Quickie Concerning US Drug Policy

While I do not necessarily condone, nor disapprove of drug use I would like to ask a simple question.

If the state of California has legalized the use of medical marijuana, and the federal government still retains the right to persecute the crime of marijuana use and possession…then who wins out exactly? Rather, does this opposing dichotomy of laws even make the least bit of sense?

This type of counter-active legislation on the state and federal level, without some intervention by either the Supreme Court or the Congress, will lead to increased legal spending and will help to choke already overloaded courtrooms.

I therefore suggest a very simple maneuver: get the legislation straight. Doesn’t matter which direction, just straighten it out either way so that this type of unnecessary spending doesn’t have to exist. Waste of money is what it is.

I’m Finally Back! Also: On the Nature of the Average of Infinite Sums

It’s been quite awhile, but this is the problem of having a life. There’s really no excuse for not being here for this long, and I apologize for it.

Also I’m loath not to put up another blog post after reading this http://www.xkcd.com/621/

Makes me feel particularly exciting…

Ok not really.

But onwards! I figured today we should go through a little thought experiment:

Let’s ask the question, “What is the average of infinitely many things?”

Now the classic definition of an average looks like this:

Given a sequence:

The average of that sequence is the series of the sequence divided by the number of terms:

Let Av represent the average of a sequence.

However, if we have an infinite sequence, then the series:

Can either converge (that is, the series has a finite value) or it diverges (the series has a value of either positive or negative infinity). Note that in this case, must either be equal to or the step must be infinitesimally small.

Noting that these are two distinct and special cases, it is clear that we will have to analyze these separately.

Convergent Series

For any convergent series:

Recall:

Therefore by the classical definition of an average, the sum of a convergent infinite series is zero. Note that here is determined to be zero because the number of terms in the sequence is infinite.

Divergent Series

For any divergent series:

Which is undefined since positive or negative infinity could converge to anywhere.

Therefore under the classical definition, the average of an infinite series is either zero or undefined.

Negation of the Convergent Sum Proof

In order to negate the proofs above, we need to find a situation that is unnecessarily true in those proofs. The easiest way to do that is to consider what a series actually is. In this case a series is the sum of infinitely many things! Those of you who know calculus should be perking up your ears by now. This is the conceptual definition of an integral.

Now of course integrals apply to functions, so in order to apply the integral in place of summation notation the following assumption has to be made:

The values of the sequence:

Are given by the function:

For any value within a specific range denoted by and , if the step in the summation notation is infinitely small.

Recall the formula for the average value of a function:

Since we have assumed that the values of the sequence $a_n$ are given by the function we can replace the summation notation with the integral:

Now if and are both finite, then it is possible to attain a meaningful average value if the integral of the function also converges:

Given:

Recall:

Observe that if and are both finite, then must also be finite:

Since the values used in this case are all finite (excluding the number of terms utilized), the final average is also finite. I also did not disprove that the average value of whatever function is being utilized is not zero, but rather since , , and are all finite values and there is no division of a finite value by infinity the average value does not have to be zero. Therefore the conclusion reached here contradicts the conclusion above, and proves that infinite sums can have meaningful average values, but only on the condition that the range within which the infinite sum exists is finite. That is to say: averages as defined by the definitions utilized can only operate on either a finite number of values or a finite range within which an infinite number of terms exists.

The Set as Represented by a Function with Relation to Averages

Interestingly enough, we should note that for ANY set consisting of real numbers, there is an equivalent function such that the average of that function is the true average of that set.

That is to say:

Given a set such that:

Where is the number of terms in that set.

There exists a function such that:

In order for this to be the case note that:

Such that

In order for this to be true then the set must follow the function , implying that they both either converge or diverge.

To show that this is true, let us create a new set such that:

Where the set consists of the all the solutions to the function .

In order for the set to follow the function $f(x)$, then both it and the set need to pass the Limit Comparison Test.

However note that we cannot simply apply the limit comparison test normally because the cardinality of the set is much higher than the cardinality of the set . That is to say that the set is much larger than set in the number of terms.

To resolve this, note that the set has certain elements that will appear in the set with a certain regularity. That is to say that:

Where is a constant value and represents the current iteration. Since also denotes the current iteration, I could have used it instead of but separating the variable and then defining it makes things more clear I think.

If we rewrite the set so that it reads:

Where x is the actual x value utilized, then we can say that:

Where

Then, by the limit comparison test:

Therefore the set follows the given function and therefore the average applies, but only in the case where is a constant for any given value of . That is to say that the values of the set $S_n$ follow the function with a regular interval rather than being simply random values from that function. Therefore it is actually possible to take the average value of a set by using the method of the average value of a function on an interval.

This Week’s Post Notice

This week’s post will not be going up as planned. I’m currently conducting some very engrossing and time-consuming research alongside my school work and as a result will not have time to formulate anything of any quality.

However, I will submit a website that all of you can visit that has a very nice mathematical explanation of the economic crisis, various credit plans, economic mathematics, general mathematics (from pre-algebra to differential equations), and covering some of the most complex topics in introductory physics:

www.khanacademy.org

Very high quality, I strongly recommend it.

Once the research is done, I should be able to resume posting as scheduled. Sorry for the long hiatus.

Posting Apologies

I did not make a post last week, nor will I be able to this week. Exams for summer classes.

Again I apologize and should back into the swing of things this weekend.

Relating Coefficients of n Length to Integers of All-Ones Digits of n Length

Today we’ll be taking a purely mathematical bent, and be exploring an interesting phenomenon concerning the coefficients of expanded form n-length polynomials and numbers consisting of all-ones digits.

For any real polynomial of the form:

Such that Z is any real number or variable defined within the real numbers; it is possible to predict the coefficients of the polynomial when it is raised to any integer power p, taking the form:

In order to predict the coefficients this procedure is followed:

First observe that Pascal’s Triangle is formed like so:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

…and so on

Pascal’s Triangle refers to the coefficients attached to each variable in a polynomial of the form:

Where p is the value of the specific row of Pascal’s Triangle.

Also note that 11 raised to a power p yields Pascal’s Triangle for any length, with addition errors:

with summation errors

In order to correct for the summation errors in the powers of 11, the following substitution can be made:

Let

Then

By raising  to differing integer powers p, the powers of Pascal’s Triangle are revealed to be the coefficients of the resultant polynomial:

And so on, again noting that the pattern becomes non-obvious at and beyond.

By noting that powers of 111 yield values that represent the powers of polynomials of length , it might be possible to attempt to employ the following substitution:

Hypothesizing that would yield the appropriate coefficients. When simplified this form is non-obvious. It is necessary to define a variable  and use y instead of  in order to make the expansion yield the appropriate coefficients in a form that is obvious:

Noting that the coefficients are non-obvious starting at .

The same pattern holds true for polynomials of length ; a new variable  must be defined and substituted in order for the expansion to yield the appropriate coefficients.

I hypothesize that this pattern holds true for any polynomial for any length.

Therefore in order to predict the appropriate coefficients for any polynomial of length n, for polynomial functions that take the form:

Let

Then

Such that it’s expansion yields:

The expansion of:

Will yield:

Such that the coefficients are appropriate prior to substitution and simplification.

Substituting and (and so on) for and generalizing the form will yield:

Which relates powers of all-ones numbers to the coefficients of polynomial functions.

This suggests that a decomposition may be possible, which would easily predict coefficients for any polynomial of length n for any integer power p. Note that a decomposition is not implied.

What this decomposition might be, I have no idea.

On the Question of the Value of Human Life

Choosing a more philosophic bent today, we will discuss the value of human life.

The question is extremely controversial and supposedly serves a number of uses in real life that are subtle, yet profound. Consider the question of a city council who must decide between constructing a new school in one section of town or constructing a new bridge in another section of town. Assuming all else is equal, or that they are purely interested in the problems as service to the public rather than interested in votes, then the question leads them to ask which is more important at that particular moment in time?

For a less abstract, and much more real life example, let us consider the problem of fishing off the coast of California. In certain sections of California during the last two decades or so a shift has occurred: there is increased agricultural activity in the region. This change has happened on a large scale, resulting in a major net increase of water being drained from the rivers and reservoirs of the California coast. This increased water uptake has resulted in less water exiting out of the mouths of the rivers and therefore resulted in much less water around river deltas. River deltas make for breeding grounds for many types of fish. Some scientists have hypothesized that the numerous impacts upon the fish populations due to the drop in water being dumped into the sea at these points has resulted in a drop in fishing uptake due to a lack of fish.

There is a confirmed drop in the fish population, which has damaged the fishing industry. Some fishermen have gone out of business and a number of fishing ports are in dire straits.  While there was an ordinance passed which required there to be a minimum amount of water exiting out of the river mouths each year, it has not been well enforced. The continually increasing agriculture has not helped matters.

For a city council member, or a state senator who must deal with this matter, he or she must consider numerous questions. One of them is the value of each population. In this example, there is no win win situation. The amount of water in the rivers each year is constant, and with the continually increasing human population and demand for potable water for uses not only for drinking and bathing and fire-fighting, but for watering of lawns, fountains, water parks, pools, and otherwise, there is continually increasing strain on the reservoirs of California.

Therefore the ideal solution, that the farmers and the fishermen simply share the required amount of water, cannot be achieved if all else remains as it is.  The situation between the two parties has resulted in a zero-sum game: if the farmers get the water, the fishemen have no fish (hypothetically speaking); if the fishermen get the water, the farmers can’t grow their crops effectively.

The standard of living of both parties is dependent upon their livelihoods, and their livelihoods are dependent upon the choices of the governing politicians. Therefore the governing politicians must consider which party is more valuable to the state, and the other party has to pay the price for their leader’s choices.

In this matter the value of human life is something that the politicians assign in the act of deciding which party will receive the water. Basically it looks like this:

If the farmers get the water:

Whereas if the fishermen get the water:

The leaders have decided that, based on various criteria that are not necessarily objective, one party is more important than the other. This implies that the lives of one party are more important than the lives of another, doesn’t it?

Further exploration of the question reveals that this may not actually be the case.

Let us make an assumption:

The value of any individual human life is infinite, or:

This implies that the value of multiple human lives N is:

Now let us assume that the value of human life, while immeasurably infinite, is also variable. This implies that the true value of a group of human lives is not calculated by the above equation; in fact it requires that the above equation be incorrect (and this is true because it now requires rectification). The new equation looks like this:

And the rectified form for a population of N size is:

Where:

Showing that the calculation involving N is simply a reversal of finding the average, and is therefore redundant, or that the value of any individual or group of individuals is not dependent on the number of individuals present.

However if we consider the real life situation that presents itself in a hospital emergency room everyday we encounter a system that seems to counteract our understanding so far: the medical practice of triage. Triage is basically the idea that medical personnel should direct their resources to those who need it most. This is an ethically charged concept that only becomes even more ethically charged when we consider the fact that in extreme situations an even more complicated version of triage in which medical supplies are stretched thin. In this scenario medical personnel may decide to deny treatment to those considered to be beyond help to focus on those in dire straits. This goes beyond first example, in which patients were merely sorted by priority. In the second level the prioritizing process actively results in the peoples’ deaths.  Following the ethics that we were taught from childhood or the ones presented on Saturday morning cartoons (for those of us born in the United States) the “correct” answer is that the medical personnel should try to save each and every patient that comes in the door no matter what. Reality proves that this is inefficient and actually results in more people dying than being saved.

Therefore the question of practicality seems to scream in defiance of our value definitions from earlier, and seems to suggest that our earlier assumption that the value of human life is infinite may be wrong.

A better way to resolve this conundrum might be to remember that when comparing the value of human life we made two assumptions:

A) That each life had an infinite value.

B) That each life had a variable value.

These two assumptions imply that each person’s life has an infinite value $V_x$ that is different in size from that of the person next to them.

Therefore when deciding on who to treat between two patients, the inequality may look like this:

Indicating that the first patient is more valuable than the second, or:

Indicating that the second patient is more valuable than the second, assuming this:

Indicating that the medical establishment must choose between one patient or the other; i.e. they cannot begin treatment on both.

Yet even then when we substitute both values in we have inequalities that don’t make sense:

and

Which both simplify to:

and

Both of these inequalities are not true. Since we do not know the relative sizes of the infinities, we cannot determine at any given time which of the two is larger, or if they are the same.

Since the comparison system falls apart upon substitution, if we assume that medical triage is based on it then the triage system would fall apart and come to indecision constantly. The system would not work. Therefore, medical triage must not depend on the individual value of human lives when considering who to treat and when.

We can extend this extreme example to the example of fishermen vs. farmers in California to reveal that the politicians cannot accurately value the lives involved in each population, and must depend on other factors.

That both medical triage and political leadership are independent ov indicates that the value of human life is moot, and that it’s true value relative to other values does not take part in actual decision making calculations from day to day. Therefore, we can conclude that the value $V_x$ is pointless, and that assigning relative $V_x$ values as hypothesized earlier doesn’t actually work.

In other words, the question of the value of life is moot and does not come into play. What matters more is efficiency and resource management rather than the “true value” of any individual. Even more deeply this seems to suggest that a person is a sum of how they affect other peoples’ lives, since those factors are the only things other people can use for analysis. No one can actually know the true value of another person’s life, and as a result it plays no part in analysis. A person’s actions are more defining than their thoughts in other words (except in the case where thought becomes action, then they are equivalent).

Apologies

Sorry for the week long hiatus.

I’ve been struggling to keep the ideas flowing weekly, but this sort of thinking and analysis doesn’t come easily. While that’s not an excuse, that is an explanation for why I have decided to abandon my attempt to describe the velocity of a bill through congress using these mathematics. While it is possible, I do not have the data nor the mathematics required to be able to do it effectively or truthfully. The ideas I have presented operate under the assumption of linearity (See “On the Nature of Proportionality”) and as a result are not only not necessarily true, but they are almost certainly not true.

The skill I require and yet do not possess is the ability to build differential equations. Hopefully I should be able to do the analysis then, but certainly cannot do it now. As a result I cannot continue the analysis in good faith. I’m sorry.

In lieu of this, I will be writing a short essay tomorrow. I’ll figure out what later tonight and post it as a comment.

Thank you all, and I’m sorry,

Ethan Mendel

On the Nature of the Veto in Relation to Bill Velocity Part I

Further explanation of the Influence Factor will have to be put off for now. I haven’t quite figured out something that might work yet. Until then: the Veto Factor (sounds like a horrible reality TV show doesn’t it?)!

We know that velocity of a bill is proportional to the veto (though the way we construct the veto may require that it be inversely proportional).

We also know the veto process is, put simply, a value that is boolean in nature. That is, the president must do one of two things: veto, or not veto.  It is clear cut.

Therefore is proportional to a boolean value (standing for the president’s effect on the veto). We therefore also know that has the a domain as follows: . That is, the boolean variable is either false or true respectively, and therefore it represents whenever the president vetoes and does not veto. The domain of is part of the definition of a boolean variable.

However the assignment of to either value in its domain is a heuristic process at this time. That is, there is not mathematical formulae that can accurately predict or account for the factors that cause a president to veto a bill, nor predict it effectively (the same could be said for the other variables in the velocity equation). This means that the value of is assigned by the person, not by a mathematical process.

Therefore we know that:

We also know that the president’s veto alters the game dramatically. The bill returns to both the House and the Senate to be revised to be placed before the president again, or to have the presidential veto overridden.

Depending on the exposure of the bill, the amount of public support or dislike, the general party line approval or disapproval, individual constants of each politician, and various other myriad factors come into play at this stage. All of these combine in a way I have yet to quite figure out.

However one thing is certain about all of them. If the president doesn’t veto, then none of the factors come into play. Therefore, the final product of the interaction of all of these factors can be multiplied by the boolean variable in order to determine if the value actually affects the final velocity or not.

How these new factors may interact will be up this Saturday or Sunday…depending on whether or not extenuating circumstances come into play.

On the Nature of Proportionality

I’ve often used this symbol in my posts:

I’ve also always said that it represents a simple relationship between two variables. Let’s say A & B. In that case I should write a proportionality that looks like this:

I would always say that the above proportionality meant that as A increased B increased, and while that is true, that’s not all the proportion says.

In actually a proportion like the one above:

Says this:

In other words, A is equal to B times a constant k, which may or may not be known.  This formula makes one incredibly dangerous and most likely untrue assumption: that the relationship between the two variables being discussed is linear.

This means that a graph of A versus B would form a straight line, and have incredibly precise and predictable mathematics governing it’s behavior. It’d also be simple.

This is simply false. With that in mind, I will continue to use proportionalities regardless. They serve an argumentative purpose here rather than a mathematical one, but they DO serve the purpose of allowing me to relate variables in a very approximate fashion, if not necessarily an accurate one.

Apologies

Sorry about the lack of postage this weekend.

I was out of town and quite simply forgot to put up something that would tell you all, still getting used to the blogging thing.

Also, I was unable to finish my analysis…don’t know quite enough math yet.

I can come up with something, but I’ll admit right now, it’s not going to be complete.

I will however still try my best.

To make up for it, this Wednesday or Thursday I will write about the meaning of proportionalities in our discusssions and why it may not actually hold true.

Last week’s post will be this week’s post. No makeup posts planned.