On largest volume simplices and sub-determinants

We show that the problem of finding the simplex of largest volume in the convex hull of $n$ points in $\mathbb{Q}^d$ can be approximated with a factor of $O(\log d)^{d/2}$ in polynomial time. This improves upon the previously best known approximation guarantee of $d^{(d-1)/2}$ by Khachiyan. On the other hand, we show that there exists a constant $c>1$ such that this problem cannot be approximated with a factor of $c^d$, unless $P=NP$. % This improves over the $1.09$ inapproximability that was previously known. Our hardness result holds even if $n = O(d)$, in which case there exists a \bar c\,^{d}-approximation algorithm that relies on recent sampling techniques, where $\bar c$ is again a constant. We show that similar results hold for the problem of finding the largest absolute value of a subdeterminant of a $d\times n$ matrix

On the convergence of the affine hull of the Chv\'atal-Gomory closures

Given an integral polyhedron P and a rational polyhedron Q living in the same n-dimensional space and containing the same integer points as P, we investigate how many iterations of the Chv\'atal-Gomory closure operator have to be performed on Q to obtain a polyhedron contained in the affine hull of P. We show that if P contains an integer point in its relative interior, then such a number of iterations can be bounded by a function depending only on n. On the other hand, we prove that if P is not full-dimensional and does not contain any integer point in its relative interior, then no finite bound on the number of iterations exists.Comment: 13 pages, 2 figures - the introduction has been extended and an extra chapter has been adde

Characterization of convection-related parameters by Raman lidar: Selected case studies from the convective and orographically-induced precipitation study

An approach to determine the convective available potential energy (CAPE) and the convective inhibition (CIN) based on the use of data from a Raman lidar system is illustrated in this work. The use of Raman lidar data allows to provide high temporal resolution measurements (5 min) of CAPE and CIN and follow their evolution over extended time periods covering the full cycle of convective activity. Lidar-based measurements of CAPE and CIN are obtained from Raman lidar measurements of the temperature and water vapor mixing ratio profiles and the surface measurements of temperature, pressure and dew point temperature provided by a surface weather station. The approach is applied to the data collected by the Raman lidar system BASIL in the frame of COPS. Attention was focused on 15 July and 25-26 July 2007. Lidar-based measurements are in good agreement with simultaneous measurements from radiosondes and with estimates from different mesoscale models

Characterization of the planetary boundary layer height and structure by Raman lidar: comparison of different approaches

Abstract. The planetary boundary layer (PBL) includes the portion of the atmosphere which is directly influenced by the presence of the earth's surface. Aerosol particles trapped within the PBL can be used as tracers to study the boundary-layer vertical structure and time variability. As a result of this, elastic backscatter signals collected by lidar systems can be used to determine the height and the internal structure of the PBL. The present analysis considers three different methods to estimate the PBL height. The first method is based on the determination of the first-order derivative of the logarithm of the range-corrected elastic lidar signals. Estimates of the PBL height for specific case studies obtained through this approach are compared with simultaneous estimates from the potential temperature profiles measured by radiosondes launched simultaneously to lidar operation. Additional estimates of the boundary layer height are based on the determination of the first-order derivative of the range-corrected rotational Raman lidar signals. This latter approach results to be successfully applicable also in the afternoon–evening decaying phase of the PBL, when the effectiveness of the approach based on the elastic lidar signals may be compromised or altered by the presence of the residual layer. Results from these different approaches are compared and discussed in the paper, with a specific focus on selected case studies collected by the University of Basilicata Raman lidar system BASIL during the Convective and Orographically-induced Precipitation Study (COPS)

Towards Lower Bounds on the Depth of ReLU Neural Networks

We contribute to a better understanding of the class of functions that is represented by a neural network with ReLU activations and a given architecture. Using techniques from mixed-integer optimization, polyhedral theory, and tropical geometry, we provide a mathematical counterbalance to the universal approximation theorems which suggest that a single hidden layer is sufficient for learning tasks. In particular, we investigate whether the class of exactly representable functions strictly increases by adding more layers (with no restrictions on size). This problem has potential impact on algorithmic and statistical aspects because of the insight it provides into the class of functions represented by neural hypothesis classes. However, to the best of our knowledge, this question has not been investigated in the neural network literature. We also present upper bounds on the sizes of neural networks required to represent functions in these neural hypothesis classes.Comment: Camera-ready version for NeurIPS 2021 conferenc

Transabdominalepelviceperineal (TAPP) anterolateral thigh flap: A new reconstructive technique for complex defects following extended abdominoperineal resection

Background: Abdominoperineal resection (APR) following radiotherapy is associated with a high rate of perineal wound complications. The anterolateral thigh (ALT) flap, combined with the vastus lateralis (VL) muscle, can cover complex perineal and pelvic anteroposterior defects. This is used for the first time transabdominally through the pelvis and the perineum (TAPP) in the infero-posterior directions; this technique has been described and illustrated in this study. Methods: Among over 90 patients who underwent perineal reconstruction between May 2004 and June 2011, six patients presented high-grade tumours invading perineum, pelvis and sacrum, thereby resulting in a continuous anteroposterior defect. ALT þ VL TAPP reconstructions were performed after extended APR and, subsequently, sacrectomy. Patients were examined retrospectively to determine demographics, operative time, complications (general and flap-related), time to complete healing and length of hospital stay. Long-term flap coverage, flap volume stability and functional and aesthetic outcomes were assessed. Results: Mean operating time of the reconstruction was 290 min. No deaths occurred. One patient presented partial flap necrosis. Another patient presented a novel wound dehiscence after flap healing, due to secondary skin dissemination of the primary tumour. Following volumetric flap analysis on serial post-operative CT scans, no significant flap atrophy was observed. All flaps fully covered the defects. No late complications such as fistulas or perineal hernias occurred. Donor-site recovery was uneventful with no functional deficits. Conclusions: The use of the ALT þ VL flap transabdominally is an innovative method to reconstruct exceptionally complex perineal and pelvic defects extending up to the lower back. This flap guarantees superior bulk, obliterating all pelvic dead space, with the fascia lata (FL) supporting the pelvic floor