Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
The time it takes an object to go through a hole through the Earth's center Newton's law of attraction is given by the differential equation $$\frac {dv}{dt} = -\frac {gr}{R}$$ where $r$ is the distance from the center of the Earth, $R$ is the radius of the Earth, and $g$ is the acceleration due to gravity. What is the... | Write it as $\frac {d^2r}{dt^2}+\frac {gr}R=0$ and you have a harmonic oscillator The angular frequency is $\sqrt {\frac {g}R}$ and the period is $2 \pi\sqrt{\frac { R}{g}}$
| {
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"timestamp": "2023-03-29T00:00:00",
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Two exercises on set theory about Cantor's equation and the von Neumann hierarchy Good evening to all.
I have two exercises I tried to resolve without a rigorous success:
*
*Is it true or false that if $\kappa$ is a non-numerable cardinal number then $\omega^\kappa = \kappa$, where the exponentiation is the ordinal ... | You really just have to apply the definitions, in both cases, and see what happens.
For the first one, recall that the definition in the case of exponentiation is as follows: $$\alpha^0=1;\ \alpha^{\beta+1}=\alpha^\beta\cdot\alpha;\ \alpha^\delta=\sup\{\alpha^\gamma\mid\gamma<\delta\}.$$
Since $\kappa$ is a limit ordin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/496591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Bounds for $\binom{n}{cn}$ with $0 < c < 1$. Are there really good upper and lower bounds for $\binom{n}{cn}$ when $c$ is a constant $0 < c < 1$? I know that $\left(\frac{1}{c^{cn}}\right) \leq \binom{n}{cn} \leq \left(\frac{e}{c}\right)^{cn}$.
| Hint: as $n$ goes to infinity, you can approximate $\binom{n}{cn}$ by Entropy function as follows:
$$
\binom{n}{cn}\approx e^{nH(c)}.
$$
where $H(c)=-c\log(c)-(1-c)\log(1-c)$
| {
"language": "en",
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If A and B are compact than also A+B Suppose we have a topological vector space $X$ and $A, B\subset X$. We define A+B to be the set of the sums $a+b$ where $a\in A$ and $b\in B$. We should prove that also A+B is compact if A and B are compact. But the union of arbitrary compact sets isn't compact in generell. Thus: wh... | The function $F(x,y): X \times X \to X$ defined by $F(x,y)=x+y$ is continuous, and $A \times B$ is compact in $X \times Y$.
The image of a compact set under an continuous function is compact.
| {
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Sum of low rank tensors How high can the sum of $k$ low rank $m\times m\times\dots \times m$ tensors of rank $t$ be? Is there a good upper bound?
| Rank $t$ means $t$ (but no more) linearly independent rows/columns. If those $t$ that are linearly independent in the first matrix are also linearly independent to $t$ of those independent in the second one, you get rank $2t$, and so on.
Therefore, your answer is $r_{MAX} = \min \{ m, kt \}$.
Without knowing anything m... | {
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Conversion Question A heavy rain fell on a city for 26 minutes, at a rate of 3.9mm/hr. If the area of the city is 244km^2, how many gallons of water fell on the city that day?
| First convert minutes into hours. So we have: $26 min = \frac{26}{60}h = \frac{13}{30}h$
Now multiply the time by the rate to find how much rain did fall.
$\frac{13}{30} \times \frac{39}{10} = \frac{507}{300}mm$.
If rain wouldn't go underground and it stayed above ground then the depth of the "pool" would be $\frac{5... | {
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Better solution to an elementary number theory problem I found the following problem about elementary number thery
Alice designed a program such that it takes an integer $n>1$, and then it factors it as $a_0^{e_0}a_1^{e_1}a_2^{e_2}\cdots a_1^{e_n}$. It then calculates $r=a_0e_0+a_1e_1\cdots+a_ne_n+1$ and repeats the p... | Lemma 1: If $ x $ is a prime, then $f(x) = x+1$.
Lemma 2: If $x = mn$ (not necessarily coprime), then $f(mn) - 1 = [f(m) - 1] + [f(n) - 1 ]$.
I consider this the crux of the function. This is easily proved (once you know it).
Now check that $f(4) = 5$, $f(6) = 6$ and $f(8) = 7$.
Lemma 3: $f(x) \leq x+1$.
Lemma 4: If $... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How prove this $\displaystyle\sum_{k=1}^{n}\sin\frac{1}{(k+1)^2}\le\ln{2}$ show that
$$\sum_{k=1}^{n}\sin\dfrac{1}{(k+1)^2}\le\ln{2}$$
I think this is nice inequality, and idea maybe use this
$$\sin{x}<x$$
so
$$\sum_{k=1}^{n}\sin{\dfrac{1}{(n+1)^2}}<\sum_{k=1}^{n}\dfrac{1}{(k+1)^2}<\dfrac{\pi^2}{6}-1\approx 0.644<\ln{... | Once you noticed that $\sin x\le x$ you do not need to know the exact value of $\sum_{k=1}^{\infty}\frac{1}{k^2}.$ Instead, you can approximate it by evaluating first few terms and estimating the tail. More precisely,
$$\sum_{k=2}^{n}\frac{1}{k^2}=\left(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}\right)+\frac{1} {5^2}+...+\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/497138",
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Calculate the possible combinations for an eight-character password Can any when one help me in this question:
Calculate the possible combinations for;
*
*An eight-character password consisting of upper- and lowercase letters and at least one numeric digit $(0–9)$?
*A ten-character password consisting of upper- an... | Hint:
(v): One character can be one of 62 (= 26[A-Z]+26[a-z]+10[0-9]) letters, and choosing one character for password is independent of choosing other characters for password.
(vi): #(An alphanumeric password containing at least one numeric digit) = #(An alphanumeric password) - #(An alphanumeric password which not co... | {
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discretization of probability measures Suppose I have given a probability measure $\nu$ over the positive reals. For a fixed $n\in\mathbb{N}$, we set $\lambda := \frac{1}{n}$ and $A_n:=\{\lambda k, k=0,\dots\}$. Now we look at a certain discretization of $\nu$ on $A_n$:
$$\nu_n(\{0\}):= \int_0^\lambda (1-nx)d\nu (x) \\... | Both sides of (1) are linear functionals of the sequence $(g(x))_{x\in A_n}$, the LHS because the support of $\nu_n$ is included in $A_n$ and the RHS because $F^n(g)$ depends on $(g(x))_{x\in A_n}$ only.
Fix some $k\geqslant0$. The coefficient of $g(k/n)$ on the LHS is $\nu_n(\{k/n\})$.
The coefficient of $g(k/n)$ on ... | {
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Show that $\frac{1}{(n+1)!}(1+\frac{1}{(n+1)}+\frac{1}{(n+1)^2}+\frac{1}{(n+1)^3}+\cdots)=\frac{1}{n!n}$ Show that $\frac{1}{(n+1)!}(1+\frac{1}{(n+1)}+\frac{1}{(n+1)^2}+\frac{1}{(n+1)^3}+\cdots)=\frac{1}{n!n}$, (here $n$ is a natural number)
Maybe easy, but I cannot see it.
Thanks in advance!
Alexander
| Since $$1+x+x^2+\cdots=\frac1{1-x}$$
Therefore,
$$1+\frac1{n+1}+\frac1{(n+1)^2}+\cdots=\frac1{1-\frac1{n+1}}=\frac1{\frac{n+1-1}{n+1}}=\frac{n+1}{n}$$
Now,
$$\frac1{(n+1)!}\cdot(1+\frac1{n+1}+\frac1{(n+1)^2}+\cdots)=\frac1{(n+1)!}\cdot\frac{n+1}{n}=\frac1{n!n}$$
| {
"language": "en",
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Expressing a $3\times 3$ determinant as the product of four factors I am attempting to express the determinant below as a product of four linear factors
$$\begin{vmatrix}
a & bc & b+c\\
b & ca & c+a\\
c & ab & a+b\\
\end{vmatrix}
=
a\begin{vmatrix}
ca & c+a\\
ab & a+b\\
\end{vmatrix}
-
bc\begin{vmatrix}
b & c+a\\
c & ... | Start by adding the 1st to the 3rd column to create a column of $a+b+c$'s. Then subtract 3rd row from 2nd & 1st ones to make two out of three entries in that column zero. Now expand wrt that column.
| {
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How to solve $e^x = 2$ I know that $\ln(x)$ is the inverse of the exponential function $a^x$.
So I thought that
$$
e^x=2 \Leftrightarrow x = \ln(2)
$$
but my calculator says $x = \ln(2) + 2 i \pi n$, where $N \in \mathbb{Z}$. What have $e^x$ and $\ln(x)$ to do with the unit circle?
| This comes from the complex analysis ideas. If we know that $x$ is real valued, then clearly $x = \log 2$. However if $x$ is allowed to be complex valued, things become trickier. We know that for any $k\in\mathbb{Z}$, $\,\,e^{i\theta + 2k\pi i} = e^{i\theta}$. You can work this out yourself with Euler's formula. So the... | {
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"timestamp": "2023-03-29T00:00:00",
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Predicate Logic Translating "All But One" I need to translate an English sentence including the phrase "all but one" into predicate logic. The sentence is: "All students but one have an internet connection." I'm not sure how to show "all but one" in logic.
I could say $\forall x ((x \neq a) \rightarrow I(x))$
$I(x)$ be... | "For all but one $\;x\;$, $\;P(x)\;$ holds" is the same as "there exists a unique $\;x\;$ such that $\;\lnot P(x)\;$ holds.
Normally the notation $\;\exists!\;$ is used for "there exists a unique" (just like $\;\exists\;$ is used for "there exists some").
If your answer is allowed to use $\;\exists!\;$, then the above ... | {
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Lemma for the construction of the reciprocity map I do not understand the highlighted part in the following proof, namley that $N(\tilde x)=1$.
To give some context, this proof is taken from Neukirch's Algebraic Number Theory, where $\tilde K$ indicates the maximal unramified extension of $K$ (and the same for $L$).
... | Ok, solved, they commute because $\sigma\in G(\tilde{L}\mid L)$ and $\tau_i\in G(\tilde L\mid\tilde K)$, which are two normal subgroups of $G(\tilde L\mid K)$ whose intersection is trivial.
| {
"language": "en",
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lcm in $\mathbb{Z}[\sqrt{-5}]$ does not exists
I need to show that lcm of $2$ and $1+\sqrt{-5}$ does not exists in $\mathbb{Z}[\sqrt{-5}]$
Getting no idea about how to start, I was thinking when does lcm cannot exists!
| Let $a=x+y\sqrt{-5}$ be an LCM of $2$ and $1+\sqrt{-5}$. Then $(a)$ is equal to the ideal $I:=(2) \cap (1+\sqrt{-5})$. It can be seen that $I$ has index $12$ in $\mathbb{Z}[\sqrt{-5}]$. But $(a)$ has index $Nm(a)=x^2+5y^2$ which is different from $12$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove $3|n(n+1)(n+2)$ by induction I tried proving inductively but I didn't really go anywhere. So I tried:
Let $3|n(n+1)(n+2)$.
Then $3|n^3 + 3n^2 + 2n \Longrightarrow 3|(n(n(n+3)) + 2)$
But then?
| Consider the binomial $(x+1)^{n+2}$. The coefficient of the $x^3$ term is
$${n+2\choose 3}={(n+2)!\over 3!(n-1)!}={n(n+1)(n+2)\over 6}$$
Every coefficient of $(x+1)^n$ is an integer for $n$ an integer, therefore $6|n(n+1)(n+2)$ and thus $3|n(n+1)(n+2)$.
Note that this mechanism can apply to any integer, including show... | {
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"timestamp": "2023-03-29T00:00:00",
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Prime and Factorization, prime divisor property Let $p$ be prime. Then if $p|ab$ then $p|a$ or $p|b$.
Proof:
Suppose $p$ does not divide $a$
Then $\gcd (a,p) = 1$ since $p$ is prime.
$$ 1 = ma + np $$
$$ b = mab +npb$$
Since $p|map$ and $p|npb$ then $p|b$
I have a problem understanding that $p|map$, can anyone show m... | Note that the initial condition is $p|ab$, so from this follows that $ab = pk$, where $k$ is some positive integer. Assuming that $gcd(p,a) = 1$, then from the Bezout Lemma follows
$$1 = ma + np$$
$$b = mab + npb$$
This is something that you've already done, now make the substitution and get:
$$b = mpk + npb$$
$$b = p(... | {
"language": "en",
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Mean of probability distribution function. The current chapter I am working on is continuous random variables. I know that the mean value of a continuous random variable is:
$$ E[X] =\int_{-\infty}^{\infty} xf(x) dx $$
That being said, my question is to find $E[X]$ of the following table:
$$
X |\hspace{4 mm} -3 \hspa... | Yes, it is right. However, and just for fun, there is an alternative definition of $E[X]$:
$$\mu_X=E[X]=\int\limits_{0}^{+\infty}{1-F_x(x)dx} -\int\limits_{-\infty}^{0}{F_x(x)dx}$$
and one can prove that is equivalent to other definition (good exercise). This definition is general for any random variable, discrete or c... | {
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Verifying finite simple groups The classification of finite simple groups required thousands of pages in journals. The end result is that a finite group is simple if and only if it is on a list of 26 sporadic groups and several families of groups.
Usually in classification theorems proving that the items on the list do... | Probably the best source for this would be the (graduate level) textbook The Finite Simple Groups by R.A. Wilson. It is under 300 pages and covers all of the finite simple groups.
It proves simplicity of all of them.
It proves existence and uniqueness of nearly all of them.
It describes interesting structure of most of... | {
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"url": "https://math.stackexchange.com/questions/498162",
"timestamp": "2023-03-29T00:00:00",
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Show that the equation $x^2+xy-y^2=3$ does not have integer solutions. Show that the equation $$x^2+xy-y^2=3$$ does not have integer solutions.
I solved the equation for $x$:
$x=\displaystyle \frac{-y\pm\sqrt{y^2+4(y^2+3)}}{2}$
$\displaystyle =\frac{-y\pm\sqrt{5y^2+12}}{2}$
I was then trying to show that $\sqrt{5y^2+1... | Note that
$$x^2+xy-y^2=(x-2y)^2+5(xy-y^2)=(x-2y)^2\qquad({\rm mod}\>5)\ .$$
But
$$0^2=0,\quad(\pm1)^2=1,\quad(\pm2)^2=-1\qquad({\rm mod}\>5)\ ,$$
which implies that $3$ is not a quadratic residue modulo $5$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/498236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Volume calculation with n-variables integrals Given $A=(a_{i.j})_{1\le i,j \le n}$ invertible matrix of size $n \times n$, and given $T$ the domain in $\mathbb{R}^n$ which is defined by the following inequality: $\alpha_i \le \sum_{j=1}^{n}{a_{i,j}x_j} \le \beta_i$.
(a). How can I calculate the volume $V(T)$?
(b). Give... | Note that $x \in T$ iff $Ax \in \prod_{i=1}^n [\alpha_i, \beta_i]$, that is $AT = \prod_{i=1}^n [\alpha_i, \beta_i]$ so we have, by the integral transformation formula
\begin{align*}
V(T) &= \int_T \,dx\\
&= \int_{AT} |\det A^{-1}|\, dy \\
&= \frac 1{|\det A|} \cdot \prod_{i=1}^n (\beta_i - \alpha_i)
\... | {
"language": "en",
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$f:\mathbb{R}\to\mathbb{R}$ is defined as $f(x)=n\forall x=n\in\mathbb{N}$ Let $f:\mathbb{R}\to\mathbb{R}$ be defined as $ f(x) := \begin{cases} x, & \text{if}\ x\in \mathbb N,\\\\ 0, & \text{else,} \end{cases} $
and $T=\mathbb{N}\cup\{n+1/n:n\in\mathbb{N}\}$.
The function $f$ is continuous on $\mathbb{N}$ with resp... | Actually, your proof is correct but not written down properly.
Let $\epsilon = \frac{1}{2}$ then we claim there is no $\delta > 0$ with
$$|f(x) - f(y)| < \epsilon \qquad \forall\ |x-y| < \delta$$
Let $\delta_0$ be such a choice and chose $N := \lceil \frac{1}{\delta_0} \rceil + 1$, then
$$\left|N - N+\frac{1}{N}\right|... | {
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"timestamp": "2023-03-29T00:00:00",
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Damped oscillation fit We have some measurement data like this:
The expected behavior of the data is a damped oscillation:
$$y=a e^{d*t} cos(\omega t+\phi) + k$$
Where:
$t$ Current time
$y$ Current deflection
$a$ Amplitude
$d$ Damping factor
$\omega$ Angluar velocity
$\phi$ Phase shift
$k$ Offset
The task is to f... | You are trying to solve the harmonic inversion problem. That website contains code and programs for it.
| {
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"timestamp": "2023-03-29T00:00:00",
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How to prove $\cos 36^{\circ} = (1+ \sqrt 5)/4$? Given $4 \cos^2 x -2\cos x -1 = 0$.
Use this to show that $\cos 36^{\circ} = (1+ \sqrt 5)/4$, $\cos 72^{\circ} = (-1+\sqrt 5)/4$
Your help is greatly appreciated! Thanks
| Hint: Look at the Quadratic Formula:
The solution to $ax^2+bx+c=0$ is $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
The equation is based on the fact that
$$
\cos(5x)=16\cos^5(x)-20\cos^3(x)+5\cos(x)
$$
and that $\cos(5\cdot36^\circ)=-1$ to get
$$
16\cos^5(36^\circ)-20\cos^3(36^\circ)+5\cos(36^\circ)+1=0
$$
Factoring yields
$$
... | {
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What is an odd function? I'm reading this term (odd function) in my numerical analysis book, but I have never heard of this. What does it mean that an function is odd ?
| A function is said to be odd if changing the sign of the variable changes the sign of the function (keeping the absolute value the same). It is even if changing the sign of the variable does not change the function. We express this mathematically as:
Odd: $f(-x)=-f(x)$
Even: $f(-x)=f(x)$
It is sometimes useful to know ... | {
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Given a covarince matrix, generate a Gaussian random variable Given a $M \times M$ desired covariance, $R$, and a desired number of sample vectors, $N$ calculate a $N \times M$ Gaussian random vector, $X$.
Not really sure what to do here. You can calculate the joint pdf given a mean, $\mu$, and covariance. So for $2 ... | To generate one vector $u\in\mathcal{R}^M$, first of all generate any vector $v$ from $\mathcal{N}(0,I)$ (or M independent normally distributed variables with mean $0$, varaince $1$).
We now need to get a matrix $L$ such that $LL^T=R$, easiest way is a cholesky decomposition. Now $u=Lv\sim\mathcal{N}(0,R)$
| {
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combinatorics: Repeating a procedure, will $0$ ever show up? On a black board we have written the numbers $1$ $2$ $...$ $50$ in a list. Each time we clear two numbers and write their difference instead. We continue this until there is only one number left. Is it possible that the number is zero?
I guess that the answer... | When we clear two numbers and write their difference two cases could happen:
If $a$ and $b$ (the two numbers we've cleared) have the same parity then $a-b$ will be even. Otherwise, $a-b$ will be odd.
Now, from $1$ to $50$ there are exactly $25$ odd numbers and $25$ even numbers. If the first case happens, then we eithe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/498830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Did I write down the derivative product rule correctly for $g(x)=(f(x))^2$ Suppose that $f(4)=5$ and $f'(4)=5$ . Use the product rule to determine the value of $g'(4)$ where $g(x)=(f(x))^2$
So I'm writing this problem as: $g'(x)=(f(x))\frac{d}{dx}f(x)+\frac{d}{dx}f(x)(f(x))$
If anybody can verify that I wrote it down... | Yes, you've written it correctly. Note that after simplification, the result is $2f(x) f'(x)$, which also agrees with what the chain rule would say in this context.
| {
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Defining a set using a predicate I encountered this notation in a proof and I'm a bit confused on the message it's trying to convey:
$ B = \{ n \in \mathbb N \mid \neg P(n) \} $
Here's the notation itself within the greater context of the proof (or proof 'fragment' more like):
...
$ P(n): $ predicate-definition
Assum... | That is exactly right. The notation $X=\{x\ |\ y\}$ means that $X$ is the set of all $x$ such that condition (or conditions) $y$ are satisfied. $x$ and $y$ can be more complex statements, as is the case in your example
| {
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Prove the limit.. $\varinjlim \sqrt{n^2+1}-n=0$.
I need to prove that this converges to 0.
Usung the definition of a sequence helps for the normal problems but for this I believe the triangle inequality is used at some point....
I let $S_n=\varinjlim \sqrt{n^2+1}-n$
Then $|S_n -S|<\epsilon \rightarrow |\varinjlim \sqr... | Try this:
$$0 < \frac{\sqrt{n^2+1}-n}{1} = \frac{(\sqrt{n^2+1}-n)(\sqrt{n^2+1}+n)}{\sqrt{n^2+1}+n} = \frac{1}{\sqrt{n^2+1}+n} < \frac{1}{2n}$$
and then prove that the transformed sequence converges to zero.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Projective space is not affine I read a prove that the projective space $\mathbb P_{R}^{n}$ is not affine (n>0): (Remark 3.14 p72 Algebraic Geometry I by Wedhorn,Gortz).
It said that the canonical ring homomorphism $R$ to $\Gamma(\mathbb P_{R}^{n}, \mathcal{O}_{\mathbb P_{R}^{n}})$ is an isomorphism. This implies that ... | First, let us review the definition of an affine scheme. An affine scheme $X$ is a locally ringed space isomorphic to $\operatorname{Spec} A$ for some commutative ring $A$. This means that if one knows one has an affine scheme $X$, then all one has to do to recover $A$ such that $X=\operatorname{Spec} A$ is to take glo... | {
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Why does $\mathbf{Var}(X) = \mathbf{Var}(-X)$ for random variable $X$? Question from UCLA Math GRE study packet, Problem Set 2, Number 4:
http://www.math.ucla.edu/~cmarshak/GREProb.pdf
Let $X$ and $Y$ be random variables. Which of the following is always true?
\begin{align}
...\\
(II) \ \mathbf{Var}(X) = \mathbf{Var}(-... | Let $X$ be a random variable and $\alpha \in \mathbb R$. We have
\begin{align*}
{\rm Var}(\alpha X) &= \def\E{\mathbb E}\E[(\alpha X)^2] - \E[\alpha X]^2\\
&= \E[\alpha^2 X^2] - \bigl(\alpha \E[X]\bigr)^2\\
&= \alpha^2 \bigl(\E[X^2] - \E[X]^2\bigr)\\
&= \alpha^2 {\rm Var}(X)
\end{align*}
In your case $\al... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding $\lim_{x\to 0}\frac{\tan x(1- \cos x)}{3x^2}$ How do I find the following limit:
$$
\lim_{x\to 0}\frac{\tan x(1- \cos x)}{3x^2}
$$
without using L'Hopital's rule? The reason I'm making a point of not using L'Hopital is that if I run the limit through Wolfram Alpha that's the method it uses, but we haven't gone ... | Hint:
$$1-\cos(x) = 2\sin(x/2)^2. $$
| {
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$|f|$ constant implies $f$ constant? If $f$ is an analytic function on a domain $D$ and $|f|=C$ is constant on $D$ why does this imply that $f$ is constant on $D$? Why is the codomain of $f$ not the circle of radius $\sqrt{C}$?
| The equation can be written as $f(z)\overline{f}(\overline{z})=C^2$.
So $\overline{f}(\overline{z})=C^2/f(z)$ is an analytic function of $z$.
That can only be analytic if it is constant.
| {
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Why is $\sin(xy)/y$ continuous? Me and my mates are crunching this question for a while now.
While we know that $\sin(xy)$ is continuous , $1 / y $ as the other part of the function clearly has a continuity gap at $y = 0 $, though the function can be continued at $y = 0$ with $f(x,0) = 0 $- why is that?
We tried some t... | The expression
$$f(x,y):={\sin(xy)\over y}$$
is at face value undefined when $y=0$, but wait: When $y\ne 0$ one has the identity
$${\sin(xy)\over y}=\int_0^x\cos(t\>y)\ dt\ .$$
Here the right side is obviously a continuous function of $x$ and $y$ in all of ${\mathbb R}^2$. It follows that the given $f$ can be extended ... | {
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How can I show that $\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}^n = \begin{pmatrix} 1 & n \\ 0 & 1\end{pmatrix}$? Well, the original task was to figure out what the following expression evaluates to for any $n$.
$$\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}^{\large n}$$
By trying out different values of $n$, I found t... | Powers of matrices occur in solving recurrence relations.
If you write
$$
\begin{pmatrix} x_{n+1} \\ y_{n+1} \end{pmatrix} =
\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix} \begin{pmatrix} x_n \\ y_n \end{pmatrix}
$$
then clearly
$$
\begin{pmatrix} x_{n} \\ y_{n} \end{pmatrix} =
\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}^... | {
"language": "en",
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Explicit Formula for a Recurrence Relation for {2, 5, 9, 14, ...} (Chartrand Ex 6.46[b])
Consider the sequence $a_1 = 2, a_2 = 5, a_3 = 9, a_4 = 14,$ etc...
(a) The recurrence relation is: $a_1 = 2$ and $a_n = a_{n - 1} + (n + 1) \; \forall \;n \in [\mathbb{Z \geq 2}]$.
(b) Conjecture an explicit formula for $a_n$... | Write out the series for $a_{n}$ to start with. We have that
$a_{n} = a_{n-1} + (n+1)\\
\quad = a_{n-2} + n + (n+1) \\
\quad = \ldots \\
\quad = a_{1} + 3 + 4 + \ldots + (n+1) \\
\quad = 2 + 3 + 4 + \ldots + (n+1) \\
\quad = \displaystyle \left(\sum_{i=1}^{n+1} i \right) - 1 \\
\quad = (n+1)(n+2)/2 - 1, \quad (\text{us... | {
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sum of irrational numbers - are there nontrivial examples? I know that the sum of irrational numbers does not have to be irrational. For example $\sqrt2+\left(-\sqrt2\right)$ is equal to $0$. But what I am wondering is there any example where the sum of two irrational numbers isn't obviously rational like an integer an... | If $a+b=q$, where $a,b\notin\mathbb{Q}$ and $q\in \mathbb{Q}$, then $a=q-b$, so just choose
a rational number whith sufficiently long period of decimals and you will get what you want.
On the other hand, this is still quite trivial, since here we just sum up $b$ and $q-b$ (in your question, $q=0$).
| {
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Value and simplify I want to find the value and simplify square root 36 ?
Square root of 36 is 6
But I would know how to find the value and simplify it .
| To find the square root of $37$ (say) involves a fair bit of calculation, and you will never get a numerically exact answer. But the situation is much different for $36$. For a rigorous proof that $\sqrt{36}=6$, all you need to do is (1) observe that $6^2=36$ and (2) note that $6$ is positive. Generally, a perfectly le... | {
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Möbius tranformation taking reals to reals can be written with real coefficients I'm working on this one (from Ahlfors' Complex Analysis):
A Fractional Linear Transformation of form $\displaystyle T(z) = \frac{a z + b}{c z + d}$ which takes the real numbers into the real numbers can be written in a way where all the co... | It turns out that this is not an answer to the OP's question. I will leave this answer nevertheless, perhaps someone might find it useful for something. This was my fault for not reading the question more carefully
I will start the answer with an exercise. We will assume throughout that $a\neq 0$, and $ad-bc\neq 0$. Th... | {
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Any hint for this calculus optimization problem? What should I use? We have a wire mesh of 1000 m to fence 2 regions, one circular and one square. Say how should the mesh should be cut to:
a) The sum of the areas of both fenced regions is maximum.
b) The sum of the areas of both fenced regions is minimum.
I don't know ... | Yes, you can use Lagrange multipliers and yes, it can be expressed as a $1$-variable problem. Your pick.
Let $x$ be the radius of the circle and $y$ the side of the square. We have the constraint
$$2\pi x+4y=1000\tag{1}.$$
We want to maximize/minimize
$$\pi x^2+y^2\tag{2}$$
subject to Condition (1).
Now use Lagrange m... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that absolute convergence implies unconditional convergence In the proof of "absolute convergence implies unconditional convergence" for a convergent series $\sum_{n=1}^{\infty}a_n$, we take a partial sum of first $n$ terms of both the original series ($S_n$) and rearranged series ($S_n'$) and compare them. Becau... | If your text actually writes
$|\sum_{i=N}^{n}a_i-\sum_{i=N}^{n}a_i'| \leq |\sum_{i=N}^{n}|a_i||$
then it is indeed mistaken. (Note also that even on its own the right hand side is strangely written: why do we need the outside absolute value?) For instance, suppose $a_N = \ldots = a_n = 0$. Then the inequality impl... | {
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Problem: What is the remainder of $a^{72} \mod 35$ if $a$ is a whole number not having $5$ or $7$ as divisors. I have the following problem:
Problem: What is the remainder of $a^{72} \mod 35$ if $a$ is a whole number not having $5$ or $7$ as divisors.
If $a$ cannot be divided by $5$ or $7$ it cannot be divided by $35$,... | Since $a$ and $35$ are coprime $(gcd (a, 35) = 1)$, use Euler's totient function:
$$a^{\phi(n)} = 1 \hspace{2 pt} mod \hspace{2 pt}n$$
So you get $$a^{\phi(35)} = a^{24} = 1 \hspace{2 pt} mod \hspace{2 pt}35$$
Thus,
$$a^{24^3} = a^{72} = 1^3 mod 35 = 1$$
So $1$ is your remainder.
Example: Set $a = 24$. http://www.calc... | {
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Prove that if $n>10$ then $\sum_{d\mid n}\phi(\phi(d))<\frac{3}5n$ Prove that if $n>10$ then $$\sum_{d\mid n}\phi(\phi(d))<\frac{3}5n,$$
where $\phi(n)$ is Euler's totient function.
| We start with the identity:
$$n=\sum_{d|n}\phi(d).$$
In order to prove it, just note that right hand side is a multiplicative function and therefore it is enough to check equality for prime power only.
Now the key point is to note that if $d|n$ then $\phi(d)|\phi(n)$ and therefore
the left hand side of our inequality... | {
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How to calculate this complex integral $\int_0^\infty \frac{1}{q+i}e^{-(q+b)^2}\text{d}q$? (Please Help) I want to carry out the following integration
$$\int_0^\infty \frac{1}{q+i}e^{-(q+b)^2}\text{d}q$$
which is trivial if calculated numerically with any value for b. But I really need to get an analytic expression fo... | $$\text{res}\left(\frac{\left(\sqrt{\pi } e^{\frac{1}{4} z (4 b+z)} \text{erfc}\left(b+\frac{z}{2}\right)\right) \left(e^{-i z} (-2 \text{Ci}(z)-2 i \text{Si}(z)-2 \log (-z)+2 \log (z)-i \pi )\right)}{}\{z,\alpha \}\right)$$ sorry my latex do not work find
| {
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"timestamp": "2023-03-29T00:00:00",
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Suppose that $G$ is a group with the property that for every choice of elements in $G$, $axb=cxd$ implies $ab=cd$. Prove that $G$ is Abelian. Suppose that $G$ is a group with the property that for every choice of elements in $G$, $axb=cxd$ implies $ab=cd$. Prove that $G$ is Abelian. (Middle cancellation implies commuta... | Let $a,b\in G$. Then you have $(bab^{-1})ba(e)=(e)ba(a)=ba^2$, so by hypothesis you can conclude that $bab^{-1}e=ea$, that is $bab^{-1}=a$, which implies $ab=(bab^{-1})b=ba$, so $G$ is abelian.
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove that $\mathbb{A}^1 - \{p_1, \dots, p_n\}$ and $\mathbb{A}^1 - \{q_1, \dots, q_m\}$ are not isomorphic for $n \neq m$ I want to prove that $\mathbb{A}^1 - \{p_1, \dots, p_n\}$ and $\mathbb{A}^1 - \{q_1, \dots, q_m\}$ are not isomorphic for $n \neq m$, where the $p$'s and $q$'s are points. This is one of those theo... | I suppose $k$ is algebraically closed or at least the points $p_i, q_j$ have coordinates in $k$. Otherwise the proof is more complicate.
If they are isomorphic, then there exists an isomorphism of $k$-algebras ($k$ is the ground field):
$$\phi: k[t, 1/(t-p_1), \dots, 1/(t-p_n)] \simeq k[t, 1/(t-q_1), \dots, 1/(t-q_m)]... | {
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Re-write $1 \cdot x$ to $x$. Given the following bi-directional re-write rules (where $1$ is a constant, $^{-1}$ is a unary operator, $\cdot$ is a binary operator, and $x,y,z$ are arbitrary terms):
$$\begin{align*}
x \cdot 1 &= x \\
x \cdot (y \cdot z) &= (x\cdot y) \cdot z \\
x \cdot x^{-1} &= 1
\end{align*}$$
we're a... | Another method of proof would be:
Using the fact that $x = (x^{-1})^{-1}$, then $x^{-1} \cdot x = 1$ from axiom 3.
\begin{align*}
&1 \cdot 1 = 1 \\
&1 \cdot (x \cdot x^{-1}) = 1 \\
&(1 \cdot x) \cdot x^{-1} = x \cdot x^{-1} \\
&((1 \cdot x) \cdot x^{-1}) \cdot x = (x \cdot x^{-1}) \cdot x \\
&(1 \cdot x) \cdot (x^{-1} ... | {
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Proof by induction in categorical terms Given a category cartesian closed $C$ and a functor $F : C \to C$, I consider the initial object in the category of $F$-algebras. This initial object $\mu F$ seems to codify an "inductive object" in $C$.
Now I'm trying to prove some property that I would "normally" prove by induc... | Martin's answer may be a bit deceptive, since it reduces to usual induction. But there is a brighter side to the story: if you only want to define a map from $\mu F$, to some object $A$, then all you need is to equip $A$ with $F$-algebra structure and let initiality of $\mu F$ apply. This is similar to defining a funct... | {
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need example a Riemann-integrable function is not continuous
every continuous function is Riemann integrable,continuity is certainly not necessary.
I dont know anything about measure.
| Define $f$ to be identically $0$ on $\Bbb{R}$, except that $f(0) = 1$. To see that this is Riemann integrable, note that the lower sums are all $0$ (suppose we're integrating on $[-1, 1]$, for clarity). But the upper sums can be made arbitrary small, by choosing small intervals around $0$.
More generally, $f$ is Riema... | {
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Why is $\lim_{x \to c}g(f(x)) = g(\lim_{x \to c}f(x))$ In this theorem (from the continuity section of the first chapter of a calculus textbook)
If $g$ is continuous at $b$, and $\lim_{x \to c}f(x)=b$, then $\lim_{x\to c}g(f(x)) = g(\lim_{x \to c}f(x))=g(b)$.
I would like an explanation (and proof) of $\lim_{x \to c}... | The theorem mentioned is one of the most useful in calculating various limits. As an example if we need to calculate the limit of $\{f(x)\}^{g(x)}$ when $x \to a$ then we normally take logs. Say the limit is $L$ then $\log L = \log(\lim_{x \to a}\{f(x)\}^{g(x)})$ and then we exchange $\log$ and the limit operation. Thi... | {
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Construct a Liapunov function for this system Construct a Liapunov function for the system (Determine the stability of $x \equiv 0$):
I have an example:$$\begin{cases}
& \mathrm { } \dot{x}= -x^3+xy^2\\
& \mathrm { } \dot{y}= -2x^2y-y^3
\end{cases} \tag{1}$$
Here's my solution:
*
*Let's try $V(x,y)=ax^2+by^2$.
... | I doubt that there is a closed-form global Liapunov function. Note that besides the centre at $(0,0)$ you have critical points at $(\pm \sqrt{3}/3,\mp 7 \sqrt{3}/9)$, which I think are saddle points.
EDIT: It looks like e.g.
$$ V(x,y) = {x}^{2}+{y}^{2}+ 8.75\,x{y}^{3}+3\,{x}^{2}{y}^{2}+ 5.25\,{x}^{3}y$$
is a Liapunov... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Letting $S(m)$ be the digit sum of $m$, then $\lim_{n\to\infty}S(3^n)=\infty$? For any $m\in\mathbb N$, let $S(m)$ be the digit sum of $m$ in the decimal system.
For example, $S(1234)=1+2+3+4=10, S(2^5)=S(32)=5$.
Question 1 :Is the following true?
$$\lim_{n\to\infty}S(3^n)=\infty.$$
Question 2 :How about $S(m^n)$ for... | I think the following result is common knowledge:
If $m$ is not a power of $10$, then for any positive integer $X$, there exists a power of $m$ which has a decimal expansion starting with $X$.
Proof idea: In other words, we should prove that $X \cdot 10^s \leq m^n < (X+1) \cdot 10^{s}$ for some positive integers $s$... | {
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How to prove: $a+b+c\le a^2+b^2+c^2$, if $abc=1$? Let $a,b,c \in \mathbb{R}$, and $abc=1$. What is the simple(st) way to prove inequality
$$
a+b+c \le a^2+b^2+c^2.
$$
(Of course, it can be generalized to $n$ variables).
| By replacing $a, b, c$ by $|a|, |b|, |c|$ if needed, we may assume they are non-negative. Then just apply Jensen inequality (or AM-GM inequality) to deduce that
$$ a+b+c = \sum_{\text{cyclic}} a^{4/3}b^{1/3}c^{1/3} \leq \sum_{\text{cyclic}} \frac{4}{6}a^{2} + \frac{1}{6}b^{2} + \frac{1}{6}c^{2} = a^2 + b^2 + c^2. $$
| {
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What is the largest value of $n$ for which $2n + 1$ is a factor of $122 + n^{2}$? Given that $n$ is a natural number, what is its largest value such that $2n + 1$ is a factor of $122 + n^{2}$?
| Note that $4\cdot(n^2+122) - (2n+1)(2n-1)=489$. Hence if $n^2+122$ is a multiple of $2n+1$, we also need $2n+1\mid 489=2\cdot 244+1$. On the other hand, $n=244$ does indeed lead to $n^2+122 = 122\cdot(2n+1)$, so the largest $n$ is indeed $244$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/501190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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solving second order differential equation Bonsoir
je cherche les solutions de l'équation differentielle de type
$$x^3y''(x)+(ax^3+bx^2+cx+d)y(x) =0$$
Merci d'avance
Good evening, I'm searching solutions of a differential equation of the type:
$x^3y''(x) + (ax^3+bx^2+cx+d)y(x) = 0.$
Thanks is advance.
| Hint:
$x^3y''(x)+(ax^3+bx^2+cx+d)y(x)=0$
$\dfrac{d^2y}{dx^2}+\left(a+\dfrac{b}{x}+\dfrac{c}{x^2}+\dfrac{d}{x^3}\right)y=0$
Let $r=\dfrac{1}{x}$ ,
Then $\dfrac{dy}{dx}=\dfrac{dy}{dr}\dfrac{dr}{dx}=-\dfrac{1}{x^2}\dfrac{dy}{dr}=-r^2\dfrac{dy}{dr}$
$\dfrac{d^2y}{dx^2}=\dfrac{d}{dx}\left(-r^2\dfrac{dy}{dr}\right)=\dfrac{d}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/501269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Dart Board Probability On a Dart board, with different areas labeled as: A, B, C, D, and each area different sizes. The probabilities of each area are: P(A)=25%, P(B)=50%, P(C)=12.5% and P(D)=12.5%
What is P(~C or B)?
I don't understand the "not C or B" term. If it is not C then it includes B already. My guess wo... | You understood it correctly. Besides recognizing that mathematically "or" includes both being true, another point was to read it as P((~C) or B) as opposed to P(~(C or B)), which would be 37.5%
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/501344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Lagrange basis functions as bases of Polynomials Space Suppose $L$ be a Vector Space of Polynomials of $x$ of degree $\leq n-1$ with coefficients in the field $\mathbb{K}$.
Define $$g_i(x) :=\prod _ {{j=1},{j\neq i}}^n \frac{x-a_j}{a_i-a_j}$$ Show that the polynomials $g_1(x), g_2(x),...,g_n(x)$ form a basis of L. Furt... | Choose arbitrary $f\in L$. Let be $$\tilde{f}(x) = \sum_{i = 1}^{n}f(a_i)g_i(x)\text{.} $$
For every $x\in \{a_1,\dots, a_n\}$ we have $f(x) = \tilde{f}(x)$, so the polynomial
$p= f - \tilde{f}$ has $n$ zeros and $\deg p \leq n-1$, so $p(x) = 0$ for every $x\in \mathbb{R}$. So $g_i$ span $L$. We know that $\dim L = n$,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
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Find zeros of this function: $$(3\tan(x)+4\cot(x))\cdot\sin(2x)$$
Do I have to multiply them and solve, or one by one, like:
$$(3\tan(x)+4\cot(x))=0$$and$$\sin(2x)=0.$$
| Observe that
if $\displaystyle(3\tan x+4\cot x)=0\implies 3\tan x+\frac4{\tan x}=0\iff3\tan^2x+4=0$ which is impossible for real $x$
If $\sin2x=0, 2x=n\pi$ where $n$ is any integer
If $n$ is even $=2m$(say) $2x=2m\pi, x=m\pi,\cot x=\cot m\pi=\frac{\cos m\pi}{\sin m\pi}=\frac{(-1)^m}0$ hence not finite
If $n$ is odd $=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/501504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Find $\lim_{x\to 0} \frac{\tan16x}{\sin2x}$ Find $\lim_{x\to 0} \frac{\tan16x}{\sin2x}$
I'm a little confused on limit trig. Am i suppose to simplify tan or do I use the derivative quotient rule?
Please Help!!!
| Recall the following limits:
*
*$\lim_{x \to 0}\dfrac{\sin(ax)}{x} = a$
*$\lim_{x \to 0}\dfrac{\tan(bx)}{x} = b$
Note that
$$\dfrac{\tan(16x)}{\sin(2x)} = \dfrac{\dfrac{\tan(16x)}{x}}{\dfrac{\sin(2x)}x}$$
Can you finish it off now?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/501609",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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} |
Find all points where the tangent line has slope 1. Let $f(x)=x-\cos(x)$. Find all points on the graph of $y=f(x)$ where the tangent line has slope 1. (In each answer $n$ varies among all integers).
So far I've used the Sum derivative rule for which I have $1+\sin(x)$. So do I put in 1 in for $x$ for sin$(x)$.
Please ... | $f(x) = x - \cos x \tag{1}$
$f'(x) = 1 + \sin x \tag{2}$
$f'(x) = 1 \Rightarrow 1 = 1 + \sin x \Rightarrow \sin x = 0 \tag{3}$
$\sin x = 0 \Leftrightarrow x = k\pi, \tag{4}$
for $k \in \Bbb Z$, the set of integers.
Hope this helps. Cheerio,
and, as always,
Fiat Lux!!!
| {
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"source": "stackexchange",
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What is the probability that the student knew the answer to at least one of the two questions? A student takes a true-false examination containing 20
questions. On looking at the examination the student and that he
knows the answer to 10 of the questions which he proceeds to answer
correctly. He then randomly answers t... | When complete, the following table shows the probabilities of getting $2$, $1$, or $0$ of the two questions right depending on the kinds of questions. $K$ refers to a question with a known answer and $U$ to a question with an unknown answer; the first letter is for the first question graded, and the second is for the s... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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Prove that $12 \mid n^2 - 1$ if $\gcd(n,6)=1$
Prove that $12 \mid n^2 - 1$ if $\gcd(n,6)=1$.
I know I have to use Fermat's Little Theorem for this but I am unsure how to do this problem.
| As is mentioned in answers above, $gcd(n,6)=1$ clearly implies that $n$ is odd. But then each of $n+1$ and $n-1$ are even, so $4$ divides $(n+1)(n-1)=n^2-1$. Thus it is only left to show that $3$ also divides $n^2-1$. But given any three consecutive integers, we know that $3$ must divide exactly one of them. Applyi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How can I solve these Modular problems? Very basic question, but how can I solve this?
$7x+9y \equiv 0 \bmod 31$ and $2x-5y \equiv 2 \bmod 31$.
| Adding to Peter's answer, if you're asked to solve this by hand, this particular modular arithmetic is easy because $31$ is prime. Multiplication and addition tables fall out very easily and the only part where you'd be required to do some computation is when you find the inverse of a number.
| {
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What is the smallest value of $x^2+y^2$ when $x+y=6$? If $ x+y=6 $ then what is the smallest possible value for $x^2+y^2$?
Please show me the working to show where I am going wrong!
Cheers
| Most reasonable and concrete solution is to find minima using derivatives.
As shown by DeltaLima.
There are many ways to find the minima graphically:
$x^2+y^2$ expression can be written as $x^2+y^2=K$(equation)(it represents parabola) and $x+y=6$ represents, a straight line. They will intersect for minimum and maximu... | {
"language": "en",
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If $x_12$, show that $(x_n)$ is convergent. If $x_1<x_2$ are arbitrary real numbers, and $x_n=\frac{1}{2}(x_{n-2}+x_{n-1})$ for $n>2$, show that $(x_n)$ is convergent. What is the limit?
The back of my textbook says that $\lim(x_n)=\frac{1}{3}x_1+\frac{2}{3}x_2$. I was thinking that if I show that the sequence is monot... | A related problem. To prove convergence, note that
$$ x_n=\frac{1}{2}(x_{n-2}+x_{n-1})\implies x_n-x_{n-1}=\frac{1}{2}(x_{n-2}-x_{n-1}) $$
$$ \implies |x_n-x_{n-1}|=\frac{1}{2}|x_{n-1}-x_{n-2}|<\frac{2}{3}|x_{n-1}-x_{n-2}|. $$
This proves that the sequence is a contraction and hence convergent by the Fixed point theore... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 1
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Exponential Function as an Infinite Product Is there any representation of the exponential function as an infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e.
$$\mathrm e^x=\prod_{n=0}^\infty a_n,$$
and by the sentence in brackets I mean that the $a_n$'s are not... | Amazingly, the exponential function can be represented as an infinite product of a product! That result was shown in the 2006 paper "Double Integrals and Infinite Products For Some Classical Constants Via Analytic Continuations of Lerch's Transendent" by Jesus Guillera and Jonathan Sondow.
It is proven in Theorem 5.3 t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 9,
"answer_id": 2
} |
Monotonic sequence and limit
For $x_n = \frac{1}{2+x_{n-1}}$ where $x_1 =1/2$, show that the
sequence is monotonic and find its limit.
What I first did was finding $x_{n+1}$, which equals $\frac{1}{2+x_n}$; then $x_{n+2}=\frac{1}{2+\frac{1}{2+x_n}}=\frac{x_n+2}{2x_n+5}$ thus it does eventually get smaller hence $x_... |
The single most useful thing to do here is to draw on a same picture the graphs of the functions $u:x\mapsto1/(2+x)$ and $v:x\mapsto x$, say for $x$ in $(0,1)$. The rest follows by inspection...
Since $u$ is decreasing from $u(0)\gt0$ to $u(\infty)=0$, $u$ has a unique fixed point, say $x^*$. Since $u(x_1)\lt x_1$, o... | {
"language": "en",
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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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What is the integral of $e^{-x^2/2}$ over $\mathbb{R}$ What is the integral of
$$\int_{-\infty}^{\infty}e^{-x^2/2}dx\,?$$
My working is here:
= $-e^(-1/2x^2)/x$ from negative infinity to infinity.
What is the value of this? Not sure how to carry on from here. Thank you.
| $$\left(\int\limits_{-\infty}^\infty e^{-\frac12x^2}dx\right)^2=\int\limits_{-\infty}^\infty e^{-\frac12x^2}dx\int\limits_{-\infty}^\infty e^{-\frac12y^2}dy=\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty e^{-\frac12(x^2+y^2)}dxdy=$$
Change now to polar coordinates:
$$=\int\limits_0^{2\pi}\int\limits_0^\infty ... | {
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"timestamp": "2023-03-29T00:00:00",
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A single gate defined in Conjunctive Normal Form This might sound like a silly question but I just want to make sure I'm not getting confused.
I understand that CNF is essentially converting the logic into AND's of OR's, so for example...
~a AND (b OR c) in CNF would be (a AND b) OR (a AND c).
My question is, what woul... | CNF stands for Conjunctive Normal Form.
A boolean expression is in CNF if it is an AND of OR expressions where every OR has nothing but literals or inverted literals as inputs. The OR expressions for CNF are commonly called clauses.
Your example:
The CNF of x AND y consists of two single-literal clauses, one is x and o... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals I've thought about the following question for a month, but I'm facing difficulty.
Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon whose area is $S$, then... | Not a complete answer, and surely not elegant, but here's an approach.
The idea is to consider "polygonal conics" in the following sense. Take the cone $K$ given by
\begin{equation}
K:\qquad x^2+y^2=z^2
\end{equation}
and slice it with the plane $z=1$.
We get a circle $\gamma$, and let's inscribe in this circle a regul... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
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normal distribution derivation In this derivation:
http://www.sonoma.edu/users/w/wilsonst/Papers/Normal/default.html
how do these equal?
$$ -k\int (x-\mu) dx = -\frac{k}{2} (x-\mu)^2$$
Isn't this the case?
$$ -k\int (x-\mu) dx = -\frac{kx^2}{2} + k\mu x$$
| The answer given comes from one antiderivative of $x-\mu$. Your answer comes from another antiderivative of $x-\mu$. The two differ by a constant, so both of them are correct antiderivatives. Neither of them is the general antiderivative of $x-\mu$.
The general antiderivative of $x-\mu$ can be written as $\frac{1}{2... | {
"language": "en",
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Calculations with permutations: show that $(1,2,3)^2(5,7)^2=(1,3,2)$ How can I show $(1,2,3)^2(5,7)^2=(1,3,2)$? And, specifically, what does $(1,2,3)^2$ and $(5,7)^2$ equal individually?
| Note that with $(5, 7)^2 = (5, 7)(5, 7)$, $\quad (5 \to 7 \to 5)$ and $(7 \to 5 \to 7),\quad $ which gives us $(5, 7)(5, 7) = (1)$, the identity permutation. Any two-cycle, squared, gives us the identity permutation: it's an order two permutation.
Since the first two squared cycles are disjoint from one another, you ca... | {
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Compound Interest -Confirming Answer- "When you get a "30 year fixed rate mortgage" on a house, you borrow a certain amount of money at a certain interest rate. You then make the same monthly payment for 30 years. At the end of that time, the loan is fully paid off. The interest on the loan is compounded monthly. Suppo... | We give a derivation of the correct formula for the monthly payments.
We will assume that the first monthly payment is made a month after the loan is issued. We also assume that the interest rate $r$ is given as a percentage. Let $P$ be the monthly payment.
The present value of a payment of $P$ made $k$ months from... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Maximize the determinant Over the class $S$ of symmetric $n$ by $n$ matrices such that the diagonal entries are +1 and off diagonals are between $-1$ and $+1$ (inclusive/exclusive), is
$$\max_{A \in S} \det A = \det(I_n)$$
?
| For the three by three case I have the following. Let $A$ be generally defined as
$$A= \pmatrix{
1& x& y\\
x& 1&z\\
y&z& 1\\
}.
$$
Then the determinant is
$$|A| =
\matrix{
1& z \\
z& 1 \\} - x\matrix{
x& z \\
y& 1 \\} + y\matrix{
x& 1 \\
y& z \\}
= 1-z^2 - x(x-zy) + y(xz-y) =$$
$$= 1-x^2-y^2-z^2 +xyz-xyz= 1-(x^2+y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/502771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to show a continuous cannot differ a characteristic function by measure $0$? How to show that there is no continuous function on $\mathbb R$ such that it differs from $\chi_{[0,1]}$, the characteristic function of $[0,1]$, by a measure (Lebesgue measure) of $0$?
| Hint: Let $f$ be such function. For any $\delta > 0$, $(-\delta, 0)$ must contain an element $x$ such that $f(x) = 0$. Similarly, $(0, \delta)$ must contain an element $x$ such that $f(x) = 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/502845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $a_0=1$, and $a_n$ is defined by $a_n=a_{n+1}+a_{n+2}$, find $a_n$. This is not a homework problem, though it is in my textbook as a practice problem that intrigues me enough to try it. I've got some idea how to solve it but I don't know how to prove my hypothesis.
The question reads exactly as follows:
Suppose $a_... | Write it in the usual way with decreasing subscripts as $$ a_{n+2} + a_{n+1} - a_n = 0. $$
Whatever you might want to call it is $$ \lambda^2 + \lambda - 1 = 0. $$ If this has distinct roots then $a_n = B \lambda_1^n + C \lambda_2^n $ for real or complex constants $B,C$ depending how it turns out.
So, $$ \lambda = ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving $ f(x) = x^2 $ is not uniformly continuous on the real line This is homework problem and the very premise has me stumped. It's in a text on PDE.
The exercise says to show that $ f(x) = x^2 $ is not uniformly continuous on the real line. But every definition I know says that it is a continuous function, and un... | Here is a more general approach to solving this problem.
Let me formulate and prove a theorem:
Theorem:
Let $E = [a,+\infty),$ function $f:E \rightarrow \mathbb{R}$ is
differentiable on $E$ and $$\displaystyle{\lim_{x \to \infty}} f'(x) =
\infty.$$ Then $f$ is not a uniformly continuous function.
Proof:
Let the f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/503093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Suppose that $f:[a,b] \to [a,b]$ is continuous. Prove that there is at least one fixed point in $[a,b]$ - that is, $x$ such that $f(x) = x$. Suppose that $f:[a,b] \to [a,b]$ is continuous. Prove that there is at least one fixed point in $[a,b]$ - that is, $x$ such that $f(x) = x$.
I haven't a clue where to even start ... | Consider the function $g(x) = f(x) - x$. Then $$g(a) = f(a) - a \ge a - a = 0$$ while $$g(b) = f(b) - b \le b - b = 0$$ So by the intermediate value theorem, ...?
| {
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"source": "stackexchange",
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Graphing Hyperbolas I know that a Hyperbola is in the form of:
$$\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1$$
But how would I graph it? I know that a Hyperbola has two asymptotes that the graph gets infinitely close to but will never touch, is there a way to find the asymptotes with that equation? and is the asymptote... | To add to Kaster's answer, there is a handy construction elaborated in this link.
In the form
$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$
the hyperbola has its fundamental rectangle with corners at $(0,b), (0,-b), (a,0), (-a,0)$ and the diagonals of this rectangle are the asymptotes. Points $(a,0), (-a,0)$ are the vertices... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Three numbers, one of the number's digit sum is equal to two other digit difference So as the title says I need three numbers witch has this quality :
one of the numbers digit sum is equal to other two number differnce e.g.
I 68
II 52
III 97
third number digit sum is 16 and its equal to I and II difference which is 16.... | Consider 3 numbers a, b and c:
$$A = 10*a1 + a2\\
B = 10*b1 + b2\\
C = 10*c1 + c2$$
(in your example: a = 68 = 10*6 + 8)
The relation you are describing:
$$\begin{cases}
a1+a2 = C-B\\
b1+b2 = C-A
\end{cases}$$
$$\begin{cases}
C = a1+a2+B\\
C = b1+b2+A
\end{cases}$$
$$a1+a2+B = b1+b2+A$$
$$a1+a2 + 10*b1+b2 = b1+b2 + 10*... | {
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Row and Column Picture of a 3 x 3 Singular Matrix (Strang P43, 2.1.32)
Suppose $\mathbf{u}$ and $\mathbf{v}$ are the first two columns of a 3 by 3 matrix $A$. Which third columns
$\mathbf{w}$ would make this matrix singular? Describe a typical column picture of $A\mathbf{x} = \mathbf{b}$ in that singular case, and a... | Your first question:
the red bracket is true because b is randomly chosen, it can be either on the plane or out of the plane of u, v, w. Furthermore, the possibility that it's out of the plane is bigger than that it's on the plane.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/503440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove irreducibility of a polynomial Let $m$ be an integer, squarefree, $m\neq 1$. Prove that $x^3-m$ is irreducible in $\mathbb{Q}[X]$.
My thoughts: since $m$ is squarefree, i have the prime factorization $m=p_1\cdots p_k$. Let $p$ be any of the primes dividing $m$. Then $p$ divides $m$, $p$ does not divide the leadin... | $x^3-m$ is reducible iff it has a factor of degree 1 iff it has a root iff $m$ is a cube. In particular, $x^3-m$ is irreducible when $m$ is squarefree.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/503511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Show that a matrix $A$ is singular if and only if $0$ is an eigenvalue. I can't find the missing link between singularity and zero eigenvalues as is stated in the following proposition:
A matrix $A$ is singular if and only if $0$ is an eigenvalue.
Could anyone shed some light?
| If 0 is an eigenvalue, then there exists a vector $v$ in your space such that $A.v = 0$. If your matrix size is 4x4 with one 0 eigenvalue and you write the image of the eigenvectors, you get:
$$(v11, v12, v13, 0)$$
$$(v21, v22, v23, 0)$$
$$(v31, v32, v33, 0)$$
$$(v41, v42, v43, 0)$$
You can see it's singular because:
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/503585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
"answer_count": 8,
"answer_id": 2
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Conditional Probability Question EDIT!!!
a) The probability that any child in a certain family will have blue eyes is 1/4, and this feature is inherited independently by different children in the family. If there are five children in the family and it is known that at least one of these children has blue eyes, what... | I know that a lot of time has passed, but FWIW here's my take.
b) Consider a family with the five children just described. If it is
known that the youngest child in the family has blue eyes, what is the
probability that at least three of the children have blue eyes?
If "it is known" implies truth, i.e. P(youngest... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/503641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Find the steady state temperature of the rod A rod occupying the interval $0 \leq x \leq l$ is subject to the heat source
$f(x) =0, $ for $ 0 < x < L/2$,
$f(x) =H $ for $ L/2 <x <L ,H>0$
(1)The rod satisfies the heat equation $$u_t = u_{xx} + f(x)$$ and its ends are kept at zero temperature. Find the steady-state tempe... | For the steady state, you are solving
$$u_t=0 \implies u_{xx}=-f(x)$$
The general solution to this, given $f$, is pretty straightforward:
$$u(x) = \begin{cases} c_1 x+c_2 & x \in \left (0,\frac{L}{2}\right)\\-\frac12 H x^2 + d_1 x+d_2 & x \in \left (\frac{L}{2},L\right) \end{cases} $$
Yes, we have different constants i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/503716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Fourier series of $f(x)=x^2$ in $x∈[0,2\pi]$ and in $x∈[−\pi,\pi]$? Fourier series - what is the difference between the Fourier series of $f(x)=x^2$ in $x∈[0,2\pi]$ and in $x∈[−\pi,\pi]$?
| Conceptually, Fourier series try to express a periodic function in terms of sines and cosines. Consider the two situations you mentioned here, letting $f(x) = x^2$.
If your periodic function is $f$ from $[0,2\pi]$, then it would be just the right part of the parabola from 0 to $2\pi$ repeated again and again.
But if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/503796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does every $p$-group of odd order admit fixed point free automorphisms?
Does every $p$-group of odd order admit fixed point free automorphisms?
equivalently,
Given an odd order $p$-group $P$, is there a group $C$ such that we can form a Frobenius group $P\rtimes C$?
Note that this is not true for $p$-groups of even... | The following family of $p$-groups provides counter examples
$$G = \langle a, b, c,d |a^{p^n}=b^{p^4}=c^{p^4}=d^{p^2}=1, [a,b]=[a,c]=b^{p^2}, [a,d]=c^{p^2}, [b,c]=a^{p^{n-2}}, [b,d]=c^{p^2}, [c,d]=c^{p^2} \rangle$$
with $p$ odd, and $n>3$.
For such a group $G$, every automorphism is central, that is $\operatorname{Au... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/503891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Hint on power sum coefficients Please do not give anything more than a tiny hint for this question.
I know that there is a well-known formula for $$\sum_{i=1}^n i^k,$$ where $k$ is any non-negative integer. I have been able to prove that in fact it is a polynomial in $n$,
$$\sum_{i=1}^n i^k = \sum_{j=0}^{k+1} a_j n^j,$... | One hint is that it is far easier to do this is you replace $i^k$ by another polynomial of degree $k$ in $i$, namely by$~\binom ik$. Check that you can find $\sum_{i=0}^n\binom ik$ easily. Then it is theoretically only a question of transforming the basis $[1,i,i^2,\ldots]$ of the polynomial functions in$~i$ to the bas... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/503986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
mean value theorem on an open interval I know that the conditions for the mean value theorem state $f$ must be continuous on $[a,b]$ and differentiable on $(a,b)$.
What happens if we change the condition to $f$ is continuous on $ (a,b)$ but not at the endpoints?
| Given $a < b$, for any $\epsilon_{a} > 0$ and any $\epsilon_{b} > 0$ such that
$\epsilon_{a} + \epsilon_{b} < b - a\quad\exists\ \xi \ni$
$$
{{\rm f}\left(b - \epsilon_{b}\right) - {\rm f}\left(a + \epsilon_{\rm a}\right)
\over
b - a - \epsilon_{a} - \epsilon_{b}}
=
{\rm f}'\left(\xi\right)\,,
\qquad\mbox{where}\qua... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/504211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Summation Index Confusion I've seen various confusing variations of the summation signs, so can anyone give me clarification on them? I understand the most common ones from calculus class, for example:
$$\sum^\infty_{i=1}\frac{1}{i}\to\infty$$
The rest is very confusing, as all the indices and whatnot are on the bottom... | TZakrevskiy’s answer covers all of your examples except the one from the multinomial theorem. That one is abbreviated, and you simply have to know what’s intended or infer it from the context: the summation is taken over all $k$-tuples $\langle a_1,\ldots,a_k\rangle$ of non-negative integers satisfying the condition th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/504262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Taylor Expanding an Integrand Just want to be sure of this: If I taylor expand an integrand about 0 then truncate it to say linear order, then integrate this truncation, does the integral evaluate to a function which is an accurate representation of the original integral, but only around 0? If so why is this?
Cheers!
| The Taylor theorem says (I skip the hypothesis): $$f(x) = f(0)+f'(0)x+\frac{f'(\xi)}{2}x^2.$$ The point is that $\xi$ depends on $x$ and in general this dependance is very nasty. You can integrate both parts of the equation with respect to $x$ on some interval, but I don't think you can get something useful from the ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/504326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Finding the Domain and Range of a function composition I'm having trouble finding the domain and range of a function composition.
$f(x) = x^2 - 3x$
$g(x) = \sqrt{x}$
$(g \circ f)(x) = g(f(x)) = \sqrt{(x^2 - 3x)}$
How do I find the domain and range of $(g \circ f)(x)$?
(I know the answer because it's in the back of the ... | The outer function $g:\ y\mapsto\sqrt{\mathstrut y}$ is defined when $y\geq 0$. The inner function $f:\ x\mapsto y:=x(x-3)$ is $\geq0$ when $x\leq0$ or $x\geq 3$, and is $<0$ when $0<x<3$. It follows that the domain of $g\circ f$ is ${\mathbb R}\setminus\>]0,3[\ $.
Since the graph of $f$ is a parabola it follows that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/504417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding time with given distance and acceleration. Help Needed A plane accelerates from rest at a constant rate of $5.00 \, \frac{m}{s^2}$ along a runway that is $1800\;m$ long. Assume that the plane reaches the required takeoff velocity at the end of the runway. What is the time $t_{TO}$ needed to take off?
I tried to... | The acceleration is $a=5$, and you start from rest at position zero on the runway. The runway length is $L = 1800$.
Your speed will be $v(t) = \int_0^t a d \tau = at$. (Starting from rest means $v(0) = 0$.)
Your position will be $x(t) = \int_0^t v(\tau) d \tau = \int_0^t a \tau d \tau = a \frac{t^2}{2}$. (Starting from... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/504494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Proving that every countable metric space is disconnected? Any hints on how I should go about this? Thanks.
| The statement is false if $|X|=1$, so countable here is probably supposed to be understood as countably infinite.
HINT: Let $X$ be the space, and let $x,y\in X$ with $x\ne y$. Let $r=d(x,y)>0$. Use the fact that $X$ is countable to show that there must be an $s\in(0,r)$ such that
$$\{z\in X:d(x,z)=s\}=\varnothing\;.$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/504588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Simplest or nicest proof that $1+x \le e^x$ The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or canonical proof? I would ideally like a proof which fits into a few lines, is acc... | For $x>0$ we have $e^t>1$ for $0<t<x$
Hence, $$x=\int_0^x1dt \color{red}{\le} \int_0^xe^tdt =e^x-1 \implies 1+x\le e^x$$
For $x<0$ we have $e^{t} <1$ for $x <t<0$
$$-x=\int^0_x1dt \color{red}{\ge} \int^0_xe^tdt =1-e^x \implies 1+x\le e^x$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/504663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "113",
"answer_count": 27,
"answer_id": 9
} |
Simplest proof that $\binom{n}{k} \leq \left(\frac{en}{k}\right)^k$ The inequality $\binom{n}{k} \leq \left(\frac{en}{k}\right)^k$ is very useful in the analysis of algorithms. There are a number of proofs online but is there a particularly elegant and/or simple proof which can be taught to students? Ideally it would r... | $\binom{n}{k}\leq\frac{n^k}{k!}=\frac{n^k}{k^k}\frac{k^k}{k!}\leq\frac{n^k}{k^k}\sum_m\frac{k^m}{m!}=(en/k)^k$.
(I saw this trick in some answer on this site, but can't recall where.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/504707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Proof: $\tan(x)$ is surjective from $(-\pi/2,\pi/2)$ onto $\mathbb R$ To prove $\tan(x)$ defined on $]-π/2;π/2[$ is injective I take the derivative of $\tan(x)$ to get $\sec(x)^2$.
This shows that $\tan(x)$ is monotonic (strictly) increasing which implies it is injective.
However how do I show it is surjective ? That e... | The above use of the intermediate value theorem uses $\overline{\mathbb R}$ to accomplish its task, but also seems to beg the question by asserting the one-sided limits of tangent at $-\frac{\pi}{2}$ and $\frac{\pi}{2}$. Here is a proof that does not use either geometry or $\overline{\mathbb R}$ The proof will however... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/504797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Open subgroups of $\mathbb{R}$ Let $G$ be a nonempty open subset of $\mathbb{R}$ (with usual topology on $\mathbb{R}$) such that $x,y\in G$ implies that $x-y\in G$. Show that $G=\mathbb{R}$.
Clearly $0\in G$. Now how to show that all real numbers are there in $G$? Please help.
| Hint: Every nonempty open subset of $\Bbb R$ contains at least one open interval.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/504845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
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