Regular Points Research Articles

Semiconvexity estimates for nonlinear integro‐differential equations

AbstractIn this paper we establish for the first time local semiconvexity estimates for fully nonlinear equations and for obstacle problems driven by integro‐differential operators with general kernels. Our proof is based on the Bernstein technique, which we develop for a natural class of nonlocal operators and consider to be of independent interest. In particular, we solve an open problem from Cabré‐Dipierro‐Valdinoci. As an application of our result, we establish optimal regularity estimates and smoothness of the free boundary near regular points for the nonlocal obstacle problem on domains. Finally, we also extend the Bernstein technique to parabolic equations and nonsymmetric operators.

Empowering Indigenous-Led Contaminant Monitoring through Collaborative Partnerships and Two-Way Capacity Sharing

Climate-driven landscape change, legacy waste and ongoing infrastructural investment are leading to concerns about contaminant release in Arctic Indigenous and Local communities. Sustainable development that considers threats posed by accelerating environmental change requires accessible, appropriate, and sustained environmental data to inform strategic decision-making. Collaborative partnerships and capacity sharing are necessary to promote resilient and sustainable environmental monitoring approaches, however effective collaboration has been hindered by mismatches in priorities and timelines between communities and research programs. We outline the development and later evolution of a community-based environmental research program focused on monitoring contamination threats posed by legacy infrastructure sourced from industry, transportation and domestic waste sites. Capacities and insights from diverse knowledge systems guided each stage of our research approach. Reflections provided by an Indigenous and non-Indigenous Early Career Researcher (ECR) share insights into different aspects of the research process. We highlight how cross-cultural partnerships and capacity sharing have enabled evolving and reflexive community-based contaminant monitoring. Our approach facilitates a structural shift from collaborative monitoring with external analysis to autonomous monitoring that supports equity in research outcomes. Appropriately considered and resourced co-development at regular points of the research process has been critical to developing a complete and effective Indigenous-led research project.

Cohomogeneity one solitons for the isometric flow of G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structures

We consider the existence of cohomogeneity one solitons for the isometric flow of G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structures on the following classes of torsion-free G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-manifolds: the Euclidean R7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^7$$\\end{document} with its standard G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structure, metric cylinders over Calabi–Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant–Salamon G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-manifolds. In all cases we establish existence of global solutions to the isometric soliton equations, and determine the asymptotic behaviour of the torsion. In particular, existence of shrinking isometric solitons on R7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^7$$\\end{document} is proved, giving support to the likely existence of type I singularities for the isometric flow. In each case, the study of the soliton equation reduces to a particular nonlinear ODE with a regular singular point, for which we provide a careful analysis. Finally, to simplify the derivation of the relevant equations in each case, we first establish several useful Riemannian geometric formulas for a general class of cohomogeneity one metrics on total spaces of vector bundles which should have much wider application, as such metrics arise often as explicit examples of special holonomy metrics.

Plant native: comparing biodiversity benefits, ecosystem services provisioning, and plant performance of native and non-native plants in urban horticulture

AbstractPlant species used in urban horticulture are frequently non-native despite evidence that urban areas are regular points of introduction, often leading to invasions and negative impacts on biodiversity. Many horticulturalists assert that non-native plants are more successful and provide equal or greater habitat and ecosystem services than native plants. However, little research has compared native and non-native plants in urban systems. We present here a systematic literature review, using the PRISMA methodology, to assess three questions: (1) Is there a difference between native and non-native plants in their ability to support faunal biodiversity in urban green spaces? (2) Is there a difference between native and non-native plants in their provisioning of urban ecosystem services? 3)Do non-native species outperform natives in urban environments in terms of survival, growth, and fitness? We extracted data from 165 total papers. One hundred and twenty studies found that native plants outperform non-native plants on the response metric evaluated in the study, 57 demonstrated mixed impacts, 56 demonstrated no differences among plant origin, and 26 found non-native plants outperformed native plants. We found overwhelming evidence that native plants support higher faunal abundance and diversity than non-native plants in urban landscapes. Relatively few studies examined ecosystem services and plant performance, but we found that native plants support higher levels and diversity of ecosystem services and that many native plants can be used for horticulture, that is, they will survive and thrive in urban conditions. Native plants provide multiple ecosystem functions in urban greenspaces, supporting urban biodiversity and provisioning ecosystem services than non-native plant species and should be prioritized in urban horticulture activities.

Counterexamples and weak (1,1) estimates of wave operators for fourth-order Schrödinger operators in dimension three

This paper is dedicated to investigating the L^{p} -bounds of wave operators W_{\pm}(H,\Delta^{2}) associated with fourth-order Schrödinger operators H=\Delta^{2}+V on \mathbb{R}^{3} with real potentials satisfying |V(x)|\lesssim \langle x\rangle^{-\mu} for some \mu>0 . A recent work by Goldberg and Green (2021) has demonstrated that wave operators W_{\pm}(H,\Delta^{2}) are bounded on L^{p}(\mathbb{R}^{3}) for all 1<p<\infty under the condition that \mu>9 and zero is a regular point of H . In the paper, we aim to further establish endpoint estimates for W_{\pm}(H,\Delta^{2}) in two significant ways. First, we provide counterexamples to illustrate the unboundedness of W_{\pm}(H,\Delta^{2}) on the endpoint spaces L^{1}(\mathbb{R}^{3}) and L^{\infty}(\mathbb{R}^{3}) for non-zero compactly supported potentials V . Second, we establish weak (1,1) estimates for the wave operators W_{\pm}(H,\Delta^{2}) and their dual operators W_{\pm}(H,\Delta^{2})^{*} in the case where zero is a regular point and \mu>11 . These estimates depend critically on the singular integral theory of Calderón–Zygmund on a homogeneous space (X,d\omega) with a doubling measure d\omega .

Decay estimates for Beam equations with potential in dimension three

This paper is devoted to studying time decay estimates of the solution for Beam equation (higher order type wave equation) with a potentialutt+(Δ2+V)u=0,u(0,x)=f(x),ut(0,x)=g(x) in dimension three, where V is a real-valued and decaying potential. Assume that zero is a regular point of H=Δ2+V, we first prove the following optimal time decay estimates of the solution operators‖cos⁡(tH)Pac(H)‖L1→L∞≲|t|−32and‖sin⁡(tH)HPac(H)‖L1→L∞≲|t|−12. Moreover, if zero is a resonance of H, then time decay of the solution operators also is considered. It is noted that a first-kind resonance does not affect the decay rates of the propagator operators cos⁡(tH) and sin⁡(tH)H, but their decay will be significantly changed for the second and third-kind resonances.

RANK OF THE DERIVATIVE OF THE PROJECTION TO SYMMETRIZED POLYDISC

The projection, also called the symmetrization mapping, from spectral ball to symmetrized polydisc is closely related to the spectral Nevanlinna-Pick interpolation problem. We prove that the rank of the derivative of the projection from the spectral unit ball to the symmetrized polydisc is equal to the degree of the minimal polynomial of the matrix at which we take the derivative. Therefore, it explains why the corresponding lifting problem is easier when the matrix base-point is cyclic since it is a regular point of the symmetrization mapping in the differential sense.

Indwelling urinary catheter use and adherence to clinical practice guidelines: A point prevalence study in adult hospital inpatients

Background Approximately 25% of hospitalised adults require an indwelling urinary catheter (IDC) during their hospital stay. IDCs expose patients to risks of infectious and non-infectious complications. Aims To identify IDC prevalence, assess adherence to clinical practice guidelines and patient-reported involvement in IDC care for adult hospital inpatients. Methods This point prevalence study was conducted in 22 wards in a single quaternary hospital. Data was collected by clinical and research nurses working in pairs on a single day. Study outcomes were reported descriptively as frequencies and percentages. Results Of 502 patients included, 77 (15.3%) had an IDC (median duration 99.6 h). The median age of patients with an IDC was 64 years (interquartile range 22–88 years), 54 (70%) were male and one-quarter ( n = 19; 25%) of IDCs were inserted at another hospital. More than half ( n = 44; 57%) of the 77 IDCs had no documented removal plan. Three patients were unavailable for review for observed clinical practices, and it was found 43% ( n = 32/74) lacked a securement device. Of 77 people with IDCs, there were 44 patient responses, and 27 (61.4%) patients did not know the reason for their catheter. Discussion Areas for improvement included securement device use, timely removal plans and patient education for the reason for the device. Regular point prevalence studies to assess use and adherence to clinical practical guidelines can improve safety outcomes for patients requiring IDCs.

Center Conditions for Nilpotent Singularities in the Plane Using Invariant Solutions

Recalling that at any regular point we always have a unique particular solution curve passing through it. In this work it is constructed such particular solution curve not passing through the nilpotent singularity but as close as we want to the singularity. By product the existence of such particular curve allows to use it to determine necessary conditions to have a center for nilpotent singularities in the plane. Several involve methods to solve the center problem are known all based in the existence of a change of variables and a scaling transformation of time bringing any differential system with a nilpotent center into a time-reversible system. Here we present a new algebraic method based on the existence of such particular solution curve not passing through the singular point and the involution associated to the nilpotent system with a center. The algebraic method needs the computation of this particular curve up to certain order, which can be done with the help of an algebraic manipulator. Finally a new algebraic method is derived computing the vanishing of a unique function which really gives a scalar method for computing the necessary conditions.

A rescaled metric entropy of C1-vector field with singularities

For a C1-vector field on a closed Riemannian manifold, we add the flow speed control to define Bowen ball, and the new Bowen balls are given after considering reparametrization. That use the idea of rescaling the size of the neighborhoods of a regular point. Then we give new definitions of metric-entropy by using the minimum number of these new types of Bowen balls covering a set and the measure of this set is more than 1−δ. We prove that these new definitions are equivalent, and we call them rescaled metric entropy. Then we prove that for a C1-vector field, the rescaled metric entropy is equal to the metric entropy of the time-1 map of the flow providing that ln⁡‖X(x)‖ is integrable.

Local smooth convergence of 𝔽-limit flows

The metric flow is introduced and extensively studied by Bamler [Compactness theory of the space of super Ricci flows, preprint (2020), arXiv:2008.09298; Structure theory of non-collapsed limits of Ricci flows, preprint (2020), arXiv:2009.03243], especially as an [Formula: see text]-limit of a sequence of smooth Ricci flows with uniformly bounded Nash entropy, in which case each regular point on the limit is a point of smooth convergence. In this note, we shall consider the [Formula: see text]-convergence of a sequence of [Formula: see text]-limit flows, and, like Bamler, show that each regular point on the limit is also a point of smooth convergence. The main result will be applied in another work of the authors [P.-Y. Chan, Z. Ma and Y. Zhang, Dimension reduction for positively curved steady solitons, preprint (2023), arXiv:2310.14020].

Optimal switching boundary control for two-dimensional piecewise linear dynamical systems

This paper deals with the optimal control design for a family of two-dimensional piecewise linear systems having a switching boundary defined by a straight line and vector fields with regular equilibrium points. Such a family is derived from discontinuous control systems characterized by exhibiting sliding motion and a single pseudo-equilibrium point at the switching boundary, which is the desired operating point of the system in the control design. The initial objective is to obtain conditions on the slope of the switching boundary so that the pseudo-equilibrium is asymptotically stable, at least locally. The second and main objective is, based on the pseudo-equilibrium stability result, to find the optimal slope for the switching boundary in order to minimize the transient response time. A numerical method for obtaining the optimal slope is presented, and the effects caused by uncertainties in the parameters are discussed. The used approach explores an innovative way of determining the optimal slope of the switching boundary based on finite-time stability conditions. The results obtained are applied to a power electronic system that describes the dynamics of a Buck Converter under a sliding mode control strategy.

On pole-skipping with gauge-invariant variables in holographic axion theories

We study the pole-skipping phenomenon within holographic axion theories, a common framework for studying strongly coupled systems with chemical potential (μ) and momentum relaxation (β). Considering the backreaction characterized by μ and β, we encounter coupled equations of motion for the metric, gauge, and axion field, which are classified into spin-0, spin-1, and spin-2 channels. Employing gauge-invariant variables, we systematically address these equations and explore pole-skipping points within each sector using the near-horizon method. Our analysis reveals two classes of pole-skipping points: regular and singular pole-skipping points in which the latter is identified when standard linear differential equations exhibit singularity. Notably, pole-skipping points in the lower-half plane are regular, while those elsewhere are singular. This suggests that the pole-skipping point in the spin-0 channel, associated with quantum chaos, corresponds to a singular pole-skipping point. Additionally, we observe that the pole-skipping momentum, if purely real or imaginary for μ = β = 0, retains this characteristic for μ ≠ 0 and β ≠ 0.

Effects of Breaking Up Sitting on Gut Hormone Responses and Subsequent Compensatory Behaviors in Physically Inactive Adults.

The effects of breaking up sitting on gut hormone responses and free-living energy compensatory behaviors are still unclear in people of Asian ethnicity. Twenty-six Asians including 13 lean individuals (Lean) and 13 individuals with centrally overweight/obesity (OW), aged between 20 and 45 yr, completed a randomized crossover study with either 5.5-h uninterrupted sitting (SIT) or 5.5-h sitting with 2-min walking at 6.4 km·h -1 every 20 min (ACTIVE) in the laboratory. Blood samples were collected at regular time points to examine postprandial glucagon-like peptide 1 (GLP-1), peptide YY (PYY), and glucose-dependent insulinotropic polypeptide (GIP) concentrations. Free-living physical activity and energy intake were recorded using wearable devices and weighed food diaries outside the laboratory until midnight. Paired t -tests were conducted to compare responses between trials. Postprandial GLP-1 and PYY incremental area under the curve values were higher in the ACTIVE trial versus SIT in both Lean and OW groups (all, P < 0.05), but there was no difference in GIP in either group (both, P > 0.05). There were no differences in free-living physical activity (volume and intensity) or energy intake (total and macronutrients) between trials in either group (all, P > 0.05), resulting in greater total physical activity over the 24-h monitoring period in ACTIVE trial versus SIT trial (both, P < 0.05). Breaking up sitting increases postprandial GLP-1 and PYY concentrations in Asians, but does not induce subsequent behavioral compensation, resulting in greater 24-h physical activity levels and lower relative energy intake, in inactive individuals irrespective of bodyweight status.

Maternal and Infant Dietary Patterns Are Not Related to Food Allergy Risk in Singapore Children: GUSTO Cohort Study

BackgroundWe previously reported that delayed allergenic food introduction in infancy did not increase food allergy risk until age 4 y within our prospective cohort. However, it remains unclear whether other aspects of maternal or infant diet play roles in the development of childhood food allergy. ObjectivesWe examined the relationship between maternal pregnancy and infant dietary patterns and the development of food allergies until age 8 y. MethodsAmong 1152 Singapore Growing Up in Singapore Towards healthy Outcomes study mother–infant dyads, the infant’s diet was ascertained using food frequency questionnaires at 18 mo. Maternal dietary patterns during pregnancy were derived from 24-h diet recalls. Food allergy was determined through interviewer-administered questionnaires at regular time points from infancy to age 8 y and defined as a positive history of allergic reactions, alongside skin prick tests at 18 mo, 3, 5, and 8 y. ResultsFood allergy prevalence was 2.5% (22/883) at 12 mo and generally decreased over time by 8 y (1.9%; 14/736). Higher maternal dietary quality was associated with increased risk of food allergy (P ≤ 0.016); however, odds ratios were modest. Offspring food allergy risk ≤8 y showed no associations with measures of infant diet including timing of solids/food introduction (adjusted odds ratio [aOR]: 0.90; 95% confidence interval [CI]: 0.42, 1.92), infant’s diet quality (aOR: 0.93; 95% CI: 0.88, 0.99) or diet diversity (aOR: 0.84; 95% CI: 0.6, 1.19). Most infants (89%) were first introduced to cow milk protein within the first month of life, while egg and peanut introduction were delayed (58.3% introduced by mean age 8.8 mo and 59.8% by mean age 18.1 mo, respectively). ConclusionsApart from maternal diet quality showing a modest association, infant’s allergenic food introduction, diet quality, and dietary diversity were not associated with food allergy development in this Asian pediatric population. Interventional studies are needed to evaluate the efficacy of these approaches to food allergy prevention across different populations.

Universality for free fermions and the local Weyl law for semiclassical Schrödinger operators

We study local asymptotics for the spectral projector associated to a Schrödinger operator -\hbar^{2}\Delta+V on \mathbb{R}^{n} in the semiclassical limit as \hbar\to0 . We prove local uniform convergence of the rescaled integral kernel of this projector towards a universal model, inside the classically allowed region as well as on its boundary. This implies universality of microscopic fluctuations for the corresponding free fermions (determinantal) point processes, both in the bulk and around regular boundary points. Our results apply to a general class of smooth potentials in arbitrary dimension n\ge 1 . These results are complemented by studying both macroscopic and mesoscopic fluctuations of the point process. We obtain tail bounds for macroscopic linear statistics and, provided n\geq 2 , a central limit theorem for both macroscopic and mesoscopic linear statistics in the bulk.

The free boundary for a semilinear non-homogeneous Bernoulli problem

In the classical homogeneous one-phase Bernoulli-type problem, the free boundary consists of a “regular” part and a “singular” part, as Alt and Caffarelli have shown in their pioneer work (Alt and Caffarelli, 1981 [1]) that regular points are C1,γ in two-dimensions. Later, Weiss (1999) [34] first realized that in higher dimensions a critical dimension d⁎ exists so that the singularities of the free boundary can only occur when d⩾d⁎.In this paper, we consider a non-homogeneous semilinear one-phase Bernoulli-type problem, and we show that the free boundary is a disjoint union of a regular and a singular set. Moreover, the regular set is locally the graph of a C1,γ function for some γ∈(0,1). In addition, there exists a critical dimension d⁎ so that the singular set is empty if d<d⁎, discrete if d=d⁎ and of locally finite Hd−d⁎ Hausdorff measure if d>d⁎. As a byproduct, we relate the existence of viscosity solutions of a non-homogeneous problem to the Weiss-boundary adjusted energy, which provides an alternative proof to existence of viscosity solutions for non-homogeneous problems.

Free boundary regularity in the fully nonlinear parabolic thin obstacle problem

Abstract We study the regularity of the free boundary in the fully nonlinear parabolic thin obstacle problem. Under the assumption of time semiconvexity, our main result establishes that the free boundary is a C 1 C^{1} graph in x near any regular free boundary point.

Bayesian modeling of post-vaccination serological data suggests that yearly vaccination of dog aged

Rabies control remains challenging in low and middle-income countries, mostly due to lack of financial resources, rapid turnover of dog populations and poor accessibility to dogs. Rabies is endemic in Cambodia, where no national rabies vaccination program is implemented. The objective of this study was to assess the short and long-term vaccination-induced immunity in Cambodian dogs under field conditions, and to propose optimized vaccination strategies. A cohort of 351 dogs was followed at regular time points following primary vaccination only (PV) or PV plus single booster (BV). Fluorescent antibody virus neutralization test (FAVNT) was implemented to determine the neutralizing antibody titer against rabies and an individual titer ≥0·5 IU/mL indicated protection. Bayesian modeling was used to evaluate the individual duration of protection against rabies and the efficacy of two different vaccination strategies. Overall, 61% of dogs had a protective immunity one year after PV. In dogs receiving a BV, this protective immunity remained for up to one year after the BV in 95% of dogs. According to the best Bayesian model, a PV conferred a protective immunity in 82% of dogs (95% CI: 75-91%) for a mean duration of 4.7 years, and BV induced a lifelong protective immunity. Annual PV of dogs less than one year old and systematic BV solely of dogs vaccinated the year before would allow to achieve the 70% World Health Organization recommended threshold to control rabies circulation in a dog population in three to five years of implementation depending on dog population dynamics. This vaccination strategy would save up to about a third of vaccine doses, reducing cost and time efforts of mass dog vaccination campaigns. These results can contribute to optimize rabies control measures in Cambodia moving towards the global goal of ending human death from dog-mediated rabies by 2030.

The Leuven Prolonged Acute Stress Test (L-PAST): A novel paradigm to induce a subjective and glucocorticoid stress response for at least ninety minutes

Laboratory stress tests typically administer stress acutely, ranging from 3 to 15 minutes. However, everyday stressors usually last longer than ten minutes (e.g., examination stressors, work stressors, and social stressors. Moreover, in some studies, it may be relevant to induce stress for a longer period to affect certain psychological or physiological parameters. To this end, we developed a novel stress test that intends to induce psychosocial stress for 90 minutes. The Leuven Prolonged Acute Stress Test (L-PAST) combines physical (hand immersion in cold water), cognitive (mental arithmetic), and psychosocial (social evaluation and feelings of failure) stress elements of two well-known laboratory stress tests, the Maastricht Acute Stress Test (MAST) and the Montreal Imaging Stress Test (MIST). Fifty healthy women were subjected to both the L-PAST and a sham (control) test in a randomized and counterbalanced manner. The stress response was determined by salivary cortisol measurements and assessment of subjective stress ratings at regular time points during the time preceding the stress period (5 min), the stress period (90 min), and the recovery period (35 min). Cognitive reactivity to failure and subjective pain levels were also assessed at various time points. The childhood trauma questionnaire (CTQ) and the perceived stress scale (PSS) were administered prior to the testing phase. As expected, linear mixed models revealed that the stress response was significantly higher during the L-PAST as indicated by a significant time point by condition interaction effect for both the cortisol response (F(10,450)=12.21, p < 0.0001, ηp2=0.11) and the subjective stress response (F(13,637)=13.98, p < 0.0001, ηp2 = 0.12). Moreover, there was a significant time point by condition interaction effect for cognitive reactivity to failure (F(13,637) = 7.97, p < 0.0001, ηp2 = 0.07) and subjective pain (F(13,637) = 38.52, p < 0.0001, ηp2 = 0.27), indicating that the levels were higher during the L-PAST at most stress induction time points. Lastly, higher CTQ scores were associated with higher subjective pain levels during the L-PAST (F(1,44)=6.05, p = 0.02). Collectively, our results confirm the efficacy of the L-PAST in inducing a prolonged subjective as well as cortisol stress response.