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dynamic – How to plot the stable and unstable manifolds of a hyperbolic fixed point of a nonlinear system of a map?

I would like to plot the stable and unstable manifold of the Hénon map x_{n+1} = 1 – a*x_n^2 + by_n, y_{n+1} = x_n. I believe that iterate the inverse map i would get the stable curve, and for the unstable i need to choose an large number of initial conditions inside the neighborhood of the saddle point along the unstable direction and then iterate the map for a few times, but i do not know how to do this things.

dynamic programming – Asymptotic complexity of Combination sum problem vs Coin change problem

I’ve been looking at the following combination sum problem:

Given a set of candidate numbers (candidates) (without duplicates) and a target number 
(target), find all unique combinations in candidates where the candidate numbers sums to 
target.

The same repeated number may be chosen from candidates unlimited number of times.

Input: candidates = (2,3,5), target = 8,
A solution set is:
(
  (2,2,2,2),
  (2,3,3),
  (3,5)
)

Most of the solutions I encountered for this problem were some variant of the following backtracking solution:-

public List<List<Integer>> combinationSum(int() nums, int target) {
    List<List<Integer>> list = new ArrayList<>();
    backtrack(list, new ArrayList<>(), nums, target, 0);
    return list;
}

private void backtrack(List<List<Integer>> list, List<Integer> tempList, int () nums, int 
remain, int start){
    if(remain < 0) return;
    else if(remain == 0) list.add(new ArrayList<>(tempList));
    else{ 
        for(int i = start; i < nums.length; i++){
            tempList.add(nums(i));
            backtrack(list, tempList, nums, remain - nums(i), i); // not i + 1 because we can reuse same elements
            tempList.remove(tempList.size() - 1);
        }
    }
}

Most of the comments of these solutions implied an asymptotic complexity of O(2^N) due to this code potentially building up all subsets in the worst case.

However, this problem statement seems very similar (if not identical) to the Coin change problem

You are given coins of different denominations and a total amount of money. Write a function 
to compute the number of combinations that make up that amount. You may assume that you have 
infinite number of each kind of coin.

Input: amount = 5, coins = (1, 2, 5)
Output: 4
Explanation: there are four ways to make up the amount:
5=5
5=2+2+1
5=2+1+1+1
5=1+1+1+1+1

The most common solution I’ve found to this problem is the following O(N*amount) dynamic programming solution:

class Solution {
  public int change(int amount, int() coins) {
    int() dp = new int(amount + 1);
    dp(0) = 1;

    for (int coin : coins) {
      for (int x = coin; x < amount + 1; ++x) {
        dp(x) += dp(x - coin);
      }
    }
    return dp(amount);
  }
}

It seems to me that the first solution is just the top down variant of this algorithm.

I can’t seem to figure out why the first solution is exponential and the second one is not.
Where have I gone wrong in my understanding?

Is it actually printing the subsets vs counting that makes the difference here?

Is there any way print all combinations whilst preserving polynomial time?

My dynamic dns behaves weird

  • I have builded a raspberry pi ubuntu all-in-one server (ftp, apache, ssh, vnc) and used dynamic dns to be able to use it remotely. When i try to connect it from lan ip it connects successfully from my connected phone. Also when i try connect it remotely from dynamic dns with cellular data it works great. However, in other possibilities such as wifi as lan connected phone to dyndns doesnt work and i dont know why? I have checked for if ufw is setted up but it says inactive for all. Looked at router settings, it allows lan by http and https, also wan by http and https. It doesnt block that website and also connects to router login page by wifi (lan) so there is no problem about that. Using port forwarding, it seems port forwarding doesn’t work for some reason on lan device. By the way the error i get is connection timed out. Maybe it is about router settings or something else i dont know… Can you explain this?

dynamic – Estimation of parameters of limit cycles for systems of high-order differential equations (n> = 3)

There is a system of differential equations:

enter image description here

Then, call the limit cycle the projection of the phase trajectory onto the plane in a pairwise combination of state variables ($x-y,y-z,x-z$).

where $x,y,z$ – state variables, $a,b,c$ – constants.

Is it possible to use Mathematica to estimate the amplitude and frequency of the limit cycle? (it is possible by approximate numerical methods, most importantly, not graphical).

how to add new dynamic dropdown to billing/shipping address on create new order

how to add new dynamic dropdown to billing/shipping address on create new order from admin? I don’t know how to change anything on that page let alone add dynamic dropdown. I tried with attribute but it does not take its value dynamically. And how do you remove fields? I tried disabling them from customer configuration but its still there.

enter image description here

post meta – Dynamic tagged image galleries

I’m creating a website for a theater company, using Elementor, ACF, CPT UI, and Jet Engine.
One custom post type, “productions,” includes an ACF gallery of production images. Another Custom Post Type, “artist bios,” includes bio and past production information for all artists (about 200 artists in total). Both of these post types use an Elementor single template.

I’ve been trying to find a way to add a gallery of tagged images (attached to the individual artists’ names) to each bio page. The goal is that, as new production images are uploaded, the actors appearing in those images could be tagged, and those tagged images would automatically appear on their bio page gallery.

I’ve tried setting up a many-to-many relationship between bio pages and media attachments, but that hasn’t been successful. I added a relationship field to media attachments, allowing you to select artist bio page(s) to link the images to, but I can’t find any way to populate a gallery based on that information.

Any suggestions? I’d like this to be able to be managed automatically and dynamically if possible, given the volume of artists and images involved.

Dynamic Program for Tree based question

I’ll assume the tree has been rooted arbitrarily.

Taking y
our idea we let $OPT(v, 1)$ to be equal to the minimum weight of a vertex cover of the subtree rooted in $v$ that includes $v$, and $OPT(v, 0)$ the one where $v$ is not included.

$OPT(v,1)$ means you the weight of that cover includes $w(v)$ and also that we have already dealt with edges that touch $v$, so all you need are MWVC for each of the subtrees of its children. So for each children $u$ of $v$, you take the minimum between $OPT(u,0)$ and $OPT(u,1)$, and you add that to $OPT(v,1)$. This can be done bottom-up.

$OPT(v,0)$ means you need to select at least of the children of $v$. Let’s say you selected its child $u$. Then you add $OPT(u,1)$, and for every other child $u’$ you add $min(OPT(u’,0), OPT(u’,1))$. Of course you don’t know the best $u$ beforehand, so you just take the minimum of the results you get by picking different $u$‘s.

I’m sure this is enough for you to code it!

Wall Climbing Dynamic program – Computer Science Stack Exchange

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This post has been edited by xetang: Today, 01:16 AM