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Dec
12
comment What's the difference between a bijection and an isomorphism?
In short, a bijection (or -jection) is a kind of function on sets. An isomorphism (or -morphism) is a kind of function that is structure preserving. Or rather one term is in the language of set theory, the other is in category theory. Underlying all isomorphisms is some bijection, but a bijection doesn't necessarily preserve structure in the category you care about. If your category is Sets, then they're the same.
Dec
8
comment Is 128 the only multi-digit power of 2 such that each of its digits is also a power of 2?
Don't forget to also rule out 8.
Dec
3
comment Is it possible to simulate a floor() function with elementary arithmetic?
@ChristianSemrau the given ops do not allow complex math (exact, not floating point) or arbitrary roots of unity either. The true answer is either 'no' because of the reason given in the top answer, or 'yes' because the domain is integers and integer division already gives the floor.
Dec
2
comment Is it possible to simulate a floor() function with elementary arithmetic?
Since all your ops are on integers, the output of each op will be integers. The default implementation of integer division and so will be to 'round to 0' which is equivalent to 'floor' on non-negative integers. Which means you don't need a special 'floor' function because you already have it by default.
Sep
29
comment Why are real numbers useful?
Use the binomial distribution.
Sep
25
comment Understanding why Hall's marriage theorem $\Leftrightarrow$ Dilworth's theorem
robertborgersen.info/Presentations/GS-05R-1.pdf gives a number of proofs around all these theorems.
Sep
25
comment Analytic Proof Of ${n\choose r}={n-1\choose r-1}+{n-1\choose r}$
I would call this an 'algebraic' proof. An 'analytic' proof would more likely be a proof using generating functions. Of course everything uses algebra to some extent, but the major category of a proof is the primary strategy.
Sep
25
comment Sum of sequence of cubes and summation on the upper index
$k^3 = 6{k \choose 3} + 3 k^2 -2k $ (I got this by expanding out ${k \choose 3}$). Now eliminate the $k^2$ using ${k \choose 2}$. Then again for the leftover $k$ term. Now, if you sum the LHS, you sum the RHS. But by the stated thm, you can calculate the sum of the RHS.
Sep
25
comment General term of $(1+x)(1+x^2)(1+x^3)…$?
If you want to calculate these without calculating coeffs of the gf directly, I think the best formula is the recursive one in the OEIS: a(n)=(1/n)*Sum_{k=1..n} b(k)*a(n-k), with a(0)=1 and b(n)= A000593(n) = sum of odd divisors of n.
Jul
28
comment Is every factorial divisible by its sum of digits?
@G.H.Faust: Add it to OEIS!!
Jul
27
comment Is every factorial divisible by its sum of digits?
what about sum of binary digits? (following the usual observation that there's nothing special about 10)
Jun
28
comment How to define the disciplines of mathematics
Everything is both similar and different from everything else. The general ideas of the above three categories solidify around their respective centers, but they cross over quite a bit.
Jun
6
comment 'Obvious' theorems that are actually false
For #4, the restriction where this _does work are given by the Fubini-Tonelli theorem
Jun
6
comment Why does differentiating a polynomial reduce its degree by $1$?
Nice... this is Sturm's theorem
Jun
6
comment Generalised Binomial Theorem Intuition
"why don't they teach it in secondary school?" - That's what math.SE is for.
Apr
10
comment (k+1)th, (k+1)st, k-th+1, or k+1?
Also, the ordinal for 301 is pronounced "three hundred and first".
Mar
27
comment Why use *λx.x* instead of *f(x)*?
One could also ask then why don't we use '(x)f' instead of 'f(x)' (postpositioning the function name makes it easier to interpret composition of functions in a left to right reading.
Mar
7
comment What is exponentiation?
It all depends on what you mean by 'mean'. If it is simply a matter of "I don't get it" despite being able to do the manipulations, one could say that there is no intention of you to 'get it', it just a mindless preservation of the rules for basic exponentiation (which I think you do 'get') to allow some calculation. If 'mean' means "I don't see a quick generalization that encompasses integers and reals" then the best generalization is to complex numbers and scaling rotation. (hmm, that doesn't explain it for reals though...OK stick with the calculus/derivative equals itself.)
Mar
5
comment Does there exist a system such that the additive identity is non-zero?
In the semiring of languages (sets of strings), union of two languages acts like addition, and the additive identity is the empty set. Multiplication is string concatenation (well a little more complicated than just that) and the multiplicative identity is the empty string.
Mar
5
comment Does there exist a system such that the additive identity is non-zero?
@Brilland: they might be tempted to call it that, but that would be misleading about the meaning of the operators. The abstract operator '⊗' acts like multiplication. The fact that it is implemented as arithmetic addition is, well not exactly irrelevant, but just not as important is the fact that it acts analogously to arithmetic multiplication.