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

### 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)

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)

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: 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! ### LaTeX runtime for NeoVim I’ve just recently migrated to using NeoVim instead of traditional Vim. One of the nice features in NeoVim (or nvim) is that it now supports asynchronous job dispatch. This makes it a bit nicer to call external previewers for instance (otherwise the previewer may block the editing). So here are the latest LaTeX runtime code that I use, modified for NeoVim. function Dvipreview() let dviviewjob = jobstart(['xdvi', '-sourceposition', line(".")." ".expand("%"), expand("%:r") . ".dvi"]) endfunction function PDFpreview() let pdfviewjob = jobstart(['evince', expand("%:r") . ".pdf"]) endfunction au BufRead *.tex call LaTeXStartup() function LaTeXStartup() set dictionary+=~/.config/nvim/custom/latextmp/labelsdictionary set iskeyword=@,48-57,_,: call SimpleTexFold() set completefunc=CompleteBib set completeopt=menuone,preview runtime custom/latextmp/bibdictionary call SetShortCuts() endfunction function SimpleTexFold() exe "normal mz" 1 set foldmethod=manual if search('\\begin{document}','nW') 1,/\\begin{document}/-1fold if search('\\section','nW') /\\section/1 endif while search('\\section','nW') .,/\\section/-1fold /\\section/1 endwhile .,$fold
endif
if search('\\begin{entry}','nW')
/\\begin{entry}/1
while search('\\begin{entry}','nW')
.,/\\begin{entry}/-1fold
/\\begin{entry}/1
endwhile
.,$fold endif exe "normal gzzv" endfunction function SetShortCuts() " Map <F2> to save and compile imap <F2> ^[:w^M:! latex -src-specials % >/dev/null^M^Mi " Map S-<F2> to save and compile as PDF " apparently <S-F2> sends the same keycode as <F12>? imap <F12> ^[:w^M:! pdflatex % >/dev/null^M^Mi " Map <F3> to Dvipreview() imap <F3> ^[:call Dvipreview()^M " Map S-<F3> to PDFpreview() " apparently <S-F3> = <F13> imap <F13> ^[:call PDFpreview()^M " Map <F4> to bibtex imap <F4> ^[:! bibtex "%:r" >/dev/null^M^Mi " Map <F5> to change the previous word into a latex \begin .. \end environment imap <F5> ^[diwi\begin{^[pi<Right>}^M^M\end{^[pi<Right>}<Up> " Map <F6> to 'escape the current \begin .. \end environment imap <F6> ^[/\\end{.*}/e^Mi<Right> " Map <F7> to search the labels dictionary for matching labels imap <F7> ^[diwi\ref{^[pi<Right>^X^K " Map <F8> to rebuild the labels dictionary imap <F8> ^[:w^M:! ~/.config/nvim/custom/latexreadlabels.sh %^M^Mi " Map <F9> to search using the bibs dictionary imap <F9> ^[diwi\cite{^[pi<Right>^X^U imap <S-Tab>C ^[diwi\mathcal{^[pi<Right>} imap <S-Tab>B ^[diwi\mathbb{^[pi<Right>} imap <S-Tab>F ^[diwi\mathfrak{^[pi<Right>} imap <S-Tab>R ^[diwi\mathrm{^[pi<Right>} imap <S-Tab>O ^[diwi\mathop{^[pi<Right>} imap <S-Tab>= ^[diWi\bar{^[pi<Right>} imap <S-Tab>. ^[diWi\dot{^[pi<Right>} imap <S-Tab>" ^[diWi\ddot{^[pi<Right>} imap <S-Tab>- ^[diWi\overline{^[pi<Right>} imap <S-Tab>^ ^[diWi\widehat{^[pi<Right>} imap <S-Tab>~ ^[diWi\widetilde{^[pi<Right>} imap <S-Tab>_ ^[diWi\underline{^[pi<Right>} endfunction  Pay attention that the control characters did not copy-paste entirely correctly in the SetShortCuts() routine. Those need to be replaced by the actual control-X sequences. The read labels shell script is simply #!/bin/sh grep '\label{'$1 | sed -r 's/.*\\label\{([^}]*)\}.*/\1/' > ~/.config/nvim/custom/latextmp/labelsdictionary


(I probably should observe the proper directory structure and dump the dictionary into ~/.local/share/ instead.)

### Gingko and proof-writing

I have recently been reminded of Lamport’s How to Write a 21st Century Proof in more than one way.

Teaching

Last semester I taught two classes. One is a “Intro to Proofs” class, and another is (supposed to be) an advanced undergraduate real analysis course. Upon reflection both of the classes could have benefitted from some inclusion of more structured proofs.

For the “Intro to Proofs” class, this is the belated recognition that “how mathematicians write and read proofs” is not always the same as “how mathematicians think about proofs”. There’s much that can be (and has been) written about this, but the short of the matter is that despite of our pretenses, mathematicians typically don’t write proofs completely rigorously. And we read proofs we don’t often check every detail, but choosing instead to absorb the “big picture”. As such, mathematical proofs that we see presented in graduate level textbooks and in journal articles are frequently really merely “sketches”: there are gaps to be filled by the reader.

What is often neglected in teaching students to read and write proofs is that these proofs or their sketches are backed up, usually, by a concrete and rigorous understanding of the subject. And that the distillation from a complete proof to what is presented on a piece of paper as the sketch is a bit of an art. In some respects a flipped classroom, especially in IBL style, is perfect for this. The students start by presenting proofs in great details, and as they collectively grow more and more may be omitted.

However, what I found disconcerting is that in a regular education, there can be fourth year undergraduate students studying for a degree in mathematics that still have not internalized this difference and are unable to successfully read and write mathematics.

This is where the involvement of more structured proof writing can become useful. Similar to how study of literature involves diagramming sentences, here we diagram proofs. A proof can be written in various levels of details. In a structured presentation these levels can be made explicit: a proof is decomposed into individual landmark statements which taken together will yield the desired result. Each statement will need justification, and the proofs also can recursively be sketched. The final writing of the proof is to prune this tree of ideas by removing the “trivial” justifications and keeping the important ones.

Instructors can demonstrate this in action by preparing detailed proofs of classical theorems in this format. I’ve found that rewriting theorems in this format forces me to re-examine assumptions and conclusions, and overall be more succinct when trying to come-up with a high level sketch. Furthermore, if lecture notes are presented in this format students will also benefit from being able to study the proofs by first obtaining a bird’s eye view of the process and then diving in various levels of detail to the nitty-gritty of the arguments.

I will try to implement this in a future undergraduate math major class and see what happens.

Research

Mathematics papers are getting longer. Especially in my field of hyperbolic PDEs. It is getting harder and harder, when reading a paper, to keep in one’s head a coherent picture of the overall argument. This is a problem that I think can be beautifully solved with more structured approach to presenting arguments.

I am not advocating re-writing proofs as pedantic as Lamport advocates. I am not even advocating the strict presentation. What I do like about the idea of structured proofs is the two-dimensionality of the presentation. In this I am also a bit influenced by Terry Tao’s “circuit-diagram” approach to diagramming proofs that he used in, among other things, his Nonlinear Dispersive Equations book and his recent Averaged Navier Stokes paper.

What I have in mind is the presentation of proofs as nested sketches, but each level written more-or-less in natural language as is currently. Each step of the proof is justified by its own “proof”. The proof can be read at different levels of details, and readers can choose to zoom in and study a portion of the proof when interested. Assumptions and conclusions of individual steps should be made clear; the tree structure of the presentation can help prevent circular arguments. (Some aspects of this is already present in modern mathematical papers: important intermediate arguments are often extracted in the form of lemmata and propositions. This proposal just makes everything more organized.)

This also can improve the refereeing experience. A paper can be rejected if an individual step can be shown to be false. Additional clarifications can be inserted if the referee feels that the paper does not go deep enough in the chain of justifications.

Technological support

This presentation of ideas does not require non-traditional media. But this presentation of ideas can be improved by non-traditional media. I’ve just run into a Web App that does some approximation of what I envisioned: Gingko. It is free to use if your usage isn’t heavy.

What I would love would be for cards to also support

• Cross referencing; currently it supports hashtags, but not referencing with a definite target card.
• Duplicating cards; currently it supports moving, but not duplicating.
• Multiple ancestry: it would be great if the same card can appear as the child of two different cards. But this can also be emulated with cross referencing support or duplication support.

### So starts the svn to git migration…

For five years now I have been a happy user of svn to manage my research work, and I probably would have remained so if it weren’t for my next job favoring git instead. So in the past few weeks I have been reading up on git and in the process discovering all sorts of things that I have been doing wrong, or at least sub-optimally. So here are just some notes on what I’ve just figured out (yay slow me!).

Each paper should be a repository

Previously I keep one single giant repository for all my research work. I’ve discovered that this is not the best idea for multiple reasons:

• Collaboration: one of the great things about version control systems is that it makes collaboration easier to manage. But your collaborators are not a static set and you probably don’t want them to peek at every one of your research ideas. The easiest way to share individual projects with only those who should be allowed to see and edit them is to have one repo for each paper. (I got away with what I did mostly because I failed to convince any of my collaborators to use a VCS beyond that built-in support in Dropbox.)
• Organisation: to keep track of papers I have them stored in subdirectories, some of which are “stuff I am working on” and some of which are “stuff that is finished from year X” and some of which are “stuff that is being refereed”. It is a bit silly that I have to do svn mv changes to “graduate” a project from one subdirectory to the next. By keeping each paper in its own (git) repository, the local directory representation of the storage is immaterial. And this makes more sense to me.
• (In)compatibility: here’s something that I changed my mind on. Previously I thought it a great idea to keep a single up-to-date bibtex file containing all the references that I can ever need, and a single up-to-date version of my custom LaTeX class and style files. The advantage of course is that I just need to issue one svn up to get the newest versions of everything. But the disadvantage is that when upgrading my class and style files, or when updating my bibtex files, I have to maintain backward compatibility. And when I do break the compatibility, it is then required that I keep a copy of the old versions of the files along with the LaTeX source that uses them, which, when you think about it, defeats the purpose of having a single up-to-date version in one repo completely.

So my new workflow, instead of one giant repository, is that I will create a repo for each paper/project. My LaTeX class and style files will be itself a separate Git repo, on which I can upgrade and develop to my hearts desire. When I start a new paper I will simply make a copy of the current version of the files (with git archive instead of git clone because I won’t need the previous versions, nor will I want to track the changes). This also allows me to set-up my “development environment” (via .gitattributes and .gitignore) quickly.

Keyword substitution is not necessary

The papers I keep in my svn repo I have been using the svn and svn-multi packages to add time-stamp and versioning information to the PDF files. Both of those packages rely on the “keyword substitution” capabilities of the svn system at commit time. Naturally when I wanted to start using git, I looked for a replacement. The obvious one is gitinfot2. One thing I don’t like is that unlike the keyword replacements, this package does not directly modified the source LaTeX file; instead it creates (via commit and checkout hooks) a supplementary file in the .git/ directory which it searches for and inserts when building the PDF file. This makes it a bit more of a hassle when uploading stuff to the arXiv, for example.

So I started reading up on how one can actually imitate keyword expansion using commit and checkout filters. And I went so far as to implement something for LaTeX. And then I read the discussion by the kernel devs on this issue, and Linus Torvalds’ comments left an impression on me. In short:

• When you are working on the code in a git repository, you don’t need this tagging since you can just “ask git”.
• Conversely, this sort of tagging is only needed when your code is ready to leave the repository (upload to arXiv or sent to non-git-using collaborators, for example).

So philosophically it is much less useful to have something that work on the working copy compared to something that works on an exported archive. And while git, by design, cannot and will not do keyword expansion on commits, it is perfectly happy to do keyword expansion when one exports the repo. Furthermore, since the export substitution can be essentially formatted arbitrarily, this moots the need for something like svn or svn-multi to parse the string generated by the RCS: we can make the string appear how we want to start with. The only hiccup is that before the substitution (i.e. when you are working in the working copy), the syntax for the export substitution is not exactly compatible with LaTeX, and requires a little mucking about with catcodes. But with that problem solved, and with the workflow now accounting for each paper as a separate repository, for arXiv uploads the easiest thing will actually be to simply issue git archive` and upload the resulting tarball.

### 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.