@Jacksoja: You should notice (if you check carefully) that D&F define $R[a]$ to mean the ring of all polynomial expressions in $a$ with coefficients in $R$, and $R(a)$ to mean all rational expressions ...
@Jacksoja it's explicitly an undergrad book on algebraic number theory (the only one I know) and it's really accessible, I'd say it's an easier read than D&F
I am learning uniform convergence. I do not know how to find the $M_n$ function for this sequence $f_n(x) = \frac{x^2}{n^2}$ as the problem says that $x$ is over the whole real line. Any ideas? I know this is not supposed to be uniform convergent, but I need to proof it using the M test.
@Jacksoja I can do that, but I'm not sure if you will get much out of it. I'm assuming familiarity with complex analysis, algebraic number theory, algebraic geometry, category theory and homological algebra, among other things
@TedShifrin Yes, you are right., But I wanted to at least try to find the $M_n$. It is for sequence of functions. THat what the book says. "Which of the following sequences of functions converge pointwise to the zero function for all x in R". Then it gives -x^2/n^2.
@TedShifrin yes, I need to find $M_n$ that does not depend on x, such that |f_n(x)|<M_n for all n. But how to find M_n? I always have time find the M_n
OK. So now understand the definition of uniform convergence. If $f_n\to 0$ uniformly, this means that for large enough $n$ all the functions will be very close ("within $\epsilon$") to $0$. You can draw graphs, even. Is that the case?
@Jacksoja: Of course people read research papers at an advanced level and present talks on what they're learning. I'm just saying that students who just read more and more books often can't do much.
@Jacksoja I agree with Ted. One book I learned a lot from is Atiyah Macdonald and I learned that much because I solved pretty much all the exercises (I asked for help on a few). The exposition itself isn't even that great, but the selection of exercises is great
This is the crux of my perpetual argument with @Jasper. He loves books ... but pays no attention to exercises or whether there even are exercises. I truly believe that until you get to super-advanced levels, what makes a top math text is the exercises.
@Jacksoja if you're intrigued by quadratic integers right now, then maybe reading Jarvis might be more motivating than continuing with D&F right now. Reading the entirety of D&F sounds a bit exhaustive, especially given that you lamented the lack of motivation before
@TedShifrin yes, I saw you said " If $f_n\to 0$ uniformly, this means that for large enough $n$ all the functions will be very close ("within $\epsilon$") to $0$" but i am not sure if this will be enough proof. But will ask teacher. I thought we are supposed to use M test here.
@MatheinBoulomenos we are only doing ring and group on DF . but my plan was to work though one complete book from start to finish just to get a complete picture of some topics, then start with real material
@TedShifrin @MatheinBoulomenos okay I see, but group ring filed,galois , linear algebra than representation theory of finite groups, is that a good road?
Well, I can do on paralell some number theory of fazer
I want to see if it is really as easy to follow as you said .
I think that is enough talking on my part , I should continue my HW ^^ thanks so much for the information and recommendation @MatheinBoulomenos @TedShifrin
For someone like me (a non-student, self-learner) it's pretty infuriating because there's no way to check that my understanding is on track or that I'm doing things correctly
But human nature is human nature. Even a self-learner is tempted to cheat a lot.
Well, most lectures are unremarkable, @Mathein, but some of us do try to give insight and show students how to approach mathematics. But, especially for a student like you, lecture is mostly redundant.
AS I explained, authors of serious math books don't want their books ruined by having them or having them available.
The undergrad diff geo book I know that has answers to all the exercises most faculty find impossible to use as a textbook if they teach the course. I don't know it well enough to comment, regardless.
the problem is that it makes the material a lot less accessible to people unless they have the means to attend a good school with professors they can talk to and ask questions to etc, being able to talk to informed people on a regular basis is invaluable
part of me also feels like if someone wants to cheat, they only cheat themselves
well, that's good philosophy, but universities still give grades that matter ... and some of us like to count homework seriously for the grade, not just base things on two exams.
But you should take advantage of serious mathematical discussions on MSE, clearly.
i understand, just saying it would be nice to see more textbooks that aren't intended for a classroom / grading setting, where practice problems and answers are provided
there are many free resources but it's also very unstructured, not everything standardized / quality-checked, could learn bad habits or slightly incorrect things conceptually, end up with gaps, unchecked assumptions, having a lot of available material doesn't always make it easier
becomes a challenge to sift through all the material and learn how to identify good material
@RyanUnger I've gotten past procrastinating on writing for research. So right now I'm instead procrastinating on rewriting a lab report guide for my students.
@user525966: I wrote 4 textbooks and made pitifully little money off them. The last one is in .pdf form available on my website and on the AMS website for free. But I'm not going to write special books for self-learners. Just isn't practical, sorry.
@user525966 even with the best possible material available, you're going to be at a severe disadvantage without access to a teacher and peers to talk to
On this paper by M. Athanassenas it is claimed that given a function $u(x,y)$ defined on the whole of $\mathbb{R}^2$, it satisfies
$$
(1+u_y^2)u_{xx} - 2u_xu_yu_{xy}+(1+u_x^2)u_{yy} = 0
$$
If and only if there exists a function $\phi(x,y)$ such that
\begin{align}
\phi_{xx} = \frac{1+u_x^2}{\sqrt{...
@user525966 there's a book I like called "Problems in Algebraic Number Theory" it's 95% problems with complete solution. Are there really no similar books like this for other areas?
Can one calculate a least norm solution for an underdetermined system using the Cholesky decomposition? All the sources I find talk about the least squares (overdetermined) solution.
Anyhow, do you agree that there's a problem with Rudin's proof? @Ryan ... My comment to the OP was that there's a good reason most of us prefer to work with smooth forms :)
@TedShifrin if we take a twice-differentiable function $f:R^2\to R$ which does not have symmetric Hessian at $0$, doesn't $\omega=df$ provide a counterexample?
In the comments of math.stackexchange.com/questions/1139603/… Daniel basically instructs the OP to show that any measurable null set $E$ is sandwiched by two other sets $A\subseteq E\subseteq B$ such that $\mu(B\setminus A) = 0$. Thus $\mu(A)\in\mathcal M^*$. I don't see how this necessitates that all subsets of $E$ are in $\mathcal M^*$.
better question based on my earlier one: Suppose I have a transformation from R^2 to itself. What does it mean if the Jacobian determinant is everywhere 1?
Oh, sorry, @Mathein. I misstated. We want a $1$-form as I described and we take the exact $2$-form $d\omega$ and ask if it is closed. Yes, for functions, it's automatic. I think that was Rudin's error in thinking.
I'm at the last part of this that involves ordinal exponentiation and $ \{ \beta < \omega_{1} \mid \omega^{\beta} = \beta \}$ being uncountable. All I know is that the first uncountable comes at $\omega_{1}$
Also my cat died unexpectedly. :(
Maybe prove by transfinite induction... Nahhh beta would have to be the limit ordinal in this case.. successor be like $\beta = \alpha +1$ and zero case is $\beta = 0$
Ahhhh I can do a proof by contradiction... Like assume that $\omega_{1}$ and then in the end thats not the case because $\omega_{1} $ is the first uncountable ordinal
Also I'm getting a new phone case for my new phone tomorrow eeee
If no one minds me asking an incredibly simple question, please give me a hand with some pre-algebra stuff from Khan Academy.
This problem:
Rob spent 25\%25%25, percent more time on his research project than he had planned. He spent an extra hhh hours on the project. Which of the following expressions could represent the number of hours Rob actually spent on the project?
Actually, let me rewrite that since the paste screwed up.
Rob spent 25% more time on his research project than he had planned. He spent an extra h hours on the project.
Which of the following expressions could represent the number of hours Rob actually spent on the project?
Well, I'd say you should try and express the sentence as an equation in some way. Like: (time spent)=(time intended to be spent)+.25(time intended to be spent)
Problem: Let $V$ be a representation of $\frak{sl}(2,\Bbb{C})$, and let $C \in End(V)$ be defined by $C = \rho (e) \rho (f) + \rho (f) \rho (e) + \frac{1}{2} \rho(h)^2$, where $e,f,h$ are the standard generators of $\frak{sl}(2,\Bbb{C})$. Show that if $V = V_k$ is an irreducible representation with highest weight $k$, then $C$ is a scalar operator: $C = c_k id$. Compute the constant $c_k$.
Question: is $V_k = \{v \mid Cv = \lambda Cv \}$?
Or is $V_k = \{v \mid \rho (x)v = \lambda \rho (x)v \}$?
Hey! I have a small question: irif.fr/~mgehrke/scriptie.pdf In the proof of Prop 6.4, there is an equality: ϕ(1)=∩ϕ(1) How is this correct? Isn't ϕ(0)=∩ϕ(1)?
@KonformistLiberal You're asking about one sentence within a proof in a published master's thesis on Stone Duality. That's sufficiently obscure that I sorta doubt you'll get a response. You may be better off emailing the author of that thesis directly.
Is a differentiable function automatically homogeneous?
Reason for the question: Let $f(u,v)$ be a differentiable function and put $h(x,y,z)=f(\frac{x}{y},\frac{y}{z})$ ($y>0, z >0$). Prove that $h$ is positive homogenous of degree 0.
Can anyone clarify if the peano axioms usually require some kind of set theory? I posted a question on MSE but some of the answers seem kind of nutty to me
At least if you're satisfied with treating the theory as a syntactic object, if you want to talk about models then some kind of set theoretic metatheory is needed
If we use $L_{PA}=\{0,S,+,\cdot\}$ as language, where $0$ is a constant symbol, $S$ is a unary function symbol and the last two are binary function symbols we can write all of the axioms
You don't need such axioms, the axioms describe how the natural numbers behave, not what they are. We know that there is a named one called $0$ because we have a constant symbol in the language, and the axiom describe the properties of this named number
when you say a set satisfies ZFC are you saying something like "our intuitive understanding of some real world concept called a set behaves the way ZFC lays out"?
@user709833 Hmm let's look at a simpler example, the theory of groups, which is a theory in the language $\{+,0\}$, where $0$ is a constant symbol and $+$ is a binary function symbol
Its axioms are the usual, $\forall x(x+0=x=0+x)$, $\forall x\forall y\forall z((x+y)+z=x+(y+z))$ and $\forall x\exists y(x+y=0)$
Note that nowhere they say what a group element is or that $0$ is a group element or that the sum of two group elements is a group elements. The axioms describe how elements of a group behave, not what they are
I think it's backward, it's not that our intuitive understanding of some real word concept called a set behaves the way ZFC lays out, it's more like we already had an intuitive understanding and ZFC tries to formalize it, so our intuitive understanding behaves the way ZFC says by construction
But I don't think that's too relevant, even the Peano axioms have plenty of models that are wildly different from the intuitive natural numbers they were trying to formalize
ah I can see how you would say it was backwards but yes I agree with your characterization, what I was trying to say is that we start with some intuitive understanding and we make ZFC to try to capture that, to say how it behaves
maybe i am confused on the usage of the word, like in physics we say the "bohr model" is a model for the real-life concept, the atom, and we say the atom behaves like such-and-such and we find it to be a useful model in certain situations
so do we say the natural numbers are a model of the peano axioms, or do we say the peano axioms are a model of the natural numbers?
