applied-math

Stable Transmission Trees on Moving Sinks of Wireless Sensor Networks

This talk was given by Yoshihiro Kaneko at SEICCGTC 2017. He’s been working on this with a bunch of students, whose names I didn’t get a chance to write down. We had a long conversation afterwards because I wanted to clarify some things that came up in the talk, but we also ended up talking about some differences between applied math and engineering ;)

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I think the name wireless sensor network is pretty self-explanatory: we have a bunch of sensors that are all collecting data, and we want to bring it all together to one place. So we allow them to communicate with each other by sending all of the data packets to a single CPU. The CPU will generally be pretty far away from most of the sensors, so it can’t connect with them directly. But it is pretty computationally expensive to set up communication channels, so we want to do it somewhat efficiently.

However, there is a fairly serious practical issue with this model, which is that the sensors closest to the CPU are then responsible for handling every data packet sent by every single sensor. If the network is large, this can lead to a backlog, or an excess of energy usage. (The latter might be a really big problem if you’re running on a charge!)

Fortunately, people have proposed many fixes for this problem, which have their own strengths and weaknesses. The one that Kaneko chooses is called the moving sink model, in which the responsibility for doing the computations moves between multiple CPUs. Note that this is not parallel processing: at any given time, only one of the CPUs is considered “active” (or the sink) and receives all the packets.

One big problem with the moving sink model is that every time the sink moves, we need to figure out how we’re re-routing the data from each sensor to get there. Remember that it’s expensive to set up communication channels, so we don’t want to have to keep re-establishing them: we’d like our network to change as little as possible.

Moving into the world of abstract graph theory, the problem looks like this: the network is a graph with a collection of distinguished vertices (corresponding to CPUs). We “move the sink” by creating a (finite) sequence \$(v_1, v_2,\dots, v_k)\$ of distinguished vertices (repeats are allowed), and for each \$v_i\$ we want to create a spanning tree \$T_i\$.

In this language, the goal is to construct the \$T_i\$ in such a way that “no two adjacent \$T_i\$ are too different from one another”. In other words, for some fixed distance function \$d\$ on the set of spanning trees and minimize the total distance between consecutive \$T_i\$:

\$\$ D = d(T_1, T_2) + d(T_2, T_3) + d(T_3, T_4) + \cdots + d(T_{k-1}, T_k). \$\$

Hmmm… distance function…… on the space of trees…… hmmm………….

But sadly to all the geometry fans out there, we went with something a little more mundane: to get the distance between two spanning trees you count the number of edges that are in one tree and not the other.

He spent the last bit of the talk describing the algorithm they created for creating the \$T_i\$ in a particular setting (the network is a grid graph, the \$v_i\$ are the boundary vertices ordered counterclockwise). His students claim to have a proof that their algorithm generates spanning trees which minimizes \$D\$, but apparently the proof involves a lot of casework and he wasn’t quite willing to confirm it yet.

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I actually asked him when we were talking if they had considered the BHV metric (which is the professional way of saying “You should consider this” to someone). They had not, but he said that it wouldn’t be likely to be useful: we want \$D\$ to represent the cost it takes to set up a communication channel. Since we expect this to be a fixed-cost operation, the edge-counting distance makes sense for this particular application. Conversely, the operation of “shrinking the tree to zero”, which is fairly common in BHV-geodesics, is pretty meaningless.

[ In the same vein: his distance was slightly different than the usual standard, because the standard method makes the trees directed and counts reversing an edge as distance 1, but he counted it as distance 0. This is because for wireless communication networks, it would be rather unusual for the communication channels to be set up one-directionally. For security and error-correcting reasons, it’s better for the receiver to acknowledge the sender and to explicitly permit an opening channel. So while this exchange is happening, you might as well set it up bidirectionally. ]

HOW DO I STUDY FOR _____________

So I think this might be the question I get asked the MOST often. People are always asking me how do I study for this or that class. So I thought I would just make a master post I could link you all to. :)

All classes

• Watch my video on how to study. This applies to almost everything you have to study.
• See below for additions to doing everything listed in that video.

Math

1. Do problems. Do all the problems. Do them again.
2. Do all the problems in your book.
3. Get another book and repeat step 2
4. Trust me 99.9% of all math classes is pattern recognition. If you can learn how to solve the problem you can ace any set of variables they throw at you.

Physics

1. See math–because physics is JUST applied math. You have to learn how to read the questions and pull out the information you need–the only way to do that is to do dozens of questions!

Micro Bio/ID

1. Flow charts–break things up by group to understand them. You have to group things to remember what’s gram positive or gram negative
2. Don’t blow off the actual micro part of micro. If you understand the virulence factors you’re more likely to understand the sx/tx
3. I had to use a lot of silly sayings to remember all the little pieces of micro. So I would remind myself about the diseases of haemophilus influenzae by saying haEMOPhilus (epiglotitus, meningitis, otitis media, pnuemonia). It was silly but it worked for me.

O Chem

1. Do all the problems. Do them again.
2. Get another book and repeat step 1
3. Flashcard the reactions you don’t understand–put the reactants on one side and the products on the back. Practice these backward and forward.
4. Draw out every step of reactions you don’t understand
5. Circle your electrons or mark whatever it is you lose track of
6. Count–count where everything went at the end to make sure you didn’t screw up.
7. Categorize. Do all members of this group react this way?? It’s easier to learn the rules and the exceptions than force memorize every individual compound’s reaction.

Gen Chem

1. See math
2. Understand real world examples. I related all of the stuff about heat to a cup of coffee. It worked for me
3. Talk through it! I had to read chemistry out loud or try to repeat it out loud in my own words to have any idea what was going on.
4. YouTube videos are absolutely perfect for gen chem!! (There’s even a whole CrashCourse series on Gen Chem that’s appropriate especially for high school level chem).

General Biology–Genetics/Immunology/Cell Biologyetc

1. You really need to watch my video
2. Cross relate–you have to integrate all your biology together to keep all that information in your head.
3. Flashcard only the stuff that can’t be understood. (Like cell markers, etc)
4. Charts! Biology is all about categorization and understanding the similarities between different groups of things. If you can simply remember the characteristics of a group it’s easy to know everything you need to about all the members of that group.

Pathology

1. Pathoma
2. Look at the pictures until you feel sick.
3. Make flashcards of the pictures so you can at least do immediate identification of what you’re looking at even if you don’t know exactly what the pathology is.
4. Integrate! How does the physiology relate to exactly what is going on with the pathology? How does the pathology predict treatment?
5. Learn some latin and greek root words. Even if you have no idea what the word means you might be able to figure it out from there. :) I’ve gotten more than one question right by just figuring out what the word meant.

Pharmacology

1. Understand the mechanism of the drug–it will really predict how it is used or what its toxicities are for
2. Flashcard the bare minimum or anything bizarre you can’t remember any other way.
3. Figure out the similarities in the names. If it sounds the same, it probably belongs in the same class.
4. Don’t learn in isolation. It’s hard to study pharmacology on its own–instead study it integrated with physiology and pathology whenever possible for the best understanding.
5. Study as case studies!! What diuretics would you give to a patient with CHF? With ESLD?

Biochemistry

1. Charts–get poster boards or tape together a ton of sheets of paper and try to write out every pathway you can to see how it all is integrated.
3. Group pathways by the “point”. Are you destroying carbohydrates or building fats? How does this compare to other pathways that do the same thing?
4. Try to rewrite the pathways from memory then see what you missed.

Anatomy

1. Spend a bunch of time with the specimens if you have access to them.
2. DRAW even if you suck at drawing
3. Learn the clinical correlations–why do you care
4. Thing about everything in relationship to one another!
5. Do questions!! Grey’s has a student question book I recommend.

I’ll probably add more to this list as I go and as more of you ask for specific subject advice, but here you go!!

When in doubt, always ask yourself “how would this be asked on a test?”. If you could write a test question about it, you should definitely know it!

And always remember that you should study for understanding and not just for a grade–always be learning and not memorizing. It’s more important you understand the material than you get the A!!

Happy studying!

4

D. R. Brillinger and M. Rosenblatt, Computation and Interpretation of kth-order Spectra

vimeo

Drawing portraits of Mathematicians using space filling curves. (Some are discrete space filling curves)

There are images here:

http://byroland.tumblr.com/post/113873435073

The program to do it was written by be (Will Gallia) and the code is online at:

https://github.com/whg/SpaceFilling

Hey! I love your blog! I love mathematics too and I want to ask which you prefer more: Pure or Applied. And why Also can I ask you for advice and help if I get stuck at some point in life. I really love your blog! STAY HAPPY!

Thank you!

So, I haven’t really done any applied math, at least not formally. In fact, I’ve spent so much time on pure math that applied math (or applied applied math, i.e., chemistry) feels kind of… icky? to me. Everything that I do in math class is precise. Something either is or is not. And then I was taking chemistry, and they were saying things like x tends to be y when z - and the implication was that it mostly did that but they weren’t willing to say it always did because there were exceptions to every rule and it was just.. so frustrating.

But also that textbook sucked. So I assume that feeling will go away once I take more and better science classes. However, for now, I prefer pure.

And yes, you can always ask me for advice! I can’t guarantee the answers will be any good, but I’ll definitely give it a shot.

There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.
—  Nikolai Lobachevsky
The highlight of Futurama math for sure is this thing that’s now known as the ‘Futurama Theorem.’ I’m descending into hyper-nerdspace now. The writer of this episode was Ken Keeler, who I mentioned earlier, who has a PhD in applied math. … And he was writing this episode where the idea was the characters are all going to switch brains, with this brain-switching machine—sort of a standard sci-fi and cartoon idea. … And we came up with this complication: If the machine switches two people’s brains, it cannot switch those same two people’s brains back. … And we were just trying to make the plot more complicated, but we realized that we had accidentally created this math problem. … Ken comes in the next morning with a stack of paper and he said, ‘I’ve got the proof,’ and he had proven that no matter how mixed up people’s brains are, if you bring in two new people who have not had their brains switched, then everybody can always get their original brain back, including those two new people. So I was very excited about this, because you rarely get to see science, let alone math, be the hero of a comedy episode of TV.
Once you go Markov, you never go Bach.

Applied Math, Harvard

"I fucking hate calculus" said Baz

Simon looks up, surprise written all over his face, “Really?”

Baz rolls his eyes, “Yes really, don’t tell me you don’t.”

Simon looks down at the book in front of him, there are eight problems left that he doesn’t even know how to start, “I mean, yeah, I hate it too, but I hate it because I don’t know how to do it. You do.”

Baz shuts his own book, “Just because I understand it doesn’t mean I don’t find it tedious and dry and any other bad adjective you can apply to maths.”

“Then why do you always volunteer to help me with it?” Simon asks.

“Well,” Baz pauses, a very light blush struggling to form across his pale nose, “because I like you.”

Honestly I disagree with the idea that everything is applied math because math was developed to describe physical phenomena, but it isn’t in and of itself physical.

Biology isn’t applied math because biology is another tool to describe physical phenomena, and while it uses math it isn’t math. Saying it’s applied math implies that it is, on some level, just math. And I don’t think that’s true.

Same with chemistry and even some branches of physics.

A TIP FOR ALL YOU AP TEST TAKERS

Okay this probably mostly applies for the math/science AP’s, but let me tell you the #1 thing that I’ve learned while studying for AP Chem.

Haha, yeah you’re funny. BUT NO.

I can’t tell you how many practice questions I got wrong just because I didn’t read one or two words of the question.

My advice to y'all is to underline EVERYTHING IMPORTANT, so that you know exactly what you’re answering.

Good luck to everyone :)

hey i was just wondering what type of engineer you are and what led you to choose it (i want to be a super kickass engineer/writer like you when i grow up)

Okay so what I got my degree in, what I tell people I do, and what I actually do are three different things so bear with me.

I got a bachelors in mechanical engineering with minors in biomedical engineering and applied mathematics. That basically qualifies me to go everything and anything. I chose that whole mess because I knew that I wanted to do something related to the medical field but wasn’t sure exactly what. The mechanical background sets you up to be the jack of all trades, which is a pro because you can do anything but there’s a thousand mechanical engineers and it’s not specialized.

That’s the point of the biomedical minor, to help me specialize and be a better candidate for medical jobs. The applied math minor is because I could.

I tell people that I’m a biomedical engineer because I pretty much am. I work for a medical company doing things on medical devices.

Now technically my job title is process engineer. Day to day I monitor and improve a manufacturing process. Now the stuff that is made goes into medical devices but really I’m responsible for the parts of the whole device. My company is really cool in that I get to work with the teams that actually design the final products.

I got into process engineering because all of my coursework was very theoretical and not very hands on. So one summer I got an internship at this place that makes seals around doors and windows for airplanes and while I did not like the company I loved the work.

Process engineering is very busy and at my company always interesting. We get to work on new products, improvements, design work, getting new equipment, and upgrading the stuff we have.

I hope that kind of explains everything. If not you can always ask more questions!

The only time math applies to life

‘Well, regular math, or applied math, is what I suppose you could call practical math,’ he said. ‘It’s used to solve problems, to provide solutions, whether it’s in the realm of economics, or engineering, or accounting, or what have you. But pure math doesn’t exist to provide immediate, or necessarily obvious, practical applications. It’s purely an expression of form, if you will – the only thing it proves is the almost infinite elasticity of mathematics itself, within the accepted set of assumptions by which we define it, of course.’
—  Hanya Yanagihara, A Little Life

Me: (a scientist) : chemistry is just applied physics and physics is just applied math
Them Art Hoes: well art is just applied paint to a canvas

• witch craft enthusiasts: if you draw some lines and connect them to stones and apply energy, they will do things for you
• me: sounds fake but okay
• microprocessors: if you take a rock and etch some lines into it and apply energy it'll do math for you
• me: FUCK

Ok, so I have no idea why I want this, but do you have any head cannons for Dazai, Chuuya, Atsushi, Aku bae, and Kunikida if they were high school students?

Ah yes my fav High School AU, enjoy! ~Admin Yuyu

• Dazai:
• Life’s a roller coaster and you gotta enjoy it while it lasts right? Wrong. I can imagine Dazai as a really studious person who takes his grades seriously.