View source for Quasiregular

==English==
{{wikipedia|Quasiregular}}

===Etymology===
From {{prefix|en|quasi|regular}}.

===Pronunciation===
* {{audio|en|LL-Q1860 (eng)-Flame, not lame-quasiregular.wav|a=US}}

===Adjective===
{{en-adj|-}}

# Having some [[regular]] characteristics.
# {{lb|en|geometry|of a polyhedron or tessellation}} That is [[semiregular]] with regular [[face]]s of precisely two types that alternate around each [[vertex]].
#* {{quote-book|en|year=1981|author={{w|Harold Scott MacDonald Coxeter|H. S. M. Coxeter}}|chapter=The Derivation of Schoenberg's Star Polytopes from Schoute's Simplex Nets|editors=Chandler Davis; Branko Grünbaum; F.A. Sherk|title=The Geometric Vein: The Coxeter Festschrift|pageurl=https://books.google.com.au/books?id=XpDbBwAAQBAJ&pg=PA151&dq=%22quasiregular+polyhedron%22%7C%22quasiregular+polyhedra%22%7C%22quasiregular+tessellation%22%7C%22quasiregular+tessellations%22&hl=en&sa=X&ved=0ahUKEwjRwt6LlqTNAhUEkJQKHdbQCZIQ6AEIlwEwFQ#v=onepage&q=%22quasiregular%20polyhedron%22%7C%22quasiregular%20polyhedra%22%7C%22quasiregular%20tessellation%22%7C%22quasiregular%20tessellations%22&f=false|page=151
|passage=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 <small><math>\begin{Bmatrix} 3 \\ 4 \end{Bmatrix}</math></small> of triangles and hexagons [9, p. 60].}}
#* {{quote-book|en|year=1996|author=Wolfgang Schueller|title=The Design of Building Structures|pageurl=https://books.google.com.au/books?id=atJRAAAAMAAJ&q=%22quasiregular+polyhedron%22%7C%22quasiregular+polyhedra%22%7C%22quasiregular+tessellation%22%7C%22quasiregular+tessellations%22&dq=%22quasiregular+polyhedron%22%7C%22quasiregular+polyhedra%22%7C%22quasiregular+tessellation%22%7C%22quasiregular+tessellations%22&hl=en&sa=X&ved=0ahUKEwjRwt6LlqTNAhUEkJQKHdbQCZIQ6AEIUDAI|page=540
|passage=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'', [https://books.google.com.au/books?id=jmtBOov50eIC&pg=PA1642&dq=%22quasiregular+polyhedron%22%7C%22quasiregular+polyhedra%22%7C%22quasiregular+tessellation%22%7C%22quasiregular+tessellations%22&hl=en&sa=X&ved=0ahUKEwjRwt6LlqTNAhUEkJQKHdbQCZIQ6AEIeDAP#v=onepage&q=%22quasiregular%20polyhedron%22%7C%22quasiregular%20polyhedra%22%7C%22quasiregular%20tessellation%22%7C%22quasiregular%20tessellations%22&f=false 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.
#* {{quote-book|en|year=2007|author=Istvan Hargittai; Magdolna Hargittai|title=Symmetry through the Eyes of a Chemist|pageurl=https://books.google.com.au/books?id=kjPTBwAAQBAJ&pg=PA89&dq=%22quasiregular+polyhedron%22%7C%22quasiregular+polyhedra%22&hl=en&sa=X&ved=0ahUKEwi5w-O35aLNAhXH2KYKHeveDIMQ6AEIOTAE#v=onepage&q=%22quasiregular%20polyhedron%22%7C%22quasiregular%20polyhedra%22&f=false|page=89
|passage=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.}}
# {{lb|en|mathematics|ring theory|of an element r of a [[ring]]}} Such that 1 − r is a [[unit]] (has a [[multiplicative]] [[inverse]]).
#* {{quote-text|en|year=1966|publisher=w:American Mathematical Society|title=Translations: Lie groups|section=Series 1, Volume 9, [https://books.google.com.au/books?id=0sJUAAAAYAAJ&q=%22quasiregular+element%22%7C%22quasiregular+elements%22&dq=%22quasiregular+element%22%7C%22quasiregular+elements%22&hl=en&sa=X&ved=0ahUKEwiKqJT4qKTNAhVG_mMKHcu4CKwQ6AEIYjAO page 215]
|passage=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).}}
#* {{quote-book|en|year=1976|author=Carl Faith|title=Algebra II: Ring Theory|pageurl=https://books.google.com.au/books?id=VI3_CAAAQBAJ&pg=PA33&dq=%22quasiregular+element%22%7C%22quasiregular+elements%22&hl=en&sa=X&ved=0ahUKEwiKqJT4qKTNAhVG_mMKHcu4CKwQ6AEIlwEwGA#v=onepage&q=%22quasiregular%20element%22%7C%22quasiregular%20elements%22&f=false|page=33
|passage=''A onesided ideal is '''quasiregular''' provided that it consists of '''quasiregular''' elements.''}}
#* {{quote-book|en|year=1989|title=Proceedings of the Royal Irish Academy: Mathematical and physical sciences|pageurl=https://books.google.com.au/books?id=9OYaAQAAIAAJ&q=%22quasiregular+element%22%7C%22quasiregular+elements%22&dq=%22quasiregular+element%22%7C%22quasiregular+elements%22&hl=en&sa=X&ved=0ahUKEwiKqJT4qKTNAhVG_mMKHcu4CKwQ6AEIuwEwHw|page=128
|passage=A norm <math>\begin{Vmatrix} \end{Vmatrix}</math> 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:{{...}}.}}
# {{lb|en|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.
#* {{quote-journal|en|year=1999|journal=Israel Mathematical Conference Proceedings|volume=12-14|pageurl=https://books.google.com.au/books?id=FenuAAAAMAAJ&q=%22quasiregular+map%22%7C%22quasiregular+maps%22&dq=%22quasiregular+map%22%7C%22quasiregular+maps%22&hl=en&sa=X&ved=0ahUKEwjRkPOYgaLNAhWFlZQKHfByDTAQ6AEIbzAS|page=61
|passage=Given two orientable Riemannian manifolds ''V''<sub>1</sub> and ''V''<sub>2</sub>, one may ask whether a non-constant '''quasiregular''' map &#x1D453; : ''V''<sub>1</sub> → ''V''<sub>2</sub> 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, [https://books.google.com.au/books?id=CwwNsOyPQV8C&pg=PA108&dq=%22quasiregular+map%22%7C%22quasiregular+maps%22&hl=en&sa=X&ved=0ahUKEwjRkPOYgaLNAhWFlZQKHfByDTAQ6AEIhwEwFw#v=onepage&q=%22quasiregular%20map%22%7C%22quasiregular%20maps%22&f=false page 108],
#*: An alternative proof that features an interesting connection with the theory of quasiconformal (or more precisely '''quasiregular''') maps proceeds as follows.
#* {{quote-book|en|year=2007|author=Luca Capogna; Donatella Danielli; Scott D. Pauls; Jeremy Tyson|title=An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem|pageurl=https://books.google.com.au/books?id=GJq0s-fuf-AC&pg=PA142&dq=%22quasiregular+map%22%7C%22quasiregular+maps%22&hl=en&sa=X&ved=0ahUKEwjRkPOYgaLNAhWFlZQKHfByDTAQ6AEImgEwGg#v=onepage&q=%22quasiregular%20map%22%7C%22quasiregular%20maps%22&f=false|page=142
|passage='''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.}}
# {{lb|en|mathematics|representation theory|topological algebra|of a representation}} That is the result of a required adjustment of an {{w|induced representation}} that would, unadjusted, give rise to (only) a {{w|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, [https://books.google.com.au/books?id=YdvrGF_S4G0C&pg=PA344&dq=%22quasiregular+representation%22%7C%22quasiregular+representations%22&hl=en&sa=X&ved=0ahUKEwjXv5Gj0KLNAhUEkpQKHRGuCOgQ6AEIJDAB#v=onepage&q=%22quasiregular%20representation%22%7C%22quasiregular%20representations%22&f=false 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
#*:: ''T''<sub>''g''</sub>&#x1D453;(''h'') = &#x1D453;(''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, [https://books.google.com.au/books?id=C89Kg-d7sgsC&pg=PA188&dq=%22quasiregular+representation%22%7C%22quasiregular+representations%22&hl=en&sa=X&ved=0ahUKEwjXv5Gj0KLNAhUEkpQKHRGuCOgQ6AEIazAO#v=onepage&q=%22quasiregular%20representation%22%7C%22quasiregular%20representations%22&f=false 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, [https://books.google.com.au/books?id=IhpqCQAAQBAJ&pg=PA6&dq=%22quasiregular+representation%22%7C%22quasiregular+representations%22&hl=en&sa=X&ved=0ahUKEwjXv5Gj0KLNAhUEkpQKHRGuCOgQ6AEIZTAN#v=onepage&q=%22quasiregular%20representation%22%7C%22quasiregular%20representations%22&f=false page 6],
#*: Note that the representation Met<sup>(2)</sup>, 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 {{m|en|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 {{pedia|Semiregular polyhedron}}) There are precisely two ''convex'' '''quasiregular''' polyhedra: the [[cuboctahedron]] and the [[icosidodecahedron]].

====Related terms====
* {{l|en|quasi-inverse}}
* {{l|en|quasiregularity}}

===See also===
* {{l|en|quasiconformal}}

{{C|en|Tesselation}}