Construction of vector-valued modular integrals and vector-valued mock modular forms

It is known that, given a vector-valued modular form of negative weight, its Fourier coefficients can be calculated based on the principal part of the form. In this paper we start with an arbitrary principal part and complete the Fourier expansion using the calculation. We show that the so-obtained function is a vector-valued modular integral of negative weight on the full modular group. Next, we construct the supplementary function associated to a vector-valued modular cusp form of positive weight. The constructions are inspired by the construction of Eichler integrals by Knopp. We conclude with a comparison of these forms and their integrals to vector-valued weak harmonic Maass forms.     

New figuring model based on surface slope profile for grazing‐incidence reflective optics

Surface slope profile is widely used in the metrology of grazing‐incidence reflective optics instead of surface height profile. Nevertheless, the theoretical and experimental model currently used in deterministic optical figuring processes is based on surface height, not on surface slope. This means that the raw slope profile data from metrology need to be converted to height profile to perform the current height‐based figuring processes. The inevitable measurement noise in the raw slope data will introduce significant cumulative error in the resultant height profiles. As a consequence, this conversion will degrade the determinism of the figuring processes, and will have an impact on the ultimate surface figuring results. To overcome this problem, an innovative figuring model is proposed, which directly uses the raw slope profile data instead of the usual height data as input for the deterministic process. In this paper, first the influence of the measurement noise on the resultant height profile is analyzed, and then a new model is presented; finally a demonstration experiment is carried out using a one‐dimensional ion beam figuring process to demonstrate the validity of our approach.     

Second generation of primaquine ureas and bis-ureas as potential antimycobacterial agents

Molecular Diversity - Here, we describe design and synthesis of twelve novel compounds bearing primaquine motif and hydroxy- or halogenamine linked by an urea or bis-urea spacer. Preparation of...     

AN ITERATIVE BEM FOR THE INVERSE PROBLEM OF DETECTING CORROSION IN A PIPE

In this paper, we consider an inverse problem of determining the corrosion occurring in an inaccessible interior part of a pipe from the measurements on the outer boundary. The problem is modelled by Laplace's equation with an unknowm term γ in the boundary condition on the inner boundary. Based on the Maz'ya iterative algorithm, a regularized BEM method is proposed for obtaining approximate solutions for this inverse problem. The numerical results show that our method can be easily realized and is quite effective.     

B(C6F5)3: an efficient catalyst for reductive alkylation of alkoxy benzenes and for synthesis of triarylmethanes using aldehydes

Tris(pentafluorophenyl)borane [B(C₆F₅)₃] has been used as an efficient catalyst for reductive alkylation of alkoxy benzenes using aldehydes as an alkylating agent in the presence of polymethylhydrosiloxane (PMHS). Various alkylated trimethoxybenzene derivatives have been prepared in good to high yields. In addition, B(C₆F₅)₃ was also used as a catalyst for the reaction of electron-rich arenes with aldehydes to obtain triarylmethanes. The use of reductive alkylation protocol for the synthesis of an isochroman and tetrahydroisoquinoline derivatives has also been demonstrated.     

Enantioselective synthesis of (−)-lasubine II

A highly enantioselective synthesis of lythraceae alkaloid lasubine II has been achieved using organo-catalyzed Mannich reaction, Maruoka allylation, and aza-Michael addition as the key steps.     

Logarithmic stability in the dynamical inverse problem for the Schrödinger equation by arbitrary boundary observation

In this paper, we investigate the inverse problem of determining the potential of the dynamical Schrödinger equation in a bounded domain from the data of the solution in a subboundary over a time interval. Assuming that in a neighborhood of the boundary of the spatial domain, the potential is known and without any assumption on the dynamics (i.e. without the geometric optics condition for the observability), we prove a logarithmic stability estimate for the inverse problem with a single measurement on an arbitrarily given subboundary.     

A series solution of the fin problem with a temperature-dependent thermal conductivity

In this work, we apply the homotopy analysis method to solve fin problems with temperature-dependent thermal conductivity. The results are compared with the solutions obtained by the Adomian decomposition method (ADM), homotopy perturbation method (HPM) and with the Taylor series expansion method as well. The results of the comparisons have shown that the ADM and HPM are just an approximation of the present study and the accuracy of the solution obtained by the present study is much better than the latter algorithms.     

Automatic alignment of a Kirkpatrick-Baez active optic by use of a soft-x-ray Hartmann wavefront sensor

We present what we believe to be the first automatic alignment of a synchrotron beamline by the Hartmann technique. Experiments were performed, in the soft-x-ray range (E=3 keV, lambda=0.414 nm), by using a four-actuator Kirkpatrick-Baez (KB) active optic. A system imaging the KB focal spot and a soft-x-ray Hartmann wavefront sensor were used alternatively to control the KB optic. The beam corrected with the help of the imaging system was used to calibrate the wavefront sensor. With both closed loops, we focused the beam into a 6.8 microm x 9 microm FWHM focal spot.