Bubbles Bad; Ripples Good

… Data aequatione quotcunque fluentes quantitates involvente fluxiones invenire et vice versa …

Category: Disseminating mathematics

A better estimate of Kempner’s series

The Kempner series recently regained some notoriety due to a Saturday Morning Breakfast Cereal comic (the last panel). The observation first appeared in a 1914 American Mathematical Monthly article, in which it was shown that the series consisting of the usual harmonic series

\displaystyle \sum_{n = 1}^{\infty} \frac{1}{n}

but with all the terms, whose decimal expansion includes the digit ‘9’, removed, in fact converges to some number below 80. The original proof is given in the Wikipedia article linked above, so I will not repeat it. But to make it easier to see the idea: let us first think about the case where the number is expressed in base 2. In base 2, all the positive integers has the leading binary bit being 1 (since it cannot be zero). Therefore there are no binary positive numbers without the bit ‘1’ in its expansion. So the corresponding series converges trivially to zero. How about the case of the bit ‘0’? The only binary numbers without any ‘0’ bits are

1, 3 = (11)_2, 7 = (111)_2, 15 = (1111)_2, \ldots, 2^n - 1.

So the corresponding series actually becomes

\displaystyle \sum_{n = 1}^\infty \frac{1}{2^n - 1} \leq \sum_{n = 1}^\infty \frac{1}{2^{n-1}} = 2

So somewhere from the heavily divergent harmonic series, we pick up a rapidly converging geometric series. So what’s at work here? Among all the n-bit binary numbers, exactly 1 has all bits not being 0. So the density of these kinds of numbers decays rather quickly: in base 2, there are 2^{n-1} numbers that are exactly n-bit long. So if a number x has a binary representation that is exactly n bits long (which means that 2^{n} \leq x < 2^{n+1}), the chances that it is one of the special type of numbers is \frac{1}{2^{n-1}} \approx \frac{2}{x}. This probability we can treat then as a density: replacing the discrete sum \sum \frac{1}{n} by the integral \int \frac{1}{x}\mathrm{d}x (calculus students may recognize this as the germ of the “integral test”) and replacing the \mathrm{d}x by the density \frac{2}{x} \mathrm{d}x, we get the estimate

\displaystyle \text{Binary Kempner series} \approx \int_1^\infty \frac{2}{x^2} = 2.

Doing the same thing with the original Kempner series gives that the chances a n-digit number does not contain the digit nine to be

\displaystyle \left(\frac89\right)\left(\frac9{10}\right)^{n-1} \approx \left( \frac{9}{10}\right)^{n}

The length of the decimal expansion of a natural number x is basically 1 + \log x. So the density we are interested in becomes

\displaystyle \left( \frac{9}{10}\right)^{1+\log x} ~\mathrm{d}x

From this we can do an integral estimate

\displaystyle \text{Kempner series} \approx 0.9 \times \int_1^\infty \left( \frac{9}{100}\right)^{\log x} ~\mathrm{d}x

The integral can be computed using that

\displaystyle a^{\log b} = b^{\log a}

to get

\displaystyle 0.9 \times \int_1^{\infty} \left( \frac{9}{100}\right)^{\log x} ~\mathrm{d}X = 0.9\times \int_1^\infty x^{\log 9 - 2} ~\mathrm{d}x = \frac{0.9}{1 - \log 9} \approx 19.66

Notice that this estimate is much closer to the currently known value of roughly 22.92 than to the original upper bound of 80 computed by Kempner.

Kempner’s estimate is a heavy overestimate because he performed a summation replacing every n-digit long number that does not contain the digit 9 by 10^{n-1}; this number can be many times (up to 9) times smaller than the original number. Our estimate is low because among the n-digit long numbers, the numbers that do not contain the digit 9 are not evenly distributed: they tend to crowd in the front rather than in the back (in fact, we do not allow them to crowd in the back because none of the numbers that start with the digit 9 is admissible). So if in the original question we had asked for numbers that do not contain the digit 1, then our computation will give an overestimate instead since these numbers tend to crowd to the back.

Abusing JabRef to manage snipplets of TeX

I use JabRef as my reference manager. In this post, however, I will discuss how we can abuse it to do some other things.

The problem

Let’s start with a concrete example: I keep a “lab notebook”. It is where I document all my miscellaneous thoughts and computations that come up during my research. Some of those are immediately useful and are collected into papers for publication. Some of those are not, and I prefer to keep them for future reference. These computations range over many different subjects. Now and then, I want to share with a collaborator or a student some subset of these notes. So I want a way to quickly search (by keywords/abstract) for relevant notes, and that compile them into one large LaTeX document.

Another concrete example: I am starting to collect a bunch of examples and exercises in analysis for use in my various classes. Again, I want to have them organized for easy search and retrieval, especially to make into exercise sheets.

The JabRef solution

The “correct” way to do this is probably with a database (or a document store), with each document tagged with a list of keywords. But that requires a bit more programming than I want to worry about at the moment.

JabRef, as it turns out, is sort of a metadata database: by defining a customized entry type you can use the BibTeX syntax as a proxy for JSON-style data. So for my lab notebook example, I define a custom type lnbentry in JabRef with

  • Required fields: year, month, day, title, file
  • Optional fields: keywords, abstract

I store each lab notebook entry as an individual TeX file, whose file system address is stored in the file field. The remaining metadata fields’ contents are self-evident.

(Technical note: in my case I actually store the metadata in the TeX file and have a script to parse the TeX files and update the bib database accordingly.)

For generating output, we can use JabRef’s convenient export filter support. In the simplest case we can create a custom export layout with the main layout file containing the single line

\\input{\file}

with appropriate begin and end incantations to make the output a fully-formed TeX file. Then one can simply select the entries to be exported, click on “Export”, and generate the appropriate TeX file on the fly.

(Technical note: JabRef can also be run without a GUI. So one can use this to perform searches through the database on the command line.)

Simulating closed cosmic strings (or yet another adventure in Julia)

One of my current research interests is in the geometry of constant-mean-curvature time-like submanifolds of Minkowski space. A special case of this is the study of cosmic strings under the Nambu-Goto action. Mathematically the classical (as opposed to quantum) behavior of such strings are quite well understood, by combining the works of theoretical physicists since the 80s (especially those of Jens Hoppe and collaborators) together with recent mathematical developments (such as the work of Nguyen and Tian and that of Jerrard, Novaga, and Orlandi). To get a better handle on what is going on with these results, and especially to provide some visual aids, I’ve tried to produce some simulations of the dynamics of closed cosmic strings. (This was also an opportunity for me to practice code-writing in Julia and learn a bit about the best practices for performance and optimization in that language.)

The code

After some false starts, here are some reasonably stable code.

function MC_corr3!(position::Array{Float64,2}, prev_vel::Array{Float64,2}, next_vel::Array{Float64,2}, result::Array{Float64,2}, dt::Float64)
  # We will assume that data is stored in the format point(coord,number), so as a 3x1000 array or something. 
  num_points = size(position,2)
  num_dims = size(position,1)

  curr_vel = zeros(num_dims)
  curr_vs = zeros(num_dims)
  curr_ps = zeros(num_dims)
  curr_pss = zeros(num_dims)

  pred_vel = zeros(num_dims)
  agreement = true

  for col = 1:num_points  #Outer loop is column
    if col == 1
      prev_col = num_points
      next_col = 2
    elseif col == num_points
      prev_col = num_points - 1
      next_col = 1
    else
      prev_col = col -1
      next_col = col + 1
    end

    for row = 1:num_dims
      curr_vel[row] = (next_vel[row,col] + prev_vel[row,col])/2
      curr_vs[row] = (next_vel[row,next_col] + prev_vel[row,next_col] - next_vel[row,prev_col] - prev_vel[row,prev_col])/4
      curr_ps[row] = (position[row,next_col] - position[row,prev_col])/2
      curr_pss[row] = position[row,next_col] + position[row,prev_col] - 2*position[row,col]
    end

    beta = (1 + dot(curr_vel,curr_vel))^(1/2)
    sigma = dot(curr_ps,curr_ps)
    psvs = dot(curr_ps,curr_vs)
    bvvs = dot(curr_vs,curr_vel) / (beta^2)
    pssps = dot(curr_pss,curr_ps)

    for row in 1:num_dims
      result[row,col] = curr_pss[row] / (sigma * beta) - curr_ps[row] * pssps / (sigma^2 * beta) - curr_vel[row] * psvs / (sigma * beta) - curr_ps[row] * bvvs / (sigma * beta)
      pred_vel[row] = prev_vel[row,col] + dt * result[row,col]
    end

    agreement = agreement && isapprox(next_vel[:,col], pred_vel, rtol=sqrt(eps(Float64)))
  end

  return agreement
end


function find_next_vel!(position::Array{Float64,2}, prev_vel::Array{Float64,2}, next_vel::Array{Float64,2}, dt::Float64; max_tries::Int64=50)
  tries = 1
  result = zeros(next_vel)
  agreement = MC_corr3!(position,prev_vel,next_vel,result,dt)
  for j in 1:size(next_vel,2), i in 1:size(next_vel,1)
    next_vel[i,j] = prev_vel[i,j] + result[i,j]*dt
  end
  while !agreement && tries < max_tries
    agreement = MC_corr3!(position,prev_vel,next_vel,result,dt)
    for j in 1:size(next_vel,2), i in 1:size(next_vel,1)
      next_vel[i,j] = prev_vel[i,j] + result[i,j]*dt
    end
    tries +=1
  end
  return tries, agreement
end

This first file does the heavy lifting of solving the evolution equation. The scheme is a semi-implicit finite difference scheme. The function MC_Corr3 takes as input the current position, the previous velocity, and the next velocity, and computes the correct current acceleration. The function find_next_vel iterates MC_Corr3 until the computed acceleration agrees (up to numerical errors) with the input previous and next velocities.

Or, in notations:
MC_Corr3: ( x[t], v[t-1], v[t+1] ) --> Delta-v[t]
and find_next_vel iterates MC_Corr3 until
Delta-v[t] == (v[t+1] - v[t-1]) / 2

The code in this file is also where the performance matters the most, and I spent quite some time experimenting with different algorithms to find one with most reasonable speed.

function make_ellipse(a::Float64,b::Float64, n::Int64, extra_dims::Int64=1)  # a,b are relative lengths of x and y axes
  s = linspace(0,2π * (n-1)/n, n)
  if extra_dims == 0
    return vcat(transpose(a*cos(s)), transpose(b*sin(s)))
  elseif extra_dims > 0
    return vcat(transpose(a*cos(s)), transpose(b*sin(s)), zeros(extra_dims,n))
  else
    error("extra_dims must be non-negative")
  end
end

function perturb_data!(data::Array{Float64,2}, coeff::Vector{Float64}, num_modes::Int64) 
  # num_modes is the number of modes
  # coeff are the relative sizes of the perturbations

  numpts = size(data,2)

  for j in 2:num_modes
    rcoeff = rand(length(coeff),2)

    for pt in 1:numpts
      theta = 2j * π * pt / numpts
      for d in 1:length(coeff)
        data[d,pt] += ( (rcoeff[d,1] - 0.5) *  cos(theta) + (rcoeff[d,2] - 0.5) * sin(theta)) * coeff[d] / j^2
      end
    end
  end

  nothing
end

This file just sets up the initial data. Note that in principle the number of ambient spatial dimensions is arbitrary.

using Plots

pyplot(size=(1920,1080), reuse=true)

function plot_data2D(filename_prefix::ASCIIString, filename_offset::Int64, titlestring::ASCIIString, data::Array{Float64,2}, additional_data...)
  x_max = 1.5
  y_max = 1.5
  plot(transpose(data)[:,1], transpose(data)[:,2] , xlims=(-x_max,x_max), ylims=(-y_max,y_max), title=titlestring)
  
  if length(additional_data) > 0
    for i in 1:length(additional_data)
      plot!(transpose(additional_data[i][1,:]), transpose(additional_data[i][2,:]))
    end
  end
  
  png(filename_prefix*dec(filename_offset,5)*".png")
  nothing
end

function plot_data3D(filename_prefix::ASCIIString, filename_offset::Int64, titlestring::ASCIIString, data::Array{Float64,2}, additional_data...)
  x_max = 1.5
  y_max = 1.5
  z_max = 0.9
  tdata = transpose(data)
  plot(tdata[:,1], tdata[:,2],tdata[:,3], xlims=(-x_max,x_max), ylims=(-y_max,y_max),zlims=(-z_max,z_max), title=titlestring)

  if length(additional_data) > 0
    for i in 1:length(additional_data)
      tdata = transpose(additional_data[i])
      plot!(tdata[:,1], tdata[:,2], tdata[:,3]) 
    end
  end

  png(filename_prefix*dec(filename_offset,5)*".png")
  nothing
end

This file provides some wrapper commands for generating the plots.

include("InitialData3.jl")
include("MeanCurvature3.jl")
include("GraphCode3.jl")

num_pts = 3000
default_timestep = 0.01 / num_pts
max_time = 3
plot_every_ts = 1500

my_data = make_ellipse(1.0,1.0,num_pts,0)
perturb_data!(my_data, [1.0,1.0], 15)
this_vel = zeros(my_data)
next_vel = zeros(my_data)

for t = 0:floor(Int64,max_time / default_timestep)
  num_tries, agreement = find_next_vel!(my_data, this_vel,next_vel,default_timestep)

  if !agreement
    warn("Time $(t*default_timestep): failed to converge when finding next_vel.")
    warn("Dumping information:")
    max_beta = 1.0
    max_col = 1
    for col in 1:size(my_data,2)
      beta = (1 + dot(next_vel[:,col], next_vel[:,col]))^(1/2)
      if beta > max_beta
        max_beta = beta
	max_col = col
      end
    end
    warn("   Beta attains maximum at position $max_col")
    warn("   Beta = $max_beta")
    warn("   Position = ", my_data[:,max_col])
    prevcol = max_col - 1
    nextcol = max_col + 1
    if max_col == 1
      prevcol = size(my_data,2)
    elseif max_col == size(my_data,2)
      nextcol = 1
    end
    warn("   Deltas")
    warn("    Left:  ", my_data[:,max_col] - my_data[:,prevcol])
    warn("    Right:  ", my_data[:,nextcol] - my_data[:,max_col])
    warn("   Previous velocity: ", this_vel[:,max_col])
    warn("   Putative next velocity: ", next_vel[:,max_col])
    warn("Quitting...")
    break
  end

  for col in 1:size(my_data,2)
    beta = (1 + dot(next_vel[:,col], next_vel[:,col]))^(1/2)
    for row in 1:size(my_data,1)
      my_data[row,col] += next_vel[row,col] * default_timestep / beta
      this_vel[row,col] = next_vel[row,col]
    end

    if beta > 1e7
      warn("time: ", t * default_timestep)
      warn("Almost null... beta = ", beta)
      warn("current position = ", my_data[:,col])
      warn("current Deltas")
      prevcol = col - 1
      nextcol = col + 1
      if col == 1
        prevcol = size(my_data,2)
      elseif col == size(my_data,2)
        nextcol = 1
      end
      warn(" Left: ", my_data[:,col] - my_data[:,prevcol])
      warn(" Right: ", my_data[:,nextcol] - my_data[:,col])
    end
  end

  if t % plot_every_ts ==0
    plot_data2D("3Dtest", div(t,plot_every_ts), @sprintf("elapsed: %0.4f",t*default_timestep), my_data, make_ellipse(cos(t*default_timestep), cos(t*default_timestep),100,0))
    info("Frame $(t/plot_every_ts):  used $num_tries tries.")
  end
end

And finally the main file. Mostly it just ties the other files together to produce the plots using the simulation code; there are some diagnostics included for me to keep an eye on the output.

The results

First thing to do is to run a sanity check against explicit solutions. In rotational symmetry, the solution to the cosmic string equations can be found analytically. As you can see below the simulation closely replicates the explicit solution in this case.

The video ends when the simulation stopped. The simulation stopped because a singularity has formed; in this video the singularity can be seen as the collapse of the string to a single point.

Next we can play around with a more complicated initial configuration.

In this video the blue curve is the closed cosmic string, which starts out as a random perturbation of the circle with zero initial speed. The string contracts with acceleration determined by the Nambu-Goto action. The simulation ends when a singularity has formed. It is perhaps a bit hard to see directly where the singularity happened. The diagnostic messages, however, help in this regard. From it we know that the onset of singularity can be seen in the final frame:

3Dtest00268

The highlighted region is getting quite pointy. In fact, that is accompanied with the “corner” picking up infinite acceleration (in other words, experiencing an infinite force). The mathematical singularity corresponds to something unreasonable happening in the physics.

To make it easier to see the “speed” at which the curve is moving, the following videos show the string along with its “trail”. This first one again shows how a singularity can happen as the curve gets gradually more bent, eventually forming a corner.

This next one does a good job emphasizing the “wave” nature of the motion.

The closed cosmic strings behave like a elastic band. The string, overall, wants to contract to a point. Small undulations along the string however are propagated like traveling waves. Both of these tendencies can be seen quite clearly in the above video. That the numerical solver can solve “past” the singular point is a happy accident; while theoretically the solutions can in fact be analytically continued past the singular points, the renormalization process involved in this continuation is numerically unstable and we shouldn’t be able to see it on the computer most of the time.

The next video also emphasizes the wave nature of the motion. In addition to the traveling waves, pay attention to the bottom left of the video. Initially the string is almost straight there. This total lack of curvature is a stationary configuration for the string, and so initially there is absolutely no acceleration of that segment of the string. The curvature from the left and right of that segment slowly intrudes on the quiescent piece until the whole thing starts moving.

The last video for this post is a simulation when the ambient space is 3 dimensional. The motion of the string, as you can see, becomes somewhat more complicated. When the ambient space is 2 dimensional a point either accelerates or decelerates based on the local (signed) curvature of the string. But when the ambient space is 3 dimensional, the curvature is now a vector and this additional degree of freedom introduces complications into the behavior. For example, when the ambient space is 2 dimensional it is known that all closed cosmic strings become singular in finite time. But in 3 dimensions there are many closed cosmic strings that vibrate in place without every becoming singular. The video below is one that does however become singular. In addition to a fading trail to help visualize the speed of the curve, this plot also includes the shadows: projections of the curve onto the three coordinate planes.

Adventures in Julia

Recently I have been playing around with the Julia programming language as a way to run some cheap simulations for some geometric analysis stuff that I am working on. So far the experience has been awesome.

A few random things …

Juno

Julia has a decent IDE in JunoLab, which is built on top of Atom. In terms of functionality it captures most of the sort of things I used to use with Spyder for python, so is very convenient.

Jupyter

Julia interfaces with Jupyter notebooks through the IJulia kernel. I am a fan of Jupyter (I will be using it with the MATLAB kernel for a class I am teaching this fall).

Plots.jl

For plotting, right now one of the most convenience ways is through Plots.jl, which is a plotting front-end the bridges between your code and various different backends that can be almost swapped in and out on the fly. The actual plotting is powered by things like matplotlib or plotlyJS, but for the most part you can ignore the backend. This drastically simplifies the production of visualizations. (At least compared to what I remembered for my previous simulations in python.)

Automatic Differentiation

I just learned very recently about automatic differentiation. At a cost in running time for my scripts, it can very much simplify the coding of the scripts. For example, we can have a black-box root finder using Newton iteration that does not require pre-computing the Jacobian by hand:

module NewtonIteration
using ForwardDiff

export RootFind

function RootFind(f, init_guess::Vector, accuracy::Float64, cache::ForwardDiffCache; method="Newton", max_iters=100, chunk_size=0)
  ### Takes input function f(x::Vector) → y::Vector of the same dimension and an initial guess init_guess. Apply Newton iteration to find solution of f(x) = 0. Stop when accuracy is better than prescribed, or when max_iters is reached, at which point a warning is raised.
  ### Setting chunk_size=0 deactivates chunking. But for large dimensional functions, chunk_size=5 or 10 improves performance drastically. Note that chunk_size must evenly divide the dimension of the input vector.
  ### Available methods are Newton or Chord

  # First check if we are already within the accuracy bounds
  error_term = f(init_guess)
  if norm(error_term) < accuracy
    info("Initial guess accurate.")
    return init_guess
  end

  # Different solution methods
  i = 1
  current_guess = init_guess
  if method=="Chord"
    df = jacobian(f,current_guess,chunk_size=chunk_size)
    while norm(error_term) >= accuracy && i <= max_iters
      current_guess -= df \ error_term
      error_term = f(current_guess)
      i += 1
    end
  elseif method=="Newton"
    jake = jacobian(f, ForwardDiff.AllResults, chunk_size=chunk_size, cache=cache)
    df, lower_order = jake(init_guess)
    while norm(value(lower_order)) >= accuracy && i <= max_iters
      current_guess -= df \ value(lower_order)
      df, lower_order = jake(current_guess)
      i += 1
    end
    error_term = value(lower_order)
  else
    warn("Unknown method: ", method, ", returning initial guess.")
    return init_guess
  end

  # Check if converged
  if norm(error_term) >= accuracy
    warn("Did not converge, check initial guess or try increasing max_iters (currently: ", max_iters, ").")
  end
  info("Used ", i, " iterations; remaining error=", norm(error_term))
  return current_guess
end

end

This can then be wrapped in finite difference code for solving nonlinear PDEs!

Riemann-, Generalized-Riemann-, and Darboux-Stieltjes integrals

(The following is somewhat rough and may have typos.)

Let us begin by setting the notations and recalling what happens without the Stieltjes part.

Defn (Partition)
Let I be a closed interval. A partition P is a finite collection of closed subintervals \{I_\alpha\} such that

  1. P is finite;
  2. P covers I, i.e. \cup P = I;
  3. P is pairwise almost disjoint, i.e. for I_\alpha, I_\beta distinct elements of P, their intersection contains at most one point.

We write \mathscr{P} for the set of all partitions of I.

Defn (Refinement)
Fix I a closed interval, and P, Q two partitions. We say that P refines Q or that P \preceq Q if for every I_\alpha\in P there exists J_\beta \in Q such that I_\alpha \subseteq J_\beta.

Defn (Selection)
Given I a closed interval and P a partition, a selection \sigma: P \to I is a mapping that satisfies \sigma(I_\alpha) \in I_\alpha.

Defn (Size)
Given I a closed interval and P a partition, the size of P is defined as |P| = \sup_{I_\alpha \in P} |I_\alpha|, where |I_\alpha| is the length of the closed interval I_\alpha.

Remark In the above we have defined two different preorders on the set \mathscr{P} of all partitions. One is induced by the size: we say that P \leq Q if |P| \leq |Q|. The other is given by the refinement P\preceq Q. Note that neither are partial orders. (But that the preorder given by refinement can be made into a partial order if we disallow zero-length degenerate closed intervals.) Note also that if P\preceq Q we must have P \leq Q.

Now we can define the notions of integrability.

Defn (Integrability)
Let I be a closed, bounded interval and f:I \to \mathbb{R} be a bounded function. We say that f is integrable with integral s in the sense of

  • Riemann if for every \epsilon > 0 there exists P_0\in \mathcal{P} such that for every P \leq P_0 and every selection \sigma:P \to I we have
    \displaystyle \left| \sum_{I' \in P} f(\sigma(I')) |I'| - s \right| < \epsilon

  • Generalised-Riemann if for every \epsilon > 0 there exists P_0 \in \mathcal{P} such that for every P \preceq P_0 and every selection \sigma: P\to I we have
    \displaystyle \left| \sum_{I' \in P} f(\sigma(I')) |I'| - s \right| < \epsilon

  • Darboux if
    \displaystyle \inf_{P\in\mathscr{P}} \sum_{I' \in P} (\sup_{I'} f )|I'| = \sup_{P\in\mathscr{P}} \sum_{I' \in P} (\inf_{I'} f )|I'| = s

From the definition it is clear that “Riemann integrable” implies “Generalised-Riemann integrable”. Furthermore, we have clearly that for a fixed P
\displaystyle \sum_{I' \in P} (\inf_{I'} f) |I'| \leq \sum_{I' \in P} f(\sigma(I')) |I'| \leq \sum_{I' \in P} (\sup_{I'} f) |I'|
and that if P \preceq Q we have
\displaystyle \sum_{I' \in Q} (\inf_{I'} f) |I'| \leq \sum_{I' \in P} (\inf_{I'} f) |I'| \leq \sum_{I' \in P} (\sup_{I'} f) |I'| \leq \sum_{I' \in Q} (\inf_{I'} f) |I'|
so “Darboux integrable” also implies “Generalised-Riemann integrable”. A little bit more work shows that “Generalised-Riemann integrable” also implies “Darboux integrable” (if the suprema and infima are obtained on the intervals I', this would follow immediately; using the boundedness of the intervals we can find \sigma such that the Riemann sum approximates the upper or lower Darboux sums arbitrarily well.

The interesting part is the following
Theorem
Darboux integrable functions are Riemann integrable. Thus all three notions are equivalent.

Proof. Let P, Q be partitions. Let |P| \leq \inf_{I'\in Q, |I'| \neq 0} |I'|, and let m be the number of non-degenerate subintervals in Q. We have the following estimate
\displaystyle   \sum_{I'\in Q} (\inf_{I'} f) |I'| - (m-1) |P| (\sup_I 2|f|) \leq \sum_{J'\in P} f(\sigma(J')) |J'| \leq \sum_{I'\in Q} (\sup_{I'} f) |I'| + (m-1) |P| (\sup_I 2|f|)
The estimate follows by noting that “most” of the J'\in P will be proper subsets of I'\in Q, and there can be at most m-1 of the J' that straddles between two different non-degenerate sub-intervals of Q. To prove the theorem it suffices to choose first a Q such that the upper and lower Darboux sums well-approximates the integral. Then we can conclude for all P with |P| sufficiently small the Riemann sum is almost controlled by the Q-Darboux sums. Q.E.D.

Now that we have recalled the case of the usual integrability. Let us consider the case of the Stieltjes integrals: instead of integrating against \mathrm{d}x, we integrate against \mathrm{d}\rho, where \rho is roughly speaking a “cumulative distribution function”: we assume that \rho:I \to \mathbb{R} is a bounded monotonically increasing function.

The definition of the integrals are largely the same, except that at every step we replace the width of the interval |I'| by the diameter of \rho(I'), i.e. \sup_{I'} \rho - \inf_{I'} \rho. The arguments above immediately also imply that

  • “Riemann-Stieltjes integrable” implies “Generalised-Riemann-Stieltjes integrable”
  • “Darboux-Stieltjes integrable” implies “Generalised-Riemann-Stieltjes integrable”
  • “Generalised-Riemann-Stieltjes integrable” implies “Darboux-Stientjes integrable”

However, Darboux-Stieltjes integrable functions need not be Riemann-Stieltjes integrable. The possibility of failure can be seen in the proof of the theorem above, where we used the fact that |P| is allow to be made arbitrarily small. The same estimate, in the case of the Stieltjes version of the integrals, has |P| replaced by \sup_{J'\in P} (\sup_{J'} \rho - \inf_{J'} \rho), which for arbitrary partitions need to shrink to zero. To have a concrete illustration, we give the following:

Example
Let I = [0,1]. Let \rho(x) = 0 if x < \frac12 and 1 otherwise. Let f(x) = 0 if x \leq \frac12 and 1 otherwise. Let Q_0 be the partition \{ [0,\frac12], [\frac12,1]\}. We have that
\displaystyle \sum_{I'\in Q_0} (\sup_{I'} f) (\sup_{I'} \rho - \inf_{I'} \rho) = 0 \cdot (1 - 0) + 1\cdot (1 - 1) = 0
while
\displaystyle \sum_{I'\in Q_0} (\inf_{I'} f) (\sup_{I'} \rho - \inf_{I'} \rho) = 0 \cdot (1-0) + 0 \cdot(1-1) = 0
so we have that in particular the pair (f,\rho) is Darboux-Stieltjes integrable with integral 0. However, let k be any odd integer, consider the partition P_k of [0,1] into k equal portions. Depending on the choice of the selection \sigma, we see that the sum can take the values
\displaystyle \sum_{I'\in P_k} f(\sigma(I')) (\sup_{I'} \rho - \inf_{I'}\rho) = f(\sigma([\frac12 - \frac1{2k},\frac12 + \frac1{2k}])) (1 - 0) \in \{0,1\}
which shows that the Riemann-Stieltjes condition can never be satisfied.

The example above where both f and \rho are discontinuous at the same point is essentially sharp. A easy modification of the previous theorem shows that
Prop
If at least one of f,\rho is continuous, then Darboux-Stieltjes integrability is equivalent to Riemann-Stieltjes integrability.

Remark The nonexistence of Riemann-Stieltjes integral when f and g has shared discontinuity points is similar in spirit to the idea in distribution theory where whether the product of two distributions is well-defined (as a distribution) depends on their wave-front sets.

Bouncing a quantum particle back and forth

If you have not seen my previous two posts, you should read them first.

In the two previous posts, I shot particles (okay, simulated the shooting on a computer) at a single potential barrier and looked at what happens. What happens when we have more than one barrier? In the classical case the picture is easy to understand: a particle with insufficient energy to escape will be trapped in the local potential well for ever, while a particle with sufficiently high energy will gain freedom and never come back. But what happens in the quantum case?

If the intuition we developed from scattering a quantum particle against a potential barrier, where we see that depending on the frequency (energy) of the particle, some portion gets transmitted and some portion gets reflected, is indeed correct, what we may expect to see is that the quantum particle bounces between the two barriers, each time losing some amplitude due to tunneling.

But we also saw that the higher frequency components of the quantum particle have higher transmission amplitudes. So we may expect that the high frequency components to decay more rapidly than the low frequency ones, so the frequency of the “left over” parts will continue to decay in time. This however, would be wrong, because we would be overlooking one simple fact: by the uncertainty principle again, very low frequency waves cannot be confined to a small physical region. So when we are faced with two potential barriers, the distance between them gives a characteristic frequency. Below this frequency (energy) it is actually not possible to fit a (half) wave between the barriers, and so the low frequency waves must have significant physical extent beyond the barriers, which means that large portions of these low frequency waves will just radiate away freely. Much above the characteristic frequency, however, the waves have large transmission coefficients and will not be confined.

So the net result is that we should expect for each double barrier a characteristic frequency at which the wave can remain “mostly” stuck between the two barriers, losing a little bit of amplitude at each bounce. This will look like a slowly, but exponentially, decaying standing wave. And I have some videos to show for that!

In the video we start with the same random initial data and evolve it under the linear wave equation with different potentials: the equations look like

\displaystyle - \partial^2_{tt} u + \partial^2_{xx} u - V u = 0

where V is a non-negative potential taken in the form

\displaystyle V(x) = a_1 \exp( - x^2 / b_1) - a_2 \exp( -x^2 / b_2)

which is a difference of two Gaussians. For the five waves shown the values of a_1, b_1 are the same throughout. The coefficients a_2 (taken to be \leq a_1) and b_2 (taken to be < b_1) increases from top to bottom, resulting in more and more-widely separated double barriers. Qualitatively we see, as we expected,

  • The shallower and narrower the dip the faster the solution decays.
  • The shallower and narrower the dip the higher the “characteristic frequency”.

As an aside: the video shown above is generated using Python, in particular NumPy and MatPlotLib; the code took significantly longer to run (20+hours) than to write (not counting the HPDE solver I wrote before for a different project, coding and debugging this simulation took about 3 hours or less). On the other hand, this only uses one core of my quad-core machine, and leaves the computer responsive in the mean time for other things. Compare that to Auto-QCM: the last time I ran it to grade a stack of 400+ multiple choice exams it locked up all four cores of my desktop computer for almost an entire day.

As a further aside, this post is related somewhat to my MathOverflow question to which I have not received a satisfactory answer.

… and scattering a quantum particle

In the previous post we shot a classical particle at a potential barrier. In this post we shoot a quantum particle.

Whereas the behaviour of the classical particle is governed by Newton’s laws (where the external force providing the acceleration is given as minus the gradient of the potential), we allow our quantum particle to be governed by the Klein-Gordon equations.

  • Mathematically, the Klein-Gordon equation is a partial differential equation, whereas Newton’s laws form ordinary differential equations. A typical physical interpretation is that the state space of quantum particles are infinite dimensional, whereas the phase space of physics has finite dimensions.
  • Note that physically the Klein-Gordon equation was designed to model a relativistic particle, while in the previous post we used the non-relativistic Newton’s laws. In some ways it would’ve been better to model the quantum particle using Schroedinger’s equation. I plead here however that (a) qualitatively there is not a big difference in terms of the simulated outcomes and (b) it is more convenient for me to use the Klein-Gordon model as I already have a finite-difference solver for hyperbolic PDEs coded in Python on my computer.

To model a particle, we set the initial data to be a moving wave packet, such that at the initial time the solution is strongly localized and satisfies \partial_t u + \partial_x u = 0. Absent the mass and potential energy terms in the Klein-Gordon equation (so under the evolution of the free wave equation), this wave packet will stay coherent and just translate to the right as time goes along. The addition of the mass term causes some small dispersion, but the mass is chosen small so that this is not a large effect. The main change to the evolution is the potential barrier, which you can see illustrated in the simulation.

The video shows 8 runs of the simulation with different initial data. Whereas in the classical picture the initial kinetic energy is captured by the initial speed at which the particle is moving, in the quantum/wave picture the kinetic energy is related to the central frequency of your wave packet. So each of the 8 runs have increasing frequency offset that represents increasing kinetic energy. The simulation has two plots, the top shows the square of the solution itself, which gives a good indication of where physically the wave packet is located. The bottom shows a normalized kinetic energy density (I have to include a normalization since the kinetic energy of the first and last particles differ roughly 10 fold).

One notices that in the first two runs, the kinetic energy is sufficiently small that the particle mostly bounces back to the left after hitting the potential.

For the third and fourth runs (frequency shift \sqrt{2} and \sqrt{3} respectively) we see that while a significant portion of the particle bounces back, a noticeable portion “tunnels through” the barrier: this caused by a combination of the quantum tunneling phenomenon and the wave packet form of the initial data.

The phenomenon of quantum tunneling manifests in that all non-zero energy waves will penetrate a finite potential barrier a little bit. But the amount of penetration decays to zero as the energy of the wave goes to zero: this is known as the semiclassical regime. In the semiclassical limit it is known that quantum mechanics converge toward classical mechanics, and so in the low-energy limit we expect our particle to behave like a classical particle and bounce off. So we see that naturally increasing the energy (frequency) of our wave packet we expect more of the tunneling to happen.

Further, observe that by shaping our data into a wave packet it necessarily contains some high frequency components (due to Heisenberg uncertainty principle); high frequency, and hence high energy components do not “see” the potential barrier. Even in the classical picture high energy particles would fly over the potential barrier. So for wave packets there will always be some (perhaps not noticeable due to the resolution of our computation) leakage of energy through the potential barrier. The quantum effect on these high energy waves is that they back-scatter. Whereas the classical high energy particles just fly directly over the barrier, a high energy quantum particle will leave some parts of itself behind the barrier always. We see this in the sixth and seventh runs of the simulation, where the particle mostly passes through the barrier, but a noticeable amount bounces off in the opposite direction.

In between during the fifth run, where the frequency shift is 2, we see that the barrier basically split the particle in two and send one half flying to the right and the other half flying to the left. Classically this is the turning point between particles that go over the bump and particles that bounces back, and would be the case (hard to show numerically!) where a classical particle comes in from afar with just enough energy that it comes to a half at the top of the potential barrier!

And further increasing the energy after the seventh run, we see in the final run a situation where only a negligible amount of the particle scatters backward with almost all of it passing through the barrier unchanged. One interesting thing to note however is that just like the case of the classical particle, the wave packet appears to “slow down” a tiny bit as it goes over the potential barrier.

Shooting a classical particle…

Here’s a small animation of what happens when you try to shoot a classical particle when there’s a potential barrier. For small initial kinetic energies, the particle bounces back. For large initial kinetic energies, the particle goes over the hump, first decelerating and then accelerating in the process.

(It may be best to watch this full screen with HD if the network supports it.)

(The NumPy code is pretty simple to write for this; and it runs relatively fast. The one for my next post is a bit more complicated and takes rather much longer to run. Stay tuned!)

Using Prezi

I was introduced to the use of Prezi for doing presentations (I know, I am slightly behind the times) from a MOOC I am taking, and so for my recent trip to Cambridge I decided to give it a try. The audience seemed to like it, and I felt that in this particular instance, the use of Prezi vastly improved my talk. On the other hand, I don’t think in the future I will do every one of my presentations as Prezis: that presentation style is suitable for some topics but traditional (be it slides or board talks) methods do shine for others. Read the rest of this entry »

Bessaga’s converse to the contraction mapping theorem

In preparing some lecture notes for the implicit function theorem, I took a look at Schechter’s delightfully comprehensive Handbook of Analysis and its Foundations (which you can also find on his website), and I learned something new about the Banach fixed point theorem. To quote Schechter:

… although Banach’s theorem is quite easy to prove, a longer proof cannot yield stronger results.

I will write a little bit here about a “converse” to the Banach theorem due to Bessaga, which uses a little bit of help from the Axiom of Choice.

Read the rest of this entry »