Graph products of groups and group spaces

An application of constructions from so-called Bass-Serre theory to group spaces in noncommutative geometry is discussed motivated by the question how to make visible the interplay of different structures coming together in the notion of a group space. A compact summary of basic notions and constructions from Bass-Serre theory up to the structure theorem is developed; it should serve as a brief guideline for further applications. Concluding prospects indicate a way for generalizing the methods from a categorical point of view.Dedicated to Professor Johannes André on the occasion of his 70th birthday     

Redistribution reactions of heteroleptic barium iodide derivatives: Synthesis and structures of trans-BaI2(DME)(triglyme), cis-BaI2(DME)(tetraglyme) and [Ba(tetraglyme)2]I2 · C7H8

The reaction between BaI₂ · 2H₂O and NaHFIP [HFIP = OCH(CF₃)₂] in a 1:1 stoichiometry gave the heterometallic compound NaBaI₂(HFIP)(H₂O)(THF)_0.5 (1). Attempts to recrystallize 1 in the presence of N- or O-donor ligands lead to redistribution reactions. Barium iodide adducts such as BaI₂(DME)₃ (2), trans-BaI₂(DME)(triglyme) (3) and cis-BaI₂(DME)(tetraglyme) (4) were isolated with DME as solvent. A similar behavior was observed for the reaction between BaI₂ · 2H₂O and NaTFA (TFA = O₂CCF₃) in a 1:1 stoichiometry in THF, and [Ba(tetraglyme)₂]I₂ · C₇H₈ (6) was isolated in the presence of excess tetraglyme. All compounds have been characterized by elemental analysis, IR and ¹H NMR as well as single crystal X-ray studies for 3, 4 and 6. Compounds 3 and 4 are covalent adducts with eight- and nine-coordinate barium, respectively. Compound 6 is an ionic compound where two tetraglyme ligands wrap the 10-coordinate barium cation in a helical fashion. The presence of DME actually allows the coordination number of barium in the mixed-ligand adducts 3 and 4 to be tuned. The average Ba–O bond lengths (2.80 for 3 to 2.87 Å for 6) reflect the coordination number of the metal. The same observation is valid for the average Ba–I bond distance, 3.442 for 3 vs. 3.536 Å for 4.     

From discrete [Y(DMF)8][Cu4(mu3-I)2(mu-I)3I2] ion pairs to extended [Y(DMF)6(H2O)2][Cu7(mu4-I)3(mu3-I)2(mu-I)4(I)]1infinity and [Y(DMF)6(H2O)3][Cu(I)7Cu(II)2(mu3-I)8(mu-I)6]2infinity arrays by H-bond templating in a confined solvent-free environment

An ionic heterometallic species [Y(DMF)(8)][Cu(4)(micro(3)-I)(2)(micro-I)(3)I(2)](1) was isolated from a solution of CuI, NH(4)I and YI(3)(Pr(i)OH)(4) in DMF-isopropoxyethanol, and was converted in a confined environment by progressive substitution of the DMF ligands with water molecules first into a 1D zig-zag structure [Y(DMF)(6)(H(2)O)(2)][Cu(7)(micro(4)-I)(3)(micro(3)-I)(2)(micro-I)(4)(I)](1infinity)(2) and finally into a 2D sheet [Y(DMF)(6)(H(2)O)(3)][Cu(I)(7)Cu(II)(2)(micro(3)-I)(8)(micro-I)(6)](2infinity)(3) by H-bond templating.     

Syntheses and Molecular Structures of Titanium Derivatives with Polymerizable Ligands. Toward Extended Arrays

The reactions between Ti(OR)₄ and allylacetatoacetate (HAAA) in 1:1 or 1:2 stoichiometry at rt gave Ti₂(OR)₆(AAA)₂ R = Et ( 1 ), iPr ( 2 ) and Ti(OR)₂(AAA)₂ R = Et ( 3 ), iPr ( 4 ) species. A monosubstituted derivative Ti₂(OiPr)₆(AMP)₂ ( 5 ) was isolated with allylmethylphenol (AMPH). 1 and 5 were characterized by single crystal X‐ray diffraction. Their molecular structures consist of dimers with the polymerizable ligands in terminal positions andbridging alkoxide ligands assembling five and six‐coordinated metal atoms, respectively. The Ti‐O bond lengths of 1 are in the range 1.76(1) to 2.11(1) Å with the variation Ti‐OEt < Ti‐μ‐OEt < Ti‐η²‐O (allylacetatoacetate). All compounds were characterized by FT‐IR and ¹H NMR. The possibility to accede to more extended arrays either by hydrolysis or by radical initiated homo‐ or co‐polymerization reactions was investigated for the allylacetato derivatives as well as for Ti(OiPr)₂(AAEMA)₂ AAEMA = [2‐(methacryloyloxy)ethylacetoacetato] for the latter reactions.     

An Elementary Method on a Minimization Problem

一、ＴｈｅＰｒｏｂｌｅｍ .Ｌｅｔｆ(ｘ) =ａｘ2 ｂｘ ｃ ａｘ2 ｐｘ ｑ,ｗｈｅｒｅａ >0 ,ｂ2 - 4ａｃ≤ 0ａｎｄｐ2 - 4ａｑ≤ 0 .ＯｕｒＰｒｏｂｌｅｍｉｓ“Ｗｈｅｔｈｅｒｄｏｅｓｍｉｎｘ∈Ｒｆ(ｘ)ｅｘｉｓ?Ｍｏｒｅｏｖｅｒ ,,ｉｆｍｉｎｘ∈Ｒｆ(ｘ)ｅｘｉｓｔｓ,ｔｈｅｎｍｉｎｘ∈Ｒｆ(ｘ)=?ａｎｄｉｎｔｈｉｓｃａｓｅｘ =?” .Ｎａｔｕｒａｌｌｙ ,ｗｅｋｎｏｗｔｈａｔｍｉｎｘ∈Ｒｆ(ｘ)ｅｘｉｓｔｓｆｒｏｍｔｈｅｋｎｏｗｌｅｄｇｅｏｆｍａｔｈｅｍａｔｉｃａｌａｎａｌｙｓｉｓ.Ａｌｓｏ,ｗｅｃａｎｇｉｖｅｔｈｅ…