# quasiregular

## English

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### Etymology

quasiregular (not comparable)

1. Having some regular characteristics.
2. (geometry, of a polyhedron or tessellation) That is semiregular with regular faces of precisely two types that alternate around each vertex.
• 1981, H. S. M. Coxeter, The Derivation of Schoenberg's Star Polytopes from Schoute's Simplex Nets, Chandler Davis, Branko Grünbaum, F.A. Sherk (editors), The Geometric Vein: The Coxeter Festschrift, page 151,
The lattice points that lie in this plane are the vertices of the regular tessellation {3, 4} of equilateral triangles, and the other points just mentioned are the vertices of the quasiregular tessellation ${\displaystyle {\begin{Bmatrix}3\\4\end{Bmatrix}}}$ of triangles and hexagons [9, p. 60].
• 1996, Wolfgang Schueller, The Design of Building Structures, page 540,
There are two quasiregular polyhedra not having identical regular faces: the cuboctahedron (dymaxion) and the icosidodecahedron.
• 2007, V. A. Blatov, O. Delgado-Friedrichs, M. O'Keeffe, D. M. Proserpio, Periodic nets and tilings: possibilities tor analysis and design of porous materials: Proceedings of the 15th International Zeolite Conference, Ruren Xu, Jiesheng Chen, Zi Gao, Wenfu Yan (editors), From Zeolites to Porous MOF Materials, page 1642,
If we allow the coordination figure to be a quasiregular polyhedron (a polyhedron with one kind of vertex and edge, but two kinds of face) there is just one possibility compatible with translational symmetry – a cuboctahedron.
• 2007, Istvan Hargittai, Magdolna Hargittai, Symmetry through the Eyes of a Chemist, page 89,
Two semiregular polyhedra are classified as so-called quasiregular polyhedra. They have two kinds of faces, and each face of one kind is entirely surrounded by faces of the other kind.
3. (mathematics, ring theory, of an element r of a ring) Such that 1 − r is a unit (has a multiplicative inverse).
• 1966, American Mathematical Society, Translations: Lie groups, Series 1, Volume 9, page 215,
The element a can be uniquely represented in the form r + t, where [rt] = 0, t is nilpotent and r is a quasiregular element of G ([1]; p. 108).
• 1976, Carl Faith, Algebra II: Ring Theory, page 33,
A onesided ideal is quasiregular provided that it consists of quasiregular elements.
• 1989, Proceedings of the Royal Irish Academy: Mathematical and physical sciences, page 128,
A norm ${\displaystyle {\begin{Vmatrix}\end{Vmatrix}}}$ is called spectral if the group of quasiregular elements is open in the associated norm topology, equivalently if it satisfies Gelfand's spectral radius formula: [] .
4. (mathematics, analysis, of a mapping from a multi-dimensional space or manifold to an equivalent space) Having certain properties in common with holomorphic functions of a single complex variable.
• 1999, Israel Mathematical Conference Proceedings, Volumes 12-14, page 61,
Given two orientable Riemannian manifolds V1 and V2, one may ask whether a non-constant quasiregular map 𝑓 : V1V2 exists.
• 1999, S. Mueller, Variational models for microstructure and phase transitions, F. Bethuel, G. Huisken, S. Mueller, K. Steffen (editors),Calculus of Variations and Geometric Evolution Problems, Springer, Lecture Notes in Mathematics, Volume 1713, page 108,
An alternative proof that features an interesting connection with the theory of quasiconformal (or more precisely quasiregular) maps proceeds as follows.
• 2007, Luca Capogna, Donatella Danielli, Scott D. Pauls, Jeremy Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, page 142,
Quasiregular maps are a generalization of quasiconformal maps where the assumption of injectivity is relaxed. Heinonen and Holopainen [138] developed nonlinear potential theory and quasiregular maps on Carnot groups.
5. (mathematics, representation theory, topological algebra, of a representation) That is the result of a required adjustment of an induced representation that would, unadjusted, give rise to (only) a quasi-invariant measure.
• 1988, I. M. Gelfand, M. I. Graev, 3: Geometry of homogeneous spaces, representations of groups in homogeneous spaces and related questions of integral geometry, Izrail M. Gelfand, Collected Papers, Volume II, page 357,
We first decompose the quasiregular representations of a complex semi-simple Lie group into irreducible ones. A representation of the group G given by the formula
Tg𝑓(h) = 𝑓(hg)
in the fundamental affine space H = G/Z is called quasiregular.
• 1998, Vladimir F. Molchanov, Discrete series and analyticity, Joachim Hilgert, Jimmie D. Lawson, Karl-Hermann Leeb, Ernest B. Vinberg (editors), Positivity in Lie Theory: Open Problems,De Gruyter Expositions in Mathematics, Volume 26, page 188,
As it is known (see [11], [13], [21], [22]), the quasiregular representation on the hyperboloid decomposes into two series of irreducible unitary representations: continuous and discrete.
• 2008, André Unterberger, Alternative Pseudodifferential Analysis: With an Application to Modular Forms, Springer, Lecture Notes in Mathematics, Volume 1935, page 6,
Note that the representation Met(2), contrary to the quasiregular representation of the same group, does not act by changes of coordinates only [] .

#### Usage notes

In geometry, the term inherits problems associated with semiregular, which is sometimes defined differently by different authors and occasionally used inconsistently by individual authors. Perhaps the most common "error" is to consider only convex polyhedra. (See Semiregular polyhedron on Wikipedia.Wikipedia ) There are precisely two convex quasiregular polyhedra: the cuboctahedron and the icosidodecahedron.