Eigenvectors and Eigenvalues.

- Transform the unit Hue disc.
- Pick the same colors. It’s eigenvectors!
- The lengths along the eigenvectors are eigenvalues.

Eigenvectors and Eigenvalues.

- Transform the unit Hue disc.
- Pick the same colors. It’s eigenvectors!
- The lengths along the eigenvectors are eigenvalues.

How I Attempted To Mathematically Model the Spread of Memes on the Internet, pt 1.

This is my first ever /actual/ Tumblr post, so first of all, let’s hope I’m doing this right.

I’ll make my introduction fairly brief as to get to the mathematics ASAP. This was part of my senior project for my mathematics B.S. (which I am very close to completing!). I presented parts of this research at the Michigan MAA Section Meeting in April of 2017. And I am *really* excited to share what I found in a more casual setting (rather than in a formal paper for my class or a conference presentation).

So let’s get to it, eh?

For the first installment of this series, I want to touch on two main things:

- Why Did I Decide To Do This?
- Things You Need To Know Before We Can Get To The Cool Stuff (I Know, I Know, But You Need To Know These Things)

My senior mathematics capstone course was on the topic of Markov chains. The main purpose of this class was for each student to complete an individual research project. I struggled for quite some time to determine what I wanted to do my project on. I eventually settled on disease modeling, but found myself disinterested with the material. I credit one visit to my advisor’s office for what sparked in me the most academic drive and passion for a project I’ve ever had.

I walked in, planning on just talking about my ideas for my disease modeling project. My advisor suggested that I could use this sort of model to work with other things, and me, being the Internet-addicted smart ass that I am, asked, “Like memes?”

So here we are.

First, I’m assuming that you are familiar with the following concepts from linear algebra:

- Matrix algebra
- Eigenvalues and eigenvectors
- Diagonalizing matrices

For the sake of time and length, I leave reviewing those as an exercise to the reader. Ha, I’ve always wanted to say that.

So I’ll begin with Markov chains.

A **Markov**** chain** is a stochastic process where the probability of being in the current state is dependent *solely* on the previous state you were in.

*What’s an example of this?*

Suppose you are playing a game on a rectangular board of 7 squares. Your piece starts on the center square. For each move, you have a probability of 1/3 of moving to the left (which I’ll denote L) and a probability of 2/3 of moving to the right (which I’ll denote R). If you get all the way to the right, you win! However, if you get all the way to the left, you lose (*sadface*). Once you reach either the leftmost square or the rightmost square, you can’t move out of them. We’ll talk about this in more detail later, but these are what we call **absorbing states**.

See below for the starting position.

Say your first four moves are R, L, R, R.

Your probability of moving to the right is still 2/3 and your probability of moving to the left is still 1/3. Much like the honey badger, the probabilities don’t care that you moved R, L, R, R already; they are still the same as before. Examining this as a Markov chain is nice because it allows us to answer two important and interesting questions:

- What is the probability of winning after a certain number of moves?
- What is the average number of moves it takes until I win?

I’ll attempt to answer question 1, and I’ll leave question 2 as an exercise.

In order to find the probability, we must first create a transition matrix. The** transition matrix**,** **$T$, is a matrix where $T_{i,j}$ is the probability of getting *to* the $i^{th}$ state *from* the $j^{th}$ state. In this example, each state is a square on our board. Our transition matrix looks something like this:

Let’s discuss!

We note that every column sums to 1. This makes sense! We *have* to move every turn, so the sum of the different probabilities of leaving each state should equate to exactly 1. We also note that $T_{1,1}$ and $T_{7,7}$ are both equal to 1. These are our **absorbing states**! Absorbing states are those states where the probability of returning to itself is exactly 1. So, you’re essentially trapped (mwa-ha-ha) in these absorbing states.

**How does this help us find the probability of winning?**

Excellent question! If we want to find the probability of being in the $i^{th}$ state after starting in the $j^{th}$ state after *n* moves, we simply find $T^n_{i,j}$. This can be done using your favorite computing software (I personally opt for SageMath), or if you’re feeling *particularly* ambitious, you can do it by hand (but I don’t recommend this, for the sake of your health).

So if I want to find the probability of winning the game in at most, say, 10 moves, I would look at $T^{10}_{7,4}$, as this entry shows the probability of ending in the 7th state (win!) from the 4th state. Additionally, I could also find the probability of losing after at most 10 moves by looking at $T^{10}_{1,4}$.

Here’s $T^{10}$ as outputted by SageMath:

Our $T_{7,4}$ entry is $\approx$ 0.7133, giving us a nice 71.33% chance of winning the game in at most 10 moves!

We can continue to raise $T$ to higher and higher powers and see if we can find the values that the probabilities approach, but there’s an easier way! We can find the **steady state distribution** of our transition matrix.

The steady state distribution is the vector that when we multiply $T$* *by this vector, we just obtain the same vector. In other words, it’s a vector $v$ of probabilities such that $vT = v$. Also, this vector $v$ is actually the eigenvector associated with the eigenvalue of 1!

For the sake of time, and since I’m assuming you’re comfortable with matrix algebra, I’ll leave the solution to you.

After finding the steady state distribution, we can determine that the overall probability of winning is 8/9 and the overall probability of losing is 1/9. Good odds!

I’ll talk more in the next post about how I viewed the spread of memes as a Markov chain, but in general, I worked *sort of backwards* from how I solved this example. In my research, I don’t know my probabilities, and my goal is to try to find them! So I use these ideas to create some ~pretty dope~ equations involving these unknown probabilities and try to match these equations to data sets I obtain from Google Trends. Of course, this will make more sense with my future posts!

**Hope you enjoyed!** Any questions/comments/concerns, feel free to send me an ask.

Best wishes and stay positive!

(I promise that wasn’t a math pun)

(Okay, maybe it was)

Eigenvectors

A linear transformation T that stays in the same vector space could have some points that stay relatively fixed. A fixed vector v would have T(*v*)=*v*. Many transformations only have the 0 vector as their only fixed point.

But, what happens in a lot is that there are certain vectors that get scaled by the transformation. That is, there exists a scalar λ such that T(*v*)= λ*v*. Since all scalars are unaffected by T, this same λ works for all scalings of *v*. If *v* has this property, it is an eigenvector for T, and λ is an eigenvalue.

A transformation can have multiple independent eigenvectors.

Another cool property is that if λ is less than 1, then T repels vectors from *v*, and if λ is greater than 1, then it attracts T.

ordinalitis replied to your post:Wait are there not like two weeks left? Do you use axler (/axler-esque approach)?

I’m not using Axler but I am somewhat influenced by him.

But generally, determinants are kind of dumb. They’re an important computational trick but that’s really all they are. The only reason to talk about them is to talk about eigenvalues, so you don’t need to define them until you do eigenvectors and eigenvalues.

And, like, I wish I’d had eight classes to spend on eigenvectors instead of five, but that’s not how things worked out this term. I had to do change of basis first, and before that I had to do linear transformations, and before that I had to do bases…and that’s basically the whole course.

anonymous asked:

8:15pm and why the hell linear algebra doesn't go after eigenvalues and eigenvectors as clearly as ordinary differential equations does (I'm so confused by math)

Sorry I got an A in calc I but after that ¯\_(ツ)_/¯ u on ur own pal

Zero is never invited to the eigenvector party.

—
Linear algebra lecturer

lesbianmooncolony asked:

assuming a starship never has to travel through an atmosphere, and never has to worry about collisions with eg. asteroids or attackers' weapons (maybe some sort of forcefield, or a perfect interceptor weapon system), are there any compelling reasons for starships to be any particular shape? (let's also assume the starship can travel at relativistic speeds)

Aha! Excellent question! I love talking about spaceships.

So most of what I know comes off *Atomic Rockets*, which has far too much love of old whitedude science fiction novels, but also a lot of useful information, and more recently, lots of conversations with arborine, who has thought/knows a great deal about space habitats and is generally wonderful.

There are a few design considerations that constrain what shape your spaceship should have…

- it needs to bear the stresses of acceleration without breaking
- it needs to dissipate the heat generated by the crew, onboard machinery and direct sunlight into the highly insulating vacuum of space
- it needs to be able to manouvre correctly and, for example, not start spinning uncontrollably when the engine does a burn
- its rotation needs to be long-term stable in flight
- it needs to be able to withstand impacts with interplanetary/interstellar debris, which although consisting only of rarified gas and small particles, may (depending on the mission) may be approaching at extremely high, even relativistic speeds
- it needs to protect equipment and crew from space radiation (from stars or cosmic rays)
- it needs to protect the crew from any radiation produced by the ship’s engines, or its power source (a big issue if you use something scary like nuclear power or antimatter annihilation!)
- if humans are on board it needs to have pressurised compartments for them to live in, and enough space that they will be comfortable for the duration of the voyage
- it needs to minimise the extent of the damage if something goes wrong, e.g. part of the ship is depressurised
- it needs to survive for the entire length of the voyage, which probably means it needs to be self-sustaining
- it needs to carry enough propellant reaction mass to give it the \(\Delta v\) for whatever it’s out there to do
- it needs to be as light as possible, because every unit of mass of spaceship also adds a lot of units of reaction mass, depending on the mission \(\Delta v\)
- if you want the crew to experience artificial ‘gravity’, it either needs to be capable of accelerating near \(g\) for nearly the entire mission, or have sections of the spaceship capable of rotation to create a centrifugal force near \(g\)

I’m sure there are a lot more of them.

So while there’s no reason for your ship to be *streamlined*, there are a bunch of constraints!

The first one depends a lot on where you position your rockets. Most designs place the propulsion at the back of the ship, *pushing* the ship forwards.

If you put the propulsion system at the back of the ship, you need your ship to be sturdy against compressive strength, which adds to the mass of the mission - compressive strength usually requires massive beams and so on, and it’s hard to get around that. When you’re accelerating, it will feel as if gravity is pushing down on everything on board towards the back of the ship, and like a building, your ship has to be strong enough to stand up. Otherwise, your engine will punch straight through your ship, or crush it, or tear off, or something similarly bad.

A tiny number of designs, like the *Valkyrie*, and in fact some of the earliest rockets ever built, have the propulsion at the front of the ship, towing the rest of the ship behind it.

If you do that, your ship needs tensile strength to not snap against the propulsion. You generally don’t need as much mass to have tensile strength as compressive strength - but putting the engine at the front does mean your crew are directly in the path of the engine exhaust! (The radiation problem is a general one for nuclear and antimatter engines, but it’s particularly severe when you’re right in the engine beam…)

The *Valkyrie* gets around this by having a shadow shield between the engine and the crew compartment, which absorbs radiation from the engine and casts a shadow over the crew compartment.

Another option is to have your engines not point directly backwards, but have two or more engines angled slightly out to the sides in opposite directions - this is apparently what the ship in the film *Avatar* does (ship’s presumably a still from the film, annotations from *Atomic Rockets*).

Putting the engines at the front on a flexible tether does make it much harder to change direction, though.

Although space is sometimes described as ‘cold’, especially in movies, that’s kind of misleading. Certainly, if not in direct sunlight, something in space without a heat source will *eventually* cool down until it’s incredibly cold… but for the same reason that we use vacuum flasks to keep tea or whatever warm, it takes a very long time.

If you’ve got something like a spaceship, which is generating a lot of heat, you need to do something to dissipate that heat. The only mechanism that heat can be lost in space, with no atmosphere to carry it away, is electromagnetic radiation. To lose heat through radiation, you need a large surface area (i.e. big flat surfaces called ‘radiators’), and to carry the heat from the rest of your ship to the radiators. You heat up the radiators and they radiate away your heat into the vacuum.

The Space Shuttle would apparently open its cargo bay doors in space to help it radiate away heat (labels once again from Atomic Rockets).

Your radiators should not have their surfaces facing each other, or else they will heat each other up, and won’t radiate heat as effectively.

Another trick to cool your ship down is to use a ‘Liquid Droplet Radiator’ (picture source - apparently the ‘ICAN-II’ design from Penn State University, no idea what that stands for). This involves using your waste heat to heat up a spray of hot liquid droplets, which travel through space, steadily cooling down. As your ship is accelerating, the droplets ‘fall’ towards the back/front of the ship (depending which way you go), and can be collected, heated up and sprayed back in the opposite direction, maintaining continuous circulation.

*Atomic Rockets* has some really weird-looking designs, including discs, triangles, and even one that looks like a spiral (using magnetic fields).

Have a look at *Atomic Rockets* - there are lots of radiator designs I haven’t covered, involving bubbles, nanotube filaments, all sorts of stuff.

If it doesn’t - if your engine is off-centre - when you do an engine burn, it will also have a nonzero moment (torque). This means your ship will start to rotate around its centre of mass when you do an engine burn. *Sometimes*, this is what you want, but usually only by a small amount! And typically a large engine burn would set you spinning wildly. So you will probably have separate small thrusters for turning.

(This gets even more complicated if your ship has flexible elements!)

Every rigid body has a thing called its moment of inertia tensor that can be calculated from its mass distribution. Its eigenvalues are called the principal moments of inertia, and its eigenvectors are called the principal axes of the body. When the engine is not burning, the ship is undergoing free precession.

I remember a series of arguments about free precession in our classical mechanics lectures that say, since a not-quite-rigid body slowly loses energy to internal stresses but the angular momentum is conserved, it precesses (the angular velocity vector moves with respect to the body coordinates, if that means anything to you) so as to end up spinning around the principal axis with the largest moment of inertia.

If your plan is that your spaceship should spin around a *different* axis, you should be worried, because it will slowly precess to spin around the major axis.

At the time the USA launched their first ever satelite, *Explorer 1*, in 1958, this was not known. The pen-shaped (shh) satellite was set spinning around its minor axis, the long axis through the middle of the spaceship. To the scientists’ surprise, the axis of rotation changed in flight, until the satellite was flipping end over end. This led to the first development of Euler rotation (the kind of thing we’re talking about) in over 200 years. But we’ve done that now, so, make sure your spaceship is spinning around its major axis if it spins!

A tiny grain of dust, when raised to relativistic speeds, can have the kinetic energy of a bullet or bomb. And, if your ship is travelling through space at relativistic speeds, every grain of dust in space will be approaching your ship at those speeds! This is bad.

One solution is to put a shield at the front of your spaceship, which will be punched up by relativistic dust, but will disperse most of its energy before it smashes anything important.

I’m going to talk about the *Valkyrie* again. I am maybe a bit too into this design, but their solution is really cool.

The* Valkyrie* sprays fluid droplets ahead of the ship. Any incoming particles can detonate harmlessly among the droplets, without touching the engine. Because the ship is constantly accelerating, the fluid droplets will ‘fall’ back down and land relatively slowly on the front of the ship, where they can be recycled. This is also used for cooling - another ‘liquid drop radiator’.

During the deceleration phase of the mission, this doesn’t work - the droplets fly off ahead of the ship instead of being recaptured. The *Valkyrie’s* solution is to grind its spent fuel tanks into fine dust and release them ahead of the ship as it decelerates. Travelling at a constant relativistic speed, the ground up fuel tanks will punch their way through any interstellar dust particles ahead of the ship, clearing a path.

Some dust will still enter the path ahead of the ship, so in addition to that trick, the *Valkyrie* extends many layers of ‘blankets’ of extremely thin material (’similar to Mylar’) ahead of the ship. Because the ship is accelerating in the opposite direction, the blankets are stretched out in front of the ship.

Obviously, nothing remotely close to this has ever been tried! So who knows if it would actually work. I haven’t seen discussion of relativistic dust in other designs on *Atomic Rockets*, though.

Even if you’re not travelling at relativistic speeds, you will (without the protection of an atmosphere) constantly be irradiated by cosmic rays and other ionising radiation from any nearby star. This is a problem even for current astronauts, who apparently hide behind water tanks during periods particularly intense radiation. And it’s often discussed as a major danger of a voyage to Mars.

To an extent, you are protected from this by the hull of your spaceship, which absorbs most kinds of radiation even if it’s quite thin. You apparently need two layers to deal with different kinds of radiation, since charged particles entering the e.g. lead you would use for gamma ray shields can create deadly X-rays. Everywhere where people will be living, long-term, needs to be protected this way.

During periods of major radiation, such as solar flares, this may not be enough. You need a particularly well-shielded part of your spaceship to hide in at these times. One method is to use the water (that you’re carrying anyway, for the crew to drink) to protect the crew at these times. You need a way for the crew to access this area, and take anything particularly vulnerable to radiation - such as food - in there with them.

Apparently people are also considering other methods, involving things like powerful electric or magnetic fields, or bubbles of plasma. I don’t know a lot about this.

To travel long distances in space, chemical rockets tend not really to be enough - you need something with a very high \(\Delta v\). Many such propulsion systems have been designed, but they tend to have the side effect of producing lots of ionising radiation.

This isn’t necessarily a huge problem. You don’t necessarily need to shield your entire reactor (which would be very heavy), but can instead use a shadow shield, like we discussed on the *Valkyrie*, to keep just your ship safe from the radiation, and let the rest of the radiation radiate away into space. But it does imply that the rest of the ship, including your thermal radiators, needs to keep itself inside the cone of safety created your shadow shield. And your shadow shield is usually still quite massive, because the main way we attenuate radiation is to make a lump of very dense materials.

Here’s an example of a NASA design which uses a shadow shield, and has to keep its radiatiors inside the shadow:

The pressure outside the spaceship is nothing. The pressure inside the spaceship is probably about 1 atmosphere. Your ship needs to be strong enough to contain that.

You definitely don’t want to have too many parts of your ship that would be particularly vulnerable to rupturing, so the walls of your pressurised areas are probably going to be nice and curved with minimal sharp corners.

One way to build a pressure vessel, the way we currently do, is to use lots of strong metal. But that adds a lot to your payload mass. So another possibility is to have an inflatable spaceship.

NASA has one such design called the TransHab. Which I assume means it can’t be boarded by cis people. It looks like this (source)…

(all of those weird CGI people are trans. no arguing)

arborine has generally talked about much larger, long-term designs: a big sphere or torus, containing a much more comfortable landscape built for many people (I will have to ask her to help me recreate the details though!). Spheres and toruses are good for inflatable shapes, because the tension is nice and even across the surface (well, this is complicated by rotation, but basically).

The need to have a pressurised hull and radiation shielding does introduce some limitations. Windows in a pressurised compartment generally need to be quite small and rounded, like aeroplane cabin windows, but when you’re far from a planet, there’s not really anything to see anyway. (It seems that, with sufficient engineering, you can make bigger windows, such as in the ISS’s observation bubble thingy. Even then, the bubble’s pretty small, and there’s plenty of structural stuff keeping that together.)

Despite all your best efforts, it seems inevitable that a pressurised compartment travelling through the vacuum of space might have something go terribly wrong. For example, something punches a hole in the hull. You need to have that not end the mission, and cause as little harm to the crew as possible.

This probably means, internally, that your ship needs to be divided into separate compartments, so that if one gets depressurised, the rest of the ship is still safe. If the walls of your ship can somehow be self-sealing, even better.

If you’re travelling on a very long voyage, carrying supplies for the entire journey just takes too much mass. You need to carry with you a functional, lasting ecosystem of plants etc. that won’t die off during the voyage.

Making a closed ecosystem is *very hard* - as far as I know, we haven’t succeeded yet. I will defer to arborine
on the details of one, as I can’t remember what she said or what was in
her extremely practical space habitats book off the top of my head.

In addition, parts of your ship will slowly fail and go wrong. Atomic Rockets claims the best NASA probes can last 40 years in space, and that may well not be enough for your purposes (even taking as much advantage as possible of relativistic time dilation). For a very long voyage like a generation ship or something, everything can probably be assumed to go wrong your spaceship to an extent needs to have the capabilities on board to build itself, and apply its repairs outside the safety of crew compartments.

Needing to recreate them en route might constrain the materials you can use, and hence
the shapes you can make.

( breathofzephyr
made an interesting point that the total computing power, data storage
etc. available to the crew of a generation ship will, unless they can
build more computers, go *down* over the course of the mission. which could have interesting social effects…)

A consequence of the rocket equation is that, for every gram (or whatever mass unit) of payload - structural elements, people, food, computers, robots, whatever - you have, you will need a certain number of grams of reaction mass to throw out the back of your rocket.

(The exception is if you are using a propulsion system such as a solar sail or a laser sail where you are propelled by collisions with material that you’re not carrying with you. Or if you’ve figured out a way to get the Bussard Ramjet, or its variants, to be practical!)

The shape of your propellant tanks depends a great deal on what kind of engine you’d use, but basically, you need a lot of pressure vessels somewhere on your ship, likely held separate from the crew modules.

The rocket equation is a very demanding thing, and unless you somehow have vast amounts of leftover \(\Delta v\), you want to take every possible opportunity to save on mass. (’loads of leftover \(\Delta v\) is like, you’re using Project Orion to travel through the solar system or something.)

This is why lots of spaceship designs are modules held together by a lightweight framework - you want to minimise heavy structural elements as much as possible.

We’ve talked quite a bit about this already, though!

For a long voyage, your crew’s bones and muscle will atrophy unless they’re given an environment with a source of ‘gravity’ similar to Earth’s.

There are two ways you can do this, which amount to accelerating some or all of your spaceship. One way is to have your engine burning so as to provide \(1 g\) acceleration throughout your voyage. By contrast, current space missions usually involve very short rocket burns, with the spaceship flying freely for most of its mission. To burn throughout the mission, your ship will need *vastly* more \(\Delta v\), and hence vastly more propellant mass per unit payload mass.

Since that’s pretty demanding, the other option is to have part of your spaceship spinning. This could be for example a toroidal ring, or a cylinder, or a sphere, or a couple of rooms on the end of long arms.

Depending on how large your rotating section is, it will have to spin more or less fast in order to maintain the amount of gravity you need. The centrifugal acceleration in a rotating reference frame is given by \(r\omega^2\), where \(r\) is the distance from the centre, and \(\omega\) is the angular speed.

Note this means the artificial ‘gravity’ in a rotating section with multiple floors, or in a spherical habitat, will vary as you get closer or further away from the centre! The further out you are, the stronger gravity will be.

A rotating habitat section has some implications about the design of the inside. ‘Level’ surfaces will be curved instead of flat, and the Coriolis effect will apply to deflect trajectories from a straight line - particularly strongly in a small, fast-spinning section.

There are lots of difficulties to overcome if only part of your spaceship is spinning. A major one is the problem of the bearing between the rotating section and the rest of the ship - it needs to be near frictionless, or your spin section will slowly spin down (and your ship will slowly spin up.

Another problem comes from spinning up your spin section without setting the rest of your spaceship rotating in the opposite direction (with possible consequences of precession, and the like). One solution is to have a flywheel inside your ship, but this implies a large mass that’s generally useless. Another is to have multiple counter-rotating spin sections, such that the net angular momentum change can be 0 without spinning the entire ship.

Or, you can just let the entire spaceship spin at once, though this makes manouevring the ship somewhat complicated because of the way gyroscopes act (it becomes forced precession, and you need to push at \(90^\circ\) from the way you actually want to turn…)

A *further* problem comes from the fact that, when your spaceship *is* accelerating, the direction of ‘down’ will change. This can be accomodated by some very complicated-looking designs involving arms that swing in and out (source), or just ignored because most of the time your ship is not accelerating.

So, yes, there are *a lot* of physical demands on the shape of a spaceship, even one that never visits an atmosphere!

Building a spaceship *is* a lot different from building on Earth, and of course nobody actually *knows* how to build a spaceship for an interstellar voyage. But this is what we think, based on the physical constraints we’ve imagined so far…

eigenvectors, eigenvalues, eigenspaces, eigenbases… eigenstuff.

Does everyone understand this? If not you are walking through the forest with a blindfold on. You are going to run into a tree, on the exam.

—
Linear analysis professor when reviewing eigenvectors