A deep take on countability, cardinality and ordering

I’ve been teaching CMSC250, Discrete Mathematics, over the past year in CS UMD. Last semester, I typed a more philosophical than mathematical post on Countability, Cardinality and Ordering, which I’m repeating here for the community’s sake.

After our ordinality lecture last Tuesday, I had a student come to me and tell me that they were not sure how to think about ordinality: they were understanding the relationship between cardinality and size, since it is somewhat intuitive even for infinite sets (at least to them!), but ordinality still appeared esoteric. That’s 100% natural, and in this post I will I’ll try to stray away from math and try to explain how I think about countability, cardinality and ordinality intuitivelyThis post has exactly zero things to do with the final, so if you want to limit your interactions with this website to the exam-specific, you may stop reading now.

Before we begin, I would like to remind you of a definition that we had presented much earlier in the semester, I believe during an online quiz: A set S is dense if between any two elements of it, one can find another element. Note something interesting: only ordered sets can be qualified as dense or not! Technically, we had not presented the notion of an ordered set when we discussed dense sets, but it is intuitive enough that people can understand it.


We say that any enumerable set is countable. Enumerable, mathematically, means that we can find a bijection from the non-zero naturals to the set. Intuitively, it means “you start from somewhere, and by sequentially making one step, no matter how long it takes, you are guaranteed to reach every single element of the set in finite time”. Whether this finite time will happen in one’s lifetime, in one’s last name’s lifetime, or before the heat death of the universe, is inconsequential to both the math and the intuition. Clearly, this is trivial to do for either the non-zero naturals or the full set of naturals: you start from either 1 or 0, and then you make one step “forward”.

However, we also saw in class that this is possible to also generalize for the full set of integers: we start from 0 and then start hopping around and about zero, making bigger hops every time. Those hops are our steps “forward”.

Those results are probably quite intuitive to you by now, and I feel that the reason for this might that both LaTeX: \mathbb{N} and LaTeX: \mathbb{Z} are non-dense sets.There are no naturals or integers between LaTeX: n and LaTeX: n+1 (LaTeX: n \in \mathbb{N}  or LaTeX: n \in \mathbb{Z} ).

Let’s stray away from LaTeX: \mathbb{Q}  for now and fast-forward to LaTeX: \mathbb{R} . We have already shown the mathematical reason, Cantor’s diagonalization, for which the set of reals is uncountable. But what’s the intuition? Well, to each their own, but here’s how I used to think about it as a student: Suppose that I start from zero just to make things easier with respect to my intuitive understanding of the real number line (I could’ve just as well started with LaTeX: -e^8). 

Then, how do I decide to make my step forward? Which is my second number? Is it 0.1? Is it -0.05? But, no matter which I pick as my second number, am I not leaving infinitely many choices in between, rendering it necessary that I recursively look into this infinite interval? Note that I have not qualified “infinite” with “countably infinite” or “uncountably infinite” yet. This was my personal intuition as a Discrete Math student about 11 years ago about why LaTeX: \mathbb{R}  is uncountable: Even if you assume that you can start from 0, there is no valid ordering for you to reach the second element in the sequence of reals! Therefore, such a sequence cannot possibly exist!

But hold on a minute; is it not the case that this argument can be repeated for LaTeX: \mathbb{Q} ? Sure it can, in the sense that between, say, LaTeX: 0 and LaTeX: \frac{1}{2}, there are still infinitely many rationals. It is only after we formalize the math behind it all that we can say that this is a countable infinity and not an uncountable one, as is the case of the reals. But still, we have to convince ourselves: why in the world is it that the fact that every one of these infinite numbers can be expressed as a ratio of integers make that infinity smaller than that of the reals?

Here’s another intuitive reason why we will be able to scan every single one of these numbers in finite time: everybody open the slide where we prove to you that LaTeX: \mathbb{Q}^{>0}  is countable using the snaking pattern. Make the crucial observation that every one of the diagonals scans fractions where the sum of the denominator and the numerator is static! The first diagonal scans the single fraction (LaTeX: \frac{1}{1}) where the sum is 2. The second one scans the fractions whose denominator and numerator sum is 3 (LaTeX: \frac{1}{2},\ \frac{2}{1}). In effect, the LaTeX: i^{th} diagonal scans the following fractions:

LaTeX: \{ \frac{a}{b} \mid (a,b \in \mathbb{N}^{\geq 1}) \land (a + b=i+1)\}

For those of you that know what equivalence classes are, we can then define LaTeX: \mathbb{Q}^{>0}  as follows:

LaTeX: \mathbb{Q}^{>0} = \bigcup_{i \in \mathbb{N}}\{ \frac{a}{b} \mid (a,b \in \mathbb{N^{\geq 1}}) \land (a + b=i+1)\}

Let’s see this in action…

LaTeX: \mathbb{Q}^{>0} = \{ \color{red}{\underbrace{ \frac{1}{1}}_{i=1}}, \color{blue}{ \underbrace{\frac{1}{2}, \frac{2}{1}}_{i=2}} , \color{brown}{\underbrace{\frac{1}{3}, \frac{2}{2}, \frac{3}{1}}_{i=3}}, \dots \}

Note that essentially, with this definition, we have defined a bijection from LaTeX: \mathbb{N^{\geq 1}} \times \mathbb{N^{\geq 1}} to LaTeX: \mathbb{Q}. We know that LaTeX: \mathbb{N}^{\geq 1} \times  \mathbb{N}^{\geq 1} is countable, so we now know that LaTeX: \mathbb{Q}^{> 0}  is also countable! 🙂

Let’s constrain ourselves now to the original challenge that we (I?) are faced with: we have selected 0 as our first element in the enumeration of both LaTeX: \mathbb{Q}  and LaTeX: \mathbb{R}  (the latter is assumed to exist), and no matter which our second element is (say it’s LaTeX: \frac{1}{2}), we have infinitely many elements in both sets between 0 and LaTeX: \frac{1}{2}But now we know that those infinites are different: in the case of LaTeX: \mathbb{Q} . we know for a fact that we will reach all of those fractions whose decimal values are in LaTeX: (0, 0.5). In the case of LaTeX: \mathbb{R} , there is no such enumeration: any enumeration we define will still leave an… uncountably infinite gap between any two elements in “sequence”.

Remember how in our lecture on Algebraic and Transcendental numbers, we gave only three examples of numbers in LaTeX: TN, yet the fact that LaTeX: TN is uncountable when LaTeX: ALG is countable guarantees that there are “many more” Transcendental numbers than Algebraic? Same thing applies here with the rationals and irrationals: given any interval of real numbers LaTeX: (r_1, r_2), there are many more irrationals than rationals inside that interval... If you define a system of whole numbers (integers), there are many more quantities that you will not be able to express as a ratio of integers. That’s why back in the day (300 B.C) when Euclid proved that LaTeX: \sqrt 2 is not expressible as such a ratio LaTeX: \frac{a}{b} (or, more accurately, that LaTeX: 2 cannot be expressed as the square LaTeX: \frac{a^2}{b^2}) his result was so unintuitive; those Hellenistic people did not have rulers. They did not have centimeters or other accepted forms of measurement. The only thing they had were shoestrings, or planks of wood which they put in line and “saw” that they were the same length, and then they measured everything else as the ratio of such “whole” lengths.


Recall something that we said when we were discussing the factorial function and its combinatorial interpretations when applied on positive integers. Bill’s explanation of why LaTeX: 0!=1 was purely algebraic: If it were LaTeX: 0, then, given the recursive definition LaTeX: n!\:=\:n\:\cdot\left(n-1\right)! for LaTeX: n\:\ge1, every LaTeX: n! would be LaTeX: 0rendering it a pretty useless operation. My explanation was combinatorial: we know that if we have a row of, say, LaTeX: n marbles, there are LaTeX: n! different ways to permute them, or LaTeX: n! different orderings of those marbles. When there are no marbles, so LaTeX: n=0there is only one way to order them: do nothing, and go watch Netflix. 

Let’s stick with Bill’s interpretation for a moment: the fact that some things need to be defined in order to make an observation about the real world work. In this case, the real world is defined as “algebra that makes some goddamn sense”. My explanation is more esoteric. You could say: “What do you mean there’s only one way to arrange zero things? I don’t understand, if there are zero things and there’s nothing to do, shouldn’t there be, like, 0 ways to arrange them?”. So, let’s stick with Bill’s interpretation to explain something that I attempted to explain to a group of students after our first lecture this semester: Why do negative numbers even exist?

Here’s one such utilitarian explanation: Because without negative numbers, Newtonian Physics, with their tremendous application in the real world, would not work. That is, the model of Newtonian Kinematics with its three basic laws, which has been empirically proven to describe very well things that we observe in the real world, needs the framework of negative numbers in order to, well, work. So, if you’re not ok with the existence of negative numbers, you had better also be able to describe to me a framework that explains a bunch of observations on the real world in some way that doesn’t use them. For example, you probably all remember the third law of Newtonian motion: For every action LaTeX: \color{red}{\vec{F}}, there exists an equal and opposite reaction LaTeX: \color{red}{-\vec{F}}:

Recall that force is a vectoral quantity since it is the case that LaTeX: \vec{F} = m \cdot \vec{a}, and acceleration LaTeX: \vec{a} is clearly vectoral, as the second derivative of transposition LaTeX: \vec{x}

The only way for Newton’s third law of motion can work is if LaTeX: \vec{F} + (-\vec{F}) = \vec{0}. This is only achievable if the two vectors have the same magnitude but exactly opposite directions. No other way. Hence the need to define the magnitudes as follows:

LaTeX: | |\vec{F}|| = \frac{1}{2} \cdot m \cdot a^2,\ | |\vec{\color{red}{-}F}|| = \color{red}{-}\frac{1}{2} \cdot m \cdot a^2

and the necessity for negative numbers becomes clear. Do you guys think the ancient Greeks or Egyptians cared much for negative numbers? They were building their theories in terms of things they could touch, and things that you can touch have positive mass, length, height…

Mathematics is not science. It is an agglomeration of models that try to axiomatize things that occur in the real world. For another example, ZFC Theory was developed in place of Cantorian Set Theory because Cantorian Set Theory can lead to crazy things such as Russel’s Paradox. Therefore, ZFC had to add more things to Set Theory to make sure that people can’t do crazy stuff like this. If we discover contradictions with the real world given our mathematical model, we have to refine our model by adding more constraints to it. Less constraints, more generality, potential for more contradictions. More constraints, less generality, less contradictions, but also more complexity.

So when discussing the cardinality of LaTeX: \mathbb{N}  and LaTeX: \mathbb{Z}  and finding it equal to LaTeX: \aleph_0, we are faced with a problem with our model: the fact that LaTeX: \color{magenta}{\mathbb{N} \subset \mathbb{Z}} (I have used the notation of proper subset here deliberately). Now, I just had a look at our cardinality slides, and it is with joy that I noticed that we don’t use the subset / superset notation anywhere. That’s gonna prove a point for us.

So, back to the original problem: intuitively understanding why the hell LaTeX: \mathbb{N}   and LaTeX: \mathbb{Z}  have the same cardinality when, if I think of them on the real number line, I clearly have LaTeX: \mathbb{N} \subset \mathbb{Z}:

LaTeX: \underbrace{\dots, -4, -3 , -2, -1, \underbrace{0, 1, 2, 3, 4, \dots}_{\mathbb{N}} }_{\mathbb{Z}}

The trouble here is that we have all been conditioned from childhood to think about the negative integers as “minus the corresponding natural”. This conditioning is not something bad: it makes a ton of sense when modeling the real world, but when comparing cardinalities between infinite sets, that is, sets that will never be counted entirely in finite time, we distance ourselves from the real world a bit, so we need a different mathematical model. To that end, let’s build a new model for the naturals. Here are the naturals under our original model:

LaTeX: 0, 1, 2, 3, \dots

This digits that we have all agreed to be using have not been around forever. The ancient Greeks used lowercase versions of their alphabet: LaTeX: \alpha, \beta, \gamma, \delta , \epsilon, \sigma \tau ', \zeta,\ \dots\ \omega  to name a total of 25 “digits”, while the Romans used a subset of their alphabet “stacked” in a certain way: LaTeX: I, II, III, IV, V, VI, \dots, X, XI\dots . These “stacked” symbols cannot be really called digits the way that we understand them, especially since new symbols appear long down the line (LaTeX: C, M) etc. These symbols we actually owe to the Arabic Renaissance of the early Middle Ages.

The point is that I can rename every single one these numbers in a unique way and still end up with a set that has the exact same properties (e.g closure of operations, cardinality, ordinality) as LaTeX: \color{red}{\mathbb{N}}. This is formally defined as the Axiom of Replacement. So, let’s go ahead and describe LaTeX: \mathbb{N}  by assigning a random string for every single number, assuming that no string is inserted twice:

LaTeX: foo, bar, otra, zing, tum, ghi,\dots

Which corresponds to our earlier

LaTeX: 0, 1, 2, 3, 4, 5,\dots

Cool! Now the axiom of replacement clearly applies to LaTeX: \mathbb{Z}  as well, so I will rewrite

LaTeX: \dots, \color{blue}{-5, -4, -3, -2, -1,}\ \color{magenta}{0, 1, 2, 3, 4, 5,}\dots


LaTeX: \dots, \color{blue}{qwerty, forg, vri, zaq,  nit,}\ \color{magenta}{bot, ware, yio, bunkm, ute, kue,}\dots

Call these “transformed” sets LaTeX: \mathbb{N}_{new} and LaTeX: \mathbb{Z}_{new} respectively. Under this encoding, guys, I believe it’s a lot more obvious that LaTeX: \mathbb{N}_{new} \not\subset \mathbb{Z}_{new} in the general case. LaTeX: \mathbb{N}_{new} \subset \mathbb{Z}_{new} under these random encodings is so not-gonna-happenish that its probability is not even axiomatically defined. Therefore, now we can view LaTeX: \mathbb{N}  and LaTeX: \mathbb{Z}  as infinite lines floating around space, lines that we have to somehow put next to each other and see whether we can line them up exactly. If you tell me that even under this visualization, the line that represents LaTeX: \mathbb{Z} _{new} is infinite in both directions, whereas that of LaTeX: \mathbb{N}_{new}  has a starting point (0), then I would tell you that I can effectively “break” the line that represents LaTeX: Z_{new} in the middle (0) and then mix the two lines together according to the mapping that corresponds to:

LaTeX: 0, 1, -1, 2, -2, 3, -3, \dots

Now we no longer have the pesky notation of the minus sign, which pulls us to scream “But the naturals are a subset of the integers! Look! If we just take a copy of the naturals and put a minus in front of them, we have the integers!”. We only have two infinite lines, that start from somewhere, extend infinitely, and it is up to us to find a 1-1 and onto mapping between them. That is, it is up to us find a 1-1 mapping between:

LaTeX: foo, bar, otra, zing, tum, ghi,\dots


LaTeX: bot, ware, nit, yio, zaq, bunkm\dots

(Note that I re-ordered the previous encoding LaTeX: \dots, \color{blue}{qwerty, forg, vri, zaq,  nit,}\ \color{magenta}{bot, ware, yio, bunkm, ute, kue,}\dots according to the “hopping” map into  LaTeX: \color{magenta}{bot}, \color{magenta}{ware}, \color{blue}{nit,} \color{magenta}{yio}, \color{blue}{zaq}, \color{magenta}{bunkm},\dots .)

Under this “visual”, you guys, it makes a lot of sense to try to estimate if the two sets have the same cardinality and, guess what, they do 🙂

Not much else to say on this topic everyone. We can have a bunch of applications of the axiom of replacement to prove, for example, that the cardinality of the integers, LaTeX: \aleph_0, is also the cardinality of LaTeX: \mathbb{N} \times \mathbb{N}LaTeX: \mathbb{Q} , etc. It is only when we start considering sets such as LaTeX: \mathbb{R} , \mathcal{P}(\mathbb{N}) and LaTeX: \{0, 1 \}^\omega  that this idea that we can be holding two infinite lines in space fails.


There’s not much to say here except that the easiest way to understand how an order differs from a set is to consider an ordering exactly as such: an order of elements! Think in terms of “first element less than second less than third less than …. “. The simplest way possible. It is then that we can prove rather easily that LaTeX: \omega \prec y \prec \zeta .

Things only become a bit more complicated when considering the ordering LaTeX: \omega + \omega:

LaTeX: 0 < \frac{1}{2} < \frac{3}{4} < \frac{5}{6} < \dots <1 < \frac{3}{2} < \frac{4}{3} <\dots <2 <\dots

Please note that this ordering is clearly not the same as LaTeX: \eta, the ordering of LaTeX: \mathbb{Q} . Between the first and the second element, for instance, there are countably many infinite rationals: LaTeX: \frac{1}{100}, \frac{2}{5}, \dots , \frac{3}{7}\dots  which are not included in the ordering. 

Finally, realize the meaning of “incomparable” orderings: a pair of orderings LaTeX: \alpha, \beta  will be called incomparable if, and only if:

LaTeX: (\alpha \npreceq \beta) \wedge (\beta \npreceq \alpha).

So please realize that this is not the same as saying, for instance, LaTeX: \beta \nprec\alpha .

I think this is all, I am bothered when I can’t explain something well to a student so I thought I’d share my views on countability in case the subject becomes easier to grasp.

Piazza sucks.

I’m in Academia. Well, at least the part of Academia that’s still related to actual teaching.

The vast majority of my collaborators in UMD as well as in other institutions use Piazza to host their courses. It’s easy, it’s fast and at the very least it looks like it has close to 100% uptime (written during a time that our UMD fork of Instructure’s Canvas has been down for hours).

However, active development on Piazza has effectively stopped since early 2015. During Fall 2016, I would look at my Android Play Store for updates on the Piazza app and for a long time the latest one would date back to February 2015. Right now, it appears that certain patches have been made as early as Feb 7,2017, but the app is still atrocious. This is just an example of the many issues that surround Piazza.

The most major issue, for which I just submitted a bug report, is the fact that there is no filesystem consistency on Piazza. If you use the “Resources” tab and post a link to the file in a discussion topic, and you want to make a change to the file, then all the other links become stale. They point to positions in an Amazon S3 filesystem. When the number of links to a file grow, this becomes a huge problem.

Furthermore: the only way to password-protect your page right now (of importance to any person who needs a private discussion forum), you need to actually send an e-mail to team@piazza.com with your password choice, which of course is then stored in cleartext in your e-mails. My response was immediately adhered to, but what happens if you need to password-protect it during a weekend? I use Piazza to communicate with my TAs, and the information conveyed is often sensitive (thoughts on midterms, recitation topics, rubrics). I don’t want students snooping around (it’s already happened once this semester).

Piazza is perfect for communication: students love it, because other students can immediately offer responses. In contrast, nobody ever uses the Canvas “Discussion” feature. However, in departments where the average course registration is in the hundreds (like us), moderating such a huge forum requires TAs dedicated to doing only that. It’s not impossible, but it’s hard. Ensuring that solutions to problems under active submission don’t leak is tough. A student can take down a post within minutes, yet a PDF with solutions can already have reached a good portion of the class.

But all of these things are simple issues of technical decisions and design. They could be met through either Bug Reports or (a)synchronous brainstorming sessions using tools like Confluence or HipChat. What really bugs me is how the Piazza team doesn’t seem to care any more about the product, shifting their entire focus into making it yet another recruitment platform, which they call “Piazza Careers“. Seriously? That’s what students need? Another recruitment platform? I’m guessing that the Piazza team had some sort of shift in their venture capital and it was required of them to transform the entire platform into another recruiting platform.

It’s a real bummer. Blackboard has been a disaster (link) and Canvas has too many issues to discuss in a blog post. Active Learning services like TopHat are breakable when people use their phones. I recently had a student pretty much admitting to me during my office hours that they never attended the lectures of a certain math course, but had a friend of theirs text them the TopHat code and the proper answers to the questions. The clickers and accompanying software offered by Turning Technologies  have multiple issues of connectivity, even though those issues mostly have to do with the ELMS-CANVAS integration and not the software or clicker device itself.

We need reliable educational software. Not another recruitment platform. Pooja Shankar, an alumni of CS UMD and the founder of Piazza, ought to be the first person to recognize this.



I had my Discrete Math students critique Charlie the Unicorn, and here is how some responded.

Title says it all. This summer I’m teaching CMSC 250, “Discrete Structures” (really, this is a misnomer; I have no idea why we don’t call it “Discrete Mathematics”), to undergraduate students in the Department of Computer Science at UMD. As one of the requirements of the course, I had them review the epic saga of Charlie the Unicorn and submit a short essay. Now I knew these kids are bright and have a sense of humor, yet once again they surpassed all expectations.

Here are anonymous excerpts of what was handed to me:

The “Charlie the Unicorn” series has taught me about the dangers of the world we live in today. Life isn’t always rainbows and unicorns and I’m pretty glad it isn’t. That world seems messed up. There are tons of two-faced people out there and it is important to read through them or else they will get you to think you are the banana king and steal your stuff.

Why yes indeed, you never know when that might happen.

The Pink and Blue Unicorns are sociopathic robbers who are unable to distinguish
reality from fantasy, as well as being able to force their fantasies onto others through either hypnosis or hallucinogenic drugs. It is obvious that these two unicorns are a threat to society and need to be put into an insane asylum and be rendered unable to create their fantasy worlds.

Ouch! So much for second chances.

While Charlie is being manipulated, they continually make fun of him and steal his belongings. These acts seem to be unprovoked and only cause them enjoyment; they gain no real reward from these acts. Each situation they get Charlie into results in a catchy song followed by the immediate death of the performer.

My favorite part of Charlie the unicorn was the part when Charlie is convinced that he is the banana king. It’s probably true that if you levitate and shine and light on someone you could probably convince them of anything.

In an attempt to make sense of this video, the only conclusion that I could come to was that this is what Jason Steele, the creator of Charlie the Unicorn, experienced while higher than a kite. I would imagine that his stoner hallucinations were best manifested in a video where he and his friends were portrayed by unicorns, so that is exactly what Steele created.

Some people were more introspective than others:

Charlie the Unicorn is a politically-themed satire lambasting both Democratic and Republican politicians alike.  In the video, Democrats are symbolized by the blue horse and Republicans, the red.  The third horse, Charlie, represents the average citizen, with his white color additionally connoting the average citizen’s relative innocence and naïveté in politics.  The blue and red horses—henceforth referred to as “the purple horses”—employ fanciful promises and extreme enthusiasm to slowly goad the white horse—who is initially reluctant—into travelling to Candy Mountain with them.  This journey represents an ordinary citizen being stirred out of political apathy by the campaigning of a compelling politician spouting ideals, hopes, and promises of a better tomorrow.  However, the motivations of the purple horses were not so noble or selfless;[…]

While others actually hinted towards inductive reasoning / rule learning:

Each adventure involves an annoying commute to the destination with the pink and blue unicorn, arriving at the destination, receiving a song, having the singer blow up, and then Charlie somehow being put into danger. From this pattern, we can build an implication relationship which Charlie quickly learned. If Charlie goes on an adventure with the pink and blue unicorn, then he will be put in danger. As far as then fourth chapter of their adventures, this rule has been valid. But we do not know for sure if it will apply for future episodes.

Or human persuasion techniques:

To me the fact that there are 3 unicorns was interesting. People tend to believe when more than 3 people start believing some idea. For example if 3 people points to the sky in the middle of the road, other people start looking at the sky since people think there must be reason the 3 people are pointing to the sky. It is called the “Power of 3”.

This person, along with the person who provided the politically themed comments, seemed to be the ones closer to what the Internet believes the videos to be about:

One thing I did find interesting throughout all the episodes is that no matter how evil the things were Pink and Blue unicorn did to Charlie were (like taking his kidney), he went on every single adventure with them. After losing my kidney or my belongings by hanging out with my friends I wouldn’t want to hang out with them anymore. I don’t know if they’re necessarily Charlie’s friends to begin with which makes me question his decisions to follow them even more. In the last episode, Pink and Blue unicorn tried to take his life, but starfish came and rescued Charlie. I honestly could not stop laughing when starfish told Charlie that he was a star and then when Charlie made the wish, starfish’s eyes burned out. I was questioning why starfish was so in love with Charlie in the third episode, but good thing he was a starfish or else Charlie wouldn’t have lived. YOLO. I wonder why Pink and Blue unicorn were able to take everything away from Charlie except for his life. Was the creator trying to tell us something there? Whatever, I’m not going to think too much into it. A+, 10/10 would watch again.

Finally, if you’re interested in finding what the Internet thinks these videos are about, (a) You have a serious problem and (b) Here you go:

Thirty Seconds Flat

Yesterday night I watched the 1995 Michael Mann crime epic Heat for the umpteenth time. It is my understanding that the movie’s not particularly appraised, and it’s definitely not among Mann’s most well-known titles. Critics and movie-goers tend to think of The Insider or The Last of the Mohicans, or maybe his Miami Vice work in the 80s, as being his defining directorial moments,  Regardless, for me the movie has attained artistic status that elevates it beyond that of a motion picture and up to par with, I don’t know, the Sistine Chapel perhaps. Think hard before you label this as sacrilege.

Chances are that if you’ve heard about the movie, you know about the diner scene, where Al Pacino and Robert de Niro, playing career cop and criminal respectively, are pitted against each other, face to face, standing firm about who they are and what they’re looking to do. It’s a marvelous scene, and if you haven’t watched it, you should.

My favorite scene, however, happens before that one. Unsurprisingly, one cannot easily find a YouTube link to it. Backdrop: L.A Homicide collectively decide on a night out with their wives. Vincent Hanna (Pacino) dances with his wife Justine Hanna (Diane Venora), both of them tipsy. At some point, Pacino gets paged (pagers, I know, right?) and his oversight is requested in a murder scene loosely connected to the main plot. “This better be earth-shattering”, he says.

Couple hours later, Pacino arrives back in the dining area, currently occupied by just Justine and another couple in a different table. Justine, obviously distraught, begins the following dialogue, which I will recite from memory, so excuse any minor discrepancies:

– I guess the earth shattered.

-So why didn’t you let Bosko take you home? (Bosko is another cop in the unit.)

– I didn’t want to ruin their night too!


– So what happened?

– Honey, you don’t wanna know.

– I’d like to know what’s behind that grim look on your face!

– I don’t do that, you know that. Come on, let’s go.

 – You never told me I was gonna be excluded.

– I told you when we first hooked up, honey, that you would have to share me with all the bad people and ugly events on this planet.

– And I bought into that sharing, because I love you. I love you fat, bald, money, no money, driving a bus, I don’t care. But you have got to be present, like a normal guy, some of the time. This is not sharing. This is leftovers.

– Oh, I see, so what I should do is come home and tell you: “Hey baby, guess what. I just walked out of a crime scene where this junkie asshole fried his baby in a microwave because it was crying too loud. So let me share that with you. And in sharing, we will somehow.. ummm.. cathartically dispel all of this heinous shit.” Right? Wrong.


This is your life. There’s a fire inside you, and it’s raging on and on day and night. It encodes what you want, what you’re looking at, and what you’re after. And the closer you get to it, the harder it burns, the harder your brain is telling you to quit and look for safety. Vince Hanna got into 3 marriages in order to lie to himself that he cares about things beyond his work. Neil McCauley (De Niro) attempts a serious relationship for the first time ever, but he knows the drill: “If you wanna be making moves on the street, never get attached to anything or anybody that you can’t walk out on thirty seconds flat after you spot the heat coming round the corner.” 

Maybe you have the opportunity to be with somebody you have feelings for, and one night you wake up, see them occupying the other side of the bed and go screw a stranger in the local dive bar.

Maybe you’re close to having the job of your dreams but you cower out and stay in your current job because it affords you safety.

Maybe you’re not asking for that hot girl’s number because you fear what will happen if she agrees to it.

Maybe you’re in a one-year long relationship you knew had practically ended a month into it.

Maybe you’re close to being one year sober, and because humanity has agreed to the Westernized division of 365 days per annum, you get shitfaced on the 364th night.

Maybe, maybe, maybe.

It doesn’t really matter. If you don’t do it, if you don’t try your utmost to touch that fire, you will always regret it. And that’s a slow death that is far worse than anything I can imagine.


As new data arrives, the covariance matrix takes notice.

The problem

I recently read a paper on distributed multivariate linear regression. This paper essentially deals with the problem of when to update the global multivariate linear regression model in a distributed system, when the observations available to the system arrive in different computer nodes, at different times and, usually, at different rates. In the monolithic, single node case, the problem’s of course been solved in closed form, since for dependent variables and design matrix X with examples in rows, the parameter vector β can be found as per:

The linear regression solution.

This is a good paper, and anybody with an interest in distributed systems and / or  linear algebra should probably read it. One of the interesting things (for me) was the authors’ explanation that, as more data arrives at the distributed nodes, a certain constraint on the spectral norm of a matrix product that contains information about a node’s data becomes harder to satisfy. It was not clear to me why this was the case and, in the process of convincing myself, I discovered something that is probably obvious to everybody else in the world, yet I still opted to make a blog post about it, because why the hell not.

When designing any data sensor, it is reasonable to assume that the incoming multivariate data tuples will all have a non-trivial covariance. For example, in the case of two-dimensional data, it is reasonable to assume that all the incoming data points will not all lie on a straight line (which denotes full inter-dimensional correlation in the two-dimensional case). In fact, it is reasonable to assume that as more data tuples arrive, the covariance of the entire data tends to increase. We will examine this assumption again in this text, and we will see that it does not always hold water.

This hypothesized increase in the data’s covariance can be mathematically captured by the spectral (or “operator”) norm of the data’s covariance matrix. For symmetric matrices, such as the covariance matrix, the spectral norm is equal to the largest absolute eigenvalue of the matrix. If a matrix is viewed as a linear operator in multi-dimensional cartesian space, its largest absolute eigenvalue tells us how much the matrix can “stretch” a vector in the space. So it gives us an essence of how “big” the matrix is in that sense, hence its incorporation into a norm formulation.

The math

We will now give a mathematical intuition about how the incorporation of new data in a sensor leads to a likelihood of increase of its spectral norm, or, as we now know, its dominant eigenvalue. For simplicity, let us assume that the data is mean centered, such that we don’t need to complicate the mathematical presentation with mean subtraction. Let λ be the covariance matrix’s dominant eigenvalue and u be a unitary eigenvector in the respective eigenspace. Τhen, from the relationship between eigenvalues and eigenvectors, we obtain:

Derivation 1

with the second line being a result of the fact that u is assumed unitary. It is therefore obvious that, in order to gauge how the value of λ varies, we must examine the 2-norm (Euclidean norm) of the vector on the right-hand side of the final equals sign.

Let’s try unwrapping the product that makes up this vector:

Derivation 2

Now, let us focus on the first element of this vector. If we unwrap it we obtain:

Derivation 3

The crimson factors really let us know what’s going on here, since the summations in the parenthesis involve the “filling up” of values in the covariance matrix that lie beyond the main diagonal. For fully correlated data, those values are all zero. On the other extreme, they are all non-zero. It is natural to assume that, as more data arrives, all those values tend to deviate from zero, since some inter-dimensional uncorrelation is, stochastically, bound to occur. On the other hand, if new data is such that it causes an increased inter-dimensional correlation, then the sum will tend towards zero, and the covariance matrix’s spectral norm will actually decrease!

The second vector element deals with the correlation between the second dimension and the rest, and so on and so forth. Therefore, the larger the values of these elements, the larger the value of the 2-norm || X’ X u ||  is going to be and vice versa.

Some code

We can demonstrate all this in practice with some MATLAB code. The following function will generate some random data for us:

function X = gen_data(N, s)
%GEN_DATA Generate random two-dimensional data.
% N: Number of samples to generate.
% s: Standard deviation of Gaussian noise to add to the y dimension.
x = rand(N, 1);
y = 2 * x + s.* randn(N, 1); % Adding Gaussian noise
X = [x,y];

This function will generate the covariance matrix of the input data and return its spectral norm:

function norm = cov_spec_norm(X)
% COV_SPEC_NORM: Estimate the spectral norm of the covariance matrix of the
% data matrix given. 
%   X: An N x 2 matrix of N 2-dimensional points.

COV = cov(X);
[~, S, ~] = svd(COV);
norm = S(1,1).^2;

Then we can use the following top-level script to create some initial, perfectly correlated data, plot it, estimate the covariance matrix’s spectral norm, and then examine what happens as we add chunks of data, with increasing amounts of Gaussian noise:

% A vector of line specifications useful for plotting stuff later
% in the script.
linespecs = cell(4, 1);
linespecs{1} = 'rx';linespecs{2} = 'g^';
linespecs{3} = 'kd'; linespecs{4} = 'mo';

% Begin with a sample of 300 points perfectly
% lined up...
X = gen_data(300, 0);
plot(X(:, 1), X(:, 2), 'b.');  title('Data points'); hold on;
norm = cov_spec_norm(X);
fprintf('Spectral norm of covariance = %.3f.\n', norm)

% And now start adding 50s of noisy points.
for i =1:4
    Y = gen_data(50, i / 5); % Adding Gaussian noise 
    plot(Y(:,1), Y(:, 2), linespecs{i}); hold on;
    norm = cov_spec_norm([X;Y]);
    fprintf('Spectral norm of covariance = %.3f.\n', norm);
    X = [X;Y]; % To maintain current data matrix
hold off;

(Note that in the script above, every new batch of data gets an inreased amount of noise, as can be seen in the call to gen_data.)

One output of this script is:

>> plot_norms
Spectral norm of covariance = 0.191.
Spectral norm of covariance = 0.200.
Spectral norm of covariance = 0.200.
Spectral norm of covariance = 0.220.
Spectral norm of covariance = 0.275.

A plot of our dataInterestingly, in this example, the spectral norm did not change after incorporation of the second noisy data. Can it ever be the case that we can have a decrease of the spectral norm? Of course! We already said that the crimson summations above, corresponding to summations over cells of the covariance matrix beyond the first diagonal, can fall closer to zero after we incorporate new data whose dimensions are more correlated with the existing data’s. Therefore, in the following run, the incorporation of the first noisy set actually increased the amount of inter-dimensional correlation, leading to a smaller amount of covariance (informally speaking).

>> plot_norms
Spectral norm of covariance = 0.177.
Spectral norm of covariance = 0.174.
Spectral norm of covariance = 0.179.
Spectral norm of covariance = 0.220.
Spectral norm of covariance = 0.248.

Another data plot.


The intuition is clear: as new data arrives in a node, observing the fluctuation of the spectral norm of its covariance matrix can tell us some things about how “noisy” our data is, where “noisiness” in this context is defined as “covariance”. I guess the question to be made here is what to expect of one’s data. If we run a sensor long enough without throwing away archival data vectors, it’s unclear whether we can expect the spectral norm to continuously increase (at least not by a significant margin). We should expect a sort of “saturation” of the spectral norm around a limiting value. This can be empirically shown by a modification of our top-level script, which runs for 50 iterations (instead of 4) but generates batches of data with standard Gaussian noise, i.e the noise does not increase with every new batch:

% Begin with a sample of 300 points perfectly
% lined up...
X = gen_data(300, 0);
norm = cov_spec_norm(X);
fprintf('Spectral norm of covariance = %.3f.\n', norm)

% And now start adding 50s of noisy points.
for i =1:50
    Y = gen_data(50, 1); % Adding Gaussian noise 
    norm = cov_spec_norm([X;Y]);
    fprintf('Spectral norm of covariance = %.3f.\n', norm);
    X = [X;Y]; % To maintain current data matrix

Notice how the call to gen_data now adds normal Gaussian noise by keeping the standard deviation static to 1. One output of this script is the following:

>> toTheLimit
Spectral norm of covariance = 0.203.
Spectral norm of covariance = 0.341.
Spectral norm of covariance = 0.388.
Spectral norm of covariance = 0.439.
Spectral norm of covariance = 0.535.
Spectral norm of covariance = 0.635.
Spectral norm of covariance = 0.677.
Spectral norm of covariance = 0.744.
Spectral norm of covariance = 0.818.
Spectral norm of covariance = 0.842.
Spectral norm of covariance = 0.881.
Spectral norm of covariance = 0.913.
Spectral norm of covariance = 0.985.
Spectral norm of covariance = 1.030.
Spectral norm of covariance = 1.031.
Spectral norm of covariance = 1.050.
Spectral norm of covariance = 1.097.
Spectral norm of covariance = 1.148.
Spectral norm of covariance = 1.154.
Spectral norm of covariance = 1.186.
Spectral norm of covariance = 1.199.
Spectral norm of covariance = 1.280.
Spectral norm of covariance = 1.318.
Spectral norm of covariance = 1.323.
Spectral norm of covariance = 1.325.
Spectral norm of covariance = 1.344.
Spectral norm of covariance = 1.346.
Spectral norm of covariance = 1.373.
Spectral norm of covariance = 1.397.
Spectral norm of covariance = 1.447.
Spectral norm of covariance = 1.436.
Spectral norm of covariance = 1.466.
Spectral norm of covariance = 1.466.
Spectral norm of covariance = 1.482.
Spectral norm of covariance = 1.500.
Spectral norm of covariance = 1.498.
Spectral norm of covariance = 1.513.
Spectral norm of covariance = 1.518.
Spectral norm of covariance = 1.518.
Spectral norm of covariance = 1.499.
Spectral norm of covariance = 1.492.

It’s not hard to see that after a while the value of the spectral norm tends to fluctuate around 1.5. Under the given noise model (Gaussian noise with standard deviation = 1), we cannot expect any major surprises. Therefore, if we were to keep a sliding window over our incoming data chunks, and (perhaps asynchronously!) estimate the standard deviation of the spectral norm’s values, we could maybe estimate time intervals during which we received a lot of noisy data, and act accordingly, based on our system specifications.

The 3-year long health saga of a medically insured PhD student


My name is Jason Filippou, and I’m a PhD student in the Computer Science department of the University of Maryland, College Park.

My intention with this blog was, and still is, to post interesting things about Computer Science, music, and maybe some latent psychological elements that I find compelling. I simply find few other items interesting at this time. However, a recent health saga which depicted a level of barbarism and medical malpractice that I never thought possible has made it useful to document my experience on a platform more friendly towards static information dissemination than Facebook, a website tuned towards posting pictures of food or other clearly time-wasting material, and upon which I’ve been documenting this stuff until very recently. I will therefore use the first post in this blog to talk about this. It is important for me to have a static account of what has happened over the 3 years that I’ve been a PhD student in the USA, in conjunction with disseminating this information online to raise awareness of the level of malpractice that even insured people can be faced with in that particular country.

An effort will be made to stick to the facts. I will also avoid mentioning specific names of medical professionals or insurance policies, unless legally advised to do so down the line. This will be a lengthy post, but one extremely worthy of reading to the end, particularly if you are interested in the health system situation of the USA, or perhaps emigrating to the USA. It is advised that it is not read during one’s meal, since it contains references to G.I-related symptoms which could potentially gross people out.

August 2012 – First blood

I arrived in the USA on the 3rd of August 2012. Within less than 24 hours, I ingested pasta sauce gone bad for months and naturally got food poisoning. I vomited twice. More unnervingly, over the next few days I started having frequent diarrhea with bloody stools, in conjunction with abdominal cramping and an urgency to visit the bathroom immediately. It should be noted that, at the time, I had not signed my insurance forms in the US, because I’d arrived two weeks early in order to acclimatize myself with College Park as much as possible before beginning my studies.

Regardless, I decided I should visit a doctor. So I visited the University Health Center in the College Park campus, where I was seen by a G.I doctor on the spot (let’s call them doctor A for brevity), with a rather small charge in my University Bursar account. The doctor, having not had any history of me in the past (I’m a Greek-Canadian dual citizen who’d lived his entire life up until August 2012 in Athens, Greece), laid out the possibilities in front of me, and those ranged from run-of-the-mill food poisoning to colon cancer. I hadn’t been in the States for more than 5 days, I was completely alone since my roommate at the time was on vacation, my parents were thousands of miles away in Greece, and I was presented with the possibility of colon cancer, a pre-existing condition that would probably make it impossible for me to successfully sign insurance forms when the time came.

It turned out that I did not have colon cancer, the final diagnosis at the time being gastrenteritis. Over the next few months, I noticed a shift in the behavior of my G.I tract; I started visiting the bathroom more frequently than I used to, and my discharges had changed in consistency, size, etc. This was not pleasing for me, but considering the fact that my diet had also changed and this can only affect one’s bathroom habits in one way or another, I was not particularly alarmed at the time.

June 2013 – The “treatment” begins

Around the end of the 2013 Spring semester, I started developing symptoms similar to those of August 2012. Thinking that this could not continue, I visited a G.I doctor in my area, let’s call them Dr. B. At that point, I was of course medically insured, let’s call the policy Insurance Policy A. The doctor asked me some questions, and theorized (without so much as requesting a stool or blood sample, for instance) that I am suffering from Irritable Bowel Syndrome and that my hemorrhoids were also inflammated. They therefore prescribed some rectal suppositories (Anucort), suggested some slight diet adjustments and instructed me to see them a month from that date. The symptoms did subside, yet a month later when I arrived at their office for the scheduled appointment, the doctor’s secretary told me that they had tried to contact me because the doctor had to cancel, yet they could not reach me on my phone “a couple of times”. At the time I owned a phone which, while not the best in the market, was endowed with state-of-the-art voicemail technology. It is therefore questionable exactly how the front office could not reach me on my phone every one of those “couple” of times. At any rate, I had to miss that appointment and re-schedule for a post-August meet, since during August I would be visiting my home country of Greece for vacation.


August 2013 – A glimmer of hope vanishes fast

During my August vacations, I started developing symptoms again. I arrived at the States on August 31, and immediately requested an appointment from Dr. B. I got an appointment on September 4, during which again no proper diagnosis was made (no rectal examination, no stool samples, nothing of the sort), yet an appointment for a colonoscopy was made for the earliest available date: November 27th. That constituted an 11-week wait for an insured patient with frequent diarrhea, bloody stools, urgency to visit the bathroom and at times severe abdominal cramping.

Instructions for properly cleansing my intestines and colon prior to the procedure were sent to me by mail couple days before 11-27. However, nowhere within the instructions was it mentioned that the day before a colonoscopy the patient should not be ingesting any food after noon. Perhaps it was assumed that this was common, universal knowledge. For a person who prior to coming to the States boasted a perfectly healthy G.I system and had never had a colonoscopy before, it was not. Therefore, I had dinner at 6pm the previous day and at 8pm started drinking my four liters of laxative. Naturally, my colon was not perfectly clean for the colonoscopy despite the laxative treatment and doctor B was not able to have a very good view of my colon. Regardless, they “diagnosed” (I have to put this in quotes, maybe they had to “diagnose” something for insurance purposes? I can’t imagine a failed colonoscopy producing anything like an actual diagnosis.) proctitis and inflammated hemorrhoids, and prescribed the same suppository.

June 2014 – A dreadful decision

Like clockwork, symptoms re-emerged some months later. Early June 2014 I called Dr. B again and requested an appointment. When I heard that the earliest appointment that they had for me would be approximately a month later, I declined the appointment and sought a referral for a different G.I doctor. I received a referral for this doctor, Dr. C, and visited them mid-June. Dr. C listened to my concerns and symptoms and once again theorized (without so much as a rectal examination or stool/blood sample) that I suffer from I.B.S and inflammated hemorrhoids. They asked details about my diet and pin-pointed certain foods about being possible culprits (red onions, creamy soups / salad dressings). They also suggested that in the mornings I have a spoonful of Benefiber with my breakfast and, since I typically had a cup of coffee in the morning and caffeine tends to quickly cause bowel movements, to wait for my first bowel movement of the day before leaving for work. Supposedly, the fast expulsion of the day’s first stools would “train” my G.I tract to cause a bowel movement when I wanted it to, instead of when it wanted to (Dr. C’s words).

Fall Semester of 2014 – False information becomes the norm

I followed Dr. C’s advice, and my symptoms subsided. However, at this point, it should be clear to the reader that my symptoms tended to have seemingly random ebbs and flows. It is therefore not safe to say that my symptoms subsided because my doctor suggested Benefiber and a 20-minute wait after my morning coffee before I took the bus to work.

For our first follow-up appointment, after 3 months, I had made progress, in the sense that I didn’t have grave symptoms such as the ones that made me want to visit a doctor immediately in the first place. However, I still exhibited a bathroom visitation pattern that was markedly different from my previous years in Greece; much more frequent visits (approx. 5 per day), lighter stool consistency, mucus on the stool sometimes, increased flatulence, the works. Dr. C focused on the flatulence and suggested, essentially, that I eat less. The flatulence, in their opinion, was a direct consequence of bacteria in my intestines eating off of the increased amount of food that I was consuming (I’d put on maybe 12lb since I came to the States).

For our next follow-up, late Fall 2014, I mentioned the same things to Dr. C, and they theorized that in addition to my I.B.S (apparently we were settled on I.B.S at the time) I might also be lactose intolerant. Throughout my adolescent life, I tended to consume either a glass of milk or a bowl of cereal on a daily basis, always as part of my breakfast. Never had I had any issues with milk. It is true, however, that many people in their 20s develop lactose intolerance, therefore we opted to run with that. So I switched from dairy milk to almond milk, and whenever faced with the prospect of eating something dairy-based, I would consume a pair of Lactaid tablets.  Symptoms did not re-emerge til the end of the year.


January 2015 – Cars, sleet and magic soup

During January 2015, my symptoms came back. Compared to previous times, I would notice a much greater need for a bathroom visit, to the point that I had a public accident once which I covered up as best as I could, and it was good enough. Blood in the stool still existed, as did abdominal pain. In fact, that period of time was particularly distressing for me because I was alone in the house, and I had to walk for 35 minutes minimum to my driving school where I was taking driving lessons. The weather at the time in Maryland was steadily below 30 degrees Fahrenheit (-1 celcius, this is considered cold if you’ve lived in Greece your entire life) and because of snow and frost on the streets, public transportation was unreliable. Being late on any given driving lesson because a bus was late was no option; I would have to repeat the lesson. So I had to rely on my own two legs for walking to my school in the freezing cold, the legs themselves depending on an abdomen that was not under severe pain. This was a dependency that was not met. Long story short, those were very painful walks that took much more than the 35 minutes that Google Maps claimed the on-foot distance was.

The symptoms almost single-handedly vanished after a visit I made to a local Vietnamese food place (one of the various “Pho” places that appear to be ubiquitous in urban areas nowadays) and had a vegetable soup that proved itself therapeutic. I mentioned the entire experience to Dr. C during our regular appointment after two months, and they appeared… pleased. I guess the fact that I had apparently found an herbal remedy of sorts must have somehow backed up their belief that this condition was something I could essentially teach my body to deal with in a somewhat self-healing, herbal fashion.


June & July 2015 – A living hell

Late June

Come June 2015, I was working under a different advisor in Maryland, and I was feeling good. Both the people that I have worked with are top representatives of their field, yet the second person appeared to be more in tune with my research goals, and I therefore opted to switch towards them. In conjunction with my symptoms disappearing for couple months (again, ebbing and flowing, it seems), I was feeling good. I had a July 15th deadline that I was working hard for and I had a very good chance of catching.

No matter. Around the end of June, the symptoms hit me the hardest they ever had. 15-20 visits to the bathroom on a daily basis. Most of those visits featuring purely watery discharges or blood. Extreme need/urgency for every one of those visits. Loss of appetite, nausea. Abdominal pain that had me rise from my office chair and head to the nearest bathroom every 20 minutes maximum, praying that the bathroom would be vacant such that I avoid accidents in the workplace. A visit to the Vietnamese soup place that I mentioned earlier did not help at all.

At this point it should also probably be mentioned that I was encountering numerous personal difficulties, concerning the mental strain and computer coding / data cleansing required to catch my deadline, a driver’s skill’s test that went really bad because the person whose car I had to drive had neglected some key elements/paperwork required to prove the driveable status of the car, as well as the fact that my new insurance provider (Insurance Policy B) was refusing payment of one of my two weekly therapist appointments completely illegally and 100% out-of-line with respect to my stated benefits. I was therefore encountering significant personal difficulties even before the symptoms hit me and my emotional state was very fragile. This detail will be important moving forward, such that the reader is able to somewhat put themselves in my shoes and grasp the amount of perseverance that has been necessary to maintain my sanity and be able to type this text as we speak.

First thing I did was call Dr. C on their cell phone around the 22nd of June and request their input on my situation. They told me to take Pepto-Bysmol since “it sounds like you had something bad to eat”. I did that for two days, to no avail. So I called them again on Thursday, June 25th, and they re-directed me to the front office to schedule an appointment.

June 30th

So I then called Dr. C’s office requesting an appointment. A lady booked an appointment for me for Tuesday, June 30th, 4pm in my doctor’s office located in City A. Because of my dependency on public transportation, I had to take an early bus and arrived at the office one hour early, a time at which I was informed that my appointment had been booked by accident in City B. I had no chance of arriving at City B on time, not even if I had a car to drive there.

I was in complete shock. It appeared to me at the time that nothing – absolutely nothing -was going my way, and surely if the reader read the italicized text a few paragraphs ago they might be inclined to feel the way I felt. After visiting the office’s bathroom, I was told to wait in the lobby such that the staff might find a suitable doctor to look me up, given the fact that my appointment had been accidentally booked by a trainee, so the fault lied with them and not myself. I remember frantically texting a friend or two through my phone while I was waiting for somebody else to see me, telling them phrases such as “Nothing’s going my way, oh God, what’s happening, why will nobody care, nobody cares”, and the like.

After about 1/2 hour, a member of the office staff announced to me that no doctor could see me, presumably because of appointment backlogging. At that point I responded: “That’s fine, I will visit the ER next door.” (The doctor’s office was in a building adjacent to a hospital and I was simply under too much pain and discomfort to let this affect me further.) I left and did just that. I checked into the ER and began a 4-hour wait for somebody to see me. My temperature and some bloodwork were both taken almost immediately after check-in, and it was determined that I had a fever. For the rest of the diagnosis, I would presumably have to wait until I moved to the top of the line and a G.I doctor could see me. My phone dying, I was worried that by the time that I was discharged I would have nobody to drive me back home, so I texted a friend, told him that I would need a ride soon but I couldn’t tell them the exact time that I would be discharged (for I did not know it!), and that I would need to shut down my phone for battery considerations. So I was dependent on somebody to pick me up after discharge, but could not confirm with them because I had no means of communication with the outside world.

At this point, it should also be mentioned that, according to Insurance Policy B, I have a co-pay of $150 for every ER visit and, in order to be admitted to a hospital for so-called “inpatient services”, one needs pre-authorization, presumably from a primary care physician or specialist. This piece of information might be important for the reader, such that they develop a full personal view of the way that I was treated.

Once I moved on to the top of the queue, I was given a bed and was administered I.V.s with natural serum and antibiotics (Ciprofloxacin and Metronidazole). Within 15 minutes or so, a doctor (Dr. D) came to talk to me about my symptoms. I explained my symptoms to them, and told them that I was visiting Dr. C for those symptoms. They appeared to have knowledge of who Dr. C was. They also explained to me that I was taking those antibiotics because I had a fever and an elevated white cell count, so we were operating under the assumption of a bacterial infection. Dr. D proceeded to tell me that the goal would now be for me to provide a urine and stool sample and, in the meantime, they would contact Dr. C to get some advice on how to proceed next. As a small reminder to the reader, the plan that day was to see Dr. C about my symptoms, but my appointment had been booked in the wrong city.

During the next hour or so, I was frequently visited by medical staff, who appeared to be somewhat confused about what I was supposed to be doing at any given point in time. For instance, after Dr. D talked to me the first time,  a nurse came over and asked me whether I’d talked to Dr. D. After some time, a nurse came to me and asked me whether I had provided my samples yet, despite the fact that I was obviously hooked on I.Vs (and therefore largely immobile) and I hadn’t been given any information or equipment (e.g urine / stool samplers) that would help me provide the samples.

After about an hour, Dr. D came over and told me that they’d talked to Dr. C and that “he would squeeze me in between appointments the next day”. I basically would have to call the front office the next morning and they would find a time for me.

As mentioned at the start of this post, an effort has been made to stick to the facts, without too much speculation or personal opinion dissemination. However, after talking to some people since this June 30th experience, it appears as it is universally interesting to many people that neither Dr. C nor Dr. D provided a pre-authorization for hospitalization. Here we have a patient with chronic symptoms that involve bloody stools, abdominal cramping, nausea, fever, and seemingly uncontrollable diarrhea, and the best that we can do for him is give him some antibiotics and a next-day appointment while we wait for his stool samples, which were finally requested after about 3 years of no real diagnosis (reminder: the only time I was requested stool samples of was during my first week in Maryland).

They then proceeded to tell me that the game plan involved me providing stool and urine samples and being discharged after this. So after about 15 minutes or so, a nurse came by, unhooked my I.Vs and told me that I would need to visit a bathroom next door, where there existed “multiple samplers” that I could use to collect my samples. Almost contemporaneously, my friend arrived and declared availability to drive me over to my place after I was done, so at least that part had been covered.

In order to collect the samples, I had to visit a very dirty bathroom that reeked of – and was filled with pools of – urine. On a shelf to the right of the door, there were many urine samplers. I did not see any samplers specialized for stool collection, I therefore assumed that they simply did not exist as a notion in the USA and  opted to just use two urine samplers. While it is true that it is phenomenally easy for guys to collect urine samples, this is not necessarily true for stool samples, particularly for a person whose stool is almost entirely watery and bloody due to illness. For obvious reasons I will have to omit the relevant details, but it’s needless to say that I was disgusted beyond belief. I slightly soiled my clothes and my hands, and ended up washing my hands and the outer surfaces of the samplers for 5 minutes straight out of pure disgust. The entire process took me 30 minutes or so.

Once I walked outside the bathroom, samplers in hand, a mobile register was waiting for me next to my bed, in order to charge me my $150 co-pay for my treatment that day. Of course, Dr. D and the hospital got a piece of a much larger pie that day, courtesy of my insurance provider:

Insurance claim for ER Visit.

June 1st

The next day I visited Dr. C. That day was going to be Dr. C’s last day in the States before a vacation that would keep them outside the country until the 22nd of July, so that would be 21 days of a separation between us. The game plan according to Dr. C was for me to undergo an antibiotic – based treatment for about 10 days and, if things did not improve, call the front office and request the first possible appointment for a colonoscopy, because “the fact that you have blood in your stool worries me”.

It becomes increasingly hard for one to stick to the facts and just the facts here. For about a year of appointments, which included complaints from me about the flaring up of my disease from time to time, chief element of which was bloody stools, Dr. C had not been worried about the bloody stools. Only after I had to admit myself to the ER did they become worried about the bloody stools. Now that I am reading this italicized paragraph again, though, it still sounds like fact, I will therefore allow it.

I was prescribed the antibiotics I was given intravenously in the ER (Ciprofloxacine and Metronidazole), to be administered twice a day in pill form. I was also prescribed Omeprazole, to be taken “as needed” in case of nausea. In conjunction with my daily dosage of Lexapro, a common SSRI which I was taking for about a year to treat my depression and generalized anxiety disorder and which was administered and closely monitored by my psychiatrist, at that point I was taking 4 medications on a daily basis, three of which I was taking without any indication that they were actually going to work, because I hadn’t had an appropriate diagnosis. I essentially did not know what I had! At least the depression/ GAD had been properly diagnosed, more than a year ago, and the FDA-approved medicine was used to treat the diagnosed disease.

It was at that point that my situation became quality-of-life destroying (not just debilitating, the word has been carefully chosen) and, as the reader will soon see, even life-threatening.

July 2-6

Starting July 1st in the evening, I started taking my prescribed antibiotics. Very fast (Thursday 3rd) it became clear that I could no longer visit the office. On top of my symptoms, which did not improve, I had a close-to-complete loss of appetite, and an erratic sleep schedule. Any attempt towards creative/analytical thinking, the cornerstone of a successful graduate student and researcher, was hampered by abdominal pain, dizziness, and > 10 daily visits to the bathroom, with the same qualitative characteristics. Clearly the antibiotics were not helping. I had to drop my deadline, which I officially did on July 6, after an e-mail to my advisor and collaborators where I told them that I basically cannot leave the house because of my symptoms.

My situation was dire; I could not work and was staying at home, slumped on a couch, being unproductive, with medication that apparently did not work.

July 7-10

On July 7th, I called Dr. C’s office again, fully aware that Dr. C would not be available for an appointment and that I’d have to schedule an appointment with somebody else. I was told that a different doctor, Dr. E, would call me back with some information for me. They did so immediately (within 5 minutes or less) and proceeded to ask me about my symptoms. They then told me that the goal right now should be for me to combat the diarrhea and proceeded to prescribe to me what might arguably be the most curious prescription of them all: Cholestyramine in powder form. An inspection of the link, as well as the informative leaflet that I examined when receiving the prescription, both have nothing to say about the drug combating diarrhea. In fact, if the reader were to follow the NIH link I just provided, they would determine for themselves that diarrhea is among the known side-effects of Cholestyramine, the drug itself typically used for dealing with high levels of cholesterol in the bloodstream.

At any rate, believing that Dr. E, being a medical professional, knew what they were doing, I proceeded to add Cholestyramine to my list of medications, which was now engrossed to 5 different prescriptions. Dr. E also proceeded to tell me that if I was still having symptoms after taking Cholestyramine, I should contact the front office to arrange a colonoscopy with any available specialist. They also informed me that, on his end, it appeared that the soonest available appointment would be on the 22nd of July, which co-incided with the date that Dr. C would be coming back. Of course, at the time, it was of much more interest to me to actually have the procedure on the 22nd, rather than have another appointment during which I would state the same things over and over and only then would I be able to arrange a colonoscopy which, who knows, could take maybe 3 months or so to schedule, like my previous one in Fall 2013. To add to my temporary relief, Dr. E told me that other specialists in the office had sooner appointments available, even next week. It appeared that I had some help after all.

To no avail. Cholestyramine caused me extreme heartburn and an inability to properly ingest any kind of food. I arrived at a point that I started sleeping with my pillow propped up, in a semi-upright position, because whenever I lied down I felt that my partly ingested food (whatever little food I was eating, anyway) was being pushed upwards on my oesophgaus and I was not keen on dying of gastro-intestinal retrogression. So at that point I started really fearing for my life.

The morning of the 8th I phoned the front office requesting the soonest possible colonoscopy appointment. At that point, I was prepared to demand a same-day appointment if need be, or, at the very least, a pre-authorization to go to the hospital. I was desperate. The person on the phone told me they’d call me back. They did, quite quickly, and told me the soonest possible appointment that they could arrange for me would be August 14. Clearly they did not have the relevant context and attempted to book me through Dr. C, after examining their post-vacation schedule. I informed them that I’d talked to Dr. E and they’d mentioned that there existed closer options than that. They told me they’d look my situation up and call me right back.

Side note: Yes, I do have an excellent memory, which is unfortunate for numerous people that choose to think that, by default, people forget. I also tend to make very detailed notes and am a Google Calendar junkie.

So they did call me and suggested an appointment for July 22nd with Dr. E. At that point in time, with Dr. E’s prescription having brought me to my knees, and the 22nd of July being 14 days away (reminder: this is July 8) I re-iterated to the person that Dr. E  told me that certain doctors had next-week appointments for colonoscopies open, and I asked them to look those options up, because “I can no longer go to work, this is very serious”. They told me they’d look it up and call me back. I received no further calls that day.The morning of the next day, Thursday 9th, I called them again. They told me they’d call me back.

The morning of Friday the 10th, I called again, and I received the same response.

At 4:55pm, I logged into Skype and called my father through Skype credit, on his cell phone. The time difference between the eastern seaboard and Greece is 7 hours, so it’s somewhat fortunate that the man was awake. I asked him whether the private insurance policy that I had in Greece and which he managed and paid for 10 years would cover me in case of hospitalization, to which he retorted that yes, it did, 100%. I told him that in light of this, I would be booking a next-day ticket to come to Greece, and if he could make arrangements such that I’m looked after as soon as possible, that would be great.

And so I did, (booked a ticket, that is) for a hefty cost.

Ticket for Athenian trip

The ticket was one-way, since I had no idea when I was going to be coming back to the States, and in what condition.

July 14-July 22

I arrived in Athens the midnight of 12 towards 13, and spent a day at my parents’. My father had made arrangements for me to visit a local clinic on Tuesday the 14th.

Useful contextual update: Sunday the 12th of July was one of the most climactic days of modern Greek history, since the Greek government was negotiating a financial care package with the Eurogroup. The possibility that Greece would leave the Eurozone (the so-called “Grexit“) was higher than ever and, given the level of collaboration between the Eurozone and the European Union itself, the status of the country as a member of the EU and the corresponding Schengen Area would also likely be jeopardized. To the day of this writing, the repercussions of that negotiation are still visible; banks have issued capital controls which do not allow people to withdraw more than a certain amount of cash on a weekly basis, and there exists massive unemployment and budget cuts across the board. The collective spirits of the Greek people are at an all-time low, and people generally want to leave. It might naturally seem, therefore, as if I was between a rock and a hard place; extreme medical malpractice in the United States on one hand, and a devastated country on the other. Surely this guy’s health is doomed. This might make the following events even more surprising for the reader, and might raise interesting questions about the efficacy of the health system of the United States, generally considered the wealthiest country in the world, when compared to that of Greece, generally considered a failed state.

Upon arriving at the clinic, I was immediately rushed into the ER, where bloodwork was gathered within 5 minutes. Two physicians arrived 5 minutes after that, and asked me questions about my symptoms. Not one hour after I arrived in the clinic, I was on a wheelchair on my way to an ultrasound. Immediately after that, I was sent downstairs for X-rays. Within 2 hours after I arrived at the clinic, I was admitted to the hospital proper, into a room of my own, with all amenities included, including a TV set and a view.

The view from the balcony

Immediately, nurses arrived, hooked me up on I.Vs and gave me both detailed instructions and suitable equipment for collecting urine and stool samples. There was no question of my bathroom reeking of urine; it was spotless. I was also told that if I had any trouble collecting my samples, I should hit one of the buttons on the bathroom wall and I would be receiving help promptly. I did not need any help.

A next day endoscopy and colonoscopy was arranged. In the meantime, my I.Vs (which contained the same antibiotics I received in the States as well as natural serum) were being changed as needed, and nurses would arrive and check my pulse and other vitals every 4 hours. Clear instructions were given to me about the colonscopy: there was absolutely no food after 12pm on Tuesday and at 4pm I started the laxative treatment. The reader can be assured that by 12pm on Wednesday, when I was wheeled towards the surgery room to get my colonoscopy, my colon was spotless.

My diet closely monitored, I had nothing to eat Wednesday and only one light soup Thursday. On that day, I also received my diagnosis: Ulcerative Colitis, along with a touch of Gastritis and Bulbitis, perhaps caused by the Cholestyramine, since I was not having issues with my stomach before taking that particular drug.

A subset of my discharge notes

In order to cause the colitis to go into remission, I needed to start an administration of a cortizone-based medication (Prezolon) and specialized medication for Ulcerative Colitis based on Mesalamine: Asacol and Salofalk. I stayed in the clinic until Wednesday the 22nd of July. By Monday the 20th, after intensive drug treatment and monitored diet, I no longer had blood in my stool. Tuesday I started eating solid food again. On Wednesday, I received my discharge documents and the G.I doctor came to me with the grim news: I would need to follow a very thorough medication schedule over the course of several months if I wanted this to go fully into remission. In detail, I would need to be taking (and am currently taking) the following medications:

  • 6 5mg pills of Presolon in the morning, and 6 at night-time. A steady 5mg per week decrease of the dosage would effectively lead us from 6 and 6 to 6 and 5, 6 and 4… all the way to the end of the 12 week period, well into the Fall semester. The cortisone treatment being as heavy as it is, the doctor warned me that there exists the danger of my body not reacting well to the decrease of the dosage, at which point I will need to have them on call to discuss what to do (and really, the only thing to do would be to go to the clinic again and re-evaluate the drug treatment with them).
  • 3 800mg Asacol tablets, administered every 8 hours.
  • 2 20mg Losec capsules before lunch and dinner.
  • Self-administration of 4gr of Salofalk every night before sleep.

Which of course led to an interesting schedule, drug-wise:

My medication schedule

It immediately became clear to me that given: (a) The number of different medications that I would need to take over the course of the following months, (b) The health risks imposed by the necessity of the cortisone administration and (c) The fact that I simply no longer trusted a single G.I doctor in Maryland with prescribing me meds or seeing me if things went awry, I would not be able to return to the USA until the disease was under control. This would undoubtedly cause me a research roadblock, jeopardize my funding (which did end up going away in its entirety) and, as of the time of this writing, has been causing me major bureaucratic issues with respect to maintaining my student status in CS UMD as well as with procuring documents that I need in order to extend my lawful leave from the Greek Army due to PhD-level studies.

Note: The medically astounding way in which I was treated in Greece is also the product of the fact that I am the holder of a private insurance policy in that country. It is common knowledge in Greece that the public healthcare system, despite having great doctors, is impoverished enough to be lacking in even elementary medical supplies, such as band-aids. I have no doubt that if I had to depend on the public healthcare system of Greece for my treatment, I would not have had this experience. My parents were smart enough to sign me up for a private healthcare plan ten years ago, when the premiums for such plans were still tractable. Ever since, my family has been paying for this plan without me ever having to use it, until now. To make ends meet, my father had to drop his own plan. We are what used to be called in pre-2009 Greece “classic middle class”, and we cannot afford maintaining such plans for all three of us (yes, I’m an only child), nor would we be able to afford such a plan if it had not been signed for 10 years ago.

Given the fact that my family has been paying money for this plan and I myself have also been paying money out of my paycheck for the aforementioned US-based Insurancy Policies A and B, I do not see why my experience with my Greek private healthcare provider does not lend itself to comparisons with Insurancy Policies A and B, merely because the public healthcare system in Greece is lacking.  In fact, if one were to compare the two countries’ public healthcare systems, it is not clear to me who would be the winner even then. Yet this is not the goal of this post. 

July 22 – Present Day & Wrapping up

I am stuck in Greece until January 2016, essentially recuperating from my disease. The disease itself is going to be there for life and will flare up whenever I stress up. If diagnosed at the proper time (Fall 2013 or even sooner), the colitis would not have me essentially miss an entire academic semester in Maryland (I’m currently trying to make research ends meet from a distance of 7 timezones, and it’s anything but easy). It would not have me pay $1200 for a ticket back to Greece, nor would it essentially limit my professional and academic prospects in the US, since my research project is now up in the air and I can’t attend meetings, internship/job interviews, can’t talk to people in person, and so on and so forth.

The aim of this post has been to make my situation known and allow the reader to draw conclusions of their own, particularly now that so many people are transforming themselves into academic Type-A personalities and are considering moving to the States to avoid the financial or other peril of their homelands. Perhaps most striking, in my opinion, is the fact that I’ve been a victim of this neglect from different G.I doctors. In fact, there only appears to exist a correlation between Drs C and E, since they work under the same firm. Drs A, B, D and the group (C, E) are all pairwise uncorrelated as far as my case is concerned, and when a common signal of neglect arrives from all those uncorrelated sources, it clearly means that the patient is faced with one of two conclusions, which do not have to be mutually exclusive:

  1. There exists a general incompetence of G.I doctors across the USA or Maryland board.
  2. There exists something in my insurance policy that makes them not like me.

The other thing that is striking is exactly that which I mentioned in #2 above: I have been a victim of such neglect while being insured. It is common knowledge in the USA that if one is not insured, then they are doomed. But what about insured people? I performed thorough research before selecting Insurance Policies A and B, and my primary factor for choosing both has been the fact that hospital inpatient services are covered 100% within in-network facilities. In the area that I live, essentially all facilities are in-network. Despite this, as mentioned in the section of the text that detailed my ER visit, I was never given the pre-authorization necessary to take advantage of this benefit.

I am currently in a state of complete shock regarding the way that I have been treated by all those medical professionals, and it has made me re-think whether I want to stay in the USA for my future, a fact that would’ve been indisputable for me two months ago. I daresay that for people that are as qualified and hard-working as me (and we are overflowing with such people in CS UMD), it constitutes a fantastic loss for the USA to lose out on us either because the local doctors are incompetent or because the insurance policies they give them are flawed in a fundamental manner, causing doctors to not want to see us.

Thanks for reading.