In this note, we prove that if $G$ is solvable and ${\rm cod}(\chi)$ is a $p$-power for every nonlinear, monomial, monolithic $\chi\in {\rm Irr}(G)$ or every nonlinear, monomial, monolithic $\chi \in {\rm IBr} (G)$, then $P$ is normal in $G$, where $p$ is a prime and $P$ is a Sylow $p$-subgroup of $G$.

The classification of trivectors(trilinear alternating forms) depends essentially on the dimension $n$ of the base space. This classification seems to be a difficult problem (unlike in the bilinear case). For $n\leq 8 $ there exist finitely many trivector classes under the action of the general linear group $GL(n).$ The methods of Galois cohomology can be used to determine the classes of nondegenerate trivectors which split into multiple classes when going from $\bar{K}$(the algebraic closure of $K$) to $K.$ In this paper, we are interested in the classification of trivectors of an eight dimensional vector space over a finite field of characteristic $3,$ $% K=\mathbb{F}_{3^{m}}.$ We obtain a $31$ inequivalent trivectors, $20$ of which are full rank. Having its motivation in the theory of the generalized elliptic curves and commutative moufang loop, this research studies the case of the forms over the 3 elements field. We use a transfer theorem providing a one-to-one correspondence between the classes of trilinear alternating forms of rank $8$ over a finite field with $3$ elements $\mathbb{F}_{3}$ and the rank $9$ class $2$ Hall generalized elliptic curves (GECs) of $3$-order $9$ and commutative moufang loop (CMLs). We derive a classification and explicit descriptions of the $31$ Hall GECs whose rank and $3$-order both equal $9$ and the number of order $3^{9}$-CMLs.

In 2015, Borovi\'{c}anin presented trees with the smallest first Zagreb index among trees with given number of vertices and number of branching vertices. The first Zagreb index is obtained from the general sum-connectivity index if $a = 1$. For $a \in \mathbb{R}$, the general sum-connectivity index of a graph $G$ is defined as $\chi_{a} (G) = \sum_{uv\in E(G)} [d_G (u) + d_G (v)]^{a}$, where $E(G)$ is the edge set of $G$ and $d_G (v)$ is the degree of a vertex $v$ in $G$. We show that the result of Borovi\'{c}anin cannot be generalized for the general sum-connectivity index ($\chi_{a}$ index) if $0 < a < 1$ or $a > 1$. Moreover, the sets of trees having the smallest $\chi_a$ index are not the same for $0 < a < 1$ and $a > 1$. Among trees with given number of vertices and number of branching vertices, we present all the trees with the smallest $\chi_a$ index for $0 < a < 1$ and $a > 1$. Since the hyper-Zagreb index is obtained from the $\chi_a$ index if $a = 2$, results on the hyper-Zagreb index are corollaries of our results on the $\chi_a$ index for $a > 1$.

Let $G$ be a simple connected graph of order $n$ and size $m$. The matrix $L(G)=D(G)-A(G)$ is the Laplacian matrix of $G$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix, respectively. For the graph $G$, let $d_{1}\geq d_{2}\geq \cdots d_{n}$ be the vertex degree sequence and $\mu_{1}\geq \mu_{2}\geq \cdots \geq \mu_{n-1}>\mu_{n}=0$ be the Laplacian eigenvalues. The Laplacian resolvent energy $RL(G)$ of a graph $G$ is defined as $RL(G)=\sum\limits_{i=1}^{n}\frac{1}{n+1-\mu_{i}}$. In this paper, we obtain an upper bound for the Laplacian resolvent energy $RL(G)$ in terms of the order, size and the algebraic connectivity of the graph. Further, we establish relations between the Laplacian resolvent energy $RL(G)$ with each of the Laplacian-energy-Like invariant $LEL$, the Kirchhoff index $Kf$ and the Laplacian energy $LE$ of the graph.

We provide some characterization theorems about just infinite profinite residually solvable Lie algebras, similarly to what C. Reid has done for just infinite profinite groups. In particular, we prove that a profinite residually solvable Lie algebra is just infinite if and only if its obliquity subalgebra has finite codimension in the Lie algebra, and we establish a criterion for a profinite residually solvable Lie algebra to be just infinite, looking at the finite Lie algebras occurring in the inverse system.

BACKGROUND AND OBJECTIVES: Land suitability analysis is a technique of attaining optimum utilization of natural available land resource. This study is the first attempt to map the potential rice suitability zone besides the existing rice cultivation zone in Imphal-Iril River catchment. The overriding objective of this study is to identify the land suitability potential zones for rice crop cultivation. The study was carried-out in Imphal-Iril River catchment, Manipur, India.METHODS: The suitability analysis was carried-out based on soil, climate and topographic parameters as the input variable using integrated geographical information system and analytic hierarchy process, a multi criteria decision based approach. To compute criteria weight for various suitability classes, pairwise comparison matrix was applied using analytical hierarchy process and the resulting weights were used for assigning criteria ranking.FINDINGS: The study result indicates that the major section of high and moderate potential suitability zones of rice is concentrated in the flatter valley regions of the catchment. The result also indicates that there is 79.15 km2 of the area which can be potentially cultivated other than the existing agriculture cover. The major patches of such zones are found in the north-western portion of the valley region in the catchment.CONCLUSION: This study clearly indicates, the potential zones lying in the foothills in the north-western which are still not under the agriculture cover have the potential to be cultivated as per the model result. The model result clearly indicates the potential of geographical information system integrated with analytical hierarchy process technique can be utilized to decide the weights of each individual parameter using experts’ opinions which can serve as a versatile tool to carry-out such kind of analysis which can aid policy makers.

BACKGROUND AND OBJECTIVES: Coronavirus-19 has affected carbon emissions, which was declared as a pandemic by World Health Organization. Unprecedented environmental effects are being caused by Bangladesh's strict lockdown policies, which were implemented to stop the spread of Coronavirus-19. However, it is still unclear how the temporary halting and restart of industrial and commercial activities will affect the environment. In this study, it has been identified how Coronavirus-19 determinants like lockdown, daily confirmed cases, and daily confirmed deaths affect greenhouse gases. METHODS: From March 18, 2020 to February 4, 2022 the data series is used for Bangladesh. To ensure that the data series were stationary, the Augmented Dickey–Fuller and Phillips–Perron tests were utilized. Johansen co-integration test was utilized to determine co-integration among variables. The Granger causality test was utilized to identify directional causes and effects between Coronavirus-19 determinants and carbon emissions and the Vector Error Correction Model was employed to determine short-run and long run connections. FINDINGS: The study finds a bidirectional relationship between lockdown, carbon emissions and daily confirmed deaths, while a unidirectional association exists among Coronavirus-19 confirmed cases according to the Vector Error Correction Model. The Granger causality test also established the relationship between variables, except for daily confirmed cases. The pandemic's onset and subsequent lockdown resulted in decreased carbon dioxide emissions. The short-run link of carbon dioxide emissions with newly confirmed cases was corroborated by the directional relationship of variables, whereas there was a long-term and short-term association between confirmed deaths and lockdown. CONCLUSIONS: The reduction in carbon emissions during the pandemic will not be long-lasting because it is anticipated that global economic activity will gradually return to the pre- Coronavirus-19 state. The directional and relational nature of lockdown offers the potential to connect carbon dioxide emissions to regular lives. During a lockdown, there is a connection between the atmosphere's changes and how natural organisms behave. Importantly, there is a room for investigation into how communities of organisms and the atmosphere would function without humans. The essential point is to stress that during the lockdown, the ecosystem is self-healing. Environmental activists and business people will find this study useful in developing future sustainable improvement strategies.

BACKGROUND AND OBJECTIVES: Waste remains an issue in tandem with the development of the local community. The quantity of waste that is stockpiled in the landfill impacts the amount of leachate, resulting in emissions and reduced landfill capacity. The main challenge for its management is choosing the most cost-effective method to minimize leachate and emissions and increase the amount of waste that is stockpiled, resulting in a longer service life of the landfill. This study aimed to select the treatment at a landfill site.METHODS: Field observations and sampling of waste composition were carried out at the Klaten Regency. Waste composition sampling was carried out over several years. Material flow analysis was used to calculate the amount of leachate, emissions, and waste in the landfills. The effectiveness and benefits of the treatment scenarios were compared.FINDINGS: The waste consists of 55 per cent organic, 24 per cent plastic, 10 per cent paper, 3 per cent wood, 2 per cent cloth, 1 per cent glass, 1 per cent metal, and 4 per cent others. The processing scenarios were determined based on this composition. Four prospective scenarios were identified: 1) waste processing with composting; 2) composting and reuse, reduction, and recycling; 3) waste to energy; and 4) the combined process of scenarios 1 – 3. All treatments carried out can reduce leachate by 5.09 – 14.32 per cent, emissions of 11.31 – 44.48 per cent, waste 14.13 – 65.97 tons/day in the landfill, and can extend the service life of the landfill by 3 – 14 years.CONCLUSION: Material flow analysis was used to calculate the waste processing, emission rate, and leachate production from the four processing scenarios. The reduction of leachate and emission was affected by the treatment used. Combined processing (scenario 2 or 4) can reduce leachate and emissions and extend service life. The selected processing alternative must also consider the benefit-cost ratio. Scenarios (2) and (4) have a benefit-cost ratio of more than 1, which means that the processing is feasible to implement. Scenario 4 has a higher investment cost; so, the scenario that can be applied to the Troketon landfill is scenario 2 with a small investment cost, capable of reducing polluters, extending the landfill's service life to more than 4 years, and a benefit-cost ratio of more than 1.

BACKGROUND AND OBJECTIVE: Air pollution is associated with population growth and economic advancement. Severe cardiovascular complications that require extensive medical service are aggravated by air pollutants. This study illustrates the trend and correlation of cardiovascular disease hospital admission with air pollutants in Sabah for the past 9 years (2010–2019). The additional information obtained from this study will be useful to enhance proper environmental management and reduce air pollution in the cities of Sabah.METHOD: Ecological study design was utilized with cardiovascular disease hospital admission and ambient air pollutants in Sabah retrospective data. Data were collected from four districts with established continuous air quality monitoring stations. Collected data were analysed spatially and statistically. Autoregressive integrated moving average modelling was implemented to forecast the cardiovascular disease hospital admission.FINDING: Kota Kinabalu recorded the highest hospital admissions for cardiovascular disease, followed by Sandakan, Tawau and Keningau. The cardiovascular disease hospital admission prevalence rate in Kota Kinabalu was 12.45 per 1,000 population, followed by Sandakan, Tawau and Keningau (4.54; 4.18; and 5.88 per 1,000 population) in 2019. The cardiovascular hospital admissions increased in Kota Kinabalu, Sandakan and Tawau. The nitrogen dioxide (<0.04 ppm), carbon monoxide (<9 ppm), ozone (<0.05 ppm) and PM10 (<100 µg/m3) gases detected are below the national standard limit levels. In the later years of the series, the ozone and fine particulate gases intensify. Carbon monoxide has the highest positive correlation with cardiovascular disease hospital admission compared to other air pollutants. The autoregressive integrated moving average (0,1,1) with carbon monoxide and ozone as external regressors is the model with minimum Akaike information criterion.CONCLUSION: The carbon monoxide concentration in ambient air illustrates a potential risk for the increasing cardiovascular disease hospital admission number in Sabah. The study findings provide evidence-based source for the healthcare management team, policymakers, and community to sustain clean and safe ambient air.

BACKGROUND AND OBJECTIVES: Chronic exposure to fine particulate matter may cause adverse health impacts on humans. The impact of fine particulate matter collected in the industrial area was explored. Therefore, this study aimed 1) to assess the levels and spatial distribution of fine particulate matter and 2) to estimate the health risks due to the exposure of fine particulate matter in the population surrounding the Maros cement industry.METHODS: Fine particulate matter measurement was carried out using direct reading HAZ-Dust Environmental Particulate Air Monitor 5000. This study used the Human Health Risk Assessment method from the United States Environmental Protection Agency to estimate health risks. The Monte Carlo Simulation model was used to estimate the uncertainty and sensitivity analysis of parameters.FINDINGS: The average fine particulate matter concentration was 23.68 micrograms per cubic meter, above the air quality guidelines of the World Health Organization. However, the Monte Carlo Simulation to assess the health risk with the 95th percentile demonstrated that children and adults are at low risk for developing adverse health effects. The result of sensitivity analysis showed that duration of exposure (27.0%) and concentration of fine particulate matter (25.7%) were the most contributing factors to health risks in adults and children, respectively. This new approach determines the critical factors with major effects on reducing the health risk of the vulnerable population.CONCLUSION: Fine particulate matter poses health risks to adults and children, despite the calculated risks are still acceptable. Thus, limiting exposure duration and maintaining fine particulate matter levels in the residential area are needed.