## Archive for June, 2008

### The Image Fulgurator

June 28, 2008

The Image Fulgurator is, roughly speaking, an analog photo camera “working in reverse”. The device enables one to project images on other people’s photographs! The following schematic illustrates quite nicely how this “projector” works:

[ schematic courtesy of Julius von Bismarck ]

The Image Fulgurator was built by German artist Julius von Bismarck. Here’s what Julius says about his creation:

The Image Fulgurator is a device for physically manipulating photographs. It intervenes when a photo is being taken, without the photographer being able to detect anything. The manipulation is only visible on the photo afterwards.

In principle, the Fulgurator can be used anywhere where there is another camera nearby that is being used with a flash. It operates via a kind of reactive flash projection that enables an image to be projected on an object exactly at the moment when someone else is photographing it. The intervention is unobtrusive because it takes only a few milliseconds. Every photo another photographer takes of an object at which the Fulgurator is also aimed is affected by the manipulation. Hence visual information can be smuggled unnoticed into the images of others.

The Image Fulgurator is equipped with a slave flash that is triggered whenever a master flash is fired. Note that if you do not use flash when taking photos, you do not risk getting your photos “fulgurated”. I would say that many people have reacted to this device in an overly negative manner. I believe that the Image Fulgurator is a good idea and that Julius deserves credit for his ingenuity.

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Related:

### Summing reciprocals of binomial coefficients

June 4, 2008

At Topological Musings, this week’s problem is:

Compute $\displaystyle \sum_{k = 0}^\infty \frac{1}{\binom{n+k}{k}}$ for $n > 1$.

Update: Todd Trimble wrote an awesome post presenting three different solutions. I particularly liked Nilay Vaish’s method.

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My solution:

$\displaystyle \frac{1}{\displaystyle\binom{n+k}{k}} = \frac{1}{\displaystyle\frac{(n+k)!}{n! k!}} = \displaystyle \frac{n! k!}{(n+k)!} = \displaystyle \frac{n (n-1)! k!}{(n+k)!}.$

Using the Gamma function (the Euler integral of the 2nd kind)

$\Gamma(z) = \displaystyle\int_0^{\infty} t^{z-1} e^{-t} dt$

we can now write

$\displaystyle\frac{1}{\displaystyle\binom{n+k}{k}} = \displaystyle \frac{n (n-1)! k!}{(n+k)!} = n\frac{\Gamma(n) \Gamma(k+1)}{\Gamma(n+k+1)},$

because $\Gamma(m+1) = m!$ for all $m \in \mathbb{N}$. From the definition of the Beta function (the Euler integral of the 1st kind)

$B(x,y)= \displaystyle\int_0^1t^{x-1}(1-t)^{y-1} dt =\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$

we then have

$\begin{array}{rl}\displaystyle\frac{1}{\displaystyle\binom{n+k}{k}} &= \displaystyle n \frac{\Gamma(n) \Gamma(k+1)}{\Gamma(n+k+1)}\\ &= n B(n, k+1)\\ &= \displaystyle\int_{0}^1 n t^{n-1} (1-t)^k dt.\\\end{array}$

Finally, summing the reciprocals of the binomial coefficients

$\begin{array}{rl} \displaystyle\sum_{k = 0}^\infty \frac{1}{\displaystyle\binom{n+k}{k}} &= \displaystyle\sum_{k = 0}^\infty \displaystyle\int_{0}^1 n t^{n-1} (1-t)^k dt\\ &= \displaystyle\int_{0}^1 n t^{n-1} \left[\displaystyle\sum_{k=0}^{\infty} (1-t)^k\right] dt\\ &= \displaystyle\int_{0}^1 n t^{n-2} dt\\ &= n \displaystyle\left(\left.\frac{t^{n-1}}{n-1}\right|_{0}^{1}\right)\\\end{array}$

and, therefore, the solution is

$\displaystyle\sum_{k = 0}^\infty \frac{1}{\displaystyle\binom{n+k}{k}} = \displaystyle\frac{n}{n-1}.$

### Variational Integrators

June 3, 2008

In my early days as an undergraduate student, I had a lot of fun writing C code to integrate ODE’s numerically, so I could simulate physical systems in a computer (integrating the pendulum ODE is a classic). However, I was not very happy to realize that traditional numerical methods very easily give results with no physical meaning.

For example, think of a simple pendulum oscillating on a vertical plane. Assuming that there are no damping forces, there’s energy conservation. However, traditional numerical methods (such as Runge-Kutta methods) do not, in general, preserve energy. The numerical simulation of the pendulum may thus exhibit non-physical energy dissipation (or amplification).

[ image courtesy of Ari Stern ]

When we integrate an ODE numerically, we obtain a discrete trajectory which we hope is “close enough” to the exact flow of the differential equation. In some problems in Physics, the continuous flow lies on a manifold, and we would like to preserve this geometric property: it would be wonderful if the discrete trajectory lied on the same manifold as the continuous flow. While thinking about numerical methods that preserve certain invariants, I became interested in the field of geometric integration.

One particular class of geometric integrators I have been interested in over the past year is the class of variational integrators. Many physical laws are formulated as variational principles from which differential equations can be derived. Instead of discretizing the differential equations, why not discretize the variational principles instead? This rather sexy thought lies at the basis of the nascent field of variational integrators.

Here’s an overview on variational integrators, by Prof. Matthew West. I have been reading some papers on the topic. This field seems to have been explored rather vigorously at Caltech, as there are/were a lot of people working on it:

and their collaborators. I have compiled a list of lecture slides, theses and papers on the area of variational integrators. Hope you find the list useful (I sure do!).

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Some lecture slides for a gentle introduction:

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A few PhD theses on variational integrators and their application:

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Some papers on the foundations of variational integrators:

Applications to Computational Mechanics:

Applications to Optimal Control Theory:

Applications to Computational Electromagnetism:

Applications to Computational Physics:

The papers I have listed so far refer to deterministic integrators. However, quite recently Prof. Houman Owhadi and Nawaf Bou-Rabee have been working on stochastic variational integrators:

I wonder if stochastic variational integrators could be used in quantitative finance. Just a wild thought…

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If you know some more papers, please let me know! Last but not least, Ganesh Swami once wrote a brief post on variational integrators.