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# Topic: Isomorphism

###### In the News (Sat 17 Aug 19)

 Isomorphism theorem In mathematics, the isomorphism theorems are 3 theorems that apply broadly in the realm of universal algebra. First we state the isomorphism theorems for groups, where they take a simpler form and state important properties of factor groups (also called quotient groups). The isomorphism theorems are also valid for modules over a fixed ring R (and therefore also for vector spaces over a fixed field). www.ebroadcast.com.au /lookup/encyclopedia/is/Isomorphism_theorem.html   (312 words)

 Isomorphism - Wikipedia, the free encyclopedia In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. Isomorphic structures are "the same" at some level of abstraction; ignoring the specific identities of the elements in the underlying sets, and focusing just on the structures themselves, the two structures are identical. In linear algebra, an isomorphism can also be defined as a linear map between two vector spaces that is one-to-one and onto. en.wikipedia.org /wiki/Isomorphism   (747 words)

 Isomorphism: Just the facts...   (Site not responding. Last check: 2007-10-22) In mathematics (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement), an isomorphism (in Greek (A native or inhabitant of Greece) isos = equal and morphe = shape) is a kind of mapping ((genetics) the process of locating genes on a chromosome) between objects. When the objects in question are groups ((chemistry) two or more atoms bound together as a single unit and forming part of a molecule), such an isomorphism is called a group isomorphism (additional info and facts about group isomorphism). In sociology (The study and classification of human societies), isomorphism is to a kind of "copying" or "imitation", especially of the practices of one organization (A group of people who work together) by another. www.absoluteastronomy.com /encyclopedia/i/is/isomorphism.htm   (558 words)

 Order isomorphism - Wikipedia, the free encyclopedia 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. Whenever two partially ordered sets are order isomorphic, they can be considered to be "essentially the same" in the sense that one of the orders can be obtained from the other just by renaming of elements. Hence, yet another characterization of order isomorphisms is possible: they are exactly those monotone bijections that have a monotone inverse. en.wikipedia.org /wiki/Order-isomorphism   (221 words)

 PlanetMath: isomorphism A morphism which is both an isomorphism and an endomorphism is called an automorphism. In the category of topological spaces and continuous maps, a continuous map is an isomorphism if and only if it is a homeomorphism. This is version 3 of isomorphism, born on 2002-02-13, modified 2005-11-30. planetmath.org /encyclopedia/Isomorphism2.html   (232 words)

 Isomorphism theorem - Wikipedia, the free encyclopedia In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms. If G and H are groups and f is a homomorphism from G to H, then the kernel K of f is a normal subgroup of G, and the quotient group G/K is isomorphic to the image of f. Let H and K be subgroups of the group G, and assume H is a subgroup of the normalizer of K. en.wikipedia.org /wiki/Isomorphism_theorem   (343 words)

 AllRefer.com - isomorphism (Chemistry) - Encyclopedia Sodium nitrate and calcium sulfate are isomorphous, as are the sulfates of barium, strontium, and lead. Isomorphous substances usually have similar chemical formulas, and the polarizability and ratio of anion and cation radii are generally comparable (see ion). Isomorphism was discovered (c.1820) by Eilhard Mitscherlich, who stated the principle that isomorphous substances have similar chemical formulas; this principle was used by J. Berzelius in determining chemical formulas and combining weights. reference.allrefer.com /encyclopedia/I/isomorph.html   (187 words)

 A.S. Luchins & E.H. Luchins: ISOMORPHISM IN GESTALT THEORY (3) Isomorphism refers not to metrical but to topological correspondences; brain processes are assumed to preserve the functional relations of symmetry, closedness, and adjacency, not the exact sizes and angles of patterns projected on the retina. The Gestalt theory is that a spatial pattern of perception is isomorphic with the spatial pattern of the underlying excitation in the brain. One system is said to be isomorphic with another in respect of their spatial relations if every point in the one corresponds to a point in the other and the topological relations or spatial orders of the points are the same in the two.... gestalttheory.net /archive/luch_iso3.html   (5784 words)

 [No title] The graph isomorphism problem consists in deciding whether two given graphs are isomorphic, i.e. While most articles related to graph isomorphism have been published in the computer science literature, the computation of the orbits of automorphism groups using partitioning techniques has received most attention in chemistry. However, in the context of chemistry because molecules are a restricted class of graphs, we have proven the problems of graph isomorphism, automorphism partitioning, and canonical labeling to be polynomial-time. www.cs.sandia.gov /~jfaulon/MICS/isomorphism/isomorphism.html   (610 words)

 A.S. Luchins & E.H. Luchins: ISOMORPHISM IN GESTALT THEORY In mathematics an isomorphism between two systems requires a one-to-one correspondence between their elements (that is, each element of one system corresponds to one and only one element of the other system, and conversely), which also preserves structures. BORING also described some criticisms of the isomorphism concept in Gestalt psychology and suggested that the future might show the validity of the criticisms, or put otherwise, the worth of the concept. It is therefore a meaningful thesis that perceptual and physical contexts are isomorphic in essential macroscopic respects, and that to this extent there is resemblance between the phenomenal and the physical world, (p. www.gestalttheory.net /archive/luch_iso1.html   (2848 words)

 Isomorphism Isomorphism is a mathematical equivalence between two or more groups. Isomorphic groups possess the same structure in the character tables, but differ in symmetry operations and selection rules. Collections of isomorphic groups are said to belong to the same abstract group, which do not in general correspond to crystallographic categories. newton.ex.ac.uk /research/qsystems/people/goss/symmetry/Isomorphism.html   (177 words)

 Slide Rule as a Teaching Tool Ordinarily, the isomorphism is used to determine a physical quantity by measuring the other physical quantities in the equation and then using the equation to solve for the missing quantity. Since addition of numbers is isomorphic to addition of distances, consider the effect of applying the distance function D to both sides of the equation. The same process could be used to show that in general the composition of two isomorphic transforms is an isomorphic transform - the mirroring of the mirroring of a domain is itself a mirroring of the domain; the translation of a translation is a translation of the original. www.geocities.com /Athens/Delphi/5136/slide.html   (3348 words)

 CASOS: Projects - Isomorphism in Organizational Language One manifestation of the institutionalization process that is of particular interest to organizational researchers and that they have theorized should be present in organizational language is isomorphism, or similarity. In addition to developing novel measures of isomorphism in texts, I break up isomorphism into a more multifaceted concept, incorporating organizational context and authorial motivations, and pointing to the key role that a text's implicit audiences have in encouraging isomorphism. The measures of isomorphism I develop include studying the networks of concepts in the texts using automated text analysis tools. www.casos.cs.cmu.edu /projects/Isomorphism   (483 words)

 RSP - Isomorphism and the Evolution of Creativity Recognizing and comparing isomorphic structures are the meat and drink of scientific enquiry and literary criticism alike, but it seems to me that the generation of novel structures—either in evolutionary terms, or in what we intuitively regard as “real” human creativity—is not necessarily a creative act. To say that A and B are isomorphic is to say that there are elements or features of A that map onto B, even though A and B may appear quite different in many respects. In linguistic or literary terms, isomorphic relationships are usually described as metaphors or analogies, and at the most general level are often known as fables, myths, legends, parables, allegories and so on. www.robinprior.net /articles/CreativeAI.htm   (3346 words)

 USS Clueless Stardate 20011019.1845 Plane geometry is not isomorphic to nagivation on the surface of a sphere. But isomorphism is more powerful than that, because not only does it permit you to switch between spaces, it also allows you to demonstrate that seemingly unrelated problems in the same space are actually identical. For example, algebraic transformations of equations are actually a series of isomorphisms; you process a complex equation into a simple one because the operations you perform in those tranformations guarantee that the result is isomorphic. denbeste.nu /entries/00001156.shtml   (957 words)

 PlanetMath: graph isomorphism A graph isomorphism is a bijection between the vertices of two graphs If an isomorphism can be constructed between two graphs, then we say those graphs are isomorphic. This is version 2 of graph isomorphism, born on 2002-02-03, modified 2002-02-03. planetmath.org /encyclopedia/GraphIsomorphism.html   (66 words)

 Gestalt Isomorphism   (Site not responding. Last check: 2007-10-22) This in turn casts doubt on the objective reality of the non-illusory objects and surfaces in the visual world within which the illusory objects appear embedded, indicating that they too are internal copies of external objects and surfaces, rather than being those objects and surfaces themselves. Isomorphism differs subtly from MÜLLER's axiom in that it states explicitly what is only implied by MÜLLER, that in the case of structured experience, equal dimensionality between percept and representation implies similarity of structure or form (KÖHLER 1947, p. In particular, all of these representations have a problem with encoding multiple surfaces at different depths, as in the perception of transparency, or encoding the volume of empty space that is perceived between the observer and a visible surface. cns-alumni.bu.edu /pub/slehar/webstuff/isomorph/isomorph.html   (6845 words)

 Boost Graph Library: Isomorphism An isomorphism is a 1-to-1 mapping of the vertices in one graph to the vertices of another graph such that adjacency is preserved. Also, if a isomorphism map named parameter is provided then an isomorphism is recorded in the map. BinaryFunction where the first argument is a vertex descriptor, the second argument is a graph, and the result type is an integer. www.boost.org /libs/graph/doc/isomorphism.html   (337 words)

 Isomorphism The idea behind an isomorphism is to realize that two groups are structurally the same even though the names and notation for the elements are different. To show that two groups are not isomorphic, we need to exhibit a structural property of one group not shared by the other. Thus the two cannot be isomorphic and belong in different isomorphism classes. www.math.csusb.edu /notes/advanced/algebra/gp/node19.html   (393 words)

 Entering Alternative Realities -- Astronautics vs Noonautics: isomorphism between launching aerospace vehicles and ... If this is the case, then it may well be that there are learnings from the isomorphism between the two approaches -- if only by using the hard images and distinctions of the aerospace enterprise to clarify some of the frequently disparaged endeavours of the subject enterprise. The approach taken in exploring the degree of isomorphism is through the language, imagery and concepts of the aerospace enterprise and how it resonates in some way, with issues faced in the subjective enterprise of engendering a new reality. The purpose of this paper, however, is to focus on the isomorphism between the complex pattern of procedures of aerospace and those of inner-space exploration as articulated in many spiritual traditions. www.laetusinpraesens.org /docs/liftin.php   (9431 words)

 Hauy on isomorphism His observations were apparently in direct opposition to the theory held by Haüy, for there could be no question about the difference in the chemical composition of the salts, and the crystals were to all intents and purposes identical in form. Further investigations were carried on by Mitscherlich, and two years later, namely in 1821, he characterized this phenomenon of analogous chemical substances crystallizing in forms which appeared to be identical as the principle of isomorphism. Many isomorphous series of artificial compounds of unusual purity were studied. www.minsocam.org /msa/collectors_corner/arc/hauyvi.htm   (1244 words)

 Brain.Save() - Isomorphism   (Site not responding. Last check: 2007-10-22) Isomorphism is a \$10 mathematical word that has been on my mind lately. The undercurrent of isomorphism is that we have many different ways of talking about the same thing, all of which are basically equivalent on the level that we're interested in. The recognition that the canonical form is on equal footing ontologically with all of its isomorphisms is a rejection of the modern neo-Platonic ideal; hence, the postmodern substitution of 'truth' for the much fuzzier idea of 'constructed convention'. hyperthink.net /blog/PermaLink,guid,46c804f0-1384-4752-9519-182d407f9253.aspx   (2095 words)

 [No title] Completion of the proof of the Wirthmüller isomorphism 5 References 9 We illustrate the force of the formal Wirthmüller isomorphism theorem of [2] * *by giving a worked example of an interesting theorem whose proof it simplifies, na* *mely the Wirthmüller isomorphism theorem in equivariant stable homotopy theory. The analogy between these isomorphisms in topology and Verdier duality was first explored by Po Hu [4], who carried out an idea of Lew* *is that these isomorphisms could be obtained using parametrized equivariant spectr* *a. The category D is triangulated, with distinguished triangles isomorphic to ca* *non- ical cofiber sequences of G-spectra. hopf.math.purdue.edu /May/WirthRev.txt   (3972 words)

 Isomorphism of Unordered Directed Trees   (Site not responding. Last check: 2007-10-22) As the trees are isomorphic they must have the same number of subtrees associated with their roots. Thus, the trees of the polynomials are isomorphic themselves, as they have isomorphic subtrees. Using the corollary, we have an immediate efficient probabilistic test for unordered tree isomorphism: test whether the difference of the polynomials associated with the trees is identically zero by evaluating it on a random input. www.reed.edu /academic/math442/tree.html   (662 words)

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