Controllability of linear systems,differential geometry curves in grassmannians and generalized grassmannians,and riccati equations

We study the linear system =Ax+Bu from a differential geometric point of view. It is well-known that controllability of the system is related to the one-parameter family of operators e^t B. We use this to give a proof of the classical controllability conditions in terms of the differential geometry of certain curves in ⁿ. We then view (t)=Im(e^t B) as a curve in appropriate Grassmannian and see that, in local coordinates, is an integral curve of the flow induced by a matrix Riccati equation. We obtain qualitative geometric conditions on that are equivalent to the controllability of the system. To get quantitiative results, we lift to a curve l' in a splitting space, a generalized Grassmannian, which has the advantage of being a reductive homogeneous space of the general linear group, GL(ⁿ). Explicit and simple expressions concerning the geometry of are computed in terms of the Lie algebra of GL(ⁿ), and these are related to the controllability of the system.James Wolper was a visiting professor in the Department of Mathematics at Texas Tech University while much of this research was conducted. He would like to express appreciation for the hospitality he received during his visit.     

On the dimerization of BH3 and the associated delocalization (resonance) energy

SCF calculations utilizing a basis of all s-type Gaussians have been performed to investigate the dimerization of BH₃ to B₂H₆. Localized molecular orbitals and pseudonatural orbitals are calculated. The dimerization energy obtained (11.5 kcal/mole) supports the assumption that the non-Hartree-Fock interactions (electron correlation) play a dominant role (Exp = 40–60 kcal/mole). The energy of the principal type of resonance structure based on the SCF-MO 's of the separated-but-distorted BH₃ systems is rigorously calculated and found to be 150 kcal/mole higher in energy than the SCF energy of B₂H₆. Thus, resonance stabilization (delocalization) is shown to be extremely important to the stability of the “three-center” bonds in B₂H₆, as has been suspected for many years.     

On the spectral representation of symmetric stable processes

The so-called spectral representation theorem for stable processes linearly imbeds each symmetric stable process of index p into L^p (0 < p ≤ 2). We use the theory of L^p isometries for 0 < p < 2 to study the uniqueness of this representation for the non-Gaussian stable processes. We also determine the form of this representation for stationary processes and for substable processes. Complex stable processes are defined, and a complex version of the spectral representation theorem is proved. As a corollary to the complex theory we exhibit an imbedding of complex L^q into real or complex L^p for 0 < p < q ≤ 2.     

Redetermination of the X-ray structure of nitroxylcobalamin: base-on nitroxylcobalamin exhibits a remarkably long Co-N(dimethylbenzimidazole) bond distance

The X-ray structures of three new crystals of nitroxylcobalamin (NOCbl) have been determined. Unlike our earlier reported structure in which NOCbl was partially oxidized (L. Hannibal, C. A. Smith, D. W. Jacobsen and N. E. Brasch, Angew. Chem., Int. Ed. 2007, 46, 5140), the O atom of the nitroxyl ligand is located in a single position with a N=O bond distance of 1.12-1.14 ?, consistent with a double bond. The Co-N-O angle is in the 118.9-120.3 ? range. The α-axial Co-N(dimethylbenzimidazole) (Co-NB3) bond distance is a remarkable 2.32-2.35 ? in length, ~0.1 ? longer than that reported for all other cobalamin structures. The change in the Gibbs free energy for the base-on/base-off equilibrium now correlates extremely well with the Co-NB3 bond distance, as observed for other cobalamins.     

Tolerance graphs

Tolerance graphs arise from the intersection of intervals with varying tolerances in a way that generalizes both interval graphs and permutation graphs. In this paper we prove that every tolerance graph is perfect by demonstrating that its complement is perfectly orderable. We show that a tolerance graph cannot contain a chordless cycle of length greater than or equal to 5 nor the complement of one. We also discuss the subclasses of bounded tolerance graphs, proper tolerance graphs, and unit tolerance graphs and present several possible applications and open questions.     

Matrix isolation study of the vacuum-ultraviolet photolysis of HCCIF2 and of HCCl2F. Infrared spectra of the parent molecules and of the ClCF2 and FCCl2 free radicals

Studies of the vacuum-ultraviolet photolysis of HCClF₂ and of HCCl₂F isolated in argon and nitrogen matrices at 14°K have led to the infrared spectroscopic identification of the free radicals ClCF₂ and FCCl₂, respectively. Support for this identification has been obtained from studies of the vacuum-ultraviolet photolysis of DCClF₂, of DCCl₂F, and of H¹³CCl₂F, as well as from studies of halogen-atom abstraction from CClF₃ and from CCl₃F by sodium atoms in an argon matrix environment. The CCl bonds of FCCl₂, like those of CCl₃ and of HCCl₂, appear to be exceptionally strong. Gas-phase infrared spectral data are also reported for DCClF₂ and for DCCl₂F, not previously studied. Vibrational assignments are proposed for these two molecules, and data supporting the reassignment of one of the CF stretching fundamentals of HCClF₂ are presented.     

Dielectrophoretically Modulated Optical Spectroscopy of Colloidal Nanoparticle Solutions in Microfluidic Channels

Optical spectroscopic techniques (e.g., extinction, scattering, and fluorescence spectroscopies) are important for the analysis of colloidal solutions of nanoparticles (NPs). They are routinely applied to plasmonic and quantum-dot NP samples assuming that these contain a single population of particles with modest size and shape dispersity. However, these spectroscopic techniques become less effective when the sample is a mixture of particles with different sizes, shapes, or composition. Here, an original microfluidic method is proposed for the optical spectroscopic analysis of colloidal NP solutions that combines periodic trapping of NPs by dielectrophoresis (DEP) with in situ optical extinction spectroscopy. The periodic trapping leads to modulation of the continuously monitored optical spectrum depending on the DEP properties of the NPs. DEP-modulated spectroscopy is demonstrated using colloidal gold NPs as small as 40 nm diameter. It is found that the DEP modulation is significantly enhanced when employing suitable microfluidic flow over a multielectrode array. Finally, it is shown that the method can identify and characterize the NP species simultaneously present in a mixture of 40 and 80 nm gold NPs, opening the way toward optical spectroscopic analysis of higher complexity NP mixtures through the combination of the DEP-modulated spectroscopy with chemometric methods.     

Acetylacetonate anchors for robust functionalization of TiO2 nanoparticles with Mn(II)-terpyridine complexes

A novel class of derivatized acetylacetonate (acac) linkers for robust functionalization of TiO2 nanoparticles (NPs) under aqueous and oxidative conditions is reported. The resulting surface adsorbate anchors are particularly relevant to engineering photocatalytic and photovoltaic devices since they can be applied to attach a broad range of photosensitizers and photocatalytic complexes and are not affected by humidity. Acac is easily modified by CuI-mediated coupling reactions to provide a variety of scaffolds, including substituted terpy complexes (terpy = 2,2':6,2'-terpyridine), assembled with ligands coordinated to transition-metal ions. Since Mn-terpy complexes are known to be effective catalysts for oxidation chemistry, functionalization with Mn(II) is examined. This permits visible-light sensitization of TiO2 nanoparticles due to interfacial electron transfer, as evidenced by UV-vis spectroscopy of colloidal thin films and aqueous suspensions. The underlying ultrafast interfacial electron injection is complete on a subpicosecond time scale, as monitored by optical pump-terahertz probe transient measurements and computer simulations. Time-resolved measurements of the Mn(II) EPR signal at 6 K show that interfacial electron injection induces Mn(II) --> Mn(III) photooxidation, with a half-time for regeneration of the Mn(II) complex of ca. 23 s.     

Virtually epimerization-free synthesis of peptide-alpha-thioesters

Under slightly basic or neutral reaction conditions peptide-alpha-thioesters are photochemically synthesized from peptide-alpha-nitroindoline precursors, either in solution, or by direct photorelease from a solid support.     

Threshold tolerance graphs

In this paper, we introduce a class of graphs that generalize threshold graphs by introducing threshold tolerances. Several characterizations of these graphs are presented, one of which leads to a polynomial-time recognition algorithm. It is also shown that the complements of these graphs contain interval graphs and threshold graphs, and are contained in the subclass of chordal graphs called strongly chordal graphs, and in the class of interval tolerance graphs.