Understanding ring strain and ring flexibility in six - and eight-membered cyclic organometallic group 14 oxides

A simple model was developed for the approximation of ring strain energies of homo- and heterometallic, six- and eight-membered cyclic organometallic group 14 oxides and the degree of puckering of their ring conformations. The conformational energy of a ring is modelled as the sum of its angular strain components. The bending potential energy functions for the various endocyclic M–O–M′ and O–M–O linkages (M, M′=Si, Ge, Sn) were calculated at the B3LYP/(v)TZ level of theory using H3MOM′H3 and H2M(OH)2 as model compounds. For the six-membered rings, the minimum total angular contribution to ring strain, ERSGmin was calculated to decrease in the order: cyclo-(H2SiO)3 (13.0 kJ mol−1)>cyclo-H2Sn(OSiH2)2O (7.0 kJ mol−1)>cyclo-H2Ge(OSiH2)2O (4.9 kJ mol−1)>cyclo-H2Si(OSnH2)2O (3.4 kJ mol−1)>cyclo-(H2SnO)3 (1.7 kJ mol−1)>cyclo-H2Si(OGeH2)2O (0.8 kJ mol−1)≈cyclo-H2Ge(OSnH2)2O (0.7 kJ mol−1)>cyclo-H2Sn(OGeH2)2O (0.1 kJ mol−1)≈cyclo-(H2GeO)3 (0 kJ mol−1). All of the six-membered rings were predicted to adopt (nearly) planar conformations (a=0.996<a<1). By contrast, all eight-membered rings were predicted to adopt strainless, but puckered conformations. The degree of puckering was predicted to increase in the order: cyclo-(H2SiO)4 (a=0.983)<cyclo-H2Sn(OSiH2O)2SiH2 (a=0.959)<cyclo-(H2SiO)2(H2SnO)2 (a=0.942)< cyclo-H2Si(OSnH2O)2SiH2 (a=0.935)<cyclo-(H2SnO)4 (a=0.916)<cyclo-(H2GeO)4 (a=0.885). The differences in ring strain and the degree of puckering were linked to the different electronegativities of Si, Ge and Sn. The results obtained are consistent with experimental ring strain energies; reactivities towards ring opening polymerizations or ring expansion reactions and observed ring conformations of cyclic organometallic group 14 oxides.

Identification of general aggregation operators by Lipschitz approximation

Aggregation operators model various operations on fuzzy sets, such as conjunction, disjunction and aver aging. The choice of aggregation operators suitable for a particular problem is frequently done by fitting the parameters of the operator to the observed data. This paper examines fitting general aggregation operators by using a new method of Lipschitz approximation.

Multi-wavelets from B-spline super-functions with approximation order

Approximation order is an important feature of all wavelets. It implies that polynomials up to degree p−1 are in the space spanned by the scaling function(s). In the scalar case, the scalar sum rules determine the approximation order or the left eigenvectors of the infinite down-sampled convolution matrix H determine the combinations of scaling functions required to produce the desired polynomial. For multi-wavelets the condition for approximation order is similar to the conditions in the scalar case. Generalized left eigenvectors of the matrix H_f; a finite portion of H determines the combinations of scaling functions that produce the desired superfunction from which polynomials of desired degree can be reproduced. The superfunctions in this work are taken to be B-splines. However, any refinable function can serve as the superfunction. The condition of approximation order is derived and new, symmetric, compactly supported and orthogonal multi-wavelets with approximation orders one, two, three and four are constructed.

Shape preserving approximation using least squares splines

Least squares polynomial splines are an effective tool for data fitting, but they may fail to preserve essential properties of the underlying function, such as monotonicity or convexity. The shape restrictions are translated into linear inequality conditions on spline coefficients. The basis functions are selected in such a way that these conditions take a simple form, and the problem becomes non-negative least squares problem, for which effecitive and robust methods of solution exist. Multidimensional monotone approximation is achieved by using tensor-product splines with the appropriate restrictions. Additional inter polation conditions can also be introduced. The conversion formulas to traditional B-spline representation are provided.