Isolating the most recent entry in a random recursive tree by random cuts

A recursive tree is constructed by starting with a root node and repeatedly adjoining new nodes to one node of the tree already constructed. Such a tree can represent, for example, the heirarchy of a workforce of a company that grows via recruiting. At times of economic depression, the company may decide to lay off participants, and in some cases it is a fair policy to relieve the last senior worker (most recent entry in the tree). If we remove an edge from such a tree then it falls into two subtrees one of which contains the most recent entry. If we continue to remove edges from the successively smaller subtrees that contain the most recent entry, we eventually isolate the most recent entry. We consider how many randomly selected edges must be removed in average before isolating the most recent entry by this procedure.     

On Commutativity of Rings With Derivations

Let R be a ring and d : R → R a derivation of R. In the present paper we investigate commutativity of R satisfying any one of the properties (i)d([x,y]) = [x,y], (ii)d(x o y) = xoy, (iii)d(x) o d(y) = 0, or (iv)d(x) o d(y) = x o y, for all x, y in some apropriate subset of R.     

Exact solutions of solid-liquid phase-change heat transfer when subjected to convective boundary conditions

A closed-form expression is presented to find the location of solid-liquid interface motion in convection-dominated solidification and melting problems. In this regard, the solutions are expressed in terms of the generalized representations of error functions,E (u, v) andF (u, v), which are useful to heat-conduction problems with convective-type boundary conditions. It is demonstrated that for constant surface temperature, the interface solution reduces to the classical Neumann solution.     

Iodine‐Catalyzed Friedlander Quinoline Synthesis under Solvent‐Free Conditions

Polyfunctional quinolines were synthesized using Friedlander method catalyzed by molecular iodine in high yields at 60 °C under solvent‐free conditions.     

Identification and Separation of Some Anions on Plain and Mixed Adsorbent Layers Using Water as Eluent

The analytical potential of water as an eluent in thin layer chromatographic separation of various anions on plain adsorbent layers (silica gel "G" alumina, and cellulose) as well as on beds containing different combinations of silica gel, alumina, or cellulose has been investigated. In addition to some important separation of anions, microgram separation of IO⁻₄ from milligram quantities of IO⁻₃, Bro⁻₃, MoO²⁻₄, and Fe (CN)³⁻₆ has been realized. The effect on pH of the sample in the separation of IO⁻₄ from accompanying ions has also been studied. The limits of detection and dilution limits of anions on alumina thin layers have been determined. NO⁻₂ in artificial seawater has been detected. The effect of CaCl₂, MgCl₂, and NaHCO₃ on some ternary separations of analytical importance has been examined.     

Steric effects on nascent vibrational distributions of the HCN product produced in CN radical reactions with ethane, propane and chloroform

Time-resolved IR emission spectroscopy (TRIES) has been used to study infrared emission in the 3400–3100 cm⁻¹ region from HCN molecules produced when CN radicals abstract a hydrogen atom from ethane, propane, and chloroform. From these observations the nascent vibrational distributions of the HCN produced were derived. The nascent vibrational population distributions of the product HCN in all of the reactions are non-statistical and inverted in both the pure CH stretch (00p) and CH stretch—bend (0np) series.     

MgBr2·OEt2 mediated protection of alcohols with hexamethyldisilazane: An efficient catalytic route for the preparation of silyl ethers under solvent-free conditions

Various types of alcohols and phenols were rapidly protected by hexamethyldisilazane in good to excellent yields at room temperature in the presence of catalytic amount of magnesium bromide ethyl etherate under solvent-free conditions. Good to excellent chemoselectivity was demonstrated for competitive protection of primary hydroxyls in the presence of secondary and tertiary alcohols. Highly selective protection of phenols in the presence of aromatic amines was also demonstrated successfully.     

Bis(2-hydroxyacetophenone)ethylenediimine as a neutral carrier in a coated-wire membrane electrode for lead(II).

A coated-wire ion-selective electrode (CWISE), based on a Schiff base as a neutral carrier, was successfully developed for the detection of Pb(II) in aqueous solution. CWISE exhibited a linear response with a Nernstian slope of 29.4 +/- 0.5 mV/decade within the concentration range of 1.0 x 10(-5) - 1.0 x 10(-1) M lead ion. CWISE has shown detection limits of 5.0 x 10(-6) M. The electrode exhibited good selectivity over a number of alkali, alkaline earth, transition and heavy metal ions. This sensor yielded a steady potential within 10 to 20 s at a linear dynamic range. The electrode was suitable for use in aqueous solutions in a pH range of 2.0 to 5.0. Applications of this electrode for the determination of lead in real samples and as indicator electrode for potentiometric titration of Pb2+ ion using K2CrO4 are reported.     

Analysis of anions by solid-state spot-tests

Trituration and capillary solid-state spot-tests for 25 anions with a number of organic and inorganic reagents have been examined. A comparative study of the corresponding reactions in solution has also been made. Incorporation of a glass-wool plug in the capillary improves the selectivity. Schemes are presented for possible use of the technique.     

Construction of solid-liquid phase diagrams in ternary systems by titration calorimetry

Titration calorimetry was used to construct the solid-liquid equilibrium line in ternary systems containing the solute and an aqueous mixed solvent by measuring the heat of dissolution of the solid solute during successive additions of the liquid solvent. The plot of cumulated heats versus the mole ratio, n_solvent/n_solute, yields two (almost) linear increases of different slopes. These two lines represent successively the enthalpy of dissolution then the enthalpy of dilution of the medium; their intersection gives the solubility and the enthalpy of dissolution of the solute. Phase diagrams have been established over the whole concentration range for o-anisaldehyde, 1,3,5-trimethoxybenzene and vanillin, in water + methanol, +ethanol, or +n-propanol at 303, 313 and 318 K.