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[–] 49 points50 points * (6 children)

It's such a basic tool that it's hard to say it has specific use. You can look up plenty of applications, some straightforward and some very clever. In many cases, they let you change variables so that a problem with n interacting variables becomes n simple problems in a single variable.

For example, if you are physically simulating the behavior of a bridge, modelling it with each beam or cable connected to other elements, you can look at how a small displacement (compressing beams a little and stretching cables) results in forces at each connection.

At first approximation this is linear, but it's a system where every element effects everything it is connect to. If you take the eigendecomposition, the eigenvectors correspond to a displacement to every element so that the resulting local force is proportional to local displacement -- across the whole structure simultaneously. Then these correspond to modes of vibration, and the eigenvalues give you the frequency.

Now to model the vibration of the bridge, instead of one big problem in n interrelated variables, you have n problems, each one behaving like the vibration of a single mass-spring system.

Edit:

Two more I can't help mention:

Look up SVD (or PCA), it's a way of taking data and using eigenanalysis to transform the data so that you need fewer variables to describe it. In a sense lets you find patterns based on the correlations between the data vectors.

Fourier analysis (and also Laplace transform) is really eigenanalysis. In the simplest case, the matrix T(x[1],x[2],...,x[n])=T(x[2],x[3],...,x[n],x[1]), is a rotation (actually also in the physical rotation sense, but here just as rotating the entries). Then T will have eigenvalues of the form r[k] = e2pi*i*k/n ,and the corresponding eigenvectors will be (1,r[k],r[k]2 , r[k]3 , ...) and the change of variables into the eigenbasis is the complex fourier transform. This is why Fourier analysis is useful in problems that are the same when you translate them, either in space or in time.

All three of these very different applications are really part of a specific case, eigenanalysis of normal transforms, or the spectral theorem. This also connects to approximation, things like audio and image and video compression, because that theorem tells you the resulting eigenvectors are orthogonal, meaning useful for least squares approximations.

[–] 28 points29 points  (4 children)

"Fourier analysis ... is really eigenanalysis."

Thank you for this!

[–] 5 points6 points  (3 children)

Agreed. The first thing I did was search for "Fourier" because as a physicist, this is usually the point where you go from knowing eigenvectors and eigenvalues as being just a particular property of matrices to having a real use.

Just in case anyone reading this wants to know its use: Fourier wave decomposition is an easy place to start by recognizing that an arbitrary (periodic) function can be represented by an integral (sum) of sine and cosine waves. Once you get familiar with the idea that a function can just as easily be a basis of a function space as a vector can be for coordinate system, you've got the beginning of an incredibly important tool figured out. Part of quantum mechanics is about finding an eigenstate (the eigenvector) of a Hamiltonian (the matrix) for a particular quantum number (the eignevalue).

[–] 2 points3 points  (1 child)

Interesting... As a physicist, the first place I wanted to go to mention the importance of eigen{vectors, values} was rotation matrices.

Use:
a) In classical mechanics, if you're given a matrix in SO(3), the eigenvectors tell you the vectors which will remain pointing in the same direction under a certain rotation, and the eigenvalues tell you by what factor these vectors are scaled.
b) In quantum mechanics, the eigenvectors of a given matrix are the states that are physically achievable when the given operation is performed as a measurement. [okay, so this isn't necessarily a rotation, but if you're looking at a 2-dimensional complex space the matrices must be contained within SU(2), which doubly-maps to SO(3)]

Sorry, my mind seems to be slowly leaking with every month I've been away from grad school but I think these are still relatively intelligent statements...

[–] 0 points1 point  (0 children)

In response to (a), I talked about Fourier series specifically because in just the mathematical context you already get that vectors are scaled by the eigenvalue. You're completely right that classical mechanics makes use of this often, but making the leap from math vectors to spatial vectors (which are still easily imaginable) isn't a huge leap. Personally I think that the leap to thinking of true abstract vectors spaces is a more important step, so that's why I brought up Fourier series.

As to (b), yes, but I didn't want to just start throwing around the details of QM. I was particularly focusing on the connection to solving for the stationary states of a Hamiltonian. Since the OP asked about just simple eigenvalues and eigenvectors, I didn't want to confuse the issue by throwing around stuff such as the SU(3)×SU(2)×U(1) symmetry group rotations which define the standard model. In this case, I think the function space case is actually easier to understand than the rotation example.

[–] 1 point2 points  (0 children)

I remember staring at the Laplace and Fourier transforms and the the various related series in college. I knew there was something deeper, I just couldn't find it.

[–] 1 point2 points  (0 children)

I have used PCA (principle component analysis) for motion detection, and it is effective.

[–] 95 points96 points  (18 children)

Eigenvalues and Eigenvectors are some of the most profound things I have ever encountered in my life, and as an applied mathematician, I can tell you they are incredibly relevant in engineering applications.

In a sentence: Eigenvectors form the orthonormal basis functions to the solution-space of a problem, given the boundary conditions and the evolution equations.

That means that ANY solution that can ever be generated for those boundary conditions and evolution equations will be some linear combination of eigenvectors, and the exact solution for a set of initial conditions will define the eigenvalues (the prefactors) to the eigenvectors.

For example, in quantum mechanics, the evolution equation is the Schrodinger Equation, and the boundary conditions are given based on the physical problem. The wavefunctions span the solution space, and any possible electron behavior is completely defined by linear combinations of the basis wavefunctions. So given an initial condition, you can evolve the system simply from the eigenvectors. This is incredibly important for using computers to solve the problem, because eigenvectors are orthonormal, which makes many things (such as projections, dot products, etc) much easier.

It takes a while to start thinking of 'functions' and 'vectors' as interchangeable, but that's really the essence of Linear Algebra after the first year.

Edit: There are examples everywhereeeee. Anything that can be fit to a characteristic equation (heat equation, laplace equation, coupled differential equations, blahblablah) will have eigenvectors and eigenvalues, which will span your solution space! THEY'RE SO IMPORTANT

[–] 21 points22 points  (1 child)

Although I share your enthousiasm, all of this is only right for linear evolution equation. It's probably the reason why nonlinear dynamics is so hard. And why the perturbation approach to quantum field theory is about linear free fields.

[–] 7 points8 points  (0 children)

True. However, if you are exploring behavior of a system close to equilibrium state, then you often can linearize the equation and explore the nature of the equilibrium.

[–] 7 points8 points * (0 children)

Another useful application - structures can be reduced to spring-lumped mass systems, with those lumped masses having translation and rotation degrees of freedom, and the springs representing the local stiffness of the structure. The stiffness relationship between the degrees of freedom, coupled with the externally applied constraints on those degrees of freedom and applied forces, form a system of linear equations. This is the basis for structural analysis everywhere from a car to a bridge to the space shuttle. A very simple example is a truss bridge, with the masses lumped at the joints, and the beams replaced with springs.

The eigenvalues of the system represent the natural frequencies of the structure. The eigenvectors represent free vibration modes - the shape the structure takes on as it vibrates (multiplied by an amplitude). Knowing (and avoiding!) the natural frequencies of a structure is a good idea if you don't want to set up resonance that will increase the amplitude of the vibration until the structure fails.

Similarly, eigenvalues can represent the loads which will cause a structure to buckle, and eigenvectors the buckling shape.

[–] 6 points7 points  (0 children)

This is an excellent description. I've never heard them described this way. So many more possibilities I understand now.

[–] 3 points4 points  (0 children)

Ph.D. Econ student here. We certainly use them when solving vector systems of differential equations, only studying discrete time models at the moment. So while I'm no mathematician, I am certainly grateful for having a basic understanding of their use. Yay for applied math!

[–] 6 points7 points  (4 children)

A great explanation. I'd only like to qualify that, in general, the eigenvectors of a system are not necessarily orthogonal.

[–] 2 points3 points  (2 children)

It takes a while to start thinking of 'functions' and 'vectors' as interchangeable, but that's really the essence of Linear Algebra after the first year.

Holy shit. Why did I never make that connection?

[–] 3 points4 points * (0 children)

If you're a beginner, a simple example is f = ekx . This is an eigenfunction of the operator D = d/dx with eigenvalue k. Thus, Df = kf is the corresponding eigenvalue equation.

It's why solving the schrödinger equation is really an eigenvalue problem, that looks like

## H |Ψ> = E |Ψ>

Where |Ψ> is a vector. E is its eigenvalue, and the operator H can take many forms. In every day, back of the envelope quantum mechanics, H is given by

## H = (-h2 /2m) d2 /dx2 + V(x)

Clearly this operator acts on a function Ψ(x). The equation now takes the more familiar form of schödinger's...but now you should never be able to "unsee" it as an eigenvalue problem.

Edit: In QM, the eigenvalues should really be called the Onlyvalues...because upon measurement, you will always measure one of the eigenvalues of the operator H

[–] 1 point2 points  (0 children)

Dual space

[–] 6 points7 points  (0 children)

In a sentence: Eigenvectors form the orthonormal basis functions to the solution-space of a problem, given the boundary conditions and the evolution equations.

Oh, of course! :P

[–] 1 point2 points  (0 children)

Eigenvectors are only required to be orthogonal if the matrix is (conjugate) symmetric IIRC.

[–] 1 point2 points  (0 children)

I think that would be a great answer if the OP had asked why a basis of a vector space is useful, not specifically a basis of eigenvectors.

How is what you said specific to eigenvectors, and not any basis of the solution space?

Though this has been addressed in the other answers. Not so much how eigenvectors are useful, but how eigenvalues are important properties of the system (resonance frequency, etc.).

[–] 1 point2 points  (0 children)

Do you know linear optimization? That is, finding the minimum (or maximum) of a linear multivariate function, subject to linear inequality constraints.

It can be shown that if linear constraints forms a shape with finite area it is a convex polygon, and the maximum (or minimum) lies on a vertex of this polygon.

If one can find all optima lying on a vertex, all other optima can be found by their linear combination.

I see you talk about the solutions being linear combination of eigenvectors.

Can one relate linear programming to eigenanalysis? I can picture a minimum on a vertex being an eigenvector of some equation and the function value there an eigenvalue, but I can't do the math that would show that

http://en.wikipedia.org/wiki/Linear_programming doesn't cite eigenstuff

[–] 0 points1 point  (0 children)

What a great explanation! Thanks!

[–] 20 points21 points  (6 children)

Since you are in comp sci you might enjoy the paper "The \$25,000,000,000 Eigenvector" which explains the use of eigenvectors in the pagerank algorithm

[–] 10 points11 points  (3 children)

Random fact: the name "PageRank" comes from Larry Page, not from "web page".

[–] 2 points3 points  (2 children)

It's a nice name because he has plausible deniability if anyone calls him out for arrogance. I'd like to think that he views it as a combination of both explanations, and at the end of the day, he has completely changed the way we search so he's entitled to some amount of bragging anyway.

[–] 1 point2 points  (1 child)

he has completely changed the way we search

Brought the spotlight onto spectral graph theory as well, I'd say. You could argue that the 1995 Spectra of Graphs was already a portrait of a maturing, slowing down field, but older textbooks for non-theoreticians dwell and zoom in on a number of centrality measures, giving eigenvector centrality a fleeting mention. (I'm an economist looking at social networks -- not as in social media, but like sociologists).

[–] 1 point2 points  (0 children)

Do you have a recommendation for an introductory graph theory book which leans heavily towards spectral theory?

[–] -1 points0 points  (0 children)

[–]Applied Math -2 points-1 points  (0 children)

Satisfied.

[–] 34 points35 points  (1 child)

Speaking from experience, if you are a computer scientist interested in machin learning and image processing, eigenvectors and related ideas of matrix decomposition and projection along basises are the single most import thing you will learn in school. They allow you to find the one dimensional spaces that most or least explain large amounts of data. A few applications:

Google's page rank is (or incorporated at one point) the values of the dominant eigenvector of a matrix of pages (calculated using the power method because the matrix is so big).

The Fiedler vector is the second smallest eigenvector of the graph laplacian and is used in the analysis of graph topology including applications in image processing where one wishes to determine what subset of pixels stands out from an image by cutting a graph composed of pixels connected to other pixels (ie picking out a picture of a cow from a picture of a field).

They, are also great for dimensionality reduction as part of machine learning (ie consider the eigen features to reduce the number of features you have to process) and as a quick way to find subsets of data in any data analysis.

They can also crop up in compression as the first N eigenvectors can allow you to recover a large amount of the input matrix.

Locally linear embedding (look up the paper in science) is a cool, relatively cutting edge method that uses eigenvectors to process data using mostly accessible math and generating very cool figures. Check it out.

[–] 7 points8 points  (5 children)

Gilbert Strang, the author of a definitive text on linear algebra, has a beautiful set of lectures on YouTube (here's the one on eigenvalues and eigenvectors). I would recommend looking at those lectures to get towards the 'why' for linear algebra.

If you understand eigenvalues mathematically, basically you know that an eigenvector is a vector v which can multiply a matrix A such that the result is parallel to v, modulated by a constant c (Av=cv).

However, the depth of eigenvalue theory is in the surprising places it appears - quantum mechanics is full of eigenvalue problems, as well as signal processing and machine learning.

One of the best ways to see a practical use of eigenvalues is in Singular Value Decomposition. This technique has applications ranging from Principal Component Analysis to image compression. Basically, using SVD, you basically assume that in the spectrum matrix, smaller eigenvalues represent smaller pieces of information. Thus, you can eliminate very small eigenvalues in order to compress files on a computer (paper).

[–] 1 point2 points  (1 child)

The lectures are really good because he focuses on the very issue OP asked about - "what does it mean?".

[–] 0 points1 point  (0 children)

yeah, it's just great to see someone who just is teaching the thing he really loves.

[–] 0 points1 point  (1 child)

Just to be certain, you mean Gilbert Strang's "Introduction to Linear Algebra", correct?

[–] 1 point2 points  (0 children)

Yeah - I would say that it's probably the best introductory text on the subject you can find.

[–] 0 points1 point  (0 children)

Saved

[–] 7 points8 points * (0 children)

I think the nicest way to think about evectors and evalues is like this. You have a vector space (lets say R2) and some matrix M acting on it. First imagine M was very simple, say M(x,y) = (2x,3y). Then we can see what M is doing to R2, namely stretching the x-direction by 2 and the y-direction by 3.

Now in general your matrix M might look pretty complicated and while you know how to calculate it for a specific vector, in general its hard to actually see what it does to R2. This is where eigenvectors help. An eigenvector is just a direction in which M looks simple - M just scales the eigenvector. With my simple matrix above, (1,0) was a simple direction and so was (0,2).

Now if you give me a complicated looking matrix M, together with a basis of eigenvalues v, w, I can tell you exactly what M does to R2 - it acts just like my simple matrix from the first paragraph. Namely, if we write x = av + bw where a and b are numbers (these exist because v, w is a basis) then Mx = aev + bfw. Here e and f are the eigenvalues of v and w. Note that this is very like the first simple matrix.

What's cool is that any symmetric matrix has a basis of eigenvectors. This means that essentially every symmetric matrix looks simple (like M from first paragraph) if it is viewed in the right way.

This has loads of applications but I'll just mention one because nobody is probably going to read this far down. Imagine I give you a matrix M and ask you to multiply it by itself 100 times. This sounds pretty horrible, given how bad it is to multiply just two matrices. However if the matrix is simple (again like in the first paragraph) then this is easy - M100 (x,y) = (2100 x, 3100 y). The cool thing to notice is that now if you have a complicated matrix M but you are given eigenvalues with it, your calculation is just as simple -

x = av + bw

M100 x = ae100 v + bf100 w.

You have changed a horrible calculation of matrices into a simple calculation of numbers. This has lots of applications in math and CS.

[–] 4 points5 points  (1 child)

It's not a complete explanation but it adds to other excellent explanations here.

Eigenvectors allow you to isolate parts of your space that don't mix with the rest. If you consider a subspace spanned by some eigenvectors, and let the operation act everything, everything in the subspace will stay in the subspace and nothing from outside will come inside the space. Assuming everything is linear of course, which is sadly rarely the case.

[–][S] 0 points1 point  (0 children)

Yes. I like this answer. It's clear, plain English.

[–] 5 points6 points  (1 child)

By choosing the largest eigenvalues, the corresponding eigenvectors give you the dominant directions. If you can pick a set of eigenvalues that are larger than rest, you are picking a subspace where most of the action occurs. In facial recognition, there is an approach that views an image as a 256 dimensional vector, and there is a 9 dimensional subspace that can discern different faces.

[–] 0 points1 point  (0 children)

The algorithm is called "eigenfaces", for the reference.

[–] 3 points4 points  (2 children)

There are already a bunch of good answers, but let me add in my view of things.

There are two main things you usually do with a matrix, call it A. One is to solve linear systems like Ax=b or least squares/norm systems if A is not square. Here, you model some system as a bunch of linearly interacting components (A describes the interactions), and you are also given an excitation of the system (the b vector), and asked what is the response of the system (the unknown vector x). This comes up everywhere in engineering for the simple reason that most systems are linear, and if they aren't, they are approximately linearizable, and we have extremely efficient methods to solve linear systems of equations.

Now, to get to your question, the other main thing you'd want to do with a matrix is solve the linear system (A-qI)x = 0, where now x and the scalar value q are unknowns. q is usually called the eigenvalue, and x is the eigenvector. I have written it this way instead of the usual way to emphasize this point: for eigenvalue problems, you are looking for a nullspace; that of the modified matrix A-qI. You can write the linear system this way as well: [A -b] * (x; 1) = 0, where the solution is the (hopefully) one vector in the nullspace of the matrix [A -b], which has one extra column than it does rows, and you want to normalize it so that the last entry is 1.

Now, when you view the eigenvalue problem as finding the nullspace of A-qI, you can easily generalize the problem to nonlinear problems where the dependence on the eigenvalue is not linear (for example A-q2 I). Just as in engineering and linear systems, one solves nonlinear eigenvalue problems repeatedly solving linear approximations to the nonlinear problem (there are other ways, but let's go with this for now).

Now, to actually answer your question: why is this at all interesting? The ability to find the "parameterized nullspace" eigenvalue problem is that it models what happens to systems when you leave them alone. The equation Ax = qx says, find me an x so that the system A produces the same x, but possibly scaled. Now obviously, if you don't excite a system, it won't do anything at all. But you might ask, is there some motion of the system that will just keep on going forever even if I never touch it? The answer is usually yes: those motions are described by the eigenvectors, and the "oscillation frequency" of those motions is given by the corresponding eigenvalues. You plug in one of these "eternal motions" as the x vector, apply A to it, and you get out the same shape of motion qx.

Real-life uses:

• In quantum mechanics, everything is actually postulated to be linear. If you want to know how electrons naturally arrange their motions inside transistors, you need to solve an eigenvalue equation (the time-independent Schroedinger equation)
• In mechanical engineering, if you want to know if a bridge is stable, you need to determine its mechanical modes (eigenvectors of motion) to make sure its oscillation frequencies are not too close to a frequency that might exist naturally.
• In graph theory, you can define the Laplacian of a graph, and its eigenvalues/vectors can tell you important things about the graph.

The concept goes beyond just these basic applications. For example, the Fourier transform can be viewed as simply a change of basis in functional space, representing functions as sums of complex exponentials instead of their pointwise-values. The complex exponentials here are eigenfunctions of a broad class of systems called linear time-invariant systems, whose analysis is the basis of much of information/communication theory. For other classes of systems, you would generalize the Fourier transform to change into a basis defined by the eigenfunctions of those respective systems. This is related to the entire field of math of spectral theory.

[–] 2 points3 points  (1 child)

I should also add that your C-code to compute QR iterations is probably really terrible, no offense. The QR iteration if applied as stated in a textbook will take FOREVER to converge in lots of cases. Modern implementations are extremely sophisticated and can essentially guarantee convergence in about 26n3 operations. It is highly unlikely that a home-grown implementation can get anywhere near that. Sadly, the only freely available code to do this is in Lapack. It may even be the only practical one in the world, that's how difficult it is to write it well.

[–] 2 points3 points  (0 children)

Questions like this are so important. Over and over again in my undergrad a topic would be introduced without any motivation and so I'd conceptualize it in a totally useless way until I finally understood why it was useful.

[–] 2 points3 points * (0 children)

Well, any time you want to do math with more than one variable (essentially the entirety of science and engineering), you need to understand linear algebra, which requires you understand eigenanalysis.

But since you're asking for examples in image processing or machine learning, here are a couple off the top of my head:

In statistics, a basic tool for analyzing data is principal component analysis. Suppose you have n data points (each of dimension p) and arrange them in some n by p matrix X, the principal components are the eigenvectors of XT X , (also known as the singular vectors of X). Look it up in wikipedia or a similar reference.

One application of eigenanalysis in computer science is spectral graph theory; you can analyze the structure of a graph by the spectrum of its adjacency matrix. To get even more specific, look at Normalized cuts, introduced in this paper:

http://www.cs.berkeley.edu/~malik/papers/SM-ncut.pdf

They take a problem in image segmentation, formulate it as a discrete optimization problem and then reformulate it so that it's reduced to computing eigenvectors.

Oh, I forgot to mention that "spectrum" is another word for the set of eigenvalues. That's not just a coincidence; the electromagnetic spectrum (the set of eigenvalues for the wave operator) is kind of important.

[–] 2 points3 points  (2 children)

Eigenvectors and Eigenvalues are amazing. Their importance to numerical analysis and control systems are two of my favorite applications. For fun I would recommend reading about eigenvalues and their relation to system dynamics. It's amazing to see how prominent of a role eigenvalues and vectors play in our physical world and how the outcome of almost everything is tied to them.

[–] 0 points1 point  (1 child)

If you could make more specific recommendations for reading material about eigenvalues and their relation to system dynamics, it'd be appreciated. I'm interested.

[–] 1 point2 points  (0 children)

I have not personally read this book, but I have a friend who has and said it was a good read. It's called The Art of Modeling Dynamic Systems its a Dover book so it is very cheap. It's the only thing I can think of that is not a textbook. All of my knowledge comes from textbooks. I looked through the book before and it covers a good amount of material with good depth. Without getting into expensive textbook recommendations that the best that I know of.

[–] 2 points3 points  (0 children)

Quantum mechanics. You need to find the eigenvalues and vectors of a given hamiltonian to find the physical states and energy of a system. They are very useful in nearly every domain of physics : you can diagonalize some matrix in mechanics to find the eigenmodes of a set of linked oscillators for instance. They are very used when you got into hamiltonian mechanics. Also, solid state physics. But I'm sure you can find a use of a good ol' eigenvalue in any physics related area.

[–] 2 points3 points  (0 children)

One interesting application is eigenfaces http://www.scholarpedia.org/article/Eigenfaces Visualizing eigenvectors this way helped me to understand the purpose better.

[–] 2 points3 points  (2 children)

I looked at your library, and I must say the code is very easy on the eyes. Great work on readability!

[–][S] 2 points3 points  (1 child)

Thank you! I'm surprised that anyone bothered to look.

[–] 0 points1 point  (0 children)

I like quick sort!

[–]Representation Theory 2 points3 points * (5 children)

The philosophy behind eigenvalues and eigenvectors is solving fixed point problems. If we have a linear transformation T from Rn to Rn, then the equation T(x) = x solves for the fixed points, which is equivalent to solving the problem (T-I)x = 0.

In linear algebra, it's not so important to fix points specifically; we can ask about fixed lines, planes, etc. Eigenvectors and eigenvalues solve the fixed line problem. That is, what lines are mapped to themselves by T? In that case, it is enough to find all vectors x that are carried to multiples of themselves. So we solve the equation T(x) = cx for some scalar c. From that comes the characteristic equation (T-cI)x = 0.

If we can find an eigenvector, the only linear thing we can do to a fixed line is to rescale it. The eigenvalue tells us how much to rescale by.

Of course, vector spaces turn up all over the place, so eigenvalues and eigenvectors appear with them. My favorite application is Binet's formula for the Fibonacci numbers, which can also be found using generating functions.

[–] 2 points3 points  (4 children)

I took linear algebra a handful of years ago. I have no idea what I just read.

[–]Representation Theory 1 point2 points  (1 child)

TL;DR - Eigenvectors denote lines that get mapped back to themselves under a linear map. Eigenvalues tell how much the lines stretch or squeeze.

[–] 1 point2 points  (0 children)

THAT I can understand. Thank you!

[–] 1 point2 points  (1 child)

Same here. I am going to khan to see if I can make sense of it later.

[–]Representation Theory 0 points1 point  (0 children)

All of that is better summarized with a picture. On the left, draw the x and y axes, and your favorite vector coming from the origin. Label it v.

In the middle, put an arrow with T over it.

On the right, draw the same picture as on the left. On top of v, extend v out a bit and label it cv. So cv is where T sends v, and the factor you stretch by is the eigenvalue c.

[–] 2 points3 points  (0 children)

Let's take a step back and look at the bigger picture.

The issue here is linearization. Think of a smooth function, like sin(x). You can approximate it pretty well in a small interval by a linear function (the tangent). You can say that smooth functions are almost linear, at least to a first approximation.

Similarly, most physical theories are linear to a first approximation, even though there may be more variables and dimensions involved. The multidimensional version of the tangent is a linear map. And linear maps can be represented by matrices, once you choose a basis.

So since most of nature is linear in a first approximation, we get a lot of linear algebra in theories that describe nature. Hence, linear algebra and matrices are everywhere when we try to describe nature.

Then why are eigenvectors and eigenvalues important? For starters, the eigenvalues are so-called invariants, i.e. independent of the basis you choose. This is good, because the basis in a linear space is kind of arbitrary. So you can't say something about the underlying physics from the matrix elements alone, because they change depending on the basis. However, the eigenvalues don't change, so you may have some deeper meaning. You can think of the eigenvalues as characteristic numbers for the theory. They have different meanings in different contexts, such as principal stresses in a material or resonance frequencies in an oscillating system.

Similarly, the eigenvectors are also invariant (taken as geometrical objects) and give additional information about the eigenvalues, such as the direction of the principal stresses or the shape of an oscillation.

Hope this helps somewhat, feel free to ask more!

[–] 5 points6 points  (0 children)

This is the example that cracked it for me: If you have a rotation matrix, the eigenvector is the axis of rotation, and the eigenvalue will be the scaling that anything that happens to lie on the eigenvector experiences when you multiply it by the matrix (1 in the case of a pure rotation matrix where there isn't any other transformation going on). You can replace matrix multiplication with multiplication by the eigenvalue for anything that happens to be on that axis of rotation, and the dot product of anything else against the eigenvector will also be scaled by the eigenvalue after matrix multiplication (this is useful computationally in a lot of cases).

[–][deleted] * (21 children)

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[–] 20 points21 points  (5 children)

Are you trying to encourage or discourage this type of question with your first few sentences?

[–][deleted]  (4 children)

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[–] 37 points38 points  (3 children)

Well, frankly its not without good reason. There are a lot more sources on the internet for low level undergrad and high school mathematics. See: Khan Academy, Paul's online notes and similar sources. And questions like those are a lot more common, if they were allowed this subreddit would be full of "explain derivative formula x" type problems that could be answered with a Google search and 5 minutes.

[–][deleted]  (2 children)

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[–] 3 points4 points  (0 children)

Only conjecture, but I suspect it is actually related to the writing of college students even more than it is to the math.

I think an explanation tailored to the questioners question can be more effective than some of the best online materials. However, it takes effort on the answerers' part, and they they may feel unappreciated if the writing style of the questioner is overly terse, or meandering, or casual, or formal or simply doesn't express exactly the right style of inquisitiveness.

Maybe as people get older they get better at asking questions that imply they value other people's time without going to far and seeming obsequious. It is a tricky balance.

[–] 5 points6 points  (0 children)

A circle would then be mapped to an ellipse. The eigenvectors determine the major & minor axes of the ellipse and the eigenvalues determine how they are naturally scaled.

For more general matrices (non-symmetric, and even rectangular matrices), this describes the singular value decomposition. It's an incredibly useful concept that, unfortunately, is rarely taught in a first course in linear algebra. For symmetric (or normal) matrices, the SVD happens to coincide with the eigenvalue decomposition.

[–][S] 2 points3 points  (0 children)

I love this answer. Yes. Thanks.

[–] 8 points9 points  (4 children)

Minor correction: What you say about eigenvectors is true only for symmetric matrices. If the the matrix is asymmetric, the eigenvectors are not orthogonal and do not form the axes of the principal ellipsoid (those would be the principal components).

[–] 6 points7 points * (2 children)

If the the matrix is asymmetric, the eigenvectors are not orthogonal

Not true. Example (fixed):

0 1

-1 0

The correct version: matrix is normal <=> eigenvectors are orthogonal.

[–] 2 points3 points  (1 child)

Oops. Though that's not an example of a normal matrix. But, you're right, normality AT A = A AT is the correct condition. Thanks.

[–] 1 point2 points  (0 children)

Sorry, I meant to write a different matrix and my mind wandered off.

[–] 4 points5 points * (1 child)

Hijacked your comment so this would get closer to the top. Here's a paper analyzing a zombie apocalypse using eigenvalues and eigenvectors.

http://mjbworks.files.wordpress.com/2011/05/zombie-math.pdf

If you can work your way through this, it might help. They're super useful when you want to but long term population distribution and you have a matrix equation governing population growth.

edit: it also gives the most 'natural' basis to work with in a lot of cases and is used in this ridiculous magical thing called PCA (principle components analysis http://en.wikipedia.org/wiki/Principal_component_analysis).

edit 2: crap wrong link lol. i'm not michael btw sorry for confusion. http://mysite.science.uottawa.ca/rsmith43/Zombies.pdf

[–] 1 point2 points  (0 children)

Anything you can do with eigen* you can do someway without them, right?

[–] 0 points1 point  (0 children)

That's an unfortunate assumption, on behalf of those who make it, that people asking about the MEANING of anything don't know how to use it. Or maybe they understand it, but not enough to explain it to others, and so ask. Perhaps it's not their major, but something they now have to add in order to do research.

Thank you for drawing attention to the 'elitist' attitude that everyone is supposed to be able to learn by the same self-imposed draconian torture methods that are 'tradition'. I've known many people who learn not to ask questions because of the attitude, which is the complete opposite of the optimum learning method. I mean, if ALL people could learn by self-study from books alone, why would we even have schools?

[–] -5 points-4 points  (1 child)

2nd to 4th year under grad? You sure? We did this at school.

[–] 1 point2 points  (0 children)

As another example: eigenvectors and eigenvalues are fundamentally important to quantum mechanics. In quantum mechanics, 'observables' (i.e. things you can measure) are represented by linear transformations, whose eigenvalues are the possible values that can be measured, and the corresponding eigenvectors describe the state of the system.

At least half of a first Quantum Mechanics/Computing course is learning to use eigenvectors and eigenvalues for this reason.

[–] 1 point2 points  (0 children)

Computer vision!

Check this out.

[–] 1 point2 points  (2 children)

So you are implementing parts of LAPACK for fun?

[–][S] 1 point2 points  (1 child)

Yep. I'm implementing LAPACK for shits and giggles.

[–] 0 points1 point  (0 children)

I love this, as long as you don't use it for anything meaningful. I always encourage people to implement toy algorithms/libraries for fun, then never use them for anything.

[–] 1 point2 points  (0 children)

Frequency domain.

[–] 1 point2 points  (0 children)

My understanding is that they simplify calculations: anything you can do with eigen* you can do someway without them (though with a lot more difficulty).

[–] 1 point2 points  (0 children)

Solving linear ordinary differential equations. If you have a system of equations \dot{x} = A*x, then the natural frequencies of the system will be the eigenvalues of A, and the modes of vibration are the corresponding eigenvectors. The eigenvalues themselves are complex numbers. The sign of the real part tells you whether the vibration is stable or unstable (negative is stable). The magnitude of the eigenvalue tells you the natural frequency of the vibration, and the angle tells you how much damping is present.

[–] 1 point2 points * (0 children)

Here's an important application that I use:

Suppose you have a dynamical system dx/dt=Ax, where x is a vector of dimension n, and A is a n*n matrix. Is the fixed point x=0 a stable attractor? Answer: it depends on the eigenvalues of A.

Here's another application: How do you raise a matrix A to a power of n?

Both of these are roughly the same idea. They show what happen if you multiply a vector by a matrix A lots of times. Hence if you really want to understand what a matrix multiplication is doing (i.e. what a linear transformation is doing), then you need to know the eigenvectors and eigenvalues.

[–] 1 point2 points * (0 children)

While certainly not a good "overall" description of eigen-vectors (which I still don't have too good of an understanding of, application-wise, despite having gone through an honors abstract linear algebra course), one sort of neat example of them I've seen is this:

If you have a system where particles move between two locations (a good example is customers moving between two pizzerias), where you represent the state of the system as a column vector (first entry is number of particles in the first location, second entry is number in second), and a transformation that says that some percent of the customers at each pizzeria will decide that they didn't have a good experience, and will switch to the other pizzeria. Ie, some matrix:

[; \begin{pmatrix} p && 1-q \\ 1-p && q \\ \end{pmatrix} ;]

where the first pizzeria will maintain a percent p of its customers (and will lose the rest to the other pizzeria), and visa versa, then the state that the system will tend to is an eigenvector of that transformation (If I'm remembering this right... which I might not be :/).

Ie, if [; A ;] is the transformation, then

[; A^n x ;]

for some initial vector [; x ;], will go to an eigenvector as [; n \rightarrow \infty ;].

I'll try and double-check that/find a source.

edit: I believe this is the page I got that from.

edit2: latex'd cause I'm bored

[–] 1 point2 points  (0 children)

Having done a small amount of quantum mechanics (well, enough to get a degree, which of course is just scraping the surface), your question made me laugh a little. :)

[–] 1 point2 points  (0 children)

In terms of physics, Eigenvalues and Eigenvectors almost never fail to represent something meaningful. The word eigen means "own", as in my own personal value, and it's very much the case in practice.

1) In classical mechanics, a coupled oscillator can be set up as an eigenvalue problem. The eigenvalues are the stable vibrational frequencies, and the eigenvectors represent the stable vibrational modes.

2)In Quantum Mechanics, the eigenvalue of schrödinger's equation is the Energy of the state. The eigenvector is the state itself. (I have a comment on here with some blue equations).

3) In classical electrodynamics, here are so many great examples of eigenvalue problems. They tend to be more abstract however.

The eigenvalues represent resonant frequencies of EM waves in waveguides - these are basically cables that carry signals, like a coaxial or more recently optical fibers.

The permittivity of space in materials can be modeled as a 3X3 matrix whose eigenvectors represent polarization states of propogating waves.

The entire field of electrostatics - i.e., finding out the steady fields of a charge distribution, is one big eigenvalue problem involving the Laplacian. The eigenvalue of interest, however, is zero.

[–] 2 points3 points  (0 children)

Also, I love this subreddit. Within an hour, there are some great explanations which I have learned so much from. Thanks all!

[–] 0 points1 point  (0 children)

[–] 0 points1 point * (0 children)

Well, I think they're damn useful in linear algebra itself (especially when combined with Cayley-Hamilton). An example would be the following:

"Let f be in end(V), dim V = n. If there exists a natural k s.t. fk = 0, show that fn = 0."

Without Eigenvalues et. al., it gets rather complicated. With them (and a few friends), it's so trivial I can do it!

You also need them to get to the Jordan normal form. And this is rather important for systems of differential equations, AFAIK. And those, in turn, are used in every science. Science, on the other hand, get's me supermarkets, and I like the fact that I do not have to hunt or stand on a field all day long, thank you very much.

[–]Math Education 0 points1 point  (0 children)

A practical application of eigenvectors is identifying multicollinearity in multiple regression. If for certain independent variables the eigenvalues are close to zero, this implies a near-singular matrix and multicollinearity requiring an elimination of the offending variable(s) from the model.

[–] 0 points1 point  (0 children)

[–]Numerical Analysis 0 points1 point  (0 children)

Eigenvalues have a wide use of applications, and in particular are a great theoretical tool when doing stability analysis for numerical methods to solve partial differential equations (just went over this in a class the other day). But again, that's just one area.

In Network Analysis, techniques like eigenvector centrality, hub and authority scores, and Google's PageRank algorithm use (dominant) eigenvectors to measure the popularity of web pages and rate citations in a bibliographic network, to name a couple examples.

In addition, eigenvalues themselves pop up in Matrix norms as well, e.g. computing the 2-norm of a square matrix A is a matter of finding the largest eigenvalue (in magnitude) of [; A* A ;], where A* is the adjoint of A.

In it's purest form, an eigenvalue of a linear operator (and thus, a matrix) is a number u such that (A - u * I) x = 0 for a vector x, called the eigenvector of A corresponding to the eigenvalue u (of course, I'm assuming you've made it this far already). In more general spaces, we look at the spectrum of a (bounded) linear operator A, which is the set of complex numbers u such that [; u \cdot I - A ;] is not invertible, and I is the identity operator. In particular, the spectrum contains values which are not necessarily eigenvalues; it just so happens that in finite dimensional vector spaces, the spectrum consists only of eigenvalues. This in itself is a very important theoretical preface to Quantum Mechanics!

So again, it all depends on the angle at which you approach this. What specifically were you looking to have answered?

[–] 0 points1 point  (0 children)

Eigenvalues and eigenvectors are one way to get a matrix to a large power.

What's an interesting application of this? How about [this ant problem I found on reddit](www.jeremyong.com/articles/3) (based on Markov Chains, but it's easy.)

[–] 0 points1 point  (0 children)

A matrix represents a linear function and it's a deep theory of linear functions that any linear function can be thought of as a three step process:

1. Rotate your space (so that the x, y, z, w... dimensions point in new ways)
2. Stretch your space along each x, y, z, w... dimension one by one including "stretching" by compacting it to zero (multiply that element of the vector by 0)

Parts (1) and (3) are like your eigenvectors. Part (2) is your eigenvalues.

The above process is known as spectral decomposition and, as noted, works on any linear transformation. Eigenvalues happen under a slight restriction of that idea where you have to have a square matrix (i.e. a linear relation between two spaces of the same dimension) which gives you a few more features of the decomposition (in particular, there's no guarantee that the number of "eigenvectors" in parts (1) and (3) are the same unless the matrix is square).

How do you use this? It's a complete characterization of every linear function that introduces the transformation bases (your eigenvectors) and weights (your eigenvalues). Many methods work by attacking not the entire matrix but instead these simpler component parts. Eigenvectors represent a rotation which "decouples" the elements of your vector. Eigenvalues can represent the "most important" components of the transformation. They can also be used to translate your linear relation to a more general setting which shares that rotation/stretching nature.

[–] 0 points1 point  (0 children)

I work in at a company that sells risk management software based on multi-factor models. Linear algebra is used extensively. One of our models uses an eigenfactor adjusted covariance matrix to account for sampling bias, we actually advertise this as a feature. Mathematically, eigenfactors are given by the eigenvectors of the factor covariance matrix; financially, they represent uncorrelated portfolios of pure factors.

Enjoy!

[–] 0 points1 point  (0 children)

Just a quick example... in quantum mechanics, the Hamiltonian (total energy of the system) has eigenfunctions of stationary states corresponding to eigenvalues which represent the energy of those states. In my experience, they are used to express systems in terms of many orthogonal pieces. This way, different pieces of the system can be solved for individually based on initial conditions. The orthonormality of the system is what allows this to happen.

[–] 0 points1 point  (0 children)

One of my favorite applications of eigenvalues is what's known as a square matrix's spectral radius. Others have already mentioned what spectrum is, but to clarify it is the set of all eigenvalues. The spectral radius is the absolute supremum of that set.

The idea that you can characteristically describe any square matrix, no matter how large, by a single real number has led to some profound results in numerical analysis.

[–] 0 points1 point  (0 children)

For one example: there is a very easy way to find a formula for the fibonacci numbers using eigenvalues.

[–] 0 points1 point  (0 children)

I just took a quick look at your library and had a look at the matrix code. Just wanted to point out that memory allocation (on the heap) is extremely expensive and not something you'd want to do for every matrix operation if performance is something you care about at all. This makes it pretty much unusable foR game development for example, which is. A shame because having a good, performant, feature rich, foss, c based mathlib would be really nice. :)

[–] 0 points1 point  (0 children)

wow, you sound like a bright kid, but I think your school has failed you utterly if they taught you how to solve eigenvalue problems but not why

[–] 0 points1 point  (0 children)

Eigenvectors of the Hamiltonian are orthogonal in Hilbert space.

[–]Representation Theory 0 points1 point  (0 children)

Here's a monster application of eigenvalues that leads to some deep theory in analysis, number theory, and physics: trace formulas.

If you have an orthogonal projection P from Rn to a subspace of Rn, it satisfies the equation P2 = P, so the eigenvalues are 0 and 1. The trace of P is the dimension of the space we project onto and is also the sum of the eigenvalues.

In general, the trace doesn't have this nice of an interpretation, but we still have that the matrix trace (sum of diagonal entries) equals the spectral trace (sum of eigenvalues with multiplicity).

One can take away the finite dimension condition under the right circumstances to get versions of the trace equalities:

Poisson Summation Formula

Selberg Trace Formula

Gutzwiller Trace Formula in Quantum Chaos

[–]Mathematical Physics 0 points1 point  (0 children)

(In quantum mechanics) Measurements or observables of a particle or group of particles are real eigenvalues of complex functions known as states or wavefunctions, acted upon by hermitian operators. So... I see that as profound.

[–] 0 points1 point  (0 children)

I think the most interesting application of eigenvalues are related to normal mode vibrations. By using eigenvalues we can make sense out of the random looking motion of molecules, mechanical systems, subatomic particles etc. Eigenvalues help us see this random looking complex motion in a simpler way. For example each normal mode has a unique frequency. Also when we look at a system in normal modes rotations and translations of the center of mass of the system disappear. Everything becomes clearer. To really see this in addition to understanding the eigenvalues we would need to understand how symmetry works too. I cannot claim to be fluent in the language of symmetry but I think it is the answer to life the universe and everything (sorry 42 doesn't cut it). This is applicable from molecular biology to quantum mechanics. I find this really fascinating. To see this better I wish I had a better understanding of higher math than just engineering level math.

[–] -1 points0 points  (0 children)

I wrote a blog post a long time ago about the Eigen-cat that may be useful.

[–] -1 points0 points  (0 children)

would love to see something simple like "very useful for facial recognition"


text:
## Department of Electrical and Computer Engineering

### ECE 559BH: SPRING 2006 Topics in Communications: Distributed Network Algorithms

Description: The course will juxtapose several topics loosely related to communication in large networks such as the Internet, and distributed coordination. The behavior of distributed network protocols, such as those for peer-to-peer networking, will be studied through the use of stochastic analysis methods. Forward error correction, such as digital fountain codes, and, more generally, network coding, are based on stochastic methods. One general application the course will focus on is simply the communication or storage of files. Another application is the problem of distributed scheduling or reservation of resources. The distributed nature of the applications, with tens, thousands, or even millions of interacting agents, suggests that ideas of game theory and mechanism design are relevant. Each student will be expected to give two 40-minute presentations of material during the semester. Grades will be based on homework (10%), classroom presentations by students (30%), a midterm and final exam (30%), and a project (30%).

Prerequisites: ( ECE 534: Random processes or Math 466: Probability II ) and CS 473: Algorithms, or consent of instructor.

Assigned Reading: Selected journal articles and portions of texts. There will be no required text, although the book R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press, 1995, is recommended.

Meeting times: 1:00-2:30 p.m. TuTh in 336 Mechanical Engineering Building

Instructor: Professor Bruce Hajek

Office hours: Wednesday and Friday, 3-4 p.m. in Room 105 CSL

Lecture Schedule
 Date Speaker(s) Topic Reference 1/17 BH Balls in bins: birthday surprise, maximum load, Poisson heuristic and associated bound Mitzenmacher and Upfal, Chapter 5 (Motwani and Raghavan, Chapter 3.6, covers much of same) 1/19 BH Balls in bins: coupon collector's problem, Poisson heuristic and associated bound See reading for 1/17 1/24 BH Chord lookup system I. Stoica, R. Morris, D. Karger, M. F. Kaashoek, and H. Balakrishnan, "Chord: a scalable peer-to-peer lookup service for Internet applications" 1/26 Srinivas Shakkottai BitTorrent file distribution system Bram Cohen, "Incentives build robustness in BitTorrent." (5 pages) Mimeo, May 22,2003, and BitTorrent protocol specification (3 pages) on BitTorrent.com D. Qiu and R. Srikant "Modeling and Performance Analysis of BitTorrent-Like Peer-to-Peer Networks, Proc. ACM SIGCOMM, Portland, OR, Sept. 2004. 1/31 BH Martingales -- conditional expectation, sigma algebras, general definition of a martingale, examples References for martingale material: basic martingale theory: G.R. Grimmett and D.R. Stirzaker's treatment of martingales (Sections 7.7-7.10 & Chapter 12). (also, James R. Norris has posted some notes from his advanced probability course.) Azuma-Hoeffding and the method of bounded differences: Mitzenmacher and Upfal, Chapter 12 and Section 4.4 of Motwani and Raghavan. 2/2 Lei Ying slides Distributed function computation R.G. Gallager, "Finding parity in a simple broadcast network," IEEE Trans. Info. Theory , vol. 34, March 88, pp. 176-180. E. Kushilevitz and Y. Mansour, "Computation in noisy radio networks," Proc. Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) , 1998, pp. 236-243. Related: N. Goyal, G. Kindler, and M. Saks, "Lower-bounds for the noisy broadcast model, and for the generalized noisy-decision trees," BH Martingales -- Azuma-Hoeffding concentration inequalities See reading for 1/31 2/7 BH Martingales -- method of bounded differences and examples See reading for 1/31 2/9 BH Martingales -- stopping times See reading for 1/31 2/14 Xiaolan Zhang notes & Ramakrishna Gummadi Random online algorithms R. Motwani and P. Raghavani, Portions of Chapter 3, and Sections 13.1 - 13.3. 2/16 Ramakrishna Gummadi (continuation) the Marker algorithm for paging See 2/14 BH Martingales -- uniform integrability and optional sampling theorems See reading for 1/31 2/21 Serdar Yuksel notes Martingale methods for stability of Markov processes Intro Markovian Processes: James Norris (sections 1.5 and 1.7) link Comparison theorem: S.P. Meyn and R.L. Tweedie, Markov Chains and Stochastic Stability, (Chapters 11-14) link 2/23 Serdar Yuksel (continued) Two applications of comparison theorem: Foster's criteria, bounding moments See 2/21 for reading BH Martinagales -- applications to algorithms See reading for 1/31 2/28 BH Martinagales -- applications to algorithms See reading for 1/31 Sichao Yang Randomized algorithms for Byzantine agreement For deterministic algorithm: M. Pease, R. Shotak, L. Lamport, Reaching Agreement in the Presence of Faults," JACM 27, 2 (April 1980) link For random algorithm: R. Motwani and P. Raghavani, Randomized Algorithms , Section 12.6 3/2 Sichao Yang Byzantine agreement (continued) See 2/28 for reading 3/7 BH LT codes - introduction, derivation of soliton and robust soliton distribution M. Luby, "LT Codes," Proc. IEEE Stoch. 2002 (On IEEE Xplore) Chapter 50 of D. J. C. MacKay, Information Theory, Inference, and Learning Algorithms, Cambridge University Press, Cambridge, 2005. link 3/9 BH LT codes - exact computatation for Poisson distribution E. N. Maneva and A. Shokrollahi, "New model for rigorous analysis of LT-codes," ArXiv, December 8, 2005. link 3/14 BH LT codes - proof of performance for finite graphs Luby's LT codes paper, see 3/7 3/16 BH LT codes -asymptotic performance Maneva and Shokrollahi (see 3/9) also discuss an asymptotic analysis based on R. W. R. Darling, J. R. Norris, "Structure of large random hypergraphs," Annals of Applied Probability 2005, Vol. 15, No. 1A, 125-152. link 3/28 BH Raptor codes Amin Sholrollahi, "Raptor Codes," To appear in IEEE Transactions on Information Theory, 2006. . link to earlier version link to December 2005 slides 3/30 midterm exam 4/4 Srinivas Shakkottai slides network coding approach to gossip S. Deb and M. M\'edard, Algebraic gossip: a network coding approach to multiple rumor mongering," Allerton Conference , 2004. Christos Gkantsidis, Pablo Rodriguez, "Network coding for large scale content distribution", IEEE/INFOCOM 2005 , Miami. March 2005. link 4/6 BH Averaging in a network with given topology by gossip S. Boyd, A. Ghosh, B. Prabhakar, D. Shah, "Gossip, analysis and applications," IEEE INFOCOM 2005 and journal version link 4/11 Xiaolan Zhang notes VCG allocation mechanism for auction of goods D. Lehmann, L.L. O’Callaghan, and Y. Shoham, “Truth revelation in approximately efficient combinatorial auctions,” J. of the Association for Computing Machinery, vol. 49, 5, September 2002, 577-602. link 4/13 Xiaolan Zhang Truthful mechanisms with approximate optimization See reading for 4/11 BH Truthful randomized auction S. Dobzinski, N. Nisan, and M. Schapira, "Truthful randomized mechanisms for combinatorial auctions" To appear in STOC 2006. link 4/18 Ramakrishna Gummadi slides The probabilistic method Mitzenmacher and Upfal, Chapter 6 4/20 Lei Ying slides The classical gossip process Section 5 of A. M. Frieze and G. R. Grimmett, "The shortest-path problem for graphs with random arc-lengths, Discrete Applied Mathematics , Vol 10, January 1985, Pages 57-77. link 4/25 Serdar Yuksel Belief propagation, with emphasis on convergence F.R. Kschischang, B. J. Frey, H.-A. Loeliger, "Factor Graphs and the Sum-Product Algorithm," IEEE Trans. Info. Theory, Feb 2001. S. C. Tatikonda, "Convergence of the sum-product algorithm," IEEE ITW April 2003. J. S Yedidia, W. T. Freeman and Y. Weiss, "Constructing Free Energy Approximations and Generalized Belief Propagation ALgorithms," IEEE Trans. Info. Theory, July 2005. 4/27 Sichao Yang Scheduling with deterministic constraints, Cruz's SCED algorithm, pricing aspects Pages 73-80 of notes for ECE 567 link 5/2 BH Coding and/or queuing in packet networks D.S. Lun, M. Medard, R. Koetter, and M. Effros, "On coding for reliable communication over packet networks" (on Arxiv.org, October 2005) link S. Bhadra and Sanjay Shakkottai, "Looking at large networks: coding vs. queueing," IEEE INFOCOM 2006, April 2006 link

Tentative summary of Topics:

• I. Distributed algorithms for information dissemination and storage (16 hours)
• Gossip algorithms (algorithms for computation and information exchange in an arbitrarily connected network, including both store-and forward and network coding approaches
• Peer-to-peer systems: principal algorithms, distributed file systems, and distributed backup
• Implementation tricks: hash functions and Bloom filters
• Digital fountain codes
• Randomized online algorithms
• Randomized algorithms for the agreement problem
• II. Distributed mechanism design, including how to make greed work in networks (11 hours)
• Mechanisms for distributed scheduling and coordination
• A few concepts from game theory and auctions
• III. Stochastic techniques (for analysis of interacting random systems, such as random algorithms or algorithms in a random graph). (16 hours)
• Martingale convergence and concentration methods
• Branching processes
• Poisson clumping method
• Moment method
• Mixing and conductance of Markov chains
• Analysis by scaling and ordinary differential equations (fluid analysis)

Selected References:

• Bloom filters and hashing functions
• Andrei Broder and Michael Mitzenmacher, Network applications of Bloom filters: A Survey", Allerton Conference, 2002. link
• J. Byers, J. Considine, M. Mitzenmacher, and S. Rost Informed Content Delivery Across Adaptive Overlay," SIGCOMM02, August, 2002. link
• M. Mitzenmacher and U. Upfal, Probability and Computing: Randomized Algorithms and Probabilistic Analysis , Cambridge University Press, Cambridge, 2005. (On reserve in Grainger Enginering Library)
• Digital fountain codes
• D. J. C. MacKay, Information Theory, Inference, and Learning Algorithms, Cambridge University Press, Cambridge, 2005. link
• Gossip algorithms
• A.M. Frieze and G.R. Grimmett, The shortest path problem for graphs with random arc-lengths," Disc. Appl. Math , 10, (1985) 57-77.
• S. Deb and M. M\'edard, Algebraic gossip: a network coding approach to multiple rumor mongering," Allerton Conference , 2004.
• Christos Gkantsidis, Pablo Rodriguez, "Network coding for large scale content distribution", IEEE/INFOCOM 2005 , Miami. March 2005. link
• lA. Dimakis, A.D. Sarwate, and M.J. Wainwright, Geographic gossip: efficient aggregation for sensor networks,'' link
• S. Boyd, A. Ghosh, B. Prabhakar, D. Shah, "Gossip, analysis and applications," IEEE INFOCOM 2005 and journal version link
• Peer to peer systems
• I. Stoica, R. Morris, D. Karger, M. F. Kaashoek, and H. Balakrishnan, Chord: A Scalable Peer-to-peer Lookup Service for Internet Applications." Proceedings of the ACM SIGCOMM 2001 Conference , August, 2001. San Diego, CA.
• Yatin Chawathe, Sylvia Ratnasamy, Lee Breslau, and Scott Shenker, Making Gnutella-like P2P Systems Scalable," SIGCOMM 2003 . August 2003. Karlsruhe, Germany.
• Alberto Blanc, Yi-Kai Liu, Amin Vahdat Designing Incentives for Peer-to-Peer Routing," Workshop on Economics of Peer-to-Peer Systems June 2004
• Stefan Saroiu, P. Krishna Gummadi, and Steven D. Gribble. Exploring the Design Space of Distributed and P2P Systems," Proceedings of the First International Workshop on Peer-to-Peer Systems (IPTPS '02). March 2002. Cambridge, MA.
• Probabilistic tools
• R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge Univ Press, 1995. (On reserve in Grainger Enginering Library)
• G.R. Grimmett and D.R. Stirzaker, Probability and Random Processes (3rd ed.) , Oxford Univ Press, 2001.
• D. Williams, Probability with Martingales, Cambridge University Priess, 1991.
• S.N. Ethier and T.G. Kurtz, Markov Processes: Characterization and Convergence , Wiley, 1986.
• D. Aldous, Probability Approximations via the Poisson Clumping Heuristic , Springer, New York, 1989.
• H. Cohn, A martingale approach to supercritical branching processes," Ann. Probab . 13, 1179 -- 1191, 1985.
• M. Jerrum and Alistar Sinclair, The Markov chain Monte Carlo method: an approach to approximate counting and integration," Chapter 12
• N. Alon and J. Spencer, The Probabilistic Method (2nd ed.), Wiley, 2000.
• Game theory and mechanism design
• Drew Fudenberg and Jean Tirole. Game Theory , MIT Press, 1991
• Noam Nisan and Amir Ronen. Algorithmic Mechanism Design," Games and Economic Behavior, 35:166-196, 2001.
• J. Shneidman, D. Parkes, and L. Massoulie, Faithfulness in internet algorithms," Proc. SIGCOMM Workshop on Practice and Theory of Incentives and Game Theory in Networked Systems (PINS'04) , Portland, OR, USA, Sept. 2004. link
• D. Lehmann, L.L. O’Callaghan, and Y. Shoham, “Truth revelation in approximately efficient combi- natorial auctions,” {\em J. of the Association for Computing Machinery,} vol. 49, 5, September 2002, 577-602.

format: plain


text:
HAL : in2p3-00023319, version 1
 arXiv : hep-ex/0410078
 Physical Review Letters 94 (2005) 161801-1-161801-7
 Measurement of the ratio of inclusive cross sections $\sigma (p\overline{p} \rightarrow Z + b jet)/ \sigma(p\overline{p}\rightarrow Z + jet)$ at $\sqrt{s}$ = 1.96 TeV
 D0 Collaboration(s)
 (2005)
 Using the data collected with the DZero detector at sqrt(s)=1.96-TeV, for integrated luminosities of about 180 pb^{-1}, we have measured the ratio of inclusive cross sections for p+pbar->Z+bjet to p+pbar->Z+jet production. The inclusive Z+b-jet reaction is an important background to searches for the Higgs boson in associated ZH production at the Fermilab Tevatron collider. Our measurement is the first of its kind, and relies on the Z->e^+e^- and Z->mu^+mu^- modes. The combined measurement of the ratio yields 0.023 +/- 0.005 for hadronic jets with transverse momenta p_T>20 GeV/c and pseudorapidities |eta|<2.5, consistent with next-to-leading order predictions of the standard model.
 Thème(s) : Physique/Physique des Hautes Energies - Expérience
 Mot(s)-clé(s) : proton-proton inclusive interactions – Z bosons – Higgs bosons – intermediate boson decay – standard model
 Lien vers le texte intégral : http://fr.arXiv.org/abs/hep-ex/0410078
 in2p3-00023319, version 1 http://hal.in2p3.fr/in2p3-00023319 oai:hal.in2p3.fr:in2p3-00023319 Contributeur : Yvette Heyd <> Soumis le : Jeudi 18 Novembre 2004, 10:49:27 Dernière modification le : Mardi 27 Juin 2006, 11:18:08


text:
International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 366-367   | 1 | 2 |

## Section 2.5.6.1. The object and its projection

B. K. Vainshteinc and P. A. Penczekg

#### 2.5.6.1. The object and its projection

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In electron microscopy (EM) and single-particle reconstruction, three-dimensional (3D) reconstruction methods are used for studying biological structures; that is, symmetric or asymmetric associations of biomacromolecules (muscles, spherical and rod-like viruses, bacteriophages, individual proteins and ribosomes) (Frank, 2006). The electron microscope is used to obtain parallel-beam two-dimensional (2D) projections (ray transform) of frozen hydrated 3D macromolecules suspended in random orientations (Fig. 2.5.6.1). The function is the 2D projection of the 3D molecular electron distribution . One can also consider one-dimensional (1D) projections of multidimensional distributions; the set of these projections is called a Radon transform. For 2D distributions, ray and Radon transforms differ only in the notation. For (d > 2)-dimensional distributions the two are different: in a ray transform the integrals are calculated over straight lines and yield (d − 1)-dimensional projections, while in Radon transforms the integrals are calculated over (d − 1)-dimensional hyperplanes and yield 1D projections. In electron microscopy, Radon transforms are not directly measurable, but can be formed computationally and used in intermediate steps of the 3D reconstruction or in alignment procedures (Radermacher, 1994).

 Figure 2.5.6.1 | top | pdf |A three-dimensional object and its two-dimensional projection . The electron beam penetrates the specimen in the direction of the z axis.

Within the linear weak-phase-object approximation of the image-formation process in the microscope [see equation (2.5.2.42) in Section 2.5.2 of this chapter], 2D projections represent line integrals of the potential of the particle under examination convoluted with the point-spread function of the microscope, s, so, using (2.5.2.43),Sincewe haveIf we omit constant terms, we obtain

In this section, we will assume that all images were collected using the same defocus setting, so the point-spread function s is constant and does not depend on the projection direction . Thus, we will concern ourselves with the inversion of the projection problemThe projection direction is defined by a unit vector and it is formed on the plane x perpendicular to . The set of various projections may be assigned by a discrete or continuous set of points on a unit sphere (Fig. 2.5.6.2).

 Figure 2.5.6.2 | top | pdf |The projection sphere and projection of along τ onto the plane . The case represents orthoaxial projection. Points indicate an arbitrary distribution of projection directions τ.

In Fourier space, the relation between an object and its projection is referred to as the central section theorem: the Fourier transformation of projection of a 3D object is the central (i.e., passing though the origin of reciprocal space) 2D plane cross section of a 3D transform perpendicular to the projection vector (Bracewell, 1956; DeRosier & Klug, 1968; Crowther, Amos et al., 1970). In Cartesian coordinates, a 3D Fourier transform isThe transform of projection along z isIn the general case of projecting along the vector , the central section theorem isFrom this theorem it follows that the inversion of the 3D ray transform is possible if there is a continuous set of projections corresponding to the motion of the vector over any continuous line connecting the opposite points on the unit sphere (Fig. 2.5.6.2) (Orlov's condition: Orlov, 1976). This result is evidenced by the fact that in this case the cross sections that are perpendicular to in Fourier space continuously cover the whole Fourier space, i.e., they yield and thereby determine .

In single-particle reconstruction, imaged objects are randomly and nonuniformly oriented on the substrate at different angles (Frank, 2006) and the distribution of their orientations is beyond our control; therefore, the practical impact of Orlov's condition is limited. In fact, it is more useful to determine a posteriori, i.e., after the 3D reconstruction of the macromolecule is computed, how well the Fourier space was covered. This can be done by calculating the distribution of the 3D spectral signal-to-noise ratio (Penczek, 2002).

### References

Bracewell, R. N. (1956). Strip integration in radio astronomy. Austr. J. Phys. 9, 198–217.
Crowther, R. A., Amos, L. A., Finch, J. T., DeRosier, D. J. & Klug, A. (1970). Three dimensional reconstruction of spherical viruses by Fourier synthesis from electron micrographs. Nature (London), 226, 421–425.
DeRosier, D. J. & Klug, A. (1968). Reconstruction of three dimensional structures from electron micrographs. Nature (London), 217, 130–134.
Frank, J. (2006). Three-Dimensional Electron Microscopy of Macromolecular Assemblies. New York: Oxford University Press.
Orlov, S. S. (1976). Theory of three-dimensional reconstruction 1. Conditions for a complete set of projections. Sov. Phys. Crystallogr. 20, 312–314.
Penczek, P. A. (2002). Three-dimensional spectral signal-to-noise ratio for a class of reconstruction algorithms. J. Struct. Biol. 138, 34–46.
Radermacher, M. (1994). Three-dimensional reconstruction from random projections: orientational alignment via Radon transforms. Ultramicroscopy, 53, 121–136.


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Let $f(x)$ be a third-degree polynomial with real coefficients satisfying $$|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.$$ Find $|f(0)|$.

Let $A = \{1, 2, 3, 4, 5, 6, 7\}$, and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Find the remainder when $N$ is divided by $1000$.

Let $f_1(x) = \frac23 - \frac3{3x+1}$, and for $n \ge 2$, define $f_n(x) = f_1(f_{n-1}(x))$. The value of $x$ that satisfies $f_{1001}(x) = x-3$ can be expressed in the form $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Given the function $f(x) = 2x^2 - 3x + 7$ with domain {$-2, -1, 3, 4$}, what is the largest integer in the range of $f$?

For every positive integer $n$, let $\text{mod}_5 (n)$ be the remainder obtained when $n$ is divided by 5. Define a function $f: \{0,1,2,3,\dots\} \times \{0,1,2,3,4\} \to \{0,1,2,3,4\}$ recursively as follows: $f(i,j) = \begin{cases}\text{mod}_5 (j+1) & \text{ if } i = 0 \text{ and } 0 \le j \le 4 \text{,}\\ f(i-1,1) & \text{ if } i \ge 1 \text{ and } j = 0 \text{, and} \\ f(i-1, f(i,j-1)) & \text{ if } i \ge 1 \text{ and } 1 \le j \le 4. \end{cases}$ What is $f(2015,2)$?

The domain of the function $f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))$ is an interval of length $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

Let $P$ be a cubic polynomial with $P(0) = k$, $P(1) = 2k$, and $P(-1) = 3k$. What is $P(2) + P(-2)$ ?

Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the prime factorization of $n>1$, then $f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\cdots (p_k+1)^{e_k-1}.$ For every $m\ge 2$, let $f_m(n)=f_1(f_{m-1}(n))$. For how many $N$ in the range $1\le N\le 400$ is the sequence $(f_1(N),f_2(N),f_3(N),\dots )$ unbounded? Note: A sequence of positive numbers is unbounded if for every integer $B$, there is a member of the sequence greater than $B$.

Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50 < f(7) < 60$, $70 < f(8) < 80$, $5000k < f(100) < 5000(k+1)$ for some integer $k$. What is $k$?

Let $f_{1}(x)=\sqrt{1-x}$, and for integers $n \geq 2$, let $f_{n}(x)=f_{n-1}(\sqrt{n^2 - x})$. If $N$ is the largest value of $n$ for which the domain of $f_{n}$ is nonempty, the domain of $f_{N}$ is $[c]$. What is $N+c$?

Let $f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))$, and $h_n(x) = h_1(h_{n-1}(x))$ for integers $n \geq 2$. What is the sum of the digits of $h_{2011}(1)$?

Let $f(x) = \log_{10} \left(\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x)\right)$. The intersection of the domain of $f(x)$ with the interval $[0,1]$ is a union of $n$ disjoint open intervals. What is $n$?

Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23, -21, -17,$ and $-15$, and $Q(P(x))$ has zeros at $x=-59,-57,-51$ and $-49$. What is the sum of the minimum values of $P(x)$ and $Q(x)$?

Suppose that $f(x+3)=3x^2 + 7x + 4$ and $f(x)=ax^2 + bx + c$. What is $a+b+c$?

Functions $f$ and $g$ are quadratic, $g(x) = - f(100 - x)$, and the graph of $g$ contains the vertex of the graph of $f$. The four $x$-intercepts on the two graphs have $x$-coordinates $x_1$, $x_2$, $x_3$, and $x_4$, in increasing order, and $x_3 - x_2 = 150$. The value of $x_4 - x_1$ is $m + n\sqrt p$, where $m$, $n$, and $p$ are positive integers, and $p$ is not divisible by the square of any prime. What is $m + n + p$?

A function $f$ has domain $[0,2]$ and range $[0,1]$. (The notation $[a,b]$ denotes $\{x:a \le x \le b \}$.) What are the domain and range, respectively, of the function $g$ defined by $g(x)=1-f(x+1)$?

The function $\displaystyle f$ has the property that for each real number $\displaystyle x$ in its domain, $\displaystyle 1\/x$ is also in its domain and $f(x)+f\left(\frac{1}{x}\right)=x$ What is the largest set of real numbers that can be in the domain of $f$?

For each $x$ in $[0,1]$, define $\begin{array}{clr} f(x) & = 2x, & \text { if } 0 \leq x \leq \frac {1}{2}; \\ f(x) & = 2 - 2x, & \text { if } \frac {1}{2} < x \leq 1. \end{array}$ Let $f^{[2]}(x) = f(f(x))$, and $f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $n \geq 2$. For how many values of $x$ in $[0,1]$ is $f^{[2005]}(x) = \frac {1}{2}$?

If $f$ is a function such that $f(f(x)) = x^2 - 1$, what is $f(f(f(f(3))))$?

Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that $P(1) = P(3) = P(5) = P(7) = a$, and $P(2) = P(4) = P(6) = P(8) = -a$. What is the smallest possible value of $a$?

Let $f(x) = x^2 + 5$, and $g(x) = 2(f(x))$. What is the greatest possible value of $f(x + 1)$ when $g(x)$ = 108?

Let $f(x) = \sqrt{2^2-x^2}$. Find the value of $f(f(f(f(f(-1)))))$.

Find all polynomials $f(x)$ such that $f(x^2) = f(x)f(x+1)$.

Let $x, y \in [-\frac{\pi}{4}, \frac{\pi}{4}], a \in \mathbb{Z}^+$, and $$\left\{ \begin{array}{rl} x^3 + \sin x - 2a &= 0 \\ 4y^3 +\frac{1}{2}\sin 2y +a &=0 \end{array} \right.$$ Compute the value of $\cos(x+2y)$

Let $f$ be a real-valued function such that $f(x) + 2f(\frac{2002}{x}) = 3x$ for all $x > 0$. Find $f(2)$.


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# Historically, how did people define multiplication for negative numbers?

Which were the first mathematical developments to state that the product of two negative numbers is a positive number, and what was their justification for this choice? I am not interested in a modern explanation for why this choice is sensible, but want to know how mathematicians from the past arrived at this convention.

Brahmagupta (598-670) was one of the first mathematicians to introduce negative numbers. He called them debts. The rule was:

The product or quotient of two debts is one fortune.

This was logical, because you have a debt of 3 by 3 people, then you have a debt by 9 people. In modern language, $3 \times -3 = -9$, however Brahmagupta wouldn't write that. So fortune times debt is debt. Fortune times fortune is a fortune. So changing fortune to debt once als changes the outcome. Therefore changing it again would result to a fortune.

• I doubt the explanation. Brahmagupta only talks of negatives as "debts", so "-3 people" does not work, and this rule appears in a long list of rules which includes $0/0=0$. Islamic mathematicians who also interpreted negatives as "losses" and accepted Brahmagupta’s rule never applied them to people or gave such a reasoning. They did motivate addition and subtraction for losses, but not multiplication. – Conifold Aug 10 '15 at 21:36
• I agree with @Conifold. The last sentence of your answer does not make any sense to me. Or am I missing something? – fdb Aug 11 '15 at 10:20
• @Conifold Indeed, I had rushed that explanation a bit too much. – wythagoras Aug 12 '15 at 19:22

Alas, we will never know. The earliest mention of negative numbers occurs in Liu Hui's c. 260 AD commentary to Chinese classic Nine Chapters on the Mathematical Art, a compendium of prescriptions for solving practical mathematical problems. The context is solution of linear systems by what we call Gaussian elimination, and the numbers are represented by red and black counting rods (contrary to the modern accounting convention red is positive):"Rods of the same name multiplied by each other make positive. Rods of different names multiplied by each other make negative". In the style of the book there is no explanation or justification for the rule. If anything, the context suggests that it was adopted because the algebra worked, not for any "intuitive" reason:"Interchanging the red and black rods in any column is immaterial. So one can make the first entries of opposite sign".

Brahmagupta's Correctly Established Doctrine of Brahma (c. 630 AD) gives the rule in the same prescription style, along with many others:"The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero. A positive divided by a positive and negative divided by a negative is positive; a zero divided by a zero is zero...". Literally, he says "fortunes" and "debts" rather than positive and negative, but while such merchant interpretation provides intuition for addition and subtraction of debts their product makes little sense, as does $0/0=0$.

Mumford comments that even the merchant interpretation is pure speculation:"It would be wonderful to know what considerations led Indian mathematicians in the late centuries BCE or the early centuries CE to these conclusions – especially for the multiplication of negative numbers. The predominately oral transmission of knowledge in the Vedic tradition – and perhaps the difficulty of preserving perishable writing materials through yearly monsoons – has not left us with any record of these discoveries. They just appear full blown in Brahmagupta’s summary. R.Mattessich has developed at length the idea that it was the highly developed tradition of accounting which led to the full understanding of negative numbers, but unfortunately no evidence for this plausible conjecture exists".

The father of algebra, Al-Khwarizmi (c. 790-840 AD), explicitly cites Brahmagupta on negatives (and zero) in the famous Al-jabr w’al Muqabala, but he has little use for them in the geometric interpretations he favors. In the whole book he mentions them only once, when explaining the identity (in modern notation) $(a-b)(c-d)=ac-ad-bc+bd$. This might be a clue as to why he accepts the Brahmagupta’s rule, it has to hold for the last term to come up with the right sign.

Mumford's essay gives a good survey of historical struggles with the nature of negatives (which continued into the middle of 19th century!) with references and attempted reconstructions of their meaning.

• "(a−b)(c−d)=ab−ad−bc+bd". I think you mean ac.... Anyway, you might want to mention that al-Kh. does not express this in algebraic notation, but (as one says) "rhetorically". – fdb Aug 11 '15 at 10:14


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# Multiple Attempts Not allowed. This test can only be taken once. Force This test can be...

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4. Walking across campus one day, you find a vial containing a powdery substance. You take it back to the lab to perform elemental analysis and you determine that it is a pure compound with the chemical formula CsoHieN a. (2.5) Calculate-SHOW YOUR WORK-the hydrogen deficiency index represented by th...
4 answers
##### The following results come from a dataset that contains information on a random sample of 534 U.S. workers surveyed in 1985. The dataset contains information about the hourly wage (in U.S_ dollars) and the sex of each of the workers surveyed, as well as information about each worker s age (categorized into 4 quartiles), union membership. job type (collapsed into 8 different categories); marital status (married O not); and occupational sector categories). You are interested in assessing the relat
The following results come from a dataset that contains information on a random sample of 534 U.S. workers surveyed in 1985. The dataset contains information about the hourly wage (in U.S_ dollars) and the sex of each of the workers surveyed, as well as information about each worker s age (categoriz...
5 answers
##### According to The Sciences magazine, the world population of 2.5 billion people in 1950 had grown to 5.5 billion by the 1990 census_ Using this data, the exponential function modeling the world population is given by where is years since 1950 Give your answers to 3 decimal places _
According to The Sciences magazine, the world population of 2.5 billion people in 1950 had grown to 5.5 billion by the 1990 census_ Using this data, the exponential function modeling the world population is given by where is years since 1950 Give your answers to 3 decimal places _...
4 answers
##### Rocall that get 3-dimcusional [OFIS 7 8 bv gling "p the opposing faces of solid cube â‚¬ Iuagine that Youl staudling It order (0 the top of T huena . which direction shoukl you look? Explain brielly. V) Wc ualerstod tic-tac-toe O" 7' Vy drawing  eended diogtam lor T2 o tl [Malle'. Draw lew props in (lillerent pvts of â‚¬ (for iustance (lower; Iucsutt , ad star) atc then Fhi extended diagram for T
Rocall that get 3-dimcusional [OFIS 7 8 bv gling "p the opposing faces of solid cube â‚¬ Iuagine that Youl staudling It order (0 the top of T huena . which direction shoukl you look? Explain brielly. V) Wc ualerstod tic-tac-toe O" 7' Vy drawing  eended diogtam lor T2 o tl [Malle&...
1 answer
##### Problem 6. (30 points). The diameters of each solid segment are dĄB= 0.75 in., dbc= 1.5...
Problem 6. (30 points). The diameters of each solid segment are dĄB= 0.75 in., dbc= 1.5 in., and dcD=0.5 in. Draw all appropriate free body diagrams and determine the values average normal stresses at points E and F. Indicate whether Tension or Compression. 8 kip 5 kip 2 kip 6 kip A D 5 kip B C...
1 answer
##### (4) What type of Measurement System did you encounter in the Experiment Vibration Table and in the Experiment Cantilever Beam. What are the important parameters of this type of measurement system...
(4) What type of Measurement System did you encounter in the Experiment Vibration Table and in the Experiment Cantilever Beam. What are the important parameters of this type of measurement system? (10 pets) (4) What type of Measurement System did you encounter in the Experiment Vibration Table and ...
1 answer
##### Analyze statement for fallacies
If you start out with a bottle of beer a day and then go on to a glass or two of wine on the weekends, you’re well on your way to becoming a hopeless drunk....
1 answer
##### Medical Abbreviations and Conversions Write the meaning of the following abbreviations, or give the appropriate abbreviation:...
Medical Abbreviations and Conversions Write the meaning of the following abbreviations, or give the appropriate abbreviation: 24. Every day 25. two times daily 26. drop 27. Nothing by mouth 28. Normal Saline 29. Intramuscular 30. as needed 31. ac 32. qid 33. cap Conversions: 9. 88 kg lb 34. 1mgmcg m...

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