Higher integrability of the gradient and dimension of the singular set for minimisers of the Mumford–Shah functional

The paper is concerned with the higher regularity properties of the minimizers of the Mumford–Shah functional. It is shown that, near to singular points where the scaled Dirichlet integral tends to 0, the discontinuity set is close to an Almgren area minimizing set. As a byproduct, the set of singular points of this type has Hausdorff dimension at most N-2, N being the dimension of the ambient space. Assuming higher integrability of the gradient this leads to an optimal estimate of the Hausdorff dimension of the full singular set. Received: 5 July 2001 / Accepted: 29 November 2001 / Published online: 23 May 2002     

Photochemical reduction of furil by pentachlorophenol: Cross relaxation and CIDNP

Photolysis of a solution of furil and pentachlorophenol yields proton spin polarized furil as the major reaction product. Dipolar (cross-) relaxation in the furil three spin system was found to transfer CIDNP between the protons, such that resonances which were emissive initially relax to enhanced absorption, and vice-versa, depending on the spin populations produced by the photochemical process.     

The preliminary investigation into the potential use of tomographic techniques for the examination of mechanical components

Tomographic techniques exploiting the measurement of transmitted gamma-radiation have been investigated as a means of providing information about the state of selected mechanical components. In this exploratory study, all samples were taken to a specially constructed laboratory rig for examination. Examples are given of the images generated.     

A complete logic forn-permutable congruence lattices

A family of logical systems, which may be regarded as extending equational logic, is studied. The equationsf=g of equational logic are generalized to congruence equivalence formulasf≡g (modx), wheref andg are terms interpreted as elements of an algebraV of some specified type. and termx is interpreted as a member of ann-permutable lattice of congruences forV. Formal concepts of proof and derivability from systems of hypotheses are developed. These proofs, like those of equational logic. require only finite algebraic processes, without manipulation of logical quantifiers or connectives. The logical systems are shown to be correct and complete: a well-formed statement is derivable from a system of hypotheses if and only if it is valid in all models of these hypotheses.     

“Living” Radical Polymerization in Tubular Reactors, 2 – Process Optimization for Tailor‐Made Molecular Weight Distributions

“Living” radical polymerization is a relatively new polymerization process that can be used to prepare resins with controlled structures. In this work, a mathematical model developed previously to describe nitroxide‐mediated “living” radical polymerizations performed in tubular reactors is used for the optimization of the process and obtainment of tailor‐made MWDs. Operating conditions and design variables are determined with the help of optimization procedures in order to produce polymers with specified MWDs. It is shown that bimodal and trimodal MWDs, with given peak locations, can be obtained through proper manipulation of the operating conditions. This indicates that the technique discussed in this work is suitable for detailed design of the MWD of the final polymer.

     

Unified approach to ion transport and structural relaxation in amorphous polymers and glasses

Kinetic data for structural relaxation in silver iodomolybdates at the glass transition temperature (Tg) are obtained by high-pressure differential scanning calorimetry (HP-DSC) and are compared with activation energies (EA) and volumes (VA) obtained earlier from conductivities below Tg. The results are fitted to an empirical equation, EA = MVA, and displayed in the form of a master plot of EA versus VA, an approach previously applied to strongly coupled systems, including polymer electrolytes and molten salts above their glass transition temperatures. The parameter M emerges as a localized modulus, expressive of interatomic forces within the medium, linking together EA,sigma, VA,sigma and EA,s, VA,s, the "apparent" activation parameters for ionic conductivity and structural relaxation, respectively. The VA and EA values for ion transport are much smaller than the corresponding values for structural relaxation. However, remarkably close agreement emerges between the "process parameters", Ms and Msigma, both close to 8 GPa, thus establishing a quantitative link between ion transport and structural relaxation in this highly decoupled system. A new EA versus VA master plot is constructed, which points the way to a unified approach to ion transport in polymers and glasses.