In mathematics, derivatives are the rate of change of a function with respect to a variable, and they are necessary for answering complex mathematical problems and differential equations. In this paper, a detailed study on the use and concept of derivative that how it comes into existence, how it can be used to calculate the differential coefficients of a function at a particular point in an effective manner, and what are the applications of the rate of change of functions in mathematics as well as in real-life situations is presented. This study will help to understand the in-depth concept of calculus to the new researchers and students of mathematics.

Key-Words / Index Term

1. Cadena, B. Travis, and S. Norman, “An evaluation of reform in the teaching of calculus,” Mathematics and Computer Education, vol. 37, no. 2, pp. 210–220, 2003.
2. M. Nahikian, W. A. Granville, P. F. Smith, and W. R. Langley, “Elements of the differential and integral calculus,” Nat. Math. Mag., vol. 16, no. 1, p. 54, 1941.
3. B. Keynes and A. M. Olson, “Redesigning the calculus sequence at a research university: issues, implementation, and objectives,” Int. J. Math. Educ. Sci. Technol., vol. 31, no. 1, pp. 71–82, 2000.
4. Rogers, Putting the differentials back into differential calculus. Washington, D.C: Mathematics Association of America, 2005.
5. Salas and E. Hille, Calculus one and several variables. New York, NY: John Wiley & Sons, 1990.

This paper aims at the importance of mathematical modelling, its growing role and its applications. It is a myth that modelling projects progress easily from working through to utilize, this is scarcely ever the situation. In computer science, the use of modelling and simulating a computer is utilized to fabricate a mathematical model which contains key boundaries of the actual model. The study aims at the use and advantageous application of mathematical modelling in the COVID pandemic such as to understand the transmission of the virus, its solution to prevent the growth of the virus among the population, calculating the infected populations and giving a slight idea on the future predicted scenarios. Thus, its role and applicability in the pandemic are discussed by framing a mathematical model case to understand the transmission of the virus at the stage of the beginning of the pandemic. Thus, the study aims to give a basic idea of mathematical modelling, its uses and its role in the recent scenarios.

Key-Words / Index Term

1. R. Waykar, “Mathematical Modelling: A way of life,” International Journal of Scientific & Engineering Research, Vol. 4, Iss. 4, pp 1336-1352, 2013.
2. R. Waykar, “Mathematical Modelling: A way of life,” International Journal of Scientific & Engineering Research, vol. 4, no. 4, 2013.
3. Zeb, E. Alzahrani, V. S. Erturk, and G. Zaman, “Mathematical model for Coronavirus disease 2019 (COVID-19) containing isolation class,” Biomed Res. Int., vol. 2020, p. 3452402, 2020.
4. ?. Nechuiviter, “Application of the theory of new information operators in conducting research in the field of information technologies,” Information Technologies and Learning Tools, vol. 82, pp. 282–296, 2021.
5. T. Smith, D. Rubin, and M. J. Turner, “Modelling dynamic response of physical systems in Simulink,” in 2018 3rd Biennial South African Biomedical Engineering Conference (SAIBMEC), 2018.
6. Mohamadou, A. Halidou, and P. T. Kapen, “A review of mathematical modelling, artificial intelligence and datasets used in the study, prediction and management of COVID-19,” Appl. Intell., vol. 50, no. 11, pp. 1–13, 2020.
7. “Mathematical Modeling,” ac.at. [Online]. Available: https://www.mat.univie.ac.at/~neum/model.html. [Accessed: 27-Dec-2020].
8. Borromeo Ferri, “Modelling problems from a cognitive perspective,” in Mathematical Modelling: Education, Engineering and Economics, C. Haines, Ed. Chichester: Horwood, 2007.
9. Blum and R. B. Ferri, “Mathematical modelling: Can it be taught and learnt?” Journal of mathematical modelling and application, vol. 1, pp. 45–58, 2009.
10. Blum, P. Galbraith, H. W. Henn, and M. Niss, Eds., “Education-Discussion document,” in Modelling and Applications in Mathematics Education, vol. 51, New York: Spring, pp. 149–171.
11. P. Daponte, “Necessary research in mathematical modelling and applications,” in Teaching and Learning Mathematics in context. Chichester: Horwood, T. Breiteig, Ed. 1993, pp. 219–227.

In the modern world, matrices are considered to be of extreme use as they can be applied in fields like Chemistry, Physics, Economics, Construction, Finance, etc. In this work, the key applications empowered by framework hypothesis in two significant fields of interest in graphic designing, in particular video game designing. The paper centers on how ma-trices and their operations play a significant role in designing and manipulating the structure on a screen to make it realistic. Among the significant applications in video games, it has a significant role in Adobe Photoshop how we put our structures in suggested dimensions and move according to our comfort in it and this is possible because of matrices. The study brings up the significant commitment made by cryptography, the security process it carries with it so that no other person in between can decode the message and the message is delivered safely to the intended person. The significance pretended by grids in addressing and handling advanced pictures is portrayed by a few illustrative applications. This paper covers many other applications in matrices of how it is used in chemistry, physics, medical science, geology, economics, and many more

Key-Words / Index Term

1. R. Camp and P. Y, “Secret codes with matrices,” Mathematics Teacher, vol. 78, no. 9, pp. 676–680, 1985.
2. Baker, Matrix Groups. London, England: Springer London, 2002.
3. A. Beauregard and J. B. Fraleigh, A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields. Boston: Houghton Mifflin, 1973.
4. I. Schiff and J. Bandhyopadhyay, “Buy Quantum Mechanics book online at low prices in India,” Amazon.in. [Online]. Available: https://www.amazon.in/Quantum-Mechanics-Leonard-I-Schiff/dp/1259062864. [Accessed: 06-Dec-2020].         
5. A, “Cayley A memoir on the theory of matrices,” Phil, vol. Trans, pp. 148 1858 17–37.
6. Arthur, The collected mathematical papers of Arthur Cayley. Charleston, SC: BiblioLife, 2009.
7. Anton and C. Rorres, “Elementary linear algebra applications version.” 1994.
8. Abraham, Elementary cryptanalysis a, a mathematical approach. Mathematical Association of America, mathematical library, 1966.
9. G. Konheim, Cryptography: A Primer, First Edition; Third Printing. Nashville, TN: John Wiley & Sons, 1981.
10. S. Blyth and E. F. Robertson, Basic linear algebra, 2nd ed. Guildford, England: Springer, 2002.
11. Anton, “Elementary Linear Algebra”, John Wiley & Sons, July 2019, ISBN: 978-1-119-62569-8.
12. Huang and S. G. Krantz, “On a problem of Moser,” Duke Math. J, vol. 78, pp. 213–228, 1995.
13. D. Kellog, “Harmonic functions and Green’s integral,” Trans. Amer. Math. Soc, vol. 13, pp. 109–132, 1912.
14. M. Rasulov and B. F. Fatulaev, Differ. Equ., vol. 38, no. 2, pp. 274–278, 2002.
15. “Doklady Mathematics,” Springer.com. [Online]. Available: https://www.springer.com/journal/11472. [Accessed: 06-Dec-2020].
16. “Introduction to Linear Algebra, 5th Edition,” Mit.edu. [Online]. Available: https://math.mit.edu/~gs/linearalgebra/. [Accessed: 06-Dec-2020].

The existence of Euler and Hamiltonian graph make it easier to solve real life problem. During the time of pandemic “Covid -19” it is very essential for each one of us to be vaccinated. Vaccination is done in the hospitals by using Eulerian and Hamiltonian graph not only prevent people from infecting but also increase the speed of vaccination. In this paper, authors have discussed the use of graph theory to effectively handle the covid – 19 pandemic situations in the allocation of patients to appropriate hospitals. The work done in this paper, may be utilized for the hospital – patient management as well as for the study by upcoming researchers to utilize the graph theory in any complicated situations.

Key-Words / Index Term

1. Sutaria, “A Hamiltonian and Eulerian cycles”, International Journal of Mathematics Trend and Technology, vol. 3, iss. 5, pp. 208-212, 2016.
2. Kumar and SHUATS, “A study on Euler Graph and it’s applications,” Int. j. math. trends technol., vol. 43, no. 1, pp. 9–15, 2017.
3. Ahmed, “Graph routing problem using Euler’s theorem and its applications,” Engineering Mathematics, vol. 3, no. 1, p. 1, 2019.
4. Fleischner, “(Some of) the many uses of Eulerian graphs in graph theory (plus some applications),” Discrete Math., vol. 230, no. 1–3, pp. 23–43, 2001.
5. S. Rahman and M. Kaykobad, “On Hamiltonian cycles and Hamiltonian paths,” Inf. Process. Lett., vol. 94, no. 1, pp. 37–41, 2005.
6. Keshavarz-Kohjerdi and A. Bagheri, “Hamiltonian paths in some classes of grid graphs,” J. Appl. Math., vol. 2012, pp. 1–17, 2012.
7. ?. Waligóra, “Application of Hamilton’s graph theory in new technologies,” World Scientific News, vol. 89, pp. 65–76, 2017.
8. A. L. T. Cherin Monish Femila, “Hamiltonian cycle and Hamiltonian path decomposition of fan graphs,” International Journal of Scientific Research in Mathematical and Statistical Sciences, vol. 6, no. 2, pp. 207–211.
9. Alhalabi, O. Kitanneh, A. Alharbi, Z. Balfakih, and A. Sarirete, “Efficient solution for finding Hamilton cycles in undirected graphs,” Springerplus, vol. 5, no. 1, p. 1192, 2016.
10. ChandraBora and B. Kalita, “Particular Type of Hamiltonian Graphs and their Properties,” Int. J. Comput. Appl., vol. 96, no. 3, pp. 31–36, 2014.

An investigation has been made on evaluating the effectiveness of aromatherapy on depression among senior citizens residing in selected oldage homes. Pre experimental one group pre-test and post-test design was used for this study. The data collections were carried out over the period of 30 days. About 40 senior citizen subjects were selected through convenience sampling technique. In pre-test, Geriatric depression scale was used for assessing the level of depression among senior citizens. Aromatherapy was given by means of applying 5 drops of Neroli oil with 10ml of coconut oil at the forehead of all the subjects once in a day for 15 days. After completion of the intervention, the post test has been conducted among subjects. Descriptive and inferential statistics were used to analyze the findings of the study. It has been observed that a significant difference (P<0.01) in the mean score level of depression among senior citizens exist. The mean score were pretest 19.7 [± 4.22] and post test15.95 [± 4.74]. There was a significant association (P<0.05) was found between the depression score and selected demographic variables like gender, education, previous occupation, chronic health problems, marital status, support system. Henceforth, it has been claimed that, the aromatherapy helps to reduce the level of depression among the senior citizens.

Key-Words / Index Term

[1] Malik, A.S. and Amin, H.U. (2017) Chapter 2 - Mental Stress, in:  Designing EEG Experiments for Studying the Brain, Academic Press, 31-46.

[2] Soysal, P., Veronese, N., Thompson, T., Kahl, K.G., Fernandes, B.S., Prina, A.M., Solmi, M., Schofield, P., Koyanagi, A., Tseng, P.T., Lin, P.Y., Chu, C.S., Cosco, T.D., Cesari, M., Carvalho, A.F., Stubbs, B. (2017)  Relationship between depression and frailty in older adults: A systematic review and meta-analysis, Ageing Research Reviews, 36, 78-87.

[3] Sharp, L.K., Lipsky, M.S. (2002) Screening for depression across the lifespan: a review of measures for use in primary care settings. Am Fam Physician., 66(6), 1001-1008.

[4] Seyyed Rasooli, A. , Salehi, F., Mohammadpoorasl, A., Goljaryan, S., Seyyedi, Z., Thomson, B. (2016)  Comparing the effects of aromatherapy massage and inhalation aromatherapy on anxiety and pain in burn patients: A single-blind randomized clinical trial, Burns, 42, 1774-1780.

[5] Ali, B., Al-Wabel, N.A., Shams, S., Ahamad, A., Khan, S.A., Anwar, F. (2015) Essential oils used in aromatherapy: A systemic review, Asian Pacific Journal of Tropical Biomedicine, 5, 601-611.

[6] Holmes, P. (1995) Neroli, International Journal of Aromatherapy, 7, 14-17.

[7] Wang, K., Zhu, R.Z., Qu, R.F., Li, Z.Y. (2012)  Comprehensive two-dimensional gas chromatography–time-of-flight mass spectrometry for the analysis of volatile components in Neroli essential oil, Mendeleev Communications, 22, 45-46.

[8] Kim, I.H., Kim, C., Seong, K., Hur, M.H., Lim, H.M., Lee, M.S. (2012) Essential Oil Inhalation on Blood Pressure and Salivary Cortisol Levels in Prehypertensive and Hypertensive Subjects, Evidence-based Complementary and Alternative Medicine : eCAM, 2012,  984203.

[9] Lee, M.K., Lim, S., Song, J.A., Kim, M.E., Hur, M.H. (2017)  The effects of aromatherapy essential oil inhalation on stress, sleep quality and immunity in healthy adults: Randomized controlled trial, European Journal of Integrative Medicine, 12, 79-86.

[10] King, I.M. (1997) King's Theory of Goal Attainment in Practice, Nursing Science Quarterly, 10, 180-185.

[11] Steflitsch, W., Steflitsch, M. (2008)    Clinical aromatherapy, Journal of Men's Health, 5, 74-85.

[12] Yesavage, J.A., Brink, T.L., Rose, T.L., Lum, O., Huang, V., Adey, M., Leirer, V.O. (1982) Development and validation of a geriatric depression screening scale: A preliminary report, Journal of Psychiatric Research, 17, 37-49.

[13] Turten Kaymaz, T., Ozdemir, L. (2017) Effects of aromatherapy on agitation and related caregiver burden in patients with moderate to severe dementia: A pilot study, Geriatric Nursing, 38, 231-237.

[14] Fowler, N.A. (2006) Aromatherapy, Used as an Integrative Tool for Crisis Management by Adolescents in a Residential Treatment Center, Journal of Child and Adolescent Psychiatric Nursing, 19, 69-76.

[15] Buckle, J. (2003) Clinical aromatherapy essential oils in practice 2nd Edition.  New York: Churchill Livingstone.

[16] Battaglia, S. (2004) The complete guide to aromatherapy 2nd Edition.  Queensland, Australia: The International Centre of Holistic Aromatherapy.

[17] Dossey, B. M. and Keegan, L. (2009) Holisticnursing: A handbook for practice (5th Edition). Sudbury MA: Jones and Bartlett.

[18] Tisserand, R. and Young, R. (2013) Essential oil safety, 2nd Edition. Edinburgh, Scotland: Churchill Livingstone Elsevier.

[19] Halcon, L. (2014) Aromatherapy fundamentals for health professionals. University of Minnesota, Center for Spirituality and Healing.