Math on traviscj/blog

loan trickery

Thu, 27 Mar 2025 10:36:04 -0500

I have this “American Express Blueprint Business Loan” product to have a line of credit for the rentals if I’m ever in a pinch.

They allow up to $21,700 with this very easy application process, you basically just slide a slider to the amount needed & select between 6, 12, 18, 24 months:

This shows a schedule like this:

Looking at this, there’s a lot of temptation to think

profitably wrong

Tue, 09 Jan 2018 08:00:00 +0000

I’ve come to realize that my career so far has been built on being “profitably wrong.” I think this is interesting because the usual approaches are being “profitably fast” (optimizing) or “profitably better” (improving), and most people think of any kind of wrongness as being a terrible thing. But sometimes the best way to optimize or improve is approximating!

The definitions of “profitably” has changed as I’ve worked on different things, as has the specific type of “wrongness”. A couple specific ways accepting “wrongness” have been profitable for me include:

dancing bull pricing structure

Thu, 04 Jan 2018 00:10:00 +0000

Dancing Bull is a great Korean BBQ restaurant that opened nearby recently. They have this pricing scheme:

Note that the unlimited bulgogi is \$23/person, but the bulgogi a la carte is \$25/grill (with each additional orders after that costing \$9). This raises an obvious question: which is the better deal?

Spoiler: it depends on both how many people you have and how hungry they are.

First, we had to clarify how it actually worked. It’s not clear from the menu, but the “\$25/grill” includes the first order of bulgogi. Introducing $P$ for the number of people, $G$ for the number of grills, and $O$ for the total number of orders, we would pay $$UL = 23\cdot P$$ for the unlimited price scheme vs $$ALC = 25\cdot G + 9\cdot ( O - G)$$ for the a la carte scheme.

snack buying

Mon, 13 Nov 2017 07:01:25 +0000

I got this text message from my father-in-law:

Ok. Soda 1.50 Chocolate bar 1.00 Gum 0.10 Jelly Bean .05

Have to buy exactly 14 items and spend $10

At least one of each.

There are 4 diff combos. (Order: soda, chocolate, gum and jelly bean)

Got 5-2-3-4 3-5-4-2

Can’t get others. Any ideas? :)

I whipped up this solution to check in python:

#!/usr/bin/python3
# created_at: 2017-11-13 13:53:19 -0800

def main():
    solutions = 0
    for si in range(1,7):
        for ci in range(1,11):
            for gi in range(1,101):
                for ji in range(1,201):
                    total = si*150 + ci*100 + gi*10 + ji*5
                    items = si + ci + gi + ji
                    if total == 1000 and items == 14:
                        print(si,ci,gi,ji, total, items)
                        solutions += 1
    print(solutions)

if __name__ == "__main__":
    main()

The core of this solution is the check

slide rules

Wed, 19 Nov 2014 00:00:00 +0000

Suppose you want to evaluate the multiplication problem $C = A\cdot B$. But you forgot your times tables, so you are stuck using your expedition watch or something. In fact, you can even make your own out of a couple of slips of paper, if you really need to.

First bit of background: One of the rules you probably learned and I definitely forgot was the rule of logs: $$ C = A\cdot B \equiv \log C = \log(A\cdot B) = \log A + \log B$$ This ends up being useful for slide rules, because you can easily add distances together by putting two things next to eachother! So that is exactly what we do. We add the logarithms by putting things next to eachother.

14 vs 1499 vs 15

Fri, 22 Aug 2014 00:00:00 +0000

Sometimes, it is tempting to see 14.99 and say “about 14,” even though we all know better. The problem with this is that by giving a 0.07% discount (14.99 vs 15), they have made you estimate a 7% discount (14 vs 15).

Nice trick!

overly-ambitious-isqo and the design of numerical codes

Wed, 23 Oct 2013 00:00:00 +0000

I have finally released the overly-ambitious-isqo project on github!

I wanted to call out two particular design concerns I had.

rich language

My first goal was to try very hard to build up the C language to very succinctly express the main algorithm in src/isqo_functor.cpp in extremely rich language. It seems like numerical code is typically implemented with loops like for (int i=0; i<N; i) and method calls like deltal(xhat, mu). I have found it much easier to reason and think deeply about codes like for (int primal_index=0; primal_index < num_primal; primal_index) and method calls like linear_model_reduction(penalty_iterate, penalty_parameter).

a really simple iterative refinement example

Thu, 17 Oct 2013 00:00:00 +0000

I wanted to have a quick example of what iterative refinement is and looks like.

Suppose you wanted to solve the (very simple!) problem of solving 5.1x=16, without dividing by decimal numbers. (Yes, this is a bit contrived–I’m not sure why you would be able to divide by an integer to get a decimal, but not divide by a decimal to get a decimal. Bear with me.)

The standard way would be, of course, to simply evaluate x = (5.1)^{-1}16=3.137254. But this is not available–you can’t find an inverse of multiplying by 5.1!

spring-mass visualization

Wed, 31 Jul 2013 00:00:00 +0000

I’m working on a paper about an algorithm for hotstarting nonlinear program solves; one application of this might be in the realm of nonlinear model predictive control. In these types of models, we first define the physical equations for the system under consideration. They are subject to some control parameters, which are just a mathematical representation of the input we could give the system. We also define an objective–something that we would like to minimize(usually something like time or energy).

some tips for project euler problems

Sat, 26 Jan 2013 00:00:00 +0000

I wanted to make a list of a few tips on solving Project Euler problems that have been helpful for me while I solve them. These are general principles, even though I do most of my Project Euler coding in Python.

Without further ado:

If the problem is asking for something concerning the number of digits, typically this indicates that the use of the $\log n$ function is warranted.
If the problem is asking for the last few digits, modulo arithmetic might speed it up considerably.
Some might consider this cheating, but looking up some small numbers in the Online Encyclopedia of Integer Sequences is occasionally pretty helpful.
Many problems boil down to: Find numbers with property $X$ and property $Y$. Two solutions are:
- Brute force: Try all numbers with tests of property $X$ and $Y$.
- Find numbers with property $X$ and filter by a test of property $Y$.
- Find numbers with property $Y$ and filter by a test of property $X$.
- Find the set of numbers with property $X$ and the set of numbers with property $Y$. Compute their intersection.

I’ve found that it’s sometimes hard to predict which one will end up being the fastest. It depends on the relative speed of the tests and the generators, and the frequency of finding numbers which have that property.

Game of Life

Tue, 20 Nov 2012 00:00:00 +0000

When I was a sophomore in high school, I was fascinated with [ Conway’s Game of Life]. I still am. I did a pretty rudamentary study of the kinds of patterns that could form from the simple rules of the game.

One thing that wasn’t available when I was a sophomore was Youtube. Two of my favorite Game of Life videos:

Another direction that Game of Life has gone recently is something that I really should have thought of, honestly. Many times I’ve thought that there’s at least some superficial relationship between Game of Life and diffusion equations. Turns out that S. Rafler has extended Game of Life to continuous domains through what he calls SmoothLife.

Perturbation Theory Problems with bvp4c

Mon, 22 Oct 2012 00:00:00 +0000

I have been watching Nathan Kutz’ lectures on Coursera. One change he made to the course since I took AMATH 581 at University of Washington was introducing the MATLAB function bvp4c. I immediately realized that this would be nice for solving boundary layer problems that arise in asymptotics.

Following my life philosophy of doing the dumbest thing that could possibly work, I tried implementing Nathan’s code for a single-layer boundary layer problem from Holmes, Chapter 2:

Numerical Recipes & Scientific Libraries

Sat, 26 May 2012 00:00:00 +0000

I attended a talk on how to use Quest, Northwestern University’s TOP500 supercomputer(At least during June 2010). Most of it was a routine introduction to MPI, but one interesting question raised was what routines we should be using in our scientific computing codes. A lot of holdouts were still using [ Numerical Recipes] for their research-level codes, which strikes me as a backwards way about it. Numerical Recipes is a starting point, and probably not the best thing to use: it has awful licensing and might not even be that reliable!

fibonacci miles and kilometers

Thu, 29 Mar 2012 00:00:00 +0000

In my running, I have been trying to keep track in kilometers. This presents a couple problems: Somehow, my mind still thinks in miles, which is weird because I do not really have /that/ good of an idea exactly how far a mile is, either. Or someone wants to know how far a 5k is. Or, when I was running the 2011 Chicago Marathon, there would be kilometer postings between the mile markers. Anyway, it is handy to convert between them, but a bit of a pain.

a trig problem solved in MATLAB

Wed, 01 Feb 2012 00:00:00 +0000

I came across this post. The basic idea is the guy wants to maximize $L_1+L_2$ constrained to this box, where $L_i$ is the length of beam $i$. It’s constrained to be a 61 cmx61 cm box, but one beam must start from 10cm up from the bottom right corner and the beams must meet at a point along the top of the box. I added the further assumption that the other beam must end in the bottom left corner.

cplex matlab interface

Wed, 01 Feb 2012 00:00:00 +0000

Just for my own reference, I’m documenting the interface to CPLEX.

CPLEX expects a problem in the form ( \begin{split} \min \qquad & g^Td + \frac12 d^TWd\ \text{subject to} \qquad & c_L \leq Ad \leq c_U\ & d_L \leq d \leq d_U \end{split} ) and is called by

cplex = Cplex('test');
cplex.Param.feasopt.tolerance.Cur = 1e-8;
if params.printLevel < 8
    cplex.DisplayFunc = [];
end
cplex.Model.sense = 'minimize';
cplex.Param.qpmethod.Cur = 1;
cplex.addCols(gk,[],bl-xk,bu-xk);
cplex.addRows(-ck, A0, -ck);
cplex.Model.Q = W;
cplex.Model.obj = g;
cplex.Model.lb = d_L;
cplex.Model.ub = d_U;
cplex.Model.lhs= c_L;
cplex.Model.rhs= c_U;
cplex.solve();

spamfunc for optimization in matlab

Mon, 30 Jan 2012 00:00:00 +0000

what spamfunc is

In developing optimization algorithms, one of the most tedious parts is trying different examples, each of which might have its own starting points or upper or lower bounds or other information. The tedium really starts when your algorithm requires first or second order information, which might be tricky to calculate correctly. These bugs can be pernicious, because it might be difficult to differentiate between a bug in your algorithm and a bug in your objective or constraint evaluation. Handily, Northwestern Professor Robert Fourer wrote a language called AMPL, which takes a programming-language specification of objective and constraints and calculates derivatives as needed. The official amplfunc/spamfunc reference is contained in Hooking Your Solver to AMPL, but I’m shooting for a more low-key introduction.

Five Critical Textbooks for (Applied!) Math & Physics Students

Sun, 04 Sep 2011 00:00:00 +0000

In the course of working through my first year at grad school, I’ve come up with five favorites for the basics of an undergrad understanding of the essential topics for applied math and physics. Without ado and in the order I’d take them off my shelf:

Mathematical Methods in the Physical Sciences - ML Boas. This is my favorite, because it contains almost every technique you need, and it has a ton and a half of problems(over 3400). It contains a good review of complex analysis, linear algebra, differential equations, and calculus, but also chapters on special functions, partial differential equations, probability, tensors, and the calculus of variations. – Hidden Gem: Chapter 4, Section 12: Differentiation of Integrals, RP Feynman’s favorite trick.
Calculus - Greenspan. A great reference on calculus.
Complex Variables and Applications - Brown & Churchill: The most readable book on complex analysis I’ve read. Not so hidden gem: Most of the solutions are given right alongside the problem–a great book for self-study. Also the material on conformal mapping and fluid flows.
Linear Algebra And Its Applications - Strang. Nice book on linear algebra theory. – Hidden Gem: Chapter 7, Section 4: Iterative Methods for Ax=b and Gershgorin’s circle theorem.
Elementary Differential Equations and Boundary Value Problems - Boyce & DiPrima. Powerhouse of differential equation knowledge. Strangely, it is the book ESAM recommended, but not the book they use for teaching their undergrads.

I’ve spent probably the most time with Boas’ Mathematical Methods in the Physical Sciences–I’ve worked nearly 1000 problems out of the book to get ready for the preliminary exams my first year. It was completely worthwhile.

Tunes U and Essential Mathematica

Sun, 13 Dec 2009 00:00:00 +0000

In a similar vein to the TED talks in the previous post, I’ve also been watching a lot of lectures from iTunes U lately. Mostly to get ready for preliminary exams, but also for their sheer awesomeness of the teaching and material. I’ve been most surprised how many people haven’t heard of this yet: Nearly everyone seems surprised when they hear about it. Anyways, here’s my list of favorites so far:

What could you possibly do with mathematics?

Mon, 24 Aug 2009 00:00:00 +0000

Recently, at a family gathering, I was confronted by the question many a college graduate is faced after telling someone I had majored in mathematics for my now-finished college degree: “But how are you going to make any money at that?”

Now, certainly it’s true: Graduate students don’t make that much. The average stipend for a grad student is roughly on par with (but still less than) unemployment checks. But that’s okay–in general, mathematicians know that they could make money other places, but they chose it anyways because of their love for the subject. Not that mathematicians make too bad of money anyway: The average in the US is about 50K, with associate professors making more than that. The particularly salary-inclined could pick up other credentials to become actuaries or work in hedge funds.

Netlib

Wed, 27 May 2009 00:00:00 +0000

Wow. Just wow.

Mission Graduate School -- Success

Wed, 25 Feb 2009 00:00:00 +0000

I’ve officially been accepted to two PhD applied math programs: Northwestern and University of North Carolina. I’ve also been accepted to the Masters of Science program at the University of Washington. Both of the PhD offers came with full fellowship offers(at least for the first year), which basically means that I just need to go and give them the best bang for their buck with my brainpower for five years, and then I’ll have my PhD. Awesome. All three schools are outstanding and have amazing faculty, as best as I can tell.

Why Do Math and Izhikevich

Tue, 06 Jan 2009 00:00:00 +0000

Professor Eric Shea-Brown has written up a nice website explaining what we’re doing with the computational neuroscience modeling. It’s currently on the Why Do Math website at this link: Brain Dynamics: The Mathematics of the Spike.

For my project, we’re starting with a simpler model that is similar in behavior but quicker computationally and (somewhat) easier to analyze mathematically, known as the Izhikevich model(after its creator, Eugene Izhikevich). His website has some amazingly cool videos and a lot of papers on what he’s been doing. His Website

Getting an AMATH Minor at the University of Washington

Sun, 26 Oct 2008 00:00:00 +0000

One of the secret gems at the University of Washington is the minor in applied math option. At least I think it’s somewhat secret. Most of the people I tell about it haven’t really heard of it. But yeah, here’s the deal: You only need to get through calculus and four more classes to get this minor! And a lot of programs require calculus anyways. I’ve taken most of the undergrad AMATH courses, so I figured I’d write something up about which ones I recommend the most.

First Couple of Days in Rio

Wed, 20 Aug 2008 00:00:00 +0000

So Monday morning was really lazy. Some people went to get a volleyball, a bunch went down to the beach again, and I let my sunburn cool off. It was also really warm and sunny, which is a nice change from Seattle! We left for IMPA around 2, got on taxis to get over there, and eventually made it. The taxi drivers are apparently pretty vicious. We took 5 cabs, 3 got charged around 15, one around 20, and one 25. So they try to rip you off.

Whack it with an X squared!

Thu, 07 Aug 2008 00:00:00 +0000

David and I were working on our Math381 model, and I was getting frustrated because the data we collected and the results from the simulation were not lining up properly, which was quite frustrating. We were hoping to see something like this:

Number of Logins from Data

Instead, we were getting stuff distributed like this:

Simulated Number of Logins

I realized that we needed some function to force a bunch of this junk further left. Recalling an old adage from Mr. Cone’s AP Chemistry class, I decided it was the right time to whack it with an X squared. This is vaguely appropriate, because rand() has a range [0,1), so squaring it should put a whole bunch of stuff further right, but not everything(ie, the first half will end up in the first quarter, the first 3/4 will end up in the first 9/16, etc). Imagine my shock when I saw this:

Dijkstra's Algorithm Paper

Wed, 30 Jul 2008 00:00:00 +0000

The week before my sister’s wedding, I was tasked with writing a paper on Dijkstra’s Algorithm for my Discrete Mathematical Modeling class. I think I might have missed the mark a little bit, but I had so much fun writing it that I’m posting it here.

I’m almost considering writing some more stuff in this style… anything anyone would like to see written about?

Here’s a link: Dijkstra’s Algorithm

Differential Equations

Mon, 12 May 2008 12:00:00 +0000

I’ve realized that a lot of people are nervous about differential equations. Which is understandable, but in general there’s some pretty straightforward ways to solve a fair number of the ones you come across. I’d really like to write some of it up.

My basic idea is going to be, show a bit about integrating factors, a bit about separation of variables, the characteristic equation, and the method of undetermined coefficients. That covers a lot of physical territory. Then some about reducing order with transformations from n-th order equations to n 1-st order equations, and basic Laplace for a sortof general method. And finally, maybe a tiny bit about numeric methods.

Cython

Fri, 09 May 2008 12:00:00 +0000

After I had finally convinced myself to get out of bed this morning to go to my ACMS seminar, I quickly checked my email and my heart sank a little. Today’s talk was on SAGE. Don’t have anything against SAGE, but I thought it was just a big pile of open source packages in a big, heavy install. Sorta cool, but worthless, in other words.

Turns out, I was pretty wrong about that. It is that, but it’s also 70k new lines of code that does a whole bunch of exciting stuff. Near the end of his talk, William Stein mentioned that they had created a new tool called Cython. (Well, extended Pyrex, but… whatever.)

Where I'm At

Sat, 03 Mar 2007 00:00:00 +0000

So it’s almost the end of Winter Quarter 07 at the University of Washington. It’s cold and rainy and lame. Last week I got turned down from the Computer Science program here, which meant I spent the weekend generally freaking out.

In any case, I then remembered that I liked the idea of an ACMS degree. The base of the major consists of the physics series, a Numerical Methods course and a Continuous modeling course from the AMATH department, and discrete models and statistics from the math department.