real analysis – Finding a metric that contracts on average in one step

I am looking at different configurations of particles on a $d$-ary tree with $n$ nodes and $lfloor frac{n}{2}rfloor$ particles. If we start at a configuration $x$, an edge is selected uniformly at random and if that edge has one particle and one empty space, then the particle and empty space are swapped. If the selected edge has two particles or no particles, then we remain at $x$. In this case, I am trying to find a metric $rho (x,y)$ between configurations $x$ and $y$ that differ along a single edge such that $(1)$ is satisfied.

$$mathbb{E} (rho(X_1,Y_1))-rho(x,y) leq 0 tag{1}$$

where $X_1,Y_1$ correspond to one move starting from $x,y$ respectively and $mathbb{E}$ is the expectation. The following figure gives an example for $d=2$. Notice that $x,y$ differ only along one edge.

enter image description here

For the above figure,

$$mathbb{E} (rho(X_1,Y_1))-rho(x,y) = -frac{1}{n-1}rho(x,y)+frac{d-1}{n-1}(rho(x,y)+rho(y,y^{+}))+frac{d}{n-1}(rho(x,y)+rho(x,x^{+}))+frac{1}{n-1}(rho(x,y)+rho(y,y^{-})) tag{2}$$

Because we decrease the distance in one move by $rho(x,y)$ only when we select the edge along which $x,y$ differ, which happens with probability $frac{1}{n-1}$ and in all other cases we increase the distance after one move. Let $x,y$ differ at height $h$ from the root. If we select another edge at height $h$ apart from the edge along which $x,y$ differ, which happens with probability $frac{d-1}{n-1}$, then $x$ remains at $x$ and $y$ moves to $y^{+}$ and that explains the second term in $(2)$. If we select an edge at height $h+1$, which happens with probability $frac{d}{n-1}$, then $x$ moves to $x^+$ and $y$ remains at $y$ and that explains the third term in $(2)$. Similarly, if we select an edge at height $h-1$, which happens with probability $frac{1}{n-1}$, then $x$ remains at $x$ and $y$ moves to $y^-$ and that explains the fourth term in $(2)$.

My attempt: I have tried $rho(x,y)=h$ and $rho(x,y)=alpha^{h}$ for $alpha>1$ a constant but both of them did not satisfy $(1)$ mainly for the reason that the distance increases in too many ways while the distance decreases only when we select one particular edge i.e. the one along which $x,y$ differ. Any thoughts about how to find $rho(x,y)$? Or how to pose this as a different problem?

Geometry Finding measure of a segment using the segments given.

So the question is: Suppose A, B, C, D, and E are points in a certain absolute geometry with AB = BC = 7, BD = BE = 3,CD = 8, A − B − C, and D − B − E. Prove AE = 8.

The book I am using is David C. Kay’s College Geometry A discovery Approach.

1]I used the definition of betweenness in a metric space to solve this problem. Betweenness A-B-C of points states “We say B is between A and C if AB+BC=AC.”

Which approach do you think would be in the right direction?

  1. Do you think I can use SSS Theorem to help prove the measure of the segment AE? Or can I not use this because we do not know if the points make a triangle.

sql – Understanding the difference in output for finding Revenue by Store in Sakila Database

I am trying to find the Revenue by Store in Sakila Database(Demo Database from MySQL).

Sakila Structure

I am doing this :

SELECT store.store_id, SUM(payment.amount) as total_revenue
FROM rental
INNER JOIN payment on rental.rental_id = payment.rental_id
INNER JOIN staff on payment.staff_id = staff.staff_id
INNER JOIN store on staff.store_id = store.store_id
GROUP BY store.store_id
ORDER BY 2

This is the result of the above query:

+--------+-----------+
|store_id|sum(amount)|
+--------+-----------+
|1       |33482.50   |
|2       |33924.06   |
+--------+-----------+

But when I am doing this :

SELECT i.store_id, SUM(p.amount) as total_revenue
FROM rental r
INNER JOIN payment p ON r.rental_id = p.rental_id
INNER JOIN inventory i ON i.inventory_id = r.inventory_id
GROUP BY i.store_id
ORDER BY 2

I am getting this result:

+--------+-----------+
|store_id|sum(amount)|
+--------+-----------+
|1       |33679.79   |
|2       |33726.77   |
+--------+-----------+

Can some one explain what is the best way to find out the Total Revenue by Store ?

linear algebra – Finding a real $3times3$ matrix with eigenvalues $1$, $i$ and $-i$ geometrically

In a problem I’m trying to find a real-valued $3times3$ matrix that has the eigenvalues $1$, $i$ and $-i$ (which must mean that the corresponding eigenvectors for $i$ and $-i$ must be complex). I already know that one way to approach this is to write out a general $3times3$ matrix of the form

$$A=
begin{pmatrix}
a & b & c \
d & e & f \
g & h & j
end{pmatrix}$$

then evaluate the determinant of $A-lambda I$ and set it equal to $0$ to form a characteristic polynomial for the eigenvalues and then choose $a,b,c,d,e,f,g,h,j$ such that the polynomial becomes $(lambda-1)(lambda^2+1)$ to give the required eigenvalues as roots.

However, I’m looking for a geometric way to find such a matrix (which is the approach hinted at in the problem). I thought maybe some kind of complex plane transformation might work, but I wasn’t sure how a $3times3$ matrix would apply in such a situation.

How can I find such a matrix geometrically, without having to do lots of algebra as in the method outlined above?

algorithms – Finding path with best distance/cost ratio from a node in a graph

We have a weighted graph, where each node is a city. And the edges between the nodes are a pair of floating-point values (distance and cost of travel).

Given a node/city, describe an algorithm that can find the path with the largest distance/cost ratio from said node/city. Note that the path can traverse multiple nodes.

I need help with this (for both directed and undirected graphs). I stumbled across this question in a lecture note from my university (I graduated but am self-studying), and have no idea how to solve this.

There is no limitation on how to store the graphs, though I only know of adjacency matrix and adjacency list.

Finding maximum possible value

Question

In class, we recently learned about inequalities between the arithmetic mean, geometric mean, harmonic mean, and quadratic mean. I started by stating that the given equation is less than or equal to (x+z+1/y)/3. However, I dont know where to go from here. How can I relate the two equations we are given to finish this problem?

differential equations – Finding the Roots of a Function

I have a system of differential equations, for x (t), y (t) and z (t), I solve the system in a simple way.
However from these solutions I have a function like this:

a(t)=(-2*y^2/z(t)^2)(y(t)^2/2+x(t)^2/2)

So I solve this equation a(t) and when I plot the graph it looks like thissee that it fluctuates a lot

I’m interested in finding out the coordinates when it intercepts the x axis, that is, it zeroes.
My code for plotting the graph was simple:

Plot(a(t), {t, 0, 50})

How can I do this?

Finding a good broker – General Forex Questions & Help

Finding a right broker is a troublesome occupation. Here are several tips to pick a standard broker. The critical thing you have to see is the true blue body of the broker. It is key bit of a broker. By then yield for leverage, edge and spreads. So in addition check beginning deposit stray pieces. Moreover, that it is so normal to deposit and withdraw your cash. Furthermore check their client leverage. I am also a forex trader. In like way, for my trading I have gotten Eurotrader as it is a regulated forex broker. The broker offers bewildering offers and relationship for their customers. 

geometry – Finding the position of a point inside a triangle, based on the position of a point in another triangle

I’m trying to translate a UV coordinate into a world coordinate. (this is not in a shader, it’s in GDScript in Godot, to bake out a texture based on some raycasts).

So I find out which triangle in the UV map the coordinate is in and then I find the corresponding triangle in the mesh and get it’s world position, that’s all good.

But then I can’t figure out how to translate the points relative position inside the UV triangle to the mesh triangle.

I’ve been trying to use the distance from the point to each of the triangle points as weights to how much of each of the mesh points I should use to find the point I’m looking for, but having no luck.

Any help would be much appreciated.

Not sure how best to illustrate this. But imagine you had a triangle where you knew the corner points a, b, and c, and a position inside it p. And you had another triangle where you knew the corner points A, B and C, how would you find a point that was relative in the same as p is to abc in triangle ABC.

enter image description here