Extension of Turán's Theorem to the 2-Stability Number

 Given a graph G with n vertices and stability number α(G), Turán's Theorem gives a lower bound on the number of edges in G. Furthermore, Turán has proved that the lower bound is only attained if G is the union of α(G) disjoint balanced cliques. We prove a similar result for the 2-stability number α₂(G) of G, which is defined as the largest number of vertices in a 2-colorable subgraph of G. Given a graph G with n vertices and 2-stability number α₂(G), we give a lower bound on the number of edges in G and characterize the graphs for which this bound is attained. These graphs are the union of isolated vertices and disjoint balanced cliques. We then derive lower bounds on the 2-stability number, and finally discuss the extension of Turán's Theorem to the q-stability number, for q>2. Received: July 21, 1999 Final version received: August 22, 2000 Present address: GERAD, 3000 ch. de la Cote-Ste-Catherine, Montreal, Quebec H3T 2A7, Canada. e-mail: Alain.Hertz@gerad.ca     

Femtosecond photoelectron spectroscopy of trans-stilbene above the reaction barrier

Femtosecond photoelectron spectroscopy was employed to study the excitation of trans-stilbene above the isomerization reaction barrier. Apart from the S₁ contribution, evidence of a second electronic state is found based on two different transients measured across the photoelectron spectrum. Time Dependent Density Functional Theory calculations on S₀, S₁, S₂ and D₀, together with simulations of the electron energy distribution support the experimental findings for selective photoelectron energies of the S₀, S₁,… electronic states.     

s-CorrPlot: An Interactive Scatterplot for Exploring Correlation

The degree of correlation between variables is used in many data analysis applications as a key measure of interdependence. The most common techniques for exploratory analysis of pairwise correlation in multivariate datasets, like scatterplot matrices and clustered heatmaps, however, do not scale well to large datasets, either computationally or visually. We present a new visualization that is capable of encoding pairwise correlation between hundreds of thousands variables, called the s-CorrPlot. The s-CorrPlot encodes correlation spatially between variables as points on scatterplot using the geometric structure underlying Pearson’s correlation. Furthermore, we extend the s-CorrPlot with interactive techniques that enable animation of the scatterplot to new projections of the correlation space, as illustrated in the companion video in supplementary materials. We provide the s-CorrPlot as an open-source R package and validate its effectiveness through a variety of methods including a case study with a biology collaborator. Supplementary materials for this article are available online.     

Elusive trimethyllanthanum: snapshots of extensive methyl group degradation in la--Al heterobimetallic complexes

Reactions of [La(AlMe4)3] and [Y(AlMe4)3] with PMe3 show that the phosphine can cleave Ln--CH3--Al linkages, separating Me3Al(PMe3). PMe3 (3 mol equiv) reacts with [Y(AlMe4)3] to give [(YMe3)n] contaminated with by-products containing phosphorus and aluminum. The La-based analog, [(LaMe3)n], is not formed selectively from the reaction of [La(AlMe4)3] with PMe3 or Et2O, which rather yields insoluble La/Al heterobimetallic products. Three multi-nuclear La-based clusters were obtained from a reaction of [La(AlMe4)3] with PMe3 (1 equiv) and identified by X-ray structure analyses. Each cluster exhibits extensive methyl group degradation and contains methylene, methine, or carbide moieties. [La4Al8(CH)4(CH2)2(CH3)20(PMe3)] has a [La4(CH)4] cuboid core supported by AlMe3, Me2AlCH2AlMe2, and PMe3 ligands. [La4Al8(C)(CH)2(CH2)2(CH3)22(toluene)] also contains a cuboid core, [La3Al(C)(CH)2(CH2)], which includes one exo cubic lanthanum atom, and is supported by AlMe3, Me3AlCH2AlMe2, (AlMe4)-, and toluene ligands. The lanthanum atoms in [La5Al9(CH)6(CH3)30] are arranged in a trigonal bipyramidal fashion with (CH) functionalities capping each face. The [La5(CH)6]3- core is formally balanced by three AlMe2 + moieties and is additionally supported by six AlMe3 ligands. The unit cell contains two independent La5 clusters, one with pseudo-C3h and the other with pseudo-D3 symmetry, as well as two molecules of the separation co-product Me3Al(PMe3).     

Self-assembly properties of alkyloxyethyl beta-glycosides with different types of carbohydrate headgroups

The effect of alkyl chain length on micelle formation in aqueous solutions of synthetic alkyloxyethyl glycosides containing an ethyl spacer with different conformations of the disaccharide headgroups was investigated. The molecular shape was systematically changed from a wedge-shaped to a rodlike geometry by changing the type of carbohydrate headgroup. The lipophilic part consists of dodecyl or tetradecyl chains. The adsorption at the liquid-air interface was investigated by surface tension measurements. The micellar phase region (L1) was studied using small-angle neutron scattering. We have observed a strong influence of the linkage between the sugar moieties in the disaccharide headgroup and the ethyl spacer on the micellar structure: the transformation from spherical to disklike aggregates was observed for compounds with a rodlike shape, but only spherical aggregates were formed by the wedge-shaped molecules.     

Conformational evolution of ubiquitin ions in electrospray mass spectrometry: molecular dynamics simulations at gradually increasing temperatures

Evidence from cross section data indicates that ubiquitin +13 ions lose their secondary and tertiary structure in mass spectrometric experiments. These transitions from the folded state into the near linear final structure occur at the experimental temperatures on time scales that are far too long for conventional molecular dynamics simulations. In this study, an approach to mass spectrometric unfolding processes is developed and a detailed application to an ubiquitin +13 ion system is presented. The approach involves a sequence of molecular dynamics simulations at gradually increasing temperatures leading to identification of major intermediate states, and the unfolding pathway. The unfolding rate at any temperature can then be calculated by a Rice-Ramsperger-Kassel (RRK) approach. For ubiquitin +13, three interesting intermediate states were found and the final near linear geometry was computed. The several relevant energy barriers calculated for the process are in the range of 7 to 15 kcal mol(-1). The unfolding time scale at 300 K was computed to be 2 ms. Cross section calculations using a hard sphere scattering model were carried out for the final structure and found to be in good accord with the results of electrospray experiments supporting the theoretical model used. The approach employed here should be applicable to any other solvent-free protein system.