Boundaries of Disk-like Self-affine Tiles

Let $T:= T(A, {\mathcal D})$ be a disk-like self-affine tile generated by an integral expanding matrix $A$ and a consecutive collinear digit set ${\mathcal D}$, and let $f(x)=x^{2}+px+q$ be the characteristic polynomial of $A$. In the paper, we identify the boundary $\partial T$ with a sofic system by constructing a neighbor graph and derive equivalent conditions for the pair $(A,{\mathcal D})$ to be a number system. Moreover, by using the graph-directed construction and a device of pseudo-norm $\omega$, we find the generalized Hausdorff dimension $\dim_H^{\omega} (\partial T)=2\log \rho(M)/\log |q|$ where $\rho(M)$ is the spectral radius of certain contact matrix $M$. Especially, when $A$ is a similarity, we obtain the standard Hausdorff dimension $\dim_H (\partial T)=2\log \rho/\log |q|$ where $\rho$ is the largest positive zero of the cubic polynomial $x^{3}-(|p|-1)x^{2}-(|q|-|p|)x-|q|$, which is simpler than the known result.Comment: 26 pages, 11 figure

Suction Drain Volume following Axillary Lymph Node Dissection for Melanoma—When to Remove Drains? A Retrospective Cohort Study

Postoperative complications such as seroma formation and wound-site infection occur following completion axillary lymph node dissection (ALND) for melanoma. We analyzed the impact of time-to-drain removal and drainage volume on seroma formation after ALND. We retrospectively analyzed data from 118 patients after completion ALND for melanoma. Primary endpoints were daily amount of drainage volume, seroma formation and time-to-drain removal. Secondary endpoints included patient-related, disease-specific and perioperative parameters as well as the number of histologically analyzed lymph nodes and surgical complications graded by the Clavien–Dindo classification (CDCL). Statistical analyses were performed using logistic regression models. Drain removal around the 8th postoperative day was statistically associated with a lower risk for the occurrence of seroma formation (p < 0.001). Patients with an increased drainage volume during the early postoperative days were more prone to develop seroma after drain removal. With 49% (CDCL I and II), most complications were managed conservatively, while only 5.9% (CDCL III) required revision surgery (CDCL overall: 55.9%). ALND is a safe procedure with a low rate of severe CDCL III type of complications. To decrease seroma evacuation, our results imply that drains should be removed around the 8th postoperative day to reduce the risk of infection, readmission or prolonged hospitalization