On metalinear ETOL systems

1980 ◽  
Vol 3 (1) ◽  
pp. 15-36
Author(s):  
Grzegorz Rozenberg ◽  
Dirk Vermeir
Keyword(s):  
Finite Index ◽  
Closure Properties ◽  
Pumping Lemma ◽  
L Systems ◽  
Structural Characterizations

The concept of metalinearity in ETOL systems is investigated. Some structural characterizations, a pumping lemma and the closure properties of the resulting class of languages are established. Finally, some applications in the theory of L systems of finite index are provided.

Synthesis and Structural Systematics of Nitrogen Base Adducts of Group 2 Salts. VII. Some Complexes of Group 2 Metal Halides With Aromatic N,N'-Bidentate Ligands

1996 ◽  
Vol 49 (1) ◽  
pp. 99 ◽  
Author(s):  
BW Skelton ◽  
AF Waters ◽  
AH White
Keyword(s):  
Room Temperature ◽  
The Novel ◽  
Bidentate Ligands ◽  
Nitrogen Base ◽  
One Dimensional ◽  
X Ray ◽  
L System ◽  
L Systems ◽  
Group 2 ◽  
Structural Characterizations

Syntheses and room-temperature single-crystal X-ray structural characterizations are recorded for a number of adducts of Group 2 halides with the aromatic N,N′- bidentate ligands L = 2,2'-bipyridine ( bpy ) and 1,10-phenanthroline ( phen ). One adduct of 1 : 1 MX2/L stoichiometry is recorded: [( bpy )Ca(μ-Br)(4/2)](∞ ׀ ∞) (1) is monoclinic, C 2/c, a l7.102(5), b 9.845(1), c 7.600(2) Ǻ, β 107.65(2)°, Z = 4 f.u .; conventional R on #1472; F ׀ at convergence was 0.057 for No 817 independent 'observed' ( I > 3σ(I)) reflections, the compound being a one-dimensional polymer… CaL (μ-Br)2CaL(μ-Br)2…, with six-coordinate calcium, cis-CaN2(μ-Br)4, on a twofold axis. 1 : 2 MX2/L systems have been described for two types of array: [( bpy )2CaI2] (2), orthorhombic, Pnca, a 16.914(4), b 13.80(1), c 9.290(4) Ǻ, Z = 4 f.u .; R 0.039 for No 1738, is discrete mononuclear, containing cis-CaN4I2 six-coordinate calcium, whereas the barium analogue is an infinite one-dimensional polymer, [( bpy )2Ba(μ-I)(4/2)](∞׀ ∞) (3), triclinic, Pī, a 11.318(2), b 11.078(2), c 9.206(4) Ǻ, α 80.87(2), β 77.52(2), γ 87.53(1)°, Z = 2 f.u .; R 0.032 for No 4154, with eight-coordinate (N2)2Ba(μ-I)4. The perchlorate analogue, [( bpy )2Ba(μ-OClO2O)(4/2)](∞ ׀ ∞) (4), although of similar stoichiometry, has a trans-rather than a cis-linked chain; it is monoclinic, P 21/c, a 6.606(2), b 18.661(7), c 19.440(8) Ǻ, β 109.45(4)°, Z = 4 f.u .; R 0.044 for No 4060, with μ-I replaced by O,O'-ClO4 bridging units and eight-coordinate barium, (N2)2BaO4. The 1 : 3 MX2/L system is represented (as its hemi( bipyridine ) 'solvate') by mononuclear [( bpy )3SrI2].0.5bpy (5), monoclinic, P 21/c, a 20.834(8), b 9.735(4), c 18.460(7) Ǻ, β 114.87(3)°, Z = 4 f.u .; R 0.039 for No 3711, containing eight-coordinate strontium, [(N2)3SrI2]; there is also an isomorphous calcium/ perchlorate analogue [( bpy )3Ca(OClO3)2].0.5bpy (6), a 21.413(6), b 9.813(4), c 18.659(2) Ǻ, β 115.67(2)°, R 0.057 for No 3090, in which unidentate O-ClO4 ligands replace the iodine about the metal atom. A 1 : 4 adduct is obtained with CaI2/phen; ionic [Ca( phen )4] I2 (7), triclinic, Pī , a 14.703(3), b 13.292(3), c 11.38(1) Ǻ, α 92.78(4), β 96.15(4), γ 105.22(2)°, Z = 2 f.u ., R 0.053 for No 3658, contains eight-coordinate Ca(N2)4 arrays, while the novel 1 : 5 adduct [ Ba ( phen )4] I2.MeCN (8) is triclinic, Pī , a 12.170(6), b 13.760(7), c 18 254(8) Ǻ, α 77.05(9), ,β 71.96(4), γ 70.83(9)°, Z = 2 f.u ., R 0.040 for No 4447, containing 10-coordinate barium, [ Ba (N2)5]. A 1 : 4 adduct formed with BaBr2 is [ BrBa( phen )4] Br.MeOH (9), triclinic, Pī , a 17.748(4), b 17.418(2), c 15.535(7) Ǻ, α 91.63(2), β 100.53(3), γ 115.39(1)°, Z = 4 f.u ., R 0.062 for No 4092, containing nine-coordinate barium, [ BrBa (N2)4].

FINITE IMAGES GENERATED BY GL-SYSTEMS

1989 ◽  
Vol 03 (03n04) ◽  
pp. 459-467 ◽  
Author(s):  
A. SAOUDI ◽  
K. RANGARAJAN ◽  
V. R. DARE
Keyword(s):  
Parallel Systems ◽  
Algebraic Characterization ◽  
Lindenmayer Systems ◽  
Closure Properties ◽  
New Device ◽  
Pumping Lemma

In this paper, we introduce a new device called GL-systems (i.e. Grammar-Lindenmayer systems) for generating finite images. GL-systems are sequential/parallel systems in which the horizontal rules form a Chomskian grammar and the vertical rules form a DTOL system. We obtain hierarchical results of various types of GL-systems. We study some properties like, closure properties, combinatorial results, pumping lemma and decidability results. An algebraic characterization for GL-systems (with variation) is presented.

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

Optimal Land Use Analysis Using Lindenmayer Grammars

2013 ◽  
Vol 8 (2) ◽  
pp. 163-175
Author(s):  
Urszula Żukowska ◽  
Grażyna Kalewska
Keyword(s):  
Land Use ◽  
Urban Development ◽  
Computer Program ◽  
L System ◽  
L Systems ◽  
The City

In today's world, when it is so important to use every piece of land for a particular purpose, both economically and ecologically, identifying optimal land use is a key issue. For this reason, an analysis of the optimal land use in a section of the city of Olsztyn, using the L-system Urban Development computer program, was chosen as the aim of this paper. The program uses the theories of L-systems and the cartographic method to obtain results in the form of sequences of productions or maps. For this reason, the first chapters outline both theories, i.e. the cartographic method to identify optimal land use and Lindenmayer grammars (called L-systems). An analysis based on a fragment of the map of Olsztyn was then carried out. Two functions were selected for the analysis: agricultural and forest-industrial. The results are presented as maps and sequences in individual steps.

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