Empowerment

Besides being an interesting problem mathematically the solution to the Minimum Links problem illustrates how knowledge and skill act as enablers. They allow us to choose the best among many possible options. Evolution has allowed the cell to acquire a "bag of tricks" which empowers it to adapt and respond to changes in its environment. Knowledge plays a similar role in society. Change alone may not be enough. We have to be selective in what we empower and avoid options which may lead to our downfall.


Consider the following set of synomyns and antonyms:

empowerment - embargo

enabler - disabler

catalyst - inhibitor

strength - weakness

able - inept

competence - incompetence


We probably all need to improve our level of competence.

Errata: Circle Proof for Fixed Angle

I found that the proof gave the wrong vector for the center of the circle and traced the error to the inverse of M. The proof is essentially the same but now checks with the previous result.

Path is a Circular Arc for Fixed Angle between Links

It was known to Euclid that the lines drawn from the ends of the diameter to a point on a circle had a 90° included angle. This can be generalized but was only assumed in the second solution to the minimum links problem. The proof is as follows. If one assumes that the link from A to the connection point, X, is a vector with magnitude a and angle α with respect to the x axis and that from X to B is of magnitude b and angle β then B is the product of a matrix, M, and a vector containing a and b as components. The included angle between the segments is assumed to be δ and if α is known then so is β.

One can find the inverse of M if α, β and δ are known.

Since β is known in terms of α and δ, a and b can be found from another matrix operating on the unit vector defined by α. From trigonometry we also know that x and y are proportional to a.

From the first row of the matrix equation for a and b one sees that the square of a is a linear combination of x and y but by the Pythagorean Theorem it is also the sum of the squares of x and y which enables us to show that the equation for the x and y is that of a circle.

Exact Solution for the Minimum Links Problem

It is possible to solve the Minimum Links Problem exactly without having to resort to using the curved lines. The center of the circle through two of the points, K, can be determined as follows with σ = 1 or -1 indicating whether the center is above or below the line joining the two points respectively as well as the radius of curvature, ρ.

One can eliminate the X·X term from the two equations for the circles to get a vector equation for a straight line. X can then be defined as a linear combination of ΔK and its normal ΔK' and the unknown parameters, λ and μ, can be found by substituting into the equations for the line and one of the circles.

Edit: This solution and the previous one are alternative solutions to Problem 1.14 in Schaum's Outline of Operations Research by Richard Bronson.

Minimizing the Length of Connecting Lines

There is a problem in Operations Research in which one is given three points and one has to find the connecting point, X, which makes the total length of the lines is a minimum.

The objective function, f, is the sum of the lengths of the lines and at the optimal point its variation is zero for all possible variations in position and as a consequence the variational coefficients are also zero.

These two conditions can be simplified to show that the sum of the unit vectors along the connecting lines is also zero. This means that the angles between these directions are all 120 degrees.





One can solve for the unknown lengths of two of the lines if the angle of one of them is assumed and in this way one can obtain two curves which intersect at the optimal point. The epsilons, ε, denote the unit vectors along the lines connecting at the optimal point.

Using a solve block one can easily find the optimal point. The function, r, solves for the unknown length of the lines for a given angle, θ, and the find function solves for the optimum angle yielding the solution, X.

The solution appears to be the intersection of two circles with one point of intersection at the optimal point and another point of intersection for which the distance from the common point, A, is zero. Circles are curves of constant curvature which may be due the angle between the two links being a constant.

"Robinson Crusoe" Economies

"Let us look more closely at the type of economy which is represented by the 'Robinson Crusoe' model, that is an economy of an isolated single person or otherwise organized under a single will. This economy is confronted with certain quantities of commodities and a number of wants which they may satisy. The problem is to obtain a maximum satisfaction. This is--considering in particular our above assumption of the numerical character of utility--indeed an ordinary maximum problem, its difficulty depending apparently on the number of variables and on the nature of the function to be maximized; but this is more of a practical difficulty than a theoretical one." --Von Neumann and Morgenstern, Theory of Games and Economic Behavior, 1944, pp. 9-10.

This quote exemplifies some of the problems that we are likely to face when dealing with the challenge of containing the oil spill in the Gulf of Mexico. It is not business as usual. We are lacking the means to achieve an end. We have to create an economy practically from scratch. Wasted effort is contrary to our goal. We have to do the best that we can under the situation. We not only have to reduce the time it takes to stop the flow of oil but we also have to try to reduce the rate at which the costs accumulate. This is probably the best plan of attack. We need to look for what is missing and strive to fill the gap.

Obama's plans for change face similar problems. We don't always know in advance what difficulties we will face. There are hazards which need to be avoided and factored into the "equation." There are barriers to progress.

What Are Our Objectives in the Gulf?

There are many of us who would like to see the federal government get more involved in mitigating the effects of the oil spill in the Gulf of Mexico. Obama seems to be reluctant to take responsibility for stopping the oil leak. Is he trying to distance himself from the problem or are there pertinent legal issues to be considered? The oil rig was outside the U.S. 3 mile limit. BP is an multinational oil company. The problem requires a lot of coordination.

From the point of view of operations research one should try to reduce the "cost" of the oil spill. One can measure cost in terms of the monetary cost or the cost to the environment. Either way one has an objective function which one tries to minimize. There are a number of methods such as the Simplex Method and Algorithm which address the best way of proceeding when constraints are present. The basic idea involved is that of gradient descent in which one follows the path which most quickly reduces the costs. In this case containing the oil spill should be given the highest priority by all parties involved. We need to end the release of oil into the environment which could mean capturing it and placing it in containment vessels. There is probably a problem with storage capacity.

If this is an international problem then it requires an international approach with someone in charge of coordinating activities among nations to get a handle on the problem. It may be that no one has the power or resources to act alone. We need to come together on this. It is the only way to get on top of the situation.

Variation on a Theme

A somewhat more stylish trident based on the same formula used before can generated by means of a color-mapped surface plot.

Design of a Symbol for "Such That"

Looking at the handwritten symbol for "such that" we see that it is not exactly the same as Poseidon's symbol of power in which the trident conveys the power to "take" or "choose." This is consistent with "such that" in that it allows us to take certain values of some variable but if we look carefully we see that it is composed of an arc and a horizontal line. It is as if a bit of the sense of "compass" has been mixed in from our palette of ideas. Compass includes the ideas of "scope" or "bounds" and so the mathematical trident suggests limits on power or constraint in its use. The etymology of "compass" indicates that it means a "stepping together" which would convey the idea that all changes take place simultaneously.

A trident formed by combining a semicircle and a horizontal line can be defined mathematically as the zeros of function defined on a plane. The equation is,

which produces a simplified trident that can be "morphed" to form a font.

The question you may ask is, "Have we just recreated the 'such that' symbol?"