Does my "feminist" card get revoked because of this post?

Today in my Facebook news feed, a friend posted this article:

His caption expressed how this action was so stupid, as this is how males naturally sit. I commented on the post with the following:

There is something to be said for the differing socialization of men and women regarding sitting. "Proper ladies" have historically always been instructed to keep their legs closed, whereas men have never been subject to such an admonition. A comical yet fictional example of how men and women are (or at least were) socialized to sit in different ways is told in The Adventures of Huckleberry Finn, when Huck dresses as a girl, and the woman whose house he visits can sniff him out based on how he sits and uses his legs while wearing a dress.

Littlemermaid Disney GIF from Littlemermaid GIFs

[Note: This gif wasn't actually included in the comment.]

The point is, men can sit however is comfortable for them without retribution, which is not necessarily the case for women. I get that the woman in this video is pointing this out--she's creating consequences for a behavior for which women already experience consequences. In other words, her thesis is that sitting with spread legs comes with consequences, regardless of gender. However, it's unclear whether she's asserting that this is how it ought to be.

All that being said, pouring bleach on a man's pants in a public space in order to make such a point is just asinine.

My relatively thoughtful analysis (if I do say so myself) went unnoticed, as the poster's response was, "I love your explanation! I didn't know where you were going with this, but I like the conclusion!"

Eyeroll.


* * *


Today was also the day that Supreme Court nominee Brett Kavanaugh testified before the Senate Judiciary Committee. As an introductory comment, I don't know if I've seen another picture that more accurately sums up the USA in 2018:

Look at the expressions of every single woman in this photo.

This half of the post is the why the post's title is what it is. Brace yourselves...

The allegations against Kavanaugh seem a lot like grasping at straws to me. People--myself included--desperately do not want to see him appointed to the Supreme Court because of his horrifyingly regressive stances. People want to find some dirt, ANY dirt, on Kavanaugh to block his nomination. So what happens? Dr. Christine Ford comes forward and states that he assaulted her at a drunken house party...in high school. Three-and-a-half decades ago.

Before I say anything else, it must be said that I believe Dr. Ford. Traumatic experiences such as sexual assault stay with you for your entire life, even if they are remembered in incomplete narratives. Asking her to recall details of the evening of the assault such as the address of the house or what she had had to drink is utterly absurd and irrelevant. I can only imagine the pain she feels seeing the perpetrator of the crime against her being considered for one of the most esteemed positions in the country.

With that said, I'm left with a feeling of, "so what?"

I am not attempting to minimizes Ford's pain. Rather, I find it unlikely that a single action that a horny, drunk, teenage male did decades ago is indicative of his character now.

I don't think it's even a question of "did he do it?" It's more of a question of "does it matter?" On the one hand, I absolutely believe that it does matter. People should have to live with the consequences of their actions. Additionally, the concept implanted into men's (and women's) heads from birth that men are entitled to women's bodies is shameful, to say the least.

But on the other hand, are we so closed-minded that we believe people cannot change? Do we really think that how a horny 17-year-old acted in high school is how he will act decades later? God knows I'm a vastly different person now than I was when I was 17; and such is the story with most sensible adults. Now, this would have been a very different issue if the allegation were from an incident five years ago or ten years ago. But this allegation is from an incident decades ago. It begs the question: where does one draw the line? How far back must one go in order for it to be acceptable to say, "That is no longer who I am"?

 

I was listening to an NPR Politics podcast last night in which the speaker reported that additional allegations against Kavanaugh had been made. These allegations claimed that while in high school, he had attended parties at which drunkenness abounded and several young women were raped. Kavanaugh was not accused of any of these rapes; he merely attended the parties at which they occurred.

Come on.

How is this allegation considered valid? Why should Kavanaugh be held responsible for what these other young men did decades ago? It is completely possible that he was unaware that this was happening at these parties; generally, sexual intercourse, non-consensual or otherwise, is not performed in the middle of a living room.

Am I really defending someone who believes that Roe v. Wade should be overturned? God, I sound like the wrong side of history.

Of course, sexual assaults against individuals tend to come to light in concentrated bursts. So I may have to rescind this entire portion of the post if more compelling evidence and incidents are revealed. But given what has been made public to this point, I do not believe that the allegations are sufficient to block him from the Supreme Court nomination. Goddammit, Republican majority, block him because he would destroy and undo years of progress!

I close with this thought-provoking cartoon.

Exponential Series

[This is a mathy post. Beware.]

I'm sure someone has already reasoned through this somewhere before, but whatever.

Consider the sequence defined by the following exponential function:

where a is a real number. (Of course, in its general form, there could be a constant term in front of a which is unaffected by the exponent. However, the constant term is irrelevant in the forthcoming analysis, so it is ignored for simplicity.) We want to examine the relationship between the n^th term of the sequence and the series formed by the first n-1 terms of the sequence. Specifically, is the following statement true?


Put into English, this inequality asserts the following: the n^th term of an exponential sequence is greater than the sum of all the previous terms in the sequence.

* * *

Proposition:


Proof: by induction on n

Base case: n = 1
The inequality simplifies to a > 1, which is true because a is greater than or equal to 2.

Inductive step:
Assume the hypothesis is true for all integers 1,...,n. Show that it holds for n + 1; that is,

This can alternatively be written as:


The left hand side (LHS) of the inequality is at least double aⁿ, since a is greater than or equal to 2. The right hand side (RHS) of the inequality is strictly less than double aⁿ, since the summation term is strictly less than aⁿ by the inductive hypothesis. Therefore, the LHS of the inequality is strictly greater than the RHS, and the proposition is proved.

* * *

It was essential that our base a be at least 2 for the inequality to be true. To demonstrate why this is so, consider the case where a = 1. The LHS of the inequality will always be the value 1 for any value of n. The RHS will always be n, since it is the summation of n exponential terms with base 1. Thus, the inequality does not hold in this case.

So the inequality is always true when a = 2, and never true when a = 1. However, how does the inequality hold up when 1 < a < 2? The answer is...it depends! Both on what the value of the base a is, as well as what the value of n is!

As an example, suppose a = 1.9.
When n = 1, the LHS is 1.9 and the RHS is 1, and the inequality is true.
When n = 2, the LHS is 3.61 and the RHS is 2.9, and the inequality is true.
When n = 3, the LHS is 6.859 and the RHS is 6.51, and the inequality is true.
When n = 4, the LHS is 13.0321 and the RHS is 13.369, and the inequality is false.

It appears that when 1 < a < 2, the inequality "flips" at a certain value of n. We will call this value of n the "critical n". In fact, the border cases of a = 1 and a = 2 each have a critical n. When a = 1, the critical n is 1, which implies that the inequality flips immediately and is never true; and when a = 2, the critical n is infinity, which implies that the inequality never flips and is always true in its originally presented form.

We want to analytically derive a formula that will reveal for what value of n the inequality no longer holds when 1 < a < 2. We can do this by using the formula for the sum of a geometric series with a finite number of terms:


Substituting this sum for the RHS of the inequality, we are attempting to find the largest integer value of n such that:


Here's a chance to brush up on your algebra skills...


Note that the inequality flips in Step 2. This occurs because the denominator in Step 1 is negative (recall that 1 < a < 2).

The critical n is the smallest integer such that the inequality in Step 7 is no longer true (or one more than the largest integer such that the inequality is still true). So the critical n can be determined by computing the RHS of the inequality in Step 7, and then taking the ceiling of that value (that is, rounding up to the nearest integer).

We see that this formula indeed holds up in the example given above when a = 1.9. The RHS of the inequality in Step 7 is 3.587, which is 4 when rounded up. Indeed, it also holds up when a = 1 and when a = 2, as the numerator is 0 and infinity in each of those cases, respectively. These round up to 1 and infinity, respectively. (Technically the RHS of the inequality in Step 7 is in the indeterminate form 0/0 when a = 1, but for the sake of the argument, we'll ignore this and only consider the numerator value.)

It is also possible to determine the smallest value of the base a for which a given critical n applies (i.e., a fixed n). Unfortunately, there is not a closed-form solution for a as for the critical n; rather, a must be found as a solution to an equation. However, some simple rearrangement reveals that this equation takes on a relatively simple polynomial structure:


We can plot the critical n as a function of a between 1 and 2. It is naturally a step function since n only takes integer values.



As expected, as a approaches 2, the critical n approaches infinity.

* * *

Conclusion: We have found the conditions under which the n^th term of an exponential sequence is greater than the sum of all the previous terms in the sequence. If the base a is 2 or greater, then this is always true. If the base a is 1 or lower, then this is never true. But for bases a between 1 and 2, this is only true for values of n up until a given "critical n". We have found a formula that can determine what this "critical n" is for a given base, as well as an implicit formula that can determine the base a at which a given n is the critical n.