Reactivity of a N-Coordinated Germylene-Borane Complex: From Ge→B to Ge→Ga Coordination

Reactivity studies of the Ge^II→B complex L(Cl)Ge⋅BH₃ ( 1 ; L=2-Et₂NCH₂-4,6-tBu₂-C₆H₂) were performed to determine the effect on the Ge^II→B donation. N-coordinated compounds L(OtBu)Ge⋅BH₃ ( 2 ) and [LGe⋅BH₃]₂ ( 3 ) were prepared. The possible tuning of the Ge^II→B interaction was proved experimentally, yielding compounds 1-PPh₂-8-(LGe)-C₁₀H₆ ( 4 ) and L(Cl)Ge⋅GaCl₃ ( 5 ) without a Ge^II→B interaction. In 5 , an unprecedented Ge^II→Ga coordination was revealed. The experimental results were complemented by a theoretical study focusing on the bonding in 1 − 5 . The different strength of the Ge^II→E (E=B, Ga) donation was evaluated by using energy decomposition analysis. The basicity of different L(X)Ge groups through proton affinity is also assessed.     

Coupled intervals in the discrete calculus of variations: necessity and sufficiency

In this work we study nonnegativity and positivity of a discrete quadratic functional with separately varying endpoints. We introduce a notion of an interval coupled with 0, and hence, extend the notion of conjugate interval to 0 from the case of fixed to variable endpoint(s). We show that the nonnegativity of the discrete quadratic functional is equivalent to each of the following conditions: The nonexistence of intervals coupled with 0, the existence of a solution to Riccati matrix equation and its boundary conditions. Natural strengthening of each of these conditions yields a characterization of the positivity of the discrete quadratic functional. Since the quadratic functional under consideration could be a second variation of a discrete calculus of variations problem with varying endpoints, we apply our results to obtain necessary and sufficient optimality conditions for such problems. This paper generalizes our recent work in [R. Hilscher, V. Zeidan, Comput. Math. Appl., to appear], where the right endpoint is fixed.     

Geometry of an Accelerated Rotating Disk

We analyze the geometry of a rotating disk with a tangential acceleration in the framework of the theory of Special Relativity, using the kinematic linear differential system that verifies the relative position vector of time-like curves in a Fermi reference. A numerical integration of these equations for a generic initial value problem is made up and the results are compared with those obtained in other works.     

Novel acrylic macromonomer with amphiphilic character derived from Triton X‐100: Radical copolymerization with methyl methacrylate and thermal properties

This article explores the synthesis of a novel methacrylic macromonomer with an amphiphilic character derived from poly(ethylene glycol) tert‐octylphenyl ether (MT) and its respective homopolymer. To know their reactivity in radical copolymerization reactions with methyl methacrylate (MMA), a model monomer (MTm) was synthesized to determine the reactivity ratios and compare them with the low molar fractions of copolymers of MT with MMA because they were difficult to isolate. They were r_MTm = 0.97 and r_MMA = 0.95. The compositional diagrams when representing the weight fraction of MT and MTm in the feed and the copolymer suggested that a clear correlation exists between the experimental points of the model monomer MTm and the macromonomer MT ones, suggesting that the length of the side poly(ethylene oxide) chain does not affect the reactivity of the methacrylic double bond in the prepared monomers for this type of polymerization reaction. The reactivity ratios of the copolymers have a tendency for the formation of random or Bernoullian copolymers. The glass‐transition temperatures (T_g's) of the prepared copolymers were determined by differential scanning calorimetry, deviated from the Fox equation, and discussed on the basis of treatments that consider the influence of the monomeric units along the copolymer chains, determining the T_g of the corresponding alternating dyads. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 1641–1649, 2003     

Killing tensors as irreducible representations of the general linear group

We show that the vector space of fixed valence Killing tensors on a space of constant curvature is naturally isomorphic to a certain highest weight, irreducible representation of the general linear group. The isomorphism is equivariant in the sense that the natural action of the isometry group corresponds to the restriction of the linear action to the appropriate subgroup. As an application, we deduce the Delong–Takeuchi–Thompson formula on the dimension of the vector space of Killing tensors from the classical Weyl dimension formula. To cite this article: R.G. McLenaghan et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Hydrophilic and hydrophobic copolymer systems based on acrylic derivatives of pyrrolidone and pyrrolidine

This article deals with the synthesis of hydrophilic methacrylic monomers derived from ethyl pyrrolidone [2‐ethyl‐(2‐pyrrolidone) methacrylate (EPM)] and ethyl pyrrolidine [2‐ethyl‐(2‐pyrrolidine) methacrylate (EPyM)] and their respective homopolymers. For the determination of their reactivity in radical copolymerization reactions, both monomers were copolymerized with methyl methacrylate (MMA), the reactivity ratios being calculated by the application of linear and nonlinear mathematical methods. EPM and MMA had ratios of r_EPM = 1.11 and r_MMA = 0.76, and this indicated that EPM with MMA had a higher reactivity in radical copolymerization processes than vinyl pyrrolidone (VP; r_VP = 0.005 and r_MMA = 4.7). EPyM and MMA had reactivity ratios of r_EPyM = 1.31 and r_MMA = 0.92, and this implied, as for the EPM–MMA copolymers, a tendency to form random or Bernoullian copolymers. The glass‐transition temperatures of the prepared copolymers were determined by differential scanning calorimetry (DSC) and were found to adjust to the Fox equation. Total‐conversion copolymers were prepared, and their behavior in aqueous media was found to be dependent on the copolymer composition. The swelling kinetics of the copolymers followed water transport mechanism case II, which is the most desirable kinetic behavior for a swelling controlled‐release material. Finally, the different states of water in the hydrogels—nonfreezing water, freezing bound water, and unbound freezing water—were determined by DSC and found to be dependent on the hydrophilic and hydrophobic units of the copolymers. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 395–407, 2003