Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems

We consider projection algorithms for solving (nonconvex) feasibility problems in Euclidean spaces. Of special interest are the Method of Alternating Projections (MAP) and the Douglas-Rachford or Averaged Alternating Reflection Algorithm (AAR). In the case of convex feasibility, firm nonexpansiveness of projection mappings is a global property that yields global convergence of MAP and for consistent problems AAR. Based on (\epsilon, \delta)-regularity of sets developed by Bauschke, Luke, Phan and Wang in 2012, a relaxed local version of firm nonexpansiveness with respect to the intersection is introduced for consistent feasibility problems. Together with a coercivity condition that relates to the regularity of the intersection, this yields local linear convergence of MAP for a wide class of nonconvex problems,Comment: 22 pages, no figures, 30 reference

Alternating Projections and Douglas-Rachford for Sparse Affine Feasibility

The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely-behaved nonconvex relaxations. In this work we consider elementary methods based on projections for solving a sparse feasibility problem without employing convex heuristics. In a recent paper Bauschke, Luke, Phan and Wang (2014) showed that, locally, the fundamental method of alternating projections must converge linearly to a solution to the sparse feasibility problem with an affine constraint. In this paper we apply different analytical tools that allow us to show global linear convergence of alternating projections under familiar constraint qualifications. These analytical tools can also be applied to other algorithms. This is demonstrated with the prominent Douglas-Rachford algorithm where we establish local linear convergence of this method applied to the sparse affine feasibility problem.Comment: 29 pages, 2 figures, 37 references. Much expanded version from last submission. Title changed to reflect new development

The Probabilities Associated with the Game Jai-alai

This paper studies the probabilities within the game Jai-alai. Before explaining the probabilities, one first needs to understand the game itself. Jai-alai is a racquetball-like game where eight ordered players compete against one another in the following manner: Player 1 plays Player 2. The winner of this game plays Player 3, while the loser goes to the end of the line behind Player 8. In the first seven games, 1 versus 2, versus 3, ? versus 8, the winner is awarded one point, and in every game after that two points. A player wins the match when he is the first to acquire seven or more points. The question involving probabilities is the following: Suppose all players are of equal ability. Is there any advantage to being in the front of the line (like players 1 and 2) versus being in the end of the line? Another way of restating the problem is calculate the probabilities each player has of winning the match given that they all are of equal ability. Two approaches are used towards solving this problem. The first method involves running simulations on a computer and tabulating these results. Then using statistics a range can be set in which the probabilities are bounded. The second approach involves modeling the game by a Markov Chain. The Markov Chain can be written in matrix form, which in turn can be manipulated so that the probabilities of the players winning the game can be calculated exactly. Also in this second approach, it was necessary to look at smaller player number versions of the Jai-alai game to detect patterns that would be in the regular Eight Player Game. Each of these methods has advantages and drawbacks. The advantage of the simulation is that it can give a fair estimate of the probabilities in a short amount of time, while the advantage of the Markov Chain is that it can give the exact probabilities

The human ARF tumor suppressor senses blastema activity and suppresses epimorphic tissue regeneration.

The control of proliferation and differentiation by tumor suppressor genes suggests that evolution of divergent tumor suppressor repertoires could influence species regenerative capacity. To directly test that premise, we humanized the zebrafish p53 pathway by introducing regulatory and coding sequences of the human tumor suppressor ARF into the zebrafish genome. ARF was dormant during development, in uninjured adult fins, and during wound healing, but was highly expressed in the blastema during epimorphic fin regeneration after amputation. Regenerative, but not developmental signals resulted in binding of zebrafish E2f to the human ARF promoter and activated conserved ARF-dependent Tp53 functions. The context-dependent activation of ARF did not affect growth and development but inhibited regeneration, an unexpected distinct tumor suppressor response to regenerative versus developmental environments. The antagonistic pleiotropic characteristics of ARF as both tumor and regeneration suppressor imply that inducing epimorphic regeneration clinically would require modulation of ARF -p53 axis activation

Carburizing of steel using cracked methane

An attempt was made to carburize steel using cracked methane, industrial terminology for methane burned in air over a catalyst to give carbon monoxide and hydrogen, as the carburizing agent. Initial tests in a laboratory scale furnace using steel in chip form showed that, the feasibility of this method was excellent. These tests also indicated that a good method of control of the carburizing would be by a system that could measure the dew point of the gases introduced into the furnace. The next step was to apply the method of carburizing to small production size lots of bearing components. The first result was satisfactory. It was necessary to try to improve the surface carbon concentration to a more acceptable level since in this test only the lower limit was attained. Several more trials were run under varying conditions and limited success was met. Before the ideal results were obtained the furnace had a major breakdown and could not be repaired. Thus the project had to be discontinued

Local Traffic Safety Analyzer – Improved Road Safety and Optimized Signal Control for Future Urban Intersections

Improving road safety and optimizing the traffic flow – these are major challenges at urban intersections. In particular, strengthening the needs of vulnerable road users (VRUs) such as pedestrians, cyclists and e-scooter drivers is becoming increasingly important, combined with support for automated and connected driving. In the LTSA project, a new system is being developed and implemented exactly for this purpose. The LTSA is an intelligent infrastructure system that records the movements of all road users in the vicinity of an intersection using a combination of several locally installed sensors e.g. video, radar, lidar. AI-based software processes the detected data, interprets the movement patterns of road users and continuously analyzes the current traffic situation (digital twin). Potentially dangerous situations are identified, e.g. right turning vehicles and simultaneously crossing VRUs, and warning messages can be sent to connected road users via vehicle-to-infrastructure communication (V2X). Automated vehicles can thus adapt their driving maneuvers. In addition, the collected data is applied to improve traffic light control depending on the current traffic situation, especially for VRUs. This abstract describes the LTSA system and its implementation in the German city of Potsdam. The current project state is presented and an outlook on next steps is given

Rough evolution equations: analysis and dynamics

In this thesis we consider Stochastic Evolution Equations driven by rough paths. The first aim is to show existence and uniqueness of mild solutions. Therefore, important basics on semigroup theory and on the theory of rough paths are introduced. Afterwards, we develop a solution theory based on heuristic considerations and use this to prove the existence of a global-in-time solution. Then, as leading example for the driving noise we consider a fractional Brownian motion which, can be lifted to a rough path, and analyze simple dynamic properties of the mild solution. We show that the solution generates a random dynamical system and investigate its long-time behavior under additional assumptions on the coefficients, i.e. we show local as well as global stability of the trivial solution.In dieser Arbeit betrachten wir Stochastische Evolutionsgleichungen getrieben durch raue Pfade. Das erste Ziel ist die Existenz und Eindeutigkeit von milden Lösungen zu zeigen. Dazu werden wichtige Grundlagen zur Halbgruppentheorie sowie zur Theorie der rauen Pfade präsentiert. Anschließend entwickeln wir, basierend auf heuristischen Betrachtungen, eine Lösungstheorie und zeigen die Existenz einer eindeutigen globalen Lösung. Als Anwendung für das treibende stochastische Rauschen betrachten wir eine fraktale Brownsche Bewegung, welche zu einem rauen Pfad geliftet wird und analysieren daran einfache dynamische Eigenschaften der milden Lösung. Wir zeigen, dass diese Lösung ein zufälliges dynamisches System generiert und untersuchen ihr Langzeitverhalten unter zusätzlichen Voraussetzungen an die nichtlinearen Koeffizienten, d.h. wir zeigen sowohl lokale als auch globale Stabilität der trivialen Lösung