Style (form of address)A style of office or form of address, also called manner of address, is an official or legally recognized form of address for a person or other entity (such as a government or company), and may often be used in conjunction with a personal title. A style, by tradition or law, precedes a reference to a person who holds a post or political office, and is sometimes used to refer to the office itself. An honorific can also be awarded to an individual in a personal capacity.
Injective objectIn mathematics, especially in the field of , the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of . The dual notion is that of a projective object. An in a is said to be injective if for every monomorphism and every morphism there exists a morphism extending to , i.e. such that . That is, every morphism factors through every monomorphism . The morphism in the above definition is not required to be uniquely determined by and .
Eternal sinIn Christian hamartiology, eternal sin, the unforgivable sin, unpardonable sin, or ultimate sin is the sin which will not be forgiven by God. One eternal or unforgivable sin (blasphemy against the Holy Spirit), also known as the sin unto death, is specified in several passages of the Synoptic Gospels, including Mark 3:28–29, Matthew 12:31–32, and Luke 12:10, as well as other New Testament passages including Hebrews 6:4–6, Hebrews 10:26–31, and 1 John 5:16.
PrinceA prince is a male ruler (ranked below a king, grand prince, and grand duke) or a male member of a monarch's or former monarch's family. Prince is also a title of nobility (often highest), often hereditary, in some European states. The female equivalent is a princess. The English word derives, via the French word prince, from the Latin noun prīnceps, from primus (first) and caput (head), meaning "the first, foremost, the chief, most distinguished, noble ruler, prince".
Injective moduleIn mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q. This concept is to that of projective modules.
Āryabhaṭa's sine tableĀryabhata's sine table is a set of twenty-four numbers given in the astronomical treatise Āryabhatiya composed by the fifth century Indian mathematician and astronomer Āryabhata (476–550 CE), for the computation of the half-chords of a certain set of arcs of a circle. The set of numbers appears in verse 12 in Chapter 1 Dasagitika of Aryabhatiya. It is not a table in the modern sense of a mathematical table; that is, it is not a set of numbers arranged into rows and columns.
BijectionIn mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set; there are no unpaired elements between the two sets. In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.
Injective hullIn mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls were first described in . A module E is called the injective hull of a module M, if E is an essential extension of M, and E is injective. Here, the base ring is a ring with unity, though possibly non-commutative. An injective module is its own injective hull. The injective hull of an integral domain is its field of fractions .
Inverse function theoremIn mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point. The theorem also gives a formula for the derivative of the inverse function. In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, giving a formula for the Jacobian matrix of the inverse.
Injective functionIn mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct elements; that is, x1 ≠ x2 implies f(x1) f(x2). (Equivalently, f(x1) = f(x2) implies x1 = x2 in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.
Weierstrass elliptic functionIn mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script p. They play an important role in the theory of elliptic functions. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice.
TangentIn geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c if the line passes through the point (c, f(c)) on the curve and has slope f(c), where f is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
