Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Covariance of order statistics (uniform case) Let $X_1, \ldots, X_n$ be uniformly distributed on $[0,1]$ and $X_{(1)}, ..., X_{(n)}$ the corresponding order statistic. I want to calculate $Cov(X_{(j)}, X_{(k)})$ for $j, k \in \{1, \ldots, n\}$.
The problem is of course to calculate $\mathbb{E}[X_{(j)}X_{(k)}]$.
The joi... | Thank you so much for posting this -- I too looked at this covariance integral (presented in David and Nagaraja's 2003 3rd edition Order Statistics text) and thought that it looked ugly. However, I fear that there may be a few small mistakes in your math on E(X_jX_k), assuming that I'm following you right. The joint de... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Derive Cauchy-Bunyakovsky by taking expected value In my notes, it is said that taking expectation on both sides of this inequality $$|\frac{XY}{\sqrt{\mathbb{E}X^2\mathbb{E}Y^2}}|\le\frac{1}{2}\left(\frac{X^2}{\mathbb{E}X^2}+\frac{Y^2}{\mathbb{E}Y^2}\right)$$ can lead to the Cauchy-Bunyakovxky (Schwarz) inequality $$\... | You can simplify your inequality as follows, for the left side:
$|\frac {XY}{\sqrt{EX^{2}EY^{2}}}|=\frac {|XY|}{\sqrt{EX^{2}EY^{2}}}$
for the right side, take the expectation:
$\frac{1}{2}E\left( \frac{X^2}{EX^2}+\frac{Y^2}{EY^2}\right)= \frac{1}{2} E \left( \frac{X^2 EY^2+X^2 EY^2}{EY^2 EX^2} \right)$
Now, $E(X^2 EY^2... | {
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Check if $\lim_{x\to\infty}{\log x\over x^{1/2}}=\infty$, $\lim_{x\to\infty}{\log x\over x}=0$ Could anyone tell me which of the following is/are true?
*
*$\lim_{x\to\infty}{\log x\over x^{1/2}}=0$, $\lim_{x\to\infty}{\log x\over x}=\infty$
*$\lim_{x\to\infty}{\log x\over x^{1/2}}=\infty$, $\lim_{x\to\infty}{... | $3$ is correct as $\log x$ grows slower than any $x^n$. So $x^{-1}$ and $x^{-\frac{1}{2}}$ will manage to pull it down to $0$. And $\displaystyle\lim_{x\to\infty}{1\over x}=\lim_{x\to\infty}{1\over \sqrt{x}}=0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Is $\sum\limits_{k=1}^{n-1}\binom{n}{k}x^{n-k}y^k$ always even? Is
$$ f(n,x,y)=\sum^{n-1}_{k=1}{n\choose k}x^{n-k}y^k,\qquad\qquad\forall~n>0~\text{and}~x,y\in\mathbb{Z}$$
always divisible by $2$?
| Hint: Recall binomial formula
$$
(x+y)^n=\sum\limits_{k=0}^n{n\choose k} x^{n-k} y^k
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/400961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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One of $2^1-1,2^2-1,...,2^n-1$ is divisible by $n$ for odd $n$
Let $n$ be an odd integer greater than 1. Show that one of the numbers $2^1-1,2^2-1,...,2^n-1$ is divisible by $n$.
I know that pigeonhole principle would be helpful, but how should I apply it? Thanks.
| Hints:
$$(1)\;\;\;2^k-1=2^{k-1}+2^{k-2}+\ldots+2+1\;,\;\;k\in\Bbb N$$
$$(2)\;\;\;\text{Every natural number $\,n\,$ can be uniquely written in base two }$$
$$\text{with maximal power of two less than $\,n\,$ }$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/401036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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determine whether an integral is positive Given a standardized normal variable $X\sim N\left(0,1\right)$, and constants $ \kappa \in \left[0,1\right)$ and $\tau \in \mathbb{R}$, I want to sign the following expression:
\begin{equation}
\int_{-\infty}^\tau \left(X-\kappa \tau \right) \phi(X)\text{d}X
\end{equation}
wh... | Note that
$$ \int_{-\infty}^\infty (X-\kappa\tau)\phi(X)\,\mathrm dX=E[X-\kappa\tau]=-\kappa\tau$$
and for $\tau>0$
$$ \int_{\tau}^\infty (X-\kappa\tau)\phi(X)\,\mathrm dX\ge\int_{\tau}^\infty (1-\kappa)\tau\phi(X)\,\mathrm dX>0.$$
So for $\tau>0$ your expression is positive.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/401109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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A problem on matrices: Find the value of $k$
If $
\begin{bmatrix}
\cos \frac{2 \pi}{7} & -\sin \frac{2 \pi}{7} \\
\sin \frac{2 \pi}{7} & \cos \frac{2 \pi}{7} \\
\end{bmatrix}^k
=
\begin{bmatrix}
1 & 0 \\
0 & 1 \\
\end{bmatrix}
$, then the least positiv... | Powers of matrices should always be attacked with diagonalization, if feasible. Forget $2\pi/7$, for the moment, and look at
$$
A=\begin{bmatrix}
\cos\alpha & -\sin\alpha\\
\sin\alpha & \cos\alpha
\end{bmatrix}
$$
whose characteristic polynomial is, easily, $p_A(X)=1-2X\cos\alpha+X^2$. The discriminant is $4(\cos^2\alp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Jacobson Radical and Finite Dimensional Algebra In general, it is usually not the case that for a ring $R$, the Jacobson radical of $R$ has to be equal to the intersection of the maximal ideals of $R$.
However, what I do like to know is, if we are given a finite-dimensional algebra $A$ over the field $F$, is it true th... | Yes, it is true. See
J.A. Drozd, V.V. Kirichenko,
Finite dimensional algebras,
Springer, 1994
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/401248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Symmetric Matrices Using Pythagorean Triples
Find symmetric matrices A =$\begin{pmatrix} a &b \\ c&d
\end{pmatrix}$ such that $A^{2}=I_{2}$.
Alright, so I've posed this problem earlier but my question is in regard to this problem.
I was told that $\frac{1}{t}\begin{pmatrix}\mp r & \mp s \\ \mp s & \pm r \end{pm... | It works because $$A^2 = \frac{1}{t^2}\begin{pmatrix}r & s\\s & -r\end{pmatrix}\begin{pmatrix}r&s\\s&-r\end{pmatrix} = \frac{1}{t^2}\begin{pmatrix}r^2+s^2 & 0\\0 & r^2 + s^2\end{pmatrix}.$$
and you want the diagonals to be 1, i.e. $\frac{r^2 + s^2}{t^2} = 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/401310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Similar triangles question If I have a right triangle with sides $a$.$b$, and $c$ with $a$ being the hypotenuse and another right triangle with sides $d$, $e$, and $f$ with $d$ being the hypotenuse and $d$ has a length $x$ times that of $a$ with $x \in \mathbb R$, is it necessary for $e$ and $f$ to have a length $x$ ti... | EDIT: This answer is now incorrect since the OP changed his question.
This is a good way of visualizing the failure of your claim:
Imagine the point C moving along the circle from A to the north pole. This gives you a continuum of non-similar right triangles with a given hypotenuse (in this case the diameter).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove that End(R) is isomorphic to R? I tried to prove this:
$R$ as a ring is isomorphic to $End(R)$, where $R$ is considered as a left $R$-module. The map of isomorphism is
$$F:R\to End(R), \quad F(r)=fr,$$
where $fr(a)=ar$. And I define the multiplication in $End(R)$ by $(.)$, where $h.g=g\circ h$ for $g,h... | It's true that: $a\mapsto ar$ is a left module homomorphism.
If we call this map $a\mapsto ar$ by $\theta_r$, then indeed $\theta:R\to End(_RR)$.
*
*Check that it's additive.
*Check that it's multiplicative. (You will absolutely need your rule that $f\circ g=g\cdot f$. The $\cdot$ operation you have given is the mu... | {
"language": "en",
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Find what values of $n$ give $\varphi(n) = 10$
For what values of $n$ do we get $\varphi(n) = 10$?
Here, $\varphi$ is the Euler Totient Function. Now, just by looking at it, I can see that this happens when $n = 11$. Also, my friend told me that it happens when $n = 22$, but both of these are lucky guesses, or educat... | Suppose $\varphi(n)=10$. If $p \mid n$ is prime then $p-1$ divides $10$. Thus $p$ is one of $2,3,11$.
If $3 \mid n$, it does so with multiplicity $1$. But then there would exist $m \in \mathbb{N}$ such that $\varphi(m)=5$, and this quickly leads to a contradiction (e.g. note that such values are always even).
Thus $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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every integers from 1 to 121 can be written as 5 powers of 3 We have a two-pan balance and have 5 integer weights with which it is possible to weight
exactly all the weights integers from 1 to 121 Kg.The weights can be placed all on a plate but you can also put some in a dish and others with the goods to be weighted.
I... | A more general result says, given weights $w_1\le w_2 \le \dots \le w_n$, and if $S_k = \sum_{i=1}^k w_k$; $S_0 = 0$, then everything from $1$ to $S_n$ is weighable iff each of the following inequalities hold:
$$S_{k+1} \le 3S_k + 1 \text{ for } k = 0,\dots,n-1$$
Note that this is equivalent to $w_k \le 2 S_k +1$ for e... | {
"language": "en",
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Calculate the limit of two interrelated sequences? I'm given two sequences:
$$a_{n+1}=\frac{1+a_n+a_nb_n}{b_n},b_{n+1}=\frac{1+b_n+a_nb_n}{a_n}$$
as well as an initial condition $a_1=1$, $b_1=2$, and am told to find: $\displaystyle \lim_{n\to\infty}{a_n}$.
Given that I'm not even sure how to approach this problem, I tr... | The answer is $a_n \to 5$ , $b_n \to \infty$.
I'm trying to prove that and I will edit this post if I figure out something.
EDIT:
I would write all this in comment instead in answer, but I cannot find how to do it.. maybe I need to have more reputation to do this (low reputation = low privileges:P)
Anyway, I still didn... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 2
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For any arrangment of numbers 1 to 10 in a circle, there will always exist a pair of 3 adjacent numbers in the circle that sum up to 17 or more I set out to solve the following question using the pigeonhole principle
Regardless of how one arranges numbers $1$ to $10$ in a circle, there will always exist a pair of thre... | We will show something stronger, namely that there exists 3 adjacent numbers that sum to 18 or more.
Let the integers be $\{a_i\}_{i=1}^{10}$. WLOG, $a_1 = 1$. Consider
$$a_2 + a_3 + a_4, a_5 + a_6 + a_7, a_8 + a_9 + a_{10}$$
The sum of these 3 numbers is $2+3 +\ldots + 10 = 54$. Hence, by the pigeonhole principle, the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/401753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the determinant of $2A+A^{-1}-I$ given the eigenvalues of $A$
Let $A$ be a $2\times 2$ matrix whose eigenvalues are $1$ and $-1$. Find the determinant of $S=2A+A^{-1}-I$.
Here I don't know how to find $A$ if eigenvectors are not given. If eigenvectors are given, then I can find using $A=PDP^{-1}$. Please solv... | \begin{pmatrix}
0 &-1 \\
-1&0
\end{pmatrix}
the Characteristic polynomial is $(x-1)(x+1)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/401821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
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Counterexample to inverse Leibniz alternating series test The alternating series test is a sufficient condition for the convergence of a numerical series. I am searching for a counterexample for its inverse: i.e. a series (alternating, of course) which converges, but for which the hypothesis of the theorem are false.
I... | If you want a conditionally convergent series in which the signs alternate, but we do not have monotonicity, look at
$$\frac{1}{2}-1+\frac{1}{4}-\frac{1}{3}+\frac{1}{6}-\frac{1}{5}+\frac{1}{8}-\frac{1}{7}+\cdots.$$
It is not hard to show that this converges to the same number as its more familiar sister.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Listing subgroups of a group I made a program to list all the subgroups of any group and I came up with satisfactory result for $\operatorname{Symmetric Group}[3]$ as
$\left\{\{\text{Cycles}[\{\}]\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left(
\begin{array}{cc}
1 & 2 \\
\end{array}
\right)\right]\right\},\left... | I have the impression that you only list the cyclic subgroups.
For $S_3$, the full group $S_3$ ist missing as a subgroup (you are mentioning that in your question).
For $S_4$, several subgroups are missing. In total, there should be $30$ of them. $14$ of them are cyclic, which are exactly the ones you listed. To give y... | {
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"timestamp": "2023-03-29T00:00:00",
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Does a closed walk necessarily contain a cycle? [HOMEWORK]
I asked my professor and he said that a counter example would be two nodes, by which the pathw ould go from one node and back. this would be a closed path but does not contain a cycle. But I am confused after looking at this again. Why is this not a cycle? Need... | I guess the answer depends on the exact definition of cycle. If it is as you wrote in your comment - a closed walk that starts and ends in the same vertex, and no vertex repeats on the walk (except for the start and end), then your example with two nodes is a cycle.
However, a definition of a cycle usually contains a c... | {
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"timestamp": "2023-03-29T00:00:00",
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Calculate the limit: $\lim_{n\to+\infty}\sum_{k=1}^n\frac{\sin{\frac{k\pi}n}}{n}$ Using definite integral between the interval $[0,1]$. Calculate the limit: $\lim_{n\to+\infty}\sum_{k=1}^n\frac{\sin{\frac{k\pi}n}}{n}$ Using definite integral between the interval $[0,1]$.
It seems to me like a Riemann integral definitio... | I think it should be
$$
\int_{0}^{1}\sin \pi xdx=\frac{2}{\pi}
$$
EDIT OK, the point here is the direct implementation of Riemanns sums with $d x = \frac{b-a}{n}, \ b=1, \ a=0$ and $\frac{k}{n}=x$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/402114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Show that $\langle(1234),(12)\rangle = S_4$ I am trying to show that $\langle(1234),(12)\rangle = S_4$. I can multiply and get all $24$ permutations manually but isn't there a more compact solution?
| Write $H$ for the subgroup generated by those two permutations. Then $(1234)(12)=(234)$, so $H$ contains certainly the elements of $\langle(1234)\rangle$, $\langle (234)\rangle$ and $(12)$, hence $\vert H\vert \geq 7$ and therefore $\vert H\vert \geq 8$ ($\vert H\vert$ must divide $24=\vert S_{4}\vert$). Since $(1234)\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/402180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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A property of uniform spaces Is it true that $E\circ E\subseteq E$ for every entourage $E$ of every uniform space?
| As noted in the comments, it clearly is not true in general.
If an entourage $E$ has the property that $E\circ E\subseteq E$, then $E$ is a transitive, reflexive relation on $X$, and the entourage $E\cap E^{-1}$ is an equivalence relation on $X$. A uniform space whose uniformity has a base of equivalence relations is n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/402231",
"timestamp": "2023-03-29T00:00:00",
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Understanding the differential $dx$ when doing $u$-substitution I just finished taking my first year of calculus in college and I passed with an A. I don't think, however, that I ever really understood the entire $\frac{dy}{dx}$ notation (so I just focused on using $u'$), and now that I'm going to be starting calculus... | $dx$ is what is known as a differential. It is an infinitesimally small interval of $x$:
$$dx=\lim_{x\to{x_0}}x-x_0$$
Using this definition, it is clear from the definition of the derivative why $\frac{dy}{dx}$ is the derivative of $y$ with respect to $x$:
$$y'=f'(x)=\frac{dy}{dx}=\lim_{x\to{x_0}}\frac{f(x-x_0)-f(x_0)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/402303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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Are there nonlinear operators that have the group property? To be clear: What I am actually talking about is a nonlinear operator on a finitely generated vector space V with dimension $d(V)\;\in \mathbb{N}>1$. I can think of several nonlinear operators on such a vector space but none of them have the requisite propert... | How about the following construction. Let $M \in R_{d}(G)$ be a $d$ dimensional representation of some Lie group (e.g. $SL(2,R)$ and $V$ is some $d$ dimensional vector space. Now define:
$$
f(x,M,\epsilon)= \frac{M x}{(1+\epsilon w.x)}
$$
where the vector $w$ is chosen so that $w.M= w$ for all $M\in SL(2,\mathbb{R})$. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can I prove this closed form for $\sum_{n=1}^\infty\frac{(4n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^n}$ How can I prove the following conjectured identity?
$$\mathcal{S}=\sum_{n=1}^\infty\frac{(4\,n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^n}\s... | According to Mathematica, the sum is
$$ \frac{3}{\Gamma(\frac13)\Gamma(\frac23)}\left( -1 + {}_3F_2\left(\frac14,\frac12,\frac34; \frac23,\frac43; -1\right) \right). $$
This form is actually quite straightforward if you write out $(4n)!$ as
$$ 4^{4n}n!(1/4)_n (1/2)_n (3/4)_n $$
using rising powers ("Pochhammer symbols... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/402432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "51",
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Intuition of Addition Formula for Sine and Cosine The proof of two angles for sine function is derived using $$\sin(A+B)=\sin A\cos B+\sin B\cos A$$ and $$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ for cosine function. I know how to derive both of the proofs using acute angles which can be seen here http://en.wikibooks.org/... | The bottom triangle is the right triangle used to compute sine and cosine of $\alpha$. The upper triangle is the right triangle used to compute sine and cosine of $\beta$, scaled and rotated so its base is the same as the hypotenuse of the lower triangle.
We know the ratios of the sides of these triangles because of th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
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$f$ and $g$ are holomorphic function , $A=\{z:{1\over 2}<\lvert z\rvert<1\}$, $D=\{z: \lvert z-2\rvert<1\}$ $f$ and $g$ are holomorphic function defined on $A\cup D$
$A=\{z:{1\over 2}<\lvert z\rvert<1\}$, $D=\{z:\lvert z-2\rvert <1\}$
*
*If $f(z)g(z)=0\forall z\in A\cup D $ then either $f(z)=0\forall z\in A$ or $g... | You can use the identity theorem, but since that theorem doesn't directly apply to products, you should explain how you use the identity theorem.
You can, for example, argue that if $f\cdot g = 0$ on some connected open set $S$, then there are $S_1,S_2$ with $f = 0$ on $S_1$, $g = 0$ on $S_2$, $S_1 \cap S_2 = \emptyset... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Exponential Growth, half-life time An exponential growth function has at time $t = 5$
a) the growth factor (I guess that is just the "$\lambda$") of $0.125$ - what is the half life time?
b) A growth factor of $64$ - what is the doubling time ("Verdopplungsfaktor")?
For a), as far as I know the half life time is $\displ... | The growth factor tells you the relative growth between $f(x)$ and $f(x+1)$, i.e. it's $$
\frac{f(t+1)}{f(t)} \text{.}
$$
If $f$ grows exactly exponentially, i.e. if $$
f(t) = \lambda\alpha^t = \lambda e^{\beta t} \quad\text{($\beta = \ln \alpha$ respectively $\alpha = e^\beta$)} \text{,}
$$
then $$
\frac{f(t+1)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/402706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Range of: $\sqrt{\sin(\cos x)}+\sqrt{\cos(\sin x)}$ Range of:
$$\sqrt{\sin(\cos x)}+\sqrt{\cos(\sin x)}$$
Any help will be appreciated.
| $$u=\cos(x); v=\sin(x)$$
$$\sqrt{\sin(u)}+\sqrt{\cos(v)}$$
$\sin(u)$ must be greater than or equal to 0, and $\cos(v)$ must be greater than or equal to 0:
$$u\in{[2\pi{k},\pi+2\pi{k}],k\in\mathbb{Z}}$$
$$v\in{[-\frac{\pi}{2}+2\pi{k},\frac{\pi}{2}+2\pi{k}]},k\in\mathbb{Z}$$
Since $v=\sin{x}$, $v\in{[-1,1]}$, $v$ will al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/402797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
a real convergent sequence has a unique limit point How to show that a real convergent sequence has a unique limit point viz. the limit of the sequene?
I've used the result several times but I don't know how to prove it!
Please help me!
| I am assuming that limit points are defined as in Section $6.4$ of the book Analysis $1$ by the author Terence Tao. We assume that the sequence of real numbers $(a_{n})_{n=m}^{\infty}$ converges to the real number $c$. Then we have to show that $c$ is the unique limit point of the sequence. First, we shall show that $c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/402847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Pointwise supremum of a convex function collection In Hoang Tuy, Convex Analysis and Global Optimization, Kluwer, pag. 46, I read:
"A positive combination of finitely many proper convex functions on $R^n$ is convex. The upper envelope (pointwise supremum) of an arbitrary family of convex functions is convex".
In order... | I think it is either assumed that the $f_i$ are defined on the same domain $D$, or that (following a common convention) we set $f_i(x)=+\infty$ if $x \notin \mathrm{Dom}(f_i)$. You can easily check that under this convention, the extended $f_i$ still remain convex and the claim is true.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/402919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Solving $\sqrt{3x^2-2}+\sqrt[3]{x^2-1}= 3x-2 $ How can I solve the equation $$\sqrt{3x^2-2}+\sqrt[3]{x^2-1}= 3x-2$$
I know that it has two roots: $x=1$ and $x=3$.
| Substituting $x = \sqrt{t^3+1}$ and twice squaring, we arrive to the equation
$$ 36t^6-24t^5-95t^4+8t^3+4t^2-48t=0.$$ Its real roots are $t=0$ and $t=2$ (the latter root is found in the form $\pm \text{divisor}(48)/\text{divisor}(36)$), therefore $x=1$ and $x=3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/402965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Inverse Laplace transformation using reside method of transfer function that contains time delay I'm having a problem trying to inverse laplace transform the following equation
$$
h_0 = K_p * \frac{1 - T s}{1 + T s} e ^ { - \tau s}
$$
I've tried to solve this equation using the residue method and got the following.
$... | First do polynomial division to simplify the fraction:
$$\frac{1-Ts}{1+Ts}=-1+\frac{2}{1+Ts}$$
Now expand $h_0$:
$$h_0=-K_pe^{-\tau{s}}+2K_p\frac{1}{Ts+1}e^{-\tau{s}}$$
Recall the time-domain shift property:
$$\scr{L}(f(t-\tau))=f(s)e^{-\tau{s}}$$
$$\scr{L}^{-1}h_0=-k_p\delta{(t-\tau)}+2k_p g(t-\tau)$$
Where $g(t)=\scr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/403042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that $n \ge \sqrt{n+1}+\sqrt{n}$ (how) Can I show that:
$n \ge \sqrt{n+1}+\sqrt{n}$ ?
It should be true for all $n \ge 5$.
Tried it via induction:
*
*$n=5$: $5 \ge \sqrt{5} + \sqrt{6} $ is true.
*$n\implies n+1$:
I need to show that $n+1 \ge \sqrt{n+1} + \sqrt{n+2}$
Starting with $n+1 \ge \sqrt{n} + \sqrt{n+1... | Hint: $\sqrt{n} + \sqrt{n+1} \leq 2\sqrt{n+1}$. Can you take it from there?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/403090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 6,
"answer_id": 3
} |
Integral over a ball Let $a=(1,2)\in\mathbb{R}^{2}$ and $B(a,3)$ denote a ball in $\mathbb{R}^{2}$ centered at $a$ and of radius equal to $3$.
Evaluate the following integral:
$$\int_{B(a,3)}y^{3}-3x^{2}y \ dx dy$$
Should I use polar coordinates? Or is there any tricky solution to this?
| Note that $\Delta(y^3-3x^2y)=0$, so that $y^3-3x^2y$ is harmonic. By the mean value property, we get that the mean value over the ball is the value at the center. Since the area of the ball is $9\pi$ and the value at the center is $2$, we get
$$
\int_{B(a,3)}\left(y^3-3x^2y\right)\mathrm{d}x\,\mathrm{d}y=18\pi
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/403162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Prove $\sin \left(1/n\right)$ tends to $0$ as $n$ tends to infinity. I'm sure there is an easy solution to this but my mind has gone blank! Any help on proving that $\sin( \frac{1}{n})\longrightarrow0$ as $n\longrightarrow\infty$ would be much appreciated.
This question was set on a course before continuity was introdu... | Hint
$$\quad0 \leq\sin\left(\frac{1}{n}\right)\leq \frac{1}{n}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/403210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
A problem from distribution theory. Let $f$, $g\in C(\Omega)$, and suppose that $f \neq g$ in $C(\Omega)$. How can we prove that $f \neq g$ as distributions?
Here's the idea of my proof.
$f$ and $g$ are continuous functions, so they will be locally integrable.
Now, take any $\phi \neq 0 \in D(\Omega)$. Let us suppose t... | Let me add a little more detail to my comment.
You are correct up to the $\int_{\Omega} \phi(x)[f(x)-g(x)]dx = 0$.
Now, for $\varepsilon > 0$ and $x_0 \in \Omega$, let $B_{\varepsilon}(x_0)$ denote the ball of radius $\varepsilon$ centered at $x_0$. Then, you can find a test function $\psi$ with the following propertie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/403298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Prove if $n^2$ is even, then $n$ is even. I am just learning maths, and would like someone to verify my proof.
Suppose $n$ is an integer, and that $n^2$ is even. If we add $n$ to $n^2$, we have $n^2 + n = n(n+1)$, and it follows that $n(n+1)$ is even. Since $n^2$ is even, $n$ is even.
Is this valid?
| We know that,
*
*$n^2=n\times n $, We also know that
*even $\times$ even = even
*odd $\times$ odd = odd
*odd $\times$ even = even
Observation $1$: As $n^2$ is even, we also get an even result in the 2nd and 4th case .
Observation $2$: In the expression "$n \times n$" both operands are same i.e. '$n$', ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/403346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "68",
"answer_count": 15,
"answer_id": 1
} |
Is There A Function Of Constant Area? If I take a point $(x,y)$ and multiply the coordinates $x\times y$ to find the area $(A)$ defined by the rectangle formed with the axes, then is there a function $f(x)$ so that $xy = A$, regardless of what value of $x$ is chosen?
| Take any desired real value $A$, then from $xy = A$, define $$f: \mathbb R\setminus \{0\} \to \mathbb R,\quad f(x) = y = \dfrac Ax$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/403401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
What is the answer to this limit what is the limit value of the power series:
$$ \lim_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \frac{x^k}{k^{k-m}}$$
where $m>1$.
| For $\lim_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \frac{x^k}{k^{k-m}}$,
the ratio of consecutive terms is
$\begin{align}
\frac{x^{k+1}}{(k+1)^{k+1-m}}\big/\frac{x^k}{k^{k-m}}
&=\frac{x k^{k-m}}{(k+1)^{k+1-m}}\\
&=\frac{x }{k(1+1/k)^{k+1-m}}\\
&=\frac{x }{k(1+1/k)^{k+1}(1+1/k)^{-m}}\\
&\approx\frac{x }{ke(1+1/k)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/403478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
When a quotient of a UFD is also a UFD?
Let $R$ be a UFD and let $a\in R$ be nonzero element. Under what conditions will $R/aR$ be a UFD?
A more specific question:
Suppose $R$ is a regular local ring and let $I$ be a height two ideal which is radical. Can we find an element $a\in I$ such that $R/aR$ is a UFD?
| This is indeed a complicated question, that has also been much studied. Let me just more or less quote directly from Eisenbud's Commutative Algebra book (all found in Exercise 20.17):
The Noether-Lefschetz theorem: if $R = \mathbb{C}[x_1, \ldots x_4]$ is the polynomial ring in $4$ variables over $\mathbb{C}$, then for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/403565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Injectivity of a map between manifolds I'm learning the concepts of immersions at the moment. However I'm a bit confused when they define an immersion as a function $f: X\rightarrow Y$ where $X$ and $Y$ are manifolds with dim$X <$ dim$Y$ such that $df_x: T_x(X)\rightarrow T_y(Y)$ is injective.
I was wondering why don't... | I think a glimpse on the wikipedia's article helps. Immersions usually are not injective because the image can appear "knotted" in the target space.
I do not think you need $\dim X<\dim Y$ in general. You can define it for $\dim X=\dim Y$, it is only because we need $df_{p}$ to have rank equal to $\dim X$ that made yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/403635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Does $x^T(M+M^T)x \geq 0 \implies x^TMx \geq 0$ hold in only one direction? I know this is true for the "if" part, but what about the "only if"?
Can you give me one example when the "only if" part does not hold? I am not quite sure about this.
I forgot to tell you that $M$ is real and $x$ arbitrary.
| Well
$x^{T}Mx \geq 0 \implies x^{T}M^{T}x \geq 0$ (by taking transpose)
Hence, $x^{T}(M^{T} + M)x \geq 0$.
So, this gives you the one side. However, you wrote the converse in your title.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/403695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
First moment inequality implies tail distribution inequality? Let $U,V$ be two continuous random variables, both with continuous CDF. Suppose that $\mathbb E V \geq \mathbb E U$. Can one conclude that $\mathbb P(V> x) \geq \mathbb P(U>x)$ for all $x\geq 0$? If not, what additional conditions are needed?
| Here is an argument with may make think it's not true: if $Y$ is a positive random variable, then $E(X)=\int_0^{+\infty}P(X>t)dt$. The fact that $P(U>x)\geqslant P(V>x)$ seems much stronger than $E(U)\geqslant E(V)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/403779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that certain elements are not in some ideal I have the following question:
Is there a simple way to prove that
$x+1 \notin \langle2, x^2+1\rangle_{\mathbb{Z}[x]}$ and
$x-1 \notin \langle2, x^2+1\rangle_{\mathbb{Z}[x]}$
without using the fact that $\mathbb{Z}[x]/\langle2, x^2+1\rangle$ is an integral domain?
Thank... | First a comment: it looks like the goal of the problem is to show that $(2,x^2+1)$ isn't a prime ideal (and hence the quotient isn't a domain), because $(x+1)^2=x^2+1+2x$ is in the ideal, and we hope to show that $x+1$ isn't in the ideal.
If $x+1\in (2,x^2+1)$, then we would be able to find two polynomimals in $\Bbb Z[... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/403858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Evaluating $\int_{0}^{1} \frac{\ln^{n} x}{(1-x)^{m}} \, \mathrm dx$ On another site, someone asked about proving that
$$ \int_{0}^{1} \frac{\ln^{n}x}{(1-x)^{m}} \, dx = (-1)^{n+m-1} \frac{n!}{(m-1)!} \sum_{j=1}^{m-1} (-1)^{j} s (m-1,j) \zeta(n+1-j), \tag{1} $$
where $n, m \in \mathbb{N}$, $n \ge m$, $m \ge 2$, and $s(m... | Substitute $x=e^{-t}$ and get that the integral is equal to
$$(-1)^n \int_0^{\infty} dt \, e^{-t} \frac{t^n}{(1-e^{-t})^m} $$
Now use the expansion
$$(1-y)^{-m} = \sum_{k=0}^{\infty} \binom{m+k-1}{k} y^k$$
and reverse the order of summation and integration to get
$$\sum_{k=0}^{\infty} \binom{m+k-1}{k} \int_0^{\infty} d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/403904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
The integral over a subset is smaller? In a previous question I had $A \subset \bigcup_{k=1}^\infty R_k$ where $R_k$ in $\Bbb{R}^n$ are rectangles I then proceeded to use the following inequality $\left|\int_A f\right| \le \left|\int_{\bigcup_{k=1}^\infty R_k} f \right|$ which I am not really certain of. Does anyone kn... | The inequality is not true in general (think of an $f$ that is positive on $A$ but such that it is negative outside $A$). But it does hold if $f\geq 0$. This is not an obstacle to you using it, because you would just have to split your function in its positive and negative part.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/403994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Normal form of a vector field in $\mathbb {R}^2$ Edited after considering the comments
Problem: What is the normal form of the vector field:
$$\dot x_1=x_1+x_2^2$$
$$\dot x_2=2x_2+x_1^2$$
Solution: The eugine values of the matrix of the linearised around $(0,0)$ system are $2$ and $1$. We, therefore, have the only reso... | I would use $y$ instead of $x$ in the normal form, since these are not the same variables. Otherwise, what you did is correct. (I don't know if the problem required the identification of a transformation between $x$ and $y$.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/404062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Proof that $1/\sqrt{x}$ is itself its sine and cosine transform As far as I understand, I have to calculate integrals
$$\int_{0}^{\infty} \frac{1}{\sqrt{x}}\cos \omega x \operatorname{d}\!x$$
and
$$\int_{0}^{\infty} \frac{1}{\sqrt{x}}\sin \omega x \operatorname{d}\!x$$
Am I right? If yes, could you please help me to i... | Let $$I_1(\omega)=\int_0^\infty \frac{1}{\sqrt{x}}\cdot \cos (\omega\cdot x)\space dx,$$
and $$I_2(\omega)=\int_0^\infty \frac{1}{\sqrt{x}}\cdot\sin (\omega\cdot x)\space dx.$$
Let $x=t^2/\omega$ such that $dx=2t/\omega\space dt$, where $t\in [0,\infty)$. It follows that
$$I_1(\omega)=\frac{2}{\sqrt{\omega}}\cdot\int_0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/404119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
What makes $5$ and $6$ so special that taking powers doesn't change the last digit? Why are $5$ and $6$ (and numbers ending with these respective last digits) the only (nonzero, non-one) numbers such that no matter what (integer) power you raise them to, the last digit stays the same? (by the way please avoid modular a... | It's because, let x be any number, if x^2 ends with x, then x raised to any positive integer(excluding zero) will end with x.
For x^2 to end with x, x(x-1) have to be a multiple of 10 raised to the number of digits in x.
(Ex: if x = 5, then 10^1. If x = 25, then 10^2)
By following this procedure, I have come up with... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/404161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 8,
"answer_id": 5
} |
Hopf-Rinow Theorem for Riemannian Manifolds with Boundary I am a little rusty on my Riemannian geometry. In addressing a problem in PDE's I came across a situation that I cannot reconcile with the Hopf-Rinow Theorem. If $\Omega \subset \mathbb{R}^n$ is a bounded, open set with smooth boundary, then $\mathbb{R}^n - \Ome... | Hopf-Rinow concerns, indeed, Riemannian manifolds with no boundary.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/404223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
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Prove that $V$ is the direct sum of $W_1, W_2 ,\dots , W_k$ if and only if $\dim(V) = \sum_{i=1}^k \dim W_i$ Let $W_1,\dots,W_k$ be a subspace of a finite dimensional vector space $V$ such that the $\sum_{i=1}^k W_i = V$.
Prove: $V$ is the direct sum of $W_1, W_2 , \dots, W_k$ if and only $\dim(V) = \sum_{i=1}^k \dim ... | Define $\ \ B_i$ a basis of $ \ \ W_i\ \ $ for $\ \ i \in {1,..,k } $. Since sum( $W_i$)$= V$ we know that $\bigcup_{i=1}^k B_i$ is a spanning list of $V$.
Now we add the condition : sum dim($B_i$)$=$sum dim($W_i$)$=$dim($V$).
This means that $\bigcup_{i=1}^k B_i$ is a basis. (Spanning list of $V$ with same dimensio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/404290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What does it mean when one says "$B$ has no limit point in $A$"? If $B$ is a subset of a set $A$, what does the sentence "$B$ has no limit points in $A$" mean? I am aware that $x$ is a limit point of $A$, if for every neighbourhood $U$ of $x$, $(U-\{x\})\cap A$ is non-empty.
Please let me know. Thank you.
| Giving an example may be helpful for you.
Example: Let $X=\mathbb R$ with usual topology. $A=\{x\in \mathbb R: x > 0\} \subseteq X$ and $B=\{1, \frac12, \frac13,... \frac1n,...\} \subset A$. It is not difficult to see that $B$ has no limit points in $A$ since the unique limit point 0 of $B$ is not in $A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/404364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Proving that ${n}\choose{k}$ $=$ ${n}\choose{n-k}$ I'm reading Lang's Undergraduate Analysis:
Let ${n}\choose{k}$ denote the binomial coefficient,
$${n\choose k}=\frac{n!}{k!(n-k)!}$$
where $n,k$ are integers $\geq0,0\leq k\leq n$, and $0!$ is defined to be $1$. Prove the following assertion:
$${n\choose k}={n\choose ... | $$
\binom{n}{n-k}=\frac{n!}{(n-k)!(n-n-k)!}=\frac{n!}{(n-k)!k!}=\binom{n}{k}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/404417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 4
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Onto and one-to-one Let $T$ be a linear operator on a finite dimensional inner product space $V$. If $T$ has an eigenvector, then so does $T^*$.
Proof. Suppose that $v$ is an eigenvector of $T$ with corresponding eigenvalue $\lambda$. Then for any $x \in V$,
$$
0 = \langle0,x\rangle = \langle(T-\lambda I)v,x\rangle
=... | If you want to know why $T^{*}$ is not onto if $T$ is not onto, just observe that the matrix of $T^{*}$ is just the conjugate transpose of the matrix of $T$ and we know that a matrix and its conjugate transpose have the same rank (which is equal to the dimension of the range space).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/404471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Find an integrable $g(x,y) \ge |e^{-xy}\sin x|$ I want to use Fubini theorem on $$\int_0^{A} \int_0^{\infty} e^{-xy}\sin x dy dx=\int_0^{\infty} \int_0^{A}e^{-xy}\sin x dx dy$$
Must verify that $\int_M |f|d(\mu \times \nu) < \infty$. I'm using the Lebesgue theorem, so far I've come up with $g(x,y)=e^{-y}$ but am not su... | Try $g(x,y)=x\mathrm e^{-xy}$, then $|\mathrm e^{-xy}\sin x|\leqslant g(x,y)$ for every nonnegative $x$ and $y$. Furthermore, $\int\limits_0^\infty g(x,y)\mathrm dy=1$ for every $x\gt0$ hence $\int\limits_0^A\int\limits_0^\infty g(x,y)\mathrm dy\mathrm dx=A$, which is finite.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Product of two functions converging in $L^1(X,\mu)$
Let $f_n\to f$ in $L^1(X,\mu)$, $\mu(X)<\infty$, and let $\{g_n\}$ be a sequence of measurable functions such that $|g_n|\le M<\infty\ \forall n$ with some constant $M$, and $g_n\to g$ almost everywhere. Prove that $g_nf_n\to gf$ in $L^1(X,\mu)$.
This is a question ... | Hint: Observe that $2M|f|$ is an integrable bound for $|g_n - g|\cdot |f|$ and the latter converges a. e. to $0$. Now apply the bounded convergence theorem.
| {
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Help $\lim_{k \to \infty} \frac{1-e^{-kt}}{k}=? $ what is the lime of $ \frac{1-e^{-kt}}{k}$ as $k \to \infty$?
Is that just equal $\frac{1}{\infty}=0$?
Does any one can help, I am not sure if We can apply L'Hopital's rule. S
| HINT:
If $t=0,e^{-kt}=1$
If $t>0, \lim_{k\to\infty}e^{-kt}=0$
If $t<0, t=-r^2$(say), $\lim_{k\to\infty}\frac{1-e^{-kt}}k=\lim_{k\to\infty}\frac{1-e^{kr^2}}k=\frac\infty\infty $
So, applying L'Hospitals' rule, $\lim_{k\to\infty}\frac{1-e^{kr^2}}k=-r^2\cdot\lim_{k\to\infty}\frac{e^{r^2t}}1=-r^2\cdot\infty=-\infty$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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definition of discriminant and traces of number field. Let $K=\Bbb Q [x]$ be a number field, $A$ be the ring of integers of $K$.
Let $(x_1,\cdots,x_n)\in A^n$. In usual, what does it mean $D(x_1,\cdots,x_n)$? Either $\det(Tr_{\Bbb K/ \Bbb Q} (x_ix_j))$ or $\det(Tr_{A/ \Bbb Z} (x_ix_j))$? Or does it always same value? I... | I'm not entirely certain what $\operatorname{Tr}_{A/\mathbb{Z}}$ is, but the notation $D(x_1,\dots,x_n)$ or $\Delta(x_1,\dots,x_n)$ usually means the discriminant of $K$ with respect to the basis $x_1,\dots,x_n$, so I would say it's most likely the former.
After all, $x_1,\dots,x_n\in A\subset K$, so it still makes sen... | {
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If $x\equiv y \pmod{\gcd(a,b)}$, show that there is a unique $z\pmod{\text{lcm}(a,b)}$ with $z\equiv x\pmod a$ and $z\equiv y\pmod b$
If $x\equiv y \pmod{\gcd(a,b)}$, show that there is a unique $z\pmod{\text{lcm}(a,b)}$ with $z\equiv x\pmod{a}$ and $z\equiv y\pmod{b}$
What I have so far:
Let $z \equiv x\pmod{\frac{a... | Put $d=\gcd(a,b)$ and $\delta=x\bmod d=y\bmod d$ (here "mod" is the remainder operation). Then the numbers $x'=x-\delta$, $y'=y-\delta$ are both divisible by$~d$. In terms of a new variable $z'=z-\delta$ we need to solve the system
$$
\begin{align}z'&\equiv x'\pmod a,\\z'&\equiv y'\pmod b.\end{align}
$$
Since $x',y',... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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find a group with the property : a)find a nontrivial group $G$ such that $G$ is isomorphic to $G \times G $
what i'm sure is that $G$ must be infinite ! but have now idea how to get or construct such group
i chose many $G$'s but all of the homomorphism was not injective
b) an infinite group in which every element ha... | For the second problem, you can use the subgroup of $\mathbb{Z}_1\times \mathbb{Z}_2\times \cdots\times \mathbb{Z}_n \times \cdot$ consisting of all sequences $(a_1,a_2,a_3,\dots)$ such that all but finitely many of the $a_i$ are $0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Determine the matrix of a quadratic function I'm given a quadratic form $\Phi:\mathbb{R}^3\longrightarrow\mathbb{R}$, for which we know that:
*
*$(0,1,0)$ and $(0,1,-1)$ are conjugated by the function
*$(1,0,-1)$ belongs to the kernel
*$\Phi(0,0,1)=1$
*The trace is $0$
From here, I know the matrix must be symmetr... | Ok, solved it, the last two equations came from knowing the vector that was in the kernel, so it should be that $f_p[(1,-0,-1),(x,y,z)]=0$, being $f_p$ the polar form of $\Phi$: $f_p=\frac{1}{2}[\Phi(x+y)-\Phi(x)-\Phi(y)]$
| {
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Interesting Problems for NonMath Majors Sometime in the upcoming future, I will be doing a presentation as a college alumni to a bunch of undergrads from an organization I was in college. I did a dual major in mathematics and computer science, however the audience that I am presenting to are not necessarily people who ... | I think geometry is the area the most attractive that a "non-mathematician" can enjoy, and I believe that's the idea Serge Lang has when he prepared his encounters with high school students and in his public dialogues, I refer her to this tow reports of These events :
The Beauty of Doing Mathematics : Three Public Dial... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can I prove $2\sup(S) = \sup (2S)$?
Let $S$ be a nonempty bounded subset of $\mathbb{R}$ and $T = \{2s : s \in S \}$. Show $\sup T = 2\sup S$
Proof
Consider $2s = s + s \leq \sup S + \sup S = 2\sup S $. $T \subset S$ where T is also bounded, so applying the lub property, we must have $\sup T \leq 2 \sup S$.
On ... | You can't assume $s$ is positive, nor can you assume $\sup S$ is positive.
Your proof also assumes a couple of other weird things:
*
*$T \subset S$ is usually not true.
*$2s + s - s \le \sup T + \sup S - 3\sup S$ is not necessarily true. Why would $-s \le -3\sup S$?
The first part of your proof is actually corre... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to calculate: $\sum_{n=1}^{\infty} n a^n$ I've tried to calculate this sum:
$$\sum_{n=1}^{\infty} n a^n$$
The point of this is to try to work out the "mean" term in an exponentially decaying average.
I've done the following:
$$\text{let }x = \sum_{n=1}^{\infty} n a^n$$
$$x = a + a \sum_{n=1}^{\infty} (n+1) a^n$$
$$... | We give a mean proof, at least for the case $0\lt a\lt 1$. Suppose that we toss a coin that has probability $a$ of landing heads, and probability $1-a$ of landing heads. Let $X$ be the number of tosses until the first tail. Then $X=1$ with probability $1-a$, $X=2$ with probability $a(1-a)$, $X=3$ with probability $a^2(... | {
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"source": "stackexchange",
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Proof identity of differential equation I would appreciate if somebody could help me with the following problem:
Q: $f''(x)$ continuous in $\mathbb{R}$ show that
$$ \lim_{h\to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2}=f''(x)$$
| Use L'Hospital's rule
$$ \lim_{h\to 0}\frac{F(h)}{G(h)}=\lim_{h\to 0}\frac{F'(h)}{G'(h)}$$
You can use the rule if you have $\frac{0}{0}$ result.
You need to apply it 2 times
$$ \lim_{h\to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2}= \lim_{h\to 0}\frac{f'(x+h)-f'(x-h)}{2h}=\lim_{h\to 0}\frac{f''(x+h)+f''(x-h)}{2}=f''(x)$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the polynomial $f(x)$ which have the following property Find the polynomial $p(x)=x^2+px+q$ for which $\max\{\:|p(x)|\::\:x\in[-1,1]\:\}$ is minimal.
This is the 2nd exercise from a test I gave, and I didn't know how to resolve it. Any good explanations will be appreciated. Thanks!
| Here's an informal argument that doesn't use calculus. Notice that $p(x)$ is congruent to $y = x^2$ (for example, simply complete the square). Now suppose that we chose our values for the coefficients $p,q$ carefully, and it resulted in producing the minimal value of $m$. Hence, we can think of the problem instead like... | {
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$y^2 - x^3$ not an embedded submanifold How can I show that the cuspidal cubic $y^2 = x^3$ is not an embedded submanifold of $\Bbb{R}^2$? By embedded submanifold I mean a topological manifold in the subspace topology equipped with a smooth structure such that the inclusion of the curve into $\Bbb{R}^2$ is a smooth embe... | It is better to view $y$ as the independent variable and $x=y^{2/3}$. Since $2/3<1$, this has infinite slope at the origin for positive $y$ and infinite negative slope for negative $y$. Hence the origin is not a smooth point of this graph, which is therefore not a submanifold.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Pigeon Hole Principle; 3 know each other, or 3 don't know each other I found another question in my text book, it seems simple, but the hardest part is to prove it. Here the question
There are six persons in a party. Prove that either 3 of them recognize each other or 3 of them don't recognize each other.
I heard the... | NOTE: in my answer I assume that the relation "know someone" is symmetric (i.e., A knows B if and only if B knows A). If this relation is not symmetric for you then, I did not really check it but I believe the statement is not true.\\\
Choose a person A at the party. The following two situations are possible:
(CASE 1) ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Sum of Random Variables... Imagine we repeat the following loop some thousands of times:
$$
\begin{align}
& \text{array} = []\\
& \text{for n} = 1: 10 000 \\
& k = 0 \\
& \text{while unifrnd}(0,1) < 0.3 \\
& k = k + 1 \\
& \text{end} \\
& \text{if k} \neq 0 \\
& \text{array} = [\text{array,k}] \\
& \text{end} \\
\end{a... | It appears you exit the loop the first time the random is greater than $0.3$. In that case, the most probable value for $k$ is $0$. It occurs with probability $0.7$. The next most probable is $1$, which occurs with probability $0.3 \cdot 0.7$, because you need the first random to be less than $0.3$ and the second to... | {
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"source": "stackexchange",
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Distribution of $pX+(1-p) Y$ We have two independent, normally distributed RV's: $$X \sim N(\mu_1,\sigma^2_1), \quad Y \sim N(\mu_2,\sigma^2_2)$$ and we're interested in the distribution of $pX+(1-p) Y, \space p \in (0,1)$.
I've tried to solve this via moment generating functions. Since $$X \perp Y \Rightarrow \Psi_X(... | Any linear combination $aX+bY$ of independent normally distributed random variables $X$ and $Y,$ where $a$ and $b$ are constants, i.e. not random, is normally distributed. You can show that by using moment-generating functions provided you have a theorem that says only normally distributed random variables can have the... | {
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Choosing a bound when it can be plus or minus? I.e. $\sqrt{4}$ My textbook glossed over how to choose integral bounds when using substitution and the value is sign-agnostic. Or I missed it!
Consider the definite integral: $$ \int_1^4\! \frac{6^{-\sqrt{x}}}{\sqrt x} dx $$
Let $ u = -\sqrt{x} $ such that $$ du = - \frac... | It is convention that $\sqrt{x} = + \sqrt{x}$. Thus, you set $u(1) = -\sqrt{1}=-1$ and $u(4) = -\sqrt{4}=-2$.
The only situation where you introduce the $\pm$ signs is when you are finding the root of a quadratic such as $y^2=x$ in which case both $y=+\sqrt{x}$ and $y=-\sqrt{x}$ satisfy the original equation.
| {
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How does one derive $O(n \log{n}) =O(n^2)$? I was studying time complexity where I found that time complexity for sorting is $O(n\log n)=O(n^2)$. Now, I am confused how they found out the right-hand value. According to this $\log n=n$. So, can anyone tell me how they got that value?
Here is the link where I found out t... | What this equation means is that the class $O(n\log n)$ is included in the class $O(n^2)$. That is, if a sequence is eventually bounded above by a constant times $n \log n$, it will eventually be bounded above by a (possibly different) constant times $n^2$. Can you prove this? The notation is somewhat surprising at fir... | {
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How do you divide a complex number with an exponent term? Ok, so basically I have this:
$$ \frac{3+4i}{5e^{-3i}} $$
So basically, I converted the numerator into polar form and then converted it to exponent form using Euler's formula, but I can have two possible solutions.
I can have $5e^{0.972i}$ (radian) and $5e^{53.1... | A complex number $z=x+iy$ can be written as $z=re^{i\theta}$, where $r=|z|=\sqrt{x^2+y^2}$ is the absolute value of $z$ and $\theta=\arg{z}=\operatorname{atan2}(y,x)$ is the angle between the $x$-axis and $z$ measured counterclockwise and in radians. In this case, we have $r=5$ and $\theta=\arctan\frac{4}{3}$ (since $x... | {
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Repeatedly assigning people to subgroups so everyone knows each other Say a teacher divides his students into subgroups once every class. The profile of subgroup sizes is the same everyday (e.g. with 28 students it might be always 8 groups of 3 and 1 group of 4). How can the teacher specify the subgroup assignments for... | The shortest number of classes is 1, with a single subgroup that contains all the students.
The second shortest number of classes is 3, which is achieved with one small subgroup and a large one. For example, ABCDEFG/HI, ABCDEHI/FG, ABCFGHI/DE.
The proof that 2 classes is impossible: let X, Y be the two largest subgrou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/406074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Category with endomorphisms only How is called a category with endomorphisms only?
How is called a subcategory got from an other category by removing all morphisms except of endomorphisms?
| Every category in which every arrow is an endomorphism is a coproduct (in the category of categories) of monoids (a monoid is a category with just one object). So a category with all morphisms endormophisms is a coproduct of monoids. I'm not aware of any specific terminology for it.
Clearly the category of all coproduc... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Invariant subspaces of a linear operator that commutes with a projection I have an assignment problem, the statement is:
Let $V$ be a vector space and $P:V \to V$ be a projection. That is, a linear operator with $P^2=P.$ We set $U:= \operatorname{im} P$ and $W:= \ker P.$ Further suppose that $T:V\to V$ is a linear ope... | Yes, that's all. $P$ doesn't have to be projection for this particular exercise.
However, we can just start out from the fact that $P$ is a projection in a solution: now it projects to subspace $U$, in the direction of $W$, and we also have $U\oplus W=V$, and $P|_U={\rm id}_U$.
Having these, an operator $T$ commutes wi... | {
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Convex homogeneous function Prove (or disprove) that any CONVEX function $f$, with the property that $\forall \alpha\ge 0, f(\alpha x) \le \alpha f(x)$, is positively homogeneous; i.e. $\forall \alpha\ge 0, f(\alpha x) = \alpha f(x)$.
| Maybe I'm missing something, but it seems to me that you don't even need convexity. Given the property you stated, we have that, for $\alpha>0$,
$$f(x)=f(\alpha^{-1}\alpha x)\leq \alpha^{-1}f(\alpha x)$$
so that $\alpha f(x)\leq f(\alpha x)$ as well. Therefore, we have that $\alpha f(x)=f(\alpha x)$ for every $\alpha>0... | {
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Show the points $u,v,w$ are not collinear Consider triples of points $u,v,w \in R^2$, which we may consider as single points $(u,v,w) \in R^6$. Show that for almost every $(u,v,w) \in R^6$, the points $u,v,w$ are not collinear.
I think I should use Sard's Theorem, simply because that is the only "almost every" statemen... | $u,v,$ and $w$ are collinear if and only if there is some $\lambda\in\mathbb{R}$ with $w=v+\lambda(v-u)$. We can thus define a smooth function
$$\begin{array}{rcl}f:\mathbb{R}^5&\longrightarrow&\mathbb{R}^6\\(u,v,\lambda)&\longmapsto&(u,v,v+\lambda(v-u))\end{array}$$
By the equivalence mentioned in the first sentence,... | {
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"timestamp": "2023-03-29T00:00:00",
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Topology of uniform convergence on elements of $\gamma$ Let $\gamma$ be a cover of space $X$ and consider $C_\gamma (X)$ of all continuous functions on $X$ with values in the discrete space $D=\{0,1\}$ endowed with the topology of uniform convergence on elements of $\gamma$. What does "topology of uniform convergence o... | In general we have a metric co-domain $(R,d)$, so we consider (continuous) functions from
$X$ to $R$, and we have a cover $\gamma$ of $X$.
A subbase for the topology of uniform convergence on elements of $\gamma$ is given by sets of the form $S(A, f, \epsilon)$, for all $f \in C(X,R)$, $A \in \gamma$, $\epsilon>0$ real... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/406403",
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Given $G = (V,E)$, a planar, connected graph with cycles, Prove: $|E| \leq \frac{s}{s-2}(|V|-2)$. $s$ is the length of smallest cycle
Given $G = (V,E)$, a planar, connected graph with cycles, where the smallest simple cycle is of length $s$. Prove: $|E| \leq \frac{s}{s-2}(|V|-2)$.
The first thing I thought about was ... | Lets use euler's theorem for this n-e+f=2 where:
n-vertices, e-edges and f-faces
Let $d_1,d_2,...,d_f$ where each $d_i$ is the number of edges in face $i$ of our grph. A cycle causes us to have a face, and according to this our smallest cycle is of size s $\Rightarrow |d_i| \geq s$
$d_1+d_2+...+d_f = 2e \Rightarrow s \... | {
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Dual space of $H^1(\Omega)$ I'm a bit confused, why do people not define $H^1(\Omega)^*$? Instead they only say that $H^{-1}(\Omega)$ is the dual of $H^1_0(\Omega).$
$H^1(\Omega)$ is a Hilbert space so it has a well-defined dual space. Can someone explain the issue with this?
| As far as I remember, one usually defines $H^{-1}(\Omega)$ to be the dual space of $H^1(\Omega)$. The reason for that is that one usually does not identify $H^1(\Omega)^*$ with $H^1(\Omega)$ (which would be possible) but instead works with a different representation. E.g. one works with the $L^2$-inner product as dua... | {
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"url": "https://math.stackexchange.com/questions/406568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can $4^x = 4^{400}+4^{400}+4^{400}+4^{400}$ have the solution $x=401$? How can $4^x = 4^{400} + 4^{400} + 4^{400} + 4^{400}$ have the solution $x = 401$?
Can someone explain to me how this works in a simple way?
| The number you're adding is being added n (number) times. Well, we can infer from this that if that number is being added n (number) times, it is multiplicating itself. Now, if a number is multiplicating itself, then we have an exponentiation!
$N_1+N_2+N_3+...+N_N = N\times N = N^2$
If you're adding a number, no matter... | {
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"answer_count": 6,
"answer_id": 5
} |
Elegant way to solve $n\log_2(n) \le 10^6$ I'm studying Tomas Cormen Algorithms book and solve tasks listed after each chapter.
I'm curious about task 1-1. that is right after Chapter #1.
The question is: what is the best way to solve: $n\lg(n) \le 10^6$, $n \in \mathbb Z$, $\lg(n) = \log_2(n)$; ?
The simplest but long... | For my money, the best way is to solve $n\log_2n=10^6$ by Newton's Method.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/406707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 2
} |
Find a maximal ideal in $\mathbb Z[x]$ that properly contains the ideal $(x-1)$ I'm trying to find a maximal ideal in ${\mathbb Z}[x]$ that properly contains the ideal $(x-1)$.
I know the relevant definitions, and that "a proper ideal $M$ in ${\mathbb Z}[x]$ is maximal iff ${\mathbb Z}[x]/M$ is a field."
I think the ma... | Hint: the primes containing $\,(x-1)\subset \Bbb Z[x]\,$ are in $1$-$1$ correspondence with the primes in $\,\Bbb Z[x]/(x-1)\cong \Bbb Z,\,$ by a basic property of quotient rings.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/406783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to find the integral of implicitly defined function? Let $a$ and $b$ be real numbers such that $ 0<a<b$. The decreasing continuous function
$y:[0,1] \to [0,1]$ is implicitly defined by the equation $y^a-y^b=x^a-x^b.$
Prove
$$\int_0^1 \frac {\ln (y)} x \, dx=- \frac {\pi^2} {3ab}.
$$
| OK, at long last, I have a solution. Thanks to @Occupy Gezi and my colleague Robert Varley for getting me on the right track. As @Occupy Gezi noted, some care is required to work with convergent integrals.
Consider the curve $x^a-x^b=y^a-y^b$ (with $y(0)=1$ and $y(1)=0$). We want to exploit the symmetry of the curve a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/406847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 2,
"answer_id": 1
} |
Evaluating the series $\sum\limits_{n=1}^\infty \frac1{4n^2+2n}$ How do we evaluate the following series:
$$\sum_{n=1}^\infty \frac1{4n^2+2n}$$
I know that it converges by the comparison test. Wolfram Alpha gives the answer $1 - \ln(2)$, but I cannot see how to get it. The Taylor series of logarithm is nowhere near thi... | Hint: $\frac 1{4n^2+2n} = \frac 1{2n}-\frac 1{2n+1}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/406918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 6,
"answer_id": 1
} |
Self-Paced Graduate Math Courses for Independent Study Does anyone know of any graduate math courses that are self-paced, for independent study?
I am a high school math teacher at a charter school in Texas. While I am quite happy with where I am right now, but my goal is to earn at least 18 graduate credits on math sub... | For a decent selection of grad courses and to whet your appetite, ocw.mit.edu : MIT open course ware.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/406960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 5,
"answer_id": 4
} |
A (not necessarily continuous) function on a compact metric space attaining its maximum. I am studying for an exam and my study partners and I are having a dispute about my reasoning for $f$ being continuous by way of open and closed pullbacks (see below). Please help me correct my thinking. Here is the problem and m... | Here is a complete proof with sequential compactness:
Suppose that $f$ has no maximum on $K$. Then there are two cases:
Case 1: $\sup_{x \in K} f(x) = \infty$.
Then for any $n \in \mathbb{N}$ there is some $x_n \in K$ such that $f(x_n) > n$. Since $K$ is compact there exists a subsequence $x_{n_k} \in K$ that converge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/407031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
How to solve $x\log(x) = 10^6$ I am trying to solve
$$x\log(x) = 10^6$$
but can't find an elegant solution. Any ideas ?
| You won't find a "nice" answer, since this is a transcendental equation (no "algebraic" solution). There is a special function related to this called the Lambert W-function, defined by $ \ z = W(z) \cdot e^{W(z)} \ $ . The "exact" answer to your equation is
$ \ x = e^{W( [\ln 10] \cdot 10^6)} \ . $ (I'm assuming yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/407112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
Integral $\frac{\sqrt{e}}{\sqrt{2\pi}}\int^\infty_{-\infty}{e^{-1/2(x-1)^2}dx}$ gives $\sqrt{e}$. How? To calculate the expectation of $e^x$ for a standard normal distribution I eventually get, via exponential simplification:
$$\frac{\sqrt{e}}{\sqrt{2\pi}}\int^\infty_{-\infty}{e^{-1/2(x-1)^2}dx}$$
When I plug this into... | This is because $$\int_{\Bbb R} e^{-x^2}dx=\sqrt \pi$$
Note that your shift $x\mapsto x-1$ doesn't change the value of integral, while $x\mapsto \frac{x}{\sqrt 2}$ multiplies it by $\sqrt 2$, giving the desired result, that is,
$$\int_{\Bbb R} e^{-\frac 1 2(x-1)^2}dx=\sqrt {2\pi}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/407237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Second pair of matching birthdays The "birthday problem" is well-known and well-studied. There are many versions of it and many questions one might ask. For example, "how many people do we need in a room to obtain at least a 50% chance that some pair shares a birthday?" (Answer: 23)
Another is this: "Given $M$ bins,... | Suppose there are $n$ people, and we want to allow $0$ or $1$ collisions only.
$0$ collisions is the birthday problem: $$\frac{M^{\underline{n}}}{M^n}$$
For 1 collision, we first choose which two people collide, ${n\choose 2}$, then the 2nd person must agree with the first $\frac{1}{M}$, then avoid collisions for the r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/407307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Permutation and Combinations with conditions Hallo :) This is a question about permutations but with conditions.
2 boys and 4 girls are to be arranged in a straight line.
In how many ways can this be done if the two boys must be separated? (The order matters)
Thank You.
| Total number of ways of arranging the people = 6!
Cases when boys are together:
(2B) G G G G
G (2B) G G G
G G (2B) G G
G G G (2B) G
G G G G (2B)
Each of the above combinations can be arranged in 2 * 4! ways.
(The factor of 2 is accommodated since the boys themselves could be interchanged, as they are on a straight line... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/407398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Smallest projective subspace containing a degree $d$ curve Is it true that the smallest projective subspace containing a degree $d$ curve inside $\mathbb{P}^n$ has dimension at most $d$? If not, is there any bound on the dimension? Generalization to varieties?
For $d=1$ this is obvious. I think for the case that the cu... | I think your observation is correct for curves.
Given a curve $C$ in $\mathbb{P}^n$ satisfying $C$ is not contained in any projective subspace of $\mathbb{P}^n$, WLOG we may assume $C$ is irreducible. Let $\tilde C$ be the normalization of $C$, then we have a regular map $\phi: \tilde C\rightarrow \mathbb{P}^n$ which i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/407501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Basis of a basis I'm having troubles to understand the concept of coordinates in Linear Algebra.
Let me give an example:
Consider the following basis of $\mathbb R^2$:
$S_1=\{u_1=(1,-2),u_2=(3,-4)\}$ and $S_2=\{v_1=(1,3),v_2=(3,8)\}$
Let $w=(2,3)$ be a vector with coordinates in $S_1$, then $w=2u_1+3u_2=2(1,-2)+3(3,-4)... | The basis for everything, unless specified, is the standard basis $\{\textbf{e}_1=(1,0),\textbf{e}_2=(0,1)\}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/407563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
An interesting problem using Pigeonhole principle I saw this problem: Let $A \subset \{1,2,3,\cdots,2n\}$. Also, $|A|=n+1$.
Show that There exist $a,b \in A$ with $a \neq b$ and $a$ and $b$ is coprime.
I proved this one very easily by using pigeon hole principle on partition on $\{1,2\},\{3,4\},\dots,\{2n-1,2n\}$.
My q... | Any number from the set $A$ is of the form $2^{k}l$ where $k\ge 0,0\le l\le (2n-1)$ and $l$ is odd.
Number of odd numbers $l\le (2n-1) $ is $n$.
Now if we select $(n+1)$ numbers from the set $A$ then there must be two numbers(among the selected numbers) with the same $l$.
That is, we must get $a,b$ with $a=2^{k_1}l$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/407648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Is $C_2$ the correct Galois Group of $f(x)= x^3+x^2+x+1$? Let $\operatorname{f} \in \mathbb{Q}[x]$ where $\operatorname{f}(x) = x^3+x^2+x+1$. This is, of course, a cyclotomic polynomial. The roots are the fourth roots of unity, except $1$ itself. I get $\mathbb{Q}[x]/(\operatorname{f}) \cong \mathbb{Q}(\pm 1, \pm i) \c... | The Galois group is the group of authomorphisms of the splitting field. It acts on the roots of any splitting polynomial (such as $f$) by permuting the roots. In your case, there are three roots, $-1, i, -i$ and the automorphisms must leave $-1$ fixed. Since the action is also free, you can view $G$ (via this action) a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/407759",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving existence of a surjection $2^{\aleph_0} \to \aleph_1$ without AC I'm quite sure I'm missing something obvious, but I can't seem to work out the following problem (web search indicates that it has a solution, but I didn't manage to locate one -- hence the formulation):
Prove that there exists a surjection $2^{\... | One of my favorite ways is to fix a bijection between $\Bbb N$ and $\Bbb Q$, say $q_n$ is the $n$th rational.
Now we map $A\subseteq\Bbb N$ to $\alpha$ if $\{q_n\mid n\in A\}$ has order type $\alpha$ (ordered with the usual order of the rationals), and $0$ otherwise.
Because every countable ordinal can be embedded into... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/407833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 2,
"answer_id": 1
} |
Using integration by parts to evaluate an integrals Can't understand how to solve this math: use integration by parts to evaluate this integrals:
$$\int x\sin(2x + 1) \,dx$$
can any one solve this so i can understand how to do this! Thanks :)
| $\int uv'=uv-\int u'v$.
Choose $u(x):=x$, $v'(x):=\sin(2x+1)$. Then $u'(x)=1$ and $v(x)=-\frac{\cos(2x+1)}{2}$. So
$$
\int x\sin(2x+1)\,dx=-x\frac{\cos(2x+1)}{2}+\int\frac{\cos(2x+1)}{2}\,dx=-x\frac{\cos(2x+1)}{2}+\frac{\sin(2x+1)}{4}+C.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/407899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
How to prove/show $1- (\frac{2}{3})^{\epsilon} \geq \frac{\epsilon}{4}$, given $0 \leq \epsilon \leq 1$? How to prove/show $1- (\frac{2}{3})^{\epsilon} \geq \frac{\epsilon}{4}$, given $0 \leq \epsilon \leq 1$?
I found the inequality while reading a TCS paper, where this inequality was taken as a fact while proving some... | On of the most helpful inequalities about the exponential is $e^t\ge 1+t$ for all $t\in\mathbb R$.
Using this,
$$ \left(\frac32\right)^\epsilon=e^{\epsilon\ln\frac32}\ge 1+\epsilon\ln\frac32$$
for all $\epsilon\in\mathbb R$.
Under the additional assumption that $ -\frac1{\ln\frac32}\le \epsilon< 4$, multiply with $1-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/407985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
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