Symbolic minimum value

I have the following polynomial expression v1[z]:

-((5 L^3 q z)/(12 J Y)) + (L^2 q z^2)/(4 J Y) - (L q z^3)/(12 J Y) 

and I would like to symbolically evaluate its minimum in 0<z<L, with q>0, L>0, J>0 and Y>0. How can I do this?

I have tried the following numerical way:

minv1 = Minimize[{v1[z] /. q -> 1 /. L -> 1 /. J -> 1 /. Y -> 1,z > 0, z < 1}, z ] 

minimum bounding ellipsoid example does not work in 12.1.1.0

I am surprised that WRI never checks previous examples before shipping new versions. The following example is mentioned and highlighted on the webpage. It does not work in 12.1.1.0 and as far as I remember it used to work in older version 12.0 perhaps. Does someone else has trouble running this example with a fresh kernel. I have filed a bug report nevertheless Case number 4696657

https://www.wolfram.com/language/12/convex-optimization/minimal-bounding-ellipsoid.html?product=mathematica

I get this output below:

enter image description here

What is the highest minimum and maximum roll possible on a (Dex) sleight of hand/pickpocket check? [closed]


FOR THE KLEPTOMANIACS OUT THERE:

Let’s say we’re attempting to pickpocket something from someone and would like to reduce our chances of failure as much as possible. What is the highest minimum roll and maximum roll reliably possible in the game? In order to make this easier to replicate for others, I’m going to offer some restrictions:

  • Things that are not largely guaranteed in most games, such as: Boons, Artifacts, and Manuals.
  • This can be any level character up to 20th level, all that matters is that it still remains the highest min and max roll for the check.
  • Must be acting alone and receive assistance only from themselves, their items, feats, racial abilities, class features, etc.

These restrictions should ensure that the check can be performed reliably any time it is made by anyone with the proposed build in nearly any campaign with the average amount of magical items and quest rewards, etc.

I will post my own findings as an answer below. Feel free to use it as a frame for further answers, or offer one all your own.

Divide first n square numbers 1^2, 2^2, ……. n^2 into two groups such that absolute difference of the sum of the two groups is minimum [closed]

lets say Given input is n = 6 (n is as large as 100000) My task is to divide {1, 4, 9, 16, 25, 36} into two groups and PRINT these two groups

Possible Solution 1: dividing groups as {1, 9, 36} and {4, 16, 25} which gives abs diff as abs(46 – 45) = 1. So the minimum difference is 1 and the two groups are {1, 9, 36} and {4, 16, 25}

Possible Solution 2: Another Possible Solution is dividing groups as {9, 36} and {1, 4, 16, 25} which gives abs diff as abs(45 – 46) = 1. So the minimum difference is 1 and the two groups are {9, 36} and {1, 4, 16, 25}.

If there are multiple solutions we can print any one. Iam trying to solve it using https://www.geeksforgeeks.org/divide-1-n-two-groups-minimum-sum-difference/ but its not working.

I know that min difference is always 0 or 1 for n >= 6 but how to divide them into two groups.

And can we extend this problem to cubes, fourth powers, so on. if so what is the strategy used

Is there a minimum requirement for a character to have line of effect with Shadow Jaunt/Stride/Blink?

I’m pretty sure this has been asked already but I can’t find the question in question, so I hope I don’t break any rules by asking again.

The Shadow X maneuvers from Shadow Hand have a line of effect and line of sight requirement to work. Let’s say I want to teleport through a hole in a wall. According to both line rules, as long as I can see though the hole and there is nothing obstructing the path though the hole, I should be able to teleport though it.

Is there a minimum size the hole has to be for my character to be able to traverse the hole?

Oriented spanning tree of a directed multi-graph with minimum weight

I have problem of finding the minimum spanning tree of a simple graph, but the result is restricted by additional two types of condition:

  • There is a known root, which is s in the following example.
  • We know directions of some edges if they are chosen. These edges are chosen yet, or the problem becomes a Steiner tree problem.

Note that numbers on edges are their weights. So we will get s -> b -> c -> a if a normal min spanning tree is applied, but the direction of edge ac is wrong. On the other hand, we cannot use Chu–Liu/Edmonds’ algorithm for spanning arborescence of directed graphs, because we don’t know and cannot infer the direction of edge bc.

We can infer some edges’ directions according to the position of the root. For example, in the example, we know s -> b and s -> a.


Oriented Spanning Tree

In the final section of spanning tree, Wikipedia, oriented spanning tree is mentioned and a paper [levine2011sandpile] is referred. The problem fits the setting. It says:

Given a vertex v on a directed multigraph G, an oriented spanning tree T rooted at v is an acyclic subgraph of G in which every vertex other than v has outdegree 1.

Note that the term "outdegree" is a bit confusing, which I think should be "indegree". But it doesn’t matter, because it just restricts the simple subgraph to be a directed tree with root being source or sink.

For edges (in the original simple graph) whose directions are unknown, we add two directed edges between two vertices with inverse directions. Then we find the oriented spanning tree of this multi-graph. But it is not clear to me how an algorithm can be implemented according to that paper.


  • Levine, L. (2011). Sandpile groups and spanning trees of directed line graphs. Journal of Combinatorial Theory, Series A, 118(2), 350-364.
  • https://en.wikipedia.org/wiki/Spanning_tree

minimum travel from point to point with incremental steps

It’s my first time to make a question here. I have a curious problem about algorithm, in the center of Cartesian plane (0,0) I need to go to another point (x,y) but I only can use horizontaly and verticaly steps and this steps increases one by one.

For example, I need to go to (1,1) point and steps are:

  • Go to (1,0), a step of 1 unit.
  • Go to (1,-2) a step of 2 units.
  • Finally, go to (1,1) a step of 3 units.

Obviously, there are several ways to go to a point from center but the problem needs the minimal.

Are there a formula or an algorithm to answer this question? Thanks for read this and for your questions.

Multi-dimensional Knapsack with Minimum Value constraints for Dimensions

In MDK, we have a vector $ W = \{W_1, W_2, …, W_d\}$ where each element corresponds to the maximum weight for the respective dimension in the knapsack.

I want to add a conditional constraint: $ V = {V_1, V_2, …, V_d}$ , where each $ i$ -th dimension in the knapsack must have a value sum greater than threshold $ V_i$ . I am not so much concerned with the total value sum.

I would like to show this problem is NP-hard. My intuition is that the additional constraint makes this problem harder than MKD and therefore is NP-hard. But clearly this doesn’t constitute a formal proof.