Complex

 

Complicated.

Why'd you have to go and make things so complicated?

I see the way you're acting like you're somebody else

Gets me frustrated.

Life's like this you,

And you fall, and you crawl, and you break

And you take what you get, and you turn it into

Honesty and promise me I'm never gonna find you faking

No, no, no

What if things are complex, complicated, and you have to consider imaginary numbers.

The formula for the hypotenuse, c, of a right triangle with sides a and b,  on a hyperbolic surface is

cosh(c)=cosh(a) * cosh(b)

Using the formula for the inverse of the hyperbolic cosine this becomes

c=ln(cosh(a) * cosh(b) ± √((cosh(a) * cosh(b))²-1))

Using the identity for  hyperbola, cosh²-sinh²=1 which also means that cosh²-1=sinh², this becomes

c=ln(cosh(a) * cosh(b) ± sinh(a) * sinh(b)).

Using Euler’s Formula which involves imaginary numbers, e^iz=cos(z)+sin(z)*i; the definitions of cosh(x) = ½(e^x+e^-x) and sinh(x) = ½(e^x‑e^-x); and letting z = ix, this becomes

cosh(-x) = sin(ix) and sinh(-x) = cos(ix).

Using the fact that cosh is symmetrical, reflects, with respect to 0, while sinh is not symmetrical with respect to 0, this becomes

cosh(x) = sin(ix) and sinh(x) = -cos(ix)

As a result,  the hypotenuse of a right triangle on a hyperbolic surface can be stated as the natural logarithm of regular circular trigonometric functions as

c=ln(sin(ia)*sin(ib) ± cos(ia)*cos(ib)).

For the case when a, or b, = π,  since cos(iπ) = -1 and sin(iπ)=0, this then means that

c=ln(0 ± cos(ai)) if b= π, or ln(0 ± cos(bi)) if a=π.

Equation 1

For the case when a, or b, =0,  since cos(i0)=1 and sin(i0)=0, this then means that

c=ln(sin(ai) ± 0) if b= 0, or ln(sin(bi) ±0)) if a=0.

Equation 2

This shows that for a hypotenuse as a complex number, if its real portion has a coefficient of zero, then the imaginary coefficient can take on any value, and there is zero standard deviation, i.e. the second eqaution.  If instead the real portion has a non-zero coefficient, then the imaginary coefficient must be zero, and there must be a non-zero standard deviation, i.e. the first equaion.
 
 
 
 
 

Friends

 

Friendship

If you're ever in a jam, here I am.
If you're ever in a mess, S.O.S.
If you're so happy, you land in jail. I'm your bail.
It's friendship, friendship, just a perfect blendship.
When other friendships are soon forgot, ours will still be hot.

Is the enemy of your enemy your friend?

As in the Cole Porter song, a friend values your User Optimal almost as much as you do in ALL things.  An enemy might share the same User Optimal as you with respect to your enemy, but that is NOT all things. He still values his User Optimal more than yours.  It is purely a transactional “friendship”.  If his User Optimal with respect to your enemy changes and is no longer shared by you, he will act in his own interests, not yours.

In this sad definition of “friendship”, today’s "friend" may be tomorrow’s enemy.  Thus when the Mujahidin in Afghanistan were fighting the Soviet Union, the United States acted liked they were our friend.  When they became the Taliban and sheltered Osama Bin Laden, they became the enemy of the United States.

When Saddam Hussein was fighting the enemy of the United States, the Islamic Republic of Iran, he was treated as our friend.  When he invaded Kuwait, he became the enemy of the United States.

The saying is that you should keep your friends close, but your enemies closer.  A distinction should be made between real friends, and purely transactional friends.  You would be wise to keep transactional friends, who are only the enemy of your enemy, almost as close as enemies. The difference is that you can trust real friends.  You can’t trust transactional friends.

Solutions

 

                                                     You Can’t Always Get What You Want

I saw her today at the reception
A glass of wine in her hand
I knew she would meet her connection
At her feet was her footloose man
No, you can't always get what you want
You can't always get what you want
You can't always get what you want
But if you try sometime you'll find
You get what you need 

What you want is your User Optimal.  What you get is a Nash Equilibrium.

By profession and training I am a traffic engineer.  When there is merging traffic ahead, “Road narrows by one lane ahead”, what are traffic engineers trained to believe will happen? 

Traffic engineers are trained to believe that everyone follows the best strategy, and there is a rolling merge in which traffic continues in the merging lane until there is a safe gap in the continuing lane (and there should always be a safe gap).  This is an System Optimal, SO, solution and has the highest capacity for the merging section.  (i.e. utilize the capacity in the ending lane as long as possible).   

What do people really do?  Each individual car will try to follow their own User Optimal, UO, strategy.  For some cars, that UO strategy will be to stay in the moving lane until the last minute and then force their way into the continuing lane, even when there is no safe gap.  To prevent this, all other cars can move into the continuing lane as soon as they see the merge notice, and ensure there is no safe gap and punish the last minute switchers, i.e. block their UO solution, by NOT letting them in.  What all cars adopt is called a Nash Equilibrium in that most cars will not follow yheir owb UO soluion, will block some from their UO solution, and will adopt a Nash Equilibrium instead. 

Traffic engineers are trained to believe that everyone will adopt the most efficient, SO, solution.  But individuals will see their own UO solution as the most efficient.  Unfortunately this leads to Nash Equilibriums, where individuals can block each other from that most efficient solution. They are called Nash Equilibriums because they were descrbed by mathematician John Nash, the subject of the movie, A Beautiful Mind. No one gets what they want. (The Blonde, if everyone goes after the Blonde, because they will block each other.)  Instead people will get what they need. (Nash Equilibriums).  https://www.youtube.com/watch?v=vCyZvfRHkC4

Fools

What A Fool Believes

But what a fool believes, he sees
No wise man has the power to reason away
What seems to be
Is always better than nothing
Than nothing at all

Are you a fool to always believe a GPS?

A GPS is marvelous device.  But while it might be marvelous, it can be wrong.  In 2006, when GPS in cars were very new, I was riding with my brother-in-law to his son’s wedding rehearsal dinner.  We were both from out of town, so he was showing off the GPS in his new car to get directions to the restaurant where the dinner was being held.  He was following the GPS’s directions like a good soldier.  But when the GPS said to make a left turn into the restaurant, I stopped him because that that was the lobby of the restaurant, and the parking lot was across the street.  The GPS knew where the restaurant was, but it didn’t account for the fact that a car had to park in the restaurant’s parking lot.

A few years later on a family vacation on the Pacific Coast, we had rented a car but taken our own portable GPS device with us.  It flawlessly predicted that we were stuck in traffic on I-5 way south of Portland, Oregon because of an accident  on the Columbia River Bridge.  It re-routed us to get off at the next exit, onto Wheatland Ferry road and onto an agricultural ferry, along with farm equipment,  to cross the Willamette River.  Because of the GPS we were able to reach our destination on-time, despite a 5 hour back up on I-5. So the GPS was very smart, correct?

The very next day, we needed some supplies and asked the GPS for directions to a store.  We followed the GPS’s directions into a forest and onto what became a logging road.  The GPS had assumed that our rental car could use this logging road to cross a mountain.  To avoid a claim with the rental car company, I demured and forgot about that store.

A GPS has lots more knowledge. But it is a fool.  It believed a restaurant’s lobby was the same as its parking lot, and a rental car was the same as a logging  truck.  Be wise and listen to a fool, but don’t always believe a fool. 

Man's Best Friend

 

I Love My Dog

I love my dog as much as I love you
But you may fade, my dog will always come through
All he asks from me is the food to give him strength
All he ever needs is love and that he knows he'll get

Except dogs don’t live as long as we do.

The editor of my favorite newsletter, Teresa Hanafin, announced the passing of her beloved Yorkshire Terrier, Brady. I sent her the following email.

"'Tis better to have loved and lost; Than never to have loved at all. 

I am sure that our West Highland Terrier is wondering what all the fuss over him is about, but then he didn't read your newsletter and see the sad news about your Yorkie. 

My condolences of course, but I will tell you the same thing that I told my niece when her beloved Cotton passed.  When Bruce Springsteen was playing at Fenway, he flashed  a picture of the late Clarence Clemons on the screen.  He said what I paraphrsed for my niece, that "if he's here and we're here, then Clarence is here".  As long as you are here, your Yorkie is still here. https://www.youtube.com/watch?v=KP_XkN2v7OM 

As to naming you next dog.... com'n there will be next one.....no one can replace Brady, but how to best honor Brady? Since I was honored to have been selected in your pet essay, in that essay I said that our West Highland terrier's name is also Brady.  I am a sports fan like you appear to be, and I can guess where the name came from ;).  Our West Highland terrier's groomer is Lisa Pacheo of Raynham, MA.  When she grooms Brady, he plays with her West Highland Terrier Pip.  Lisa and her husband, the former chief of police of Raynham, (why are former Chiefs of Police either like Andy Taylor or Bull Conners?) show her terrier.  She, well Pip, won best of Breed at the 2015 Westminster Dog Show.   As a result we learned that Pip's American Kennel Club, AKC name was Great Expectations. Thus, if our Brady was AKC registered then his name would obviously be the Grtaetst Of All Time,  G.O.A.T.  So to honor your Brady, may I suggest that, if its a male, the names Bill (Russell), Red (Arnold Auerbach) or Bobby (Orr).  If its a female maybe Simone (Biles) or Mia (Hamm)."

Polling II

 

Fugue of Tin Horns

I got the horse right here,
The name is Paul Revere,
And here's a guy that says if the weather's clear,
Can do,
Can do,
This guys says the horse can do,
If he says the horse can do,
Can do,
Can do,
Can do.

Polls are touting.

I am not trying to look down my nose here. At various points in my career, I have been a pollster and have touted. The problem is that I understand how polls work, what they do, and how they should be used. And one way that they should NOT be used is in horseracing to predict the next single outcome. A series of outcomes, the house odds, sure that is good touting. The next outcome? Are you crazy!

The golden age of polling has passed. Every response in a poll has to be properly expanded to represent a universe. Once telephone polling could be certain that the telephone number defined geography, which could be used to expand the poll response, But that land line telephone, which once determined geography, and thus other expansion characteristics, is no more. My parent’s landline phone determined the geography, not only in the area code, the first three digits, but also in the Exchange, the next three digits. My parent’s phone was 401-941-XXXX which meant that it was 401-WIlliams 1-XXXX, such that the area code not only told you the portion of a state, but the Exchange told you the town and portion of town served by that Exchange and suffix. My old land line phone was 508-339-XXXX and my sister, who lives two miles away in the same town is 508-337-XXXX. Thus not only do we have the same exchange, EDgewood or 33, but we have different suffixes within that exchange. And besides people used to answer land line phones!

I said former landline phones because my family cut the cord years ago and no longer has a landline. My cell phone has a Boston area code because I worked in Boston when I bought that cell phone. My son who now lives in California has a south suburban Boston area code because he lived in my home when he got his cell phone. My wife has a Los Angeles area code, because that son bought her cell phone. So phone numbers being used to expand the respondent’s answers? And answering cell phones calls from an unknown caller doing a poll? As if!!!

So polls today are no longer as easy to do as they once were. And anyway people confuse the expansion process with the results. This includes the US Supreme Court which confused the process of Affirmative Action with its results. https://dbeagan.blogspot.com/2022/08/affirmative-action.html

LOL. You’re correct Supreme Court.😜  I won’t give preference to race, income, or ethnicity when admitting students. I will instead give preference to polo players, who ski in Europe. That will assure I have not given any preferences to rich WASPs!?!

Polls to tell you the general direction? An honest tout. Polls to pick the winner in a horse race? A dishonest tout.  A poll that says Trump is ahead of Biden?  You be the judge.

Certainty

 

Blue Christmas

I'll have a blue Christmas for certain
And when that blue, blue heartache starts hurtin'|
You'll be doing alright with your Christmas of white
But I'll have a blue, blue Christmas

How certain are you?

Ranked choice voting is about compromising, finding a common ground to achieve certainty.  Winner takes all voting is only about dominance, not certainty or compromise. 

Take a voting slate of four candidates, (A, B, C and D) and ranked ballots from eight voters to choose the most acceptable candidate to those voters. The ballots by voter are as in the table below.

Voter
	1	2	3	4	5	6	7	8
1st Choice	A	C	B	C	B	A	C	D
2nd Choice	B	B	A	A	A	B	A	C
3rd Choice	C	A	C	B	D	D	B	A
4th Choice	D	D	D	D	C	C	D	B

From only the First choices, Candidate A has 2/8 of votes, Candidate B has 2/8 of the votes, Candidate C has 3/8 of the votes and Candidate D has 1/8 of votes.  Candidate C is 100% the most dominate candidate if only First Place votes are considered.  However certainty is not the outcome with the highest total.  It is the certainty of the outcome with the highest total multiplied by the probability of that outcome.  In that case Candidate C is has a certainty of less than 24% if only First place finishes are considered.

Candidate Points					Candidate certainty
A	B	C	D	Points	A	B	C	D
2	2	3	1	8	18.8%	18.8%	23.4%	10.9%
8	7	7	2	24	44.4%	41.3%	41.3%	15.3%
16	14	13	5	48	66.7%	62.0%	59.2%	28.0%
24	22	21	13	80	63.0%	59.8%	58.1%	40.8%

If the First and Second place votes are counted, with 2 points for First place and one point for Second place, for a total of 24 possible points, then Candidate C is no longer the dominant candidate. In fact that Candidate is now tied with Candidate B.  The most dominate is Candidate A and the certainty that this is the preferred Candidate of all voters is over 44%.  If 1 point is awarded for Third place, 2 points awarded for Second place, and 3 points are awarded for First place, then the dominant candidate is still Candidate A, but the certainty that this is also the preferred Candidate has increased to almost 67%. Increasing this to 1 Point for Fourth place, 2 points for Third Place, 3 points for Second place and 4 points for First place does not change the dominant candidate from being Candidate A but the certainty of that Candidate has decreased to 63%.

A "winner takes all" contest with no consideration of seeding, handicaps, or different points for finishes other than first, can determine dominance, but it can not determine certainty.  Before it is suggested that this is a complicated system by those who favor “winner take all of first place”, it is observed that this  is how Sports Polling, the Heisman Trophy, Hall of Fame inductions, favorite restaurants, etc. are determined.  Even in playoff systems such as the NCAA March Madness, or professional sport playoffs, seedings are used, not mere dominance. And most playoffs are series of games to increase certainty and overcome the “Any given Sunday” randomness of “Winner takes all”. Even political conventions once chose candidates based on something other than plurality takes all.  It is only in recent years in one political party that “winner takes all” state primary primaries award a state’s delegates on the basis of plurality alone.  That is a way to show dominance, but it is NOT a way to show certainty. Ranked Choice voting may sound more complicated, but it is definitely more Certain.

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