de France region; this dialect was a predecessor to Modern French.
DiphthongA diphthong (ˈdɪfθɒŋ,_ˈdɪp- ; , ), also known as a gliding vowel, is a combination of two adjacent vowel sounds within the same syllable. Technically, a diphthong is a vowel with two different targets: that is, the tongue (and/or other parts of the speech apparatus) moves during the pronunciation of the vowel. In most varieties of English, the phrase "no highway cowboy" (noʊ_ˈhaɪweɪ_ˈkaʊbɔɪ) has five distinct diphthongs, one in every syllable.
BlackBlack is a color that results from the absence or complete absorption of visible light. It is an achromatic color, without hue, like white and grey. It is often used symbolically or figuratively to represent darkness. Black and white have often been used to describe opposites such as good and evil, the Dark Ages versus Age of Enlightenment, and night versus day. Since the Middle Ages, black has been the symbolic color of solemnity and authority, and for this reason it is still commonly worn by judges and magistrates.
Black braneIn general relativity, a black brane is a solution of the equations that generalizes a black hole solution but it is also extended—and translationally symmetric—in p additional spatial dimensions. That type of solution would be called a black p-brane. In string theory, the term black brane describes a group of D1-branes that are surrounded by a horizon. With the notion of a horizon in mind as well as identifying points as zero-branes, a generalization of a black hole is a black p-brane.
Black holeA black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole. The boundary of no escape is called the event horizon. Although it has a great effect on the fate and circumstances of an object crossing it, it has no locally detectable features according to general relativity.
Black catA black cat is a domestic cat with black fur that may be a mixed or specific breed, or a common domestic cat of no particular breed. The Cat Fanciers' Association (CFA) recognizes 22 cat breeds that can come with solid black coats. The Bombay breed is exclusively black. All-black fur pigmentation is slightly more prevalent in male cats than female cats. Most black cats have golden irises due to their high melanin pigment content. In popular myths, witches are believed to be associated with black cats.
Chain ruleIn calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g. More precisely, if is the function such that for every x, then the chain rule is, in Lagrange's notation, or, equivalently, The chain rule may also be expressed in Leibniz's notation. If a variable z depends on the variable y, which itself depends on the variable x (that is, y and z are dependent variables), then z depends on x as well, via the intermediate variable y.
