graph theory: is this a characterization of a tournament score?

For all $ n in mathbb N $, $ Delta_n = (0,2,3, …, n) $,$ D_n = X + X ^ 2 + …. X ^ $ Y $ I_n = mathbb N cap (0, n) $

For all
$ f in mathbb N ^ {I_n} $ strictly growing, we say that $ P = Sigma_ {i in I_n} a_i.X ^ {s_i} en mathbb Z (X) $ is 1-equivalent to $ P = Sigma_ {i in I_n} a_i.X ^ {$} Y 2 equivalents to $ P = Sigma_ {i in I_n} to _ { sigma (i)}. X ^ s_i + D_ {s_n} $ For any $ sigma $ permutation of $ I_ {s_n} $

$ Z subset mathbb Z (X) $ is the kind of $ 0 $ according to him $ 3-equivalence $ Witch is defined as the equivalence ratio generated by these two superior equivalence relationships.

We say that $ P = Sigma_ {i in I_n} a_i.X ^ {i} en mathbb Z (X) $,have braquetting property yes for every positive integer $ k leq n $
we have

$ Sigma_ {i in I_k} a_i geq 0 $ Y $ Sigma_ {i in I_n} a_i = 0 $

We say he has the property score If any $ Q $ that is to say $ 3 equivalent $ to $ P $ have the braquetting property

One can show that the elements of $ Z $ have the property score, and the question is the reciprocal:

Is any $ P in mathbb Z (X) $ which has the property of punctuation, in Z?

Note that $ (a_0, a_1 + 1, a_2 + 2, …, a_n + n) $ it is the list of exits of a complete digraph (tournament) if and only if $ a_0 + a_1X + a_2X ^ 2 + …. + a_nX ^ n $ have the property score

graph theory: what is the best method to classify many multiple feature elements into multi feature containers?

• There are 100 items to classify in just over 100 containers, where no container has more than one item and not all containers are necessarily full, but each item must be placed.
• There are certain items that are of "quality" high enough to be placed in any container, and there are also items that may not be of sufficient quality for certain exclusive high quality containers. There are no exclusive low quality containers.
• Containers are only open during certain intervals throughout the day and, therefore, certain items are not compatible with certain containers because items are not available during open container schedules. When a container is not open, the item cannot be placed inside.
• The items also have preferences on the containers in which they are placed, and similarly, the containers have preferences on the items to be stored.

Assuming that all the required information about each "item" and "container" is known, what is the best method to classify large quantities of items into approximately the same number of containers for each item to have a container, there are no locations to do That an item is incompatible with a container (and vice versa) and maximizes total "happiness" in each pair of items and containers?

Algorithms: minimum route coverage in a directed acyclic graph

Given a weighted directed acyclic graph G = (V, D, W) and a set of arches RE & # 39; of rewhere the weights of W They are at the vertices. The problem is partitioning Sun on a minimum number of paths separated from vertices that cover all vertices of Sun subject to restrictions that:

  1. the weight of each route is maximum k.
  2. each route must include at least one edge of D & # 39;

What is the complexity of this problem?

Complexity of time – Algorithm 1 Remove node vk from graph G represented as an adjacency matrix A

The function accepts an adjacency matrix A, which represents a graph G, and an integer k, and returns an adjacency matrix A representing the graph G which is the result of eliminating the k-th node vk of G.

Question: Provide an implementation, in pseudocode, of the NodeRemoval function. Give time the complexity of its implementation. Is it the best possible complexity for this task? Briefly describe your reasoning

Consider the following: Algorithm 1 Remove node vk from graph G represented as an adjacency matrix A

Require: A ∈ {0, 1} (power) n × n, k ∈ N, k ≤ n

Make sure: A 0 ∈ {0, 1} (power) (n – 1) × (n – 1)

1: NodeRemoval function (A, k)

two:. . .

3: return A

4: final function

graph – Matlab, rotate a scatter marker

I find myself doing graphics in matlab with the scatter command
This is discrete information and I am using markers of different colors to differentiate between different samples.
I would like to know if it is possible to add some way to add angles to these markers, to stop rotating the marker '>'

Longer trajectory in an acyclic graph directed with restrictions

Given a directed weighted acyclic graph G = (V, D, W) and a subset of edges RE & # 39; of re. The problem is finding the longest path in Sun that goes through exactly one edge of RE & # 39;.

What is the complexity of this problem?

differential equations: following maximum graph with NDSolve results

I want to produce a next maximum plot, that is, I want to draw all the local maximums $ i_ {t + 1} $ against the local maximu $ i_t $ to verify chaotic behavior.

In particular, I have the following code:

(*Parameters*)eps = 1.4434; m = 0.3; c11 = 0.1732; maxCellMeasure = 
pde11 := 
  D(pp(t, x), t) == 
   0.05*Laplacian(pp(t, x), {x}) + 
    pp(t, x)*(1 - c11*pp(t, x) - z(t, x)/(1 + pp(t, x)^2));
pde21 := D(z(t, x), t) == 
   0.05*Laplacian(z(t, x), {x}) + 
    z(t, x)*(eps*pp(t, x)/(1 + pp(t, x)^2) - m);
(*Initial conditions*)
lo = 98;
hi = 102;
domlen = 200;
ic11(x_) := Which(x > lo && x < hi, 6, True, 0);
ic21(x_) := Which(x < hi && x > lo, 0.5, True, 1/c11);
(*Numerical approximation using NDSolve with zero-flux boundary 
{solp, solz} = 
 Monitor(NDSolve({pde11, pde21, z(0, x) == ic11(x), 
    pp(0, x) == ic21(x)}, {pp, z}, {t, 0, 1000}, {x, 0, domlen}, 
   EvaluationMonitor :> (monitor = Row({"t = ", CForm(t)}))), monitor)

Therefore, first I need to find the local maximums of say solp and then I need to plot all the maximums $ i_ {t + 1} $ against $ i_t $.

I think the solution should be quite simple. I checked EventLocator, but I couldn't find out how to apply it in my code.

Thanks for any help!

Number of elements in a set of all possible discrete secondary graphs of a simple graph

Yes $ G $ it's a simple chart with $ n $ vertices, then the number of all possible discrete subgraphs of $ G $ is $ 2 ^ n $ (since each vertex of $ G $ it will be a discrete sub graphic of $ G $) But, I'm not sure if it's true?

machine learning: what is a good way to find an optimal or optimal state in a graph without heuristics or values?

What is a good way to find the optimal solution in a graph without heuristics or knowledge of the value of that state until we are there? Is there a good way to maybe create an estimate? I have examined things like the iteration of values ​​and the iteration of policies, as well as Q-learning, but what happens if another problem is to evaluate the true reward of a state that is very expensive?

graph theory – Iterated inverse structures: polynomial representation of entire pre-image partitions in Sigma Matrices (reference request)

I am studying iterated image function structures in a finite set.

The main structure of interest to me, the Sigma Matrix, is derived from a matrix that lists the preimage sets of wise elements at an increasing inverse depth. This intermediate matrix is ​​the "preimage matrix" (shown as $ P $ down)

$ P = left ( begin {array} {cccc}
f ^ 1 (x_ {1}) and f ^ {2} (x_ {1}) and cdots & f ^ {n) (x_ {1}) \
f ^ – 1 (x_ {2}) & ddots \
vdots \
f ^ – 1 (x_ {n}) & & & f ^ {- n} (x_ {n})
end {array} right) $

Then we look at the sizes of such matrix elements.
Give the "sigma" matrix $ Sigma $ with tickets

$ Sigma = left ( begin {array} {cccc}
mid f ^ – 1 (x_ {1}) mid & cdots & mid f ^ {- n} (x_ {1}) mid \
& ddots \
vdots \
mid f ^ -1 (x_ {n}) mid & & mid f ^ {n} (x_ {n}) mid
end {array} right) $

  • Result 1: Each column forms an integer partition of n = dom (f) since each column of the sigma matrix has sum n.

Let each column of $ Sigma $ be represented by its own polynomial where: yes $ col (1) = (a, b, c) $ then the associated polynomial for $ Sigma_ {X1} $ is
$ y_1 = ax 0 + bx 1 + cx 2

  • Result 2: Now taking all the columns, for any sigma matrix size (in any size domain) and deriving all the polynomials, they have a solution in $ (1, n) $.

Question: There It seems Always be one and only solution for these systems of equations. I would appreciate a reference to this type of exploration

For more information, see my last question about sigma matrices: previous question