Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How can I find maximum and minimum modulus of a complex number? I have this problem. Let be given complex number $z$ such that
$$|z+1|+ 4 |z-1|=25.$$
Find the greastest and the least of the modulus of $z$.
I tried with minimum.
Put $A(-1,0)$, $B(1,0)$ and $M(x,y)$ present of $z$.
We have $O(0,0)$ is the midpoint of th... | For maximum $|z|$, we have
\begin{align}
|5z|&=|(z+1)+4(z-1)+3|\\
&\le|z+1|+4|z-1|+|3|\\
&\le25+3\\
|z|&\le \frac{28}{5}
\end{align}
with the equality holds if and only if $\displaystyle z=\frac{28}{5}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2314488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Integration of $1/(1+a \csc^2(x))$
Integration of $$\int_{0}^{\frac{(M-1)\pi}{M} }\frac{1}{1+\alpha \csc^2(x)} dx,$$ where $ \alpha $ is a constant.
I tried taking $\cot(x) = t$, then differentiating it w.r.t $dx$ we get, $-\csc^2(x)dx = dt$. And as we know that, $\csc^2(x)= \cot^2(x) +1$, so tried substituting the... | We can simplify it as $\displaystyle1-\frac{a}{\sin^2 (x)+a} $.
Then by using $\displaystyle \sin (x)=\frac {\tan x}{\sec x},\sec^2x=\tan^2x+1$, we have the next term as $\\\displaystyle\frac {a\sec^2 (x)}{(a+1)\tan^2 (x)+a} $. Now let $\tan (x)=t \implies\sec^2 (x)dx=dt $.
Thus the next part changes to the integral ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Tight integral inequality We define the integral
$$
J_k = \int_0^\pi (a+bx) \frac{\sin^3x}{1 + k \cos^2x}\,\mathrm{d}x\,,
$$
and
$$
J = J_1 = \int_0^\pi (a+bx) \frac{\sin^3x}{1 + \cos^2x}\,\mathrm{d}x\,,
$$
where $a,b \in \mathbb{R}$.
Prove that $J_{1/k} / J$ is independent of $a,b$ and that $$ J_{1/3}
>J \cdot (\l... | I dont know any explicit simplification but the function is such that the integral for any k can be calculated so we first calculat the integral for general k. Let $J_k=T $ thus $T=a\int _0 ^{\pi} \frac{\sin^3 (x}{1+k\cos^2 (x)}+M $ where $M $ is the next part of the integral. Thus $M=b\int _0 ^{\pi} (\pi-x)\frac {sin^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2314694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve for $x$ in $\cos(2 \sin ^{-1}(- x)) = 0$
Solve $$\cos(2 \sin ^{-1}(- x)) = 0$$
I get the answer $\frac{-1}{ \sqrt2}$
By solving like this
\begin{align}2\sin ^{-1} (-x )&= \cos ^{-1} 0\\
2\sin^{-1}(- x) &= \frac\pi2\\
\sin^{-1}(- x) &= \frac\pi4\\
-x &= \sin\left(\frac\pi4\right)\end{align}
Thus $x =\frac{-1}{\... | You need to solve $\cos \left(2 \arcsin(-x) \right) = 0$. Let $y = 2 \arcsin(-x)$ then $\cos y = 0$ so $y = \pi/2 \pm n\pi$. Then,
$$
2 \arcsin(-x) = \frac{pi}{2} \pm n\pi
$$
which implies
$$
x = -\sin \left( \frac{\pi}{4} \pm \frac{n\pi}{2} \right)
$$
Can you simplify this?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does arctan(1/n) summed not converge? My question is as follows: WolframAlpha tells me that the sum series S doesn't converge, but why?
$$S=\sum_{n=1}^{\infty}\arctan(\frac{1}{n})$$
$$\lim_{n\to \infty} \arctan(\frac{1}{n})=0$$
So S (slowly) stops growing when n gets larger and larger. So why doesn't it converge? ... | For values of $x$ near $0$, $\arctan(x)$ ~ $x$.
IF you know calculus, this is because the rate of change of $\arctan(x)$, $\displaystyle \frac{1}{1+x^2}$, approaches $1$,the rate of change of $x$, as $x$ approaches $0$.
It is a well known fact that the harmonic series or $\displaystyle \frac{1}{x}$, that is $1+\frac{1}... | {
"language": "en",
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Proposition $5.2$, Section $2.3$ (Cyclic Groups) in Dummit and Foote's Abstract Algebra I'm trying to give an alternative proof of a proposition in Dummit and Foote's Abstract Algebra, but I am unable to complete the details. The proposition is:
\begin{align*}
\text{Let $G$ be a group, and let $x \in G$ and let $a \in... | It's not restrictive to assume $a>0$, as $|x^a|=|x^{-a}|$.
Let $y=x^a$. If $y^m=1$, then $n\mid am$, because $x^{am}=y^m=1$.
Conversely, if $n\mid am$, then $am=nq$ and
$$
y^m=x^{am}=x^{nq}=1
$$
In particular, $|y|$ is the minimal $m$ such that $n\mid am$.
If $d=\gcd(n,a)$, then $n=dn'$ and $a=da'$. Suppose $n\mid am$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2315243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Name for logical predicates which act like homomorphisms from set union to logical conjunction? Is there a name for predicates defined over sets with the property that:
$$P(S\cup Q)\iff P(S)\land P(Q)$$
For example the predicate $P(Q)=``Q\text{ is empty"}$ would be one such predicate because:
$$S\cup Q=\emptyset\iff (S... | These are semilattice homomorphisms. A semilattice is a set with a binary operation that is associative, commutative, and idempotent (i.e., $x \vee x = x$). As in algebra, a semilattice homomorphism is a function that preserves the operation. Examples of semilattices include:
*
*subsets of a given set, where the ope... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2315353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Boolean algebra, logic expression minimization $$(A\land \neg A)\lor((A \land B) \lor(A\land B\land \neg C ))$$
I'm trying to minimize this Boolean expression with boolean algebra but i cant minimize it completly can i get some help?
| It can be simplified as follows:
\begin{array}{l}
& (A \land \neg A) \lor ((A \land B) \lor (A \land B \land \neg C)) & \text{ Given }\\
& F \lor ((A \land B) \lor (A \land B \land \neg C)) & \text{ Complement }\\
& (A \land B) \lor (A \land B \land \neg C) & \text{ Identity }\\
& (A \land B) \land (T \lor \neg C) & \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2315427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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What annual installment will discharge a debt of $\large\bf{₹}$ $1092$ due in $3$ years at $12\%$ Simple Interest? Question : What annual installment will discharge a debt of $\large\bf{₹}$$1092$ due in $3$ years at $12\%$ Simple Interest?
Options
a. $\large\bf{₹}$$300$
b. $\large\bf{₹}$$225$
c. $\large\bf{₹}$$400$
d.... | You have a debt which has a value of $1092$ in three years. You can say that this value is the future value of the debt.
You pay immediately an installment of $x$. This payment has to be compounded two years to get the value at the beginning of the third year. The factor is $1.24 (=1+0.12\cdot 2)$. The next (second) ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A question about inflection point Does the function
$$
f(x)=
\begin{cases}
x^2 & x\leq1\\
2-x^2 & x>1
\end{cases}
$$
has an inflection point at $x=1$, or it hasn't because $f$ isn't differentiable at $x=1$? Thanks!
| Surprise surprise! A function doesn't have to be differentiable in order to have an inflection point. Simply put, an inflection point is a point at which $f(x)$ switches concavity.
$\displaystyle \lim_{x \to 1^-}f''(x)=2$ and $\displaystyle \lim_{x \to 1^+}f''(x)=-2$
We can easily see that $f(x)$ is continuous at $x=1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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proof problem on proving $x_{1}=1,x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{2}{x_{n}}\right)$ has a limit Sequence $\{x_{n}\}$ is defined by $$x_{1}=1,x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{2}{x_{n}}\right)$$, as for proving whether the sequence has a limit, one of my friends told me his proof as following:
First, assuming ... | For $x>0$ we have $x+\frac2x\geq 2\sqrt{x} \sqrt{\frac2x}=2\sqrt2$. So $x_n\ge \sqrt2$. Further $\frac{x_{n+1}}{x_n}=\frac12\left(1+\frac2{x_n^2}\right)\le 1.$
So the sequence converges. Its limit is already known.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to evaluate $\lim_{n \rightarrow \infty}\left(\frac{(2n)!}{n!n^n} \right)^{1/n}$? Find$$\lim_{n \rightarrow \infty}\left(\frac{(2n)!}{n!n^n} \right)^{1/n}$$ is there some trick in this questions. seems it must simplify to something but I am unable to solve it.
| Using Stirling's approximation, just for the sake of it and for reference (and to give an alternative approach to that of the excellent answer by lab bhattacharjee). Caveat: overly detailed.
$$
n! \operatorname*{\sim}_{n\to\infty} \sqrt{2\pi n}\left(\frac{n}{e}\right)^n \tag{Stirling's approximation}
$$
yields
$$
\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2315943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Count the number of bases of the vector space $\mathbb{C}^3$. Problem: Consider the set of all those vectors in $\mathbb{C}^3$ each of whose coordinates is either $0$ or $1$; how many different bases does this set contain?
In general, if $B$ is the set of all bases vectors then,
$$B=\{(x_1,x_2,x_3),(y_1,y_2,y_3),(z_1,... | Knowing that none of x,y,z can be (0,0,0) there are only 7 choices for each. Since they must be different you only have $\binom{7}{3}=35$ choices to make.
This is small enough to sort through by hand.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2316042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Holomorphic Function constant in $\mathbb{P}^1(\mathbb{C})$ I want to show that a holomorphic function $f: \mathbb{P}^1(\mathbb{C}) \to \mathbb{C}$ is constant. $\mathbb{P}^1(\mathbb{C})$ is the projective line. I'm not very sure how to solve that. I have the idea to start with the Maximum-Principle. For that I need a ... | Preliminary remark : This does not makes sense to talk about metric as the metric of $\Bbb C$ can't be extended to $P^1(\Bbb C)$ as one would have $d(0, \infty) = \infty$ but distance between two points is always finite. On the other hand, it is well known that $S^2 \cong P^1(\Bbb C)$ so it is indeed compact.
Now, if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2316163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Given any two non zero vectors $x,y$ does there exist symmetric matrix $A$ such that $y = Ax$?
Let $x,y$ be non-zero vectors from $\mathbb{R}^n$. Is it true, that there exists symmetric matrix $A$ such that $y = Ax$.
I was reasoning the following way. Having an equation $y = Ax$ for some symmetric matrix $A$ is equiv... | It's not true that if you have more variables than equations, the system is always solvable! That's like saying a matrix with more columns than rows is always onto, which is also false.
There might be a slicker solution, but the first thing I thought of was this: Let $d = ||x||$. Then there exists a unitary matrix $U$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do I find matrix B when adj(B) is given? (a) find matrix B such that $adj(B)=A$, with $A$ given by :
$$A:=\left(\begin{array}{rrrr}
1 & 2 & 5 & 4 \\
0 & -1 & -2 & -1 \\
-1 & 1 & 3 & 0 \\
0 & 2 & 5 & 3 \\
\end{array}\right)$$
(b) For the same matrix $A$, find all complex matri... | Building up on Bye_World's comment, note that you can use the fact - if $A$ is $n×n$, then $\lvert (adj(A))\rvert = \lvert A\rvert^{n−1}$ to find the determinant of B and once you have the determinant of $B$ just plug it in your inverse formula
$$B^{-1}=\frac{1}{\lvert B \rvert}adj(B)$$
to get $B^{-1}$ and then find i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2316470",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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How to learn a great number of theorems by heart? Imagine you have ten definitions and you want to learn them by heart. It is easy - definitions are somehow unique. But, imagine 40 (60,100,1000) theorems that all look somehow similar and are all important. How would you learn them by heart? What the word "learn" mean h... | There is really no point in memorizing $1000$ theorems. For one thing, different expositions of the same subject will organize the theorems somewhat
differently. A particular theorem in textbook A might correspond to parts of several different theorems in textbook B, or might just be an exercise in textbook C.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2316619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Let $X=l^2$ and $A$ be the set of all positve elements of $X$, Then show that $Cone(A-x)$ is not a subspace. Let $X=l^2 = \{x=\{x_i\}_{i=1}^{\infty} ~ | \quad \sum_{i=1}^{\infty} |x_i|^2 < \infty \}$ and $A = \{ x=\{x_i\}_{i=1}^{\infty} \in X ~ | \quad x_i >0 ~ \forall i \}$ and take $a \in A.$
My questions (I hav... | 2nd Edit: Again you were completely right, I edited my answer. Hopefully I got it right this time.
3d Edit: I added an answer to the second question.
First question
Let $y \in C$. We try to look for a sufficient and necessary condition that $-y \in C$. Without loss of generality we assume that $y = x-a$ for some $x \in... | {
"language": "en",
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Existence of Inverse Mapping : $f: \;P(\Bbb R) \rightarrow \Bbb R $ where $f(x) = {1\over x^2}-x\cdot arctanx+{1 \over 2}log(1+x^2) $ I'd like to show that below function holds the inverse mapping:
$f: \;P(\Bbb R) \rightarrow \Bbb R $ where $f(x) = {1\over x^2}-x\cdot arctanx+{1 \over 2}log(1+x^2) $
To show the existen... | Hints (Every monotone function has a inverse mapping):
Since $$f'(x)=-\frac{1}{x^3}-\arctan x,$$
we can conclude that $f'(x) > 0$ whenever $x<0$; $f'(x) < 0$ whenever $x>0$, which leads that $f(x)$ has a inverse mapping.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2316962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Non-negative operator & self-adjoint operator I am wondering how to show that:
if $A$ is a non-negative operator, then $A$ is self-adjoint.
Def. 1. $A$ is non-negative if $\langle Ax,x \rangle \geq 0$ for $\forall x\in H$, where $H$ is a Hilbert space.
Def. 2. $A$ is self-adjoint if $A = A^*$.
| For any linear $A:H \rightarrow H $, we have
$\langle A^{*}x, x\rangle = \langle x, Ax\rangle =\overline {\langle Ax, x \rangle},$
But, for $A $ non-negative, then $\langle Ax, x\rangle$ is real, so
$\langle Ax, x\rangle = \overline {\langle Ax, x \rangle}$
i.e. $\langle Ax, x\rangle =\langle A^{*}x, x\rangle $
$\imp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2317097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Function class $C^{\infty}$ Show that exists function $ g \in \mathcal{C}^{\infty} \ [\mathbb{R},[0,1]) $ such that $g(x)=0$ for $|x|\le 1/2$ and $g(x)=1$ for $|x|>1$.
So, I have :
$f(x)=\begin{cases} \exp(-1/x) &\text{for } x > 0\\0 &\text{for } x<0 \end{cases}$
And now:
$g(x) = \frac{f(x)}{f(x) + f(1-x)}$
However, I ... | You should not be convinced, it isn't true. In particular, if x= 1/2 then 1- x= 1/2 so that $f(x)= f(1-x)= e^{-1/(1/2)}= e^{-2}$ and then $g(1/2)= \frac{e^{-2}}{2e^{-2}}= \frac{1}{2}$, not 0.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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multiply $2^{(a-1)b}$ by $2^b$ and get $2^{ab}$? How is this so? I’m reading How To Prove It and in the following proof the author is doing some basic algebra with exponents that I just don’t understand. In Step 1.) listed below he is multiplying $2^b$ across each term in (1 + $2^b$ + $2^{2b}$ +···+$2^{(a-1)b}$) and g... | Using the laws of exponents, $2^{(a-1)b}2^b = 2^{(a-1)b+b} = 2^{ab}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2317435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Simple property of Dirac's $\delta$-function. I'm on Page 63 of R. Shankar's "Principles of Quantum Mechanics".
I'm trying to do Exercise 1.10.1 by proving that $\displaystyle{\delta(ax) = \frac{\delta(x)}{|a|}}$, where $a \in \mathbb R \backslash\{0\}$.
We know that $\displaystyle{\int_{-\infty}^{+\infty}\delta(t)~\m... | You could make an even stronger argument:
For all $A \subset \mathbb{R},$ we have $$\int_A \delta(ax)dx = \int_A \frac{\delta(x)}{|a|} dx$$
Now you can use this type of result.
| {
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"url": "https://math.stackexchange.com/questions/2317543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show which of $6-2\sqrt{3}$ and $3\sqrt{2}-2$ is greater without using calculator How do you compare $6-2\sqrt{3}$ and $3\sqrt{2}-2$? (no calculator)
Look simple but I have tried many ways and fail miserably.
Both are positive, so we cannot find which one is bigger than $0$ and the other smaller than $0$.
Taking the fi... | $$
6-2√3 \sim 3√2-2\\
8 \sim 3√2 +2√3 \\
64 \sim 30+12√6\\
34 \sim 12√6\\
17 \sim 6√6\\
289 \sim 36 \cdot 6\\
289 > 216
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2317625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 1
} |
Contour integration with variable epsilon I've been wracking my brain on a particular contour integration which I thought I solved easily enough, but the answer says differently. It asks to prove the following for any $\xi \in \Bbb{R}$:
$$\int_{-\infty}^{\infty} \frac{e^{-2\pi i x \xi}}{(1+x^2)^2}dx=\frac{\pi}{2}(1+2\p... | Let $z=x+iy$. Note that
$$
|{e^{-2\pi i z \zeta}}|=e^{2 \pi\zeta y}.
$$
Hence if $\zeta\geq 0$, the contour should be the semicircle in the lower plane and if $\zeta< 0$, the contour should be the semicircle in the upper plane to induce the fact that the integral go to zero over the arc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2317732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Relation between Lipschitz and differentiablity
Let $f:\Bbb R \to \Bbb R$ be a Lipschitz function. Show that the set of all reals at which $f$ is differentiable in non-empty.
I know that if $f$ is differentiable and derivative is bounded then it is Lipschitz. I know that the converse is FALSE. For example $f(x)=|x|$... | The set would be better than being merely non-empty.
If $f:\mathbb{R}\to\mathbb{R}$ is Lipschitz, then it differentiable almost everywhere in $\mathbb{R}$ by Rademacher's theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2317844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How does one prove that $\|x\|_2\le \|x\|_1$ I intuitively understand that $\|x\|_2 = \sqrt{x_1^2+\dots+x_n^2}\le \sqrt{x_1^2}+\dots+\sqrt{x_n^2}=|x_1|+\dots+|x_n|=\|x\|_1$. But the thing I'm concerned about is how to prove that $\sqrt{x_1^2+\dots+x_n^2}\le \sqrt{x_1^2}+\dots+\sqrt{x_n^2}$?
| I find it easier to argue in the following way: the inequality is true when $x=0$, so assume that $x\neq 0$. Then by the homogeneity of the norms we can assume that $||x||_1=1$. Therefore $|x_j|\leq 1$ for all $j$, so
$$ ||x||_2^2=|x_1|^2+\dots+|x_n|^2\leq |x_1|+\dots+|x_n|=1$$
and taking square roots yields the desir... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2318091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to define a finite topological space? I want to develop a simple way to define topologies on finite sets
$X=\{1,2,\dots,n\}$ for computational experiments.
Does any function $c:X\to \mathcal P(X)$, such that $x\in c(x)$,
define a closure operator on $X$?
The idea is that $c$ should define a closure operator by... | Starting with an arbitrary function $c:X\to\mathcal P(X)$ such that $\forall x\in X:x\in c(x)$, this should be an algorithm to transform $c$ to a function that defines a closure operator, due to the accepted answer:
$0:\quad$ $\mathrm{ready}\leftarrow\mathrm{true}$
$1:\quad$ $i\leftarrow 1$
$2:\quad$ $j\leftarrow 1$
$3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2318236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 5
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Show that the function $x \rightarrow ||x||^2, \mathbb{R}^p\rightarrow \mathbb{R}, p\ge 1$ is not uniformly continuous. Show that the function $x \rightarrow ||x||^2, \mathbb{R}^p\rightarrow \mathbb{R}, p\ge 1$ is not uniformly continuous.
What i figured was that for $\forall \epsilon >0, \exists \delta>0, \forall x,y... | Consider the sequences $x_n=(n+\frac 1n,n+\frac 1n,...,n+\frac 1n)$ and $y_n=(n,n,...n)$ in $\Bbb R^p$.
Then $||x_n-y_n||=\underbrace{\sqrt{(n+\frac 1n -n)^2+(n+\frac 1n-n)^2+\cdots+(n+\frac 1n-n)^2}}_{p-\text{times}}=\frac{\sqrt p}n \Rightarrow \lim||x_n-y_n||=\lim \frac {\sqrt p}n =0.$
But $|f(x_n)-f(y_n)|=|||x_n||^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2318419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Reformulating a Euclidean distance minimization problem into a semidefinite program The following minimization problem is a Euclidean distance form of a single-facility location problem
$$\min \quad \sum_j \sqrt {(x-a_j)^2+(y-b_j)^2}$$
where $(x,y)$ and $(a_j,b_j)$ are the coordinates of the new facility and current f... | It's SOCP representable, in fact it almost doesn't get more SOCP than this.
For instance, minimizing $||q|| + ||p||$ is equivalent to minimizing $s + t$ subject to $||q||\leq t, ||p||\leq s$ which is an SOCP in standard form.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2318507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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When are Lipschitz constant uniformly bounded? I have a sequence of Lipschitz functions $f_i$ which converges uniformly to a Lipschitz function $f_0$. I also can make it the case that the domain for these functions are compact. Are these conditions enough to guarantee that the Lipschitz constants for $f_i$ are bounded?... | Let $f_i : [0,1] \to \mathbb R$ be a piecewise linear function that is $0$ on $[0,1-1/i^2]$ and then increases from $0$ to $1/i$ on the interval $[1-1/i^2, 1]$. These functions are convex and uniformly convergent to $0$, but the Lipschitz constant of $f_i$ is $i$.
If you want the convergence to imply a uniform Lipschi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2318610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $\underset {x \in [0,1]} {\sup} f_n(x) \rightarrow \underset {x \in [0,1]} {\sup} f(x)$ as $n \rightarrow \infty$.
Let $\{f_n\}$ be a sequence of continuous functions converging uniformly to a function $f$ on $[0,1]$. Then show that $\sup\limits{x \in [0,1]} f_n(x) \rightarrow \sup\limits_{x\in[0,1]} f(x)$ a... | You have
$$ f(x) = f_n(x) + (f(x) -f_n(x)) \leq \sup_{y\in [0,1]} f_n(y) + \sup_{y\in [0,1]} (f(y) -f_n(y)). $$
Taking supremum over $x\in [0,1]$ gives
$$ \sup_{x\in [0,1] } f(x) \leq \sup_{y\in [0,1]} f_n(y) + \sup_{y\in [0,1]} (f(y) -f_n(y)). $$
Taking the $\liminf$ on both sides yields by uniform convergence
$$ \sup... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2318728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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ray class groups in $\Bbb{Q}$ I study class field theory from the book "Primes of the form $x^2+ny^2$", D. Cox. I want to find ray class groups in $\Bbb{Q}$.
Let $K$ be a number field, $\mathfrak{m}$ be a modulus of $K$. In the book, $I_K(\mathfrak{m})$ is defined as the group of all fractional ideals of $K$ coprime t... | The ray class group modulo $8$ corresponds to the ray class field modulo $8$, which is the maximal totally real abeliab extension of ${\mathbb Q}$ with conductor $8$, i.e., the maximal real subfield of ${\mathbb Q}(\zeta_8)$, namely ${\mathbb Q}(\sqrt{2})$. This is a quadratic extension, as is confirmed by the fact tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2318893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Is a finitely generated submodule of a direct sum of a field of fractions of a Dedekind domain projective? This is a question out of Donald Passman's "A Course in Ring Theory".
Let $R$ be a Dedekind domain with field of fractions $F$, and let $V$ be a finitely generated $R$ submodule of $F^n:=F\oplus\dots\oplus F$. P... | Even though $F$ itself is not free, your approach is still valid:
*
*First, note that $F$ is the directed union of its finitely-generated, free submodules. Namely, if $x=\frac{a}{b}\in F$ then $x\in U_b := R\frac{1}{b}\subset F$ which is free since $F$ is torsion-free. Moreover, $U_b\subset U_{bb^{\prime}}$ for any... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Separable quotient of a non-separable normed space I want to find an example of a non-separable normed space $X$ and a closed subspace $M$ of $X$ such that $X/M$ is separable.
The first think came to my mind is $\ell^{\infty}$. Can I find a closed subspace $M$ of $\ell^{\infty}$ so that $\ell^{\infty}/M$ is isometrica... | $\ell_1$ is not isomorphic to a quotient of $\ell_\infty$. Indeed, the latter one is a Grothendieck space, hence so is every its quotient. By the Eberlein-Smulyan theorem, a separable space is Grothendieck if and only if it is reflexive. Certainly, $\ell_1$ is not reflexive.
As a matter of fact, $\ell_p$ is a quotient ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Conformal Map wanted I am in search of a conformal map that will stretch the rectangle $P_1 =\lbrace(x,y) : (-W < x < W , -L < y < L )\rbrace$ to the entire real plane $P_2 = \lbrace(u,v) : u,v \in \mathbb{R}\rbrace$, where the sides of $P_1$ are mapped to infinity. For example, this transformation:
$$ u = \frac{xy}{W^... | Unfortunately, no such map exists: If there were a conformal bijection from your rectangle to the entire plane, the inverse mapping (or its complex conjugate) would be a bounded, entire function. But a bounded, entire function is constant by Liouville's theorem from elementary complex analysis.
Generally, no proper ope... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2319452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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if $A$ is $2 \times 2$ matrix find $\lim_{n \to \infty} A^n$ if $A$ is $2 \times 2$ given by
$$A= \begin{bmatrix}
1& \frac{\theta}{n} \\
-\frac{\theta}{n} & 1
\end{bmatrix}$$
matrix find $$\lim_{n \to \infty} A^n$$
I tried like this:
$$A-I=\begin{bmatrix}
0& \frac{\theta}{n} \\
-\frac{\theta}{n} & 0
\end{bmatrix}$... | Let p(t) be a polynomial function with variable t.
Let $ p(t)=\det(tI-A) $.
$$p(t)=t^2 -2t+1+ \frac{θ^2 }{n^2}$$
By Cayley-Hamilton theorem, p(matrix A)= zero matrix
Set p(t)=0, t= 1 (+ or -) θi/n, and these are the eigenvalues.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2319550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Self-functions of the class of ordinals growing fast Let $\lambda$ be a regular ordinal and let $S:=\{\beta:\beta<\lambda\}$. I wonder if it is possible to find a (non-decreasing, but this should not be a problem) function
$$
f\colon S\to S
$$
such that, for any $\alpha\in S$, there exists $\beta\in S$ such that
$$
\be... | There is no such function; it is a corollary of the Fodor's lemma. Assume that there is a such $f$, so for each $\xi<\lambda$ we can choose $\beta_\xi< \xi$ such that $\xi<f(\beta_\xi)$. The function $g(\xi) = \beta_\xi$ is regressive, and the set $\lambda$ is stationary.
Hence, there is a stationary subset $S\subseteq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2319706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Calculate convergence of random variables We are given $X_1,X_2,...$which are all independent random variables and have $Exp(\ln n)$ distribution. Our task is to show that this random variables converge to 0 with probability but not almost surely. I am hitting a wall with this one. What should be my approach here?
| Hint: Recall the definition of convergence in probability: $X_n \xrightarrow{p} 0$ means that for each $\epsilon > 0$, $\mathbb{P}(X_n > \epsilon) \to 0$ as $n \to \infty$. Can you compute $\mathbb{P}(X_n > \epsilon)$ explicitly?
To show that there isn't almost sure convergence, show that for some $\epsilon > $, $\mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2319804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Haar measure and SU(2) I have some very basics question on the Haar measure on $SU(2)$.
What I understood from definition of Haar Measure is that it is a measure that ensure me to have the property :
$$ \int f(gh) d \mu(g) = \int f(h) d \mu(h) $$
Where $g$ and $h$ are element of a Lie group. And $f$ a function from the... | The answer to your first question is no. It would mean that the group SU(2) were commutative, which it is not.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2319965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Definite integral integration by parts Can we write the integration by parts for definite integral the following way:
$$\int^a_b f(x)g(x)dx=f(x)\int^a_b g(x)dx-\int^a_b \left[ \dfrac{df(x)}{dx}\int^a_b g(x)dx \right]dx $$
My book gives the following formula for definite integral integration by parts:
$$\int^a_b f... | Integration by parts is defined by
$$\int f(x) \, g(x) \, dx = f(x) \int g(u) \, du - \int f'(t) \left(\int^{t} g(u) \, du \right) \, dt.$$
When applying limits on the integrals they follow the form
$$\int_{a}^{b} f(x) \, g(x) \, dx = \left[f(x) \int g(u) \, du\right]_{a}^{b} - \int_{a}^{b} f'(t) \left(\int^{t} g(u) \,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2320051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Computing of $\int_{-1}^1\frac{e^{ax}dx}{\sqrt{1-x^2}}, \: a \in \mathbb{R}$ I would like to find Fourier series for $f(x) = e^{ax}$ using Chebyshev polynomials. And first step is computation following integral.
How to compute $$\int_{-1}^1\frac{e^{ax}dx}{\sqrt{1-x^2}}, \: a \in \mathbb{R}$$
| By setting $x=\sin\theta$ we have
$$ I(a)=\int_{-1}^{1}\frac{e^{ax}}{\sqrt{1-x^2}}\,dx = \int_{-\pi/2}^{\pi/2}\exp\left(a\sin\theta\right)\,d\theta \tag{1}$$
and we may expand the exponential function as its Taylor series at the origin. Since the integral of and odd integrable function (like $\sin^3$ or $\sin^5$) over ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2320165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The group $\mathbb Z_n$ is isomorphic to a subgroup of $GL_2(\mathbb R)$. I need to prove the following:
The group $\mathbb Z_n$ is isomorphic to a subgroup of $GL_2(\mathbb R)$.
How can I prove this?
$\mathbb Z_n$ is of order $n$ so it is isomorphic to a subgroup of $GL_n(\mathbb R)$.
I know this is true but here i... | Consider the group of those matrices of the form\begin{pmatrix}\cos\left(\frac{2k\pi}n\right)&-\sin\left(\frac{2k\pi}n\right)\\\sin\left(\frac{2k\pi}n\right)&\cos\left(\frac{2k\pi}n\right)\end{pmatrix}($k\in\{0,1,\ldots,n-1\}$). In other words, it is the subgroup of $GL_2(\mathbb{R})$ generated by the matrix$$\begin{pm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2320275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Swapping hands in a generalized clock Consider a generalized clock,
where the minute hand
goes n times as fast as the hour hand,
where n is a positive integer.
The standard clock has
n=12 (sometimes n=24).
As which times can
swapping the hour and minute hands
result in a legal time?
In particular,
for each hour h
from ... | Let the hour hand be pointing at an exact value $H \in [0,1]$ ($0$ represents "12 o'clock" or the angle zero. $.5$ represents "6 o'clock" or the angle 180 or $\pi$ and $1$ represents 360, "12 o'clock" or $2\pi$ or "full circle"). Then the minute hand must be pointing at a precise value $M = \{nH\}$. i.e. the fraction... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2320374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Can every compact subset of $\mathbb R^n$ be realized as a manifold with boundary? I think it should be true since in Lee's introduction to smooth manifolds, he referred to a differential form on a compact subset $D \subset \mathbb R^n$. It is easy to provide charts for $\operatorname{Int} D$, but for $\partial D$, how... | No. Just look at the Cantor set in $\mathbb R$ as an example. It has no neighborhoods homeomorphic to any Euclidean space (or half space), but it is compact.
For a less pathological example, consider
$$\left\{\frac1n:n\in \mathbb N\right\}\cup\{0\}$$
This is compact, but no neighborhood of $0$ is homeomorphic to Euclid... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2320464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Other than using the units digit division method, how would I demonstrate $999^{2016}$ 's remainder when it is divided by 5?
Using the other than the units digit division method, how would I demonstrate $999^{2016}$ 's remainder when it is divided by 5?
I have decided that the best method in solving the problem was t... | $$(x+y)^n=\sum_{k=0}^n\binom{n}{k}x^ky^{n-k}$$
so
$$(1000-1)^n=\sum_{k=0}^n\binom{n}{k}1000^k(-1)^{n-k}$$
all terms except the last is divisible by $1000$, thus $(1000-1)^n\equiv(-1)^n\mod5$
Actually, it's clear that $(1000-1)^n\equiv(-1)^n\mod1000$, so the last $3$ digits of $999^n$ are $999$ when $n$ is odd, and $00... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2320545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Help Solving $F_N = \frac{1}{\sqrt5}((\frac{1+\sqrt5}{2})^N - (\frac{1-\sqrt5}{2})^N)$ by Induction I have recently got stuck on an induction problem in my textbook.
It is a big one so major kudos to anybody that can help me out.
The question states to prove this formula inductively:
$$F_N = \frac{1}{\sqrt5}((\frac{1+\... | The link to the other thread about Binet's formula is full of overly complicated answers for a problem that is explained by a variant of a very, very simple identity, one that I have repeated time and time again:
$$a^{n+1} - b^{n+1} = (a+b)(a^n - b^n) - ab(a^{n-1} - b^{n-1}).$$ This is trivially proven by expanding th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2320674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\lim\limits_{x\to \infty} x\big(\log(x+1) - \log(x-1)\big) =e^2$ need to find the value of
$$\lim\limits_{x\to \infty} x(\log(x+1) - \log(x-1))$$
$x(\log(x+1) - \log(x-1))=x(\log{ x+1\over x-1}) = \log({x+1\over x-1})^x = \log(1+{2 \over x-1})^x = \log(1+{2 \over x-1})^{x-1} + \log(1+{2 \over x-1})$
If we take $\lim\... | The reasoning is very correct...the answer is not: apparently you forgot you had logarithms and instead of $\;e^2\;$ the answer should be $\;\log e^2=2\;$ :
$$\lim_{x\to\infty}\color{red}\log\left(1+\frac2{x-1}\right)^x=\lim_{x\to\infty}\left[\color{red}\log\left(1+\frac2{x-1}\right)^{x-1}+\color{red}\log\left(1+\frac2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2320827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Extracting a direction gradient from a set of points I have a matrix containing a set of points:
[
100,100,40,50,30,
30,100,20,20,30,
10,20,45,30,22,
102,200,10,0,10
10,20,20,30,40
]
Is there a way we can retrieve a directional gradient (vector) from this matrix? The vector should be pointing toward the region that co... | You can use standard finite difference methods to estimate the derivative in each direction, and therefore determine the gradient. So for example, the gradient at $(3,3)$ can be estimated component-wise with
$$\frac{\partial f(x,y)}{\partial x} \approx \frac{f(x+1,y)-2f(x,y)+f(x-1,y)}{1^2}$$
$$\frac{\partial f(x,y)}{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2320953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Products of sparse sets of primes Let $S$ be a set of primes such that $\prod_{p \in S} (1 - 1/p)^{-1}$ converges, so the sum of the reciprocals of the products of these primes converges.
If if $n_S$ is the largest factor of $n$ that is a product of elements of $S$, then this condition is $$\sum_{n_S = n} \frac{1}{n} =... | I think the $\log n$ here is a bit of a red herring; I suspect it was made deliberately weaker than necessary in order to retain the strong analogy with Lemma 14. (As we'll see in a moment, it would still be true with the RHS replaced by $\log \log \log n$.)
Let $S^*$ denote the set of values of $n_S$ (in other words ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2321069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Integer Partitions asymptotic behaviour Let $ P(n) $ be the number of partitions of number $n$.
Prove that $ P(n)$, grows faster than any polynomial from $n$.
I am looking for an elementary (rather bijective) proof of the fact.
| The number of ordered partitions of $n$ into exactly $k$ parts is, by the usual combinatorial "stars and bars" argument, $\binom{n-1}{k-1}$: imagine placing $n$ objects in a row and adding $k-1$ dividers in the $n-1$ spaces between them.
As a result, a lower bound for the number of unordered partitions of $n$ into $k$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2321179",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Irreducibility of $X^4 + a^2$ with odd $a$ For my abstract algebra course I have to decide whether
$$(*)\qquad X^4 + a^2, \; a \in \mathbb{Z}\;\text{odd}$$ is irreducible over $\mathbb{Q}[X]$ and $\mathbb{Z}[X]$.
Since the degree of the polynomial is 4 and thus even I should be able to split it into two polynomials ... | Hint: the roots of $x^4 + a^2$ are
$$ \pm\sqrt{a}\frac{1 \pm i}{\sqrt{2}}. $$
Separate these into conjugate pairs to find the factorization of $x^4 + a^2$ over $\mathbf{R}[x]$.
If you want to save time, or perhaps check your work, the answer is
$$ x^4 + a^2 = (x^2 - \sqrt{2a} \cdot x + a)(x^2 + \sqrt{2a} \cdot x + a). ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2321285",
"timestamp": "2023-03-29T00:00:00",
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Example of a random variable $X$ that is an $\mathscr{F}_t$-local martingale, but not an $\mathscr{F}_t^X$-local martingale. This is a problem from Ethier and Kurtz' Markov Processes. The book introduces some theorems on local martingales but they all involve the process being right continuous. I think this problem mus... | Because $E|X_t|=E|\xi|=\infty$ for $t\ge 1$, $\{X_t\}$ is not a martingale (with respect to any filtration).
Define $\tau_n:=n$ on $\{|\xi|\le n\}$ and $\tau_n=0$ on $\{|\xi|>n\}$. Then $\{\tau_n\}$ is an increasing sequence of $(\mathcal F_t)$ stopping times (note that $\xi$ is $\mathcal F_0$-measurable) with limit $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2321385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Basis of the image of the linear transformation $f:\mathbb{R^4}\rightarrow\mathbb{R^3}$? I'm asked to find a basis of the image of the linear transformation $f:\mathbb{R^4} \rightarrow \mathbb{R^3}$ defined as $f(v) = (v_1-v_3+v_4,2v_1+v_2+2v_3+v_4,3v_1-v_2+v_4)$.
I found the matrix of the linear transformation $Af = ... | If it's not completely obvious which column in a linear combination, then just row reduce?
$\begin{pmatrix}
1 & 0 & -1 & 1\\
2 & 1 & 2 & 1\\
3 & -1 & 0 & 1 \\
\end{pmatrix} \to \begin{pmatrix}
\color{red}{1} & 0 & -1 & 1\\
0 & \color{red}{1} & 4 & -1\\
0 & 0 & \co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2321528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Locally constant sheaf on a simply connected space I was reading an article and at some point the writer claims that
1)A locally constant sheaf on a simply connected topological space is a constant sheaf.
2) $H^{i}(U,\mathcal{F})=0 \hspace{0.1cm}\forall i>1$ where U is a homotopically trivial open set and $\mathcal{F}... | A more abstract way of reformulating the result is that the category of local systems ( = locally constant sheaves) on $X$ with stalk $M$ ($M$ is a $k$-vector space or a module) is equivalent to the category of representations $\rho : \pi_1(X) \to GL(M)$. In particular, if $X$ is simply connected then every locally co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2321629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Rigorous nature of combinatorics Context: I'm a high school student, who has only ever had an introductory treatment, if that, on combinatorics. As such, the extent to which I have seen combinatoric applications is limited to situations such as "If you need a group of 2 men and 3 women and you have 8 men and 9 women, h... | What you phrased as "can you find a situation where the statement is incorrect" is better known as proof-by-contradiction in the realm of propositional logic. Proof by contradiction is a fairly common approach to proving statements about combinatorics and other areas of study in discrete mathematics--typically after a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2321757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "84",
"answer_count": 7,
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find points of ramification, hurwitz formula Hi I have the following question:
$$f(z)=4z^2(z-1)^2/(2z-1)^2$$
considered as a meromorhic function over $\mathbb{C}_{\infty}$
has as zeros:
$z=0$, $ord_0(f)=2$
$z=1$, $ord_1(f)=2$
and as poles:
$z=1/2$, $ord_{1/2}(f)=2$
$z=\infty$, $ord_{\infty}(f)=2$
Now, considering the ... | You haven't found all of the ramification points. You've found it has multiple zeroes and poles, but it might also be ramified at values other than the zeroes and poles. You can find the other ramification points by finding the zeroes of the derivative $f'(z)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2321831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Integrating a function over a square using polar coordinates Say we have a function $f(x,y)$ over the unit circle. To integrate with polar coordinates we replace the x and y in $f(x,y)$ with $r\cos\theta$ and $y\sin\theta$ to get $f(r,\theta)$ and we integrate $f(r,\theta)rdrd\theta$ for $r$ between $0$ and $1$ and $\t... | For each of the four sides that make up the square, we will have $0 \le r \le p\sec(\theta-c)$, for suitable values of $p$ and $c$, and $\theta$ ranging suitably, as follows:
Notice how $r=\sec(\theta)$ is the equation of a straight line with perpendicular angle $0$ (vertical), one unit to the right of the origin, so $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2322034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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proving a differentiable function $f: \Bbb R \to \Bbb R$ and a constant $c>0$ with $f'(x) \geq c$ for all $x \in \Bbb R$. is a bijection We have a differentiable function $f: \Bbb R \to \Bbb R$ and a constant $c>0$ with $f'(x) \geq c$ for all $x \in \Bbb R$.
Show that $f$ is a bijection from $\Bbb R$ to $\Bbb R$.
From ... | From $c>0$ and $f(x) \geq f'(0)+ cx$ for $x \ge 0$ we get
$ \lim_{x \to \infty}f(x)= \infty$.
From $c>0$ and $f(x) \le f'(0)+ cx$ for $x \le 0$ we get
$ \lim_{x \to -\infty}f(x)= -\infty$.
Now we derive , by the intermediate value theorem: $f( \mathbb R)= \mathbb R$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2322134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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How would you prove that this is a group isomorphism? Let $$M_{a} = \begin{pmatrix} 1 & a & \frac{a^2}{2} \\ 0 & 1 & a \\ 0 & 0 & 1\end{pmatrix}$$
where $a \in \mathbb{R}$
and let the function $\phi : \mathbb{R}\rightarrow G$ where $\phi(a) = M_a$ and $G$ is the set containing $M_a$.
I've shown this is a group homomo... | I suppose you meant $\;G=\text{Im}\,\phi\le GL(3,\Bbb R)\;$ , and to answer your question: you simply have to prove the easy following fact
$$\ker\phi=\{0\}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2322229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integrating product of linear functions over a triangle Given a triangle with vertices $N=\{A,B,C\}$, where the triangle is defined to be the convex hull of those vertices, the nodal basis function is defined to be $$\phi_P(x) = \begin{cases} 1, x = P \\ 0, x \in N \setminus \{P\} \end{cases}$$
where $P \in \{A,B,C\}$
... | For the triangle $T_*$ with vertices $A_0=(0,0)$, $A_1=(1,0)$, $A_2=(0,1)$ one has $\phi_1(x,y)=x$ and $\phi_2(x,y)=y$. This implies
$$\int_{T_*}\phi_1(x,y)\phi_2(x,y)\>{\rm d}(x,y)=\int_0^1\int_0^{1-x} x y\>dy\>dx=\ldots={1\over24}\ .$$
From "general principles" (see achille hui's comment) it then follows that for an ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2322328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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On the definition of the Zariski Tangent space I have the following (relating to the definition of the Zariski tangent space) in my notes.
Let $m_P$ be the ideal of $P$ in $k[V]$, and $M_P$ as the ideal $\langle X_1,...,X_n\rangle\subset k[X_1,...,X_n]$. Then $m_P=M_P/I(V)$. Then
$$m_P/m_P^2=M_P/(M_P^2+I(V))$$
I'm ha... | Let $R$ be a ring, $J$ an ideal of $R$, and $\pi$ be the canonical projection $R \to R/J$.
Consider an ideal $I$ of $R$. I find $\pi(I)$ — the ideal of $R/J$ generated by the image of $I$ — to be a much more convenient notion than its explicit construction va cosets $(I+J)/J$.
(this is related to the fact that, for $a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2322503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$u \in C^\infty(\mathbb{R^n})$ with compact support $\implies$ $f(x) = ⨏_{\partial B(0,\vert x \vert)} u(t) \ dt\in C^\infty$ with comp. s.? $\newcommand{\avint}{⨍}$
Let $u \in C^\infty(\mathbb{R^n})$ with compact support.
How can we prove using direct calculations or Fourier transform methods that $$f(x) = \avint_{\pa... | Let $u \in C_c^\infty(\mathbb R^n)$ and let $\sigma$ be the measure on $S^{n-1}$. Then we set
$$f(x) = \frac{1}{\sigma(\partial B(0,|x|))} \int_{\partial B(0,|x|)} u(t) \, dt$$
It's obvious from the definition that $f(x)$ only depends on $|x|$, i.e. $f(x) = \hat f(|x|)$ where
$$
\hat f(r)
= \frac{1}{r^{n-1} \sigma(S^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2322628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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In a triangle $a:b:c =4:5:6$, then $3A+B$ equals to? In the above question $a,b,c$ are sides of triangle and $A,B,C$ are angles. The correct answer is $\pi$ but I am getting $\pi - C$.
| HINT:
We have $\dfrac a4=\dfrac b5=\dfrac c6=k$(say)
$\implies a=4k$ etc.
Use cosine formula
$$\cos A=\dfrac{b^2+c^2-a^2}{2bc}=\cdots=\dfrac{45}{60}>\dfrac12\implies0<A<60^\circ$$
and $$\cos B=\dfrac9{16}$$
$$\cos3A=-\dfrac9{16}$$
Now use How do I prove that $\arccos(x) + \arccos(-x)=\pi$ when $x \in [-1,1]$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2322703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Diophantine equation without elementary solution but with simple non elementary solution Is there some example of an diophantine equation that satisfies:
*
*No solution is known using elementary methods.
*It is simple to solve using non elementary methods (e.g. using number fields).
My goal is to find good motiva... | I think a good example is given by the equation $$x^2-6y^2=1$$ This equation has infinitely many integer solutions $(x,y)=(a_n,b_n)$ determined by $$a_n+b_n\sqrt6=(5+2\sqrt6)^n\space\space........ (n\ge1)$$ Here the calculation of the number $5+2\sqrt6$ is obviously "fundamental" and it is not obvious at all that there... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2322820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
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How to calculate limit as x approaches infinity of a^x/b^x? I'm trying to calcululate $\lim_{x\to\infty}\frac{a^x}{b^x}$.
I've done a few examples on Wolfram Alpha, and it seems if $a>b$ it goes to infinity and if $a<b$ it goes to $0$, but I am not sure how to prove it.
L'Hopital is normally what I would try, but it do... | Hint: review properties of exponents, especially that the quotient of two numbers each raised to the the $x$ power is equal to the quotient of the two numbers itself raised to the $x$ power.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2322876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Absolute value equation infinite solutions $$|3-x|+4x=5|2+x|-13$$
One of the solutions is $[3,\infty)$
I'm not familiar with interval solutions for absolute equations.
How to solve for this interval?
| One way to solve a problem like this is by graphing both sides of the equation.
You can do this in, for example, Desmos here. An image of the linked graph, showing that the two lines coincide for $x \geq 3$ (and showing that they coincidence at one other $x$-value, too):
Another way to solve these sorts of problems is... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Convergence/Divergence of some series If I have $\sum_{n=2}^{\infty} \frac {1}{n\log n}$ and want to prove that it diverges, can I use following?
$$\frac {1}{n\log n} \lt \frac {1}{n}$$
*
*$\sum_{n=1}^\infty \frac 1 n$ diverges, but the limit of $\frac 1 n$ equals to zero so the comparasion I think isn't useful.
*O... | Bertrand's series $\displaystyle \sum\limits_{n\ge 2}\frac 1{n^\alpha\ln(n)^\beta}$ converges only for $(\alpha>1)$ or $(\alpha=1,\beta>1)$.
This is generally proved using the comparison to an integral as other answers have shown.
But in general for series of the kind $\displaystyle \sum\limits_{n\ge 2}\frac 1{n^{\alph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2323143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 7,
"answer_id": 2
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Variance when playing a game with a fair coin I am having a hard time with this question for some reason.
You and a friend play a game where you each toss a balanced coin. If the upper faces on
the coins are both tails, you win \$1; if the faces are both heads, you win \$2; if the coins
do not match (one shows head an... | Variance is the mean of the squares minus the square of the mean.
$$
0.25\cdot1^2+0.25\cdot2^2+0.5\cdot(-1)^2-0.25^2=1.6875
$$
For independent events, the variance of a sum is the sum of the variances, so the variance for $50$ events is
$$
50\cdot1.6875=84.375
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2323223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the distinct number of arrangements of the symbols in the string###@@\$\$\$%%%% that begin and end with % I have tried to solve this by excluding two % and performing permutation for non-distinct objects. As I excluded I left with 10 elements so n=10 and then considered non-distinct elements like # (n1=3),@(n2=2),... | So what I'm assuming: $3$ hashtags, $2$ at's, $3$ dollar signs, $4$ percentage signs
You first fix two percentages, so you are arranging $3$ hashtags, $2$ at's, $3$ dollar signs, and $2$ percentages
This can be done in $\displaystyle \frac{(3+2+3+2)!}{3!2!3!2!}=\boxed{25200}$ ways.
Basically, arrange $10$ objects, and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2323330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Is $\infty^0= 1$? for given integral $\int_e^\infty \frac{1}{x(\log x)^p}dx$,
I had derived above integral = $\int_1^\infty {u^{-p}}du = [ {1\over -p+1}u^{-p+1}]_1^\infty = {1\over -p+1}[\infty^{-p+1} -1]$
I need to characterize the range of $p \in \Bbb R$ which makes the given integral converges, however, when p =1, t... | In general improper integrals like $\int_a^{\infty}{f(x)dx}$ can be defined as $\lim_{b \to \infty}F(x) - F(a)$ where $F(x)$ is the anti derivative of $f(x)$.
In this case how ever the anti derivate for the case $p=1$ is not the one deduced from formula for other values of $p$. So you have to take it as special case a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2323389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to find number of solutions to $2^x=x^2$ without Graphing Find number of real solutions to $2^x=x^2$ without plotting graph:
I considered $f(x)=2^x -x^2$
$$f'(x)=2^x \ln 2-2x=0$$
we get again a transcendental equation. Any good approach please
| $a_{0}2^x+b_{0}$ may have at most one zero. Before that it is negative, after that zero, if it exists, it is positive.
Based on just this remark, $a_{1}2^x+b_{1}x$ may have at most two zeros, since its derivative $a_{0}2^x+b_{0}$ is forcing it to behave similar to a parabola, if the derivative has one zero. Again, base... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2323500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Is Borel-field different from $\sigma$-field? My mathematical statistics book denotes $\sigma$-field as following:
Let $\Bbb B$ be the collection of subsets of $\Bbb C$ where $\Bbb C$ denotes sample space which is the collection of all possible events. Then $\Bbb B$ is $\sigma$-field if
(1) $\emptyset \in \Bbb B$ and ... | A sigma field on a non-emptyset $X$ is a collection $\mathcal{F}\subseteq 2^X$ that contains $\emptyset$, is closed under complementation and is closed under countable unions.
A Borel field is a sigma field $\mathcal{F}$ that is defined on a topological space $(X, \mathcal{T})$ such that $\mathcal{T} \subseteq \mathcal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2323671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Where does $xe^x$ solution come from when the characteristic polynomial is square? When solving the differential equation $y'' + ay' + by = 0$ (with constant, real coefficients $a$ and $b$, although they could be complex if you like), you do it by setting up the characteristic equation $r^2 + ar + b = 0$, finding its s... | The simple direct approach gives the solution $xe^x$ without much hassle. Let the equation be $$y''-2y'+y=0$$ and let $z=y'-y$ so that the equation can be written as $$z'-z=0$$ The above equation on multiplying with $e^{-x} $ gives $$(ze^{-x}) '=0$$ or $$ze^{-x} =c_1$$ so that $$y' - y=z=c_1e^x$$ Again multiplying by $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2323733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 2
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Graduations of volume on the side of a cone I am trying to put graduated volume markings (every 10 liters) on the side of a cone.
Specifically this is the conical section of a wine tank. I know the dimensions of the whole cone (h: 108cm, r: 103.5cm, l: 150.3cm, V: 1220L) but I am having trouble figuring out the volumes... | Think of the cone as situated vertex-down with its axis vertical, with slant neight $\ell$ (in cm) measured from the vertex. As you suspected, neglecting the bit of cylinder near the vertex, the volume held by the tank from the vertex to a slant height $\ell$ is proportional to $\ell^{3}$, so that
$$
V = k\ell^{3} = \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2323847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Finding intersection angle at intersection point of two curves I've got two curves:
$$(x,y) = (t^2,t+1), \quad t\in\mathbb{R}$$
$$5x^2 + 5xy + 3y^2 -8x -6y + 3 = 0$$
I've found the intersection points:
$$(0,1) , (1,0)$$.
But I can't figure out how to the the angle between the two curves at these intersection points? ... | plugging $$x=t^2,y=t+1$$ in the given equation we get
$$5t^4+5t^2(t+1)+3(t+1)^2-8t^2-6(t+1)+3=0$$
which can be simplified to $$5t^3(t+1)=0$$
can you solve it?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2323998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Decomposition of set of roots for a Lie algebra and its Cartan subalgebra Consider a finite dimensional complex semi-simple Lie algebra $L$ with Cartan subalgebra $H$ (i.e. every $h\in H$ is $ad$-nilpotent). Denote $\Phi=\Phi(L,H)$ the set of roots.
Assume $\Phi=\Phi_1\cup\Phi_2$ for non empty $\Phi_i$ and $(\alpha,\be... | It is obvious that $L=L_1\oplus L_2$ as vector spaces. To check that it holds as Lie algebras, you need to show that $[L_1,L_2]=0$. But this follows immediately from the orthogonality assumption, and the fact that $\alpha+\beta\notin\Phi$ for all $\alpha\in\Phi_1$ and $\beta\in\Phi_2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2324086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What are the total number of ways in which $n$ distinct objects can be put into two different boxes so that no box remains empty? I have came across this problem which is my textbook.According to the book the answer is: $2^n - 2$. But i don't understand how they got to that answer. Can someone help me out?
| There are $2^n$ different subsets that can be taken from a set of $n$ objects. Pick any subset and put it in the first box. Then put the rest in the other box. That makes $2^n$ ways to put $n$ object in $2$ boxes. But you have the restriction that neither box can be empty. So you can't choose the empty set for box... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2324221",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to estimate condition number based on SVD of submatrix? Given an $m\times n$ ($m\geq n$) real valued matrix, $A$, its SVD, and an $n$-dimensional real valued vector, $x$, is there a computationally efficient way to accurately estimate the condition number of the matrix, $B$, constructed by appending $x$ as an addit... | I was able to find the answer to my question by reading this, which also provides a great list of references. I'll leave out the detailed derivation that is presented in those works, and summarize the answer. Keep in mind, I'm only interested in computing the condition number of the updated matrix, and not all the sing... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2324341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$e^{2\sqrt{2} \pi i}=(e^{2 \pi i})^{\sqrt{2}}=1$? This is probably stupid. But this true?$$e^{2\sqrt{2} \pi i}=(e^{2 \pi i})^{\sqrt{2}}=1$$
I feel like this is wrong but I cannot see how. Any help is appreciated. Thank you
| The "obvious" identity $(a^b)^c=a^{bc}$ does not hold in complete generality.
It does hold for all complex numbers $a$ if $b$ and $c$ are restricted to be integers (with a minor caveat for $a=0$ if $b$ and $c$ and negative integers).
It does hold for all positive real numbers $a$ if $b$ and $c$ are restricted to be rea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2324412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
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Show that if $f:X\to Y$ is a continuous function if and only if the graph of $f$ is a closed subset of $X\times Y$
Given a function $f:X\to Y$, we define the graph of $f$ as the set
$$G(f)=\{(x,f(x)),x\in X\}$$ Show that if $X$ is compact then $f$ is a
continous function if and only if $G(f)$ is a closed subset of... | Your purported equivalence can fail in both directions, if we don't add extra assumptions on $Y$.
The implication $G(f)$ closed implies $f$ continuous can fail for compact $X$:
Let $X = [0,1]$ in the cofinite topology; this is a compact space.
Let $Y = [0,1]$ in the discrete topology.
Define $f(x) =x$ from $X$ to $Y$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2324482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Calculus BC problem So I as attempting to solve this problem and just got stuck overall. Since this is just practice an explanation is much more valuable than the answer itself. Thanks!
Here is the problem
| Hint:
The slope of the line is $m=\dfrac{\sin(K)-0}{K-0}=\dfrac{\sin(K)}K$, then its equation is $$y=\dfrac{\sin(K)}Kx$$
Now, observe that
\begin{align*}\text{Yellow area }&=\int_0^K\left(\sin x-\dfrac{\sin(K)}Kx\right)dx\\&=1-\cos(K)-\frac{\sin(K)}{2K}K^2\\&=1-\cos(K)-\tfrac K2\sin(K)
\end{align*}
And
\begin{align*}
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2324563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
On negation of lipschiz continuity Let $f: [a,b] \to R$ be continuous function which is not Lipschitz continuous.
Can we say there exist $x \in [a,b] $ and strictly monotone sequences, $\{x_n\}_{n=1}^{\infty} \subseteq [a,b] $ and $\lambda_{n} \in \mathbb{R^+} $ such that $x_n \to x$ and $\lambda_{n} \to + \infty $, ... | Consider the function
$$
f(x) =
\begin{cases}
x \sin (1/x), &\text{if}\ x\neq 0,\\
0, & \text{if}\ x = 0.
\end{cases}
$$
This function is continuous in $\mathbb{R}$, it is not Lipschitz continuous, and it has continuous derivative in $\mathbb{R}\setminus\{0\}$, hence around each point $x\neq 0$ it is locally Lipschitz... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2324713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Is $\sum_{n=0}^{\infty} \frac{n^2}{(b-n^2)(a-n^2)}$ expressible in terms of trigonometric functions I recently ran into the sum
$$S=\sum_{n=0}^{\infty} \frac{n^2}{(\alpha-n^2)(\beta-n^2)}.$$
Mathematica gives it in terms of the Digamma function as
$$S=\frac{-\alpha \psi ^{(0)}(1-\alpha )+\alpha \psi ^{(0)}(\alpha +1)... | $$S=\sum_{n=0}^{\infty} \frac{n^2}{(b-n^2)(a-n^2)}=\frac 1 {a-b}\left(\sum_{n=0}^{\infty} \frac{a}{n^2-a}-\sum_{n=0}^{\infty} \frac{b}{n^2-b}\right)$$ leading to $$S=\frac{\pi \sqrt{b} \cot \left(\pi \sqrt{b}\right)-\pi \sqrt{a} \cot \left(\pi
\sqrt{a}\right)}{2 (a-b)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2324796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Why radius of convergence is $\frac{1}{r}=\liminf_{n\to \infty }\left|\frac{a_{-n-1}}{a_{-n}}\right| ?$ Let consider the series $$\sum_{n\in\mathbb Z}a_nz^n.$$
We denote $R$ the radius of $\sum_{n=0}^\infty a_nz^n$ and $r$ the radius of $\sum_{n=-\infty }^{-1}a_nz^n$, i.e.e the series converge absolutely if $r<|z|<R$. ... | This is wrong:
$$\frac{1}{r}=\liminf_{n\to \infty }\left|\frac{a_{-n-1}}{a_{-n}}\right|$$
A better formula is
$$\frac{1}{r}=\liminf_{n\to \infty }\left|\frac{a_{-n}}{a_{-n-1}}\right|$$
But as Daniel Fischer points out in a comment, this is not generally correct. It is correct if $|a_{-n}|$ is monotonic.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2324900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Poisson-processes and it's arrival times I am currently studying for my non-life insurance exam and have the following problem:
Let $S(t) = \sum_{i=1}^{N(t)} (X_i + T_i)^2$, where $X_i$ are i.i.d. r.v. with density $f(x)$ and $T_i$ are the arrival times of the homogeneous possion process $N(t)$ with intensity $\lambda ... | http://www.maths.qmul.ac.uk/~ig/MAS338/PP%20and%20uniform%20d-n.pdf
Using the well-known result about the symmetric functional of the arrival times
(Theorem 1.2), we have
$$ \begin{align} E[S(t)] &= E\left[\sum_{i=1}^{N(t)} (X_i + T_i)^2\right] \\
&= E\left[\left[\sum_{i=1}^{N(t)} (X_i + T_i)^2\Bigg|N(t)\right]\right] ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2325035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Is $|x| \cdot |x| = |x^2| = x^2$?
Is $|x| \cdot |x| = |x^2| = x^2$ ?
I'm very sorry if this question is a duplicate but I couldn't find anything about it (most likely because it's wrong..). But I'm not sure if this is correct so I need to ask you.
$$|x| \cdot |x| = |x^2| \text{ should be alright}$$
Now my confusion s... | You are thinking it too hard. You could just look at the definition of the absolute value
$$
|x|:=\begin{cases}
x,&x\geq 0\\
-x,&x<0
\end{cases}
$$
and check on your own that $|x|^2=|x^2|=x^2$.
In general, we have $|a|\cdot|b|=|ab|$, which is true also for complex numbers; but the identity $|x^2|=x^2$ is not necessar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2325138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 0
} |
Find this limit. Compute the value of the limit :
$$
\lim_{x\to\infty}{\frac{1-\cos x\cos2x\cos3x}{\sin^2x}}
$$
I've tried simplifying the expression to
$$
\lim_{x\to\infty}\frac{-8\cos^6x+10\cos^4x-3\cos^2x+1}{\sin^2x}
$$
But I don't know what to do after this.
| Using the identities $\displaystyle \sin^2(x)=\frac{1-\cos(2x)}{2}$, $\displaystyle \cos(4x)=2\cos^2(2x)-1$, and $\displaystyle \cos(x)\cos(3x)=\frac12(\cos(2x)+\cos(4x))$, we obtain
$$\begin{align}
\frac{1-\cos(x)\cos(2x)\cos(3x)}{\sin^2(x)}&=\frac{1-\frac{\cos(2x)\left(\cos(2x)+\overbrace{(2\cos^2(2x)-1)}^{=\cos(4x)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2325217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 1
} |
Understanding the third Sylow theorem I am trying to understand the following theorem:
Let $G$ be a finite group, $p$ a prime number, and let's suppose $|G|=p^ns$ s.t. $p$ doesn't divide $s$
Let $n_p$ be the number p-sylow subgroups of G then we have
$$\begin{cases}n_p |s\\n_p \equiv 1 \mod p\end{cases}$$
Now my pr... | Maximal means "not contained in anything else" not "everything else is contained in it."
Look at $S_3$ where you have three Sylow-$2$ subgroups, $\{(1), (ij)\}$ for any $1\le i\ne j\le 3$.
Or if you prefer a simpler example of where "maximal" means this, look at a case with some sets: consider $\{\varnothing, \{1\},\{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2325384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Derivative of the logit function I have plotted a logit function and its derivative. My first question is that how can I interpret the derivative graph of the logit function and second, why in logit function, the second derivative becomes the logit function itself?
| The second derivative of the logit function is not equal to itself. Look:
$$l(x)=\ln\bigg(\frac{x}{1-x}\bigg)$$
$$l(x)=\ln(x)-\ln(1-x)$$
Then differentiate:
$$l'(x)=\frac{1}{x}+\frac{1}{1-x}$$
Then differentiate again:
$$l''(x)=-\frac{1}{x^2}-\frac{1}{(1-x)^2}$$
The second derivative of the logit function is a complete... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2325498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
"X is distributed as ..." notation A question about notation:
We sometimes use $\sim$ to denote "distributed as" e.g. if $X$ is Gaussian we write $X \sim N(\mu, \sigma^2)$.
Is it acceptable to use the "~" notation for an arbitrary distribution? e.g. can we write
$$X \sim \begin{cases} \frac{3}{2}x^2, & x \in [-1,1] \... | To the best of my knowledge, it is not that rare to see $X\sim f_X(x)$ as a shorthand for $X$ is distributed according to $f_X(x)$. However, I've never encountered in any formal settings your version ($\sim$ followed by brackets etc.).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2325592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Why can I put the limit sign in equations Why can I put the limit sign in both sides of an equation?
Is there a rigorous way to prove that if:
$f(x)=g(x)$
Then
$\lim \limits_{x\to x_0}f(x) =\lim \limits_{x\to x_0}g(x)$
Thanks.
| Well, if $f(x)=g(x)$, then $f(x)=g(x)=f(x)$.
Therefore, we have $\displaystyle \lim_{x \to x_0}[f(x)]=\lim_{x \to x_0}[g(x)]$, because we have an identical expression under a different name.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2325661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find the minimum and maximum distance between the ellipsoid of ecuation $x^2+y^2+2z^2=6$ and the point $P=(4,2,0)$ I've been asked to find, if it exists, the minimum and maximum distance between the ellipsoid of ecuation $x^2+y^2+2z^2=6$ and the point $P=(4,2,0)$
To start, I've tried to find the points of the ellipsoid... | hint
The ellipse can be parametrized as
$$x_e =\sqrt {6}\sin (\phi)\cos (\theta) $$
$$y_e=\sqrt{6}\sin (\phi)\sin (\theta ) $$
$$z_e=\sqrt {3}\cos (\phi) $$
the square of the distance from a point of the ellipse to the point $(4,2,0) $ is
$$D^2=(x_e-4)^2+(y_e-2)^2+z_e^2$$
$$=3\sin^2 (\phi)+23-8x_e-4y_e $$
You can find ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2325729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
When in $b-$base representation of a number all of $0, 1, 2, ..., b-1$ exist? If $a,b \ge 2$ are given, prove that there is a positive integer $m$ such that $ a \mid m$ and in $b-$base representation of $m$ all of $0, 1, 2, ..., b-1$ exist.
My teacher gave this to me, but I think his statement was wrong. It was like th... | Pick $r$ with $b^r>a$. Let $A$ be the number that, in base $b$, uses all digits. Then one of the numbers $b^rA,\ldots, b^rA+a-1$ will do.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2325896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Suppose that $\lim_{x\to \infty} f'(x) = a$. Is it true that $\lim_{x\to \infty} {f(x)\over x} = a$ Suppose that $\lim_{x\to \infty} f'(x) = a$. Is it true that $\lim_{x\to \infty} {f(x)\over x} = a$
If so, can you prove it? Thanks!
| HINT
Let assume $a \gt 0$. Then, because $\lim_{x\to \infty} f'(x) = a \gt 0$ there is $M \gt 0$ such that $f'(x) \gt 0$ for all $x \ge M$. Therefore $f$ is strictly increasing on $[M, +\infty)$ and unbounded (otherwise $\lim_{x\to \infty} f'(x) =0$). It follows that $\lim_{x\to \infty} f(x) = +\infty$ and we can apply... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2326042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
equation of ellipse after projection If I have the intersection of $x+z=1$ and $$x^2 +y^2 +z^2=1$$ which is a circle in $O'xyz$. Then I do a projection of this circle on the $O'xy$ plane, it'll be an ellipse. How can I then find the equation of this ellipse?
| If $(x,y,z)$ is a point on the circle, then $(x,y,z)$ satisfy both $x+z=1$ and $x^2+y^2+z^2=1$. It's equations are
$$\begin{cases} z=1-x \\ x^2+y^2+(1-x)^2=1\end{cases}$$
Its projection has $z$-coordinate equals $0$ and keep the $x$ and $y$-coordinates. So the equation of the projection on the $xy$-plane is
\begin{alig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2326177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Is the tangential velocity the same as finding the tangent vector? Sorry for stupid question. I am calculating the tangent vector for a vector function and the problem also asks for arc length, speed and unit tangent vector. I did OK but when I hear the term tangential velocity of an object in physics is that the same ... | The tangential velocity is usually the length of the tangent vector. It depends on the parametrization, if you, e.g., pass slower through the curve the length of the tangent vectors reduces.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2326287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Evaluate the following Determinant of $12$th degree polynomial Evaluate $$\Delta=\begin{vmatrix}
\frac{1}{(a+x)^2} & \frac{1}{(b+x)^2} & \frac{1}{(c+x^2)}\\
\frac{1}{(a+y)^2} & \frac{1}{(b+y)^2} & \frac{1}{(c+y)^2}\\
\frac{1}{(a+z)^2} & \frac{1}{(b+z)^2} & \frac{1}{(c+z)^2}\\
\end{vmatrix}$$
My Try: I have taken all... | It's obvious that we have a factor $(a-b)(b-c)(c-a)(x-y)(y-z)(z-x)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2326385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proof or counterexample: Let $A$ be a square matrix, then:
*
*If $A$ is diagonalizable, then so is $A^2$
I answered yes. I argued that since $A$ is diagonalizable there exists an eigenbasis, and since $A^2$ has the same eigenvectors than $A$, and its eigenvalues are those of $A$ squared, there is also an eigenbasis... | Suppose $A$ is diagonalisable. Then there is a basis of eigenvectors $e_i$ with eigenvalues $\lambda_i$, not necessarily distinct. With respect to this basis, $A^2 e_i = \lambda_i^2 e_i$, so $e_i$ is also a basis of eigenvectors for $A^2$, and hence $A^2$ is diagonal in this basis.
(Or one can use the existence of a un... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2326522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Angle chasing with two tangent circles. The bigger circle $\Omega$ is tangent to the smaller circle $\omega$.
Also, $GE=2CG$.
We have to find $\angle DEC$.
MY WORK SO FAR.
I proved using the Alternate Segment Theorem that:
$$GF\parallel ED$$
And that,
$$\angle DCH=\angle HCE=45°$$
Also,
$$GF=GH$$
| The fact that there are points $G$ and $E$, $G$ on the small circle, and $E$ on the big circle such that $\vec{CE}=3 \vec{CG}$ means that the circles are homothetic with the homothety with center $C$ and scale factor 3 (http://www.cut-the-knot.org/Curriculum/Geometry/Homothety.shtml). Thus, we also have, for the center... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2326623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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