Some jacobi matrices with decaying potential and dense point spectrum

We discuss doubly infinite matrices of the formM _ij= _i,j+1+ _i,j–1+V _{i ij} as operators on ². We present for each >0, examples of potentialsV _n with |V _n|=O(n ^–1/2+) and whereM has only point spectrum. Our discussion should be viewed as a remark on the recent work of Delyon, Kunz, and Souillard.Research partially supported by USNSF under grant MCS 81-20833     

Dipolare (1:1)-Addukte aus der Reaktion von 3-Amino-2H-azirinen mit 1,3,4-Oxadiazol- und 1,3,4-Thiadiazol-2(3H)-onen

Dipolar 1:1 Adducts from the Reaction of 3-Amino-2H-azirines with 1,3,4-Oxadiazol- and 1,3,4-Thiadiazol-2(3H)-ones 3-Amino-2H-azirines 1 react with 5-(trifluoromethyl)-1,3,4-oxadiazol-2(3H)-one ( 2 ) as well as with different 5-substituted 1,3,4-thiadiazol-2(3H)-ones ( 5a–e ) in 2-propanol at room temperature to give dipolar 1:1 adducts of type 3 and 6 , respectively, in reasonable-to-good yields (Schemes 3 and 6, Tab. 1 and 2). The structure of two of these dipolar adducts, 6a and 6e , which are formally donor-acceptor-stabilized azomethin imines, have been elucidated by X-ray crystallography (Figs. 1-4). In the reaction of 2 and sterically crowded 3-amino-2H-azirines 1c–e with a 2-propyl and 2-propenyl substituent, respectively, at C(2), a 4,5-dihydro-1,2,4-triazin-3(2H)-one of type 4 is formed as minor product (Scheme 3 and Table 1). Independent syntheses of these products proved the structure of 4 . Several reaction mechanisms for the formation of compounds 3 and 4 are discussed, the most likely ones are described in Scheme 4: reaction of 2 as an NH-acidic compound leads, via a bicyclic zwitterion of type A , to 3 as well as to 4 . In the latter reaction, a ring-expanded intermediate B is most probable.     

Theoretical calculation of electrode potentials: Electron-withdrawing compounds

The electrode potential of 2,3-dicyanobenzoquinone in aqueous solution has been calculated relative to parabenzoquinone using a thermodynamic cycle approach that includes accurate gasphase ab initio calculations and calculation of differences in free energies of hydration using the free-energy perturbation method. The discrepancy between the calculated and experimental electrode potential is disappointingly large (99 mV) compared to previous studies using this approach. This, along with the experimental evidence, suggests that the experimental value itself is too large and that theoretical approaches may indeed be as reliable as experimental ones for determining redox properties of molecules such as 2,3-dicyanobenzoquinone. In the light of this discrepancy we have examined the variation of the results with the basis set, inclusion of electron correlation and changes in the parameters used in the molecular dynamics free-energy simulations. The results are shown to be dependent upon the torsional parameters and especially dependent upon the basis set or semiempirical method used to obtain the electrostatic potential-derived charges. The best charge set was determined using the ab initio criteria of completeness—as far as it can be applied to large molecules—and also by studying the effect of hydration on these charges. This was done by allowing the solvent to perturb the wave function prior to the electrostatic potential determination. Thus, 3-21G and 6-31G * basis sets were found to give satisfactory results. Similar results were obtained using semiempirical and ab initio geometries.     

Ca(1)(-)(x)Na(2)(x)Al(2)B(2)O(7): a structure with tunable density of Na(+) vacancies

A series of samples with the composition Ca(1)(-)(x)Na(2)(x)Al(2)B(2)O(7) (0 < x < or = 1) was investigated and a hexagonal structure with unusually large range of homogeneity (at least from x = 0.01 to 0.95) was revealed. The hexagonal phase consists of [Al(2)B(2)O(7)](infinity)(2)(-) lamellae stacked along the c axis, as in CaAl(2)B(2)O(7) and Na(2)Al(2)B(2)O(7). Nevertheless, the configuration and stacking sequence of the [Al(2)B(2)O(7)](infinity)(2)(-) lamellae are different in these three structures. In the hexagonal structure of Ca(1)(-)(x)()Na(2)(x)()Al(2)B(2)O(7), Ca and half Na cations (Na1) statistically occupy the same crystallographic site which is located between the [Al(2)B(2)O(7)](infinity)(2)(-) lamellae, the other half Na cations (Na2) distribute in the planes bisecting the [Al(2)B(2)O(7)](infinity)(2)(-) lamellae. Depending on the composition, the site occupation factor of Na2 site can vary in the same range as x, leading to a tunable density of Na(+) vacancies in the structure. The AlO(4) tetrahedra and BO(3) triangles in the structure tilt in appropriate ways to improve the bond valence sum of Na2 cations which are not sufficiently bonded to the anions.     

Infrared bounds,phase transitions and continuous symmetry breaking

We present a new method for rigorously proving the existence of phase transitions. In particular, we prove that phase transitions occur in (·) ₃ ² quantum field theories and classical, isotropic Heisenberg models in 3 or more dimensions. The central element of the proof is that for fixed ferromagnetic nearest neighbor coupling, the absolutely continuous part of the two point function ink space is bounded by 0(k ^–2). When applicable, our results can be fairly accurate numerically. For example, our lower bounds on the critical temperature in the three dimensional Ising (resp. classical Heisenberg) model agrees with that obtained by high temperature expansions to within 14% (resp. a factor of 9%).Research supported by USNSF under grants GP-38048 and MPS-74-13252A. Sloan Fellow; also in the Department of Physics