Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Alternating sum of combinatorics I have tried to do this exercise using index degradation, using Newton's binomial, if you can give me some idea to solve this problem I will appreciate
Let $a,b\in\mathbb{Z}^+$ such that
$${1999 \choose 0}-{1999 \choose 2}+{1999 \choose 4}-\dots-{1999 \choose 1998}=a^b$$
the question is... | Visualization
I will first explain how to visualize the back propagation of Pascal's identity, just in case you haven't seen this method before. Let's look at these identities graphically on Pascal's triangle:
Pascal's identity:
$$\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$$
Whenever we see a number in Pascal's ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3718866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to find the distance along a sphere from an angle? Imagine there are two points on planet Earth and a light is shone from one to the other by reflecting off an object 500km up (think of this as a mirror oriented parallel to the surface right below it). Let us assume the Earth is a perfect sphere. As the distance b... | Analytical geometry calculation. You want intersection between ( meridians of ) sphere and cone of semi-vertical angle $\theta$ from a celestial mirror at $O$.
$$ z^2+r^2=R^2 \tag1$$
and a ray ( generator of cone ) from the mirror
$$ r \cot \theta-z = R+h\tag2$$
Eliminate $z$ between (1),(2)
$$ r^2( 1+\cot^2 \theta) -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3719032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
Discrete-time Input-to-State Stability From a famous control paper titled 'Input-to-state stability for discrete-time nonlinear systems', the following holds true: given a discrete-time system
$$x(k+1)=f(x(k),u(k))$$
let $V(x)=|x|^2$ where $|\cdot|$ indicates L2 norm, given that
$$V(x(k+1))-V(x(k))\leq-aV(x(k))+b|u|^2,... | OK, after some further reading it seems that there is a general form of the solution:
$$
|x(k)|\leq\left(1-\left(1-\rho\right)a\right)^{\frac{k}{2}}|x(0)|+\left(\frac{b}{a\rho}\right)^\frac{1}{2}|u|_\infty,
$$
given that $0<a<1$ and for an arbitrary $\rho$ such that $0<\rho<1$.
Let me know if this is not correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3719154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Transforming a summation over multiple variables with a condition Given a general sum with multiple variables from some finite set, I can transform it to be a nested sum or a product of distinct sums over one variable each:
$$ S = \sum_{\substack{a,b,c, d}} f(a)f(b)f(c)f(d) = \left(\sum_a f(a)\right)\left(\sum_b f(b)\r... | From
$$
a+b = c +d,
$$
we obtain
$$
d = a+b-c,
$$
and then
$$
\sum_a \sum_b \sum_c \sum_d f(a) f(b) f(c) f(d) = \sum_a \sum_b \sum_c \sum_{a+b-c} f(a) f(b) f(c) f(a+b-c).
$$
In the right-hand side, the first three summation symbols can of course be interchanged freely without affacting the answer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3719298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Evaluating $\lim_{n \to \infty} \frac{(2n)!}{2^n (n!)^2}$
Can you please explain how should I evaluate this limit
$$\lim_{n \to \infty} \frac{(2n)!}{2^n (n!)^2}$$
I know the solution is $\geq1$ but I don't know how I can just simplify like this but I stuck here $$\frac{2n(2n-1)(2n-2)...1}{2^n(n(n-1)(n-2)...1)}$$ I do... | Let $a_n=\frac{(2n)!}{2^n (n!)^2}$ and consider the limit of the ratio
$$\frac{a_{n+1}}{a_n}=\frac{(2n+2)!}{2^{n+1} ((n+1)!)^2}\cdot \frac{2^n (n!)^2}{(2n)!}=\frac{(2n+2)(2n+1)}{2 (n+1)^2}\to \ ?$$
Do you know any theorem about limit of sequences which involves such ratio?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3719439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Let $m\in\mathbb N$ be such that $m>1$ and $n$ be another natural number such that $n\mid(m^2+1).$ If $n>m,$ then how to prove that $n>m+\sqrt m.$ (Basically I have no clue how to approach this problem but still tried to write few lines based on my little knowledge..)
We have: $n>m>1.$ Let $n=m+x$ with $x\le\sqrt m.$ T... | Yes, just go a further step with your proof: Since $n \mid (x^2+1)$, it must be that $x^2 + 1 \geq n \gt m$, therefore $x^2 \geq m$, so $x \geq \sqrt{m}$.
But if $x = \sqrt{m}$, then from the above we know $m+1 \geq n \gt m$, therefore $n = m+1$ and $n = m+\sqrt{m}$, so $\sqrt{m}=1$, $m=1$, contradicting the assumption... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3719573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Ways to write $n=p^k$ as a product of integers Let's say that $F(n)$ is the number of ways to write $n$ as a product of integers greater than $1$. For example, $F(12)=4$ since $12=2\cdot 2 \cdot 3$, $12=2\cdot 6$, $12=3\cdot 4$ and $12=12$.
Given $n=p^k$ where $p$ is a prime number, what is the value of $F(n)$? I know ... | As already answered, the number is given by the number of partitions.
We add two more facts about it.
1 - By the fundamental theorem for finite abelian groups the number of abelian groups of order $n=p_1^{n_1}\dots p_k^{n_k}$ is the product of the partition numbers of $n_i$.
2- For the partition function $p(n)$ there a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3719661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find number of ordered pairs $(a, b)$ satisfying $a^2+b^2=2^3\cdot3^4\cdot5\cdot17^2$ Find number of ordered pairs $(a, b)$ satisfying $a^2+b^2=2^3\cdot3^4\cdot5\cdot17^2$. By rearranging the terms, I found a pair (918, 306). But I wonder if there is a systematic way to solve for the number of pairs? Any hint will be a... |
Not a 'real' answer, but it was too big for a comment.
I wrote and ran some Mathematica code:
In[1]:=Length[FullSimplify[
Solve[{a^2 + b^2 == 2^3*3^4*5*17^2, b > a}, {a, b}, Integers]]]
Running the code gives:
Out[1]=12
Looking for the solutions, we can see:
In[2]:=FullSimplify[
Solve[{a^2 + b^2 == 2^3*3^4*5*17^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3720150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What are the last two digits in the ordinary decimal representation of $3^{400}$ My question is to find the last two digits of $3^{400}$ and I found that the last digit is $1$ but I'm not sure about the other one. Any help would be appreciated, thank you for your time.
Edit:
I've found that
$3^{\phi(100)} = 1 \pmod{100... | The multiplicative group of integers mod 100 has order of $\phi (100) = 100 \cdot \frac 12 \cdot \frac 45 = 40$
$a^{40} \equiv 1 \pmod {100}\\
(a^{40})^{10} \equiv 1 \pmod {100}$
The last two digits are $01$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3720307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Counter example for a matrix $A^3=A$ but $\mbox{rank}(A) \neq \mbox{tr}(A^2)$ If $A$ is diagonalizable then $\mbox{rank}(A) = \mbox{tr}(A^2)$.
How to find a counter example for a matrix $A^3=A$ but $\mbox{rank}(A) \neq \mbox{tr}(A^2)$
| There is not any counterexample, regardless of the underlying field or whether $A$ is diagonalisable. When $A^3=A$, $\operatorname{rank}(A)$ must be equal to $\operatorname{tr}(A^2)$. (When the field has characteristic $p>0$, this identity should be understood as one over $\mathbb F$. That is, we actually mean $\varphi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3720581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Behaviour of sine sums I was considering the following function
$$f_a(x)=\sum_{n=0}^{\lfloor x\rfloor}\sin^2(an)$$
and, as expected, $f_a(x)\approx x/2$ for every $a$ (except $2\pi$ and similar). This is because the function $\sin^2(ax)$ is "on average" equal to $1/2$. That is: we are adding $x$ numbers all close to $1... | The function has a closed form, found with a sum formula similar to the sine-sum formula you cite $$\sum_{n=0}^k\cos(k \varphi) = \frac{\sin(k\varphi/2)\sin((k+1)\varphi/2)}{\sin(\varphi/2)}$$
thus
$$
f_a(x) = \sum_{n=0}^{\lfloor x \rfloor}\sin^2(2an) = \sum_{n=0}^{\lfloor x \rfloor}\frac12 - \sum_{n=0}^{\lfloor x \rfl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3720695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Evaluation of $S_{k,j}=\sum_{n_1,\ldots,n_k=1}^\infty\frac{n_1\cdots n_j}{(n_1+\cdots+n_k)!}$ for $0\leqslant j\leqslant k>0$ For a positive integer $k$, and an integer $j$ with $0\leqslant j\leqslant k$, the problem of evaluating $$S_{k,j}=\sum_{n_1,\ldots,n_k=1}^\infty\frac{n_1\cdots n_j}{(n_1+\cdots+n_k)!}$$ appears... | I'm using the Cauchy integral formula ($n$ is a nonnegative integer):
$$\frac{1}{n!}=\frac{1}{2\pi\mathrm{i}}\oint_C\frac{e^z\,dz}{z^{n+1}},$$
where $C$ is, say, the circle $|z|=r$, with $r>1$ to ensure the convergence:
\begin{align*}
S_{k,j}&=\frac{1}{2\pi\mathrm{i}}\sum_{n_1,\ldots,n_k=1}^{\infty}n_1\cdots n_j\oint_C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3721015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Special types of prime numbers We know that there is infinitely many of prime by the proof of Euclidean.and we know also that there are infinitely many of primes in arithmetic progression by the Dirichlet theorem
My question is, are there some special types of prime number that it has proven that there is infinitely m... | *
*Famous Dirichlet theorem: Let $\ a\ $ and $\ b\ $ be relatively prime natural numbers. Then, there are infinitely many primes $\ p\equiv a\mod b.$
*Fascinating Friedlander–Iwaniec theorem: there are infinitely many primes $\ p=a^2+b^4\ $ where $\ a\ $ and $\ b\ $ are natural numbers.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3721135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Sum of series $\sum_{n=1}^{\infty}\frac{1}{n(n+\frac{1}{2})}$ $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n(n+\frac{1}{2})}$ i am trying to solve an integral and this ended up being last part I need, Wolfram alpha gives $4-2\ln{4}$ but I don't know how they actually got that value. can anyone help or give a starting poin... | Hint: $a_n = \dfrac{4}{2n} - \dfrac{4}{2n + 1}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3721381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Showing $\frac{d\theta }{ d \tan \theta}=\frac{ 1}{ 1+ \tan^2 \theta}$ I suppose that
$$
\frac{d\theta }{ d \tan \theta}=\frac{d \arctan x }{ d x}= \frac{1}{1+x^2}=\frac{ 1}{ 1+ \tan^2 \theta}
$$
So is
$$
\frac{d\theta }{ d \tan \theta}=\frac{ 1}{ 1+ \tan^2 \theta}
$$
correct? And
$$
\frac{d (\theta) }{ d \tan \fr... | Follows is the way I look at the derivation of
$\dfrac{d\theta}{d\tan \theta} = \dfrac{1}{1+ \tan^2 \theta} \tag 0$
and related identities; starting with
$\dfrac{d\theta}{d\tan \theta} = \dfrac{1}{\dfrac{d\tan \theta}{d\theta}} \tag 1$
we use the definition $\tan \theta = \sin \theta / \cos \theta$ and the quotient rul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3721507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Transform expression for Taylor series $f^{(k)}(x) = \frac{(-1)^{k+1}}{2^k}\cdot\prod^{k-1}_{j=1}(2j-1)\cdot\frac{1}{\sqrt{x^{2k-1}}}$
For $x_0=1$ calculate Taylor series.
So the above expression has already been proven by induction.
$T_{f,x_0}= \sum^\infty_{k=0}\frac{f^{(k)}(x_0)}{k!}\cdot(x-x_0)^k$
$\begin{align}
T_{... | I have the feeling that there is something in particular when suddenly $k!$ disappears.
Starting from the beginning
$$f^{(k)}(x) = \frac{(-1)^{k+1}}{2^k}\prod^{k-1}_{j=1}(2j-1)\frac{1}{\sqrt{x^{2k-1}}}=\frac{(-1)^{k+1}}{2^k\sqrt{x^{2k-1}}}\prod^{k-1}_{j=1}(2j-1)$$
Now
$$\prod^{k-1}_{j=1}(2j-1)=\frac{2^{k-1} }{\sqrt{\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3721675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
integral calculation mistake I try to solve this:
$
\int_{0}^{\pi /2}\sin x \cos x\sqrt{1+\cos^{2}x } dx
$
This is what I do:
$ \cos x = t; -\sin x dx = dt; -\sqrt{1-u^{2}} dt $
$ \frac{-\sqrt{1-t^{2}}*t*\sqrt{1+t^{2}}}{-\sqrt{1-t^{2}}} dt $
$-\int_{0}^{1} t * \sqrt{1+t^{2}} dt$
$1+t^2 = a; 2tdt = da; tdt = da/2$
$-\i... | Your second substitution is: $a=1+t^2$. If $t \in [0,1]$, then $a \in [1,2]$ and not $a \in [0,2].$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3721874",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Finding Mass of Object Given Density I need to find the mass of an object that lies above the disk $x^2 +y^2 \le 1$ in the $x$-$y$ plane and below the sphere $x^2 + y^2 + z^2 = 4$, if its density is $\rho(x, y, z)=2z$.
I know that the mass will be $\iiint_R 2z$ $dV$, and I just need to determine the region $R$ which bo... | Try to do it using cylindrical coordinates instead.
You can find the limits of $z$ by solving the sphere and a cylinder (extended into z - axis from disc) of radius $r$.
$$(x^2+y^2) + z^2 = 4$$
$$r+z^2 = 4$$
$$z = \sqrt{4-r^2}$$
$$\iiint_V \rho(x,y,z)\,dx\,dy\,dz = \int_0^1\int_0^{2\pi} \int_0^{\sqrt{4-r^2}} 2z\,dz\,(r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3722056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove there exists $\alpha \ge 0$ s.t $\int_0^\alpha f(x)dx =\int_0^\infty g(x)dx$ given that $f,g\ge 0$, $F(x)$ diverges and $G(x)$ converges This is one of the problems we got as an assignment:
if $f(x),g(x)$ are two integrable functions on $[0,t]$ for any $0<t\in \Bbb{R}$.
and suppose that:
*
*$f(x)\ge 0,\ g(x)\ge... | If $\int_0^{\infty}g(x)dx=0$,then clearly $\alpha=0$
If not , then let $\int_0^{\infty}g(x)dx=L\gt 0$ (since $g\ge 0$)
Now consider $F(x)=\int_0^{x}f(t)dt$ .
Then $F(0)=0$ and $F(x)$ is continous and $\displaystyle\lim_{x\to \infty}F(x)=\infty$ (Why? Because $F(x)$ is increasing and thus the above limit diverges to i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3722149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How can I say a set has measure $1$? Suppose $(\Omega,\mathscr{E},\mathbb{P})$ is a measure space such that $\mathbb{P}(\Omega)=1$.
Suppose $A_i \in \mathscr{E}$ for $1 \leq i \leq n$ where $n \in \mathbb{N}$.
Suppose $\mathbb{P}(\bigcup_{1 \leq i \leq n} A_i)=1$.
Can I conclude that there exists $1 \leq i \leq n$ such... | No. There seems to be no inclusion hypothesis in your statement.
Consider $\Omega=[0,1]$ with the Borel $\sigma$-algebra generated by the open sets and the the usual probability $dx$. Then you can have
$$A_j = \left(\frac{j-1}{n},\frac{j}{n}\right)\,.$$
Clearly the union has probability one but they are all disjoint an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3722279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Finite-index normal subgroup acts amenably on compact Hausdorff space $X$ Let $\Gamma$ be a discrete group acting on a compact Hausdorff space $X$. The action is called (topologically) amenable if there exists a net of continuous maps $m_i: X\rightarrow \text{Prob}(\Gamma)$ such that for each $s \in \Gamma$, $$\lim_{i\... | This is a corollary of the following fact, found as Proposition 5.1.11 in Brown & Ozawa's "$C^*$-Algebras and Finite-Dimensional Approximations".
Let $\Gamma$ be a discrete group, $N$ be a normal subgroup and $\overline \Gamma=\Gamma/N$. If $Y$ is a compact amenable $\overline \Gamma$-space and $X$ is a $\Gamma$-space... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3722411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Show that if $X$ and $Y$ are independent with the same exponential distribution, then $Z= |X - Y|$ has the same exponential distribution $$P\left(Z\le z\right)=P\left(\left|X-Y\right|\le z\right)=P\left(-z\le X-Y\le z\right)=P\left(Y-z\le X\le Y+z\right)$$
This means that, because $\space f(x,y)=f(x)f(y) \space$ as the... | The main error arises from not considering when the interval $[y-z, y+z]$ is not a subset of $[0,\infty)$. That is to say, when $y < z$, then $y-z < 0$. So you need to take this into account when integrating over $x$. The second error is in writing the integrand as the square of the marginal density $f_X(x)$, when t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3722528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Using Uniqueness Result for Analytic Functions
I am reviewing for an Analysis qual and stumbled upon this question. In particular, I am having difficulties with part (ii). My attempt is the following:
Using the hint, let $\Omega = \mathbb{C}$, $S=\{1/n : n\in \mathbb{N}\}$, and $g(z)=z^2$. We have that since $S \subse... | First, if $f$ is any entire function, $\overline{f(\bar{z})}$ is always entire, because $\overline{\sum_{n=0}^\infty a_n \bar{z}^n} = \sum_{n=0}^\infty \bar{a_n}z^n,$ which converges exactly when $\sum_{n=0}^\infty a_n z^n$ does.
As $f$ is holomorphic at $0$ and not uniformly $0$, there is a unique integer $n \geq 0$ s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3722756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Prove that if $n\times n$ matrix $A$ is nonsingular, then it is invertible. I know that nonsingularity means having unique soln. $x$ for every RHS b in the system $Ax= b$. But how do I use this definition to come up with a proof for the problem ? I really have tried but to no avail.
| Recall that
$$\mathbf{M}^{-1}=\frac{1}{\det(\mathbf{M})}\operatorname{adj}(\mathbf{M})$$
Simply show that $\operatorname{adj}(\mathbf{M})$ always exists (this is not difficult), and thus $\mathbf{M}^{-1}$ always exists if $\det(\mathbf{M}) \neq 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3722868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
If $f\in L^1(\Bbb R)$ is continuous then $f\in C_0(\Bbb R)$ Suppose $f \in L^1(\Bbb R)$ is continuous. Then is it necessarily true that $f\in C_0(\Bbb R)$? It is easily seen to be true if $f$ is uniformly continuous, but I can't see whether it is true if $f$ is merely continuous. Thanks in advance.
| You can even have continuous unbounded real maps that are Lebesgue integrable. See here for an example.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3723018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Complex polarization identity proof getting stuck towards the end w.r.t. the imaginary part I'm working on a homework problem regarding the proof for the polarization identity for complex scalars. I've taken a look at another question on this community (Polarization Identity for Complex Scalars) and have tried working ... | Hint: For $z \in \mathbb{C},$ $$z-\overline{z}=2i\,\Im{z},$$ where $\Im z$ denotes the imaginary part of $z.$
Btw, once you have shown that $$ \Vert x + y \Vert^2 - \Vert x - y \Vert^2=4\Re \langle x,y\rangle,$$ then replacing $y$ by $iy:$
$$ \Vert x + iy \Vert^2 - \Vert x - iy \Vert^2=4\Re\langle x,iy\rangle=4\Im \lan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3723388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is it possible to define the notion of a Submanifold of Euclidean space through properties of its tangent cone? The tangent cone to a set $\mathcal S \subset\Bbb R^n$ at a point $x \in \Bbb R^n$ is the set of all vectors $w \in \Bbb R^n$ for which there exists sequences $x_i \in \mathcal S$ and $\tau_i> 0$, with $x_i\t... | I don't think so The subsets of $\mathbb{R}^n$ with that property seems to include much, much much more than submanifolds.
As a counterexample in $\mathbb{R}^2$, consider a the union of a circle and one of its tangent lines, which intersect at a point $p$.
Since these constructions are local, there is no issue at any p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3723491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating matrix equation A $2x2$ matrix $M$ satisfies the conditions $$M\begin{bmatrix}
-8 \\
1
\end{bmatrix} = \begin{bmatrix}
3 \\
8
\end{bmatrix}$$
and
$$M\begin{bmatrix}
1 \\
5
\end{bmatrix} = \begin{bmatrix}
-8 \\
7
\end{bmatrix}... | We know that
$$M\begin{pmatrix}
-8 & 1 \\
1 & 5
\end{pmatrix} = \begin{pmatrix}
3 &-8\\
8 & 7
\end{pmatrix}.$$
Multiply that equation from right by
$$
\begin{pmatrix}
-8 & 1 \\
1 & 5
\end{pmatrix}^{-1}=\frac{1}{-41}\begin{pmatrix}
5 & -1 \\
-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3723590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
What are the definable subsets of $(\mathbb{R}, +)$? Consider the structure $(\mathbb{R},+)$. What are the subsets of that structure that are definable without parameters? I conjecture there are only four, namely $\emptyset$, $\mathbb{R}$, $\{0\}$, and $\mathbb{R} - \{0\}$. Is this correct?
| Turning the comment thread above into an answer:
Most of the time it's easier to analyze the orbit relation of a structure than its definable sets per se. This is the relation $a\sim b$ iff there is an automorphism sending $a$ to $b$. Since automorphisms preserve $\models$, we know that every definable set is closed wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3723727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
"infinitely oscillating" manifold Find out whether $M:=\{\left(x\cdot\cos\left(\frac{1}{x}\right),\,x\cdot\sin\left(\frac{1}{x}\right)\right)|\,x>0\}$ is a submanifold of $\mathbb{R}^2$ or not.
My guess is that it's not since I've tried to construct a homeomorphism from $\mathbb{R}$ to $M$ $\left(x\mapsto \left(x\cdot\... | It is a submanifold.
Consider the map $f : (0,\infty) \to \mathbb R^2, f(x) = (x\cos(\frac{1}{x}),x\sin(\frac{1}{x}))$. We have $M = f((0,\infty))$. The map $f$ is injective since $\lVert f(x) \rVert = \sqrt{x^2\cos^2(\frac{1}{x}) + x^2\sin^2(\frac{1}{x})} = \sqrt{x^2} = x$ which implies $f(x_1) \ne f(x_2)$ for $x_1 \n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3723891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding volume of the region enclosed by $x^2+y^2-6x=0, z = \sqrt{36-x^2-y^2}$ and $z=0$ I'm trying to calculate the volume of the region enclosed by the cylinder $x^2+y^2-6x=0$, the semicircle $z = \sqrt{36-x^2-y^2}$ and the plane $z=0$. I tried moving the region towards -x so that the cylinder has it's center at zero... | If you take cylindrical coordinates $x = r \cos \theta, y=r \sin \theta, z=z$ with Jacobian $J=r$, then you'll have
$$\int_{0}^{2 \pi}\int_{0}^{6 \cos \theta}\int_{0}^{\sqrt{36-r^2}}rd\theta drdz$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3724012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
AHSME Challenge Math Question Cowboy A cowboy is $4$ miles south of a stream which flows due east. He is also $8$ miles west and $7$ miles north of his cabin. He wishes to water his horse at the stream and return home. The shortest distance (in miles) he can travel and accomplish this is:
a. $4 + \sqrt{185}$
b. $16 $
... | Your answer proposes the following trajectory for the cowboy:
But that's not optimal: it's better for the cowboy to be moving east the whole time, not just after getting water.
To see the optimal path, imagine the cowboy's evil twin, who is 8 miles further north of the cowboy: his position is the reflection of the cow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3724171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Calculating gradient, how to do this?
Given $f(x,y)$, a function that has continuous partial derivatives in
every point.
such that $\nabla f(0,-18)=-2i+3j$
We define a new function $g(x,y)=f(xy+x^2,xy-y^2)$ calculate $\nabla
g(3,-3)$
How I tried to solve this?
I need to find:
$$\nabla g(3,-3) = g_x'(3,-3)i+g_y'(3,-3... | We can make use of the multivariable chain rule (see this answer) which states that given a function $h: \mathbb{R}^n \to \mathbb{R}^m$ and a function $f: \mathbb{R}^m \to \mathbb{R}$ we have
$$
\nabla (f \circ h) = \left(h'\right)^T \cdot \left(\nabla f \circ h\right)
$$
where "$\cdot$" represents matrix multiplicatio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3724288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Power series approximation for $\ln((1+x)^{(1+x)}) + \ln((1-x)^{(1-x)})$ to calculate $ \sum_{n=1}^\infty \frac{1}{n(2n+1)} $ Problem
Approximate $f(x) = \ln((1+x)^{(1+x)}) + \ln((1-x)^{(1-x)})$ and then calculate $ \sum_{n=1}^\infty \frac{1}{n(2n+1)} $
My attempt
Let
$$f(x) = \ln((1+x)^{(1+x)}) + \ln((1-x)^{(1-x)}) ... | $$ f(x) = \sum_{n=0}^\infty \frac{2x^{n+2} + (-1)^n x^{n+1} - x^{n+1} }{n+1} $$
Supposing the above is right. We want to change the $n+2$'s to $n+1$'s. To do this, write, by letting $m+1 = n+2$,
$$ \sum_{n=0}^\infty \frac{2x^{n+2}}{n+1} = \sum_{m=1}^\infty \frac{2x^{m+1}}{m} = \sum_{n=1}^\infty \frac{2x^{n+1}}{n},$$
w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3724442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
General form for this problem I encountered this problem by Polya about counting the number of ways that a dollar can be changed. Suppose that there are pennies (worth 1), nickels (worth 5), dimes (worth 10), quarters (worth 25) and fiftycent coins (worth 50). The number of ways to change a dollar (worth 100) can be wr... | If you have a closed formula for your generating function $D(z)$, all you have to do to obtained its coefficients differentiate several times and evaluate at zero.
This works since if
$$ D(z) = a_0 + a_1 z + a_2 z^2 + ...$$
then we have that
$$a_k = \frac{D^{(k)}(0)}{k!},$$
and this is easy enough to compute for finite... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3724627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Show that $ x\cdot\cos(x)+\sin(x)/2=\sum_{n=2}^\infty (-1)^n\cdot\frac{2n}{n^2-1}\cdot\sin(nx)$ when $x\in [-\pi,\pi]$ To show this, I have used definitions for $\cos(x)$ and $\sin(x)$:
$$x\cdot \cos(x)+1/2\sin(x)=x\cdot \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}\cdot x^{2n}+1/2\cdot \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!... | Hint:
$$\dfrac{2n\sin nx}{(n+1)(n-1)}=\dfrac{\sin nx}{n-1}-\dfrac{\sin nx}{n+1}$$
Now $(-1)^n\dfrac{\sin nx}{n-1}$ is the imaginary part of $$\dfrac{(-1)^ne^{inx}}{n-1}=e^{ix}\cdot-\dfrac{(-e^{ix})^{n-1}}{n-1}$$
Now $$\sum_{n=2}^\infty-\dfrac{(-e^{ix})^{n-1}}{n-1}=\ln(1+e^{ix})=\ln (e^{ix/2})+\ln\left(2\cos\dfrac x2\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3724730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
System of differential equations depending on parameter This is the first time to get a question like this:
How to solve the system $y'=3by+(1-2b)z$, $z'=by+z+e^{4x}$, where $b\in\mathbb{R}$?
Any help is welcome.
| Hint:
Substracting both differential equations gives us:
$$(y-z)'-2b(y-z)=-e^{4x}$$
This DE can easily be solved. It's a first linear DE.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3724937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Am I misapplying the chain rule when differentiating $x^{5x+7}$ with respect to $x$? The problem I am attempting to solve is:
\begin{align}
y=x^{5x+7} \\
\text{Find $\frac{dy}{dx}$}
\end{align}
Here is my working so far:
$$\begin{align}
\text{let }u &= 5x+7 \\
\frac{dy}{dx}&=\frac{dy}{du} \cdot \frac{du}{dx} \\
\frac{d... | If $u= 5x+7$ and $y = x^{5x+7} = x^u$ then if you attempt to solve $\frac {dy}{dx}=\frac{dy}{du}\frac {du}{dx}$ then you must solve $\frac {dy}{du} = \frac {dx^u}{du}$ but $x$ is dependent on $u$ so you'd have to solve $\frac {dx^u}{du} = \frac {dx^u}{dx}\frac {dx}{du}$ but to solve $\frac {dx}{du}$ we have $x$ is depe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3725023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Geometric approach to $\lim_{n\to\infty}\left(\frac{x_{n+1}}{x_n}\right)^n$ where $x_1=1$ and $x_{n+1}=\sqrt{1+x^2_n}$? I was solving a question :
Let $x_1=1$ and $x_{n+1} = \sqrt{1+x^2_n } \ \ \forall \ \ n\in \mathbb{N}$
Then evaluate $$\lim_{n \to \infty} \left( \frac{x_{n+1}}{x_n} \right)^n$$
The way I did it wa... | You can define as $\alpha_n$ the angle of the Pythagorean triplet $(1, x_n, x_{n+1})$ between the sides of length $x_n$ and $x_{n+1}$. Its value comes from
$$
\tan\alpha_n = \frac{\sin\alpha_n}{\cos\alpha_n} = \frac{1/x_{n+1}}{x_n/x_{n+1}} \implies \alpha_n = \cot^{-1} x_n.
$$
From your drawing, one can see (as you sug... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3725296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 1,
"answer_id": 0
} |
Proof of Lagrange's Mean Value Theorem? The Mean value theorem for Mapping says:
Let $f(x,y)$ be differentiable in $D$. (D is open and connected).
For every $p=(x_1,y_1), q=(x_2,y_2)$ there exists a point $s \in [p,q]$ such that:
$$f(q)-f(p) = \nabla f(s)*(q-p)= f_x(s)(x_2-x_1)+f_y(s)(y_2-y_1)$$
Note: the interval $[... | Assuming$[p,q]\subseteq D$ (see @LurchedSawyer 's answer):
Let $g(t)=f((1-t)p+tq)$.
Then $$g'(t)=\nabla f((1-t)p+tq)*(q-p)$$ by the chain rule. Using the usual 1 dimensional mean value theorem you may find a value $u\in [0,1]$ such that $$g'(u)=g(1)-g(0)=f(q)-f(p)$$
Setting $$s=(1-u)p+uq$$ gives:$$f(q)-f(p)=g'(u)=\nab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3725463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Exterior derivative of a vector field The exterior derivative of a scalar function is
$d f(x,y,z) = (
\frac{\partial f}{\partial x} dx
+ \frac{\partial f}{\partial y} dy
+ \frac{\partial f}{\partial z} dz
)$
Am I correct in assuming then that
$d\left( F_x(x,y,z) e_x + F_y(x,y,z) e_y + F_z(x,y,z) e_z \right)$
would be... | $$d(F_xdx + F_ydy + F_zdz) = dF_x \wedge dx + dF_y \wedge dy + dF_z \wedge dz$$ and use that $$dF = \frac{\partial F}{\partial x}dx + \frac{\partial F}{\partial y}dy + \frac{\partial F}{\partial z}dz$$ according to your first claim.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3725638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that $ \sum_{i=1}^{N} a_i \leq \sqrt{N \sum_{i=1}^{N}a_i^2} $ Prove that $ \sum_{i=1}^{N} a_i \leq \sqrt{N \sum_{i=1}^{N}a_i^2} $. Well i choose $u=(1,\ldots,1)$ and $v=(a_1,\ldots,a_N)$ whit $a_i$ positive and. Apply $u \circ v \leq |u||v|$. With this now i want to prove that $\sum_{i,j}^{N}\frac{\partial u}{\p... | Recall the Cauchy-Schwarz inequality for sums:
$$\left( \sum_1^N a_n b_n \right)^2 \le \left( \sum_1^N a_n^2 \right) \left( \sum_1^N b_n^2 \right)$$
Let $b_n = 1$. Then this simplification results:
$$\left( \sum_1^N a_n \right)^2 \le \left( \sum_1^N a_n^2 \right) N$$
Taking the square root of each side gives the desire... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3725762",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Calculate the residue of $\exp\left(\frac{z+1}{z-1}\right)$ in every point of $\mathbb{C}$ I have to calculate the residue of $\exp\left(\frac{z+1}{z-1}\right)$ in every point of $\mathbb{C}$.
So I tried to compute the Laurent Series expansion $\forall z_0 \in \mathbb{C}$.
For $z_0 = 0$ we obtain that $f(z)=\sum_{k \g... | The only singularity of $f$ is at $z=1$, but that's an essential singularity. I get
$$f(z)=\sum_{n=0}^\infty\frac{(z+1)^n}{n!(z-1)^n}$$
but that's not a Laurent series as it stands.
But also
$$f(z)=\exp\left(1+\frac{2}{z-1}\right)=e\exp\left(\frac{2}{z-1}\right)
=e\sum_{n=0}^\infty\frac{2^n}{n!(z-1)^n}.$$
Now that is a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3726093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is there a natural way to totally order the set of unlabeled binary trees on $n$ nodes? Let $C_n$ be the $n^{th}$ Catalan number. There are $C_n$ unlabeled binary trees having $n$ internal nodes. I want to totally order these trees in some (hopefully not too complicated) natural manner. Perhaps there is some "standar... | For $k = 0$ to $n - 1$ (or vice versa) enumerate all the possible left subtrees with $k$ nodes. For each possible left subtree with $k$ nodes, enumerate all the possible right subtrees with $n - k - 1$ nodes. Recursively, this gives for every positive integer $n$ a fairly natural way to totally order the set of all unl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3726215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to prove that this function all over the positive integers gives us this sequence? On the first hand, I have this sequence : $0,1,1,2,2,2,3,3,3,3,...$ which is the sequence where an $n$ positive integer appears $n+1$ times consecutively.
On the other hand, I have this function : $a_n=\lfloor\frac{\sqrt {1+8n}-1}{2}... | Idea:
In the sequence, $a_n$ becomes $m$ when $n=\sum\limits_{i=0}^m i=\dfrac{m(m+1)}2$; i.e., $m^2+m-2n=0$.
Solving this quadratic for $m$, we get $m=\dfrac{-1+\sqrt{1+8n}}2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3726350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Maximal tamely ramified abelian extension of $\mathbb{Q}_p$ is finite over the maximal unramified extension $\mathbb{Q}_p^{nr}$? I came across an curious exercise in Neukirch's algebraic number theory book. Exercise 2 page 176 (Chapter II section 9) asks the following
Prove that the maximal tamely ramified abelian exte... | $V$ is a tamely ramified extension of $Q_p$ and it contains $Q_p^{nr}=\bigcup_{p\ \nmid\ m} Q_p(\zeta_m)$ so $$V=\bigcup_j Q_p^{nr}(p^{1/n_j})$$ for some $p\nmid n_j$ and $n_j | n_{j+1}$.
Since any subextension of an abelian extension is abelian
It suffices to find for which $n$ we have $Q_p^{nr}(p^{1/n})/Q_p$ is abel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3726660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Easy proof that $\{(x,y) \in \mathbb{R}^2 : y=\tan(x) \}$ is a closed set. I'd like to know whether there's an "easy" proof that $$A:= \{(x,y) \in \mathbb{R}^2 : y=\tan(x) \} .$$
I've tried to prove that its complement is open, but given an $(x,y)$ such that $\tan(x) \neq y$, it's a bit of a grind to find (in the gener... | The graph of a continuous function $f: X \to Y$ is closed in $X \times Y$, assuming $Y$ is hausdoff (you can google this result or even better, prove it). The $\tan$ function is not continuous however but it is continuous on each interval $(-\frac{\pi}{2}+ n \pi , \frac{\pi}{2}+ n \pi )$.
Hence, you only must check tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3726817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Given $V \in \mathbb{R}^{n\times(n-r)}$, why $V^TAV = 0$ implies $\operatorname{rank}(A) \leq r$? I am doing a problem from convex optimization written by Stephen P Boyd. I am having trouble understanding the solution.
The original problem statement and solution is as follow:
2.13 Conic hull of outer products. Conside... | If ${\cal R} V = \ker(A+B)$ then $(A+B)V x = 0$ for all $x$ hence $(A+B)V=0$.
Hence it follows that $V^T(A+B)V = 0$.
Note that if $A$ is symmetric positive semi definite then using the spectral decomposition we can write $A = C^T C$ for some $C$.
So, if $V^TAV = 0$ then $(CV)^T(CV) = 0$ and so $CV =0$ and so $C^TCV=AV ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3726932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
show this $\tan{x}\tan{y}\tan{z}=1$ in an acute.$\Delta ABC$,if $x,y,z$ such$$\cos{A}=\cos{y}\sin{z},\cos{B}=\cos{z}\sin{x},\cos{C}=\cos{x}\sin{y}$$show that
$$\tan{x}\tan{y}\tan{z}=1$$or $$\sin{x}\sin{y}\sin{z}=\cos{x}\cos{y}\cos{z}$$
I want use $$\cos^2{A}+\cos^2{B}+\cos^2{C}+2\cos{A}\cos{B}\cos{C}=1$$
so we have
$$... | Let
$$
\tan x\tan y\tan z=t
$$
then
$$
\sin x\sin y\sin z=t\cos x\cos y\cos z
$$
so we get
$$
\sum_{cyc}\cos^2x-\sum_{cyc}\cos^2x\cos^2y+2t\cos^2x\cos^2y\cos^2z=1
$$
then
$$
\begin{aligned}
& (2t-1)\cos^2x\cos^2y\cos^2z \\\\[1ex]
=& 1-\sum_{cyc}\cos^2x+\sum_{cyc}\cos^2x\cos^2y-\cos^2x\cos^2y\cos^2z \\\\
=& \left(1-\cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3727103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Lemma 4.1.3 in Category Theory in Context: Adjoint Functors I'm having hard time proving this lemma, so hints would be greatly appreciated.
The naturality of collections of isomorphisms is defined as follows:
I was able to prove the right direction. i.e. showing that if the collection is natural, the left square comm... | I’ll only show naturality in $D$, but I hope this helps. Suppose we have $k: d \to d’$. By assumption, we have some isomorphism of sets $D(Fc, d) \cong C(c, Gd)$. Hence consider a morphism $f^{\#}: Fc \to d$. Then this corresponds to some $f^{\flat}: c \to G(d)$, which we may compose with
$G(k)$ to obtain $G(k) \circ f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3727208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Order of an element divides $m$ when $a^m \equiv 1 \pmod n$ https://brilliant.org/wiki/order-of-an-element/
I was referring to the above link for order of an element and in basic properties while proving property $1$ due to minimality of d, $d \le gcd(m,d)$ is written.
Is it because $mx+dy\ge d$ i.e. $gcd(m,d)\ge d$ bu... | The order is defined to be the smallest positive $p$ such that $a^p \equiv 1 \pmod{n}$ holds.
Since we have shown that $a^{gcd(d.m)} \equiv 1 \pmod{n}$ and we know that $gcd(d,m)>0$, then we must have $d \le gcd(d,m)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3727345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Does $\operatorname E[e^{-X}Y]\le c$ imply $\operatorname E[Y]\le c\operatorname E[e^X]$? Quick question: Let $X,Y$ be real-valued random variables, $Y$ being nonnegative, such that $\operatorname E[e^{-X}Y]\le c$ for some $c\ge0$. Can we somehow show that $\operatorname E[Y]\le c\operatorname E[e^X]$? It's clearly tru... | This is equivalent to asking for the inequality $E[ZW]\le E[Z]E[W]$ to hold for all $Z\ge0$ and $W> 0$ (set $W=e^X$ and $Z=e^{-X}Y$). In other words, is it true that if $Z,W>0$, then $\operatorname{Cov}(Z,W)\le 0$?
This is not the case, see for instance $U$ uniformly distributed on $(0,1]$, $Z=U^3$ and $W=U^5+1$. We ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3727512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Convert sum of $N$ bits, $k$ shift right, to propositional formula. This problem is somehow related to multiplication circuits
Input: $N$ bits, integer $K$
*
*Output: a propositional formula that is satisfiable if and only if the sum of the bits, K shift right mode 2 equals 1
Example:
*
*N = {1,0,1,1}, K = 1 -> ou... | I found an exponential solution for it, I hope that other answers will come out with something more efficient or prove that it is hard to find a better answer.
I will start with the example of N = {1,0,1,1}, K = 1. It's easy to solve it on the paper by summing those numbers and deleting the last digit. $1+0+1+1 = 11$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3727795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is the Jackson integral of $e_q(x)$ , $e_q(x)$ itself? Some info-
*
*Jackson integral (q-analog of standard integration) simply defined as -$$\int f(x){\mathrm d}_qx=(1-q)x\sum_{k=0}^\infty q^kf(q^kx)$$
*q-exponential defined as $$e_q(x)=\sum_{n=0}^\infty \frac{x^n}{[n]_q!} =
\sum_{n=0}^\infty \frac{x^n (1-q)^n}{(q... | Write out $e_q(q^kx)$ explicitly and then interchange the summations:
$$ \begin{array}{ll} \displaystyle \int e_q(x)\,\mathrm{d}_qx & \displaystyle =(1-q)x\sum_{k=0}^\infty q^k e_q(q^kx) \\[5pt] & \displaystyle =(1-q)x\sum_{k=0}^\infty q^k \sum_{n=0}^\infty \frac{(q^kx)^n}{[n]_q!} \\[5pt] & \displaystyle = (1-q)x\sum_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3727934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Rotation Lemma explanation. I'm reading the following rotation lemma on graphs. There's this statement in the proof: "If $y \in V(P)$, rotate $P'$ along the edge $\{v,y\}$", which I don't see how it could happened when applied to the example I've drawn.
Rotations: Suppose $P - v_0v_1...v_t$ is a path in $G$. Suppose $... | If you rotate a path along the last edge, this does nothing. You get the original path back.
In the example you drew, rotating $P'$ along the edge $\{v,y\}$ gives the path $P'$ back again, and it's still true that the endpoint of the resulting path (namely, $v$) is a neighbor of $y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3728266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
X Hausdorff: every point has precompact neighborhood iff X has a basis of precompact open sets Claim Suppose $X$ is Hausdorff, then:
Every point $p\in X$ has a precompact neighborhood in $X$ $\Longleftrightarrow$ $X$ has a basis of precompact open subsets
Question: It's not clear to me why it's necessary for X to be Ha... | $X=(\mathbb{R}\times\mathbb{Z})/\sim$, where $\sim$ is the equivalence relation generated by
$(x,n)\sim (x,m)$ for all $n,m\in \mathbb{Z}$ and all $x\ne 0$.
$X$ is non-Hausdorff. $X$ is locally compact but does not have a basis of precompact open subsets. This comes from "Introduction to Topological Manifolds" by J.M.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3728390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Unable to Solve a quiz question asked in mathematics exam ( Quantitative Aptitude) I am self studying for an exam and I am unable to solve this quiz question.
Adding it's image ->
I tried by finding numbers in the sentences but couldn't find and I think that's a wrong approach.
Can anyone please tell how to solve this... | Numbers are spelled out in each phrase. The first has eleven (Tinselevent), the second nine. Look for the other two. There is no excuse for this being called mathematics.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3728507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Direction vector from 2 orthogonal angles I have an acoustic sensor that measures angle on a single axis. If the sensor is pointing upwards, then I can measure an angle $\theta$ that has the range $-\pi/2 < \theta < \pi/2$. A value of $\theta = 0$ means that an object (sound source) is directly above it. In other words... |
Does this figure help you? The "perspective of sensor" triangles give you the angles you need.
EDIT:
The boxed equations are the ones you want, if i've interpreted your question correctly. (See new image)
You can figure out a direction vector, because my $\beta$ and $\alpha$ are the co-latitude and azimuthal angles in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3728684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Equivalence definition of Lower Integral let $f$ be a real-valued bounded function on $[a,b]$.
For all partition $P:x_0,...x_N$ of $[a,b]$, define $m_k(f,P)=\inf_{x_{i-1}\le x\le x_i}(f(x))$ and $m^*_k(f,P)=\inf_{x_{i-1}< x< x_i}(f(x))$ (which does not include the endpoints) for all $k=1,...,N$.
The lower integral of $... | For a given partition $P = (x_0,x_1, \ldots x_n)$, let $L(f,P) = \sum_{k=1}^nm_k(x_k - x_{k-1})$ denote the usual lower Darboux sum and let $L^*(f,P) = \sum_{k=1}^nm_k^*(x_k - x_{k-1})$ denote the lower sum with infima taken over open subintervals.
You already have shown that $L(f,P) \leqslant L^*(f,P)$ which implies t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3728829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding an irreducible polynomial over the rationals I'm very confused by a homework question.
"Find the irreducible polynomial for $ \sin{2\pi/5}$ over Q.
I found that $16t^{4}-20t^{2}+5=0$ but this is not monic? This is also irreducible by Eisenstein, but minimal polynomials are always monic?
| $p(t)=t^{4}-5/4t^{2}+5/16$ is the minimal polynomial of $\sin 2\pi/5$ over $\mathbb Q$. It is irreducible in $\mathbb Q[x]$ according to Gauss's lemma as $16t^{4}-20t^{2}+5$ is irreducible in $\mathbb Z[x]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3729140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Taylor series for $\ln(\frac{1-z^2}{1+z^3})$ Taylor series for $\ln(\frac{1-z^2}{1+z^3})$
I've tried to $$\ln(1-z^2)-\ln(1+z^3)=\sum (-1)^{3n-1}z^{2n}-\sum(-1)^{4n-1}z^{3n}$$
I didn't manage to make it one series any help is good
| Starting from @J.G.'s answer, all symplify to
$$\log\left(\frac{1-z^2}{1+z^3}\right)=\sum_{n=2}^\infty \frac{2 \cos \left(n\frac{\pi }{3}\right)-1}{n} z^n$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3729239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Computing $\lim_{x\rightarrow 0}{\frac{xe^x- e^x + 1}{x(e^x-1)}}$ without L'Hôpital's rule or Taylor series This limit really stamped me because i'm not allowed to use L'Hôpital's rule or Taylor's series, please help!
I think the limit is $\frac{1}{2}$, but i don't know how to prove it without the L'Hôpital's rule or T... | How about using the Cauchy's mean value theorem (L'Hospital rule can be seen as a specialization of this). Let $f(x)=xe^x-e^x+1$ and $g(x)=xe^x-x$, then $f(0)=0=g(0)$ and by the (generalize) mean value theorem, there is $c_x$ between $0$ and $x$ such that
$$f'(c_x)(g(x)-g(0))=g'(c_x)(f(x)-f(0)).$$ This can be expressed... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3729502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Maximum number of acute triangles in a regular convex polygon triangulated into $n-2$ triangles by its diagonals. I have been reading about convex polygons, and I found the following:
We say that a simple polygon is convex if all its interior angles are less than $\pi$. If $P$, a regular convex polygon, is divided int... | To reword that sentence,
If $P$ is a convex polygon with $n$ sides then you can use several diagonals of $P$ to subdivide $P$ into $n-2$ triangles.
In fact we can be still more precise than that:
If $P$ is a convex polygon with $n$ sides then you can use $n-3$ diagonals of $P$ to subdivide $P$ into $n-2$ triangles.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3729637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
When is $\mathbb Z[\sqrt d]$ a Euclidean or principal ideal domain? Let $d$ be an integer $\neq 1$ such that $d$ is not divisible by the power of any prime.Consider the ring $\mathbb Z[\sqrt d]$.My question is when is this ring a Euclidean domain,and when it is not an ED,when is this ring a PID?Is there any criterion t... | *
*There is no simple rule that classifies when $\mathbf Z[\sqrt{d}]$ is a PID or Euclidean when $d$ is squarefree and positive, and there is no reason to expect a simple rule. (But note, as Gerry Myerson points out, that $\mathbf Z[\sqrt{d}]$ is the "wrong ring" to be thinking about when $d \equiv 1 \bmod 4$ since th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3729808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How is a real number a proper subset of $ℚ$! (dedekind cut)? I just started studying some set theory using "Classic Set Theory: For Guided Independent Study" and i've been stuck at the 8th and 9th page for 2 days lol. It says:
A Dedekind left set is a subset of $r$ of $ℚ$ with the following properties:
*
*$r$ is a pr... | Well, $\sqrt2$ is the set of all rational numbers $x$ which are negative or that $x^2<2$. This set is certainly not empty, it contains all the negative rationals, etc.
But wait, you might say, this is somehow circular. How do you know to define it by $x^2<2$? Well, $\sqrt\cdot$ is not an integral part of our language. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3729946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Inequality of arithmetic mean of two sets If $a,b>0$ and $Q=\{x_1, x_2, x_3,..., x_a\}$ a subset of the natural numbers $1, 2, 3,..., b$ such that, for $x_i+x_j<b+1$ with $1 ≤ i ≤ j ≤ a$, then $x_i+x_j$ is also an element of Q. Prove that:
$ \frac{x_1+x_2+x_3+...+x_a}{a} ≥ \frac{b+1}{2}$ so basically, you have to prov... | Hint: For each $i$, show that $ x_i + x_{a-i} \geq b+1$.
Proof by contradiction. Suppose $x_i + x_{a-i} < b+1$, which satisfies the condition in the problem.
What does it make sense to consider next?
Do some work to reach a contradiction.
Corollary: $ 2\sum x_i \geq a (b+1)$, and the result follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3730083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
On ranges of nuclear operators Consider nuclear (trace class) operators acting on a separable infinite-dimensional Hilbert space. Does there exist a nuclear operator $A$ such that, for any other nuclear operator $B$, $\mathrm{ran}(B) \subset \mathrm{ran}(A)$?
It is known that a nuclear operator cannot have a closed ran... | No. This fails even if you require compact and not trace-class.
Note first that the range of any compact operator has an orthonormal basis. This follows from the polar decomposition, as we can write $T=V|T|$ with $|T|$ positive and $V$ a partial isometry. As $|T|$ is self-adjoint and compact, its range has an orthonorm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3730217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that there exists and angle $\alpha$ and $r \in \Bbb R$ such that $a\cos x + b\sin x = r\cos\alpha$ Let's say that we have an expression $a\cos x + b\sin x$ where $a \in \Bbb R$ and $b \in \Bbb R$.
I was learning about finding the minimum and maximum values of an expression of this form for some given value of $a... | We begin by observing that
$$a\cos x+b\sin x =\sqrt{a^2+b^2}\left\{\frac{a}{\sqrt{a^2+b^2}}\cdot\cos x +\frac{b}{\sqrt{a^2+b^2}}\cdot\sin x\right\}$$
Now, define $\phi\in[0,2\pi)$ such that
$$\cos\phi=\frac{a}{\sqrt{a^2+b^2}}\text{ and }\sin\phi=\frac{b}{\sqrt{a^2+b^2}}$$
Note that such a value of $\phi$ is unique. The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3730356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Why do we use the rank of a matrix to compute the Schatten norms? The Schatten norm is defined as
$$\|A\|_{S_P} = \left(\sum_{i=1}^{r(A)}\sigma_i^P(A)\right)^{\frac{1}{P}}$$
where $r(A)$ represents the rank of the matrix $A$. Why do we use the rank of the matrix to compute the Schatten $p$-norm?
| If you assume (as you do, although you don't say) that the singular values are ordered from bigger to smallest, you have
$$
\sum_{i=1}^{r(A)}\sigma_i^P(A)=\sum_{i=1}^{n}\sigma_i^P(A),
$$
since the singular values $\sigma_{r(A)+1},\ldots,\sigma_n\}$ are zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3730443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove that $\prod_{k=2}^{99}\frac{k^{3}-1}{k^{3}+1}$ is greater than $\frac{2}{3}$. I have to prove that the product
$$\prod_{k=2}^{99}\frac{k^{3}-1}{k^{3}+1}$$
is greater than $\displaystyle\frac{2}{3}$.
I've tried to write $k^{3}-1$ as $(k-1)(k^{2}+k+1)$ or another ways but I couldn't finish it.
| $\begin{array}\\
f(n)
&=\prod_{k=2}^{n}\dfrac{k^{3}-1}{k^{3}+1}\\
&=\dfrac{\prod_{k=2}^{n}(k^{3}-1)}{\prod_{k=2}^{n}(k^{3}+1)}\\
&=\dfrac{\prod_{k=2}^{n}(k-1)(k^2+k+1)}{\prod_{k=2}^{n}(k+1)(k^2-k+1)}\\
&=\dfrac{\prod_{k=1}^{n-1}k}{\prod_{k=3}^{n+1}k}\dfrac{\prod_{k=3}^{n+1}((k-1)^2+(k-1)+1)}{\prod_{k=2}^{n}(k^2-k+1)}
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3730561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Geometry Proof to Find Maximum area of $\triangle PIE$
Circle $\omega$ is inscribed in unit square $PLUM,$ and points $I$ and $E$ lie on $\omega$ such that
$U, I,$ and $E$ are collinear. Find, with proof, the greatest possible area for $\triangle PIE.$
I'm not sure if there is a solution possible without trigonometry... | Let $O$ be the center of circle $\omega.$ Let $X$ be the foot of the altitude from $P$ to $IE,$ and let $Y$ be the foot from $O$ to $IE.$ Denote line segment $\overline{YO}$ as length $d,$ the radius as $r,$ and $\angle XUP$ as $\theta.$
$\textbf{Claim:}$ The greatest area of $\triangle PIE$ is $\frac{1}{4}.$
$\textbf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3730695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Prove that $ \int_{\gamma} \frac{1}{z} dz = 0$ Let $\displaystyle\gamma$ be a closed curve exactly located in $A =\mathbb C \setminus\{z\in\mathbb C: Re(z)\leq 0\}$.
I found a similar problem here :
Find $\int_{\gamma}\frac{dz}z$, but they concluded that the value of the contour integral is $\displaystyle i\pi$.
How do... | If you know the Cauchy's integral theorem you can use that $\gamma$ is closed and $\frac{1}{z}$ is holomorphic in $A$, so the theorem say that the integral is zero.
However I think the easy way is notice that in $A$ the principal branch of the complex logarithm is well defined, so you can use that $\frac{1}{z} = \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3730795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Is the interior of the union of $n$ closed balls equal to the union of the interiors of the $n$ closed balls? I am reading "Calculus on Manifolds" by Michael Spivak.
I am solving problem 1-22 on p.10 now.
If the following equality holds, I can solve the problem.
Let $B_1, \dots, B_n$ be closed balls in $\mathbb{R}^m$.
... | I think your intuition is correct in one sens
this a counterexample take two intervals $[1,2]$and $[2,3]$
it's clear that the interior of the union is not equal to the union of the interior even if your sets are closed
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3730921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How many equivalence classes will there be? Consider the subset $T\subseteq \mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$ where the three numbers will be the corner angles (in degrees)
of a (real) triangle. For example $(30, 70, 80)\in T$ but
$(10, 30, 50) \not\in T$ (since $10 + 30 + 50 < 180$), and
$(−10, 20, 170) \n... | For $(a,b,c)$ to represent angles of a triangle, we should have $a+b+c=180$. Two triangles are equivalent iff their largest angles are the same. For example, $(61,60,59) \sim (61,61,58)$.
Note that at least one of $a,b$ or $c$ must be at least $60$ for the triangle to exist.
So to count the number of equivalence classe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3731078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to find area of rectangle inscribed in ellipse. In an ellipse $4x^2+9y^2=144$ inscribed is a rectangle whose vertices lies on the ellipse and whose sides are parallel with the ellipse axis.
Longer side which is parallel to major axis, relates to the shorter sides as $3:2$. Find area of rectangle.
I can find the val... | Let the top-right corner be at $(x,y)$. Squaring the aspect ratio, we have the system
$$\begin{cases}4x^2+9y^2=144,\\4x^2=9y^2\end{cases}$$
the solution of which is $x^2=18,y^2=8$.
Area $$4\sqrt{18\cdot8}=48.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3731491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
What does $\int_{-\infty}^{\infty}\sum_{i=0}^{\infty}\frac{x^i}{i!}dx$mean? I was reading a research paper and I came across this sentence which I didn't really understand.
We first cast Eq.1 into its standard form: $$\int_{-\infty}^{\infty}\sum_{i=0}^{\infty}\frac{x^i}{i!}dx$$
$$T^2-944T+155184-h=0$$
Eq. 1 is stated ... | I don't know the context, but $$\sum_{i=0}^{\infty} \frac{x^i}{i!} = e^x$$
because the sum is the Taylor series of $e^x$.
If this is the case, the integral has a more clear meaning.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3731598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Question about finding the value of an infinite sum What is the value of:
$$\sum_{k=0}^\infty \frac{1}{(4k+1)^2}?$$
I realised that
$$\sum_{n=2,4,6,8,...} \frac{1}{n^2} + \sum_{n=1,3,5,7,...} \frac{1}{n^2} = \sum_{n \geq 1 } \frac{1}{n^2}$$
$$\sum_{n \geq 1 } \frac{1}{4n^2}+\sum_{n \geq 0 } \frac{1}{(2n+1)^2} = \sum_{n... | The sum can be expressed in terms of Catalan's constant. In the following video, the sum is computed in a step by step manner: https://www.youtube.com/watch?v=r2OJtsHNDZA.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3731838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find $\sum_{n=-\infty}^{\infty} (0.5)^n$ We have: $$L=\lim_{n\to\infty} \sum_{k=-n}^n \frac1{2^k}.$$ The limit surely diverges or tends to $\infty.$ But I can't think of a proper way to show this.
Please suggest, how can I show that $L=\infty\,?$ Thanks in advance.
| $$\sum_{k=-n}^n\frac1{2^k}\ge2^n$$
just by the term $k=-n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3731964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
Determinant of $2 \times 2$ block matrix whose diagonal blocks are zero $\Bbb A$ is an $n\times n$ matrix, and $\Bbb B$ is an $m × m$ matrix.$\space$What is the determinant of matrix $\Bbb C$?
\begin{equation*}
\mathbb{C}=
\begin{pmatrix}
\begin{array}{@{}c|c@{}}
\begin{matrix}
0
\end{matrix}
& \mathbb{A} \\
\h... | This determinant can be computed by the Laplace expansion theorem (the generalized form). Fix the first $1, 2, \ldots, n$ rows, and let columns range over $(j_1, j_2, \ldots, j_n) \in \{1, 2, \ldots, n + m\}$. Since the square minor is non-zero only if $(j_1, j_2, \ldots, j_n) = (m + 1, m + 2, \ldots, m + n)$, it foll... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3732109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Explanation for behaviour of graph of $y=x^2e^{-x^2}$ (Maxwell-Boltzmann distribution) Consider the function
$$y=x^2e^{-x^2}$$
The graph initially behaves as a parabola then in later part exponential part of it dominates; i.e., the graph looks exponential after maximum of the curve.
Actually this graph is related to Ma... | You actually gave the mathematical explanation. The graph is below. Over the range $[-1,1]$ the exponential doesn't change that much-it is $1$ at the center and $\frac 1e \approx 0.3679$ at the ends. That is less than a factor $3$. The parabola is $0$ at the middle and $1$ at the ends, an infinite ratio. It domina... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3732230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
A proof of existence of canonical divisors I am confused with the proof of Lemma 1.5.10 in Algebraic Function Fields and Codes by Henning Stichtenoth.
Let $0\ne\omega\in\Omega_F$. Then there is a uniquely determined divisor $W\in M(\omega)$ such that $A≤W$ for all $A\in M(\omega)$.
$\Omega_F$ is the set of Weil differ... | Note that $\alpha'_P$ is the $p$th component of the adele $\alpha'$ and that $\nu_P(0) = \infty$ by definition. Therefore, $\nu_Q(\alpha'_Q) = \nu_Q(0) = \infty > -\nu_Q(W)$ since $\nu_Q(W)$ is necessarily finite. Therefore, you get that $\alpha'\in \mathcal A_F(W)$. This same reasoning also gives you that $\alpha''\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3732383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Nice inequality ,Prove that $\Gamma\Big(\frac{\sin(x)}{x}\Big)\leq \frac{\pi}{\pi-x}$ Playing with geogebra I get :
Let $0\leq x<\pi$ then we have :
$$\Gamma\Big(\frac{\sin(x)}{x}\Big)\leq \frac{\pi}{\pi-x}$$
Where we have the Gamma function .
I have tried to use the Wendel inequality to prove that the ratio of the... | Over $[0,\pi]$
$$ \Gamma\left(1+\frac{\sin x}{x}\right) \leq 1 \leq \frac{\pi \sin(x)}{x(\pi -x)}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3732511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Probability question - Normal and Uniform In a factory there are $2$ machines that create tubes (They are independent from each other) Length of the tubes of machine A is distributed normally with an expectancy of $101$ cm. and a Variance of $102$. ($\mu = 101, V[A]=102$)
The tubes of the second machine is distribute... | *
*what is requested is a conditional probability.
Then to get a you have to solve
$$\frac{0.6755\cdot a}{ 0.6755\cdot a +(1-a)\cdot \frac{2}{3}}=0.9$$
leading to $a\approx 0.899$
I want to underline that "Variant" in Statistics has no a nice meaning.
We have "variance" and "standard deviation".
If you 102 is expres... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3732714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Convolution with Dirac delta How to solve this expression:
$$\int_{-\infty}^{\infty} \left[ \delta(k-k_0)f(k)\right]*f(k)dk=?$$
Here $\delta$ represents the Dirac delta function and $*$ represents the convolution over the $k$ variable.
What I think:
$$\int_{-\infty}^{\infty} \left[ \delta(k-k_0)f(k)\right]*f(k)dk=\int_... | $$\left[\delta\left(k-k_0\right)\ f(k)\ *\ f(k)\right](x)=f\left(k_0\right)\ f\left(x-k_0\right)$$
$$\int\limits_{-\infty}^{\infty}\left[\delta\left(k-k_0\right)\ f(k)\ *\ f(k)\right](x)\ dx=\int\limits_{-\infty}^{\infty}f\left(k_0\right)\ f\left(x-k_0\right)\ dx$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3732854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proving $\prod_{i = 1}^n X_i \xrightarrow[n \to \infty]{(\mathbb{P})} 0 \iff \prod_{i = 1}^\infty \mathbb{E}[\sqrt{X_i}] = 0$ I am having a hard time proving the following :
Let $(X_i)_{i \geq 1}$ be a sequence of independent random variables which take their value in $\mathbb{R}^{+*}$ and such that $\mathbb{E}[X_i]... | Declare a positive $\varepsilon$.
For any integer $n$, you can control $E[\sqrt{(\prod_i X_i)}]$ by separating the events $\prod_i X_i < \varepsilon$ and $\prod_i X_i \geq \varepsilon$.
The first term is bounded by $\sqrt{\varepsilon}$. For the second term, use Cauchy-Schwarz : $E[1_{\prod_i X_i \geq \varepsilon } \, .... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3733000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is This Proof for "If $\sup A < \sup B$, show that there exists an element $b\in B$ that is an upper bound for $A$" correct? This question is from Understanding Analysis (Stephan Abbot) Exercise $1.3.9$. The Question is If $\sup A < \sup B$, show that there exists an element $b \in B$ that is an upper bound for $A$. M... | Your proof is correct but be careful with quantifiers. $(\forall a \in A ,b \in B\, a \geq b)$ is not true. We only know that for each $b \in B$ there exists $a \in A$ such that $a \geq b$, but in general this is not true for every element $a \in A$.
Here is another proof which I find be more natural:
Let $c \in ]\sup... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3733157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What is the true status of the Lehmer totient problem? The Lehmer-totient problem : For a prime number $\ n\ $ we have $\ \varphi(n)=n-1\ $. In particular, we have $\ \varphi(n) \mid n-1\ $. Is there a composite number $\ n\ $ with $\ \varphi(n)\mid n-1\ $ ?
It can easily be shown that such a number must be a Carmicha... | A quick search through arXiv yields the following results:
(1) This proof for the Lehmer-totient problem has been withdrawn:
On the Lehmer's problem involving Euler's totient function (Huan Xiao)
(2) I highly doubt the validity of the following proof, although it is not yet withdrawn:
An analytical proof for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3733276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Does $\left\lVert a-b \right\rVert_\infty < \epsilon \implies (\int a-b)< \int \epsilon$? $a,b$ are continuous functions.
I think this is true as constants are continuous functions but I'm not sure if it holds for all cases.
Edit: Integrated over two real numbers
| Only if the measure space is finite, otherwise you'll have only $\int (a-b) \le \int \varepsilon$ because both sides may be infinite.$\,$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3733362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If R is a ring, and A has all the sets in R and it's complements, is A an algebra? (Halmos Measure Theory question) The question I have is related to problem 4.5 in chapter 1 of Halmos' text. Some definitions related to the question are the following. If $X$ is a set then a ring $\textbf{R}$ is a non-empty class of sub... | Note that $E$ and $F^c$ are in $\mathbf{R}$, hence so is $F^c-E=F^c\cap E^c = (F\cup E)^c = (E\cup F)^c$. But if $(E\cup F)^{c}\in\mathbf{R}$, then $E\cup F\in A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3733486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Linear Least Squares with Monotonicity Constraint I'm interested in the multidimensional linear least squares problem: $$\min_{x}||Ax-b||^2$$
subject to a monotonicity constraint for $x$, meaning that the elements of $x$ are monotonically increasing: $x_0 \leq x_1$, $x_1 \leq x_2$, ... , $x_{n-1} \leq x_n$.
I basically... | Your idea to reformulate the problem so that the variables are $x_0$ and $y_i = x_i - x_{i-1}$ for $i =1, \ldots, n$ will work. Let $y$ be the vector whose components are $x_0, y_1, \ldots, y_n$ and
define
$$
M = \underbrace{\begin{bmatrix} 1 & 0 & \cdots & 0 \\
1 & 1 & \cdots & 0 \\
\vdots & & & \vdots \\
1 & 1 & \cdo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3733659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Spreading tickets in a lottery actually diminishes your chances? Here is the scenario: There is a lottery running for $n$ terms, which means that it is repeated. In each term, there are a total of $T$ tickets and one prize. You currently own $t$ tickets and your dilemma is to either use all of your tickets in one go or... | Suppose $n=2$. Using $x$ tickets in the first lottery and $t-x$ in the second yields a probability of winning $1-(1-\tfrac{x}{T})(1-\tfrac{t-x}{T})$. The maximum in the range $0\leq x \leq t$ is at the bounds, so it is better to attend in only one lottery, either the first or the second.
This is true in general for $n>... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3733941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Constructing a right triangle with a given hypotenuse segment and given point of tangency for its incircle
Given a hypotenuse $AB$ and an arbitrary point $C$ on $AB$. How to construct a right triangle with the given hypotenuse $AB$ such that point $C$ is the point of tangency of the inscribed circle?
My attempt: F... |
Let $ABG$ be the triangle you want to construct, $F$ its incenter. In the circle $\Gamma$ with diameter $AB$, let $D$ be the endpoint of the diameter perpendicular to $AB$ and on the opposite side of $AB$ to $G$.
Since $AGB$ is a right triangle, $G$ lies on $\Gamma$. Because $GF$ bisects $\angle AGB$, it meets $\Gamma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3734067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Proving angles in a circle are equal
$A$, $B$, $R$ and $P$ are four points on a circle with centre $O$.
$A$, $O$, $R$ and $C$ are four points on a different circle.
The two circles intersect at the points $A$ and $R$.
$CPA$, $CRB$ and $AOB$ are straight lines.
Prove that angle $CAB$ = angle $ABC$.
Not really sure ho... | Since $AB$ is a diameter, as you've already stated, this means $\measuredangle ARB = 90^{\circ}$ as well. As such, you also have $\measuredangle ARC = 90^{\circ}$. Thus, in the circle on the left side, you have $AC$ is its diameter. This means $\measuredangle COA = 90^{\circ}$ (note you could also get $\measuredangle C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3734203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Open and Closed subset of $\mathbb{R}^n$ Let $A,B\subseteq \mathbb{R}^n$, define $A+B=\{a+b : a\in A,b\in B\}$ Then which of the following is/are true?
$(1)$ If $A$ and $B$ are open, then $A+B$ is open.
$(2)$ If $A$ open and $B$ close, then $A+B$ closed.
$(3)$ If $A$ is closed and $B$ open, then $A+B$ is open.
$(4)$ If... | Your answers seem right to me.
Here is a more general way to look at (1) and (3).
Claim 1. If $A$ is open and $B$ is arbitrary then $A+B$ is open.
Proof. Note that $A+B=\bigcup_{b\in B}A+\{b\}$. Each set $A+\{b\}$ is open (by your argument for "open + closed is open"; but this specific case is easier.) Now $A+B$ is a u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3734380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Localization commutes with Hom for finitely presented modules I am trying to solve an exercise given in Vakil's Algebraic Geometry notes. Suppose $M$ is a finitely presented $A$-module. The $M$ fits inside an exact sequence $A^q\rightarrow A^p\rightarrow M\rightarrow 0$. I'd like to understand why in this case we get a... | As $S^{-1}$ is a functor we have a map
$$\mathrm{Hom}_A(M,N)\rightarrow \mathrm{Hom}_{S^{-1}A}(S^{-1}M,S^{-1}N)$$
and as multiplication by an element $s\in S$ gives an isomorphism in the module $\mathrm{Hom}_{S^{-1}A}(S^{-1}M,S^{-1}N)$ this map extends to a map
$$\tag{$\star$} S^{-1}\mathrm{Hom}_A(M,N)\rightarrow \math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3734488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Interesting Partition Questions There is a good question here.
My question is;
"x is a positive integer and
$\lfloor x\rfloor$ denote the largest integer smaller than or equal to $x$. Prove that $\lfloor n / 3\rfloor+1$ is the number of partitions of $n$ into distinct parts where each part is either a power of two or t... | Let’s use a generating function.
If $p(n)$ is the number of partitions of $n$ into numbers of the form $2^k$ or $3\cdot 2^k$, then we have the following generating function:
$$\sum_{n=0}^\infty p(n)x^n = \prod_{k=0}^\infty (1+x^{2^k})(1+x^{3\cdot 2^k})$$
Recall the following identity, which follows from the fact that e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3734626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Finding zeroeth coefficient of a Chebyshev polynomial expansion Let $v_\theta = (\cos\theta,\sin\theta)$ be a unit vector in the plane. I have a kernel $p(\theta,\theta') = p(v_\theta\cdot v_{\theta'})$ that satisfies
$$\int_0^{2\pi} p(v_\theta\cdot v_{\theta'})\,d\theta' = 1\;\;\;(*)$$
for all $\theta\in [0,2\pi]$. I ... | As it turns out, using orthonormality was a a red herring, and the solution is actually quite simple. Choosing $v_\theta = (1,0)$, we compute
\begin{align*}
1 &= \int_0^{2\pi} p(v_\theta\cdot v_{\theta'})\,d\theta'\\
&= \sum_{k=0}^\infty \int_0^{2\pi} p_kT_k(v_\theta\cdot v_{\theta'})\,d\theta'\\
&= \sum_{k=0}^\infty \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3734763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Variance of a discrete random variables that takes on 2 values. Suppose I have a random variable that takes on a value of 10 with p(x=10)=.7 and a value of 20 with p(x=20) = .3.
The E(X) = .7(10)+.3(20) = 13.
The variance would be the expected value of the differences from the mean for each x.
Var(X) = .7(13-10)^2 + .3... | Consider a Bernoulli random variable $Y$ with parameter $p=0.3$: i.e., $\Pr[Y=1]=p$, $\Pr[Y=0]=1-p$. It is known (and easy to verify) that $\operatorname{Var}[Y] = p(1-p)$.
Set $b=20$, $a=10$. Note that $X$ has the same distribution as $(b-a)Y+b$ (can you see why?). Therefore,
$$
\operatorname{Var}[X] = \operatorname{V... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3734864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Curvature of Fernet curve on a sphere The question is, how to prove that the curvature of any Frenet curve on a sphere with radius $R$ is bigger or equal to $1/R$.
I have managed to prove so far that the Gauss curvature of the sphere $x^2+y^2+z^2=R^2$ is $1/R^2$, but I don't know if this helps at all
| Suppose $\alpha(s)$ is a unit speed curve lying in the sphere of radius $R$ centered at the origin. Then
$\alpha(s) \cdot \alpha (s) = R^2, \tag 1$
whence
$\dot \alpha(s) \cdot \alpha(s) = 0; \tag 2$
since
$\dot \alpha(s) = T(s), \tag 3$
the unit tangent vector to $\alpha(s)$, (2) becomes
$T(s) \cdot \alpha (s) = 0; \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3735022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Rank of $A^n$ and $A^{n+1}$ Suppose $A$ is a $n\times n$ real matrix.
Then is it always true that Rank($A^n$) = Rank ($A^{n+1}$) for a matrix $A$?
This doubt came while solving the attached question:
If A is a 10×10 real matrix, then which of the following is true:
*
*rank($A^8$)=rank($A^9$)
*rank($A^9$)= rank($A^{1... | Yes, it is always true. One argument is to use Jordan form: a matrix $A$ is necessarily similar to a block diagonal matrix of the form
$$
\pmatrix{M & 0\\0& N},
$$
where $M$ is invertible and $N$ is nilpotent. Since $N^n = 0$, the ranks of $A^n,A^{n+1}$ must be equal to the ranks of
$$
\pmatrix{M & 0\\0& N}^n = \pmatr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3735153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.