Recent content by 1MileCrash

You've made a choice to express the formulas in terms of the square's side length, but there's nothing wrong with expressing the formulas in terms of some other measurement of the square. The generalization is found by changing the measurement of the polygon that we express the formulas in...

I thought you guys might appreciate this. A lot of people notice that the derivative of area of a circle is the circle's circumference. This can be generalized to all regular polygons in a nice way.

Alright. And then, what remains after being factored has its largest value at k=6, and its value is smaller than any (k+1), and so I may write < k!(k+1), completing the induction.

Homework Statement Show that n^3 < n! for all n >= 6. Homework EquationsThe Attempt at a Solution We see that for the base case of n = 6, the claim holds. Suppose that k^3 < k! for some natural number k >= 6. Consider that: (k+1)^3 = k^3 + 3k^2 + 3k + 1 < k! + 3k^2 + 3k + 1 [By induction...

So, as all of you know, it is common in mathematical proof to begin a statement within the proof with one of those phrases such as "then," or "therefore," or "and so," or "hence", "thus" etc. But sometimes, for flavor, they can get a little more smug, such as, "indeed," - my topology...

You don't know crackpots until you've known John Gabriel. Think epic NPD and Dunning-Kruger.

It's not really suggesting that it is an "improvement", I'm merely asking the question "what happens if we relax our axioms." We don't have to call this new object a group any more, it doesn't matter. Immediately, Lagrange's Theorem will no longer work, for example, and G/{} would be a quotient...

Clearly, if the empty set were considered a subgroup, it would also be considered a group..

That's a pretty interesting "intuitive" explanation, but the fact does follow immediately from the distributive property.

Isn't a vector whose components are vectors, a pretty natural way to think of a 2nd order tensor?

One of the professors at my school has this as a main part of her research (looking at her publications, it appears frequently in the form of fractional differential equations). I've never done much reading into it, and it's well beyond my knowledge as well, but the example in the Wiki article...

So, subspaces of vector spaces, and subgroups of groups, are not allowed to be empty. This is because "there exists an identity element". We could include the empty set in these substructures but have the definition otherwise unchanged. I'm curious as to what the consequences of such would be...

If p is an automorphism on G, and H and K are subgroups of G, does p(H intersect K) = p(H) intersect p(K)? If so, how can I show this? EDIT: nevermind

AMenendez: the integral is not a very close approximation, it is exact. The integral of sin(x) from 0 to 42 doesn't give me a very close approximation, it gives me the exact area under the curve. A Riemann sum gives an approximation to the area of the curve. The limit of the Riemann sum is the...

https://proofwiki.org/wiki/Intersection_with_Normal_Subgroup_is_Normal Here is a very fast proof of a well-known theorem. What is proofwiki taking ":" to mean? I take : to mean "such that", but it doesn't make any sense that way here.