Memory Minimization

As I grow older, I’m starting to forget things (such as all the math I ever learned) … but that’s not the reason for the title of this post. A somewhat interesting question popped up on Mathematics StackExchange. It combines a basic sequencing problem (ordering the processing of computational tasks) with a single resource constraint …

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If This and That Then Whatever

I was asked a question that reduced to the following: if \(x\), \(y\) and \(z\) are all binary variables, how do we handle (in an integer programming model) the requirement “if \(x=1\) and \(y=1\) then \(z=1\)”? In the absence of any constraints on \(z\) when the antecedent is not true, this is very easy: add …

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Update Error: Wrong Architecture

Yesterday I ran into one of those mystifying glitches that, will infrequent, serve as a reminder that Linux is not for the faint of heart. When I booted my desktop system (Linux Mint 18.1 Serena), the system tray icon for the software updater was displaying a red “X” that indicates it tried and failed to …

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Von Neumann and Realpolitik

“Right, as the world goes, is only in question between equals in power, while the strong do what they can and the weak suffer what they must.” — Thucydides, Melian Dialogue. Von Neumann, Feynman, and Ulam. Adventures of a Mathematician (Ulam): … Once at Christmas time in 1937, we drove from Princeton to Duke University …

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Another Absolute Value Question

Someone asked whether it is possible to eliminate the absolute value from the constraint \(Lx\le |y| \le Ux\) (where \(L\ge 0\) and \(U>0\) are constants, \(x\) is a binary variable, and \(y\) is a continuous variable). The answer is yes, but what it takes depends on whether \(L=0\) or \(L>0\). The easy case is when …

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Fischetti on Benders Decomposition

I just came across slides for a presentation that Matteo Fischetti (University of Padova) gave at the Lunteren Conference on the Mathematics of Operations Research a few days ago. Matteo is both expert at and dare I say an advocate of Benders decomposition. I use Benders decomposition (or variants of it) rather extensively in my …

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Oppenheimer on Bohr (1964 UCLA)

Oppenheimer on Bohr (1964 UCLA) I came across this 1964 UCLA talk by Oppenheimer, on his hero Niels Bohr. Oppenheimer: Mathematics is “an immense enlargement of language, an ability to talk about things which in words would be simply inaccessible.” I find it strange that psychometricians usually define “verbal ability” over a vocabulary set that …

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MIP Models in R with OMPR

A number of R libraries now exist for formulating and solving various types of mathematical programs in R (or formulating them in R and solving them with external solvers). For a comprehensive listing, see the Optimization and Mathematical Programming task view on CRAN. I decided to experiment with Dirk Schumacher’s OMPR package for R. OMPR …

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Better Estimate, Worse Result

I thought I might use a few graphs to help explain an answer I posted on Quora recently. First, I’ll repeat the question here: In parametric optimization with unknown parameters, does a better estimator of the parameter always lead to better solutions to the actual problem? Here we’re trying to minimize f(x,θ) with respect to …

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Finding the Kernel of a Matrix

I’m working on an optimization problem (coding in Java) in which, should various celestial bodies align the wrong way, I may need to compute the rank of a real matrix and, if it’s less than full rank, a basis for its kernel. (Actually, I could get by with just one nonzero vector in the kernel, …

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Enforcing Simultaneous Arrivals

I’m recapping here an answer to a modeling question that I just posted on a help forum. (Since it will now appear in two different places, let’s hope it’s correct!) The original poster (OP) was working on a routing model, in which vehicles (for which I will use and if needed as indices) are assigned …

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MathJax Whiplash

Technology is continuously evolving, and for the most part that’s good. Every now and then, though, the evolution starts to look like a random mutation … the kind that results in an apocalyptic virus, or mutants with superpowers, or something else that is much more appealing as a plot device in a movie or TV …

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Oracle Java 8 Update

For quite a while, I was getting security nags from Firefox every time a web site wanted to run a Java applet. Firefox would tell me I needed to upgrade to the latest version of Java. That would have been fine, except that I was already running the latest Java (1.8.0_66 as of this writing). …

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The cult of genius?

In one of his early blog posts, Terence Tao (shown above with Paul Erdos in 1985) wrote Does one have to be a genius to do maths? The answer is an emphatic NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other …

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David Donoho interview at HKUST

A long interview with Stanford professor David Donoho (academic web page) at the IAS at HKUST. Donoho was a pioneer in thinking about sparsity in high dimensional statistical problems. The motivation for this came from real world problems in geosciences (oil exploration), encountered in Texas when he was still a student. Geophysicists were using Compressed …

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Ambigrams & Math: In one embeddable ebook

Over the past two years Gaurav Bhatnagar and I have written five columns for the Math education journal At Right Angles  on the topics of mathematics and visual wordplay, specifically Ambigrams. In this five articles we have explored everything from symmetry to self-similarity, fractals to paradoxes. I have posted these articles on this website as they have been …

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Paradoxes & Ambigrams: Article 2 of 2

A few months ago I had posted about publication of the first of two articles on mathematics, visual wordplay and paradoxes. The second article (part of our series on Art and Math co-authored with my friend Gaurav Bhatnagar and published by At Right Angles) is now available. You can download all of the articles in the series Of Art & Math by following the links below …

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Optimizing Part of the Objective Function

A somewhat curious question showed up on a forum today. The author of the question has an optimization model (I’ll assume it is either a linear program or mixed integer linear program) of the form \begin{alignat*}{2} & \textrm{maximize} & & \sum_{i=1}^{N}x_{i}\\ & \textrm{s.t.} & & x\in\mathcal{X} \end{alignat*} where the feasible region $\mathcal{X}$ is presumably polyhedral. …

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Selecting the Least Qualifying Index

Something along the following lines cropped up recently, regarding a discrete optimization model. Suppose that we have a collection of binary variables $x_i \in B, \, i \in 1,\dots,N$ in an optimization model, where $B=\{0, 1\}$. The values of the $x_i$ will of course be dictated by the combination of the constraints and objective function. …

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Erdös with a non-kosher side of Bacon

Paul Erdös was a prolific and important mathematician. He also had hundreds of collaborators from around the world who coauthored papers with him. Years ago, Casper Goffman explained an idea, called the Erdös number, that describes the “collaborative distance” between Erdös and someone else, where that distance is defined by the smallest number of steps based …

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John Nash, dead at 86

The original title of this post was For this you won a Nobel (Memorial) Prize? But see sad news at bottom. A Beautiful Mind: Nash went to see von Neumann a few days after he passed his generals? He wanted, he had told the secretary cockily, to discuss an idea that might be of interest …

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Formulating Optimization Models

Periodically, on OR Exchange and other forums, I encounter what are surely homework problems involving the construction of optimization models. “The Acme Anvil Corporation makes two types of anvils, blue ones and red ones. Blue anvil use 185 kg. of steel and have a gross revenue of $325 each; red anvils …” Really? Does anyone …

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Partitioning with Binary Variables

Armed with the contents of my last two posts (“The Geometry of a Linear Program“, “Branching Partitions the Feasible Region“), I think I’m ready to get to the question that motivated all this. Let me quickly enumerate a few key take-aways from those posts: The branch and bound method for solving an integer linear program …

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Making Math Fun

I was a math major from undergrad to doctorate, so obviously I think math is fun. Equally obvious to me (especially after teaching a variety of mathematical topics to college students), not everyone shares that opinion, which is too bad. Recently, though, I came across an organization devoted to making math fun for small children, …

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Analogies between Analogies

As reported by Stan Ulam in Adventures of a Mathematician: “A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between …

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Linearize That!

For whatever reason, I’ve seen a bunch of questions posted on various fora boiling down to “How do I linearize <insert grossly nonlinear function>?” Whether by coincidence or due to some virtual viral epidemic, I’ve seen three or four in the past week that involved logarithms. So, without further ado, here is the quick solution: …

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The Reciprocal Normal Distribution

A recent question on OR-Exchange dealt with the reciprocal normal distribution. Specifically, if k is a constant and X is a Gaussian random variable, the distribution of Y=k/X is reciprocal normal. The poster had questions about approximating the distribution of Y with a Gaussian (normal) distribution. This gave me a reason (excuse?) to tackle something …

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Nifty papers I wrote that nobody knows about (Part 4: Complex Langevin equation)

This is the last installment of the “Nifty Papers” series. Here are the links to Part1, Part2, and Part 3. For those outside the computational physics community, the following words don’t mean anything: “The Sign Problem“ For those others that have encountered the problem, these words elicit terror. They stand for sleepless nights. They spell despair. They make grown men …

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Happy Teacher’s Day (new ambigrams)

September 5 is Teacher’s Day in India. It is celebrated on the birthdate of Sarvepalli Radhakrishnan, Indian philosopher and statesman who was also the first Vice-President and the second President of India. He famously said, “teachers should be the best minds in the country.” To celebrate this day, here are three new ambigram designs (see image below). …

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A Musical Random Walk

I just read a blog post here at MSU about The Infinite Jukebox, a web tool that will analyze a song (either one you upload or one from their library) and map out links between segments of the song that have approximately the same beat. You then see a visualization of the song as a …

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Why math ed sucks (not just in India)

My friend Hartosh Bal (author of A Certain Ambiguity, a mathematical novel) has a piece in Caravan Magazine titled “Why Fields medalists are unlikely to emerge from the Indian educational system.” He mentions the fact that of the three winners of the Field’s medal (the highest accolade in mathematics) are Brazilian, Iranian and Canadian respectively. The …

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Vijay Iyer, polymath & a fantastic example of trans-disciplinary creativity

Vijay Iyer,  (http://vijay-iyer.com/) is an Indian-American jazz pianist and composer. He is a MacArthur Genius grant winner and is currently Franklin and Florence Rosenblatt Professor of the Arts at Harvard University and is widely regarded as being one of the most innovative composers and musicians today. His music transcends national boundaries through creatively bringing together traditional …

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