Tag: math

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Single Component Vectors

Sports score couplets are vectors. They have at least one component and a particular order. The specific order isn’t the criteria; that there is an order is the criteria.

However that doesn’t mean that vectors require two or more components. A single component can be a vector.

Example: score differentials. Last week JMU lost 38-45 to Georgia Southern. The differential was -7. That differential is a vector.

It’s got at least one component, the 7, with a particular order. The order is the order of the two teams. The differential could be +7, with direction reversed but magnitude the same, provided the order is reversed. +7 would be the Georgia-JMU differential. Single component, particular order.

Put another way, a vector is an amplitude and a direction. The 7 is the amplitude, the +/- is the direction.

This is why all negative numbers are vectors. They address a state of change, and the order of the components changing is the order of this small vector.

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Complex Vectors

The complex number i or j is a pseudo-unit vector. It’s purpose is to enable vector operations on scalars. Put another way, it’s an adapter so you can plug scalars into vector operations.

To elaborate, start with Aristotle. He came up with an idea of four causes: material, formal, efficient, and final. These causes are effective descriptions of what something is. Here is a fairly good description.

Think for a moment of a phone charger, USB to wall plug. When you think of what that is, you don’t think of transistors and bridge rectifiers, transformers, and coils. You describe it by what it does and how you use it. This is Aristotle’s formal cause. The thing is an adapter so you can plug your phone into the wall. The material cause, the circuitry and plastic, is honestly mostly irrelevant and makes you take a long detour around a short mental trip. Think of a phone adapter. What is it? It’s an adapter.

That’s what i is and how it should be considered. The mathematical definition, sqrt(-1) is the material cause. That’s what makes it up. But to almost everyone who uses it, what i is is an adapter. It lets anyone do vector math on scalars.

What’s a scalar? What’s a vector?

Numbers come in two basic groups: scalars and tensors. A scalar is a number by itself. A tensor is any concatenation of more than one scalar together. Vectors are a particular type of tensor, a bunch of scalars put together, in a certain order. For your curiosity, they’re organized with one index in particular order. Which particular order doesn’t make them vectors or not; they just need an order.

Let’s think sports. The score of one team is a scalar. Yesterday, 9/24, JMU scored 32 points against Appalachian State (American College Football). The 32 is a scalar. It’s a number by itself.

Clearly, there’s some utility in that single number. The JMU coaches know if they’re doing well or not. The ASU coaches know if their defense is doing well or not. It’s a high number, so it was probably an exciting game. The number is useful but exists by itself. It is a scalar.

But it doesn’t tell me who won the game. For that, I need another number: the other team’s score. ASU got 28, so the final score was 32-28. 32-38 is a vector. It’s got two scalars, 32 and 28, put together in a certain order.

What order? JMU/ASU. Could you flip it around? Easily. But you need to be careful so the numbers retain their meaning.

Two numbers together tell you something that one number doesn’t: who won the game. What’s more, the vector 32-28 tells you something that would be difficult to grab with the numbers unordered. It’s trivially obvious that you can do something with the scores as a vector that is difficult or impossible with the scores as independent scalars, and the punchline here is pretty simple. Vectors are super useful in a bunch of circumstances.

Give me one step of trust: vectors are super useful in many ways. Velocities, scores, electrical currents, and a million other things are more usefully addressed as vectors.

Vectors are so useful that people wanted to do vector math on scalars. Which, if you think about it, doesn’t really make a lot of sense, but it turns out it’s also super useful. So a group of enterprising mathematicians figured out a way to do it.

Complex numbers let you do vector math on scalars. It’s what ‘i’ does. It’s what ‘i’ is. It’s the thing, like your phone charger, you plug your phone into one end, stick the other end into your wall outlet, and they work together. Complex formulations let you do vector math on scalars.

And seriously, that’s it. Don’t worry about the sqrt(-1). It’s as irrelevant to most functions as the bridge rectifier inside your charger is to whether or not it charges your phone. Yeah, it’s there, and yeah, it needs to be there, but if you’re just plugging in your phone, it’s not worth a lot of time. It’s certainly not worth aggravation and frustration.

Complex numbers are just stuck with two really terrible names: imaginary or complex numbers. They’re neither moreso than any other numbers.

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Geothermal Energy 2

A great peculiarity is that radioactive decay seems utterly immune to influence. High magnetic fields, electric fields, movement, and spin don’t seem to change the rates of radioactive decay at all. Nothing does. Particle interactions, including some really high energy photons, can cause a radioactive particle to decay differently right now but that’s usually a different decay. There doesn’t seem to be any particular way of changing the halflife of some whatever.

In numbers, the halflife of U238 is 4.468 billion years. If one sample of U238 has a hundred nuclei, in 4.468 by that sample will have fifty nuclei. We can’t change that without seriously altering the decay path. If an experimentalist whacked those one hundred nuclei with high energy neutrons, one could get some other decay path and decay reaction, meaning in far less than 4.468 by the sample would have only fifty or less nuclei. But the natural decay path is more or less immutable. Exposing the sample to a high magnetic field doesn’t change it. Putting the sample in zero g doesn’t change it. Putting it at the center of the Earth doesn’t change it. Whacking the sample with high energy neutrons doesn’t change the characteristics of the existing natural decay path (ie decay reaction); whacking the sample with neutrons may shift the decay reaction to something else. So instead of decaying into Thorium 234, we could smack it until it decays into something else.

So what does this mean?

It means the radioactive elements in the Earth’s core are naturally decaying in some fashion, releasing heat at some fixed rate, and there’s not much we can do to change that. This energy and the process of harnessing it are both called geothermal energy, sort of like how football is the game, the physical ball, and the activity.

So how good an idea is taking this fixed supply of energy for our purposes?

There’s a finite amount of sunlight that hits the Earth, right about 1300w/m^2 at the top of the atmosphere, and about 1kw/m^2 at the surface of the Earth. It’s coming whether we use it or not. Solar and wind are both fundamentally solar energy, as wind is caused by temperature differentials in the air, caused by sunlight hitting the heterogeneous Earth.

There is no consequence on the Sun by us harnessing solar energy. It’s close to free. To be more accurate, the energy is free; building stuff to harness the energy has costs, and so when we talk about the cost of solar energy, we talk about the costs of building the stuff. And maintaining the stuff. And hiring someone to go out to the solar panel farm to clean away the bird crap, pull the squirrel nests out of the wiring box, replace the wiring the squirrels ate, and replace the broken solar panels, etc.

If we take energy out of the Earth, would we be cooling the Earth? Is it radiant free like sunlight, or could we cause serious negative consequences? What happens to plate tectonics if we cool some part of the crust by 200k in a localized area?

Probably nothing, but that same probably nothing that people thought about greenhouse gases.

Remember the source of the energy, radiant heat from radioactive decay, is (as far as we can tell, but I find this kinda sus) utterly invariant.

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Interest Rate Policy and Differential Equations

There are these things called differential equations. They’re very common everywhere. They haven’t really penetrated the general consciousness of economics though, and that’s leading to the low interest rate problem.

Suppose something is related to and directly caused by something else. I have a scale. On the scale is a basket. I put apples in the basket and read the scale. There are no strings or springs or other weird stuff, and each apple weighs half a pound (a little more than 200-ish grams). I zero out the weight of the basket.

The output of the scale, the numbers on the display, is a function of the number of apples in the basket. If I put one apple in the basket, the scale will read 1/2 pound. Two apples: 1 pound. Three apples: 1 1/2 pounds. Etc. It’s exactly what it sounds like, and this isn’t a trick example. This is called a function, and in math terms, we would write f(x) = y where y is what the scale display displays, and x is the number of apples. So again, x=1, y=1/2, because each apple weighs half a pound. It’s exactly what it sounds like.

Every human being on Earth with a scale knows how this works. The trick is that a function doesn’t have to take inputs that are so concrete. The function input can be time, and that’s the real kicker.

Suppose you’ve got a house with a rain barrel (a rain barrel is just a barrel under the drain spout so when rain falls on the roof, it runs into the gutters, from the gutters it runs into the drain spout, and the drain spout empties into the rain barrel. Rain barrels usually have a lid on them so when it’s not raining, the water in the barrel can’t evaporate away. So all the water in the barrel comes from the drain spout). Now suppose it rains. Every second, some water is going to fall into the rain barrel through the drain spout. Suppose it’s not raining that hard, so every second a quart of water (a little less than a liter) runs down the drainspout into the barrel. One could write that as a function, f(x) = y where y is the total water in the barrel, and x is time in seconds.

One would need to pay attention to whether or not there was any water in the barrel to begin with, as well as whether or not the rain is constant. If it starts raining harder, the function f will change. If it lightens up, it will change the other way. Wind and temp may change things as well, as would the roof springing a leak. We’re going to ignore all of that for the moment.

But the basic math is pretty simple and intuitive. If one quart a second is pouring into the barrel, after one second, the barrel will have one quart. After two seconds, it will have two quarts. Three seconds: three quarts. Etc. Again, this is exactly what you think it is.

Now suppose the rain is getting harder. It started out with a drizzle, and the rain got steadily harder and harder. Maybe in the first second 0.1 quarts poured out the drain spout. In the next second, 0.2 quarts poured out. Third second: 0.3 quarts. So on and so forth. Obviously the function f will be changing. Initially, if nothing else changes, the math will just get more complicated. There may be more steps, but those steps will just be more of the same. You might have some multiplication or something, maybe a few more multiplications, but the math is basically the same kind of math, you’re just doing more of it. It is a little easier to make a mistake or forget to carry the one, but it’s generally the same stuff.

Here’s how to think about it: there’s only one function. That function may change, but there’s only one function. So the amount of rain that pours into the barrel between second 0 and second 1 might be different than between second 8 and second 9, but it’s just one function.

Now imagine the roof has two parts. One part is a little higher than the other. They’re connected, but each part of the roof has it’s own drainspout. For simplicity, say both spouts feed into the same barrel. We could easily write two functions, f and g, where f is the lower roof and g is the upper roof.

With me?

If the two roofs are separate, things are a little more complicated but not much. You just have to keep track of two functions. Maybe the upper roof is a little bigger than the other, more area, so it drains more water. So g(x) is a little bigger than f(x) at the same time. But as long as the rooves are separate we still only have two functions, and they don’t depend on each other. They are separate, the functions are ‘separable’ (math term), and again, you might have a little more math to do, but the math is all pretty straight forward.

Bad news. We didn’t clean the drainspout. It’s clogged. The upper drainspout g will carry some water, but if too much water lands on the upper roof, it overflows and spills onto the lower roof, and goes out the lower drainspout, f.

We now have a differential equation.

A differential equation is (often) non separable. You can’t talk about f(x) without talking about g(x). The ‘difference’ is inherent to the equation, hence ‘differential equation.’ The word differential means ‘showing a difference.’

Interest rates are what banks charge you or each other when they lend you or each other money. These interest rates change with time. That’s what it sounds like too. So last year, I got quoted a mortgage rate of 4.7% in Denver, CO, and earlier this year, I got quoted 3.7%. The interest is a function of time. Remember that f(x)=y? Same deal. The interest rate is y, and the x is time in years, but that’s the same as seconds, just with bigger numbers.

CHANGE in interest rates would be written as f'(x). See that little single quote? It’s called a prime, and it means we’re talking about how interest rates change. F'(x) is called the first derivative of f(x).

If we talk about how the water coming out of the drainspout changes with time, we’re talking about the first derivative. Since we aren’t talking about any derivatives, we can just call f'(x) the derivative. If it starts raining harder, f(x) goes up which means f'(x) is positive. If it starts raining lighter, f(x) goes down which means f'(x) is negative. See how that works? A negative f'(x) doesn’t mean the rain is going back up the drain spout. I guess that is possible, but it’s not what we’re doing here. A negative f'(x) means less water is coming out the drainspout every second, and a positive derivative (f'(x)) means more water is coming out the drainspout every second. If the derivative is zero, the amount of water coming out the drainspout isn’t changing. That’s why the derivative is called the rate of change.

Back to interest rates, the rate of change in interest rates has been negative in Denver over the last year. They went down. Last year I got quoted 4.7% and now I got quoted 3.7%. (I’m trying to figure out a way to make this work, but I’m a grad student and I have no money. Buy Mara, please. Pretty please. I wanna buy a house, or at least a condo)

If interest rates go up, the rate of change is positive. If interest rates go down, the rate of change is negative.

Still with me?

It’s sort of like running long distance. It’s simple to understand, but doing it is hard.

Negative interest rate changes, f'(x), are good for the economy. Negative interest rates, f(x), are bad. If f'(x) is negative, economy good! If f(x) is negative, economy bad. Conversely, if f'(x) is positive, economy bad. If f(x) is positive, well, actually there’s a middle ground that’s good but if they get too high, that’s bad too.

This is the problem. Cutting rates, what the Central Banks are addicted to, is good, but the problem is that cutting rates, making f'(x) negative, results in low actual rates, f(x), and that’s terrible. That’s tanking Japan, trying to tank Europe, and is a terrible concern in the US. Yes, in the short term, if you’re going from high rates to low, the cutting is good for the economy. But in the long term you get stuck with low rates, almost a decade now for Europe and longer for Japan, and it kills the economy. It kills the banks, and bank health is the poop of an economy. You may not like it, but if the poop is unhealthy, everything’s unhealthy.

But economics as a whole doesn’t recognize that these two things, f'(x) and f(x), affect the health of the economy differently. F'(x) negative is good, so central banks, CBs, keep cutting. But then f(x) is low, and there’s no way out. The lower bound of zero is a little fuzzy, so Europe can go -0.5% or so, but they can’t hit -3% or every bank in the continent will explode. And so they’re stuck, because they can’t go back any more and low interest rates plus a positive interest change (f(x) and f'(x) respectively) are both bad for the economy.

The best thing that happened for the US during the Trump administration, and the reason we outperformed the EU and Japan, was the Fed (US CB) raised interest rates off the floor starting in 2016. But they cut again with the pandemic, and now we’re stuck again.