Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Prove that $ a^4b^4+ a^4c^4+b^4c^4\le3$
Let $a,b,c>0$ and $a^3+b^3+c^3=3$. Prove that
$$ a^4b^4+ a^4c^4+b^4c^4\le3$$
My attempts:
1) $$ a^4b^4+ a^4c^4+b^4c^4\le \frac{(a^4+b^4)^2}{4}+\frac{(c^4+b^4)^2}{4}+\frac{(a^4+c^4)^2}{4}$$
2) $$a^4b^4+ a^4c^4+b^4c^4\le3=a^3+b^3+c^3$$
| By AM-GM and Schur we obtain
$$\sum_{cyc}a^4b^4=\frac{1}{3}\sum_{cyc}(3ab)a^3b^3\leq\frac{1}{3}\sum_{cyc}(1+a^3+b^3)a^3b^3=$$
$$=\frac{1}{3}\sum_{cyc}(a^3b^3+a^6b^3+a^6c^3)=\frac{1}{9}\sum_{cyc}a^3\sum_{cyc}a^3b^3+\frac{1}{3}\sum_{cyc}(a^6b^3+a^6c^3)=$$
$$=\frac{1}{9}\sum_{cyc}(4a^6b^3+4a^6c^3+a^3b^3c^3)=$$
$$=\frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2589946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Australian Math Competition Geometry Problem
$PQRS$ is a rectangle with a centre $C$. $PQ$ has length $4$ and $PS$ has length 12. The circles meet $PS$ at $U$ and $V$ with both having radius $1$. $PU$ has length $1$ and $PV$ has length $4$. What is $PW$?
I’ve tried this problem for days and tried to find answers els... | Draw a rectangle where C is extended straight downwards until it reaches PS (call this point of intersection A) and one line from C to the left until it reaches PQ. This rectangle has side lengths 2 and 6, because if C is the center of rectangle PQRS, then it should divide the rectangle's sides into halves as well. Als... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2590046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\sum_{n=1}^\infty f_n$ converges uniformly on $\mathbb{R}$ where $f_n$ is defined piecewise.
Prove that the series of functions $\sum_{n=1}^\infty f_n$ converges
uniformly on $\mathbb{R}$, where $$f_n: \mathbb{R} \to \mathbb{R}: x
\mapsto \begin{cases}0 \quad x \neq n \\\frac{1}{x} \quad x
=n\end{cases... | You are mostly there. With the $f$ you have found as the limiting function, for any $x\in \mathbb{R}$,
$$
\left|\sum_{k=1}^nf_k(x)-f(x)\right|\leq \frac{1}{n+1}
$$
since if $x\not\in \mathbb{N}$ or if $x\in \{ 1,\dots,n\}$ the difference is zero. Otherwise, the worst this difference could be is the difference at $x=n+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2590134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Calculate the amount of series We know that $\sum_{k=1}^\infty \frac{1}{k^2+1} = \frac{1}{2}(\pi \coth(\pi) - 1)$. Now, how do we calculate the series $\sum_{k=1}^\infty \frac{1}{(k + x)^2+1}$ for $x\geq 0$?
| One may write
$$
\begin{align}
\frac{2i}{(k + x)^2+1}&= \frac{1}{k+x-i}-\frac{1}{k+x+i}
=\!\left(\frac{1}{k}-\frac{1}{k+x+i}\right)-\left(\frac{1}{k}-\frac{1}{k+x-i}\right)
\end{align}
$$ yielding
$$
\sum_{k=1}^\infty \frac{1}{(k + x)^2+1}=\frac1{2i}\psi(x+1+i)-\frac1{2i}\psi(x+1-i),\qquad \text{Re}(x+1)>-1,\tag ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2590235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Non uniformly integrable sequence I am looking for a sequence of random variable $X_1, X_2, \cdots$ on $([0,1], \mathbb B, \lambda)$ such that
$$X_n\to 0 \quad \text{a.s.}$$
$$EX_n\to 0$$
but such that $(X_n)_n$ is not uniformly integrable.
I already showed that the sequence can't have $X_n\geq 0$ for all $n$. Can we m... | Consider $$X_n := n 1_{(0,1/n)}-n 1_{(1/n,2/n)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2590350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Curl of a cross product with constant vector If $\mathbf a$ is a constant vector in the 3-dimensional space and $\mathbf s=x\mathbf e_x+y\mathbf e_y +z\mathbf e_z$, I want to show that
$$\nabla \land \left(\mathbf a \land \mathbf s\right) = 2\mathbf a. $$
I have done as follows:
$$\nabla \land \left(\mathbf a \land \m... | We have the general formula $$\nabla \wedge ({\bf a}\wedge {\bf s}) = {\bf a}(\nabla \cdot {\bf s}) - {\bf s}(\nabla \cdot {\bf a}) + ({\bf s}\cdot \nabla){\bf a} - ({\bf a}\cdot \nabla){\bf s}.$$Now let's see each piece.
*
*Since ${\bf s} = (x,y,z)$, clearly $\nabla \cdot {\bf s} = 3$.
*Since ${\bf a}$ is constan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2590478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$i^*(\omega)$ vanish on S $\Leftrightarrow$ A is tangent to $S$ If $\omega\in \Omega^2(\mathbb{R}^3)$ is a 2-form: $\omega=f\;dx\wedge dy+g\;dx\wedge dz+h\;dy \wedge dz$, and $S$ is a surface in $\mathbb{R}^3$, prove that $i^*(\omega)$ vanish on $S$ $\iff$ $A=h\frac{\partial}{\partial x}-g\frac{\partial}{\partial y}+f\... | By duality, $\omega$ corresponds to the vector field $(h,-g,f)$. Then $\iota^\ast(\omega)$ measures exactly the normal component of $(h,-g,f)$ with respect to $S$. With this in mind, there's nothing else to do.
Explicitly, we identify $$F_1\partial_x + F_2\partial_y +F_3 \partial_z \leftrightarrow F_1 \;{\rm d}y\wedge ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2590590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A stronger form of Bernoulli's inequality Let $x\geq 0$ be fixed. By Bernoulli's inequality we know that for all $n \in \mathbb{N}_0$,
$$
(1+x)^n \geq 1+nx
$$
Now, let $\alpha \geq 0$ be fixed as well. By a limit argument, we know that
$$
(1+x)^n \geq 1+\alpha nx
$$
for large $n$. But how large does $n$ have to be? Mor... | If $n\ge 2$ then $$(1+x)^n\ge 1+nx+\frac{n(n-1)}{2}x^2.$$So you just need $\frac{(n-1)}{2}x>\alpha-1.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2590661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
When does $\arcsin^2x+\arccos^2x=1$ for $x$ real or comlex? let $\arcsin$ is the compositional inverse of $\sin$ and $\arccos$ the compositional inverse of $\cos$ , my question here is it possible to find $x$ for which : $\arcsin^2x+ \arccos^2x=1$.
Note: $x$ is a real number or complex
| You want $u^2+v^2=1$ and $\sin u=\cos v$. This means:
Letting $z=e^{iv}$ you want $z+\frac{1}{z}=2\sin u$ or $z=\frac{2\sin u\pm \sqrt{4\sin^2 u-4}}{2}=\sin u \pm i\cos u$.
If $u=\frac{\pi}{2}-w$ then this is $z=e^{\pm iw}$ or $v=2\pi k \pm\left( \frac{\pi}{2}-u\right)$.
Case 1: If $u+v=2\pi k +\frac{\pi}{2}=S_+$ the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2590793",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Are fundamental representations of Lie algebras faithful? Let $L$ be a semisimple algebra, $\alpha_1, \cdots, \alpha_l$ a base of the root system $\Phi$, $\omega_1, \cdots, \omega_l$ the dual basis relative to the inner product (such that
$ \langle \omega_i, \alpha_j \rangle = \delta_{ij}$). Is the irreducible represe... | To summarize the discussion in the comments: this is true iff $L$ is simple. Writing $L$ as the direct sum of its simple factors $L_i$, the simple roots of $L$ are the disjoint union of the simple roots of each $L_i$, and similarly for the fundamental weights. So the fundamental representations of $L$ are the fundament... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2590902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Using the weak law of large numbers to find the limit of $\sum\limits_{r = an}^{bn} {n \choose r } p^r (1-p)^{n-r}$ I need to find the limit of $\sum\limits_{r}^{} {n \choose r } p^r (1-p)^{n-r}$ such that $ an < r < bn $ in the cases $p < a$, $ a < p < b$, and $b < p$.
I know I need to consider the sum of $n$ identica... | Let $X_n$ be binomial with parameters $p$ and $n$. The desired sum is $P(a<X_n/n<b)$. But by the WLLN, $X_n/n\xrightarrow{p} p$, so ...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2591006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to prove $\cos(n!) \neq 1$ without using $\pi$ is irrational Prove
$[\forall n \in \mathbb{N}, \cos(n!) \neq 1]$,
without using $\pi$ is irrational.
Using $\pi \in (\mathbb R-\mathbb Q)$, I can prove it...
Thanks everyone!
| If $\cos(n!) = 1$ for some $n\in\mathbb N,$ then $n! = 2\pi m$ for some $m\in\mathbb N,$ so $\pi = m/(n!)$ and thus $\pi$ is rational.
Conversely if $\pi$ is rational then $\pi = m/\ell$ for some $m,\ell\in\mathbb N,$ and $\ell$ is a divisor of $n!$ for some $n\in\mathbb N,$ so then $\cos(n!)=1.$
Therefore the only way... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2591123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Computing the sum $\sum_\limits{n=2}^\infty \left(\frac{1}{(n-1)!}-\frac{1}{n!}\right)\frac{1}{n+1}$ I have come across an infinite series, but I have no clue on how to compute its sum.
$$\sum_{n=2}^\infty \left(\frac{1}{(n-1)!}-\frac{1}{n!}\right)\frac{1}{n+1}$$
It should have something to do with the Taylor expansi... | \begin{align*}
\sum_{n=2}^\infty \left (\frac{1}{(n-1)!}-\frac{1}{n!}\right)\frac{1}{(n+1)}
&=\sum_{n=2}^\infty \left (\frac{n}{(n+1)!}-\frac{1}{(n+1)!}\right)\\
&=\sum_{n=2}^\infty \left (\frac{n+1-1}{(n+1)!}-\frac{1}{(n+1)!}\right)\\
&=\sum_{n=2}^\infty \frac{1}{n!}-2\sum_{n=2}^\infty \frac{1}{(n+1)!}\\
&=-2+\sum_{n=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2591213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
if $f(x,y) = \int_{x}^{y}p(p-1)(p-2)dp$ then calculate $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ How do i take derivative of this function : $f(x,y) = \int_{x}^{y}p(p-1)(p-2)dp$.
For single variable I can evaluate but this involves two variables.
Any hint please I am stuck here.
I want to cal... | $$\int_x^yp(p-1)(p-2)\,dp=\int_x^y (p^3-3p^2+2p)\,dp=-\frac{x^4}{4}+x^3-x^2+\frac{y^4}{4}-y^3+y^2$$
Now differentiate with respect to whatever variable you want.
In general, if we are given a continuous function $f(p)$ and we want to differentiate the integral
$$G(x,y)=\int_x^yf(p)\,dp$$
as a function of $y$, say, the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2591357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Solving the inequality ${x^2+2x+2^{|a|}\over x^2-a^2} > 0$ How do I solve the inequality ${x^2+2x+2^{|a|}\over x^2-a^2} > 0$?
My idea is that for $a\ne 1$, the numerator will always be positive. So the inequality reduces to $
{1 \over x^2-a^2 } > 0
$
My doubt is in this part. If we factorise the denominator, we get
$... | The problem is that in your solution you can have the $2$ intervals ovelapping, i.e. $-a > a$ which holds for all $a<0$
Hence in order for the intervals to not overlap, you have to use the modulus
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2591453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Higher-order derivative of $1/(1+e^x)$ Let
$$
f(x) = \frac{1}{1+e^x}
$$
1) Is there a closed formula for the $k$-derivative $f^{(k)}(x)$?
2) Is there a simple argument which shows for $k \geq 1$ that
$$
\lim_{x \to \pm \infty} f^{(k)}(x) = 0 ?
$$
Thanks!
| Write $e^x=:u$ for short. Then there is a sequence $(p_n)_{n\geq0}$ of polynomials such that
$$f^{(n)}(x)={p_n(u)\over(1+u)^{n+1}}\qquad(n\geq0)\ .$$
It is easy to show that the $p_n$ satisfy the recursion
$$p_0=1,\qquad p_{n+1}=(u+u^2)\, p_n'-(n+1) u\, p_n\qquad(n\geq0)\ .$$
E.g., one obtains
$$p_6(u)=u (-1+57 u-302 u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2591565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Trigonometric limit mistake Question: $$\lim_{x\to 0}\frac{\tan x-\sin x}{x^3}$$
The answer, by L'Hopital's rule as well as wolfram and desmos is $\frac{1}{2}$
Here's what I did:
$$\lim_{x\to0}({\tan x \over x}\times{1\over x^2}-{\sin x \over x}\times{1\over x^2})$$
$$\lim_{x\to0}({1 \over x^2}-{1 \over x^2})=0$$
Im no... | Note that there is no split of the limit like $$\lim f(x) - \lim g(x) $$ here. Rather the expression under limit has been split as a difference based on laws of algebra and this is perfect.
The mistake happens in the next step and is very common and that is replacing the expressions $(\sin x) /x$ and $(\tan x) /x$ by $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2591683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Embedding a Riemann surface which is diffeomorphic to a punctured disc in $\mathbb{C}$ How do we prove that any Riemann surface which is diffeomorphic to a punctured disc can be holomorphically embedded in $\mathbb{C}$?
The reason I am thinking about this is because I was trying to classify complex structures on a pun... | Such a Riemann surface $X$ has fundamental group $\Bbb Z$.
By the uniformisation theorem,
its universal cover $U$ is either $\Bbb C$ or the upper half plane, and $X$
is the quotient of $U$ by an automorphism $\gamma$ of infinite order. Up to inner automorphisms, if $U=\Bbb C$ then $\gamma$ is conjugate to $z\mapsto z+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2591786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How can I solve this rather simple looking integral equation? I was working on a physics problem and I have reduced it down to a simple integral equation with two boundary conditions:
$$\int_0^{l-t}y(x, t) dx = lh$$
With the conditions:
$$y(0, t) = y(l-t, t) = h$$
I am looking for $y(x, t)$. $l$, $h$ and $t$ are positi... | Has this something to do with FEM?
I would do:
at $x=0$, $$y(x, t)=h$$
at $x=\frac{1}{2} \cdot(l-t)$, $$y(x, t)=h+\frac{2ht}{l-t}$$
at $x=(l-t)$, $$y(x, t)=h$$
Is this what you are looking for?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2591899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Eigenvector and eigenvalue of $\mathbf A+\lambda\mathbf I$ where $\mathbf A=\mathbf{vv}^\top$
Given $\mathbf v=\begin{bmatrix}v_1\\v_2\\\vdots\\v_n\end{bmatrix}$ and $\mathbf A=\mathbf{vv}^\top$, find the eigenvectors and eigenvalues of $\mathbf A+\lambda\mathbf I$.
My current work progress is:
Since $(\mathbf A+\lam... | Hint: Consider a vector orthogonal to $v$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2592045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Lemma 2.8 - Elements of integration by Bartle I'm self studying Measure Theory by Bartle's book and I have a doubt in the proof of a lemma. Before I say what is the lemma and what is my doubt, I would like to say that the book is available here if anyone wants to see and I would like to introduce some notation accordin... | This equality is valid for $\alpha <0$, so
\begin{align}\{ x \in X \ ; \ f_1(x) > \alpha \}&=\{ x \in X \ ; \ f(x) > \alpha \}\cup\{ x \in X \ ; \ f_1(x) =0 \}\\[1ex]
&=\{ x \in X \ ; \ f(x) > \alpha \}\cup (A\cup B)\\[1ex]
&=\{ x \in X \ ; \ f(x) > \alpha \}\cup B
\end{align}
since $\;A\subset \{ x \in X \ ; \ f(x) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2592170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
the set of all matrices which represent orthogonal projection in $M_n(C)$ is closed The set of all matrices which represent orthogonal projection in $M_n(C)$ is closed. I can not identify the set. Can you please help me to do so?
| The set you are looking for is the following:
$$\mathrm{Gr}_n(\mathbb{C}):=\{A\in\mathcal{M}_n(\mathbb{C})\textrm{ s.t. } A^2=A,{}^\intercal\overline{A}=A\}.$$
It is closed, since it is given as an intersection of closed sets. Indeed, the two following maps $A\mapsto A^2-A$ and $A\mapsto{}^\intercal{\overline{A}}-A$ ar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2592250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Complex analysis on real integral Use complex analysis to compute the real integral
$$\int_{-\infty}^\infty \frac{dx}{(1+x^2)^3}$$
Attempt
I think I want to consider this as the real part of
$$\int_{-\infty}^\infty \frac{dz}{(1+z^2)^3}$$
and then apply the residue theorem. However, I am not sure how that is the complex... | It is much faster to apply Feynman's trick. For any $a>0$ we clearly have
$$ \int_{-\infty}^{+\infty}\frac{dx}{a+x^2} = \frac{\pi}{\sqrt{a}} \tag{1} $$
and by applying $\frac{d^2}{da^2}$ to both sides:
$$ \int_{-\infty}^{+\infty}\frac{2\,dx}{(a+x^2)^3} = \frac{3\pi}{4a^2\sqrt{a}} \tag{2} $$
hence by evaluating $(2)$ at... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2592350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
The $4 x^3 = y^2 +27$ solutions. Is it true that if
$$4 x^3 = y^2 +27$$
then
$$y=3k$$
I tried the Fermat theorem for $x^3$, but seems I am missing something.
EDITED:
$x$ and $y$ are integers.
| You are correct that you could use Fermat's theorem for $x^3+y^3=z^3$.
Starting with your elliptic curve
$$4x^3=y^2+27$$
we borrow Yong Hao Ng's answer by multiplying it by $16$ to get
$$(4x)^3=(4y)^2+432$$
$$X^3-432=Y^2\tag1$$
This is a special case of a well-known family. The problem of finding two rational cubes e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2592454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
What is the coefficient of $ a^8b^4c^9d^9$ in $(abc+abd+acd+bcd)^{10}$? I am new to Binomial Theorem and I want to find out the coefficient of $ a^8b^4c^9d^9$ in $$(abc+abd+acd+bcd)^{10}$$ How to find that?
| Let us set $abc=D, abd=C, acd=B, bcd=A$. Then $a^8 b^4 c^9 d^9 =A^2 B^6 C D$ and the problem boils down to finding the coefficient of $A^2 B^6 C D$ in $(A+B+C+D)^{10}$, i.e. to counting the anagrams of the word $AABBBBBBCD$. They are
$$ \frac{10!}{2!6!}=\color{red}{2520}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2592558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Connection compatible with a volume form Let $M$ be a smooth, orientable $n$-manifold and $\eta$ a volume form on $M$. Does there exist a connection $A$ on $TM$ such that
$$\tag{$*$}D\eta=0,$$
where $D$ is the appropriate covariant derivative associated to $A$ on $\Omega^n(M)$? Can one assume $A$ to be symmetric?
I ha... | If $\eta$ vanishes somewhere, then the only way it can be parallel (for any connection at all) is if it vanishes identically. Thus I will assume $\eta$ is non-vanishing.
Let $g$ be a Riemannian metric on $M$ and $\omega$ the volume form of $g.$ Since $\omega,\eta$ are smooth non-vanishing sections of a line bundle, the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2592624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Non real roots and Gauss Lucas I am stuck on this problem:
a) let a be a complex number, M and M' the images of a and $\bar a$. Study the sign of Im{$\frac{1}{z-a}+\frac{1}{z-\bar a}$} with the position of the point P image of z d to the real axis and to the circle of diameter MM'.
b) Be A a polynomial in R[X]. Be $\... | This is Jensen's theorem.
Let $p\in{}\mathbb{R}[x]$. It is known that if $a$ is a complex root of $p$, Then $\bar a$ is also a root. In this case, we will say that the disk of diameter $aa^*$ (i.e. its diameter is the line between $a$ and $\bar a$) is a Jensen Disk of $p$ (example: http://mathworld.wolfram.com/JensenDi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2592744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$\sqrt{I}=R \iff I=R$ Let $R$ be a commutative ring with identity and $I$ and ideal. I have a proof that $\sqrt{I}=R \iff I=R$ and I am wondering if its correct:
"$\Leftarrow$":Obvious, since $I\subset \sqrt{I}$
"$\Rightarrow$":We know that $\sqrt{I}=\bigcap{p_i}$ where $p_i$ is prime and $p_i\supset I$ for all $i$. Be... | It's much simpler to use the definition: if $\sqrt{I}=R$, then $1\in\sqrt{I}$, so $1^n\in I$, for some $n$.
Your proof is incorrect. From $\bigcap_i p_i=R$ you argue that $I$ is maximal, which is wrong. You should instead argue that there is no prime ideal containing $I$. Therefore $I$ is not a proper ideal.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2592888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
how to solve the logarithmic equation which has both n and logn How to solve this logarithmic equation? $8n^2 = 64n\log n$, ($\log n$ here is base 2)
I have tried to convert it to $n-8\log n = 0$, but how to solve the latest?
| This doesn't have any solutions using elementary functions. But, using the Lambert W function, we get: $$n = -\frac {8}{\ln 2} \operatorname{W} \left (-\frac {\ln 2}{8} \right)$$ and $$n = -\frac {8}{\ln 2} \operatorname{W}_{-1} \left (-\frac {\ln 2}{8} \right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2593003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A few general questions on the Penrose transform Let us consider the Bateman or Whittaker's pioneering examples of a Penrose transform. Starting from a holomorphic function on an open subset of twistor space, they constructed a solution to the Laplace equation (in one case in dimension $4$ and in the other case in dime... | I found the explanation in Huggett and Tod's book, "Introduction to Twistor theory", to be very clear and down to earth (it was recommended to me by D. Calderbank, and I thank him for this reference). Using homogeneous coordinates on (projective) twistor space, a la Penrose, and using the integral formula, one can then... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2593068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Compute $\lim_{x \to \infty} \frac{\log(x)}{x^a}$ How can I compute $\lim_{x \to \infty} \frac{\log(x)}{x^a}$ for some $a \in \mathbb R$ with $x^a := e^{a \log(x)}$? Can you give me a hint?
I want to use only the basic properties of limits, like the linearity, multiplicativity, monotonicity, the Sandwich property and c... | Set $y:= a \cdot \log(x)$ than it follows for $a>0$ that
$$\lim_{x \rightarrow \infty} \frac{\log(x)}{x^{a}}= \lim_{x \rightarrow \infty} \frac{\log(x)}{e^{a \log(x)}} = \frac{1}{a} \lim_{y \rightarrow \infty} \frac{y}{e^{y}}=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2593289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
If $\sum_{i=1}^\infty i^2\mathbb{P}(i\leq X_nI am studying probability theory myself, so I have been asking questions a lot recently. Please help.
This question comes from Rosenthal's 3.6.13
Let $X_1, X_2,\dots$ be defined jointly on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$, with $\sum_{i=1}^\infty i... | The event $X_n \ge n$ i.o. is indeed a tail event. You can tell because if you change any finite number of the $X_n,$ it won't change the truth value of $X_n \ge n$ i.o. By the Kolmogorov 0-1 law, $\mathbb{P}(X_n \ge n \text{ i.o})$ is either 0 or 1. Therefore, it suffices to show that it is not equal to 1.
So assume... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2593371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove that equality is correct
Prove that
$$\int_{0}^{1} \frac{\ln \left ( x^2+x+1 \right )}{x}\mathrm dx=\frac{\pi^2}{9}.$$
As I understand, I can do this:
$$\large 1- x^3 = (1-x)(1+x+x^2) \Rightarrow x^2 +x+1 = \frac{1-x^3}{1-x}, $$
this gives
$$f(x)= \frac{1}{x} \ln \left(\frac {1-x^3}{1-x}\right) =\frac{1}{x}... | You recognized a crucial fact, i.e. that $x^2+x+1$ is a cyclotomic polynomial. For any $m\geq 1$ we have
$$ \int_{0}^{1}\frac{-\log(1-x^m)}{x}\,dx = \sum_{n\geq 1}\int_{0}^{1}\frac{x^{mn-1}}{n}\,dx = \frac{1}{m}\sum_{n\geq 1}\frac{1}{n^2} = \frac{\zeta(2)}{m}\tag{1}$$
hence
$$ \int_{0}^{1}\frac{\log\Phi_3(x)}{x}\,dx =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2593518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Integrable function has bounded variation Let $f:[0,1] \to \mathbb{R}$ be Lebesgue integrable. Define $F(x) = \int_0^x f$. Prove $F(X)$ has bounded variation on $[0,1]$. What does this imply for differentiability of $F$ and why?
Attempt
$F$ is monotone (increasing) very obviously. So if $F$ is of bounded variation and ... | Hint: If $0\le a < b\le 1,$ then $|F(b)-F(a)| \le \int_a^b|f|.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2593632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove: For $\triangle ABC$, if $\sin^2A + \sin^2B = 5\sin^2C$, then $\sin C \leq \frac{3}{5}$. We have a triangle $ABC$. It is given that $\sin^2A + \sin^2B = 5\sin^2C$. Prove that $\sin C \leq \frac{3}{5}$.
Let's say that $BC = a$, $AC=b$, $AB=c$.
According to the sine law,
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \fra... | Since $A + B + C = \pi$, we have $\sin (A + B) = \sin C$.
Also notice that
$$\cos^2 A + \cos^2 B = 1 - \sin^2A + 1 - \sin^2B = 2 - 5\sin^2(A + B)$$
Proceeding by CSB, we get:
\begin{align}
\sin(A + B) &= \sin A \cos B + \sin B \cos A \\
&\le \sqrt{\sin^2 A + \sin ^2 B} \sqrt{\cos^2A + \cos^2B} \\
&= \sqrt{5} \sin(A + B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2593739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Prove that $(-1)^n\sum_{k=0}^{n}{{{n+k}\choose{n}}2^k}+1=2^{n+1}\sum_{k=0}^{n}{{{n+k}\choose{n}}(-1)^k}$ Define:
$$A_n:=\sum_{k=0}^{n}{{n+k}\choose{n}} 2^k,\quad{B_n}:=\sum_{k=0}^{n}{{n+k}\choose{n}}(-1)^k$$
I've found that (based on values for small $n$) this identity seems to be true:
$${\left(-1\right)}^nA_n+1=2^{n+... | Using formal power series we have the Iverson bracket
$$[[0\le k\le n]] = [z^n] z^k \frac{1}{1-z}.$$
We then get for $A_n$
$$\sum_{k\ge 0} [z^n] z^k \frac{1}{1-z} {n+k\choose k} 2^k
= [z^n] \frac{1}{1-z}
\sum_{k\ge 0} z^k {n+k\choose n} 2^k
\\ = [z^n] \frac{1}{1-z} \frac{1}{(1-2z)^{n+1}}.$$
This yields for $1+(-1)^n A_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2593862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Differentiate a Variable Limit Ito Integral Consider a variable limit integral $I(t)=\int\limits_0^{\phi(t)}M(s)dW(s)$,
where $\phi(t)$ is an increasing deterministic function with $\phi(0)=0$, the integrand $M(t)$ is stochastic, and $W(t)$ is a standard Brownian motion. Assume that $M(t)$ and $W(t)$ are adapted to fi... | I think you cannot write it in this form, but using random time change, you can write it in another form as follows. By Theorem 8.5.7 of Oksendall's book, you can write
$$I_t = \int_0^t M_{\phi(s)}\sqrt{\phi'(s)}d\tilde B_s,$$
where $\tilde B_t = \int_0^{\phi(t)} \sqrt{(\phi^{-1})'(s)}dW_s$ is another Brownian motion (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2593960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to find the limit:$\lim_{n\to \infty}\left(\sum_{k=n+1}^{2n}\left(2(2k)^{\frac{1}{2k}}-k^{\frac{1}{k}}\right)-n\right)$ How to find the limit:$$\lim_{n\to \infty}\left(\sum_{k=n+1}^{2n}\left(2(2k)^{\frac{1}{2k}}-k^{\frac{1}{k}}\right)-n\right)$$
I can't think of any way of this problem
Can someone to evaluate this?... | For small $x>0$, we have $1+x\leq\exp(x)\leq 1+x+x^{2}$, then for large $k$,
\begin{align*}
1+\dfrac{\log 2}{k}-\left(\dfrac{\log k}{k}\right)^{2}\leq 2(2k)^{1/2k}-k^{1/k}\leq 1+\dfrac{\log 2}{k}+\dfrac{(\log 2k)^{2}}{2k^{2}},
\end{align*}
and for large $n$,
\begin{align*}
\left(\sum_{k=n+1}^{2n}2(2k)^{1/2k}-k^{1/k}\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2594050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Sobolev functions vanish in a ball Assume $u\in H^1(\mathbb R^N)$ and $u=0$ a.e. in $B_1$.
Does it hold that $u\in H_0^1(\mathbb R^N\setminus \overline B_1)$?
I tried to find smooth functions with compact support to approximate $u$.
For example $\rho_n*u$, with $\rho_n$ being mollifiers.
However the support of these s... | I take it that $H^1_0(\Omega)$ is the closure of $C^\infty_c(\Omega)$?
Hint: For $r>0$ define $$f_r(x)=f(rx).$$Since $C_c(\mathbb R^d)$ is dense in $L^2$ it follows that if $f\in L^2(\mathbb R^d)$ then $$\lim_{r\to1}||f-f_r||_{L^2}=0.$$Applying this to $f$ and to $f'$ shows that the same is true with $H^1$ in place of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2594160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
To solve $ \frac{1}{a}+\frac{1}{a+b}+\frac{1}{a+b+c}=1$ in natural numbers. I really stuck on the following 7th grade problem (shame on me).
The problem asks to solve an equation in natural numbers (that is to find all possible such natural $a,b,c~$ that)
$$ \frac{1}{a}+\frac{1}{a+b}+\frac{1}{a+b+c}=1$$
I have fou... | The above mentioned equation shown below has parametric solution given below:
$\frac{1}{a}+\frac{1}{a+b}+\frac{1}{a+b+c}=1$
$(a,b,c) =[w(-1-k),w(1+2k),w(k)]$
where, $w=(k+3)/[2k(k+1)]$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2594282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
single valued function and multi valued function square root function is a multi valued function. for example 16 have two square root 4 and -4. but it is violation of the definition of function that each element of domain has one and only one image. then how it is a function. please clear my dout.
| $f:[0,\infty)\to[0,\infty)$ defined by $f(x)=\sqrt{x}$ assigns to each $x$ a single value, the positive square root. This is a function.
Similarly, $-f$ is a function.
Functions cannot output more than one value for a given input.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2594410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
$K\subseteq X$ compact, C closed, can I say something about $K \cap C$? If C is closed then C is compact as well, but the intersection of two compact set is not compact in general. It would be nice to have this intersection to be compact though, any idea if this is true?
| $K \cap C$ is closed in $K$, by the definition of subspace topology, so compact. A closed subset of a compact space is always compact, no extra assumptions needed.
$K \cap C$ need not be closed if $K$ is not closed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2594566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to color a map, so that one color covers the maximum area. Given a graph $G$ where every node resembles a number. How would one color this graph, so that no two connected nodes have the same color and that one color has the maximum of area? Area is defined by the number inside the node.
For this example problem I ... | I believe I worked out a solution to the example problem and a decent method to get there. Copy of the problem:
First I decided that it wasn't needed to label all of them with the same color. There's is only one color we're actually interested in. So we can label every node with R (Red) or O (Other).
For the 8 we have... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2594656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Convergence of Series $\sum_{n=1}^{\infty} \left[ \frac{\sin \left( \frac{n^2+1}{n}x\right)}{\sqrt{n}}\left( 1+\frac{1}{n}\right)^n\right]$
Consider the series $\displaystyle{ \sum_{n=1}^{\infty} \left[
\frac{\sin \left( \frac{n^2+1}{n}x\right)}{\sqrt{n}}\left(
1+\frac{1}{n}\right)^n\right]}$ . Find all points at wh... | Pointwise convergence: Fix $x.$ Our series equals
$$\tag 1 \sum_{n=1}^{\infty} \frac{\sin(nx+x/n)-\sin (nx)}{\sqrt n}(1+1/n)^n + \sum_{n=1}^{\infty} \frac{\sin (nx)}{\sqrt n}(1+1/n)^n.$$
Now by the MVT,
$$|\sin(nx+x/n)-\sin (nx)| = |(\cos c)(x/n)| \le |x/n|.$$
So in the first series in $(1),$ the sum of the absolute va... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2594753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Evaluating $\lim_{a,b\to + \infty} \iint_{[0,a]\times[0,b]}e^{-xy} \sin x \,dx\,dy$ I'm trying to calculate the following:
$$\lim_{a,b\to + \infty} \iint_{[0,a]\times[0,b]}e^{-xy} \sin x \,dx\,dy$$
Trying to calculate by definition didn't get me far. Any ideas how to attack this problem?
| Let $I=\int_0^b\sin xe^{-xy}dx$. Let $u=\sin x,\,dv=e^{-xy}dx$. Then $$I=\int_0^b\sin xe^{-xy}dx=\frac{\sin xe^{-xy}}{-y}\bigg|_0^b+\frac{1}{y}\int_0^b\cos xe^{-xy}dx.$$
Now apply by parts once again with $u=\cos x,\, dv=e^{-xy}dx$
we get$$I=\int_0^b\sin xe^{-xy}dx=\frac{\sin xe^{-xy}}{-y}\bigg|_0^b-\frac{\cos xe^{-xy}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2594858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Show that $e^{1-n} \leq \frac {n!}{n^n}$ How can I show that for a $n \in \mathbb N$
$$e^{1-n} \leq \frac {n!}{n^n}$$
I tried using the binomial theorem like this
$$n^n \le (1+n)^n = \sum_{k=0}^n \binom nk n^k \le \sum_{k=0}^\infty \binom nk n^k = \sum_{k=0}^\infty \frac{n!}{k!(n-k)!} n^k \le \sum_{k=0}^\infty \frac{n!... | Using induction we see that for $n=1$, the inequality holds. Assume that it holds for some number $k$.
Then, using $\left(1+\frac1k\right)^k<e$, we find that
$$\begin{align}
\frac{(k+1)!}{(k+1)^{k+1}}&=\frac{k!}{k^k\left(1+\frac1k\right)^k}\\\\
&\ge \frac{e^{1-k}}{e}\\\\
&=e^{1-(k+1)}
\end{align}$$
And we are done!... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2594928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
} |
Solving $\frac{1}{|x+1|} < \frac{1}{2x}$ Solving $\frac{1}{|x+1|} < \frac{1}{2x}$
I'm having trouble with this inequality. If it was $\frac{1}{|x+1|} < \frac{1}{2}$, then:
If $x+1>0, x\neq0$, then
$\frac{1}{(x+1)} < \frac{1}{2} \Rightarrow x+1 > 2 \Rightarrow x>1$
If $x+1<0$, then
$\frac{1}{-(x+1)} < \frac{1}{2} \Righ... | So from first case where $x\in(-1,\infty)$, you get the permissible via of $x$ to be $$(-1,\infty)\cap(-\infty,1)=(-1,1)$$
In the second case where $x\in(-\infty,-1)$ you get the permissible via of $x$ to be $$(\infty,-1)\cap\left(-\infty,-\frac13\right)=(-\infty,-1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2595010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Show that $2x^5+3x^4+2x+16$ has exactly one real root It's clear that this function has a zero in the interval $[-2,-1]$ by the Intermediate Value Theorem. I have graphed this function, and it's easy to see that it only has one real root. But, this function is not injective and I'm having a very hard time proving that ... | Exploiting Descartes law of signs is the way to go, anyway there is a (longer) alternative which consists in studying the variations of $f$ starting by its second derivative which is easily factorisable.
$f(x)=2x^5+3x^4+2x+16 $
$f'(x)=10x^4+12x^3+2=2(x+1)(5x^3+x^2-x+1)$
$f''(x)=40x^3+36x^2=4x^2(10x+9)$
So we can start ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2595102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Find $f(2^{2017})$ The function $f(x)$ has only positive $f(x)$. It is known that $f(1)+f(2)=10$, and $f(a+b)=f(a)+f(b) + 2\sqrt{f(a)\cdot f(b)}$. How can I find $f(2^{2017})$?
The second part of the equality resembles $(\sqrt{f(a)}+\sqrt{f(b)})^2$, but I still have no idea what to do with $2^{2017}$.
| $f(2a) = f(a+a) = f(a) + f(a) + 2\sqrt{f(a)f(a)} = 4f(a)$
By induction $f(2^k)=4f(2^{k-1}) = 4^2f(2^{k-1}) = 4^kf(1)$.
$f(1) + f(2) = f(1) + 4f(1) = 5f(1) = 10$.
So $f(1) = 2$.
So $f(2^k) = 4^kf(1) = 2*4^k$
$f(2^{2017} = 2*4^{2017}=2^{4035}$
====
What what be interesting is what other values are.
$f(3) = f(1) + f(2) + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2595273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Is this a combinatorial identity: $ \sum_{k=1}^{n+1}\binom{n+1}{k} \sum_{i=0}^{k-1}\binom{n}{i} = 2^{2n} $? $$
\sum_{k=1}^{n+1}\left(\binom{n+1}{k} \sum_{i=0}^{k-1}\binom{n}{i}\right) = 2^{2n}
$$
This is my first question, please feel free to correct/guide me. While solving a probability problem from a text book l redu... | The combinatorial interpretation provided by Lord Shark is really nice and elementary, but there also is a brute-force way to proving such identity.
$$ \sum_{k=1}^{n+1}\left[\binom{n+1}{k}\sum_{i=0}^{k-1}\binom{n}{i}\right]=\sum_{a=0}^{n}\left[\binom{n+1}{a+1}\sum_{b=0}^{a}\binom{n}{b}\right]=\sum_{0\leq b\leq a\leq n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2595398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
What does $2^π$ : the multiplication of $2 \pi$ many times? This is rather an intuitive question, in the sense that indeed a real number raised to the power of an irrational doesn't make sense but I wanted to know what does it mean intuitively. If $2^7=2*2*2*2*2*2*2, 2^{\frac{1}{2}}=\frac{1}{\sqrt{2}}$ and $a^b=a*a*a*a... | Start with $2^3=8$. Then think about what $2^{3.1}$ is:
$$2^{3.1} = 2^{31/10} = \sqrt[10]{2^{31}} \approx 8.574187700.$$
Then consider
$$2^{3.14} = 2^{314/100} = \sqrt[100]{2^{314}} \approx 8.815240927.$$
Since the sequence $3, 3.1, 3.14, 3.141, \ldots$ approaches $\pi,$, the
sequence
$$2^3, 2^{3.1}, 2^{3.14}, 2^{3.1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2595558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 0
} |
Factoring the polynomial $3(x^2 - 1)^3 + 7(x^2 - 1)^2 +4x^2 - 4$ I'm trying to factor the following polynomial:
$$3(x^2 - 1)^3 + 7(x^2 - 1)^2 +4x^2 - 4$$
What I've done:
$$3(x^2 -1)^3 + 7(x^2-1)^2 + 4(x^2 -1)$$
Then I set $p=x^2 -1$ so the polynomial is:
$$3p^3 + 7p^2 + 4p$$
Therefore: $$p(3p^2 + 7p + 4)$$
I apply Cro... | Taking $ (x^2 - 1) $ as common, you get:
$$ (x^2 - 1) (3(x^2 - 1)^2 + 7(x^2 - 1) + 4) $$
$$ = (x+1)(x-1)( 3 (x^2-1)^2 + 3(x^2 - 1) + 4(x^2 - 1) + 4 )$$
$$ = (x+1)(x-1)( 3(x^2 - 1) + 4 ) ( x^2 - 1 + 1 ) $$
$$ = (x+1)(x-1)( 3x^2 - 3 + 4)(x^2) $$
$$ = x^2(x+1)(x-1)(3x^2 + 1) $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2595704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Derivative of $e^{2x^4-x^2-1}$ with limit definition of derivative Let $f:\mathbb{R}\to\mathbb{R}$ be defined as
$f(x) = e^{2x^4-x^2-1}$.
I have to find the derivative using the defintion:
$$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$
My approach:
$$
\begin{align}
&\lim_{h \to 0} \frac{\exp({2(x+h)^4-(x+h)^2-1})-\ex... | Hint
Factor the term $e^{2 x^4 - x^2 - 1}$ in each of the two terms of the difference. Then you get the product
$$\frac{f(x+h)-f(x)}{h}=e^{2 x^4 - x^2 - 1} \times \frac{e^{hA(x,h)}-1}{h}$$
Where $A(x,h)=h(8x^2 -x) +h^2 B(x,h)$ with $A,B$ polynomials.
Then as $e^{y} -1 \approx y$ around $0$, you get
$$e^{2 x^4 - x^2 - 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2595841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Subgroups and ideals of integer numbers. Let $(\mathbb{Z},+)$ be the additive group of integers, and $(\mathbb{Z}, +, \cdot)$ the ring of integers. By definition, every ideal of $(\mathbb{Z}, +, \cdot)$ is a subgroup of $(\mathbb{Z},+)$. Is the opposite true? Is every subgroup of $(\mathbb{Z},+)$ also an ideal of $(\ma... | The equivalence follows from the following two (easy to prove) results:
*
*The ideals of a commutative ring $R$ are precisely the $R$-submodules of $R$ as a module over itself.
*The Abelian groups are essentially $\mathbb Z$ modules, in the sense that the notions of group homomorphisms, subgroups, quotients, map to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2595987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Proving Identity for Derivative of Determinant For a square matrix $A$ and identity matrix $I$, how does one prove that $$\frac{d}{dt}\det(tI-A)=\sum_{i=1}^n\det(tI-A_i)$$ Where $A_i$ is the matrix $A$ with the $i^{th}$ row and $i^{th}$ column vectors removed?
| Here is one way to see this:
Note that the map $\phi(t_1,...,t_n) = \det ( \sum_k t_k e_k e_k^T -A)$ is smooth, and if $\tau(t) = (t,....,t)$ then $f(t)=\det (tI-A) = \phi(\tau(t))$.
In particular, $f'(t) = \sum_k {\partial \phi(t,....,t) \over \partial t_k}$.
If we adopt the notation $\det B = d(b_1,...,b_n)$, where $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2596098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Convex envelop of Tr(XY)? How would one go about calculating the convex envelop of $f(X,Y) = Tr(XY)$, where both $X \in R^{ n \times n}$ and $Y \in R^{ n \times n}$ define the domain of $f$ and are both symmetric PSD? I am trying to calculate a global under-estimator of $f$.
| It's well known that $\mbox{tr}(XY) \geq 0$ for all PSD pairs $(X,Y)$. Unfortunately, you can't do any better than that in finding the convex envelope. Let $g(X,Y)$ be the convex envelope of $f(X,Y)$. I claim that $g(X,Y)=0$.
Take any pair $(X,Y)$ in the domain of $f$. The pairs $(2X,0)$ and $(0,2Y)$ are also in th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2596238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$Re(f)=Re(g)$ implies $f(z)=g(z)+ic$ Let $f$ and $g$ be analystic in a region $G$. If $\Re(f)=\Re(g)$ in $G$, then prove
$$f(z)=g(z)+ic \tag{for all $z \in G$}$$
where $c$ is a real constant.
By cauchy riemann we have
$$\int \frac{\partial}{\partial y}Re(g)=-\int\frac{\partial}{\partial x}Re(g)$$
Hence, $Im(f)$ must be... | Notice that $f-g$ has values in $i\mathbb{R}$ and if $f-g$ is non-constant, then its image must be open in $\mathbb{C}$, which is a contradiction. Whence the result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2596329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Complex exponential squared does not equal sinusoid squared? I just noticed this today and I'm a bit confused by it.
If we represent cos(x) as the real part of exp(ix), then I always thought that we could then say that cos(x)^2 is equal to the real part of exp(ix)exp(ix)=exp(2*i*x). However, this clearly cannot be cor... | You are not violating a rule, rather, you are making up a rule which does not exist. You are assuming that the real part of $z^2$ is (the real part of $z$), squared. But in fact if we write $z=a+ib$ then
$$z^2=a^2-b^2+2iab\ ,\quad \Re(z^2)=a^2-b^2$$
while
$$\Re z=a\ ,\quad (\Re z)^2=a^2\ ,$$
and these are not equal (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2596415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Explicit formula for a reversible function $f: [0,1] \rightarrow \mathbb{R}$ There is a bijection between $[0,1]$ and $\mathbb{R}$ (because they have a same cardinality). Can we write an explicit formula for such a function? (or at least a reversible function $f$ whose domain is $[0,1]$ and its range cover all real num... |
or at least a reversible function $f$ whose domain is $[0,1]$ and its range cover all real numbers?
I'm not sure how is that different...
Anyway I assume that by "explicit" you mean "there is an algorithm that for a given $x$ it can calculate $f(x)$ using a given set of elementary operations (whatever that means)". O... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2596549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How do I calculate the average expected value of the dice bowl game? I am trying to better understand the "dice bowl game" from Goldratt's The Goal.
In the book, five kids (named Andy, Ben, Chuck, Dave and Evan) each start with an empty bowl, and a 6-sided die. There is a box of matches next to Andy.
For 10 rounds, eac... | Let $R_i, i = 1, 2, \ldots n$ be the die value of the $i$-th kid, where $n$ is the total number of kids participating in the game. Assuming the die are fair, we have $R_i$ are i.i.d. discrete uniform random variables on $\{1, 2, 3, 4, 5, 6\}$. Let $T_i$ be the number of matches taken from the previous bowl (or matchbox... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2596663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $b_n$ is convergent, then $a_n$ is also convergent
Let $(a_n)_{n\geq 1}$ and $(b_n)_{n \geq 1}$ be two sequences of real numbers such that $$b_n=a_{n+2}-5a_{n+1}+6a_{n}, \: \forall n \geq 1$$
Prove that if $(b_n)$ is convergent, then $(a_n)$ is also convergent.
I defined $c_n=a_{n+1}-2a_n$ and the relation becam... | Setting $b_n = 1$ (which is clearly convergent) and $a_1 = a_2 = 1$, we get
$$
a_n = \frac16(3\cdot 2^n - 3^n + 3)
$$
(WolframAlpha calculation), which doesn't converge.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2596747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
} |
What does the vector space R^[0,1] mean? While reading a book Linear Algebra Done Right, I came to knew that a vector space $\mathbf{R}^n$ represents a space with dimensions as $(x_1, x_2, ...,x_n)$, but there were other vector spaces that I could not understand.
There was a statement as Ref: 1.35
The set of continuou... | The set $\mathbb{R}^{[0,1]}$ is the set of all functions from $[0,1]$ into $\mathbb R$ and the set $\mathbb{R}^\mathbb{R}$ is the set of all functions from $\mathbb R$ into itself.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2596832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
How can we visualize that $2^n$ gives the number of ways binary digits of length n? Like if we have to find the number of ways can be represented in bits up to 4 places. We use $2^4$, but why do we use this method?
| Because for each bit you have 2 different simbols.
Then if $n$ denotes the number of bits, this means that we have to make a choice between $1$ or $0$ for $n$ times is:
$$\underbrace{2\cdot2\cdot2\cdot\dots\cdot2}_{n\text{-times}}=2^n$$
This happens because every choice is independent from the others.
If you want you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2596918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Is my reasoning correct (regarding minimal distance on graph)
Question: Given two point $A(0,10)$ and $B(30,20)$, find the point $P$ on x axis for which sum of distances from given points to the required point is minimum
Now i could form a lengthy equation and use differentiation but instead I decided to do somethi... | Consider the "conjugate" of $B$: $B'=(30,-20)$.
Note that $P$ is the intersection of $x$-axis with the line that joins the points $A$ and $B'$ (by triangular inequality)
This line is $y=-x+10$ and $P=(10,0)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2597019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
If $i$ is a root, then $-i$ is also a root? I have the following question:
Prove that i is a roof of the equation $g(z) = 0$, where $g(z) = z^3 - 3z^2 + z - 3$. Find the other roots of this equation.
The answer says without any working out:
Since i is a root, then -i is a root. (The answers are $[z - i], [z + i]$, and ... | Any polynomial with real coefficients with root $r = a+ib$ also has root $\bar{r} = a-ib$. This is known as the conjugate root theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2597154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Function $u$ solves the heat equation $\Longrightarrow$ $\langle x, \nabla u \rangle + 2t u_t$ solves the heat equation I am baffling with this homework problem:
Assume that a smooth function u solves the heat equation $u_t − \Delta u = 0.$ Show that also the function
$v(x, t) = \langle x, ∇u(x, t) \rangle + 2tu_t (x, ... | For $\langle \cdot, \cdot \rangle $ denoting the Euclidian scalar product you get
$$v_t = \nabla u \cdot \boldsymbol{x}_t + \boldsymbol{x} \cdot \partial_t \nabla u + 2 (u_{tt} + u_t) = \boldsymbol{x} \cdot \partial_t \nabla u + 2 (tu_{tt} + u_t) \tag{1}$$
Use Green's Vector identity for the $\Delta \langle \boldsymbol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2597242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Does the sum of exponents of a symbol in a word of a free group have a specific name in the literature? Let $A$ be a non-empty set and let $F(A)$ be the free group it generates. An element of $F(A)$ is of the form
$$w = a_1^{\varepsilon_1}a_2^{\varepsilon_2}\cdots a_{n-1}^{\varepsilon_{n-1}}a_n^{\varepsilon_n}$$
where ... | If $B$ is a subset of $A$, I would call projection from $F(A)$ onto $F(B)$ the morphism $\pi_B: F(A) \to F(B)$ defined, for each $a \in A$, by
$$
\pi_B(a) =
\begin{cases}
a &\text{if $a \in B$}\\
1 &\text{otherwise}
\end{cases}
$$
In particular, your morphism would be the projection from $F(A)$ onto $F(b)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2597474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How would I solve the following congruence? $5x \equiv 118 (\mod 127)$
This is what I have done so far:
$127 = 25*5+2$
$2 = 127-25*5$
$25 = 12*2+1$
$1 = 25-12*2$
$1 = 25-12(127-25*5)$
I am a little stuck on how to continue.
I know I am supposed to write it in a form such that $5v+127w = 1$ but I am not exactly sure how... | To solve it as a naive, start from your second line of solution:
$$127-25 \times5=2 \tag{1}$$ Now$$118/2=59$$
Multiply (1) by 59:-
$$59\times 127-5\times 1475=118$$
All the solutions will be of the form $$x=59+\frac{1475k}{1}\tag{2.}$$and
$$y=-1475+\frac{59k}{1} \tag{3.}$$for $127x+5y=118$.
Or you can adopt extended e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2597599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 8,
"answer_id": 4
} |
Fundamental theorem of algebra applied to $y=x^2$ The fundamental theorem of algebra seems to indicate that there should be two roots for $y=x^2;$ however, applying the quadratic formula leads to just $0.$ Considering that $+0$ and $-0$ make no sense, what is the reasoning behind this superficial contradiction?
| I don't know what exact statement of the Fundamental Theorem of Algebra you're using. The Fundamental Theorem of Algebra guarantees at least one root. Sometimes you'll see it stated that for a polynomial degree $n$, there are $n$ roots (some with multiplicity), but multiplicity is defined precisely by how many times th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2597682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
$\frac{1}{1+x^2}$'s Taylor expansion about a point $a\in\mathbb{R} $, given by $f(x) = \sum_{n = 0} ^ \infty$ $a_n (x -a)^n$. Radius of convergence? Let $f(x) = \frac{1}{1+x^2}$. Consider its Taylor expansion about a point $a \in \mathbb{R}$ given by
$$f(x)=\sum_{n = 0}^\infty a_n(x-a)^n.$$
What is the radius of conver... | I do not know if there is a simple proof by real analytic methods but it is elementary when you consider the complex function $\frac 1 {1+z^{2}}$. The largest disc around a on which the function is analytic has radius $(1+a^{2})^{1/2}$, the distance from a to the points $i,-i$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2597799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Uniform convergence of $\int_0^{+\infty}\frac{1-\cos{\alpha x}}{x}e^{-\lambda x}dx$ Please help me to solve of the following problem:
Let $\alpha, \lambda>0$. Prove uniform convergence of improper
integral with respect to $\alpha$:
$$\int_0^{+\infty}\frac{1-\cos{\alpha x}}{x}e^{-\lambda x}dx$$
My attempt:
I think... | Condition 2 is not fulfilled because, loosely speaking, on average $\sin ^2(\frac{\alpha x}2)$ equals $\frac 12$ and the integral of $\frac 1{2x}$ diverges for $x\rightarrow\infty$.
It might be easier to fall back on the definition of uniform convergence and find an upper bound for $\int_B^\infty \frac{1-\cos \alpha x}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2598032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Product of 2-digit numbers
31 people are dancing in a circle formation. Each people’s age is a 2-digit number, and for each digit we know that the units’ digit is equal to the tens’ digit of the person who is in the clockwise position, while the tens’ digit is equal to the units’ digit of the person who is in the coun... | It's possible. To restate the goal, we want a cycle of digits $d_1 \to d_2 \to d_3 \to \ldots \to d_{31} \to d_1$, such that
$$
(d_1 d_2) (d_2d_3) (d_3d_4) \ldots (d_{31} d_1)
$$
is a perfect square.
We can build this up by combining small cycles. First,
$$2 \to 7 \to 2$$
is a nice move, as it gives $27 \cdot 72$ -- di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2598144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
$2^p -1$ then $p$ is a prime I have been trying to solve this following problem:
If $2^p-1$ is a prime then prove that $p$ is a prime, where $p \geq 2$.
which way should I go to prove this? Using fermat's or Bezout's theorem? or is it something else?
This is what I managed to come up with,I am confused about it and ... | If $p=mn$, where $m>1$ and $m>1$ then $2^{mn}-1$ is divided by $2^m-1$ and by $2^{n}-1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2598242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Path in a directed graph Let $G$ be a directed graph with $2^k$ vertices where there is exactly one edge between each two vertices. Prove that that regardless of the directions (orientations) of the edges there exist a path in $G$ which goes through $k+1$ unique vertices.
I know that there are $\binom{2^k}2$ edges, but... | Hint: Try splitting the graph in half, and consider induction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2598338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove that $XY$ crosses the midpoints $\triangle ABC$ has altitudes $AD$, $BE$, $CF$. The reflections of $E$, $F$ in $H$ are $E'$, $F'$. The circle $DE'F'$ intersects $BE$, $CF$ at $X$, $Y$. Prove that $XY$ goes through the midpoints of $AB$, $AC$.
I can show that $XY$ is parallel to $BC$ by simply angle-chasing. $EYF... | To illustrate the idea of the solution, consider the following picture first, where the mid points $B_1$, $C_1$ of $CA$, $AB$, and the reflection $D'$ of $D$ in $H$ are also present.
We want to show that the points $X,H,D,B_1$ are on a cycle. This would be enough to conclude, since from here we are allowed to write th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2598445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
If $5|P(2),2|P(5)$ then which of the followings divides $P(7)$?
Suppose that $P(x)$ is a polynomial with integer coefficients.If $5|P(2),2|P(5)$ then which of the followings divides $P(7)$?
a)10
b)7
c)3
d)4
e)8
$5|P(2),2|P(5)\Rightarrow5-2|p(5)-p(2)$ but this doesn't help.I don't know any property of poly... | Meta-solution
There are no restriction on the degree of your polynomial.
If you take the constant polinomial $P(x)=10$ the conditions required are satisfied.
So the only coherent answer is $a)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2598540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Continuous surjective map Does there exist a continuous surjective map from $\{(x, y)\in \mathbb{R^2} | x^2-y^2=1\}$ to $\mathbb{R}$?
Any help would be greatly appreciated. I was trying to work with taking one branch of the hyperbola and restricting the function to it. But I cannot understand how to proceed much furthe... | Yes! Ascribe points of hyperbola with $x>0$ to positive numbers and points with $x<0$ to negative ones. Also map $(1,0)$ and $(-1,0)$ to 0.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2598621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Bridgeland-Stability Condition: Why is the Harder-Narasimhan filtration unique? I'm trying to understand Bridgeland's notion of stability condition on a triangulated category as defined in Definition 1.1 of the paper Stability conditions on triangulated categories.
See also Definition 3.3 in the same paper.
The decompo... | The short answer is that the filtrations are unique because they arise as factorization of an orthogonal, multiple factorization system: see here §4.1.
The even shorter answer is that you can prove it for triangulated categories, but it's unsatisfying.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2598710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Show $f$ is not differentiable at $x = 0$.
Define $f(x) = \begin{cases}
x & x\in \mathbb{Q} \\
0 & x \in \mathbb{R}\setminus\mathbb{Q}
\end{cases}$
Is $f$ differentiable at $0$? Is $g(x) = xf(x)$ differentiable at $0$?
Is my reasoning correct? Is there a simpler way to prove $f'(0)$ does not exist? I... | For $h \neq 0$ you have
$$\frac{f(h)-f(0)}{h-0}=\begin{cases}
1 & \text{for } h \in \mathbb Q\\
0 & \text{for } h \in \mathbb{R}\setminus\mathbb{Q}
\end {cases}$$
So $\lim\limits_{h \to 0} \frac{f(h)-f(0)}{h-0}$ can’t exist as a limit is unique. Hence $f$ is not differentiable at $0$.
$g$ is differentiable at $0$ and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2598902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Probability and Combinatorial Group Theory. If this is too broad or is otherwise a poor question, I apologise.
I learnt recently that the probability that two integers generate the additive group of integers is $\frac{6}{\pi^2}$.
What other results are there like this?
I'm looking for any results of probability appli... | For a finite group you can compute the probability that any two random elements will commute by looking at certain conjugacy classes.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2598998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 1
} |
How to prove a problem is #P-complete? What are the major steps in proving a problem is #P-complete?
For example, I know that showing a problem is NP-complete requires (i) showing the problem is in NP by giving a polytime verification algorithm, (ii) showing an existing NP-hard problem is polytime reducible to the give... | A good treatment of the complexity of counting problems may be found in Moore and Mertens and Arora and Barak. In summary, the common approach to proving #P-completeness is through reduction. Given a known #P-complete problem $P_1$, if we can find an appropriate reduction of $P_1$ to $P_2$, then $P_2$ is also #P-comp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2599133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
To use trigonometric identities to find the value of $\tan\alpha$ I'm stuck on this problem:
If $\alpha+\beta=\frac{\pi}{2}$ and $\beta+\gamma=\alpha $ then find the value of $\tan\alpha$
I have tried isolating $\alpha$ but ended up getting $\tan\alpha = \frac{1}{\tan\beta}$ and $\tan\beta=\frac{1 - \tan\frac{\gamma}... | $$\alpha+\beta=\dfrac\pi2$$
$$\alpha-\beta=\gamma$$
Add to find $\alpha$ in terms of $\gamma$
Then apply $$\tan(x+y)$$ formula
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2599247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Understanding cutting plane proofs I am trying to understand the following definition:
Def: A cutting plane proof from the system $Ax\leq b$ for an inequality $c^Tx\leq d$ is a sequence of inequalities $c_i^T\leq d_i$, $(i = 1,\ldots, k)$ with the following properties
i) every $c_i$ is integral,
ii) $c_k = c$ and $d_k ... | Suppose $Ax\le b$ is the following system:
$$(1): -x_1-3x_2\le -3\\
(2): 8x_1+3x_2\le 24\\
(3): -2x_1+x_2\le 1 $$
The solution set is represented by the following graph:
Suppose we want to derive $c^Tx\le b$ where $c^T=(0,1),b=3$, i.e., $x_2\le 3$ from $Ax\le b$.
From the picture, the problematic point is the intersec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2599391",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
On weak star convergent sequences of functions that are not strongly convergent anywhere Can we find example of a sequence of functions $g_n:(0,1)\rightarrow\mathbf{R}$ such that $g_n$ converges weak-star in ${\rm L}^{\infty}(0,1)$ as $n\rightarrow+\infty$, but such that for every measurable set $A\subseteq (0,1)$ such... | Remark on previous comment by pozz: I point out that constant functions actualy have bounded oscillation property. In this previous comment we need to take into account that ${\rm sup}_{J^n_i}g_n\geq b$ is never satisfied, provided $g_n=c$, and provided we choose $b>c$. So there exists no sub-interval at all which sati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2599543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
The Cartesian product of two regular graphs is regular Consider two regular graphs $G_1$ and $G_2$ of degrees $d_1$ and $d_2$, respectively. I would like to prove that the Cartesian product $G_1 \square G_2$ is regular of degree $d_1 + d_2$.
Currently I have a proof that revolves around the following facts:
*
*The a... | There's a far more elementary proof, using just the definition of the Cartesian product. Let $G_1 = (V_1, E_1)$ be $d_1$-regular and let $G_2 = (V_2, E_2)$ be $d_2$-regular. Recall that the Cartesian product is defined as $ G_1 \square G_2 = (V_1 \times V_2, E) $ where $\times$ is the ordinary Cartesian product (on set... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2599645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Intersection of two cylinders What is the easiest way to find the area of the surface created when the cylinder $$x^2+z^2=1\text { intersects the cylinder }y^2+z^2=1.$$
I have used double integrals and also line integrals to find the area of the surface created when the cylinder $x^2+z^2=1$ intersects $ y^2+z^2=1.$
... | \begin{align}
R &= \{ 0<x<1,-x<y<x,z=\sqrt{1-x^2} \} \\
S &= 8\iint_R dA \\
z &= \sqrt{1-x^2} \\
z_x &= -\frac{x}{\sqrt{1-x^2}} \\
z_y &= 0 \\
dA &= \sqrt{1+z_x^2+z_y^2} \, dx \, dy \\
&= \frac{1}{\sqrt{1-x^2}} \, dx \, dy \\
S &= 8\int_{0}^{1} \int_{-x}^{x} \frac{1}{\sqrt{1-x^2}} \, dy \, dx \\
S &= ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2599749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solving a system of third order homogenous ODEs I am trying to solve this third order system of homogenous ODEs.
*
*$x'''=2x+y$
*$y'''=x+2y$
Initial conditions are given as well.
Higher order systems weren't covered in the lectures hence I am a bit lost. Nor can I find any similar questions asked on this site. I h... | Because of the symmetric cyclic/circulant nature of the system matrix, you can decouple the system by Fourier transform. Set $u=x+y$, $v=x-y$ to get the system
$$
u'''=3u\\
v'''=v
$$
which can be both solved with standard methods for linear ODE with constant coefficients.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2599883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
find the limits $\lim_{x\to \infty} \lfloor f(x) \rfloor=? $ let $f(x)=\dfrac{4x\sin^2x-\sin 2x}{x^2-x\sin 2x}$ the fine the limits :
$$\lim_{x\to \infty} \lfloor f(x) \rfloor=? $$
$$f(x)=\dfrac{4x\sin^2x-\sin 2x}{x^2-x\sin 2x}=\dfrac{2\sin x(2x\sin x-\cos x)}{x(x-\sin2x)}$$
what do i do ?
| A more heuristic argument is as follows: The key observation is that $\sin u$ is always between $-1$ and $1$, no matter what $u$ is. Consider the impact of this on the numerator. We have
\begin{align}
\text{numerator} & = 4x \times \text{(something between $-1$ and $1$)} - \text{(something between $-1$ and $1$)} \\
&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2599997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Equivalent of a remainder I want to find en equivalent of the rest of the convergent series for $n \in \mathbb{N}^{*}$
$$
R_n=\sum_{k=n+1}\frac{1}{k^2\ln\left(k\right)}
$$
I used that $\displaystyle x \mapsto \frac{1}{x^2\ln\left(x\right)}$ is positive, decreasing and tending to $0$ for $x \in \left[2,+\infty\right[$.
... | Since $\log n$ is approximately constant on short intervals and $\sum_{k>n}\frac{1}{k^2}$ behaves like $\frac{C}{n}$, it is reasonable to expect that $R_n \sim \frac{C}{n\log n}$. And by the Stolz-Cesàro theorem we have
$$ \lim_{n\to +\infty}\frac{R_n}{\frac{1}{n\log n}}=\lim_{n\to +\infty}\frac{R_n-R_{n+1}}{\frac{1}{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2600081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Resolvent estimate self-adjoint operator Let $A:D(A)\longrightarrow H$ be an unbounded self-adjoint (or normal) operator on a Hilbert space $H$.
Then we know that $\sigma(A) \neq \emptyset$ and
$$\|(\lambda-A)^{-1}\|=\frac{1}{d(\lambda,\sigma(A))}, \quad \forall \lambda \in \rho(A),$$
where $d(\lambda,\sigma(A))=\min_{... | The following is exact:
$$
\|A(\lambda I-A)^{-1}\|=\sup_{\mu\in\sigma(A)}\left|\frac{\mu}{\lambda-\mu}\right|
% \\ = \sup_{\mu\in\sigma(A)}\left|-1+\frac{\lambda}{\lambda-\mu}\right|.
$$
If $\sigma(A)=\mathbb{R}$ and $\lambda=i$, then the above gives
$$
\|A(\lambda I-A)^{-1}\| = \sup_{\mu\in\mathbb{R}}\frac{|\m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2600213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Wrting $\operatorname{Eq}\bigl(\prod_{i\in I}X_i\rightrightarrows\prod_{j\in J}Y_j\bigr)$ as a limit of a single diagram Let $D\colon \mathcal{I}\longrightarrow\mathcal{C}$ be a small diagram in the category $\mathcal{C}$. It is rather well known that a limit of $D$ can be computed as the equalizer of the two maps $$\p... | A more elementary example.
Take the category of vector spaces over your favourite field.
Writing $V^*$ for the dual of $V$,
$$\text{lim}_i(V_i^*)\cong(\text{colim}_iV_i)^*,$$
so every limit $\text{lim}_iV_i$ of finite dimensional vector spaces is a dual, $(\text{colim}_iV_i^*)^*$, and so can’t have countably infinite ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2600315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to calculate $\lim_{n\to \infty } \frac{n^n}{n!^2}$?
How to calculate $\lim_{n\to \infty } \frac{n^n}{n!^2}$?
I tried to solve this by using Stirling's formula but I got stuck at $\lim\limits_{n\to \infty } \frac{e^{2n}}{n^{n+1}}$. Any help?
| Most of the time you do not need the exact Stirling approximation, just use the simplest:
Factorial Inequality problem $\left(\frac n2\right)^n > n! > \left(\frac n3\right)^n$
It ensues that $\dfrac{(n!)^2}{n^n}>\left(\dfrac n9\right)^n\to\infty$ so the inverse is going to $0$.
Anyway regarding the point where you ar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2600424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Integral of $\int_{-\infty}^\infty\frac{1}{t}e^{-\frac{t^2}{4}}\int_0^{\frac{t}{2}}e^{u^2}dudt$ How to calculate the following integral:
$$\int_{-\infty}^\infty\frac{1}{t}e^{-\frac{t^2}{4}}\int_0^{\frac{t}{2}}e^{u^2}dudt\ \ ?$$
I don't understand how to calculate. Please help. Thank you.
| With the substitution $u\to\frac{t}2x$, the inner integral becomes
$$\int_0^{\frac{t}{2}}e^{u^2}\,du=\frac{t}2\int^1_0e^{\frac{t^2}{4}x^2}\,dx,$$
implying
$$\int_{-\infty}^\infty\frac{1}{t}e^{-\frac{t^2}{4}}\int_0^{\frac{t}{2}}e^{u^2}\,du\,dt=\frac12\int_{-\infty}^\infty\int^1_0 e^{-\frac{t^2}{4}(1-x^2)}\,dx\,dt=\int_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2600528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Definition of canonical isomorphism between two vector spaces I'm reading "Linear Algebra via Exterior Products" by Winitzki. The part leading up to the canonical isomorphism definition firstly proves the result that an $n$-dimensional vector space $V$ over $\mathbb{R}$ is isomorphic to $\mathbb{R}^n$. Suppose the isom... | Consider the vector space of linear and constant polynomials with real coefficients:
$$V=\{a+bx: a,b \in \mathbb R\}.$$
Let's also consider the bases $B_1=\{1,x\}$ and $B_2=\{x,x+1\}$. If you express in each basis the polynomial $f(x)=2-3x$ you get
$$f(x)=2\cdot 1+(-3)\cdot x$$
and
$$f(x)=(-5)\cdot x+2\cdot (x+1).$$
Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2600640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Product of cdf and pdf of normal distribution.
A continuos random variable $X$ has the density
$$
f(x) = 2\phi(x)\Phi(x), ~x\in\mathbb{R}
$$
then
(A) $E(X) > 0$
(B) $E(X) < 0$
(C) $P(X\leq 0) > 0.5$
(D) $P(X\ge0) < 0.25$
\begin{eqnarray}
\Phi(x) &=& \text{Cumulative distribution function of } N(0,1)\\
\phi(x) &=& ... | This question requires no calculations.
You should not integrate anything to answer it.
The Key
If
$\phi(x)$ is the density function of a distribution $D$,
and
$\Phi(x)$ is the cumulative distribution function of $D$,
then
the density $f(x) = 2\phi(x)\Phi(x)$
corresponds to the distribution of a random variable $X$ def... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2600776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 0
} |
Showing measurability of composite function Let $p:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous map and $p':\mathbb{R}\rightarrow\mathbb{R}$ with $p'(x) = \left\{
\begin{array}{lr}
a & : x=0\\
p(x) & : x\neq 0.
\end{array}
\right.$
I want to show that $p'$ is Lebesgue measurable, i.e. for... | You have:
$$p^{-1}(L) = p^{-1}(\{a\} \cup L \setminus \{a\})= p^{-1}(\{a\}) \cup p^{-1}(L \setminus \{a\}) = \{0\}\cup p^{-1}(L \setminus \{0\}),$$
which enables to conclude as $\{b\}$ is the complement of the open set $(-\infty,b) \cup (b, \infty)$ for any $b \in \mathbb R.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2600897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
If $\bar f(x)$ irreducible in $\mathbb{Z}_2[x] \Rightarrow f(x)$ irreducible in $\mathbb{Q}[x]$? The example I'm working with is $\bar f(x) = x^4+x^3+\bar 1 \in \mathbb{Z}_2[x]$ , which I know is irreducible in $\mathbb{Z}_2[x]$. The text that I'm reading from seems to imply that my "if, then" statement in my title abo... | Not quite. For example, $2x^2 + x$ is irreducible in $\mathbb{Z}_2[x]$, but not in $\mathbb{Z}[x]$.
Here is a correct statement: if $f\in \mathbb{Z}[x]$, and the leading coefficient of $f$ is not divisible by $2$, then $\overline{f}$ irreducible implies ${f}$ irreducible over $\mathbb{Q}$.
To see this, suppose that $f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2601024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Irreducible polynomials over an integrally closed domain Let $A$ be an integrally closed domain, with quotient field $K_A$. My main question is the following:
Question. Does any non constant irreducible polynomial of $A[X]$ stays irreducible in $K_A[X]$ ?
Of course, this is true for monic non constant polynomials, so m... | Ok, finally the answer to the main question (and subquestion $2$) is no.
$A=\mathbb{Z}[i\sqrt{13}]$ is integrally closed, and $P=2X^2+2X+7$ is irreducible if I am not mistaken. However $2P=(2X+1+i\sqrt{13})(2X+1-i\sqrt{13})$, so in particular $P$ is not irreducible in $K_A[X]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2601124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Triangular Numbers and Perfect Squares
Prove that $n$ is a triangular number if and only if $8n+1$ is a perfect square.
I proved the easier part first (I think), that is, if $n$ is a triangular number then $8n+1$ is a perfect square.
I don't know where to start from for the other part, please help.
By the way, this w... | *
*If $n$ is a triangular number, then $n = \frac{k(k+1)}{2}$ and $8n+1 = 4k(k+1)+1 = (2k+1)^2$.
*If $8n+1$ is a perfect square, then $8n+1 = (2m+1)^2 \implies n = \frac{4m^2+4m+1-1}{8} = \frac{m(m+1)}{2}$ (because $8n+1$ is odd so it must be square of an odd number).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2601270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the greatest possible value for the difference in ratings. Seven experts evaluate the picture. Each of them makes an assessment - an integer number of points from 0 to 10 inclusive. It is known that all experts have put different points. According to the old rating system, the rating of the picture is the arithme... | Let the seven ratings be $a,b,c,d,e,f,g$ where, WLOG, $a<b<c<d<e<f<g$.
The old system would give $\frac{a+b+c+d+e+f+g}{7}$, whereas the new one would give $\frac{b+c+d+e+f}{5}$
We would like to maximize the absolute difference between the two, which is given by
\begin{align}
\ & |\frac{a+b+c+d+e+f+g}{7}-\frac{b+c+d+e+f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2601359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.