@TedShifrin I might be overstepping here, but do you happen to have a suggested exercise list for the chapters of your text? I'd like to do all the questions for every chapter, but as we all know that isn't possible.
@dc3rd I actually did assign this problem. You can find my problem sets online (I can give you the link), but the computational stuff was WeBWork, online, so the problem sets are more theoretical. I don't know your background or goals.
Background: Did Spivak last year along with Insel's Linear Algebra, doing the second half of this concurrently with your book. Going to be in my 4th year of University in the fall. Took a year and half off because my math foundations have not been satisfactory for what I want to do.
Goals: In the end statistical modelling, but I would like to get a masters as well and who knows perhaps a PhD because I do enjoy the theoretical components of everything.
I'm glad you're putting forth the effort to improve your math foundations. The majority of students I've seen wanting to go into stats/DS have no interest in doing that
Spivak, Insel (Linear Algebra), Rice (Math Stats) [done it already] ----> Shifrin (then Munkres), Insel, Differential Eqns, Num Theory, Stats courses (regressions, etc) --> Topology, Groups, Rudin (or something similar was also thinking Royden) ---> Upper Year Stats courses that are built off of everything above
Stats/DS does not use differential equations at all. If you intend on going into high-performance computing, you'll find numerical analysis useful, but otherwise, you won't use it.
@Clarinetist Yes. For whatever reason I'm extremely particular and want to make sure I know WHY I may be able to apply such and such a technique. I'm not into just depending on the formula and chugging along. What if I want to alter the expressions and things of that nature.
@Clarinetist well I always had the view that whatever modelling I am going to be doing isn't going to be static and as such Diff Eqns might be handy for those scenarios
After you get through Insel's and Ted's text, I would suggest diving more deeply into statistical theory combining linear algebra and probability, something like Foundations of Linear and Generalized Linear Models by Agresti.
There's some other things I probably forgot to inlude, but that's the skeleton more or less
@Clarinetist I have Agresti's book here actually, but I know I'm not ready for it yet. You don't think maybe I should do the measure theory from Rudin or Royden first before I approach these higher level Stats books?
Stochastic modeling requires some background in stochastic processes - so you'd probably need something like Royden's text, a more probability-focused measure theory text, and then probably Stochastic Differential Equations by Oskendal
You DEFINITELY don't need measure theory before Agresti
If you want an alternative text, another one is Plane Answers to Complex Questions by Christiansen
Ok...well I'm also working through Kutner's Linear Regressions book at the moment. Definitely is a gem compared to the book I had to use for a past linear regressions course
(but be sure to start at the appendices before you start on Ch. 1)
Kutner is a classic, but avoids linear algebra for the most part (as far as I recall; I don't own that text but have skimmed it)
You're probably slightly confused by what I'm suggesting. I'll give you the spiel I told a Math PhD student recently
Stats does not build on itself like math does.
You do not need to take pre-calculus based stats, for example, before taking calculus-based stats. In fact, you could do measure-theoretic stats without knowing any calculus-based stats.
There are essentially three "levels" of stats. None of them build on each other, but are entirely dependent on your math background. Talk to any decent stats department with a graduate program: they will care more about your math background than any stats classes you have taken.
But there would be large gaps in what I'm missing from the big picture right? So for example I sat in on a grad level math stats course and the discussion of a sample space was far different from my 2nd year course
My personal opinion: I didn't really understand stats until my master's level courses (i.e., Agresti, Casella & Berger). Undergrad-level stats is too watered down for practical use.
Everything you thought you knew about sample spaces you will throw out when you hit PhD-level probability.
I see. Well now I have something to concern me because I in essence have scorched earth with respects to my math stream of things and was really banking on my performance from my higher level stats courses to compensate for the earlier poor math showing (I took math courses before I was mathematically mature enough and suffered the consequences)
hence why we have a "year and a half" break from school....lol
My alma mater, for example, strongly emphasized good performance in real analysis coming in.
Also, there is no shame in doing a MS before you pursue a PhD. That's essentially the route I went, because I was (and have) been working full-time on top of my education.
Anyway, my two cents. Hope that helps. I had to learn the hard way when it came to understanding how to structure my stats curricula to prepare me for industry work. No one really guides you through this stuff.
If you're intending on entering industry with a MS, Agresti and Casella & Berger will be the foundations of your work. I use Agresti's Categorical Data Analysis nearly every day (a higher-level text than the previous Agresti).
FYI I absolutely hate C&B, but it's better than the alternatives. You already know the material from Rice, so you have a decent foundation coming in already.
I have heard good things about Hogg and Tanis' text, but I've not obtained a copy nor read it.
If you want an absolute headache, see Bickel and Doksum's Mathematical Statistics, Vol. I
@dc3rd For stat grad work you definitely need some analysis strength, but you don't need Munkres topology. That is clearly overkill. I'm not sure you need more linear algebra than what's in my book (@Clarinet Do statisticians care about Jordan canonical form?). You definitely want some measure theory to do serious graduate probability, so Royden would be more useful than Rudin.
You need a bit of facility with fiddling with unions and intersections of sets, but you definitely do need a whole lot of point set topology.
@TedShifrin Regarding Jordan Canonical Form, no, it's not important at all. The eigendecomposition, Cholesky decomposition, and singular value decomposition are much more important.
@Clarinetist stats/ DS is making major inroads to physics and yes it is starting to be used on differential equation type systems, its a new area, but already strong results in it. deepmind protein folding is a top example, and ML is being used on quantum computing problems increasingly, etc
Actually before I took the decision to take this year off and thought I could rapidly plug my gaps I was reading and "attempting" Athreya's Measure Theory and Probability and encountered Borel sets, so it was from there I assumed that topology would be important even at the master's level
Remember how I said that when you hit PhD-level probability, you're going to throw out any understanding you had of sample spaces? That's the main reason.
anyway also on that topic, DiffEq is used heavily in engr ofc, and DS+ML increasingly being used in engr areas etc. so yeah, think DiffEq is one of the top math courses to take esp if one has any applied interests. and heck even if you dont theres a lot of deep theory in DiffEq etc
Right. And as much as I love Athreya's text... I've actually bought about 20 other books on that topic since I enrolled in that program about 5-6 years ago (just graduated last year)... I've not found a single book on that topic at that level that was suitable for self-study
I'm going through the second semester right now through a similar text, but I refer to Athreya all of the time as a reference.
@TedShifrin I would agree with that. You may want to also consider, in addition to the intro book by Strang, Linear Algebra and Learning from Data by Strang, better suited as a second text
We are given that $\forall x,y, f(x+y) = f(x) + f(y).$
a) We claim that $\forall i \in \mathbb{Z}_{\ge2}, \forall x_i \in \mathbb{R}, f(x_1 + \dots + x_n) = f(x_1) + \dots + f(x_n).$
Let $x_1,x_2,\dots,x_n \in \mathbb{R}.$ Using our given, let $x_1 = x$ and $x_2 = y$. Then $f(x_1 + x_2) = f(x_1) + f(x_2).$ Assume that it holds for some positive integer $k \ge 2$. Then $f(x_1 + \dots + x_k) = f(x_1) + \dots + f(x_k)$. We now note that $x_1,x_2,\dots,x_n$ are just real numbers, so define $z = x_1 + x_2 + \dots + x_k$ Then using our given, $f(z + x_{k+1}) = f(z) + f(x_{k+1}) = f(x_1 + \dots…
b) For the longest time, the wording confused me. I thought there was one universal $c$ which worked for all $x$, in other words asking me to prove the claim that $\exists c \in \mathbb{R}, \forall x \in \mathbb{Q}, f(x) = cx.$ Either way...
So after that wording confusion, I was unsure of what to do. So then I just continued onto #17:
We are given that $\forall x, y \in \mathbb{R}, f(x + y) = f(x) + f(y)$ and $\forall x, y\in \mathbb{R}, f(x\cdot y) = f(x) \cdot f(y),$ and $\exists x \in \mathbb{R}, f(x) \neq 0.$
a) We claim that $f(1) = 1.$ Since $f(x+y) = f(x) + f(y),$ notice that $f(1 + 0) = f(1) + f(0) = f(1) \Rightarrow f(1) = f(1) + f(0) \Rightarrow f(0) = 0.$ Then, using the other property, $f(1 \cdot 1) = f(1) \cdot f(1) \Rightarrow f(1) = f(1)^2 \Rightarrow f(1)(f(1) - 1)) = 0.$ Thus, $f(1) = 0$ or $f(1) - 1 = 0$. If $f(1) = 0$, then $f$ is the map $x \mapsto 0,$ since using definition two $f(x \cdot 1) = f(x) …
For problem 12 in the derivatives chapter, I have this:
a) $a'(t) = L(a(t))$ I assume one might mistake and write $L(t)$, but $t$ is time and $L$ maps distance to speed limit, not time spent on road to speed limit.
b) We are given that $a'(t) = L(a(t)),$ and $b(t) = a(t-1).$ We wish to show that $b'(t) = L(b(t)).$ Taking the derivative of $b(t) = a(t-1),$ we have $b'(t) = a'(t - 1) = L(a(t - 1)) = L(b(t)).$
c) We are told that $a(t) = b(t) + c,$ where $c \ge 0$ is some distance. Since $a'(t) = L(a(t))$, we see that $a'(t) = b'(t) = L(a(t)) = L(b(t) + c).$ $B$ will always travel at the speed limit if $c = 0.$
Here is my work on question 8 in the limits chapter:
a) It can exist. Define $f(x) = 0$ if $x < 0$ and $f(x) = 1$ if $x \ge 0$. Define $g(x) = 1$ if $x < 0$ and $g(x) = 0$ if $x \ge 0$. We see that $\lim_{x \to 0+} f(x) = 1, \lim_{x \to 0^-} f(x) = 0$, so $\lim_{x \to 0} f(x)$ does not exist. Similarly, $\lim_{x \to 0^+} g(x) = 0, \lim_{x \to 0^-} g(x) = 1.$ Therefore, $\lim_{x \to 0} g(x)$ does not exist either. However, notice that $\forall x \in \mathbb{R}, f(x) + g(x) = 1.$ Thus, $\lim_{x \to 0} (f(x) + g(x)) = 1,$. Using the same definitions for $f,g$, notice that $\forall x \in \math…
Am I? In that case... We know that if $\lim_{x \to a} f(x) = L_1$ and $\lim_{x\to a} g(x) = L_2$, then $\lim_{x \to a} (f(x) + g(x)) = L_1 + L_2$, then
Whenever you're proving that a certain limit exists assuming other limits exist, rarely do you want to use the raw "$\epsilon$" in cases for which you know the limit exists.
I know this is minor, but how is it that you justify this formally?
$$
\begin{equation}
\begin{split}
| x - a | < \delta &\Rightarrow |f(x) - l| < \epsilon \\
&\Rightarrow |f(x) - l| < \frac{\epsilon}{2}
\end{split}
\end{equation}
$$
To better illustrate what I mean, consider as a counter-exam...
The main point is this: since that inequality is true for EVERY $\epsilon > 0$, it is also true for any reasonable transformation of $\epsilon$ that also results in a positive number.
Ted is pointing you in the right direction, assuming you have that theorem available.
So then since the sum limit exists, we have that $\forall \epsilon > 0, \exists \delta_c > 0, 0 < |y - a| < \delta_c \Rightarrow |f(y) - L_1| + |g(y) - L_2| < \epsilon$
Heading to bed soon, but will chime in anyway: there is nothing deep to think about here (i.e., don't complicate this). What do you have available as the assumptions in part (b)?
Okay. Define $P$ to be the limit of f exists, $Q$ to be limit of g does exist, R to be the lim of the sum to exist. Our theorem states that $P \land Q \implies R$. As you wrote, $F \implies R$ is always true, so maybe?
The thing is you need to show that the set of those ' pythagorean tuples hypotenuse kind of thing' are unbounded. It's not too difficult, there's a one-line argument.
if $f : X \rightarrow \mathbb{R}$ is measurable, why is $g : X \times \mathbb{R} \rightarrow \mathbb{R}$ defined by $g(x,t) = \chi_{f^{-1}(t, +\infty)}(x)$ measurable?
More specifically, I want to know why the proof of this theorem goes through:
Let $\phi(t) = v[0,t)$ where $v$ is a borel measure on $\mathbb{R}_{+}$. Then for a random variable $f : (\Omega,\Sigma) \rightarrow (\mathbb{R}_{+} , \mathcal{B})$ we have $\int_{\Omega} \phi(f) d \mu = \int_{0}^{\infty} \mu(\{x : f(x) > t \}) v(dt)$.
The proof starts with $\int_{0}^{\infty} \mu(\{x : f(x) > t \}) v(dt) = \int_{0}^{\infty} \int_{\Omega} \chi_{f^{-1}(t,+\infty)}(x) d \mu(x) dv(t)$. Now we apply tonelli. At the very least to apply tonelli we should have that $\chi_{f^{-1}(t, + \infty)}(x) = g(x,t) : \Omega \times \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$ is measurable, but I can't seem to prove this, even just manually checking the definiton
I also supose we require $v$ to be a $\sigma$-finite borel measure ?
@Curio They have direction, so going from [1,1] to [1,2] is a different path then [1,2] to [1,1]. But I'm not sure what paths they allow. do they allow [1,1] to [2,3]? (i.e. top left to bottom middle?) do they allow [1,1] to [3,3] (skipping [2,2])?
I'm assuming not, paths can only go from [i, j] to [i ± 1, j ± 1] because then I do get 40
@dc3rd often, you need a minimum of topological structure on the space to prove existence of the (conditional) measures that you want to use
And to prove measurability of certain maps
Ted is right though, the necessary topology will mostly be supplemented by your lecturers/books. And the point-set topology that topologists care about is not the same as the point-set topology that probabilists/statisticians care about
@porridgemathematics I have no clue what that proof is trying to do, but this should be straightforward. It's only necessary to check that $\{(x,t):f(x)>t\}$ is measurable and this is the preimage of $(0,\infty)$ under the measurable function $(x,t)\mapsto f(x)-t$
the proof is actually a bit slick, once you apply tonelli you get $\int_{0}^{\infty} \int_{\Omega} \chi_{f^{-1}(t,+\infty)}(x) d \mu(x) dv(t) = \int_{\Omega} \int_{0}^{\infty} \chi_{f^{-1}(t,+\infty)(x) dv(t) d\mu(x) = \int_{\Omega} \int_{[0,f(y))} 1 dv(t) d\mu(x) = \int_{\Omega} v[0,f(y)) d\mu(x) = \int_{\Omega} \phi(f) d \mu$
anyway @Thorgott answered my question, it comes down to the measurability of that function, since $\chi_{f^{-1}(t,+\infty)}(x) = 1$ if and only if $f(x) -t > 0$ if and only if $(x,t) \in h^{-1}(0,+\infty)$ where $h(x,t) = f(x) - t$ which is measurable as he explains
so the function is really $\chi_{h^{-1}(0,+\infty)}(x,t)$
A countable locally finite graph G is called amenable if inf{|del K|/|K| : K in G a finite subgraph} = 0, where |K| is number of vertices, and |del K| is number of edges going between points in K and points in K^c (the "boundary edges")
For example, Z^n is amenable (as are Cayley graphs of amenable groups). Groups satisfying linear isoperimetric inequality breaks amenability, so these are hyperbolic groups, for example.
To be clear my cigar is homeomorphic to the plane. The graph is $(\Bbb Z/4) \times \Bbb N$ with the bottom four vertices connected to a vertex on the mouth end of the cigar
I'm just making the circles fixed radius as you go off to infty
One possible way to state approximation I suppose would be spectral, that is, maybe spectrum of the adjacency matrix of K_n converges to the spectrum of G
This makes sense, yeah? Take a large n >> 1. Enumerate the vertices of G such that the first |K_n| vertices are from K_n. Then the adjacency matrix of K_n gives a |K_n| x |K_n| submatrix of the adjacency matrix of G such that it is almost a block diagonal form
Not all terms off-(block)-diagonally are zero, but there are a very few nonzero terms, exactly |del K_n| many
Which, relative to the size of the block |K_n|, is as small as you like.
So the question boils down to if you have an almost-block-diagonal self-adjoint infinite matrix like this, is the spectrum of the block close to the spectrum of the big infinite matrix?
This is the linear algebra. Should be true but I have not checked
Something like this seems right except I don't know the spectrum of the bottom right bit.
The spectrum of $\begin{pmatrix} A & C \\ C^T & B \end{pmatrix}$ is very close to the spectrum of $A \oplus B$ when $C$ is small. (Straightforward computation; pay attention to what happens to kernel instead of arbitrary eigenvalue since the latter is a variation on the former)
For instance if $\lambda$ is not an eigenvalue of $A$ or $B$ and $\|C\|$ is (a factor of like 1/5 or something) smaller than $|(A-\lambda)^{-1}|$ and $|(B-\lambda)^{-1}|$ then $\lambda is not an eigenvalue of this resulting matrix.
The spectrum of everything relevant are discrete sets I believe
we're looking at spectrum of I - (adjacency matrix), the graph Laplacian, which always have discrete spectra. So maybe also convergence pointwise, eg, looking at the bottom of the spectrum, convergence there, then the next one, etc
Guys, I want to show that if $(X,\mathcal O_X)$ is an algebraic variety, then $(U,\mathcal O_X\vert_U)$ is too. I checked that $(U,\mathcal O_X\vert_U)$ is a $k$-space. Now for $x\in U$, there exists some open $V\subset X$ with $x\in V$ and closed $Y\subset\mathbb A^n$ such that $\phi\colon V\to Y$ is an isomorphism of varieties.
Iif we restrict $\phi$ to $U$, then this is still a morphism (since we have a morphism iff each component function of $\phi$ is regular, which is a local property). However, now we have $\phi\colon V\cap U\to Y$, but $\phi(V\cap U)$ isn't necessarily closed in $\mathbb A^n$, so I'm not sure how to get an iso.
What is the spectrum? Surely it convergences to the spectrum of the Laplacian of Z, which is tridiagonal infty x infty with 1's along the main diagonal and -1's along the super and sub, but going both sides
Spectrum is just {1, -1} with some multiplicities, or what?
You're right though that in general it's not clear why the spectrum of the complement of the Folner set doesn't matter
This is just not a counterexample. Maybe there is some
In any case I'd argue by Mike's statements at least one direction is clear, the limit of the spectra of the Folner sets sits inside the spectrum of the full thing
When dealing with groups if $F_n$ is a Folner sequence then $K_n=\{ab^{-1}\mid a,b\in F_n\}$ does cover the whole thing. Is it still Folner? Not sure. Can something similar be done with graphs?
A sequence of closed sets $A_n$ of $\Bbb R$ converges to $B$ if for every $R > 0$, $A_n \cap [-R, R]$ converges to $B \cap [-R, R]$ in the Hausdorff metric on $[-R, R]$
Suppose $x_{n} \in \mathbb{N}$ for all $n$ and $\left(x_{n}\right)$ converges to $x .$ Show that there is an integer $N \in \mathbb{N}$ so that $x_{n}=x$ for all $n \geq N$
@MatheusSousa Forget all the other complicated junk. Thorgott wants you to tell him what happens to $i^k$ as $k$ increases, and then $(-1)^k$ as k increases, then to $(-1)^k$ as k increases.
This should make clear what these things (to the power of 4n-k) are.
(I won't be around to discuss this more, just wanted to clarify his hint)
@MikeMiller it happens that $i^k$, $(-i)^k$ and $(-1)^k$ increase absolute value when k increases, but the negative terms oscillate between negative and positive when k increases
On page 18 of this paper, the author states that there is a duality (correspondence?) between semilattices (i.e., abelian semigroups of idempotents) and totally disconnected locally compact Hausdorff spaces.
Given a semilattice $E$, consider the space of characters
$$\widehat{E} := \{\chi : E \to...
@MatheusSousa The thing about absolute value is totally false. These all have absolute value 1. I don't follow what you mean about negative terms but it sounds like it's on the right track.
So I haven't done calculus in enough time- if $\partial f /\partial y = 0$ for $f(x,y)$ will all the mixed partials (including higher order ones) also be zero?
I know this is minor, but how is it that you justify this formally?
$$
\begin{equation}
\begin{split}
| x - a | < \delta &\Rightarrow |f(x) - l| < \epsilon \\
&\Rightarrow |f(x) - l| < \frac{\epsilon}{2}
\end{split}
\end{equation}
$$
To better illustrate what I mean, consider as a counter-exam...
