Keys and Lockdowns

When an outbreak occurs, the FIRST thing you need to do is isolate it. “Lockdown” is one, usually effective way to do this. It should be undertaken along with tracing to determine whence it originated and who might be susceptible. Until those are determined, the only way to stop the spread is isolation. Full stop.

Innoculation (whether by vaccine or exposure) is, in a sense, a form of isolation. Vaccines, however, take YEARS to develop. The vaccine itself may take relatively little time (months or even weeks) to create in a lab. Determining whether it is safe enough, and effective enough to distribute to everyone is what takes the bulk of that time. With CoVID-19, exposure and infection can result in serious illness and death, and it is not yet clear that any immune response the body may have developed to fight it will last more than a year. Someone once infected could in theory be infected again.

Even after that is determined, a manufacturing process that can produce billions of doses will not be set up overnight. You’ll read of companies that are doing this now. This does not mean a vaccine is imminent; it’s simply smart logistics, not waiting until the last minute to set up and test the process so that it will be ready once the (safe, effective) vaccine is available.Do NOT expect a vaccine to be available within the next two years, in spite of all the hyperventillating stories you read or hear. There is a slim chance it could be ready sooner. SLIM.

The best thing we can do right now to avoid general lockdowns is come up with inexpensive tests that do not need to be adminstered or evaluated by a medic. These tests exist today.

They are called ANTIGEN tests. Currently they cost between $5 and $10 each, as opposed to PCR tests which cost $30 or more each. Antigen tests can be as simple as a card you lick or spit on, wait 15-20 minutes, and see if the card changes color. Depending on the color, the test will tell you if you are negative, or if further testing is needed.

Here’s the thing: antigen tests have a relatively high false positive rate, whereas PCR testing gives a very low false positive. In other words, if a PCR test says you have CoVID-19, you very likely (almost certainly) do have it. On the other hand, a positive result on an antigen test may mean you have it, but there’s at least a 10% chance you don’t. BUT — the antigen tests are at least as good at telling you you DO NOT have CoVID-19 as PCR tests are at telling you you do have it.

So, you distribute these cards widely, and freely — to everyone. Each day, you wake up, spit on the card, wait 20 minutes. If the card doesn’t change (or the change indicates you don’t have it, you go on about your day. If the card indicates you may have it, you (and whoever you’ve had contact with who is likewise not testing negative will go get a PCR test to confirm whether or not you really have CoVID-19. If it turns out you DO have it, you go into quarantine and start getting immediate treatment to lessen (and hopefully shorten) the disease’s impact on you. No further locking down of entire cities would be required.

To do this requires governments doing what we create them and elect them to do: manage our common resources for the benefit of all. Orginsing the resources necessary to do this is something government should be doing. Spending money to make those testing cards freely available to everyone is something government should be doing. That is what government can do to keep the economy going and maintain a semblance of “normalcy”, and avoid or altogether eliminiate the need for widespread lockdowns.

Collateral Damage

Ellen stops off at the grocery store on her way in to work to pick up a few things she’ll need. She pauses by the hand sanitizer and thinks, “I just washed my hands a little while ago. I don’t need to bother.”

She peruses the aisles that have the items she’s after – among these a box of tissues. She’s about to pick up her usual brand but notices another is on sale, so she puts the one down to pick up the other, compare them. She decides to go with her first choice and puts down the “cheaper” one.

Unbeknownst to her, Lachlan was looking at those same tissues a few minutes before. Lachlan didn’t know it at the time (and won’t know it for another few days) but he’s infected with CoVID-19. He doesn’t know what all this mask fuss is about and though he wears one, it tends to slip down around his chin, he frequently scratches his nose, touches his face … and boxes of tissues he’s picking up to decide which one to get.

Ellen takes her items to the checkout, pays, and on her way out does use the hand sanitizer, since she has after all been picking up things other people have been touching. Ellen’s smart and she’s trying to do the right thing.

Mark comes into the grocery store a few minutes after Ellen. They don’t know each other, and they never will. Mark is doing the week’s shopping for himself and his wife Margot, who’s at home recovering from her latest round of chemo. In better times Margot would have gone with Mark. Grocery shopping is one of the hundreds of little things they love doing together. But, today Margo must stay home. She’s “neutropoenic” meaning her immune cells are nearly gone, having been wiped out by the latest round of chemo. They’ll come back as her stem cells reconstitute her immune system. In the mean time, she needs to avoid coming in contact with germs that could make her sick, germs that a normal healthy immune system disposes of thousands of times a day without us being the least bit aware.

Mark has “box of tissues” on his list, and he heads down the aisle and picks up a couple of their usual brand, which he’s pleased to see is on sale this week. Mark pushes his cart to the checkout, pays, dowses his hands with sanitizer, and heads to the car where he loads up the several bags of groceries and supplies. When he gets home, Margot asks, “Did you remember the tissues?”. “Of course, here.”, he says, opening and handing her the box.

A few days later, we find Mark sitting outside the ICU. Margot is inside, fighting for her life. Margot “spiked” a fever a few hours after Mark got home from the store. Mark brought her to the ED, as he had once or twice before. Trips to the ED are not uncommon for neutropoenic patients, and these all had a happy ending after a round of broad-spectrum antibiotics. This time was different. Each time before they’d tested her for CoVID-19 and both times she’d tested negative. This time she tested positive.

Margot never met or saw Lachlan, nor did Mark, nor did Ellen.

A Gentle Introduction to Quantum Computing

A very good, truly gentle intro to quantum computing.  A basic understanding of probability and complex numbers is required.  But, if you’re truly interested in gaining a basic understanding of QC’s mathematics, you’ll likely already be familiar with those.

ABSTRACT: Quantum Computing is a new and exciting field at the intersection of mathematics, computer science and physics. It concerns a utilization of quantum mechanics to improve the efficiency of computation. Here we present a gentle introduction to some of the ideas in quantum computing. The paper begins by motivating the central ideas of quantum mechanics and quantum computation with simple toy models. From there we move on to a formal presentation of the small fraction of (finite dimensional) quantum mechanics that we will need for basic quantum computation. Central notions of quantum architecture (qubits and quantum gates) are described. The paper ends with a presentation of one of the simplest quantum algorithms: Deutsch’s algorithm. Our presentation demands neither advanced mathematics nor advanced physics.

Noson Yanofsky et al.

 

Paris and Nicole’s Mayhem

In their frequent shopping sprees Paris and Nicole often have trouble checking the bill. To address their less than impressive numeracy skills they decide to return to school. After a year of maths training the teacher wants to test the class of 25 students and lines them up in a queue such that Paris and Nicole are standing next to each other. The teacher then writes a whole number on the board and the first person in the queue says “That number is divisible by 1.” Then the second person in the queue says “That number is divisible by 2,” and so on till the final student in the queue says “That number is divisible by 25.” After all this the teacher exclaims “Well done! Except for Paris and Nicole everyone made a correct statement.”  Find where Paris and Nicole were standing in the queue.

 

So, we’re looking for a number that can be divided by all numbers from 1 through 25 EXCEPT two of them, that are also consecutive. The number 1 will divide anything, so, that’s trivial. We can also come up with a number that can be divided by ALL of the numbers by simply taking the product of all of those numbers. This doesn’t quite give us our answer, and it also results a number that is much bigger than we really need since some factors would be repeated. (e.g. any number for which 4 is a factor will also have 2 as a factor, so if we don’t need to include both 4 (i.e. 2 x 2) and 2 as factors. Take this a bit further, you see that by including 16 as a factor (2 x 2 x 2 x 2) we include 2, 4, and 8 as well.)

 

Let’s create such a product, expressed in a form that is prime-factored, though assuming all of the students’ answers were correct:
2: prime 
3: prime 
4: 2 x 2 
5: prime 
6: 2 x 3 
7: prime 
8: 2 x 2 x 2 
9: 3 x 3 
10: 2 x 5 
11: prime 
12: 2 x 2 x 3 
13: prime 
14: 2 x 7 
15: 3 x 5 
16: 2 x 2 x 2 x 2 
17: prime 
18: 2 x 3 x 3 
19: prime 
20: 2 x 2 x 5 
21: 3 x 7 
22: 2 x 11 
23: prime 
24: 2 x 2 x 2 x 3 
25: 5 x 5
Combining these (e.g. the factored forms of 4 and 8 are implicit in the factorised 16) we get
2 x 2 x 2 x 2 x 3 x 3 x 5 x 5 x 7 x 11 x 13 x 17 x 19 x 23 = 26771144400. 
This number is divisible by all of the numbers from 1 to 25. It should be easy to see that by removing, say, 13, we get a number that is still divisable by all of the other numbers from 1 through 25. (13 x 2, the smallest of the other numbers, equals 26, which falls outside range of factors.) Removing 13 we get
2 x 2 x 2 x 2 x 3 x 3 x 5 x 5 x 7 x 11 x 17 x 19 x 23 = 2059318800,
But, we need TWO CONSECUTIVE numbers. Suppose we stick with 13 and try removing 12 as well. But, 12 = 2 x 2 x 3, so, we’d have to either remove both of the 3’s as factors, or at least 3 of the 2’s to have no 12. Suppose we remove both 3’s to give us
2 x 2 x 2 x 2 x 5 x 5 x 7 x 11 x 17 x 19 x 23
Now we’d have no 13, no 3, no 6, no 9, no 15 … already we have more than two numbers that don’t divide whatever was written on the board. So, 12 won’t work. How about 14? Same problem. You’d need to remove either all of the 2’s or the 7, but, removing all of the 2’s gets rid of all even numbers as valid factors, and removing the 7 gets rid of 14 and 21 as well. So neither Paris nor Nicole occupy the 13th place in line.

Let’s try 17. Like 13, when you remove this as a factor from the original factorization, above, you get a number that is divisible by all other numbers except for 17. We then need to look at 16 and 18 as possible second candidates. 16 = 2 x 2 x 2 x 2. Take out just one of those 2’s and we can still keep every other product in which 2, 4, or 8 is a factor, so removing just one 2 results in the loss of only 16 as a factor. Looks like Nicole and Paris were the 16th and 17th students in the line. For completeness we can look at 18 and see that it doesn’t work since 18 = 2 x 3 x 3, so we’d need to drop all of the 2’s which means we’d have a product with more than two wrong answers in the list.

So, our answer is
2 x 2 x 2 x 3 x 3 x 5 x 5 x 7 x 11 x 19 x 23 = 787386600.
To see if I got the right answer, I googled the problem and found this site (page 3) with the exact same problem and a solution. They get the same places for Paris and Nicole as I did (16 & 17) but their number (that the teacher wrote on the board) is different: 2362159800. This looks like it’s about 3 times what my answer is, and in fact it is exactly 3 times my answer. In other words, they have an extra “3” in their factorization, for whatever reason. So, both answers are correct, but mine is the SMALLEST number that is a factor of all numbers from 1 through 25 excepting 16 and 17.

Radical Maths

This little puzzle popped up the other day and I thought it would make a good, simple example of how to deal with square roots.

First, a little terminology:  The word for root in latin is radix from which we get words like radius and radical. Mathematicians refer to the square root symbol √ as the radical symbol, and the stuff we’re taking the square root of they’ll refer to as what’s under the radical.

We start with this.

\sqrt{x + 15} + \sqrt{x} = 15

We can get rid of square root symbols by just squaring what’s under them, which will leave just the terms under the radical and nothing else.   Since this is an equation, we’ll need to do this to both sides:

(\sqrt{x+15} + \sqrt{x})^2 = 15^2 = 225

When we multiply this out, it looks like a real mess:

x + 15 + 2(\sqrt{x+15}\sqrt{x}) + x = 225

We can combine the two x’s to make this

2x + 15 + 2(\sqrt{x+15}\sqrt{x}) = 225

But, it isn’t all that much simpler.  Let’s try another approach.   Having two different radicals on the same side of the equation is what makes this messy, so, let’s move one of them to the other side of the equals sign.   We can move the \sqrt{x} term by subtracting it from both sides.

\sqrt{x + 15} = 15 - \sqrt{x}

Now let’s square both sides of the equation like we did before and see what we get:

x + 15 = 225 - (2)(15)\sqrt{x} + x = 225 - 30\sqrt{x} + x

I know, it doesn’t look all that much simpler, but, notice how there is a solitary x term on both sides?  We can get rid of that by subtracting x from both sides:

x + 15 -x = 225 - 30\sqrt{x} + x -x

leaving us with

15 = 225 - 30\sqrt{x}

It’s looking easier already!   Let’s subtract 15 from both sides, and then add 30\sqrt{x} to both sides.  We get

30\sqrt{x} = 225 - 15 = 210

Now we have simple terms for both left and right hand sides of the equation.   Divide both sides by 30 to get

\sqrt{x} = 210 / 30 = 7

Square both sides (yes, again) to get rid of the radical, and we end up with

x = 49

Let’s see if this is right.  Plug 49 in for x in the original equation:

\sqrt{49 + 15} + \sqrt{49} = 15

We can add the 49 and 15 under the first radical to get

\sqrt{64} + \sqrt{49} = 15

The square root of 64 is 8 (8 x 8 = 64) and we already have square root of 49 being 7, so

8 + 7 = 15

So, the answer checks out!

Totally RADICAL, eh?