Positivity and strong ellipticity

We consider second-order partial differential operators $H$ in divergence form on \Ri^d with a positive-semidefinite, symmetric, matrix $C$ of real $L_\infty$-coefficients and establish that $H$ is strongly elliptic if and only if the associated semigroup kernel satisfies local lower bounds, or, if and only if the kernel satisfies Gaussian upper and lower bounds.Comment: 9 page

Second-order operators with degenerate coefficients

We consider properties of second-order operators $H = -\sum^d_{i,j=1} \partial_i \, c_{ij} \, \partial_j$ on \Ri^d with bounded real symmetric measurable coefficients. We assume that $C = (c_{ij}) \geq 0$ almost everywhere, but allow for the possibility that $C$ is singular. We associate with $H$ a canonical self-adjoint viscosity operator $H_0$ and examine properties of the viscosity semigroup $S^{(0)}$ generated by $H_0$. The semigroup extends to a positive contraction semigroup on the $L_p$-spaces with $p \in [1,\infty]$. We establish that it conserves probability, satisfies $L_2$~off-diagonal bounds and that the wave equation associated with $H_0$ has finite speed of propagation. Nevertheless $S^{(0)}$ is not always strictly positive because separation of the system can occur even for subelliptic operators. This demonstrates that subelliptic semigroups are not ergodic in general and their kernels are neither strictly positive nor H\"older continuous. In particular one can construct examples for which both upper and lower Gaussian bounds fail even with coefficients in C^{2-\varepsilon}(\Ri^d) with $\varepsilon > 0$.Comment: 44 page

Pemetrexed plus Platinum as the First-Line Treatment Option for Advanced Non-Small Cell Lung Cancer: A Meta-Analysis of Randomized Controlled Trials

To compare the efficacy and toxicities of pemetrexed plus platinum with other platinum regimens in patients with previously untreated advanced non-small cell lung cancer (NSCLC). Methods: A meta-analysis was performed using trials identified through PubMed, EMBASE, and Cochrane databases. Two investigators independently assessed the quality of the trials and extracted data. The outcomes included overall survival (OS), progression-free survival (PFS), response rate (RR), and different types of toxicity. Hazard ratios (HRs), odds ratios (ORs) and their 95% confidence intervals (CIs) were pooled using RevMan software. Results: Four trials involving 2,518 patients with previously untreated advanced NSCLC met the inclusion criteria. Pemetrexed plus platinum chemotherapy (PPC) improved survival compared with other platinum-based regimens (PBR) in patients with advanced NSCLC (HR = 0.91, 95% CI: 0.83–1.00, p = 0.04), especially in those with non-squamous histology (HR = 0.87, 95% CI: 0.77–0.98, p = 0.02). No statistically significant improvement in either PFS or RR was found in PPC group as compared with PBR group (HR = 1.03, 95% CI: 0.94–1.13, p = 0.57; OR = 1.15, 95% CI: 0.95–1.39, p = 0.15, respectively). Compared with PBR, PPC led to less grade 3–4 neutropenia and leukopenia but more grade 3–4 nausea. However, hematological toxicity analysis revealed significant heterogeneities. Conclusion: Our results suggest that PPC in the first-line setting leads to a significant survival advantage with acceptable toxicities for advanced NSCLC patients, especially those with non-squamous histology, as compared with other PRB. PPC could be considered as the first-line treatment option for advanced NSCLC patients, especially those with non-squamous histology

Coverage and area spectral efficiency in downlink random cellular networks with channel estimation error

We investigate the impact of channel estimation on the performance of downlink random cellular networks. First, we derive a new closed-form expression for the coverage probability under certain practical conditions. We show that the coverage probability is dependent on the user and base station (BS) densities solely through their ratio for arbitrary pilot-training length. Next, we derive the optimal pilot-training length that maximizes the area spectral efficiency (ASE) in several asymptotic regimes, and capture the dependence of this optimal length on the ratio between the user and BS densities. The ASE loss due to training is shown to be less significant in small cell networks with a larger base station density. © 2013 IEEE