Laser cleaning of stone by different laser pulse duration and wavelength

The present work focuses on the main phenomenological features of stone cleaning by lasers. These are the removal rate, cleaning degree, and chromatic appearance of the treated surface associated with different conservation problems and laser parameters. A set of three different outdoor stone conservation problems were investigated here. The measurement of the ablation rates were carried out on encrusted stone artifacts and two sets of standards in order to derive general behaviors through repeatable measurements. The analysis of the irradiation tests provided quantification of the different efficiencies, cleaning degree, and chromatic appearances associated with the fundamental harmonic of Q-switching, long Q-switching, and short free-running Nd:YAG lasers (1064 nm), as well as with the second harmonic of Q-switching sources (532 nm).     

On 2-Detour Subgraphs of the Hypercube

A spanning subgraph H of a graph G is a 2-detour subgraph of G if for each x, y ∈ V(G), d _H (x, y) ≤ d _G (x, y) + 2. We prove a conjecture of Erdős, Hamburger, Pippert, and Weakley by showing that for some positive constant c and every n, each 2-detour subgraph of the n-dimensional hypercube Q _n has at least clog₂ n · 2 ⁿ edges. József Balogh: Research supported in part by NSF grants DMS-0302804, DMS-0603769 and DMS-0600303, UIUC Campus Reseach Board #06139 and #07048, and OTKA 049398. Alexandr Kostochka: Research supported in part by NSF grants DMS-0400498 and DMS-0650784, and grant 06-01-00694 of the Russian Foundation for Basic Research.     

Unprecedented acid-catalyzed Wurtz-type coupling of meso-bromoindodicarbocyanine dyes

meso-Bromosubstituted indodicarbocyanine dyes produce dimeric molecules, together with the debrominated species, under acid catalysis. The relationship between the dimerization and hydrodebromination routes depends on the aryl substitution of the dyes. A possible reaction mechanism is also proposed.     

On the complexity of the (r|p)-centroid problem in the plane

In the (r∣p)-centroid problem, two players, called leader and follower, open facilities to service clients. We assume that clients are identified with their location on the Euclidean plane, and facilities can be opened anywhere in the plane. The leader opens p facilities. Later on, the follower opens r facilities. Each client patronizes the closest facility. In case of ties, the leader’s facility is preferred. The goal is to find p facilities for the leader to maximize his market share. We show that this Stackelberg game is \(\varSigma_{2}^{P}\) -hard. Moreover, we strengthen the previous results for the discrete case and networks. We show that the game is \(\varSigma_{2}^{P}\) -hard even for planar graphs for which the weights of the edges are Euclidean distances between vertices.     

Synthesis of magnesium dicyclopentadienide catalyzed by titanium and vanadium derivatives

Synthesis of magnesium dicyclopentadienide from metal and cyclopentadiene in THF is effectively catalyzed by the derivatives of vanadium and titanium. The kinetics of the synthesis, as well as thermodynamic parameters of reagents adsorption on the magnesium surface, have been determined. The process catalyzed by titanium derivatives is accompanied by the quantitative hydrogenation of cyclopentadiene to cyclopentene whereas the catalysis by vanadium derivatives leads to hydrogen gas evolution. A number of reaction intermediates including Cp₂TiCl, Cp₂TiH₂MgCl, Cp₂Ti(cyclo‐C₅H₇) and Cp₂V, were identified by the electron spin resonance method. The equilibrium constants, enthalpies and entropies of adsorption of metal halides on the magnesium surface were calculated. Copyright © 2001 John Wiley & Sons, Ltd.     

The singular values of the imbedding operators of some classes of analytic functions of several variables

We discuss three results. The first exhibits the order of decrease of the s-values as a function of the C R-dimension of a compact set on which we approximate the class of analytic functions being studied. The second is an asymptotic formula for the case when the domain of analyticity and the compact set are Reinhart domains. The third is the computation of the s-values of a special operator that is of interest for approximation theory on one-dimensional manifolds. Bibliography: 6 titles.Translated fromProblemy Matematicheskogo Analiza, No. 12, 1992, pp. 115–127.     

Bernstein and Gelfand widths of certain classes of analytic functions. II

The Gelfand widths of the unit ball of H²(ν) (the weighted Hardy space) with respect to the metric of the space L_∞(T_r) are considered (here T_r is the circle of radius r centered at the origin), as well as the Bernstein widths of the unit ball of H^∞ with respect to the metric of the space L₂(T_r, μ). Asymptotic formulas for the widths in question are established for arbitrary measures ν, μ. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 134–140. Translated by S. V. Kislyakov.     

An Ore-type analogue of the Sauer-Spencer Theorem

Two graphs G₁ and G₂ of order n pack if there exist injective mappings of their vertex sets into [n], such that the images of the edge sets do not intersect. Sauer and Spencer proved that if Δ (G₁) Δ (G₂) < 0.5n, then G₁ and G₂ pack. In this note, we study an Ore-type analogue of the Sauer–Spencer Theorem. Let θ(G) = max{d(u) + d(v): uv∈E(G)}. We show that if θ(G₁)Δ(G₂) < n, then G₁ and G₂ pack. We also characterize the pairs (G₁,G₂) of n-vertex graphs satisfying θ(G₁)Δ(G₂) = n that do not pack. This work was supported in part by NSF grant DMS-0400498. The work of the first author was also partly supported by grant 05-01-00816 of the Russian Foundation for Basic Research.