Click chemistry in materials synthesis. III. Metal‐adhesive polymers from Cu(I)‐catalyzed azide–alkyne cycloaddition

1,2,3‐Triazole‐based polymers generated from the Cu(I)‐catalyzed cycloaddition between multivalent azides and acetylenes are effective adhesive materials for metal surfaces. The adhesive capacities of candidate mixtures of azide and alkyne components were measured by a modified peel test, using a customized adhesive tester. A particularly effective tetravalent alkyne and trivalent azide combination was identified, giving exceptional strength that matches or exceeds the best commercial formulations. The addition of Cu catalyst was found to be important for the synthesis of stronger adhesive polymers when cured at room temperature. Heating also accelerated curing rates, but the maximum adhesive strengths achieved at both room temperature and high temperature were the same, suggesting that crosslinking reaches the same advanced point in all cases. Polytriazoles also form adhesives to aluminum, but copper is bound more effectively, presumably because active Cu(I) ions may be leached from the surface to promote crosslinking and adhesion. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 5182–5189, 2007     

Elliptic cross section without warping under torsion

The object of this paper is the pure torsion of the nonhomogeneous anisotropic elastic beam. The results of Saint-Venant’s theory of uniform torsion are used to prove a nonwarping property of elliptic cylinders.     

Young's inequality in trace-class operators

 If a and b are trace-class operators, and if u is a partial isometry, then , where ∥⋅∥₁ denotes the norm in the trace class. The present paper characterises the cases of equality in this Young inequality, and the characterisation is examined in the context of both the operator and the Hilbert–Schmidt forms of Young's inequality. Received: 20 December 2001 / Revised version: 11 July 2002 / Published online: 10 February 2003 Mathematics Subject Classification (2000): 47A63, 15A60     

Highly edge‐connected detachments of graphs and digraphs

Let G = (V,E) be a graph or digraph and r : V → Z₊. An r‐detachment of G is a graph H obtained by ‘splitting’ each vertex ν ∈ V into r(ν) vertices. The vertices ν₁,…,ν_r(ν) obtained by splitting ν are called the pieces of ν in H. Every edge uν ∈ E corresponds to an edge of H connecting some piece of u to some piece of ν. Crispin Nash‐Williams 9 gave necessary and sufficient conditions for a graph to have a k‐edge‐connected r‐detachment. He also solved the version where the degrees of all the pieces are specified. In this paper, we solve the same problems for directed graphs. We also give a simple and self‐contained new proof for the undirected result. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 67–77, 2003     

Characterization of Cds clusters in zeolite-A grown in alkaline solution

CdS clusters were synthesized in A type zeolite by reaction in alkaline aqueous solution at temperatures from 30 to 70 °C. The optical properties of the samples were studied by diffuse reflectance and photoluminescence spectroscopy. Their crystalline structure and morphology were studied by X-ray diffraction and by transmission and scanning electron microscopy. We found that at lower temperatures the CdS clusters are encapsulated in the zeolite cages. We compared the properties of these clusters with those encapsulated in the cages of zeolites X and Y, prepared by similar methods. CdS clusters smaller than the CdS exciton diameter are also formed outside the cages in the zeolite matrix. The size of these clusters increases with temperature producing a red-shift of the absorption edge in the optical absorption spectra.     

Weakly nonlocal irreversible thermodynamics

Weakly nonlocal thermodynamic theories are critically revisited. A relocalized, irreversible thermodynamic theory of nonlocal phenomena is given, based on a modified form of the entropy current and new kind of internal variables, the so called current multipliers. The treatment is restricted to deal with nonlocality connected to dynamic thermodynamic variables. Several classical equations are derived, including Guyer‐Krumhansl, Ginzburg‐Landau and Cahn‐Hilliard type equations.     

The Calogero–Bogoyavlenskii–Schiff Equation in 2+1 Dimensions

We use the classical and nonclassical methods to obtain symmetry reductions and exact solutions of the (2+1)-dimensional integrable Calogero–Bogoyavlenskii–Schiff equation. Although this (2+1)-dimensional equation arises in a nonlocal form, it can be written as a system of differential equations and, in potential form, as a fourth-order partial differential equation. The classical and nonclassical methods yield some exact solutions of the (2+1)-dimensional equation that involve several arbitrary functions and hence exhibit a rich variety of qualitative behavior.     

Extrinsic Killing spinors

 Under intrinsic and extrinsic curvature assumptions on a Riemannian spin manifold and its boundary, we show that there is an isomorphism between the restriction to the boundary of parallel spinors and extrinsic Killing spinors of non-negative Killing constant. As a corollary, we prove that a complete Ricci-flat spin manifold with mean-convex boundary isometric to a round sphere, is necessarily a flat disc. Received: 2 February 2002; in final form: 1 August 2002 / Published online: 1 April 2003 Mathematics Subject Classification (1991): 53C27, 53C40, 53C80, 58G25 The authors would like to thank Lars Andersson for helpful discussions and for bringing to our knowledge the information regarding Remark 4. We are also grateful to the referee for pointing out that Corollary 5 and Corollary 6 are only valid when the boundary is at least 2-dimensional. Research of S. Montiel is partially supported by a Spanish MCyT grant No. BFM2001-2967