MATHS MAGAZINE ARTICLE

MATHS The golden ratio and symmetry, Every construction’s backbone is Geometry Algebra and statistics help us all, In creating structures like a school building, home or mall Mathematics has made everything possible, by identifying coordinates and making everything traceable It helps us make combinations and find the probability It goes by proof and enhances our ability To analyse and solve a problem, To learn to share by dividing candies and being awesome. We see mathematics in all little and big forms And it helps us find the probability of thunderstorms Symmetry always helps us all And makes beautiful all creatures great and small Trigonometry is used to find the distance between rivers We also use mathematics to find the height of pillars From counting stars and birds To learning area, volume and law of Surds. In real world it has imaginary numbers With maths in hand we can always create wonders It is just to use it in the right way All ideas are proved, none is stray It never stops but goes on till infinity Look around you and you'll find all examples of mathematics in your vicinity. Aishanya Sangwan IX-K

GEOMETRY Here we are again, Reviewing maths from 1 to 10. Starting from a point, a long way to go, Stay tuned, for this journey won’t be slow. It has no sides or lengths to say, In fixed positions it shall stay. It is represented by a tiny dot, A POINT on paper that we plot. They go on and on but never intersect, Like in Physics each other they reflect. They have no ends they are infinite vines, In Maths we call them PARALLEL LINES. It has a start but not an end, It doesn’t cross over or bend. Like a shining sun at the beach’s bay, What we have here is called a RAY. Two Rays meet at a point, Like in our body we have joints They stay apart and never tangle, Together, they both form an ANGLE. We took an angle to cut its ray, We added a line on its way. What we got was not a bangle, But with 3 sides we call it a TRIANGLE. 4 equal sides it has with him, Whom we can not move, touch or trim. Just like himself, he has a brother, SQUARE and RECTANGLE, quite similar to each other. Shapes, Lines, Angles and so much more, On paper they could seem like a hefty chore However, here and there MATHS is all around, And between these concepts a poem found. By: Muskaan Saxena 10 F

The Universe of Mathematics: In Verse In ancient times There were many great sages Brahmagupta, Bhaskar, Aryabhatta to name some Whose theorems will be remembered for times to come. The Greeks were not far behind Euclid, Pythagoras and Hypatia come to mind Gave us concepts, which to this day Are used to find our way. In the centuries of middle age The period when renaissance was a rage Mathematics grew by leaps and bounds Cantour, Euler, Newton made the concepts sound. A young boy named Terry Tao Inspired by the great Indian C. R. Rao Was well ahead of his time And found new patterns in the numbers Prime. Of all the great names I have said Based on the concepts till now I have read Euler is my favourite by a long run As he made mathematics so much fun. If you add the number of faces and vertices And the number of edges you subtract For a polyhedron that doesn’t intersect The answer is always two Which I will show by examples a few. In case of a cube, there are faces six Edges 12 and vertices 6 Now let’s do the sum and find out The answer is 2 without any doubt. You may try it with any other polyhedron May it be tetra or octahedron Thus we may like to conclude

That there can be only five kinds of Platonic solids Was proven by Euler Dude. LOMASH SHARMA Class X-M

ACTIVITIES OF MATHEMATICS DEPARTMENT DURING THE SESSION 2020-2021 The Mathematics Society of Delhi Public School R.K. Puram organised a series of Inter-section competitions for students of classes VI-XII which was wherein the students participated with great enthusiasm. “Design your Superhero” was an opportunity for students to bring their wildest imagination to structure. The second runner up for the event were Arnav Sakshya of VIC, Ridaan Sethi of VIH ,and Aditri Shree Dwivedi of VI I, first runner up by Advika Aggarwal of VIB,Shriya Sawhney of VIH and Reyansh Wadhwa of VIJ and the winners were Kamya Agarwal of VIA,Kashvi Singh of VI I and Baarika Ayekpam of VI K. “Symphonies of Mathematics” created a beautiful harmony between Mathematics and Music. The second runner up for the event were Rakchit Anand of VII I, first runner up by Anika Goel of VII C, Shivin Sareen of VII D, Darsh Sikka of VII I,Abhaya Trivedi of X O and Riddhi Gupta of X J and the winners were Inayat Kaur Bajaj of VII D, Poorvi Shankar of VII H, Atishay Jain of X F and Muskan Saxena of X F. “Powerpoint Presentation” was a way for participants to have full reign of how they wanted to present their enthusiasm. The second runner up for the event were Arnab Kumar Mullick and Manya of XI V, first runner up by Shashwat Raj of XI I and the winners were Aarushi Gupta of XI M along with Stuti Rastogi and Sunidee Jaiswal of XII N. The special mentions of this event were Atharv Agarwal and Rishit Gupta of XII M along with Arnav Kumar and Manhar Bansal of XII U. And lastly, “Magazine Cover Design” gave the freedom to design how they wanted their magazines to look.The second runner up for the event were Ananta Jha of XII C and Vartika Verma of XII P, first runner up by Trisha Ghosh of XII K and the winner was Sarthak Gupta of XII D. The special mentions of this event were Shamayita Biswas & Krish Kejiriwal of XI F. The event also included an intriguing Guest Speaker Programme conducted by Dr. Jonaki Ghosh on ‘Discovering the Beauty of Doing Mathematics’ for the students of classes XI and XII. Even the teachers of the Mathematics Department attended a workshop on Best Teaching Practices. On 22nd December, on the occasion of National Mathematics Day, a presentation prepared by Arnav Goel and Samridh Gupta, members of the Mathematical Society, was shown to students of all classes by their respective mathematics teachers.

Hands-on activities like sudokus and puzzles were given to the students of classes VI-VIII. Even with the pandemic,competitive exams like the International Foundation Mathematics Olympiad, SOF Mathematics Olympiad and Aryabhatta saw participation on a large scale from students from all classes. Maanya Sareen and Ananya Gupta of class XI secured international rank 1 in IFO Mathematics Olympiad whereas Arnab Goyal and Ashmit Nangia of class XI secured ranks 9 and 11 in 38th Aryabhatta Inter-School Competition. An Inter School Competition Sci-π National Science and Mathematics Festival was held on 29th December 2020 in which 62 schools all over the country participated. Competitions like Ekphrasis, which evaluated participants on how they portrayed the picture as a poem, Tira Comica, a comic strip designing competition, Experimathics, which looked over demos done by students on any mathematics or science concept and Innovative Lernen, on how to learn Mathematics or Science. The Festival ended with Lancers International School, Gurgaon and Delhi Public School, Sushant Lok, Gurgaon as the overall winners.













Mathematical Modelling of COVID-19 The Novel Coronavirus aka COVID 19, is a deadly infectious disease that originated from Wuhan in China, and was recognised as a global pandemic by the World Health Organization on 11th March 2020. Belonging to the family including SARS and MERS, the virus attacks the respiratory system of the human body. Due to the high clinical severity as well as transmissivity, it becomes a very crucial task to model each and every step of the pandemic and one that can be used to predict the spread of a pathogen in the society, the duration of the infection, the recovery period and so on. THE SIR MODEL The SIR Model, also known as a compartmental disease model, is a mathematical epidemiological model that computes the theoretical number of people infected by a contagious illness in a closed population as a function of time. The name SIR derives from the fact that at any point of time the complete population can be divided into three sets, Susceptible (people prone to catching the illness), Infected (people currently infected by the illness and can transmit it to others) and Removed (people that have recovered from the illness, gained immunity or died from the illness). As the pandemic progresses, the number of people in each set keep changing with respect to time. The rate of this change can be described by time-dependent ordinary differential equations as follows: (Where S, I and R are the sets of the population explained previously and N is the total population)

The key value governing the time evolution of these equations is the so-called epidemiological threshold or the basic reproduction number R0= β . γ After solving by integration we arrive at the following equation with which we can plot our model. However for a much accurate result, the values of both ꞵ and ������ are estimated by minimizing the MSE (Mean Square Error) between the actual and predicted values. THE LOGISTICS MODEL The logistics model is based on the logistic function, giving a sigmoid curve representing the rate of growth of the number of cases. The logistic curve is generally used to model population growth in a closed biological system, but its uses can be extended to epidemiology as well. A standard logistic function looks like the following: For our purposes of epidemic modelling, we can use the following variant of the logistic function, giving an equation describing the rate of change in the number of cases.

Where C represents the number of cases, r represents the rate of the infection (No. of people infected/day) and K represents the final/limiting epidemic size Solving the equation by isolating the variables and then integrating, we arrive at the equation giving our logistics model. Where, To estimate the final size of the epidemic (K) we can make use of Aitken's Δ2 Process/Shanks Transformation. Therefore if C1,C2,C3…….Cn are the total cases at time t1,t2,t3…….tn respectively, and the predicted values for K from this data are K1,K2,K3…….Kn. To achieve convergence faster, we can apply iterated Shanks Transformation (Given K>Cn). RESULTS OF THE MODELS 1. After completing modelling for all three countries, we collected all the graphs and noticed three distinct variations in the graphs for all the countries. The first country we modelled, i.e. India presented a general trend in the acceleration as well as the deceleration in the number of cases.

Due to this reason the graphs obtained for India show minimal error between the predicted and actual values indicating that India is following the standard shape of both the models. 2. The second variation we can see is that of the United States of America, in this case from the graphs given below it can be easily seen that more than two 'apparent' COVID waves are present. The best way to contain such waves is to enforce lockdowns as well as increase testing on a local scale, this is a very important factor that can lead to the control of these waves.

3. The third and the last case is that of China, where the cases has already approached the plateau/ending stage. This can be seen from the graph given below where we can see that the number of cases have flatlined. We also notice a small jump in the number of cases corresponding to an increase in the testing of COVID cases by China during the month of February.

HOW CAN WE STOP COVID During December and January, the current bed of roses situation of COVID-19 in India can change very easily before approaching the plateau stage. This is because of the fact that Coronavirus belongs to the SARS family of viruses (SARS-CoV-2) and is a respiratory illness thriving in winters. Experiments have also revealed the fact that coronavirus prefers cold and dry conditions, especially out of direct sunlight. During winter the virus thrives inside people's homes, because during those times houses are heated to around 20-25 0C and the air is not properly ventilated i.e. it is dry. These conditions can lead to a major spike in the number of cases. However these spikes can easily be kept in control by ramping up the testing for coronavirus, following social distancing rules strictly, and practicing respiratory and hand hygiene measures. The fact that most people stay indoors during winters can also be a major factor in this spike, this is because a fall in the Vitamin-D levels can lead to decreased immunity. This is why it is also advised by medical officials to stay in direct sunlight for at least half an hour daily. ADITYA MITTAL 11-E ASHMIT NANGIA 11-E KAVYA AGRAWAL 11-E