@ShineOnYouCrazyDiamond You can look at Hilbert's tenth problem, if that's what you're looking for.
> Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm which, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns), can decide whether the equation has a solution with all unknowns taking integer values.
> Hilbert's tenth problem has been solved, and it has a negative answer: such a general algorithm does not exist. This is the result of combined work of Martin Davis, Yuri Matiyasevich, Hilary Putnam and Julia Robinson which spans 21 years, with Matiyasevich completing the theorem in 1970. The theorem is now known as Matiyasevich's theorem or the MRDP theorem.
Let S be the set of all bitstrings of length 3. Let R be the relation defined on S by aRb iff a XOR b 000 for a, b e S Here, XOR is the bitwise exclusive OR operator. (a) Is R reflexive? (b) Is R symmetrie? (e) Is R antisymmetric? (d) Is R transitive? (e) Is R an equivalence relation? Explain your answers.
Clearly it is reflexive since for exor same bit give vealue 0 so aRa but what about symmetry?
Set up a induction using $A$ of any finite cardinality, where the base case is 0. Then by induction, there exists $B$ that does not biject to any finite cardinality
so it seems let $[]_p$ be the preclosure operator with preservation of unions thrown away. Then given $A$ a natural number, if for all $B=[A]_p$ there exists a preclosure $[B]_p$ then there exists an infinite set (because it will fail to be bijective to any naturals)
Reading that RudyRucker book (currently in Chapter 4) seemed to help me understand why closed sets are called closed: They are precisely those sets which you cannot extend into any set that is more than itself
Therefore, a closure operator differs from a preclosure operator in that the extension is done in one step, such that after cl(X) is computed, it is already closed, but $[X]_p$ can be anything except each step brings it closer to cl(X)
But to say an actual infinite is "the first instance where something that is built from a rule eventually violate it under extension" is false. This is because of the existence of counterexamples like these:
Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of "proof". I receive responses like: "surely if Collatz is true up to $20×2^{58}$, then it must always be true?"; and "the sequence of number of edges on a complet...
@MatheinBoulomenos Rereading the wiki article, I see now the alternative description as the product, so I'm good on that now. But I don't understand your comment about the Chinese remainder theorem. What version of it are you applying to get this infinite product? The only statements and proofs I'm aware of require a finite product... If it is to work, I guess there is some intermediate step I'm not seeing.
@LeakyNun OK that certainly sounds reasonable. The main problem here is, as I was commenting earlier, a severe ignorance about how the projective limit works. Thanks!
I planning on getting around to category theory applied to commutative algebra someday... but right now it's lined up behind differential calculus and Lie algebra needed to state Noether's theorems.
@rschwieb you take a big product $\prod X_\alpha$ of objects indexed by a directed poset and then define projections to each factor that are somehow compatible with each other, and then the projective limit of that system is the set of all sequences in your product that satisfy some junk
If you google this one, you're going to get sickened by all the plaigiarization in the world going on. Everyone is stuck on the obvious case of $n = 2$. I want for $n$ variables (general linear).
So given an equation of the form
$$
D = Ax + By + \dots + Cz \tag{1}
$$
in finitely many variabl...
@ÍgjøgnumMeg I remember learning about cones and co-cones, and the special case of directed unions. But the whole thing about it commuting with the direct product is one of those pithy category things that I'm not handy with :)
@Shine in the case where you only have 2 variables $a_1 X + a_2 Y = b$ you have a system $a_1X \equiv \bar{b} \bmod a_2$ and $a_2Y \equiv \bar{b} \bmod a_1$
of linear congruences
alternatively one can just look for common factors on each side
>If $f$ is a continuous, even function such that $\int^3_0f(x)dx=-4$, then what is $\int^3_{-3}5(f(x)+1)dx$?
I was given this problem to solve, but I am not sure how to do it. How does the first integral help me find the second? What information does it give me?
gives these numbers: "Today, these itinerant teachers make up a whopping 75 percent of college instructors, with their average pay between $20,000 and $25,000 annually."
"A full course load for professors teaching at most Canadian universities is four courses a year. Depending on the faculty, their salary will range between $80,000 and $150,000 a year. A contract faculty person teaching those same four courses will earn about $28,000."
so contract/adjunct wages are still substantially lower, though not quite to the same extent
"According to figures provided by the Laurier Faculty Association, 52 per cent of Laurier students were taught by [contract academic staff] in 2012, up from 38 per cent in 2008. But of all the money the university spends during the year, less than four cents out of every dollar goes towards [contract academic staff] salaries. So the university spends less than 4 per cent of its budget to teach more than 50 per cent of its students."
anyways, what this establishes is that (adjunct pay) ~ (convenience store clerk pay), not <
> If you do want the big bucks, get an MBA, EngMBA or collab. program like a Law/MBA or MD/MBA. If you are only interested in an MBA, get one at a top business school....
interestingly, places like Germany don't have MBA programs.
@ÍgjøgnumMeg That comment displays horribly and I love it.
For reference, the number is (1237^103-1)/103 = 31713329297374332576894527165641966424631436761194411709592668486282299302630325315666637210628266399360431900619608528566764989896329675261932560474061176540538672215400517809802091028919080025379941402827808704922731160367673086713025323015760133812704050712567629023040055603812772486290056631432711359366015570884
If you google this one, you're going to get sickened by all the plaigiarization in the world going on. Everyone is stuck on the obvious case of $n = 2$. I want for $n$ variables (general linear).
So given an equation of the form
$$
D = Ax + By + \dots + Cz \tag{1}
$$
in finitely many variabl...
