Drinfeld algebra deformations, homotopy comodules and the associahedra

The aim of this work is to construct a cohomology theory controlling the deformations of a general Drinfel'd algebra and thus finish the program which began in [13], [14]. The task is accomplished in three steps. The first step, which was taken in the aforementioned articles, is the construction of a modified cobar complex adapted to a non-coassociative comultiplication. The following two steps each involves a new, highly non-trivial, construction. The first construction, essentially combinatorial, defines a differential graded Lie algebra structure on the simplicial chain complex of the associahedra. The second construction, of a more algebraic nature, is the definition of a map of differential graded Lie algebras from the complex defined above to the algebra of derivations on the bar resolution. Using the existence of this map and the acyclicity of the associahedra we can define a so-called homotopy comodule structure (Definition 3.3 below) on the bar resolution of a general Drinfel'd algebra. This in turn allows us to define the desired cohomology theory in terms of a complex which consists, roughly speaking, of the bimodule and bicomodule maps from the bar resolution to the modified cobar resolution. The complex is bigraded but not a bicomplex as in the Gerstenhaber-Schack theory for bialgebra deformations. The new components of the coboundary operator are defined via the constructions mentioned above. The results of the paper were announced in [12].

     

Hardy and BMO spaces associated to divergence form elliptic operators

Consider a second order divergence form elliptic operator L with complex bounded measurable coefficients. In general, operators based on L, such as the Riesz transform or square function, may lie beyond the scope of the Calderón–Zygmund theory. They need not be bounded in the classical Hardy, BMO and even some L ^p spaces. In this work we develop a theory of Hardy and BMO spaces associated to L, which includes, in particular, a molecular decomposition, maximal and square function characterizations, duality of Hardy and BMO spaces, and a John–Nirenberg inequality. S. Hofmann was supported by the National Science Foundation.     

Manifold Reconstruction in Arbitrary Dimensions Using Witness Complexes

It is a well-established fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions. Specifically, it has been proved that the witness complex coincides with the restricted Delaunay triangulation on curves, and is still a subset of it on surfaces, under mild sampling conditions. In this paper, we prove that these results do not extend to higher-dimensional manifolds, even under strong sampling conditions such as uniform point density. On the positive side, we show how the sets of witnesses and landmarks can be enriched, so that the nice relations that exist between restricted Delaunay triangulation and witness complex hold on higher-dimensional manifolds as well. We derive from our structural results an algorithm that reconstructs manifolds of any arbitrary dimension or co-dimension at different scales. The algorithm combines a farthest-point refinement scheme with a vertex pumping strategy. It is very simple conceptually, and it does not require the input point sample to be sparse. Its running time is bounded by c(d)n ², where n is the size of the input point cloud, and c(d) is a constant depending solely (yet exponentially) on the dimension d of the ambient space. Although this running time makes our reconstruction algorithm rather theoretical, recent work has shown that a variant of our approach can be made tractable in arbitrary dimensions, by building upon the results of this paper. This work was done while S.Y. Oudot was a post-doctoral fellow at Stanford University. His email there is no longer valid.     

Peptide mimotopes of phomopsins: Identification, characterization and application in an immunoassay

Peptide mimotopes of plant-associated toxins offer the potential for improving analytical and diagnostic methodologies as well as providing candidates for potential protective vaccines against plant poisoning diseases. Monoclonal antibody (mAb) C₃C₁₁, which recognizes the antimicrotubule phomopsin mycotoxins, was used to isolate peptide mimics of phomopsin A from a random 15-mer phage display peptide library. A total of 46 clones were isolated that showed specific reactivity with the mAb. Amino acid sequence analysis revealed four different types of mimotope sequences, all of which contained a common motif V-A-L/V-C. Of the 46 clones isolated, 44 contained the motif V-A-L-C while 2 contained the V-A-V-C motif. All four types of phage clones inhibited the reactivity of the mAb with phomopsin A in a competition ELISA. The clone with the mimotope sequence CT VALCNMYFGAKLD demonstrated the strongest binding. It was further shown that synthetic peptides containing these mimotope amino acid sequences were able to inhibit the mAb-phomopsin A interaction, indicating that the peptide mimotopes were responsible for the specific binding, independent of the phage framework. The results also suggest that the mimotope peptides bind to mAb C₃C₁₁ at the same site as phomopsin A. The application of recombinant phage particles carrying phomopsin mimotopes in immunoassay was evaluated and the results demonstrated approximately 100-fold increase in sensitivity in comparison with a conventional immunoassay using a chemically linked phomopsin-horseradish peroxidase conjugate.     

A strategic r&d planning model with an application to an energy research organization

This paper describes a two-stage strategic planning model for determining the optimal R&D portfolio in the presence of both exploratory and developmental research projects. An example is illustrated in the context of energy research. Research portfolios are analysed using a Bayesian approach in which probability assessments are updated as new information generated by exploratory research becomes available. Useful insights are obtained through sensitivity analysis. It is found that exploratory research tends to exhibit a lumpy characteristic.     

Persistence stability for geometric complexes

In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris–Rips, ?ech and witness complexes) built on top of totally bounded metric spaces. Using recent developments in the theory of topological persistence, we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov–Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips and ?ech complexes built on top of compact spaces.     

Geometry on Probability Spaces

Partial differential equations and the Laplacian operator on domains in Euclidean spaces have played a central role in understanding natural phenomena. However, this avenue has been limited in many areas where calculus is obstructed, as in singular spaces, and in function spaces of functions on a space X where X itself is a function space. Examples of the latter occur in vision and quantum field theory. In vision it would be useful to do analysis on the space of images and an image is a function on a patch. Moreover, in analysis and geometry, the Lebesgue measure and its counterpart on manifolds are central. These measures are unavailable in the vision example and even in learning theory in general. There is one situation where, in the last several decades, the problem has been studied with some success. That is when the underlying space is finite (or even discrete). The introduction of the graph Laplacian has been a major development in algorithm research and is certainly useful for unsupervised learning theory. The approach taken here is to take advantage of both the classical research and the newer graph theoretic ideas to develop geometry on probability spaces. This starts with a space X equipped with a kernel (like a Mercer kernel) which gives a topology and geometry; X is to be equipped as well with a probability measure. The main focus is on a construction of a (normalized) Laplacian, an associated heat equation, diffusion distance, etc. In this setting, the point estimates of calculus are replaced by integral quantities. One thinks of secants rather than tangents. Our main result bounds the error of an empirical approximation to this Laplacian on X.     

Dynamics of Atomic and Molecular Hydrogen Elimination from Hydrocarbons at VUV Excitation

Elimination of atomic hydrogen (H) and molecular hydrogen (H₂) are important elementary chemical processes in photochemistry and combustion chemistry. Recently, unique and sensitive detection techniques for atomic and molecular hydrogen detection were developed in our laboratory. Using the advanced molecular beam methods, we have studied the photodissociation of a few typical hydrocarbons at 157 nm excitation, especially their atomic and molecular hydrogen elimination processes. In this report, we will briefly describe the results from photodissociation of propane, ethylene, propyne and methanol at 157 nm excitation. These molecules represent different classes of hydrocarbons such as alkane, alkene, alkyne and alcohol. Through careful studies on differently deuterated compounds, clear pictures of selective atomic and molecular hydrogen elimination processes can be constructed for all of the above compounds. These results will help us to understand the dissociation dynamics of the small hydrocarbon molecules.     

Design of surface active soluble peptide molecules at the air/water interface

Surface active molecules collect at interfaces and have the potential to be used for water evaporation reduction. The objective of this work is to design surface active soluble peptides that collect at the air/water interface using molecular simulations. Rotational isomeric state Monte Carlo (RISMC) sampling together with a solvation model that we recently invented, the AAD solvation model [Gu, C.; Lustig, S.; Trout, B. J. Phys. Chem. B 2006, 110 (3), 1476-1484] was applied to calculate the adsorption free energy of the peptide molecule at the air/water interface. The results were validated by both molecular dynamics simulations with an explicit solvent model and surface tension measurements on synthesized peptides. It was demonstrated that this approach is able to give a reasonable prediction of surface activity with an approximately 50% hit rate in terms of designed surface active molecules actually being surface active. The relationship between the chemical composition and the surface morphology is also discussed.     

Synthesis and application of chiral N-heterocyclic carbene-oxazoline ligands: iridium-catalyzed enantioselective hydrogenation

Two libraries of enantiomerically pure imidazolium salts bearing an oxazoline unit were synthesized. Deprotonation of the imidazolium salts and complexation of the resulting oxazoline-carbene ligands to iridium(I) was achieved in one step by mixing the imidazolium salts with NaOtBu and [(eta(4)-cod)IrCl](2) in THF at room temperature. The air-stable complexes were purified by flash chromatography. All complexes were analyzed by two-dimensional (2D) NMR methods and one compound from each family was characterized by X-ray structure analysis. The two libraries of iridium complexes were successfully tested in the asymmetric hydrogenation of unfunctionalized and functionalized olefins. Enantioselectivities of up to 90 % ee were obtained with trans-alpha-methylstilbene. Upon complexation of imidazolium salt 15 p with R(1) = phenyl, C-H bond activation of the phenyl ring gave rise to iridium(III) complex 17, which was fully characterized by NMR spectroscopy and X-ray structure analysis. Complex 17 proved to be catalytically inactive in the hydrogenation.