Uncomputability and undecidability in economic theory

Economic theory, game theory and mathematical statistics have all increasingly become algorithmic sciences. Computable Economics, Algorithmic Game Theory[Noam Nisan, Tim Roiughgarden, Éva Tardos, Vijay V. Vazirani (Eds.), Algorithmic Game Theory, Cambridge University Press, Cambridge, 2007] and Algorithmic Statistics[Péter Gács, John T. Tromp, Paul M.B. Vitányi, Algorithmic statistics, IEEE Transactions on Information Theory 47 (6) (2001) 2443-2463] are frontier research subjects. All of them, each in its own way, are underpinned by (classical) recursion theory - and its applied branches, say computational complexity theory or algorithmic information theory - and, occasionally, proof theory. These research paradigms have posed new mathematical and metamathematical questions and, inadvertently, undermined the traditional mathematical foundations of economic theory. A concise, but partial, pathway into these new frontiers is the subject matter of this paper. Interpreting the core of mathematical economic theory to be defined by General Equilibrium Theory and Game Theory, a general - but concise - analysis of the computable and decidable content of the implications of these two areas are discussed. Issues at the frontiers of macroeconomics, now dominated by Recursive Macroeconomic Theory (The qualification ‘recursive’ here has nothing to do with ‘recursion theory’. Instead, this is a reference to the mathematical formalizations of the rational economic agent’s intertemporal optimization problems, in terms of Markov Decision Processes, (Kalman) Filtering and Dynamic Programming, where a kind of ‘recursion’ is invoked in the solution methods. The metaphor of the rational economic agent as a ‘signal processor’ underpins the recursive macroeconomic paradigm.), are also tackled, albeit ultra briefly. The point of view adopted is that of classical recursion theory and varieties of constructive mathematics.     

Donor-acceptor heteroleptic open sandwiches

A series of donor-acceptor heteroleptic open sandwiches with formula CpM-M'Pyl (M = B, Al, Ga; M' = Li, Na; Cp = cyclopentadienyl; Pyl = pentadienyl) has been designed in silico using density functional theory. The most stable complexes are those containing boron as a donor atom. A molecular orbital analysis shows that the s character of the lone pair located at the group 13 element is mainly responsible for the complex stabilization. It is also found that the surrounding medium has a similar effect on these sandwiches such as in the "classical" donor-acceptor complexes, showing a decrement in the group 13 element-alkaline metal bond lengths.     

C-H bond activation and subsequent C-C bond formation promoted by osmium: 2-vinylpyridine-acetylene couplings

Complex OsH2Cl2(PiPr3)2 promotes the C-H activation of 2-vinylpyridine and subsequently couples the activated substrate with a second 2-vinylpyridine and two acetylene molecules. In the absence of 2-vinylpyridine, the activated substrate is coupled with an acetylene unit to afford a 2-butadienylpyridine derivative.     

Probing particle size distributions in natural surface waters from 15 nm to 2 microm by a combination of LIBD and single-particle counting

We present a technique for measuring colloid size distributions between 15 nm and 2 microm at concentrations relevant to natural surface waters. Two particle-measuring methods are combined: laser-induced breakdown detection (LIBD), which allows the quantification of colloid size distributions below 400 nm, and a commercial single-particle counter that extends the accessible size range up to two mum. Centrifugation was used in order to separate micrometer sized particles for the LIBD measurement. The feasibility is demonstrated on water of Lake Brienz (Switzerland) and the River Pfinz (Germany) and the particle size distributions follow Pareto's law even down to 15 nm in both cases.     

Can the disproportion of oxidation state III be favored in RuII-OH2/RuIV=O systems?

Three new Ru-aqua complexes containing a mixed carbene and pyridylic ligands with general formulas [Ru(CNC)(bpy)(H2O)](PF6)2 (1) (CNC is 2,6-bis(butylimidazol-2-ylidene)pyridine; bpy is 2,2'-bipyridine) and cis-/trans-[Ru(CNC)(nBu-CN)(H2O)](PF6)2 (cis-2 and trans-2) (nBu-CN is 2-(butylimidazol-2-ylidene)pyridine) have been prepared and structurally characterized both in the solid state (monocrystal X-ray diffraction analysis for 1 and for the related complex trans-[Ru(Br)(CNC)(nBu-CN)](PF6)) and in solution (for all of them) through NMR. The electrochemical properties of these three Ru-aqua complexes have been investigated by cyclic voltammetry, differential pulse voltammetry and Coulombimetric techniques. It is found that, for complex 1 at pH 7, the difference between the IV/III and the III/II redox couples (DeltaE1/2) is 50 mV, which is the smallest ever reported for this type of complex. On the other hand, for complexes cis-2 and trans-2, the oxidation state III is unstable with respect to disproportionation to II and IV. The reactivity of their Ru=O species has been tested toward cis-beta-methylstyrene oxidation, and it has been compared to [Ru(O)(trpy)(bpy)]2+. An inverse correlation between the degree of cis/trans-epoxide isomerization and DeltaE1/2 is found. In particular, for complexes cis-2 and trans-2, which have a DeltaE1/2 < 0, the epoxidation is highly stereoselective, yielding only cis-epoxide.     

Furanoside thioether-phosphinite ligands for Pd-catalyzed asymmetric allylic substitution reactions: Scope and limitations

A series of readily available thioether-phosphinite ligands has been tested in the Pd-catalyzed allylic substitution reactions of several acyclic and cyclic allylic substrates (S1-S7). This series of ligands have been designed to uncover their important structural features and to determine the scope of the thioether-phosphinite ligands in these catalytic reactions. Systematic variation of the electronic and steric properties at the thioether moiety provide useful information about the ligand parameters that control the enantiodiscrimination. By carefully selecting the ligand parameters, good enantioselectivities with high activities were obtained for hindered linear substrates S1 and S2 (ee’s up to 95%) and for unhindered cyclic substrates S4 and S5 (ee’s up to 91%).     

Self-assembly of a tetranuclear Ni4 cluster with an S = 4 ground state: the first 3d metal cluster bearing a mu4-eta2:eta2-O,O carbonate ligand

Reaction of nickel(II) acetate with H(3)L (2-(5-bromo-2-hydroxyphenyl)-1,3-bis[4-(5-bromo-2-hydroxyphenyl)-3-azabut-3-enyl]-1,3-imidazolidine) yields [Ni(2)L(OAc)(H(2)O)(2)].3MeCN.2H(2)O (1.3MeCN.2H(2)O), crystallographically characterized. 1 is unstable in solution for a long time and hydrolyzes to give [Ni(2)L(o-OC(6)H(3)BrCHO)(H(2)O)].2.25MeCN.H(2)O (2.2.25MeCN.H(2)O). In addition, 1 uptakes CO(2) from air in a basic methanol/acetonitrile solution, yielding [[Ni(2)L(MeOH)](2)(CO(3))].1.5MeOH.MeCN.H(2)O (3.1.5MeOH.MeCN.H(2)O). The X-ray characterization of 3 reveals that it is a tetranuclear nickel cluster, which can be considered as the result of a self-assembly process from two dinuclear [Ni(2)L](+) blocks, joined by a mu(4)-eta(2):eta(2)-O,O carbonate ligand. The coordination mode of the carbonate anion is highly unusual and, to the best of our knowledge, it has not been described thus far for first-row transition metal complexes or magnetically studied until now. Magnetic characterization of 1 and 3 shows net intramolecular ferromagnetic coupling between the metal atoms in both cases, with S = 2 and S = 4 ground states for 1 and 3, respectively.     

Barrier height homogeneity for 4.5 kV 4H-SiC Schottky diodes

4.5 kV SiC Schottky diodes have been fabricated using Ni as the Schottky contact. A manufacturing yield of 40% is reached for the bigger area diodes (1.6×1.6 mm²) and of 70% for the smaller ones (0.4×0.4 mm²). The measured variations of barrier height and ideality factor with temperature do not agree with the thermionic model. This has been interpreted in terms of barrier height inhomogeneities using the Werner model. We extracted an average barrier height and its standard deviation . These two parameters are almost independent of the diode size. The variation of the barrier height distribution with field has also been investigated and shows a dependence similar to that of Schottky diodes realized from other semiconductor materials.     

Asymmetric Rh-catalyzed hydrogenation using a furanoside phosphite-phosphoroamidite and diphosphoroamidite ligand library

A furanoside phosphite-phosphoroamidite and diphosphoroamidite ligand library L1-L5a-f was tested in the asymmetric Rh-catalyzed hydrogenation of α,β-unsaturated carboxylic acid derivatives and enamides. Enantioselectivity depended strongly on the ligand parameters. High enantioselectivities were obtained in the reduction of dimethyl itaconate (up to >99% ee), α-dehydroamino acid esters (up to 99% ee) and several enamides (up to 92% ee). Kinetic and NMR studies on the intermediates of the catalytic cycle of the reaction indicate that the [Rh(P(1)-P(2))(substrate)](+) species is the resting state of the reaction and that the rate dependence is first order in rhodium and hydrogen pressure and zeroth order in the substrate.     

Complex surface chemistry of 4-mercaptopyridine self-assembled monolayers on Au(111)

The adsorption of 4-mercaptopyridine on Au(111) from aqueous or ethanolic solutions is studied by different surface characterization techniques and density functional theory calculations (DFT) including van der Waals interactions. X-ray photoelectron spectroscopy and electrochemical data indicate that self-assembly from 4-mercaptopyridine-containing aqueous 0.1 M NaOH solutions for short immersion times (few minutes) results in a 4-mercaptopyridine (PyS) self-assembled monolayer (SAM) with surface coverage 0.2. Scanning tunneling microscopy images show an island-covered Au surface. The increase in the immersion time from minutes to hours results in a complete SAM degradation yielding adsorbed sulfur and a heavily pitted Au surface. Adsorbed sulfur is also the main product when the self-assembly process is made in ethanolic solutions irrespective of the immersion time. We demonstrate for the first time that a surface reaction is involved in PyS SAM decomposition in ethanol, a surface process not favored in water. DFT calculations suggest that the surface reaction takes place via disulfide formation driven by the higher stability of the S-Au(111) system. Other reactions that contribute to sulfidization are also detected and discussed.