Some Numerical Data for Single-Server Queues Involving Deterministic Input Arrangements

The steady-state parameters of the bulk input queue D _c /M/1 and the Erlang service queue D/E _c /1 have been tabulated for C = 1(1)6(2)12(4)20 and 25, 50 and 100 and for ρ = 0·1(0·1)0·9. The tabulation includes the mean waiting time, idle time and queue size. In addition the queue D/E _c /1 has been compared with the queue M/E _c /1 to indicate the gains to be achieved by regularizing the arrival mechanism for a given E _c service facility.     

Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case

Consider ?Δ + λV with V short range at a value λ₀ where some eigenvalue e(λ) → 0 as λ ↓ λ₀. We analyze two questions: (i) What is the leading order of e(λ), i.e., for what α does e(λ) ～ c(λ ? λ₀)^α? (ii) Is e(λ) analytic at λ = λ₀ and, if not, what is the natural expansion parameter? The results are highly dimension dependent.     

On the mechanism of allylic alkylations catalyzed by palladium

Reults are reported indicating the intermediacy of a symmetrical π-allyl species in the alkylation of an allylic lactone. A previous report indicating that another mechanism was operative is disputed.     

An Unexpected α‐Oxidation of Cyclic Ketones with 1,4‐Benzoquinone by Enol Catalysis

The first direct and asymmetric α‐aryloxylation of cyclic ketones via enol catalysis has been achieved using quinones as the reaction partners. Catalytic amounts of a phosphoric acid promote the exclusive formation of α,α‐disubstituted ketones from the corresponding α‐substituted ketones in good yields and enantioselectivities (up to 96.5:3.5 er). Preliminary mechanistic experiments suggest that this reaction proceeds via a proton‐coupled electron transfer (PCET) followed by radical recombination.     

SuFEx Chemistry of Thionyl Tetrafluoride (SOF4) with Organolithium Nucleophiles: Synthesis of Sulfonimidoyl Fluorides,Sulfoximines, Sulfonimidamides,and Sulfonimidates

Thionyl tetrafluoride (SOF₄) is a valuable connective gas for sulfur fluoride exchange (SuFEx) click chemistry that enables multidimensional linkages to be created via sulfur–oxygen and sulfur–nitrogen bonds. Herein, we expand the available SuFEx chemistry of SOF₄ to include organolithium nucleophiles, and demonstrate, for the first time, the controlled projection of sulfur–carbon links at the sulfur center of SOF₄‐derived iminosulfur oxydifluorides (R¹?N=SOF₂). This method provides rapid and modular access to sulfonimidoyl fluorides (R¹?N=SOFR²), another array of versatile SuFEx connectors with readily tunable reactivity of the S?F handle. Divergent connections derived from these valuable sulfonimidoyl fluoride units are also demonstrated, including the synthesis of sulfoximines, sulfonimidamides, and sulfonimidates.     

A New Portal to SuFEx Click Chemistry: A Stable Fluorosulfuryl Imidazolium Salt Emerging as an “F−SO2+” Donor of Unprecedented Reactivity,Selectivity, and Scope

Sulfuryl fluoride, SO₂F₂, has been found to derivatize phenols in all kinds of environments, even those in highly functional molecules. We now report that a solid fluorosulfuryl imidazolium triflate salt delivers the same “F?SO₂ + ” fragment to Nu?H acceptor groups in the substrates. However, this triflate salt is a far more reactive fluorosulfurylating agent than SO₂F₂ and displays selectivity preferences of its own. Moreover, the new azolium triflate reagent reacts once with primary amines and anilines before the reaction stops. On the other hand, with triethylamine and two equivalents of the “F?SO₂ + ” donor present, it proceeds on to the bis(fluorosulfuryl)imides in good yield—two important conversions that we have never seen with sulfuryl fluoride as the electrophile.     

The Structure of Trivalent Graphs with Minimal Eigenvalue Gap

Let G be a connected trivalent graph on nvertices (n10) such that among all connected trivalentgraphs on n vertices, G has the largest possiblesecond eigenvalue. We show that G must be reduced path-like, i.e. G must be of the form: where theends are one of the following:(the right-hand end block is the mirror image of one of the blocks shown)and the middle blocks are one of the following: This partially solves a conjecture implicit in a paper of Bussemaker,obelji, Cvetkovi, and Seidel [3].