Delicious!!! Once again a pedagogical Rosetta Stone. Thank you , Carl

Thanks Arben, I will get back to you on that FM question. I have to find my notes on when I last tried to do it.

"If I'm reading your (original) post correctly, you could also do pattern-matching or selection by going directly into the cosine terms, extracting the coefficients of the omegas with CoefficientList and grabbing only those cosine terms for which the absolute value of the difference is 1"

That sounds kinda like what I was originally trying to do but unsuccessfully. Could you show an example?

Arben,

First thank you for teaching this class. This has been very helpful.

I am running into some unexpected behavior. I just downloaded Version 14, and when I do TreeForm of a list I get:

but when I do the same thing using Plus, I get:

I was expecting to get the same thing but with List replaced with Plus. I am a bit puzzled.

Many thanks for your help. Michael O'Connor

Thank you, Arben!!!

That's a really neat approach. Had not yet thought along the path of creating a list of all the terms. But entirely consistent with thinking about the heads and changing them. That's a powerful analytical tool.

Your solution has a lot of pedagogical value. I am going to be staring at it a bit.

If you are on a roll, I've got another one coming as soon as I can figure out how to ask it properly. Let me take a crack and if it needs more let me know: It has to do with a problem in FM theory, where recognizing that Cos (B Sin [w t]) can be expanded as a Fourier series whose coefficients are an ordinary Bessel Function of the first kind

= Jo(B) + 2 Sum (J2n(B) Cos (2 n w t) )

Then you have to truncate the series to get the coefficients of interest. There is a similar expansion for Sin(B Sin (w t). You can sample an FM signal and do an FFT and just see the spectrum, but recognizing the Bessel Function behavior provides an intuitive understanding of FM theory (bandwidth, mod index, S/N properties etc) that even when FM was really popular few people grasped. Now it's beginning to be a lost art. What I never figured out how to do was use WL pattern matching and mapping tools to "recognize" and derive this classical solution. It has other analytical advantages for other problems too.

Try removing the semicolon

From yesterday's lesson, does anyone understand why

    larger[x_, y_] := If[x > y, x, y];
    Fold[larger, {3, 1, 2, 6, 5}]

returns larger[larger[larger[larger[3, 1], 2], 6], 5] and not 6?

map mystery

Can someone explain what is going on in the expressions below the tree, or would this be too advanced for the level of this course (which, if it is, is OK with me)?

(See attached png file)

Attachments:

Map example.png

Arben - Thank you very much for the detailed explanation. I am finding this material somewhat difficult but rich and rewarding when something sticks.

Hi Phil,

I think this can be better understood and handled through contacting customer support:
https://www.wolfram.com/support/contact/email

Good catch! I've fixed this in our repo and it will be fixed in our next deployment.

I asked in the BigMarker Q&A panel how to upload a WL Notebook to my Wolfram Cloud account. The reply was to use the "upload" button in the upper right corner of the Wolfram Cloud window. This doesn't work; I get the error message, "To access this feature, subscribe to a plan." Apparently, the Basic plan isn't good enough.

I can upload files through Mathematica on another machine through the File->Save to Cloud function. I don't understand why that works but simply doing an "upload" in cloud does not. Is there a good reason for keeping a random user from uploading any notebook to a Wolfram Cloud account through the web? If not, please file this as a bug.

One note for the staff answering questions: when they are answering a question like this, they should try to perform the function with the same level of privilege/access as a normal customer. In this case, the Wolfram Cloud "upload" function should have been tried on a Basic plan.

I can heartily recommend this. While I am in no position to judge Wolfram's contribution to Physics, the ideas are fascinating and are something I would like to pursue.

If I am hearing him correctly, he is attempting to make sense of Physics by looking at it as a system that is computable by the application of rules for re-writing. He seems to be saying the system evolves what look like small black holes and "elementary" particles from vacuum energy. Of course he is talking to a professional physicist here so we hear him respond to some good questions.

I am stricken by the similarity of the method Wolfram is proposing and the spirit of the the approach we seem to be taking in grappling with Mathematica. Very stimulating!

https://www.youtube.com/live/ITJ3AF3TK5M?si=rECCBuBAAVWa_Im4

I'm with you there—I immediately thought that the following image, from this blog post, was very striking:

The notion of uncuttable is really helpful, thank you. I’ve now learnt to turn to inbuilt functions in order to better understand what’s going on and just the idea of syntactic convenience is helpful in getting over what at times has felt like contradictions in terms. I remember Prof Richard J. Gaylord saying once that Wolfram, at times, behaves strange, and never quite knew what was meant by that (likely the pitfalls of intuition).

I think the best way to see what is going on, on your own machine, is to look at y//FullForm and z//FullForm. They are quite different.

However if you insert a space between a and x in your expression for z, then y and z should be the same.

[If you are somehow stripping out spaces, then that will cause you problems. Mathematica really uses the spaces as Arben says.

I am already finding that when I am getting odd results, using FullForm, TreeForm etc can make things clearer.

Yes, I also found that I couldn't use the code output by those boxes in situ. Entering it into a new code cell works fine.

I think the apparent confusion regarding the word atom is as a result of an error in the evolution of our scientific knowledge. It is only relatively recently that we discovered that 'atoms' of, say, radium are not really atoms in the logical sense of the word. Then we repeated the mistake, calling some of the sub-atomic bits "fundamental" only to find...

In a parallel universe where the Wolfram language was developed in 1880, the concepts of atoms in Mathematica and Chemistry would have been consistent. Although some physicists maintained that the atoms of the chemical elements were fictional conveniences for the sake of calculation and that, if they even existed they would be too small to ever be known. Then came 1905 and Einstein's analysis of Brownian motion.

Nice! I hadn't thought (until today!) about the derivation of the word atom. So, 'tom' as in 'tomography'.

ax is on variable not two variables a * x is and expression equal to a space x

There are sub-atomic parts

Please see the answers downthread—the easiest way I know to explain this is that if you say ax, how is Wolfram Language to know that you actually, truly, really meant a*x? It can't.

If you were to type myVar^2, you ostensibly would not expect to get m^2 * y^2 * V^2 * a^2 * r^2, so why expect differently for ax^2?

(Since tone is never conveyed well over text, I want to be clear that the intended tone here is "Socratic" rather than "snippy" ;).)

Hate to redundant, I still can't get by this. If this is a real error, other errors I get may be me or Mathematica - can't tell. Btw, $Version returns: 14.0.0 for Mac OS X ARM (64-bit) (December 13, 2023).

Yeah, it's just atoms have parts as far as atoms are concerned, as apposed to atomic expressions. The same signifier ("atom") points to different signified concepts depending on the context, hence counterintuitive. Essentially, atomic parts are not the same in the case of the fundamental concept in chemistry and physics and atomic expressions and therefore not covered by Part in the normative sense. So an error message is returned.

While working with the exercises for Day 3 (Syntax and Expressions), I got the following error message while attempting Exercise 4 (ListLinePlot for airline passenger data), namely "An improperly formatted option was encountered. The left-hand side of the option was not a symbol or string* My solution(s) worked fine initially, but then went wonky all of sudden. In fact every plot in the notebook threw the same error message, whether options were specified or not. (Also all of the dynamic Hints and Solutions boxes disappeared, but that's not my main concern right now). Please see attached notebook..

Attachments:

Day03-SyntaxAndE...nb

It can happen even after a decade of using the language—I was once thwarted for an entire day by an errant curly/("""smart""") quote breaking a string expression when I first start working here!

In Wolfram Language, the structure of expressions and how they are indexed can sometimes be counterintuitive if you're not familiar with the underlying representation. Let's break down your example to understand it better.

Consider the expression exp = 1 + x^2. In Wolfram Language, this is internally represented as Plus[1, Power[x, 2]]. The head of this expression is Plus, and it has two arguments: 1 and Power[x, 2].

exp[[2, 1]] returns x because it's accessing the first part (x) of the second argument (Power[x, 2]) of exp. exp[[2, 2]] returns 2 because it's accessing the second part (2) of the second argument (Power[x, 2]) of exp. Now, regarding exp[[1, 1]], the confusion arises because 1 is an atomic expression in Wolfram Language (it does not have a head and arguments structure like Power[x, 2] does). Atomic expressions include things like numbers, symbols, and strings. When you try to access a part of an atomic expression, it results in an error or an unexpected result because atomic expressions are considered as having no parts.

So, exp[[1, 1]] is trying to access a part of the atomic expression 1, which doesn't have any parts, hence the issue. In Wolfram Language, the parts of atomic expressions are not accessible in the same way as the parts of composite expressions.

• GPT 4

I know you touched on this, but in exp = 1 + x^2, exp[[2, 1]] returns x, yet exp[[1, 1]] is longer than the depth of the object, why again?

To be absolutely clear: this is equivalent to asking why var^2 does not return v^2 a^2 r^2. If you don't type a space or an asterisk between two letters, there's no way for the system to know that you're talking about two separate symbols.

Steven - I get the correct answer when using explicit multiplication:

    y = a*x^2 + b*x + c

Why is this? Arben?

I don't believe this is a space issue. The notebook shows input (w/o spaces) and the incorrect output.

Steven/Arben - I too am getting the INCORRECT result. I am also running Mathematica 14.0.0 on Mac silicon. (This might also explain some of the results I have been getting!) I create a notebook and set y = ax^2 + bx + c. I then try Expand[y^2] and get the following:

    ax^4 + 2 ax^2 bx + bx^2 + 2 ax^2 c + 2 bx c + c^2

I have attached my notebook and an image making it clear what element is incorrect in Mathematica's answer.

Attachments:

IMG_0260.jpg

problem.nb

I thought I understood this material during the lecture; most things made sense. Some parts of Problem 1 however just don't seem right. For example, why does pat3 give the following result?

exprs = {1, h[], h[a], h[b], h[a, b], h[a, a], h[a, b, c], g[h[a, b]]}
pat3 = x_
Cases[ exprs, pat3]
    {1, h[], h[a], h[b], h[a, b], h[a, a], h[a, b, c], g[h[a, b]]}

I see no way to get this result.

Thanks in advance,

In today's post-lecture questions "pat2 = _h" Case tells me that pat2 matches the following elements in exprs:

    {h[], h[a], h[b], h[a, b], h[a, a], h[a, b, c]}

Does this mean that pat2 = h = h? It seems like it, but this seems odd to me in that we are PATTERN MATCHING and, it seems to me, elements in one order should not necessarily be equivalent to elements in a different order.

Comment?

Wonderful explanation! I'm going to give this more thought. I'm trying to get my head around Wolfram engine.

question = "If Mathematica is a term rewriting system, when I write a programme in Mathematica, am I writing a term rewriting rule, and is the programme a function?";

If[question==={"yes", "yes"}, "At what point does a rule become a system?","Help!"]

Hey Coe, could you be a little clearer in your description of your inputs, their actual outputs, and the outputs you would expect? In the meantime, let me explain the meaning of each line you've included here.

Alphabet[][[;;10;;4]]

This is equivalent to Alphabet[][[1;;10;;4]]. This means "starting with the list of all of the letters in the (Latin) alphabet, take the first through the tenth elements in steps of 4." I.e., that's elements 1,1+4,1+4,+4===1,5,9. (You don't go to 13 since the initial list only has 10 elements.) And indeed, that's what this code does—cf. MapIndexed[#2[[1]] -> #1 &, Alphabet[]] or Thread[Range@26 -> Alphabet[]] to see it directly.

Actually, I see what you were initially asking. I think the above sufficiently clarifies it, but "in order":

• Alphabet[] provides a list containing the characters of the Latin alphabet
• Appending [[;;10]] is the same as appending [[1;;10]] and is extracting the 1st through 10th parts of the list containing the alphabet
• Similarly, appending [[;;4]] is equivalent to appending [[1;;4]] and is extracting the 1st through 4th parts of the list containing the alphabet

Doing [[a;;b;;c]] is not the same as doing [[a;;b]][[;;c]]. The "sandwich" construction specifically means "go from a to b in jumps of size c"; it's different from "chaining" ;;/Span commands through [[ ]]/Part.

Hi arben, Yes this is the tip of iceberg. I suggest you hash table as something to know. O(1) as a search algorithm. Hash tree and topology is a rich context to know

The link in the email I received doesn't work. How do I navigate to the materials folder? I looked around and didn't see an obvious link.

Hi everybody—I've added today's "filled-out" notebook to the materials folder. See you tomorrow!

I can't make it at this time on MWF, but can on TTh.