One to chelate them all: investigation of a versatile, bifunctional chelator for 64Cu, 99mTc, Re and Co

We describe the synthesis of the dip (di-picolyl-carboxylate) bifunctional chelator system, capable of fast coordination of Cu(2+), (64)Cu(2+) and Co(2+), as well as the [M(CO)(3)](+)-core (M = (99m)Tc, Re); it displays a variety of binding modes--tridentate when protected, tetradentate when deprotected. Syntheses of both the benzyl-nitro derivative and the benzyl-amino derivatives are described. The latter was coupled to biotin to show the viability of the system for functionalization with biomolecules. Besides coordination chemistry with stable isotopes, we also present labelling data with (64)Cu and (99m)Tc, as well as in vitro stability studies.     

Greedy Approximation of High-Dimensional Ornstein–Uhlenbeck Operators

We investigate the convergence of a nonlinear approximation method introduced by Ammar et?al. (J. Non-Newtonian Fluid Mech. 139:153–176, 2006) for the numerical solution of high-dimensional Fokker–Planck equations featuring in Navier–Stokes–Fokker–Planck systems that arise in kinetic models of dilute polymers. In the case of Poisson’s equation on a rectangular domain in ?², subject to a homogeneous Dirichlet boundary condition, the mathematical analysis of the algorithm was carried out recently by Le Bris, Lelièvre and Maday (Const. Approx. 30:621–651, 2009), by exploiting its connection to greedy algorithms from nonlinear approximation theory, explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173–187, 1996); hence, the variational version of the algorithm, based on the minimization of a sequence of Dirichlet energies, was shown to converge. Here, we extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le Bris et al. to a technically more complicated situation, where the Laplace operator is replaced by an Ornstein–Uhlenbeck operator of the kind that appears in Fokker–Planck equations that arise in bead–spring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a high-dimensional Cartesian product configuration space D=D ₁×?×D _N contained in ? ^Nd , where each set D _i , i=1,…,N, is a bounded open ball in ? ^d , d=2,3.     

How to find groups? (and how to use them in Erdős geometry?)

Geometric questions which involve Euclidean distances often lead to polynomial relations of type F(x, y, z)=0 for some F ∈ ?[x, y, z]. Several problems of Combinatorial Geometry can be reduced to studying such polynomials which have many zeroes on n×n×n Cartesian products. The special case when the relation F = 0 can be re-written as z = f(x, y), for a polynomial or rational function f ∈ ?(x, y), was considered in [8]. Our main goal is to extend the results found there to full generality (and also to show some geometric applications, e.g. one on “circle grids”). The main result of our paper concerns low-degree algebraic sets F which contain “too many” points of a (large) n×n×n Cartesian product. Then we can conclude that, in a neighborhood of almost any point, the set F must have a very special (and very simple) form. More precisely, then either F is a cylinder over some curve, or we find a group behind the scene: F must be the image of the graph of the multiplication function of an appropriate algebraic group (see Theorem 3 for the 3D special case and Theorem 27 in full generality).     

[11C]-labeling of some caffeine derivatives for mapping adenosine A2a receptors by PET technique

[¹¹C]-labeled form of ten A_2a adenosine receptor specific 8-styryl-7-methyl-xanthine derivatives ([¹¹C]-caffeines) were synthesised by N-methylation of the corresponding 8-styryl-xanthine derivatives using [¹¹C]-methyl iodide in optimized reaction conditions. The results show that the [¹¹C]-methylations take place with excellent radiochemical yields (35–93%), and can be utilised easily in online preparations. These labeled ligands may facilitate the positron emission tomographic (PET) investigation of adenosine A_2a receptors.     

On the Pósa-Seymour conjecture

Paul Seymour conjectured that any graph G of order n and minimum degree at least contains the k^th power of a Hamilton cycle. We prove the following approximate version. For any ϵ ≥ 0 and positive integer k, there is an n₀ such that, if G has order n ≥ n₀ and minimum degree at least $(\frac{k}{k+1} + \epsilon )n$, then G contains the k^th power of a Hamilton cycle. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 167–176, 1998     

Generalized pattern functions

Presented by I. Rosenberg.