Hi Mike—please check out my answer in the thread for the current Study Group: https://community.wolfram.com/groups/-/m/t/2972173

Hey Phil—that's a totally reasonable solution to that problem, yes. The only issue is that the autograder used there is quite a complicated object, and it doesn't always get everything right. (For quizzes and other assessments, we try to update it whenever we find one of these false negatives, but it's not always possible.)

I thought that Counts[StringContainsQ[StringTake[WordList[], 1], "q"]][True] might be taken a little bit better, but no dice :)

Arben, Thanks for your description of the quiz festivities. I wonder how Wolfram knows to calculate the grade on the download file if it isn't via a live wire. I used your manual copy from my downloaded quiz to the one I thought was the real thing, and neither saved my answers. Bob

Hi Robert—thanks for your kind words! I think that the giest of what Phil said is correct—probably you ought to submit the quiz directly online. Let me know if that doesn't work, and we'll look into it!

WRT Wolfram Study Groups, there's one other administrative detail that could make a difference.

In one of the lecture Q&A sessions last week, Cassidy noted that videos of the lectures will remain online as long as Wolfram Research has a contract with BigMarker (i.e., a long time). However, the lectures are inaccessible unless one has access to the URL for each lecture. Those "recording available" URLs are unique for every lecture and each is e-mailed once to each participant. They come with a tag of "bu=" plus a very long sequence of hexadecimal numbers, which implies the URLs are tagged and unique for each participant.

I have started a practice of assembling all URLs for WL Notebooks and BigMarker Videos in an archive for each SG that I take. As I see it, Wolfram Research could do one of three things for Study Group participants to help them organize an archive of the materials for a course:

1. Make a specific recommendation during Session 1 of a course to suggest participants to create a document archiving links to the the training materials for the Study Group. The archive would be useful for review for the quiz, and useful for longer-term review of the material.

2. Provide a WL Notebook with functionality for creating, editing, and saving an archive of the materials for a course.

3. 2-3 business days after course completion, send an e-mail to course participants providing a document with links to all course materials. This document could be bundled with the "reminder" e-mail about the quiz deadline for the course.

Hmmm. After writing those three, I suppose #1 makes the most sense. #2 would be complicated to make a robust functionality, and #3 may be difficult for Wolfram U to automate adding URLs with any user-specific tags needed in the URLs. If generic URLs could be provided for the videos, then drafting the archive for this e-mail becomes much easier.

CC @Jamie Peterson

Yep, I get you! How about this? (It's been a long time since I've heard or thought about Larmour frequencies, so let me know if this is not right.)

Thanks Larry; I'm glad it was helpful! hopefully it paves a path forward for you to analyze and visualize other systems. I'm happy that we were able to to introduce and re-introduce you to various functionality, too!

I figured out my problem. Sorry for the last msg.

Hi Peter—there's no issue, it's just a warning that the notebooks were made in a newer version so there might be some functionality that doesn't work because it doesn't exist in your installation.

Tuesday's example of the Hamitonian classical mechanics was in 1 dimensional phase space. How should the code be modified to model n-dimensions?

Help me with the 2d Hamiltonian: H[q1,q1,p1,p1,t]. q1'=dH/dp1, q2'=dH/dp2, p1'=-dH/dq1, p2'=-dH/dq2 all partial derivatives. Hspace i R^2d+1.

Hi, I am trying to work with the notebook from the "Differential Equations" lecture yesterday. For some reason, when I run the DSolve code, it's giving me this error message: I'm not sure if it's because I ran the code out of order but I tried deleting the notebook and redownloading it and it's still showing this. Any idea of how I can fix this? I'm new to mathematica so I'm sure I'm missing something basic.

I also don't understand why the numbering on the In[#] and Out[#] is continuous throughout all the different notebooks I have. Is there a way to make them separate? Thanks so much!

Hi Phil, thanks for the suggestion; I will Reload a Community page before I post next time. It's a good practice in general as I also missed the latest post by Arben Kalziqi.

There were two questions for previous presenters to which I've just received answers:

A question for Lusine, who gave the talk on number theory:

I would be interested in what is the title of the book about number theory and when the new online course is expected to be launched on number theory mentioned during this session.

Lusine notes that the book is not yet complete and the title has not been finalized, but it will probably have a rather traditional name like Introduction to Number Theory.

A question for Jaebum, who gave the talk on linear algebra and graphs:

I would appreciate if you could mention some references (e.g. books, articles, web pages) that contain already accomplished research topics in Wolfram language related to graphs.

Jaebum has provided three examples:

1. Analyzing Narratives in Social Networks
2. Operator Calculus on Graphs, Theory and Applications in Computer Science
3. Functional Networks with Applications

Thanks!

Great thanks!

Indeed, I wind up doing that myself pretty frequently. The only thing to keep in mind is that f[{x,y,z}]!=f[x,y,z], of course, but if you're defining your own functions it isn't a big problem.

However! If you take advantage of Apply (@@) and MapApply(@@@) and understand how slots work, you can mix and match and get the best of both worlds.

Hello Peter,

Thank you for these interesting questions.

1. The PerformanceGoal option for AsymptoticDSolveValue is set to "Quality" by default, and the "Speed" setting is only used to disallow the use of exact methods based on DSolve, for computing an asymptotic approximation. You will probably not need this option very often but the following rather artificial example illustrates its use in AsymptoticDSolveValue:

   In[1]:= deqns = {y'[x] + Sqrt[x]  y'[x]^2 == 1, y[1] == 1};
   
   In[2]:= AsymptoticDSolveValue[deqns, y[x], {x, 0, 3}]
   
   Out[2]= 1/6 (11 + 5 Sqrt[5]) - 2 Sqrt[x] - x + (2 x^(3/2))/3 - x^2 + 
    2 x^(5/2) - (14 x^3)/3
   
   In[3]:= AsymptoticDSolveValue[deqns, y[x], {x, 0, 3}, 
    PerformanceGoal -> "Quality"]
   
   Out[3]= 1/6 (11 + 5 Sqrt[5]) - 2 Sqrt[x] - x + (2 x^(3/2))/3 - x^2 + 
    2 x^(5/2) - (14 x^3)/3
   
   In[4]:= AsymptoticDSolveValue[deqns, y[x], {x, 0, 3}, 
     PerformanceGoal -> "Speed"] // Head
   
   Out[4]= AsymptoticDSolveValue
   

2. The leading order approximation for the singular perturbation problem is computed using a simple "closed form" for it based on a discussion in Chapter 9 of Bender and Orszag which works well for second-order linear ODEs for which the coefficient of y'[x] is positive on the given interval. However, as you have noted, the boundary condition at x = 1 is only satisfied asymptotically with this formula. One of the "later" perturbation methods in AsymptoticDSolveValue uses the WKBJ method which yields an approximation that is more complicated but does satisfy the boundary conditions (and also works for higher-order approximations). This method can be accessed as follows:

   In[1]:= eqns = {ep y''[x] + c Derivative[1][y][x] + 
        y[x] == 0, y[0] == 0, y[1] == 1/2};
   
   In[2]:= psol = 
    AsymptoticDSolveValue[eqns, y[x], x, {ep, 0, 1}] /. c -> 2
   
   Out[2]= E^(1/2 + 2/ep - x/2)/(2 (-E + E^(2/ep))) - E^(
    1/2 + 2/ep + (-2 x + (ep x)/2)/ep)/(2 (-E + E^(2/ep)))
   
   In[3]:= psol /. x -> 0
   
   Out[3]= 0
   
   In[4]:= psol /. x -> 1 // Simplify
   
   Out[4]= 1/2
   
   In[5]:= dsol = DSolveValue[eqns /. c -> 2, y[x], x]
   
   Out[5]= -((
    E^(1/ep + Sqrt[1 - ep]/
      ep) (E^((-(1/ep) - Sqrt[1 - ep]/ep) x) - 
       E^((-(1/ep) + Sqrt[1 - ep]/ep) x)))/(
    2 (-1 + E^((2 Sqrt[1 - ep])/ep))))
   
   In[6]:= Limit[dsol/psol, ep -> 0, Assumptions -> 0 < x < 1]
   
   Out[6]= 1
   

   Hope this helps.

A proposed solution to the exercise.

Attachments:

20230526 JD Spar...nb

Here's a way to get the SpaceArray without the dreaded index scaffolding:

Two comments:

In past classes, I had been opening the Wolfram Language & System Documentation Center in a browser window. I accidentally opened the docs center in a Wolfram Language window. Opening the help docs in the Wolfram Language app is vastly superior! You can run a line or 2 of code immediately right in the documentation window. Also, you're not thrown by the distraction of browser windows in your Wolfram explorations.

I am not a mathematician; I had difficulty brushing off the cobwebs in the recent Wolfram Calculus Study Group. OTOH, I remembered enough about matrix math to remember that the diagonals next to the main diagonal of a matrix were highly useful -- and there must be a way to access them. There had to be a built-in library function to get to them! It took me a couple of days to find Band. If I had looked in the "See Also" section of the docs for SparseArray, I would have found Band much much faster. There is a brilliance in the design of the WL; great minds have laid out a wonderful... array... of functions.

I think it's OK to use Abs to see the "distance" from the diagonal. OTOH, your code ends of applying that test to every cell (or, at least, the calculated value of the coordinates of every cell) in the matrix. You're doing a bunch of superfluous computation; it should be slower:

I'm gonna give that a No Pass.

Dear Arben,

Just missed your reply with the solution of Part 1 (no browser refresh).

I joined the Study Group in the second week. Apparently, I posted my solution as someone else (Vitaliy Kaurov, WOLFRAM Research) which is very weird (I used a VPN). The impersonated post is below. I reported the issue to Wolfram Support.
[Update] WR has restored my post attribution.

Is it this Arben for the Part 1 code?

Normal@SparseArray[{{i_, j_} /; (Abs[i - j] <= 1) -> 
     PrimePi[i + j]}, {10, 10}] // MatrixForm

Yes José, that's basically the idea—the only difference is that I used Abs instead of checking whether it was between ±1. I'll check out your notebook when I have the chance, hopefully today.

Hi Alexander, in the details section of the documentation of FindEquationalProof, it clarifies this somewhat. In cases where the function is able determine it is not possible to prove the given theorem from the axioms, it will return a Failure object. However, as you mention, in full generality it is not possible to decide this and the only option is to let the program run to see if it terminates. For this reason there is also a TimeConstraint option to impose an artificial limit on how long the function will keep running. You can also manually stop any ongoing evaluation by clicking Abort Evaluation under the Evaluation menu in the user interface.

Hi Phil—hopefully John and Charles were able to answer your question in the webinar chat today. To wit, they mentioned that certain trigonometric simplifications were available via FullSimplify which wouldn't be tried with Simplify, and that this could result in simpler results at the cost of slower speeds. John also mentioned that there are trig-specific simplification functions like TrigExpand which can work several orders of magnitude faster than either of the more general simplification functions.

Hi Phil! That's a lovely suggestion; I'll see if I can work it in.