Convergence of optimal stochastic bin packing

Consider independent identically distributed random variables (X_i) valued in [0,1]. Let B(n) be the optimal (minimum) number of unit size bins needed to pack n items of size X₁, X₂,…,X_n. We prove that there exists a numerical constant C such that for t > 0,

$\text{Pr(∣B(n)?E(B(n))∣>t}\text{n}\text{)≤ C exp(? t).}$

The constant C does not depend on the distribution of X.     

Crystallization of chiral compounds. 1. Spectroscopic,thermochemical, and crystallographic investigation of homochiral and racemic glycidyl p-toluenesulfonate

Crystalline specimens of homochiral and racemic glycidyl p-toluenesulfonate were studied by IR spectroscopy, differential scanning calorimetry, and X-ray diffraction analysis. The melting phase diagram of glycidyl p-toluenesulfonate was constructed. The stacking effect in the crystals of the racemic sulfonate is responsible for a more dense molecular packing, with the result that a heterochiral type of crystallization becomes more favorable.     

Copolymer of Di (methacryloyloxymethyl) naphthalene and divinylbenzene as a column packing for high-performance liquid chromatography

Summary A new porous polyaromatic ester packing was synthetized for high performance liquid chromatography. The relationship between retention and chain length of the members of homologous series of alkylbenzenes, N-alkylanilines, alkylarylethers, alkylbenzoates and alkylarylketones on this new stationary phase using different eluents was investigated. Using the alkylarylketone scale the retention indices of the homologues and test compounds were calculated. The results were compared with those obtained for poly (styrene-divinylbenzene) polymers. For both types of packing the first members of each homologous series gave non-linear behaviour. The methylene group index increments are different for the studied homologous series; thus there is no simple additivity of the retention indices. The efficiency of the porous polymeric columns is a function of the capacity factor of the solute and the organic component of the eluent.     

Solvent intercalation in cobalt dimethylglyoximato complexes: A miniature host-guest system

When crystallized from appropriate solvents, the complex aqua-bis(dimethylglyoximato)nitrocobalt (III) may incorporate solvent molecules, thus forming a variety of mixed crystals. In the resulting host-guest crystals, the space groupP2_l/m and the packing motif of the pure host compound are retained. Lattice constantsa andb remain essentially unaltered upon intercalation, whereasc and the monoclinic angle depend largely on the clathrated guest. Space filling and intermolecular contacts are discussed.Dedicated to Professor Peter Paetzold at the occasion of his 60th birthday. A preliminary account of this work has been given at the Spring Meeting of the British Crystallographic Association, Newcastle upon Tyne, 1994.     

[Hg3]4+ Cation in Inorganic Crystal Structures

The crystal-chemical structure of the minerals and inorganic compounds kuznetsovite [Hg₃](AsO₄)Cl and terlinguaite [Hg₃][HgO₂]Cl₂, as well as [Hg₃]₃(AsO₄)₄ and [Hg₃]₂[HgO₂](PO₂ and their analogs, is considered from the viewpoint of the packing of large supramolecular atomic groups. The cationic layer of polyhedra around the large [Hg₃]⁴⁺ cations, alternating with the anionic layer of [HgO₂]^2-, (AsO₄)^3-, (PO₄)^3-, Cl^-, plays the major role in structure formation. Segregation of mercury cations of different valences possibly reflects the solid-state transformations occurring in nature and accompanied by mercury liberation.     

Splitless injection of up to hundreds of microliters of liquid samples in capillary GC: Part II,experimental results

Sample evaporation in splitless injection of large volumes is rapid: depending on the experiment, results indicate that 200 μl of hexane, for instance, evaporates in 2–10 s, producing vapor at a rate of many hundreds of milliliters per minute. A 60 × 4 mm packed bed of 20–35 mesh Tenax TA enabled injection of 200 μl volumes of all solvents tested, and even 1 ml injections were possible provided they were performed over a period of 30 s. Retention of volatile sample components depends on the sample solvent, the injection volume, and the injection speed, but only little on the injector temperature. Losses of n-tridecane varied from hardly 15 % (when dissolved in pentane) to ca 60 % (ethyl acetate); losses of n-heptadecane were usually below 20 %. The column temperature during injection should be at least ca 20–30°C above the standard solvent boiling point.     

Catalytic activity of silica in ozone formation in electrical discharges

The catalytic action of granular silica packing on ozone formation has been observed under discharge conditions. Using a glass ozonizer with a metal high-voltage electrode, at a frequency of 400 Hz it was possible to obtain much higher ozone concentrations in the presence q( silica than without packing, with the same total energy consumption. The dependence between ozone production and energy efficiency is considered, and conditions of the optimum ozonizer run are discussed. It is shown that in the all-glass ozonizer with a narrow discharge gap, the ozone concentration may be as high as 6.5% (ca. 130g O₃/m³) when silica packing is applied.     

New crystal structures of β-[Ni(NCS)2(4-methylpyridine)4] clathrates with furan,tetrahydrofuran, methylene chloride,benzene + ethanol and methylcellosolve as guest molecules

Single crystal X-ray structures of clathrates of -[Ni(NCS)₂(4-methylpyridine)₄] with furan, tetrahydrofuran, benzene + ethanol, methylene chloride, and methylcellosolve as guests molecules are reported. The location of the guest molecule in the partially decomposed clathrate with methylene chloride was defined by X-ray diffraction and compared with the fully occupied one. The host lattices of all clathrates studied are tetragonal (I4₁/a) and do not differ significantly from typical -phase clathrates of [M(NCS)₂(4-methylpyridine)₄] (M = divalent metal cation). Arrangements of guest molecules represent different types of packing: one type of guest molecule occupies both possible types of positions, two different guest molecules occupy different positions, with only one type of positions occupied by one type of guest molecule. Possible stoichiometries of clathrates with -type lattices are discussed.Presented at the Sixth International Seminar on Inclusion Compounds, Istanbul, Turkey, 27–31 August 1995.     

Stochastic on-line knapsack problems

Different classes of on-line algorithms are developed and analyzed for the solution of {0, 1} and relaxed stochastic knapsack problems, in which both profit and size coefficients are random variables. In particular, a linear time on-line algorithm is proposed for which the expected difference between the optimum and the approximate solution value isO(log^3/2 n). An(1) lower bound on the expected difference between the optimum and the solution found by any on-line algorithm is also shown to hold.Corresponding author.Partially supported by the Basic Research Action of the European Communities under Contract 3075 (Alcom).Partially supported by research project Models and Algorithms for Optimization of the Italian Ministry of University and Scientific and Technological Research (MURST 40%).     

Aurifeuillian factorizations and the period of the Bell numbers modulo a prime

We show that the minimum period modulo of the Bell exponential integers is for all primes and several larger . Our proof of this result requires the prime factorization of these periods. For some primes the factoring is aided by an algebraic formula called an Aurifeuillian factorization. We explain how the coefficients of the factors in these formulas may be computed.