On Descartes’ rule for polynomials with two variations of signs

Lithuanian Mathematical Journal - For sequences of d + 1 signs + and ? beginning with a + and having exactly two variations of sign, we give some sufficient conditions for the (non)existence...     

Stereoselective Synthesis of 2-Ethylidene-3-aminonitriles

2-(l-acetoxyethyl) acrylonitrile 1 can be stereoselectively converted into 2-ethylidene -3-aminonitriles 4 in good yields in the presence of primary or secondary arnines in THF at room temperature.     

Structural and Physicochemical Studies of [4-ClC6H4CH2NH3]4Li2P6O18·4H2O

The crystal structure of a new organic cation cyclohexaphosphate, [4-ClC ₆ H ₄ CH ₂ NH ₃ ] ₄ Li ₂ P ₆ O ₁₈ .4H ₂ O, is reported. It crystallizes in the triclinic system (space group P-1) with the following unit-cell parameters: a = 9.628(8), b = 12.801(9), c = 19.528(6) Å, α = 78.60(4)°, β = 83.00(5)°, β = 89.98(4)°, Z = 2, and V = 2341(3)Å ³ . The structure has been solved using direct methods and refined by least-squares analysis [R ₁ = 0.043, wR ₂ = 0.108]. The structure can be described as infinite anionic layers with composition of [Li ₂ (P ₆ O ₁₈ )(H ₂ O) ₄ ] ^{4 ?} and parallel to the ac plane. The organic groups are located in the accessible voids. The molecules are stabilized by O─H…O and N─H…O types of intermolecular hydrogen bonds in the unit cell in addition to Van der Waals forces.     

Stereoselective Synthesis of Some Dialkyl (E)-2-Bromomethylene Glutarates

Reaction of 2-bromo-2-bromomethyl glutaric acid esters with tetraalkylammonium fluoride in HMPA, conveniently provides the corresponding dialkyl (E)-2-bromomethylene derivatives in good yields.     

Efficient and green syntheses of novel γ-aminophosphonate and phosphine oxide derivatives

Two synthetic protocols leading to novel γ-aminophosphonate and phosphine oxide derivatives, by reductive amination of γ-phosphonylketones, are reported. The first method involved a two-step procedure. Imine intermediates were first isolated from the p-toluenesulfonic acid-catalyzed reaction of primary amines with γ-ketophosphonates and phosphine oxides, then reduced with NaBH₄ in refluxing ethanol. The second method consists of a one-pot procedure which includes the condensation of γ-ketophosphonates and phosphine oxides with primary amines, in the presence of molecular sieves, followed by reduction with NaBH₄. These methods offer significant advantages over prior reports, such as efficiency, generality, and good yields. Furthermore, they are green protocols avoiding hazardous hydrides and solvents.     

Log-Polynomial Period Functions for Hecke Groups

In this article we shall determine the automorphic integrals of positive and negative integral weight associated with the full modular group and some Hecke groups. This will be done by using the Hecke Correspondence. We will also give a characterization of multiplier systems of real weight for Hecke groups.     

Indecomposable decomposition of tensor products of modules over Drinfeld doubles of Taft algebras

In this paper, we study the tensor product structure of the category of finite dimensional modules over Drinfeld doubles of Taft Hopf algebras. Tensor product decomposition rules for all finite dimensional indecomposable modules are explicitly given.