Two-weight estimates for singular and strongly singular integral operators

We give a Pontryagin-Thom type construction for Stein factorizations of fold maps of 3-manifolds into the plane. As an application, we compute the cobordism group of Stein factorizations of fold maps of oriented 3-manifolds into the plane and the oriented cobordism group of fold maps of 3-manifolds into the plane. It turns out that these two groups are isomorphic to Z ₂ ⊕ Z ₂. We have the analogous results about bordism groups as well.      

Symmetry Reduced Dynamics of Charged Molecular Strands

The equations of motion are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible continuous filamentary distributions of interacting rigid charge conformations. The new feature is that these equations are nonlocal when the screened Coulomb interactions, or Lennard–Jones potentials between pairs of charges, are included. The nonlocal dynamics is derived in the convective representation of continuum motion by using modified Euler–Poincaré and Hamilton–Pontryagin variational formulations that illuminate the various approaches within the framework of symmetry reduction of Hamilton’s principle for exact geometric rods. In the absence of nonlocal interactions, the equations recover the classical Kirchhoff theory of elastic rods. The motion equations in the convective representation are shown to arise by a classical Lagrangian reduction associated to the symmetry group of the system. This approach uses the process of affine Euler–Poincaré reduction initially developed for complex fluids. On the Hamiltonian side, the Poisson bracket of the molecular strand is obtained by reduction of the canonical symplectic structure on phase space. A change of variables allows a direct passage from this classical point of view to the covariant formulation in terms of Lagrange–Poincaré equations of field theory. In another revealing perspective, the convective representation of the nonlocal equations of molecular strand motion is transformed into quaternionic form.     

Poly[3-(3,4-dihydroxyphenyl)glyceric acid], a new biologically active polymer from Symphytum asperum Lepech. and S. caucasicum Bieb. (Boraginaceae)

Two high-molecular water-soluble preparations with high anticomplementary, antioxidant, antilipoperoxidant and antiinflammatory activities were isolated from the roots of Symphytum asperum and S. caucasicum. Their main chemical constituent was found to be poly[oxy-1-carboxy-2-(3,4-dihydroxyphenyl)ethylene], according to IR and NMR spectroscopy. The Symphytum high-molecular preparations can modulate in vitro B- chronic lymphocytic leukaemia (B-CLL) cells apoptosis and cell cycle progression.     

Criteria for the Boundedness and Compactness of Operators with Power-Logarithmic Kernels

In the present work, necessary and sufficient conditions are given in terms of a nonnegative Borel measure which ensure the boundedness and compactness of operators with power-logarithmic kernels from L ^p (0, a) to L ^p (0, a) (or to L ^q (0, a)), where 0 < a < , 1 < p, q < , > 1/p and 0.     

New labdane diterpenes from the stem bark of Larix laricina

Two new labdane diterpenes, cis-19-hydroxyabienol (1) and 8alpha-hydroxy-12Z,14-labdadien-19-al (2), along with another labdane described for the first time in the genus Larix, 19-acetoxy-13S-hydroxy-8(17),14-labdadiene (3) and a stilbene, 3-methoxy-3,3',5'-trihydroxystilbene (4), were isolated from the stem bark of Larch (Larix laricina). Their structures were established by standard chemical and spectroscopic methods. Compounds 2 and 3 were shown to be moderately cytotoxic against A-549, DLD-1 and normal skin fibroblast cell lines, WS1. Compound 1 was found to be selectively active against colon carcinoma cell lines, DLD-1.