In 1836 Sturm published two papers in Liouville’s J. Math. Pures et Appl., giving his famous comparison and oscillation theorems. In its simplest form, Sturm’s comparison result is Theorem Let q_1(x) ≤ q_2(x) be not identically equal on [a, b]. Let u be a nontrivial solution to u” + q_1(x)u = 0 on [a, b], and let v be a nontrivial solution to v”+ q_2(x)v = 0 on [a, b]. Suppose that u(a)=u(b) =0. Then v has at least one zero in (a, b). One may ask if there is a discrete version of this. There is, where the second derivative operator is replaced by the usual finite difference approximation, as has been observed by many people. Furthermore, it continues to hold for more general tri-diagonal operators. The second derivative operator may also be approximated by a discretization which includes next nearest neighbor interactions. Does the above theorem hold for such penta-diagonal operators? Not in general, as we show by example. In another direction, being the main point of this paper, one can replace the second derivative operator by an integral operator, bounded or unbounded, resulting in a nonlocal diffusion-like operator. Can one prove a similar comparison theorem? Here we discuss the case where u” is replaced by Lu:= ∫ J(x-y)[u(y)-u(x)]dy, where J≥ 0 is continuous, even, with compact support and J(0)>0 . After this, we discuss the case when the nonlocal diffusion operator is replaced by a scaled version L_ϵu:= ∫ _Ω1/ϵ^n+2J(x-y/ϵ)[u(y)-u(x)]dy, for ϵ>0 and small, and where Ω⊂ℝ^n is smoothly bounded, and J is radially symmetric. This operator converges, in some sense, to a multiple of the Laplacian, as ϵ→ 0 , and so the question arises as to whether or not a Sturm-like comparison theorem holds with this nonlocal diffusion operator.

Science is too important to be left solely to scientists, and so the public need to be involved in the design, funding, delivery and implementation of health research, and in discussions about the ethics of research. Since the 1960s, the United Kingdom and many other countries have included scientists from outside health care in various roles in health care research, as well as nonscientists, ordinary citizens, patients and carers. In the last 20 years, these roles have increased in number and range, but significant challenges remain in ensuring that research is always conducted in an ethical fashion. Errors arise when it is assumed that research is ethical because it has passed a single test rather than being subject to constant vigilance; when academic training on its own is regarded as sufficient to guarantee ethical conduct; when pontification about sophisticated dilemmas ignores fundamental matters of equity and helpfulness and when there is an absence of curiosity about the value positions of others (Boaz et al., 2016). We argue in this chapter that in every setting, citizens have the potential to contribute to ethical debates, whether they assist in establishing priorities for research funding, serve as research funding co-applicants, take the lay member places on Research Ethics Committees and Steering Committees, collect and analyze data or co-author academic papers.

Abstract Background Recovery Colleges are a relatively recent initiative within mental health services. The first opened in 2009 in London and since then numbers have grown. They are based on principles of personal recovery in mental health, co-production between people with lived experience of mental health problems and professionals, and adult learning. Student eligibility criteria vary, but all serve people who use mental health services, with empirical evidence of benefit. Previously we developed a Recovery College fidelity measure and a preliminary change model identifying the mechanisms of action and outcomes for this group, which we refer to as service user students. The Recovery Colleges Characterisation and Testing (RECOLLECT) study is a five-year (2020–2025) programme of research in England. The aim of RECOLLECT is to determine Recovery Colleges’ effectiveness and cost-effectiveness, and identify organisational influences on fidelity and improvements in mental health outcomes. Methods RECOLLECT comprises i) a national survey of Recovery Colleges, ii) a prospective cohort study to establish the relationship between fidelity, mechanisms of action and psychosocial outcomes, iii) a prospective cohort study to investigate effectiveness and cost-effectiveness, iv) a retrospective cohort study to determine the relationship between Recovery College use and outcomes and mental health service use, and v) organisational case studies to establish the contextual and organisational factors influencing fidelity and outcomes. The programme has been developed with input from individuals who have lived experience of mental health problems. A Lived Experience Advisory Panel will provide input into all stages of the research. Discussion RECOLLECT will provide the first rigorous evidence on the effectiveness and cost effectiveness of Recovery Colleges in England, to inform their prioritising, commissioning, and running. The validated RECOLLECT multilevel change model will confirm the active components of Recovery Colleges. The fidelity measure and evidence about the fidelity-outcome relationship will provide an empirically-based approach to develop Recovery Colleges, to maximise benefits for students. Findings will be disseminated through the study website (researchintorecovery.com/recollect) and via national and international Recovery College networks to maximise impact, and will shape policy on how Recovery Colleges can help those with mental health problems lead empowered, meaningful and fulfilling lives.

We analyse a quasi-one-dimensional Poisson–Nernst–Planck system for ionic flow through a membrane channel with nonzero but small permanent charge. The system includes three ion species, two cations with the same valences and one anion. The cross-sectional area of the channel is included in the system, which provides certain information of the geometry of the three-dimensional channel. This is critical for our analysis. We treat the model as a boundary value problem of a singularly perturbed system of differential equations. Under the framework of geometric singular perturbation theory, together with specific structures of the model, the existence and local uniqueness of solutions to the boundary value problem for small permanent charges is established. Furthermore, treating the permanent charge as a small parameter, via regular perturbation analysis, we are able to derive an approximation of the I – V (current–voltage) relations explicitly, from which the permanent charge and channel geometry effects on ionic flows are analysed in detail. Critical potentials are identified and their roles in characterising the ionic flow properties of interest are studied. Some critical potentials can be experimentally estimated, and this provides an efficient way to adjust the boundary pot to observe distinct dynamics of ionic flows. Mathematical analysis indicates that a short and narrow filter within which the permanent charge is confined is expected in order to optimise the effect of permanent charges. This is consistent with the typical structure of an ion channel.

A quasi-one-dimensional Poisson–Nernst–Planck system for ionic flow through a membrane channel is studied. Nonzero but small permanent charge, the major structural quantity of an ion channel, is included in the model. The system includes three ion species, two cations with the same valences and one anion, which provides more correlations/interactions between ions compared to the case included only two oppositely charged particles. The cross-section area of the channel is included in the system, which provides certain information of the geometry of the three-dimensional channel. This is crucial for our analysis. Under the framework of geometric singular perturbation theory, more importantly, the specific structure of the model, the existence and local uniqueness of solutions to the system for small permanent charges is established. Furthermore, treating the permanent charge as a small parameter, through regular perturbation analysis, we are able to derive approximations of the individual fluxes explicitly, and this allows us to examine the small permanent charge effects on ionic flows in detail. Of particular interest is the competition between two cations, which is related to the selectivity phenomena of ion channels. Critical potentials are identified and their roles in characterizing ionic flow properties are studied. Some critical potentials can be estimated experimentally, and this provides an efficient way to adjust/control boundary conditions (electric potential and concentrations) to observe distinct qualitative properties of ionic flows. Mathematical analysis further indicates that to optimize the effect of permanent charges, a short and narrow filter, within which the permanent charge is confined, is expected, which is consistent with the typical structure of an ion channel.

Introduction The BRIGhTMIND study aims to determine the clinical effectiveness, cost-effectiveness and mechanism of action of connectivity guided intermittent theta burst stimulation (cgiTBS) versus standard repetitive transcranial magnetic stimulation (rTMS) in adults with moderate to severe treatment resistant depression. Methods and analysis The study is a randomised double-blind controlled trial with 1:1 allocation to either 20 sessions of (1) cgiTBS or (2) neuronavigated rTMS not using connectivity guidance. A total of 368 eligible participants with a diagnosis of current unipolar major depressive disorder that is both treatment resistant (defined as scoring 2 or more on the Massachusetts General Hospital Staging Score) and moderate to severe (scoring >16 on the 17-item Hamilton Depression Rating Scale (HDRS-17)), will be recruited from primary and secondary care settings at four treatment centres in the UK. The primary outcome is depression response at 16 weeks (50% or greater reduction in HDRS-17 score from baseline). Secondary outcomes include assessments of self-rated depression, anxiety, psychosocial functioning, cognition and quality of life at 8, 16 and 26 weeks postrandomisation. Cost-effectiveness, patient acceptability, safety, mechanism of action and predictors of response will also be examined. Ethics and dissemination Ethical approval was granted by East Midlands Leicester Central Research Ethics Committee (ref: 18/EM/0232) on 30 August 2018. The results of the study will be published in relevant peer-reviewed journals, and then through professional and public conferences and media. Further publications will explore patient experience, moderators and mediators of outcome and mechanism of action. Trial registration number ISRCTN19674644

There have been calls for greater inclusion of Indigenous and local knowledge (ILK) in applied ecosystems research and ecological assessments. The Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services (IPBES) Global Assessment (GA) is the first global scale assessment to systematically engage with ILK and issues of concern to Indigenous peoples and local communities (IPLC). In this paper, we review and reflect on how the GA worked with ILK and lessons learned. The GA engaged in critical evaluation and synthesis of existing evidence from multiple sources, using several deliberative steps: having specific authors and questions focus on ILK; integrating inputs from ILK across all chapters; organizing dialogue workshops; issuing calls for contributions to identify other forms and systems of knowledge; and encouraging IPLC to be key stakeholders and contributors. We identify content areas where attention to ILK was particularly important for questions in applied ecology. These include: (a) enriching understandings of nature and its contributions to people, including ecosystem services; (b) assisting in assessing and monitoring ecosystem change; (c) contributing to international targets and scenario development to achieve global goals like the Aichi Biodiversity Targets and the Sustainable Development Goals and (d) generating inclusive and policy-relevant options for people and nature. However, challenges in engaging different knowledge systems were also encountered. Policy implications. The Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services (IPBES) Global Assessment (GA) demonstrated the importance of Indigenous peoples and local communities (IPLC) to global biodiversity conservation and ecosystem management. Initiatives seeking to engage Indigenous and local knowledge (ILK) can learn from the experience of the GA. Successfully bringing ILK into assessment processes and policy arenas requires a deliberate framework and approach from the start that facilitates recognition of different knowledge systems, identifies questions relevant at various scales, mobilizes funding and recognizes time required and engages networks of stakeholders with diverse worldviews. In turn, fostering inclusion of ILK and partnering with IPLC can help future assessments understand how natural and cultural systems co-produce each other, identify trends of change through diverse biocultural indicators and improve sustainable development goals and policies.

We study a quasi-one-dimensional steady-state Poisson-Nernst-Planck type model for ionic flows through a membrane channel with three ion species, two positively charged with the same valence and one negatively charged. Bikerman's local hard-sphere potential is included in the model to account for ion sizes. The problem is treated as a boundary value problem of a singularly perturbed differential system. Under the framework of a geometric singular perturbation theory, together with specific structures of this concrete model, the existence and uniqueness of solutions to the boundary value problem for small ion sizes is established. Furthermore, treating the ion sizes as small parameters, we derive an approximation of individual fluxes, from which one can further study the qualitative properties of ionic flows and extract concrete information directly related to biological measurements. Of particular interest is the competition between two cations due to the nonlinear interplay between finite ion sizes, diffusion coefficients and boundary conditions, which is closely related to selectivity phenomena of open ion channels with given protein structures. Furthermore, we are able to characterize the distinct effects of the nonlinear interplays between these physical parameters. Numerical simulations are performed to identify some critical potentials which play critical roles in examining properties of ionic flows in our analysis.

Objective: To examine the feasibility of conducting a fully powered randomized controlled trial (RCT) of Individual Placement and Support (IPS). IPS is a form of supported employment which aims to put people into open employment quickly and in accordance with their preferences. It is delivered by employment specialists collocated within clinical teams, and provides time unlimited support for the individual and their employer, along with welfare benefits counselling. Method: A feasibility cluster RCT of treatment as usual (TAU) plus IPS versus TAU alone was conducted over 12 months among patients with offending histories in a community forensic setting in the UK. The feasibility criteria were to achieve 50% recruitment rate; 50% completion rate for IPS; 50% completion rate of all outcome measures; and 80% acceptability rating for IPS. The primary efficacy outcome was the proportion of people in open employment at 12 months. The secondary outcomes were other vocational and educational activities; Brief Psychiatric Rating Scale; Rosenberg's Self-esteem Scale; Client Service Receipt Inventory; quality of life using the SF12-v2 and EQ5-D3; Social Functioning Questionnaire; Work Limitation Questionnaire; and reoffending. Results: Participants' mean age was 39.2 years. The majority were male (88.9), White British (72.2), and single (72.2%). Over 72% had no higher qualification beyond secondary education; mean years in education was 10.4. Over one third had schizophrenia, one fifth had depression, and the rest had personality disorder as their primary diagnosis. Participants had a lifetime average of 7.5 convictions for 15.5 offences. The recruitment rate of all referrals was 38.3% (IPS n = 11; TAU n = 7). Completion rate for IPS was 54.5, with 45.5% acceptability rating. Completion rates for outcome measures for the groups at baseline and 12 months ranged from 22.2 to 100%. The proportion of people in open employment at 12 months were 9.1 and 0% for IPS and TAU respectively. Conclusion: It is not feasible to conduct a full RCT of IPS in community forensic settings in the UK owing to recruitment and retention difficulties. Conducting a trial of this kind requires a large pool of patients from multiple sites and longer IPS implementation and recruitment periods than those of this study. Clinical Trial Registration: www.ClinicalTrials.gov, identifier NCT02442193.

Purpose In care homes concerns about abuse have established a culture where all information pertaining to a person must be shared, and little attention is paid to privacy in its broader sense. The purpose of this paper is to take a human rights perspective and consider how information governance may impact on the health, well-being and quality of life of residents. It proposes a proactive approach and presents a template for a privacy impact assessment which services could use to improve their approach to privacy, protecting the human rights of those in their care, contributing to their independence and improving outcomes. Design/methodology/approach A review of historical and current thinking about the value of privacy in human services and wider society leads to a series of challenges to the way in which privacy is upheld in residential care services. Findings Recent preoccupations with data privacy have led to a myopic neglect of broader considerations of privacy. Whilst it continues to be important to protect the confidentiality of personal data and to ensure that residents are protected from abuse, human services that provide 24 hour care in congregated settings must not neglect broader components of privacy. Research limitations/implications - Privacy impact assessments have been widely used to check whether data privacy is being upheld. The broader concept that might be termed "Big Privacy" is introduced within which data privacy is but one section. It is suggested that big privacy is severely compromised in residential care settings, thus denying residents their human right to privacy. The extent of such violation of rights should be investigated. Practical implications - Having set out the potential reach of the human right to privacy, important work needs to be done to find out how privacy might be upheld in the real world of congregate residential care. Some service providers may have solutions to the organisational challenges, have addressed staff training needs and revised risk assessment strategies so that privacy is upheld alongside other rights. Social implications - Nearly half a million people live in congregate residential care settings in England, and deprivation of privacy is argued to be a significant deprivation of human rights. Occasional tragedies and scandals in congregate settings create pressure for increasing the level of surveillance, and the right to privacy is sacrificed. This paper offers a challenge to this process, arguing that competing rights need to be balanced and privacy is an essential component of a decent quality of life. Originality/value Personal growth and development depends to some extent on choice and control over access to privacy. Recent changes in the law regarding data protection have narrowed our thinking about privacy until it is a small concept, largely concerned with data handling. This paper invites consideration of big privacy, and invites congregate residential care settings to consider how a deep and broad definition of privacy could transform these services.

Objective: Recovery colleges are widespread, with little empirical research on how they work and the outcomes they produce. This study aimed to coproduce a change model characterizing mechanisms of action (how they work) and outcomes (their impact) for mental health service users who attend recovery colleges. Methods: A systematized review identified all publications about recovery colleges. Inductive collaborative data analysis of 10 key publications by academic researchers and coresearchers with lived experience informed a theoretical framework for mechanisms of action and student outcomes, which was refined through deductive analysis of 34 further publications. A change model was coproduced and refined through stakeholder interviews (N=33). Results: Four mechanisms of action for recovery colleges were identified: empowering environment (safety, respect, and supporting choices), enabling different relationships (power, peers, and working together), facilitating personal growth (for example, coproduced learning, strengths, and celebrating success), and shifting the balance of power through coproduction and reducing power differentials. Outcomes were change in the student (for example, self-understanding and self-confidence) and changes in the student's life (for example, occupational, social, and service use). A coproduced change model mapping mechanisms of action to outcomes was created. Conclusions: Key features differentiate recovery colleges from traditional services, including an empowering environment, enabling relationships, and growth orientation. Service users who lack confidence, those with whom services struggle to engage, those who will benefit from exposure to peer role models, and those lacking social capital may benefit most. As the first testable characterization of mechanisms and outcomes, the change model allows formal evaluation of recovery colleges.

Background Patient and Public Involvement (PPI) in mental health research is increasing, especially in early (pre-funding) stages. PPI is less consistent in later stages, including in analysing qualitative data. The aims of this study were to develop a methodology for involving PPI co-researchers in collaboratively analysing qualitative mental health research data with academic researchers, to pilot and refine this methodology, and to create a best practice framework for collaborative data analysis (CDA) of qualitative mental health research. Methods In the context of the RECOLLECT Study of Recovery Colleges, a critical literature review of collaborative data analysis studies was conducted, to identify approaches and recommendations for successful CDA. A CDA methodology was developed and then piloted in RECOLLECT, followed by refinement and development of a best practice framework. Results From 10 included publications, four CDA approaches were identified: (1) consultation, (2) development, (3) application and (4) development and application of coding framework. Four characteristics of successful CDA were found: CDA process is co-produced; CDA process is realistic regarding time and resources; demands of the CDA process are manageable for PPI co-researchers; and group expectations and dynamics are effectively managed. A four-meeting CDA process was piloted to co-produce a coding framework based on qualitative data collected in RECOLLECT and to create a mental health service user-defined change model relevant to Recovery Colleges. Formal and informal feedback demonstrated active involvement. The CDA process involved an extra 80 person-days of time (40 from PPI co-researchers, 40 from academic researchers). The process was refined into a best practice framework comprising Preparation, CDA and Application phases. Conclusions This study has developed a typology of approaches to collaborative analysis of qualitative data in mental health research, identified from available evidence the characteristics of successful involvement, and developed, piloted and refined the first best practice framework for collaborative analysis of qualitative data. This framework has the potential to support meaningful PPI in data analysis in the context of qualitative mental health research studies, a previously neglected yet central part of the research cycle.

We derive a spectral collocation approximation to the fractional Laplacian operator based on the Riemann-Liouville fractional derivative operators on a bounded domain Ω = [a, b]. Corresponding matrix representations of (−△) α/2 for α ∈ (0,1) and α ∈ (1,2) are obtained. A space-fractional advection-dispersion equation is then solved to investigate the numerical performance of this method under various choices of parameters. It turns out that the proposed method has high accuracy and is efficient for solving these space-fractional advection-dispersion equations when the forcing term is smooth.

Objective: Recovery Colleges are widespread, with little empirical research on their key components. This study aimed to characterize key components of Recovery Colleges and to develop and evaluate a developmental checklist and a quantitative fidelity measure. Methods: Key components were identified through a systematized literature review, international expert consultation ( n = 77), and semistructured interviews with Recovery College managers across England ( n = 10). A checklist was developed and refined through semistructured interviews with Recovery College students, trainers, and managers ( n = 44) in 3 sites. A fidelity measure was adapted from the checklist and evaluated with Recovery College managers ( n = 39, 52%), clinicians providing psychoeducational courses ( n = 11), and adult education lecturers ( n = 10). Results: Twelve components were identified, comprising 7 nonmodifiable components (Valuing Equality, Learning, Tailored to the Student, Coproduction of the Recovery College, Social Connectedness, Community Focus, and Commitment to Recovery) and 5 modifiable components (Available to All, Location, Distinctiveness of Course Content, Strengths Based, and Progressive). The checklist has service user student, peer trainer, and manager versions. The fidelity measure meets scaling assumptions and demonstrates adequate internal consistency (0.72), test-retest reliability (0.60), content validity, and discriminant validity. Conclusions: Coproduction and an orientation to adult learning should be the highest priority in developing Recovery Colleges. The creation of the first theory-based empirically evaluated developmental checklist and fidelity measure (both downloadable at researchintorecovery.com/recollect ) for Recovery Colleges will help service users understand what Recovery Colleges offer, will inform decision making by clinicians and commissioners about Recovery Colleges, and will enable formal evaluation of their impact on students.

In a general Banach space we consider gradient-like dynamical systems with the property that there is a manifold along which solutions move slowly compared to attraction in the transverse direction. Conditions are given on the energy (or Lyapunov functional) that ensure that solutions starting near the manifold stay near for a long time or even forever. The abstract results are then applied to a finite dimensional case and used to show the super slow motion of interfaces for the vector Allen-Cahn and Cahn-Morral systems.

We study a quasi-one-dimensional steady-state Poisson–Nernst–Planck model with a local hard-sphere potential for ionic flows of two oppositely charged ion species through a membrane channel. Of particular interest are qualitative properties of ionic flows in terms of individual fluxes Jk without assuming electroneutrality boundary conditions. For the first order approximation Jk1 (in diameter of charged particle) of Jk, our result shows that the quantities ∂VJk1 and∂Vλ2Jk1 that play critical roles in characterizing ion size effects on ionic flows can be both negative depending sensitively on boundary concentrations and relative ion valences while they are always positive under electroneutrality conditions. This new observation indicates that, for fixed ion species, opposite finite size effects on ionic flows can occur determined further by boundary concentrations. Numerical simulation is performed to detect some critical potentials closely related to ion size effects. Numerical simulation results are consistent with our analytical predictions.

We present a systematic study of entire symmetric solutions u : ℝ^n →ℝ^m of the vector Allen–Cahn equation Δ u - W_u(u) = 0 for all x ∈ℝ^n, where W:ℝ^m →ℝ is smooth, symmetric, nonnegative with a finite number of zeros, and where W_u= (∂ W / ∂ u_1,…,∂ W / ∂ u_m)^⊤ . We introduce a general notion of equivariance with respect to a homomorphism f:G→Γ ( G, Γ reflection groups) and prove two abstract results, concerning the cases of G finite and G discrete, for the existence of equivariant solutions. Our approach is variational and based on a mapping property of the parabolic vector Allen–Cahn equation and on a pointwise estimate for vector minimizers.

Book Emerging issues in nonlinear elliptic equations: singularities, singular perturbations and non local problems 18.06.2017 24.06.2017 — Będlewo Organizers Oscar Agudelo Pavel Drabek Michał Kowalczyk Angela Pistoia Piotr Rybka Ahmad Sabra List of participants Oscar Agudelo Thomas Bartsch Peter Bates Luca Battaglia Bartosz Bieganowski Piotr Biler Vladimir Bobkov Lucio Boccardo Denis Bonheure Jean-Baptiste Casteras Maya Chhetri Jorge Cossio Tomasz Dłotko Giorgio Fusco Petr Girg Wojciech Górny Massimo Grossi Ky Ho Isabella Ianni Lakshmi Sankar Kalappattil Agnieszka Kałamajska Lukas Kotrla Wojciech Kryszewski Włodzimierz Laskowski Rafael López Soriano Mohan Kumar Mallick Jarosław Mederski Roberta Musina Monica Musso Tokinaga Namba Francesco Petitta Angela Pistoia Tonia Ricciardi Matteo Rizzi Carlos Román Andrzej Rozkosz Bernhard Ruf Federica Sani Abhishek Sarkar Sarath Sasi Ratnasingham Shivaji Pieralberto Sicbaldi Mikołaj Sierzęga Panayotis Smyrnelis Philippe Souplet Andrzej Szulkin Cristina Tarsi Susanna Terracini Anoop Thazhe Veetil Giusi Vaira Juncheng Wei Dariusz Wrzosek Anna Zatorska-Goldstein