On the existence of two-dimensional Euler flows satisfying energy-Casimir stability criteria

The energy-Casimir stability method, also known as the Arnold stability method, has been widely used in fluid dynamical applications to derive sufficient conditions for nonlinear stability. The most commonly studied system is two-dimensional Euler flow. It is shown that the set of two-dimensional Euler flows satisfying the energy-Casimir stability criteria is empty for two important cases: (i) domains having the topology of the sphere, and (ii) simply-connected bounded domains with zero net vorticity. The results apply to both the first and the second of Arnold’s stability theorems. In the spirit of Andrews’ theorem, this puts a further limitation on the applicability of the method. © 2000 American Institute of Physics.

Trinuclear Cu(II) complexes containing peripheral ketonic oxygen bridges and a mu(3)-OH core: steric influence on their structures and existence

Two tridentate Schiff bases, HL1(6-amino-3-methyl-1-phenyl-4-azahex-2-en-1-one), and HL2 (6-atnino-3,6-dimethyl-1-phenyl-4-azahex-2-en-1-one) on reaction with Cu(II) perchlorate in the presence of triethyl amine yielded two new trinuclear copper(II) complexes, [(CuL1)(3)(mu(3)-OH)](ClO4)(2) (1) and [(CuL2)(3)(mu(3)-OH)](ClO4)(2) center dot 0.75H(2)O (2), whereas another tridentate ligand HL3 (7-amino-3-methyl-1-phenyl-4-azahept-2-en-1-one) underwent hydrolysis under the same reaction conditions to result in the formation of a mononuclear complex, [Cu(bn)(pn)ClO4] (3) [where bn = 1-benzoylacetonate and pn = 1,3-propanediamine]. All three complexes have been characterized by X-ray crystallography. For both 1 and 2 the cationic part is trinuclear with a [Cu3OH] core held by three carbonyl oxygen bridges between each pair of copper(II) atoms. The structure of 3 is a monomer with a chelating 1,3-propanediamine and a benzoyl acetone moiety. Magnetic measurements of I and 2 have been performed in the 2-300 K temperature range. The experimental data could be satisfactorily reproduced by using an isotropic exchange model, H = -J(S1S2 + S2S3 + S1S3), yielding as best fit parameters: J = -25.6 cm(-1), g = 2.21 for 1 and J = 11.2 cm(-1), g = 2.10 for 2. The EPR spectra at low temperature could be indicative of spin frustration in complex 1. (C) 2006 Elsevier Ltd. All rights reserved.

Potential energy surface and moleculardynamics of MbNO: existence of an unsuspected FeON minimum

Ligands such as CO, O2, or NO are involved in the biological function of myoglobin. Here we investigate the energetics and dynamics of NO interacting with the Fe(II) heme group in native myoglobin using ab initio and molecular dynamics simulations. At the global minimum of the ab initio potential energy surface (PES), the binding energy of 23.4 kcal/mol and the Fe-NO structure compare well with the experimental results. Interestingly, the PES is found to exhibit two minima: There exists a metastable, linear Fe-O-N minimum in addition to the known, bent Fe-N-O global minimum conformation. Moreover, the T-shaped configuration is found to be a saddle point, in contrast to the corresponding minimum for NO interacting with Fe(III). To use the ab initio results for finite temperature molecular dynamics simulations, an analytical function was fitted to represent the Fe-NO interaction. The simulations show that the secondary minimum is dynamically stable up to 250 K and has a lifetime of several hundred picoseconds at 300 K. The difference in the topology of the heme-NO PES from that assumed previously (one deep, single Fe-NO minimum) suggests that it is important to use the full PES for a quantitative understanding of this system. Why the metastable state has not been observed in the many spectroscopic studies of myoglobin interacting with NO is discussed, and possible approaches to finding it are outlined.

On the existence of extreme waves and the Stokes conjecture with vorticity

This is a study of singular solutions of the problem of traveling gravity water waves on flows with vorticity. We show that, for a certain class of vorticity functions, a sequence of regular waves converges to an extreme wave with stagnation points at its crests. We also show that, for any vorticity function, the profile of an extreme wave must have either a corner of 120° or a horizontal tangent at any stagnation point about which it is supposed symmetric. Moreover, the profile necessarily has a corner of 120° if the vorticity is nonnegative near the free surface.

Robot free will

We introduce the perspex machine which unifies projective geometry and Turing computation and results in a supra-Turing machine. We show two ways in which the perspex machine unifies symbolic and non-symbolic AI. Firstly, we describe concrete geometrical models that map perspexes onto neural networks, some of which perform only symbolic operations. Secondly, we describe an abstract continuum of perspex logics that includes both symbolic logics and a new class of continuous logics. We argue that an axiom in symbolic logic can be the conclusion of a perspex theorem. That is, the atoms of symbolic logic can be the conclusions of sub-atomic theorems. We argue that perspex space can be mapped onto the spacetime of the universe we inhabit. This allows us to discuss how a robot might be conscious, feel, and have free will in a deterministic, or semi-deterministic, universe. We ground the reality of our universe in existence. On a theistic point, we argue that preordination and free will are compatible. On a theological point, we argue that it is not heretical for us to give robots free will. Finally, we give a pragmatic warning as to the double-edged risks of creating robots that do, or alternatively do not, have free will.