# Irreducible case

## Definitions

• Webster's Revised Unabridged Dictionary
• Irreducible case (Alg) a particular case in the solution of a cubic equation, in which the formula commonly employed contains an imaginary quantity, and therefore fails in its application.
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## Usage

### In literature:

IRREDUCIBLE CASE, name given to a cubic equation which cannot be solved by the rule of CARDAN (q. v.).
"The Nuttall Encyclopaedia" by Edited by Rev. James Wood
In case the rupture cannot be returned, it is called irreducible and is a more serious form.
"The Home Medical Library, Volume II (of VI)" by Various
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### In science:

In this case, A is simple if and only if f is irreducible.
On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field
For this purpose let us consider the case the c1 (E ) = c2 (E ) = 0.3 It is known that there is a one-to-one correspondence between irreducible representations of π1 (M ) and stable Higgs bundles with vanishing Chern classes, see .
Higgs Bundles and Four Manifolds
The resulting behavior of the topological complexity, the irreducible braid length, differs strongly from the Abelian case.
Random walks on the braid group B_3 and magnetic translations in hyperbolic geometry
Since these two expressions are irreducible, this case is ruled out.
Number Operator Algebras
These nonunit sequences have rank one, in the terminology of the previous cases, and their only divisors are associates and units; therefore they are irreducibles.
Factorization of integers and arithmetic functions
Again, in the semisimple case the decomposition of W λ ⊗ W µ into irreducibles is given by the Littlewood-Richardson rule.
The representation theory of the Ariki-Koike and cyclotomic q-Schur algebras
Therefore, in the regular case, we may identify the set {χ} in 6.1 with a set of irreducible Gs -equivariant local systems on N a .
Induced and simple modules of double affine Hecke algebras
Also in the present case the reason is that we want to obtain a description in terms of irreducible symmetric spaces.
Random matrix theory and symmetric spaces
We will use the following criterion for the irreducibility in the case p = 2.
Classification of simple modules over degenerate double Affine Hecke algebras of type A
It makes no odds whether one adopts the Heisenberg or the Schr¨odinger viewpoint, it is still the case that joint (and irreducible) properties of subsystems are being used to carry information in the protocols.
Nonlocality and information flow: The approach of Deutsch and Hayden
However, the situations considered in are rather simple and in each case the irreducibility of the restricted space is straightforward.
Generating connected acyclic digraphs uniformly at random
Let us now check the irreducibility in the nondegenerated case.
Exchangeable Fragmentation-Coalescence processes and their equilibrium measures
In the remaining case (i.e., ck = 0, νDisl ≡ 0 and there exists k > 0 such that νCoag ({x ∈ S ↓ : Pi=k i=1 xi < 1}) = 0) the situation is slightly different in that Pn is not the irreducible class.
Exchangeable Fragmentation-Coalescence processes and their equilibrium measures
This agrees with results already known for irreducible ﬁnite type Artin groups by Cohen and Paris which generalised a much earlier Theorem of Artin in the case of the braid groups.
Automorphisms and abstract commensurators of 2-dimensional Artin groups
Final conclusion in case rak > 1: up to scalar multiples, ek is a unique lowest weight/root vector associated to g1 so g1 is irreducible.