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Diamond Theorem
In mathematics, diamond theorem may refer to: * Aztec diamond theorem on tilings * Diamond isomorphism theorem on modular lattices * Haran's diamond theorem on Hilbertian fields * Second Isomorphism Theorem for Groups * Cullinane diamond theorem on the Galois geometry Galois geometry (so named after the 19th-century French mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or ''Galois field''). More narrowly, ''a'' Ga ... of graphic patterns See also * Diamond (other) {{mathdab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aztec Diamond Theorem
In combinatorial mathematics, an Aztec diamond of order ''n'' consists of all squares of a square lattice whose centers (''x'',''y'') satisfy , ''x'', + , ''y'', ≤ ''n''. Here ''n'' is a fixed integer, and the square lattice consists of unit squares with the origin as a vertex of 4 of them, so that both ''x'' and ''y'' are half-integers. The Aztec diamond theorem states that the number of domino tilings of the Aztec diamond of order ''n'' is 2''n''(''n''+1)/2. The Arctic Circle theorem says that a random tiling of a large Aztec diamond tends to be frozen outside a certain circle. It is common to color the tiles in the following fashion. First consider a checkerboard coloring of the diamond. Each tile will cover exactly one black square. Vertical tiles where the top square covers a black square, is colored in one color, and the other vertical tiles in a second. Similarly for horizontal tiles. Knuth has also defined Aztec diamonds of order ''n'' + 1/2. They are identical wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diamond Isomorphism Theorem
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition, ;Modular law: implies where are arbitrary elements in the lattice, ≤ is the partial order, and ∨ and ∧ (called join and meet respectively) are the operations of the lattice. This phrasing emphasizes an interpretation in terms of projection onto the sublattice , a fact known as the diamond isomorphism theorem. An alternative but equivalent condition stated as an equation (see below) emphasizes that modular lattices form a variety in the sense of universal algebra. Modular lattices arise naturally in algebra and in many other areas of mathematics. In these scenarios, modularity is an abstraction of the 2nd Isomorphism Theorem. For example, the subspaces of a vector space (and more generally the submodules of a module over a ring) form a modular lattice. In a not necessarily modular lattice, there may s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haran's Diamond Theorem
In mathematics, the Haran diamond theorem gives a general sufficient condition for a separable extension of a Hilbertian field to be Hilbertian. Statement of the diamond theorem Let ''K'' be a Hilbertian field and ''L'' a separable extension of ''K''. Assume there exist two Galois extensions ''N'' and ''M'' of ''K'' such that ''L'' is contained in the compositum ''NM'', but is contained in neither ''N'' nor ''M''. Then ''L'' is Hilbertian. The name of the theorem comes from the pictured diagram of fields, and was coined by Jarden. Some corollaries Weissauer's theorem This theorem was firstly proved using non-standard methods by Weissauer. It was reproved by Fried using standard methods. The latter proof led Haran to his diamond theorem. ;Weissauer's theorem Let ''K'' be a Hilbertian field, ''N'' a Galois extension of ''K'', and ''L'' a finite proper extension of ''N''. Then ''L'' is Hilbertian. ;Proof using the diamond theorem If ''L'' is finite over ''K'', it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism Theorems
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences. History The isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noether in her paper ''Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern'', which was published in 1927 in Mathematische Annalen. Less general versions of these theorems can be found in work of Richard Dedekind and previous papers by Noether. Three years later, B.L. van der Waerden published his influential ''Moderne Algebra'' the first abstract algebra textbook that took the groups-rings-fie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cullinane Diamond Theorem
Cullinane may refer to: * Cullinane (name). including a list of people with the name * Cullinane, Queensland, a locality in the Cassowary Coast Region, Australia * Cullinane Corporation (Cullinet), a former software company from Westwood, Massachusetts *Cullinane College, Wanganui Cullinane College is an integrated, Co-Educational Secondary school in Whanganui, New Zealand for students in Year 9 to Year 13. Cullinane College was founded in 2003, through the combining of Sacred Heart College (founded in 1880 and operated by ..., a college in New Zealand See also * '' Cullinane v McGuigan'', a New Zealand case relating to breach of contract {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Geometry
Galois geometry (so named after the 19th-century French mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or ''Galois field''). More narrowly, ''a'' Galois geometry may be defined as a projective space over a finite field. Objects of study include affine and projective spaces over finite fields and various structures that are contained in them. In particular, arcs, ovals, hyperovals, unitals, blocking sets, ovoids, caps, spreads and all finite analogues of structures found in non-finite geometries. Vector spaces defined over finite fields play a significant role, especially in construction methods. Projective spaces over finite fields Notation Although the generic notation of projective geometry is sometimes used, it is more common to denote projective spaces over finite fields by , where is the "geometric" dimension (see below), and is the order of the finite field (or Galois fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
