Eigenvalues of the Dirac Operator on Fibrations over S1

We consider the Dirac operator on fibrations overS ¹ which have up to holonomy a warped product metric. Wegive lower bounds for the eigenvalues on M and if the Diracoperator on the typical fibre F has a kernel, we calculatethe corresponding part of the spectrum on M explicitly.Moreover, we discuss the dependence of the spectrum of theholonomy and obtain bounds for the multiplicity of the eigenvalues.     

Tomographic Description of a Quantum Wave Packet in an Accelerated Frame

The tomography of a single quantum particle (i.e., a quantum wave packet) in an accelerated frame is studied. We write the Schrödinger equation in a moving reference frame in which acceleration is uniform in space and an arbitrary function of time. Then, we reduce such a problem to the study of spatiotemporal evolution of the wave packet in an inertial frame in the presence of a homogeneous force field but with an arbitrary time dependence. We demonstrate the existence of a Gaussian wave packet solution, for which the position and momentum uncertainties are unaffected by the uniform force field. This implies that, similar to in the case of a force-free motion, the uncertainty product is unaffected by acceleration. In addition, according to the Ehrenfest theorem, the wave packet centroid moves according to classic Newton’s law of a particle experiencing the effects of uniform acceleration. Furthermore, as in free motion, the wave packet exhibits a diffraction spread in the configuration space but not in momentum space. Then, using Radon transform, we determine the quantum tomogram of the Gaussian state evolution in the accelerated frame. Finally, we characterize the wave packet evolution in the accelerated frame in terms of optical and simplectic tomogram evolution in the related tomographic space.     

Formation and cleavage of C-H, C-C, and C-O bonds of ortho-methyl-substituted anisoles by late transition metals

2,6-Dimethyl-substituted anisoles can be converted into the corresponding 2-ethyl-6-methylphenols in a several-step reaction mediated by a TpMe2Ir(III) complex; use of the 13C-enriched anisoles, ArO13CH3, shows that the 13C label distributes across the two ethyl sites with a preference for the terminal position.     

Self-assembled pentamers and hexamers linked through quadruple-hydrogen-bonded 2-ureido-4[1H]-pyrimidinones

The preorganization of bifunctional 2-ureido-4-pyrimidinones mediated by either 1,3-substituted adamantane or meta-substituted phenylene ring linkers leads to the preferred formation of stable pentameric (1)(5) and hexameric (2)(6) assemblies, respectively. Despite the high binding constant of the 2-ureido-4-pyrimidinone dimers and the highly preorganized structure of the monomer, the predominant formation of cycles (1)(5) and (2)(6) in solution occurs only within a specific concentration range.     

Concise approach to the "higher sugar" core of the nucleoside antibiotic hikizimycin

A highly productive synthesis of phenylthio glycoside 33 is described which constitutes a fully functional surrogate for the hikosamine core of hikizimycin 1, a complex nucleoside antibiotic endowed with promising anthelmintic properties. The chosen approach to this undecose derivative starts from mannofuranose 7 which was one-carbon homologated to alkyne 8 in one step on treatment with lithio (trimethylsilyl)diazomethane. Alkynyl iodide 12 derived from 8 was combined with the tartrate-derived aldehyde 17 by a Nozaki-Hiyama-Kishi reaction that can either be performed using overstoichiometric amounts of CrCl2 or by means of a catalytic manifold based on the turnover of a cat. CrCl2/chlorosilane/manganese redox couple. Semi-hydrogenation of the resulting alkyne 18 to (Z)-olefin 19 required the use of Pd/C as the catalyst, whereas conventional Lindlar reduction was unsatisfactory. Attempted cis-dihydroxylation of alkene 22 (formed from 19 by a Mitsunobu reaction with phthalimide) by using catalytic amounts of OsO4 and NMO as the stoichiometric oxidant essentially failed, whereas a stoichiometric osmylation afforded the stable osmate ester 26 a as a single diastereomer. Since the use of OsO4 in stoichiometric amounts deemed inappropriate for a total synthesis project, recourse was taken to catalytic "Blitz dihydroxylation" with RuO4 in the presence of FeCl2.4H2O as co-catalyst. Application of these conditions to alkene 30 bearing a free aldehyde function at the terminus of the "higher sugar" chain furnished pyranose 32 in good yield and excellent diastereoselectivity, which was converted into the targeted thioglycoside 33 on treatment with PhSSPh/Et3P. It is particularly noteworthy that the conformational constraints of the acyclic substrate 30 enforce the dihydroxylation to violate Kishi's empirical rule for transformations of this type.     

Atomic-scale spin-polarized scanning tunneling microscopy applied to Mn3N2(010)

Atomic-scale spin-polarized scanning tunneling microscopy is demonstrated in the case of the unique surface spin structure of Mn3N2(010) at 300 K. We find that the surface spin structure is manifested as a modulation of the normal atomic row height profile. The atomic-scale spin-polarized image is thus shown to contain two components, one the normal, nonpolarized part, and the other the magnetic, spin-polarized part. A method is presented for separating these two spatially correlated components, and the results are compared with simulations based on integrated local spin density of states calculated from first principles.     

Reaction of lithium enediolates with perfluoroketene dithioacetals. Synthesis of α-trifluoromethyl γ-dicarboxylic acid derivatives

The reaction of perfluoroketene dithioacetals with dianions of carboxylic acids proceeds through the substitution of the vinylic fluoride. The preparative value of this reaction depends strongly on the reaction and work-up conditions, the optimisation of which led to use LDA as a base and multiple extraction techniques. The overall process may be considered as a formal synthesis of α-trifluoromethyl γ-dicarboxylic acid derivatives.     

On the Continuity of the Optimal Value in Parametric Linear Optimization: Stable Discretization of the Lagrangian Dual of Nonlinear Problems

This paper is focused on the stability of the optimal value, and its immediate repercussion on the stability of the optimal set, for a general parametric family of linear optimization problems in ⁿ. In our approach, the parameter ranges over an arbitrary metric space, and each parameter determines directly a set of coefficient vectors describing the linear system of constraints. Thus, systems associated with different parameters are not required to have the same number (cardinality) of inequalities. In this way, discretization techniques for solving a nominal linear semi-infinite optimization problem may be modeled in terms of suitable parametrized problems. The stability results given in the paper are applied to the stability analysis of the Lagrangian dual associated with a parametric family of nonlinear programming problems. This dual problem is translated into a linear (semi-infinite) programming problem and, then, we prove that the lower semicontinuity of the corresponding feasible set mapping, the continuity of the optimal value function, and the upper semicontinuity of the optimal set mapping are satisfied. Then, the paper shows how these stability properties for the dual problem entail a nice behavior of parametric approximation and discretization strategies (in which an ordinary linear programming problem may be considered in each step). This approximation–discretization process is formalized by means of considering a double parameter: the original one and the finite subset of indices (grid) itself. Finally, the convex case is analyzed, showing that the referred process also allows us to approach the primal problem.Mathematics Subject Classifications (2000) Primary 90C34, 90C31; secondary 90C25, 90C05.     

Coupling of internal alkynes in tp(me2)ir derivatives: selective oxidation of a noncoordinated double bond of the resulting iridacycloheptatrienes

The reaction of different Tp(Me2)Ir derivatives and dimethylacetylene dicarboxylate (DMAD) allows the preparation of three different metallacycloheptatriene complexes and an unusual allyl-terminated metallacycle. The C atoms of distant C=C bonds in the metallacycles, including aromatic ones, can be converted selectively to the corresponding keto functionality under mild conditions.