hello, how does one use the cauchy-schwarz inequality to minimize a squared distance? specifically, i'd like to use the cauchy-schwarz inequality to compute the distance between two parallel hyperplanes
Yeah, I find myself trying to parse a sentence for a minute and trying to be fair that maybe they do understand the concept and are just communicating ineffectively
I can understand why people go for the "if I don't understand it, that's X points off" option
when left to my own devices in calculational classes, i was not a picky grader if the approach was solid. near full credit even if you were off by a minus sign or factor of 2 every once in a while. but when the 'little stuff' is so bad you can't do a single problem correctly, it shouldn't be like, okay, 60/100 because you always got a little more than halfway on each problem.
which is effectively what that instructor was doing.
people do introduce all kinds of relations to distinguish things but i have not seen people do that with the notion of equality. it's too useful to have and you can use some funny symbol or decorate a common symbol to mean something nonstandard.
some treatments of "infinity" do not give actual meaning to the symbol. calculus treatments of limits are often like this. so the kinds of equalities people write to motivate calculation may not make literal sense at the time they are introduced.
semi: who?
just kidding. i thought it would be funny to say that.
it wasn't a very good initial estimate but it was a starting point. newton was able to do a bit better
(specifically, what they were able to estimate is how long it took light to go from the Earth to the Sun. combine that with the actual distance and you get the speed of light)
I say this because $f(0)=0$ (it can be concluded so in this case) and then by subsequent usage of Young's inequality.
@robjohn noted :).
@leslietownes Just before an exercise on proving Young's inequality, it was also asked to prove that: if f is increasing on [a,b], a<b then $\int_a^bf^{-1}=bf^{-1}(b)-af^{-1}(a)-\int_{f^{-1}(a)}^{f^{-1}(b)} f$
But not sure if this is a generalization. I tried to use this in finding the largest value of a stated in the earlier question but it didn't seem to work.
A novice question, for anyone who cares to answer: Is 'Sum of Squared Deviations' the same thing as 'Sum of Squares Error'? (aka 'Residual Sum of Squares')
Hello im reading some lecture notes where they take a supremum of functions over some set which I am not familiar with : they write $\sup_{f\in \text{mlb.}}$, but dont clarify the notation for the set mlb - what could this stand for
Yeah, they're important; they show up everywhere. E.g., Sofic group conjecture, taking a direct limit of the $S_n$ gives you group of all permutation on $\Bbb{N}$ with finite support.
Conjugacy classes are important to think about too.
No, I don't. I think it has to do with the number of irreducible representations of the group, but I don't know exactly.
I read that: if in the cycle form of a member of a conjugacy class of $S_n$ has distinct odd numbers then conjugacy class of the corresponding element for $A_n$ splits in two equal halves.
if this makes sense.
I'm trying to prove that one.
The statement is stronger: it is if and only if actually.
By cycle form: I mean, the length of cycles in non decreasing order. e.g. (123) in $S_5$ has cycle form: 1,1,3 (1 is for 1 cycle; sum is =5).
say you have a structure with addition and mulitplication where the dynamics between these operations are of interest. you combine these operations into a single binary operation. Can the single binary operation encode useful information about the dynamics of the first structure?
geocalc: maybe? at that level of generality it seems like anything could be possible. although it isn't clear to me why the number of operations of interest would be more interesting than whatever information they were 'encoding' (e.g. what you would necessarily gain by going from two to one)
note e.g. that all of commutative algebra can be thought of as investigation between the relation between addition and multiplication, as could e.g. the goldbach conjecture. a narrower setting than 'addition and multiplication' might focus the issue
throughout the first chapter of these notes math.columbia.edu/~mnutz/docs/EOT_lecture_notes.pdf the author uses the notation $\sim$. Im not sure what it means? Its not an equivalence relation. Because he writes for instance for two measures $g,f$ $g \sim f $
I have seen that notation for random variables : i.e $X\sim Y$ if they have the same distribution
another related question I have is, if a binary operation, say addition, is not well-defined in a particular set, then I think the addition of elements won't necessarily form a nice algebraic structure...but well-defined just means not nice properties. You can still add elements if you wish. Is there a formal argument that says that if addition is not well-defined in a space, then you cannot get an algebraic structure?
geocalc at that level of generality i don't know what to say. it is true that if you assume various axioms ("structure") this can impose restrictions on how operations that those axioms refer to can be defined. the classic example is the ring axioms defining 0 purely in terms of + but the distributive law forcing a value of 0*0.
at the other end if you want a ring to have a certain thing you can often use a quotient object to create one. e.g. if i want something with x^2 = 2 i could consider the polynomial ring R[x] and mod out by the ideal generated by x^2 - 2. and you can more generally force other relations to hold literally in a quotient of a 'free' object. but sometimes it can be difficult to identify what that quotient object is or whether distinct things in the original ring remain distinct in the quotient.
geo i would defer to any homebrew definition in a paper but it likely means just f(x+y) isn't necessarily f(x) + f(y) (where you have some addition on the domain and codomain already)
hmmm... I mean the calculus/analysis classes I've taken so far seemed... I don't know how to put it. But while I learned a lot, I didn't really get what analysts do all day
For example, I have a professor who studies graph theory. And I kind of understand what he does all day. He told me some of the problems he was working on and I understood what the questions he was asking were
now there are other professors who study things like algebraic geometry, commutative algebra, harmonic analysis, etc, etc. I have no idea what they do all day
One problem with mathematics — compared, say, with biology — is that the field is very old. Although hard classic number theory questions can often be phrased without much background needed, most math research questions are past the level of introductory graduate courses, let alone what we typically teach the best undergraduates.
a weird flip side of that is the methods used in some of the fields where the problems can be expressed most simply (some number theory, some graph theory, maybe combinatorics generally) are also fields where the research techniques do not resemble much of what anybody sees in undergrad
at least, the parallels between undergrad analysis, geometry, and algebra and their research equivalents are a little closer
how do you define the multiplication of equivalence classes? they all turn out to be the same thing, but the proof might look different, depending on the definition.
ok. well the multiplication of classes is basically defined that way. it takes a little work to see that the class [xy] does not depend on which representative x you choose from [a] and which representative y you choose from [b], but it doesn't. so (having checked that detail) you can take [a][b] = [ab] to be the definition of multiplication of equivalence classes.
the answer is assuming this to be known. i don't see a proof of it.
@TedShifrin then how does an undergraduate figure out what areas they're interested in? Aren't you kind of supposed to know that before you apply to grad school?
Not really, @Derivative. But see what undergrad courses you enjoy the most and do best at. My standard suggestion to students was to focus on a handful of hard problems you solved and enjoyed struggling with.
Also read articles in the American Math Monthly or some comparable publication.
A good undergrad at a university will take some graduate classes.
@RandomVariable That looks like that should work. It doesn't answer your previous query, but it looks like what I was thinking about yesterday, but never got a chance to work on.
What is the maximum number of functions with a given y-coordinate and with an x-coordinate that depends on the y-coordinate in some way?
For example: $f(x)=\frac{1}{x}$ has coordinates $(x,\frac{1}{x}).$ And I believe this is the only function that satisfies the constraints.
So for the given y-...
i didn't downvote it, but i don't understand the question. "with a given y-coordinate" meaning what, exactly? (x^3, 1/x) parametrizes the graph of a function that is not the graph of f(x) = 1/x
if you're thinking about which functions from $\mathbb{R} \setminus \{0\}$ to itself have graphs that can be reparametrized as curves of the form $t \mapsto (f(t), 1/t)$, $t \neq 0$ for some $f: \mathbb{R} \setminus \{0\} \to \mathbb{R} \setminus \{0\}$, i think it's the bijections
although the 'formulas' in that first coordinate might get ugly if the bijection doesn't have a nice formula for its inverse
I think I just expressed that question poorly. I think my initial motivation was investigating two different parametrisations $t \mapsto (f(t),1/t),$ taking a mellin transform of both functions obtained by the parametrisations, and taking the inverse transforms, substituting $t=n^s$, summing over $n$, and calculating the resulting inverse Mellin transform after the substitution
to see if they were different
and in this case they're not different because, for example (t,1/t) and (t^3,1/t) have the same distribution arising from their mellin transforms
so I guess that means it's not a useful method for extending the domain of the series you get after substituting t=n^s
there's already a better method to extend such a p-series
I'm just exploring like when mellin inversion is a useful method in continuation of a given series. I don't really understand it completely
Would it be brighter? Different color? Gravitational lensing? Would black holes exist?
