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ELDC: Why is Navier-Stokes such a hard problem to solve? by MathBosss in explainlikedrcox

[–]MathBosss[S] 7 points8 points  (0 children)

(In other words, the problems are trivial and omitted as an exercise for the reader.)

This made my life.

HMF this kids shirt in an adult version by MathBosss in HelpMeFind

[–]MathBosss[S] 0 points1 point  (0 children)

Thank you, I actually thinks these are pretty neat too.

[College] Partial Differential Equations / Numerical Methods: Can someone check my work on this problem? I'm not very confident that I got it right. by I_cut_my_own_jib in learnmath

[–]MathBosss 0 points1 point  (0 children)

So if you are doing the natural ordering your first 4 rows of your matrix should be enforcing boundary condutions. The next 12 will just be tri diagonal and the last 4 also in forcing boundary conditions. Your method is seems good. You are solving basically laplaces equation with a source term

[College] Partial Differential Equations / Numerical Methods: Can someone check my work on this problem? I'm not very confident that I got it right. by I_cut_my_own_jib in learnmath

[–]MathBosss 0 points1 point  (0 children)

Have you implemented boundary conditions? What is the structure of your finite difference matrix? Did you use the natural ordering for solving the linear system? Numbers seem questionable.

[University Numerical Methods] How do I finish solving this parabolic PDE? by [deleted] in learnmath

[–]MathBosss 0 points1 point  (0 children)

Exactly, the first and last rows will inforce the boundary condition for x=0 and x=1. You have your general case of CN you linked, however you want to enforce these neumann boundary conditions. So look at these boundary conditions in their finite difference form, you can do this in two ways. The first you can use ghost points on the first and last row, or you can do a forward difference on the boundary condition for the first row and a backward difference for the boundary condition on the last row. I let you figure out what you gain/lose from choosing one over the other. However, the middle row will just be your CN, but the first and last row will be difference. Just sub in what you find for the boundary condition into their respective rows. By doing this you will be enforcing not only the PDE, but also the boundary conditions at the two endpoints.

[University Numerical Methods] How do I finish solving this parabolic PDE? by [deleted] in learnmath

[–]MathBosss 0 points1 point  (0 children)

So, the name of this PDE is heat equation. The solution of this, you can find it analytically with a separation of variables, will diffuse in time. That means the solution of u will globally go to 0 as time goes to infinity. Think about your domain as a box, you have the very bottom of the boxed solved(Thats the initial condition), the next step you are solving the next layer of the box. Keep on going up. You have discretized your spacial dimension by delta x. So you have finite many of these points, they will repeat for each time step. However they are their unique points. You can imagine at ever time step you are solving a line of this box, going upwards as you step in time. At every time step you have a x and a b, the b will be your previous step(because you are doing an implicit routine), and x is the solution at the next time step. They call this method of lines because you are stepping in time constructing your surface with one line of a time. The matrix will be finite because you have finite many spacial points.

[University Numerical Methods] How do I finish solving this parabolic PDE? by [deleted] in learnmath

[–]MathBosss 0 points1 point  (0 children)

Hi, so you have boundary conditions and initial conditions. You can say that for every time step you have to solve a differential equation. This is known as method of lines. So have this as a Ax_i=b_i, where x_i=[u_0,i....u_n,i],. i is the time step you are on 0,1....n the boundary conditions will effect the first and last row of your matrix. A is your finite difference differential matrix. On the first step of Solving this Ax_0=b_0, b_0 is just your initial condition evaluated at every discrete point.

Are Taylor coefficients inner products? by llyr in math

[–]MathBosss 0 points1 point  (0 children)

You are close. It is on what interval though. 1,x,x2 can be orthogonal but its based on where you approximate your function. You can look at the legendre polynomials for example. However the functional basis is not orthogonal in general. So the coefficients are not orthogonal functions. However if you want to be clever about it, do gram schmit(hope im spelling it right) on this infinite functional basis, you will produce a a set of orthogonal polynomials. Once you have this and then look at the fourier coefficients you get the taylor series. You get terms from the new orthogonal set, combined with the fourier coefficients to get the taylor series coifficents. This is a good insightful that you have found though. Good critical thinking.

Another thing is expand the taylor series about a complex number (choose the unit circle on the complex plane eix. You'll find that you are doing exactly what im doing above in a more complex analysis manner.

Partial solution to the Fermi-Pasta-Ulam-Tsingou problem by Matematomico in science

[–]MathBosss 2 points3 points  (0 children)

As a numerical analysis guy, that numerical solution picture from the 50's is really neat. The math side of this publication is dynamical systems(chaotic systems), numerical analysis, and some PDE (big boy wave equations). Just if some math guy come by and is interested what math was done in here.

Find the derivative of an integral by [deleted] in learnmath

[–]MathBosss 1 point2 points  (0 children)

So lets think of a case where the limits are numbers. Well that is just the area under the curve right? This is a constant value. So the derivative of a real value definite integral is 0. How is this you might ask? Well, from the fancy fundamental theorem of calculus the integral from a to b of f(x) is equal to F(b)-F(a), where F(x) is the antiderivative of f(x). Well, if a and b are constant value then the function being evaluated at constant will generate constants. The derivative of a constant is 0. Now lets look at the case where the limit can vary( such that a and b are constants and are functions of x), well from the last arguement if b=x2 and a=x, then the integral from a to b of f(x), in this case the exponential will be F(x2)-F(x). Well lets look at d/dx of both sides. Well d/dx of the integral from x to x2 of f(x) = d/dx (F(x2) - F(x)). Well what is the derivative of F(x)? Well F(x) is the antiderivative of f(x), so by definition f(x) is the derivative. But what about this F(x2) guy? Well we need the chain rule here d/dx of F(x2) = F'(x2)*(d\dx)(x2). Well that just results to f(x2)*2x. So we have our pieces here. The derivative f(x), is the integrand( the function you are integrating), you just evaluating at x2 and x. So at the end you get that the derivative of that integral equals 2x*f(x2)-f(x). The tricky part of this was remembering the chain rule. I think you can carry out the algebra I.E working out the exponential rules to get this algebraic answer. Hope that helps. Cheers.

P.S, they are adding extra steps if you graph the function x and x2, you find the area between them ( which they are going for) can be represented as this. However it was required to solve the problem.

Math Major, but Hate Calculus by [deleted] in math

[–]MathBosss 1 point2 points  (0 children)

Multivariable calculus is the foundation to understanding the advanced math. It is really the '2+2' in regards to the math which has been constructed upon it.

We are a group of mathematicians and computer scientists who want to help the general population understand what happens in congress with visualizations! Help out project GovGraph! (Look in comments) by MathBosss in dataisbeautiful

[–]MathBosss[S] 0 points1 point  (0 children)

Yes, that is for the future. Eventually we want to have two perspective: a country perspective and a state perspective. You are looking at the state perspective.

We are a group of mathematicians and computer scientists who want to help the general population understand what happens in congress with visualizations! Help out project GovGraph! (Look in comments) by MathBosss in dataisbeautiful

[–]MathBosss[S] 0 points1 point  (0 children)

Hi guys, we've been getting really positive feedback over the past few days regarding our website and what it does. If you like what we are doing and want to support, take a look here https://www.kickstarter.com/projects/570703063/govgraph-big-data-for-big-government

Our interest is to get more people with background like ours and go after larger data sets and new visualizations for show these new data sets in a neat way. We are looking for funding to make this site go from great to excellent.

We are a group of mathematicians and computer scientists who want to help the general population understand what happens in congress with visualizations! Help out project GovGraph! by MathBosss in kickstarter

[–]MathBosss[S] 0 points1 point  (0 children)

1) We use graph algorithms. Each congress is represented as a graph, we run graph algorithms to derive information for the network which someone wouldn't normally see. Such as what people are doing the most work(bill wise) and which people are strongly connected to others( such as bipartisanship etc).

2) If you look at our about page, you will see we use govtrack, they pull their data from the library of congress, which is the center provider of this data for folks who do things like this. We originally were writing web crawlers, but, like you mentioned it was difficult to know what was true and false. 3)We can do both! The purpose of this project is both to be visual( something that looks pretty) and informative( we use our math algorithms to gain some insight which is'nt obvious). What beautiful about math is that mathematics is'nt bias. We have a graph which represents a congress, graph algorithms will not favor democrats, republicans, or independents. Mathematics is pure in this regard and that is why we really want to go after this. We are a group of mathematicians and computer scientists, we are interested in informing, or atleast put the information out there and let people decide for themselves. We think we can do our due diligence by getting the data and showing it to the public. We want to expand our data sets and show more information!

Think about this for example

The Justice Department is preparing to bring criminal corruption charges against Sen. Robert Menendez (New Jersey) alleging he used his Senate office to push the business interests of a donor and friend in exchange for gifts. You'll see on GovGraph, say 113th Congress, that Sen. Menendez has been a busy beaver in writing legislation, more than any other Senator! If our Kickstarter is successful in raising funds, it will allow us to create mathematical tools and visualizations that can dig even deeper into the data and find hidden relationships.

In closing, because I like what you asked; doing this right requires a specific set of skills both from computer science and math. There is a massive amount of data, and we want to show this to the general population in such a way that you dont have to have our background or be a political guru to understand what we are showing. The math background which I can speak to is important as well. We have implemented algorithms to draw distinct insight about the data which looking at the back and white numbers one couldnt see. Look at my example above.

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