A remark on partial linear spaces of girth 5 with an application to strongly regular graphs

We derive a lower bound on the number of points of a partial linear space of girth 5. As an application, certain strongly regular graphs with=2 are ruled out by observing that the first subconstituents are partial linear spaces.     

Estimation of density functions by order statistics

The properties of the empirical density function,f _n(x) = k/n( _j ⁺ – _j-1 ⁺ ) if _j-1 ⁺ < x ⁺ where _j-1 ⁺ and _j ⁺ are sample elements and there are exactlyk – 1 sample elements between them, are studied in that practical point of view how to choose a suitablek for a good estimation. A bound is given for the expected value of the absolute value of difference between the empirical and theoretical density functions.     

Mixed finite element approximation of the vector potential

Summary We consider a mixed finite element approximation of the three dimensional vector potential, which plays an important rôle in the simulation of perfect fluids and in the calculation of rotational corrections to transonic potential flows. The central point of our approach is a saddlepoint formulation of the essential boundary conditions. In particular, this avoids the wellknown Babuka paradox when approximating smooth domains by polyhedrons. Using piecewise linear/piecewise constant elements for the vector potential/the boundary terms, we obtain optimal error estimates under minimal regularity assumptions for the solution of the continuous problem.     

An asymptotic finite element method for improvement of solutions of boundary layer problems

Summary A scheme that uses singular perturbation theory to improve the performance of existing finite element methods is presented. The proposed scheme improves the error bounds of the standard Galerkin finite element scheme by a factor of O(ⁿ⁺¹) (where is the small parameter andn is the order of the asymptotic approximation). Numerical results for linear second order O.D.E.'s are given and are compared with several other schemes.     

Comparison of the Hartree-Fock,Møller-Plesset,and Hartree-Fock-Slater method with respect to electrostatic properties of small molecules

Summary The results of various quantum chemical calculations, the Hartree-Fock (HF) method, the Møller-Plesset perturbation theory (MP2), and the Hartree-Fock-Slater (HFS) method are compared. Atomic charges, dipole moments, topological properties of the electron density distribution and polarizabilities, and first hyperpolarizabilities are calculated. Atomic charges obtained with the HFS method are found to be very close to those calculated with the MP2 method, from which we conclude that the HFS method describes to some extent electron correlation effects. Performing an MP2 calculation after an HF calculation improves the molecular dipole moments considerably, yielding values close to the experimental ones. HFS calculations are computationally less demanding than MP2 and yield comparable results for the electron density distributions, dipole moments and polarizabilities.     

Rearrangement during cyclisation of some 3-(2-pyrrolyl) propionic acids and related compounds

Cyclisation of the title compounds leads to mixtures of the expected 4H-cyclopental[b]pyrrol-4-ones (1), the corresponding 6-ones (3) by a single rearrangement, and the cyclopenta[c]pyrrol-4-ones (4) by a double rearrangement, the proportions depending upon the substituents.The ¹H NMR spectra of 2-methyl-4H-cyctopenta[b]thiophen-6-one (3f) shows 6J long range $\text{CH}{}_3\text{CH}{}_2$ coupling, but this is absent in the corresponding pyrroles (3c,d). The ¹³C spectra of 1 and 3 cannot be interpreted on the basis of substituent chemical shifts in pyrroles and thiophens, and are clearly -CH = CH-X (X = NMe, NPh, S) bridged derivatives of cyclopent-2-enone.     

Symmetric Skorohod topology onn-variable functions and hierarchical Markov properties ofn-parameter processes

Summary We introduce a new Skorohod topology for functions of several variables. Since ann-variable function may be viewed as a one-variable function with values in the set of (n–1)-variable functions, this topology is defined by induction from the classical Skorohod topology for one-variable functions. This allows us to define the notion of completen-parameter symmetric Markov processes: Such processes are, for any 1pn, rawp-parameter Markov processes (in the sense of our previous paper [17]) with values in the space of (n–p)-variable functions. We prove, for these processes and their Bochner subordinates, a maximal inequality which implies the continuity of additive functionals associated with finite energy measures. We finally present several important examples.     

Symmetric balanced ternary designs with ϱ1 = 1 or 2

Summary We prove the following two non-existence theorems for symmetric balanced ternary designs. If ₁ = 1 and 0 (mod 4) then eitherV = + 1 or 4₂ – + 1 is a square and (4₂ – + 1) divides ² – 1. If ₁ = 2 thenV = ((m + 1)/2) ² + 2,K = (m ² + 7)/4 and = ((m – 1)/2)² + 1 wherem 3 (mod 4). An example belonging to the latter series withV = 18 is constructed.     

Relationship between tetrahedron shape measures

Tetrahedron shape measures are used for evaluating the quality of tetrahedra in finite element meshes. Three shape measures, theminimum solid angle _min theradius ratio , and themean ratio , are discussed in this paper. A new formula for the computation of a solid angle of tetrahedron is derived. For different shape measures andv (with values 1), we establish a relationship between andv of the form wherec ₀,c ₁,e ₀, ande ₁ are positive constants. This means that if one measure approaches zero for a poorly-shaped tetrahedron, so does the other. Combined with the property that each measure attains a maximum value only for the regular tetrahedron, this means that the shape measures are equivalent.This work was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.