A Statement of Purpose (SOP) can be referred to as a document including all the blanket statements about an applicant’s life story. In other words, an SOP encompasses every relevant positives-and-negatives of the applicant’s academic life. An SOP, otherwise known as the Statement of Intent or Academic Statement, can be one, two, or three pages in length. The page-limit, ultimately, depends on the university or grad school that you are applying to; the lower the limit, the more concise and succinct your SOP has to be to cover all relevant minutiae in a single page. On the contrary, the longer your SOP is, the more difficult it is to hold the reader’s attention. …


Why do we need to create tables using commands and environments in LaTeX when someone already took the toll to generate the built-in functions that give out a beautiful user-interface (UI) to create tables to our satisfaction, like in Microsoft Word?

Literally, we are just a few clicks away from creating our own tables. Well, it is not always about coding, at times, it is about creating with one’s own satisfaction; also, sometimes complex tables, which make our MS word document fall apart all of a sudden, need special attention to not let the document go full bizarre. Although it is said that LaTeX tables are for technical documents, I personally think creating tables in LaTeX can help an individual perceive the underlying mechanism (codes and built-in functions) of how your clicks on the UI do justice to the document for creating beautiful tables. So, in simpler words, clicks on the UI are tantamount to the codes being interpreted or compiled in the background. So, you will get to see some of these codes in this article. …


So to initiate, we need to declare the type of document we want to have our project printed in; that is, whether we want an article or a book-type document.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}

Then, we need some packages to help the LaTeX compiler understand the lines we are going to write. \usepackage{amsmath}- this is the package we need to use some commands, such as double integral, triple integral, and many more. AMS stands for American Mathematical Society.

There are 2 ways to write mathematical expressions in LaTeX:
1. In-line Mode: $ . . . . . $ or \(….\)
2. Display Mode: $$ . . . . . . …


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(Prologue: In order to understand how mathematical formalism in Quantum Mechanics works, we have to get a good grasp on the Hilbert Space, and for that, we need to know what Space is in terms of mathematics before jumping onto the Hilbert Space. Although this is a detour chapter that mostly ignores the charming word ‘Quantum’, this will not fail to mesmerize you, for sure.)

Space is what we see around us. In a general context, space means some kind of distance; this can be verified when your fiance says, “I need some space. Don’t call or text me.” Jokes aside, space really means distance, at least from a layman’s perspective. But, mathematics does not concur with this notion. In mathematics, the term space is a set of objects which follows some ground rules and postulates or properties. …


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Quantum Mechanics- a hard nut to crack? Let’s break it down to its root words- Quantum and Mechanics. Let’s start with Mechanics; this is related to the mathematical formalism of things- from a grain of sand to the moon and so on- with which the whole universe can be described. Quantum basically refers to the smallest discrete unit of any physical property, such as energy or matter. So, a combination of these two words results in something that describes how everything in this world works; it is like a panacea for everything in this universe. In other words, QM describes how and why atoms and sub-atomic particles behave in certain manners. Moreover, it is a great way to perceive how the microscopic world affects the macroscopic world- the world in which we live and see. …


HAL/S (High-order Assembly Language/Shuttle) is a programming language developed for NASA in the early 1970s by Intermetrics. HAL/S is regarded as one of the then real-time, high-level programming languages for onboard programs of Spaceflights. Almost 85% of the programs used in the Space Shuttle programs were written in HAL/S.

At the time of the emergence of HAL/S, almost every program of onboard spacecraft was written either in assembly languages or in interpreted languages. But the slowness of interpreted languages and the difficulty of writing programs in assembly languages made NASA come up with a new high-order language that can be used in future projects as well. …


Constants in big O notations are usually ignored because they don’t add much to the complexity when the input size is very large. But, they do have significant meaning when the input is small or when we want to make a comparison between different algorithms. Let’s discuss these two scenarios in this short-but-somewhat-useful treatise.

In my previous write-up, Big O Notation, I tried to present big O notations in a fathomable way. When we determine O( ) for an algorithm, we tend to ignore constants since when the input is very large, these constants don’t matter much.

Constants matter for the following cases:

Small Input Size:

Let’s take T(n) = 5n² + 6n + 7, where T(n) becomes 2127 if n = 20. But, if n = 2 X 10⁴, then T(n) becomes 2000120007. Furthermore, if n becomes larger and larger, then T(n) will also be larger but the constant term (7 in this case) makes a lesser and lesser contribution to T(n). If the constant was 10 instead of 7, then T(n) would be 2130 instead of 2127 (for n = 20), and 2000120010 instead of 2000120007 (for n = 2 X 10⁴). In the first case where n is small, the constant term makes a somewhat significant contribution, but in the latter case, both values are quite similar (2000120007 and 2000120010). So, this is why constants are neglected while computing Big O for an algorithm since Big O is concerned with large inputs. …


When we write an algorithm to get something done, the compiler takes up some resources from our computer to execute the code. These resources are not limitless and they should be checked so that minimal resources are taken within each execution. In this short article, we are going to traverse through the beautiful concepts of how we can speed up our algorithms if the input size is large for which the compilation takes a significant amount of time.

There’s a caveat here; we are not dealing with improving our code’s runtime, rather we are more concerned with how our code’s runtime will vary if and only if the input is large. To exemplify this, let’s consider we want to develop an algorithm that can sort all the netizens of a densely-populated country on the basis of their annual income. This is a huge task since the input size might be in the order of millions or billions. So, now we need to think about the Time Complexity of our code so that we don’t end up losing much of the resources (Time and Space- here we will accentuate on Time). Moreover, this algorithm might be a 10-line code but that doesn’t mean we can ignore the total time needed to run this code for all the netizens since the code is very short. …


Think about a box comprised of people from a conservative cultural background that has been undergoing some changes due to the advent of external stimuli, such as, but not limited to, a foreign norm. Well, those people will start disseminating their own opinion on this new culture; some will accept and some not. This can hamper personal relationships among people within the community. For example, many will move from one social group or circle to another only to justify their stance. Don’t be so skeptical, it’s just a metaphor. …


We all know that one couch-potato who advertently does not tend to move out from the couch because of his exigency to binge-watch some trendy Netflix show. That’s fine, and I oftentimes make myself turn into this potato to absorb those fancies as well. Well, you know how to kickstart that person (assuming you are not the couch-potato).

So, you are applying Newton’s laws of motion, although it sounds enormously weir, to make someone move. Duh? Classical Mechanics!! You are kicking the atoms and molecules to move to a new position. They don’t move as much as they should’ve been since you are, presumably, a macho-person. But, in the atomistic framework, the atoms move in a different direction depending on the force applied to them. …

About

Afnan Mostafa

Science-savvy

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