Measurable Taylor’s theorem: an elementary proof

OMS-Vol. 7 (2023), Issue 1, pp. 321-324 Open Access Full-Text PDF
Gianluca Viggiano

Abstract:The Taylor expansion is a widely used and powerful tool in all branches of Mathematics, both pure and applied. In Probability and Mathematical Statistics, however, a stronger version of Taylor’s classical theorem is often needed, but only tacitly assumed. In this note, we provide an elementary proof of this measurable Taylor’s theorem, which guarantees that the interpolating point in the Lagrange form of the remainder can be chosen to depend measurably on the independent variable.

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On natural approaches related to classical trigonometric inequalities

OMS-Vol. 7 (2023), Issue 1, pp. 299-320 Open Access Full-Text PDF
Abd Raouf Chouikha

Abstract:In this paper, we establish sharp inequalities for trigonometric functions. We prove in particular for \(0 < x < \frac{\pi}{2}\) and any \(n \geq 5\) \[0 < P_n(x)\ <\ (\sin x)^2- x^3\cot x < P_{n-1}(x) + \left[\left(\frac{2}{\pi}\right)^{2n} - \sum_{k=3}^{n-1} a_k \left(\frac{2}{\pi}\right)^{2n-2k}\right] x^{2n} \] where \(P_n(x) = \sum_{3=k}^n a_k x^{2k+1}\) is a \(n\)-polynomial, with positive coefficients (\(k \geq 5\)), \(a_{{k}}=\frac{{2}^{2\,k-2}}{\ \left( 2\,k-2 \right) ! } \left( \left| {B}_{ 2\,k-2} \right| +{\frac { \left( -1\right) ^{k+1}}{ \left( 2\,k-1 \right) k}} \right),\) \( B_{2k} \) are Bernoulli numbers. This improves a lot of lower bounds of \( \frac{\sin(x)}{x}\) and generalizes inequalities chains. Moreover, bounds are obtained for other trigonometric inequalities as Huygens and Cusa inequalities as well as for the function \[g_n(x) = \left(\frac{\sin(x)}{x}\right)^2 \left( 1 - \frac{2\left(\frac{2 x}{\pi}\right)^{2n+2}}{1-(\frac{2x}{\pi})^2}\right) +\frac{\tan(x)}{x}, \ n\geq 1 \].

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Exploiting quadratic \(\varphi(\delta_{1},\delta_{2})-\)function inequalities on fuzzy Banach spaces based on general quadratic equations with \(2k\)-variables

OMS-Vol. 7 (2023), Issue 1, pp. 287-298 Open Access Full-Text PDF
Ly Van An

Abstract:In this manuscript, our primary focus revolves around extending the inequalities associated with the Quadratic \(\varphi(\delta_{1},\delta_{2})-\)function. Our approach involves leveraging the general quadratic functional equation encompassing \(2k\)-variables within the context of the fuzzy Banach space. Our main contribution lies in the expansion of these inequalities, representing a significant result within this study.

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The bounds for topological invariants of a weighted graph using traces

ODAM-Vol. 6 (2023), Issue 3, pp. 30 – 34 Open Access Full-Text PDF
Emre Sevgi, Gül Özkan Kizilirmak and Serife Büyükköse

Abstract: In this paper, we obtain the bounds for the Laplacian eigenvalues of a weighted graph using traces. Then, we find the bounds for the Kirchhoff and Laplacian Estrada indices of a weighted graph. Finally, we define the Laplacian energy of a weighted graph and get the upper bound for this energy.

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A note on the characterization of claw-free and paw-free graphs

ODAM-Vol. 6 (2023), Issue 3, pp. 26 – 29 Open Access Full-Text PDF
Phillip Mafuta and Josiah Mushanyu

Abstract: A number of results on claw-free, paw-free graphs have been presented in the literature. Although the proofs of such results are elegant, sound and valid, it has gone unnoticed that all the results about claw-free, paw-free graphs in the literature are a consequence of a result by Olariu [1]. The note, apart from covering the aforementioned gap, also provides an alternate proof to a result by Faudree and Gould found in [2] in that, an unnoticed consequence resulted in the characterization of claw-free, paw-free graphs.

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On the super edge-magicness of graphs with a specific degree sequence

ODAM-Vol. 6 (2023), Issue 3, pp. 22 – 25 Open Access Full-Text PDF
Rikio Ichishima and Francesc-Antoni Muntaner-Batle

Abstract: A graph \(G\) is said to be super edge-magic if there exists a bijective function \(f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left( G\right) \right\vert \right\}\) such that \(f\left(V \left(G\right)\right) =\left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert \right\}\) and \(f\left(u\right) + f\left(v\right) + f\left(uv\right)\) is a constant for each \(uv\in E\left( G\right)\). In this paper, we study the super edge-magicness of graphs of order \(n\) with degree sequence \(s:4, 2, 2, \ldots, 2\). We also investigate the super edge-magic properties of certain families of graphs. This leads us to propose some open problems.

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Lattice path coronoids

ODAM-Vol. 6 (2023), Issue 3, pp. 1 – 21 Open Access Full-Text PDF
Tricia Muldoon Brown

Abstract: Coronoids are nice chemical structures that may be represented mathematically in the planar hexagonal lattice. They have been well-studied both for their chemical properties and also their enumerative aspects. Typical approaches to the latter type of questions often include classification and algorithmic techniques. Here we study one simple class of coronoids called hollow hexagons. Notably, hollow hexagons may be represented with a collection of partitions on the set \(\{2,3,4,6\}\). The hollow hexagons are used to classify another family of primitive coronoids, which we introduce here, called lattice path coronoids. Techniques from lattice path enumeration are used to count these newly-defined structures within equivalence classes indexed by enclosing hollow hexagons.

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Unveiling the status of wash in secondary schools located in Saurashtra region of Gujarat-India

TCMS-Vol. 3 (2023), Issue 3, pp. 29 – 37 Open Access Full-Text PDF
Eshwar Gupta, Dipesh parmar, Monika Patel, Jitesh Mehta and Dhaval padaliya

Abstract:In developing countries, Diseases related to inadequate water, sanitation, and hygiene are a huge burden. It is estimated that 88% of diarrheal disease is caused by unsafe water supply and inadequate sanitation and hygiene. The future generation of a nation spends most of its time in schools, and since the status of the WASH component in school is strongly linked to Learning and health. Therefore, Objectives of the study was to assess the status of availability, accessibility, functionality, usability, and quality of WASH standards in secondary schools located in a Municipal corporation area of Saurashtra region. Methods: The study was carried out in 34 secondary schools in the study city for a period of 18 months. Data was collected using a pre-validated, standard tool in the form of a questionnaire prepared by WHO and UNICEF and analysed. Results: Out of 34 schools, the majority (24) of the schools had piped water supply as their primary source of drinking water, 70.59% had hand washing stations available, but the availability of both soap and water was observed only in 3 government and 11 non-government schools. Menstrual hygiene management like Covered bins facilities were found only in 18 schools, with a higher proportion in non-government schools; 73.52% of schools had adequate distribution of students per toilet, but the norms for girls per toilet were only met in non-government schools. A wide gap was found between the availability of toilet facilities, usability, and functionality. Conclusions: While the availability of the structural component of WASH was satisfactory in most of the schools (with a better proportion in non-government ones), its functionality, usability, and quality were not uniform among schools.

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The correlation of quadriceps angle with respect to bicondylar distance of femur and body height measurement in north indian adult human population

TCMS-Vol. 3 (2023), Issue 3, pp. 21 – 28 Open Access Full-Text PDF
Sumit Kumar, Sudha Srivastava, Pawan Kumar Mahato

Abstract:Background: Before taking the dimension, the persons were positioned such that the hips and knees were in extension, the quadriceps muscles were relaxed, and the legs and ankles were neutral. Following mark¬ing, one of the arms of the manual goniometric was placed such that the ASIS and the patellar middle point were combined, and the other arm was placed so that the tuberose as tibia and the patellar centre were collective. The dimension value was recorded after the manual goniometry was positioned. \textbf{Aim and Objective:} The objective of the study is to find out the bilateral variation in the mean Q- angle (Quadriceps angle) values in both male and female. This study is to determine the correlation between Q- angle (quadriceps angle) values, Bicondylar distance and body height measurement. Material and Method: The present study was done on 500 adult human subjects (300 male and 200 female) of north Indian origin. The Q angle is precise with a full circle universal manual goniometry which is made of clear steel with the subject standing in the upright weight-bearing position. Result: The dimension of Quadriceps angle values in males from right side (mean 13.25) and from left side (mean 13.41) , the Q- angle values in females from right side (mean 16.97) and from left side (mean 16.35) (P<0.0001). The effect shows that the Q- angle was more prominent in females than in males. The difference of Q-angle with dominant side revealed that left side of Q- angle value (both males and females) was more often higher than right side. Conclusion and Discussion: The Quadriceps angle values of both males and females decreases with increases in body height which shows a positive correlation with body height and condylar distance. Positive correlation exists between Quadriceps angle and Body height.

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