Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Compute average distance between numbers $0$ to $10$ I'd like to calculate the average difference between two numbers, each between $0$ and $10$. I calculated this for integers and came up with an average distance of $4$.
My method: there are $10$ ways to obtain a difference of $1$, $9$ ways to obtain a difference of ... | Every pair of numbers between $0$ and $10$ can be associated with a unique point $(x,y)$ in the square $[0,10] \times [0, 10]$. For such a point, the function $f(x,y)=|x-y|$ measures the difference between the two numbers. To find the average value of that function, one computes
$$ \frac{1}{100}\int_0^{10} \int_0^{10... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $(2+ \sqrt5)^{\frac13} + (2- \sqrt5)^{\frac13}$ is an integer When checked in calculator it is 1. But how to prove it?
Also it is not a normal addition like $x+ \frac1x$ which needs direct rationalization. So I just need something to proceed.
| Not a full solution but if you use the equality: $(x+y)^3 = x^3 + y^3 + 3x^2y + 3y^2x$, you can reach the equation $z^3 + z = 4$ where $z$ is the value of the above expression. This leads to $z = 1$ as a solution. I cubed your formula and simplified. I do not, however, know how to solve $z^3 + z = 4$. Cubing both s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2082836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 7
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Interpreting Division as subtraction Basically, Division is the inverse of multiplication, see this instance:
5 × 4,
This basically means, adding five four times,
5, 5, 5, 5 added together is 20;
If division is the inverse of multiplication and multiplication comes from repeated addition, how can we interpret divis... | If you want to interpret "$20$ divided by $4$" in terms of repeated subtraction, what you would do is repeatedly subtract $4$ from $20$ until you get $0$:
$$16, 12, 8, 4, 0$$
The number of times you had to subtract $4$ is exactly the quotient $\frac{20}{4}$. This same principle carries over to the quotient $\frac{6!}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2082911",
"timestamp": "2023-03-29T00:00:00",
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Screen reader help for "the region {(x, y)|x + y ≤ 1}" (Series of screen reader related questions on basic symbology)
"the probability that x, y are in A where A is the region {(x, y)|x + y ≤ 1}."
How should I write this out the bolded section to that a screen reader will read it correctly?
| $A$ is
the region in the plane consisting of the points whose $x$- and $y$-coordinates satisfy the inequality $x+y\le 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2082972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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How to find the convergence of this sequence? $x_n=1+\frac{1}{x_{n-1}}$ The sequence ($x_n$) is defined by $x_0=1$ and $x_n=1+\frac{1}{x_{n-1}}$ for $n\in ℕ^*$, then how could I find the limit of it? By trying some, I've found that the answer is $\frac{1+\sqrt5}{2}$, but I'm looking for a "legit" way to find it.
| In the limit, you have the following continued fraction:
$$x = \lim\limits_{n \to \infty} x_n = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \ddots}}}}.$$
If you're familiar with the golden ratio you'll recognise this is $\varphi$ immediately, but if not, notice that
$$x = 1+\frac{1}{x},$$
which can be rearr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2083106",
"timestamp": "2023-03-29T00:00:00",
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Prove $\cos{\frac{\pi}{n}}\notin\mathbb Q$ for any postive integer $n\ge 4$ Let $n\ge 4$ postive integer,show that
$$\cos{\dfrac{\pi}{n}}\notin\mathbb Q$$
Now I have solve for a case:
Assmue that
$$\cos{\dfrac{\pi}{n}}=\dfrac{q}{p},(p,q)=1,p,q\in N^{+}$$ use Chebyshev polynomials?
$$T_{n}(\cos{x})=\cos{(nx)}$$
so we ha... | An elementary proof. From deMoivre's Theorem, if $k\in \Bbb N$ then $\cos kx=\sum_{(0\le 2j\le k)}\binom {k}{2j}(-1)^j(\cos x)^{k-2j}(1-\cos^2 x)^j.$ So if $\cos (\pi/n)\in \Bbb Q$ and $m|n$ then $\cos (\pi/m)\in \Bbb Q.$ So it suffices to show $\cos (\pi/m)\not \in \Bbb Q$ when (i)$\,m=4,$ or (ii) $\,m=6,$ or (iii) $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2083214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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According to this $0!$ Should be not defined, but why it is not? According to my knowledge factorial means that the number is multiplied by the numbers preceding it up to $1$ . That's it .
$5!=5×4×3×2×1=120$
$4!=4×3×2×1=24$
$3!=3×2×1=6$
$n!=n×(n-1)×\cdots \times 2×1$
And similarly
$0!=0×(-1)×(-2)\times\cdots\times1$
Bu... | My favorite is the following. We define $n!$ to be the number of ways $n$ distinct objects can be ordered. Now, how many ways can you order $0$ objects? Well, you can either say that (1)"you can't", (2) "there are zero ways", or (3) "there is one way to organize zero objects". If you choose (1) you're just being defian... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2083280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Interesting log series Calculate the sum of series:
$$\sum_{k=1}^{\infty} k\left(\log\left(\frac{k+1}{\sqrt{(k+1)^2+1}}\right)-\log\left(\frac{k}{\sqrt{k^2+1}}\right)\right)$$
| We write
$$ k\left(\log\left(\frac{k+1}{\sqrt{(k+1)^2+1}}\right)-\log\left(\frac{k}{\sqrt{k^2+1}}\right)\right) = (k+1) \log\left(\frac{k+1}{\sqrt{(k+1)^2+1}}\right)-k \log\left(\frac{k}{\sqrt{k^2+1}}\right) +\log\left(\frac{k+1}{\sqrt{(k+1)^2+1}}\right).$$
The sum of the first two terms is telescoping and we obtain t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2083396",
"timestamp": "2023-03-29T00:00:00",
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Asymptotes and focus of a conic? This is the conic $$x^2+6xy+y^2+2x+y+\frac{1}{2}=0$$
the matrices associated with the conic are:
$$
A'=\left(\begin{array}{cccc}
\frac{1}{2} & 1 & \frac{1}{2} \\
1 & 1 & 3 \\
\frac{1}{2} & 3 & 1
\end{array}\right),
$$
$$
A=\left(\begin{array}{cccc}
1 & 3 \\
3 & 1
\end{array}\right),
$$
... | These two examples are from a book that does not assume linear algebra... Both good and bad, as they are giving concrete methods, which are a bit much to memorize. Worth doing both ways, really, get them to agree; this way, then linear algebra and coordinate changes (translation to center followed by rotation). They d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2083475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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General solution of linear equation = particular + general homogeneous solution
Let $x_p$ be the particular solution of $Ax = b$ and $x_h$ be the solution to
the homogeneous system $Ax = O$. All the solutions of $Ax = b$ are of the form $x_p + x_h$
Proof:
Let $x$ be the solution of $Ax = b$, then $A(x − x_p) = Ax − ... | Yes, presented more clearly
$$\overbrace{{\rm if}\ \ ax_p = b}^{{\rm particular\ solution\ }{\large x_p}}\!\! {\rm then}\ \ ax=b \!\iff\! a(x\!-\!x_p) = 0\!\iff\! \ \underbrace{x-x_p = x_h\ \ {\rm and} \overbrace{ a\, x_h = 0}^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{\rm homogeneous\ solution}\ \large x_h}}_{\large \bb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2083582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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finding the center of a circle (elementary geometry) I am considering the problem of finding the center of a given circle C of radius r. I currently know 3 elementary ways (ruler+compass construction) of doing this :
- choose 2 points on the circle, draw the bisector, and draw another bisector involving a third point o... | Based on the described construction and using the OP's notations, we have the following figure:
It is enough to demonstrate that $DG=r$, the radius of the circle.
By the construction, the triangle $ABD$ is equilateral and then $AB=BD=AF$. $BDG$ is an isosceles triangle because the triangles $ABG$ and $ADG$ are congru... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2083673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Summation symbol with just one letter below it? How to verbally state.
I'm used to seeing summation capital sigmas with a "start" at bottom and a "stop" at top and would say the sum from i to 100 or whatever. But in this case, there is just a y at bottom.
How would I write this out so a screen reader would be able to... | It depends on context. Sometimes it means "sum over all possible values of $y$" (where which values of $y$ are allowed should be clear from the context), but sometimes it means an indefinite sum or "anti-difference", analogous to an indefinite integral or antiderivative:
$$ \sum_y f(x,y) = g(x,y)\ \text{where}\ g(x,y+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2083774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Simple inequality ( or false) Given $a$, $b$ and $c > 0$ such that $abc=1$, prove that
$$\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b} \leq 8abc$$
I don't know if its true or not but i need help in both cases.
Thanks
| If $a$ and $b$ are "small" and very very close together (but $a$ slightly larger), then $c$ is "large" and $c/(a-b)$ can be made arbitrarily large.
$a/(b-c)$ is negative but a it can be a "small" negative (as $c$ is "large" compared to $a$ and $b$. $b/(c-a)$ is positive (but small). So that sum can be arbitrarily l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2083887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving $\lim\limits_{x \to \infty} \frac{\sin(x)}{x-\pi}$ using L'Hôpital's rule
I know how to solve this using the squeeze theorem, but I am supposed to solve only using L'Hôpital's rule
$$\lim\limits_{x \to \infty} \frac{\sin(x)}{x-\pi}$$
I tried:
$$\lim\limits_{x \to \infty} \frac{\sin(x)}{x-\pi} = \lim\limits_{x... | I bet that your problem is, in fact, to compute
$$\lim _{x \to \color{red} \pi} \frac {\sin x} {x - \pi} ,$$
so either you have mistyped $\infty$ instead of $\pi$, or there is a typo in the text where you took this from.
In this case, L'Hospital's theorem could be used, but it's not necessary, because
$$\lim _{x \to \p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2083962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Does an equilibrium solution have to be a constant? I'm having difficulty understanding equilibrium solutions. Consider trying to find the equilibrium solution(s) of the following ODE:
\begin{equation}
y'=x^2-y^4
\end{equation}
Firstly, I set $y'=0$ and solved for y:
\begin{align}
0&=x^2-y^4 \\
y^4&=x^2 \\
y&=\pm\sqrt{... | What you found out is not a solution. Just try to substitute in the equation.
By definition, an equilibrium point is a constant solution of the equation. In this case, substituting $y=c$ leads to $x^2=c^2$ and so there are no solutions (note that by definition a solution is a $C^1$ function defined on some open interva... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2084060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Trigonometry and Quadratics If $\tan A$ and $\tan B$ are the roots of $x^2+px+q=0$, then prove that
$$\sin^2(A+B)+p \sin(A+B) \cos(A+B) + q \cos^2(A+B) = q$$
My Attempt:
Using the sum and product formulae we have,
$q=\tan A\tan B, $ $-p=\tan A+\tan B$
And,
$\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B} \Right... | You can proceed easily after finding $\tan (A+B)$.
So we have $$p =\tan (A+B)(q-1) $$ So substituting into the LHS we get, $$\sin^2 (A+B)+ (q-1)\tan (A+B)\sin (A+B)\cos (A+B) +q \cos^2 (A+B) = \sin ^2 (A+B) +(q-1)\sin ^2(A+B) +q\cos ^2 (A+B) =q\sin ^2 (A+B) +q\cos ^2 (A+B) =q $$ Hope it helps.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2084202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What's wrong with this sequential reasoning for a formula for a_n? The following sequence is given. Write a formula for $a_n$.
$\displaystyle\{1,2,4,7,11,16\}$
Note the arithmetic formula.
$\displaystyle a_n=a_1+\left(n-1\right)d$
Define $d$ as $d_n$, where $d_n$ is the difference between term $n$ and its succeeding te... | Following your notations :-
$d_n$ is the difference between the $(n+1)th$ and $nth$ term $\implies d_n=a_{n+1}-a_n$.
Also, $d_n=d_1+(n-1)g$ and $d_1=1$ and $g=1\implies d_n=n$.
Thus $a_{n+1}-a_n=n\implies a_{n+1}=a_n+n, \ n\geq 1$.
The mistake you were making is to assume that $a_n=a_1+(n-1)d$, which is true with a fi... | {
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Why does $\log x$ go to $-\infty$ so slowly? While checking convergence of integrals, I always have to remind myself that $\log x$ goes to infinity very slowly. This is almost like and axiom to me. I don't intuitively get why it goes to infinity so slowly. On a graph, it sure doesn't look that way. Any thoughts?
More p... | Because $\log(x^{\delta})=\delta\log(x)$, which is essentially $\log(x)$, and
$$\log(\log(x))\leftrightarrow \log(x)\leftrightarrow x\leftrightarrow e^x\leftrightarrow e^{e^x}$$
truly represent different growth behaviors.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2084384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Doing definite integration $\int_0^{\pi/4}\frac{x\,dx}{\cos x(\cos x + \sin x)}$ We have to solve the following integration
$$
\int_0^{\pi/4}\frac{x\,dx}{\cos x(\cos x + \sin x)}
$$
I divided both in Nr and Dr by $\cos^2 x$.
But after that I stuck.
| \begin{align}
\int_0^{\pi/4}\frac{x\,dx}{\cos x(\cos x + \sin x)} &= \int_0^{\pi/4}\frac{x\,dx}{\cos^2(x)+\cos(x)\sin(x)}\\ &=\int_0^{\pi/4}\frac{2x\,dx}{\cos(2x)+\sin(2x)+1} \\&=\frac{1}{2}\int_0^{\pi/2}\frac{t\,dt}{\cos(t)+\sin(t)+1}
\end{align}
By $t \to 1-t$
$$I =\frac{1}{2}\int_0^{\pi/2}\frac{(\pi/2-t)\,dt}{\cos(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2084474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 2
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Quadratic equation system $A^2 + B^2 = 5$ and $AB = 2$ Given a system of equations
$A^2 + B^2 = 5$
$AB = 2$
what is the correct way to solve it?
I see immediately that the answers are
*
*$A=1, B=2$
*$A=2, B=1$
*$A=-1, B=-2$
*$A=-2, B=-1$
but I don't understand the correct way of getting there. I have tried to is... | My favorite way:
Multiply the first equation by $A^2$ ($\ne0$):
$$A^4+A^2B^2=5A^2.$$
As $A^2B^2=4$, you get a biquadratic equation:
$$A^4-5A^2+4=0.^*$$
Then
$$A^2=\frac{5\pm3}{2}=1\text{ and }4,$$ giving the four solutions
$$A=-2,-1,1,2$$
and corresponding $B=2/A$:
$$B=-1,-2,2,1.$$
$^*$Following @Hurkyl's argument, yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2084539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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"answer_id": 5
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Why does taking the derivative with respect to time of an integral (with respect to tau) result in a function of time?
I am having trouble understanding why line 3 to line 4 works, specifically when the derivative of u with respect to time is taken.
u is an integral of a function of tau with respect to tau, but taking... | For any continuous function $f$, we have that
$$\frac{d}{dx} \left(\int_a^x f(y)dy \right) = f$$
this is called the Fundamental theorem of calculus.
In your case, you have $t$ in the upper bound of the function. for every $t$, $\int_0^t f(\tau)d\tau$ is some number, so you have a function of $t$ (not of $\tau$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is the following proof of convergence valid? A problem from Stephen Abbott's Understanding Analysis:
Exercise 2.5.4.: Assume $(a_n)$ is a bounded sequence with the property that
every convergent subsequence of $(a_n)$ converges to the same limit $a \in \mathbb{R}$. Show that $(a_n)$ must converge to $a$.
Is the follow... | Your proof has two problems: first, the problem does not state that every subsequence converges, but rather that if a given subsequence has a limit then this limit must be $a$.
Second, since there are infinitely many subsequences of $\{a_n\}$, there are infinitely many $n_J$, and so the $N$ in your proof is not necessa... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A trick to simplify $z\ln z+\overline{z}\ln\overline{z}$?
Q: Given complex conjugates $z = a+bi$ and $\overline z = a-bi$, what would be substitution needed to find $R$,
$$R= z\ln z+\overline{z}\ln\overline{z}\tag1 $$
such that $R$ is an expression without imaginary numbers?
This question arose as a special case ... | With a notation that should be obvious,
$$R=(x+iy)(\log r+i\theta)+(x-iy)(\log r-i\theta)=2(x\log r-y\theta)=2\left(x\log\sqrt{x^2+y^2}-y\,\arccos\frac xr\right).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2084925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Decrease rate of the shadow
If a man of height $6ft$ moves with $5ft/sec$ velocity towards a lamp hanging at $15ft$ height, then at what rate his shadow will decrease?
My Work: Let the initial distance between the man and the lamp post be $x$ and the length of shadow be $y$.
Then, $\frac{x}{9} = \frac{y}{6} \implies... | Anatoly has given a simple solution,
but in case you are expected to use calculus, here's another simple solution.
Starting from $y = \dfrac23 x,\;\; \dfrac{dy}{dt} = \dfrac23 \dfrac{dx}{dt} = \dfrac 23\times -5 = -\dfrac{10}{3}$ ft/s,
the negative sign showing decreasing length of shadow.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2085091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Linear transformation and matrix representation in $\mathbb R^{2\times 2}$ After spending some time on the next problem, I still don't get it. It consists of a,b and c and I think I solved a (though I might have done something wrong so any corrections are appreciated)
The real trouble starts at b, I really don't know h... | When $M = \pmatrix{a&b\\c&d}$, we can find $L$ by multiplying out $MX$ and $XM$ and adding the results. Thus
$$
MX + XM = \pmatrix{a&b\\c&d}\pmatrix{x_{11}&x_{12}\\x_{21}&x_{22}} +
\pmatrix{x_{11}&x_{12}\\x_{21}&x_{22}}\pmatrix{a&b\\c&d} \\
LX= \pmatrix{ax_{11}+bx_{21} & ax_{12}+bx_{22}\\cx_{11}+dx_{21} & cx_{12}+dx_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2085216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding the numbers $n$ such that $2n+5$ is composite. Let $n$ be a positive integer greater than zero. I write
$$a_n =
\begin{cases}
1 , &\text{ if } n=0 \\
1 , &\text{ if } n=1 \\
n(n-1), & \text{ if $2n-1$ is prime} \\
3-n, & \text{ otherwise}
\end{cases}$$
The sequence goes like this $$1,1,2,6,12,-2,30,42,-5,72,90... | Ignore $n = 0, 1$ since they're kind of irrelevant. Then $a_n \leq 0$ for all $n$ unless $2n - 1$ is prime, by the definition. In that case $a_n = n(n - 1)$. So,
$$4a_n + 1 = 4(n^2 - n) + 1 = (2n-1)^2$$
So, then $\sqrt{4a_n + 1} = 2n - 1$, which is prime.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Analytical solution to linear coupled ODEs? I have this set of equations and I am trying to find $X(t)$ and $Y(t)$ analytically with initial values known such as $X(0)=X_0$ and $Y(0)=Y_0$. How should I approach to solve it?
$dX/dt=a \times X(t) + b \times Y(t)$
$dY/dt=c \times Y(t) + d \times X(t)$
| Write your system of equations in matrix form:
$$\begin{pmatrix} \frac{dX}{dt} \\ \frac{dY}{dt} \end{pmatrix} =\begin{pmatrix} a & b \\ d & c \end{pmatrix} \begin{pmatrix} X \\ Y \end{pmatrix}$$
You can find the two eigenvalues $\lambda_1$ and $\lambda_2$ by letting $\det(A-\lambda I)=0$, and then evaluate correspondin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2085397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Linear Algebra: coordinates and basis I'm asked to show that if $f,g \in \mathrm{Sig}_3$ (sig stands for signal) have the coordinates $x,y \in \mathbb{R^4}$ with respect to the basis $(v_0, v_1, v_2, v_3)$, then
$$\left<f,g\right>=\pi(x_1y_1+x_2y_2+x_3y_3+x_4y_4)$$
where
$$v_0=\frac{1}{\sqrt2},
v_1=\cos x,
v_2=\cos(2... | The question is a bit hard to understand because it's written in a weird order. Usually the question goes (environment) (specifics) (statement to be proved). If I understand your problem well, I think it would be better written this way:
Let $f$ and $g$ be elements of $\mathrm{Sig}_3$, which has inner product
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2085567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Proving $a\cos x+b\sin x=\sqrt{a^2+b^2}\cos(x-\alpha)$
Show that $a\cos x+b\sin x=\sqrt{a^2+b^2}\cos(x-\alpha)$ and find the correct phase angle $\alpha$.
This is my proof.
Let $x$ and $\alpha$ be the the angles in a right triangle with sides $a$, $b$ and $c$, as shown in the figure. Then, $c=\sqrt{a^2+b^2}$. The le... | We can prove this by using properties of complex numbers:
$$ \begin{align}a\cos(x) + b\sin(x) &= a \cos(x) + b \cos\left(x-\frac{\pi}{2}\right)\\
&=a \operatorname{Re}\left\{e^{ix}\right\} + b \operatorname{Re}\left\{e^{i(x-\pi/2)}\right\}\\
&= \operatorname{Re}\left\{ae^{ix} + be^{i(x-\pi/2)}\right\}\\
&= \opera... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2085687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Transformation between random variables If given realization of X and Y, which are i.i.d. Gaussian random variables.
We need to generate random variables C and D which are jointly Gaussian with zero mean and covariance matrix like
$
\begin{bmatrix}
2 & -1 \\
-1 & 2
\end{bmatrix}$
using X and Y. How to pro... | from the covariance matrix $\sigma^2_C=2, \sigma^2_D=2, \sigma_{CD}=\sigma_{DC}=-1$.
Hence, $\rho_{CD}=\frac{-1}{2}$. The desired transform is
$$\begin{align}
C&=\sigma_CX\\
D&=\sigma_D(\rho_{CD}X+\sqrt{1-\rho^2_{CD}}Y)
\end{align}$$
It is assumed that $X$ and $Y$ are zero-mean. Otherwise, the same transformation shoul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2085791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Find coordinates of point $C$ lying on both the planes such that the perimeter of $\triangle ABC$ is minimum $A(3\mid −1\mid −1)\quad$ is a point in the plane $x+y+z=1.$
$B(3\mid 1\mid 0)\quad$ is a point in the plane $2x−y−z=5.$
Find coördinates of point C lying on both the planes such that the perimeter of △ABC is mi... | The third point is
$C(2\mid t\mid -1-t).$
Using the distance formula, calculate $\overline{AC}\text{ then }\overline{BC}.$
Their sum is a function in t. $\sqrt{2+2t+2t^2}+\sqrt{3+2t^2}=y$
Calculate $\frac{dy}{dt}=\frac{1+2t}{\sqrt{2+2t+2t^2}}+\frac{2t}{\sqrt{3+2t^2}}=0$
$t=-\frac{2-\sqrt2}2$
$C\left(2\mid-\frac{2-\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2085908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Intuition behind $\zeta(-1)$ = $\frac{-1}{12}$ When I first watched numberphile's 1+2+3+... = $\frac{-1}{12}$ I thought the sum actually equalled $\frac{-1}{12}$ without really understanding it.
Recently I read some wolframalpha pages and watched some videos and now I understand (I think), that $\frac{-1}{12}$ is just ... | The values of $\zeta$ for negative integers can be directly calculated from the Bernoulli numbers, from:
$$\zeta(-n)=(-1)^n\frac {B_{n+1}}{n+1}$$
and $B_2=\dfrac 16$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2086035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 7,
"answer_id": 6
} |
Game Theory- altruistic preferences How would I answer this question?
John is an altruistic person who cares about his own income as well as the income of his best
friend Ryan. The value he attaches to each unit of his own income is the same as the value of
3 units of Ryan’s income. For example, he is indifferent betwe... | Your question steps out of game theory at the moment you say "John is an altruistic person"
However, I think you can make your way back to game theory by rewriting an "altruistic income". Take the usual income, and then recalculate the one of John by applying the rule of your choice (in that case, 3 times original inco... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2086127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can we find a subfield $F$ of $\mathbb C$, such that ${ F}^{*p}={ F}^{*}$ for any prime $p\neq 5$ and ${F}^{*5}\neq F^{*}$? We know for field $\mathbb R$, ${\mathbb R}^{*p}={\mathbb R}^{*}$, for any prime $p\neq 2$ and ${\mathbb R}^{*2}\neq {\mathbb R}^{*}$. (${\mathbb R}^{*}$ is the multiplicative group of $\mathbb R... | Here's a possible construction.
If $F$ is a countable subfield of $\mathbb{C}$, we can form the field $F'$ by splitting all polynomials of the form $X^p-a$ for $p\neq 5$ a prime and $a\in F$. There are countably many such polynomials, so order them in some way and split them successively.
Now, iterate this procedure. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to find the $f(3)=y$ with given points? If an quadratic equation $f(x)$ has max value $4$ when $x = 0$ and $f(-2)=0$, what is the value of $f(3) =$ ?
| making the ansatz $$f(x)=ax^2+c$$ we get easily $$c=4$$ from $$f(-2)=0$$ we get $$a=-1$$ and our searched function is $$f(x)=-x^2+4$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2086463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Prove by tableau method - Axioms of equality Example:
Let $T^*$ be a theory with axioms of equality. Prove by tableau method that
(a) $T^* \vDash x=y \rightarrow y=x \qquad $ (symmetry of =)
(b) $T^* \vDash (x=y \, \land \,y=z) \rightarrow x=z \qquad$ (transitivity of =)
Hint: To show (a) apply the axiom of equality (... | You have to use both equality axioms :
$\forall x \ (x=x)$ --- reflexivity
$\forall x \ \forall y \ (x=y) \to (\phi \to \phi')$ --- where $\phi'$ is obtained from $\phi$ by replacing $x$ in zero or more places by $y$ : replacement axiom.
Specifically, the instance of the replacement axiom needed is :
$\forall x \ \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2086567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find all positive integer solutions to the equation $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}$ Find all positive integers to the equation $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}$
Multiply both sides with $(abc)^2$ to get $(bc)^2 + (ac)^2 = (ab)^2$.
I then tried some pythagorean triples and nothing worked so I assu... | Rewrite the equation as
$$
\frac{c^2}{a^2}+\frac{c^2}{b^2}=1
$$
and set $c/a=x/z$, $c/b=y/z$, where $x$ and $y$ are coprime.
Then $(x,y,z)$ is a primitive Pythagorean triple, so, without loss of generality,
$$
x=u^2-v^2,\quad y=2uv,\quad z=u^2+v^2
$$
where $\gcd(u,v)=1$, one among $u$ and $v$ being even.
Thus
$$
\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2086656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Analyzing a function with cube roots I am analyzing this function:
\begin{align}
f(x) = \sqrt[3]{ x } - \sqrt[3]{x+1}
\end{align}
I proved it has a horizontal asymptote at y=0. However, on the graph, it has a minimum for x=-0.5 but I can't find that. First derivative has no zeroes, so the function has no stacionary poi... | You must've taken the first derivative wrong:
$$f'(x)=\frac13(x^{-2/3}-(x+1)^{-2/3})$$
Setting $f'(x)=0$,
$$0=x^{-2/3}-(x+1)^{-2/3}$$
$$x^{-2/3}=(x+1)^{-2/3}$$
$$\pm x=x+1$$
$$x=-1/2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2086754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Direct proof of the level four eta identity Of Somos' over 6200 eta identities, the one of smallest level (maybe the simplest?) is the eta identity of level $4$ $$\eta(z)^{16} \eta(4z)^8 + 16 \eta(4z)^{16} \eta(z)^8 = \eta(2z)^{24}.$$
As with all identities among modular forms, we can prove it by checking the first few... | I am surprised at the idea that this needs "real" proof. The identity is the very first one in my list and the very first formulation of it is the "aequatio identica satis abstrusa" straight from Jacobi's Fundamenta Nova Theoriae Functionum Ellipticarum, page 90 equation (14) published in 1829. There is a reference to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2086969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Normalizer of upper triangular group in ${\rm GL}(n,F)$ The following question has already appeared on mathstack:
If $B$ is the subgroup of ${\rm GL}(n,F)$ consisting of upper triangular matrices then normalizer of $B$ in ${\rm GL}(n,F)$ is $B$ itself.
I know a proof of this using Bruhat decomposition of ${\rm GL}(n,... | $\DeclareMathOperator{\GL}{GL}$$\newcommand{\Span}[1]{\left\langle #1 \right\rangle}$$\newcommand{\Set}[1]{\left\{ #1 \right\}}$Let $e_0, e_1, \dots, e_{n-1}$ be a basis with respect to which $B$ is upper-triangular, and write
$$
V_i = \Span{ e_j : j \ge i}.
$$
Allow me to use row vectors, so that the group $G = \GL(n,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2087062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
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How to integrate $\tan(x) \tan(2x) \tan(3x)$? We have to integrate the following
$$\int \tan x \tan 2x \tan 3x \,\mathrm dx$$
In this I tried as
First split it into sine and cosine terms
Then used $\sin 2x =2\sin x \cos x$
But after that got stuck
| Notice
$$\tan3x=\frac{\tan2x+\tan x}{1-\tan2x\tan x}\Rightarrow \tan3x\tan2x\tan x=\tan3x-\tan2x-\tan x$$
then
$$\int \tan3x\tan2x\tan x \mathrm{d}x = \int \tan3x - \tan2x - \tan x \ \mathrm{d}x$$
the answer will follow.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2087119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Followup question on first-order formulas in first-order languages and open formulas. This is a followup question to my question here. I will reproduce the contents of my original question as follows.
For any first-order formula $X$ in the first-order language $\langle 0, S, \le\rangle$ (possibly with free variables) ... | No, it does not. The set $\{1\}$ is definable in that structure - it is the set of $x$ such that $$x\not=0\wedge \forall y(y\le x\implies [y=0 \vee y=x]).$$ However, it's not hard to show that it is not definable by a quantifier-free formula (there aren't very many sets which are definable by a quantifier-free formula ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2087230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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How to calculate $\lim\limits_{n \to \infty } \left(\frac{n+1}{n-1}\right)^{3n^{2}+1}$? I have $\lim_{n \to \infty } \biggr(\dfrac{n+1}{n-1}\biggr)^{3n^{2}+1}$
and I use this way: $\lim_{n \to \infty } \frac{n+1}{n-1}=|\frac{\infty}{\infty}|=\lim_{n \to \infty } \frac{1+\frac{1}{n}}{1-\frac{1}{n}}=1$ and $\lim_{n \to \... | To apply your formula, notice that
$$\left(\frac{n+1}{n-1}\right)^{3n^2+1}=\left(1+\frac2{n-1}\right)^{3n^2+1}$$
Let $u=\frac{n-1}2$ or $n=2u+1$,
$$=\left(1+\frac1u\right)^{12u^2+12u+4}=\left(\left(1+\frac1u\right)^u\right)^{12(u+1)}\left(1+\frac1u\right)^4$$
$$\to e^{12(u+1)}\to e^\infty=\infty$$
Thus, the limit is in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2087394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
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Maximum value of $|x-y|$ Given $x,y\in\mathbb R$ such that
$$5x^2+5y^2-6xy\ =\ 8$$
find the maximum value of $|x-y|$.
My attempt
$5x^2 - 6yx + (5y^2-8)\ =\ 0$
$x\ =\ \dfrac{6y\pm\sqrt{(6y)^2-4(5)(5y^2-8)}}{10} = \dfrac{6y\pm\sqrt{160-64y^2}}{10} = \dfrac{3y\pm2\sqrt{10-4y^2}}{10}$
$5y^2 - 6xy + (5x^2-8)\ =\ 0$
$y\ =\ \... | You tried to solve for both $x$ in terms of $y$ and $y$ in terms of $x$.
Try expressing everything as either a function of $x$ or of $y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2087496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Calculate the center point between three pointsTh I have three positions, with an $X$ and $Y$ value and a value that indicates the distance to of each of the individual points (not in the same measurement). So to clarify my question:
I have three $XY$-positions and three values indicating the distance to the point (not... | Weighted ( by reciprocal strength $s$) average for each vertex
$$ \bar{ x} = \dfrac {x_1 s_1 + x_2 s_2 +x_3 s_3}{s_1+s_2+s_3} $$
$$ \bar{ y} = \dfrac {y_1 s_1 + y_2 s_2 +y_3 s_3}{s_1+s_2+s_3} .$$
where $$ s_i= \dfrac{1}{Strength_i} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2087625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Prove $\int_{B(x,r)}|\nabla u|^2\leq \frac{C}{r^2}\int_{B(x,2r)}|u|^2$ if $-\Delta u(x)+f(x)u(x)=0.$ Let $\Omega $ a smooth domain of $\mathbb R^d$ ($d\geq 2$), $f\in \mathcal C(\overline{\Omega })$. Let $u\in \mathcal C^2(\overline{\Omega })$ solution of $$-\Delta u(x)+f(x)u(x)=0\ \ in\ \ \Omega .$$
Assume that $f(x)\... | Suppose that $\varphi : \bar{B}(x,2r) \to [0,\infty)$ is Lipschitz and $\varphi =0$ on $\partial B(x,2r)$. Note that since $\varphi$ is Lipschitz, it is differentiable almost everywhere by Rademacher's theorem. Then
$$
0 \le \int_{B(x,2r)} \varphi^2 f u^2 = \int_{B(x,2r)} \varphi^2 u \Delta u = \int_{B(x,2r)} - \nab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2087724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Solving the integral of a function with modulus I would like to solve the following:
$$\int e^{-|x|}dx$$
The integral $\int e^{-x}dx = -e^{-x} +C$ is trivial, but the one with the modulus on $x$ seems to have a catch, as typing it into the integral calculator re-writes the integral and gives:
$$\frac{x}{|x|}\int e^{-|x... | This integral $$\int e^{-|x|}dx$$ is an even integral, such that we could define it as the amalgam of two integrals, each defined in a specific domain.
$$I_{1}=\int e^{-x}dx, x\geq 0 $$
$$I_{2}=\int e^{x}dx, x\leq 0 $$
As the question states, both these integrals are trivial.
$$I_{1}=- e^{-x} , x\geq 0 $$
$$I_{2}=e^{x}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2087944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
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Using the pigeon principle I was given the next question:
I need to prove that any subset with $25$ elements of the set $\{1,2,....150\}$, contain least $4$ elements, $x$, $y$, $z$, $t$ such that $x+y=z+t$.
Now my idea is to use the pigeon principle and set the pigeons to be the number of the possible pairings and th... | Your idea is fine.
We have ${25\choose2}=300$ pigeons and as the sums can only range from $1+2=3$ to $149+150=299$, there are only $297$ cages. Hence there are two different pigeons $\{x,y\}$ and $\{z,t\}$ in the same cage, i.e., $x+y=z+t$. Note that the pairs cannot overlap as for example $y=z$ would imply $x=t$ cont... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2088050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Heine-Borel theorem. I'm interested in one question about Heine-Borel theorem.
We know that if S is bounded and closed then it's compact.
Standard proof using fact that some segment is compact. And because of S is bounded it's have left and right bound of segment. But my question is my S should be closed? We could take... | Let $S=(0,1)$ and consider the cover $\{(1/n,1) : n \ge 2\}$. The union of any finite subset of the cover is of the form $(1/n,1)$ for some $n$, so no finite subset can cover $S$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2088157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Find largest dividing integer within range Is there a neat way to find the largest integer that divides another integer fully, within a range. As an example, I would like to find the largest integer smaller than 1131 that divides 3500 completely.
So far I have just tried by breaking up 3500 into its prime components a... | You could look for the smallest integer greater than $\dfrac{3500}{1131}\approx 3.09$ that divides $3500$ exactly
This is obviously $4$ (there is no smaller integer greater then $3.09$ and $4$ does divide $3500$ as it divides $100$) so the answer to your original question is $\dfrac{3500}{4}=875$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2088229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Hyperbolic substitution for $\int\frac{dx}{x\sqrt{1-x^2}}$
Substitute $x=\sinh\theta$, $\cosh\theta$ or $\tanh\theta$. After integration change back to $x$.$$\int\frac{dx}{x\sqrt{1-x^2}}$$
Substituting $x=\tanh\theta$, we have
$$\begin{align}
\int\frac{dx}{x\sqrt{1-x^2}}&=\int\frac{\text{sech}^2\,\theta\,d\theta}{\ta... | For positive $x$, another substitution that does not involve integrating $\csc{x}$ or $\operatorname{csch}{x}$ is a hyperbolic secant substitution (similar to a secant/cosecant substitution for $x^2-1$), where $x=\operatorname{sech}{\theta}$, $dx=-\operatorname{sech}{\theta}\tanh{\theta}\,d\theta$, and $$\sqrt{1-x^2}=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2088318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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how to convert /understand /reason the summation conversion? How to interpret / reason / understand the summation conversion for the below equation ?
$$\sum_{m = 1}^\infty 2^{-2m} = \frac{1}{4}\sum_{m = 0}^\infty 4^{-m}$$
| There are two steps here. First, pulling down the $-2$ in the exponent:
$$\sum_{m=1}^\infty 2^{-2m} = \sum_{m=1}^\infty (\frac{1}{4})^m$$
Next, shifting the lower index to zero (by replacing $m$ everywhere with $m+1$) and pulling out a factor of $\frac{1}{4}$:
$$\sum_{m=1}^\infty (\frac{1}{4})^m = \sum_{m=0}^\infty (\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2088454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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initial conditions, pde Together with a differential equation and initial conditions arise the notion of a well-posed problem. My question is, how does one choose the type of initial conditions to be implemented in the problem or is it due to the nature of the differential equation? Any reference to this answer will be... | To be specific, let us consider the differential equation $x''(t)+x(t)=0$ for the SHM; for which in order to start oscillations you need to release the mass bob (tied to a thin thread with other end clamped) from extreme left i.e. $x(0)=-a$. Moreover, assuming that no external force is provided at the start i.e. $x'(0)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2088569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 0
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If $f(x+1)+f(x-1)= \sqrt{2}\cdot f(x)$, find the period of $f(x)$. If $f(x+1)+f(x-1)= \sqrt{2}\cdot f(x)$, then the period of $f(x)$ is?
I tried replacing $x$ with $x-1$ and stuff, but didn't lead me to anything...
| Note that $\sqrt{2} = \frac{2}{\sqrt{2}}$. Hence,
$$f(x + 1) + f(x - 1) = \frac{1}{\sqrt{2}}f(x) + \frac{1}{\sqrt{2}}f(x)$$
$$\begin{align}f(x + 1) &= \frac{1}{\sqrt{2}}f(x) + \frac{1}{\sqrt{2}}\left(f(x) - \sqrt{2}f(x - 1)\right)\\
&=\frac{1}{\sqrt{2}}f(x) + \frac{1}{\sqrt{2}}\left(-f(x - 2)\right)\\
&= \frac{1}{\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2088809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Prove that triangle is isosceles, triangle that is inscribed in a circle. $AD$ is a median to side $BC$ in triangle $ABC$ that is inscribed in a circle that his center is in $(5,6)$.
Given: $ D (9,2)$ and the centroid of the triangle is $(6,5$).
There is not so much information given about the circle, thus I can't draw... | Sketch of the proof
Let $G$ be the centroid and $O$ the circumcentre. We know that $A$, $G$ and $D$ are collinear since they are in the median to the side $BC$.
We know that the line through $O$ and $D$ is the perpendicular bisector of the side $BC$.
Since $D$ $G$ and $O$ are collinear (the three points are in the line... | {
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Using big Oh notation. How to show that $O(x^{1/2} e^{ (log (\frac{x}{c}))^{1/2}})=O(x^{1/2+\epsilon})$ for any $\epsilon>0$?
| Let $f(x)=e^{(\log(x/c))^{1/2}}=e^{\sqrt{\log(x/c)}}$
Then we have $f(x)^{\sqrt{\log(x/c)}}=e^{(\log(x/c))}=x/c$
So we can see that $f(x)^{\sqrt{\log(x/c)}}=x/c$ which means $f(x)=(x/c)^{\frac{1}{\sqrt{\log(x/c)}}}<x^{\epsilon}$ since $\frac{1}{\sqrt{\log(x/c)}}\rightarrow 0$.
Combining all together yields $x^{1/2}e^{(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2089015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If $n={P_1}^{r_1}.{P_2}^{r_2}.\dots.{P_k}^{r_k}$ then $f(n)={r_1}^{P_1}.{r_2}^{P_2}.\dots.{r_k}^{P_k}$then which one is true? If $n={P_1}^{r_1}.{P_2}^{r_2}. \dots .{P_k}^{r_k}$ then $f(n)={r_1}^{P_1}.{r_2}^{P_2}.\dots.{r_k}^{P_k}$then which one is true?
1.The function is one-to-one.
2.The function is surjective.
3.If ... | (4) is wrong. Note that primes $p$ are mapped to $1$, now let $p$ a prime and $m = n = p$. Then
$$ f(m)f(n) = f(p)f(p) = 1 \not\ge f(p^2) = 2^p $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2089145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that the sequence $x_{n+1}=\frac{1}{2}\left(x_n+\frac{1}{x_n}\right)$ is not increasing Let $x_1\ge1$ and $x_{n+1}=\frac{1}{2}\left(x_n+\frac{1}{x_n}\right)$. Prove that the sequence $\{x_n\}$ is not increasing, that is, $x_{n+2}\le x_{n+1}$.
My attempt was as follows: we have to prove that $1\le \frac{x_{n+1}}{x... | The reasoning seems correct until the end when you write
$$x_n+\frac{1}{x_n}\geq 2$$ Now here you should prove that $$x_n+\frac{1}{x_n}\geq 2$$
Which is trivial by AM-GM.
Here's a similar approach note that by AM-GM
$$x_{n+1}=\frac{1}{2}(x_n+\frac{1}{x_n})\geq 1$$
hence we have that $x_n\geq 1$.
Now $$\frac{x_n}{x_{n+1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Compactness of the open and closed unit intervals In the article by Tao it's explained that the compactness can be formulated in the most general way as:
(All open covers have finite subcovers) If $`V_\alpha:\alpha\in\mathcal{a}`$ is any collection of open sets which covers $X$, then there must exist a finite sub-colle... | showing it is not possible for $(0,1)$ is easy, just take the open cover $(0,1-\frac{1}{n})$.
Showing it is possible for $[0,1]$ is not so easy, you may want to look here for proofs that do not go through the whole Heine-Borel theorem.
| {
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When is a spherical polygon convex suppose we have a polygon on the sphere $\mathbb{S}^2$, that is, a domain with piecewise geodesic boundary. Is it true that this domain is convex if and only if all angles of the polygon are in $(0,\pi)$?
A subset $U\subset\mathbb{S}^2$ is called convex if for any pair of points $p,q\... |
Is it true that this domain is convex if and only if all angles of the polygon are in $(0,π)$?
No, according to your definitions, this is not true. Take three quarter slices of a sphere. That gives you an inner angle of $\frac34\cdot2\pi>\pi$, but for every pair of points the plane spanned by these two points and the... | {
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Is there anything better than an exact equivalence? It's a basic principle of category theory that any "good" concept should be preserved by equivalence of categories. However, since Abelian categories are by definition $\mathbf{Ab}$-enriched, we require a bit more structure for our functors – we almost always like the... | To see that any equivalence of abelian categories is additive and exact, use the following observations:
*
*If $F\colon \mathcal{A} \to \mathcal{B}$ and $G\colon \mathcal{B}\to \mathcal{A}$ give an equivalence, then $F$ is both left- and right-adjoint to $G$.
*Any left adjoint functor is right exact (as it preserve... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why can't we separate fraction powers in euler's formula? I am sure there is something wrong i am doing but i just want someone to point it to me.
Why can't we say that $e^{\frac {\pi i}{3}} = \left(e^{\pi i}\right)^{\frac {1}{3}} = (-1)^{\frac {1}{3}} = -1$
Why do we have to calculate it by the formula as it is (which... | It is simply the case that $(a^b)^c$ is not always equal to $a^{bc}$ if $a$ is not positive or if $b$ and/or $c$ are complex.
The problems here are that
$$x=(-1)^{1/2}\implies x^2=-1\implies x^4=1$$
However, when you say that $1^{1/4}=1$, you cause a misconception. Usually, this is perfectly fine, but when you think a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the coefficient of $x^m$ in $(x+x^2+x^3+.....+x^i)^n$ EDIT:
My doubt may be silly but if the expansion of $(1-x)^{-n}$ is infinite, how come wolfram displayed 11 terms?
Wolfram result
Can anybody help to solve the whole equation?
I'm trying to solve this question.
I have an equation $(x+x^2+x^3+x^4+x^5+x^6)^2$... | This GitHub repository solved my problem.
| {
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"source": "stackexchange",
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Evaluate $\int_0^\infty \frac{\ln\left ( 2x+1 \right )}{x\left ( x+1 \right )}\mathrm dx$ How to evaluate
$$\int_{0}^{\infty }\frac{\ln(2x+1)}{x(x+1)}\,\mathrm dx?$$
I tried
$$\frac{\ln(2x+1)}{x(x+1)}=\ln(2x+1)\left (\frac{1}{x}-\frac{1}{1+x} \right)$$
but I don't know how to go on.
| $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Basic proof that $GL(2, \mathbb{Z})$ is not nilpotent I need to show that $GL(2,\mathbb{Z})=\left\{ \begin{pmatrix} a&b \\ c&d \end{pmatrix} \mid a,b,c,d \in \mathbb{Z}, ad-bc = \pm 1\right\}$ is not nilpotent. I have seen this question but the answer given there is too advanced for where I am currently in my studies.
... | One possibility is to show that there is a proper, self-normalizing subgroup,
$$
\left\{ \begin{pmatrix} a&b \\ 0&d \end{pmatrix} : a, d = \pm 1, b \in \mathbb{Z}\right\}
$$
being perhaps the easiest candidate.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Equation of a plane $Ax+By+Cz=D$ I have question abous parametrization and plane equation.How can I write the equation of a plane in the form $Ax+By+Cz=D$ if it is given as
$$x=2s+3t$$
$$y=s-2t$$
$$z=3s+t$$.
| hint: $x - 2y = 7t, 3y - z = -7t \implies (x-2y) + (3y-z) = 0$
| {
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Find the matrix $(X^{-1} A X)^n$ Given that $$A=\begin{bmatrix}1&1\\ 0&1\end{bmatrix}$$ $$X=\begin{bmatrix}2&1\\ 3&2\end{bmatrix}$$
I computed that $$X^{-1} A X = \begin{bmatrix}7&4\\ \:\:\:\:-9&-5\end{bmatrix}=Y$$
Now the usual way would be to find the eigenvalues and corresponding eigenvectors and then calculate the... | Hint:
first prove, by induction, that
$$
A^n=\begin{bmatrix}
1&n\\0&1
\end{bmatrix}
$$
Then use the fact (easy to prove):
$$
(X^{-1}AX)^n=X^{-1}A^nX
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Number of ordered triples with given condition We are given that there are $6250k$ ordered triples $(m,n,p)$ where $m,n,p$ are integers such that $$1≤m≤100,\;1≤n≤50,\;1≤p≤25\quad \& \quad 3\,|\,2^m+2^n+2^p$$ Find $k$.
With the given range of values of $m, n,p$, I guess the possible numbers of ordered triples without... | Given that $1≤m≤100,\;1≤n≤50,\;1≤p≤25$ and $ 3\,|\,2^m+2^n+2^p$ where $m,n,p$ are integers.
Now if $2^m+2^n+2^p$ is divisible by $3$, then each of $2^m$, $2^n$, $2^p$ is divided by $3$.
Remainder of ${2^m}$ when it is divisible by $3$ is $(-1)^m$.
So remainder of ${2^m+2^n+2^p}$ when it is divisible by $3$ is ${(-1)... | {
"language": "en",
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Quaternion Algebras and isomorphisms I was reading a book on quaternion algebras and found the following exercise and was unable to do it and it has been really frustrating me,
Let $(a,b)_F$ be a quaternion algebra with $i^2=a$ , $j^2=b$ and $ij=-ji$
Prove that $(a,b)_F$ , $(b,a)_F$ and $(ac^2,b)_F$ are isomorphic and... | Let's show that $(a,b)_F$ and $(b,a)_F$ are isomorphic. Let's say $(a,b)_F$ is generated by $i,j$ with
$$ i^2=a, \quad j^2=b, \quad ij=-ji $$
and $(b,a)_F$ is generated by $k$ and $\ell$ satisfying
$$ k^2=b, \quad \ell^2=a, \quad k\ell=-\ell k. $$
To define an isomorphism $\phi:(b,a)_F\to(a,b)_F$ it suffices to specify... | {
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A polynomial question. I came across this question in my weekly test paper and didn't know how to tackle it. The question goes as follows:
$a_0,a_1,a_2....a_{34}$ are the coefficients of $x^0,x^1,x^2......x^{34}$ of the polynomial obtained on opening the parenthesis of $(1+x+x^2)^{17}$, then which is true?
$1)~a_1+a_2+... | Hint: if you plug $x=1$ in, you will get the sum of the coefficients.
The coefficients may be represented using Viete's formulas.
| {
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"timestamp": "2023-03-29T00:00:00",
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Find the sum of first $n$ terms of the series: $\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+\cdots +\frac{1}{n\times(n+1)}$
I have the series $$\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+\cdots +\frac{1}{n\times(n+1)}$$
I know the following formulas: $$1+2+3+\cdots +n=\frac {n (n+1)}{2}\ta... | Try to observe that $$\frac{1}{n\times (n+1)}=\frac{1}{n}-\frac{1}{n+1}$$ $\therefore$ The given series can be written as $$1-\frac12+\frac12-\frac13+\frac13+\cdots -\frac{1}{n}+\frac1n-\frac{1}{n+1}$$ Each term will cancel out other term except $1$ and $\frac{1}{n+1}$ .
$\therefore$ $$=1-\frac1{n+1}$$ $$=\frac{n}{n+1}... | {
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"timestamp": "2023-03-29T00:00:00",
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Difficulty solving problem regarding 2 parallel vectors I have been asked to solve the following question, and I am not sure where to begin with it :
Given two vectors $a$ and $b$ , where the magnitude of $b$ is double the
magnitude of $a$ and the angle between the two vectors is $60°$, we may
define two further vecto... | $$c\cdot d=(a-2b)\cdot (14a-2b)=14|a|^2-30a\cdot b+4|b|^2$$
Use that
$$|b|=2|a| \quad (1)$$
and then
$$a\cdot b=|a||b|\cos 60º \rightarrow a\cdot b=2|a|^2\cdot1/2=|a|^2 \quad (2)$$
Remember that $c \perp d \Leftrightarrow c \cdot d=0$.
Using $(1)$ and $(2)$ we get:
$$c\cdot d=14|a|^2-30a\cdot b+4|b|^2=14|a|^2-30|a|^2+... | {
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Find the numbers in A.P. those sum is $24$ and product is $440$
If the sum of three numbers in A.P. is $24$ and their product is $440$, find the numbers.
My Approach: Let the numbers be $a,a+d,a+2d$
So, according to question $$3a+3d=24$$ $$a+d=8$$ and $$a(a+d)(a+2d)=440$$ $$8a+ad=55$$ I can’t proceed from here.
Ple... | Let the numbers be : $a-d, a, a+d$
Then According To Question. $$a-d+a+a+d=24$$ $$a=8$$ and $$(a-d)(a)(a+d)=440$$ $$(8-d)(8)(8+d)=440$$ $$(8-d)(8+d)=55$$ $$64-d^2=55$$ $$-d^2=-9$$ $$d=\pm3$$ $\therefore$ the A.P. is $$5,8,11$$ $$**OR**$$ $$11,8,5$$
| {
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"timestamp": "2023-03-29T00:00:00",
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How to factorize this $\sqrt{8 - 2\sqrt{7}}$? When I was at high school, our teacher showed us a technique to simplify square roots like this $\sqrt{8 - 2\sqrt{7}}$ that I forgot.
It was something like 8 = 7+1; 7 = 7*1; and using them we could represent $\sqrt{8 - 2\sqrt{7}}$ in simpler form. I would be happy if you c... | You could approach this by setting $\sqrt{8 - 2\sqrt{7}} = \sqrt a \pm \sqrt b$ for some $a,b$. Then, squaring both sides, we have:
$$8 - 2\sqrt{7} = a \pm 2\sqrt{ab} + b,$$
so that $8=a+b$ and $-2\sqrt7 = \pm 2\sqrt{ab}$.
In other words, the $\pm$ sign must be $-$, and we now have a system of two equations $\{8=a+b, 7... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Covering compact subsets of plane with certain rectangles Suppose $\{F_n\}$ is sequence of compact subsets of the half-plane $\mathbb H=(0,+\infty)\times(-\infty,+\infty)$ with positive area such that $F_n\subset F_{n+1}$ and $\cup_n F_n=\mathbb H$. Do there exist a sequence of positive numbers $\{a_n\}$ and positive c... | No. Let $c_n$ be a monotone increasing sequence. For convenience let $d_n = \frac{c_n}{c_n-1}$. Then set
$$F_n = \bigg( [c_n^{-1},c_n]\times[-c_n, c_n] - [d_n^{-1},d_n]\times[-d_n, d_n] \bigg) \cup \{1\}\times [-c_n, c_n] $$
Each $F_n$ is closed, bounded, so compact. It looks like a large rectangle with a smaller rec... | {
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"timestamp": "2023-03-29T00:00:00",
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Proving $ \sum_{n=1}^{\infty} nz^{n} = \frac{z}{(1-z)^2}$ for $z \in (-1, 1)$ I do not know where to start, any hints are welcome.
| Without differentiation
$$\sum_{n=1}^\infty nz^{n-1} = 1+2z+3z^2+\cdots \tag 1$$
$$\sum_{n=1}^\infty nz^{n} = z+2z^2+3z^3+\cdots \tag 2$$
Subtract (2) from (1)
$$\sum_{n=1}^\infty nz^{n-1}(1-z) = 1+z+z^2+\cdots$$
$$\sum_{n=1}^\infty nz^{n}= \frac{z}{(1-z)^2}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Limit I don't know how to start solving. This is the limit:
$$\lim_{n\to\infty}\frac{2^{n-1}-4^n}{3^n\sin n + 4^{n+2}}$$
I have a solution and the steps and I still haven't understood how it's done, here's the proposed solution:
$$\lim_{n\to\infty}\frac{2^{n-1}-4^n}{3^n\sin n + 4^{n+2}}=\lim_{n\to\infty}\frac{\frac12(\... | Some intuition for how to come up with this solution:
As $n$ gets very large, $4^n$ is much much bigger than $2^n$. A common trick to "prove" this, or tease this fact out, is to divide the numerator and denominator by the highest order term. This is legal since it amounts to multiplying by 1, and $4^n$ is never 0 for a... | {
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Wolfram Alpha gives different answers for doing the same thing to both sides of this equation. I have the equation $10^x=2i$ (I'm doing this because I wanted to take the log of $2i$) I was trying to solve it by hand first, but I was running into errors, so I wanted to make sure when I square both sides of the equation,... | It is not WolframAlpha's fault. To analyze the problem, let us look at a far simpler problem:
$$e^x=-1$$
We all know the solution to this should be given by $x=(2n+1)\pi i$ for $n\in\mathbb Z$. But upon squaring both sides,
$$e^{2x}=(-1)^2=1$$
The solution to $e^{2x}=1$ is different, as it includes, say, $x=0$, which... | {
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$4x^2-5xy+4y^2=19$ with $Z=x^2+y^2$, find $\dfrac1{Z_{\max}}+\dfrac1{Z_{\min}}$ I have no idea how to approach this question at all. I've tried to find the maximum and minimum of the quadratic but i am too confused on what to do afterwards.
| Without Lagrange multipliers (but with trigonometry). Use polar coordinates:
$x=\sqrt Z\cos\theta$
$y=\sqrt Z\sin\theta$
$xy=Z\sin\theta\cos\theta=\frac{Z}{2}\sin(2\theta)$
From your equation you get $\frac{1}{z}=\frac{4}{19}-\frac{5\sin(2\theta)}{38}$
As $\sin$ takes values between $-1$ and $1$:
So $\frac{1}{z_{min}}+... | {
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"answer_id": 1
} |
Maximum of $f(x) =(1+\cos x)\cdot \sin(\frac{x}{2})$ on $x \in (0, \pi)$ I've attempted to solve this question by using $$f(x) = 2\sin\frac{x}{2}\cos^2\frac{x}{2} \leq \frac{(2\sin\frac{x}{2}+\cos^2\frac{x}{2})^2}{2}$$ but it results in the wrong answer every time. Is there another way to solve this question and is the... | By AM-GM $$\left(1+\cos{x}\right)\sin\frac{x}{2}=2\cos^2\frac{x}{2}\sin\frac{x}{2}=2\left(\sin\frac{x}{2}-\sin^3\frac{x}{2}\right)=$$
$$=-2\left(-\sin\frac{x}{2}+\sin^3\frac{x}{2}+\frac{1}{3\sqrt3}+\frac{1}{3\sqrt3}\right)+\frac{4}{3\sqrt3}\leq$$
$$\leq-2\left(-\sin\frac{x}{2}+3\sqrt[3]{\sin^3\frac{x}{2}\cdot\frac{1}{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2091339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Is the Euclidean valuation on a Euclidean domain unique? I have just started studying Euclidean domain which is defined as follows :
An integral domain $R$ is called a Euclidean domain if there exists a function $\delta : R \setminus {\{0\}} \longrightarrow \mathbb {N_{0}}$ which satisfies the following properties.
(1)... | Another example (to show non-uniqueness): On the ring of integers, define $\delta (x) = x^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2091438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
How to show that this algorithm needs $\Omega(n \log n)$ comparisons How can I show, that every algorithm based on pairwise comparisons for sorting a total ordered set consisting of $n$ elements requires in the $\Omega(n \log n)$ comparisons in the average case.
The hint I was already given is to show that for all $x \... | Suppose that we have $n$ elements that need to be sorted and we can only do pairwise comparisons. There are $n!$ ways to arrange these elements, and if we assume the elements are unique, then there is only one way to sort them.
Let's think of each pairwise comparison as a yes/no question. Each pairwise comparison r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2091519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Find the value of $ 99^{50}-\binom{99}{1}(98)^{50}+\binom{99}{2}(97)^{50}-\cdots \cdots +99$ Find the value of $\displaystyle 99^{50}-\binom{99}{1}(98)^{50}+\binom{99}{2}(97)^{50}-\cdots \cdots +99$
Binomial identity:
$\displaystyle (1-x)^{99} = \binom{99}{0}-\binom{99}{1}x+\binom{99}{2}x^2-\cdots \cdots -\binom{99}{99... | It is simply zero. Let $\delta$ be the (difference) operator mapping $p(x)$ into $p(x)-p(x-1)$: if $p(x)$ is a polynomial with degree $d\geq 1$, then $(\delta p)(x)$ is a polinomial with degree $d-1$. Since
$$(\delta^n p)(x) = \sum_{k=0}^{n}(-1)^k \binom{n}{k} p(x-k) \tag{1}$$
if $p(x)$ is a polynomial with degree $<n$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2091656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Domino game maximum achievable points Given is a domino parts set with typical 28 parts from [ | ] to [6|6].
[ | ] [ |1] [ |2] [ |3] [ |4] [ |5] [ |6]
[1|1] [1|2] [1|3] [1|4] [1|5] [1|6] [2|2]
[2|3] [2|4] [2|5] [2|6] [3|3] [3|4] [3|5]
[3|6] [4|4] [4|5] [4|6] [5|5] [5|6] [6|6]
As every domino player knows, one can o... | I made a program to simulate the game and got the following result:
The middle row shows the placement of the dominoes, the bottom row shows the order of placement of each domino and the upper row shows the total score as each domino is placed. The total score after the placement of the last domino can be seen to be $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2091748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Function linear in one variable and quadratic concave on another, is concave in jointly? I have a function $f(x,y,z)$. $f$ is convex with respect to $x$, and linear with respect to $y$. Also, it is quadratic concave with respect to $z$. I want to solve the following optimization problem:
\begin{align}
\min_{x\in C... | No, you may not. Suppose $f(x,y,z) = -yz^2$, with $D_y = \mathbb{R_+}$, then the Hessian is:
$$\begin{pmatrix}0 & -2z \\ -2z & -2y\end{pmatrix}$$
which is indefinite for $z \neq 0$, since the eigenvalues are $-y \pm \sqrt{y² + 4z²}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2091845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Approximating a function in $H^1\cap L^\infty$ by $H^1_0\cap L^\infty$ functions $\Omega$ is a bounded smooth domain. Take $u \in H^1(\Omega) \cap L^\infty(\Omega)$. Is it possible to find a sequence $u_n$ such that $u_n \in H^1_0(\Omega) \cap L^\infty(\Omega)$ and $u_n \to u$ in $L^\infty(\Omega)$?
It seems so, since ... | *
*One cannot approximate $u \in H^1(\Omega) \cap L^\infty(\Omega)$ in $H^1$ norm by $H^1_0(\Omega)$ functions unless $u$ is already in $H^1_0(\Omega)$. This is because $H^1_0(\Omega)$ is a closed subspace of $H^1(\Omega)$.
*The function $u\equiv 1$ is in $ H^1(\Omega) \cap L^\infty(\Omega)$ but cannot be approximate... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2091926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove by contradiction that any prime number is either in the from of $ab+1$ or $ab+5$. It says:
Prove by contradiction that any prime number is either of the form of $ab+1$ or $ab+5$.
And this was all. But it seems both belong to $\mathbb{N} \cup \{0\}$. Because otherwise it wouldn't hold for $2$ which is prime.
Wha... | HINT: Consider modulus 6. A number $p $ taken modulus 6 can only be $0, 1, 2, 3, 4$ or $5$. What happens if $p $ is either $0, 2$ or $4$ mod 6? What if it is $3$ mod 6?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2092046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Is recursion a type of differential equations Unfortunately, this question has a very bland answer; however, it occurred to me that recursive functions, like ones to program and calculate factorials for instance, are very similar to differential equations whose results depend on initial conditions. Is recursion essenti... | Discrete Differential equations are called "Difference Equations" and are mostly expressed by $$k_{n+1} - k_{n} = f(n, k) \quad n\in \mathbb{N}$$
These give the difference between two consecutive values.
Recursive Expressions ware mostly stated by $$k_{n+1} = f(n, k)$$ It is always possible to make this into a differe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2092162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to tackle this question related to gcd. Suppose $a,b$ be two positive integers such that $a = {p_{1}}^{r_{1}}{p_{2}}^{r_{2}}...{p_{n}}^{r_{n}}$ and $b = {p_{1}}^{s_{1}}{p_{2}}^{s_{2}}...{p_{n}}^{s_{n}}$, where $p_{i}$ are distinct primes and $r_{i} , s_{i} \in \mathbb N_{0}$ for $i = 1,2,...,n$ where $\mathbb N_{0}... | Let $x$ be a common divisor.
Let $q$ be prime and $q|x$ then $q|a$ and $q|b$. The only prime factors of $a$ and $b$ are $p_i$. So $q = p_i$ for some $i$.
Let $k$ be the highest power of $q=p_i$ that divides $x$. That is $q^k|x$ but $q^{k+1} \not \mid x$. Then $q^k|a$ and $q^k|b$. If $k > r_i$ then $p_i^k\not \mid ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2092275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving that a series diverges Let $(a_n)$ positive sequence where $$\dfrac{n-1}{n} \leq \dfrac{a_{n+1}}{a_n} \leq \dfrac {n}{n+1}$$
Show that $\sum_{n=1}^{\infty} a_n$ diverges.
I already found out that
$\sum_{n=1}^{\infty} (-1)^na_n$ is convergent and that $\lim a_n =0$.
I'm not sure how to approach this one.
| For all $n \ge 1$,
$$a_{n+1} \ge \frac{n-1}{n}a_n \ge \frac{(n-1)(n-2)}{n(n-1)}a_{n-1}\ge \cdots \ge \frac{(n-1)(n-2)\cdots 2\cdot 1}{n(n-1)\cdots 3\cdot 2}a_2 = \frac{1}{n}a_2$$
Since $\sum \frac{1}{n}a_2$ diverges, by comparison the series $\sum a_n$ diverges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2092340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Is finding the minmax easier than finding the Nash Equilibrium? In one of the asnwers in game theory, I came across the answer that finding a Nash Equilibrium in a bimatrix game has a complexity level of PPAD-complete.
However I couldn't find whether finding the minimax(or maximin for that matter) for a game is a lowe... | The problem of finding maxmin or minmax strategies is (presumably) easier than PPAD. They can be found, and thus zero-sum games can be solved, by Linear Programming, which can be done (provably) in polynomial time, by the Ellipsoid method, or more practically by Interior Point methods (or non-provably but practically e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2092459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Compute ${11 \choose 1} + {11 \choose 3} + ... + {11 \choose 11}$? I did this with brute force, and got 1024.
What is the faster method of solving this?
| $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2092649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 5
} |
Calculate $\int_{0}^{\infty }e^{-x^{2}-x^{-2}}dx$ How to calculate
$$\int_{0}^{\infty }e^{-x^{2}-x^{-2}}dx$$
I have no idea where to start.Is it connect with the euler-poisson integral?
| In fact,let see the more general form
$$I=\int_{0}^{\infty }e^{-\alpha ^{2}\left ( x^{2}+x^{-2} \right )}\,\mathrm{d}x$$
let $x\rightarrow x^{-1}$, we have
$$I=\int_{0}^{\infty }e^{-\alpha ^{2}\left ( x^{2}+x^{-2} \right )}\,\mathrm{d}x=\int_{0}^{\infty }x^{-2}e^{-\alpha ^{2}\left ( x^{2}+x^{-2} \right )}\,\mathrm{d}x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2092754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Find probability that the sum of two dice is not 6 and not 5? When rolling two fair dice, what is P(sum of two dice is not 6 and not 5)?
Calculation:
First, I found the probability of two numbers that would roll a sum of 6:
(1,5) (2,4) (3,3) (4,2) (5,1) = 5/36 (each probability is 1/36)
Then, I found the probabilit... | The probability that the sum of two dice is 6 and 5 is 9/36.
So the probability that the sum of two dice is not 6 and 5 is 1- 9/36=27/36.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2092879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Which number is bigger than the others? Which number is bigger?
$2^{431},3^{421},4^{321},21^{43},31^{42}$
My attempt:
$4^{321}=2^{642}>2^{431},4^{321}=2^{642}>2^{640}=32^{128}>31^{42}$
$3^{421}>3^{420}=27^{140}>21^43$
But I don't know how to compare $4^{321}$ and $3^{421}$ Any hints?
| $4^{321}=2^{642}=2^{11*58+4}=16*2048^{58}<27*2187^{58}=3^3*3^{7*58}=3^{409}<3^{421}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2092994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Solving irrational inequalities: $\sqrt{x^2-7x+10}+9\log_4{x/8}\geq 2x+\sqrt{14x-20-2x^2}-13$
Solving the following inequalities: $$\sqrt{x^2-7x+10}+9\log_4{x/8}\geq 2x+\sqrt{14x-20-2x^2}-13$$
I have been solving some questions on inequalities lately and I have come across this question, but I can't figure out a ... | If $x \in \mathbb R $, then we have that the expressions $\sqrt {x^2-7x-10} \text { and } \sqrt {20+14x-2x^2}$ should be positive, that is, $\geq 0$. But one expression is $\sqrt {2}i $ times the other implying that both expressions are zero, that is, $x=2$ and $x=5$ can satisfy this inequality.
But however $x=5$ doe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2093159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Polynomial factorization into irreducibles over $\mathbb{Q}[x]$ I need to find irreducible factors of $f(x)=x^4+3x^3+2x^2+1$ in $\mathbb{Q}[x]$ and explicitely prove that these factors are indeed irreducible.
I believe we can't reduce $f(x)$ any further but I have to prove that this is the case.
I have already shown th... | $f$ has degree $4$ and assumes prime values for these $9 > 2 \cdot 4$ points and so must be irreducible:
$$
\begin{array}{rl}
n & f(n) \\
-8 & 2689 \\
-7 & 1471 \\
-4 & 97 \\
-3 & 19 \\
1 & 7 \\
3 & 181 \\
5 & 1051 \\
6 & 2017 \\
7 & 3529 \\
\end{array}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2093272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
When is $\int_{0}^1|x^{a-b-1}\cos(x^{-b})|\ dx<\infty$?
Let $a,b>0$. Consider the function
$$
f(x)=x^{a-b-1}\cos(x^{-b}),\quad x\in(0,1].
$$
When is $f$ absolutely integrable on $[0,1]$?
If $a>b>0$, it is not hard to show that $f$ is absolutely integrable on $[0,1]$. For the case $0<a\leq b$, is it true that
$$
... | $\left|\cos y\right|$ is a continuous, bounded, $\pi$-periodic, non-negative function with mean value $\frac{2}{\pi}$.
It follows that
$$ \int_{1}^{M} y^{-a/b}\left|\cos y\right|\,dy \approx \frac{2}{\pi}\int_{1}^{M} y^{-a/b}\,dy \tag{1} $$
and the RHS of $(1)$ converges as $M\to +\infty$ iff $\frac{a}{b}>1$, i.e. $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2093423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Where is the hypothesis that $X$ and $Y$ are path-connected being used?
The above is from Hatcher's book.
I am curious exactly where is the hypothesis that $X$ and $Y$ are path-connected being used in the proof? I don't see it being invoked explicitly, nor can I see where it is implicitly used.
Also, a second question... | There is confusion in the literature between a space and a space with base point. Grothendieck wrote to me in 1983:"... the choice of a base point, and the $0$-connectedness assumption, however innocuous they
may seem at first sight, seem to me of a very essential nature. To make an analogy, it would
be just impossible... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2093508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How can I solve this? $$ \int_{-\infty}^{\infty} \frac{x^2e^{-\alpha x^2}}{x^2+b^2} dx $$
| A 'sort-of' elementary way to proceed [without complex analysis] is as follows. Write
$$
\frac{1}{x^2+b^2}=\int_0^\infty ds\ e^{-s(x^2+b^2)}
$$
and exchange the order of integrals to get
$$
I= \int_0^\infty ds\ e^{-s b^2}\int_{-\infty}^\infty dx\ x^2 e^{-(\alpha+s)x^2}
$$
$$
=\partial_\alpha\int_0^\infty ds\ e^{-s b^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2093579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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