Photochemical Reaction of Decacarbonyldirhenium with Thiophene and Crystal Structure of Tetradecacarbonylhydridotrirhenium

The photochemical reaction of Re₂(CO)₁₀ with thiophene in hexane solution was investigated under vacuum. Three rhenium clusters: H₂Re₃(CO)₁₂, HRe₃(CO)₁₄ and Re₃(CO)₁₄(OH)₄, were isolated. The structure of Fellmann-Kaesz cluster Complex HRe₃(CO)₁₄ was determined by use of the X-ray diffraction method. The three rhenium atoms form a plane of symmetry and L: Re₁Re₂Re₃ is 107°. The ten carbonyl groups bonded to the two terminal rhenium atoms Re₁ and Re₃, are staggered with respect to the central rhenium atoms. The bond lengths are 3.10 Å for Re₂-Re₃ and 3.34 Å for Re₁-Re₂. The bridging hydride is between Re₁ and Re₂.     

The existence of non-topological solitons in the self-dual Chern-Simons theory

In the recently discovered (2+1)-dimensional relativistic Chern-Simons model, self-duality can be achieved when the Higgs potential density assumes a special form for which both the asymmetric and symmetric vacua are ground state solutions. This important feature may imply the coexistence of static topological and non-topological vortex-like solutions inR ² but the latter have been rather elusive to a rigorous construction. Our main purpose in this paper is to prove the existence of non-topological radially symmetricN-vortex solutions in the self-dual Chern-Simons model. By a shooting method, we obtain a continuous family of gauge-distinctN-vortex solutions. Moreover, we are also able to classify all possible bare (or 0-vortex) solutions.     

A note on differentiability of the cluster density for independent percolation in high dimensions

The cluster density function of independent percolation in ad-dimensional lattice is considered. For eachn, it is shown that(p) has finitenth leftderivative at critical probabilityp _c ifd is sufficiently large. This result agrees with the Bethe lattice approximation, where thenth one-sided derivative of(p) is bounded atp _c for alln.