In this paper, the module algebra structures of $X_{q}(A_{1})$ on quantum polynomial algebra $\C_{q}[x,y,z]$ are investigated, and a complete classification of $X_{q}(A_{1})$-module algebra structures on $\C_{q}[x,y,z]$ is given
We investigate the Berezin-Toeplitz operators that operate on the geometric quantized space corresponding to the $SO(3)$-Witten-Chern-Simons theory. We conjecture that the $SO(3)$-Berezin-Toeplitz operators quantized from the A-polynomial annihilate the corresponding $SO(3)$-knot states.
We prove that any fusion category over $\mathbb{C}$ with exactly one non-invertible simple object is spherical. Furthermore, we classify all such categories that come equipped with a braiding.
We make a first step towards categorification of the dendriform operad, using categories of modules over the Tamari lattices. This means that we describe some functors that correspond to part of the operad structure.
Purpose: The success of an IMRT program depends upon the accuracy of the Multi‐leaf Collimator's (MLC). The usual QA and calibration of MLC's are carried out at a gantry angle of 0°. We have developed a tool, which efficiently and accurately, measures the MLC offsets from desired position at different gantry angles. Materials/Methods: Data was acquired on a Siemens Oncor linear accelerator, which was equipped with an EPID for a period of 6 weeks. The geometric corrections are applied to the images at the imaging workstation. The EPID was exposed to 5 strips of width equal to 2cm. Data was obtained at various gantry angles. The data was compared with the template and later evaluated with software developed in Matlab 7.1. We compared the results from two edge detection algorithms, ‘Sobel’ and ‘Canny’. Data was obtained to verify the accuracy of the auto initialization process. Results: Time require to obtain and analyze MLC strip data at gantry angles of 0°, 90°, 180°, 270° was 12 minutes. It was observed that the overall effect gantry angle on error distribution is not significant, i.e. it's within ± 2mm. Effect of leaf position on error distribution, Pre and post AI, and weekly drifts were evaluated. It was observed that the percentage of leaves with a cumulative error less than 1mm at gantry angle of 0 and 180 was 80 %, while that for gantry angle of 90 and 270 was 72%. Conclusion: The tool helps the physicist and service engineer regarding the frequency and magnitude of AI/Calibration required. Currently the service engineer enters the leaf offsets manually to correct the encoder, which is stored in an EPROM. We plan to enter the offsets directly from software presented here to the EPROM via the controller card.
In the paper we formulate and verify a difference counterpart of the Macdonald-Mehta conjecture and its generalization for the Macdonald polynomials. Namely, we determine the Fourier transforms of the polynomials multiplied by the Gaussian, which is closely connected with the new difference Harish-Chandra theory.
We construct a certain solution to the Witten--Dijkgraf--Verlinde--Verlinde equation related to the small quantum cohomology ring of flag variety, and study the t-deformation of quantum Schubert polynomials corresponding to this solution.