Economic Equilibrium: Optimality and Price Decentralization

Mathematical economics has a long history and covers many interdisciplinary areas between mathematics and economics. At its center lies the theory of market equilibrium. The purpose of this expository article is to introduce mathematicians to price decentralization in general equilibrium theory. In particular, it concentrates on the role of positivity in the theory of convex economic analysis and the role of normal cones in the theory of non-convex economies.     

Synthesis and characterisation of the platinum complexes [PtCl(CClPAr)(PPh3)2] and [PtCl(CClPAr′)(PPh3)2] as potential intermediates in the preparation of phosphaalkynes

Oxidative addition reactions of Cl₂CPR (R = 2,4,6-tris(trifluoromethyl)phenyl (Ar) or 2,6-bis(trifluoromethyl)phenyl (Ar′) with Pt(PPh₃)₄ yield the cis and trans (at platinum) complexes [PtCl(ClCPAr)(PPh₃)₂] and [PtCl(ClCPAr′)(PPh₃)₂]. All starting materials and intermediates have been characterised by NMR spectroscopy. The crystal and molecular structures of the trans-platinum complexes have been determined by single-crystal X-ray diffraction at low temperature.     

A concise total synthesis of the azaphenanthrene alkaloid eupolauramine.

A six-step total synthesis of the azaphenanthrene alkaloid eupolauramine 1 has been achieved using combinational metalation-cyclization tactics. The synthetic route involved first the construction of the azaisoindolinone 9 by aryne-mediated cyclization of he phosphorylated pyridocarboxamide 7 and subsequent dephosphorylation. Metalation of 9 followed by connection of the hydroxybenzyl appendage and E(1)CB anti-elimination allowed the formation of the halogenoarylmethylene azaisoindolinone 4 in the exclusive E-form. Oxidative radical cyclization gave rise to the azaphenanthrene skeleton and regioselective bromination of 3 induced the incorporation of the bromine atom at the 6-position of the azaphenanthrene lactam. Ultimate replacement of the bromine atom of 2 by the methoxy functionality by sequential transmetalation, in situ oxidation, and O-methylation of the phenolic derivative 14 completed the synthesis of the target natural product eupolauramine.     

Nonlinear behaviour of the instability of steady natural convection in a vertical air layer

In this paper we deal with the flow of natural convection between two vertical planes with horizontal temperature gradient. We show that periodic steady flow occur when Rayleigh number increases. Critical values are obtained numerically and a nonlinear analysis is presented according to Center manifold method.     

Cooling of hot electron-hole plasmas in the presence of screened electron-phonon interactions

We have computed for several semiconductors, (GaAs, CdSe, CdS), the energy loss-rate in hot electron-hole plasmas, at high density, when occurs the screening of the electron-phonon interactions. We show that the long range interactions (piezoelectrical and polar) are quickly reduced when the plasma density is raised, and that the energy loss-rate diminishes therefore significantly (one order of magnitude).     

Simultaneous production of K1o (892) and Δ++(1236) in the reaction K+p→K+π+π−p from 4.3 to 5.0 GeV/c

The reaction $\text{K}{}^{\mathrm{+}}\text{d}\text{→}\text{K}{}^{\mathrm{1o}}\text{(892) Δ}{}^{\mathrm{++}}\text{(1236)}\text{n}{}_{\mathrm{s}}$ from a K⁺d experiment with the CERN 2 m bubble chamber at 4.6 GeV/c is compared with the reaction $\text{K}{}^{\mathrm{+}}\text{p}\text{→}\text{K}{}^{\mathrm{1o}}\text{(892)Δ}{}^{\mathrm{++}}\text{(1236)}$ from a K⁺p experiment at the same energy. In addition, predictions of an absorption model, a Regge pole model and a quark model for the production and decay characteristics of the reaction $\text{K}{}^{\mathrm{+}}\text{p}\text{→}\text{K}{}^{\mathrm{1o}}\text{(892)Δ}{}^{\mathrm{++}}\text{(1236)}$ are tested with the combined statistics of the K⁺d experiment and three K⁺p experiments between 4.3 and 5.0 GeV/c.     

A precise study of the rotational spectrum of formaldehyde H212C17O and H213C17O

The pure rotational spectra of H₂¹²C¹⁷O and H₂¹³C¹⁷O have been investigated in the frequency region between 8 and 360 GHz in the ground vibrational state. For both isotopic species the ¹⁷O nuclear quadrupole coupling constants and spin-rotation constants have been obtained. From both Q- and R-branch transitions a set of rotational constants and several distortion constants could be derived employing Watson's formalism in A reduction. The obtained rotational constants are in Megahertz:

     

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	H212C17O	H213C17O
$\text{A}$	281 965.0 (30)	281 987.3 (19)
$\text{B}$	37 812.287(45)	36 776.790(25)
$\text{C}$	33 214.523(31)	32 412.920(19)