Leonetti and Luca [‘On the iterates of the shifted Euler’s function’, Bull. Aust. Math. Soc., to appear] have shown that the integer sequence $(x_n)_{n\geq 1}$ defined by $x_{n+2}=\phi (x_{n+1})+\phi (x_{n})+k$, where $x_1,x_2\geq 1$, $k\geq 0$ and $2 \mid k$, is bounded by $4^{X^{3^{k+1}}}$, where $X=(3x_1+5x_2+7k)/2$. We improve this result by showing that the sequence $(x_n)$ is bounded by $2^{2X^2+X-3}$, where $X=x_1+x_2+2k$.

Gibbons are often difficult to observe in dense forest habitats using traditional ground-based methods. This makes it challenging to estimate group sizes and, in turn, population sizes. This has proven to be a key constraint on accurate monitoring of the last remaining population of the Critically Endangered cao vit gibbon Nomascus nasutus. However, new technologies are beginning to circumvent the problems associated with traditional methods. We hypothesized that, by using an unoccupied aerial vehicle (UAV) equipped with thermal and standard (RGB) cameras, we could obtain more accurate group size counts than by using ground-based observations, as fewer gibbons would be missed. We tested this during a population survey of the cao vit gibbon, finding that the thermal video footage revealed additional individuals that were not counted by ground-based surveyors. Statistically, there was strong evidence (93% probability) that UAV-derived counts were higher (by 41%) than concurrent ground-based counts. We recorded six primate groups of three species (cao vit gibbon, rhesus macaque Macaca mulatta and Assamese macaque Macaca assamensis), including 24 gibbons across four groups (c. 20% of the global population). The RGB video footage also revealed seven female gibbons, two of which were carrying infants, providing vital group composition data. These data have contributed directly to a more accurate population survey of the species than would have been possible using direct observation only. We anticipate more widespread use of UAVs in the study of gibbons and other threatened species, leading to a more robust evidence base for their conservation.

Objectives: Antimicrobial resistance (AMR) has emerged as a major concern in Vietnam, mainly due to the inappropriate use of antibiotics. Appropriate antibiotic management enables us to minimize the likelihood of antibiotic resistance and the spread of resistant bacteria. We evaluated vancomycin and colistin resistance and related factors in the intensive care unit (ICU) of Hue Central Hospital, a national hospital in central Vietnam. Methods: Using a cross-sectional descriptive study, we enrolled 362 patients who were prescribed antibiotics and were admitted to the ICU in 2019. Pathogens isolated from 473 routine clinical samples were subjected to antimicrobial susceptibility testing following the recommendations in the Clinical & Laboratory Standards Institute M100, 28th Edition. Colistin testing was performed using the broth microdilution method. Statistical significance was determined using the Fisher exact test. Results: The most commonly identified microorganisms were Acinetobacter baumannii (31.5%), Klebsiella pneumoniae (31.2%), Pseudomonas aeruginosa (12%), and Staphylococcus aureus (8.9%). All isolates of A. baumannii, K. pneumoniae, and P. aeruginosa tested with colistin were nonresistant. Moreover, >65% of A. baumannii isolates were resistant to all antibiotics except colistin. S. aureus had the highest resistance rate to erythromycin (80.6%), but no vancomycin-resistant isolates were identified. Factors associated with resistance to at least 1 antibiotic tested included length of stay (OR, 5.32; 95% CI, 1.47–19.17; P = .017), duration of antibiotics therapy (OR, 5.25; 95% CI, 1.46–18.95; P = .017), and the use of tracheal intubation and ventilator (OR, 3.08; 95% CI, 1.09–8.72; P = .038). Conclusions: These data indicated that although the vancomycin and colistin resistance rate is low, patients with longer length of stay, longer time on antibiotics, and invasive ventilation were at higher risk of AMR infection. Decreasing device use and strong antibiotic stewardship program at the hospital would help to reduce AMR infections.

Objectives: SARS-CoV-2 is a novel and highly infectious virus. An effective response requires rapid training of healthcare workers (HCWs). We measured the change in knowledge related to COVID-19 and associated factors before and after training of HCWs in Vietnam. Methods: A quasi-experimental design was used to evaluate HCW knowledge related to prevention and control of SARS-CoV-2 before and after attending a 2-day training-of-trainers course. Between June and September 2020, 963 HCWs from 194 hospitals in 21 provinces received the training. HCW knowledge was assessed using a 20-item questionnaire consisting of multiple-choice questions at the beginning and closing of the training course. A participant received 1 point for each correct answer. He or she was considered to have improved knowledge the posttest score was higher than the pretest score with a score ≥15 on the posttest. We applied the McNemar test and logistic regression model to test the level of association between demographic factors and change in knowledge of COVID-19. Results: Overall, 100% of HCWs completed both the pretest and posttest. At baseline, only 14.7% scored ≥15. Following the training, 78.4% scored ≥15 and 64.3% had improved knowledge according to the predetermined definition. Questions related to the order of PPE donning and doffing and respiratory specimen collection procedures were identified as having the greatest improvement (44.6% and 60.7%, respectively). Being female (OR, 1.5; 95% CI, 1.1–2.0), having a postgraduate degree (OR, 2.5; 95% CI, 1.4–4.4), working in a nonmanager position (OR, 1.5; 95% CI, 1.1–2.1), previous contact with a COVID-19 patient (OR, 1.5; 95% CI, 1.1–2.0), and working in northern Vietnam (OR, 2.0; 95% CI, 1.4–2.6), were associated with greater knowledge improvement. Conclusions: Most HCWs demonstrated improved knowledge of COVID-19 prevention and control after attending the training. Particular groups may benefit from additional training: those who are male, leaders and managers, those who hold an undergraduate degree, and those who work in the southern provinces.

We extend a result of Lieb [‘On the lowest eigenvalue of the Laplacian for the intersection of two domains’, Invent. Math. 74(3) (1983), 441–448] to the fractional setting. We prove that if A and B are two bounded domains in $\mathbb R^N$ and $\lambda _s(A)$, $\lambda _s(B)$ are the lowest eigenvalues of $(-\Delta )^s$, $0<s<1$, with Dirichlet boundary conditions, there exists some translation $B_x$ of B such that $\lambda _s(A\cap B_x)< \lambda _s(A)+\lambda _s(B)$. Moreover, without the boundedness assumption on A and B, we show that for any $\varepsilon>0$, there exists some translation $B_x$ of B such that $\lambda _s(A\cap B_x)< \lambda _s(A)+\lambda _s(B)+\varepsilon .$

Acute myocarditis is one of the common complications of coronavirus disease 2019 (COVID-19) with a relatively high case fatality. Here reported is a fulminant case of a 42-year-old previously healthy woman with cardiogenic shock and refractory cardiac arrest due to COVID-19-induced myocarditis who received veno-arterial (VA) extracorporeal membrane oxygenation (ECMO) after 120 minutes of cardiopulmonary resuscitation (CPR). This is the first adult case of cardiac arrest due to COVID-19-induced myocarditis supported by ECMO that fully recovered with normal neurological functions. The success of the treatment course with full recovery emphasized the potential role of ECMO in treating these patients.

Although compelling evidence from observational studies supports a positive association between consumption of cereal fibre and CVD risk reduction, randomised controlled trials (RCT) often target viscous fibre type as the prospective contributor to lipid lowering to reduce CVD risk. The objective of our study is to compare the lipids-lowering effects of viscous dietary fibre to non-viscous, cereal-type fibre in clinical studies. RCT that evaluated the effect of viscous dietary fibre compared with non-viscous, cereal fibre on LDL cholesterol and alternative lipid markers, with a duration of ≥ 3 weeks, in adults with or without hypercholesterolaemia were included. Medline, EMBASE, CINAHL and the Cochrane Central Register were searched through October 19, 2021. Data were extracted and assessed by two independent reviewers. The generic inverse variance method with random effects model was utilised to pool the data which were expressed as mean differences (MD) with 95 % CI. Eighty-nine trials met eligibility criteria (n 4755). MD for the effect of viscous dietary fibre compared with non-viscous cereal fibre were LDL cholesterol (MD = –0·26 mmol/l; 95 % CI: –0·30, −0·22 mmol/l; P < 0·01), non-HDL cholesterol (MD = –0·33 mmol/l; 95 % CI: –0·39, −0·28 mmol/l; P < 0·01) and Apo-B (MD = –0·04 g/l; 95 % CI: –0·06, −0·03 g/l; P < 0·01). Viscous dietary fibre reduces LDL cholesterol and alternative lipid markers relative to the fibre from cereal sources, hence may be a preferred type of fibre-based dietary intervention targeting CVD risk reduction.

Iodine is essential for thyroid hormone synthesis. Individuals adhering to vegan and vegetarian diets have been found to be vulnerable to iodine deficiency. Yet, iodine has not been monitored in these groups across time. This study aims to investigate iodine status, intake and knowledge in vegans, vegetarians and omnivores and determine changes between 2016–2017 and 2019. Dietary intake (µg/d) was estimated by 3-d food diaries and iodine FFQ. Urinary iodine concentration, analysed by inductively coupled plasma mass spectrometry, assessed iodine status according to WHO criteria. Iodine knowledge was scored by an adapted questionnaire. IBM SPSS was used for statistical analysis. Ninety-six adults (18–60 years) were recruited in October 2016–2017 (vegans: 12; vegetarians: 5; omnivores: 43) and June 2019 (vegans: 7; vegetarians: 10;omnivores: 19). Median dietary iodine was below the Reference Nutrient Intake for all groups. Vegans and vegetarians had the lowest iodine intake. Vegans had significantly lower iodine intake than omnivores (2016–2017, P = 0 032; 2019, P = 0 001). Omnivores had the highest iodine status (2016–2017, 79 4 µg/l; 2019, 72 4 µg/l) and vegans the lowest (2016–2017, 31 2 µg/l; 2019, 12 2 µg/l). Iodine knowledge was poor but did not differ between dietary groups (2016–2017, P = 0 219; 2019, P = 0 532). Vegans and vegetarians continue to be at risk of iodine deficiency. Further, iodine intake in the UK is poor independent of dietary choice. Iodine education is needed along with research into improving iodine nutrition at national level.

One of the three Critically Endangered and endemic primate species in Viet Nam, Delacour's langur Trachypithecus delacouri, is restricted to the north of the country. The largest remaining population is in Van Long Nature Reserve, Ninh Binh Province, and the second largest is in nearby Kim Bang Protection Forest, Ha Nam Province, with other populations believed to be too small for recovery. The population in Van Long has been well studied but the status of that of Kim Bang has not previously been adequately evaluated. To address this, a survey was conducted during 10 August–7 October 2018. In total, we recorded 13 groups comprising a total of at least 73 individuals, almost doubling the highest number reported in previous studies. We documented six new groups in Lien Son and Ba Sao communes in Kim Bang. Our findings indicate there is a high likelihood that Delacour's langur population in Kim Bang will recover and could be important for the long-term conservation of this Critically Endangered species. However, immediate and appropriate conservation measures need to be implemented to protect the population from major anthropogenic threats, namely poaching and habitat destruction, detected during our survey.

Livestock production has increased in many emerging economies, but productivity is often substantially impaired by infectious diseases. The first step towards improved livestock health and productivity is to map the presence of livestock diseases. The objective of this review was to summarize studies conducted on such diseases in an emerging economy, Vietnam, and thereby identifying knowledge gaps that may inform the design of surveillance and control programs. Few studies were found to evaluate the distribution of infectious livestock diseases other than avian influenza. Also, many regions with dense livestock populations had received little attention in terms of disease investigation. A large proportion of the studies dealt with zoonoses and food-borne infections which might be due to funding agencies priorities. On the contrary, studies targeting infections that affect livestock and their productivity were few. We think that this limitation in scientific reports on infectious diseases that only affect livestock productivity is a common phenomenon in low and lower middle income countries. More science-based data on such diseases would help policymakers to prioritize which livestock diseases should be subject to animal health programs aimed to support rural livelihoods and economic development.

We study the fractional parabolic Lichnerowicz equation

$$ \begin{align*} v_t+(-\Delta)^s v=v^{-p-2}-v^p \quad\mbox{in } \mathbb R^N\times\mathbb R \end{align*} $$

$p>0$

$ 0<s<1 $

$p>1$

$0<p\leq 1$

$$ \begin{align*} (-\Delta)^s u=u^{-p-2}-u^p \quad\mbox{in }\mathbb R^N \end{align*} $$

$p>0$

$0<s<1$

We first establish a family of sharp Caffarelli–Kohn–Nirenberg type inequalities (shortly, sharp CKN inequalities) on the Euclidean spaces and then extend them to the setting of Cartan–Hadamard manifolds with the same best constant. The quantitative version of these inequalities also is proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli–Kohn–Nirenberg type inequalities with the same sharp constant as in the Euclidean space of the same dimension. Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Kristály (J. Math. Pures Appl. 119 (2018), 326–346) to a larger class of the sharp CKN inequalities.

For a vibration system, the best designed spring is compliant to a desired vibration mode while it is robust to other undesired modes. There are several types of spring design for displacing the proofmass along the x and y axes, however, very few designs of spring compliant to the z axis are introduced. Therefore, we propose a z axis microactuator in which the suspending spring is designed so that it is only compliant to vibration along the z axis. The suspending spring consists of straight beam stages mechanically coupled with each other via frames which are symmetrically designed around a center plate. The operation characteristics of the microactuator is investigated by theoretical expresses and numerical simulation. The frequency split between the z axis mode and undesired modes can obtain more than 45%. The operation frequency can be modified in a wide range, from 68 kHz to 400 kHz, by changing the dimensional parameters of spring beams. The spring beams can be lengthened to increase displacement in the z axis while the mode cross-talk is still suppressed. Compared to the previously reported researches, the current microactuator shows robustness to undesired vibration modes, which is potential for integration in low mode cross-talk multi-axis micro-stages and low-noise mechanical sensors.