Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
In a triangle, prove that $ \sin A + \sin B + \sin C \leq 3 \sin \left(\frac{A+B+C}{3}\right) $
Prove that for any $\Delta ABC$ we have the following inequality:
$$ \sin A + \sin B + \sin C \le 3 \sin \left(\frac{A+B+C}{3}\right) $$
Could you use AM-GM to prove that?
| I'm sure there's a different way to approach this question, but here's one way using the graph of $\sin x $:
Consider $3$ points on the graph for $x\in(0,\pi)$. They are $(A,\sin A)$, $(B,\sin B)$, and $(C,\sin C)$ as shown. $A,B,C$ are such that $A+B+C=\pi$.
(Ignore the fact that $A,B$ and $C$ are angles of a triangl... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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How come the time complexity of Binary Search is log n I am watching this professor's video on Binary Search but when he reached here, I am a bit lost. How come he came up the time coomplexity is log in just by breaking off binary tree and knowing height is log n
https://youtu.be/C2apEw9pgtw?t=969 . and then the time c... | First, it is important to note that the running time of an algorithm is usually represented as function of the input size. Then, we 'measure' the complexity by fitting this function into a class of functions. For instance, if $T(n)$ is the function describing your algorithm's running time and $g\colon\mathbb{N}\to\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3224473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
$2f(x)=f(y) \Rightarrow 2f(tx)=f(ty)$ Find all continuous and strictly monotonic function $f:[0,\infty)\to \Bbb R$ such that:
*
*If there is a pair $(x,y)\neq (0,0)$ such that $2f(x)=f(y)$ then $2f(tx)=f(ty)$ for all $t>0$;
*There is at least one pair $(x,y)$ where the above condition hold.
What I got so far:
*
... | I found another solution and I think it is insteresting to post here. It was inspired in change of variable sugested by @Yuri Negometyanov.
Let's set $tx=2^z$ and $ty=2^w$, where $z,w \in \Bbb R$ and $t>0$. We also call $g:\Bbb R \to \Bbb R$ such that $g(s)=f(2^s)$, and then we get
$$2f(tx)=f(ty)\to 2f(2^z)=f(2^w)\to 2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Easiest way to solve $2^{\sin^2x}-2^{\cos^2x}=\cos 2x$ I'm trying to find best shortcut to crack this task
$$2^{\sin^2x}-2^{\cos^2x}=\cos 2x$$
I tried first to go through using trigonometric identities
$$\cos^2x+\sin^2x=1 \quad\text{and}\quad 2\sin^2x=1-\cos 2x$$
After substitution I reached this equation
$$2^{1-\c... | HINT.-You have immediately a solution if you note that for $x=\dfrac{\pi}{4}$ one has $\sin(x)=\cos(x)$ and $\cos(2x)=0$. The solution is done by $$x=\dfrac{(2n+1)\pi}{4};\space\space n\in\mathbb Z$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Compute polynomial $p(x)$ if $x^5=1,\, x\neq 1$ [reducing mod $\textit{simpler}$ multiples] The following question was asked on a high school test, where the students were given a few minutes per question, at most:
Given that,
$$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$
and,
$$Q(x)=x^4+x^3+x^2+x+1$$
what is the re... | I would have thought that bright students, who knew $1+x+x^2+\cdots +x^{n-1}= \frac{x^n-1}{x-1}$ as a geometric series formula, could say
$$\dfrac{P(x)}{Q(x)} =\dfrac{x^{104}+x^{93}+x^{82}+x^{71}+1}{x^4+x^3+x^2+x+1}$$
$$=\dfrac{(x^{104}+x^{93}+x^{82}+x^{71}+1)(x-1)}{(x^4+x^3+x^2+x+1)(x-1)}$$
$$=\dfrac{x^{105}-x^{104}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3224765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "63",
"answer_count": 6,
"answer_id": 5
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Show that $\min\{X_{1},X_{2},\ldots,X_{n}\}$ is sufficient for $\mu$ when $\sigma$ is fixed
Let $X_{1},X_{2},\ldots,X_{n}$ be a sample from a population with density $p(x,\theta)$ given by
\begin{align*}
p(x,\theta) = \frac{1}{\sigma}\exp\left\{-\left(\frac{x-\mu}{\sigma}\right)\right\}
\end{align*}
if $x\geq \mu$ a... | By Neyman-Fisher Lemma, $T(\boldsymbol{x})$ is a Sufficient Statistics for $\mu$ if and only if there exist two non negative functions $g(T(\boldsymbol{x}), \mu)$ and $h(\boldsymbol{x})$ such that:
$$L(\boldsymbol{x}, \mu) = g(T(\boldsymbol{x}), \mu)h(\boldsymbol{x})$$
Note that I am directly considering only $\mu$ as ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3224872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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How to take the derivative of minimum of a norm? Suppose $f: \mathbb{R}^n \rightarrow \mathbb{R}$ where $f$ is the following:
$$
f(z) =
\begin{cases}
0 \,\,\,\, z \in C \\
\min_{x\in C} \frac{1}{2} \|x-z\|_2^2 \,\,\,\ z\notin C
\end{cases}
$$
where $C$ is a closed convex $C$ in $\mathbb{R}^n$ and $x$ is a point in $C$... | Actually the answer is
$$f^\prime(z) = z - x,$$
and not $x - z$. Here $x$ is a point of $C$ nearest to $z$.
First of all, let us denote by $x(u)$ a point of $C$ nearest to $z + u$. We will show that $\|x(u) - x\| \to 0$ as $u\to 0$ (i.e., function mapping $z$ to its nearest point is continuous). Indeed, assume for cont... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3224980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Understanding numerical methods for nonlinear hyperbolic equation For the linear advection $u_t + au_x = 0$, we have the explicit Lax-Friedrichs scheme
$$ u_k^{n+1} = \frac{1}{2} (u_{k+1}^n + u_{k-1}^n) - a\frac{\Delta t }{2 \Delta x } (u_{k+1}^n - u_{k-1}^n) $$
But, if we replace this equation with $u_t + ( f(u) )_x =... | Both conservative and non-conservative Lax-Friedrichs schemes are identical. To see this, one injects the expression of the numerical flux in $(\text B)$, which gives $(\text A)$.
When considering smooth solutions, one can either use $u_t + f'(u) u_x = 0$ or $u_t + f(u)_x = 0$ to develop numerical methods that will co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3225126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Solving pde on a strip
Let $\Omega = \lbrace x_1,x_2 \in \mathbb{R}^2 | x_1>0, 0<x_2<1\rbrace$. Solve $$
\begin{aligned}
u_t(\boldsymbol{x},t) &= \nabla^2 u(\boldsymbol{x},t) , \qquad \boldsymbol{x}\in \Omega, t> 0,\\
u(\boldsymbol{x},0) &= g(\boldsymbol{x}) , \\
u_{x_1}(0,x_2,t) &= 0 ,\\
u(x_1,0,t) &= u(x_1,1,t) = 0,... | Ignoring the initial condition, the generic separated solution has the form
$$
A(n,s)\sin(n\pi x_2)\cos(s x_1)e^{-(s^2+n^2\pi^2)t} \\
n=1,2,3,\cdots,\;\;\; s \ge 0.
$$
These are required because $\sin(n\pi x_2)$ vanishes at $x_2=0,1$, and because $\cos'(sx_1)=0$ at $x_1=0$. $A(n,s)$ is an unknown ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3225252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
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Probability of X being a trick coin (heads every time) after heads is flipped k amount of times
A magician has 24 fair coins, and 1 trick coin that flips heads
every time.
Someone robs the magician of one of his coins, and flips it $k$ times
to check if it's the trick coin.
A) What is the probability that the coi... | $$P(trick|H_k)=\frac {P(trick \cap H_k)} {P(H_k)}=\frac {P(H_k|trick).P(trick)} {P(H_k)}.$$
Now,
$$\begin{align}P(trick)&= \frac{1}{24}\\
P(H_k|trick)&=1\\
P(H_k)&=P(H_k|trick)\cdot P(trick)+P(H_k|fair)\cdot P(fair)\\
&=1\cdot\frac{1}{24}+\frac{1}{2^k}\cdot\frac{23}{24}\end{align}$$
Hope this helps
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $2^{2^{2^{\cdot^{\cdot^{2}}}}} \mod 9 = 7$
Prove that $\underbrace{2^{2^{2^{\cdot^{\cdot^{2}}}}}}_{2016 \mbox{
times}} \mod 9 = 7$
I think that it can be done by induction:
Base:
$2^{2^{2^{2}}} \equiv 2^{16} \equiv 2^8 \cdot 2^8 \equiv 2^4 \cdot 2^4 \cdot2^4 \cdot2^4 \equiv 7^2 \cdot 7^2 \equiv 4 \cdot ... | Just because we are working in modulo $9$, that doesn't mean that the exponents work in modulo $9$. In general, $2^{9k}\not\equiv 1$. Actually, if $k$ is odd, $2^{9k}\equiv -1$. Which is exactly what goes wrong here: You are getting $-7\equiv 2$, rather than $7$.
You can, of course, try to take this into account in you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3225521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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(Tridiagonal) Inverse of a matrix Given this $n \times n$ matrix:
$$ A= \left(\begin{matrix}a_1&a_1&...&a_1\\a_1&a_2&...&a_2\\\vdots& &\ddots &\vdots\\a_1&a_2&...&a_n\end{matrix}\right) $$
How can I show that the inverse of this type of matrices are tridiagonal?
Help would be nice! Thank you!
| I would prove explicit formulae. For $n>1$,
$$(A^{-1})_{k,k+1}=(A^{-1})_{k+1,k}=\frac{1}{a_k-a_{k+1}}\qquad(1\leqslant k<n)\\(A^{-1})_{k,k}=\frac{a_{k-1}-a_{k+1}}{(a_k-a_{k-1})(a_k-a_{k+1})}\qquad(1\leqslant k\leqslant n)$$
with $a_0=0,a_{n+1}=\infty$ in the latter one, so that
$$(A^{-1})_{1,1}=\frac{a_2}{a_1(a_2-a_1)}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Geometric interpretation of ranks of matrices gathering coefficients of 3 affine and associated vectorial planes Given three planes $\pi_{1}, \pi_{2},\pi_{3}$ $\subseteq \mathbb{R}^{3}$ with their respect cartesian equation of the form $A_{i}x + B_{i}y + C_{i}z + D_{i} = 0$, we can determine their relative position in ... | This array displays the different possible cases :
Let us first recall that the rank of a matrix $M$ can be considered at least from 2 angles of attack: (a) rank of its columns' space or (b) max. size of a non zero determinant of a submatrix.
Remark; there is a third characterization: (c) by using rank-nullity theorem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3225833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is $\langle (-y)/(x^2+y^2),x/(x^2+y^2) \rangle$ a conservative vector field? If it is,I found two potential function which is $\arctan(y/x)$ and $-\arctan(x/y)$.
But I don't know which is true.
| Briefly: no, it's not a conservative vector field on its entire domain $\mathbb{R}^2 \setminus \{0\}$. The vectors of the vector field form counterclockwise circles about the origin, and the line integral of any counterclockwise circle about the origin is $2\pi$.
On any simply connected subset of its domain, however, t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3225917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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ABC triangle with $\tan\left(\frac{A}{2}\right)=\frac{a}{b+c}$ I have a triangle ABC and I know that $\tan\left(\frac{A}{2}\right)=\frac{a}{b+c}$, where $a,b,c$ are the sides opposite of the angles $A,B,C$. Then this triangle is:
a. Equilateral
b. Right triangle with $A=\pi/2$
c. Right triangle with $B=\pi/2$ or $C=\pi... | One way to see that c) is the correct answer is as follows. Draw yor triangle $ABC$. Construct the angle $A/2$ by extending $BA$ until a point $M$ such that $MA=b$. Note that $MA=AC=b$ so, $MAC$ is isosceles, so you have $\angle CMA=A/2$. Look at the triangle $MBC$ You have $\angle CMB=A/2$ with $CB=a$ and $MB=b+c$. So... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3226067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Derivative equals 1 Consider the following curve:
$$
x^{2}+y^{2}=2 x y+16
$$
By differentiating both sides with respect to $x$
$$
\frac{d y}{d x}(2 y-2 x)=2 y-2 x
$$
$$
\frac{d y}{d x}=\frac{2 y-2 x}{2 y-2 x}=1
$$
Are there any reasons that explain why the derivative at each point is equal to 1?
Update: I think the ... | $$
x^2+y^2=2xy+16\implies\\
x^2-2xy+y^2=16\implies\\
(x-y)^2=16\implies\\
|x-y|=4\implies\\
x>y,\ y=x-4\\
x<y,\ y=x+4.
$$
So, the original implicit function $x^2+y^2=2xy+16$ is equivalent to the two functions $y=x-4$ and $y=x+4$ for different values of $x$ and $y$. Graphically, those are nothing more than two lines of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3226204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Gödel's theorem vs unprovable mathematical results As an answer to this question, Peter Smith wrote:
Indeed, it is a fairly gross misunderstanding of what Gödel's theorem says to
summarize it as asserting that "there exist mathematical results that
cannot be proven"
It made me realize that I don't understand the ... | You're understanding Godel correctly - you're misunderstanding the misunderstanding. :P The following will just reaffirm things you already know, but readers may find it useful:
The issue is in understanding what "cannot be proven" means. All too frequently someone will make the implicit error of assuming that this is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3226336",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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Show that $B(X,Y^*)$ and $B(Y,X^*)$ are isometrically isomorphic.
If $X$ and $Y$ are normed spaces then we define. $$B(X,Y)= \{ f:X\rightarrow Y | f :\text{ f is a linear operator and bounded }\}$$
$X^*= \{f:X \rightarrow \mathbb{R} | \text{ f is a linear operator and bounded }\}$
I only know Hanh-Banach Theorem. An... | Given a bounded operator $T:X \to Y^{*}$ define $S: Y \to X^{*}$ by $(Sy)(x)=(Tx)(y)$. It is fairly routine to very that this is an isometric isomorphism. I will be glad to provide details if you get stuck.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3226499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Number of ones in Binary matrix multiplication Consider a binary matrix $\mathbf A_n$ corresponding to values $0$ to $2^n-1$ where each row represents a length $n$ binary representation of a real number. For example, for $n=3$ we have
$\mathbf A_3=\begin{bmatrix}
0 & 0 & 0\\
1 & 0 & 0\\
0 &... | By induction:
You've checked the base case manually.
Suppose true for $n$. In the case $n+1$, we select nonzero $v \neq w$. $v$ differs from $w$ in at least one entry, say the $i$th entry. Consider the vectors $\tilde{v}, \tilde{w}$ that have their $i$th entry deleted:
\begin{align}
\tilde{v} &= (v_1, \dots, v_{i-1}, v... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to factorized this 4th degree polynomial? I need your help to this polynomial's factorization.
Factorize this polynomials which doesn't have roots in Q.
$ \ f(x) = x^4 +2x^3-8x^2-6x-1 $
P.S.) Are there any generalized method finding 4th degree polynomials factor?
| You can always find the roots of a fourth degree polynomial. The formulas are not pretty, but they do exist.
Using the formulas, or, like me, using Wolfram Alpha, can show you that the roots of the polynomial are $1\pm\sqrt{2}$ and $-2\pm \sqrt{3}$ so the polynomial is
$$\begin{align}f(x)&=(x-1-\sqrt{2})(x-1+\sqrt{2})(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Whether $\Bbb{R}^3\setminus \{0\}$ and $\Bbb{R}^3 \setminus \{0,1 \}$ are homeomorphic or not? I am thinking about whether the two spaces $\Bbb{R}^3 \setminus \{0 \}$ and $\Bbb{R}^3 \setminus \{0,1 \}$ are homeomorphic or not?
I guess they are not homeomorphic but cannot find out the proper reason. Till now I have come... | $\Bbb R^3 - \{0\}$ deformation retracts to $S^2$ , whereas $\Bbb R^3 - \{0,1\}$ is homotopic (actually deformation retracts) to $S^2 \lor S^2$ but $S^2$ is clearly not homotopic to $S^2 \lor S^2$ since, $H_2 (S^2) \cong \Bbb Z$ but $H_2 (S^2 \lor S^2) \cong \Bbb Z^2$ , thus $\Bbb R^3 - \{0\}$ is not homotopic to $\Bbb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3226849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Derivative rules I have some problems when I have to derive: I don't know where I have to begin and where I have to stop. For example, if I have $$f(x)=(x^2 +1)(x^2 +3)$$ I know that I have to use the product rule so I get $$f'(x)=(x^2 +1)'(x^2 +3)+(x^2 +1)(x^2 +3)'$$ and the resolution is $$f'(x)=4x^3 +8x$$. But why c... | Write $g(x) = x^2+1$ and $h(x) = x^3+3$.
Then your $f$ is
$$ f(x) = g(x)h(x)$$
So the product rule says
$$
f'(x) = g'(x)h(x) + g(x)h'(x).
$$
The second thing you wrote would be equivalent to
$$
f'(x) = g'(x)h'(x)
$$
and this is just not how the derivative works.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3227026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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$A\in M_2(\mathbb{R}), \lambda \in \mathbb{R}$ such that $A^{2}-\lambda A+\lambda ^{2}I_{2}=O_2$, $A^{2018}=?$ $A\in M_2(\mathbb{R}), \lambda \in \mathbb{R}$ such that $$A^{2}-\lambda A+\lambda ^{2}I_{2}=O_2$$
I need to find $A^{2018}$.
From that equation I know that $det(A)=\lambda^{2}$ and $tr(A)=\lambda$.
Also I g... | $$A^{2}-\lambda A+\lambda ^{2}I_{2}=O_2 \implies A^2=\lambda A-\lambda ^2 I \implies A^3=-\lambda ^3 I$$
Note that $$2018=3(672)+2$$
Thus $$A^{2018} = (-\lambda ^3 I)^{672} A^2 =\lambda ^{2017} A -\lambda^{2018} I = \lambda ^{2017}( A -\lambda I)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3227218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to get the probability of a defective item We are given three boxes as follows: Box $1$ has $10$ light bulbs of which $4$ are defective. Box $2$ has $6$ light bulbs of which $1$ is defective. Box $3$ has $8$ light bulbs of which $3$ are defective. If a box is selected at random and then a bulb is drawn. (a). What i... | $P(\text{defective}) = P(b_1)P(\text{defective}|b_1) + P(b_2)P(\text{defective}|b_2) + P(b_3)P(\text{defective}|b_3)$
where $b_1$ means box 1 and so on. The idea is: you first have to chose a box $b_1,b_2,b_3$ at random, then you have to choose a bulb inside that box. Can you put together the numbers?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3227366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Topology Filters Let $X$ be a set and $\mathcal F$ a filter on $X$ with a countable filter basis $\mathcal B$. Prove that $\mathcal F$ has a countable filter basis $\mathcal B´$ which is nested, i.e. $\mathcal B´$={$B_n$} with $n\in \Bbb N$ such that $\ B_{n+1}\subset B_n$ for all n.
I am thinking on takin $B´_n=B_1\ca... | You're quite right: define $C_n = \bigcap_{i=1}^n B_i$ which are all in $\mathcal{F}$ too (as filters are closed under finite intersections) and is clealry nested as $C_{n+1} \subseteq C_n$ for all $n$.
If $A \in \mathcal{F}$ then some $B_k \subseteq A$ (definition of the $\{B_n: n \in \mathbb{N}\}$ being a base) and t... | {
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"url": "https://math.stackexchange.com/questions/3227487",
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"source": "stackexchange",
"question_score": "3",
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What formula to chose a nonlinear formula? Suppose I have a formula
$$f(x) = x,$$
where $0 \leq x \leq 255$
Now I want to have a formula
$$f(x) = y,$$
where $0 \leq x \leq 255$, where $f(0) = 0$, $f(255) = 255$ and e.g., $f(128) = 150$ (the value of $150$ might vary).
All other values should be interpolated.
So act... | To satisfy the requirements:
$$f(128)=150, f(255)=255$$
However, $f(0)=0$ can't be satisfied by this method. I assumed that f(1)=1, however
you can change the numbers but not use zero, otherwise, there would be no inverse.
An example of the curve looks like this Curve
I can provide more info about the derivation if y... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Conic formulation with binary variables in Gurobi I have a constraint of the following form
$$x^2 \leq yz$$
where $z$ is binary, $y \geq 0$, and $x$ is free. Can Gurobi handle this constraint?
| This is a (mixed integer) rotated quadratic cone. This can be handled in Gurobi. See for instance http://www.gurobi.com/documentation/8.1/refman/c_grbaddqconstr.html and https://www.gurobi.com/documentation/8.1/examples/qcp_py.html .
Alternatively, presuming there is a known upper bound for y, the right-hand side can ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Looking to create a ranked list for PUBG teams PUBG is a video game played with multiple teams in each game. I am looking to create a rating system so that as teams play each other, they will gain/lose rating based on their placement (1st being the best), how many teams were playing (making it more difficult to survive... | The Weng-Lin rating systems are perfect for your needs. There are several free and open source implementations in different languages by the name openskill.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3227854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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What is p(Evidence) exactly in a bayesian model? I'm having a hard time intuitively understanding what this means in a machine learning context. When using the variables $A$ or $B$ or some trivial example, it all makes sense, but when looking at machine learning formulas where there are real variables its harder to see... | Let's take your dice example to try to illustrate the issue. Here $T$ is your uncertain parameter and $t$ a value it can take, while $X$ is your observation and $x$ a particular value it can take.
*
*Suppose you have a $t$-sided fair die, but you do not know what value $t$ has. You do have a prior distribution for... | {
"language": "en",
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Rudin, theorem 2.30 intuition behind could you show 1-2 real examples that is related to this theorem?
2.30 Theorem: Suppose ⊂. A subset of is open relative to if and only if =∩ for some open subset of .
| Take $\mathbb{R}=Y\subset X=\mathbb{R}^2$. Then any open set $E$ in $\mathbb{R}$ is given by a countable union of open intervals, $\Omega=\bigcup_{j\in J}(a_j,b_j)$.
Each $(a_j,b_j)$ can be rewritten as $B\left(\frac{a_j+b_j}{2},\frac{b_j-a_j}{2}\right)\cap \mathbb{R}$, i.e. a ball in $\mathbb{R}^2$ of radius half the... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does $f_n\to0$ pointwise + $f_n$ integrable + $f_n$ uniformly bounded imply $\int f_n\to 0$? If $\{f_n:[0,1]\to\mathbb{R}\}_{n\ge 1}$ is a sequence of Riemann-integrable functions that converge pointwise to the zero function and $\{f_n\}_{n\ge 1}$ is uniformly bounded by $M$ can we prove that $\int_{[0,1]}f_n\to0$ ?
It... | It is true, by Lebesgue's dominated convergence theorem. Just take $M>0$ such that$$(\forall n\in\mathbb N)\bigl(\forall x\in[0,1]\bigr):\bigl\lvert f_n(x)\bigr\rvert\leqslant M.$$Then, if you define $g(x)=M$ for each $x\in[0,1]$, $g$ is integrable and$$(\forall n\in\mathbb N)\bigl(\forall x\in[0,1]\bigr):\bigl\lvert f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How many inner products exist in $R^n$? I want to know how many inner products exist in $R^n$ making it a Hilbert space. I know they all can be corresponded to the Euclidean inner product by some Isometry/Unitary function, but I want to know more explicit formula for this function, and if it is twice differentiable or ... | Given an inner product $\langle \cdot, \cdot \rangle$ in $\mathbb{R}^n$, consider a basis $\{
v_i \}_{i=1}^n$ of $\mathbb{R}^n$ and define $g_{ij} = \langle v_i, v_j \rangle$.
Take $u=\sum_{i=1}^n \alpha_i v_i$ and $v=\sum_{j=1}^n \beta_j v_j$.
Then
$$(1) : \langle u, v \rangle = \sum_{i, j = 1}^n g_{ij} \alpha_i \be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3228740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is H2 the same as the 4th alternating group
Would H2=A4 as if we apply sigma^5 to both sides we get sigma^12=1 which is true for all elements of the 4th alternating group by Lagrange's theorem.
| $H_2\subset A_4$, by definition. Conversely, let $\sigma \in A_4$. Then $\sigma ^{12}=1$ (since $\mid A_4\mid=12$) $\implies \sigma ^7=\sigma ^{-5}\implies \sigma \in H_2$. Thus $A_4\subset H_2$. Thus $H_2=A_4$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to find $\frac{∂f}{∂y}$ for Functions of the form $f(x,y,z(y))$? This is the first time I'm using math.stackexchange. Please excuse me and correct me if I'm not doing things in the right format.
So my question is this: given a function of the form $f(x,y,z(y))$, and suppose we want to find $\frac{∂}{∂y} f(x,y,z(y))... | Don't confuse the function $f(x,y,z)$ and the function $F(x,y)=f(x,y,z(y))$.
The first is a function of three variables $x,y,z$ while the second is a function of two variables $x,y$.
People familiar with multi variables calculus loosely use a common symbol for both without making mistake. For one not yet familiar wit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3229134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Convergence (to zero) for PDF of normal distribution. I need to prove that the PDF converges to zero when $n\to\infty$; that is,
$$\lim_{n\to \infty}f_n(x) =\lim_{n\to\infty} \frac{1}{\sqrt{2\pi n^{-3}}}\exp\left(-\frac{(x-\frac{1}{n})^2}{2n^{-3}}\right)\to 0$$
I have tried using L'Hopital and differentiate $3$ times a... | $$\log f_n(x) \propto -3 \log n -\frac{(x-\frac{1}{n})^2}{2n^{-3}} \longrightarrow -\infty$$
So $$f_n(x) \longrightarrow e^{-\infty} = 0$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proof continuous function $g$ defined on $[0,1]$ has a fixedpoint $x \in [0,1]$ Claim: For the continuous function $g:[0,1]\rightarrow[0,1] \exists x \in [0,1]: g(x_1) = x_1$
Proof:
Case 1: If g(0) = 0 and/or g(1) = 1 then g has at least one fix point on one of the ends of the intervall. This means $\exists x_1\in[0,1]... | Your proof is correct!
Geometrically, this says, any continuous function $g$ from $[0,1]$ to itself MUST intersect the diagonal $y=x$, and that intersecting point is the fixed point
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Using chain rule to differentiate $f(x)=a(x)b(x)$? Why can I not apply the chain rule to a product in the following way.
If we have some product:
$$f(x)=a(x)b(x)$$
Consider the multiplication of b by a as another’s function so that:
$$f(b(x))=ab$$
So that
$\frac{df}{dx} = f’(b)b’(x)$
Something feels very wrong. But I ... | You can't use the chain rule because this is not a composition. Just calling it one does not make it one. For example, suppose $a(x)=x+1$ and $b(x)=x^2$. Then your first line says
$$
f(x) = x^2(x+1) = x^3+ x^2
$$
so
$$
f(b(x)) = f(x^2) = (x^2)^3 + (x^2)^2
$$
so not the same as "$ab$".
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 4
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If $a^{m}+1\mid a^{n}+1$ then prove that $m\mid n$.
Let $a$ be an integer, $a\ge2$. If $a^{m}+1\mid a^{n}+1$ then prove that $m\mid n$.
Actually I know a similar proof which is, $a^{m}-1\mid a^{n}-1 \iff m\mid n$, but I can't prove this. I also need some examples of the question.
Can't seem to find any correlation be... | $U_{k} = a^{k} + 1$ are terms of a Lucas sequence; hence, $U_{m} | U_{n}$ iff $m | n$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to get sum of $\frac{1}{1+x^2}+\frac{1}{(1+x^2)^2}+...+\frac{1}{(1+x^2)^n}$ using mathematical induction Prehistory: I'm reading book. Because of exercises, reading process is going very slowly. Anyway, I want honestly complete all exercises.
Theme in the book is mathematical induction. There were examples, where w... | Hint:
For a geometric series with ratio $q$:
$$q+q^2+\dots+q^n=\frac{q(1-q^n)}{1-q}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3229983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Spivak Calculus Chapter 2,Number 18, Connection with Number 17 In Spivak's Calculus, Problem 17 has you $(1)$ verify that all natural numbers are factorable into a product of purely prime numbers and $(2)$ has you show that $n^{\frac{1}{k}}$ is irrational unless $n=m^k$ for some natural number $m$.
Problem 18 asks
Pr... | If $k,n\in\mathbb N$, consider the polynomial $x^k-n$. Problem 17 is equivalent to the assertion that if this polynomial has a rational root, then it is actually an integer one.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$x_1 x_2 x_3 x_4 + x_2 x_3 x_4 x_5 +......+ x_n x_1 x_2 x_3 = 0$ then what is $n$? Can anyone please help me to understand what is the following problem saying?[!
Each of the numbers $x_1,x_2,\cdots,x_n,n>4$, is equal to $1$ or $-1$. Suppose
$$x_1x_2x_3x_4+x_2x_3x_4x_5+\cdots +x_nx_1x_2x_3=0$$
then,
$(1)$ $n$ ... | Let $$I_1 = x_{1}x_{2}x_{3}x_4$$ $$I_2 = x_{2}x_{3}x_4x_5$$ $$\vdots $$ $$I_n = x_nx_{1}x_{2}x_{3}$$ Since each $I_k\in\{-1,1\}$ and $I_1+I_2+...+I_n=0$ we must have equaly $-1$ and $1$ so $n$ must be even.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Equilateral triangles on the sides of a triangle We have a triangle.
We then construct three points outside of the triangle by drawing three equilateral triangles on the sides of the original triangle.
Now we want to do the opposite: from the three points we constructed, we want to construct the original triangle. (Wit... | Denote the points like on this picture:
Consider the following composition of rotations: $I= R_{C',60^\circ}\circ R_{A',60^\circ}\circ R_{B',60^\circ}$. The classification of isometries says that $I$ is a central reflection, but also $I(A)=A$, so $I$ is the central reflection with respect to $A$: $I=S_A$. Thus $A$ can... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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splitting accommodation costs between people when some of them stay for fewer days We're 4 people and we are staying 7 nights, for a total cost of 546.
One of us however is leaving 1 day earlier.
Initially I thought the problem was very simple.
I reasoned that the 3 of us staying for the full period should pay $C_1$, a... | If we both buy a £10 meal every day that we are on holiday and you stay for 6 nights while I stay for 7 then your meal costs will be 6/7 of mine. But if the daily cost isn't constant, that doesn't apply.
The constant cost per day assumption doesn't apply to your situation: the daily cost goes up when there are fewer p... | {
"language": "en",
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How to check if functions are linear? I know that a function is linear if $f(ax+by) = af(x)+bf(y)$, but whilst doing some exercises I came across to this one where it says that I have to check if the following functions are linear. I don't really understand how should I proceed.
\begin{align*}
f: \mathbb{K}^{3} &\to ... | Conditions of linearity of function is $(1)f(x+y)=f(x)+f(y)$ and $f(cx)=cf(x)$ where $c$ is scalar. Alternatively we may write $f(cx+dy)=cf(x)+df(x)$, $c$, $d$ are scalars. If a function satisfy these properties we declared it as linear. As $f(x) = tx$ is linear.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Inverse function theorem. Let $f:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$, $C^{1}$, such that $det(f'(x))\neq 0,\ \forall\ x\in \mathbb{R}^{n}$ and $f^{-1}(K)$ is a compact set for all $K\subset \mathbb{R}^{n}$ compact. Prove that $f$ is a surjective function.
If anyone can help, I'll be grateful.
| You know $f$ is open, so $f(\mathbb{R}^n)$ is open. It suffices to prove $f(\mathbb{R}^n)$ is closed.
Let $y\in\overline{f(\mathbb{R}^n)}$. Choose $y_n\in f(\mathbb{R}^n)$, $y_n\to y$. Since $\overline{B_r}(y)$ is compact, we know $f^{-1}(\overline{B_r}(y))$ is compact. If $x_n\in\mathbb{R}^n$ be such that $f(x_n)=... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Degree 2 Recurring monic polynomials Consider a monic polynomial $x^2+ax+b=0$, with real coefficients. If it has real roots $p$ and $q$, such that $p\leq q$, then you construct a new monic polynomial as $x^2+px+q=0$.
If this polynomial has real roots $p_1$ and $q_1$, $p_1\leq q_1$ you again construct a monic polynomia... | If you start with the polynomial $a=b=0$, then the polynomial will always be $x^2$ and so the process goes on forever. However, in all other cases, it will terminate in finitely many steps.
First, suppose $b=0$ and $a\neq 0$. If $a<0$, the next polynomial is $x^2-a$ which does not have real roots. If $a>0$, the next... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3230835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Categorical general topology -- reference request I am looking for literature describing general topology in terms of category theory. I would prefer literature which does not assume too much familiarity with category theory, but would appreciate any coherent refrences.
| I recall G. Preuß' book Theory of Topological Structures: An Approach to Categorical Topology and some works by Horst Herrlich (references at that page) as well.
There have been some conferences on "categorical topology" of which there are proceedings (e.g. this one or this one etc.; this is all pretty old and in vogu... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the difference between these two combinatorics problems? So the first problem is
"In how many ways can we arrange the letters in the word Alabama."
and the second questions is
"In how many ways can we arrange three Mathematics books, five English
books, four Science books and a dictionary?"
I can solve ... | The second question implicitly asks for the books on each subject to be kept next to each other. Very very poorly worded, I know. Thus we have $3!$ ways for the maths, $5!$ for english, $4!$ for science, $1!$ for dictionary, and $4!$ ways to rearrange, and we multiply these.
| {
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Let $f:[a,\infty)\rightarrow \mathbb{R}$ be a uniformly continuous function. $\int_{a}^{\infty} f$ converges.Prove that $\lim_{x\to\infty} f(x)=0$
Let $f:[a,\infty)\rightarrow \mathbb{R}$ be a uniformly continuous function in that range. $\int_{a}^{\infty} f$ converges. Prove that $\lim_{x\to\infty} f(x)=0$
Hint: Use ... | First of all, note that for all $n\in\mathbb{N}$
$$\lim_{x\rightarrow\infty}F_n(x)=n\left[\lim_{x\rightarrow\infty}\left(\int_a^{x+\frac{1}{n}}f(t)\mathrm{d}t- \int_a^xf(t)\mathrm{d}t\right)\right]=n\left[\int_a^{\infty}f(t)\mathrm{d}t-\int_a^{\infty}f(t)\mathrm{d}t\right]=0.$$
Now let $\varepsilon>0$ be arbitrary. By ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3231173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Surface integral of hyperboloid using polar coordinates fails? I am trying to find the surface area of the hyperboloid $x^2 + y^2 − z^2 = 1$ where $0\le z \le 1 $. My book goes ahead making hyperbolic substitutions, however I don't understand why the simple approach fails.
$$\mathbf n= \langle 2x, 2y, -2z \rangle$$
$$=... | When doing a surface integral, as is mentioned above in the comments, it is necessary to choose the "correct" normal vector. You start by choosing a parameterization of your surface, of the form $$r(u,v) = (x(u,v), \, y(u,v), \, z(u,v)) $$
In other words, express the $x,y,$ and $z$ coordinates as functions of two param... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove that there exists an integer greater than x such that any polynomial $f(x)$ will be strictly non-negative and get large? Hi I am taking a number theory class and so far I have been proving modular congruences, modular arithmetic, and prime properties. There is this theorem that came up in the textbook and apparen... | One should note that the field of interest here (presumably) is $\mathbb{R}$, so that $f(x)$ is a real polynomial. Notice first that for $a_n<0$, this fails completely, because we can just choose $f(x)=-x$ and it does not have the property.
If we assume that $a_n>0$, then write
$$ \frac{f(x)}{x^n}=a_n+a_{n-1}x^{-1}+\cd... | {
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$ \lim_{x\to \frac{1}{{\sqrt 2}^+}} \frac{\cos ^{-1} \left( 2x\sqrt{1-x^2}\right)}{x-\frac{1}{\sqrt{2}}}$
$\displaystyle \lim_{x\to {1\over \sqrt{2}^+}} \dfrac{\cos ^{-1} \left( 2x\sqrt{1-x^2}\right)}{x-\dfrac{1}{\sqrt{2}}}$
I have tried substituting $x$ for $\sin \theta$, doing the calculations and ended up with -$2... | One may use l'Hopital rule, then
$$\lim_{x\to {1\over \sqrt{2}^+}} \dfrac{\cos ^{-1} \left( 2x\sqrt{1-x^2}\right)}{x-\dfrac{1}{\sqrt{2}}} = \lim_{x\to {1\over \sqrt{2}^+}}\dfrac{-2(1-2x^2)}{\sqrt{1-x^2}\sqrt{1-4x^2-4x^4}} = \lim_{x\to {1\over \sqrt{2}^+}}\dfrac{-2(1-2x^2)}{\sqrt{1-x^2}(2x^2-1)}=2\sqrt{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3231558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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Continuous function as difference of convex functions Can every continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ be written as the difference of two convex functions?
If not, can every twice continuously differentiable function $f:\mathbb{R} \rightarrow \mathbb{R}$ be written as the difference of two convex fu... | For $f \in C^{2}$ the result is true but your construction does not work. Start with $g_1(x)=\int_0^{x}(f''(t))^{+}dt$ and then take $g(x)=\int_0^{x} g_1(t) dt$. Simialry define $h_1$ and $h$ and see that $f(x)=f(0)+xf'(0)+g-h$. [Here $x^{+}=\max \{x,0\}$ and $x^{-}=-\min\{x,0\}$]. Note that $f(0)+xf'(0)+g$ and $h$ are... | {
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Balls between boxes equation I have a practice question I don't know how to answer past part (a)
Consider nonnegative integer solutions of the equation $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 26$
(a) How many different solutions are there? [3 marks]
(b) How many solutions also satisfy: for every $i \in \{1, 2, 3\}, x... | The task in part b and part c is to find a way to reduce the problem to part a.
Say for part b, here's a hint. Let:
$$x_i^{'} = x_i - 1, \quad \forall i$$
Now if we find the non-negative solutions to below problem:
$$\sum_{i=1}^{6} x_{i}^{'} = 32$$
we get all positive solutions to the original problem asked.
Which i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof verification for $(A-B)-C = (A-C)-(B-C)$
Let $A$, $B$, and $C$ be sets.
Prove that $(A-B)-C=(A-C)-(B-C)$.
I attempted to prove this with the following. I am very new to proof writing, and I feel I may have glossed over something or have done something just completely invalid. Any assistance with revisions or a ... | The main idea is good, but the way you write it can be improved.
In the first part, after taking an element $x$, your goal is to show that $x \in (A - C) - (B - C)$.
$x \in A$ and $x \notin C$ could be written as $(A - C)$
I see what you are trying to say, but this is not mathematically correct. "$x \in A$ and $x \no... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Must every right-inverse of a linear transformation be a linear transformation? Let T be a linear transformation $\mathbb{R}^2 \rightarrow \mathbb{R}^3$. Let S be the right-inverse of T. Does S have to be linear transformation?
Thanks in Advance.
| No
(Actually a linear map $\mathbb R^2 \to \mathbb R^3$ cannot have a right inverse, since having a right inverse is equivalent to being surjective, and linear maps have the dimension of the range at most the dimension of the domain, so there are no surjective linear maps $\mathbb R^2 \to \mathbb R^3$)
BUT ANYWAY. . . ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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System of first order linear differential equations
The solutions of a homogeneous system of linear differential equations of the first order $$\frac{d \vec x}{d t} = A \vec x$$ are $$\vec x = e^{ \lambda t} \vec v $$ where $ \lambda $ are the eigenvalues of A and $\vec v$ are the associated eigenvectors.
So the fir... | Consider the matrix
\begin{align}
A =&\
\begin{pmatrix}
1 & -1 & 1 & 0\\
-1 & 2 & 0 & 1\\
1 & 2 & 0 & 0 \\
-2 & -1 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
-2 & 0 & 0 & 0\\
0 & -1 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
1 & -1 & 1 & 0\\
-1 & 2 & 0 & 1\\
1 & 2 & 0 & 0 \\
-2 & -1 & 0 & 0
\end{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that a directed graph with no cycles has at least one node of indegree zero How would I show this? I know a directed graph with no cycles has at least one node of outdegree zero (because a graph where every node has outdegree one contains a cycle), but do not know where to go from here.
| Suppose that there exists a graph with no cycles and there are no nodes of indegree $0$. Then each node has indegree $1$ or higher. Pick any node, since its indegree is $1$ or higher we can go to its parent node. This node has also indegree $1$ or higher and so we can keep doing this procedure until we arrive at the no... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
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Construct a rank-3 matroid using rank-2 flat Let $E$ be a finite set with size bigger or equal to 3. Let $L$ be a collection of subsets of $E$ such that:
*
*$2 \leq |A| < |E|$ for any $A \in L$
*$|A \cap B| \leq 1$ for any $A, B \in L$
Now show that there exists a unique simple rank-3 matroid $M$ on $E$ such that $... | Because $M$ is supposed to be simple, every singleton $\{e\}$ for $e \in E$ must be a flat, and these are the flats of rank $1$. This shows uniqueness.
The conditions stated on $L$ are not enough to ensure such $M$ exists. Let $E = \{1,2,3,4\}$ and let $L = \{\{1,2\}, \{1,3\}\}$. The sets $E$ and $L$ satisfy your condi... | {
"language": "en",
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"source": "stackexchange",
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Question about associated primes and annihilator I am trying to solve the following exercise:
Let $R$ be noetherian and $M$ a finitely genererated $R$-module. Show that $\mathrm{Ass}(R/\mathrm{Ann}(M)) \subseteq \mathrm{Ass}(M)$ and both sets have the same minimal elements. Show that in general the inclusion is not an... | Let me demonstrate the inclusion using the equality you proved.
Let $p \in Ass (R/ ann (M))$. Then there exists $x$ in $R$ such that $p = ann(M) :_R x$. You showed is that $ann(M) :_R x = 0 :_R xM$. Since $M$ is finitely generated, you may choose a finite generating set $\{m_1,\dots,m_n\}$ of $M$. Then
$0:_R xM = \cap... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Nonvanishing vector field on an odd sphere
This is an exercise I am somewhat confused about. Here $X$ looks like a vector field on $\mathbb{R}^{2n}$, not $S^{2n-1}$. Then how should I interpret $X$ to make it a vector field on the sphere? Could anyone please explain?
| The sphere $S^n$ is an imbedded submanifold of $R^{n+1}$ under the inclusion map $i$. Hence, at every point $p\in S^n$, the differential $di$ establishes a one-to-one correspondence between the tangent space $T_pS^n$ and a subspace of $T_p\mathbb R^{n+1}\equiv \mathbb R^{n+1}$. Suppose that $\gamma(t)=(x_1(t),...,x_{n+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3232799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proving goal having the form $P \lor Q$, is it redundant to separate into two cases? In Velleman's How to Prove It, the strategy given for proving goal of the form $P \lor Q$ goes like this:
If $P$ is true, then clearly $P \lor Q$ is true. Now suppose $P$ is false.
[Proof of Q goes here]
Thus, $P \lor Q$ is true.
I f... | Any statement in any proof is redundant if you consider it ‘too obvious’.
In this case, maybe you consider it obvious that $P \implies P \lor Q$ (and I wouldn’t blame you). It does follow directly from the definition of $\lor$, but it’s nice to be especially clear when writing a proof and so it doesn’t hurt to add it i... | {
"language": "en",
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For given $ab\leq n$, do there exist $a'\geq a$ and $b'\geq b$ such that $a'b'=n$? For example, given $a=3$, $b=3$, and $n=14$, no such $a',b'$ exists. On the other hand, for $a=3$, $b=3$, and $n=12$, we can use $a'=3$ and $b'=4$.
Is there a simple formula that can help determine the answer to this question, rather tha... | In reverse, it's related to the formula for markup and margin. No possibility of such a factorization exists, with $ab=n$. In fact: $$\sqrt{n}<a\land \sqrt{n}<b \implies n<ab<a'b'$$
Which is then a contradiction. One way would be:$${n\over ab}=(1+b_\text{markup})(1+a_\text{markup})$$ where you use percentage markup th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Closed form solution for constant exponent in sum I am trying to solve for $\alpha$ in the following equation:
$$ 0.80 = \frac{1}{3} \left( X_1^\alpha + X_2^\alpha + X_3^\alpha \right)$$
Right now I just use Excel and solver to find a numerical solution to problems like these. Seems like there should be a way to obtai... | Considering that you look for the zero of function
$$f(\alpha)=\sum_{i=1}^n X_i^\alpha - nk$$ without loss of generality, suppose that the $X_i$ are such that $X_1\leq X_2\leq \cdots \leq X_n$.
So the solution is bounded
$$ \alpha_{min}=\frac{\log (k)}{\log (X_1)}\leq \alpha \leq \frac{\log (k)}{\log (X_n)} =\alpha_{m... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Integration of $\int_\pi^{+\infty}e^{-st}(t-\pi)\,dt$ How can i calculate this integral?
$$
\int_\pi^{+\infty}e^{-st}(t-\pi)\,dt
$$
I tried many different things but i was hopeless.
I know that integrating by parts may be the way to go but i am truly lost.
| Integration by parts will give you (choosing $f = t-\pi$ and $g' = e^{-st}$)
$${\displaystyle\int}\left(t-{\pi}\right)\mathrm{e}^{-st}\,\mathrm{d}t=-\dfrac{\left(t-{\pi}\right)\mathrm{e}^{-st}}{s}-{\displaystyle\int}-\dfrac{\mathrm{e}^{-st}}{s}\,\mathrm{d}t +C$$
which is
$${\displaystyle\int}\left(t-{\pi}\right)\mathr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Discrete Exponential Martingale - Properties This question is about the discrete exponential martingale.
Let $(Y_n)_n$ be a sequence of independent and identically distributed random variables
with $m_{Y}(t) :=\mathbb{E}\left[e^{t Y_{1}}\right]<\infty, t \in \mathbb{R}$. I want to show that
\begin{align*}
M_{n} :=\frac... | The candidate martingale has perhaps been miscopied. As per the comment by @Sesame, if instead one defines $M_n$ by
$$M_n = \frac{\exp(t\sum_{i=1}^n Y_i)}{m_Y(t)^n},$$
then the usual computations carry through to show this is a martingale.
Summarizing the steps presented in the comment, we compute
$$\mathbb{E}(M_{n+1}|... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Tensor product of exact complexes is exact Let $M_\circ = \dots \to M_n \dots \to M_0 \to 0$ and $N_\circ = \dots \to N_n \dots \to N_0 \to 0$ be exact complexes of modules over a ring $A$ such that each module is flat.
Is it then true that $(M\otimes N)_\circ = \dots M_n\otimes_AN_n \dots \to M_0\otimes_A N_0 \to 0$ i... | No. For instance, let both complexes be $0\to A\to A\to 0$ but with one of them in degrees $0$ and $1$ and the other in degrees $1$ and $2$. Then the tensor product will be nonzero only in degree $1$ and so will not be exact.
| {
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How to compute the MLE of the zero truncated poisson? $$P(X = x ) =
\frac{\theta ^ x e^{- \theta} }{x ! \left ( 1 - e^{- \theta} \right )},\ x=1,2,\cdots,\ 0<\theta<\infty$$
Then the likelihood of $(x_1,\cdots,x_n)$ is
$$\mathcal L(\theta \mid \boldsymbol y) = \prod_{i=1}^n \frac{e^{-\theta}}{1-e^{-\theta}} \frac{\t... | Your calculation is right so far. As you said it does not exist a closed form. We can use the Lambert W function to obtain a solution for $\theta$
$\frac{e^\theta}{1-e^{\theta}} + \frac{\bar x}{\theta}=0$
Multiplying both sides by $(1-e^{\theta})$ and $\theta$
$\theta\cdot e^\theta+\bar x\cdot (1-e^{\theta})=0$
$e^\the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3234144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Playing with squares Extending from particular examples I've found that $$n^2=\sum_{i=1}^{i=n-1} 2\, i+n$$
this is that for any square of side $n$ the area can be calculated in a simple way.
Example
For a square of side $7$, the result is: $2×1+2×2+2×3+\cdots + 7=49$
Question
Is there any way to prove this generally t... | $$\sum_{i=1}^{i=n-1} 2\, i\,+n=2\left(\dfrac {n (n-1)}2\right)+n=n^2$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 3
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How much ways there are to produce sum = 21 with 4 different natural numbers? 0 isn't natural number , and the sum way is important (e.g. 3+5+6+7 is different from 5+6+7+3).
I got that I have for a+b+c+d = 21 , 2024 options.
I think I need to sub the invalid numbers , so I need to relate for the cases :
a = 0 , b = 0, ... | $$
\eqalign{
& N = {\rm No}{\rm .}\,{\rm of}\,{\rm sol}{\rm .}\left( {a + b + c + d = 21\quad \left| {\;0 < a \ne b \ne c \ne d} \right.} \right) \cr
& \quad \Downarrow \cr
& N = 4!\;{\rm No}{\rm .}\,{\rm of}\,{\rm sol}{\rm .}\left( {a + b + c + d = 21\quad \left| {\;0 < a < b < c < d} \right.} \right) \cr... | {
"language": "en",
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Manifold parallelizable equivalent condition
This is an exercise from Loring Tu's Introduction to Manifolds that I am stuck at. I know that a tangent bundle is trivial if it is isomorphic to the product bundle $M \times \mathbb{R}^{n}$. Here $n$ is the dimension of the (smooth of course) manifold $M$.
I think I have t... | If there is a smooth frame $X_1,\dots,X_n$, then each tangent vector $V$ is uniquely written as
$$V=v^1X_1+\dots+v^nX^n.$$
What can you say about the map $(x,V)\mapsto(x,v^1,\dots,v^n)$?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Coordinate transformations I have two scalar functions of $x$ and $y$ that I can define:
$$f(x,y)=x^2+y^2\qquad
\text{and}\qquad
g(x,y)=x^2 + \sin^2(x) y^2.$$
Is it true that there is literally no coordinate change that will take one to the other?
| FWIW, note that $$\mathrm{d}f\wedge\mathrm{d}g ~=~ h(x,y)\mathrm{d}x\wedge\mathrm{d}y,$$ where $$h(x,y)~:=~ 2y\{2x(\sin^2(x)-1)-y^2\sin(2x) \} ~=~ -4y\cos(x)\{x \cos(x)+y^2\sin(x) \}.$$ The pair $(f,g)$ are by definition functionally independent within the set $$\Omega~:=~\{(x,y)\in \mathbb{R}^2| h(x,y)\neq 0\}.$$ By t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculate $\int e^{2x}(\cos x)^3 dx$
Calculate $$\int e^{2x}(\cos x)^3 dx$$
My try:
*
*Firsty I tried to use integration by parts but then I got:
$$\int e^{2x}\cos^3(x) dx=...=\frac{1}{2}\cos^3(x) e^{2x}+\frac{3}{2}\left(\int e^{2x} \sin(x) dx-\int e^{2x} \sin^3(x) dx \right)$$So my calculation have not many sens ... | Linearise $\cos^3 x$ first: $\;\cos 3x=4\cos^3x-3\cos x$, so $\;\cos^3x=\frac14(\cos 3x+3\cos x)$, whence
$$\mathrm e^{2x}\cos ^3 x=\tfrac14\operatorname{Re}\Bigl(\mathrm e^{(2+3i)x}+3\mathrm e^{(2+i)x}\Bigl)$$
so calculate $\;\frac14\displaystyle\int\bigl(\mathrm e^{(2+3i)x}+3\mathrm e^{(2+i)x}\bigl) \mathrm dx$ and t... | {
"language": "en",
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Identity in a composition algebra Let $A$ be a real composition algebra ($A=\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$). I would like to prove that $$ |\lambda|=1 \implies(\lambda u) \overline{(\lambda v)}=u\overline{v}$$
In a composition algebra we have that $\lambda \bar{\lambda}=|\lambda|^2$ but $A$ may be non... | This is not true (in the noncommutative case). For instance, in the quaternions, taking $\lambda=u=i$ and $v=j$ we get $$(\lambda u)\overline{(\lambda v)}=i^2\overline{(ij)}=k\neq -k=i\overline{j}=u\overline{v}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3234928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Characterization of interior point of convex set using normal cones There is a theorem saying that for any convex set $Q$, $x\in \text{int } Q \Leftrightarrow N_Q(x)=\{0\}$. I'm trying to prove the backward direction, and my argument is as follows: If $N_Q(x)=\{0\}$, then equivalently any nonzero vector cannot be in th... | Assume that $N_{Q}(x) = \{0\}$. For the sake of contradiction, take $x \in \mathbf{bd}(Q)$, the set's boundary. Then, the supporting hyperplane theorem implies that $\exists v \in \mathbb{R}^d, v \neq 0$ such that
$$
\langle v, x \rangle \geq \langle v, y \rangle, \; \forall y \in Q \implies
\langle v, y - x \rangle \l... | {
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Put trivariate PDF in terms of bivariate PDFs Let there be the random variables $X$, $Y$, and $Z$. Let all the bivariate PDFs $f_{X, Y}$, $f_{X, Z}$, and $f_{Y, Z}$ be known.
Can we write the unknown trivariate PDF $f_{X, Y, Z}$ in terms of the known bivariate PDFs?
| Here is an example, obtained by tweaking a 2D counterexample: Let $(X,Y,Z)$ be such that
$$
f_{X,Y,Z}(x,y,z)=
\begin{cases}
2 \phi(x)\phi(y)\phi(z) & xyz>0\\
0 & \text{otherwise}
\end{cases}
$$
where $\phi$ is the standard 1D Gaussian pdf. Then the bivariates $f_{X,Y}, f_{Y,Z}, f_{Z,X}$ are all standard 2D Gaussians, ... | {
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If $\space$ $\forall$ $x \in \Bbb R$, $\space$ $f(f(x))=x^2-x+1$. Find the value of $f(0)$. If $\space$ $\forall$ $x \in \Bbb R$, $\space$ $f(f(x))=x^2-x+1$. Find the value of $f(0)$.
I thought that making $f(x)=0$ implies that $f(0)= 0^2 - 0 + 1 = 1$, but i think that this isn't correct, because the $x$ in $f(f(x))$ i... | Note that the only fixed point of $ff$ is $1$. Since $ff(0)=1$ is fixed by $ff$, we have $fff(0)$ must also be a fixed point of $ff$. This gives $fff(0)=1$ and so $f(0)\in(ff)^{-1}(1)=\{0,1\}$. But $0$ isn't a fixed point of $f$ (since it isn't fixed under $ff$), so $f(0)=1$.
| {
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Can the minimum of two consecutive prime gaps become arbitary large? Here :
https://oeis.org/A023186
the so-called "Lonely primes" are shown.
Let $$[a,b,c]$$ be a triple of consecutive primes and define $$d:=\min(c-b,b-a)$$
My question :
Can we prove that $d$ can become arbitary large ? In other words, can we prove ... | If you read that OEIS page you linked to, it says
Erdős and Suranyi call these reclusive primes and prove that there are an infinite number of them.
(Emphasis mine.) Each lonely prime is the $b$ of a consecutive prime triple $[a, b, c]$ for which $d = \min(b-a, c-b)$ is larger than for any previous prime triple. Sinc... | {
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Partial Differentiation of an equation using implicit differentiation confusion I wanted to ask a question about implicit differentiation in partial differentiation.
When I was at school, I remember partial differentiation as something like this:
When you have a function composed of $x$ and $y$'s and you run into a $... | You're right that $N$ is not a constant; in fact, it's shorthand for a function $N(N_A, N_B) = N_A + N_B$.
$N$ is substituted for convenience. If you want to differentiate the expression $F/kB$ with respect to $N_A$, one option is to replace $N$ with $N_A+N_B$ wherever it occurs. $N$ is defined to be $N_A+N_B$ so thi... | {
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If $f:A\rightarrow \mathbb{R}$ is an integrable function on $B$, do $\int_Af$ exists? Let $A$ and $B$ be open sets of $\mathbb{R}^n$ such that $A-B$ has measure zero. If $f:A\rightarrow \mathbb{R}$ is an integrable function on $B$, can we ensure that $\int_Af$ exists?
I am trying show that $\int_Af$ exists, where
$$A=\... | You are right. You can use that the integral over $A$ is equal to the integral over $A\cap B$ plus the integral of $A\setminus B$ that is zero (and that $f\geq 0$). So $f$ is an integral function over A.
| {
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"answer_id": 0
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Determine this limit $\lim_{n\to\infty} \frac{\ln\left(\frac{3\pi}{4} + 2n\right)-\ln\left(\frac{\pi}{4}+2n\right)}{\ln(2n+2)-\ln(2n)}.$ how can I determine the following limit? $$\lim_{n\to\infty} \frac{\ln\left(\frac{3\pi}{4} + 2n\right)-\ln\left(\frac{\pi}{4}+2n\right)}{\ln(2n+2)-\ln(2n)}.$$
This question stems from... | Using the rules of logarithm so write $$\frac{\ln\left(\frac{\frac{5\pi}{4}+2n}{\frac{\pi}{4}+2n}\right)}{\ln\left(\frac{2n+2}{2n}\right)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3235939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Index of Fibonacci primes and Lucas primes. For an integer $n\geq 0$ let $F_n$ denote the $n$th Fibonacci number and let
$L_n$ denote the $n$th Lucas number.
It is known
that $F_n$ is prime only if $n$ is prime or $n=4$.
According to Wikipedia
it is known that $L_n$ is prime only if $n$ is $0$, prime or a power of $2... | Your question may not yet be answerable because we don't know if there exist an infinite number of Fibonacci (or Lucas primes); Moreover, we don't even know if there exist an infinite number of composite Fibonacci (Lucas) numbers.
One thing (among many) that we do know is that $F_{2n} = F_{n}L_{n}$ with the $gcd(F_{n}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Triangle inequality for angles in Euclidean space Is there any simple proof of the following statement: for all vectors $ v,w,u\in V\setminus\{0\} $, where $ V $ is a Euclidean space, inequality
$$ \angle(u,v)\le\angle(u,w)+\angle(w,v)$$
holds.
Unfortunately, couldn't find anything useful in books or Google. I've seen... | That inequality is only true if you are careful about the numerical value assigned to an angle and how you add angles.
The dot product definition gives signed angles.
If you measure angles by the (nonnegative) great circle arclength they cut off on the unit sphere what should happen when the sum wraps around to more t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3236181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Fluids - vector identity confusion Trying to prove Kelvin's Circulation Theorem, but struggling to see why the following equality holds:
$$\textbf{u} \cdot (d\textbf{l}\cdot \nabla)\textbf{u} = d\textbf{l} \cdot \nabla \left(\frac{1}{2} u^2 \right)$$
I don't really have much idea where to start. I can take that gradi... | \begin{align}
d\textbf{l} \cdot \nabla \left(\frac{1}{2} u^2 \right)
&\;=\; \sum_{\mu\, \nu} dl_{\mu} \, \frac{\partial}{\partial x_{\mu}} \, \frac{1}{2} u_{\nu}u_{\nu}\\
&\;=\; \sum_{\mu\, \nu} dl_{\mu} \, u_{\nu}\, \frac{\partial}{\partial x_{\mu}} u_{\nu}\\
&\;=\; \sum_{\mu\, \nu} u_{\nu} \, dl_{\mu} \, \frac{\parti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3236285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If ${a_k}$ is bounded, then $\sum_{k=0}^{\infty}{a_k}z^{k}$ defines an analytic function on the open unit disk. Is this conclusion true, If ${a_k}$ is bounded, then $\sum_{k=0}^{\infty}{a_k}z^{k}$ defines an analytic function on the open unit disk.
I tried to construct counterexamples but looks like the statement is ... | Use M-test. $|z| <1$ implies $\sum |a_k||z^{k}|\leq M\sum|z|^{k} =M\frac 1 {1-|z|}<\infty$ where $M=\sup \{|a_k|:k\geq 1\}$. This implies that the sum is analytic in the open unit disk.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3236542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Cycle notation for cyclic orders Is there a convenient cycle notation for cyclic orders (https://en.wikipedia.org/wiki/Cyclic_order)?
For example:
Definition. A set of four elements $a, b, c, d$ of a cyclically ordered set is a 4-cycle $[a, b, c, d]$ if $[a,b,c] \land [c,d,a]$.
Using transitivity it is easy to show tha... | I am not aware of any standard notation, but your notation $[a, b, c, d]$ makes sense and could be extended to $[a_1, a_2, \dotsm, a_n]$ for any $n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3236651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Question regarding partially ordered sets I have encountered few questions while reading the book 'Modern Algebra'.
Let $\mathbb Q$ be the set of rational numbers.
Let
$B = \{ x : x\in\mathbb Q,\sqrt2 < x < \sqrt3 \}$.
How it can be shown that -
*
*$B$ has infinite number of upper and lower bounds.
*$\inf B$ and ... |
Why $\sqrt2$ and $\sqrt 3$ can not be taken as lower and upper bounds?
Because the exercise is implicitly asking about an inf and sup that are in $\mathbb Q$. The exercise is perhaps not stating this very clearly, but remember that infimum and supremum is always about how a set fits into a certain ordered superset.
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How many toys can be chosen? There are 3 red, 5 blue, 2 yellow and 4 green toys in the box. In how many different ways can 6 toys be chosen if one of them should be blue and the other one - yellow?
I came up with a solution but i am not sure if it is right.
2 toys out of 6 should be of a specific colour. Then, only 4 t... | Given the clarifications in comments – the toys are distinct, order doesn't matter, exactly one is blue and exactly one is yellow – we have
*
*$5$ ways to choose the blue toy and $2$ ways to choose the yellow one
*$\binom74=35$ ways to choose the other four toys
Thus there are actually $5×2×35=350$ valid selectio... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Covering map from sphere with six points removed to doubly-punctured complex plane
$X$ is $S^2\subset \mathbf{R}^3$ with its intersection points with the coordinate axes removed.
Show that the following map is a covering map. $$\begin{align*}p:X&\longrightarrow \mathbf{C}-\{0,1\} \\ (x,y,z)&\longmapsto \left(\frac{x+i... | $\left(\dfrac{x+iy}{1-z} \right)^4 = 1$ means that $\dfrac{x+iy}{1-z} \in \{ 1, -1, i, -i \}$. Let us assume $\dfrac{x+iy}{1-z} = \pm1$. Then $x + iy = \pm(1-z)$ which implies $y = 0$ and $x = \pm(1-z)$. Since $x^2 + z^2 = 1$, we get $(1-z)^2 + z^2 = 1$ which implies $z =0$ or $z=1$. For $z=0$ we get $x = \pm 1$, hence... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3237091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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$f(x)= \frac{\sqrt{x^2-1}}{x+\log x}$ in the set $E=[1,+ \infty)$ I have the function $f(x)= \frac{\sqrt{x^2-1}}{x+\log x}$ in the set $E=[1,+ \infty)$
(i) I have to prove that $\forall y_0 \in [0,1) $exists only one $x_0 \in E $ such that $f(x_0)=y_0$
(ii) to discuss the uniform continuity of f in E.
Supposing that th... | 1) first show that $f$ is a strictly increasing function; hence, (i) follows.
2) note that $f$ is also bounded and continuous. Now show that any bounded continuous increasing function is uniformly continuous. Hence $f$ is also uniformly continuous.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3237240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Counting strategies Exam Question How many different ways can people finish in
i) a $4$ person race, ii) a $6$ person race, iii) a $10$ person race
What I did:
$4^4 = 256$
$6^6 = 46,656$
$10^{10}$
as there are $4$ people and therefore 4th place 3rd place 2nd place and 1st place so $4$ to the power of $4$.
I don't kno... | Just because there are $4$ ways to choose the first person and $4$ ways to choose the second person doesn't mean there are $16$ ways to choose both; some combinations such as $AAAA$ are't allowed (but are still counted).
Instead, try looking at the factorial function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3237411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Factoring primes in $\mathbb{Q}(\sqrt{-1},\sqrt{5})$
Show that $2,5$ are the only primes which ramify in $L:=\mathbb{Q}(\sqrt{-1},\sqrt{5})$, and that their ramification indices are both $2$.
Obviously $K_1:=\mathbb{Q}(\sqrt{-1})$ and $K_2:=\mathbb{Q}(\sqrt{5})$ are both Galois and it's easy to check that:
$$K_1K_2=... | I'll make my comment and answer and add the case of $5$.
See what happens to $(2)$ in $_2$ and then from there see what happens to $K_1K_2$. From $\mathbb{Q}$ to $K_2$ we have $e=1$ and $f=2$ and $g=1$ (since $^2−x−1$ is irreducible mod 2). Then going from $_2$ to $_1_2$ a degree 2 extension we know $=1$ or $e=2$ but i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3237501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Integral with log of absolute value of sine Show that
$$\int_{-\pi/3}^{\pi/3} \log \vert 8 \sin(t/2) (1 + \sin t)^2 \vert dt = 0.$$
WolframAlpha claims that this is true. I've tried manipulating the integrand a bunch and various trig identities, but it hasn't simplified things. It looks vaguely like Jensen's formula, b... | Note that (using $t \to -t$ on the negative interval):
$\int_{-\pi/3}^{\pi/3} \log \vert 8 \sin(t/2) (1 + \sin t)^2 \vert dt =2\pi \log 2+2\int_{0}^{\pi/3}\log {\sin(t/2)}dt+ 4\int_{0}^{\pi/3}\log {\cos t}dt$
Using now $t/2 \to t$ we get
$\int_{0}^{\pi/3}\log {\sin(t/2)}dt=2\int_{0}^{\pi/6}\log {\sin t}dt=2\int_{\pi/3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3237621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Reasoning behind integrating f(x)/g(x)? I understand the method to integrate this function would be:
$\int{\frac{x^2+1}{x^4-x^2+1} \thinspace dx}$
Divide all terms by $x^2$:
$= \int{\frac{\frac{x^2}{x^2}+\frac{1}{x^2}}{\frac{x^4}{x^2}-\frac{x^2}{x^2}+\frac{1}{x^2}} \thinspace dx}$
=$ \int{\frac{1+\frac{1}{x^2}}{x^2-1+\... | Here's another way to do it. I would go for this way if I don't immediately see a trick.
It's a big theorem that all rational functions have elementary antiderivatives. The general way to integrate a rational function is to factor it into quadratics and linears (this is always possible by FTA), and use partial fraction... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3237793",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to solve this inequation? So the inequation is this $x^{2016}-1<0 $
My initial idea was to transform it like this $x^{2016}>x^0$ and then to look four cases:
$1.$ when $x \lt 0\lt 1$
$2.$ when $x \gt 1$
$ 3.$ when $-1\lt x\lt 0$
$4.$ when $x \lt-1$
Is this the proper way to do it?
| As others have said, a far simpler method would be to just do the following:
$x^{2016}-1<0 \Rightarrow x^{2016}<1 \Rightarrow \ln|x^{2016}|<\ln|1|\Rightarrow\ln|x^{2016}|<0\Rightarrow \ln|x|<0\Rightarrow |x|<1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3237873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
The dual space of $C(Y,\mathbb R)$ when $Y$ is a complete and separable metric space Just to be confirmed what is the dual space of $C(Y,\mathbb R)$ i.e the vector space of all continuous functions $f: Y\to \mathbb R$ when $Y$ is complete and separable metric space? Is it the same when $Y$ is compact, which is the spac... | If you are also interested in the space $C_b(Y, \mathbb{R})$ of bounded continuous functions endowed with the topology of uniform convergence, which is normed by
\begin{equation*}
||f|| := \sup_{y \in Y} |f(y)|, \quad f \in C_b(Y, \mathbb{R}),
\end{equation*}
then the assumption that $Y$ is complete separable metric sp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3238053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Paul J Nahin. Story of minus one, I'm stumped on page 5. every time I try to read this book and follow the logic I fail on just page 5 where it states near the bottom
$${1 \over x } + 14 x + \sqrt{{1\over x^2} + 196 x^2 } = 12,$$
which is easily put into the form given above,
$$ 172 x = 336 x^2 +24.$$
(From An Imag... | Let $\frac{1}{x}+14x=y$ then we can rewrite the equation as $y+\sqrt{y^2-28}=12 \rightarrow \sqrt{y^2-28}=12-y$ Squaring both parts we get: $y^2-28=144-24y+y^2$ or $24y=172$ or $$24(\frac{1}{x}+14x)=172$$ Can you finish?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3238196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Wikipedia's proof of Jensen's inequality I think there is a glitch in the proof by induction. The proof is still valid, but they add an unnecessary assumption:
In the induction step, they choose one of the $\lambda_i$'s that is strictly positive (I guess by that, they mean nonzero). Since the sum of the $\lambda_i$'s i... | Of course, $\lambda_1=0$ is valid, but not so interesting:
$$ \varphi\left(\sum_{i=2}^{n+1}\lambda_i x_i \right) ~\leqslant~ \varphi\left( \sum_{i=2}^{n+1} \lambda_i x_i \right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3238316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
The limit of the interval endpoints depending on $n$ For $n \in \mathbb{N_0}$ consider the sequence of intervals of the the following from:
\begin{align}
A_k &:= [ k, k + 1 ), \quad k \in \mathbb{N_0} \\
\frac{A_k}{2} &:= \left[ \frac{k}{2}, \frac{k + 1}{2} \right), \quad k \in \mathbb{N_0} \\
. \\
. \\
. \\
\frac{A_k... | For the decreasing part, it's enough to show that $\frac{k_n+1}{2^n}$ is also the endpoint of one of the intervals for any $m> n$. Because then it's obvious that $\frac{k_m+1}{2^m}$ is either that or something smaller. And in fact, if $m>n$, you have:
$$\frac{k_n+1}{2^n}=\frac{2^{m-n}(k_n+1)}{2^m}=\frac{\left(2^{m-n}k_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3238421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A polynomial algorithm to determine whether a finite group is nilpotent Does there exist a polynomial (in respect to the order of the group) algorithm that given a Cayley table of a finite group determines, whether a group is nilpotent or not?
There do exist polynomial algorithms, that determine, whether a group is nil... | A simple (but inefficient) way to test nilpotency in polynomial time would be to compute its lower central series and see whether the final group is trivial or not.
Each commutator subgroup computation takes $O(n^2)$ time, and since the orders of the groups in this central series do not increase, they must stabilise in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3238510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
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