Lattices contained in planar orders are planar

The covering graph of a lattice which is contained in an ordered set with a planar covering graph is itself planar.Presented by R. W. Quackenbush.     

On the numerical solution of axisymmetric domain optimization problems by dual finite element method

Shape optimization of an axisymmetric three-dimensional domain with an elliptic boundary value state problem is solved. Since the cost functional is given in terms of the cogradient of the solution, a dual finite element method based on the minimum of complementary energy principle is used. © 1994 John Wiley & Sons, Inc.     

Polarization transforming properties of Dove prisms

We analyze the polarization changes introduced by a rotated Dove prism on the linearly polarized light, using the Jones calculus and the exact ray trace analysis. The state of polarization changes from the linear to a mildly elliptical one when a plane wave front passes through a rotated Dove prism: its semi-major axis is nearly parallel to the input plane of polarization, for any angle of prism rotation. The interferogram contrast remains high for all shearing angles in spite of polarization changes when the Dove prism is incorporated into a rotational shearing interferometer. These results are confirmed experimentally.     

Heat Kernel Asymptotics of Zaremba Boundary Value Problem

The Zaremba boundary-value problem is a boundary value problem for Laplace-type second-order partial differential operators acting on smooth sections of a vector bundle over a smooth compact Riemannian manifold with smooth boundary but with discontinuous boundary conditions, which include Dirichlet boundary conditions on one part of the boundary and Neumann boundary conditions on another part of the boundary. We study the heat kernel asymptotics of Zaremba boundary value problem. The construction of the asymptotic solution of the heat equation is described in detail and the heat kernel is computed explicitly in the leading approximation. Some of the first nontrivial coefficients of the heat kernel asymptotic expansion are computed explicitly. This revised version was published online in July 2006 with corrections to the Cover Date.     

Asymptotic approximation of functions and their derivatives by Müller’s Gamma operators

We obtain the complete asymptotic expansion of the image functions of Müller’s Gamma operators and of their derivatives. All expansion coefficients are explicitly calculated. Moreover, we study linear combinations of Gamma operators having a better degree of approximation than the operators themselves. Using divided differences we define general classes of linear combinations of which special cases were recently introduced and investigated by other authors.     

Insertion of benzothiazolium compounds into montmorillonite interlayers

Benzothiazolium compounds exhibit pronounced antimicrobial activities and stimulation effects on plant growth. When applied to fields they can pollute soil colloids. In contact with the soil they can potentially interact with a clay fraction giving rise to clay organocomplexes. Model intercalation complexes were prepared using monoionic montmorillonite and a variety of water-soluble benzothia zolium salts. It was found that the adsorption into the silicate interlayer space proceeds via a cation exchange process. Substantial differences were observed in the extent of the reaction between non-substituted andN-substituted species. The sulphur atoms present in the benzothiazolium molecules repel the surface oxygen atom: this is considered to be the factor responsible for prevention of insertion of the guest cations perpendicular to the layers.     

Cyclic trihydroxamic acid derivatives

In the crystal structures of two cyclic trihydroxamic acid derivatives containing the same substructure unit, viz. 1,3,5‐trihydroxy‐1,3,5‐triazinane‐2,4,6‐trione dihydrate, C₃H₃N₃O₆·2H₂O, (I), and 1,3,5‐benz­yloxy‐1,3,5‐triazinane‐2,4,6‐trione, C₂₄H₂₁N₃O₆, (II), there is no significant difference in the geometric parameters. In (I), there are 11 hydrogen bonds of the O—H⋯O type inter­connecting the mol­ecules in a three‐dimensional network, while in (II) there are only two weak C—H⋯O hydrogen bonds. The results of IR spectroscopic analysis are in good agreement with the crystallographic study.     

Separation of vibrational-electronic Hamiltonian in crude representation

A separation of the vibrational-electronic Hamiltonian in crude representation is presented. The separation is done in the following way: first we quantize only the electronic states on the unperturbed Hamiltonian level, then using the perturbation theory, we treat the terms where we have electronic operators to obtain new harmonic potential energy. Then we quantize a-posteriori the nuclear motion and as a result the only perturbation term is due to anharmonicity.     

A multivalued version of Sharkovskiĭ’s theorem holds with at most two exceptions

A multivalued version of Sharkovskiĭ’s theorem is formulated for M-maps on linear continua and, more generally, for triangular M-maps on a Cartesian product of linear continua. This improves the main result of [AP1] in the sense that our multivalued analogue holds with at most two exceptions. A further specification requires some additional restrictions. For instance, 3- orbits of m-maps imply the existence of k-orbits for all k ? \mathbbNk \in {\mathbb{N}} , except possibly for k ?k \in {4, 6}. It is also shown that, on every connected linearly ordered topological space, an M-map with orbits of all periods can always be constructed. This demonstrates that Baldwin’s classification of linear continua in terms of admissible periods [Ba] is useless for multivalued maps.