Hom-polytopes

We study the polytopes of affine maps between two polytopes—the hom-polytopes. The hom-polytope functor has a left adjoint—tensor product polytopes. The analogy with the category of vector spaces is limited, as we illustrate by a series of explicit examples exhibiting various extremal properties. The main challenge for hom-polytopes is to determine their vertices. A polytopal analogue of the rank-nullity theorem amounts to understanding how the vertex maps behave relative to their surjective and injective factors. This leads to interesting classes of surjective maps. In the last two sections we focus on two opposite extremal cases—when the source and target polytopes are both polygons and are either generic or regular.     

A new look at the crystallization of polyethylene. II. Crystallization from the melt at low supercoolings

This article is part of the general project laid out in Part I (ref. 9) and is concerned with obtaining information on primary (unthickened) crystals of polyethylene formed at low supercoolings. For this, a technique had to be devised by which crystallization could be speeded up so as to eliminate or at least reduce lamellar thickening. Indeed we were able to increase the rate of crystallization by an order of magnitude using a technique which we have called enhanced self-nucleation. Using this technique we find that when viewed under an optical microscope, spherulites crystallize uniformly over the field of view, and not, as is usual, by a radial growth process. Isothermal crystallization in bulk linear polyethylene has been studied by means of the enhanced self-nucleation technique as a function of crystallization time by using Raman LAM and melting points to assess variations of fold length Data have been obtained at very much shorter times than before. At short times, we find a constant fold length; at longer times the crystals thicken linearly with the logarithm of time. Values of the initial fold length for crystallization temperatures between 118 and 130°C are presented. Associated with the thickening at short times we find an induction time which increases with temperature.     

{Zn3(Et2O)2[(EtO)PO2(C6H5NH)]6.2THF}: a trinuclear zinc amidophosphate with an hourglass structure

The reaction of ZnMe2 and the N-substituted phosphoramidic monoester [Et2NH2][(EtO)PO2(C6H5NH)] produces the trinuclear zinc cluster Zn(3)(Et2O)2[(EtO)PO2(C6H5NH)]6.2THF, demonstrating that the P-N bond can survive under mild solvothermal reaction conditions.     

Synthetic studies of organodi-π-cyclopentadienyl- titanium (III) derivatives

Two synthetic routes to compounds of the type π-Cp₂Ti^IIIR (R=CH₃, CH₂Si(CH₃)₃, C₆F₅) have been investigated: (a) chemical reduction of π-Cp₂Ti^IV(R)Cl by zinc or aluminum metal in tetrahydrofuran, and (b) conventional organometallic syntheses using organo-lithium or -magnesium reagents and [π-Cp₂Ti^IIICl]₂. The preferred route is via an organolithium reagent, since chemical reduction gives a mixture of products. Green, monomeric complexes (R = CH₂Si(CH₃)₃, C₆F₅) were isolated and characterised. From the reaction of π-Cp₂Ti^IVCl₂ and trimethylsilylmethyllithium in a 1/1 ratio, π-CpTi^IV [CH₂Si(CH₃)₃]₃ was obtained. Unlike π-Cp₂Ti^IIIC₆F₅, π-Cp₂Ti^IIICH₂Si(CH₃)₃ does not form a blue complex with molecular nitrogen.     

Formation,Structural Characterization,and Calculated NMR Chemical Shifts of Selenium‐Nitrogen Compounds from SeCl4 and ArNHLi (Ar = supermesityl,mesityl)

Supermesityl selenium diimide [Se{N(C₆H₂^tBu₃‐2, 4, 6)}₂; Se{N(mes*)}₂] can be prepared in a good yield from the reaction of SeCl₄ and (mes*)NHLi. The molecule adopts an unprecedented anti, anti‐conformation, as deduced by DFT calculations at PBE0/TZVP level of theory and supported by ⁷⁷Se NMR spectroscopy and a crystal structure determination. An analogous reaction involving (C₆H₂Me₃‐2, 4, 6)NHLi [(mes)NHLi] unexpectedly lead to the reduction of selenium and afforded the selenium diamide Se{NH(mes)}₂ that was characterized by X‐ray crystallography and ⁷⁷Se NMR spectroscopy. The Se‐N bonds of 1.847(3) and 1.852(3) Å show normal single bond lengths. The <NSeN bond angle of 109.9(1)° also indicates a tetrahedral AX₂E₂ bonding arrangement around selenium. Two N‐H···N hydrogen bonds link the Se{NH(mes)}₂ molecule with two discrete (mes)NH₂ molecules. In the solid state selenium diamide adopts the anti‐conformation, whereas in solution the presence of both syn‐ and anti‐isomers could be observed. PBE0/TZVP calculations of the shielding tensors of 28 different types of selenium‐containing molecules, for which the ⁷⁷Se chemical shifts are unambiguously known, were carried out to assist the spectral assignment of Se{N(mes*)}₂ and Se{NH(mes)}₂.     

The Preparation and X-Ray Structure of [AsPh4][Ph2P(S)NSiMe3] · 0.5 THF

The reaction of Ph₂P(S)N(SiMe₃)₂ with potassium tert-butoxide in a 1:1 molar ratio produces K[Ph₂P(S)NSiMe₃], which was converted to the AsPh₄⁺ salt by metathesis with [AsPh₄]Cl. The X-ray crystal structure of [AsPh₄][Ph₂P(S)NSiMe₃] · 0.5 THF consists of noninteracting AsPh₄⁺ and Ph₂P(S)NSiMe₃^? ions with d(P? S) = 1.980(4) Å and d(P? N) = 1.555(8) Å. The PNSi bite angle in the anion is 136.3(5)°. Electrophilic attack by Ph₂P(S)Cl occurs at the sulfur atom of Ph₂P(S)NSiMe₃^?. The oxidation of the anion with iodine produces a disulfide which regenerates K[Ph₂P(S)NSiMe₂] upon treatment with potassium tert-butoxide.     

Preparation and x-ray structures of alkali-metal derivatives of the ambidentate anions [tBuN(E)P(mu-NtBu)2P(E)NtBu]2- (E = S, Se) and [tBuN(Se)P(mu-NtBu)2PN(H)tBu)]-

The ambidentate dianions [(t)BuN(E)P(mu-N(t)Bu)(2)P(E)N(t)Bu](2)(-) (5a, E = S; 5b, E = Se) are obtained as their disodium and dipotassium salts by the reaction of cis-[(t)Bu(H)N(E)P(mu-N(t)Bu)(2)P(E)N(H)(t)Bu] (6a, E = S; 6b, E = Se), with 2 equiv of MN(SiMe(3))(2) (M = Na, K) in THF at 23 degrees C. The corresponding dilithium derivative is prepared by reacting 6a with 2 equiv of (t)BuLi in THF at reflux. The X-ray structures of five complexes of the type [(THF)(x)()M](2)[(t)BuN(E)P(mu-N(t)Bu)(2)P(E)N(t)Bu] (9, M = Li, E = S, x = 2; 11a/11b, M = Na, E = S/Se, x = 2; 12a, M = K, E = S, x = 1; 12b, M = K, E = Se, x = 1.5) have been determined. In the dilithiated derivative 9 the dianion 5a adopts a bis (N,S)-chelated bonding mode involving four-membered LiNPS rings whereas 11a,b and 12a,b display a preference for the formation of six-membered MNPNPN and MEPNPE rings, i.e., (N,N' and E,E')-chelation. The bis-solvated disodium complexes 11a,b and the dilithium complex 9 are monomeric, but the dipotassium complexes 12a,b form dimers with a central K(2)E(2) ring and associate further through weak K.E contacts to give an infinite polymeric network of 20-membered K(6)E(6)P(4)N(4) rings. The monoanions [(t)Bu(H)N(E)P(mu-N(t)Bu)(2)P(E)N(t)Bu)](-) (E = S, Se) were obtained as their lithium derivatives 8a and 8b by the reaction of 1 equiv of (n)BuLi with 6a and 6b, respectively. An X-ray structure of the TMEDA-solvated complex 8a and the (31)P NMR spectrum of 8b indicate a N,E coordination mode. The reaction of 6b with excess (t)BuLi in THF at reflux results in partial deselenation to give the monolithiated P(III)/P(V) complex [(THF)(2)Li[(t)BuN(Se)P(mu-N(t)Bu)(2)PN(H)(t)Bu]] 10, which adopts a (N,Se) bonding mode.     

Syntheses and structures of an unsolvated tetrakisimidophosphate (Li3(P(NBut)3(NSiMe3))2 and the face-sharing double-cubane (Li2(THF)[P(O)(NBut)2(NHBut)])2

The treatment of Me3SiN=P(NHBut)3 with three equivalents of LiBun in toluene produces (Li3(P(NBut)3(NSiMe3)))2 comprised of a Li6N6 cyclic ladder capped on the two hexagonal faces by mu 3-PNSiMe3 groups; the corresponding reaction of O=P(NHBut)3 yields the face-sharing double-cubane (Li2(THF)P(O)(NBut)2(NHBut))2 with a central Li2O2 ring.