Tree 3-spanners in 2-sep directed path graphs: Characterization, recognition, and construction

A spanning tree T of a graph G is called a treet-spanner, if the distance between any two vertices in T is at most t-times their distance in G. A graph that has a tree t-spanner is called a treet-spanner admissible graph. The problem of deciding whether a graph is tree t-spanner admissible is NP-complete for any fixed t≥4, and is linearly solvable for t=1 and t=2. The case t=3 still remains open. A directed path graph is called a 2-sep directed path graph if all of its minimal a−b vertex separator for every pair of non-adjacent vertices a and b are of size two. Le and Le [H.-O. Le, V.B. Le, Optimal tree 3-spanners in directed path graphs, Networks 34 (2) (1999) 81-87] showed that directed path graphs admit tree 3-spanners. However, this result has been shown to be incorrect by Panda and Das [B.S. Panda, Anita Das, On tree 3-spanners in directed path graphs, Networks 50 (3) (2007) 203-210]. In fact, this paper observes that even the class of 2-sep directed path graphs, which is a proper subclass of directed path graphs, need not admit tree 3-spanners in general. It, then, presents a structural characterization of tree 3-spanner admissible 2-sep directed path graphs. Based on this characterization, a linear time recognition algorithm for tree 3-spanner admissible 2-sep directed path graphs is presented. Finally, a linear time algorithm to construct a tree 3-spanner of a tree 3-spanner admissible 2-sep directed path graph is proposed.     

α-Azido ketones. Part 7: synthesis of 1,4-disubstituted triazoles by the “click” reaction of various terminal acetylenes with phenacyl azides or α-azidobenzo(hetera)cyclanones

Copper(I)-catalyzed alkyne-azide 1,3-cycloaddition reaction (CuAAC, Sharpless–Meldal reaction) of various α-azido ketones such as substituted 2-azidoacetophenones, 2-azidobenzosuberone and 3-azido(thio)chromanones with terminal alkynes was studied. The reaction resulted in the formation of the expected 1,2,3-triazoles in moderate to good yields although the reactivity was somewhat lower than in the case of simple azides. Reaction of ethynylchromones as alkynes gave interesting dichromonoid systems bridged by a triazole unit.     

Three‐Way Cooperativity in d8 Metal Complexes with Ligands Displaying Chemical and Redox Non‐Innocence

Reversible proton‐ and electron‐transfer steps are crucial for various chemical transformations. The electron‐reservoir behavior of redox non‐innocent ligands and the proton‐reservoir behavior of chemically non‐innocent ligands can be cooperatively utilized for substrate bond activation. Although site‐decoupled proton‐ and electron‐transfer steps are often found in enzymatic systems, generating model metal complexes with these properties remains challenging. To tackle this issue, we present herein complexes [(cod_?H)M(μ‐L^2?) M (cod_?H)] (M=Pt^II, [ 1 ] or Pd^II, [ 2 ], cod=1,5‐cyclooctadiene, H₂L=2,5‐di‐[2,6‐(diisopropyl)anilino]‐1,4‐benzoquinone), in which cod acts as a proton reservoir, and L^2? as an electron reservoir. Protonation of [ 2 ] leads to an unusual tetranuclear complex. However, [ 1 ] can be stepwise reversibly protonated with up to two protons on the cod_?H ligands, and the protonated forms can be stepwise reversibly reduced with up to two electrons on the L^2? ligand. The doubly protonated form of [ 1 ] is also shown to react with OMe^? leading to an activation of the cod ligands. The site‐decoupled proton and electron reservoir sources work in tandem in a three‐way cooperative process that results in the transfer of two electrons and two protons to a substrate leading to its double reduction and protonation. These results will possibly provide new insights into developing catalysts for multiple proton‐ and electron‐transfer reactions by using metal complexes of non‐innocent ligands.     

Validation of numerical solvers for liquid metal flow in a complex geometry in the presence of a strong magnetic field

Following the magnetohydrodynamic (MHD) code validation and verification proposal by Smolentsev et al. (Fusion Eng Des 100:65–72, 2015), we perform code to code and code to experiment comparisons between two computational solvers, FLUIDYN and HIMAG, which are presently considered as two of the prospective CFD tools for fusion blanket applications. In such applications, an electrically conducting breeder/coolant circulates in the blanket ducts in the presence of a strong plasma-confining magnetic field at high Hartmann numbers, \(\textit{Ha}\) (\(\textit{Ha}^2\) is the ratio between electromagnetic and viscous forces) and high interaction parameters, \(\textit{N}\) (\(\textit{N}\) is the ratio of electromagnetic to inertial forces). The main objective of this paper is to provide the scientific and engineering community with common references to assist fusion researchers in the selection of adequate computational means to be used for blanket design and analysis. As an initial validation case, the two codes are applied to the classic problem of a laminar fully developed MHD flows in a rectangular duct. Both codes demonstrate a very good agreement with the analytical solution for \(\textit{Ha}\) up to 15, 000. To address the capabilities of the two codes to properly resolve complex geometry flows, we consider a case of three-dimensional developing MHD flow in a geometry comprising of a series of interconnected electrically conducting rectangular ducts. The computed electric potential distributions for two flows (Case A) \(\textit{Ha}=515\), \(\textit{N}=3.2\) and (Case B) \(\textit{Ha}=2059\), \(\textit{N}=63.8\) are in very good agreement with the experimental data, while the comparisons for the MHD pressure drop are still unsatisfactory. To better interpret the observed differences, the obtained numerical data are analyzed against earlier theoretical and experimental studies for flows that involve changes in the relative orientation between the flow and the magnetic field.     

Photophysics of Supercomplexes. Adduct between Ru(bpy)(CN)(4)(2-) and the [32]ane-N(8)H(8)(8+) Polyaza Macrocycle

The formation of a supercomplex between the Ru(bpy)(CN)(4)(2-) (bpy = 2,2'-bipyridine) complex and the [32]ane-N(8)H(8)(8+) macrocycle (1) has been studied in water and in acetonitrile. In acetonitrile, supercomplex formation is accompanied by (i) large hypsochromic shifts in the absorption spectrum (color changes from deep violet to yellow) and in the emission spectrum, (ii) large anodic shifts in standard oxidation (0.73 V) and reduction (0.37 V) potentials, (iii) typical shifts of (1)H-NMR signals for the macrocycle N-bound protons and the complex bipyridine protons, and (iv) a large increase in the MLCT excited-state lifetime of the complex. In water, the spectral shifts and the changes in standard potential are much less pronounced, but supercomplex formation is evidenced by (13)C-NMR (and (1)H-NMR) and by emission lifetime changes. In both solvents, supercomplex formation is complete in 1:1, 1.0 x 10(-4) M solutions, indicating very large stability constant values. A structure of the supercomplex with the macrocycle bound in a "boat" conformation to the four cyanide ligands of the complex, plausible in terms of molecular models, is consistent with all the experimental data. In water, the supercomplex further associates with added negative species containing carboxylate functions, as shown by partial reversal of the lifetime changes. When the added species is also a potential electron transfer quencher (such as, e.g., Rh(dcb)(3)(3-), dcb = 4,4'-dicarboxy-2,2'-bipyridine), however, association is not accompanied by quenching. This behavior is attributed to the structure of the supercomplex-quencher adduct, in which the macrocycle acts as an insulating spacer between the excited complex and the quencher.     

Riesz Transform for a Flow Laplacian on Homogeneous Trees

Journal of Fourier Analysis and Applications - We introduce and study the notion of weak semi-greedy systems—which is inspired in the concepts of semi-greedy and branch semi-greedy systems...     

Development of tBu-phenyl Acetamide Appended Thiacalix[4]arene as “Turn-ON” Fluorescent Probe for Selective Recognition of Hg(II) Ions

Journal of Fluorescence - Herein, a novel N-(4-(tert-butyl)-phenyl)-2-chloroacetamide functionalized thiacalix[4]arene architecture, viz TCAN2PA has been synthesized and the sensing behaviour...