ho.history overview – Has any open/difficult problem in ordinary mathematics been solved only/mostly by appeal to set theory?

We know that many (if not all) mathematical notions can be reduced to the talk of sets and set-membership. But it nevertheless sounds like a grueling task (if at all possible) to actually get advanced results in many branches of ordinary mathematics if we only work with sets and set-membership relation in our language, or otherwise only rely on set theory. To put it differently: it seems that in order to get results in many branches of mathematics one might not need to be very familiar with set theory at all, let alone being able to translate everything to the language of sets or to heavily rely on set theory.

I’m wondering if there are cases where an open/a difficult problem in other branches of mathematics (e.g., number theory or real analysis) has been solved mostly/only because of the insight that set theory has offered, directly or indirectly (say, through branches that heavily appeal to set theory, such as model theory). Even a historical incident will be helpful: a problem of the sort that was first solved thanks to set theory, but later on more accessible solutions have been found that don’t deal much with sets.

Thank you very much!

network flow – Integral solutions to circulation problem

Suppose we have a circulation problem (with only one commodity), where all lower bounds, upper bounds, and costs are integers. Are we guaranteed that if there is a solution, then there is an integral solution? Is there an algorithm that can find one in polynomial time?

For standard network flow, we know that there is a max flow that is integral, and there are polynomial-time algorithms that can find such an integral flow. I’m wondering if the same is true for circulation problems, which are a generalization of max-flow.

sparse arrays – Insufficient memory problem in Mathematica

I am trying to initialize a 48600 x 48600 packed array from a sparse matrix and a diagonal matrix in Mathematica 8. If S1 is a sparsearray of that size then and S2 is list of 48600 length, then I am using the command

H=Developer`ToPackedArray(S1+DiagonalMatrix(S2));

This returns an error.

General::nomem:
   The current computation was aborted because there was insufficient memory
    available to complete the computation.

This is impossible because the RAM I have is 512GB, plus when I do the same thing in a 256GB Mathematica 11.0.2 machine, it goes through, but then I need to perform other operations(diagonalization, multiplication) on this so I actually need the 512 GB RAM.

Additionally the same machine of 512GB RAM returns an error when I try to make the sparsearry go to normalform and then pack it, as,

In:-S1packed=Developer`ToPackedArray(Normal(S1));

SparseArray::ntb:


 Cannot convert the sparse array SparseArray(<463320>, {48620, 48620})
     to an ordinary array because the 2363904400
     elements required exceeds the current size limit.

Out(1)= SystemException(SparseArrayNormalLimit,

>    Normal(SparseArray(<463320>, {48620, 48620})))

Even this operation goes through in Mathematica11 256 GB RAM, but not here. Apart from the obvious answer install Mathematica 11 , which I cant since its an institute machine, anybosy know anything about increasing max memory usage/size limit of normalform from SparseArray in Mathematica?

Update:- I was testing both the systems with

T1=RandomReal(1,{50000,50000});

The 512 GB Mathematica 8 machine gives,

RandomReal::arrsize: 
   In array dimensions {50000, 50000} at position 2 of 
    RandomReal(1, {50000, 50000}), the product of the first 2 dimensions is 
    2500000000, which is not a machine integer.

It’s a 64 bit machine so it’s impossible that the largest machine integer is actually less than this.

While for the 256GB Mathematica 11.0.2 machine it goes through(takes a lot of RAM though). What’s going on?

combinatorics – How to perform a reduced knapsack problem

I have a problem statement that seems to be a reduced version of the knapsack problem, but I don’t know how do it in Mathematica.

The problem is as follows:
Given a set, S, of integers (e.g {a,b,c,…}) and a specific integer T, find all the possible combinations of the elements of S that sum exactly to T. (e.g returns {a,d,e} and {e,f} because a+d+e=T and e+f=T).

entities – Problem regarding saving field data programmatically

I’m developing a module for Drupal 9 which creates nodes. In order to achieve that, I added the field types, field formatters, field widgets, a form display, a view display as well as configuration for the fields inside the mymodule/config/install directory (e.g. field.field.node.foo.field_bar.yml and field.storage.node.field_bar.yml). Then, based on some business logic, I’m adding new nodes in PHP as follows (for simplicity, here is only one custom field):

use DrupalnodeEntityNode;
$node = Node::create((
  'type' => 'foo',
  'title' => 'Example',
  'status' => 1,
  'langcode' => 'en',
  'field_bar' => 'Content of bar',
));
$node->save();

Creating nodes and setting the title works fine. For some reason, I can see the values for field_bar in the database (table node__field_bar contains rows where field_bar_value is set as intended) but they don’t appear inside the corresponding text input when I’m editing a node. Also if I want to output the field’s value, it is empty. However, when I just create a new foo item using Drupal’s GUI, everything seems to work as intended. Therefore, I think the error is rather in the above code than in the field’s setup (field type class, yml configuration files, etc.). Feel free to correct me if this assumption is wrong. Thanks in advance!

dnd 5e – Would there be a problem to incorporate Pathfinder-2e’s “Step” action in place of DND 5e’s “Disengage” action?

Pathfinder’s Step:

You carefully move 5 feet. Unlike most types of movement, Stepping doesn’t trigger reactions, such as Attacks of Opportunity, that can be triggered by move actions or upon leaving or entering a square.
You can’t Step into difficult terrain, and you can’t Step using a Speed other than your land Speed.

DND 5e’s Disengage:

If you take the Disengage action, your Movement doesn’t provoke Opportunity Attacks for the rest of the turn.

My intention would be to create for DND 5e something like:

If you take the Step action, you can move 5 feet to an unoccupied space not in difficult terrain without triggering any opportunity attacks.

As Disengage applies for a creature’s full movement, I think giving the creature an extra 5 feet of movement when taking the Step action makes up for the loss. This would of course apply for NPCs and PCs, so I don’t see this giving an unfair advantage to one group over the other, except for maybe those with the ability to disengage as a bonus action, like Goblins or Rogues; switching that for the Step action about may be a slight nerf to the class/creature/race.

I find the Pathfinder Step makes more sense than the DND Disengage, as I can’t make sense of Disengage working for more than the enemies directly threatening the character: a creature that takes the disengage action could then run there full movement, potentially avoiding a dozen or more attacks.

nt.number theory – univariate integer version of Hilbert’s 17th problem

Let $f(x)$ be a polynomial with integer coefficients such that $f(x)geqslant 0$ for all real $x$. Is it necessarily true that $f$ is a sum of squares of polynomials with integer coefficients?

This is true if $deg f=2$ and true if the coefficients of the squared polynomials are allowed to be real (or rational).

Association and TemplateApply problem in associations section of The Wolfram Language:Fast introduction to programmers

I am going over The Wolfram Language:Fast introduction to programmers. In the associations section, following question and answer is given

Which of the following applies a template to make a string with appetizer and dessert from the association

 meal=<|"appetizer" →"nachos", "salad" → "spinach", "dessert" → "chocolate"|>
 TemplateApply["The appetizer is `appetizer` and the dessert is `dessert`.", meal]

But in my mathematica notebook, I am getting errors from the above two lines of code.

Incompatible elements in Join …. cannot be joined.

But according to web site, this is correct answer. Where is my mistake?

Problem 8.8(a) Apostol introduction to analytic number theory

I am self studying analytic number theory from Tom Apostol and couldn’t solve this particular question which is 8(a) of chapter 8 on page 175.

It’s image:enter image description here

Attempt: I thought of proving that for every $a_i$ that satisfies $aequiv a_{i} (mod k_i) $ will satisfy $a_{i} equiv 1 (mod k_j) $ .

But I couldn’t. Also, I realized that this will not prove existence of $a_i$ .

So, I am struck on this problem and need your help.

Thanks