Rough setIn computer science, a rough set, first described by Polish computer scientist Zdzisław I. Pawlak, is a formal approximation of a crisp set (i.e., conventional set) in terms of a pair of sets which give the lower and the upper approximation of the original set. In the standard version of rough set theory (Pawlak 1991), the lower- and upper-approximation sets are crisp sets, but in other variations, the approximating sets may be fuzzy sets. The following section contains an overview of the basic framework of rough set theory, as originally proposed by Zdzisław I.
Fuzzy setIn mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics , decision-making , and clustering , are special cases of L-relations when L is the unit interval [0, 1].
Universal setIn set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory include a universal set. Many set theories do not allow for the existence of a universal set. There are several different arguments for its non-existence, based on different choices of axioms for set theory. In Zermelo–Fraenkel set theory, the axiom of regularity and axiom of pairing prevent any set from containing itself.
Neutral theory of molecular evolutionThe neutral theory of molecular evolution holds that most evolutionary changes occur at the molecular level, and most of the variation within and between species are due to random genetic drift of mutant alleles that are selectively neutral. The theory applies only for evolution at the molecular level, and is compatible with phenotypic evolution being shaped by natural selection as postulated by Charles Darwin.
Set-builder notationIn set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy. Defining sets by properties is also known as set comprehension, set abstraction or as defining a set's intension. Set (mathematics)#Roster notation A set can be described directly by enumerating all of its elements between curly brackets, as in the following two examples: is the set containing the four numbers 3, 7, 15, and 31, and nothing else.
Ring homomorphismIn ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is a function f : R → S such that f is: addition preserving: for all a and b in R, multiplication preserving: for all a and b in R, and unit (multiplicative identity) preserving: Additive inverses and the additive identity are part of the structure too, but it is not necessary to require explicitly that they too are respected, because these conditions are consequences of the three conditions above.
Inverse elementIn mathematics, the concept of an inverse element generalises the concepts of opposite (−x) and reciprocal (1/x) of numbers. Given an operation denoted here ∗, and an identity element denoted e, if x ∗ y = e, one says that x is a left inverse of y, and that y is a right inverse of x. (An identity element is an element such that x * e = x and e * y = y for all x and y for which the left-hand sides are defined.
HomomorphismIn algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word homomorphism comes from the Ancient Greek language: ὁμός () meaning "same" and μορφή () meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).
Group homomorphismIn mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h : G → H such that for all u and v in G it holds that where the group operation on the left side of the equation is that of G and on the right side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure".
G2 phaseDISPLAYTITLE:G2 phase G2 phase, Gap 2 phase, or Growth 2 phase, is the third subphase of interphase in the cell cycle directly preceding mitosis. It follows the successful completion of S phase, during which the cell’s DNA is replicated. G2 phase ends with the onset of prophase, the first phase of mitosis in which the cell’s chromatin condenses into chromosomes. G2 phase is a period of rapid cell growth and protein synthesis during which the cell prepares itself for mitosis.
Module homomorphismIn algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if M and N are left modules over a ring R, then a function is called an R-module homomorphism or an R-linear map if for any x, y in M and r in R, In other words, f is a group homomorphism (for the underlying additive groups) that commutes with scalar multiplication. If M, N are right R-modules, then the second condition is replaced with The of the zero element under f is called the kernel of f.
G1 phaseDISPLAYTITLE:G1 phase The G1 phase, gap 1 phase, or growth 1 phase, is the first of four phases of the cell cycle that takes place in eukaryotic cell division. In this part of interphase, the cell synthesizes mRNA and proteins in preparation for subsequent steps leading to mitosis. G1 phase ends when the cell moves into the S phase of interphase. Around 30 to 40 percent of cell cycle time is spent in the G1 phase.
