Applied math Archives - The Silent Knight

Truth In Advertising: An Unborn Baby’s Beating Heart

Above is one version of a common billboard along the interstates in Florida and across America. The problem for me is that the spokesman didn’t look anything like that when he was 18 days from conception; he would have been barely visible and a lot less cuddly. Below, I’ve taken the liberty to replace him with his earlier self, which would have been about half the size of a pea or just able to fit through the pupil in the eye of his older self as shown in the background of the revised billboard. His heart would have been larger than a grain of sand. He still has no eyes, arms, or lungs, and I’ve found no information on brain function. I don’t think he’s feeling any pain. Now ain’t that cute?

His Odds Of Success

While I have touched on this subject before (in my article Save Your Birthday – Vote For Clinton), this time I dug deeper in order to put things into a more serious perspective. Here’s what I found:

According to ABORT73.COM, abortions have been decreasing slightly every year for the last decade. 879 thousand occurring in the United States in 2017.
The Statistica website says that the number of births in this country has been fairly stable. There were about 3.86 million of them in 2017.
The attrition rate of fertilized human eggs is not as well known. Comparing articles on spontaneous abortion in humans and Early embryo mortality in natural human reproduction, it looks like only 25% of fertilized eggs actually make it to birth (± at least 10%).

Here’s what that means: if you start with 18 fertilized human eggs, 13 will die from natural causes before birth, and one embryo or fetus would be aborted. Both forms of attrition are “front-loaded”; 2/3 of abortions happen by the 8th week^A1, while 2/3 of fertilized eggs will die in their first 11 weeks^A2.

To see the Note click here.To hide the Note click here.

A common mistake would be to compare the number of abortions with the number of births directly but to compare apples to apples, as they say, you need to use the same reference point. Here’s the math:

3.86 million births divided by 25% (or 0.25) gives 15.4 million fertilized eggs. That would be the common reference point. Divide that by 879 thousand abortions and you get the total number of fertilized eggs there are for every abortion (17.6 – let’s round to 18). To find the number of those fertilized eggs that would die naturally, simply multiply the same 18 by 75% (or 0.75).

One thing that is not clear is “what are the odds that the aborted “child” would have died from natural causes anyway?” The easy answer (and my best guess so far) is 75%. That means there are a lot of self-righteous people out there (many being middle-aged white males with nothing invested) who are willing to throw a woman in jail for long periods of time without even caring about the circumstances of her case to protect something that probably wasn’t going to make it anyway. I guess for the pious, any excuse is a good one for beating up on your neighbor.

At least that’s the way I see it. Any comments?

Please Help! The Special Theory Of Relativity Is Haunting (or is it Taunting) Me!

A long time ago (when I was in the sixth or seventh grade) in a galaxy far, far away (namely Southern California), I was introduced to Einstein’s Special Theory of Relativity by way of a story about two astronauts on different spacecraft watching a bouncing-light-beam clock, and I was really impressed. But upon further review as I was chewing my cud (See definitions of “ruminate”), as I was wont to do walking home from school, it didn’t seem to make as much sense. I developed some questions but didn’t know where to get answers, and as life pressed on my attention wandered elsewhere, and everyone lived happily ever after. . .

Until recently. In the last year, the subject has come up several times, the questions seem to be the same, and I still don’t know where to turn.

The Story

Although not exactly as I remember it, www.dummies.com¹ describes a similar thought experiment in the second section, “UNIFYING SPACE AND TIME”, with a spacecraft traveling at ½ the speed of light, but doesn’t give much explanation. A more detailed explanation can be found in The Star Garden².

The Problem

“Time Dilation”, Section 7.2.2 of Reference 2 concludes

“The time between heartbeats is also slower, and so from the perspective of a stationary person, a moving person appears to be living their life at a slower rate. Conversely, from the perspective of the moving person, the stationary person seems to live their life as if it is being fast-forwarded. If they travel fast enough, then they could see the stationary person age before their eyes.”

My problem with that conclusion is that based on the second paragraph of Section 7.1 at the beginning the article, which states

“there’s no such thing as absolute speed or velocity, and something can only be said to be moving at a constant velocity relative to something else. In the same way, something can only be said to be stationary relative to something else”,

how do we really know which astronaut is supposed to be aging before our eyes? What if we put a bouncing-light clock on each spacecraft? Would it explode?

The authors of Reference 2 seem to address this issue at the bottom of the next section, 7.2.3, where they say

“The twin paradox

The twin paradox asks why the astronaut can consider themselves to be moving and the Earth to be stationary, when Galileo’s relativity shows that there’s no such thing as absolute velocity. Why can’t the astronaut consider themselves to be stationary while the Earth moves away at tremendous speeds?

The answer is acceleration. Galileo’s relativity applies to inertial – that is non-accelerating – reference frames. The fact that the astronaut must have accelerated before getting to such high speed means that they know they are the one that is moving.”

To me, this sounds bogus; any acceleration before or after the experiment should be immaterial. Let’s have three observers; one person remains on Earth while two astronauts board different spacecraft, each leaving the Earth in opposite directions and reaching similar stable speeds well in excess of ½ the speed of light (meaning their relative speed would exceed the speed of light in a non-relativistic world). Each of the observers has their own bouncing-light clock. If you start counting after their speed stabilizes, exactly how do each of the three observers see the ages change for the other two?

One Last Question

A question that one might ask in each of these scenarios is “how does the state of the bouncing light in one spacecraft become known to the other observers?” Reference 1 states that Amber, on a different spacecraft, would see the bouncing light travel further between bounces, as if Amber had super X-ray vision and/or was otherwise experiencing the light beam in real time. How does that work? If she had to wait for reflected light rays from the event to reach her eyes, would that affect the apparent outcome in any way?

One phenomenon that may or may not have anything to do with the solution to this problem involves ocean waves. In deep water, a wave’s speed is nearly proportional to the square root of its wavelength^A,

To see the Note click here.To hide the Note click here.

$S \cong 1.251 \sqrt{W}$

where S is the wave’s speed (measured in meters per second) and W is its wavelength (in meters).
For shallow water waves, the speed is proportional to the square root of the depth.

$S \cong 3.1 \sqrt{d}$

where d is the water’s depth (in meters).

but in all cases, it is much less than the speed of light. If an observer were to watch the crest of a wave as it moved along a seawall, or along any imaginary line that wasn’t along the wave’s direction of travel (directly away from a point source, or in the direction of the wind, or perpendicular to the wavefront, etc.), then the apparent speed would be greater than the calculated or expected speed, and as the angle of that reference line approached 90° to the direction of travel, the apparent speed would approach infinity.

To see the Note click here.To hide the Note click here.

$A = \frac{S}{\cos \theta}$

where A is the apparent speed, S is the expected speed, and θ is the angle between the reference line and the direction of travel.

which is well above the speed of light. As far as I know, this has no implications or gives no reason for hope for wannabe time travelers.

In Summary

So now you see my dilemma. To repeat the title plea, please help me understand. A crucial early step in solving any problem may be asking the right questions. Finding those should be as important, and in some cases may be as difficult as answering them. So let’s get started. Thank you for your help. If I do figure it out without your help, I’ll let you know.

The Walmart Donation-Matching Gimmick/Scam

The other day, I saw an advertisement/announcement on TV stating that for every dollar you contributed to certain causes, Walmart would give two dollars. You can get the details on their web page, Walmart and the Walmart Foundation Announce up to $20 Million Toward Hurricane Harvey Relief and Recovery. I was impressed for about thirty seconds.

History

Many companies have done matching-donation campaigns, whereby if you give a dollar, they will give a dollar. It can be effective in promoting more charity by the public in some communities. It too can be a gimmick. It could just be an excuse to limit their donation – “We can afford to budget one million dollars, but if you can’t find matchers, I’ll just pocket the difference”. Usually they mention a cap or maximum donation that they are willing to make. That’s where the real deceit comes in.

Donation Limits

These limits sound reasonable for ensuring the donor company doesn’t get in over their heads by misjudging popular support and becoming committed to a donation they cannot afford. But these companies have already done their homework. For example, if the Green Cross (names in these examples have been changed just to keep you from thinking these principles have limited applicability) Christmas campaign raised ten million dollars last year and has grown as much as 20% year to year, with many years much less, an observer would have reason to be cynical if the donation-matching company has a donation limit of much less than about 15 million dollars.

Typically, the donor company will impose a limit of one or two million dollars. I’m sure their maximum donation will be much less than the cost their advertising company would charge for similar exposure. But when the donation-matching company imposes such a low limit, their actual final donation will always be equal to that limit. Reaching the limit will NOT be advertised, and the deception of the public will begin.

The ethical thing would have to just announce they were making a simple donation of one million dollars. But if you can’t fool suckers into coughing up more money (mistakenly thinking their donation is worth twice as much to the charity), where is the sport in that. At least this time it is for a good cause.

Walmart Doubles Down

Walmart’s new wrinkle takes this scam to the next level. But all it means is that the owners of Walmart will reach their company donation limit sooner, and will be able to laugh at the stupidity of even more Americans. That’s called arrogance.

Suppose that in this year’s campaign, the Green Cross is expecting ten million Americans to each donate a dollar (I’m just making the math simple). Under their old plan, Acme Widgets would make their generous four-million-dollar-donation by announcing they will match your donation dollar-for-dollar (for up to four million of their dollars). The first four million Americans bring in eight million dollars for the Green Cross (including Acme’s portion). If the remaining six million Americans donate expecting to see their money go further, they are being misled. They will bring in just six million dollars, and the Green Cross total for that year will be $14 million – a clear victory for the Green Cross (and Acme).

Under their new plan, Acme will now make their budgeted four-million-dollar-donation by announcing they will match your donation with, say, four dollars for each of yours (for up to one million of your dollars). Now Acme reaches their limit much faster. The first million Americans, with matching funds, help the Green Cross bring in five million dollars. The last nine million donors (up 50% from last year) will make a contribution that they mistakenly believe is being amplified. They will contribute only their own nine million dollars, bringing the Green Cross the same $14 million as last year. Acme will contribute the same amount. And the same amount will come from individual donors, 90% of whom were deceived.

Disclaimers

I am not suggesting here that donating to charity is bad; quite the contrary. I believe helping others is a good idea, an idea that is even supported in the Bible. But there are a few things one should consider:

Are you donating for the right reasons? Donations with strings attached aren’t really donations. You should not donate expecting to get anything in return (not even a reserved seat in the Hereafter). Your rewards should be internal.
Similarly, don’t assume, think, or claim the recipient of your donation will place a higher value on it than you do; that’s just fraud. I’ve recently heard complaints from members of relief organizations accepting donations for victims of Hurricane Harvey about people donating expired or opened food, worn-out clothes, etc. You are not fooling anybody.
On the other hand, are you giving within your means? Creating a hardship for one person to ease another – the saying for that is “Robbing Peter to pay Paul” (a Biblical reference, but not from the Bible), is not recommended. Even the airlines advise you to put the oxygen mask on yourself before tending to your children.
Finally, not all charities are created equal. And not all charities are the best stewards of your money. I suggest doing your homework;
After checking out a charity’s website, you can research potential donees at places like Charity Navigator.

What Next?

As time permits, I will be discussing some ways advertisers use math to deceive their customers. Stay tuned.

Not Quite Clear On The Concept (Part 1)

Earlier this month, the Catholic archdiocese of Newark, New Jersey, decreed that after four years of Catholic Youth Organization (CYO) basketball together, the St. John’s 5^th grade team (nine boys and two girls) would not be allowed to play the last two games of the season with girls on the team^A.

First, A Little Math

The maximum of any subset cannot be greater than the set maximum. This means that if the largest member of your weight-watching group is, say 400 pounds, then as people leave the group, that maximum will not get instantaneously larger. It could remain 400 pounds for a while, but will eventually get smaller as people continue to leave.

To see the Note click here.To hide the Note click here.

The average group weight, on the other hand, could go up or down depending on how people are selected for removal from the group, but averages aren’t generally used to measure greatness.

Similarly, in sports you cannot raise the level of competition by restricting participation. This means that you can’t say your team is the best in the universe if any member of the universe was barred from competing. Consequently, the only logical reason for restricting membership would be to protect the members from unfair competition. A team would only ban girls if they thought their boys weren’t ready for real competition.

Motives

As we all know, a group’s stated reasons for an action may differ from their real reasons. I suspect the archdiocese’s advertised reason for the decree is to protect girls from competition they can’t handle. But for that argument to have any credibility at all, at least two new conditions would have to be in effect:

There would actually have to be a girl’s team if you want anybody to believe that their interests are really your first priority.
You would protect a “weaker” group by banning the unfair competition from that group, not banning the allegedly weaker competition from the “stronger” or open group. The later option will rightly cause others to question your motives. “Who are you really protecting?”

The required game forfeitures would be further evidence of their true motive. A team should be required to forfeit a game only if they won using an unfair advantage. You would not make a boxer forfeit all the matches he won with one hand tied behind his back. Obviously, the other boys’ teams not only considered the girls a threat, but most likely the sole reason for the team’s success.

A Happy Ending

On hearing the decree, the St. John’s 5^th grade team immediately and unanimously decided to stick with their teammates and forfeit the season.

The girls, understandably, felt bad and offered to sacrifice themselves^A. St. John’s athletic director honorably rejected that offer. (In the body of that article, it suggests that the league director had already cancelled St. John’s season, making the athletic director’s gesture moot.)

A new Cardinal reversed the ban and allows St. John’s to play^A.

A Not-So-Happy Ending: Politics Trumps Logic

I just read about a different, but logically related case in Texas^A. A girl was taking testosterone to become a boy and wanted to compete with boys. New rules for the state of Texas required her to compete as a girl. She won their state wrestling championship. Not everybody was happy with her. I’ll leave the application of principle and subsequent comments to the reader.

Simple English: The Problem With The “If” Statement

You are probably very proud of your grasp of English (unless you live in South Florida, in which case you may not give a damn). And yet I have seen plenty of people whose lack of understanding about basic structures like the “If” statement

To see the Note click here.To hide the Note click here.

The next article in this series will discuss how misunderstandings about the conjunction “or” have caused so much trouble.

cause them to make terrible assumptions.

An Example

Suppose a young child is misbehaving to the point that the caregiving parent decrees “If you don’t knock that off, I’m going to paddle you” (This is an old example; I’m sure nobody would ever actually do that today 😉 ). As young children have been known to do, for whatever reason, the child continues with its behavior. The parent repeats the statement, with added emphasis. Nothing changes. The parent soon throws their hands up and says “wait until (your other parent) gets home”.

The parent’s first decree, like all “if” statements, had two parts; a condition and a consequence (joined by the conjunction “if”), with the understanding that if the condition is true, then the consequence will occur. It’s simple enough that even a young child can understand it. If the condition is met and the consequence is not accomplished, then the statement would be considered false. In short, the child knew that the parent was lying.

Now suppose the non-caregiving parent comes home, sees the objectionable behavior, makes a similar decree, and then the first parent points out that they had already made that decree to no avail. The child, for whatever reason, stops the objectionable behavior. To everybody’s surprise, the second parent paddles the child. Although the child and many of you listeners may think bad thoughts about this parent, one thing you can’t call him/her is a liar.

The Problem

As you can see here, the problem with the “if” statement is that is incomplete in the sense that it only addresses what happens when the condition is true, remaining completely silent to the possibility that the condition could be false. This allows most people to make the assumption that if the condition is false, the opposite of the consequence must occur. As the young child in our example learned, that assumption would be a mistake.

The Solution

Don’t make stupid assumptions. As your lawyer would tell you, get it in writing. In the above example, since the second parent didn’t make any promises about what would happen if the behavior did stop, s/he can’t be accused of lying. If this example bothers you, I’m sure the second parent told the child afterward that the paddling was for not obeying the first parent, in which case we would be unable to judge the truthfulness of their claim until after the right set of conditions are met following some later episode of misbehavior (guesstimating any change in likelihood of that future misbehavior based on recent events will be left as an exercise for the reader). To lawyers, mathematicians, and the like, the parent’s explanation doesn’t matter to this case and is unnecessary.

P.S.

Logicians have named operators (or functions) fulfilling all sixteen patterns of truthfulness or falsehood of expressions based on the truthfulness or falsity of two variables, such as the condition and consequence of the “if” statement described above. Engineers call the statement that yields the results you thought the “if” statement provided the “exclusive nor” function, “nor” being short for not or, meaning “giving the opposite results than the ‘or’ function”. Some refer to it as logical equality. In English, it would be represented by a sentence including the phrase “If, and only if”, such as “I will ground you for the rest of your life if, and only if, you do not stop screaming this very second”. If that type of statement had been the norm in this household, the non-caregiving parent, upon hearing the lack of results achieved by the other parent, was still free to add other (most likely “or”) clauses (to be discussed in a later article) to his/her decree. If you are now totally confused, please do not sign any document containing more than six words before consulting an attorney, or at least a mathematician. On second thought, in cases like this, I would stick with the lawyer.

How America’s Cup Committee Stole Victory From Kiwis

The America’s Cup, a yacht race between a single defender and a single challenger and first run in 1851 by the Royal Yacht Squadron in England for a race around the Isle of Wight, is known as the oldest international sporting trophy^A. The referenced Wikipedia article provides an excellent history of the contest, and notes that the Americans held the cup continuously from the first race until 1983, the longest winning streak in the history of sport. The cup was last contested in San Francisco in 2013, but this wasn’t your father’s America’s Cup. The vessels were now 72-foot catamarans (boats with two hulls) costing well over ten million dollars each with rigid “sails” and foils (short underwater wings) that lifted the entire hull out of the water. Their top speed was over 50 miles an hour, or about twice the wind speed. The whole race was held within sight of spectators ashore and instead of taking all day, as was customary, was required to be finished in less than forty minutes to fit snugly between commercials in a one-hour television format. The series also turned out to have one of the greatest comebacks in sports. Stu Woo of the Wall Street Journal did an outstanding job of explaining that competition^A. Stu failed to mention a minor change in the rules that bought the American team just enough time to complete the changes necessary to turn things around.

The 2013 America’s Cup was originally billed as a best-of-seventeen series.

To see the Note click here.To hide the Note click here.

October 7, 2018 Update: Notified of a broken link, I see all official mention of the best-of-seventeen aspect of the competition has been scrubbed. After looking through multiple websites, I did find it mentioned in the third-to-last paragraph of a third-party article, “The Miracle on San Francisco Bay” at www.slate.com.

A Best-of-T series (where T represents the total number of games to be played and is always an odd number) is widely used in sports championships, although before September 2013 I had never heard of T being larger than seven. In English (for those of you who are not sports fans) it means that the two teams will compete exactly T times (for which we will use seventeen in the rest of our examples) and then count up the points to see who won. One team can declare themselves the winner and send everybody home early if they can accumulate more of a lead than the other team can overcome in the remainder of the seventeen games. In a simple world, meaning a world without ties (in those sports that allow them) or penalty points, that would be nine wins in a best-of-seventeen series (round up(17 ÷ 2)). Yacht racing, as the references in the first paragraph suggested, is not a simple world. Before the series started the American team was penalized two points for cheating in an earlier round of competition, meaning that their first two wins wouldn’t really count (except to keep points away from their competitor). And although the calendar continued to list seventeen races (with the caveat “if needed” as appropriate) well into the competition^D, all of the experts were still saying that it would take nine wins for the Kiwis to take home the trophy. The truth is that after the seventeenth race had been sailed, if New Zealand had only eight wins, then the Americans would have won nine races (17 – 8 = 9), but would only have had seven points (9 – 2 = 7), meaning that New Zealand would have still taken home the trophy. That is what a best-of-seventeen series really means. And, as you can see near the bottom of the Final Scorecard at the end of Stu Woo’s Wall Street Journal article, that is exactly what happened. But just before New Zealand earned their eighth victory in Race 11, the race committee declared that because the American team’s penalty shouldn’t affect New Zealand, they would still need nine wins to take home the trophy. Shortly thereafter Races 18 & 19 were added to the schedule. As we now know, thanks to a brilliant turn-around by the American sailors, New Zealand never got that ninth win; the American’s got that point in the nineteenth and final race. Competition doesn’t get any better than that. Had the Kiwis been able to count up to seventeen or had their English been good enough to understand the meaning of “best of seventeen races”, the ending of this story may have been completely different.