@GFauxPas I mean the ordering is just furnished by the index, so you don't need to worry about that too much. I've seen both and neither causes a problem
Ah I mean series is another story, but like, that's where you determine the order of the sum just by the index, so the notation for the sequence still seems to me as being somewhat immaterial
But yeah I dunno, some mix of my having other things to do and not being particularly interested in bio made this probably the class that I was least engaged with, at least in college and possibly ever
Let $X$ be a subset of $\Bbb R$ so that every continuous function $X \to \Bbb R$ can be extended to a continuous function $\Bbb R \to \Bbb R$. What can we say about $X$?
Semi I was feeling rushed in my homework for analysis so for one of the problems i extended the function to $\mathbb C \to \mathbb C$ so I could use the identity theorem on it
was deriving the power series for arctan by integrating a geometric series, but the geometric series was only valid for $r < 1$, but I made a complex argument that it should hold for $r = 1$ as well
@orbit-stabilizer But yeah E is dense in [0, 1]. Or, as you say, that means every element of [0, 1] is a limit point of E. What does that say about g - h?
Let $f\left(x\right)=\arctan\left(x\right)-\frac{n}{x}+1$ and let $a_n$ be the solution of the equation $f(x)=\frac{\pi}{2}$ on $(0,\infty)$ Prove that $\arctan\left(\frac{1}{a_n}\right)=\frac{n}{a_n}-1$
@BalarkaSen okay looking at the rational sequences, the functions must agree on each point in the sequence, and since they're cts, they must agree on their limits as well?
@orbit-stabilizer so a set $S$ is connected if there is no $A$ and $B$ such that $A \cup B \supseteq S$, $A \cap \overline B = \varnothing$, and $B \cap \overline A = \varnothing$?
@LeakyNun: Two subsets $A$ and $B$ of a metric space $X$ are said to be seperated if both $A\cap \overline{B} = \emptyset$ and $\overline{A} \cap B = \emptyset$.
Kolmogorov & Fomin: Elements of the Theory of Functions and Functional Analysis
But it does all the Banach space theory immediately after sets/metric spaces so it doesn't work as much with L^p, and generally feels less comprehensive
Traditionally the class used Brezis but our prof felt like it didn't do stuff like spectral theory enough, and treated functional analysis as a bunch of tools that you had to slog through in order to do PDEs, especially those related to fluid dynamics
Dumb question: Suppose $x \in [k,k+1)$ for some $k \in \Bbb{N}$. I need to find $N \in \Bbb{N}$ such that $(x - \frac{1}{n}, x + \frac{1}{n}) \subseteq [k,k+1)$ for every $n \ge N$.
In studying for an upcoming prelim, I came across this problem:
Classify all groups of order $182 = 2*7*13$.
Now, the standard tricks here are to look at Sylow's theorems or semi-direct products. Let $n_p$ denote the number of Sylow $p$-subgroups. We can conclude that $n_7 =1$ since $n_7 |...
Hey! I need help. I'm trying to find a stackexchange site that will help me figure out a clever way to teach solving system of linear equations by substitution because I'm somehow failing with my explanation. I have done so many examples with them. I'm beginning to think that their cognitive ability is not up to the task or I'm also thinking I might suck at reaching them in this topic. :(
So it is an 8th grade class just so you know. In a way I do think the topic is advanced but you know state standards. I teach them we need to rearrange one equation of the two for one variable and insert it into the other and boom it is just like solving a regular linear equation from there.
Okay so, this may be a bit strange, but have you tried explaining it in a sort of, "what can we guarantee from each" way?
Keep in mind that I haven't done any pedagogical research so anything backed up by cognitive studies should immediately take precedence over what I'm saying
I'm worried it might be too advanced and that is why there is so much torture but it could also mean I just suck at getting the explanation across. I feel like I have explained it myself real well but to 8th graders and how they perceive things they most likely don't feel the same way.
But when I first learned it, I found it helpful to think about it by looking at the equation and saying well, if we know this equation should be true, then we can rearrange it to guarantee something else is true
prentice hall course 3 mathematics... i probably made a mistake and am teaching them too much at one time. The book jumps from slope-intercept form without doing any rearranging to solving systems of equations that do require rearranging.
Now this and another equation both have to be true, this allows us to plug one thing in another. If students have a conceptual problem, that might be a way to approach it
I wouldn't expect this to be too advanced, like I think this is basically standard fare for 8th grade, but maybe
In studying for an upcoming prelim, I came across this problem:
Classify all groups of order $182 = 2*7*13$.
Now, the standard tricks here are to look at Sylow's theorems or semi-direct products. Let $n_p$ denote the number of Sylow $p$-subgroups. We can conclude that $n_7 =1$ since $n_7 |...
