Center of mass on line of symmetry: questions from a little guy

August 27, 2026 math and physics play No comments , ,

[Click here for a PDF version of this post]

Motivation

My friend’s son is an exceptional little guy. I think he’s about 12 or so, but he’s already half way through our old 1st year calculus book. When they visited, he was intensely dissatisfied with an “obvious” statement that the center of mass necessarily lies on the axis of symmetry (when there is such a symmetry). I didn’t take the time when he was here to reason through that statement in a systematic way. It turns out that it’s pretty simple to do so.

A uniform mass with symmetry

Suppose that we have equal density thin plate, with surface density \( \rho \), with rotational symmetry about some axis, as illustrated in fig. 1.

fig. 1. An flat mass with rotational symmetry

In the plot example, I’ve plotted a positive function \( g(x) = \Abs{f(x)} \), and rotated it, plotting \( -g(a-x) \) below the axis, with the whole things bounded, in this case by \( x \in [-2, 5] \). We don’t actually need those bounds if the function is integrable everywhere (we just need a finite mass, i.e.: a finite area).

The problem of computing the center of mass coordinates is to find the values
\begin{equation}\label{eqn:centerOfMassRotationalSymmetry:20}
\begin{aligned}
\bar{x} M &= \int \sigma x dA \\
\bar{y} M &= \int \sigma y dA,
\end{aligned}
\end{equation}
where \( M = \int \sigma dA \). We are also only interested in \( \bar{y} \), since we want to show that the center of mass lies on the axis of symmetry. In symbols, with the surface mass density \( \sigma \) constant, that is
\begin{equation}\label{eqn:centerOfMassRotationalSymmetry:40}
\bar{y}
= \inv{A} \int y dx dy.
\end{equation}
Let’s ignore the bounds in the example, and perform the integral for all \( x \). If the area is bounded, then that integral will be zero outside of some region. That is
\begin{equation}\label{eqn:centerOfMassRotationalSymmetry:60}
\begin{aligned}
\bar{y}
&= \inv{A} \int_{x = -\infty}^\infty dx \int_{y = -g(a-x)}^{g(x)} y dy \\
&= \inv{A} \int_{x = -\infty}^\infty dx \inv{2} \evalrange{ y^2 }{-g(a-x)}{g(x)} \\
&= \inv{2 A} \int_{x = -\infty}^\infty dx \lr{ g^2(x) – \lr{ -g(a-x) }^2 } \\
&= \inv{2 A} \int_{x = -\infty}^\infty g^2(x) dx – \inv{2 A} \int_{x = -\infty}^\infty g^2(a-x) dx
\end{aligned}
\end{equation}
A change of variables \( u = a – x \) in the second integral gives
\begin{equation}\label{eqn:centerOfMassRotationalSymmetry:80}
\begin{aligned}
\inv{2 A} \int_{x = -\infty}^\infty g^2(a-x) dx
&=
\inv{2 A} \int_{u = \infty}^{-\infty} g^2(u) (-du) \\
&=
\inv{2 A} \int_{u = -\infty}^{\infty} g^2(u) du,
\end{aligned}
\end{equation}
which is exactly the first integral (other than a change of dummy variables.) That means
\begin{equation}\label{eqn:centerOfMassRotationalSymmetry:100}
\bar{y} = 0.
\end{equation}
This wasn’t the most general function, as we used a well behaved function (not a relation), that isn’t multivalued for any value of \( x \), but this argument would still work (requiring superposition arguments to generalize it.)

Customized MathJax now reinstalled on the blog.

August 27, 2026 Incoherent ramblings No comments ,

The way that I previously installed my customized MathJax was as a sub-directory of mathjax-latex/.  A side effect of this is that a purposeful or accidental upgrade of the mathjax-latex (wordpress) plugin, will obliterate that MathJax/ download (and default.js customization).

I’m using GoDaddy wordpress hosting (I probably should have used an amazon ec2 instance, which was more flexible and probably would have been cheaper). GoDaddy now offer ssh (not just sftp) access, which makes this reconfiguration much easier.  I couldn’t find a zip for the Mathjax distro anymore, but this worked:

cd html/wp-content
git clone https://github.com/mathjax/MathJax.git
cd MathJax
git checkout 2.7.9
cd config
vim default.js

You can see what my default.js looks like with:

curl https://peeterjoot.com/wp-content/MathJax/config/default.js

that has a big LaTeX Macros block, equivalent to a subset of my peeters_macros.sty

Installing the customized MathJax tree this way means that I need only set the mathjax-latex CDN path to:

https://peeterjoot.com/wp-content/MathJax/MathJax.js

Screwed up my blog’s LaTeX (mathjax) plugin.

August 27, 2026 Incoherent ramblings No comments

I’ve accidentally upgraded the mathjax plugin to the latest version, which wiped out the customizations I had.  This effectively means that all math content on this blog will not render properly until I figure out how to rebuild the customizations (custom CDN url, …) I put in place years ago.

Enterprise COBOL 6.5 TYPEDEF and “User Defined Functions”

June 28, 2026 COBOL , , , , , , , ,

I continually disparage and make fun of COBOL, a “language” deserving of all levels of mockery and condescension.  My last LinkedIn mockery of COBOL paragraphs, to my great amusement, resulted in a COBOL defender saying it’s not a programming language, but a report writing language.  That’s an even stronger case than I tried to make!  Even people that like the language cannot defend it.

I’d like to share some features that IBM has recently retrofitted into their Enterprise COBOL Language Reference that actually go a very long way into making COBOL into a real programming language.  I have to hand it to IBM for adding these features.  Hopefully, these will also make it into ISO COBOL, should there ever be a refresh of that standard.

I’m not saying that COBOL is no longer the worst programming language in the universe.  However, it could be improved quite a bit by two specific fairly innocuous seeming features, if used well:

  • New TYPEDEF/TYPE keywords, and qualified member access.
  • User Defined Functions (UDFs)

TYPEDEF/TYPE

Anybody who has used a real programming language takes it for granted that one can make type definitions.  COBOL has been around since 1959, and IBM’s Enterprise COBOL 6.5 was released in 2024 — it took about 65 years before a mainstream version of COBOL was available with a basic type mechanism (Microfocus and TypeCOBOL both did it earlier, but I’m sure that neither of these have much of the COBOL market compared to IBM’s compiler.)

Here’s an example of a declaration in COBOL, taken from the NIST suite

If you want a second instance of such a variable, you have to copy this and give every symbol a different name.  That’s such a common pattern in COBOL that the #include like statement (COPY) has a REPLACING keyword that can be used to change a boilerplate prefix or suffix into a name that is specific to the new use.  Here’s an example of a copybook fragment that is meant to be used with COPY REPLACING:

where the #include site may look like:

and the result, after preprocessing, would be:

When you have two instances of the same type in COBOL, you have no way of knowing if that’s the case.  There’s no real type information — instead, you have to know that the membership is all identical.  For large structures, this can be very unintuitive, to say the least.

Enterprise COBOL 6.5 provides the TYPEDEF mechanism taken from TypeCOBOL & MicroFocus (according to Claude.)  Instead of a COPYbook that has to be copied with REPLACING, to define a type, one can do so in a structured fashion.  Example:

And at the use point:

Unlike conventional COBOL where every instance of “the same type” has a different name and different member names, this new declaration mechanism gives the same name to each member.  You can do that in conventional COBOL, and write something like:

but this new COBOL TYPEDEF comes with a (double-colon) qualified member access syntax. Here’s an example:

Conventional COBOL qualified access would look like:

with verbosity that obscures any meaning, typical of most COBOL code.  In this specific case, since the COPY REPLACING guaranteed different names for all fields, the use case could have implied membership access:

If you go looking for L-IN1-RE or L-IN2-RE, you will never find it, because it exists only as a member of the copybook that has been processed with REPLACING.  This is a great example of a COBOL software development problem.  Even when you have the COBOL source code, understanding the code is a reverse engineering exercise.

This TYPE/TYPEDEF syntax is nice, but it’s really just syntactic sugar.  It doesn’t actually introduce any notion of strict typing to the language.

There are some minor exceptions to this lack of type safety.  For example gcobol will not permit ZERO to be assigned to USAGE POINTER type.  Instead you have to assign NULL.  However, for the most part you can still convert anything to anything, and most of the time you’ll never get any sort of warning or error for doing so.

Despite the weak nature of this type mechanism, I think that it’s actually a very important feature.  When you have a typical production COBOL program with 10000 lines of global variables (WORKING-STORAGE), you now have the capability of running an analysis that retroactively extracts the underlying implicit type representation, only known to the original programmers long since dead.  You can now assign names to types that are common to a given program, and even better, names to entities that are common to a software suite.  The act of reverse engineering COBOL program behaviour from the source code can then be made a little bit easier.

It only took 65 years to jam this little bit of sanity into the COBOL programming language.

User Defined Functions

The next little bit of sanity added to the language is a mechanism to define a function.  If you say, wait a sec, no programming language doesn’t have functions, are you sure that COBOL doesn’t have functions.

IBM’s Enterprise COBOL 6.4, released in 2022 (according to Claude), added user defined functions, so how does this differ from the other function like constructs available in COBOL?  A COBOL programmer may characterize the COBOL paragraph as a function.  You can call such a “function” using the PERFORM statement.  Here’s an example of a paragraph IN-4, called twice in the paragraph IN-3:

There’s a key observation to make here.  Notice how the caller does not pass any parameter to the callee.  That’s not just because this particular “function” takes no parameters, but it’s because that it NOT POSSIBLE to pass parameters.  COBOL paragraphs are only function like in that they can be called, but

  • they cannot pass parameters,
  • cannot return anything,
  • cannot have local variables,
  • and may or may not, depending on the whim of the author, implicitly fall through to whatever code happens to follow them.

That last point is probably why, in this case, that there is a GO TO.  Without that GO TO, the IN-4 paragraph may actually be executed twice. But you can’t actually know if that will be the case unless you know how you got to IN-3.  If IN-3 was “called” implicitly, due to IN-2 before it finishing and falling through, then that same loop of two could be performed by:

In that case, IN-4 will be executed once by IN-3, then IN-3 will fall through to IN-4 and it will execute a second time.  The tricky thing about this example is that you can’t look at this code and know how many times IN-4 will be called.  If IN-3 was called with PERFORM, then IN-4 will be called once, but if IN-3 was “called” by fall-through or GO TO, then IN-4 will be called twice.

Basically, the language,  and all programs written in it, are sufficient to make any programmer have a terrible, horrible, no good, very bad day.

Given that paragraphs are “functions” that cannot have parameters, nor return codes, then how is is justifiable to call them functions.  That justification is only possible because you can use global variables to simulate parameters and returns.  Here’s an example:

These are real and imaginary complex “functions”, each relying on you to copy values into specific “input” global variables, and get “return” values from specific output global variables.

What you will often see in production code is that programmers use naming conventions to simulate parameters, perhaps like this, for example:

A naming convention like PARAGRAPH-<something>-IN, and PARAGRAPH-<something>-OUT, can make it more obvious that the expected use case for the code is to MOVE things to the -IN variables before a PERFORM and to grab stuff out of the -OUT variables when it’s done.  You don’t have any guarantee that this is anything more than a naming convention, and any paragraph or section could potentially change the “parameter” and “return” variables of any other paragraph or section (a section is a collection of one or more paragraphs).

It is at least conceptually possible to take a well structured COBOL program, that uses a naming convention like this to model parameters, and translate it to a sane language — but you need an audit to make sure that the convention is actually followed before any sort of automatic translation can occur.  And even if a program follows a function like naming convention like this, that doesn’t mean that each one, doesn’t also have a thousand other side effects that have to be figured out.  With all variables usually accessed without qualification, it is very hard to look at COBOL code and have any sort of idea what data structures it is operating on.

Somebody that knows COBOL may ask.  What about nested subprograms and external program calls.  Aren’t those “function” like.  That I would agree with.  A subprogram or external program is very much like a void function that passes parameters all by reference.  When used in a structured way, this could model function in a modern programming language that has a set of inputs and a set of outputs, or input, outputs and mixed ins/outs.

Here’s an example of that, using the COMPLEX-REAL example above:

This might be equivalent to the following C++:

struct complex{
   double re_;
   double im_;
};

void crreal(complex * in, double * out) {
   *out = in->re_; 
}

You could think of this as a function with a return *out, where *in is agreed to be used in a read only fashion.

As an abstract entity, the UDF has the characteristics of a PROGRAM, specifying a FUNCTION-ID instead of a PROGRAM-ID, and also requiring a RETURNS.  One might look like:

This is probably not an ideal recoding in as a UDF, as it is possible to use BY VALUE parameter passing (and I assume BY VALUE return).  However, it illustrates the rough idea.  One of the other differences between the UDF and a program (or subprogram) is that a UDF can be prototyped.  Here’s an example of that:

Typical of COBOL, the verbosity is horrendous.  This isn’t a one like prototype like you would have in C, and is much harder to read and understand.  You can, however, put all the bloated PROTOTYPEs for your library functions in a copybook, and have your program COPY that.  The fact that it can be prototyped is a big improvement over a COBOL program, which is completely untyped.  You can have a COBOL program that takes an int by value, but pass it a parameter by reference, or vice-versa ; or that is supposed to take 3 parameters, but is passed only one, or is passed 5.  The call site has no way of knowing if the type or nature of the parameters matches the implementation.

With a program being far superior to a paragraph as a function-like entity, you have to wonder why it is not a common COBOL paradigm.  I suspect that a big part of that is the expense of a paragraph vs. a PROGRAM.

At LzLabs, a call to a program, subprogram or otherwise, was not cheap.  A big part of that was WORKING-STORAGE related.  WORKING-STORAGE is something like static storage in C, if static storage persisted across multiple invocations of a program (until “rununit” termination).  Contrast that to LOCAL-STORAGE which is more “stack” like.  I don’t know whether an IBM Enterprise COBOL subprogram call is any cheaper than a call to an entry function.  It was not typical to see customer code that made use of PROGRAMs as function calls, except for very specific use cases.

I also don’t know if the new IBM User Defined Functions are cheap enough that programmers would opt to use them instead of paragraphs.  I saw only one or two programs out of thousands (both from one customer) that used this feature, but it was still a very new feature, and relative to the age of COBOL, it still is.

At least conceivably, a UDF that doesn’t use WORKING-STORAGE (only LOCAL-STORAGE and LINKAGE-SECTION), and if used only with pass by value parameters, could, theoretically, be as cheap as a C function call.  I don’t know if that’s the case on the mainframe, but it’s at least possible.  You could imagine a “fastcall” convention with pass by registers for the parameters for a UDF, instead of the usual indirect “PARM” mechanism.  Regardless of the cost, you could use this tomodel procedural programming in a way that can be translated to another language.

You could imagine that it would be possible to find paragraphs for which a set of WORKING-STORAGE variables are always re-written when the paragraph is executed (i.e.: find the set of variables that are effectively local to a paragraph), and then factor a paragraph out into a UDF with well defined semantics, with inputs, outputs, and local variables, reducing that giant set of “10000” working storage variables in the original program.  If you repeat that process across an entire program, also extracting named representations for the implicit types used, would you be able to systematically “find” the structure of the program, and then have a candidate for translation to something not as intrinsically evil as COBOL?

Stream of consciousness dump: new job, claude terminal, vscode, vim mode, …

May 4, 2026 Incoherent ramblings , , , ,

This post is a fairly random stream of consciousness dump, including bits on all of the following:

  • Job transition, and hints of what I’m going to be working on
  • Vscode, and vim mode deficiencies
  • AI development integration

Job change

My last day at Lemurian was two weeks ago, after 11 months there. I enjoyed much of the work, which was well suited for my skills and aptitudes, and learned a lot, but ultimately it wasn’t the right fit for me. I miss a number of the people I worked with, and wish them and Lemurian continued success.

The new job is at a just-formed startup, with a handful of developers, no product yet, and no Canadian corporate presence (so I’m working as a contractor for now). Budget is tight and there’s no additional hiring capacity at the moment, so this isn’t a “come join us” post — it’s more of a “here’s what I’m up to.”

That new job is really fun so far. In one week of 14+ hour days (with extensive Claude generation of boilerplate tablegen/parser/builder), I have most of an MLIR language dialect + parser/builder built for a specific programming language. There’s lots more to do, and the interesting and tricky parts will start soon, once I have infrastructure in place. The work is going to be lots of fun; the team is great, really capable, and knows how to build and get the job done. I don’t know how long it will take us to get the product built, and we are starting from absolutely nothing, but the pieces are coming together quickly.

Vscode

At Lemurian, most people used VSCode for their development. I’ve been using tmux+vim+cscope+ctags+perl so long that I found it hard to make the switch. I’d start in VSCode in the morning and, in frustration, end up back in the terminal in short order.

As an example, suppose that you want to filter the current function through a script (say, clang-format). In vi, you can do something like:

:,/^}/!clang-format

or if your function is indented (by four spaces, say):

:,/^    }/!clang-format | sed 's/^/    /;s/ *$//'

VSCode has a vim mode, but its pattern selector is broken, so you can’t do the last (I don’t remember if you can do filtering either, but let’s stick to the selection issue). To do that line selection in vim mode VSCode, the line range selection part has to be written with the following clunky expression:

:,/^\s\s\s\s}/

It was little quirks like that which repeatedly ejected me from VSCode back to the terminal, or I’d run both VSCode and terminal. It was a strange workflow.

There were a few things that I liked about VSCode:

1. The right-click for type definitions was really nice. I do this the hard way, with memorization, grep, cscope, ctags, and other ad-hoc methods.
2. The autocomplete was nearly magical sometimes.
3. The integration with Claude Code or other AI agents works fabulously, giving the agent full context for the repo and project.

Points 1 and 2 may eventually draw me back to VSCode, despite how slow I feel working in that environment. On the other hand, the AI tools are so good now that English is becoming a programming language. Do I really need to go through the pain of figuring out how to be effective in VSCode?

Claude terminal

For point 3, at the new job, I’ve now got a great replacement: Claude terminal!

Running that terminal application (in an ssh session to my development VM), I can grant the AI tooling access to the repo, as I could in VSCode.  However, I’m able to run the claude session in a tmux window (tmux rename-window claude; claude), and away we go.

A development model that works really well is to describe the desired task, have a conversation about it, and have it produce a design document and execution plan.  I review that, choose which parts I want to do myself, and throw the tooling at the rest, then review that work thoroughly.  The result is staggeringly fast iterations.  Some spectacularly large systematic tasks that may have taken months to do unassisted, can be done in hours.

One gotcha: I’d like to be able to enter multi-line prompts in the terminal UI application, but haven’t figured out how. Claude claims that shift-enter works, but perhaps only in the web app? I’ve resorted to writing some prompts in a file in a separate tmux window and then asking Claude to read that file.