Also, starting with any compact real form of a semisimple Lie algebra g its complexification as a real Lie algebra of twice the dimension splits into g and a certain solvable Lie algebra ( the Iwasawa decomposition ), and this provides a canonical bicrossproduct quantum group associated to g. For su ( 2 ) one obtains a quantum group deformation of the Euclidean group E ( 3 ) of motions in 3 dimensions.

* The Iwasawa decomposition G = KAN of a semisimple group G as the product of compact, abelian, and nilpotent subgroups generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix ( a consequence of Gram – Schmidt orthogonalization ).

In mathematics, the Iwasawa decomposition KAN of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix ( a consequence of Gram-Schmidt orthogonalization ).

Then the Iwasawa decomposition of is

and the Iwasawa decomposition of G is

For a general semisimple Lie group, the decomposition is the Iwasawa decomposition of G as G = KAN in which K occurs in a product with a contractible subgroup AN.

Before that he worked on Lie groups and Lie algebras, introducing the general Iwasawa decomposition.

Iwasawa worked with so-called-extensions: infinite extensions of a number field with Galois group isomorphic to the additive group of p-adic integers for some prime p. Every closed subgroup of is of the form, so by Galois theory, a-extension is the same thing as a tower of fields such that.

More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a p-adic Lie group.

Combining this with the p-parity theorem by and the announced proof of the main conjecture of Iwasawa theory for GL ( 2 ) by, they conclude that a positive proportion of elliptic curves over Q have analytic rank zero, and hence, by, satisfy the Birch and Swinnerton-Dyer conjecture.

However, this same year Iwasawa became sick with pleurisy, and was unable to return to his position at the university until April 1947.

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It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa, in the 1950s, as part of the theory of cyclotomic fields.

Iwasawa decompositions also hold for some disconnected semisimple groups G, where K becomes a ( disconnected ) maximal compact subgroup provided the center of G is finite.

* Iwasawa, Kenkichi: On some types of topological groups.

Principal ideal domains are thus mathematical objects which behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements ( so an analogue of the fundamental theorem of arithmetic holds ); any two elements of a PID have a greatest common divisor ( although it may not be possible to find it using the Euclidean algorithm ).

This decomposition of a flow variable into a mean value and a turbulent fluctuation was originally proposed by Osborne Reynolds in 1895, and is considered to be the beginning of the systematic mathematical analysis of turbulent flow, as a sub-field of fluid dynamics.

According to Vandervert, in the emotion-driven prodigy ( commonly observed as a " rage to master ") the cerebellum accelerates the streamlining of the efficiencies of working memory in its manipulation and decomposition / re-composition of visual-spatial content into language acquisition and into linguistic, mathematical, and artistic precocity.

In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into a product of matrices.

The multiway mathematical methods applied to these sorts of problems include PARAFAC, trilinear decomposition, and multiway PLS and PCA.

The mathematical models that describe chemical reaction kinetics provide chemists and chemical engineers with tools to better understand and describe chemical processes such as food decomposition, microorganism growth, stratospheric ozone decomposition, and the complex chemistry of biological systems.

In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition.

In mathematical statistics, Wold contributed the Cramér – Wold theorem characterizing the normal distribution and developed the Wold decomposition in time series analysis.

# The updating parameter for the multiverse is analogous to time via a mathematical isomorphism, but the updating parameter involves a decomposition across alternate universes.

A matrix containing word counts per paragraph ( rows represent unique words and columns represent each paragraph ) is constructed from a large piece of text and a mathematical technique called singular value decomposition ( SVD ) is used to reduce the number of columns while preserving the similarity structure among rows.

In the mathematical field of geometric topology, a Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.

In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces.

In the mathematical subfield of numerical analysis the symbolic Cholesky decomposition is an algorithm used to determine the non-zero pattern for the factors of a symmetric sparse matrix when applying the Cholesky decomposition or variants.

In fluid dynamics and the theory of turbulence, Reynolds decomposition is a mathematical

In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum decomposition of the tensor product of two irreducible representations into irreducible representations, in cases where the numbers and types of irreducible components are already known abstractly.

Secondly, a function may be defined in terms of some presentation or decomposition of a mathematical object ; for instance, the Euler characteristic of a cell complex is defined as the alternating sum of the number of cells in each dimension.

* Föllmer – Schweizer decomposition in mathematical finance

* Tucker decomposition, mathematical decomposition for tensors

Multipole moments in mathematics and mathematical physics form an orthogonal basis for the decomposition of a function, based on the response of a field to point sources that are brought infinitely close to each other.

The decomposition process is aided by shredding the plant matter, adding water and ensuring proper aeration by regularly turning the mixture.

* Oxygen — for oxidizing the carbon, the decomposition process.

The curing time allows fungi to continue the decomposition process and eliminating phytotoxic substances.

* Hybrid Physical-Chemical Vapor Deposition ( HPCVD ) – This process involves both chemical decomposition of precursor gas and vaporization of a solid source.

The decomposition process itself is called a Fourier transform.

Functional decomposition refers broadly to the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed ( i. e., recomposed ) from those parts by function composition.

In general, this process of decomposition is undertaken either for the purpose of gaining insight into the identity of the constituent components ( which may reflect individual physical processes of interest, for example ), or for the purpose of obtaining a compressed representation of the global function, a task which is feasible only when the constituent processes possess a certain level of modularity ( i. e., independence or non-interaction ).

Direction of causal flow is upward. For a multivariate function, functional decomposition generally refers to a process of identifying a set of functions such that

Saltpetre harvesting was recorded by Dutch and German travelers as being common in even the smallest villages and was collected from the decomposition process of large dung hills specifically piled for the purpose.

Storage facilities need to be carefully designed to keep the potatoes alive and slow the natural process of decomposition, which involves the breakdown of starch.

In addition to removing water, the distillation process leads to the formation and decomposition of numerous aroma compounds, fundamentally altering the composition of the distillate from its source.

The decomposition process uses oxygen and deprives the deeper waters of oxygen which can kill fish and other organisms.

Naturally occurring bacteria in Everglades peat and muck assist with the process of decomposition under water, which is generally very slow, partially due to the low levels of dissolved oxygen.

Glacial polish, striations and grooves commonly found on erratics of this size have for the most part been effaced by the normal process of decomposition called weathering.

Tanning is the process of treating skins of animals to produce leather, which is more durable and less susceptible to decomposition.

To speed up the decomposition process, fallen leaves can be shredded, for instance by using a rotary lawn mower.

* Anaerobic digestion, ( AD ) is the naturally occurring process of anaerobic decomposition harnessed and contained to manage waste and / or release energy.

After the signature of the Ohrid Agreement in August 2001, and the decomposition of the NLA in September, Ahmeti was engaged in the political process of the implementation of this Agreement.

As part of the process of decomposition, the myosin heads are degraded by the enzymes, allowing the muscle contraction to release and the body to relax.

The process of diagenesis is also important in the decomposition of bone tissue.

The thought of a long, slow decomposition process is unappealing to some ;

These containers slow the decomposition process by ( partially ) physically blocking decomposing bacteria and other organisms from accessing the corpse.

Alexander's work describes a process of decomposition, in which the designer has a problem ( perhaps a commercial assignment ), selects a solution, then discovers new, smaller problems resulting from the larger solution.

The resulting aeration of the soil leads to the oxidation of its organic components, such as peat, and this decomposition process may cause significant land subsidence.