A Variant of the Mountain Pass Theorem and Variational Gluing

none2siThis paper surveys some recent work on a variant of the Mountain Pass Theorem that is applicable to some classes of differential equations involving unbounded spatial or temporal domains. In particular its application to a system of semilinear elliptic PDEs on $R^n$ and to a family of Hamiltonian systems involving double well potentials will also be discussed.openMontecchiari, Piero; Rabinowitz, Paul H.Montecchiari, Piero; Rabinowitz, Paul H

Non-negative solutions of a sublinear elliptic problem

In this paper the existence of solutions, $(\lambda,u)$, of the problem $$-\Delta u=\lambda u -a(x)|u|^{p-1}u \quad \hbox{in }\Omega, \qquad u=0 \quad \hbox{on}\;\;\partial\Omega,$$ is explored for $0 < p < 1$. When $p>1$, it is known that there is an unbounded component of such solutions bifurcating from $(\sigma_1, 0)$, where $\sigma_1$ is the smallest eigenvalue of $-\Delta$ in $\Omega$ under Dirichlet boundary conditions on $\partial\Omega$. These solutions have $u \in P$, the interior of the positive cone. The continuation argument used when $p>1$ to keep $u \in P$ fails if $0 < p < 1$. Nevertheless when $0 < p < 1$, we are still able to show that there is a component of solutions bifurcating from $(\sigma_1, \infty)$, unbounded outside of a neighborhood of $(\sigma_1, \infty)$, and having $u \gneq 0$. This non-negativity for $u$ cannot be improved as is shown via a detailed analysis of the simplest autonomous one-dimensional version of the problem: its set of non-negative solutions possesses a countable set of components, each of them consisting of positive solutions with a fixed (arbitrary) number of bumps. Finally, the structure of these components is fully described.Comment: 25 pages, 11 figure

Remodeling of the Metabolome during Early Frog Development

A rapid series of synchronous cell divisions initiates embryogenesis in many animal species, including the frog Xenopus laevis. After many of these cleavage cycles, the nuclear to cytoplasmic ratio increases sufficiently to somehow cause cell cycles to elongate and become asynchronous at the mid-blastula transition (MBT). We have discovered that an unanticipated remodeling of core metabolic pathways occurs during the cleavage cycles and the MBT in X.laevis, as evidenced by widespread changes in metabolite abundance. While many of the changes in metabolite abundance were consistently observed, it was also evident that different female frogs laid eggs with different levels of at least some metabolites. Metabolite tracing with heavy isotopes demonstrated that alanine is consumed to generate energy for the early embryo. dATP pools were found to decline during the MBT and we have confirmed that maternal pools of dNTPs are functionally exhausted at the onset of the MBT. Our results support an alternative hypothesis that the cell cycle lengthening at the MBT is triggered not by a limiting maternal protein, as is usually proposed, but by a decline in dNTP pools brought about by the exponentially increasing demands of DNA synthesis

Impact of anti‐immigrant rhetoric and policies on frontline health and social service providers in Southeast Michigan, U.S.A

Rising hostility towards immigrants characterised the 2016 Presidential election in the United States (US) and subsequent policy priorities by the new presidential administration. The political shift towards aggressive policies targeting undocumented immigrants is far‐reaching and extends into other communities that convive con—or coexist with—immigrant communities. Our study aims to examine the rippling effects of these anti‐immigrant policies and rhetoric on health and social service providers in Southeast Michigan who predominantly serve Latino immigrants. Between April and August 2018, we conducted in‐depth individual interviews in two Federally Qualified Health Centers and a non‐profit social service agency at a county health department. We interviewed 28 frontline health and social service providers. After coding and thematic analyses, we found that staff members’ experiences in supporting immigrant clients was congruent with definitions of secondary trauma stress and compassion fatigue, whereby exposure to clients’ trauma combined with job burden subsequently impacted the mental health of providers. Major themes included: (a) frontline staff experienced a mental and emotional burden in providing services to immigrant clients given the restrictive anti‐immigrant context; and (b) this burden was exacerbated by the increased difficulties in providing these services to their clients. Staff described psychological and emotional distress stemming from exposure to clients’ immigration‐related trauma and increased mental health needs. This distress was exacerbated by an increased demand to meet clients’ needs, which involved explaining or translating documents into English, assisting with legal paperwork, referring clients to mental health resources, addressing increased transportation barriers, and reestablishing trust with the community. Our findings add qualitative data on the mental health implications for frontline providers who support Latino immigrant clients impacted by immigration and highlights the need for further research and resources that address the workplace‐related stress generated by heightened immigration enforcement.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/163387/2/hsc13012.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/163387/1/hsc13012_am.pd

A qualitative study on the impact of the 2016 US election on the health of immigrant families in Southeast Michigan

Abstract Background Given the anti-immigrant rhetoric and policy proposals by President Donald Trump during the 2016 presidential campaign and afterwards, his election to president in November 2016 and subsequent policy changes has affected immigrant families. In this study, we aim to better understand how post-election policy change may have impacted the health and well-being, including health and social service utilization, of Latino immigrants in Southeastern Michigan. Methods We conducted 28 in-depth interviews with frontline staff at two Federally Qualified Health Centers and a non-profit agency. These staff had intimate knowledge of and insights into the lived experiences of the mixed-status immigrant families they serve. The interviews were audio recorded, transcribed, and analyzed thematically. Results Our findings show three major themes: (1) An increased and pervasive fear of deportation and family separation among mixed-status immigrant clients, (2) The fear of deportation and family separation has resulted in fractures in community cohesion, and (3) Fear of deportation and family separation has had an impact on the healthcare utilization and health-related behaviors of mixed-status families. Staff members report that these three factors have had an impact on physical and mental health of these immigrant clients. Conclusions These results add to previous literature on the effect of immigration policies on the health and provide key insights for interventions to improve the health of immigrants within this socio-political environment.https://deepblue.lib.umich.edu/bitstream/2027.42/152129/1/12889_2019_Article_7290.pd

Author Correction: Cross-ancestry genome-wide association analysis of corneal thickness strengthens link between complex and Mendelian eye diseases.

Emmanuelle Souzeau, who contributed to analysis of data, was inadvertently omitted from the author list in the originally published version of this Article. This has now been corrected in both the PDF and HTML versions of the Article

Adaptive energy minimisation for hp-finite element methods

This article is concerned with the numerical solution of convex variational problems. More precisely, we develop an iterative minimisation technique which allows for the successive enrichment of an underlying discrete approximation space in an adaptive manner. Specifically, we outline a new approach in the context of hp-adaptive finite element methods employed for the efficient numerical solution of linear and nonlinear second-order boundary value problems. Numerical experiments are presented which highlight the practical performance of this new hp-refinement technique for both one- and two-dimensional problems