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1h
answered Is this a valid way of solving modular equations?
1h
revised How do I solve the following differential equation
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2h
answered How do I solve the following differential equation
2h
comment How do I solve the following differential equation
This is not correct. Airy functions are solutions to $y''(x) = x y(x)$, not $y''(x) = x^2 y(x)$.
May
20
comment Constructing the natural numbers without set theory.
Let us continue this discussion in chat.
May
20
comment Constructing the natural numbers without set theory.
@goblin I am aware of ETCS (and its distinction from material set theory), but I don't see how we need ETCS or any other set theory to construct a type theory. All I see is a rewrite system which stands just fine on its own, and never talks about sets, material or otherwise. There is of course some degree of similarity between ETCS and type theory (e.g. neither has a global membership relation), but neither needs the other. And once again, functions in ETCS are very different from those in type theory.
May
20
comment Constructing the natural numbers without set theory.
@goblin Yes, but then how is my distinction between a type-theoretic function and a set-theoretic one false? $S$ cannot be constructed as a set of ordered pairs, because the elements of the pairs would involve $S$ itself. So the type-theoretic notion of a function clearly is of a very different nature, especially when we extend it to dependent function types.
May
20
comment Constructing the natural numbers without set theory.
@goblin I'm not sure what you mean. If you look at a particular type theory (e.g. Calculus of Inductive Constructions) there is no appeal to any set theory; it is defined as a rewrite system that is independent of either an external set theory or logic. ZFC is often appealed to for proofs of consistency (relative to ZFC of course), but this is irrelevant to the fundamental construction.
May
20
comment Constructing the natural numbers without set theory.
@DavidWheeler There is no appeal to set theory, only type theory. A type theoretic function is of a very different nature than a set theoretic one (it is not constructed as a left total and right unique relation).
May
20
comment Constructing the natural numbers without set theory.
$0 \in \mathbb{N}$! Blasphemy!
May
20
answered Constructing the natural numbers without set theory.
May
20
comment Proving by Contradiction
@Kingle Because if $n = 4(k^2 + k + l^2 + l) + 2$, then the above formula gives $4|(4(k^2 + k + l^2 + l) + 2 + (4(k^2 + k + l^2 + l) + 2)4)$ which doesn't help us. However I was wrong too; $n = -(k^2 + k + l^2 + l)$. We need the negative sign so it will cancel with the $4(k^2 + k + l^2 + l)$ term.
May
20
revised Proving by Contradiction
added 1 character in body
May
19
answered Proving by Contradiction
May
18
comment “Vectors aren't really numbers” - how sound is that statement?
@immibis The complex numbers can be ordered if you only consider addition and subtraction. However our usual definition of order for fields includes compatibility with multiplication, so the usual rules like $a \geq 0$ and $b \geq 0$ implies $a b \geq 0$ hold. The lexicographic order on the complex numbers doesn't respect positivity when multiplication is involved, and in fact no order does (which is what oxeimon was referring to).
May
15
comment Why does $tr({A^*}A) = \sum\limits_{i = 1}^n {{\sigma _i}^2} $?
@AndreasH. The order is not important, since $A A^*$ and $A^* A$ have the same eigenvalues.
May
15
comment calculation of binary power like $a^b$ where a,b are binary numbers
Let us continue this discussion in chat.
May
15
comment calculation of binary power like $a^b$ where a,b are binary numbers
$0010^{0111} = (0010*0010)^{0011} * 0010 = 0100^{0011} * 0010 = (0100*0100)^{0001} * 0100 * 0010 = 10000 * 0100 * 0010 = 1000000 = 128_{10}$. If you found that hard to follow, that is because binary is not a good way to do calculations by hand; its much easier to just compute $2^7$.
May
15
comment calculation of binary power like $a^b$ where a,b are binary numbers
I'm not sure why you see the need to do it by hand with binary numbers; binary is mostly useful when doing things by computer, but here it goes:
May
15
comment calculation of binary power like $a^b$ where a,b are binary numbers
What this algorithm does is give an efficient way of computing an integer power of anything (even matrices or other groups/associative operations), using only multiplication.