Generating Functions in the Knapsack Problem

The knapsack problem with Boolean variables and a single constraint is considered. Combinatorial formulas for calculating and estimating the cardinality of the set of feasible solutions and the values of the functional in various cases depending on given parameters of the problem are derived. The coefficients of the objective function and of the constraint vector and the knapsack size are used as parameters. The baseline technique is the classical method of generating functions. The results obtained can be used to estimate the complexity of enumeration and decomposition methods for solving the problem and can also be used as auxiliary procedures in developing such methods.     

Combinatorial chemistry of natural products: Solid phase synthesis of D- and L-cycloserine derivatives

An efficient methodology for a solid phase synthesis of D- and L-cycloserine derivatives is described. Fmoc-D-cycloserine 4 and its L-enantiomer 5 prepared by a selective amine acylation of bis-silylated parent compounds are immobilized on Sasrin or 2-chlorotrityl linker resins using Mitsunobu-type reaction or direct tritylation, respectively. The resulting Fmoc-cycloserine resins 7, 10, and 11 are deprotected with piperidine in DMF or DCM to generate immobilized cycloserine reagents with a primary amino group exposed for various synthetic transformations. An example of the parallel D-cycloserine library synthesis on a reaction plate is described.     

Synthesis and transformations of 1-(1,3-butadien-1-yl)benzotriazoles

1-[α-(Phenylthio)alkyl]benzotriazoles were converted into the corresponding 1-(1,3-butadien-1-yl)benzotriazoles in good yields by one-pot sequential reactions with (i) lithium diisopropylamide, (ii) allyl bromide or cinnamyl chloride, and (iii) potassium t-butoxide. Diels-Alder and hetero [4 + 2] cycloadditions of 1-(1,3-butadien-1-yl)benzotriazoles and some transformations of their α-lithio derivatives were studied.     

Formation of a stable bivalent state of Yb ions in Na4Y6F22:Ce3+, Yb3+ crystal under intense UV irradiation

The evolution dynamics of absorption spectra induced in samples of Na₄Y₆F₂₂:Ce³⁺, Yb³⁺ crystal by radiation resonant with 4f-5d transitions of Ce³⁺ ions was studied and analyzed. It was found that at least two types of color centers with different life times are induced in the studied crystal. It is established that the group of absorption bands in the UV spectral range that demonstrate long-term stability after excitation is caused by the 4f ¹³–4f ¹²5d transitions of bivalent ytterbium ions. The sequence of processes that lead to the reduction of ytterbium ions from the trivalent to the bivalent state is proposed.