Hamilton's principle,covariant Hamiltonian,and canonical formalism in four and five dimensions

A discussion of the 1950s and 1960s on the existence of an explicit covariant canonical formalism is renewed. A new point of view is introduced where Hamilton's principle, based on the existence of a Hamiltonian, is postulated independently from the Lagrange formalism. The Hamiltonian is determined by transformation properties and dimensional considerations. The variation of the action without constraints leads to an explicit covariant canonical formalism and correct equations of motion. The introduction of the charge as a fifth momentum gives rise to a reformulation of classical relativistic point mechanics as a five-dimensionalU(1) gauge theory with a theoretically invisible extra dimension. A generalization to other gauge groups is given. The inversion of the proper time is introduced as a new particle-antiparticle symmetry that allows one to show that in the five-dimensional classical theory all particles have positive energy.     

First quantization of mass and charge

The proper time is introduced as a parameter into the wave functions of relativistic quantum theory by first quantization of the mass. The classical limit is shown to be given by a recently developed canonical formulation of classical relativistic mechanics. The adjoint spinor is redefined with the help of a sign operator to remove a discrepancy between the classical and quantum actions in the behavior under time inversion. This results in positive energy densities for the Dirac theory. The inclusion of this sign operator into the definition of the probability current then removes negative probabilities from the theory. A five-dimensional formulation with first quantized charge is given.     

Convex ordering criteria for Lévy processes

Modelling financial and insurance time series with Lévy processes or with exponential Lévy processes is a relevant actual practice and an active area of research. It allows qualitatively and quantitatively good adaptation to the empirical statistical properties of asset returns. Due to model incompleteness it is a problem of considerable interest to determine the dependence of option prices in these models on the choice of pricing measures and to establish nontrivial price bounds. In this paper we review and extend ordering results of stochastic and convex type for this class of models. We also extend the ordering results to processes with independent increments (PII) and present several examples and applications as to α-stable processes, NIG-processes, GH-distributions, and others. Criteria are given for the Lévy measures which imply corresponding comparison results for European type options in (exponential) Lévy models.     

Free isotropic-nematic interfaces in fluids of charged platelike colloids

Bulk properties and free interfaces of mixtures of charged platelike colloids and salt are studied within the density-functional theory. The particles are modeled by hard cuboids with their edges constrained to be parallel to the Cartesian axes corresponding to the Zwanzig model. The charges of the particles are concentrated in their center. The density functional is derived by functional integration of an extension of the Debye-Hückel pair distribution function with respect to the interaction potential. For sufficiently small macroion charges, the bulk phase diagrams exhibit one isotropic and one nematic phase separated by a first-order phase transition. With increasing platelet charge, the isotropic and nematic binodals are shifted to higher densities. The Donnan potential between the coexisting isotropic and nematic phases is inferred from bulk structure calculations. Nonmonotonic density and nematic order parameter profiles are found at a free interface interpolating between the coexisting isotropic and nematic bulk phases. Moreover, electrically charged layers form at the free interface leading to monotonically varying electrostatic potential profiles. Both the widths of the free interfaces and the bulk correlation lengths are approximately given by the Debye length. For fixed salt density, the interfacial tension decreases upon increasing the macroion charge.     

Alkaline Earth Transition Metal Antimonides AT4Sb12 (A = Ca,Sr, Ba; T = Fe,Ru, Os) with LaFe4P12-Structure

The title compounds were prepared by reaction of CaSb₂, SrSb₂, or BaSb₃ with the transition metals and antimony in sealed silica tubes. They crystallize with the cubic LaFe₄P₁₂-structure, which was refined from single-crystal X-ray data of CaFe₄Sb₁₂, SrRu₄Sb₁₂, and BaRu₄Sb₁₂ to residuals of R = 0.014, 0.016, and 0.014, respectively. The thermal parameters of the alkaline earth ions increase with decreasing ionic size. The Sb? Sb distances are greater in the iron compound than they are in the two ruthenium compounds. This is rationalized to be due to a larger portion of electrons in antibonding Sb? Sb states in the iron compound.     

Reactions of Ru3(CO)12 with Diphosphenes — A New Route to 50‐Electron Ru3P2 nido‐Clusters

The reaction of the symmetric diphosphene 2, 4, 6‐(CF₃)₃‐C₆H₂‐P=P‐C₆H₂‐2, 4, 6‐(CF₃)₃ 4 with Ru₃(CO)₁₂ led to the 50‐electron Ru₃P₂ nido‐cluster Ru₃(CO)₉[μ‐P‐C₆H₂‐2, 4, 6‐(CF₃)₃]₂ 5 , which in solution at room temperature displays hindered rotation of the aromatic rings about the C(aryl)—P bonds. The structure of 5 was determined by X‐ray crystal structure analysis; its Ru₃P₂ centre forms a distorted square pyramid with one ruthenium atom at the apex. One of the two C₆H₂(CF₃)₃ groups is also appreciably distorted. Temperature‐dependent ¹⁹F NMR studies of the [A₃M₃X]₂ spin system (A = M = CF₃, X = ³¹P) of 5 indicated a rotational barrier ΔG^≠ of 82.3 kJ mol^‐1 at 141 °C. The same Ru₃P₂ core was obtained by the reaction of the unsymmetric diphosphene Mes*‐P=P‐Mes 11 with Ru₃(CO)₁₂; hindered rotation about the C(aryl)—P bonds was also observed, in this case.     

Chromium-mediated aldol and homoaldol reactions on solid support directed towards an iterative polyol strategy

Chromium-Reformatsky and chromium-homoaldol reactions run under neutral and mild reaction conditions. They are highly chemoselective, tolerant towards most common functional groups, and are not prone to retroaldol reactions. Initial studies directed to transfer these homogeneous chromium-mediated solution-phase reactions to solid phase are presented. The main objective was to develop a methodology to aid a combinatorial iterative strategy to polyols (polyketides) on solid phase. A general reactivity problem was observed with polystyrene based resins compared to the solution-phase reactions, independent if the electrophilic (aldehyde) or nucleophilic (bromide) end of the polyol chain was supported to the resin. A complicated penetration, or loss of the polar solvent environment after penetration into the resin, might be responsible for the reduced reactivity. Application of either a soluble polystyrene resin or a polystyrene resin with a polar polyethylene glycol tether resulted in improved yields.     

Structure and reactivity of xanthocorrinoids. Part V. Formation of trans-diol derivatives of 5,6-dihydrocobyrinic acid from xanthocorrinoids under acidic conditions

On treatment with H₂SO₄/MeOH, epimerization of hexamethyl cis-5,6-dihydroxycobyrinate c,8-lactam ( 3 ) takes place quantitatively at C(6), yielding the corresponding trans-diol 4 . The corresponding lactone 7 , whose structure has been established by X-ray analysis, is obtained from xanthocorrinoids 5 and 6 under similar conditions.     

Structure and Reactivity of Xanthocorrinoids. Part III. The first example of a pinacol-type rearrangement in the corrin series

The transformation of the c-acetic-acid chain of hexamethyl Coα, Coβ-dicyanocobyrinate into an ethyl group (→ 2 ) as well as the synthesis of the pentadecaalkyl-cobalticorrin 6d from commercial cyanocobalamin are described. On reaction of 2 or 6d with O₂ in the presence of ascorbic acid, migration of the CH₃ group at C(5) to the vicinal position C(6) takes place concomitantly with the introduction of a carbonyl group at C(5).