where can i learn the theory of equations for elementary set theory
specifically, (A n X) U (B N X^c) = {} where A and B are fixed subsets of a universal set and X is a subset of a universal set restrained by the equation it appears in. find a solution
it also does not help that finite sums of algebraic is algebraic, so transcendence happens at the infinite limit
Somehow in going to the infinite limit, the difference between the series and any nth partial sum is bounded above by all conceivable rationals that is parametrised by n. If there is a way to resolve this, then we can pin down the occurrence of transcendence
well I am not sure, but diophatine approximation is one of the commonly used techniques in transcendental number theory, while the other is to come up auxiliary functions so that the roots of them fall in convenient places
Nothing much other than the intersections are all algebraic
i am not sure if anything more can be said as if your s < 1/5, then you cannot guaranteed to get surds as the roots will be beyond 5th degree and 5th degree polynomials don't necessary have surd solutions
That equation reminds of some kind of generalised conics in $L^p$ space, thus presumably some kind of algebraic rotation can be defined on those points
but I don't know of any examples from my head which maps algebraics to algebraic in some natural fashion
mathien night have some ideas of common examples of such maps
I'm working on designing a drone system with some kids from school and one of the parameters of the challenge is that we can't use GPS. We decided that Light Detection and Ranging (LiDAR) was the best option and I came across this:
Is anyone familiar with these kinds of equations? I find this interesting, but am still trying to wrap my around it a bit.
Hm... well, I guess it's a little straightforward...
> WE TEND TO MEET ANY NEW SITUATION BY REORGANIZING. IT CAN BE A WONDERFUL METHOD FOR CREATING THE ILLUSION OF PROGRESS WHILE PRODUCING CONFUSION, INEFFICIENCY, AND DEMORALIZATION. -- PETRONIUS ARBITER, 210 B.C.
Or in other words: Permutation of sets produces nothing new
@CaptainQuestion Perpendicular to the x=0 plane (i.e. the y-z plane) is just any plane with normal vector in $\{(0,a,b)\mid a,b\in \Bbb R, (a,b)\ne(0,0)\}$ so any plane of the form $ay+bz=d$ for $d\in\Bbb R$. Requiring it to pass through $(2,-1,0)$ says that $-a=d$ so any plane of the form $a(y+1)+bz=0$ (with $(a,b)\ne (0,0)$).
@AlexanderGruber Hi. Do you find it OK to post such a link at my user profile of the site, indicating the right spirit to approach the mathematical problems? youtube.com/watch?v=upwyWKzozII&t=5s
It could be inspiring, and it also reveals a true fact: the ground of mathematics is a very tough one. You want to be a super professional.
However, the clip I admit could be wrongly understood and eventually considered rude. That's why I asked.
Anyway, I prepare to celebrate the new year, but not for too long since I have a lot of research to do. I've always found these days perfect for research (toward the end of the year).
I have born in leap day march 18.03.2000 who to calculate my age march 18 leap day in Gregorian clander please say the formula to calculate the age on leap day march 18 is leap day what is my real age?
If I have a compact manifold X with a cohomology class $0\neq \alpha \in H^1(x;\mathbb Z)$. What does the following mean? "Let $\pi:\bar X \to X$ be the corresponding infinite cyclic covering"?
Is there a good resource/reference for citation conventions in math papers? For example, if Smith wrote a textbook with the standard definition of a "frumious" widget, and I'm writing a paper on semiwidgets with an analogous definition, do I need to call it out explicitly, like "following section 4 of Smith, we say that a semiwidget is "frumious" if [almost identical definition]"?
I am shifting inside a dilogarithm and inside a logarithm, and I would like to use the identity: wolframalpha.com/input/…
The integration boundaries are between 0 and 1, regardless, this seems to constrain what I call $\alpha$ in the link above
$\alpha$ is real in almost all cases that I could see, but may have an imaginary part of a multiple of the shift that I envisioned to apply the identity
Suppose we call the constant $c$, then I would like to shift by $ic$ for example. The denominator may have a $nic$ in it, with $n$ part of the real integers except zero (in which case the identity can be applied trivially)
@1010011010 in the complex numbers, instead of doing substitutions, we chose a different path in the complex plane along which we integrate. Using the cauchy integral formula/ the resiude theorem often such a substitution is valid, but in genereal it is not
@1010011010 Maybe you could try to see what happens if you chose a suitable contour for integration. As you want to integrate over a bounded set, you will probably get correction summands from closing the contour. This may lead to an ugly calculation. You should ask this as a question on the main site.
Perhaps I should note that by shifting my integration coordinates by the above, I obtain a real expression inside the integral, only the variable is complex
But I'm not sure whether one can treat the variable temporarily as real
I'm aware that I'll run into some ugly expressions. I expected no less given the context of my motivation for doing the integral in the first place
conjecture: if $\operatorname{char}(K) \nmid n$ and $\mu_n \subseteq K$, then for $a \in K$ there is $ef=n$ and $b \in K$ such that the factorization of $t^n-a$ into primes is $\prod_{i=1}^f (t^e - \zeta^{ie} b)$ where $\zeta$ is a primitive $n^{\text{th}}$ root of unity
for the above case $K = \Bbb Q(i)$, $n = 4$, $a=-4$, we have $e=1$ and $f=4$ and $b=1+i$, i.e. $t^4-(-4) = (t-(1+i)) (t-i(1+i)) (t-i^2(1+i)) (t-i^3(1+i))$
conjecture: if $\operatorname{char}(K) \nmid n$ and $\mu_n \subseteq K$, then for $a \in K$ there is $ef=n$ and $b \in K$ such that the factorization of $t^n-a$ into primes is $\prod_{i=1}^f (t^e - \zeta^{ie} b)$ where $\zeta$ is a primitive $n^{\text{th}}$ root of unity
@Astyx well anyway here's the (elementary-)number-theoretic version of the conjecture: if a partition of $\{1, 2, \cdots, n\}$ into $f$ boxes is stable under $i \mapsto i+1$ (map $n$ to $1$) then the partition must be in the form $\{ \{ i + jf \mid 1 \le j \le e \} \mid 1 \le i \le f\}$
i.e. if $n=9$ and $f=3$ then it must be {{1,4,7},{2,5,8},{3,6,9}}
do you think the conjecture/theorem above has a name
conjecture: if a partition of {1,2,⋯,n} into f boxes is stable under i↦i+1 (map n to 1) then the partition must be in the form {{i+jf∣1≤j≤e}∣1≤i≤f} i.e. if n=9 and f=3 then it must be {{1,4,7},{2,5,8},{3,6,9}}
A, B, C, D, ..., and so on are fixed subsets of a universal set, U. X is a subset of U and is restrained only by its particular equation. Find the condition(s) under which a certain equation has a solution and obtain all the solutions.
Ex) X intersection C = D only has a solution if C + D is a subset of X, and X is a subset of D. where "+" represents symmetric difference
i read "set theory and logic" by robert stoll and the author talks about such things but it's seemingly obscure and i'd like to see other literature on the topic
@Jasper I'm sure you will. Any difficulty can be surpassed, even if it's hard and often seems impossible.
@Jasper You never need confirmation you can, you have to know within yourself you can and you just do it. That's the attitude, to be extremely powerful, in all circumstances.
Always winner, always you know you'll do it no matter what.
Probably there are more computer-illiterate people in these fields.
