|
Isomorphism (other)
Isomorphism or isomorph may refer to: * Isomorphism, in mathematics, logic, philosophy, and information theory, a mapping that preserves the structure of the mapped entities, in particular: ** Graph isomorphism a mapping that preserves the edges and vertices of a graph ** Group isomorphism a mapping that preserves the group structure ** Order isomorphism a mapping that preserves the comparabilities of a partially ordered set. ** Ring isomorphism a mapping that preserves both the additive and multiplicative structure of a ring ** Isomorphism theorems theorems that assert that some homomorphisms involving quotients and subobjects are isomorphisms * Isomorphism (sociology), a similarity of the processes or structure of one organization to those of another * Isomorphism (crystallography), a similarity of crystal form * Isomorphism (Gestalt psychology), a correspondence between a stimulus array and the brain state created by that stimulus * Cybernetic isomorphism, a recursive property ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived . The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Isomorphism
In graph theory, an isomorphism of graphs ''G'' and ''H'' is a bijection between the vertex sets of ''G'' and ''H'' : f \colon V(G) \to V(H) such that any two vertices ''u'' and ''v'' of ''G'' are adjacent in ''G'' if and only if f(u) and f(v) are adjacent in ''H''. This kind of bijection is commonly described as "edge-preserving bijection", in accordance with the general notion of isomorphism being a structure-preserving bijection. If an isomorphism exists between two graphs, then the graphs are called isomorphic, often denoted by G\simeq H. In the case when the isomorphism is a mapping of a graph onto itself, i.e., when ''G'' and ''H'' are one and the same graph, the isomorphism is called an automorphism of ''G''. Graph isomorphism is an equivalence relation on graphs and as such it partitions the class of all graphs into equivalence classes. A set of graphs isomorphic to each other is called an isomorphism class of graphs. The question of whether graph isomorphism can be dete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a bijection between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished. Definition and notation Given two groups (G, *) and (H, \odot), a ''group isomorphism'' from (G, *) to (H, \odot) is a bijective group homomorphism from G to H. Spelled out, this means that a group isomorphism is a bijective function f : G \to H such that for all u and v in G it holds that f(u * v) = f(u) \odot f(v). The two groups (G, *) and (H, \odot) are isomorphic if there exists an isomorphism from one to the other. This is written (G, *) \cong (H, \odot). Often shorter and simpler notations can be used. When the relevant group operations are understood, they are omitted and one writes G \co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Isomorphism
In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that either of the orders can be obtained from the other just by renaming of elements. Two strictly weaker notions that relate to order isomorphisms are order embeddings and Galois connections. The idea of isomorphism can be understood for finite orders in terms of Hasse diagrams. Two finite orders are isomorphic exactly when a single Hasse diagram ( up to relabeling of its elements) expresses them both, in other words when every Hasse diagram of either can be converted to a Hasse diagram of the other by simply relabeling the vertices. Definition Formally, given two posets (S,\le_S) and (T,\le_T), an order isomorphism from (S,\le_S) to (T,\le_T) is a bijective function f from S to T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Isomorphism
In mathematics, 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 that preserves addition, multiplication and multiplicative identity; that is, : \begin f(a+b)&= f(a) + f(b),\\ f(ab) &= f(a)f(b), \\ f(1_R) &= 1_S, \end for all ''a'', ''b'' in ''R''. These conditions imply that additive inverses and the additive identity are also preserved. If, in addition, is a bijection, then its inverse −1 is also a ring homomorphism. In this case, is called a ring isomorphism, and the rings ''R'' and ''S'' are called ''isomorphic''. From the standpoint of ring theory, isomorphic rings have exactly the same properties. If ''R'' and ''S'' are s, then the corresponding notion is that of a homomorphism, defined as above except without the third condition ''f''(1''R'') = 1''S''. A homomorphism between (unital) rings need not be a ring homomorphism. The composition of two rin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism Theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship among quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and 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- fields ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism (sociology)
In sociology, an isomorphism is a similarity of the processes or structure of one organization to those of another, be it the result of imitation or independent development under similar constraints. The concept of institutional isomorphism was primarily developed by Paul DiMaggio and Walter Powell. The concept appears in their 1983 paper ''The iron cage revisited: institutional isomorphism and collective rationality in organizational fields''. The term is borrowed from the mathematical concept of isomorphism. Isomorphism in the context of globalization, is an idea of contemporary national societies that is addressed by the institutionalization of world models constructed and propagated through global cultural and associational processes. As it is emphasized by realist theories the heterogeneity of economic and political resource or local cultural origins by the micro-phenomenological theories, many ideas suggest that the trajectory of change in political units is towards homog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism (crystallography)
In chemistry, isomorphism has meanings both at the level of crystallography and at a molecular level. In crystallography, crystals are isomorphous if they have identical symmetry and if the atomic positions can be described with a set of parameters (unit cell dimensions and fractional coordinates) whose numerical values differ only slightly. Molecules are isomorphous if they have similar shapes. The coordination complexes tris(acetylacetonato)iron (Fe(acac)3) and tris(acetylacetonato)aluminium (Al(acac)3) are isomorphous. These compounds, both of ''D''3 symmetry have very similar shapes, as determined by bond lengths and bond angles. Isomorphous compounds give rise to isomorphous crystals and form solid solutions. Historically, crystal shape was defined by measuring the angles between crystal faces with a goniometer. Whereas crystals of Fe(acac)3 are deep red and crystals of Al(acac)3 are colorless, a solid solution of the two, i.e. Fe1−xAlx(acac)3 will be deep or pale pi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism (Gestalt Psychology)
The term isomorphism literally means sameness (iso) of form (morphism). In Gestalt psychology, Isomorphism is the idea that perception and the underlying physiological representation are similar because of related Gestalt qualities. Isomorphism refers to a correspondence between a stimulus array and the brain state created by that stimulus, and is based on the idea that the objective brain processes underlying and correlated with particular phenomenological experiences functionally have the same form and structure as those subjective experiences. Isomorphism can also be described as the similarity in the gestalt patterning of a stimulus and the activity in the brain while perceiving the stimulus. More generally, this concept is an expression of the materialist view that the properties of mind and consciousness are a direct consequence of the electrochemical interactions within the physical brain. A commonly used example of isomorphism is the phi phenomenon The term phi phenomenon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cybernetic Isomorphism
Cybernetics is the transdisciplinary study of circular causal processes such as feedback and recursion, where the effects of a system's actions (its outputs) return as inputs to that system, influencing subsequent action. It is concerned with general principles that are relevant across multiple contexts, including in engineering, ecological, economic, biological, cognitive and social systems and also in practical activities such as designing, learning, and managing. Cybernetics' transdisciplinary character has meant that it intersects with a number of other fields, leading to it having both wide influence and diverse interpretations. The field is named after an example of circular causal feedback—that of steering a ship (the ancient Greek κυβερνήτης (''kybernḗtēs'') refers to the person who steers a ship). In steering a ship, the position of the rudder is adjusted in continual response to the effect it is observed as having, forming a feedback loop through which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Viable System Model
The viable system model (VSM) is a model of the organizational structure of any autonomous system capable of producing itself. It is an implementation of viable system theory. At the biological level, this model is correspondent to autopoiesis. A viable system is any system organised in such a way as to meet the demands of surviving in the changing environment. One of the prime features of systems that survive is that they are adaptable. The VSM expresses a model for a viable system, which is an abstracted cybernetic (regulation theory) description that is claimed to be applicable to any organisation that is a viable system and capable of autonomy. Overview The model was developed by operations research theorist and cybernetician Stafford Beer in his book ''Brain of the Firm'' (1972). Together with Beer's earlier works on cybernetics applied to management, this book effectively founded management cybernetics. The first thing to note about the cybernetic theory of organizations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorph (gene)
In Muller's classification, an isomorph is described as a gene mutation that expresses a nonsense point mutant, with expression identical to the original allele. Therefore, in respect to the relationships between the original and mutated genes, it is difficult to ascertain the effects of dominanceness and/or recessiveness. ;Muller's classification of mutant alleles See also *Allele *Mutation *Muller's morphs Hermann J. Muller (1890–1967), who was a 1946 Nobel Prize winner, coined the terms amorph, hypomorph, hypermorph, antimorph and neomorph to classify mutations based on their behaviour in various genetic situations, as well as gene interac ... References {{reflist Mutation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
