Parametric Presburger arithmetic: complexity of counting and quantifier elimination

We consider an expansion of Presburger arithmetic which allows multiplication by k parameters $t_1,\dots ,t_k$. A formula in this language defines a parametric set $S_{\mathbf{t}}?{\mathbb{Z}}^d$ as $\mathbf{t}$ varies in ${\mathbb{Z}}^k$, and we examine the counting function $|S_{\mathbf{t}}|$ as a function of t . For a single parameter, it is known that $|S_t|$ can be expressed as an eventual quasi‐polynomial (there is a period m such that, for sufficiently large t, the function is polynomial on each of the residue classes mod m). We show that such a nice expression is impossible with 2 or more parameters. Indeed (assuming $\mathbf{P}\ne \mathbf{NP}$) we construct a parametric set $S_{t_1,t_2}$ such that $|S_{t_1,t_2}|$ is not even polynomial‐time computable on input $(t_1,t_2)$. In contrast, for parametric sets $S_{\mathbf{t}}?{\mathbb{Z}}^d$ with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that $|S_{\mathbf{t}}|$ is always polynomial‐time computable in the size of t , and in fact can be represented using the gcd and similar functions.     

Syntheses and structures of magnesium and zinc boraamidinates: EPR and DFT investigations of Li, Mg, Zn, B, and In complexes of the [PhB(NtBu)2].- anion radical

The first magnesium and zinc boraamidinate (bam) complexes have been synthesized via metathetical reactions between dilithio bams and Grignard reagents or MCl2 (M = Mg, Zn). The following new classes of bam complexes have been structurally characterized: heterobimetallic spirocycles {(L)mu-Li[PhB(mu-NtBu)2]}2M (6a,b, M = Mg, L = Et2O, THF; 6c, M = Zn, L = Et(2)O); bis(organomagnesium) complexes {[PhB(mu3-NtBu)2](MgtBu)2(mu3-Cl)Li(OEt2)3} (8) and {[PhB(mu3-NtBu)2](MgR)2(THF)2} (9a, R = iPr; 9b, R = Ph); mononuclear complex {[PhB(mu-NDipp)2]Mg(OEt2)2} (10). Oxidation of 6a or 6c with iodine produces persistent pink (16a, M = Mg) or purple (16b, M = Zn) neutral radicals {Lx-mu-Li[PhB(mu-NtBu)2]2M}. (L = solvent molecule), which are shown by EPR spectra supported by DFT calculations to be Cs-symmetric species with spin density localized on one of the bam ligands. In contrast, characterization of the intensely colored neutral radicals {[PhB(mu-NtBu)2]2M}. (5c, M = In, dark green; 5d, M = B, dark purple) reveals that the spin density is equally delocalized over all four nitrogen atoms in these D2d-symmetric spirocyclic systems. Oxidation of the dimeric dilithio complex {Li2[PhB(mu4-NtBu)2]}2 with iodine produces the monomeric neutral radical {[PhB(mu-NtBu)2]Li(OEt2)x}. (17), characterized by EPR spectra and DFT calculations. These findings establish that the bam anionic radical [PhB(NtBu)2].- can be stabilized by coordination to a variety of early main-group metal centers to give neutral radicals whose relative stabilities are compared and discussed.     

Bis(1 degree-amino)cyclodistib(III)azanes: the first structural characterization of cis and trans isomers of a single cyclodipnict(III)azane

The dichlorocyclodistib(III)azane [ClSb(mu-NtBu)]2 (1) has been shown to exist as the cis isomer in the solid state. A series of bis(1 degree-amino)cyclodistib(III)azanes [R'NHSb(mu-NtBu)]2 (2, R' = tBu; 3, R' = Dipp; 4, R' = Dmp) has been prepared by the reaction of 1 with 2 equiv. of LiNHR'. On the basis of NMR solution spectra, all three derivatives are formed as a mixture of cis and trans isomers. In the case of 3, the structures of both the cis and trans isomers have been determined by X-ray crystallography; cis-3 adopts an endo, endo arrangement for the amido protons of the DippNH groups. Isomerization of trans-3 into cis-3 occurs slowly in solution. Deprotonation of 2 with 2 equiv. of nBuNa or trans-3 with nBuLi produces [Na2Sb2(mu-NtBu)4] (5) and [Li2Sb2(mu-NtBu)2(mu-NDipp)2] (6), whose solvated cubane structures were established by X-ray crystallography. In contrast, the reaction of cis-3 with 2 equiv. of nBuLi produces the tricyclic compound [Li2Sb(mu-NtBu)2(mu-NDipp)(mu-NHDipp)] (7).     

Syntheses, X-ray structures and AACVD studies of group 11 ditelluroimidodiphosphinate complexes

Reactions of Na(tmeda)[N((i)Pr(2)PTe)(2)] with CuCl, AgI or AuCl (in the presence of PPh(3)) in THF produced the coinage metal ditelluroimidodiphosphinate complexes {Cu[N((i)Pr(2)PTe)(2)]}(3), (5), {Ag[N((i)Pr(2)PTe)(2)]}(6) (6) and Au(PPh(3))[N((i)Pr(2)PTe)(2)] (7), respectively. Complexes 5, 6 and 7 were characterized in the solid state by X-ray crystallography. Complex 5 is trimeric and exhibits a highly distorted Cu(3)Te(3) ring. In contrast, the Ag(I) complex 6 is a hexamer, and forms a twelve-membered Ag(6)Te(6) ring. The replacement of the (i)Pr groups on phosphorus by Ph results in an intriguing structural change to a tetramer with a boat-shaped Ag(4)Te(4) ring in {Ag[N(Ph(2)PTe)(2)}(4).2THF (8). The gold(I) complex 7 is monomeric. Aerosol-assisted chemical vapour deposition (AACVD) of compounds 5, 6 and 7 yields CuTe, Ag(7)Te(4), AuTe(2) and Au films, respectively. The films were grown at temperatures of 300-500 degrees C and characterized by X-ray powder diffraction (XRD), scanning electron microscopy (SEM) and energy dispersive analysis of X-rays (EDAX).     

A square‐planar tellurium(II) complex with Te,Te′‐chelating ligands

While exploring the chemistry of tellurium‐containing dichalcogenidoimidodiphosphinate ligands, the first all‐tellurium member of a series of related square‐planar E^II(E′)₄ complexes (E and E′ are group 16 elements), namely bis(P,P,P′,P′‐tetraphenylditelluridoimidodiphosphinato‐κ²Te,Te′)tellurium(II) (systematic name: 2,2,4,4,8,8,10,10‐octaphenyl‐1λ³,5,6λ⁴,7λ³,11‐pentatellura‐3,9‐diaza‐2λ⁵,4λ⁵,8λ⁵,10λ⁵‐tetraphosphaspiro[5.5]undeca‐1,3,7,9‐tetraene), C₄₈H₄₀N₂P₄Te₅, was obtained unexpectedly. The formally Te^II centre is situated on a crystallographic inversion centre and is Te,Te′‐chelated to two anionic [(TePPh₂)₂N]⁻ ligands in an anti conformation. The central Te^II(Te)₄ unit is approximately square planar [Te—Te—Te = 93.51 (3) and 86.49 (3)°], with Te—Te bond lengths of 2.9806 (6) and 2.9978 (9) Å.     

B-scanning and holography in ophthalmic diagnosis

The ultrasonic holographic apparatus now undergoing evaluation at Moorfields Eye Hospital and described in the last issue is capable of obtaining A-scan, B-scan and holographic displays of the living human eye. The clinical application of the machine is described and some early results presented. These emphasize the complementary nature of B-scan and holographic displays and are used to discuss the extent to which holography may be a valuable diagnostic technique in ophthalmology.