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?"

Does LaTeX Contain a "such that" Symbol?

The phrase such that is commonly used in mathematics to designate conditions that a statement is subject to. The handwritten symbol for such that is similar to a right parenthesis with a strikethrough and can be created using MathType thus,

such that symbol created with MathType

Math textbook often use a symbol similar to a backwards epsilon to represent such that but it is a flipped version of the symbol used to represent set membership. In LateX, a math typesetting language, the names for these symbols are \ni and \in respectively.

such that in a 1967 Addison-Wesley math book

People have tried to come up with ways to better represent such that in print. In TeX one can use a macro to combine a \ni with a horizontal line and this is used as a definition. LaTex apparently lacks this capability.

The current usage can potentially lead to confusion. The problem is that one is using a symbol from set theory to represent a control operator. One expression is subject to the control of another. This relationship suggests that the symbol used may be a simplified representation of a trident which was a symbol for power in the religion of ancient Greece. The trident as a symbol is directional and it is clear from context which expression exercises control. A parenthesis with strikethrough would therefore be a better symbol for such that.

Efforts are underway to develop a new version of LaTeX. Perhaps they could include strikeout text within a math statement so that one can create a unique such that symbol. From the computer science perspective it would be better to have a separate "entity," an amperstand-number combination, to designate a sideways trident like symbol for such that.

I should note that both MathType and Wolfram Alpha use a LaTeX \backepsilon or ∍, Extended HTML entity &8717;, for such that. The entity corresponding to \ni or ∋ is ∋. (If you see square boxes instead of symbols, try viewing this page in another browser such as Google Chrome or Apple's Safari.)

Are We Lacking in Leadership?

There is some indication that the country is lacking in leadership. By leadership I mean the anticipation of events, preparedness and readiness to meet the challenges. Leadership requires a certain degree of foresight.

Government can fail through inaction and over reaction. I do not think that high office is just an honorary position. There is a job to be done which requires a set of skills. Responding to a crisis after the fact and then throwing money at a problem is not the economical approach. The country's elections have been a race between the Tortoise and the Hare with the roles often trading places. Critics rule by letting a crisis develop and then going around and telling people that they are bad. Revolutionaries tend to favor more change than is necessary. Proper leadership requires a response fitting the situation.

We can't expect to get ahead by waging a war of attrition. This has gone on too long. The movers and doers need a freer rein.