Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How can I interpret $\max(X,Y)$? My textbook says:
Let $X$ and $Y$ be two stochastically independent, equally distributed
random variables with distribution function F. Define $Z = \max (X, Y)$.
I don't understand what is meant by this. I hope I translated it correctly.
I would conclude that $X=Y$ out of this. And ... | What's the problem? Max is the usual maximum of two real numbers (or two real-valued random variables, so that we can define, more explicitely, that
$$
Z = \begin{cases} X & \text{if $X \ge Y$} \\
Y & \text{if $Y \ge X$} \\ \end{cases}
$$
So your conclusion is most surely wrong! There is no base f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/278732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Some weird equations In our theoreticall class professor stated that from this equation $(C = constant)$
$$
x^2 + 4Cx - 2Cy = 0
$$
we can first get:
$$
x = \frac{-4C + \sqrt{16 C^2 - 4(-2Cy)}}{2}
$$
and than this one:
$$
x = 2C \left[\sqrt{1 + \frac{y}{2C}} -1\right]
$$
How is this even possible?
| Here's the algebra:
$$x^2 + 4Cx - 2Cy = (x+2C)^2-4C^2 - 2Cy = 0 $$
Thus:
$$
(x+2C)^2 = 4C^2 + 2Cy = 2C(2C+y).
$$
Take square roots:
$$
x_1 = -2C + \sqrt{2C(2C+y)} =\frac{-4C + \sqrt{16C^2 +8Cy}}{2}$$
and
$$
x_2 = -2C - \sqrt{2C(2C+y)}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/278780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Prove that any function can be represented as a sum of two injections How to prove that for any function $f: \mathbb{R} \rightarrow \mathbb{R}$ there exist two injections $g,h \in \mathbb{R}^{\mathbb{R}} \ : \ g+h=f$.
Could you help me?
| Note that it suffices to find an injection $g$ such that $f+g$ is also an injection, as $f$ can then be written as the sum $(f+g)+(-g)$. Such a $g$ can be constructed by transfinite recursion.
Let $\{x_\xi:\xi<2^\omega\}$ be an enumeration of $\Bbb R$. Suppose that $\eta<2^\omega$, and we’ve defined $g(x_\xi)$ for all ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/278838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Proving convergence of sequence with induction I have a sequence defined as $a_{1}=\sqrt{a}$ and $a_{n}=\sqrt{1+a_{n-1}}$ and I need to prove that it has an upper bound and therefore is convergent. So i have assumed that the sequence has a limit and by squaring I got that the limit is $\frac{1+\sqrt{5}}{2}$ only $ \mat... | Let $f(x) = \sqrt{1+x}-x$. We find that $f'(x) = \frac 1 {2\sqrt{1+x}} -1 <0$. This means that $f$ is a strictly decreasing function. Set $\phi = \frac{1+\sqrt 5}{2}$.
We now that $f(\phi)=0$. We must then have that $f(x)>0$ if $x<\phi$ and $f(x)<0$ if $x>\phi$. So $a_{n+1}>a_n$ if $a_n< \phi$ and $a_{n+1} < a_n$ if $a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/278902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Finding a topology on $ \mathbb{R}^2 $ such that the $x$-axis is dense The problem is the following
Put a topology on $ \mathbb{R}^2$ with the property that the line $\{(x,0):x\in \mathbb{R}\}$ is dense in $\mathbb{R}^2$
My attempt
If (a,b) is in $R^2$, then define an open sets of $(a,b)$ as the strip between $d$ and ... | The following generalizes all solutions (EDIT: not all solutions, just those which give a topology on $\mathbb{R}^2$ homeomorphic to the standard topology). It doesn't have much topological content, but it serves to show how basic set theory can often be used to trivialize problems in other fields. A famous example o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/278968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Four times the distance from the $x$-axis plus 9 times the distance from $y$-axis equals $10$. What geometric figure is formed by the locus of points such that the sum of four times the distance from the $x$-axis and nine times its distance from $y$-axis is equal to $10$?
I get $4x+9y=10$. So it is a straight line... | If $P$ has coordinates $(x,y)$, then $d(P,x\text{ axis})=|y|$ and $d(P,y\text{ axis})=|x|$. So your stated condition requires $4|y|+9|x|=10$, the graph of which is shown below.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/279077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Neighborhoods of half plane Define $H^n = \{(x_1, \dots, x_n)\in \mathbb R^n : x_n \ge 0\}$, $\partial H^n = \{(x_1, \dots, x_{n-1},0) : x_i \in \mathbb R\}$.
$\partial H^n$ is a manifold of dimension $n-1$: As a subspace of $H^n$ it is Hausdorff and second-countable. If $U \subseteq \partial H^n$ is open in $H^n$ wit... | Suppose $U'$ is open in $\mathbb R^n$ and $U' \ni x \in \partial H^n$. Then $U'$ must contain an open ball $B$ around $x$ and so there must exist a point $y \in B$ with $y_n < 0$ and therefore $U' \not \subset H^n$.
For the more general argument, suppose $\phi: U' \to \mathbb R^n$ is a homeomorphism. Then there is an o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/279198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Do we have such a direct product decomposition of Galois groups? Let $L = \Bbb{Q}(\zeta_m)$ where we write $m = p^k n$ with $(p,n) = 1$. Let $p$ be a prime of $\Bbb{Z}$ and $P$ any prime of $\mathcal{O}_L$ lying over $p$.
Notation: We write $I = I(P|p)$ to denote the inertia group and $D = D(P|p)$ the decomposition gro... | You can write your $L$ as the compositum of $\mathbb{Q}(\zeta_{p^k})$ and $\mathbb{Q}(\zeta_{n})$. Since $(p,n)=1$, the two are disjoint over $\mathbb{Q}$, and so the Galois group of $L$ is isomorphic to the direct product of the two Galois groups, one of which is $E$. Let's call the other subgroup $H$. Now, every elem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/279264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Confusion related to integral of a Gaussian I am a bit confused about calculating the integral of a Gaussian
$$\int_{-\infty}^{\infty}e^{-x^{2}+bx+c}\:dx=\sqrt{\pi}e^{\frac{b^{2}}{4}+c}$$
Given above is the integral of a Gaussian. The integral of a Gaussian is Gaussian itself. But what is the mean and variance of this ... | The question is only meaningful if $\Im{b} \ne 0$. Let's say that, rather, $\Re{b} = 0$ and $b = i B$. Now you can assign a mean/variance to the resulting Gaussian. This, BTW, is related to the well-known fact that a Fourier transform of a Gaussian is a Gaussian.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/279326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Compute $\lim_{n\to\infty} \left(\sum_{k=1}^n \frac{H_k}{k}-\frac{1}{2}(\ln n+\gamma)^2\right) $ Compute
$$\lim_{n\to\infty} \left(\sum_{k=1}^n \frac{H_k}{k}-\frac{1}{2}(\ln n+\gamma)^2\right) $$
where $\gamma$ - Euler's constant.
| We have
\begin{align}
2\sum_{k=1}^n \frac{H_k}{k} &= 2\sum_{k=1}^n \sum_{j=1}^k \frac{1}{jk} \\
&= \sum_{k=1}^n \sum_{j=1}^k \frac{1}{jk} + \sum_{k=1}^n \sum_{j=1}^k \frac{1}{jk} \\
&= \sum_{k=1}^n \sum_{j=1}^k \frac{1}{jk} + \sum_{j=1}^n \sum_{k=j}^n \frac{1}{jk}, \text{ swapping the order of summation on the second ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/279380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 1,
"answer_id": 0
} |
A numerical inequality. If $a_1\ge a_2\ge \cdots \ge a_n\ge 0$, $b_1\ge b_2\ge \cdots \ge b_n\ge 0$ with $\sum_{j=1}^kb_j=1$ for some $1\le k\le n$. Is it true that $2\sum_{j=1}^na_jb_j\le a_1+\frac{1}{k}\sum_{j=1}^na_j$?
The above question is denied.
Give a simple proof to a weaker version?
Under the same condtion, ... | No, it isn't true. Let $a_j=b_j=1$ for $1\leq j\leq n$. Then $\sum_{j=1}^1b_j=1$, hence $k=1$, thus we've satisfied the hypotheses. However, $$2\sum_{j=1}^na_jb_j=2n\geq1+n$$ with equality if and only if $n=1$.
EDIT: For the second case, let $a_j=1/10$ for all $j$, $b_1=1$ and $b_j=10$ for $j\geq 2$. Then we end up wit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/279491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Evaluating: $\int_0^{\infty} \frac{1}{t^2} dt$ I try to evaluate following integral $$\int_0^{\infty} \frac{1}{t^2} dt$$
At first it seems easy to me. I rewrite it as follows.$$\lim_{b \to \infty} \int_0^{b} \frac{1}{t^2} dt$$ and integrate the $\frac{1}{t^2}$. I proceed as follows:
$$\lim_{b\to\infty} \left[
-t^{-1}\... | You have done something strange with the lower limit. In fact, your integral is generalized at both endpoints, since the integrand is unbounded near $0$.
You get
$$\int_0^\infty \frac{1}{t^2}\,dt = \lim_{\varepsilon \to 0^+} \int_\varepsilon^1 \frac{1}{t^2}\,dt + \lim_{b \to\infty} \int_1^b \frac{1}{t^2}\,dt$$
and whil... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/279547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
How to evaluate $\lim_{n\rightarrow\infty}\int_{0}^{1}x^n f(x)dx$, How to evaluate $\lim\limits_{n\rightarrow\infty}\int_{0}^{1}x^n f(x)dx$, well, i did one problem from rudins book that if $\int_{0}^{1}x^n f(x)dx=0\forall n\in\mathbb{N}$ then $f\equiv 0$ by stone weirstrass theorem. please help me here.
| Assuming $f$ is integrable you can use dominated convergence. If $f$ is positive, monotone convergence works too. In either case, the limit is $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/279622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Linear transformation satisfies $T^n=T$; has eigenvalues? I have a linear transformation $T:V\to V$ over a (finite, is needed) field $F$, which satisfies $T^n=T$.
Prove that $T$ has a eigenvalue, or give a counter example.
Thanks
| This is false. The matrix $$A = \left(\begin{matrix}0 & -1\\1 & 0 \end{matrix}\right)$$ satisfies $A^4 = 1$ (thus $A^5 = A$) but its characteristic polynomial $X^2+1$ has no real roots, so $A$ has no real eigenvalues.
For an example over a finite field, consider
$$A = \left(\begin{matrix}0 & 1\\1 & 1 \end{matrix}\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/279671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Applications of Residue Theorem in complex analysis? Does anyone know the applications of Residue Theorem in complex analysis? I would like to do a quick paper on the matter, but am not sure where to start.
The residue theorem
The residue theorem, sometimes called Cauchy's residue theorem (one of many things named af... | Other then as a fantastic tool to evaluate some difficult real integrals, complex integrals have many purposes.
Firstly, contour integrals are used in Laurent Series, generalizing real power series.
The argument principle can tell us the difference between the poles and roots of a function in the closed contour $C$:
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/279737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 1
} |
Proving $\gcd(a, c) = \gcd(b, c)$ for $a + b = c^2$ I am trying to prove that, given positive integers $a, b, c$ such that $a + b = c^2$, $\gcd(a, c) = \gcd(b, c)$. I am getting a bit stuck.
I have written down that $(a, c) = ra + sc$ and $(b, c) = xb + yc$ for some integers $r, s, x, y$. I am now trying to see how I c... | I don't see a way to proceed using the approach you suggest - this doesn't mean that there isn't one (it's often a good method to work through). But I don't see it yet. But you can directly show that the same primes to the same powers divide each:
Consider a prime $p$. Suppose that $p^\beta \mid \mid \gcd(a,c)$, so tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/279795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Question about limit of a product
Is it possible that both $\displaystyle\lim_{x\to a}f(x)$ and $\displaystyle\lim_{x\to a}g(x)$ do not exist but $\displaystyle\lim_{x\to a}f(x)g(x)$ does exist?
The reason I ask is that I was able to show that if $\displaystyle\lim_{x\to a}f(x)$ does not exist but both $\displaystyle... | Yes. As a simple example, I'll work with sequences. Let $f(n) = (-1)^n$ and $g(n) = (-1)^{n}$. Then $f(n)g(n) = 1$, so $1 = \lim_{n \to \infty} f(n)g(n)$, but neither of the individual limits exist.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/279851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
prove the divergence of cauchy product of convergent series $a_{n}:=b_{n}:=\dfrac{(-1)^n}{\sqrt{n+1}}$ i am given these series which converge. $a_{n}:=b_{n}:=\dfrac{(-1)^n}{\sqrt{n+1}}$ i solved this with quotient test and came to $-1$, which is obviously wrong. because it must be $0<\theta<1$ so that the series conver... | $\sum_{n=0}^\infty\dfrac{(-1)^n}{\sqrt{n+1}}$ is convergent by Leibniz's test, but it is not absolutely convergente (i.e. it is conditionally convergent.)
To show that the Cauchy product does not converge use the inequality
$$
x\,y\le\frac{x^2+y^2}{2}\quad x,y\in\mathbb{R}.
$$
Then
$$
\sqrt{n-k+1}\,\sqrt{k+1}\le\frac{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/279890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
What is limit of: $\displaystyle\lim_{x\to 0}$$\tan x - \sin x\over x$ I want to search limit of this trigonometric function:
$$\displaystyle\lim_{x\to 0}\frac{\tan x - \sin x}{x^n}$$
Note: $n \geq 1$
| Checking separatedly the cases for $\,n=1,2,3\,$, we find:
$$n=1:\;\;\;\;\;\;\frac{\tan x-\sin x}{x}=\frac{1}{\cos x}\frac{\sin x}{x}(1-\cos x)\xrightarrow [x\to 0]{}1\cdot 1\cdot 0=0$$
$$n=2:\;\;\frac{\tan x-\sin x}{x^2}=\frac{1}{\cos x}\frac{\sin x}{x}\frac{1-\cos x}{x}\xrightarrow [x\to 0]{}1\cdot 1\cdot 0=0\;\;(\te... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/279967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
extension of a non-finite measure For a finite measure on a field $\mathcal{F_0}$ there always exists its extension to $\sigma(\mathcal{F_0})$.
Can somebody give me an example of a non-finite measure on a field which cannot be extended to $\sigma(\mathcal{F_0})$.
It would be better if somebody can point in the proof (... | Every measure can be extended from a field to the generated $\sigma$-algebra. The classical proof by Caratheodory does not rely on the measure being finite, so there is no such example. As Ilya mentioned in a comment, the extension may not be unique. Here is an explicit example:
Let $\mathcal{F}$ be the field of subset... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/280022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Why is there always a basis for the Cartan orthonormal relative to the Killing form? I'm trying to understand a step in a proof:
Let $\mathfrak{g}$ be semi-simple (finite dimensional) Lie-algebra over $\mathbb{C}$, $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra and let $\kappa:\mathfrak{g}\times\mathfrak{g}\to\m... | The Killing form is symmetric and non-degenerate(Cartan's criterion). For such bilinear forms you can always diagonalize it via a proper basis. So in particular over $\mathbb{C}$ you should be able to find an orthonormal basis.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/280090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Number Theory and Congruency I have the following problem:
$$2x+7=3 \pmod{17}$$
I know HOW to do this problem. It's as follows:
$$2x=3-7\\
x=-2\equiv 15\pmod{17}$$
But I have no idea WHY I'm doing that. I don't really even understand what the problem is asking, I'm just doing what the book says to do. Can someone expla... | The $\bmod{17}$ congruence classes are represented by $0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16
$. You're trying to find out which of those classes $x$ belongs to. That's why you're doing that.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/280157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 3
} |
Lower bounds on numbers of arrangements of a Rubik's cube Last night, a friend of mine informed me that there were forty-three quintillion positions that a Rubik's Cube could be in and asked me how many there were for my Professor's Cube (5x5x5).
So I gave him an upper bound:
$$8!~3^8\cdot24!~2^{24}/2^{12}\cdot12!~2^{1... | Chris Hardwick has come up with a closed-form solution for the number of permutations for an $n\times n\times n$ cube: http://speedcubing.com/chris/cubecombos.html
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/280276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt[k+1]{k}}} \geq \frac{n^2+3n}{2n+2}$ How to prove the following inequalities without using Bernoulli's inequality?
*
*$$\prod_{k=1}^{n}{\sqrt[k+1]{k}} \leq \frac{2^n}{n+1},$$
*$$\sum_{k=1}^{n}{\frac{1}{\sqrt[k+1]{k}}} \geq \frac{n^2+3n}{2n+2}.$$
My proof:
*
... | The inequality to be shown is
$$(n+1)^{n+1}\geqslant(n+3)^n,
$$
for every positive integer $n$.
For $n = 1$ it is easy. For $n \ge 2$, apply AM-GM inequality to $(n-2)$-many $(n+3)$, 2 $\frac{n+3}{2}$, and $4$, we get
$$(n+3)^n < \left(\frac{(n-2)(n+3) + \frac{n+3}{2} + \frac{n+3}{2} + 4}{n+1}\right)^{n+1} = \left(n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/280360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Finding the general solution of a quasilinear PDE This is a homework that I'm having a bit of trouble with:
Find a general solution of:
$(x^2+3y^2+3u^2)u_x-2xyuu_y+2xu=0~.$
Of course this should be done using the method of characteristics but I'm having trouble solving the characteristic equations since none of the equ... | Follow the method in http://en.wikipedia.org/wiki/Characteristic_equations#Example:
$(x^2+3y^2+3u^2)u_x-2xyuu_y+2xu=0$
$2xyuu_y-(x^2+3y^2+3u^2)u_x=2xu$
$2yu_y-\left(\dfrac{x}{u}+\dfrac{3y^2}{xu}+\dfrac{3u}{x}\right)u_x=2$
$\dfrac{du}{dt}=2$ , letting $u(0)=0$ , we have $u=2t$
$\dfrac{dy}{dt}=2y$ , letting $y(0)=y_0$ , ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/280411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
how to find inverse of a matrix in $\Bbb Z_5$ how to find inverse of a matrix in $\Bbb Z_5$
please help me explicitly how to find the inverse of matrix below, what I was thinking that to find inverses separately of the each term in $\Bbb Z_5$ and then form the matrix?
$$\begin{pmatrix}1&2&0\\0&2&4\\0&0&3\end{pmatrix}$$... | Hint: Use the adjugate matrix.
Answer: The cofactor matrix of $A$ comes
$\color{grey}{C_A=
\begin{pmatrix}
+\begin{vmatrix} 2 & 4 \\ 0 & 3 \end{vmatrix} & -\begin{vmatrix} 0 & 4 \\ 0 & 3 \end{vmatrix} & +\begin{vmatrix} 0 & 2 \\ 0 & 0 \end{vmatrix} \\
-\begin{vmatrix} 2 & 0 \\ 0 & 3 \end{vmatrix} & +\begin{vmatrix} 1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/280522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
} |
Can you provide me historical examples of pure mathematics becoming "useful"? I am trying to think/know about something, but I don't know if my base premise is plausible. Here we go.
Sometimes when I'm talking with people about pure mathematics, they usually dismiss it because it has no practical utility, but I guess t... | Group theory is commonplace in quantum mechanics to represent families of operators that possess particular symmetries. You can find some info here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/280530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "166",
"answer_count": 34,
"answer_id": 29
} |
Non-Abelian simple group of order $120$ Let's assume that there exists simple non-Abelian group $G$ of order $120$. How can I show that $G$ is isomorphic to some subgroup of $A_6$?
| A group of order 120 can not be simple. Let's assume that there exists simple non-abelian group $G$ of order 120. Then we know the number Sylow 5-subgroups of $G$ is 6. Hence, the index of $N_{G}(P)$ in $G$ is 6 ($P$ is a Sylow 5-subgroup of $G$). Now there exists a monomorphism $\phi$ of $G$ to $S_{6}$. We claim that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/280657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
How do i apply multinomial laws in this question? the Question is i assume i have 15 students in class
A grade obtain probaiblity = 0.3
B grade obtain probability =0.4
C grade obtain probability = 0.3
and I have this question
What is the probabilty if we are given 2 students at least obtain A ?
Do I need apply ... | We reword the question, perhaps incorrectly.
If a student is chosen at random, the probabilities she obtains an A, B, and C are, respectively, $0.3$, $0.4$, and $0.3$.
If $15$ students are chosen at random, what is the probability that at least $2$ of the students obtain an A?
The probability that $a$ students get an ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/280749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
An inequality about sequences in a $\sigma$-algebra Let $(X,\mathbb X,\mu)$ be a measure space and let $(E_n)$ be a sequence in $\mathbb X$. Show that $$\mu(\lim\inf E_n)\leq\lim\inf\mu(E_n).$$
I am quite sure I need to use the following lemma.
Lemma. Let $\mu$ be a measure defined on a $\sigma$-algebra $\mathbb X$.
... | For every $i\geq 1$ we have that
$$
\bigcap_{n=i}^\infty E_n\subseteq E_i
$$
and so for all $i\geq 1$
$$
\mu\left(\bigcap_{n=i}^\infty E_n\right)\leq \mu(E_i).
$$
Then
$$
\mu(\liminf_n E_n)=\lim_{i\to\infty}\mu\left(\bigcap_{n=i}^\infty E_n\right)\leq \liminf \mu(E_i),
$$
using the fact that if $(x_n)_{n\geq 1}$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/280840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Examples of familiar, easy-to-visualize manifolds that admit Lie group structures I have a trouble learning Lie groups --- I have no canonical example to imagine while thinking of a Lie group. When I imagine a manifold it is usually some kind of a $2$D blanket or a circle/curve or a sphere, a torus etc.
However I have ... | Think of $SO_2$ as the group of $2\times 2$ rotation matrices:
$$ \left[\begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array} \right]$$
or the group of complex numbers of unit length $e^{i\theta}$.
You can convince yourself directly from definitions that either of these objects is a group un... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/280896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 3
} |
When do regular values form an open set? Let $f:M\to N$ be a $C^\infty$ map between manifolds.
When is the set of regular values of $N$ an open set in $N$?
There is a case which I sort of figured out:
*
*If $\operatorname{dim} M = \operatorname{dim} N$ and $M$ is compact, it is open by the following argument (fixe... | The set of critical points is closed. You want that the image of this set under $f$ be closed. What about demanding that $f$ is closed? A condition that implies that $f$ is closed is to demand that $f$ is proper (i.e. preimages of compact sets are compact).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/280965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 1
} |
Studying $ u_n = \int_0^1 (\arctan x)^n \mathrm dx$ I would like to find an equivalent of:
$$ u_n = \int_0^1 (\arctan x)^n \mathrm dx$$
which might be: $$ u_n \sim \frac{\pi}{2n} \left(\frac{\pi}{4} \right)^n$$
$$ 0\le u_n\le \left( \frac{\pi}{4} \right)^n$$
So $$ u_n \rightarrow 0$$
In order to get rid of $\arctan x$... | Another (simpler) approach is to substitute $x = \tan{y}$ and get
$$u_n = \int_0^{\frac{\pi}{4}} dy \: y^n \, \sec^2{y}$$
Now we perform an analysis not unlike Laplace's Method: as $n \rightarrow \infty$, the contribution to the integral is dominated by the value of the integrand at $y=\pi/4$. We may then say that
$$u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/281017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Computing conditional probability combining events I know $P(E|A)$ and $P(E|B)$, how do I know $P(E|A,B)$? Assuming $A,B$ independent.
| Consider this example:
*
*$A$ = first coin lands Head,
*$B$ = second coin lands Head,
*$E_1$ = odd number of Heads,
*$E_2$ = even number of Heads.
The first two conditional probabilities are both $1/2$. The third is $0$ for $E_1$ and $1$ for $E_2$.
(Also: Welcome to Math.SE, and please do not try to make y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/281099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Book Recommendation for Integer partitions and $q$ series I have been studying number theory for a little while now, and I would like to learn about integer partitions and $q$ series, but I have never studied anything in the field of combinatorics, so are there any prerequisites or things I should be familiar with befo... | George Andrews has contributed greatly to the study of integer partitions. (The link with his name will take you to his webpage listing publications, some of which are accessible as pdf documents.) Also see, e.g., his classic text The Theory of Partitions and the more recent Integer Partitions.
You can pretty much "jum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/281153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 1
} |
Equation of the Plane I have been working through all the problems in my textbook and I have finally got to a difficult one.
The problem ask
Find the equation of the plane.The plane that passes through the points $(-1,2,1)$ and contains the line of intersection of the planes $x+y-z =2$ and $2x-y+3z=1$
So far I found... | The following determinant is another form of what Scott noted: $$\begin{vmatrix}
(x-x_0)& (y-y_0)& (z-z_0)\\1& 1& -1\\2& -1& 3
\end{vmatrix}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/281191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
On a special normal subgroup of a group Let $G$ be a group such $H$ is a normal subgroup of $G$ and $Z(H)=1$ and $Inn(H)=Aut(H)$. Then prove there exists a normal subgroup $K$ of $G$ such that $G=H\times K$.
| First, define the map $\psi: G \to Inn(G)$ by $\psi_g(x) = gxg^{-1}$. Since $H$ is a normal subgroup, $\psi_g(H) = H$, so $Inn(G) = Aut(H) = Inn(H)$. From $Z(H) = 1$, we know that $Inn(H) \cong H$. Compose this isomorphism with $\psi$ to get a surjective map $G \to H$. The kernel of this map is $K$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/281245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Connectedness of $\beta \mathbb{R}$ I am not very familiar with Stone-Čech compactification, but I would like to understand why the remainder $\beta\mathbb{R}\backslash\mathbb{R}$ has exactly two connected components.
| I finally found something:
For convenience, let $X= \mathbb{R} \backslash (-1,1)$.
First, we show that $(\beta \mathbb{R})\backslash (-1,1)= \beta (\mathbb{R} \backslash (-1,1))$. Let $f_0 : X \to [0,1]$ be a continuous function. Obviously, $f_0$ can be extend to a continuous function $f : \mathbb{R} \to [0,1]$; then ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/281325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
How does one show that the set of rationals is topologically disconnected? Let $\mathbb{Q}$ be the set of rationals with its usual topology based on distance:
$$d(x,y) = |x-y|$$
Suppose we can only use axioms about $\mathbb{Q}$ (and no axiom about $\mathbb{R}$, the set of reals). Then how can we show that $\mathbb{Q}$ ... | The rationals is the union of two disjoint open sets $\{x\in\mathbb{Q}:x^2>2\}$ and $\{x\in\mathbb{Q}:x^2<2\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/281377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
How do i prove that this given set is open? Let $X$ be a topological space and $E$ be an open set.
If $A$ is open in $\overline{E}$ and $A\subset E$, then how do i prove that $A$ is open in $X$?
It seems trivial, but i'm stuck.. Thank you in advance.
| Is enough to show that $A=G\cap E$ for an open set $G\subset X$.
Use the hypothesis and the fact that $A=G\cap \overline{E}\Rightarrow A=G\cap E$ (why?).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/281459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
What is $\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$ for $z \in \mathbb C$? What is $\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$ for $z \in \mathbb C^*$? I need it to determine the type of the singularity at $z = 0$.
| We have $a_n=\frac{1}{n} \to 0$ as $n\to \infty$ and
$$
\lim_{n \to \infty}\Big|a_n\sin\Big(\frac{1}{a_n}\Big)\Big|=\lim_{n \to \infty}\frac{|\sin n|}{n}=0,
$$
but
$$
\lim_{n \to \infty}\Big|ia_n\sin\Big(\frac{1}{ia_n}\Big)\Big|=\lim_{n \to \infty}\frac{e^n-e^{-n}}{2n}=\infty.
$$
Therefore $\lim_{z \to 0}|z\sin(z^{-1})... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/281518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Showing that $\displaystyle\int_{-a}^{a} \frac{\sqrt{a^2-x^2}}{1+x^2}dx = \pi\left (\sqrt{a^2+1}-1\right)$. How can I show that $\displaystyle\int_{-a}^{a} \frac{\sqrt{a^2-x^2}}{1+x^2}dx = \pi\left(\sqrt{a^2+1}-1\right)$?
| Let $x = a \sin(y)$. Then we have
$$\dfrac{\sqrt{a^2-x^2}}{1+x^2} dx = \dfrac{a^2 \cos^2(y)}{1+a^2 \sin^2(y)} dy $$
Hence,
$$I = \int_{-a}^{a}\dfrac{\sqrt{a^2-x^2}}{1+x^2} dx = \int_{-\pi/2}^{\pi/2} \dfrac{a^2 \cos^2(y)}{1+a^2 \sin^2(y)} dy $$
Hence,
$$I + \pi = \int_{-\pi/2}^{\pi/2} \dfrac{a^2 \cos^2(y)}{1+a^2 \sin^2(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/281587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 2
} |
Can that two double series representations of the $\eta$/$\zeta$ function be converted into each other? By an analysis of the matrix of Eulerian numbers(see pg 8) I came across the representation for the alternating Dirichlet series $\eta$:
$$ \eta(s) = 2^{s-1} \sum_{c=0}^\infty \left( \sum_{k=0}^c(-1)^k \binom{1-s}{c-... | For $|z|<1$ and any $s$
$$-Li_s(-z) (1+z)^{1-s}= \sum_k z^k (-1)^{k+1}k^{-s}\sum_m z^m {1-s\choose m}=\sum_c z^c \sum_{k\le c} (-1)^{k+1}k^{-s}{1-s\choose c-k}$$
Interpret $-Li_s(-e^{2\pi it}) (1+e^{2\pi it})^{1-s}$ as $\lim_{r\to 1^-}-Li_s(-r e^{2\pi it}) (1+r e^{2\pi it})^{1-s}$.
With enough partial summations we ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/281711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Complex numbers straight line proof
Prove that the three distinct points $z_1,z_2$, and $z_3$ lie on the
same straight line iff $z_3 - z_2 = c(z_2 - z_1)$ for some real number
$c$ and $z$ is complex.
I know that two vectors are parallel iff one is a scalar multiple of the other, thus $z$ is parallel to $w$ iff $z... | If $z_k=x_k+iy_k$ for $k=1,2,3$
As $z_3-z_2=c(z_2-z_1),$
If $c=0, z_3=z_2$ and if $z=\infty, z_2=z_1$ so $c$ non-zero finite number.
$\implies x_3-x_2+i(y_3-y_2)=c\{x_2-x_1+i(y_2-y_1)\}$
Equating the real & the imaginary parts, $$x_3-x_2=c(x_2-x_1),y_3-y_2=c(y_2-y_1)$$
So, $$\frac{y_3-y_2}{x_3-x_2}=\frac{y_2-y_1}{x_2-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/281784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Book recommendations for commutative algebra and algebraic number theory Are there any books which teach commutative algebra and algebraic number theory at the same time. Many commutative algebra books contain few chapters on algebraic number theory at end. But I don't need that. I'm seaching for book which motivates c... | There's no law against reading more than one book at a time!
Although algebraic number theory and algebraic geometry both use commutative algebra heavily, the algebra needed for geometry is rather broader in scope (for alg number theory you need to know lots about Dedekind domains, but commutative algebra uses a much w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/281863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 2,
"answer_id": 1
} |
What is a 'critical value' in statistics? Here's where I encountered this word:
The raw material needed for the manufacture of medicine has to be at least $97\%$ pure. A buyer analyzes the nullhypothesis, that the proportion is $\mu_0=97\%$, with the alternative hypothesis that the proportion is higher than $97\%$. He ... | A critical value is the point (or points) on the scale of the test statistic beyond which we reject the null hypothesis, and is derived from the level of significance $\alpha$ of the test.
You may be used to doing hypothesis tests like this:
*
*Calculate test statistics
*Calculate p-value of test statistic.
*Compa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/281940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How is the number of points in the convex hull of five random points distributed? This is about another result that follows from the results on Sylvester's four-point problem and its generalizations; it's perhaps slightly less obvious than the other one I posted.
Given a probability distribution in the plane, if we kn... | Denote the probability for the convex hull of the five points to consist of $k$ points by $x_k$. The convex hull has five points if and only if the five points form a convex pentagon, so $x_5=p_5$.
Now let's determine the expected number of subsets of four of the five points that form a convex quadrilateral in two diff... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/282147",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Derivative of $\sqrt{\sin (x^2)}$ I have problems calculating derivative of $f(x)=\sqrt{\sin (x^2)}$.
I know that $f'(\sqrt{2k \pi + \pi})= - \infty$ and $f'(\sqrt{2k \pi})= + \infty$ because $f$ has derivative only if $ \sqrt{2k \pi} \leq |x| \leq \sqrt{2k \pi + \pi}$.
The answer says that for all other values of $x$,... | I don't know if you did it this way, so I figured that I would at least display it.
\begin{align}
y &= \sqrt{\sin x^2}\\
y^2 &= \sin x^2\\
2yy' &= 2x \cos x^2\\
y' &= \frac{x \cos x^2}{\sqrt{\sin x^2}}
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/282279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Logarithm as limit Wolfram's website lists this as a limit representation of the natural log:
$$\ln{z} = \lim_{\omega \to \infty} \omega(z^{1/\omega} - 1)$$
Is there a quick proof of this? Thanks
| $\ln z$ is the derivative of $t\mapsto z^t$ at $t=0$, so
$$\ln z = \lim_{h\to 0}\frac{ z^h-1}h=\lim_{\omega\to \infty} \omega(z^{1/\omega}-1).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/282339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
} |
At what speed should it be traveling if the driver aims to arrive at Town B at 2.00 pm? A car will travel from Town A to Town B. If it travels at a constant speed of 60 km/h, it will arrive at 3.00 pm. If travels at a constant speed of 80kh/h, it will arrive at 1.00 pm. At what speed should it be traveling if the drive... | The trip became $120$ minutes ($2$ hours) shorter by using $\frac34$ of a minute per kilometer ($80$ km/hr) instead of $1$ minute per kilometer ($60$ km/hr.) Since the savings from going faster was $\frac14$ of a minute per kilometer, the trip must be $480$ kilometers long, so it took $8$ hours at $60$ km/hr, and we s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/282402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Upper bound for the absolute value of an inner product I am trying to prove the inequality
$$
\left|\sum\limits_{i=1}^n a_{i}x_{i} \right| \leq \frac{1}{2}(x_{(n)} - x_{(1)}) \sum\limits_{i=1}^n \left| a_{i} \right| \>,$$
where
$x_{(n)} = \max_i x_i$ and $x_{(1)} = \min_i x_i$, subject to the condition $\sum_i a_i = ... | Hint:
$$
\left|\sum_i a_i x_i\right| = \frac{1}{2} \left|\sum_i a_i x_i\right| + \frac{1}{2} \left|\sum_i a_i \cdot (-x_i)\right| \>.
$$
Now,
*
*What do you know about $\sum_i a_i x_{(1)}$ and $\sum_i a_i x_{(n)}$? (Use your assumptions.)
*Recall the old saw: "There are only three basic operations in mathematics: ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/282462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Simple combinations - Party Lamps [IOI 98]
You are given N lamps and four switches. The first switch toggles all lamps, the second the even lamps, the third the odd lamps, and last switch toggles lamps $1, 4, 7, 10, \dots $
Given the number of lamps, N, the number of button presses made (up to $10,000$), and the state... | The naive solution works in this way: There are four buttons we can push. We need to account for at most $10000$ button presses. Let's make it easier and say we only have to account for at most three button presses. Then our button-press 1 is either the first button, the second one, the third one, or the fourth one. Si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/282530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Let $a,b$ and $c$ be real numbers.evaluate the following determinant: |$b^2c^2 ,bc, b+c;c^2a^2,ca,c+a;a^2b^2,ab,a+b$| Let $a,b$ and $c$ be real numbers. Evaluate the following determinant:
$$\begin{vmatrix}b^2c^2 &bc& b+c\cr c^2a^2&ca&c+a\cr a^2b^2&ab&a+b\cr\end{vmatrix}$$
after long calculation I get that the answer w... | Imagine expanding along the first column. Note that the cofactor of $b^2c^2$ is $$(a+b)ac-(a+c)ab=a^2(c-b)$$ which is a multiple of $a^2$. The other two terms in the expansion along the first column are certainly multiples of $a^2$, so the determinant is a multiple of $a^2$. By symmetry, it's also a multiple of $b^2$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/282655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 4
} |
Combinatorics Problem: Box Riddle A huge group of people live a bizarre box based existence. Every day, everyone changes the box that they're in, and every day they share their box with exactly one person, and never share a box with the same person twice.
One of the people of the boxes gets sick. The illness is spread ... | Just in case this helps someone:
(In each step we must cover a $N\times N$ board with $N$ non-self attacking rooks, diagonal forbidden).
This gives the sequence (I start numbering day 1 for N=2) : (2,4,4,6,8,8,8,10,12,12,14,16,16,16)
Updated: a. Brief explanation: each column-row corresponds to a person; the numbered... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/282740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "40",
"answer_count": 4,
"answer_id": 1
} |
A function with a non-negative upper derivative must be increasing? I am trying to show that if $f$ is continuous on the interval $[a,b]$ and its upper derivative $\overline{D}f$ is such that $ \overline{D}f \geq 0$ on $(a,b)$, then $f$ is increasing on the entire interval. Here $\overline{D}f$ is defined by
$$
\overli... | Probably not the best approach, but here is an idea: show taht MVT holds in this case:
Lemma Let $[c,d]$ be a subinterval of $[a,b]$. Then there exists a point $e \in [c,d]$ so that
$$\frac{f(d)-f(c)}{d-c}=\overline{D}f(e)$$
Proof:
Let $g(x)=f(x)-\frac{f(d)-f(c)}{d-c}(x-c) \,.$
Then $g$ is continuous on $[c,d]$ and he... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/282889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
} |
Trace and Norm of a separable extension. If $L | K$ is a separable extension and $\sigma : L \rightarrow \bar K$ varies
over the different $K$-embeddings of $L$ into an algebraic closure $\bar K$ of $K$, then
how to prove that
$$f_x(t) = \Pi (t - \sigma(x))?$$ $f_x(t)$ is the characteristic polynomial of the l... | First assume $L = K(x)$. By the Cayley-Hamilton Theorem, $f_x(x) = 0$, so $f_x$ is a multiple of the minimal polynomial of $x$ which is $\prod_\sigma (t-\sigma(x))$. Since both polynomials are monic and have the same degree, they are in fact equal.
For the general case, choose a basis $b_1,\ldots,b_r$ of $L$ over $K(x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/282966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Why is this function entire? $f(z) = z^{-1} \sin{z} \exp(i tz)$ In problem 10.44 of Real & Complex Analysis, the author says $f(z) = z^{-1} \sin{z} \exp(i tz)$ is entire without explaining why. My guess is that $z = 0$ is a removable singularity, $f(z) = 1$ and $f'(z) = 0$, but I cannot seem to prove it from the defini... | Note that $u:z\mapsto\sin(z)/z$ is indeed entire since $u(z)=\sum\limits_{n=0}^{+\infty}(-1)^nz^{2n}/(2n+1)!$ has an infinite radius of convergence. Multiplying $u$ by the exponential, also entire, does not change anything.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/283030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Multilinear Functions I have a question regarding the properties of a multilinear function. This is for a linear algebra class. I know that for a multilinear function,
$f(cv_1,v_2,\dots,v_n)=c⋅f(v_1,v_2,\dots,v_n)$
Does this imply
$f(cv_1,dv_2,\dot,v_n)=c⋅d⋅f(v_1,v_2,\dots,v_n)$?
It is for a question involving a multil... | If I interpret your question correctly, then the clue is in $f:\mathbb{R}^2\times\mathbb{R}^2\times\mathbb{R}^2\to\mathbb{R}$, which seems to imply that the function is trilinear, that is, in three inputs of two dimensions each. In that case,
$\begin{align*}f((2,3),(-1,1),(7,4))
=&2\cdot -1\cdot 7\cdot f((1,0),(1,0),... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/283080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Application of fundamental theorem of calculus I have this problem:
$$ \frac{d}{dx} \left( \int_{\sqrt{x}}^{x^2-3x} \tan(t) dt \right) $$
I know how found the derivative of the integral from constant $a$ to variable $x$ so:
$$ \frac{d}{dx} \left( \int_a^x f(t) dt \right) $$
but I don't know how make it between two var... | First we work formally: you can write your integral, say $F(x)=\int_a^{g(x)}f(t)\,dt-\int_a^{h(x)}f(t)\,dt$, where $f,g$ and $h$ are the functions appearing in your problem, and $a\in\mathbb R$ is constant. Next, you can apply chain rule together with fundamental theorem of calculus in order to derivate the difference ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/283210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
How I study these two sequence?
Let $a_1=1$ , $a_{n+1}=a_n+(-1)^n \cdot 2^{-n}$ , $b_n=\frac{2 a_{n+1}-a_n}{a_n}$
(1) $\{\ {a_n\}}$ converges to $0$ and $\{\ {b_n\}}$ is a cauchy sequence .
(2) $\{\ {a_n\}}$ converges to non-zero number and $\{\ {b_n\}}$ is a cauchy sequence .
(3) $\{\ {a_n\}}$ converges to $0$ and $\... | We have $b_n=\frac{2 a_{n+1}-a_n}{a_n}=2\frac{a_{n+1}}{a_n}-1$. For very large values of $n$, since $a_n\to2/3$ we have $a_{n+1}\sim a_n$. So $b_n\to 2-1=1$ so it is Cauchy as well.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/283294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How I can find this limit?
If $a_n=(1+\frac{2}{n})^n$ , then find $$\lim_{n \to \infty}(1-\frac{a_n}{n})^n$$.
Trial: Can I use $$\lim_{n \to \infty}a_n=e^2$$ Again $$\lim_{n \to \infty}(1-\frac{a_n}{n})^n=\exp(-e^2)$$ Please help.
| Due to
$$(1-\frac{a_n}{n})^n=\left[\left(1-\frac{a_n}{n}\right)^{\frac{n}{-a_n}}\right]^{\frac{-a_n}{n}n}=\left[\left(1-\frac{a_n}{n}\right)^{\frac{n}{-a_n}}\right]^{-a_n}.$$
$$\lim_{n\to\infty}\left(1-\frac{a_n}{n}\right)^{\frac{n}{-a_n}}=e$$
and $$\lim_{n\to \infty}(-a_n)=-e^2$$
Let $A_n=\left(1-\frac{a_n}{n}\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/283396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
} |
Solving linear first order differential equation with hard integral I'm try to solve this differential equation: $y'=x-1+xy-y$
After rearranging it I can see that is a linear differential equation:
$$y' + (1-x)y = x-1$$
So the integrating factor is $l(x) = e^{\int(1-x) dx} = e^{(1-x)x}$
That leaves me with an integral ... | A much easier way without an integrating factor:
$y′=x−1+xy−y$
$y′=x-1+y(x-1)$
$y′=(x-1)(1+y)$
$\frac{dy}{(1+y)} = (x-1)dx$
$ln|1+y| = x^2/2 -x + C$
And you can do the rest
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/283450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Prove that given a nonnegative integer $n$, there is a unique nonnegative integer $m$ such that $(m-1)^2 ≤ n < m^2$ Prove that given a nonnegative integer $n$, there is a unique nonnegative integer $m$ such that $(m-1)^2 ≤ n < m^2$
My first guess is to use an induction proof, so I started with n = k = 0:
$(m-1)^2 ≤ 0 <... | Its too late to answer the question but if it helps you can prove it by contradiction also.
Assume that there exists a k such that k is less than m.
so
(k−1)2≤n< k2
The smallest k which is possible is k = m-1. Then we have
(m-2)2 ≤ n< (m-1) 2
which is contradicting the assumed statement. so the solution has a uniqu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/283515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
$\lim_{x \to 0} \frac {(x^2-\sin x^2) }{ (e^ {x^2}+ e^ {-x^2} -2)} $ solution? I recently took an math exam where I had this limit to solve
$$ \lim_{x \to 0} \frac {(x^2-\sin x^2) }{ (e^ {x^2}+ e^ {-x^2} -2)} $$
and I tought I did it right, since I proceeded like this:
1st I applied Taylor expansion of the terms to th... | How is $\frac{0+o(x^2)}{0+o(x^2)}$ zero?
You need to expand to a degree high enough to keep something nontrivial after cancellation!
Note that $\sin(x^2)=x^2-\frac12 x^4+o(x^6)$ and
$e^{\pm x^2}=1+\pm x^2+\frac 12 x^4+o(x^6)$, hence
$$f(x)= \frac{\frac12 x^4 + o(x^6)}{x^4+o(x^6)}=\frac{\frac12 + o(x^2)}{1+o(x^2)}\to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/283585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 3
} |
How many combinations of coloured dots (with restrictions)? My friend is designing a logo. The logo can essentially be reduced to 24 coloured dots arranged in a circle, and they can be either red or white. We want to produce a individual variation of this logo for each employee. That, if I have worked it out right, (si... | If rotations of the circle are allowed, you need to apply Pólya's coloring theorem. The relevant group for just rotations of 24 elements is $C_{24}$,
whose cycle index is:
$$
\zeta_{C_{24}}(x_1, \ldots x_{24}) = \frac{1}{24} \sum_{d \mid 24} \phi(d)x_d^{24 / d}
= \frac{1}{24} \left( x_1^{24} + x_2^{12} + 2 x_3^{8} + 2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/283662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Weak convergence of a sequence of characteristic functions I am trying to produce a sequence of sets $A_n \subseteq [0,1] $ such that their characteristic functions $\chi_{A_n}$ converge weakly in $L^2[0,1]$ to $\frac{1}{2}\chi_{[0,1]}$.
The sequence of sets
$$A_n = \bigcup\limits_{k=0}^{2^{n-1} - 1} \left[ \frac{2k}... | Suggestions:
*
*First consider the case where $g$ is the characteristic function of an interval.
*Generalize to the case where $g$ is a step function.
*Use density of step functions in $L^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/283737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Homogeneous system of linear equations over $\mathbb{C}$ I have two systems of linear equations and I need to verify if they are indeed the same system, and if they are I must rewrite each equation as a linear combination of the others. | In B, multiply 2nd equation by $i$, add to 1st equation (so $x_3$ disappears), solve for $x_1$ in terms of $x_2$ and $x_4$, substitute this into either original equation of B, solve for $x_3$ in terms of $x_2$ and $x_4$, compare with your answer for A.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/283853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Have I justified that $\forall x \in \mathbb{R}$, $x > 1 \rightarrow x^2 > x$ Have I justified that $\forall x \in \mathbb{R}$, $x > 1 \rightarrow x^2 > x$
Here is what I would do if this were asked on a test and I was told to "justify" the answer.
Let $x \in \mathbb{R}$
Assume $x$ is greater than $1$.
Then $x * x > x$... | It indeed does fly. If you multiply both sides of an inequality by a positive quantity, the inequality is preserved.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/283909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Partial Derivatives involving Vectors and Matrices Let $Y$ be a $(N \times 1)$ vector, $X$ be a $N \times M$ matrix and $\lambda$ be a $M \times 1$ vector.
I am wondering how I can evaluate the following partial derivative.
\begin{align}
\frac{\partial (Y-X\lambda)^T (Y-X\lambda)}{\partial \lambda_j}
\end{align}
where... | See the entry on Matrix Calculus in Wikipedia, or search for "matrix calculus" on the internet. In your particular case,
$$
\frac{\partial (Y-X\lambda)^T (Y-X\lambda)}{\partial\lambda^T}=-2(Y-X\lambda)^T X
$$
and hence the partial derivative w.r.t. $\lambda_j$ is the $j$-th entry of the row vector $-2(Y-X\lambda)^T X$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/283981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Simple proof for uniqueness of solutions of linear ODEs? Consider the system of linear ODEs $\dot{x}(t)=Ax(t)$, $x(0)=x_0\in\mathbb{R}^n$. Does anyone know a simple proof showing that the solutions are unique that does not require resorting to more general existence/uniqueness results (e.g., those relating to the Picar... | Since the students are engineers, why don't you want to show them explicit solutions, which surely they'd need to see anyway? If we knew about a matrix exponential $e^{At}$, then to show $x(t) = e^{At}x_0$ let's look at the $t$-derivative of $e^{-At}x(t)$, which is
$$
e^{-At}x'(t) + (-Ae^{-At})x(t) = e^{-At}Ax(t) - A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/284061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
A question about polynomial rings This may be a trivial question. We say an ideal $I$ in a ring $R$ is $k$-generated iff $I$ is generated by at most $k$ elements of $R$. Let $F$ be a field. Is it true that every ideal in $F[x_1,x_2,....,x_n]$ is $n-$generated. (This is true when $n=1$, because $F[x_1]$ is a PID)
Seco... | Since Qiaochu has answered your first question, I'll answer the second: yes, every ideal $I\subset F[x_1,x_2,x_3,...]$ is generated by a countable set of elements of $F[x_1,x_2,x_3,...]$.
Indeed, let $G_n\subset I_n$ be a finite set of generators for the ideal $I_n=I\cap F[x_1,x_2,x_3,...,x_n]$ of the noetherian ring... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/284127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
How many 8-character passwords can be created with given constrains How many unique 8-character passwords can be made from the letters $\{a,b,c,d,e,f,g,h,i,j\}$ if
a) The letters $a,b,c$ must appear at least two times.
b) The letters $a,b,c$ must appear only once and $a$ and $b$ must appear before $c$.
So for the first... | For the first:
count the number of passwords that do not satisfy the condition, then subtract from the total number of passwords
For the second:
Lay down your 5 "non-a,b,c" letters in order. There are $7^5$ ways to do this.
Then you have to lay down the letters a,b,c in the 6 "gaps" between the 5 letters (don't forget ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/284259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Derive the Quadratic Equation Find the Quadratic Equation whose roots are $2+\sqrt3$ and $2-\sqrt3$.
Some basics:
*
*The general form of a Quadratic Equation is $ax^2+bx+c=0$
*In Quadratic Equation, $ax^2+bx+c=0$, if $\alpha$ and $\beta$ are the roots of the given Quadratic Equation, Then,
$$\alpha+\beta=\frac{-b}... | Here $$-\frac ba=\alpha+\beta=2+\sqrt3+2-\sqrt3=4$$ and
$$\frac ca=\alpha\beta=(2+\sqrt3)(2-\sqrt3)=2^2-3=1$$
So, the quadratic equation becomes $$x^2-4x+1=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/284338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
How many possible arrangements for a round robin tournament? How many arrangements are possible for a round robin tournament over an even number of players $n$?
A round robin tournament is a competition where $n = 2k$ players play each other once in a heads-up match (like the group stage of a FIFA World Cup). To accomm... | This is almost the definition of a "$1$-factorization of $K_{2k}$", except that a $1$-factorization has an unordered set of matchings instead of a sequence of rounds. Since there are $2k-1$ rounds, this means that there are $(2k-1)!$ times as many tournaments, according to the definition above, as there are $1$-factor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/284416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Relation between the notions connected and disconnect, confused In the textbook "Topology without tears" I found the definition.
$(X, \tau)$ is diconnected iff there exists open sets $A,B$ with $X = A \cup B$ and $A \cap B = \emptyset$.
In Walter Rudin: Principles of Analysis, I found.
$E \subseteq X$ is connected iff ... | First, note that one should (in both versions) add that $A,B$ should be nonempty.
If $A,B$ are open and disjoint, then also $\overline A$ and $B$ are disjoint as $\overline A$ is the intersection of all closed sets containing $A$, thus $\overline A$ is a subset of the closed set $X\setminus B$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/284507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Combinatorial Correctness of one-to-one functions Let $\lbrack k\rbrack$ be the set of integers $\{1, 2, \ldots, k\}$. What is the number of one-to-one functions from $m$ to $n$ if $m \leq n$? My answer is: $\dfrac{n!}{(n-m)!}$
My reasoning is the following:
We have an $m$-step, independent process:
Step 1: choose the... | It is OK, modulo minor problems. You don't have to select the $m$'s, just go with the natural order 1, 2, ... Take a look at the notation suggested by Knuth et al in "Concrete Mathematics", it really does clean up much clutter.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/284576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If a and b are relatively prime and ab is a square, then a and b are squares.
If $a$ and $b$ are two relatively prime positive integers such that $ab$ is a square, then $a$ and $b$ are squares.
I need to prove this statement, so I would like someone to critique my proof. Thanks
Since $ab$ is a square, the exponent of... | Yes, it suffices to examine the parity of exponents of primes. Alternatively, and more generally, we can use gcds to explicitly show $\rm\,a,b\,$ are squares. Writing $\,\rm(m,n,\ldots)\,$ for $\rm\, \gcd(m,n,\ldots)\,$ we have
Theorem $\rm\ \ \color{#C00}{c^2 = ab}\, \Rightarrow\ a = (a,c)^2,\ b = (b,c)^2\: $ if $\rm\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/284636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
} |
$\binom{n}{n+1} = 0$, right? I was looking at the identity $\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}, 1 \leq r \leq n$, so in the case $r = n$ we have $\binom{n}{n} = \binom{n-1}{n-1} + \binom{n-1}{n}$ that is $1 = 1 + \binom{n-1}{n}$ thus $\binom{n-1}{n} = 0$, am I right?
| This is asking how many ways you can take $n$ items from $n-1$ items - there are none. So you are correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/284791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Question about theta of $T(n)=4T(n/5)+n$ I have this recurrence relation $T(n)=4T(\frac{n}{5})+n$ with the base case $T(x)=1$ when $x\leq5$. I want to solve it and find it's $\theta$. I think i have solved it correctly but I can't get the theta because of this term $\frac{5}{5^{log_{4}n}}$ . Any help?
$T(n)=4(4T(\fra... | An alternative approach is to prove that $T(n)\leqslant5n$ for every $n$. This holds for every $n\leqslant5$ and, if $T(n/5)\leqslant5(n/5)=n$, then $T(n)\leqslant4n+n=5n$. By induction, the claim holds.
On the other hand, $T(n)\geqslant n$ for every $n\gt5$, hence $T(n)=\Theta(n)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/284848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
$AB-BA=I$ having no solutions The following question is from Artin's Algebra.
If $A$ and $B$ are two square matrices with real entries, show that $AB-BA=I$ has no solutions.
I have no idea on how to tackle this question. I tried block multiplication, but it didn't appear to work.
| Eigenvalues of $AB \text{ and }BA$ are equal.Therefore, 0 must be the eigenvalue of $AB-BA$. The product of all eigenvalues is the determinant of the operator. Hence, $$|AB-BA|=|I| \implies 0=1, \text{ which is a contradiction }$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/284901",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "37",
"answer_count": 6,
"answer_id": 5
} |
How to find finite trigonometric products I wonder how to prove ?
$$\prod_{k=1}^{n}\left(1+2\cos\frac{2\pi 3^k}{3^n+1} \right)=1$$
give me a tip
| Let $S_n = \sum_{k=0}^n 3^k = \frac{3^{n+1}-1}{2}$. Then
$$3^{n}- S_{n-1} = 3^{n} - \frac{3^{n}-1}{2} = \frac{3^{n}+1}{2} = S_{n-1}+1.
$$
Now by induction we have the following product identity for $n \geq 0$:
$$
\begin{eqnarray}
\prod_{k=0}^{n}\left(z^{3^k}+1+z^{-3^k}\right) &=& \left(z^{3^{n}}+1+z^{-3^{n}}\right)\pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/284971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
Irreducibility preserved under étale maps? I remember hearing about this statement once, but cannot remember where or when. If it is true i could make good use of it.
Let $\pi: X \rightarrow Y$ be an étale map of (irreducible) algebraic varieties and let $Z \subset Y$ be an irreducible subvariety.
Does it follow that $... | Hmm...what about $\mathbb{A}^1 - 0 \rightarrow \mathbb{A}^1$ - 0, with $z \mapsto z^2$? Then the preimage of 1 is $\pm 1$, which is not irreducible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/285117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Finding solutions using trigonometric identities I have an exam tomorrow and it is highly likely that there will be a trig identity on it. To practice I tried this identity:
$$2 \sin 5x\cos 4x-\sin x = \sin9x$$
We solved the identity but we had to move terms from one side to another.
My question is: what are the thing... | The intended techniques all follow from using the sine and cosine addition formulas and normalization, which you should have seen before.
However, I wanted to point out that a more unified approach to the general problem of proving trig. identities is to work in the complex plane. For instance $e^{ix}=\cos(x)+i\sin(x)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/285189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
limit of the sum $\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n} $ Prove that : $\displaystyle \lim_{n\to \infty} \frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+\cdots+\frac{1}{2n}=\ln 2$
the only thing I could think of is that it can be written like this :
$$ \lim_{n\to \infty} \sum_{k=1}^n \frac{1}{k+n} =\lim_{n\to \inf... | We are going to use the Euler's constant
$$\lim_{n\to\infty}\left(\left(1+\frac{1}{2}+\cdots+\frac{1}{2n}-\ln (2n)\right)-\left(1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n\right)\right)=\lim_{n\to\infty}(\gamma_{2n}-\gamma_{n})=0$$
Hence the limit is $\ln 2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/285308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 7,
"answer_id": 1
} |
a theorem in topology Is there anyone know there is a theorem in topology which states that a compact manifold "parallelizable" with N smooth independent vector fields. must be an N-torus? and why the vector field here is parallel to manifold ?
| I think you are talking about a theorem due to V.I. Arnold: you can find more details in "Mathematical methods of classical mechanics", chapter 10. Here is the statement.
Theorem: Let $M$ be a n-dimensional compact and connected manifold and let $Y_{1},...,Y_{n}$ be smooth vector fields on M, commuting each other. If, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/285354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why is $\lim_{x \to 0} {\rm li}(n^x)-{\rm li}(2^x)=\log\left(\frac{\log(n)}{\log(2)}\right)$? I'm trying to give at least some partial answers for one of my own questions (this one).
There the following arose:
$\hskip1.7in$ Why is $\lim_{x \to 0} {\rm li}(n^x)-{\rm li}(2^x)=\log\left(\frac{\log(n)}{\log(2)}\right)$?
... | $$
\begin{align}
\lim_{x\to0}\int_{2^x}^{n^x}\frac{\mathrm{d}t}{\log(t)}
&=\int_{x\log(2)}^{x\log(n)}\frac{e^u}{u}\mathrm{d}u\\
&=\lim_{x\to0}\left(\color{#C00000}{\int_{x\log(2)}^{x\log(n)}\frac{e^u-1}{u}\mathrm{d}u}
+\color{#00A000}{\int_{x\log(2)}^{x\log(n)}\frac{1}{u}\mathrm{d}u}\right)\\
&=\color{#C00000}{0}+\lim_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/285406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Reflection around a plane, parallel to a line I'm supposed to determine the matrix of the reflection of a vector $v \in \mathbb{R}^{3}$ around the plane $z = 0$, parallel to the line $x = y = z$. I think this means that, denoting the plane by $E$ and the line by $F$, we will have $\mathbb{R}^{3} = E \oplus F$ and thus ... | The definition is exactly as stated in the question.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/285490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Intermediate Value Theorem guarantee I'm doing a review packet for Calculus and I'm not really sure what it is asking for the answer?
The question is:
Let f be a continuous function on the closed interval [-3, 6]. If f(-3)=-2 and f(6)=3, what does the Intermediate Value Theorem guarantee?
I get that the intermediate ... | Since $f(-3)=-2<0<3=f(6)$, we can guarantee that the function has a zero in the interval $[-3,6]$. We cannot conclude it has only one, though (it may be many zeros).
EDIT: As has already been pointed out elsewhere, the IVT guarantees the existence of at least one $x\in[-3,6]$ such that $f(x)=c$ for any $c\in[-2,3]$. No... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/285536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
When the spectral radius of a matrix $B$ is less than $1$ then $B^n \to 0$ as $n$ goes to infinity Hello how to show the following fact?
When the spectral radius of a matrix $B$ is less than $1$ then $B^n \to 0$ as $n$ goes to infinity
Thank you!
| There is a proof on the Wikipedia page for spectral radius.
Also there you will find the formula $\lim\limits_{n\to\infty}\|B^n\|^{1/n}$ for the spectral radius, from which this fact follows. However, the Wikipedia article's author(s) used the result in your question to prove the formula.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/285603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Show that $\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}$
Consider a function $f$ on non-negative integer such that $f(0)=1,f(1)=0$ and $f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)$ for $n \geq 2$. Show that
$$\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}$$
Here
$$f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)$$
$$\implies f(n)=(n-1)(f(n-... | Let
$$ g(n) = \sum_{k=0}^n(-1)^k\frac{n!}{k!} \tag{1} $$
then
\begin{align}
g(n) &= \sum_{k=0}^n(-1)^k\frac{n!}{k!} \\
&= n\sum_{k=0}^{n-1}(-1)^k\frac{(n-1)!}{k!} + (-1)^n\frac{n!}{n!} \\ \\ \\
&= ng(n-1)+(-1)^n \\ \\
&= (n-1)g(n-1) +g(n-1)+(-1)^n \\
&= (n-1)g(n-1)+\Big((n-1)g(f-2)+(-1)^{n-1}\Big) + (-1)^n\\
&= (n-1)g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/285672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
maximum modulus principle on $\lbrace z : |f(z)| \geq \alpha \rbrace$
Let $f(z)$ be an entire function that is not identically constant. Show that
$$\lbrace z : |f(z)| \geq \alpha \rbrace = \text{cl }\lbrace z : |f(z)| > \alpha \rbrace.$$
This question was in our exam and hinted that we had to apply the maximum modu... | Let's prove this by showing inclusion in both directions.
Let $w$ be a limit point of $E = \{z : |f(z)| > a\}$. This means that there is a sequence $\{z_k\} \subset E$ so that $z_k \to w$ as $k \to \infty$. Since $f$ is continuous, it follows that $|f(w)| \ge a$ and $\operatorname{cl}(E) \subset \{z : |f(z)| \ge a\}$.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/285725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What is the homology group of the sphere with an annular ring? I'm trying to compute the homology groups of $\mathbb S^2$ with an annular ring whose inner circle is a great circle of the $\mathbb S^2$.
space X
Calling this space $X$, the $H_0(X)$ is easy, because this space is path-connected then it's connected, thus $... | If I understand your space correctly, it seems you could do a deformation retraction onto $S^2$, and hence $H_1(X)=H_1(S^2)=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/285793",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
vanishing of higher derived structure sheaf given a field $k$ and a proper integral scheme $f:X\rightarrow \operatorname{Spec}(k)$, is it true that $f_{*}\mathcal{O}_{X}\cong \mathcal{O}_{\operatorname{Spec}(k)}$?
Consider the normalization $\nu:X_1\rightarrow X$,let $g:X_1\rightarrow \operatorname{Spec}(k)$ be the st... | The isomorphism $f_{*}O_X=k$ holds if $k$ is algebraically closed. Otherwise, take a finite non-trivial extension $k'/k$ and $X=\mathrm{Spec}(k)$ you will get a counterexample.
A sufficient condition over an arbitrary field is $X$ is proper, geometrically connected (necessary) and geometrically reduced. This amounts t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/285855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
True/false question: limit of absolute function I have this true/false question that I think is true because I can not really find a counterexample but I find it hard to really prove it. I tried with the regular epsilon/delta definition of a limit but I can't find a closing proof. Anyone that
If $\lim_{x \rightarrow a... | Let $f$ be constant $1$ and $A:=-1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/285917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Need help with an integration word problem. This appears to be unsolvable due to lack of information. I'm not sure I understand what to do with what's given to me to solve this. I know it has to do with the relationship between velocity, acceleration and time.
At a distance of $45m$ from a traffic light, a car traveli... | Hint: Constant acceleration means that the velocity $v(t)=v(0)+at$ where $a$ is the acceleration. The position is then $s(t)=s(0)+v(0)t+\frac 12 at^2$. You should be able to use these to answer the questions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/285975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Linear independence in Finite Fields How can we define linear independence in vectors over $\mathbb{F_{2^m}}$ ?
Let vectors $v_1,v_2,v_3$ $\in$ $\mathbb{F_{2^m}}$,
If $v_1,v_2,v_3$ are linearly independent,then
$\alpha_1v_1+\alpha_2v_2+\alpha_3v_3$=0 if and only if $\alpha_1=\alpha_2=\alpha_3=0$ and
$\alpha_1,\alpha_2... | Linear independence is defined the same way in every vector space:
$\{v_i\mid i\in I\}$ is a linearly independent subset of $V$ if $\sum_{i=1}^n \lambda_i v_i=0$ implies all the $\lambda_i=0$ for all $i$, where the $\lambda_i$ are in the field.
In short, you definitely would not take the $\lambda_i$ from $F^m$. You are... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/286027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to efficiently compute the determinant of a matrix using elementary operations? Need help to compute $\det A$ where
$$A=\left(\begin{matrix}36&60&72&37\\43&71&78&34\\44&69&73&32\\30&50&65&38\end{matrix} \right)$$
How would one use elementary operations to calculate the determinant easily?
I know that $\det A=1$
| I suggest Gaussian Elimination till upper triangle form or further but keep track of the effect of each elementary.
see here for elementary's effect on det
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/286080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Limit question with absolute value: $ \lim_{x\to 4^{\large -}}\large \frac{x-4}{|x-4|} $ How would I solve the following limit, as $\,x\,$ approaches $\,4\,$ from the left?
$$
\lim_{x\to 4^{\large -}}\frac{x-4}{|x-4|}
$$
Do I have to factor anything?
| Hint: If $x \lt 4, |x-4|=4-x$. Now you can just divide.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/286140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Function in $L^1([0,1])$ that is not locally in any $L^{\infty}$ Can we find a function such that $f\in L^1([0,1])$ and for any $0\leq a<b\leq 1$ we have that $||f||_{L^{\infty}([a,b])}=\infty$?
| Yes, we can. Consider $\{r_j,j\in\Bbb N\}$ an enumeration of rational numbers of $[0,1]$ and
$$f(x):=\sum_{j=1}^{+\infty}\frac{2^{—j}}{\sqrt{|x-r_j|}}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/286222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Nondeterministic finite automaton proof I am having a really hard time working the problem below out.
I am not sure I am even on the right direction with this logic . Swapping the accept and reject states alone is not sufficient to accept all string of the language ~ L. We would need to swap the transition directions ... | $\bar L$ is the complement of $L$, that is, $\bar L$ is the set of strings that are not in $L$.
Hint: make a nondeterministic automaton that accepts every string, and so that if you switch the accepting and rejecting states it still accepts every string.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/286295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.