Game-theoretical control with continuous action sets

Motivated by the recent applications of game-theoretical learning techniques to the design of distributed control systems, we study a class of control problems that can be formulated as potential games with continuous action sets, and we propose an actor-critic reinforcement learning algorithm that provably converges to equilibrium in this class of problems. The method employed is to analyse the learning process under study through a mean-field dynamical system that evolves in an infinite-dimensional function space (the space of probability distributions over the players' continuous controls). To do so, we extend the theory of finite-dimensional two-timescale stochastic approximation to an infinite-dimensional, Banach space setting, and we prove that the continuous dynamics of the process converge to equilibrium in the case of potential games. These results combine to give a provably-convergent learning algorithm in which players do not need to keep track of the controls selected by the other agents.Comment: 19 page

Dark-bright gap solitons in coupled-mode one-dimensional saturable waveguide arrays

In the present work, we consider the dynamics of dark solitons as one mode of a defocusing photorefractive lattice coupled with bright solitons as a second mode of the lattice. Our investigation is motivated by an experiment which illustrates that such coupled states can exist with both components in the first gap of the linear band spectrum. This finding is further extended by the examination of different possibilities from a theoretical perspective, such as symbiotic ones where the bright component is supported by states of the dark component in the first or second gap, or non-symbiotic ones where the bright soliton is also a first-gap state coupled to a first or second gap state of the dark component. While the obtained states are generally unstable, these instabilities typically bear fairly small growth rates which enable their observation for experimentally relevant propagation distances

A quantitative framework for exploring exit strategies from the COVID-19 lockdown

Following the highly restrictive measures adopted by many countries for combating the current pandemic, the number of individuals infected by SARS-CoV-2 and the associated number of deaths steadily decreased. This fact, together with the impossibility of maintaining the lockdown indefinitely, raises the crucial question of whether it is possible to design an exit strategy based on quantitative analysis. Guided by rigorous mathematical results, we show that this is indeed possible: we present a robust numerical algorithm which can compute the cumulative number of deaths that will occur as a result of increasing the number of contacts by a given multiple, using as input only the most reliable of all data available during the lockdown, namely the cumulative number of deaths

Quick or cheap? Breaking points in dynamic markets

We examine two-sided markets where players arrive stochastically over time. The cost of matching a client and provider is heterogeneous, and the distribution of costs – but not their realization – is known. In this way, a social planner is faced with two contending objectives:(a) to reduce the players’ waiting time before getting matched; and (b) to reduce matching costs. In this paper, we aim to understand when and how these objectives are incompatible. We identify two regimes dependent on the ‘speed of improvement’ of the cost of matching with respect to market size. One regime results in a quick or cheap dilemma without ‘free lunch’: there exists no clearing schedule that is simultaneously optimal along both objectives. In that regime, we identify a unique breaking point signifying a stark reduction in matching cost contrasted by an increase in waiting time. The other regime features a window of opportunity in which free lunch can be achieved. Which scheduling policy is optimal depends on the heterogeneity of match costs. Under limited heterogeneity, e.g., when there is a finite number of possible match costs, greedy scheduling is approximately optimal, in line with the related literature. However, with more heterogeneity greedy scheduling is never optimal. Finally, we analyze a particular model where match costs are exponentially distributed and show that it is at the boundary of the no-free-lunch regime We then characterize the optimal clearing schedule for varying social planner desiderata

Orientational order and glassy states in networks of semiflexible polymers

Motivated by the structure of networks of cross-linked cytoskeletal biopolymers, we study the orientationally ordered phases in two-dimensional networks of randomly cross-linked semiflexible polymers. We consider permanent cross-links which prescribe a finite angle and treat them as quenched disorder in a semi-microscopic replica field theory. Starting from a fluid of un-cross-linked polymers and small polymer clusters (sol) and increasing the cross-link density, a continuous gelation transition occurs. In the resulting gel, the semiflexible chains either display long range orientational order or are frozen in random directions depending on the value of the crossing angle, the crosslink concentration and the stiffness of the polymers. A crossing angle $\theta\sim 2\pi/M$ leads to long range $M$-fold orientational order, e.g., "hexatic" or "tetratic" for $\theta=60^{\circ}$ or $90^{\circ}$, respectively. The transition is discontinuous and the critical cross-link density depends on the bending stiffness of the polymers and the cross-link geometry: the higher the stiffness and the lower $M$, the lower the critical number of cross-links. In between the sol and the long range ordered state, we always observe a gel which is a statistically isotropic amorphous solid (SIAS) with random positional and random orientational localization of the participating polymers.Comment: 20 pages, added references, minor changes, final version as published in PR

Radiationless energy exchange in three-soliton collisions

We revisit the problem of the three-soliton collisions in the weakly perturbed sine-Gordon equation and develop an effective three-particle model allowing to explain many interesting features observed in numerical simulations of the soliton collisions. In particular, we explain why collisions between two kinks and one antikink are observed to be practically elastic or strongly inelastic depending on relative initial positions of the kinks. The fact that the three-soliton collisions become more elastic with an increase in the collision velocity also becomes clear in the framework of the three-particle model. The three-particle model does not involve internal modes of the kinks, but it gives a qualitative description to all the effects observed in the three-soliton collisions, including the fractal scattering and the existence of short-lived three-soliton bound states. The radiationless energy exchange between the colliding solitons in weakly perturbed integrable systems takes place in the vicinity of the separatrix multi-soliton solutions of the corresponding integrable equations, where even small perturbations can result in a considerable change in the collision outcome. This conclusion is illustrated through the use of the reduced three-particle model.Comment: 11 pages, 14 figures, submitted for publicatio

Diagnosis of Single- or Multiple-Canal Benign Paroxysmal Positional Vertigo according to the Type of Nystagmus

Benign paroxysmal positional vertigo (BPPV) is a common peripheral vestibular disorder encountered in primary care and specialist otolaryngology and neurology clinics. It is associated with a characteristic paroxysmal positional nystagmus, which can be elicited with specific diagnostic positional maneuvers, such as the Dix-Hallpike test and the supine roll test. Current clinical research focused on diagnosing and treating various types of BPPV, according to the semicircular canal involved and according to the implicated pathogenetic mechanism. Cases of multiple-canal BPPV have been specifically investigated because until recently these were resistant to treatment with standard canalith repositioning procedures. Probably, the most significant factor in diagnosis of the type of BPPV is observation of the provoked nystagmus, during the diagnostic positional maneuvers. We describe in detail the various types of nystagmus, according to the canals involved, which are the keypoint to accurate diagnosis

Cavernous Hemangioma of the Rib: A Rare Diagnosis

Hemangioma of the rib is an uncommon benign vascular tumour. A case of rib hemangioma in a 29-year-old woman is presented. Chest roentgenogram and computed tomography revealed a mass along the inner surface of the 7th left rib with bone destruction. She underwent resection of the 7th rib. The pathologic diagnosis was cavernous hemangioma. Hemangiomas of the rib are rare tumours but should be kept in mind in the differential diagnosis of rib tumours