X-ray and neutron reflectometry study of glow-discharge plasma polymer films

Radio-frequency glow-discharge plasma polymer thin films of allylamine (AA) and hexamethyldisiloxane (HMDSO) were prepared on silicon wafers and analyzed by a combination of X-ray photoelectron spectroscopy (XPS), atomic force microscopy (AFM), X-ray reflectometry (XRR), and neutron reflectometry (NR). AFM and XRR measurements revealed uniform, smooth, defect-free films of 20-30 nm thickness. XPS measurements gave compositional data on all elements in the films with the exception of hydrogen. In combination with XRR and NR, the film composition and mass densities (1.46 and 1.09 g cm(-)(3) for AA and HMDSO, respectively) were estimated. Further NR measurements were conducted with the AA and HMDSO films in contact with water at neutral pH. Three different H(2)O/D(2)O mixtures were used to vary the contrast between the aqueous phase and the polymer. The amount of water penetrating the film, as well as the number of labile protons present, was determined. The AA film in contact with water was found to swell by approximately 5%, contain approximately 3% water, and have approximately 24% labile protons. The HDMSO polymer was found to have approximately 6% labile protons, no thickness increase when in contact with water, and essentially no solvent penetration into the film. The difference in the degree of proton exchange within the films was attributed to the substantially different surface and bulk chemistries of the two films.     

Simpler Tests for Semisparse Subgroups

The main results of this article facilitate the search for quotients of regular abstract polytopes. A common approach in the study of abstract polytopes is to construct polytopes with specified facets and vertex figures. Any nonregular polytope may be constructed as a quotient of a regular polytope by a (so-called) semisparse subgroup of its automorphism group W (which will be a string C-group). It becomes important, therefore, to be able to identify whether or not a given subgroup N of a string C-group W is semisparse. This article proves a number of properties of semisparse subgroups. These properties may be used to test for semisparseness in a way which is computationally more efficient than previous methods. The methods are used to find an example of a section regular polytope of type {6, 3, 3} whose facets are Klein bottles. Received February 15, 2005     

A one-pot procedure for methylenating carbonyl compounds using the Nysted reagent and titanocene dichloride

The combination of the Nysted reagent and titanocene dichloride methylenates aldehydes and ketones to give alkenes, and in a microwave-assisted process, esters and lactones give enol ethers. The methylenating agent in this one-pot procedure is presumed to be titanocene methylidene, which is the same reactive intermediate as that generated from Tebbe, Petasis and Grubbs reagents, each of which have to be prepared before use.     

Eulerian abstract polytopes

The classical theory of regular convex polytopes has inspired many combinatorial analogues. In this article, we examine two of them, the eulerian posets and the abstract regular polytopes, and see what the overlap between the concepts is. It is shown that a section regular polytope is eulerian if and only if it is spherical, or it has even rank and is locally spherical. Equivelar polytopes of rank less than 4 are eulerian, and some progress is made towards a characterisation of equivelar eulerian posets in higher rank. In particular, necessary conditions are given for an equivelar quotient of a cube or a torus to be eulerian.     

Quotients of Some Finite Universal Locally Projective Polytopes

Abstract. This paper classifies the quotients of a finite and locally projective polytope of type {4,3,5} . Seventy quotients are found, including three regular polytopes, and nine other section regular polytopes which are themselves locally projective. The classification is done with the assistance of GAP, a computer system for algebraic computation. The same techniques are also applied to two finite locally projective polytopes respectively of type {3,5,3} and {5,3,5} . No nontrivial quotients of the latter are found.     

Locally projective polytopes of type

This paper attempts to classify the locally projective section regular n-polytopes of type {4,3,…,3,p}, that is, to classify polytopes whose facets are cubes or hemicubes, and the vertex figures are spherical or projective polytopes of type {3,…,3,p}, with the facets and vertex figures being not both spherical. Spherical or projective (n−1)-polytopes of type {3,…,3,p} only exist when p4, or p=5 and n−14, or n−1=2. However, some existence and non-existence results are obtained for other values of p and n. In particular, a link is derived between the existence of polytopes of certain types, and vertex-colourability of certain graphs. The main result of the paper is that locally projective section regular n-polytopes exist only when p=4, or when p=5 and n=4 or 5.     

Fractional Trigonometry and the Spiral Functions

This paper develops two related fractional trigonometries based on the multi-valued fractional generalization of the exponential function, the R-function. The trigonometries contain the traditional trigonometric functions as proper subsets. Also developed are relationships between the R-function and the new fractional trigonometric functions. Laplace transforms are derived for the new functions and are used to generate solution sets for various classes of fractional differential equations. Because of the fractional character of the R-function, several new trigonometric functions are required to augment the traditional sine, cosine, etc. functions. Fractional generalizations of the Euler equation are derived. As a result of the fractional trigonometry a new set of phase plane functions, the Spiral functions, that contain the circular functions as a subset, is identified. These Spiral functions display many new symmetries.     

A Frequency-Domain Approach to Optimal Fractional-Order Damping

In this paper, we will consider the single term optimal fractional-order damper for an otherwise undamped oscillator. First, we will find the single term damper that minimizes the time domain integral of the squared step error (2-norm) and the integral of the time-weighted squared error (Hilbert–Schmidt–Hankel norm). Next we will consider a more intuitive frequency domain approach that insures the maximally flat magnitude response. Time and frequency domain plots are given for comparison with the integer-order solutions. Further improvements in performance are shown to be possible using multiple active fractional-order dampers.