@Drathora Not saying there is anything wrong with explaining in detail. I've done it a lot of times in chat. I've also done it in one go. Sometimes it's easier to see the whole thing at once. Not sure which is the case here.
If you think about it, for every path to $n+1$ there are $f_m$ paths to add to our "list"
So take a path $p$ from $1$ to $n+1$. Then if there are $f_m$ ways from $n+1$ to $n+m$, then there are $f_m$ paths from $1$ to $n+m$ that have $p$ as a sub-path
Or if you prefer "begin with $p$"
And since there are $f_{n+1}$ unique paths $p$ to $n+1$ we can choose, that means that the total number of paths that go through $n+1$ is equal to the multiplication
Since for every $p$ we add $f_m$ paths to our list, and there are $f_{n+1}$ paths that we could have chosen as $p$
I have the following question: Consider one vector of responses x taken from a population following Np(µ,I) where µ is known to satisfy µ'µ = 1. Find the maximum likelihood estimator of µ.
That being said, this is multivariate, so what is the first thing you think of when you are asked to maximize (or minimize) some multivariate function with a constraint on one of the variables? @talisa
Other asker mysteriously deletes their question and asks a new one on the same topic (multivariate analysis). Definitely seems like an online exam.
created a stats.SE account to vote up the answer sot aht the question cant be deleted. Got "Thanks for the feedback! Votes cast by those with less than 125 reputation are recorded, but do not change the publicly displayed post score."
@FranklinPezzutiDyer If $Rf$ is the Riesz transform of $f$, where $f\in L^2\cap C^0$ and $f$ has the modulus of continuity $\rho$, then $Rf$ has modulus of continuity $$ \tilde \rho(r) = C \int_0^r \frac{\rho(t) }t \ dt + Cr \int_r^\infty \frac{\rho(t)}{t^2} \ dt $$
I don't find anything wrong in asking for help, whether it's for an exam or something else. If the material in the test was actually in the notes and I hadn't studied or done further research that's understandable. But it wasn't so I tried my luck haha
The nature of stackexchange makes it very easy to get answers without doing any of the work. On this principle I think it's bad to consult it as a reference for classwork, let alone examination. To your benefit, you asked in the chat where the culture is a little bit more about assisting rather than spoonfeeding (though not much better).
In formal logic, is a proposition a statement which has a determined truth value, or is it merely something that can be said to be true or false, but not both.
I remember in my first class on logic it was "a statement with a well-defined truth value", but this seems like a definition which can be interpreted in various ways.
I know a random metric on a graph when appropriately scaled converges Gromov-Hausdorff to some some deterministic shape, and now I have a sequence of random metrics on a specified graph converging weakly to another random metric, and now I want to take "fiberwise Gromov-Hausdorff limit" after scaling and see what happens at the limit there
I am getting totally confused by symbols so I will just try to write it in words like this lmao
Let's look at a simpler example, the language of groups has the symbols $1,\cdot$, where the former is a constant symbol and the latter a binary function symbol
So formulas in this language are things like $\forall x\exists y(x\cdot y=1\land y\cdot x=1)$
The formula $\forall x\forall y(x\cdot y=y\cdot x)$ is a formula, because it is written down following the "grammar rules" telling us how to put together a formula, but it is true in some models and false in others
But once you fix a group it is either commutative or not
I've just found out that there is highschools in USA that teaches multivariable calculus (among the others)
How is that even possible
I've finished math undergrad in croatia last year. We have multivariable calc in 2nd year in both semesters. One for differentiating other semester for integrating.
And those 2 subjects are one of the hardest generally
both to pass and to learn
Now I'm wondering how can a 17 year old actually attend something like that ?
Considering the fact you need to have a relatively good knowledge in real analysis on the first hand
@domocar1 it's also worth mentioning that (multi and single variable) calculus courses are not actually the pinnacle of mathematics. It's mostly just mundane computations and methods, not requiring knowledge but just rote skill and repeated practice.
With that in mind, it's unsurprising that they could teach it to a high school student.
Whenever proofs crucially use thinking in terms of inequalities, it is hard
Many analysis statements are in this sense soft, because there's rarely any inequality-thinking inside the proof. Uniform limit of continuous functions is continuous is soft
The inequalities aren't fundamentally inequalities, they are just "how much blah is close to blah". For crucial inequalities, the sign $<$ is not important, but the distinction between "$x < y$" or "$y < x$" is.
The "very close" to argument you gave earlier is different from situations where you need bounds (like squeezing). Those things needn't necessarily be close, but one dominates the other.
Arthur Mattuck (emeritus at MIT) wrote a nice little analysis book, and his whole thing was to do proofs with $\approx_\epsilon$ rather than inequalities.
language is integral to doing mathematics, because proper definitions are and the way you define uniform continuity is in terms of inequalities (don't mention uniform spaces)
But he is basing the usual estimates on the philosophical difference Balarka and I were drawing. It is more intuitive and more accessible than a Rudin-type course.
I agree with Balarka that the notion of uniform continuity/convergence can be stated verbally with no mention of inequalities ... This is not true of all estimates in analysis.
I used to draw "epsilon fences" when I taught limits and analysis ideas. Granted, there's an implicit inequality when you stay within the fence ... but to stay within the fence everywhere at once is an intuitive notion.
but regardless, whatever you could reasonably call hard analysis will include more than quantitative inequalities and I don't see any reason to draw the line between soft/hard analysis at how you are using inequalities
I can give an elaborate example right now because I am trying to prove $\lim \mu_n = \mu$ for some constants $\mu_n, \mu$ and my proof is breaking up into two clear parts: $\limsup \mu_n \leq \mu$ and $\liminf \mu_n \geq \mu$. The first is "soft" in the sense that it follows from measure theoretic thinking. The second is turning out to be hard and I'm in dismay
Language definitely is crucial in mathematics. My observation is that at elementary levels (starting with little kids and going through middle of undergraduate) the people who struggle the most struggle because of language.
@TedShifrin in your experience, how do Ph.D.s usually choose topics for their theses? In particular, how much does the thesis supervisor dictate their chosen topic?
Is it relatively rare for the student's topic to diverge from the main supervisor's research interests? On the level of, say, the supervisor being interested in Poisson geometry, and the student pursuing a topic in low-dimensional symplectic geometry, or something like that. Kind of like, still in the same "subfield" classification, but different flavour, tools, and problems.
@BalarkaSen Although that was his 2nd PhD on some functional analysis thing just to get a job in US because he left Hungary. He was already famous for extremal graph theory
I'm sure Adams enjoyed this oddball student amidst his homotopical crew
I was thinking that this maybe boundary type question, where if minimum distance is less than 2 there will be no point on circle that will lie at a distance of 2 unit.
When is ODE taken in the US? Do you get any other experience with differential equations before that? I've heard that it's after calc 3 and sometimes its own course or lumped into Calc 4.
Interesting. The reason I ask is that both of my friends graduated without running into it. One majored in CS and the other in Civil Engineering. They did Calc 3 but no ODE.
Figured that'd be something that would be inevitable. Both took linear algebra though.
Makes sense. I can't imagine that a typical civil engineer is doing differential equations. But for CS, I can imagine that linear algebra is imperative.
I was mathematically illiterate for a long time, but I've been working my butt off to learn these things. MIT's OpenCourseWare and KhanAcademy have been invaluable. Having two friends that serve as tutors is nice too.
Oh Ted, just saw that you answered my question yesterday, "RREF doesn't guarantee that a system is maximally simplified, since different assignments between variables and columns yields different RREFs, correct?" with "Incorrect. Renumbering the variables and changing the columns of the matrix is a different matrix." I'm aware of this; what I meant to ask was, suppose you start with an underdetermined system, pick an arbitrary assignment from variables to columns, simplify via RREF, and convert
the result back to system form. Are you guaranteed that the result is maximally simplified, or could you get a better result by repeating the entire process with a different assignment between variables and columns?
"Is there a criterion, a clue that makes me think that certain integrals can also be solved through complex analysis and how to solve them?"
When I can't solve an integral, I use the numerical methods.
Thank you all very much.
"Is there a criterion, a clue that makes me think that certain integrals can also be solved through complex analysis and how to solve them?"
When I can't solve an integral, I use the numerical methods.
Thank you all very much.
