Combining Strong Convergence, Values Fast Convergence and Vanishing of Gradients for a Proximal Point Algorithm Using Tikhonov Regularization in a Hilbert Space

In a real Hilbert space $\mathcal{H}$. Given any function $f$ convex differentiable whose solution set $\argmin_{\mathcal{H}}\,f$ is nonempty, by considering the Proximal Algorithm x_{k+1}=\text{prox}_{\b_k f}(d x_k), where $0<d<1$ and (\b_k) is nondecreasing function, and by assuming some assumptions on (\b_k), we will show that the value of the objective function in the sequence generated by our algorithm converges in order $\mathcal{O} \left( \frac{1}{ \beta _k} \right)$ to the global minimum of the objective function, and that the generated sequence converges strongly to the minimum norm element of $\argmin_{\mathcal{H}}\,f$, we also obtain a convergence rate of gradient toward zero. Afterward, we extend these results to non-smooth convex functions with extended real values

Can p63 serve as a biomarker for giant cell tumor of bone? A Moroccan experience

<p>Abstract</p> <p>Background</p> <p>Multinucleated giant cell-containing tumors and pseudotumors of bone represent a heterogeneous group of benign and malignant lesions. Differential diagnosis can be challenging, particularly in instances of limited sampling. The purpose of this study was to evaluate the contribution of the P63 in the positive and differential diagnosis of giant cell tumor of bone.</p> <p>Methods</p> <p>This study includes 48 giant cell-containing tumors and pseudotumors of bone. P63 expression was evaluated by immunohistochemistry. Data analysis was performed using Epi-info software and SPSS software package (version 17).</p> <p>Results</p> <p>Immunohistochemical analysis showed a P63 nuclear expression in all giant cell tumors of bone, in 50% of osteoid osteomas, 40% of aneurysmal bone cysts, 37.5% of osteoblastomas, 33.3% of chondromyxoide fibromas, 25% of non ossifiant fibromas and 8.3% of osteosarcomas. Only one case of chondroblastoma was included in this series and expressed p63. No P63 immunoreactivity was detected in any of the cases of central giant cell granulomas or langerhans cells histiocytosis. The sensitivity and negative predictive value (NPV) of P63 immunohistochemistry for the diagnosis of giant cell tumor of bone were 100%. The specificity and positive predictive value (PPV) were 74.42% and 59.26% respectively.</p> <p>Conclusions</p> <p>This study found not only that GCTOB expresses the P63 but it also shows that this protein may serve as a biomarker for the differential diagnosis between two morphologically similar lesions particularly in instances of limited sampling. Indeed, P63 expression seems to differentiate between giant cell tumor of bone and central giant cell granuloma since the latter does not express P63. Other benign and malignant giant cell-containing lesions express P63, decreasing its specificity as a diagnostic marker, but a strong staining was seen, except a case of chondroblastoma, only in giant cell tumor of bone. Clinical and radiological confrontation remains essential for an accurate diagnosis.</p> <p>Virtual slides</p> <p>The virtual slide(s) for this article can be found here: <url>http://www.diagnosticpathology.diagnomx.eu/vs/1838562590777252</url>.</p