Spectra of expanding maps on Besov spaces

A typical approach to analysing statistical properties of expanding maps is to show spectral gaps of associated transfer operators in adapted function spaces. The classical function spaces for this purpose are H\"older spaces and Sobolev spaces. Natural generalisations of these spaces are Besov spaces, on which we show a spectral gap of transfer operators.Comment: 19 pages. Updated introduction and main theorem, added references and added Appendix A. Accepted for publication in DCDS-

An Lk1∩LkpL^1_k\cap L^p_k approach for the non-cutoff Boltzmann equation in R3\mathbb{R}^3

In the paper, we develop an $L^1_k\cap L^p_k$ approach to construct global solutions to the Cauchy problem on the non-cutoff Boltzmann equation near equilibrium in $\mathbb{R}^3$. In particular, only smallness of $\|\mathcal{F}_x{f}_0\|_{L^1\cap L^p (\mathbb{R}^3_k;L^2(\mathbb{R}^3_v))}$ with $3/2<p\leq \infty$ is imposed on initial data $f_0(x,v)$, where $\mathcal{F}_x{f}_0(k,v)$ is the Fourier transform in space variable. This provides the first result on the global existence of such low-regularity solutions without relying on Sobolev embedding $H^2(\mathbb{R}^3_x)\subset L^\infty(\mathbb{R}^3_x)$ in case of the whole space. Different from the use of sufficiently smooth Sobolev spaces in those classical results by Gressman-Strain and AMUXY, there is a crucial difference between the torus case and the whole space case for low regularity solutions under consideration. In fact, for the former, it is enough to take the only $L^1_k$ norm corresponding to the Weiner space as studied in Duan-Liu-Sakamoto-Strain. In contrast, for the latter, the extra interplay with the $L^p_k$ norm plays a vital role in controlling the nonlinear collision term due to the degenerate dissipation of the macroscopic component. Indeed, the propagation of $L^p_k$ norm helps gain an almost optimal decay rate $(1+t)^{-\frac{3}{2} (1-\frac{1}{p})_+}$ of the $L^1_k$ norm via the time-weighted energy estimates in the spirit of the idea of Kawashima-Nishibata-Nishikawa and in turn, this is necessarily used for establishing the global existence.Comment: 38 page

Binocular stereo-navigation for three-dimensional thoracoscopic lung resection

BACKGROUND: This study investigated the efficacy of binocular stereo-navigation during three-dimensional (3-D) thoracoscopic sublobar resection (TSLR). METHODS: From July 2001, the authors’ department began to use a virtual 3-D pulmonary model on a personal computer (PC) for preoperative simulation before thoracoscopic lung resection and for intraoperative navigation during operation. From 120 of 1-mm thin-sliced high-resolution computed tomography (HRCT)-scan images of tumor and hilum, homemade software CTTRY allowed sugeons to mark pulmonary arteries, veins, bronchi, and tumor on the HRCT images manually. The location and thickness of pulmonary vessels and bronchi were rendered as diverse size cylinders. With the resulting numerical data, a 3-D image was reconstructed by Metasequoia shareware. Subsequently, the data of reconstructed 3-D images were converted to Autodesk data, which appeared on a stereoscopic-vision display. Surgeons wearing 3-D polarized glasses performed 3-D TSLR. RESULTS: The patients consisted of 5 men and 5 women, ranging in age from 65 to 84 years. The clinical diagnoses were a primary lung cancer in 6 cases and a solitary metastatic lung tumor in 4 cases. Eight single segmentectomies, one bi-segmentectomy, and one bi-subsegmentectomy were performed. Hilar lymphadenectomy with mediastinal lymph node sampling has been performed in 6 primary lung cancers, but four patients with metastatic lung tumors were performed without lymphadenectomy. The operation time and estimated blood loss ranged from 125 to 333 min and from 5 to 187 g, respectively. There were no intraoperative complications and no conversion to open thoracotomy and lobectomy. Postoperative courses of eight patients were uneventful, and another two patients had a prolonged lung air leak. The drainage duration and hospital stay ranged from 2 to 13 days and from 8 to 19 days, respectively. The tumor histology of primary lung cancer showed 5 adenocarcinoma and 1 squamous cell carcinoma. All primary lung cancers were at stage IA. The organs having metastatic pulmonary tumors were kidney, bladder, breast, and rectum. No patients had macroscopically positive surgical margins. CONCLUSIONS: Binocular stereo-navigation was able to identify the bronchovascular structures accurately and suitable to perform TSLR with a sufficient margin for small pulmonary tumors