On the equations for the movement and deformation of a reaction front

Summary The equations for the movement of a burning interface are derived when the oxidant has to diffuse through a mantle of ash to reach the reaction zone and the heat generated there must be conducted away either through the fuel or the ash. Since interest centres on whether a small perturbation of the plane interface will disappear or whether it will develop into a deeper perturbation, the moving interface and the characteristic lines of its motion are chosen as an orthogonal coordinate network. The resulting equations are simplified for the case of a dimple and an equation obtained for its curvature on the centreline. The physical interpretation of the equations and results of some calculation suggest that in most cases the dimple is likely to flatten out and it is concluded that the deep pits that have been found are more likely to be caused by local inhomogeneities or transitions between steady states.     

Carbon magnetic resonance spectra of representative organometallic π-propene complexes

The ¹⁹C magnetic resonance spectra of silver, rhodium and iron complexes of propene are reported and discussed.     

Sard theorems for Lipschitz functions and applications in optimization

We establish a “preparatory Sard theorem” for smooth functions with a partially affine structure. By means of this result, we improve a previous result of Rifford [17, 19] concerning the generalized (Clarke) critical values of Lipschitz functions defined as minima of smooth functions. We also establish a nonsmooth Sard theorem for the class of Lipschitz functions from R^d to R^p that can be expressed as finite selections of C^k functions (more generally, continuous selections over a compact countable set). This recovers readily the classical Sard theorem and extends a previous result of Barbet–Daniilidis–Dambrine [1] to the case p > 1. Applications in semi-infinite and Pareto optimization are given.     

"Switching on" the properties of single-molecule magnetism in triangular manganese(III) complexes

The reaction between oxide-centered, triangular [MnIII3O(O2CR)6(py)3](ClO4) (R = Me (1), Et (2), Ph (3)) compounds and methyl 2-pyridyl ketone oxime (mpkoH) affords a new family of Mn/carboxylato/oximato complexes, [MnIII3O(O2CR)3(mpko)3](ClO4) [R = Me (4), Et (5), and Ph (6)]. As in 1-3, the cations of 4-6 contain an [MnIII3(mu3-O)]7+ triangular core, but with each Mn2 edge now bridged by an eta1:eta1:mu-RCO2- and an eta1:eta1:eta1:mu-mpko- group. The tridentate binding mode of the latter causes a buckling of the formerly planar [MnIII3(mu3-O)]7+ core, resulting in a relative twisting of the three MnIII octahedra and the central O2- ion now lying approximately 0.3 A above the Mn3 plane. This structural distortion leads to ferromagnetic exchange interactions within the molecule and a resulting S = 6 ground state. Fits of dc magnetization data for 4-6 collected in the 1.8-10.0 K and 10-70 kG ranges confirmed S = 6 ground states, and gave the following D and g values: -0.34 cm(-1) and 1.92 for 4, -0.34 cm(-1) and 1.93 for 5, and -0.35 cm(-1) and 1.99 for 6, where D is the axial zero-field splitting (anisotropy) parameter. Complexes 4-6 all exhibit frequency-dependent out-of-phase (chi" M) ac susceptibility signals suggesting them possibly to be single-molecule magnets (SMMs). Relaxation rate vs T data for complex 4 down to 1.8 K obtained from the chi" M vs T studies were supplemented with rate vs T data measured to 0.04 K via magnetization vs time decay studies, and these were used to construct Arrhenius plots from which was obtained the effective barrier to relaxation (Ueff) of 10.9 K. Magnetization vs dc field sweeps on single-crystals of 4.3CH2Cl2 displayed hysteresis loops exhibiting steps due to quantum tunneling of magnetization (QTM). The loops were essentially temperature-independent below approximately 0.3 K, indicating only ground-state QTM between the lowest-lying Ms = +/-6 levels. Complexes 4-6 are thus confirmed as the first triangular SMMs. High-frequency EPR spectra of single crystals of 4.3CH2Cl2 have provided precise spin Hamiltonian parameters, giving D = -0.3 cm(-1), B40 = -3 x 10(-5) cm(-1), and g = 2.00. The spectra also suggest a significant transverse anisotropy of E > or = 0.015 cm(-1). The combined work demonstrates the feasibility that structural distortions of a magnetic core imposed by peripheral ligands can "switch on" the properties of an SMM.     

Copper inverse-9-metallacrown-3 compounds showing antisymmetric magnetic behaviour

The use of phenyl 2-pyridyl ketoxime (PhPyCNO)/X(-)"blends" (X(-) = OH(-), Cl(-), ClO(4)(-)) in copper chemistry has yielded trinuclear clusters that have been characterized as inverse-9-metallacrown-3 accommodating one or two anions. The magnetic behaviour has shown a large antiferromagnetic interaction. The discrepancy between the Brillouin curve and the experiment has been assigned to the influence of the antisymmetric interaction. By introducing in the magnetization formula the antisymmetric terms derived from the fitting of the susceptibility data the simulated curve become almost superimposable to the experimental one. From the EPR findings it has been shown that the compound [Cu(3)(PhPyCNO)(3)(OCH(3))(Cl)(ClO(4))] () having isosceles magnetic symmetry or lower (delta not equal 0), the antisymmetric exchange is important (G not equal 0) and DeltaE > hv. The structures of the two complexes have been determined by single-crystal X-ray crystallography.     

Coercivity conditions and variational inequalities

Received November 17, 1997 / Revised version received August 6, 1998?Published online March 16, 1999     

On the structure of the theory of viscoplasticity

In various formulations of plasticity, there is evident a structure embracing several features, including inviscidity, a yield condition, and a constitutive inequality. By means of these features the constitutive equations of plasticity are derived. In the present paper we introduce a viscoplastic counterpart of the constitutive inequality of plasticity, and we consider its physical significance. We also present a theory of viscoplasticity having a structure similar to that of plasticity and its relation with the Hohenemser-Prager prototype of viscoplastic constitutive relations is considered.