Periodic Points Research Articles

Computing parametrised large intersection sets of 1D invariant manifolds: a tool for blender detection

AbstractA dynamical system given by a diffeomorphism with a three-dimensional phase space may have a blender, which is a hyperbolic set $$\Lambda $$ Λ with, say, a one-dimensional stable invariant manifold that behaves like a surface. This means that the stable manifold of any fixed or periodic point in $$\Lambda $$ Λ weaves back and forth as a curve in phase space such that it is dense in some projection; we refer to this as the carpet property. We present a method for computing very long pieces of such a one-dimensional manifold so efficiently and accurately that a very large number of intersection points with a specified section can reliably be identified. We demonstrate this with the example of a family of Hénon-like maps $$\mathcal {H}$$ H on $$\mathbb {R}^3$$ R 3 , which is the only known, explicit example of a diffeomorphism with proven existence of a blender. The code for this example is available as a Matlab script as supplemental material. In contrast to earlier work, our method allows us to determine a very large number of intersection points of the respective one-dimensional stable manifold with a chosen planar section and render each as individual curves when a parameter is changed. With suitable accuracy settings, we not only compute these parametrised curves for the fixed points of $$\mathcal {H}$$ H over the relevant parameter interval, but we also compute the corresponding parametrised curves of the stable manifolds of a period-two orbit (with negative eigenvalues) and of a period-three orbit (with positive eigenvalues). In this way, we demonstrate that our algorithm can handle large expansion rates generated by (up to) the fourth iterate of $$\mathcal {H}$$ H .

Fixed point and periodic point theorems

Fixed point and periodic point theorems

Hyperuniformity in two-dimensional periodic and quasiperiodic point patterns

Hyperuniformity in two-dimensional periodic and quasiperiodic point patterns

Implementation and effectiveness outcomes of Community Health Advocacy Teams to improve long-lasting insecticide net distribution and use in six districts in Ghana: A one-group pre-post-test study.

Malaria remains a leading cause of illness and death especially among children and pregnant women in Ghana. Despite the efforts made by the National Malaria Elimination Programme (NMEP), including distribution of Long-Lasting Insecticide Nets (LLINs) to households through periodic Point Mass Distribution (PMD) campaigns and continuous channels (antenatal, schools and postnatal), there is a gap between access and use of LLINs in Ghana. An effective and functional community-based group that would seek to improve the effectiveness of LLIN distribution before, during, after PMD Campaigns and continuous distribution at the community level could help address this gap. This paper assesses the implementation outcomes and short-term effectiveness of the pilot implementation of co-created community health advocacy teams (CHAT) intervention in Ghanaian communities to plan and implement campaigns to increase LLIN use. The study employed a one-group pre-post study design and measured implementation outcomes (acceptability, appropriateness, and feasibility) and effectiveness outcomes (LLIN awareness, LLIN access, willingness to purchase LLIN, and LLIN use) among 800 community households. The CHAT intervention was implemented for four months across six districts in the Eastern and Volta regions of Ghana. The data were downloaded directly from REDCap and analyzed statistically (descriptive and McNemar test of association) using SPSS 22 software. After the implementation period, the majority of respondents in all six districts indicated that the CHAT intervention was acceptable (89.8%), appropriate (89.5%), and feasible (90%). Also, there was a significant association between baseline and end-line assessment on all four effectiveness outcome measures. Household members' awareness of, access to, willingness to purchase, and use of LLINs increased significantly over the four-month period that the CHAT intervention was implemented. The study concludes that CHAT is an acceptable, appropriate, and feasible intervention for supporting the National Malaria Programme in LLIN PMD and for engaging in Social and Behaviour Change Communication activities through the continuous channels of distribution. Additionally, the CHAT demonstrates short-term effectiveness outcomes in terms of creating LLIN awareness, providing access to LLIN, and encouraging Ghanaian community members to be willing to purchase and use LLINs. Although the activities of CHAT members were largely voluntary, integration into the existing primary health care system will make it sustainable.

Sources of Social Support and Trauma Recovery: Evidence for Bidirectional Associations from a Recently Trauma-Exposed Community Sample

Although the association between post-traumatic stress disorder (PTSD) and social support is well documented, few studies have tested the causal pathways explaining this association at several points in the acute post-trauma recovery period or examined whether the association varies for different sources of social support. To address these gaps, 151 community individuals (mean age = 37.20 years, 69.5% women) exposed to trauma within the previous 6 months were recruited to complete measures of PTSD and social support from intimate partners, friends, and relatives four times in 1 year. In line with recent recommendations for research on social support and PTSD symptoms, random intercept cross-lagged panel modeling (RI-CLPM) was used to examine dynamic changes between PTSD severity and social support over time. The pattern of RI-CLPM cross-lagged coefficients indicated that positive deviations from one’s expected stable level of total social support (across all sources) sped up the recovery of PTSD symptoms at the end of the post-trauma year, and more severe PTSD symptoms than expected based on one’s expected stable level of PTSD started eroding social support midway through the assessment year. When specific sources of social support were analyzed separately, the association between within-person increases in social support from friends at any given time point accelerated the recovery from PTSD across the entire year. Among participants with intimate partners (n = 53), intimate partner support did not predict PTSD symptoms, but more severe PTSD symptoms at any given time point predicted less support at the following time point. Results from this longitudinal study provide additional support for the bidirectional relationship between PTSD and social support over time and suggest that perceived social support from friends may be especially helpful during trauma recovery.

Shadowing for nonuniformly hyperbolic maps in Hilbert spaces

We prove a shadowing lemma for nonuniformly hyperbolic maps in Hilbert spaces. As applications, we firstly prove that the positive Lyapunov exponents of a hyperbolic ergodic measure µ can be approximated by positive Lyapunov exponents of atomic measures on hyperbolic periodic orbits; secondly, give an upper estimation of metric entropy by using the exponential growth rate of the number of such periodic points that their atomic measures approximate µ and their positive Lyapunov exponents approximate the positive Lyapunov exponents of µ.

Effectiveness of erector spinae plane block in children with pectus excavatum deformity during minimally invasive thoracoplasty with osteosynthesis of the sternocostal complex: a prospective randomized comparative study

BACKGROUND: Despite the widespread use of regional blocks in children, ensuring adequate perioperative pain control during minimally invasive thoracoplasty with osteosynthesis of the sternocostal complex remains a problem. The introduction of ultrasound navigation has allowed specialists to increasingly give preference to peripheral blockades; paravertebral anesthesia has proven to be effective in thoracic surgery. However, safer interfascial blocks, including erector spinae plane block, are rarely used in pediatrics, and there is no evidence base for the effectiveness of the method.
 OBJECTIVE: Our aim was to compare the effectiveness of erector spinae plane block with paravertebral blockade during minimally invasive thoracoplasty with osteosynthesis of the sternocostal complex in children.
 MATERIALS AND METHODS: A prospective randomized comparative study of children who underwent minimally invasive thoracoplasty with osteosynthesis of the sternocostal complex was conducted from September 2022 to April 2023. According to the inclusion criteria, the selected 39 children were divided into groups: I — paravertebral blockade, and II — erector spinae plane block. The mean arterial pressure, heart rate during the perioperative period stages, postanesthesia recovery by Aldrete scale and pain intensity by the Wong–Baker scale, and stress response markers (cortisol and lactate) were assessed.
 RESULTS: No significant differences were observed in the study groups when comparing mean arterial pressure and heart rate at any of the measurement points of the perioperative period (p 0.05). Further, no significant differences were noted between the groups when comparing cortisol and lactate levels (p 0.05). The awakening level was characterized by a short recovery period and did not differ between the groups (p 0.05). The intensity of pain during the observation period did not exceed 5 points on the verbal rating scale in both groups, and the severity of pain was significantly higher in the group with blockade of the erector spinae plane block only 2 hours after surgery (p 0.05).Postoperative nausea and vomiting were more common in children with erector spinae plane block (p 0.05). No other adverse events were recorded.
 CONCLUSION: Erector spinae plane block is an effective and simple method of regional anesthesia for minimally invasive thoracoplasty with osteosynthesis of the sternocostal complex and can be used as an alternative to paravertebral blockade in children.

Synchronizing dynamical systems: Their groupoids and 𝐶*-algebras

Building on work of Ruelle and Putnam in the Smale space case, Thomsen defined the homoclinic and heteroclinic C ∗ C^\ast -algebras for an expansive dynamical system. In this paper we define a class of expansive dynamical systems, called synchronizing dynamical systems, that exhibit hyperbolic behavior almost everywhere. Synchronizing dynamical systems generalize Smale spaces (and even finitely presented systems). Yet they still have desirable dynamical properties such as having a dense set of periodic points. We study various C ∗ C^\ast -algebras associated with a synchronizing dynamical system. Among other results, we show that the homoclinic algebra of a synchronizing system contains an ideal which behaves like the homoclinic algebra of a Smale space.

Trends in air toxics cancer risk in Southern California, 1998-2018

Air toxics are an important category of air pollutants that are known to cause adverse health effects, including increased cancer risk. Regulatory efforts at federal, state, and local levels have aimed to decrease air toxics emissions over the past several decades. This study evaluated trends in air toxics cancer risks in Southern California using data from 1998 to 2018. We estimated air toxics cancer risk for each of four iterations of the South Coast Air Quality Management District’s Multiple Air Toxics Exposure Study, which included at least one year of measurements at 10 stations and air toxics modeling for each iteration. Cancer risks were calculated using the measured and modeled air toxics concentrations averaged over a one to two year period and multiplied by the corresponding cancer potency factor and combined exposure factor that accounted for multiple exposure pathways and children’s increased sensitivity to the health effects of air pollution. We examined temporal trends in overall air toxics cancer risks and evaluated changes in the air toxics species that contributed most to cancer risk in the region. Both measurement and modeling results show that air toxics cancer risk in Southern California decreased by more than 80% between 1998 and 2018, including a decrease of about 50% from 2012 to 2018. Diesel particulate matter was the main risk driver, followed by benzene, 1,3-butadiene, and formaldehyde. We found that more densely populated communities showed larger decreases than sparsely populated areas. The substantial decrease in air toxics levels over this 20-year period points to the success of air pollution policies aimed at addressing air toxics emissions and can inform future policy efforts to further reduce air toxics health impacts.

Positively ep-expansive dynamical systems

In this paper we investigate the basic properties of those dynamical systems which are positively expansive on a dense set containing all eventually periodic points and show how the dynamical behavior of these systems fits within the theoretical framework of usual (positive) expansiveness.

Control of chaos in Darcy–Bénard axisymmetric convection in a cylindrical enclosure using a uniform vertical cross-flow

The linear and weakly nonlinear stability analyses of Darcy–Bénard convection of a Newtonian fluid experiencing a uniform vertical cross-flow is investigated in the paper for various aspect ratios. Making use of the Maclaurin series representation, an expression for axial eigenfunctions is obtained with the radial eigenfunction being a Bessel function of first kind. These eigenfunctions are influenced by the Peclet number, Pe, the non-dimensional number that signifies the rate of vertical cross-flow. The modified-Vadasz-Lorenz model obtained in this paper has newly defined non-dimensional parameters that capture the influence of vertical cross-flow. From the linear stability analysis, it is found that the effect of introducing vertical cross-flow is to stabilize the system. Using a weakly nonlinear stability analysis, the closed-form expression of the Hopf-Rayleigh number as a function of Pe is obtained. Furthermore, the behavior of the modified-Vadasz–Lorenz model is analyzed using the largest Lyapunov exponent and the bifurcation diagram. This gives information about the intensity of chaos and occurrence of the periodic motion. We observe that the influence of vertical cross-flow is to increase the value of the Hopf–Rayleigh number and thereby to delay the onset of chaos. Furthermore, the appearance of the first periodic point is preponed due to the vertical cross-flow. As the rate of vertical cross-flow increases, the intensity of chaos decreases, thereby indicating that the effect of introducing vertical cross-flow is to suppress chaos.

Comparison of hemodynamic and analgesic effects of interscalene block with bupivacaine versus bupivacaine-dexmedetomidine combination for shoulder arthroscopy under general anesthesia

Background: The interscalene block in shoulder arthroscopy is a well-established procedure. Many drugs have been used as adjuvants to local anesthetics to increase the quality of block in regional anesthesia, with variable results. Aims and Objectives: The present study was designed to evaluate the effect of dexmedetomidine as an adjuvant to bupivacaine during interscalene block in terms of intraoperative hemodynamic changes and postoperative analgesia. Materials and Methods: Thirty patients, aged 20–55 years of either sex, American Society of Anesthesiologists physical status I-II, scheduled for shoulder arthroscopic surgery for <2 h, were allocated into two equal groups to receive inj. bupivacaine (0.25%) with inj. dexmedetomidine 1 μg/kg (Group BD, n=15) or inj. bupivacaine (0.25%) with 1 mL normal saline (Group BS, n=15), total volume 20 mL in each case. After settlement of the interscalene block, both groups received general anesthesia as per standard protocol. Hemodynamic parameters (heart rate [HR] and systolic blood pressure [SBP]) were assessed in the intraoperative period, and postoperative pain was assessed using a visual analog scale score in the postoperative period at stipulated time points. Results: The mean values of HR and SBP were considerably low at all observed time points in the intraoperative period (i.e., at 10 min, 30 min, 60 min, 90 min, and120 min), in comparison with the saline group (P<0.0001). VAS scores between the two groups were considerably lower only at 8, 12, and 18 h in patients receiving dexmedetomidine (P<0.001). Conclusion: Dexmedetomidine as an adjuvant with bupivacaine can achieve a better hemodynamic profile (lower HR and SBP) in the intraoperative period. Also, the use of dexmedetomidine provides better postoperative analgesia profiles in the later part of the postoperative period (8–18 h), and the effect usually wanes by 24 h.

A dynamical analogue of a question of Fermat

Given a quadratic polynomial with rational coefficients, we investigate the existence of consecutive squares in the orbit of a rational point under the iteration of the polynomial. We display three different constructions of $1$-parameter quadratic polynomials with orbits containing three consecutive squares. In addition, we show that there exists at least one polynomial of the form $x^2+c$ with a rational point whose orbit under this map contains four consecutive squares. This can be viewed as a dynamical analogue of a question of Fermat on rational squares in arithmetic progression. Finally, assuming a standard conjecture on exact periods of periodic points of quadratic polynomials over the rational field, we give necessary and sufficient conditions under which the orbit of a periodic point contains only rational squares.

Results in Cone Metric Spaces and Related Fixed Point Theorems for Contractive Type Mappings With Application

In this article, we prove some new fixed point theorems for contractive type mappings in the setting of complete cone metric spaces and provide examples to illustrate the concepts and results developed in the article. We consider some consequences of our results to establish fixed point theorems in the context of cone metric spaces. As an application of our results, periodic point results for the contractive type mappings are proved.

Transcriptome analysis of the harmful alga Heterosigma akashiwo under a 24-hour light-dark cycle

The photoperiod, which is defined as the period of time within a 24-hour time frame that light is available, is an important environmental regulator of several physiological processes in phytoplankton, including harmful bloom-forming phytoplankton. The ichthyotoxic raphidophyte Heterosigma akashiwo is a globally distributed bloom-forming phytoplankton. Despite extensive studies on the ecological impact of H. akashiwo, the influence of the photoperiod on crucial biological processes of this species remains unclear. In this study, gene expression in H. akashiwo was analyzed over a 24-hour light-dark (14:10) treatment period. Approximately 36 % of unigenes in H. akashiwo were differentially expressed during this 24-hour treatment period, which is indicative of their involvement in the response to light-dark variation. Notably, the number of differentially expressed genes exhibited an initial increase followed by a subsequent decrease as the sampling time progressed (T0 vs. other time points). Unigenes associated with photosynthesis and photoprotection reached their peak expression levels after 2–4 h of illumination (T12-T14). In contrast, the expression of unigenes associated with DNA replication peaked at the starting point of the dark period (T0). Furthermore, although several unigenes annotated to photoreceptors displayed potential diel periodicity, genes from various photoreceptor families (such as phytochrome and cryptochrome) showed unique expression patterns. Collectively, our findings offer novel perspectives on the response of H. akashiwo to the light-dark cycle, serving as a valuable resource for investigating the physiology and ecology of this species.

Approximating fixed points of weak enriched contractions using Kirk’s iteration scheme of higher order

In this paper, we introduce two types of weak enriched contractions, namely weak enriched F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{F}$\\end{document}-contraction, weak enriched F′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{F^{\\prime}}$\\end{document}-contraction, and k-fold averaged mapping based on Kirk’s iterative algorithm of order k. The types of contractions introduced herein unify, extend, and generalize several existing classes of enriched and weak enriched contraction mappings. Moreover, K-fold averaged mappings can be viewed as a generalization of the averaged mappings and double averaged mappings. We then prove the existence of a unique fixed point of the k-fold averaged mapping associated with weak enriched contractions introduced herein. We study necessary conditions that guarantee the equality of the sets of fixed points of the k-fold averaged mapping and weak enriched contractions. We show that an appropriate Kirk’s iterative algorithm can be used to approximate a fixed point of a k-fold averaged mapping and of the two weak enriched contractions. We also study the well-posedness, limit shadowing property, and Ulam–Hyers stability of the k-fold averaged mapping. We provide necessary conditions that ensure the periodic point property of each illustrated weak enriched contraction. Some examples are presented to show that our results are a potential generalization of the comparable results in the existing literature.

Starting Point of the Limitation Period for a Private Damages Action

Starting Point of the Limitation Period for a Private Damages Action

Topological entropy, sets of periods, and transitivity for circle maps

UDC 517.9 Transitivity, the existence of periodic points, and positive topological entropy can be used to characterize complexity in dynamical systems. It is known that, for every graph that is not a tree and any ε > 0 , there exist (complicated) totally transitive maps (then with cofinite set of periods) such that the topological entropy is smaller than ε (simplicity). To numerically measure the complexity of the set of periods, we introduce a notion of the <em>boundary of cofiniteness.</em> Larger boundary of cofiniteness corresponds to a simpler set of periods. We show that, for any continuous degree one circle maps, every totally transitive (and, hence, robustly complicated) map with small topological entropy has arbitrarily large (simplicity) boundary of cofiniteness.

Psychiatric and Substance Use Disorders and Their Association With Clinical Outcomes in Diabetic Ketoacidosis and Hyperosmolar Hyperglycemic Syndrome

BackgroundDiabetic Ketoacidosis (DKA) and hyperosmolar hyperglycemic state (HHS) are life-threatening conditions that send nearly 180,000 patients to the ICU each year, with mortality rates up to 5-10%. Little is known about the impact of concurrent psychiatric disorders on specific DKA/HHS outcomes. Identifying these relationships offers opportunities to improve clinical management, treatment planning, and mitigate associated morbidity and mortality. MethodsWe conducted a retrospective review including adult DKA/HHS admissions within a large Massachusetts hospital system from 2010 – 2019. We identified patients admitted inpatient for DKA or HHS, then filtered by ICD-9-CM, ICD-10-CM codes for psychiatric diagnoses that were present in patients EMR at any point in this observational period. Outcomes included number of inpatient admissions for DKA/HHS, age of death, rates of discharging against medical advice (AMA) from any inpatient admission, and ESRD/dialysis status. Multivariate regression was conducted using R software to control for variables across patients and evaluate relationships between outcomes and concurrent psychiatric disorders. Significance was set at p <0.05. Results7756 patients were admitted for DKA or HHS, 66.9% of which had a concurrent psychiatric disorder. 54.5% of these patients were male, 70.4% White, and an average age of 61.6 years. This compares with 26.1% with concurrent psychiatric condition within the general diabetes population, 52.1% of which were male, 72.1% White, and an average of 68.2 years . A concurrent psychiatric disorder was associated with increased odds of rehospitalization (aOR= 1.62 95%Cl 1.35 - 1.95, p<0.001), of being diagnosed with end-stage renal disease and on dialysis (aOR = 1.02 95% Cl 1.002 – 1.035, p = 0.02), and in leaving AMA (aOR = 6.44 95%Cl 4.46 – 9.63, p <0.001). The average age of death for those with a concurrent psychiatric disorder had an adjusted mean difference in years of -7.5 years (95% Cl -9.3 5.8) compared to those without a psychiatric disorder. Conclusion66.9% of patients with DKA/HHS have a concurrent psychiatric disorder. Patients with a concurrent psychiatric disorder admitted for DKA/HHS were more likely to have multiple admissions, to leave AMA, to be on renal dialysis, and have a lower age of mortality.

Displacement field reconstruction in landslide physical modeling by using a terrain laser scanner - Part II: Application and large strain/displacement and water effect analysis

Deformation analysis is fundamental in geotechnical modeling. Nevertheless, there is still a lack of an effective method to obtain the deformation field under various experimental conditions. In this study, we introduce a process–based physical modeling of a pile–reinforced reservoir landslide and present an improved deformation analysis involving large strains and water effects. We collect multi–period point clouds using a terrain laser scanner and reconstruct its deformation field through a point cloud processing workflow. The results show that this method can accurately describe the landslide surface deformation at any time and area by both scalar and vector fields. The deformation fields in different profiles of the physical model and different stages of the evolutionary process provide adequate and detailed landslide information. We analyze the large strain upstream of the pile caused by the pile installation and the consequent violent deformation during the evolutionary process. Furthermore, our method effectively overcomes the challenges of identifying targets commonly encountered in geotechnical modeling where water effects are considered and targets are polluted, which facilitates the deformation analysis at the wading area in a reservoir landslide. Eventually, combining subsurface deformation as well as numerical modeling, we comprehensively analyze the kinematics and failure mechanisms of this complicated object involving landslides and pile foundations as well as water effects. This method is of great significance for any geotechnical modeling concerning large-strain analysis and water effects.