This may be obvious to some, but I wasn’t taught the math lingo. I’d like to know:
a) What is discrete math? For every “x” there is one and only one “y”, correct???
b) What does “non-recursive” mean?
c) What is a logarithmic spiral? Is that a golden spiral? And if so, where does the “log” come into play?
d) What is an implicit formula? Is there an explicit form? I recall from ODE (Ordinary Diffy’Q) there is a general solution and a y-specific solution. But, I cannot recall implicit vs. explicit forms of equations. What is y = mx^2 + b? vs. y = Ax^2 + Bx + C?
The two questions here don’t seem to have anything to do with each other.
Discrete math is mathematics which studies objects of a discrete, rather than continuous, nature. That is, things like integers, or finite trees, or other such finitary structures, as opposed to things like so-called (sigh, what a terrible name) “real” numbers, or arbitrary smooth curves, or such things.
As for “x” and “y”… these are just names which can be used however you like. If “y” is defined to be a function of “x”, then for every choice of value for x, there is one corresponding value for y. But that’s not the only way these names are ever used. You could define x to be a function of y instead, or both to be completely independent variable quantities.
Something is called “recursive” if it is defined in some sort of self-referential way; e.g., I might say “A power of 2 is an integer which is either 1, or 2 times a power of 2”. Here I’ve used “power of 2” in the definition of “power of 2” itself; it is what we would call a recursive definition.
“Non-recursive” just means not recursive.
A logarithmic spiral is one where the distance from the center grows by the same factor any time you turn by the same angle. In other words, the distance from the center is an exponential function of the total angle by which you’ve turned from some starting point. In other words, the total angle by which you’ve turned from some starting point is a logarithmic function of the distance from the center; hence the name.
The golden spiral is the particular special kind of logarithmic spiral where the distance from the center grows by the golden ratio with every quarter turn (where the golden ratio is the unique positive ratio whose square is itself plus one). In other words, the golden spiral is the particular logarithmic spiral whose distance from the center after a half turn is the same as its distance from the center after a quarter turn plus its distance from the center right now.
An implicitly defined function is one where, instead of explicitly describing how to go from input values to output values, you just describe some relationship which has to hold between the input values and the output values.
For example, f(x)[sup]3[/sup] + x[sup]2[/sup] = 5 is an implicit definition of f(x) in terms of x. In this case, we can rewrite this more explicitly as f(x) = (5 - x^2)[sup]1/3[/sup].
Some times, you can turn an implicit definition into an equivalent, more explicit definition. Some times, you can’t (depending on what you’re willing to count as “explicit”).
That’s true for ordinary differential equations, but not everything is a differential equation. In particular, the two you’re about to list aren’t differential equations; there’s no mention of differentiation (i.e., derivatives) in them.
Those are both explicit definitions of y as a function of x. They tell us exactly how to compute what y is, one we know what x is.
(In fact, your first one is just a special case of your second one, where A = m, B = 0, and C = b.)
IIRC, “For every ‘x’ there is one and only one ‘y’” is the 6th grade definition of a function*. It’s also another way of describing the Vertical Line Test. That’s where you graph something and pass a vertical line through it. If the vertical line never touches two points on the graph at once, it’s a function.
*I say ‘6th grade definition’ because even though I’m a math major and should know better, I’ve since forgotten most of my math skills and don’t want to get myself into trouble by saying it IS the definition since I’m sure there’s something more complicated that I’m forgetting. But back in junior high, that definition will suffice.
Does one actually encounter ‘non-recursive’ used in this way? The only context I’d expect the term would be that of something like non-recursive sets, functions etc., where ‘recursive’ means basically ‘computable’.
It’s a perfectly good 12th grade definition of a function as well. If you want to get fancy then you can say it’s an informal definition of a surjection.
Ah, I didn’t think of that. “Non-recursive” does get used the way I mentioned (if you Google “non-recursive”, it’ll recommend the related searches “non-recursive quicksort”, “inorder traversal non recursive”, “non recursive merge sort”, “non recursive tree traversal”, “non recursive fibonacci”, and “recursively defined”), but I suppose, in a mathematics context, yours is more likely to be the relevant definition.
So, Jinx:
Another (completely different, only historically related) definition of the word “recursive” is “Something which a computer can be programmed to do”. There’s some push to stop using the word “recursive” in this sense and just say “computable” instead (see here), but for the time being, it has not yet taken over.
So you just have to be wary that there are these two completely different senses of the word “recursive” out there and use context to clarify.
It’s a perfectly fine definition of y being a function of x, as you note.
I’m not sure why you mention surjections, though: one might also say x is a surjective injective multivalued partial function of y, I suppose, but that’s just another (rather roundabout) way of saying y is a function of x. The function f for which y = f(x) needn’t be a surjection, though. Saying “For every x, there is one and only one corresponding y” doesn’t imply “For every y, there is some corresponding x”.
If you want to get fancy then you can say it’s an informal definition of a surjection.