Group (mathematics)In mathematics, a group is a non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inverse element. Many mathematical structures are groups endowed with other properties. For example, the integers with the addition operation is an infinite group, which is generated by a single element called 1 (these properties characterize the integers in a unique way).
Venetian paintingVenetian painting was a major force in Italian Renaissance painting and beyond. Beginning with the work of Giovanni Bellini (c. 1430–1516) and his brother Gentile Bellini (c. 1429–1507) and their workshops, the major artists of the Venetian school included Giorgione (c. 1477–1510), Titian (c. 1489–1576), Tintoretto (1518–1594), Paolo Veronese (1528–1588) and Jacopo Bassano (1510–1592) and his sons. Considered to give primacy to colour over line, the tradition of the Venetian school contrasted with the Mannerism prevalent in the rest of Italy.
Group theoryIn abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
Dihedral groupIn mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The notation for the dihedral group differs in geometry and abstract algebra. In geometry, D_n or Dih_n refers to the symmetries of the n-gon, a group of order 2n. In abstract algebra, D_2n refers to this same dihedral group.
Doge of VeniceThe Doge of Venice (doʊdʒ ) sometimes translated as Duke (compare the Italian Duca), was the chief magistrate and leader of the Republic of Venice between 726 and 1797. Doges of Venice were elected for life by the Venetian nobility. The doge was neither a duke in the modern sense, nor the equivalent of a hereditary duke. The title "doge" was the title of the senior-most elected official of Venice and Genoa; both cities were republics and elected doges.
Reductive groupIn mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation that has a finite kernel and is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n).
Republic of VeniceThe Republic of Venice or Venetian Republic was a sovereign state and maritime republic in parts of present-day Italy (mainly northeastern Italy) that existed for 1100 years from AD 697 until AD 1797. Centered on the lagoon communities of the prosperous city of Venice, it incorporated numerous overseas possessions in modern Croatia, Slovenia, Montenegro, Greece, Albania and Cyprus. The republic grew into a trading power during the Middle Ages and strengthened this position during the Renaissance.
HistoryHistory (derived ) is the systematic study and documentation of the human past. The period of events before the invention of writing systems is considered prehistory. "History" is an umbrella term comprising past events as well as the memory, discovery, collection, organization, presentation, and interpretation of these events. Historians seek knowledge of the past using historical sources such as written documents, oral accounts, art and material artifacts, and ecological markers.
Solvable groupIn mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup. Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable (note this theorem holds only in characteristic 0).
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.
