Synthesis of N,4-diaryl substituted β-lactams via Kinugasa cycloaddition/rearrangement reaction

The methodology of construction of N,4-diaryl substituted β-lactam framework, based on the Kinugasa cycloaddition/rearrangement sequence is presented. The series of protected chiral propargyl alcohols was treated with diaryl nitrones to afford mainly the cis-I adduct, providing direct access to the highly-functionalized azetitidin-2-one derivatives with a well-defined stereochemistry. Under the optimized reaction conditions, the unprotected chiral propargylic alcohols were also found to be suitable precursors of β-lactams. The absolute configuration of adducts was determined by CD or HPLC-CD technique, which was shown to be reliable method of determination of the configuration at C-4 of 4-aryl-substituted azetidin-2-ones. Epimerization of the cis adduct to the respective trans isomer could be easily done by the oxidation of hydroxyl group next to the four-membered β-lactam ring to the ketone, followed by a base-mediated epimerization of the malonyl fragment.     

Solid state H NMR study of molecular dynamics and domain sizes in PBT with the fullerene derivates: Decylamine-C60 and tetracyanoethylene oxide-C60

The solid-state ¹H NMR on-resonance and off-resonance techniques have been performed to study molecular dynamics and domain sizes in heterogenous nanocomposites based on poly(butylene terephthalate) and nanoparticles C₆₀ modified by n-decylamine (DA) or tetracyanoethylene oxide (TCNEO), respectively. The spin-lattice off-resonance relaxation times in the rotating frame T_1ρ^off as well as the second moment M₂ of the absorption line were analysed as a function of temperature. To determine the size of heterogeneities and characterize the morphology of the nanocomposites, the ¹H NMR spin-diffusion experiment designed by Goldman-Shen [1,2] was performed. The results from the measurements allow us to investigate the spin-diffusion phenomenon and relaxation behavior of the new nanocomposites.     

Generalized Tractability for Multivariate Problems Part II: Linear Tensor Product Problems, Linear Information, and Unrestricted Tractability

We continue the study of generalized tractability initiated in our previous paper “Generalized tractability for multivariate problems, Part I: Linear tensor product problems and linear information”, J. Complex. 23:262–295, 2007. We study linear tensor product problems for which we can compute linear information which is given by arbitrary continuous linear functionals. We want to approximate an operator S _d given as the d-fold tensor product of a compact linear operator S ₁ for d=1,2,…, with ‖S ₁‖=1 and S ₁ having at least two positive singular values. Let n(ε,S _d ) be the minimal number of information evaluations needed to approximate S _d to within ε∈[0,1]. We study generalized tractability by verifying when n(ε,S _d ) can be bounded by a multiple of a power of T(ε ⁻¹,d) for all (ε ⁻¹,d)∈Ω⊆[1,∞)×ℕ. Here, T is a tractability function which is non-decreasing in both variables and grows slower than exponentially to infinity. We study the exponent of tractability which is the smallest power of T(ε ⁻¹,d) whose multiple bounds n(ε,S _d ). We also study weak tractability, i.e., when . In our previous paper, we studied generalized tractability for proper subsets Ω of [1,∞)×ℕ, whereas in this paper we take the unrestricted domain Ω ^unr=[1,∞)×ℕ. We consider the three cases for which we have only finitely many positive singular values of S ₁, or they decay exponentially or polynomially fast. Weak tractability holds for these three cases, and for all linear tensor product problems for which the singular values of S ₁ decay slightly faster than logarithmically. We provide necessary and sufficient conditions on the function T such that generalized tractability holds. These conditions are obtained in terms of the singular values of S ₁ and mostly asymptotic properties of T. The tractability conditions tell us how fast T must go to infinity. It is known that T must go to infinity faster than polynomially. We show that generalized tractability is obtained for T(x,y)=x ^1+ln y . We also study tractability functions T of product form, T(x,y)=f ₁(x)f ₂(x). Assume that a _i =lim inf  _x→∞(ln ln f _i (x))/(ln ln x) is finite for i=1,2. Then generalized tractability takes place iff

Polynomial approximation of rational Bézier curves with constraints

We present an efficient method to solve the problem of the constrained least squares approximation of the rational Bézier curve by the polynomial Bézier curve. The presented algorithm uses the dual constrained Bernstein basis polynomials, and exploits their recursive properties. Examples are given, showing the effectiveness of the algorithm.     

Multifractal analysis of the electronic states in the Fibonacci superlattice under weak electric fields

We report the effect of substitution of Cr by V and Mn in the spin- \frac12\frac{1}{2} 1 2 dimer compound Sr₃Cr₂O₈. In the pentavalent Cr site, V ion has zero effective moment, while Mn remains in the spin-1 state. For both V and Mn substitutions, the gap in the spin excitation spectra remains almost unchanged in the dilute doping regime. The low temperature Curie tail, expected to be originated from free or quasi-free magnetic ions, is found to be larger in case of V doped samples. However, inter-dimer interaction is found to be stronger in the Mn doped samples. The underlying magnetic behaviour of these doped samples is presumably related to the effect of broken dimers due to random substitutions.