Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Seeking Name of Theory for multiple integral Consider two definite integrals:
\begin{equation}
I_1 = \int_{R_1} f(x) \:dx\qquad I_2 = \int_{R_2} g(y) \:dy
\end{equation}
Then,
\begin{equation}
I_1 \cdot I_2 = \left[ \int_{R_1} f(x) \:dx \right] \left[ \int_{R_2} g(y) \:dy\right]
\end{equation}
Which under certain c... | This is just linearity of the integral. It's automatic as long as the product $I_1\cdot I_2$ itself is meaningful (i.e. we need $I_1, I_2, I_1I_2$ to exist in some sense, so we shouldn't have e.g. $I_1=0, I_2 = \infty$)
Edit: After thinking a little more, it occurs to me that we also need the "inner" integral in your ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3320552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Solving $2\left(\sqrt{2s-16}-\sqrt{s}\right)-8=0$ I am trying to solve the equation
$$2\left(\sqrt{2s-16}-\sqrt{s}\right)-8=0$$
Using regular method I found two roots of this namely $32(2+\sqrt{3})$ and $32(2-\sqrt{3})$. But when I tried to confirm them only $32(2+\sqrt{3})$ worked as a root whereas $32(2-\sqrt{3})$ ... | Whenever you square an equation you get additional roots. For example $x=1$ has a unique solution but $x^{2}=1^{2}$ has two solutions $x=1$ and $x=-1$. Your 'regular method' involves squaring so you got an extra root. After getting the two values for $s$ you have to go back to the given equation and keep only the one t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3320682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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The set of all conjugation classes of group $G$ form a partition of $G$ So the property I would like to proof is the same as stated in the title:
Consider the group $G,\cdot$. The set of all conjugation classes of group $G$ form a partition of $G$
So to prove this, one needs to show that the union of conjugation classe... | Hint:
You can prove this by showing that the relation $\sim$ on $G$ defined by:$$g\sim h\iff \exists x\in G\;[gx=xh]$$is an equivalence relation.
The equivalence classes (they form a partition of $G$) are exactly the conjugation classes.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3320789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Conjugate diameters of ellipse How to find the length of major and minor axis of ellipse given the length of two conjugate diameters and the angle between them?
I am aware about how to construct the ellipse using the above given facts(not by Rytz's Construction). I would like to know, independent of what method of cons... | Here's a geometric construction: if $MN$ and $DE$ are conjugate diameters, draw line $QQ'$ through $N$ perpendicular to $DE$ (see diagram below). Points $Q$ and $Q'$ must be chosen such that $NQ=NQ'=OD$. Major axis $IR$ is the bisector of angle $\angle QOQ'$ and minor axis $TS$ is perpendicular to it.
Their lengths can... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3320901",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What p.d.f. over angles is equivalent to a uniform distribution over a hypersphere? Eric Weisstein's Sphere Point Picking points out that sampling uniformly from each angle $\phi$ and $\theta$ in spherical coordinates does not sample from the uniform sphere because it clusters near the poles. I am interested in which d... | If you're using hyperspherical coordinates, the area element of the $n$-sphere can be written $$ \sin^{n-1}(\phi_1)\sin^{n-2}(\phi_2)\ldots \sin(\phi_{n-1})d\phi_1\ldots d\phi_n$$ where $\phi_1,\ldots\phi_{n-1}$ range from $0$ to $\pi$ and $\phi_n$ ranges from $0$ to $2\pi.$
So this gives you the pdf's of the angles ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3321025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Is there a matrix that can be used to find the transpose of a matrix? Let $A$ be a general $n\times n$ invertible matrix. Let $T^A$ be the "transposer" matrix i.e. $T^A A = A'$. (Does that $T^A$ multiplied by $A$ equal the transpose of $A$?) Then does $T^A$ depend on the matrix $A$: is $T^A = T^B$ for all invertible Ma... | There is no general transposer matrix.. For this, e.g. note that the only transposer for the identity in any dimension is the identity as the equation
$$T^II=I$$
needs to be fulfilled, but $T^II=T^I$.
However, for e.g. for non-symmetric matrices $A$, the identity is definitely not the transposer, as $A^\top\neq A$, but... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3321172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Prove the zeros of a polynomial all lie in an annulus. I am working on a problem. It has two parts:
(a) Let $c_{0}>c_{1}>\cdots c_{n}>0$. Show that the polynomial $P(z):=c_{0}+c_{1}z+\cdots+c_{n}z^{n}$ has no zeros inside the closed unit disc.
(b) Show that the zeros of polynomial $P_{n}(z):=1+\frac{z}{2}+\frac{z^{2}... | Consider the polynomial $R_n(z)=(z-1)P_{n}(z)=-1+\dfrac{z}{2}+\dfrac{z^{2}}{6}+\cdots+\dfrac{z^{n}}{n(n+1)}+\dfrac{z^{n+1}}{n+1}$.
We will show that the Cauchy bound $\rho_n=1+\gamma_n$ of $R_n$, satisfies $\gamma_n \to 0$ as $n \to \infty$ hence we are done since all the roots $w_{k,n}$ of $R_n$ (hence of $P_n$) sati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3321265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Show that if vectors $(\overline{v},\overline{w}) \in V$ are linearly independent then they are not parallel Problem
Show that if vectors $(\overline{v},\overline{w}) \in V$ are linearly independent and neither of them is zero vector then they are not parallel
Attempt to solve
vectors $\overline{v},\overline{w}$ are li... | By the contrapositive, if they are parallel, then there must exist a scalar $\alpha$ such that $\bar{v} - \alpha \bar{w} = 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3321379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Evaluating $\lim_{k\to\infty}\sum_{n=1}^{\infty} \frac{\sin\left(\pi n/k\right)}{n}$ Recently, I was asked by a friend to compute the limit of the following series
$$\displaystyle{\lim_{k\to\infty}}\sum_{n=1}^{\infty} \frac{\sin\left(\frac{\pi n}{k}\right)}{n}$$
Having seen a similar problem to this before, Difficul... | The Riemann Sum would be
$$
\begin{align}
\lim_{k\to\infty}\sum_{n=1}^\infty\frac{\sin\left(\frac{\pi n}k\right)}{n}
&=\lim_{k\to\infty}\sum_{n=1}^\infty\frac{\sin\left(\frac{\pi n}k\right)}{n/k}\frac1k\\
&=\int_0^\infty\frac{\sin(\pi x)}x\,\mathrm{d}x\\
&=\int_0^\infty\frac{\sin(x)}x\,\mathrm{d}x\\[3pt]
&=\frac\pi2\ta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3321478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Why $a_{-1}$ term of Laurent series may not be residue? Suppose $f(z)$ is analytic on $0<|z-z_0|<R$. And we find a Laurent's series for $f(z)$ on annulus $r<|z-z_0|<R$ where $r$ may not be $0$. Then it is said that $a_{-1}$ of such Laurent's series may not be residue unless $r=0$ (Residue is defined as $Res(f,z_0)=\fra... | Don't confuse $$\int_{|z| = r+\epsilon} f(z)dz= 2i\pi a_{-1}, \qquad \int_{|z| = \epsilon} f(z)dz = 2i\pi b_{-1}$$
where $f$ is assumed to be analytic on $|z| \in (0,2\epsilon)$ and $|z| \in (r,R)$ and $a_n,b_n$ are the Laurent coefficients of the expansion on each annulus.
If $f$ is analytic on $|z|\in (0, R)$ then $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3321549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Calculate number of elements in a set How many natural numbers are there below 1000 that are multiples of 3 or that contain 3 in any digit of the number?
My effort : Here we need to calculate union of two set. First set is natural number which are multiple of 3. So it's cardinality will be the nearest integer of 1000/3... | How many numbers have $3$ as a first digit? How many number that do not have $3$ as a first digit have $3$ as a second? How many that do not have $3$ in one of the first two positions have $3$ in the third? Add these together to answer your question. You can't just add this to $333$ because all the ones that are mu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3321630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Simplify $ \frac{ \sqrt[3]{16} - 1}{ \sqrt[3]{27} + \sqrt[3]{4} + \sqrt[3]{2}} $ Simplify
$$ \frac{ \sqrt[3]{16} - 1}{ \sqrt[3]{27} + \sqrt[3]{4} + \sqrt[3]{2}} $$
Attempt:
$$ \frac{ \sqrt[3]{16} - 1}{3 + \sqrt[3]{4} + \sqrt[3]{2}} = \frac{ \sqrt[3]{16} - 1}{ (3 + \sqrt[3]{4}) + \sqrt[3]{2}} \times \frac{ (3 + \sqrt[... | Let $x=\sqrt[3]2$ then we have $${x^4-1\over x^2+x+3}={x^6-x^2\over x(x^3+x^2+3x)}={4-x^2\over x(2+x^2+3x)}= {(2-x)(2+x)\over x(x+2)(x+1) }$$
$$ = {2-x\over x^2+x}= {(2-x)(x-1)\over x(x+1)(x-1)}= {(2-x)(x-1)\over x^3-x}$$
$$= {(2-x)(x-1)\over 2-x} = x-1$$
Edit: but other solution is much nicer then this one.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3321716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Find the kernel of a ring homomorphism $f(x)\to f(\sqrt 2)$ Let $$\phi:\mathbb Z[x]\longrightarrow \mathbb R$$, where $$\phi(f(x))=f(\sqrt 2)$$
find $$\operatorname{Ker}(\phi)=\{f(x)\in \mathbb Z[x]\mid f(\sqrt2)=0\}$$
1st) I wanted to use isomorphism theorem since we have $\mathbb R$-field so $Z[x]/\ker\phi\simeq \mat... | $\newcommand\Ker{\operatorname{Ker}}$First note that $x^2-2\in\Ker\varphi$.
Conversely, if $f\in\Ker\varphi$, write $f(x)=(x^2-2)q(x)+r(x)$ with $r(x)=0$ or $\deg r(x)<2$.
Since $f(\sqrt 2)=0$ you get $r(\sqrt 2)=0$, but since $\sqrt 2$ is irrational and $\deg r<2$ we get $r=0$.
This proves $(x^2-2)|f(x)$.
Consequently... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3321829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Linear differential equation with driving In class we are solving the linear differential equation with driving given by
$$
\frac{dx}{dt} = -\gamma x + f(t)
$$
The professor first transformed to a new variable $y$,
$$
y(t) := x(t) e^{\gamma t}
$$
and then calculate the differential equation for $y$:
$$
\frac{dy}{dt} = ... | Differentiate $y(t) = x(t) e^{\gamma t}$ with the product rule:
$$y'(t)=x'(t)e^{\gamma t}+x(t) \gamma e^{\gamma t}.$$
Since $x'(t)=- \gamma x(t)+f(t)$, we get
$$y'(t)=(- \gamma x(t)+f(t))e^{\gamma t}+x(t)\gamma e^{\gamma t}=f(t)e^{\gamma t}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3321929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How the components of alternating link diagram $D$ are boundaries of the regions of one color after performing positive smoothing at all crossings? I am reading the proof of the Proposition 5.3. of the chapter "The Jones Polynomial of an Alternating Link" from the book "Introduction to knot theory" by "Lickorish". I ha... | Here's what this is meant to mean:
Take a look at a region of the alternating knot diagram, which is a disk. Due to it being an alternating diagram, regions come in only two types, depending on what the incident crossings look like:
I am calling them type R and type L. Notice that around a crossing, type R and type ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3322019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to solve $M - EV - (EV)^T = 0$ against $E$? I have a matrix equation:
$$M - EV - (EV)^T = 0$$
which i want to solve against $E$. Is this possible? How to do that?
Remarks: matrices are square, $E$, $M$ are symmetric, $V$ is invertibile.
Regards,
Marek
| We can write this equivalently as
$$
EV + (EV)^T = M.
$$
Because the (linear) operator $E \mapsto EV + (EV)^T$ is not invertible, this equation will have infinitely many solutions. In particular, we can always take $E = \frac 12 MV^{-1}$. Indeed, plugging this $E$ in yields
$$
EV + (EV)^T = \frac 12 MV^{-1}V + (\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3322171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving that the equivalence class generates the group (mod 125). The Question:
Let n be a positive integer and let $G_n = \left\{[a] ∈ \mathbb{Z}_n ; \text{gcd}(a,n) = 1\right\}$ be the group of invertible elements in (Zn,·), where ”·” represents the product (mod n).
Prove that $(G_{125},·)$ is a group with 100 eleme... | Hint:
$2^{50} \equiv 1 \implies 2^{25}\equiv \pm 1 $
In general if $x^2\equiv 1\pmod{p^e}$, where $p$ is an odd prime,
then $x\equiv \pm 1 \pmod{p^e}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3322309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Computing $\int\frac {du}{\sqrt{u^2 + s^2}} =\log \lvert (u + \sqrt{u^2 + s^2}) \rvert$ with a substitution Can someone please show me where I am going wrong? It seems there is a contradiction in the formula for the the following integral.
$$\int\frac {du}{\sqrt{u^2 + s^2}} = \log \bigl\lvert u + \sqrt{u^2 + s^2}\bigr... | You have a faster way, using some hyperbolic trigonometry:
Set $t=\dfrac us$, so that $\mathrm du=s\,\mathrm dt$. The integral becomes, supposing $s>0\,$:
$$\int\frac {\mathrm du}{\sqrt{u^2 + s^2}}=\int\frac {\not\! s\,\mathrm dt}{\not \!s\sqrt{ t^2 + 1}}=\int\frac {\mathrm dt}{\sqrt{ t^2 + 1}}=\operatorname{argsinh}t$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3322560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Time series prediction for three constrained variables (x+y+z=1) How can I PREDICT time-series for three non-negative variables that sum to 1? Say, x+y+z=1. I have historical data for x ,y, z , t. Based on historical data, I can create an ARIMA model for each variable individually, and make predictions for the future. ... | tl;dr: Reduction of variables is a generic method to apply constraints. We show this below and it is easy because ARIMA is indifferent to affine linear transformations. However, at the end, we discover that we have just run ARIMA on two of the original variables and reconstructed the predictions of the third using th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3322653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Prove that for $n\in\mathbb{N}$, $\sum_{k=1}^{\infty}\left[\frac{n}{5^{k}}\right]=15\iff\left[\frac{n}{5}\right]=13.$ How to show that the following relation? : for $n\in\mathbb{N}$, $$\sum_{k=1}^{\infty}\left[\frac{n}{5^{k}}\right]=15\iff\left[\frac{n}{5}\right]=13.$$ It's not obvious to me. Can anyone help me? Thank ... | It's simple.
Let us assume $\left[\frac{n}{5}\right]=13$ to be true.
Since $\left[\frac{n}{5}\right]=13$,
$\Rightarrow\frac{n}{5}\in[13,14)$
$\Rightarrow n\in[13*5,14*5)$
$\Rightarrow n\in[65,70)$
Let us consider,
$$\sum_{k=1}^{\infty}\left[\frac{n}{5^{k}}\right]=\left[\frac{n}{5}\right]+\left[\frac{n}{5^2}\right]+\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3322763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Fourier sine expansion for $x(\pi - x)$ I'm trying to find the Fourier sine expansion for the function $f(x)$ = $x(\pi - x)$ for the interval $0 \leq x \leq \pi$.
I think I am supposed to find $\sum_{k=1}^{n} b_k \sin{(kx)}$ where $b_k = \frac{2}{\pi}$ $\int_{0}^{\pi} x(\pi - x) \sin{(kx)} dx$. However this is turning... | Your formula for $b_k$ is correct in this example. Compute the integral correctly, and you will see that you obtain $b_k=0$ when $k$ is even, while $$b_{2j+1}={8\over\pi(2j+1)^3}\qquad(j\geq 0)\ .$$
This leads to
$$f(x)=\sum_{j=0}^\infty b_{2j+1}\sin\bigl((2j+1)x\bigr)={8\over\pi}\sum_{j=0}^\infty{\sin\bigl((2j+1)x\big... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Can $e^x$ be expressed as a linear combination of $(1 + \frac x n)^n$? Can $e^x$ be expressed as a linear combination of $(1 + \frac x n)^n$? In other words, does there exist an infinite sequence $(a_k)_{k \in \mathbb N_0}$ such that $$e^x = a_0 + \sum_{1 \leq k < \infty} a_k \left(1 + \frac x k\right)^k$$
for all $x \... | Let $R_n(z)=\sum_{1 \leq k \le n} a_k \left(1 + \frac z k\right)^k$. Assuming the hypothesis we will show:
1: $R_n(z) \to e^z-a_0$ uniformly in the disc $|z| \le \frac{1}{2}$
2: $\sum_{1}^{\infty}{\frac{a_k}{k^q}}=0, q \ge 1$ arbitrary integer
3: $a_k=0, k \ge 1$
We use that if $|z| \le \frac{1}{2}, k \ge 1$, $|(1+\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3323005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 1,
"answer_id": 0
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How to find the shaded region
Find the area of the blue shaded region of the square in the following image:
[Added by Jack:]
The area of the triangle in the middle of the square is given by
$$
4.8\times 6=28.8\ (cm^2)
$$
Other than this, it seems difficult to go further with the given information. It seems that on... | Let us start with a square $ABCD$ and construct on the sides $AB$, $BC$, $CD$, $DA$ points $E,F,G,H$ so that $AE=BF=CG=DH$:
Let $X$ be the intersection $X=AG\cap BH$. Similar points $Y,Z,W$ obtained by rotation around the center of the square were also drawn. This realizes the situation from the given problem. We want... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3323175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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Evaluating the integral: $ I = \int e^{\frac xa} \sin x \, \mathrm dx$
Evaluating the integral:
$$ I = \int e^{\frac xa} \sin x \, \mathrm dx \tag {1}$$
This question was asked in CBSE Board 12th Grade (India). So, here was the approach I made.
Proposition 1: $$ for, \, y= u(x), \forall \, x \in \mathbb{R} $$
$$ \in... | There are several issues here.
*
*Your $(\ast)$ should read $\int e^{x/a}u(x)dx=e^{x/a}(au-a^2u^\prime+a^3u^{\prime\prime}-\cdots)+C$.
*We have $\int e^{x/a}\sin xdx=e^{x/a}(a\sin x-a^2\cos x-a^3\sin x+\cdots)+C$. Thanks to the powers of $a$, you can use a geometric series, $\frac{a}{1+a^2}e^{x/a}(\sin x-a\cos x)+C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3323330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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For any positive integer $a$, prove that there exists infinitely many composite $n$ such that $a^{n-1}\equiv 1\mod n$. Studying for an upcoming comprehensive exam we stumbled upon the following problem:
Prove that for every positive integer $a$, there exists infinitely many composite integers $n$ such that $a^{n-1}\equ... | First: If $p$ is a prime and
$$n=\frac{a^{2p}-1}{a^2-1} =1+a^2+a^4+...+a^{2p-2}$$
Then
$$ 1+a^2+a^4+...+a^{2p-2}=0 \pmod{n} \\
a^2( 1+a^2+a^4+...+a^{2p-2})=0 \pmod{n} \\
a^2+a^4+...+a^{2p-2}+a^{2p}=0 \pmod{n} \\
$$
Subtracting the first and last relation you get
$$1=a^{2p} \pmod{n}$$
Now, if you now chose $p,n$ so tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3323510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How does one derive the following formula of integration? $$\int_0^\infty\frac{\exp{\left(-\frac {y^2}{4w}-t^2w\right)}}{\sqrt {\pi w}}dw=\frac{\exp(-ty)}t$$ for $t$ and $y$ positive. This integral is useful in the following context: suppose we are given $$\int_0^\infty tf(t){\exp{\left(-t^2w\right)}}dt$$ (a function o... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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\new... | {
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Taylor series of $\ln(1+x+x^2+...+x^{10})$ I have to find Taylor series of $\ln(1+x+x^2+...+x^{10})$.
I have a clue that I write like the two difference of logarithm but I do not how.
Any help?
| Recall the formula for a finite geometric series:
$$1+x+x^2 + \cdots + x^n = \frac{1-x^{n+1}}{1-x}$$
Take $n=10$ and you can apply this to your problem fairly readily.
| {
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equation simplification. $(5y-1)/3 + 4 =(-8y+4)/6$ Simplification of this equation gives two answers when approched by two different methods.
Method 1 Using L.C.M( least common multiple)
$(5y-1)/3 + 4 =(-8y+4)/6$
$(5y-1+12)/3 = (-8y+4)/6$
$5y-11 = (-8y+4)/2$
$(5y-11)2= (-8y+4)$
$10y-22 = -8y+4$
$18y=26$
$y = 26/18=13/... | From the second step transitioning to the third step of your work, you incorrectly computed $5y-1+12$ to be $5y-11$ when it should be equal to $5y+11$. An obvious careless sign error.
| {
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Intersection of ideals $(x,y),(y,z)$ and $(x,z)$ in $K[x,y,z]$ I have to prove that the intersection of ideals $(x,y),(y,z)$ and $(x,z)$ is equal to the ideal generated by $xy ,yz$ and $xz$. I am unable to prove it using the definitions only.
| Since $xy, yz, xz \in (x,y)\cap (y, z) \cap (x, z)$, clearly $(xy, yz, xz) \subset (x,y)\cap (y, z) \cap (x, z)$. Suppose now that $p(x,y,z)\in (x,y)\cap (y, z) \cap (x, z)$. Fix a monomial term, $m(x,y,z)$ of $p(x,y,z)$. Since $p(x,y,z)\in (x,y)$ we have $p(x,y,z)=xp_1(x,y,z)+yp_2(x,y,z)$ for some polynomials $p_1$ a... | {
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Why does $x_0 + \sin(x_0) \approx \pi$, when computing it multiple times? Why does $x_0 + \sin(x_0) \approx \pi$ when computing this multiple times on the calculator?
So for any value of up to $\approx 6.25$, doing the following operation ($x_0 + \sin(x_0)$, then the newly obtained value (let's call it $x_1$) is repla... | You start with $x_0 \in \mathbb R$ and define recursively for $n \ge 0$
$$x_{n+1} = x_n + \sin x_n .$$
What can we say about the convergence of the sequence $(x_n)$?
If it converges to some $x \in \mathbb R$, then necessarily
$$x = \lim x_{n+1} = \lim (x_n + \sin x_n) = \lim x_n + \lim \sin x_n = \lim x_n + \sin (\lim ... | {
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Help to find complex Fourier series coefficient of this periodic function I'm having big trouble finding the complex Fourier series coefficient of the following periodic function
$$\frac{a-b\cos\varphi}{\sqrt{a^2+b^2-2ab\cos\varphi}}$$
Mathematica is unable to compute it!!
| You might combine the Legendre expansion (wlog $a>b$)
$$
\frac1{\sqrt{a^2+b^2-2ab\cos\varphi}}=\frac1a\sum_{n=0}^\infty\left(\frac ba\right)^nP_n(\cos\varphi)
$$
with the Fourier expansion (Gradshteyn-Ryzhik 8.826)
$$
P_n(\cos\varphi)=\\\frac{2^{n+2}n!}{\pi(2n+1)!!}\left(\sin(n+1)\varphi+\frac11\frac{n+1}{2n+3}\sin(n+3... | {
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solve sum of square roots using a single functiom using Newton's Method I have to estimate $n= \sqrt{3} + \sqrt{7}$ using Newton's Method of approximation but I have to determine a single function which can be used to estimate $n$.
P.S: The function should not involve radical expressions (nth root of constants or vari... | Considering that radical roots appear in conjugate pairs, you could construction a rational function that has the root $n=\sqrt{7}+\sqrt{3}$ as follows,
$$[x-(\sqrt{7}+\sqrt{3})][x-(\sqrt{7}-\sqrt{3})] = x^2 +4 -2\sqrt{7}x$$
$$ (x^2 +4 -2\sqrt{7}x)(x^2 +4 + 2\sqrt{7}x) = (x^2+4)^2-28x^2 $$
Thus, the function
$$f(x)=(x^... | {
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How does the product rule work for a double derivative, $\frac{\partial^2 }{\partial x^2}$? I have this function $$A_x=A_0\frac{1}{1+e^{-az}}e^{ik(z-ct)}$$ where $A_0$ is a constant and $a$ and $k$ are constants with dimensions of inverse length and $z \in \mathbb{R}$
I wish to compute $$\frac{\partial^2 A_x}{\partial ... | The product rule is slightly different for double derivatives. For simplicity, let $f$ and $g$ be twice continuously differentiable functions of $x$. Then,
$$\frac{d^2}{dx^2}fg = \frac{d}{dx}\left(\frac{d}{dx}fg\right) = \frac{d}{dx}\left(f'g + fg'\right) = \frac{d}{dx}f'g + \frac{d}{dx}fg' = f''g + f'g' + f'g' + fg'' ... | {
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Determining the truth of $\forall \; n \in Z, \exists \; a,b \in Z$ : $n=4a+5b$ $\implies$ $n^2 = 5a+4b-1$ The question at proving $\forall n \in Z,$ : ($\exists a,b \in Z$ $n=4a+5b$) $\implies$ ($\exists a,b \in Z$ $n^2 = 5a+4b-1$) originally asked one thing but then it was corrected to ask something quite different. ... | If $n=4a+5b$ and $n^2+1=5a+4b$, then $n^2+n+1=9(a+b)$, which implies $9$ divides $n^2+n+1$. This in turn implies $9$ divides $4n^2+4n+4=(2n+1)^2+3$. But this is impossible, since if $9$ divided $(2n+1)^2+3$, then $3$ would have to divide $2n+1$, in which case $(2n+1)^2+3$ leaves remainder $3$ when divided by $9$.
| {
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eigenvector transformation Say you have the following matrix A in $R^2 \rightarrow R^2$:
$
\begin{bmatrix}
7 & -10 \\
5 & -8
\end{bmatrix}
$
Thus the eigenvalues/eigenvectors are: 2 $\begin{bmatrix} 2 \\ 1 \end{bmatrix}$ and -3 $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$.
Thus the eigenspace matrix is $\begin{bmatrix} 2 & 1... | You need to write $(2,3)$ as a linear combination of the eigenvectors.
In this case, $(2,3) = -(2,1) + 4 (1,1)$, so
$$A \begin{bmatrix}2 \\ 3 \end{bmatrix} = - A \begin{bmatrix}2 \\ 1 \end{bmatrix} + 4 A \begin{bmatrix}1\\ 1 \end{bmatrix}$$
and then you can use the fact that these are eigenvectors to easily complete ... | {
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probability picking parts no replacement In a bin containing 30 parts, 27 parts are good and 3 parts are defective.
a) What is the probability that if you select 3 parts randomly, without replacing the parts in the bin, from the bin that you will have 1 defective part?
I thought of $\dfrac{\dbinom{27}{1}\dbinom{26}{1}\... | Close, but as you seek the probability for obtaining $2$ from the $27$ good parts and $1$ from the $3$ defective parts, when selecting any $3$ from all $30$ parts (without replacement or bias), it is: $$\dfrac{\dbinom{27}{2}\dbinom{3}{1}}{\dbinom{30}{3}}$$.
| {
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Help computing $\lim_{n\to \infty}\int_{0}^{1}f_{n}(x)\,dx$ We have the function $f(x)=2x(1-x)$ and we define $f^{\circ 2}=f\circ f$, and $f^{\circ n}=f\circ f^{\circ(n-1)}$ for $n>2$. We need to compute $$\lim_{n\to\infty}\int_{0}^{1}f_{n}(x)dx$$.
To do that, I found that $f^{\circ n}(x)\leq \frac{1}{2}$ for all $x\i... | You can verify that for $x\in (0,1)$,
$$\left|f(x)-f\left(\frac 1 2\right)\right| = \left|f(x)-\frac 1 2\right| \leq \|f^\prime\|_{\infty}\left|x-\frac 12\right|=2\left|x-\frac 12\right|$$
Thus
$$\left|f_n(x)-\frac 1 2\right|\leq 2^n\left|x-\frac 12\right|^n$$
So you can now see that for all $x\in (0,1)$, you have poi... | {
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Four touching circles and one common tangent
Given four touching circles and one common tangent, show that $$\angle BAD = \dfrac{1}{2}(\angle DO_1A + \angle AO_2B)$$
It is done by looking at triangles $\triangle ADO_1$ and $\triangle ABO_2$.
Now I am trying to prove that a circle can be circumscribed around $ABCD$.... | Hint: As you proved $$<BAD = {1\over 2}(<AO_2B+ <AO_1D)$$
similary we have also: $$<BCD = {1\over 2}(<CO_3B+ <CO_4D)$$
so $$<BAD +<BCD = {1\over 2}(<AO_2B+ <AO_1D)+{1\over 2}(<CO_3B+ <CO_4D)$$ $$={1\over 2}360 = 180$$
| {
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10 + 67 × 2= ?, This question is causing a ruckus among my friend Help!
How would you solve 10 + 67 × 2?
I solved by putting a bracket between 10 and 67. Then multiplying by 2
Like this (10 + 67)2 = 154
However, others insist it's 144 and I'm wrong.
| If they meant $(10 + 67) \times 2$, they should have written that. Otherwise one should assume the standard order of operations is meant. Go to Wolfram Alpha and put in 10 + 67 * 2. It will answer 144 because there is nothing to indicate the addition should be done first.
Another possibility is that your friend is tryi... | {
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Combination patterns - how to calculate? I have a bag of 9 different colored balls, and I want to calculate the probabilities of all the patterns there can be. If I picked 4 balls (with replacement), I could get the following patterns:
*
*all 4 are the same color (eg RRRR)
*3 are the same color, and 1 is different... | In the event that order matters:
*
*All balls same color: $9$
*Three balls same color, one ball different: $\binom{4}{3}\cdot 9\cdot \binom{1}{1}\cdot 8 = 4\cdot 9\cdot 8 = 288$
*Two balls same color, remaining two balls same color: $\binom{9}{2}\binom{4}{2}= 216$
*Two balls same color, remaining two balls differ... | {
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Continuous Bijection of Top Spaces It is well known that a continuous bijection of compact hausdorff topological spaces is a homoemorphism. I am wondering, is it true that a continuous bijection of compactly generated spaces is a homemorphism?
I am also interested in other generealizations of this theorem.
Thanks so mu... | Here is a negative result that precludes many generalizations you might consider (though it doesn't quite address your case where you require both the domain and the codomain to be compactly generated).
Theorem: Let $X$ be a regular Hausdorff space such that every continuous bijection $X\to Y$ to a Hausdorff space $Y$ ... | {
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solve the Lagrange multiplier equations the equations :
$$\left\{\begin{array}{l}{x+6 y+4 \lambda x=0} \\ {6 x+2 y+\lambda y=0} \\ {4 x^{2}+y^{2}-25=0}\end{array}\right.$$
I've done many transformations,but I still can't get the answer.
the last I did this:
$$\lambda=\left(6 \frac{x}{y}+2\right)=\left(\frac{1}{4}+\frac... | Consider the first two equations, under this slightly different form:$$\left\{\begin{array}{l}(1+4\lambda)x+6y=0\\6x+(2+\lambda)y=0.\end{array}\right.$$Suppose that this homogeneous system has exatly one solution; then this solution is $(x,y)=(0,0)$, which is not a solution of the third one.
But\begin{align}\text{The s... | {
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solve the functional equation $f(x+t)-f(x-t)=4xt$ I think this question might be related with arbitrary functions, but I’m not sure. I also tried to set $t$ to different values but couldn’t get it to work.
I tried to set $t=x$ and end up with $f(2x)=f(0)+4x^2$, $f(x)=f(0)+x^2$.
| Set $x=t$ and we get $$f(2x)-f(0)= 4x^2$$ so $f(x) =x^2+a$ where $a= f(0)$.
Check: If we now put this in to starting equation we get: $$ (x+t)^2+a-(x-t)^2-a = 4xt$$ which is always true. So $\boxed{f(x) =x^2+a}$ for all real $a$.
| {
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Explicit description of quotient ring of $\mathbb{Z}[x]$ I am studying for a prelim and I stumbled on this problem:
Describe explicitly the elements in the quotient ring $\dfrac{\mathbb{Z}[x]}{(3,x^3-x+1)}$. First of all I don't see why the ideal $(3,x^3-x+1)$ is a maximal ideal in $\mathbb{Z}[x]$. If there is anyone w... | You can solve this problem by a two step process. First, let $J = (3, x^3-x+1)$, and let $I = (3)$. These are ideals of $\mathbb Z[x]$ with $I \subset J$.
The third isomorphism theorem says that
$$\mathbb Z[x]/(3,x^3-x+1) = \mathbb Z[x]/J \cong \frac{\mathbb Z[x]/I}{J/I}$$
In other words, the ring you are looking f... | {
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Volume of a frustum given the bottom radius and the top cone height.
A cone with base radius 12 cm is sliced parallel to its base, as shown,
to remove a smaller cone of height 15 cm. If the height of the smaller
cone is three-fourths that of the original cone, what is the volume of the remaining frustum?
I set t... | Hint:
The smaller cone is homothetic of the larger cone in a homothety with centre the vertex of the cones and ratio $3/4$. So the height $h$ of the smaller cone is $h=3/4$ of the height $H$ of the larger cone, its base area is $b=9/16B$ and its volume is $v=27/64V$. So the remaining frustum has volume
$$\mathcal V=V-... | {
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Does de morgans law apply to literals within brackets? This is the example I have, however, in the second line de morgans law does not turn the literals x,z, and y into not x,z,y. Why is this? Does deMorgans law not apply to those literals for some special reason?
≡ ∃x, y, ¬(¬(x < y) ∨ ∃z (x < z ∧ z < y)) (by p ⇒ q ≡ ¬... | In the language of my first order logic text, $x,y,z$ are individual variables, not propositional variables. They don't represent truth values, they are merely arguments for the predicate variables that do return truth values. Perhaps it's clearer to write the sentence without the infix inequality symbols:
$$\exists... | {
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Prove there exists $2011$ consecutive amazing integers Recently, I have found this problem:
We call a positive integer $n$ amazing if there exists positive integers $a, b, c$
such that the equality $$n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)$$
holds. Prove that there exists $2011$ consecutive positive int... | Note that if $n=d^2k$, with $d+2|k$, then with $c=d$, $b=\frac{dk}{d+2}$, then $a=bc$, $(a,b)(c,ab)=bc=d^2\frac{k}{d+2}$, $(b,c)(a,bc)=d^3\frac{k}{d+2}$, and $(c,a)(b,ac)=bc=d^2\frac{k}{d+2}$, so the sum is $n$, which is thus amazing.
So consider a sequence $\delta_1 \geq 6$ and $\delta_{i+1}=\prod_{k=1}^i{(\delta_k^2-... | {
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Doubt in Hoffman and Kunze Section 5.2 (existence of determinant) I am trying to read Hoffman Kunze's book on linear algebra and I have a doubt in a particular result, (Theorem 1) of Section 5.2. Specifically, the theorem states:
Let $n > 1$ and let $D$ be an alternating $(n - 1)$-linear function on
$(n - 1)\times (... | As you have observed, $D_{ij}(A)$ is linear in every row except the $i$th row, and $D_{ij}(A)$ is independent of row $i$. On the other hand, $A_{ij}$ is independent in every row except the $i$th row, and $A_{ij}$ is linear in row $i$. Thus, $A_{ij}D_{ij}(A)$ is linear in every row!
| {
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Does any manifold admit a complete distance metric?
Let $M$ be a smooth manifold. Is there a metric $d$ on $M$ compatible with the original topology such that $(M,d)$ is a complete metric space? Note that in this question I always mean distance metric, not a Riemannian metric.
My initial idea was to start with a cove... | The Hopf-Rinow Theorem states that a (connected) Riemannian manifold $(M, g)$ is complete if and only if the metric $d$ on $M$ induced by $g$ is complete. (Here, $d$ is the Riemannian distance determined by $g$: $$d(x, y) := \inf_\gamma \int_\gamma ds = \int_{t_0}^{t_1} g(\gamma'(t), \gamma'(t)) \,dt ,$$ where $\gamma$... | {
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Inductively simplify specific Vandermonde determinant From Serge Lang's Linear Algebra:
Let $x_1$, $x_2$, $x_3$ be numbers. Show that:
$$\begin{vmatrix} 1 & x_1 & x_1^2\\ 1 &x_2 & x_2^2\\ 1 & x_3 &
x_3^2 \end{vmatrix}=(x_2-x_1)(x_3-x_1)(x_3-x_2)$$
The matrix presented above seems to be the specific case of Vander... | The general proof is not difficult.
From the definition of a determinant (sum of products), the expansion must be a polynomial in $x_1,x_2,\cdots x_n$, of degree $0+1+2+\cdots n-1=\dfrac{(n-1)n}2$, and the coefficient of every term is $\pm1$.
On another hand, the determinant cancels whenever $x_j=x_k$, so that the poly... | {
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Find value of $(\cos\frac{2\pi}{7})^ {\frac{1}{3}} + (\cos\frac{4\pi}{7})^ {\frac{1}{3}} + (\cos\frac{8\pi}{7})^ {\frac{1}{3}} $ This question was on my list. I was trying to apply the $n$-th roots of unity, but other ideas are welcome. I also tried Newton's sums, but it's not working.
I searched around here and I didn... | This cubic-root sum, which has the inscrutable value $\sqrt[3]{(5-3\sqrt[3]{7})/2}$, was discovered over a hundred years ago. Outlined below is an elementary evaluation of the sum.
Note that $\cos\frac{2\pi}7$, $\cos\frac{4\pi}7 $ and $\cos\frac{8\pi}7 $ are the roots of
$$x^3+\frac{1}{2}x^2-\frac{1}{2}x-\frac{1}{8}=0$... | {
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Calculate how many ways you can give $7$ children $7$ identical candies My try:
Calculate how many ways you can give $7$ children $7$ identical candies if each child got at most 2 candies.
$$x_1+x_2+x_3+x_4+x_5+x_6+x_7=7 \text{ for } x_i \in \left\{ 0,1,2\right\}$$
$$[t^7](1+t+t^2)^7[t^7](\frac{1-t^3}{1-t})^7=[t... | As was commented, the only issue is the sign of the middle term must be negative. To see it, express the first binomial as a sum too:
$$[t^7](1-t^3)^7 \sum {n+6 \choose 6}t^n=[t^7]\sum_{k=0}^7{7\choose k}(-t^3)^k \sum_{n=0}^{\infty} {n+6 \choose 6}t^n=\\
\underbrace{{7\choose 0}{13\choose 6}}_{k=0,n=7}-\underbrace{{7\c... | {
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"answer_id": 0
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Investigating Continuity of Dirichlet and related functions: An $\epsilon-\delta$ approach I have trouble proving discontinuity of the Dirichlet function, using the $\epsilon-\delta$ approach.
The function is defined as follows:
$$ f(x) = \left\{\begin{array}{l l} 1 &\text{if }x \in \mathbb{Q} \\ 0 & \text{i... | For Dirichlet function, do you understand that $\mathbb{Q}$ is dense in $\mathbb{R}$? If so, you will find the function is discontinuous for every point just by $\epsilon-\delta$ approach. For the second function, you can just pick any $\epsilon$, then pick $\delta=\epsilon$, for $x\in(-\delta,\delta)$, if $x$ is ratio... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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10-digit numbers with constraints
How many 10-digit numbers can be made by using the digits {5,6,7} (all of them) and with the additional constraints that no two consecutive digits must be the same and also that the first and last digits of the number must be the same?
I am trying to find a solution by using combinat... | I have another solution. We call a number satisfying the condition of the problem "accepted". Let $a(n)$ denote the number of accepted numbers with $n$ digits. Imagine an accepted number with $n-2$ digits and put two empty places at the left side of the number with $n-2$ digits. You can easily make two different accept... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Understand the rank of $ \begin{bmatrix} A&b\\ b^{*}&0 \end{bmatrix}$
Let $A \in M_n(C)$ and $b$ be a column vector of n complex complements. Denote $\widetilde A = \begin{bmatrix}
A&b\\
b^{*}&0
\end{bmatrix} $ If $rank(\widetilde A)=rank(A)$, which of the following is true?
(a) $Ax = b$ has infinitely many solution... | Hint: If both matrices have the same rank (column rank = row rank), then $b$ lies in the column space of $A$ and so there is a linear combination of the columns of $A$ which gives $b$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to choose k in Poisson distribution?
Problem: In a factory, the probability of a screw being defective is p
= 0.015. What is the probability that a box of 100 screws does not contain a defective one?
One way to answer this problem is to use the Poisson formula the following way:
a) $P (\textrm{There are no defe... |
"The Poisson distribution can be applied to systems with a large
number of possible events, each of which is rare"
wiki, Poisson distribution
This is not the case if $p=0.98$. Basically the random variables is binomial distributed. This variable can be approximated by the Poisson distribution if $n$ is large and $p... | {
"language": "en",
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Angles between vectors of center of two incircles I have two two incircle between rectangle and two
quadrilateral circlein. It's possible to determine exact value of $\phi,$ angles between vectors of center of two circles.
| There are constraints that $A$ and $B$ must fullfill, i.e., the following system of equations coming from Pythagoras theorem applied to certain right triangles :
$$\begin{cases}(B-2)^2+2^2=(A/2)^2\\(B-1)^2+(A/2)^2=(B+1)^2\end{cases}$$
giving $A=8 \sqrt{2}$ and $B=8$.
If now we take equations as in the partial solution ... | {
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"source": "stackexchange",
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An entire function satisfies $f(az+b)=f(z)$ Here is a problem that I got stuck on while preparing for an upcoming exam:
If $a,b\in \mathbb{C}$ and $f:\mathbb{C}\to\mathbb{C}$ is non-constant and entire with $f(az+b)=f(z)$ for all $z\in \mathbb{C}$, prove that there exists a positive integer $n$ such that $a^n=1$.
I p... | Hint: if $a\neq 1$ then $g(z):=f\big(z+b/(1-a)\big)$ satisfies $g(az)=g(z)$.
| {
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Problem with $x^{6} - 2 = 0$ compute roots in $\mathbb{C}$
I have problem with simple equation $x^{6} - 2 = $ compute roots in
$\mathbb{C}$
I will try compute roots of $x^{6} - 2 = (x^{3}-\sqrt{2})(x^{3}+\sqrt{2})=(x-2^{1/6})(x^{2}+2^{1/6}x+2^{1/3})(x^{3}+\sqrt{2})$, but this not looks good.
Maybe is better soluti... | As commented above, you used $a^3-b^3=(a-b)(a^2+ab+b^2)$. You can also use $a^3+b^3=(a+b)(a^2-ab+b^2)$ and easily solve the quadratic equations:
$$x^{6} - 2 = (x^{3}-\sqrt{2})(x^{3}+\sqrt{2})=\color{red}{(x-2^{1/6})}\color{green}{(x^{2}+2^{1/6}x+2^{1/3})}\color{blue}{(x+2^{1/6})}\color{purple}{(x^{2}-2^{1/6}x+2^{1/3})}... | {
"language": "en",
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"source": "stackexchange",
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"answer_id": 5
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Stabiliser of a Subset of Center I have no clue for the following problem:
Let $G$ be a finite group, $p$ a prime number, $S$ a Sylow $p$ subgroup of $G$. Let $N$ be the normalizer of $S$ inside $G$. Let $X, Y$ two subsets of $Z(S)$ (center of $S$) such that $\exists g \in G, gXg^{-1}= Y$. Then we need to show that $... | Because $X$ is contained in $Z(P)$, it follows that $N_G(X)$ contains $P$. That means that $N_G(Y) = N_G({}^gX)$ must contain ${}^gP$. But it also contains $P$, since $Y$ is central in $P$.
Now, notice that both $P$ and ${}^gP$ are Sylow $p$-subgroups of $N_G(Y)$. Can you take it from there?
| {
"language": "en",
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Line Graph Doubt The line graph $L(G)$ of a simple graph $G$ is defined as follows:
There is exactly one vertex $v(e)$ in $L(G)$ for each edge $e$ in $G$.
For any two edges $e$ and $e'$ in $G$, $L(G)$ has an edge between $v(e)$ and $v(e')$, if and only if $e$ and $e'$ are incident with the same vertex in $G$.
Which ... | HINT: Let $G$ be a clique on the $n$ vertices $\{x_1,x_2,\ldots, x_n\}$ for $n \ge 4$. Then edges $e_1 = x_1x_2$ and $e_2=x_3x_4$ do not share a vertex. So are $v(e_1)$ and $v(e_2)$ adjacent to each other in $L(G)$?
If you want to look at this another way, $L(K_n)$ has $\frac{n(n-1)}{2}$ vertices. But for each $e \in K... | {
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Are (∀x∈A)(∃y∈B)(x≤y) and (∃y∈B)(∀x∈A)(x≤y) the same? Do the statements $$(∀x∈A)(∃y∈B)(x≤y)$$ and $$(∃y∈B)(∀x∈A)(x≤y)$$ mean the same, even though the first two brackets are reversed?
P.S: Lets say, I have a sentence: There is no number from A, so it would be bigger than all numbers from B.
| As Scientifica mentions in their answer, translating the logical statement to a sentence helps.
If you can simplify the sentence towards more natural language, do so.
Moreover, it is often instructive to to consider a special case.
And even better, combine the two and write sentences about a special case.
If $A=B=\math... | {
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Find the difference of the shaded areas from two overlapping squares I have a question from a school problem-solving homework sheet.
Here is the diagram of the problem:
Question: Two squares, A and B of the side lengths 6 cm and 5 cm respectively, overlap each other partially. Find the difference of the two shaded a... | The difference of the two shaded areas is just the area difference of the two squares, i.e. $36 - 25 = 11\space \text{cm}^2$. The overlapping area simply cancels out in the difference.
An obvious case is where the smaller square is right inside the larger one.
| {
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Name of a property in Graph Theory A multigraph is a graph which allows for more than one edge between a pair of nodes in a graph. What would be the name of a graph which allows for more than one type of node. For example, buyers and sellers. I've heard multimodal, but I don't know if that is correct.
| I believe what you are describing is referred to in network science as a heterogeneous graph. A heterogeneous graph is a graph $G = (V, E)$ such that the vertex set $V$ is a disjoint union $V = V_1 \cup V_2 \cup \dots V_n$ where each $V_i$ denotes some set of vertices that share a common label. If $G$ is a directed gra... | {
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Probability of knowing the source of a sound after hearing it I am given this problem by my professor and I can't figure a reasonable answer, though I know it'll be easy.
Suppose to hear a sound on your roof at midnight. It could've been a thief or an animal, respectively with probability
$$ P(S \mid X=\text{thief})= 0... | You are correct up to guessing $P(S)=1$. The probability of hearing a sound is given by the probability of hearing a sound and $X$ being a thief added to the probability of hearing a sound and $X$ being an animal. Hence we have
$$P(S)=0.001\times0.8+(1-0.001)\times0.3=0.3005$$
| {
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$2\pi$-periodic $L^2$ functions on $R^1$ approximated by its Fourier series I'm reading section 4.26 in Big Rudin, but I have two questions.
Suppose $f$ is in $L^1(T)$. This means $f$ is the class of all complex, $2\pi$-periodic, and Lebesgue measurable functions on $R^1$ for which the norm $$||f||_p=\{\frac{1}{2\pi}\i... | Note $$\hat{s}_N(m) = \sum_{\lvert n\rvert \le N} \hat{f}(n)\cdot\frac{1}{2\pi}\int_{-\pi}^\pi e^{i(n-m)t}\, dt = \sum_{\lvert n\rvert \le N} \hat{f}(n)\delta_{nm}$$ where $\delta_{nm}$ equals $1$ when $n = m$ and equals $0$ otherwise. If $\lvert m\rvert > N$, then $\delta_{nm} = 0$ for every $n$ with $\lvert n \vert \... | {
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Does $(f_n)=(n\sin(\frac{x}{n})-x)$ converge uniformly on $[-a,a]$ for $a\geq0$? I'm trying to solve the next problem: Let $\left(f_{n}\right)_{n\in\mathbb{N}}$
be a sequence of functions such that $f_{n}\colon\mathbb{R}\to\mathbb{R}$
is given by $f_{n}\left(x\right)=n\sin\left(\frac{x}{n}\right)-x$,
for all $n\in\math... | It is known that $\lim\limits_{y \to 0}\frac{\sin y}{y} =1$.
Take $\epsilon >0$. You can pick up $\delta >0$ such that $\left\vert \frac{\sin y}{y} -1 \right\vert \le \epsilon$ for $\vert y \vert \le \delta$. For $n >a/\delta$ and $\vert x \vert \le a$, you have $\left\vert \frac{x}{n} \right\vert \le \delta$ and there... | {
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Limit of a function in which square roots are involved $$\lim_{x \to 0}\frac{x+2-\sqrt{2x+4}}{3x-1+\sqrt{x+1}}$$
Could someone please help me solve this problem.
I tried multiplying by a unity factor but I end up stuck.
| You may use $$\sqrt{1+x}=1+\frac x2+O(x^2)$$
First,
$$\sqrt{2x+4}=2\sqrt{1+\frac x2}=2(1+\frac x4)+O(x^2)=2+\frac x2+O(x^2)$$
And
$$\frac{x+2-\sqrt{2x+4}}{3x-1+\sqrt{x+1}}
=\frac{x+2-2-\frac x2+O(x^2)}{3x-1+1+\frac x2+O(x^2)}
\\=\frac{\frac x2+O(x^2)}{\frac72x+O(x^2)}=\frac17+O(x)\underset{x\to0}\longrightarrow\frac17
... | {
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Are there any infinite dimensional division algebras? Appart from the finite dimensional division algebras like $\mathbb{R, C, H, O}$
Are there any infinite dimensional division algebras? (Especially any "exceptional" ones?)
I was thinking maybe the ring over polynomials might be a division algebra if you include negat... | The rational functions (in one variable) provide an example of such a ring ... after we mod out by the appropriate equivalence relation, namely "equality off a finite set" (so that e.g. "$x+3$" and "${(x-2)(x+3)\over x-2}$" are the same thing). And the rational functions are only the tip of a much larger iceberg of "we... | {
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Universal property of product topology, unique up to homeomorphism
Let $(X_j, \tau_j)_{j\in J}$ be a family of topological spaces. Let $X=\prod_{j\in J} X_j$ provided with the product topology and let $pr_k: X\to X_k$ be the projection on the $k$-th coordinate, then has $(X,\tau, (pr_j)_{j\in J})$ the following univer... | You know that $\alpha: X \to P$ satisfies $$\forall j: \beta_j \circ \alpha = \text{pr}_j\tag{1}$$
and $\beta: P \to X$ satisfies $$\forall j : \text{pr}_j \circ \beta= \beta_j \tag{2}$$
now using $(1)$ and $(2)$ we get that for any $j$:
$$\text{pr}_j \circ (\beta \circ \alpha) = (\text{pr}_j \circ \beta) \circ \alpha... | {
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How to prove 1111......11 (91 digits) is a prime or composite number?
How to prove $1111......11$ ($91$ digits) is a prime or composite number?
My Approach:
$1111......11$ can be expressed as $10^{0}+10^{1}+10^{2}+...…..+10^{90}$
Using summation of a geometric progression formula,
$$10^{0}+10^{1}+10^{2}+...…..+10^{90... | Note that$$91=13(7)$$ so you may factor $$1111111$$ and your number is $$1111111(10^{12\times {7}} + 10^{11\times 7}+10^{10\times 7}+...1)$$
So it is composite.
| {
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Why does the general solution of $y'=y$ not covering $y=0?$ It is regarding the ODE $y'=y.$
Usually we try to find its general solution using separation of variables as
$\frac{dy}{y}=dx\implies\log y=x+c\implies y=e^{x+c}$ ($c$ being arbitrary constants).
Please tell me why does the general solution does not cover th... | Generally, when solving ODEs, you are looking for a non-trivial solution; i.e. $y$ that is not identically zero. Note that the trivial solution ($ y = 0$) is quite often a valid solution, but it just isn't interesting from either a mathematical or a practical point of view. Moreover, as others have mentioned, the metho... | {
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Integral $\int^{\infty}_0 \exp\left[-\left(4x+\frac{9}{x}\right)\right] \sqrt{x}\,dx$
How do I evaluate $$\displaystyle\int^{\infty}_0 \exp\left[-\left(4x+\dfrac{9}{x}\right)\right] \sqrt{x}\;dx?$$
To my knowledge the following integral should be related to the Gamma function.
I have tried using the substitution $t^... | It looks like a tricky integral, however Feynman's trick deals with it nicely.
$$I=\int^{\infty}_0 \exp\left(-\left(4x+\dfrac{9}{x}\right)\right) \sqrt{x}dx\overset{\sqrt x\to x}=2\int_0^\infty \exp\left(-\left(4x^2+\frac{9}{x^2}\right)\right)x^2 dx$$
Now consider the following integral:
$$I(t)=2\int_0^\infty \exp\lef... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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For non-negative $a$ and $b$ with $a+b \leq c$ for a small constant $c$, what is the minimum of $\cos a + \cos b$?
Let $a,b \geq 0$ with $a+b \leq c$ for a small constant $c$ between $0$ and $1$.
What is the minimum of $\cos(a) + \cos(b)$?
I conjecture it is $\cos(0)+\cos(c) = 1 + \cos(c)$ but I have no proof for thi... | This is not an answer but it strengthen your guess. Suppose $a+b=2k\leq c$ and let $a=k-\varepsilon$ and $b=k+\varepsilon$ then:
$$f(a,b) :=\cos(a)+\cos(b)=2\cos(k)\cos(\varepsilon),$$
Now one can see that when $\varepsilon\to 0$, $f(a,b)$ increases and for $\varepsilon\to k$, $f(a,b)$ decreases. So probably the minimu... | {
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Residue field at the generic point is equal to the fraction field of the section I want to show that:
If $X$ is a non-empty integral scheme and $U \subset X$ is an affine, non-empty, open subset of $X$ then
$$ K_{X, \eta} = \text{Frac}(\mathcal{O}_X(U)).$$
Here $K_{X, \eta}$ is the residue field at the generic point $\... | Your reason for $s=0$ is correct. It is interesting that the map $\mathcal{O}_X(U)\to K_{X,\eta}$ is injective even when $U$ is not affine. What you have done for the affine case is generalized into two facts:
*
*Let $X$ be a scheme (in fact this is true for a locally ringed space). Let $f\in \mathcal{O}_X(U)$. The ... | {
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Visual Intuition for the Sum of a FINITE Geometric Series I'm interested in intuitive visual explanations for the sum of a finite geometric series.
I know there are some pretty "intuitive" explanations out there (including some on this site), but I haven't seen any that provide a visual intuition.
If anyone here knows... | I think it is Matteo's method in disguise , since you want it for finite sum . (Very sorry for bad drawing)
k is common difference (r).
Assume you want to calculate $$S=2+2^2+2^3+2^4$$.
Now imagine a rectangle of side 1 & 2 . Now take two more such rectangles and put them on adjacent positions they form a big rectangl... | {
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"question_score": "7",
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For how many pairs of positive integers n and m is the statement $mn - 8m + 6n =0$ true?
Each interior angle of a regular polygon with $n$ sides is $\frac{3}{4}$
of each interior angle of a second regular polygon with $m$ sides.
How many pairs of positive integers $n$ and $m$ are there for which this
statement is ... | $$mn-8m+6n=0$$
$$m(n-8)=-6n$$
$$m=\frac {6n}{8-n}$$
The positive integral solutions are $$(m,n)=(2,2),(6,4),(18,6),(42,7),(10,5) $$
| {
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Define $f : L^2 \rightarrow \mathbb{R}$ by $f(x) = \sum_{n=1}^{\infty} \frac{x_n}{n}$. Is $f$ continuous? Problem I came across studying for qualify exams. I think I have it, if $x^k$ is a sequences of sequences in $L^2$ and it converges to $x$ in $L^2$, then $d(x^k_n, x_n) \rightarrow 0$ as $k \rightarrow \infty$ for ... | Hint:
$$|f(x) - f(y)| \le \frac{\pi}{\sqrt 6} \left( \sum_{k=1}^\infty (x_k - y_k)^2 \right)^{1/2}.$$
| {
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How can I solve prove that $8(1-a)(1-b)(1-c)\le abc$ with the conditions below? There was a homework about inequalities (that why I ask a bunch of inequality problems). But I couldn't solve the following:
If $0<a,b,c<1$ and $a+b+c=2$, prove that $8(1-a)(1-b)(1-c)\le abc$
I tried many times, and finally I used Muirhea... | Also, we can use Muirhead here.
Let $a+b-c=z$, $a+c-b=y$ and $b+c-a=x$.
Thus, $$x=a+b+c-2a=2(1-a)>0.$$
Similarly, $y>0$ and $z>0$ and we need to prove that
$$8xyz\leq(x+y)(x+z)(y+z)$$ or
$$\sum_{cyc}(x^2y+x^2z-2xyz)\geq0,$$ which is true by Muirhead.
Also, we can use AM-GM:
$$\prod_{cyc}(x+y)\geq\prod_{cyc}2\sqrt{xy}=8... | {
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What does this particular graph notation mean? I am trying to understand this paper about video summarization using a graph algorithm.
Section 4.1 of the linked paper, describes two graphs, with the same node set, while the edges differ, as they have different defining equations for edge weight. And the paper later com... | Unfortunetaly their notation is not clear at all.
They say,
We negate values to transfer the difference into similarity and normalize matrix G
So it would seem like G is a matrix. Then they define,
We construct a graph G(V,W)
Which is already a bit strange because G(V,W) can denote a graph, and G some matrix. The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3329785",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Range of $f(x,y)=\frac{4x^2+(y+2)^2}{x^2+y^2+1}$ I am trying to find the range of this function:
$$f(x,y)=\frac{4x^2+(y+2)^2}{x^2+y^2+1}$$
So I think that means I have to find minima and maxima. Using partial derivatives gets messy, so I was wondering if I could do some change of variables to make it easier computatio... | Idea: $$f(x,y) =\frac{4x^2+4y^2+4+(y+2)^2-4y^2-4}{x^2+y^2+1}$$
$$=4+\frac{-3y^2+4y}{x^2+y^2+1}$$
$$\leq 4+\frac{-3y^2+4y}{y^2+1}$$ if $-3y^2+4y\geq 0$ (else it is reversed),
$$= 1+\frac{4y+3}{y^2+1} =:g(y)$$
So you have to find a maximum value of $g$ on $[0,{4\over 3}]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3329898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
prove the absolute value of the integral from $a$ to $b$ of $f$ is less or equal than integral from $a$ to $b$ of the absolute value of $f$ if $f$ is integrable on $[a,b]$ , then
$$\bigg\lvert\,\int_a^b{f(x) dx}\,\bigg\rvert \leq \int_a^b{\big\lvert\,f(x)\,\big\rvert\,dx}$$
I know how this works because of the area of ... | Hint: if $\forall x \in [a,b], f(x) \leq g(x)$, then $\int_a^b f(x)dx \leq \int_a^b g(x)dx$
How to actually do it:
We apply that statement in the following two cases
*
*$f$ and $\lvert\,f\,\rvert$: So $\int_a^b f(x)dx \leq \int_a^b \lvert\,f(x)\,\rvert\,dx$
*$-f$ and $\lvert\,f\,\rvert$ So $\int_a^b -f(x)dx = -\int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3330078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Find the minimum $n$ such that $x^2+7=\sum_{k=1}^n f_k(x)^2$ where $f_k(x)\in \mathbb{Q}[x]$ Recently, I have found this problem:
Find the minimum $n \in N$ such that $x^2+7=f_1(x)^2+f_2(x)^2+\cdots+f_n(x)^2$ where $f_1(x),+f_2(x),+\cdots+f_n(x)$ are polynomials with rational coefficients.
I have tried to solve this... | Let's assume you can do with only two polynomials. Then we have:
$$
p = \sum_{i=1}^{k_1}{p_nx^n}
\\
q = \sum_{i=1}^{k_2}{q_nx^n}
$$
with
$$
p^2+q^2 =x^2 +7
$$
Therefore, if we can do it with only two polynomials, then there's a solution for
$$x^2+7 = (ax+b)^2 + (cx+d)^2
\\
x^2+7 = (a^2+c^2)x^2+(2ab+2cd)x + (b^2+d^2)
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3330146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
confusion about negative trigonometric identities Given that $\cos A=1/2$ and $\cos A$ and $\sin A$ have the same sign, find the value of $\sin(-A)$. If the question is referring to the first quadrant, where all trigonometric identities are positive, why is the value of $\sin (-A)$: $-\sqrt{3}/2$? Is it because this ru... | Remember the negative angle identities:
$$\sin(-A)=-\sin(A)$$
$$\cos(-A)=\cos(A)$$
To find the value of A, we have to solve the equation $\cos^{-1}(A)=\frac{1}{2}$ (Remember your 30-60-90 triangle)
Assuming that this is over the interval $[0, 360]$, you can get either $60^{\circ}$ or $300^{\circ}$ (Remember your refer... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3330254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Software for 3D graphing complex functions? (I am new to this forum.)
I am interesting in seeing what some equations look like when they are plotted 3-dimentionally, with one axis real numbers, the second axis imaginary numbers (thus the complex plane), and the third axis real numbers. Is such software available eithe... | Mathematica wouldn't be your worst option, but it will cost you a fair bit unless you're a student. You won't find anything as fully featured as Mathematica for free, but there are free alternatives that will at least be able to do what you need them to.
I would google free Mathematica alternatives and pick one that s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3330328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Finding all real solutions of $x-8\sqrt{x}+7=0$ Finding all real solutions of $x-8\sqrt{x}+7=0$.
Man, I tried subtituting $x=y^2$ but IDK things got complicated. What is the best way to figure this out? Thanks!
| Let $y=\sqrt{x}$, therefore $y^2=x$
$y^2-8y+7=0$ therefore $(y-7)(y-1)=0$ hence $y=1,7$.
Therefore $x=1,49$. These are indeed the only solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3330413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 0
} |
How much different between $D_{KL}[P(X)||Q(X)]$ and $D_{KL}[Q(X)||P(X)]$? While Kullback Leibler divergence is not symmetric: $D_{KL}[P(X)||Q(X)] \neq D_{KL}[Q(X)||P(X)]$, how much different $D_{KL}[P(X)||Q(X)]$ from $D_{KL}[Q(X)||P(X)]$?
Does the quantity $D_{KL}[P(X)||Q(X)] - D_{KL}[Q(X)||P(X)]$ have some interesting... | Let $\epsilon \in (0,1)$ and suppose that $P(X)=(1-\epsilon)\delta_{0}+\epsilon\delta_{1}$ and $Q(X)=\frac 12\delta_{0}+\frac 12\delta_{1}$.
Then $D_{KL}[P(X)||Q(X)]=(1-\epsilon)\ln(2(1-\epsilon))+\epsilon \ln(2\epsilon)$ and $$D_{KL}[Q(X)||P(X)] = \frac 12 \ln(\frac 1{2(1-\epsilon)})+\frac 12\ln(\frac{1}{2\epsilon})$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3330531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Question about affine transformation and one-one function. An Affine transformation is a function $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $f(v) = Av+b $, where $\det A \neq 0 $ and $b \in \mathbb{R}^2.$
My professor give me a task, I need to prove that a function $f$ is Affine transformation iff $f$ is one... | I think you have mis-interpreted the question. What is true is if $f(v)=Av+b$ for some square matrix $A$ and some vector $b$ then $f$ is one-to-one iff $\det(A) \neq 0$.
Proof: If $\det(A) \neq 0$ and $Av+b=Aw+b$ then $A(v-w)=0$ and this implies $v=w$ because $A$ is non-singular (and its kernel is $\{0\}$).
If $\det(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3330625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Help calculate this limit,about double factorial. $$
\lim_ {n\to\infty} \dfrac {\left [\left (2n-1\right)!! \right] ^ {1/ {2n}}} {\left [\displaystyle\prod_ {k=1} ^ {n} (2k-1)!! \right] ^ {1/ {n^2}}}$$
| The numerator is $\int_0^1 x\log(x)dx=-1/4$
so $\lim_{n\to\infty}\frac{1}{n}(\sum_{k=1}^n \frac{2k-1}{2n-1}\log(\frac{2k-1}{2n-1}))=-\frac14$
so $\lim_{n\to\infty}(\prod_{k=1}^n(\frac{2k-1}{2n-1})^{2k-1})^{\frac{1}{n(2n-1)}}=e^{-\frac14}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3330761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to show that $\left \lceil x\right \rceil-\left \lceil y \right \rceil=\left \lceil x-y \right \rceil-1$ with $x,y \in R^+$ Recently, I have found this inequality: $\left \lceil x \right \rceil + \left \lceil y \right \rceil\geq \left \lceil x+y \right \rceil$. Im' trying to understand also if this similar equality... | Note that $\lceil p\rceil-\lceil q\rceil-(\lceil p-q\rceil-1)=\lceil x\rceil-\lceil y\rceil-(\lceil x-y\rceil-1)$ when $p=x-\lceil x\rceil$ and $q=y-\lceil y\rceil$.
Thus, we just need to consider when $-1<x,y\le 0$.
Since $\lceil x\rceil,\lceil y\rceil=0$ here, we need to show $0\ge\lceil x-y\rceil-1$ and this is triv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3330969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Handle a function of itself I have a tricky question. I want to handle a function y(x) defined in this way:
$y(x)=f(y(x))g(x)$
Here, $f(y(x))$ and $g(x)$ are smooth functions: is there any way/method to express $y(x)$ as a (complicated) function of $x$ solely?
Maybe it is impossible if $f$ and $g$ are generic.
Edit:
If... | You can find many functions to satify your conditions.
For example,on $x\in[0,\infty)$ define $$y(x)=x^2,f(x)=\sqrt x, g(x)=x$$
We get $$f(y(x))g(x) =x\sqrt {y(x)} = x^2= y(x)$$
You may design other examples with little effort.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3331264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
How many groups of order at most $25$ are "pleasant" (abelian, with every non-identity element having prime order)?
A group $G$ is called pleasant if it is abelian and every non-identity element $g$ in $G$ has prime order. Up to isomorphism, how many pleasant groups are there of order at most $25?$
Options: $0, 9, 16,... | Notice that by the fundamental theorem of finite abelian groups, we have that if $\Gamma$ is abelian and finite that $$\Gamma \cong \bigoplus_{i=1}^n \mathbb{Z}_{p_i^{e_i}}^{k_i} $$ where each of the $p_i$ are primes. Now, if any of the $e_i$ were bigger than one, so $\mathbb{Z}_{p^\alpha}$ is a subgroup of $\Gamma$ wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3331377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
Complete metric Suppose $M_1=(\mathbb{R}^2,g_s)$ is the plane with standard flat metric, it is a complete manifold. Now if I delete point origin, $M_2=(\mathbb{R}^2\setminus\{0\},g_s)$ is obviously not complete. However, when punctured plane is given a different metirc, it becomes complete, such as
$$g=\frac{1}{|x|^2}g... | Write $g_s$ in polar coordinates: $g_s=dr^2 + r^2\,d\theta^2$. Then, with the change of variables $s=\ln r$, it holds that $g=ds^2 + d\theta^2$. This is the standard metric on the cylinder $\mathbb{R}\times S^1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3331665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Why do we need the covariant derivative along a curve - why are linear connections not sufficient? I can't figure out why we need the definition of a 'covariant derivative along a curve', i.e. I can't see why we can't use a 'linear connection' even when the vector fields are not extendible.
I'm reading Lee's book on Ri... | A quick answer to the Title.
One of the important and powerful tool in studying Differential geometry and Riemannian geometry is understanding the behavior of geodesics. And what is the geodesic?
There are two key properties
satisfied by straight lines in $\Bbb R^n$, either of which serves to characterize them
u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3331751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Pairs of integer pairs with same lcm, gcd and mean The problem is to find all pairs of two distinct pairs(up to permutation) of integer(!) numbers $(a, b)$ and $(c, d)$ s.t. $$\operatorname{lcm}(a, b) = \operatorname{lcm}(c, d)$$
$$\gcd(a, b) = \gcd(c, d)$$ and
$$\frac{a + b}{2} = \frac{c + d}{2}$$
It is easy to show ... | Given $\operatorname{lcm}(x,y)\cdot\operatorname{gcd}(x,y)=|xy|$, we have $\operatorname{lcm}(a,b)\cdot\operatorname{gcd}(a,b)=\operatorname{lcm}(c,d)\cdot\operatorname{gcd}(c,d)$, so $|ab|=|cd|$. Given $\frac{a+b}{2}=\frac{c+d}{2}$, we have $c=(a+b)-d$. When substituted into $|ab|=|cd|$, this gives the two quadratics ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3331849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Subgroup of $S_n$ generated by $(1,2,\cdots,n)$ and $(1,2,\cdots,m)$. I'm working on the following problem:
Let $G$ be the subgroup of $S_n$ generated by $(1,2,\cdots,n)$ and
$(1,2,\cdots,m)$ where $1<m<n$. Show $G$ is $S_n$ if either $m$ or $n$
is even, and otherwise, $G$ is $A_n$.
I know that $G$ is primitive ... | I'll be using right actions (so for $x,y\in S_n$, $xy$ means apply $x$ then $y$) and the notation $x^y=y^{-1}xy$.
I will assume the following (which can be proved by induction):
$A_n=\langle(1,2,3),(2,3,4),\ldots,(n-2,n-1,n)\rangle$
Let $\sigma=(1,\ldots,n)$, $\tau=(1,\ldots,m)$, so
$$\tau^\sigma=(2,3,\ldots,m+1)$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3331933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Need for basis for a topology Why did we need to define a basis for a topology?
I read that it is difficult to specify topology on a bigger set, so we define topology on a smaller collection.
Now I want to ask that what were the requirements to accomplish this plan?
When this question was in front of us, what actually ... | The biggest advantage to talking about bases is that they control the entirety of a space's topological information while being easy to work with. The prototypical example would be something like open balls or open rectangles in $\mathbb{R}^n$. Open subsets of $\mathbb{R}^n$ can be very strange, but open balls are simp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3332043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Tangent to a 2-Dimensional curve, in 3 Dimensions For line to be tangent to a given curve, they should pass through a common point and both should have same slope at that point. But how do we compare slope in 3D?
E.g.=> For a circle in 2-Dimensions, we know which lines will be it's tangents. But what if I ask, which l... | You can define slope at a point of a curve just as you did in calculus 1 - take the limit of approximating secant lines. The only difference this time is that the limiting object will be a line pointing in, well, any direction. So, at this point, it is best to use vectors.
Two curves in space will be tangent to eachoth... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3332150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Almost uniform convergence implies convergence in measure Let $(A,\mathcal{F},\mu)$ be finite measure space and $\{f_n\}$ a sequence of finite real measurable functions so that $f_n\rightarrow f$ a.e. We say $f_n\rightarrow f$ almost uniformly if $\epsilon>0$, there is $E\subseteq A$ such that $f_n \rightarrow f$ unifo... | This is proved more succinctly by a direct proof.
Given $\varepsilon > 0,$ let $E$ be as in the definition of almost-uniform convergence. Then there is some $N$ such that $n \geq N$ implies
$$\mu(E^c \cap \{|f_n-f|\geq\varepsilon\}) = 0$$
What does this imply for $\mu(|f_n-f|\geq \varepsilon)?$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3332265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Counterexamples concerning the central limit theorem The central limit theorem states that, if $X$ is a random variable with finite variance $\sigma^2$ and expected value $\mu$, and if $(X_n)$ is a sequence of independent random variables identically distributed like $ X $, then
\begin{equation}
Z_n = {\frac{{\overline... | This counterexample is from the book Limit Distributions for Sums of Independent Random Variables by Gnedenko and Kolmogorov.
Let $X$ have density $\begin{cases}
0 &\text{if} |x|\geq \frac 1e \\
\frac{1}{2|x|\log^2(|x|)} &\text{if} |x|< \frac 1e
\end{cases}$
The authors argue that $f_n$ the density of $\sum_{i=1}^n X_i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3332370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 0
} |
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