Abstractly, a hierarchy can be modelled mathematically as a rooted tree: the root of the tree forms the top level, and the children of a given vertex are at the same level, below their common parent.

Abstractly, this is equivalent to stabilising a flag: upper triangular matrices are precisely those that preserve the standard flag, which is given by the standard ordered basis and the resulting flag < math > 0 <

Abstractly, these are the left adjoint and right adjoint, respectively, to the inclusion functors of in D. In addition, the truncation functors fit into a triangle, and this is in fact the unique triangle satisfying the third axiom above:

Abstractly, an array reference is simply a procedure of two arguments: an array and a subscript vector, which could be expressed as ; but many languages provide special syntax like ; similarly an array element update is abstractly something like, but many languages provide syntax like.

Abstractly, we can say that D is a linear transformation from some vector space V to another one, W. We know that D ( c ) = 0 for any constant function c. We can by general theory ( mean value theorem ) identify the subspace C of V, consisting of all constant functions as the whole kernel of D. Then by linear algebra we can establish that D < sup >− 1 </ sup > is a well-defined linear transformation that is bijective on Im D and takes values in V / C.

Abstractly programmable toolpath guidance began with mechanical solutions, such as in musical box cams and Jacquard looms.

Abstractly, we can identify the chain rule as a 1-cocycle.

Abstractly, 1 / 8-schisma tuning may be considered the analog, among schismatic tunings, of 1 / 4-comma meantone among meantone tunings, as it also has pure interval ratios of 2: 1 and 5: 4, though with much more accurate interval ratios of 3: 2 and 6: 5 ( less than a quarter of a cent off from just intonation ) than its meantone counterpart.

Abstractly, a parametric equation defines a relation as a set of equations.

Abstractly, this circle is a cyclic group of order twelve, and may be identified with the residue classes modulo twelve.

Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group.

Abstractly, in trying to prove a proposition P, one assumes that it is false, and that therefore there is at least one counterexample.

Abstractly it is assumed that the economy consists only of capitalist production and that the capitalist economy is equal to capitalist society.

For compact groups, the Peter – Weyl theorem explains how one may get harmonics by choosing one irreducible representation out of each equivalence class of representations.

Weyl brought the early period of the development of the theory of Lie groups to fruition, for not only did he classify irreducible representations of semisimple Lie groups and connect the theory of groups with quantum mechanics, but he also put Lie's theory itself on firmer footing by clearly enunciating the distinction between Lie's infinitesimal groups ( i. e., Lie algebras ) and the Lie groups proper, and began investigations of topology of Lie groups ( Borel ( 2001 ), ).

Hermann Weyl used Lie's work on group theory in his papers from 1922 and 1923, and Lie groups today play a role in quantum mechanics.

From 1923 to 1938, Weyl developed the theory of compact groups, in terms of matrix representations.

Non-compact groups and their representations, particularly the Heisenberg group, were also streamlined in that specific context, in his 1927 Weyl quantization, the best extant bridge between

Tits proved that all irreducible spherical buildings ( i. e. with finite Weyl group ) of rank greater than 2 are associated to simple algebraic or classical groups.

The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula.

In mathematics, the Peter – Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian.

Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras.

Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac – Moody algebras.

which may be viewed as dealing with " continuous symmetry ", is strongly influenced by the associated Weyl groups.

This could recently be used on the specific cases and explains e. g. the numerical coincidence between certain coideal subalgebras of these quantum groups to the order of the Weyl group of the Lie algebra.

During this period he established as his special area the study of the discrete series representations of semisimple Lie groups, which are analogues of the Peter – Weyl theory in the non-compact case.

# Generalised spaces with structure groups and connections, Cartan connection, holonomy, Weyl tensor

Dirac and Weyl spinors are complex representations while Majorana spinors are real representations.

The Dirac, Lorentz, Weyl, and Majorana spinors are interrelated, and their relation can be elucidated on the basis of real geometric algebra.

This example shows how the Hermite polynomials and Laguerre polynomials are interrelated through the Wigner – Weyl transform.

As noted by Weyl, Formal logical systems also run the risk of inconsistency ; in Peano arithmetic, this arguably has already been settled with several proofs of consistency, but there is debate over whether or not they are sufficiently finitary to be meaningful.

In 1911 Weyl published Über die asymptotische Verteilung der Eigenwerte ( On the asymptotic distribution of eigenvalues ) in which he proved that the eigenvalues of the Laplacian in the compact domain are distributed according to Weyl law.

The quotient group is the symmetric group, and this construction is in fact the Weyl group of the general linear group: the diagonal matrices are a maximal torus in the general linear group ( and are their own centralizer ), the generalized permutation matrices are the normalizer of this torus, and the quotient, is the Weyl group.

These reflections generate a Coxeter group W, called the Weyl group of A, and the simplicial complex A corresponds to the standard geometric realization of W. Standard generators of the Coxeter group are given by the reflections in the walls of a fixed chamber in A.

The 4 × 4 gamma matrices used in Peskin & Schroeder ( Weyl representation ) are

These are permuted by the action of the Weyl group, and it is a theorem that this action is simply transitive.

In vacuum regions ( those points of spacetime free of matter ), this inability to tilt all the light-cones so that they are all parallel is reflected in the non-vanishing of the Weyl tensor.

* Two elements in T are conjugate if and only if they are conjugate by an element of W. That is, the conjugacy classes of G intersect T in a Weyl orbit.

According to Hermann Weyl, the assumption that space is made of finite and discrete units is subject to a further problem, given by the " tile argument " or " distance function problem ".

Removing the hyperplanes defined by the roots of Φ cuts up Euclidean space into a finite number of open regions, called Weyl chambers.

* The normalizer of T is closed, so the Weyl group is finite

The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula.

That is, by the Peter – Weyl theorem the irreducible unitary representations ρ of G are into a unitary group ( of finite dimension ) and the image will be a closed subgroup of the unitary group by compactness.

The Peter – Weyl theorem extends many results about representations of finite groups to representations of compact groups.

Note that if the functions f and g are polynomials, the above infinite sums become finite ( reducing to the ordinary Weyl algebra case ).