Menger curvature as a knot energy

Motivated by the suggestions of Gonzalez and Maddocks, and Banavar et al. to use geometrically defined curvature energies to model self-avoidance phenomena for strands and sheets we give a self-contained account, aimed at non-experts, on the state of art of the mathematics behind these energies. The basic building block, serving as a multipoint potential, is the circumradius of three points on a curve. The energies we study are defined as averages of negative powers of that radius over all possible triples of points along the curve (or via a mixture of averaging and maximization). For a suitable range of exponents, above the scale invariant case, we establish self-avoidance and regularizing effects and discuss various applications in geometric knot theory, as well as generalizations to surfaces and higher-dimensional submanifolds.     

Multi-walled carbon nanotubes as electrode materials for electrochemical studies of organometallic compounds in organic solvent media

Abstract  

Films of vertically aligned multi-walled carbon nanotubes (MWCNT) were selectively synthesized on silicon dioxide substrate by catalytic chemical vapor deposition using either benzene or acetonitrile as carbon source and ferrocene (1% w/w) as catalyst. The MWCNT were extensively characterized by using scanning electron microscopy, transmission electron microscopy, thermogravimetric analysis, and Raman spectroscopy. In order to examine the prospective application of the fabricated MWCNT films for the detection of electro-active compounds in organic solvent media, electrochemical studies of the oxidation of cobaltocene (CoCp₂) to cobaltocenium cation (CoCp₂⁺) (Cp = cyclopentadienyl anion) in acetonitrile were performed on these films. For this purpose, cyclic voltammetry and electrochemical impedance spectroscopy were employed. The electrochemical parameters for the CoCp₂^+/0 couple in acetonitrile were derived and compared with those obtained using a conventional glassy carbon electrode. The results demonstrate that the synthesized MWCNT films are promising electrode materials for the electrochemical detection of electro-active species in organic solvents. The MWCNT film formed upon decay of benzene has higher capacitance, less Warburg impedance, and less charge transfer resistance, and consequently it provides faster electron transfer kinetics.     

Local rotundity structure of generalized Orlicz–Lorentz sequence spaces

We give some criteria for extreme points and strong U-points in generalized Orlicz–Lorentz sequence spaces, which were introduced in [P. Foralewski, H. Hudzik, L. Szymaszkiewicz, On some geometric and topological properties of generalized Orlicz–Lorentz sequence spaces, Math. Nachr. (in press)] (cf. [G.G. Lorentz, An inequality for rearrangements, Amer. Math. Monthly 60 (1953) 176–179; M. Nawrocki, The Mackey topology of some F-spaces, Ph.D. Dissertation, Adam Mickiewicz University, Poznań, 1984 (in Polish)]). Some examples show that in these spaces the notion of the strong U-point is essentially stronger than the notion of the extreme point. This paper is related to the results from [A. Kamińska, Extreme points in Orlicz–Lorentz spaces, Arch. Math. 55 (1990) 173–180] (see Remark 1).     

Modeling techniques for analysis and interpretation of electron magnetic resonance (EMR) data for transition ions at low symmetry sites in crystals—A primer for experimentalists

Electron magnetic resonance (EMR) studies of paramagnetic centers exhibiting monoclinic and triclinic local site symmetry have gained renewed importance, since such centers occur often in various technologically important materials and biological systems. The intricate low symmetry aspects, which arise for such centers, bear on meaningful interpretation of EMR data and their correlation with structural data. This review provides a primer for experimentalists who wish to utilize efficiently the modeling techniques for analysis and interpretation of EMR data for transition ions, especially ions located at low symmetry sites in crystals. This requires proper understanding of the low symmetry effects observable in EMR spectra as well as related theoretical questions concerning, e.g., (i) existence of physically equivalent zero-field splitting (ZFS) parameter sets, (ii) clear definitions of the axis systems, (iii) proper forms of spin Hamiltonians, and (iv) distinction between apparent and actual low symmetry cases. The question (i) involves consideration of the orthorhombic standardization, which provides basis for standardization of monoclinic and triclinic ZFS parameters. Thus, the aspects pertaining to orthorhombic site symmetry are also outlined. To solve other questions several modeling techniques have been utilized and related computer packages have recently been developed in our group: (1) the superposition model calculations of the zero-field splitting parameters (ZFSPs) in arbitrary symmetry, (2) the procedure for diagonalization of the 2nd-rank ZFSPs and transformation of respective 4th- and 6th-rank ZFSPs, (3) the pseudosymmetry axes method for approximation of the 4th- and 6th-rank ZFSPs to higher symmetry, and (4) the closeness factors and norm ratios for quantitative comparisons of various ZFSP sets. These modeling techniques enable deeper analysis and interpretation of the low symmetry aspects involved in the fitted and theoretical ZFSPs. The computer packages facilitate extracting useful structural information inherent in monoclinic and triclinic ZFSP sets. Illustrative examples taken from recent studies of low symmetry ion-host systems are discussed.     

Subobject Transformation Systems

Subobject transformation systems STS are proposed as a novel formal framework for the analysis of derivations of transformation systems based on the algebraic, double-pushout (DPO) approach. They can be considered as a simplified variant of DPO rewriting, acting in the distributive lattice of subobjects of a given object of an adhesive category. This setting allows a direct analysis of all possible notions of dependency between any two productions without requiring an explicit match. In particular, several equivalent characterizations of independence of productions are proposed, as well as a local Church–Rosser theorem in the setting of STS. Finally, we show how any derivation tree in an ordinary DPO grammar leads to an STS via a suitable construction and show that relational reasoning in the resulting STS is sound and complete with respect to the independence in the original derivation tree. Research partially supported by EU IST-2004-16004 SEnSOria and MIUR PRIN 2005015824 ART. The third author acknowledges the support of EPSRC grant EP/D066565/1.     

On some geometric and topological properties of generalized Orlicz–Lorentz sequence spaces

Generalized Orlicz–Lorentz sequence spaces λ_φ generated by Musielak‐Orlicz functions φ satisfying some growth and regularity conditions (see [28] and [33]) are investigated. A regularity condition δ^λ ₂ for φ is defined in such a way that it guarantees many positive topological and geometric properties of λ_φ. The problems of the Fatou property, the order continuity and the Kadec–Klee property with respect to the uniform convergence of the space λ_φ are considered. Moreover, some embeddings between λ_φ and their two subspaces are established and strict monotonicity as well as lower and upper local uniform monotonicities are characterized. Finally, necessary and sufficient conditions for rotundity of λ_φ, their subspaces of order continuous elements and finite dimensional subspaces are presented. This paper generalizes the results from [19], [4] and [17]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Normal- and Reversed-Phase Thin-Layer Chromatography of Selected Calixarenes and Pyrene

JPC – Journal of Planar Chromatography – Modern TLC - The chromatographic behavior of calix[4]arene, 4-tert-butyl-calix[4]arene tribenzoate, 4-tert-butylcalix[5]arene, and pyrene have...