How can I find all the ring homomorphisms from $(\mathbb{Z}, +, \cdot)$ to $(\mathbb{Z}, +, \cdot)$ where $+$ and $\cdot$ are the usual addition and multiplication on the integers? So far I can only see that the idenity map on $\mathbb{Z}$ is a homomorphism
More specifically if I let $S$ be the set of all ring homomorphims from $(\mathbb{Z}, +, \cdot)$ to itself I need to write all the elements of $S$ down and then prove that the list of elements if complete
It holds that $C(\mathbb{R})$ is a Banach space, or not? Then $f_n(x)\in X\subseteq C(\mathbb{R})$. So a Cauchy sequence $\{f_n\}$ of $C(\mathbb{R})$ converges to $f\in C(\mathbb{R})$.
It is left to show that if $f_n$ is $T$-periodic then $f$ is $T$-periodic.
Suppose that $f$ is not $T$-periodic, then $f(t)\neq f(t+T)$ for $t\in \mathbb{R}$.
Since $f_n$ is $T$ periodic, we have that $f_n(t)=f_n(t+T)$. We take the limit $n\rightarrow\infty$ and get $f(t)=f(t+T)$, a contradiction.
That means that a Cauchy sequence $\{f_n\}$ of $X$ converges to $X$, and so $X$ is complete.
@Mary Star I think this is something you do in any calculus class, showing that Cauchy sequences of continuous functions converge uniformly to a continuous function.
@Dami @Alessandro I have a question for you guys, when you guys are studying for some (specifically) pure maths test, do you guys memorize any ideas for long proofs before hand? Or do you guys just rederive the proofs on your own?
Like yesterday in my algebra test, we were asked to prove that if $R$ is a unital commutative ring, then $R/I$ is a field if and only if $I$ is a maximal proper ideal of $R$, and the proof took me 6 pages to write out and there were some tricks that were used in the proof that I would'nt be able just to derive on my own in say an hour or so
But then again my lecturer's just walk up to the board and are like "Okay dokey I'm just gonna rederive the whole proof by hand" without missing a trick just off the top of their heads
So what I do to prep for a test is basically try and derive the simpler proofs on my own by hand and for the longer proofs I try and make notes of the general ideas and then rely on myself to fill out the nitty-gritty details in the test
It kinda depends. Fundamentally, some proofs have a central idea, and if you're experienced enough in the subject, you sorta associate the theorem with that idea in your mind. What you mention above is sorta like that
I'd you're familiar with algebra and work in it, the fact that intermediate ideals correspond to ones in the quotient is second nature, it's almost the first thing that comes to mind when thinking about the quotient
You may hear a professor say something to the effect of "We do the only thing it even makes sense to try"
That's the business at play. Once you have certain associations that you're really used to in your mind, you make them immediately, and then bam the pieces have exactly one way that could hope to work. But earlier on it's much trickier
But yeah as for other proofs, some just aren't at all of that sort
Sard's theorem is one of those things that people have to either memorize the proof of, not care, or do but with a reference close at hand. And yeah you're welcome
I think I'll take a nap at this point, have a good morning everyone!
Assume R/I is a field. Let J be an ideal between I and R. If it contains an element not in I, say x, then x+I is not zero in R/I, so it has an inverse, i.e. y+I such that xy+I=1+I, i.e. xy-1 in I. xy-1 is in J and x is in J, so 1 is in J, so J=R.
Let I be a maximal ideal. For any non-zero x+I in R/I, (x)+I is an ideal properly containing I, so (x)+I=R, so xy+i=1 for some i in I and y in R, so xy+I=1+I, i.e. x+I is invertible. Therefore, R/I is a field.
Lemmas used: 1. x+I is zero in R/I iff x in I. 2. 1 in J iff J=R. 3. (x)+I is an ideal.
I don't think I need 6 pages to prove those lemmas...
Part of the reason why it took 6 pages long was because we have to motivate every thing we did
Like if x is in I, then for us to say -x is in I we'd have to say something like, "since I is an ideal and thus also a subgroup it follows that -x is in I"
@LeakyNun $λ^2v-5λv+6v=0$, so assuming $v\ne 0$, since we are interested in eigenvectors, $λ^2-5λ+6=0$, so only possible eigenvalues are $2$ and $3$. How do we know that these are eigenvalues indeed?
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Given $f(x)=\frac{ln(x)}{x^s}$, where both x,s are real numbers and $x>0$ and $s>1$. How do I determine the intervals of monotonicity of $f$? I found $f'(x)=x^{-s-1}(1-s\cdot ln(x))$ which means that $f'(x)=0\implies x=e^\frac{1}{s}$. I want to check when $f'(x)>0$ and $f'(x)<0$. Normally I would plug some number larger and some smaller number into the function to see if the function is decreasing or increasing. How can I do it with this function?
I'm just a bit confused on the task. I tried plugging in $e^{\frac{2}{s}}$ and $e^{\frac{0}{s}}$, to check if the intervals around the critical point $e^{\frac{1}{s}}$ is decreasing or increasing or neither. But I can't really make sense of the results and evaluate it properly.
Definition. For a real valued function $f$ and an interior point $x$ of its domain, the uppper derivative of $f$ at $x$ denoted by $\overline{D}f(x)$ is defined as follows: $$\overline{D}f(x)=\lim_{h\rightarrow0}\left[ \sup \left \{\frac{f(x+t)-f(x)}{t}: 0<|t|\leq h \right \} \right]$$
I am ...
Why may we assume that each interval in $\mathcal{F}$ is contained in $\mathcal{O}$? What warrants this reduction?
Why is statement (4) true? If $x \in E - \bigcup_{k=1}^n I_k$, then $x \in E$ and $x \notin I_k$ for every $k=1,...,n$. Given some $\epsilon > 0$, there exists $I \in \mathcal{...
Corollary 5: Let $\{f_n\}$ be a sequence of nonnegative integrable functions on $E$. Then
$$\lim_{n \to \infty} \int_E f_n = 0 ~~~~~~(5)$$
if and only if
$$f_n \to 0 \mbox{ in measure on } E \mbox{ and } \{f_n\} \mbox{ is uniformly integrable and tight over } E ~~~~~(6)$$
H...
@LeakyNun evaluating $x=1$ gives $f'(1)=1$ so f is increasing in the interval $(0,e^\{1}{s})$. Now I simply need to figure a way to evaluate the other interval next to the critical point $e^\frac{1}{s}$ (like the idea you got with $1=e^0>e^\frac{1}{s}$. Is that correctly understood?
@LeakyNun I just noticed something. Wouldn't evaluating $x=1$ mean that $f$is increasing on the interval given that $(e^\frac{1}{s}),\infty$? As $1>e^\frac{1}{s}$?
I just noticed something. Wouldn't evaluating $x=1$ mean that $f$ is increasing on the interval $(e^\frac{1}{s},\infty)$? given that $1>e^\frac{1}{s}$?*
@LeakyNun I tried to evaluate at $x=e^s$ as $e^s>e^\frac{1}{s}$.
So $f'(e^s)=\frac{1-s\cdot ln(e^s)}{e^{s^{-s-1}}}<0$. Because the denominator is >1 and if s=1 then $s\cdot ln(e^s)=0$, but as $s>1$ and log is always increasing for $ln(s), s>1$ then $s\cdot ln(e^s)>1\implies f'(e^s)<0$.
Hey guys, in joint probability P(x,y) is the same thing as P(y,x) right ? The ordering here doesn't matter because they both mean "the probability of x and y happening together at the same time", yes ?
I mean that any map that squares to the identity should only have eigenvalues $\pm 1$
For instance, suppose you were talking about functions. Then the map $P$ such that $Pf(x)=f(-x)$ will have eigenvalues $\pm 1$ corresponding to even/odd functions of x.
In linear algebra, if
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1
,
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,
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l
{\displaystyle A_{0},A_{1},\dots ,A_{l}}
are
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wrooong liiink
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that's for matrices, of course, but the extension to operators is obvious enough
that's a topic I only know at the level of the definition, though
I'm probably being captain obvious, but you can look at the edit history :P
searching for matrix pencil stuff brings up some applications to the Schrodinger equation with a periodic potential, so I guess that's how I'm familiar with it
