Achyut Bharadwaj, Saee Patil, Valentio Iverson, Danny LiAugust 2021
Acknowledgements # This work was carried out as part of a research project at PROMYS-2021. I am a grateful recipient of the Mehta Fellowship to the PROMYS programs in 2021 and 2022.
Definition and Preliminaries Notation # Let us define a polynomial $f(x)$ to be $$f(x) = \sum_{i = 0}^n a_i x^i$$ for some $a_i \in \mathbb{C}$, for all $0 \le i \le n$ and $n \in \mathbb{N}_0$. We define the degree of a polynomial to be the largest non-negative integer $n$ such that $a_n \not= 0$ and let degree of the polynomial $f(x) = 0$ be $0$.
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Achyut Bharadwaj, Saee Patil, Valentio Iverson, Danny LiAugust 2021
Acknowledgements # This work was carried out as part of a research project at PROMYS-2021. I am a grateful recipient of the Mehta Fellowship to the PROMYS programs in 2021 and 2022.
Preliminary Definitions # Finding Patterns # (1) $$1,4,7,10,\cdots\;\;?$$ Here’s a harder one: (2) $$3,4,6,9,13,\cdots\;\;?$$ Notations # (3) $$\Delta\left(f\right)(x)=f(x+1)-f(x)$$ (4) $$\Delta^{n+1}(f)=\Delta\left(\Delta^n(f)\right)$$ What is $\Delta\left(f\right)$ when $f$ is $1,4,7,10,\cdots$?
What is $\Delta\left(f\right)$ when $f$ is $3,4,6,9,13\cdots$?
What is $\Delta^2(f)$ when $f$ is $3,4,6,9,13\cdots$?
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Achyut BharadwajJanuary 2020
In this article, we will be going through some of the basic concepts required to understand Kepler’s laws.
Angular Displacement and Velocity # Angular Displacement # Just as displacement is defined as the change in position of an object, angular displacement is defined as the angle swept by an object travelling in a circular path.
Let’s look at an example. Say, an object travels from point $A$ to point $B$. In this process, say it covers an angle of $\theta$ (see Fig.
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