Prox-regularity of rank constraint sets and implications for algorithms

We present an analysis of sets of matrices with rank less than or equal to a specified number $s$. We provide a simple formula for the normal cone to such sets, and use this to show that these sets are prox-regular at all points with rank exactly equal to $s$. The normal cone formula appears to be new. This allows for easy application of prior results guaranteeing local linear convergence of the fundamental alternating projection algorithm between sets, one of which is a rank constraint set. We apply this to show local linear convergence of another fundamental algorithm, approximate steepest descent. Our results apply not only to linear systems with rank constraints, as has been treated extensively in the literature, but also nonconvex systems with rank constraints.Comment: 12 pages, 24 references. Revised manuscript to appear in the Journal of Mathematical Imaging and Visio

Software design for the Tritium System Test Assembly

The control system for the Tritium Systems Test Assembly (TSTA) must execute complicated algorithms for the control of several sophisticated subsystems. It must implement this control with requirements for easy modifiability, for high availability, and provide stringent protection for personnel and the environment. Software techniques used to deal with these requirements are described, including modularization based on the structure of the physical systems, a two-level hierarchy of concurrency, a dynamically modifiable man-machine interface, and a specification and documentation language based on a computerized form of structured flowcharts

Hardy's inequality and curvature

A Hardy inequality of the form \int_{\tilde{\Omega}} |\nabla f({\bf{x}})|^p d {\bf{x}} \ge (\frac{p-1}{p})^p \int_{\tilde{\Omega}} \{1 + a(\delta, \partial \tilde{\Omega})(\x)\}\frac{|f({\bf{x}})|^p}{\delta({\bf{x}})^p} d{\bf{x}}, for all $f \in C_0^{\infty}({\tilde{\Omega}})$, is considered for $p\in (1,\infty)$, where ${\tilde{\Omega}}$ can be either $\Omega$ or $\mathbb{R}^n \setminus \Omega$ with $\Omega$ a domain in $\mathbb{R}^n$, $n \ge 2$, and $\delta({\bf{x}})$ is the distance from ${\bf{x}} \in {\tilde{\Omega}}$ to the boundary $\partial {\tilde{\Omega}}.$ The main emphasis is on determining the dependance of $a(\delta, \partial {\tilde{\Omega}})$ on the geometric properties of $\partial {\tilde{\Omega}}.$ A Hardy inequality is also established for any doubly connected domain $\Omega$ in $\mathbb{R}^2$ in terms of a uniformisation of $\Omega,$ that is, any conformal univalent map of $\Omega$ onto an annulus

Sharp Trace Hardy-Sobolev-Maz'ya Inequalities and the Fractional Laplacian

In this work we establish trace Hardy and trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for domains satisfying suitable geometric assumptions such as mean convexity or convexity. We then use them to produce fractional Hardy-Sobolev-Maz'ya inequalities with best Hardy constants for various fractional Laplacians. In the case where the domain is the half space our results cover the full range of the exponent $s \in (0,1)$ of the fractional Laplacians. We answer in particular an open problem raised by Frank and Seiringer \cite{FS}.Comment: 42 page

Geometric Approach to Pontryagin's Maximum Principle

Since the second half of the 20th century, Pontryagin's Maximum Principle has been widely discussed and used as a method to solve optimal control problems in medicine, robotics, finance, engineering, astronomy. Here, we focus on the proof and on the understanding of this Principle, using as much geometric ideas and geometric tools as possible. This approach provides a better and clearer understanding of the Principle and, in particular, of the role of the abnormal extremals. These extremals are interesting because they do not depend on the cost function, but only on the control system. Moreover, they were discarded as solutions until the nineties, when examples of strict abnormal optimal curves were found. In order to give a detailed exposition of the proof, the paper is mostly self\textendash{}contained, which forces us to consider different areas in mathematics such as algebra, analysis, geometry.Comment: Final version. Minors changes have been made. 56 page

Explanation for the increase in high altitude water on Mars observed by NOMAD during the 2018 global dust storm

The Nadir and Occultation for MArs Discovery (NOMAD) instrument on board ExoMars Trace Gas Orbiter (TGO) measured a large increase in water vapor at altitudes in the range of 40‐100 km during the 2018 global dust storm on Mars. Using a three‐dimensional general circulation model, we examine the mechanism responsible for the enhancement of water vapor in the upper atmosphere. Experiments with different prescribed vertical profiles of dust show that when more dust is present higher in the atmosphere the temperature increases and the amount of water ascending over the tropics is not limited by saturation until reaching heights of 70‐100 km. The warmer temperatures allow more water to ascend to the mesosphere. Photochemical simulations show a strong increase in high‐altitude atomic hydrogen following the high‐altitude water vapor increase by a few days

Adjuvant chemotherapy in upper tract urothelial carcinoma (the POUT trial): a phase 3, open-label, randomised controlled trial

Background: Urothelial carcinomas of the upper urinary tract (UTUCs) are rare, with poorer stage-for-stage prognosis than urothelial carcinomas of the urinary bladder. No international consensus exists on the benefit of adjuvant chemotherapy for patients with UTUCs after nephroureterectomy with curative intent. The POUT (Peri-Operative chemotherapy versus sUrveillance in upper Tract urothelial cancer) trial aimed to assess the efficacy of systemic platinum-based chemotherapy in patients with UTUCs. Methods: We did a phase 3, open-label, randomised controlled trial at 71 hospitals in the UK. We recruited patients with UTUC after nephroureterectomy staged as either pT2–T4 pN0–N3 M0 or pTany N1–3 M0. We randomly allocated participants centrally (1:1) to either surveillance or four 21-day cycles of chemotherapy, using a minimisation algorithm with a random element. Chemotherapy was either cisplatin (70 mg/m²) or carboplatin (area under the curve [AUC]4·5/AUC5, for glomerular filtration rate <50 mL/min only) administered intravenously on day 1 and gemcitabine (1000 mg/m²) administered intravenously on days 1 and 8; chemotherapy was initiated within 90 days of surgery. Follow-up included standard cystoscopic, radiological, and clinical assessments. The primary endpoint was disease-free survival analysed by intention to treat with a Peto-Haybittle stopping rule for (in)efficacy. The trial is registered with ClinicalTrials.gov, NCT01993979. A preplanned interim analysis met the efficacy criterion for early closure after recruitment of 261 participants. Findings: Between June 19, 2012, and Nov 8, 2017, we enrolled 261 participants from 57 of 71 open study sites. 132 patients were assigned chemotherapy and 129 surveillance. One participant allocated chemotherapy withdrew consent for data use after randomisation and was excluded from analyses. Adjuvant chemotherapy significantly improved disease-free survival (hazard ratio 0·45, 95% CI 0·30–0·68; p=0·0001) at a median follow-up of 30·3 months (IQR 18·0–47·5). 3-year event-free estimates were 71% (95% CI 61–78) and 46% (36–56) for chemotherapy and surveillance, respectively. 55 (44%) of 126 participants who started chemotherapy had acute grade 3 or worse treatment-emergent adverse events, which accorded with frequently reported events for the chemotherapy regimen. Five (4%) of 129 patients managed by surveillance had acute grade 3 or worse emergent adverse events. No treatment-related deaths were reported. Interpretation: Gemcitabine–platinum combination chemotherapy initiated within 90 days after nephroureterectomy significantly improved disease-free survival in patients with locally advanced UTUC. Adjuvant platinum-based chemotherapy should be considered a new standard of care after nephroureterectomy for this patient population. Funding: Cancer Research UK

Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem