A Notebook

6 Things You Probably Don't Know about Colour

2013-04-12T02:26:00.002-04:00

Colour is a strage phenomenon that is fascinating to both children and adults. It's something almost every one of us reading this post would intuitively understand -- or do we really? This post asks six questions about colour, and explores the phenomenon from a mix of physiological, mathematical, and information visualization perspectives. See if any of the answers surprise you. (Disclaimer: content is rather technical. Author is a pure math grad, after all!)

1. Why are there three primary colours?

To understand this, we need to understand how we perceive colour physiologically. Visible light is electromagnetic radiation with wavelength ranging between 380nm and 740nm. Our eyes contain two different types of photoreceptor cells: cones and rods. The rod cells are very sensitive, so it is very useful for night vision. During normal daylight, though, they are overstimulated and do not contribute to vision. The cone cells is mostly responsible for colour vision, and there are three types of these cells, called S-cones, M-cones, and L-cones. Each type responds differently to light of different wavelengths: in particular S responds most strongly to blue light, M to green and L to red.

Response to light of various wavelengths, normalized

In this way, the distribution of light striking a particular area in our eyes are translated into three different signals. In other words, we perceive the colour space as a three-dimensional space. Linear algebra tells us, therefore, that there are three basis elements.

Incidentally, some humans are born without one of the three cone types, so they perceive colour as a two-dimensional space. This is of the ways people can be colour blind. Some other animals such as birds have four types of cone cells, and this is known as tetrachromacy. The mantis shrimp actually have sixteen types of cone cells!

2. Does the rainbow contain all perceivable colours?

You can probably just name some colours, like pink, that are not in the rainbow, but for the more mathematically inclined reader, here's another explanation:

Each patch of light on a rainbow has light waves of just a single wavelength, and that wavelength varies continuously from 380nm to 740nm. So think of the rainbow as a mapping from [380nm, 740nm] to the three dimensional colour space. But [380, 740] is a one-dimensional object, so following this result, there must be some colour that is not in the rainbow.

3. Can two colours look the same but be different?

Let's rephrase the question as such: can two different patches of light be composed of different combinations of wavelengths, yet be perceived to be the same? The answer is yes. A patch of light can be composed of various amount and distributions of light of different wavelengths. This is actually an infinite dimensional space, but it gets projected down to the 3D colour space when we perceive it. So some information is lost.

There are actually exhibits in science centers showing two different yellow light sources. The two yellows appear identical, but one of them is a "pure" yellow (single wavelength), and another a mixture of two or more different wavelengths. When you observer the light source through some coloured transparency that absorbs one of the "non-yellow" light, the two lights would appear different.

The two yellows may appear distinct to birds and mantis shrimps.

4. Can two colours be the same but look different?

You've probably seen the optical illusion shown here. The two shades of grey in A and B are the same shade of gray, but they appear different. What's happening?

This is where we begin moving away from the physiology of colour and towards other ways our brain has evolved to help us survive. The brain does a lot of post-processing to help us better understand the outside world, and this means that we automatically correct for artifacts like shading.

What's less obvious from this optical illusion is that your perception of colour depends on other colours nearby. For example, the grey bar below appears to be a gradient, even though it is not. This is one of the reasons why overuse of colour to represent information is discouraged from an information visualization perspective: we can interpret the information differently depending on surrounding colours.

5. Does the RGB space consist of all possible colours?

In other words, can you obtain all possible colours by mixing red, green, and blue? First of all, note that the RGB space we've been referring to is an additive colour model (as opposed to subtractive and others). Think of the additive model as mixing paint or adding light waves to be reflected, and subtractive space as absorbing certain wavelengths of light. Because we can only add light, the answer is actually no.

Chromaticity is an objective specification of the quality of a colour, regardless of its luminance (or how much light there is). On the left is a visualization of the chromaticity space, and the right the range of colours available to a typical computer monitor. Of course, everything outside of the triangle on the left side of the screen is not well represented since you're probably reading this on a monitor.

6. Why do printed colours look different from colours on a monitor?

If you ever printed a colour image, you may notice discrepancies between a printed image and the same image displayed on a monitor. Typically in printing, a CMYK colour space is used, and the possible printable colours are not identical to disable colours on the screen. Here is an example of a set of colours displayable on a screen and printable with in:

Notice that there are areas where they do not overlap. There are different ways to fix this issue, such as shrinking the RGB triangle inwards or some other continuous mapping between the two areas. Either way, the resulting may not be the same.

There are actually other issues with RGB colour space, and people do use other colour spaces for various reasons. One issue with the RGB colour space is that the amount of red, green, and blue doesn't directly tell us that much about a colour. How muted is it? How bright is it? From an information visualization perspective this matters a lot. When choosing different colours for several objects that you would like to draw equal attention to, it makes sense to make them all of similar brightness and saturation. If you want to highlight a single element, changing its brightness or saturation is one method. This is one of the reasons that HSL or HSV space is often useful.

Other Reading

Stephen Few has more to say about colour in information visualization. The following are excellent.
http://www.perceptualedge.com/articles/visual_business_intelligence/rules_for_using_color.pdf
http://www.perceptualedge.com/articles/b-eye/choosing_colors.pdf

Although this post links a lot to Wikipedia, I learned most of the information here from Information Visualization, Second Edition: Perception for Design, which is also fascinating (and is linked below as well). I actually know very little about colour, so if you have any further questions, it's likely beyond my power to answer.

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Using Data to Optimize Jobmine Hiring

2013-03-05T21:41:00.001-05:00

Two years ago I wrote a series called Mining Jobmine that explored data from Waterloo's co-op job postings. It came as a surprise to find this analysis useful for myself. This post highlights how I used the results from that analysis to write our job posting, and how my startup Polychart hired our first two University of Waterloo co-op students.

What The Data Said

There were three main learnings from the Mining Jobmine analyses that were applicable. First is to keep job postings short, as data showed that shorter job postings attract a higher number of applicants. The second is to focus on describing the company and the role, rather than the ideal candidate. The third is to avoid using "ought" words. Both of these correlate to application levels.

Our Job Posting?

We kept our job posting to around 200 words. We described the company and the role, and while we did use some "ought" words -- well, I couldn't resist it. We also added a little bit of humour that ideal candidates would appreciate (but would confuse non-ideal candidates), and a link to our product and demo -- because who wouldn't love working on a slick product?

This was our job posting:

Polychart is a startup in the KW region building a really cool, drag-and-drop data visualization tool. We are looking for a software co-op to speed up our roadmap. See our website at http://polychart.com/ and our demo video at https://www.youtube.com/watch?v=tbvx90KDouY for what we have to date.

You will be writing code (surprise, surprise!). This includes adding features to Polychart, testing said features, and releasing it to users. Because we're a startup, you will play an important role in the development team, have a huge impact on the company, and learn a TON.

The ideal candidate will have some non-school related experience in software development, preferably web development. Specific technologies we use include:

- the usual web stuff like HTML, less/CSS
- a ton of CoffeeScript/JavaScript, JQuery, Knockout, Raphael
- Python/Django/Tornado
- MySQL and likely other databases
- git and github
- linux

Bonus points if you have some experience in statistics, have an opinion on pie charts, and on vim vs emacs.

You should be able to reverse a linked list.

Please include in your resume your github account, twitter profile, and other relevant social media profiles. Also, please talk about projects you've done outside of school. Don't write a cover letter unless there is something not in your resume you'd like to talk about.

How did we do? Well, although no one wanted to show us their social media profiles, we did receive 55 applications for 2 advertised positions. Compared to startups and companies our size, we've done very well.

The Rest of the Hiring Process

First, we had to screen the 55 applicants. The goal of the first screening is to see if there's a chance an applicant can code. A whopping 25 of them did not appear to fit the bill -- these were sometimes students from other faculties with no programming background. We didn't discriminate against first and second years, and were more lenient on those with higher GPA's.

Those successfully passing the first screening received the following programming challenge:

The goal of this challenge is to find the index of a given word in a dictionary [alphabetically sorted listed of words]. The only way to interface with the dictionary is through a function `lookup(index)`, which either returns the word at that index, or `false` if the index is out of bounds. Words are represented as strings. The indices of the dictionary are a contiguous set of non-negative integers starting with 0.
Please write your solution in either JavaScript or CoffeeScript, and include in your solution:

The code

A brief description of your solution, and why it is (or isn’t) the most efficient

How you verified that your code is correct

How much time you spent

We chose this problem because it is short but difficult to find the solution through an online search. If you are able to google a solution then you probably know a thing or two about algorithms and will meet our requirements anyways. I advised applicants not to spend over an hour on it. Around 25 out of the 30 applicants responded with a solution. Less than half of them did reasonably well.

While the puzzle helped to determine who did and did not know how to code, there were many candidates that had a plausible solution that had one or more areas of improvement. We probably weighed the code quality a little too much, since those without perfect solutions ended up doing just as well (or better!) in the interviews.

We only interviewed with 9 candidates to give each interview 40 minutes. This is to have ample time to sell the position to students, and to make sure not to run over time. We also advised students not to dress up for the interviews. We found out later that little gestures like that really helps students.

Because we already had students' solutions to puzzles, I wasn't sure whether asking additional technical questions was necessary. They were. Those with near perfect, textbook solutions to the puzzle tended not to do as well when asked to code quickly. (And yes, I brought a laptop for candidates to type on, again to make their lives easier.)

The end results were hires whose background and history either resonated with or complimented ours (diversity is very important). We're very excited to have both of them onboard!

Polychart is Hiring!

We are looking for a Director of Engineering to bring on as a partner -- a technologist who sees the potential in Polychart and wants to build a great company with us. Companies are built by people, and so a good way to judge how well a company will do is to look at their hiring process. This was ours. If you think Polychart will be a good fit, shoot me an email!

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A defence of Waterloo

2013-02-24T14:35:00.001-05:00

A couple weeks ago Khan from Khan's academy did a thought experiment to reinvent higher education. He concluded that University of Waterloo's education system comes close to what higher education should be like. The cornerstone of Waterloo education is the co-op system, which requires students to complete six, four-month length internships prior to graduation.

Those most critical of Khan were none other than Waterloo's own students and alumni. Objections ranged from a sarcastic "Apparently Khan has never attended a lecture at UW" to a more serious challenge: that the university is too focused on career development and not enough on "learning for its own sake" -- learning for the purpose of fulfillment, pleasure, and other personal purposes.

Ironically, I think that the focus on career development actually puts students in a better position to learn for its own sake. It's easy to forget that historically, learning for pleasure has always been a privilege of the rich. Even with the more accessible, modern higher education, there is a tension between choosing a major that one enjoys and choosing a more useful course of study.

Most undergraduate students are concerned about their ability to make a living post-graduation, and rightly so. Being able to sustain oneself is not only the responsible thing to do, but it also brings much personal fulfillment. Not many people want to end up as the waitress or barista with a liberal arts major and only the thousands of dollars of student debt to show for it.

Waterloo's early focus on career development means that students become secure about their future ability to sustain themselves. With that need satisfied comes the privilege of learning for its own sake. Waterloo students even have the luxury of choosing a major that is perhaps different from their career choice, but nevertheless gaining the required skills for employment through internships. (I majored in pure math, for example, and now run a data visualization software startup.)

Underlying this entire discussion is the debate about what higher education should be for. Even if one believes that higher education should promote learning for its own sake, one should be realistic about student needs that come before personal development. Being able to sustain oneself is important, and if our universities ignore that, fewer people will be able to pursue higher education responsibly.

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To get lucky, do lots of work

2013-01-15T22:37:00.000-05:00

I don't watch TV or listen to the radio, but Ira Glass from "This American Life" never fails to inspire. In the below clip (5 min) from "Ira Glass on Storytelling", he talks about how as a beginner, your abilities will fall short of your "taste". The work you do doesn't seem to have that something special that you'd like it to have, and you know that the work is not "good enough". It is only after years of work that your skills will catch up to your taste.

What he describes resonates with me in a lot of what I do, and I really like his advice for people in this situation -- just do a lot of work. Produce a lot. Gain a lot of experience. Try different things.

A couple of days ago some blog or another pointed to the below, newer clip (24min). He gives an interesting perspective on journalism, and more interestingly, about luck. Glass talks about how as a journalist, recording an interesting interview is really about luck. In order to harness it, one needs to talk to a lot of people. Likewise, in order to "get lucky" and produce great work, one needs to produce a ton of work, most of them probably mediocre.

It sounds very simple and reasonable: the more times you roll a dice the higher your chances of getting a six at least once. This is why I'm skeptical of people claiming that so-and-so was successful because of "luck". Sure, luck can be involved, but luck can be very predictable. The person who pursues every opportunity, takes every chances, and produces a lot of work will be more "lucky" than the persons who don't.

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Impact

2012-12-30T20:54:00.001-05:00

“Impact” is a key term used by both startups and big companies to attract talent. Both claim that joining them would be the best way to “changing the world”. But “impact” and “changing the world” can mean two very different things: setting direction, or creating value.

In a startup, you have heavy influence on the direction of the company and thus how society as a whole allocates resources. If you’re the founder or the CEO, the idea is yours and you’re setting the direction. However, the amount of actual value created per person is not yet certain.

In a large, established company, the company’s activity is known to be something with a high yield, one that creates a lot of value for society. As an employee, while you did not set the direction, you’re doing something that guarantees the creation of value.

Both of these influence the world. The former is important because it can further society in new and previously unthought of ways, even though there is little guarantee of success. The latter is important because it is where value is actually created, even though there is little room for innovation.

So who is changing the world more? The Microsoft or Google employee whose code is routinely rolled out to hundreds of millions of people, or the founders of startups creating new and exciting yet-to-be-released products?

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Goals vs. Resources

2012-12-20T22:56:00.003-05:00

Should you define your goals based on the resources available to you, or find the resources for the goals that you choose?

That's what I think book So Good They Can't Ignore You really boils down to. More often than not, successful people got to where they are by taking full advantage of the resources that are available to them. They did not choose a goal blindly, without having access to the resources to support it. Instead, they let their goals be influenced by what is available around them.

If you're equally interested in two fields, and have more resources, connections, and skills in one of the two, then it's obvious which one to pursue. But what if you're more interested in the one with less resources? No resources? Wouldn't your passion guide you through?

Perhaps, but the odds are against you. The sad thing about all the success stories we hear where "passion" won out is that we don't know how many people failed doing the exact same thing. Failures don't make a good story, so it doesn't get talked about.

Then again, "success" can mean different things to different people.

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Startup Girl Survival Guide

2012-12-07T13:36:00.001-05:00

This is a guest post I wrote for VeloCity's guest blog. When Kim asked me to write about being a startup girl back in September, I wasn't awfully thrilled about the idea. There were many of themes that I was only beginning to come to term with, so it look a lot of thinking. By far this was the hardest blog posts I've ever written. (Yes, even harder than the blog post about Galois Theory.) You can also read the post at http://velocity.uwaterloo.ca/guest-bloggers/startup-girl-survival-guide, and check out my startup Polychart when you're done! Enjoy!

It feels like I’m doing a disservice talking about being a startup girl. True, I’m one of the few female technical founders in the VeloCity program, and as a pure & applied math grad, a programmer, data scientist, entrepreneur and a go player, there are many facets of my life where I’ve felt a distinct lack of the double-x chromosome. Despite all of that, I was fortunate to have a very supportive (and female) programming teacher in high school, and really awesome co-op and VeloCity mentors. They helped me make it through without feeling like a startup girl is all that different from any other startup founders.

But are we really that different? We face the same challenges as anyone else, balancing between getting the product out, talking to users, innovating, shipping, selling, building, and marketing. We go through the same emotional roller coaster ride as everyone else.

I don’t think that we are really different. I do think, though, that there are challenges we face that our male counterparts do not. That’s what this guide is about, advice from one startup girl to another.

1. Be confident. Confidence is crucial for all startup founders, and especially key when you could be mistaken for the secretary or the supportive girlfriend who’s helping out her co-founder boyfriend. It should be your priority to take the lead to introduce yourself, and to take a more assertive role in conversations. Confidence building is very personal, but there are tools out there that can help, rejection therapy being a great one (that I haven’t, but should have, finished). Understanding the imposter syndrome helped me a lot personally.

2. Build a Reputation. Just by having a female name, women are perceived to be less competent. How, then, can we build trust and make sure we are heard? One solution for this is to build a reputation by being outspoken about your work. Keeping a blog is a great way of doing that. Writing about projects you’re working on, sharing insights, and contributing to the community gives people a chance to understand what you’re capable of. It also gives a chance for people you meet to do “background checks” on you, and see the best of your talents.

3. Find Mentors. Lack of mentors is a frequent complaint from women in technology. Entire blog posts have been written about why this is the case, and the most interesting one hypothesizes the following: older and more experienced men don’t want to be seen as the “dirty old man” that is helping a young woman for a less than altruistic reason. This is a difficult stereotype to overcome. One way around it is to join a program with a mentorship component. VeloCity is one, as are other incubators and accelerators.

4. Find Peers. I don’t necessarily mean being in a community of other startup girls, but I do find it important to be amongst peers — people who are in similar positions, with whom there is mutual respect. This should happen naturally as you gravitate towards people with similar interests, but in a male dominated industry, groups can be difficult to break into. Like in any situation, confidence is key. The culture of the group may change because you’re a female, but let your confidence and actions speak for themselves. As with all social groups there are always challenges to entry, look at this as an opportunity, not a problem.

5. Affirmative Action. There, I said the “A” words. I used to get very worked up about affirmative action and how unfair it is to men, and even pondered whether my team won the VVF because I am a startup girl. Not any more. While affirmative action programs are right in our faces, the subtle ways we are discouraged are not. It’s a fair compensation.

6. Don’t think about being a startup girl. If you look for discrimination, you’ll find it everywhere. As a startup girl, you’re better off spending the emotional energy on the startup itself.

Don’t get me wrong, I do believe that as a society, we need to understand and eliminate the extra barriers that prevent startup girls from succeeding. There are a number of characteristics that women must have that men don’t need to be successful in this business, and it’s not fair. For now though, I think the best way for women to be successful is to be confident, be relentless, and break through those barriers.

Don't follow your passion

2012-12-03T16:39:00.001-05:00

Cal Newport argues in his book, So Good They Can't Ignore You, that "follow your passion" might just be terrible advice. This is a much needed challenge to the typical North American "you can do whatever you put your mind to" philosophy. While Newport's tone is not at all pessimistic, it's hard not to be reminded of this comic:

No one in their right mind (at least in North America) would tell a kid that they can't be what they want to be. This optimism can be healthy and can encourage innovation, but can also trivialize the amount of hard work required to build any kind of great career. Great careers are rare and in order to have one, one must have skills that are equally rare and valuable. These skills take many years to develop.

Newport studies people who are passionate about what they're doing, to find out how they got there. For one, these people took little steps (as opposed to big ones) to get to where they are. They focused on gaining valuable skills at each step, and when their abilities outgrew their current job, they took the next logical step.

These people did not identify a passion and follow it. Newport believes that focusing on passion is harmful. He believes it convinces people that there is a magic "right" job waiting for them somewhere, and the moment they find it they'll recognize it. In reality, this kind of certainty is rare, and questions like "Is this who I really am?" and "Do I love this?" rarely reduce to a clear yes-or-no response.

Among the many suggestions he makes, one that stood out was to try different projects, each one not taking more than a couple of months, and see how you and the world feel about them. This feels like the start of a "lean career" movement: try a lot of different projects, see what works and what doesn't, and iterate. He also suggest that perhaps it's not so much about what you do, but about how you do it and the attitude you take. I certainly know people who seem to enjoy everything they do, and do it all with an intensity that makes you think that this is their life passion.

So Good They Can't Ignore You is a book with some refreshing ideas: ones that finally put more emphasis on skills and hard work, and less on passion and dreams.

PS: Sivers has a great set of notes for this book that covers the main points (and is also more concise than the book itself).

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Best for the company

2012-07-12T23:10:00.002-04:00

A while ago I heard of someone saying something to the effect of this,

I don't believe that the decision to do X would benefit anyone, but it's the best for the company.

No matter what Romney says, corporations are not people. Corporations do not have values or desires the same way people do. When we say "Y is the best decision for the company", it is ambiguous whether we mean Y is the best thing for its employees, its customers, its investors, or its other stakeholders.

Maybe this ambiguity is intentional. Maybe we ourselves are confused. Or maybe we are too afraid to acknowledge who we're really trying to benefit at everyone else’s expense.

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Why women may be thought of as bad programmers

2012-07-10T23:51:00.002-04:00

I got my first tech-related internship back in the summer of 2009. The position was a very competitive one with a company in the heart of San Francisco. While I was ecstatic, I was also confused.

I hadn’t done very well on the interview. I screwed up the first question about dropping two eggs from a 100-story building (couldn’t get away from thinking binary search), and messed up a different question about the number of bits required to store some large number (mixed up “bits” and “bytes” and tried to calculated log_2 in my head, only to get it wrong).

Nor did I have much experience. I had barely heard of a version control system, had never really gotten accustomed to the command prompt, and hadn’t even written a single line of production code (well... beyond my silly tournament signup page, which refused to let you sign up if you had an underscore in your email).

Needless to say I probably left very bad first impressions on my coworkers. I was the silly little girl asking questions like, “What was that command you use to get into that other machine?”

So, here’s the hypothesis: companies want to hire more female programmers, either because they are constantly accused of being sexist or because they actually value having a gender-balanced team. So they lower the hiring bar for women, and end up letting in people who may not be as experienced. The rest of the team only sees that a woman was hired and she has very little experience as a programmer, and concludes that women are bad programmers.

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Value of Re-Posts

2012-07-03T23:18:00.000-04:00

Prior to 2009 all of the posts here were written for my own pleasure. I didn't consider writing for a wider audience to be worthwhile, because there were already a surplus of mediocre bloggers and z-list blogs. If there was something I could think of to say, it was probably already said in much more eloquent words by a more accomplished person. Would I be adding any value by just saying the same things in different words?

Then in late 2010, there was a discussion about a biologist who published a paper in 1994 reinventing the trapezoid rule in calculus. While it points to a serious flaw in the peer-review system, John D. Cook had this to say,

The paper reinventing the trapezoid rule has been cited 75 times. It must have filled a need. Yes, the author was ignorant of basic calculus. But apparently a lot of other doctors are just as ignorant of calculus. The author did the medical profession a service by pointing out a simple way to estimate the area under a glucose-response curve. The technique was not original, and should not have been published as original research, but it was valuable.

The message contained in a blog post is not the only factor that determines its worth. What matters more is how much the message is spread and how it affects its readers. This is true for other communication mediums. The endless retweets, reposts, etc. does serve the value of making an idea reach more people.

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Startups vs passion

2012-07-02T23:59:00.000-04:00

One thing I dislike about the startup world is that we sometimes take the word "startup" to be synonymous with doing what one loves and pursuing one's passion. Founders talk about how their love and passion are the reasons why they started a startup, and the rest of the world concludes that anyone who is not doing a startup is not doing what they love.

It is true that a lot of love and passion needs to be poured into a startup. As Steve Jobs puts it, "because [doing a startup] is so hard, that if you don't [have the passion], any rational person will give up."

What's not true is that startups are the one and only way to pursue one's passion. By definition a startup's purpose is to find a repeatable and scalable business model. If you don't want to find a repeatable and scalable business model, you don't want to do a startup.

Instead, your passion may involve a business model that is not scalable or not repeatable. It might not be a business at all. Perhaps your passion can be better packaged as one of the following:

open source project
consulting firm
blog
lifestyle business
non-profit
vacation
academia
etc...

The nice thing about startups is that there are fair amounts of resources and support systems around building them. How do you know if a startup is right for you? Instead of saying "I want to do a startup in X", say "I want to find a repeatable and scalable business model in X", and see if it makes you wince.

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Ikea shopping and λ-calculus

2012-07-01T12:34:00.000-04:00

Yesterday I was at Ikea with a friend of mine (whose startup, Flockwire, has desks right beside mine), picking furniture for my first ever unfurnished apartment.

I was spending way too much time trying to pick the right shower curtain, when two thoughts simultaneously crossed my mind.

The first was a recent story I heard about why Alonzo Church picked the greek letter λ for his calculus. It is from Dana Scott's talk in the Turing Award webcast,

...but many years later I asked John Addison, Church's son-in-law if he would ask Church where the lambda came from. So he wrote him a post card [...] "Dear Prof Church, Russell had the iota operator, Hilbert had the epsilon operator, why in the world did you choose lambda for your operator?" So Church didn't write a letter, he just annotated the post card and sent it back, and he put in the margins, "Eeny, meeny, miny, moe."

The second was the following clip from Fight Club,

On one hand, a shower curtain is a shower curtain. There is really no difference between two shower curtains of a similar quality, just as the letter λ is no better or worse than any other greek letter. On the other hand, whichever shower curtain I pick will be sharing my bathroom for at least a year, so as silly as it sounds, I did want to choose one that "defined me as a person".

It's funny to imagine Church asking the question, "which greek letter best defines me as a person?" The letter Church chose is now attached to his name -- not for one year, not for two years, but indefinitely.

I ended up choosing the curtain by bringing two to checkout, and making the final decision under time pressure. It was much easier that way.

(It feels so strange, even blasphemous, comparing λ-calculus to my shower curtain.)

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Microsoft and Income Distribution

2012-06-30T23:59:00.000-04:00

Last year Forbes published an article titled "9.2% Unemployment? Blame Microsoft." The premise was the following:

Over the past twenty years, the technology industry, led by companies like Microsoft, have given us powerful databases, operating systems, networks and software applications that have made it easier for us to accomplish more tasks than we did before with less people. And it’s not just Microsoft who you can blame.

The article included the following graphic, which shows that despite a decrease in manufacturing jobs, manufacturing output has risen from 2000 until the 2008 crisis.

It shouldn't be surprising that we're becoming more and more efficient, and that there are fewer unskilled job options available. The question is, with US unemployment rate at about 8%, and 12 million americans unemployed, (and Canada only doing slightly better,) is this increased technological efficiency part of the root of our current economic situation? What if these people just aren't needed in any modern economy, healthy or not? What if we just don't need many people to work to support everyone?

I have one group of friends who are complaining about the economy, and another group of friends who studied one of the STEM fields and have many competing offers for six-figure salaries right out of school. By and large the first group would balk at hearing something about executives being paid millions of dollars to lead a company to lose money, about the University of Waterloo President's $500,000 2011 salary, and heck, even about how much Facebook interns make.

But what if it really is demand and supply? What if, because of technological advancement and other factors, these CEO's skills and work are more valuable than 100 or even 1000 workers? What would it do to society if a quarter of the population (plus some robots) can produce more than enough for the rest of the world to consume? What, then, would be a "fair" way to distribute income that would take advantage of this surplus?

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Who you know vs. what you know

2012-06-29T21:00:00.001-04:00

Back in high school, people told me that in real life, success will be determined more by who you know and less by what you know.

I think that these people have been correct in that success is largely determined by who you know. But they failed to mention that who you know in turn depends on what you know. What you know, what you learn, what you do and how interesting your projects are determines how interesting you are as a person, and whether someone might want to get to know you.

So the moral of the story is: to meet interesting people, do interesting things.

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Small decisions

2012-06-28T23:27:00.000-04:00

We normally think that big decisions are what makes the biggest impact on our lives. We thus spend a lot of time deciding where school to go to, which job offer to take, and where to live. These decisions are so-called "big decisions" because they end up affecting a sizeable chunk of our lives for a large period of time in somewhat predictable ways.

Small decisions are rarely thought about. These are decisions like whether to send that email now or in an hour, whether to ask to present your findings in tomorrow's meeting or next week's, and whether you follow up with someone you met last night or put it off.

There's only a small chance that any single small decision would affect much of anything. Most of the time, the email receives no response, the presentation is uneventful, and no useful lead comes up. But because we face these small decisions on a regular basis, sheer number of these events mean that there is still a lot to be gained from being just a little more proactive.

I found that reaching out just a little bit further on a regular basis makes it more likely for those one-off "lucky" events to happen: a blog post goes viral, a grant is received, a new friend gained. In fact many of the wonderful things in my life happened by making a small decision that required being a little bit out of my comfort zone.

So in the end, which school you go to, which job offer you take and where to live might affect your life much less that what you do there, which opportunities you seek and how much you push yourself.

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Reading and Writing

2012-05-01T13:09:00.004-04:00

It is quite unfortunate that English teachers are the ones who teach young children about the importance of reading. The only things I ever saw English teachers read were novels, and I found it difficult to believe that reading novels is that important.

Fast forward a couple of years, almost everything I learn is from books, blog posts, or other forms of written media. Reading has been the easiest way to get inside the best minds.

Since I've founded the data visualization startup Polychart, I've been spending a considerable amount of time learning enough business-ish concepts to get by. Books I've recently enjoyed include:

The Lean Startup by Eric Ries: a book that everyone in the startup world talks about and references. The case studies included are interesting.
The Personal MBA by Josh Kaufman: a really dense book about many aspects of business; claims to contain all the fundamental principles taught in an MBA program.
The Art of Profitability by Adrian Slywotzky: thoughts about profitability in an almost novel-like format. It's hard not to read many chapters in one sitting, even though you're not supposed to.
Ignore Everybody by Hugh MacLeod: fun read but really only learned one thing from it, which is to separate one's work from one's hobby. This goes against the conventional wisdom that you should follow your passions and make it your career.
Purple Cow by Seth Godin: again a fun read but it seems like the ideas in this book are already well-received enough that they are no longer surprising.
Don't Just Roll The Dice by Neil Davidson: a short read on software pricing. Page 37 onwards contain a lot of not-so-obvious psychological nuances about how people react to pricing. There is also a free PDF version.

It is also quite amazing how much time I'm spending writing. I spend maybe 10-20% of my time on email, and at least 30% of my time writing. This includes writing things like documentations, blog posts, emails, and other necessary documents.

So if you're still in grade school, don't neglect to practise reading and writing.

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Pure and applied math

2012-03-07T17:34:00.001-05:00

I had several questions about my majoring in pure math and applied math. To people not acquainted with the different branches of mathematics, this is the picture they have in mind:

Here, pure math is thought to be the kind of math that has no foreseeable application, and applied math consists of everything else. Fortunately (for me), the actual picture looks more like this:

Applied math is about modelling systems (e.g. physical systems) using calculus and tools derived from calculus. Pure math can be thought of as a combination of two sub-fields: algebra and analysis. Algebra deals with properties of sets objects that can be operated on (i.e. added/subtracted/composed), and analysis provides a rigorous basis for calculus (i.e. a lot of epsilon-delta proofs). Since applied math relies on calculus, analysis is a topic of interest to applied mathematicians as well. Thus pure and applied math intersects.

A lot of times, we consider other branches of math that are "applied" as their own branches. In that case, the picture looks more like this:

Computer science uses many ideas from pure math, and some computer science theorems and proofs read like theorems and proofs from algebra. Machine learning, a computer science discipline, is really just another name for statistical learning. There are other topics in mathematics, but I don't know about them enough to come up with a full taxonomy.

Moral of the story is: pure math and applied math do intersect. In fact, if you pick any two branches of math, there is probably an overlap.

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SOPA/UBB and NDAA

2012-01-16T13:03:00.002-05:00

It's pretty awesome to see ordinary people speak up for the Internet to prevent the passage of SOPA. The whole "black-out" campaigns and boycotts were quite nicely played out. It reminds me of a year ago when telecom companies in Canada tried to introduce Usage Based Billing to charge Canadians more for internet use, and squeeze out independent ISP's. Canadians won in a similar manner: by making their voices heard on the internet.

Yes this is great, but there's something unsettling as well. All this activism to stop the politicians from mingling around with the beloved Internet, and dead silence on NDAA, which allows indefinite detention without trial to be codified into law. Rights and freedoms? Sure, take them! But god help you if you try to charge more for Internet.

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Reword your resolutions

2011-12-31T00:04:00.000-05:00

I had various degrees of (lack of?) success setting goals. Most of the goals I had were pretty standard and dryly worded: "one tweet per day", "finish reading data viz books", etc. Even though these are things I really want to do, just reading the list was a little draining.

But one goal was different. It was written:

Meet Colbert.

This was in reference to the Hacking Education contest. Entering the contest was really an excuse to practise playing with data and writing about the results in my new data blog. Colbert wasn't even relevant. Even then, those two words made me smile. It excited me, so much that I just had to work on it... Although I didn't end up meeting Colbert, I accomplished what I was really after.

It's so much easier to stick to goals that make you smile, that motivate you, and that capture the desired result rather than the hard work required (even though the hard work is just as important). So if you're setting goals, make them interesting. Make them playful. Make them funny. Figure out what excites you, then use that as a proxy for what you actually want to accomplish (e.g. "meeting colbert" as a proxy for "start a data blog and write stuff in it").

Happy 2012 and if you have some resolutions in mind, good luck!

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Galois Theory in 1500 Words

2011-12-13T19:52:00.000-05:00

[This is a very brief overview of Galois theory. It's not meant to be rigorous, so apologies in advance for leaving out a lot of details and avoiding some delicacies. It was meant to be intuitive and light on math, but it turned out to be neither. If you aren’t familiar with field and group theory, well... proceed at your own risk...]

For a long time, people wondered whether it is possible to write down something like the "quadratic formula" for cubic, quartic and quintic polynomials with integer coefficients. We now know that for cubic and quartic polynomials, this is possible. But for degree 5 polynomials and beyond, it isn't. A proof of this was scribbled down hastily by Galois the night before his duel. Galois linked together field theory and group theory in a beautiful way to answer this very question.

Galois’s Approach: The Big Idea

What does writing down a “formula” for roots of a polynomial really mean? For one, we’d be writing down the roots in terms of rational numbers and a combination of +, -, x, ÷, and radicals (taking n-th roots). This is a very limited set of operations, and certainly not all real numbers can be written this way -- π clearly can’t be written this way. We say that π is not solvable in radicals.

Are the roots of polynomials with integer coefficients solvable in radicals? Those roots aren’t just any real number, and certainly π is not a root of any polynomial with integer coefficients. Yet Galois showed that there are some degree-5 polynomials with roots that are not solvable in radicals. To see how he did this, we first need some terminology about fields, field extensions, and groups.

Fields

The set of rational numbers Q is an example of a field: a set of things you can add, subtract, multiply and divide. We can “extend” Q into bigger fields by adjoining things to it. For example, L=Q(√2) -- pronounced "Q adjoined root two" -- is defined to be the smallest field that contains both Q and √2, and is closed under +, -, x, and ÷. Here elements of Q(√2) are exactly numbers that can be written in the form (a+b√2)/(c+d√2) where a, b, c, d are integers. We call such a field L a (field) extension over Q (written L/Q).

When we're adjoining √2 to Q to construct Q(√2), what we're really doing is adjoining to Q a root of the polynomial f(x)=x² -2. We could do this with other higher-degree polynomials. Let p(x) be a polynomial with integer coefficients (and no repeating roots). We define the splitting field K of p(x) to be the smallest field containing both Q and all the roots of p(x). For example, the splitting field of p(x)=x²+1 has roots i and -i so a splitting field K of p(x) is K=Q(i,-i)=Q(i). (The last equality is true because 0 and i are in Q(i), so -i=0-i is also in Q(i)).

Conversely, we call an extension K/Q a Galois extension if it is the splitting field of some polynomial p(x). From before, Q(i)/Q is a Galois extension.

Q-Fixing Automorphisms

An automorphism F of a field K is an isomorphism from K to itself, where the algebraic structure is preserved -- specifically, F(a+b)=F(a)+F(b), F(ab)=F(a)F(b). In the case that K is an extension of Q, we’re more interested in automorphisms of K that has F(x)=x for all x in Q (or that F fixes elements of Q). An automorphism F of K is a Q-fixing automorphism if it has this property.

There are two Q-fixing automorphisms of L=Q(√2): the identity automorphism (call it e) that takes each element of Q(√2) to itself, and an automorphism (call it t) that takes anything from Q to itself, and √2 to -√2. It is possible to show that there is exactly one such automorphism t.

Groups

A group is a set with a "composition" operation • with an identity element. From above, the set of Q-fixing automorphisms of K, denoted G(K/Q) is a group with • being function composition, and e being the identity element. Observe that we do not require • to be commutative (so a•b may not be the same as b•a in this case).

Two groups are isomorphic if they have the same algebraic structure -- i.e. they’re essentially the same group except the elements have different names. It is useful to see whether G(K/Q) is isomorphic to a group that we know and understand. Two important classes of groups that are well understood are cyclic groups and permutation groups.

Examples of Groups

The cyclic group C(n) of order n is the set {0,1,2,3,..,n-1}, with • being addition mod n. From the example of L=Q(√2), L has G(L/Q)={e,t} with • being function composition. This is actually isomorphic to C(2)={0, 1}.

Permutation groups consist of functions that permute some "letters". We'll use S(n) to denote the group of permutations of n letters. For example, S(3) is all the permutations of letters {a, b, c}. A function F that takes a→b, b→a, c→c is one such permutation (and hence an element of S(3)).

The Fundamental Theorem of Galois Theory

Galois noticed that for a Galois extension K/Q, there is a link between "subfields" of K containing Q, and "subgroups" of G=G(K/Q). The quoted words mean exactly what you might think -- a subfield of K is a field L that is contained in K (and is closed under +, -, x, ÷). For example, Q is a subfield of R, and Q is a subfield of Q(√2). A subgroup of G is a group H that is contained in G (that is closed under • and contains the identity element). For example, the subset {0,2,4} is a subgroup of C(6) = {0,1,2,3,4,5}.

More concretely, there is a 1-1 correspondence between subfields of L and subgroups of H:

For a subfield L of K containing Q, there is a subgroup H of G corresponding exactly to the automorphisms that fix all of L -- i.e. f(x)=x for all x in L, not just Q.
The reverse is true as well: if H is a subgroup of G, then there is some subfield L of K containing Q that is fixed by all of H.

In particular, whenever L/Q is itself a Galois extension, H is a normal subgroup of G. Think of H being normal as H being well-behaved enough that the quotient G/H is another group. We won't get into what this means, but this group G/H turns out to be isomorphic to G(L/Q).

For a concrete example, let’s take K=Q(i, ∛2). It’s possible to show that K/Q is a Galois extension, and that G=G(K/Q) is isomorphic to S(3). In particular, the subfields of K are Q(i) and Q(∛2), and they correspond to subgroups of S(3) isomorphic to C(3) and S(2). We saw before that Q(i) is a Galois extension, and C(3) happens to be normal in S(3).

Linking Back to the Big Idea

Suppose p(x) is a degree 5 polynomial, and that it has a root x that is solvable in radicals. Then really, x is in some field K containing Q, where K can be "built up" from Q by successively adjoining (n√α), the n-th root of α, for some n and α. For example, take x=√(2+ √5). We set L=Q(√5) and K=L(√(2+√5)) to "build up" K in this manner.

Solvable Fields

In general, we call an extension K/Q solvable if K=K0⊇K1⊇K2⊇...⊇Q, where each K(i-1)=Ki(n√α) for some n and α. This is exactly the construction we had a paragraph ago. As another example, K=Q(i, ∛2) is a solvable extension since Q(i, ∛2)=Q(i)(∛2)⊇Q(i)⊇Q is in the desired form (recall i=√(-1)).

For a polynomial p(x) in Q, the splitting field K of p(x) is the smallest field containing all the roots of p(x), so the roots of p(x) are solvable in radicals if and only if K is solvable.

Solvable Groups

We can actually assume (ignoring some subtleties) that each K(i-1)/Ki from above is a Galois extension. In this case each G(K(i-1)/Ki) is actually isomorphic to C(n) for some n.

Thus in order for a field extension K/Q to be solvable, G=G(K/Q) must be in a particular form: there has to be a chain of subgroups, G=G0⊇G1⊇G2⊇...⊇{e}, where each Gi is a normal subgroup of G(i-1) and Gi/G(i-1)=C(n) for some n, and {e} is the trivial group with just the identity element. In the case of K=Q(i, ∛2), the chain of subgroups looks like G(K/Q)=S(3)⊇C(3)⊇{e}.

A Quintic Formula Cannot Exist

To recap, roots of p(x) being solvable in radicals requires the splitting field K of p(x) to be a solvable field, which in turn requires G(K/Q) to be a solvable group.

But with a little group theory, we can show that S(5) is not solvable. Further, any quintic polynomial with two non-real roots has Galois group S(5). These last facts require some more concepts to develop, but in any case -- not all roots of quintic polynomials are solvable in radicals.

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Visualizing 4+ Dimensions

2011-12-05T22:21:00.001-05:00

When people realize that I study pure math, they often ask about how to visualize four or more dimensions. I guess it's a natural question to ask, since mathematicians often have to deal with very high (and sometimes infinite) dimensional objects. Yet people in pure math never really have this problem.

Pure mathematicians might like you to think that they're just that much smarter. But frankly, I've never had to visualize anything high-dimensional in my pure math classes. Working things out algebraically is much nicer, and using a lower-dimensional object as an example or source of intuition usually works out -- at least at the undergrad level.

But that's not a really satisfying answer, for two reasons. One is that it is possible to visualize high-dimensional objects, and people have developed many ways of doing so. Dimension Math has on its website a neat series of videos for visualizing high-dimensional geometric objects using stereographic projection. The other reason is that while pure mathematicians do not have a need for visualizing high-dimensions, statisticians do. Methods of visualizing high dimensional data can give useful insights when analyzing data.

"But there aren't any more direction to draw the fourth axis..."

A professor would draw the usual x, y and z-axis on the 2D chalkboard, then announce that there aren't any more directions remaining to draw another axis until someone invents a 3D chalkboard. Then I would die a little.

Of course there is! How did we pick the direction to draw the z-axis on a 2D surface? We picked a direction not parallel to the x and y-axis, kind of at random (because any such direction would work). In doing so, we decided on a projection of the 3D space onto the 2D surface, like this:

There are some ambiguities as to which points in the 3D space these black points actually represent. In fact, each point above has an entire line projecting to it. Sometimes, authors would disambiguate points by dropping a line parallel to the z-axis down to the x-y plane, like this.

So, as we said before, we add a 4th dimension by drawing a t-axis not parallel to any existing axis. Here each point have an entire plane projecting to it, so we drop down another a line parallel to the t-axis to disambiguate. We get this.

So voila, we get a projection of 4D space onto the 2D surface of your screen. The following image from Wikipedia showing hypercubes of dimensions 1-4 uses the same method as above:

Technically speaking, this method can work for any number of dimensions, but things gets messy and confusing very quickly. Even in the 4D plot above it's challenging to wrap your head around what's happening -- dropping verticals parallel to all the other axes might help a bit, but even in 5D this would get pretty messy. Perhaps that's why the images of the hypercube stopped at dimension 4.

So, are there nicer ways of visualizing even higher dimensional data?

Parallel Coordinates, star plots, etc

Yes, there are many, and they are often used in statistics. One simple technique is to map certain dimensions to other features of each "point" -- its shape, colour, size, etc. This is so often done that you probably wouldn't even think of it as a visualization technique, but it works for visualizing a few more dimensions. There are other techniques, too:

Radial plots have all the axes in a circle, like below, with each "point" or observation drawn as a polygon with vertices determined by its value along each axes. Sometimes it is better to make separate plots for each "point", since overlapping lines can make the chart messy. Again this works well for visualizing a handful of dimensions, and small number of "points" or observations.

Parallel coordinate plots are like unwrappings of the above. A "point" becomes a series of line segments connecting its values along each dimension. This chart is considerably less messy, so a larger number of "points" can be plotted on a singe graph.

Then there are goofier things like Chernoff's faces, which maps dimensions to features of faces. The idea is that since we are biologically hardwired to tell apart faces, we'd be able to easily tell which data points are similar to each other and which ones are different.

All of these plots make it relatively easy to find clusters within the data. However, it is difficult to find geometric properties: Can you imagine what points on a tetrahedron in 3D would look like here? A sphere? Thus from a mathematical standpoint, these graphics do not preserve anything that is usually of geometrical importance (angels, lengths, etc) -- stereographic projection fare better, but algebraic techniques are often quite powerful.

Visualizing High Dimensional Data is a Statistician's Problem, not a Mathematician's

To conclude, it is possible to visualize high dimensional objects. However, from the point of view of a pure mathematician, such visualizations are usually less helpful compared to algebraic techniques and intuitions on how low-dimensional object behave. Thus the problem of visualizing high-dimensions is a statistician's problem. Statisticians have much more to gain from visualizing their high-dimensional data.

So if you ever want to ask this question to a pure mathematician, ask a statistician instead. They'll be able to give you a better answer.

[PS: As @chlalanne pointed out, statisticians have built pretty good tools for visualizing high dimensional data. GGobi is a pretty powerful one. It interfaces with R through the rggobi package.]

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It turns out

2011-11-29T20:28:00.001-05:00

So it turns out that Douglas Adams had this to say,

Incidentally, am I alone in finding the expression ‘it turns out’ to be incredibly useful? It allows you to make swift, succinct, and authoritative connections between otherwise randomly unconnected statements without the trouble of explaining what your source or authority actually is. It’s great. It’s hugely better than its predecessors ‘I read somewhere that...’ or the craven ‘they say that...’ because it suggests not only that whatever flimsy bit of urban mythology you are passing on is actually based on brand new, ground breaking research, but that it’s research in which you yourself were intimately involved. But again, with no actual authority anywhere in sight.

I'm convinced that every professor knows about this, and chuckle to themselves every time they use this phrase.

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Bubble Sort

2011-11-13T12:31:00.001-05:00

Bubble sort is the crappiest sorting algorithm out there. For the longest time I found it perplexing why it was taught in school, when insertion sort and selection sort are more intuitive, (slightly) more efficient, and easier to implement. Researchers mentioned in the Wikipedia page find it mind-boggling as well.

Then bubble sort came up in the most unlikely circumstances, in Galois theory, when proving that the roots of the polynomial x^5+15x+5 cannot be written using +, -, *, / and radicals (taking the n-th root). The key idea is that one can sort a list when equipped with two operations: rotating the entire list, and flipping two elements (at fixed indices).

This means that one can also obtain any permutation of a list using the same two operations. The fact that this can be done isn't difficult to show, but it isn't immediately obvious either (at least to me).

So the takeaway is this: bubble sort tells us that it is possible to permute a list by means of rotation and flipping two elements.

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Coding Feels Good

2011-11-06T23:46:00.002-05:00

It has been several weeks since I just sat down and coded for several hours. It felt good. The satisfaction of building something useful and beautiful is part of the reason why it felt good, but not all of it. There's something meditation-like about the way it makes you focus.

The bulk of my last few weeks were spent studying math. Fourth year pure math forces you to focus, in a way. It slams something hard right in your face, and if your mind isn't already in a state of complete focus, it'll go right over your head. Coding isn't like that. Coding is gentle, it draws you into that focused state slowly and steadily. You can start slow if you'd like, then dig deeper and harder until you're knee deep rewiring the inner workings of your creation. You suddenly find your hands dirty. There are smudges on your face.

It is almost therapeutic. Especially after you stop doing it for a few weeks.

At the end of first year, I threw a fit when a good friend tried to teach me to use svn and Google Projects. I was convinced that wouldn't have to code anymore after CS136. Fast forward a couple of years, and I realize how you can't really do much without knowing how to code. I think that programming is the new literacy. Knowing how to code is almost like knowing magic spells -- useful spells that can automate things, interact with people far away, and manipulate large amount of data.

And spell-casting feels pretty darn good.

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