Twin peaks

We study random labelings of graphs conditioned on a small number (typically one or two) peaks, that is, local maxima. We show that the boundaries of level sets of a random labeling of a square with a single peak have dimension 2, in a suitable asymptotic sense. The gradient line of a random labeling of a long ladder graph conditioned on a single peak consists mostly of straight line segments. We show that for some tree‐graphs, if a random labeling is conditioned on exactly two peaks then the peaks can be very close to each other. We also study random labelings of regular trees conditioned on having exactly two peaks. Our results suggest that the top peak is likely to be at the root and the second peak is equally likely, more or less, to be any vertex not adjacent to the root.     

Functional limit theorems for random regular graphs

Consider $d$ uniformly random permutation matrices on $n$ labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree $2d$ on $n$ vertices. We consider limit theorems for various combinatorial and analytical properties of this graph (or the matrix) as $n$ grows to infinity, either when $d$ is kept fixed or grows slowly with $n$ . In a suitable weak convergence framework, we prove that the (finite but growing in length) sequences of the number of short cycles and of cyclically non-backtracking walks converge to distributional limits. We estimate the total variation distance from the limit using Stein’s method. As an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix. A key step in this latter derivation is an extension of the Kahn–Szemerédi argument for estimating the second largest eigenvalue for all values of $d$ and $n$ .     

Photoselective excited state dynamics in ZnO-Au nanocomposites and their implications in photocatalysis and dye-sensitized solar cells

Improving the performance of photoactive solid-state devices begins with systematic studies of the metal-semiconductor nanocomposites (NCs) upon which such devices are based. Here, we report the photo-dependent excitonic mechanism and the charge migration kinetics in a colloidal ZnO-Au NC system. By using a picosecond-resolved F?rster resonance energy transfer (FRET) technique, we have demonstrated that excited ZnO nanoparticles (NPs) resonantly transfer visible optical radiation to the Au NPs, and the quenching of defect-mediated visible emission depends solely on the excitation level of the semiconductor. The role of the gold layer in promoting photolytic charge transfer, the activity of which is dependent upon the degree of excitation, was probed using methylene blue (MB) reduction at the semiconductor interface. Incident photon-to-current efficiency measurements show improved charge injection from a sensitizing dye to a semiconductor electrode in the presence of gold in the visible region. Furthermore, the short-circuit current density and the energy conversion efficiency of the ZnO-Au NP based dye-sensitized solar cell (DSSC) are much higher than those of a DSSC comprised of only ZnO NP. Our results represent a new paradigm for understanding the mechanism of defect-state passivation and photolytic activity of the metal component in metal-semiconductor nanocomposite systems.     

Tailor-made chiral pyranopyrans based on glucose and galactose and studies on self-assembly of some crystals and low molecular weight organogel (LMOG)

InCl₃ catalyzed reactions of 2-C-acetoxymethylglycal derivatives with different phenolic compounds resulted in the formation of sugar based pyranoarenopyrans in good yields and moderate to excellent diastereoselectivity in favor of 10aR- or 12aR- or 14aR-products. One of the synthesized compounds, viz. (2R,3R,12aR)-2,3,5,12a-tetrahydro-2-methoxymethyl-3-methoxypyrano[2,3-b]naphtho[1,2-e]pyran gelated polar solvents like MeOH, EtOH and non-polar solvents like pentane, hexane, heptane, octane, pet. ether, etc. The SEM picture of the corresponding hexane xerogel exhibited a rare type of microtubular gel assembly, whereas the SEM pictures of MeOH and pet. ether xerogels showed different types of three-dimensional network. Study of the MeOH gel by Fluorescence spectroscopy, ¹H NMR and X-ray powder diffraction analysis indicated that the supramolecular assembly in the MeOH gel can be attributed to π-stacking. The crystal packing of (2R,3R,10aR)-3,10a-dihydro-2-methoxymethyl-3-methoxy-7-methyl-2H,5H-pyrano[2,3-b][1]benzopyran, a benzopyran analogue of the above organogelator was stabilized by C-H?O and C-H?π hydrogen bonds forming one-dimensional columns parallel to the [001] direction, whereas the corresponding benzopyran 3-epimer showed only C-H?O hydrogen bonds among the molecules.     

Construction of Quaternary Stereocenters by Nickel‐Catalyzed Heck Cyclization Reactions

A nickel‐catalyzed Heck cyclization for the construction of quaternary stereocenters is reported. This transformation is demonstrated in the synthesis of 3,3‐disubstituted oxindoles, which are prevalent motifs seen in numerous biologically active molecules. The method shows broad scope, proceeds in synthetically useful yields, and provides a rare means to construct stereochemically complex frameworks by nonprecious‐metal catalysis.     

Microwave-assisted synthesis and notable applications of Schiff-base and metal complexes: a comparative study

Research on Chemical Intermediates - Schiff base and metal complexes are useful entities with practical applications in every discipline. Since their discovery (1864), several synthetic procedures...     

Embedding optimal transports in statistical manifolds

We consider Monge-Kantorovich optimal transport problems on ? ^d , d ≥ 1, with a convex cost function given by the cumulant generating function of a probability measure. Examples include theWasserstein-2 transport whose cost function is the square of the Euclidean distance and corresponds to the cumulant generating function of the multivariate standard normal distribution. The optimal coupling is usually described via an extended notion of convex/concave functions and their gradient maps. These extended notions are nonintuitive and do not satisfy useful inequalities such as Jensen’s inequality. Under mild regularity conditions, we show that all such extended gradient maps can be recovered as the usual supergradients of a nonnegative concave function on the space of probability distributions. This embedding provides a universal geometry for all such optimal transports and an unexpected connection with information geometry of exponential families of distributions.     

Synthesis and characterization of two Cu(II) complexes with a new pyrazole-based Schiff base ligand: crystallography,DNA interaction and antimicrobial activity of Ni(II) and Cu(II) complexes

Reaction of Cu(II) nitrate with a new pyrazole-based Schiff base ligand, 5-methyl-3-formylpyrazole-N-(2′-methylphenoxy)methyleneimine (MPzOA), afforded two types of Cu(II) complexes at different reaction temperatures, [Cu(MP_zOA)(NO₃)]₂ (1) and [Cu(3,7,11,15-tetramethylporphyrin)(H₂O)](NO₃)₂ (2), reported together with a Ni(II) complex, [Ni(MPzOA)₂(H₂O)₂]Br₂ (3). The compounds are characterized by single crystal X-ray structure analyses along with several physico-chemical and spectral parameters. Complex 1 is authenticated as a bis(μ-pyrazolato)dicopper(II), while 2 is a porphyrinogen and 3 is a distorted octahedral complex. Structural analyses of the complexes reveal that 1 crystallized in monoclinic P2₁/n space group while 2 and 3 crystallized in monoclinic C2/c space group. DNA-binding studies of the complexes have shown that the complexes interact with CT-DNA. DNA-cleavage studies with plasmid DNA have shown that 1 and 2 induce extensive DNA cleavage in the presence of H₂O₂ as an additive, whereas there is no change in degradation of super-coiled DNA by 3 in the presence of additive. The antimicrobial studies of the complexes against Escherichia coli DH5α bacteria strain indicated that all the complexes were capable of killing E. coli with different LD50 values.     

Tunable Strategy for the Asymmetric Synthesis of Sulfoglycolipids from Mycobacterium tuberculosis To Elucidate the Structure and Immunomodulatory Property Relationships

We developed a versatile asymmetric strategy to synthesize different classes of sulfoglycolipids (SGLs) from Mycobacterium tuberculosis. The strategy features the use of asymmetrically protected trehaloses, which were acquired from the glycosylation of TMS α-glucosyl acceptors with benzylidene-protected thioglucosyl donors. The positions of the protecting groups at the donors and acceptors can be fine-tuned to obtain different protecting-group patterns, which is crucial for regioselective acylation and sulfation. In addition, a chemoenzymatic strategy was established to prepare the polymethylated fatty acid building blocks. The strategy employs inexpensive lipase as a desymmetrization agent in the preparation of the starting substrate and readily available chiral oxazolidinone as a chirality-controlling agent in the construction of the polymethylated fatty acids. A subsequent investigation on the immunomodulatory properties of each class of SGLs showed how the structures of SGLs impact the host innate immunity response.     

Synthesis of a sugar-functionalized iridium complex and its application as a fluorescent lectin sensor

Syntheses of two [Ir(bipy)(Ppy)₂] complexes functionalized with mannose and galactose are reported. The complexes were characterized by NMR and mass spectrometry. Fluorescence studies have shown enhanced interaction of the mannose-functionalized complex with mannose specific lectin concanavalin A whereas the galactose-functionalized complex did not show any interaction. The binding constant (2.9 × 10⁵ mol⁻¹) was determined from the fluorescence spectroscopic data.