A \(k\)-plane tree is a tree drawn in the plane such that the vertices are labeled by integers in the set \(\{1,2,\ldots,k\}\), the children of all vertices are ordered, and if \((i,j)\) is an edge in the tree, where \(i\) and \(j\) are labels of adjacent vertices in the tree, then \(i+j\leq k+1\). In this paper, we construct bijections between these trees and the sets of \(k\)-noncrossing increasing trees, locally oriented \((k-1)\)-noncrossing trees, Dyck paths, and some restricted lattice paths.

TEMO = topological effect on molecular orbitals was discovered by Polansky and Zander in 1982, in connection with the eigenvalues of molecular graphs. Eventually, analogous regularities were established for a variety of other topological indices. We now show that a TEMO-type regularity also holds for the Sombor index (\(SO\)): For the graphs \(S\) and \(T\), constructed by connecting a pair of vertex-disjoint graphs by two edges, \(SO(S) < SO(T)\) holds. Analogous relations are verified for several other degree-based graph invariants.

This study focused on line integral and its applications. The study was designed to show the areas where line integral is applicable and point out the role of line integral in solving practical problems. The study found out that space curves, and the concepts of scalar and vector fields are basic concepts to deal line integral. Also, the study found out that line integral is used to calculate mass, center of mass and moments of inertia of a wire, work done by a force on an object moving in a vector field, magnetic field around a conduct, voltage generated in a loop, length of a curve, area of a region bounded by a closed curve, and volume of a solid formed by rotating a closed curve about the \(x-\)axis.

Enhancement of heat and mass transfer heat over rotating plates in industrial processes is a major area of research recently due to several attempts to find cost-effective means. In this study, the flow of Williamson fluid is considered because of its ability to exhibit pseudo-plastic and shear-thinning properties. A theoretical analysis of the effect of Coriolis force and the angle inclination on the magnetohydrodynamic flow of Williamson fluid is considered. The flow is modelled by including Coriolis force and angle of inclination in the Navier-Stokes equation. By adopting a suitable similarity transformation, the system of governing partial differential equations is reduced to a system of ordinary differential equations which are solved using bvp4c solver in MATLAB. The simulations are depicted as graphs and it is found that velocity increases with increasing Coriolis force while it decreases as the magnetic field strength and inclination angle increases. Also, the local skin friction reduces as the rotation increases. Hence, to boost heat and mass transfer in the flow of fluid over a rotating inclined plate in a magnetic field, it is recommended that rotation should be increased and magnetic field strength should be reduced.

In this paper, by using \(t\)-norms, we introduce fuzzy subalgebras and fuzzy \(d\)-ideals of \(d\)-algebra and investigate some properties of them. Moreover, we define the cartesian product and intersection of fuzzy subalgebras and fuzzy \(d\)-ideals of \(d\)-algebra. Finally, by homomorphisms of \(d\)-algebras, we consider the image and pre-image of them.

Water scarcity comes with its attendant socio-economic effects. This paper evaluated the potential water supply and sanitation facilities in Dagbolu-Nigeria to boost economy. Facilities survey was carried out for the study including information from water, sanitation and health institutions across the State. GIS approach was used to depict the groundwater recharge potential of the study area.15.3% of the populace have access to potable water while 32.5% have access to good sanitation. Water-related diseases in the area are preponderantly malaria (81.2%), diarrhoea (8.41%), typhoid fever (3.40%), dysentery (3.22%) and cholera (2.76%). Annual loss due to unproductive downtime sickness in the Nigeria is estimated to be N 414,763,442,768:00. Improve management through effective policies of water resources leading to good water supply, hygiene and sanitation will enhance development or our underdeveloped countries.

Wastewater discharged by dye manufacturing and textile finishing industries has become an environmental concern. The textile dyeing plants utilize a variety of synthetic dyes and dump massive amounts of dyeing effluent because the uptake of these dyes by fabrics is very low. The plant’s photosynthetic activity is significantly harmed by this highly colored textile dyeing effluent, impairs aquatic life because of its low light penetration and oxygen consumption. Owing to the presence of heavy metallic materials and chlorine in synthetic dyes, it could also be harmful to some aquatic creatures. Therefore, these textile wastewaters need to be treated before their discharge. Various techniques for dealing with textile dyeing effluent have been discussed in this paper. Treatment techniques presented in this study include oxidation methods, physical methods, and biological methods. Also, the paper is prepared to compile all the updated data on textile dyeing effluents’ characterization and their impact on the environment from various journals and websites and some from personal communication with some factories. Since an extensive range of synthetic dyes, namely, azo dye, vat dye, reactive dye, disperse dye, is widely used in the textile industry, some of the dyeing effluents’ physicochemical parameters surpassed their standard limits. Hence, these days, the proper monitoring and corrective steps such as the elimination process have become the most thoughtful tasks globally, particularly the developing and transition economies. It is crucial to take immediate action to minimize environmental emissions due to the discharge of untreated textile dye waste.

Prevalence of Obstructive sleep apnoea (OSA) has been increasing owing to increase in its risk factors. The pathophysiology of OSA is related to narrowing or obliteration of upper airway. Neck circumference (NC) is a measure which predicts the neck fat and hence narrowing of airway. Modified Berlin’s Questionnaire is a validated tool to assess the risk of OSA in Indian setting. The present study was designed to determine the association of NC with the risk of OSA in young adults. 1500 subjects of both genders between 18 to 30 years of age were recruited, their NC was measured and they were administered screening questionnaire first. Those who gave one or more positive response were administered modified Berlin questionnaire and risk of OSA was assessed and it was compared with NC. There was a significant increase in risk of OSA in subjects with higher NC with a cut off of 36 cm in males and 32 cm in females. NC is an important anthropometric measurement to predict risk of OSA.

Background: The aim of the study is to evaluate effect of type 1 tympanoplasty on the quality of life of CSOM patients.
Materials and Methods: In this observational study, ninety- two patients of chronic suppurative otitis media with perforations of either genders were included. The chronic otitis media-5 (COM-5) questionnaire was administered to all patients. Air bone gap, physical suffering scores, hearing loss score, caregiver’s concern scores, emotional distress score and activity limitation scores were recorded.
Results: Side involved was right in 58 and left in 34. Location of perforation was anterior quadrant in 15, posterior quadrant in 7 and central perforation in 70 cases. Pre- operative physical suffering scores was 3.74 and post- operative score was 2.10, hearing loss score was 3.24 and 1.28, caregiver’s concern scores was 2.34 and 1.12, emotional distress score was 1.92 and 1.04 and activity limitation scores was 1.62 and 1.10 respectively. Pre-op mean air bone gap was 32.2 db and post- op mean air bone gap was 18.4 db. The difference was 13.8 decibels.
Conclusion: Type 1 tympanoplasty is common surgical procedure performed in patients with chronic suppurative otitis media and it found to be effective in improving quality of life in these patients.

Aim: A clinicopathologic investigation study of lichen planus.
Methods: This prospective and observational study was carried out in the Department of Oral Pathology & Microbiology, S.B.Patil Dental College \& Hospital, Bidar, Karnataka for a period of 1 year. 90 cases of Lichen Planus (LP) were included in this study. A complete clinical history was obtained, including duration, place of start, symptoms, medication history, and family history. A thorough general checkup, systemic examination, and dermatological examination were performed. The form and location of skin lesions, as well as the existence of any other related disorders, were all documented. Mucosa, hair, nails, palms, soles, and extremities involvement were all noted as concomitantly affected. To assess the histological diagnosis, haematoxylin and eosin stained sections of lesional tissue biopsies were generated. Once the histopathology slide was complete, each slide was attentively examined, and the findings were thoroughly analyzed and appraised.
Results: Out of 90 patients 55(61.11%) were male and 35(38.89%) were female and male to female ration were 1:1.57. Most of the patients belong to 30-40 years of age group (38.89%), followed by 20-30 years (26.67%), below 20 years (22.22%) and 12.22% above 40 years. 45 individuals had LP for less than 5 months before presenting to our out-patient service. The length varied from 5 to 11 months in 22 cases. The duration was longer than 24 months in 15 individuals. The length varied from 11 to 17 months in 5 individuals and from 17 to 24 months in the remaining 3. The classical type was encountered in 51 (56.67 percent) of the 90 Lichen Planus(LP) patients, followed by the hypertrophic type in 23 (25.56 percent) patients, linear LP and lichen planopilaris (LPP) in 4 (4.44 percent) patients each, oral and annular 2(2.22 percent) patients each, and follicular, bullous LP and LP pigmentosus in 1 (1.11 percent) patients each. Of the 90 patients in our research, 40 showed oral signs of LP, and four of them were diagnosed with just oral LP devoid of cutaneous indications of LP. Buccal mucosa was usually implicated in the oral cavity in 30 (75%) patients, lips in 10 (25%) individuals, and tongue in 7 (17.5%) patients. In our study, only two clinical kinds of oral LP were identified: the reticulate type in 35 (87.5 percent) of the subjects and the erosive variation in the other 5 people (12.5 percent ).
Conclusion: The current study sheds light on the clinicopathological behaviour of lichen planus patients according on their gender. Further research in this area will aid in the exploration of this uncharted territory, allowing for faster diagnosis and better treatment. LP was usually noticed in people in their forties and fifties. In our analysis, classical lesions were the most prevalent, followed by hypertrophic and linear types.