Nonlinear Models of Suspension Bridges: Discussion of the Results

In this paper we present several nonlinear models of suspension bridges; most of them have been introduced by Lazer and McKenna. We discuss some results which were obtained by the authors and other mathematicians for the boundary value problems and initial boundary value problems. Our intention is to point out the character of these results and to show which mathematical methods were used to prove them instead of giving precise proofs and statements.     

Equilateral Sets in <Emphasis Type="Italic">l</Emphasis><Stack><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript></Stack>

We show that for every odd integer p 1 there is an absolute positive constantc_p, so that the maximum cardinality of a set of vectors in Rⁿ such that the l_p distance between any pair is precisely 1, is at most c_p n log n. We prove some upper bounds for other l_p norms as well.     

On the Generalization of the Courant Nodal Domain Theorem

In this paper we consider the analogue of the Courant nodal domain theorem for the nonlinear eigenvalue problem for the p-Laplacian. In particular we prove that if u_λn is an eigenfunction associated with the nth variational eigenvalue, λ_n, then u_λn has at most 2n−2 nodal domains. Also, if u_λn has n+k nodal domains, then there is another eigenfunction with at most n−k nodal domains.     

Cycles of Nonzero Elements in Low Rank Matrices

Dedicated to the memory of Paul Erdős We consider the problem of finding some structure in the zero-nonzero pattern of a low rank matrix. This problem has strong motivation from theoretical computer science. Firstly, the well-known problem on rigidity of matrices, proposed by Valiant as a means to prove lower bounds on some algebraic circuits, is of this type. Secondly, several problems in communication complexity are also of this type. The special case of this problem, where one considers positive semidefinite matrices, is equivalent to the question of arrangements of vectors in euclidean space so that some condition on orthogonality holds. The latter question has been considered by several authors in combinatorics [1, 4]. Furthermore, we can think of this problem as a kind of Ramsey problem, where we study the tradeoff between the rank of the adjacency matrix and, say, the size of a largest complete subgraph. In this paper we show that for an real matrix with nonzero elements on the main diagonal, if the rank is o(n), the graph of the nonzero elements of the matrix contains certain cycles. We get more information for positive semidefinite matrices. Received September 9, 1999 RID="*" ID="*" Partially supported by grant A1019901 of the Academy of Sciences of the Czech Republic and by a cooperative research grant INT-9600919/ME-103 from the NSF (USA) and the MŠMT (Czech Republic).     

Structural analysis of lignin by resonance Raman spectroscopy

Resonance Raman (RR) spectroscopy, combined with Kerr gated fluorescence rejection in the time domain, has recently elucidated lignin structure with unique sensitivity and selectivity. This promises structural studies of fluorescent natural macromolecules, such as lignin, which were previously not possible. Such studies rely on an improved understanding of the RR spectral behavior of lignin, which is today scarcely understood. We explain for the first time this behavior by a semi-empirical theory, and observe its pertinent features for lignin in vascular plants. We have used well-defined oxidative treatments as means of probing lignin structural elements, and show that RR sensitivity and selectivity depend crucially on excitation wavelength. Through the theory we relate these results to basic structural aspects of lignin. Spectra obtained by blue light laser excitation (400 nm) are dominated by low redox potential syringyl lignin groups, whereas lower photon energy (500 nm) decreases the selectivity markedly. RR bands depend on molecular structure but also on molecular environment. Thus charge transfer donor-acceptor interactions within lignin reduce the intensity of bands associated with electron rich moieties. New possibilities for basic and selective structural information on fluorescent natural materials, such as lignin, have thus appeared.     

Copper-catalyzed N-arylation of carbamate-protected hydrazones with organobismuthanes

Efficient copper(II) acetate-catalyzed N-arylation of carbamate-protected hydrazones was achieved under mild reaction conditions with organobismuthanes.     

Crystal structures of lanthanide(III) complexes with cyclen derivative bearing three acetate and one methylphosphonate pendants

A series of lanthanide(III) complexes formulated as M[Ln(Hdo3ap)].xH(2)O (M = Li or H and Ln = Tb, Dy, Er, Lu, and Y) with the monophosphonate analogue of H(4)dota, 1,4,7,10-tetraazacyclododecane-1,4,7-triacetic-10-methylphosphonic acid (H(5)do3ap), was prepared in the solid state and studied using X-ray crystallography. All of the structures show that the (Hdo3ap)(4-) anion is octadentate coordinated to a lanthanide(III) ion similarly to the other H(4)dota-like ligands, i.e., forming O(4) and N(4) planes that are parallel and have mutual angle smaller than 3 degrees . The lanthanide(III) ions lie between these planes, closer to the O(4) base than to the N(4) plane. All of the structures present the lanthanide(III) complexes in their twisted-square-antiprismatic (TSA) configuration. Twist angles of the pendants vary in the range between -24 and -30 degrees, and for each complex, they lie in a very narrow region of 1 degree. The coordinated phosphonate oxygen is located slightly above (0.02-0.19 Angstroms) the O(3) plane formed with the coordinated acetates. A water molecule was found to be coordinated only in the terbium(III) and neodymium(III) complexes. The bond distance Tb-O(w) is unusually long (2.678 Angstroms). The O-Ln-O angles decrease from 140 degrees [Nd(III)] to 121 degrees [Lu(III)], thus confirming the increasing steric crowding around the water binding site. A comparison of a number of structures of Ln(III) complexes with DOTA-like ligands shows that the TSA arrangement is flexible. On the other hand, the SA arrangement is rigid, and the derived structural parameters are almost identical for different ligands and lanthanide(III) ions.     

Classical trajectories for two ring-shaped potentials

The present paper deals with the classical trajectories for two superintegrable systems: a system known in quantum chemistry as the Hartmann system and a system of potential use in quantum chemistry and nuclear physics. Both systems correspond to ring-shaped potentials. They admit two maximally superintegrable systems as limiting cases, viz., the isotropic harmonic oscillator system and the Coulomb–Kepler system in three dimensions. The planarity of the trajectories is studied in a systematic way. In general, the trajectories are quasi-periodic rather than periodic. A constraint condition allows to pass from quasi-periodic motions to periodic ones. When written in a quantum mechanical context, this constraint condition leads to new accidental degeneracies for the two systems studied. © 1992 John Wiley & Sons, Inc.     

Synthesis and rearrangements of aminosubstituted ferra- and ruthenatricarbaboranes

A room-temperature reaction between the [7-tBuNH-nido-7,8,9-C3B8H10]- anion (1a) and [Cp*RuCl]4 leads to the ruthenatricarbollide [1-Cp*-12-tBuNH-1,2,4,12-RuC3B8H10] (2) (yield 85%). Analogously, the room-temperature photochemical reaction of 1a with [CpFe(C6H6)]PF6 gives the previously reported iron complex [1-Cp-12-tBuNH-1,2,4,12-FeC3B8H10] (3) (yield 82%). Both reactions are associated with extensive polyhedral rearrangement, which occurs under very mild conditions and brings the carbon atoms to positions of maximum separation within the framework. Compounds 2 and 3 were also surprisingly obtained via complexation of the isomeric [8-tBuNH-nido-7,8,9-C3B8H10]- (1b) anion. Complex 2 rearranges further to [1-Cp*-10-tBuNH-1,2,4,10-RuC3B8H10] (4) upon refluxing in xylene (145 degrees C). Density functional theory calculations at the B3LYP/SDD level were used to estimate relative stabilities of these metallacarborane isomers. Compounds 2 and 4, along with the 11-vertex closo compounds [1-Cp*-1,2,3,10-RuC3B7H10] (5) and [1-Cp*-10-tBuNH-1,2,3,10-RuC3B7H9] (6), were also isolated from the reaction between [Cp*RuCl2]2 and 1a in boiling xylene. The structure of 2 was established by an X-ray diffraction study, and the constitution of all compounds was determined unambiguously by multinuclear NMR spectroscopy, mass spectrometry, and elemental analyses.     

Voltammetry of osmium-modified DNA at a mercury film electrode: application in detecting DNA hybridization

Mercury film electrodes (MFE) have recently been used in nucleic acid electrochemical analysis as alternatives to the classical mercury drop ones. DNA modified with osmium tetroxide, 2,2'-bipyridine (Os,bipy) can be detected with a high sensitivity at mercury electrodes via measurements of a catalytic osmium signal. In this paper we show that mercury film on a glassy carbon electrode can be used in voltammetric analysis of Os,bipy-modified DNA. Application of the MFE as a detection electrode in double-surface electrochemical DNA hybridization assay involving osmium labeling of target DNA is demonstrated.