## Calvin and Hobbes

Calvin’s view on Mathematics…

## Plagiarism Checker

What is plagiarism? It is “the practice of taking someone else’s work or ideas and passing them off as one’s own.”

In this age of blogs, Facebook pages and Twitter, plagiarism is an unwanted but common practice. Recently, I applied for an internship on Content Writing at a company. They selected me and a few other people and I received a mail with a bunch of terms and conditions that must be met by me to get a stipend. One of the rules said that I must check whatever I wrote on a particular plagiarism checker on the net, viz. this, and my article must be 95% unique. Fair enough, I thought, but I was curious about how the checker works, etc. So, I clicked on the link and went to the site. I copied a short story I had written not too long ago and added it there. It came out to be 88% unique. Uh-oh. Then I copied an article from this blog on Mr Perelman. That came out to be 40% unique. What the! Then I copied the Windows 10 vs Ubuntu blog. And this came out to be 4% unique. Oh my God!

But then I realized, of course! It found my blog post on the net, that must be the problem. Then I delved into how they were checking plagiarism. They were searching Google for phrases. They were using exact search, of course, with inverted commas attached. But the problem was, even if Google couldn’t find anything, Google was removing the inverted commas and searching the same phrase, and of course finding millions of results and saying did you mean this? The checker, however, did not realize that this was happening and it was counting the “suggested” search results too.

Thankfully, the company agreed to check the plagiarism themselves and not use the checker. But this made me realize something. Isn’t the Internet getting a bit too saturated?

I believe the checker has been improved since this happened because this post came out to be 97% unique and I can live with that.

## J S Milne

James S Milne is a professor of Mathematics at the University of Michigan, Ann Arbor. He mainly works in arithmetic geometry. He has written a few books on the subject as well as some related fields. He hails from New Zealand.

His personal website can be found here.

His site has the most awesome notes on several topics in Algebra, including Group Theory, Ring Theory, Fields and Galois Theory, Algebraic Number Theory, Algebraic Geometry, etc. Here’s a link. I read a few pages of his ANT notes in my first year and I loved it. I am thinking of completing that in the coming summer.

## Mid-Sem

My mid-semester exams are going on now. 5 exams in 5 days. Having given Algebra (Galois Theory), Graph Theory and (General) Topology exams in the first 3 days, I thought I would get some rest yesterday. But that was before I saw last year’s Statistics question paper. Thankfully, tonight I can rest easy. Tomorrow’s exam is Computer Science. That too, only on Octave/MATLAB.

I think the exams in my college are way way different than in other colleges. To prevent cheating, our question papers and answer scripts (!) are colored. Five colors in five days: pink, yellow (extremely bright), whitish green, sky blue and white (Thank God!). One of my classmates, S requested our Dean, Prof. Archimedes, not to give use the yellow shade because it literally hurts to look at it constantly for 3 hours. Prof. Archimedes is now sure that S used to cheat in exams and is now unable to do so because of the colors. We, on the other hand, are sure that Prof. Archimedes loves to play with colorful stuff like all other kids.

There is a small auditorium in our department in the college. That is the venue for our exams. There are 4 wall fans in 4 corners, which make such a lot of noise that they are practically useless. There are six (!) air-conditioners. Usually when I enter the exam hall 5 minutes before an exam, I see all three AC’s on my opposite side have temperatures of 18-20 degrees. Among the three on my side, two are usually on about 25 degrees while the one directly above my head is usually at 30+ degrees. God knows who does this and why.

On Monday, the exam had started when I realized the lights were all switched off. Apparently, Prof. Young, one of the invigilators was also aware of this but he just couldn’t find the switches. I sometimes wonder how most of the people in my colleges are geniuses in their fields but in daily life….

Having said all that, it’s funny when there’s a tree outside the window and sometimes monkeys can be seen in the tree. It’s extremely helpful when I can’t make head or tail of the question paper but I don’t want to leave the hall too early.

## Windows 10 vs Ubuntu 14.04.3 LTS

I used to be a hardcore fan of Windows. I had Windows 98 in my desktop. I upgraded it to Windows XP and I loved it. I got a laptop with XP and loved it. The laptop got infected and I formatted it and upgraded to Windows 7. (In the middle I used Windows Vista on my father’s laptop and managed to not hate it). I got a new laptop with Windows 8.1 and loved the change. Start Menu or no Start Menu, it didn’t matter to me. And then, sometime in July last year, came the little notification saying I could upgrade to Windows 10. I was hesitant for a few days. In the meanwhile, P upgraded to Windows 10. It was quite beautiful, it seemed modern. The basic stuff was still the same but the exterior looked nice.

In August, I upgraded my Windows. It was beautiful. And it was buggy. Way too buggy. The Start Menu refused to open half the time. The taskbar stopped responding. Audio stopped working sometimes. And even if I ignored the bugs (which was impossible at a point of time), updates were forced down my throat at least once a week. And there was no way I could turn them off. But I persevered. I would never leave Windows, I had once sworn. So, I braved the storm. Now, it’s February 2016. It’s been 6 months. The bugs have decreased. But still, at odd times, the Start Menu doesn’t work. Cortana is a joke. A really big joke. And the disk usage by internal stuff is often too much. When I start Windows, often it requires about 5 minutes for me to be able to use the OS properly with Avast running scans by itself and Windows checking for updates and everything.

Did I accept Windows? The answer is partially no. What does that mean? Well, in January, my Computer Science II Course started at college. And my instructor advised us to use Ubuntu. Now, I have a problem with working on computers in the lab. I just don’t like sitting at a different computer everyday or finding my files deleted or my complex directory system removed by some joker. So, in mid-January, I added an OS to my laptop. Ubuntu 14.04.3 LTS (The LTS tag made me prefer this to Version 15.something). And I’ll never regret my decision.

Ubuntu is fast. Blazing fast. I allocated only 20 GB to it. It’s still amazingly fast. It starts up and shuts down in at most 15 seconds. Everything is way more simple and way more easy. Of course, it has its drawbacks. I can’t play GTA V on Ubuntu. I find the Microsoft Office to be way better than all other offices. And some other stuff such as design, etc. (Although, I love the Ubuntu font)

In short, I will say that perfect way to go is use Ubuntu and Windows side-by-side. They are good in their own ways. Windows is absolutely necessary to play the heavy games. But when you want to write a paper or work out an online course, you don’t want a notification saying Windows needs to restart your computer. But hey, it’s your computer. Do whatever you want.

## Integer Lattice Graphs

An integer lattice graph can be defined as a graph $G=(V,E)$ such that $V = \mathbb{Z}^n$ and $E = \{vw|v,w\in V, ||v-w||=1\}$, i.e., two vertices are adjacent when exactly one of their co-ordinates differ by $1$ while the other co-ordinates are all equal. It is a particularly easy graph to visualize. It is 2n-regular. [Clearly, given a vertex, to get one of its neighbours, you have to change exactly one of the co-ordinates. You may either increase it by $1$ or decrease it by $1$ and this may be done for all n co-ordinates. Thus, there are $2n$ neighbours for every vertex.]

It becomes a bit more interesting when you consider the finite integer lattice graph. As you can guess, the vertex set is now defined by $V = \{-n, -(n-1), ..., 0, ..., n\}^d$, where $d$ is a +ve integer. The edge set has the same definition. This graph, while still quite easy to visualize is no longer regular. And this makes way for an interesting problem:

Find the average degree of a finite integer lattice graph.

How do we do this? Well, it is clear that the interior points still have degree $2d$. However, everything changes for the points whose co-ordinates have at least one $n$ or $-n$.

We have to tackle this part carefully. Let $v\in V$. Let $k$ be the number of co-ordinates whose absolute value is not $n$. Then, $0\leq k\leq d$. To find the degree of $v$, we have to find out the number of neighbours of $v$. To go to a neighbour from $v$, we have to change exactly one of the co-ordinates as before. For the $k$ co-ordinates which are strictly between $-n$ and $n$, we can either increase or decrease by $1$ as before. However, for the other $d-k$ co-ordinates, we can only decrease by $1$ (when the co-ordinate is $n$) or increase by $1$ (when the co-ordinate is $-n$, but we cannot do both. So, now we can obtain a neighbour of $v$ in $d+k$ ways. Hence, $deg(v)=d+k$.

Is this enough? No. Just the degree won’t do. We also need to know the exact number of vertices of each degree between $d$ and $2d$. How many vertices can have exactly $d-k$ co-ordinates equal to $n$ or $-n$ and the remaining not equal to $\pm n$? The answer is clearly ${n\choose k}2^{d-k} (2n-1)^k$ [Note that there are $2n + 1$ possibilities for each co-ordinate].

Then, we have

$\sum_{v\in V} deg(v) = \sum_{k=0}^{d} (d+k){n\choose k}2^{d-k} (2n-1)^k$

$\Rightarrow \sum_{v\in V} deg(v) = d \sum_{k=0}^{d} {n\choose k}2^{d-k} (2n-1)^k+\sum_{k=0}^{d} k{n\choose k}2^{d-k} (2n-1)^k$

$\Rightarrow \sum_{v\in V} deg(v) = d(2n+1)^d + (2n-1)d\sum_{k-1=0}^{d-1} \frac{(d-1)!}{((k-1)!(d-k)!}2^{d-k} (2n-1)^{k-1}$

$\Rightarrow \sum_{v\in V} deg(v) = d(2n+1)^d + (2n-1)d(2n+1)^{d-1} = 4nd(2n+1)^{d-1}$

Hence, the average degree of the graph is $\frac{4nd}{2n+1}$.

That was interesting, wasn’t it? The result isn’t very surprising if you try putting $d=2$.

Phew! I am literally sick and tired of writing $\LaTeX$ now. The next such post will be at least a week later.

Acknowledgements: Prof Young, who gave this problem in an assignment