Linear semigroups with coarsely dense orbits

Let $S$ be a finitely generated abelian semigroup of invertible linear operators on a finite dimensional real or complex vector space $V$. We show that every coarsely dense orbit of $S$ is actually dense in $V$. More generally, if the orbit contains a coarsely dense subset of some open cone $C$ in $V$ then the closure of the orbit contains the closure of $C$. In the complex case the orbit is then actually dense in $V$. For the real case we give precise information about the possible cases for the closure of the orbit.Comment: We added comments and remarks at various places. 14 page

POD for optimal control of the Cahn-Hilliard system using spatially adapted snapshots

The present work considers the optimal control of a convective Cahn-Hilliard system, where the control enters through the velocity in the transport term. We prove the existence of a solution to the considered optimal control problem. For an efficient numerical solution, the expensive high-dimensional PDE systems are replaced by reduced-order models utilizing proper orthogonal decomposition (POD-ROM). The POD modes are computed from snapshots which are solutions of the governing equations which are discretized utilizing adaptive finite elements. The numerical tests show that the use of POD-ROM combined with spatially adapted snapshots leads to large speedup factors compared with a high-fidelity finite element optimization

The discontinuous Galerkin method for fractional degenerate convection-diffusion equations

We propose and study discontinuous Galerkin methods for strongly degenerate convection-diffusion equations perturbed by a fractional diffusion (L\'evy) operator. We prove various stability estimates along with convergence results toward properly defined (entropy) solutions of linear and nonlinear equations. Finally, the qualitative behavior of solutions of such equations are illustrated through numerical experiments

Expansion in SL_d(Z/qZ), q arbitrary

Let S be a fixed finite symmetric subset of SL_d(Z), and assume that it generates a Zariski-dense subgroup G. We show that the Cayley graphs of pi_q(G) with respect to the generating set pi_q(S) form a family of expanders, where pi_q is the projection map Z->Z/qZ

Fractional-order operators: Boundary problems, heat equations

The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential methods. The second half takes up the associated heat equation with homogeneous Dirichlet condition. Here we recall recently shown sharp results on interior regularity and on $L_p$-estimates up to the boundary, as well as recent H\"older estimates. This is supplied with new higher regularity estimates in $L_2$-spaces using a technique of Lions and Magenes, and higher $L_p$-regularity estimates (with arbitrarily high H\"older estimates in the time-parameter) based on a general result of Amann. Moreover, it is shown that an improvement to spatial $C^\infty$-regularity at the boundary is not in general possible.Comment: 29 pages, updated version, to appear in a Springer Proceedings in Mathematics and Statistics: "New Perspectives in Mathematical Analysis - Plenary Lectures, ISAAC 2017, Vaxjo Sweden

Expansion in perfect groups

Let Ga be a subgroup of GL_d(Q) generated by a finite symmetric set S. For an integer q, denote by Ga_q the subgroup of Ga consisting of the elements that project to the unit element mod q. We prove that the Cayley graphs of Ga/Ga_q with respect to the generating set S form a family of expanders when q ranges over square-free integers with large prime divisors if and only if the connected component of the Zariski-closure of Ga is perfect.Comment: 62 pages, no figures, revision based on referee's comments: new ideas are explained in more details in the introduction, typos corrected, results and proofs unchange

Evaluation of outreach services for primary care and mental health; assessing the impact

Objectives: This paper reports an evaluation, carried out for London Health Libraries, of the impact of outreach services to primary care and mental health workers in thirteen different settings. The main aims of the project were to identify the impact being made by the service, and to produce best practice guidelines for outreach services in this kind of ‘difficult’ community setting. Methods: Methods used were: analysis of documents (all 13 services); analysis of any evaluation already performed by or for the service (all 13 services); interviews with outreach librarians (11 services); questionnaire survey of a representative sample of users (8 services, with 66 returned questionnaires, 35% response rate). The services evaluated were very diverse, in terms of setting, structure, functions and activities, and extent and nature of self-evaluation and reporting. The evaluation was therefore largely qualitative, in order to deal with the lack of a consistent ‘template’ for analysis. Emphasis was placed on trying to identify critical incidents , where it could be shown unambiguously that the outreach services made a difference to practice. Study limitations included the difficulty of summarising and comparing very different situations and diverse services, difficulty in identifying critical incidents, and an inability to study ‘non-users’. Findings: Service recipients felt better informed, more up- to-date, more aware of resources, more confident and supported in their work, and saved time. Services contributed to a richer information environment. Direct impacts, demonstrably improved patient care, cost savings etc., were more difficult to establish

Training medical specialists to communicate better with patients with medically unexplained physical symptoms (MUPS). A randomized, controlled trial

Background Patients with medically unexplained physical symptoms (MUPS) are prevalent 25-50% in general and specialist care. Medical specialists and residents often find patients without underlying pathology difficult to deal with, whereas patients sometimes don't feel understood. We developed an evidence-based communication training, aimed to improve specialists' interviewing, information-giving and planning skills in MUPS consultations, and tested its effectiveness. Methods The intervention group in this multi-center randomized controlled trial received a 14-hour training program to which experiential learning and feedback were essential. Using techniques from Cognitive Behavioral Therapy, they were stimulated to seek interrelating factors (symptoms, cognitions, emotions, behavior, and social environment) that reinforced a patient's symptoms. They were taught to

Boundedness of Pseudodifferential Operators on Banach Function Spaces

We show that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\mathbb{R}^n)$ and on its associate space $X'(\mathbb{R}^n)$, then a pseudodifferential operator $\operatorname{Op}(a)$ is bounded on $X(\mathbb{R}^n)$ whenever the symbol $a$ belongs to the H\"ormander class $S_{\rho,\delta}^{n(\rho-1)}$ with $0<\rho\le 1$, $0\le\delta<1$ or to the the Miyachi class $S_{\rho,\delta}^{n(\rho-1)}(\varkappa,n)$ with $0\le\delta\le\rho\le 1$, $0\le\delta0$. This result is applied to the case of variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$.Comment: To appear in a special volume of Operator Theory: Advances and Applications dedicated to Ant\'onio Ferreira dos Santo

The fundamental left-right asymmetry in the Germanic verb cluster

Cinque (2005, 2009, 2014a) observes that there is an asymmetry in the possible ordering of dependents of a lexical head before versus after the head. A reflection on some of the concepts needed to develop Cinque’s ideas into a theory of neutral word order reveals that dependents need to be treated separately by class. The resulting system is applied to the problem of word order in the Germanic verb cluster. It is shown that there is an extremely close match between theoretically derived expectations for clusters made up of auxiliaries, modals, causative ‘let’, a main verb, and verbal particles. The facts point to the action of Cinque’s fundamental left-right asymmetry in language in the realm of the verb cluster. At the same time, not all verb clusters fall under Cinque’s generalization, which, therefore, argues against treating all cases of restructuring uniformly