Integrability of the continuum wavelet kernel

We state and prove sufficient conditions for the absolute integrability of the inverse of the continuum wavelet kernel.

     

Glycopeptide and Oligosaccharide Libraries

Despite the burgeoning interest in the various biological functions and consequent therapeutic potential of the vast number of oligosaccharides found in nature on glycoproteins and cell surfaces, the development of combinatorial carbohydrate chemistry has not progressed as rapidly as expected. The reason for this imbalance is rooted in the difficulty of oligosaccharide assembly and analysis that renders synthesis a rather cumbersome endeavor. Parallel approaches that generate series of analogous compounds rather than real libraries have therefore typically been used. Since generally low affinity is obtained for interactions between carbohydrate receptors and modified oligosaccharides designed as mimetics of natural carbohydrate ligands, glycopeptides have been explored as alternative mimics. Glycopeptides have been proven in many cases to be superior ligands with higher affinity for a receptor than the natural carbohydrate ligand. High-affinity glycopeptide ligands have been found for several types of receptors including the E-, P-, and L-selectins, toxins, glycohydrolases, bacterial adhesins, and the mannose-6-phosphate receptor. Furthermore, the assembly of glycopeptides is considerably more facile than that of oligosaccharides and the process can be adapted to combinatorial synthesis with either glycosylated amino acid building blocks or by direct glycosylation of peptide templates. The application of the split and combine approach using ladder synthesis has allowed the generation of very large numbers of compounds which could be analyzed and screened for binding of receptors on solid phase. This powerful technique can be used generally for the identification and analysis of the complex interaction between the carbohydrates and their receptors.     

Volume of a small Extrinsic Ball in a Submanifold

For a submanifold M^p R, we determine a two-term asymptoticformula for vol (M^p B(x)) for x M^p as 0. The second termis a quadratic curvature invariant of the second fundamentalform of the imbedding. Imbedded spheres are characterized amongcompact hypersurfaces by this term.     

Probabilistic and combinatorial aspects of the card‐cyclic to random insertion shuffle

Consider a permutation σ ∈ S_n as a deck of cards numbered from 1 to n and laid out in a row, where σ_j denotes the number of the card that is in the j‐th position from the left. We study some probabilistic and combinatorial aspects of the shuffle on S_n defined by removing and then randomly reinserting each of the n cards once, with the removal and reinsertion being performed according to the original left to right order of the cards. The novelty here in this nonstandard shuffle is that every card is removed and reinserted exactly once. The bias that remains turns out to be quite strong and possesses some surprising features. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 362–390, 2015     

Finite time blow-up for the inhomogeneous equation

We consider the inhomogeneous equation

where , and , and give criteria on , and which determine whether for all and all the solution blows up in finite time or whether for and sufficiently small, the solution exists for all time.

     

Comparison Theorems for the Spectral Gap of Diffusion Processes and Schrodinger Operators on an Interval

The spectral gaps and thus the exponential rates of convergenceto equilibrium are compared for ergodic one-dimensional diffusionson an interval. One of the results may be thought of as thediffusion analogue of a recent result for the spectral gap ofone-dimensional Schrödinger operators. The similaritiesand differences between spectral gap results for diffusionsand for Schrödinger operators are also discussed.     

The effect of instanton-anti-instanton interactions on current correlation functions in massless QCD

The leading order effects, away from the Dilute Gas Approximation, of instanton-antiinstanton interactions are studied in massless QCD. Propagation and current correlation functions are computed in the interacting dilute gas. While no mass is generated, power law corrections are achieved for the correlation function entering intoe ⁺ e ^? annihilation.