Explicit uniform bounds on integrals of Bessel functions and trace theorems for Fourier transforms

Explicit and partly sharp estimates are given of integrals over the square of Bessel functions with an integrable weight which can be singular at the origin. They are uniform with respect to the order of the Bessel functions and provide explicit bounds for some smoothing estimates as well as for the L² restrictions of Fourier transforms onto spheres in which are independent of the radius of the sphere. For more special weights these restrictions are shown to be Hölder continuous with a Hölder constant having this independence as well. To illustrate the use of these results a uniform resolvent estimate of the free Dirac operator with mass in dimensions is derived.     

Index transforms with the squares of Kelvin functions

New index transforms, involving squares of Kelvin functions, are investigated. Mapping properties and inversion formulas are established for these transforms in Lebesgue spaces. The results are applied to solve a boundary value problem on the wedge for a fourth order partial differential equation.     

A generalization of the Leibnitz rule of successive differentiation

This note generalizes the well-known Leibnitz rule of successive differentiation for the product of two functions to a similar result for the product of three functions.     

Euler–Goursat-like formula via Laplace–Borel duality

The Goursat formula for the hypergeometric function extends the Euler–Gauss relation to the case of logarithmic singularities.     

A general poisson approximation theorem

We study the rate of convergence in a limit theorem due to Kabanov-Liptser-Shiryayev. We show how the probabilities P(N _t= k) can be computed from the compensator, when it is deterministic.     

Bessel functions of third order and the distribution of cyclic planar motions with three directions

The exact distribution of a cyclic planar motion with three directions is explicitly derived in terms of Bessel functions of order three (suitably combined). The absolutely continuous part of the distribution is proved to satisfy suitable boundary conditions and some of its properties are analyzed. The transformations converting the governing equations of order three is presented and its solutions (used here) derived by applying the Frobenius method.     

Theory of third-harmonic generation using double half-Gaussian hollow beams and self-phase-matching

We first (to best of out knowledge) use the slowly varying envelope approximation (SVEA) to obtain the equation of third harmonic (TH) generated by focusing the double half-Gaussian hollow beams into an atomic medium. The two-dimension intensity distribution is achieved by simulating, and it is similar to that generated by the 0th order Bessel beams. In addition to the similarities between the experimental setups, we demonstrate the accuracy of the equation. Compared with that generated by the Bessel beams, the pattern of the TH intensity generated by the double half-Gaussian hollow beams has salient features.     

Tuning the Reactivity of a Substrate for SNAP-Tag Expands Its Application for Recognition-Driven DNA-Protein Conjugation

Recognition-driven modification has been emerging as a novel approach to modifying biomolecular targets of interest site-specifically and efficiently. To this end, protein modular adaptors (MAs) are the ideal reaction model for recognition-driven modification of DNA as they consist of both a sequence-specific DNA-binding domain (DBD) and a self-ligating protein-tag. Coupling DNA recognition by DBD and the chemoselective reaction of the protein tag could provide a highly efficient sequence-specific reaction. However, combining an MA consisting of a reactive protein-tag and its substrate, for example, SNAP-tag and benzyl guanine (BG), revealed rather nonselective reaction with DNA. Therefore new substrates of SNAP-tag have been designed to realize sequence-selective rapid crosslinking reactions of MAs with SNAP-tag. The reactions of substrates with SNAP-tag were verified by kinetic analyses to enable the sequence-selective crosslinking reaction of MA. The new substrate enables the distinctive orthogonality of SNAP-tag against CLIP-tag to achieve orthogonal DNA-protein crosslinking by six unique MAs.     

The Almost Split Sequences for Trivial Extensions of Hereditary Algebras

Let A be a basic hereditary artin algebra and R = A Q be the trivial extension of A by its minimal injective cogenerator Q. We construct some right （left） almost split morphisms and irreducible morphisms in modR through the corresponding morphisms in modA. Furthermore, we can determine its almost split sequences in modR.