Chiral polymers containing a carbohydrate moiety: synthesis

This paper reports the preparation of poly(acetal-ethers) by two alternative synthetic pathways (alkaline and acid-catalyzed conditions). Polycondensations of methyl 2,3:4,6-di-O-salicylidene-alpha-D-mannopyranoside (3) (mixtures of endo-H and exo-H dioxolan-2-yl-diastereomers) with 1,4-dibromobutane (4) (method I a and I b) were performed in solvents (DMF, butyl acetate/DMF, DMSO) and were catalyzed by K2CO3/KI or KOH. A similar polymer (6) was formed by the reaction of methyl alpha-D-mannopyranoside (1) and 1,4-bis(2-formylphenoxy)butane (7), catalyzed by p-toluenesulfonic acid (method II). Regardless of conversion or initial comonomer feed ratios, the composition of the polycondensates depended on the reaction conditions, leading to the formation of macrocyclic [1 + 1] (5) and [2 + 2] compounds, which were macromolecules with diverse molecular weights and optical properties. The regioselective polycondensation was examined by 1H NMR spectroscopy of selected polymers. In the case of 5-membered cyclic acetal units, mixtures of the endo-H and exo-H dioxolan-2-yl system, diastereomers were formed in the polymer chain. The macrocycles and linear oligomers were identified by NMR and electrospray mass spectrometry (ESI-MS). Thermodynamically controlled reactions for making macrocycles as well as oligomers in the absence of templates are also discussed.     

Poisson deformations of symplectic quotient singularities

We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety.In particular, let V be a finite-dimensional complex symplectic vector space and G⊂Sp(V) a finite subgroup. Our main result says that the so-called Calogero-Moser deformation of the orbifold V/G is, in an appropriate sense, a versal Poisson deformation. That enables us to determine the algebra structure on the cohomology of any smooth symplectic resolution X?V/G (multiplicative McKay correspondence). We prove further that if is an irreducible Weyl group and , then no smooth symplectic resolution of V/G exists unless G is of types .     

Holomorphic functions and quasiconformal mappings with smooth moduli

We discuss the implication , where f is a holomorphic function (resp., a quasiconformal mapping) on a domain (resp., ) and Λ_ω(G) is the Lipschitz space associated with a majorant ω.     

Genericity of wild holomorphic functions and common hypercyclic vectors

Let T_α be the translation operator by α in the space of entire functions defined by . We prove that there is a residual set G of entire functions such that for every f∈G and every the sequence is dense in , that is, G is a residual set of common hypercyclic vectors ( functions) for the family . Also, we prove similar results for many families of operators as: multiples of differential operator, multiples of backward shift, weighted backward shifts.     

The converse theorem for Minkowski's inequality

Let (Ω, Σ, μ) be a measure space and ?, ψ : (0, ∞) → (0, ∞) some bijective functions. Suppose that the functional _?,ψ defined on class of μ-integrable simple functions χ : Ω → [0, ∞), μ({? : χ(?) > 0} > 0, by the formula

     

An alternate proof of Cohn's four squares theorem

While various techniques have been used to demonstrate the classical four squares theorem for the rational integers, the method of modular forms of two variables has been the standard way of dealing with sums of squares problems for integers in quadratic fields. The case of representations by sums of four squares in was resolved by Götzky, while those of and were resolved by Cohn. These efforts utilized modular forms. In previous work, the author was able to demonstrate Götzky's theorem by means of the geometry of numbers. Here Cohn's theorem on representation by the sum of four squares for is proven by a combination of geometry of numbers and quaternionic techniques.     

Proprieétés du degré des morphismes de Belyi

Properties of the degree of Belyi functions. A famous theorem of Belyi characterizes the curves defined over a number field by the existence of an element of its function field with certain ramification properties. In this article we are interested in the degree of these functions. We define the Belyi degree of a curve defined over a number field and the Belyi degree of a point on such a curve. We prove finiteness results concerning these invariants. We give an explicit upper bound for the Belyi degree of a point on the projective line, depending on the height and on the degree of its field of definition.     

Projective spaces with Cayley-Klein metrics

Projective metrics were first introduced by A. Cayley and F. Klein who constructed analytical models over the field of complex numbers. The aim of this paper is to give for the first time a purely synthetic definition of all projective spaces with Cayley-Klein metrics and to develop the synthetic foundation of projective-metric geometry to a level of generality including metrics over arbitrary fields of characteristic 2.     

On conformally flat contact metric manifolds

In the present paper we classify the conformally flat contact metric manifolds of dimension satisfying . We prove that these manifolds are Sasakian of constant curvature 1.