Achyut Bharadwaj, Saee Patil, Valentio Iverson, Danny LiAugust 2021
Acknowledgements
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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
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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 $0$. Degree
of a polynomial $f$ will be denoted as $\text{deg}(f)$.
...Achyut Bharadwaj, Saee Patil, Valentio Iverson, Danny LiAugust 2021
Acknowledgements
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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
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Finding Patterns
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(1) $$1,4,7,10,\cdots\;\;?$$
Here’s a harder one:
(2) $$3,4,6,9,13,\cdots\;\;?$$
Notations
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(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 $3,4,6,9,13\cdots$?
...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
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Angular Displacement
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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. 1)
...