i never did this in a calculus class for fear that people would ape the wrong things. they won't know what the example is trying to convey vs. what it's not.
my first 'in the abstract' there should be replaced with something like 'as a matter of principle.' you aren't specifying a number of sentences as much as a kind of coverage.
one time as a postdoc i was mentoring someone for some kind of 'honors' project. everything i got was garbled over and over. it was very confusing because they were one of the best students in the class i taught but absolutely nothing they sent me on the project made sense.
in my case it turned out the student was working 30 hours/week on top of 2 courses and this honors thing and life stuff was getting in the way. i had a very unpleasant conversation where i said, this is not an evaluation of you, but nothing i've seen is close to being an honors project, please preserve yourself and try again next semester.
don't say "so it actually seems to me like you should have fewer distractions."
i had to pull back from some work stuff earlier in the year because a family member attempted suicide in a way that led to long term disability, this is just how my humor works.
Many other MSE users including mathlove, BillyJoe, and JyrkiLahtonen have been helpful in developing this formula.
Let $p_n$ denote the $n$th prime number. Let $a \gt p_n$ and $b \lt p_{n+1}^2$ be any such integers. Their oddness or divisibility does not matter as in my previous posts, which mak...
@Semiclassical you mean for the first one (since it is C(x) and not C[x], which I did not know there is difference), then omega is basically, in general, is a ratio of two polynomials in x, whose coefficients can be complex? Like this (3*x^2+2*x^2)/x ? or (3*I*x+2)/(3*x) etc...
@Semiclassical and what about the second one please? Did I read it right? The above is from the original Kovacic's paper I am reading now. I am confused about the use of "algebraic" there. Since this mean omega is a ROOT of a polynomial, right? Then how could the root be function of x?
@Semiclassical Yes. He wrote the second paper to add more examples. I am trying to understand the algorithm. but wanted to get over little bit of the math notation used. I am not pure math student.
In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A natural example of a differential field is the field of rational functions in one variable over the complex numbers,
C
(
t
)
{\displaystyle \mathbb {C} (t)}
, where the derivation is differentiation with respect to t.
Differential algebra refers also to the area of mathematics...
I wish there is a place or book which translate some of these expressions commonly used in math, to plain English that even engineering students can understand :)
@Semiclassical thanks. Will go over this link you show. look useful. Yes, I know omega will end up being solution to linear ODE, from the algorithm itself. It is just translating the math to something more easy to understand, so I can code the algorithm is the issue. But will go the ref you showed now. It should help. thanks.
@Nasser the trouble you'll run into is that differential algebra assumes a working knowledge of field theory in abstract algebra, including the theory of field extensions
@Semiclassical Yes. I am just trying to implement the algorithm now. it has 3 cases. The first one is the easiest one. But just want to learn a little bit of the theory first. But it is hard, since I do not have the pure math background needed.
@Semiclassical there is another topic called 'Lie symmetry group for solving ODE" which is also very useful. But I know little about Lie symmetry yet. But it can also be used to solve ODE's in way one can't using standard methods. Lie symmetry and Kovacic algorithm are both implemented in Maple and Mathematica to solve ODE's but I do not know to what extent.
The main author of Maple dsolve command has number of papers on Lie symmetry.
I read this last night. It is cool that by just looking at the term called "r" in the paper, one can quickly check if there is Liouvillian solution or not.
Let $p_n$ denote the $n$th prime number. Let $a \gt p_n$ and $b \lt p_{n+1}^2$ be any such integers. Their oddness or divisibility does not matter as in my previous posts, which makes this formula particularly nice.
An inclusion-exclusion based & modular-arithmetical counting formula for the twi...
please, I want to learn more about n root and cube root do you have any good books about that, and please Does the Domain of Cube root is R not like square root R^{+} Thanks because I'm confusing in this question when I want to calculate the limit in the left side of 0. math.stackexchange.com/questions/4299150/…
@AbstractSpacecraft BTW, you may annoy the room regulars by excessively promoting your posts. Three mentions (including two one-boxes) within a few hours during a slow period is probably a bit much. ;)
@AbstractSpacecraft I had a look at your Python code in some of your twin prime questions. I didn't look very closely for bugs, and it appears that you've resolved those issues, but I did see several things that can be optimised.
Eg, instead of for i in range(m): if i<a: continue you should do for i in range(a, m):. Also, if a & b are integers, a // b is much more efficient than floor(a / b).
@AbstractSpacecraft Well, kind of. It looks neat, but it's kind of hiding the mess in things like the CRT stuff. ;) But maybe there's room for some optimisation there...
Another thing I noticed is floor(sqrt(x)) inside a multiply nested loop. That's a bit slow. You could make a table for that. Another option is to create a loop that avoids doing the square root calculation. I wrote a little demo...
Yes, I used that formula (in math though) $-\lfloor x \rfloor = \lceil - x \rceil$ converting the $a$ expression to floor from ceil. I thought it would be used in relation to $\pi(x)$, but it turns out not
I was introduced to the term half life as the time it takes for the number of radioactive nuclei to become half of its initial value in a radioactive sample.
But there is a question in "Concept of Physics by HC Verma}" which says that a free neutron decays with a "half life" of 14 minutes. Now th...
I'm reading lecture notes in which my professor is arguing that a certain bounded linear operator $T$ from a Banach space to itself has $0$ as a spectral value. He claims it suffices to find a sequence of points $x_n$ such that $||x_n|| =1$ but $||Tx_n|| \to 0$.
Why does this suffice to show $0$ is a spectral value of $T$?
write y_n = T x_n. if 0 is not in the spectrum of T then T^{-1} exists and is bounded by some standard theorem. but then 1 = ||x_n|| = ||T^{-1} y_n|| <= ||T^{-1}|| ||y_n|| is bounded above by a sequence that converges to zero.
given the level of the question i might spend a moment or two commenting on why T^{-1} is bounded. maybe this is baked into your definition of 'spectrum.'
if not it probably falls out of any of the holy trinity of functional analysis theorems.
i saved time in my response by not using chatjax. this is how i beat other people to the punch.
@user193319 i saw your post on the fourier transform of something in L^2 \ L^1 the other day. did anyone answer?
@leslietownes Yes, I was able to figure it out. Thanks for asking. It turns out that that that homework was based on two sets of lectures, instead of the usual one lecture per homework. But I still did fine on the homework.
many classes include at least some general result about integral operators with square-integrable kernels providing bounded operators from L^2 to L^2. because it is easy. then the fourier transform has a kernel that is obviously not square integrable.
you can get L^2-boundedness but maybe not isometry-ness from the hausdorff young inequality.
Yeah, we need bounds on it, for example a positive lower bound would prove twin primes as $b = p_{n+1}^2 - 1$ grows
What's neat about it is that if it is growing with $n$ then likely it is always growing so that you can set it equal to zero and try to find a contradiction
Instead of just growing sometimes
has to do with the size of interval $(p_n, p_{n+1}^2)$
Intuitively there are plenty of twin primes that show up when you square the next prime
The verfication codes I wrote show a growing quantity
$$ f(a,b) = b - a + \sum_{1 \ \neq \ d \ \mid \ p_n\#} (-1)^{\omega(d)}\sum_{2 \ \nmid \ c \ \mid \ d}\left( \left\lfloor\dfrac{b - x_{c,d}}{d}\right\rfloor + \left\lfloor\dfrac{x_{c,d} - a}{d}\right\rfloor \right) $$
@TedShifrin
Isn't that a beauty - look at the symmetry!
It turns out that this approach does not automatically also work for higher gap sizes - you'd have to do another derivation for higher than gap 2
Ramanujan must have put the formula on my tongue whilst I slept
@monoidaltransform I'm not sure where you're getting this question, but the usual issue is whether the map is continuous. Here's a typical theorem: If $\bar M\subset N$ is an immersed submanifold. If $f\colon X\to N$ is smooth with $f(X)\subset\bar M$ and $f$ is continuous as a map into $\bar M$, then $f$ is smooth as a map to $\bar M$.
@AbstractSpacecraft Don't just ping me randomly. I am not interested in the twin prime conjecture.
@monoidal I used $\bar M$ so that you could then put $X=M$ for your use, I think.
The remark I made is the interesting situation. There are various ways to see what he claims; the most geometric is to use a tubular neighborhood of the embedded submanifold. But it's just using a usual chart that flattens out the submanifold.
right for the immersed situation, I feel it's harder. Because for the embedded case, I can invoke the fact that characteristic property of the subpace topology
she went and got crackers out of a cabinet. these were put away, which resulted in the retort: "you can't just tell me that you put away my crackers."
then she told my wife to stop telling her to have lunch. "when a kid tells you to stop, STOP" she said. i wonder if she may have heard this at day care.
