Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How to prove that ${l \choose a_1,...,a_n}\le n^{l-1} $ , when $a_1+...+a_n=l$. In the proof of (Corollary 8 chap. 3 ) in the book "Sobolev Spaces on Domains" by Burenkov the following inequality is used : given $a_1,...,a_n \in \mathbb{N}$ such that $a_1+...+a_n=l$, then
$${l \choose a_1,...,a_n}\le n^{l-1} $$
where ... | Assuming that $a_1,\ldots,a_n$ are distinct integers,
$$n\binom{l}{a_1,\ldots,a_n}\\=\binom{l}{a_1,a_2,\ldots,a_{n-1},a_n}+\binom{l}{a_2,a_3\ldots,a_{n},a_1}+\ldots+\binom{l}{a_n,a_1,\ldots,a_{n-2},a_{n-1}}$$
but the last sum is less than the sum of any multinomial coefficient $\binom{l}{x_1,x_2,\ldots,x_{n-1},x_n}$ wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1804703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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How is the entropy of the normal distribution derived? Wikipedia says the entropy of the normal distribution is $\frac{1}2 \ln(2\pi e\sigma^2)$
I could not find any proof for that, though. I found some proofs that show that the maximum entropy resembles to $\frac{1}2+\ln(\sqrt{2\pi}\sigma)$ and while I see that this ca... | You have already gotten some good answers, I thought I could add something more of use which is not really an answer, but maybe good if you find differential entropy to be a strange concept.
Since we can not store a real or continuous number exactly, entropy for continuous distributions conceptually mean something diff... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1804805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
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Suppose $0
Let a and b be real numbers, and suppose $0<a<b$.
Prove for all $n\geq 2$, $0< \sqrt[n]a< \sqrt[n]b$.
Proof: Suppose there exists an $n\geq 2$ such that $0 \geq \sqrt[n]a \text{ or } \sqrt[n]a \geq\sqrt[n]b$.
case $0 \geq \sqrt[n]a $:
Then $0 \geq a$, which contradicts our assumption that $0<a<b$.
case ... | You can prove this by showing that $\sqrt[n]{x}$ is monotonically increasing when $n\geq 2$.
Let $f(x)=\sqrt[n]{x}$, then $f'(x)=\frac{1}{n}x^{\frac{1-n}{n}}>0$ when $n\geq 2$, so $f(x)$ is monotonically increasing when $n\geq 2$. Also, $f(x)>0$ when $x>0$. Therefore, for all $n \geq 2, 0<\sqrt[n]{a}<\sqrt[n]{b}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1804905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove that $\frac {e^{b^2-1}}{b^2}$ ≥ 1 How to prove that $$\frac {e^{b^2-1}}{b^2} \ge 1?$$ Use logarithm or limit or what? Or do we have to use it as a conclusion to prove it backwards? And how to prove it forwards, that is, without assuming this is right.
| First, let's make a substitution: $x=b^2$
The expression now becomes $\displaystyle\frac{e^{x-1}}{x}$.
Next, let's take the derivative of this expression:
$\displaystyle \frac{d}{db}\frac{e^{x-1}}{x}=\frac{(x-1)e^{x-1}}{{x^2}}$
We know that local maxima/minima of this expression occur at values of $x$ for which $\displ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1804985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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What's the most efficient algorithm to check the number of cycles of length 4 in an undirected graph? What's the most efficient algorithm to check the number of cycles of length 4 in an undirected, unweighted graph?
| I don't know whether this is the fastest possible, but this is how I'd do it if I had to do it fast: Use a hash map that maps pairs of vertices to the number of paths of length $2$ between them. Iterate over the vertices, adding $1$ to the entry for each pair of neighbours of the vertex. Then iterate over the entries, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1805067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Group extension that doesn't realize a coupling Let $E$ be an extension of $N$ by $G$:
$$N \hookrightarrow E \twoheadrightarrow G$$
If $N$ is abelian, then $E$ uniquely defines an action of $G$ on $N$. More generally, it defines a unique class $\chi$ on:
$$\text{Out}(N) = \text{Aut}(N)/\text{Inn}(N)$$
We call the pair ... | I know one example of this, but it might not be the smallest.
Let $N = {\rm SL}(2,9)$, which is isomorphic to a double cover $2.A_6$ of $A_6$. Then ${\rm Out}(N) \cong C_2 \times C_2$. Let $G = {\rm Out}(N)$ with $\chi$ the identity map. Then there is no extension $E$ that induces this coupling.
In the ATLAS of Finite ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1805162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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How to evaluate $\int_{0}^{1} \frac{\ln x}{x+1} dx$ I want to evaluate:
$$\int_{0}^{1} \frac{\ln x}{x+1} dx$$
If I was asked I would to evaluate:
$$\int_{0}^{1} \frac{\ln x}{x-1} dx$$
That would be easy because if I use the Taylor series for $\ln x$ centered at $1$ then things will cancel out and leave me with a easy i... | We know that $$\int_0^1 x^a~dx=\frac{1}{a+1}$$
Then we can differentiate with respect to $a$:
$$\int_0^1 x^a \ln x ~dx=-\frac{1}{(a+1)^2}$$
Now we can use the geometric series:
$$\sum_{a=0}^\infty (-1)^a x^a=\frac{1}{1+x},~~~|x|<1$$
$$\sum_{a=0}^\infty (-1)^a \int_0^1 x^a \ln x ~dx=- \sum_{a=0}^\infty (-1)^a \frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1805230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do curves consist of points? According to Euclid, a point is something which has no dimensions. And we know that all curves of any type consists of points. Now this thing bothers me because if a point has no dimensions, i.e. in other words there is nothing, then how is it possible to draw any curve?
The thing I co... | A curve is completely determined by two facts:
*
*Knowledge of all of the points lying on the curve
*Knowledge that the curve is drawn on the Euclidean plane
When it's said that a curve is made out of points, one really means to include in the latter fact too, or something similar (e.g. a topology or a metric on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1805301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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"answer_id": 2
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Showing that $\text{exp}(\left \lvert z \right \rvert/(\left\lvert z \right\rvert - 1))\leq \left \lvert 1 + z \right\rvert$ Suppose $z \in \mathbb{C}$ with $\left\lvert{z}\right\rvert < 1$. I want to prove that that
$$\exp\left(\frac{|z|}{|z| - 1}\right)\leq |1 + z|.$$
I tried writing $z$ in cartesian form as $x + iy... | Let $z = re^{i\theta}$, $0 < r < 1$, $\theta \in [0,2\pi]$.
The desired inequality becomes
$$
\exp\left(\frac{r}{r-1}\right) \leq |1+re^{i\theta}|.
$$
The RHS is:
$$
|1+re^{i\theta}| = \sqrt{(1+re^{i\theta})(1+re^{-i\theta})} = \sqrt{1+2r\cos\theta + r^2}.
$$
Now the desired inequality is equivalent to
$$
\frac{r}{r-1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1805376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Given a matrix $A$ with $\operatorname{tr} (A) = 0$, prove that there is a B such that $\forall 1\leq i\leq n :(B^{-1}AB)_{i,i}=0$ I've tried using some matrices $B^{-1}$ that switch the rows, but the $B$ at the end placed the elements back in the diagonal (in different order) so I couldn't find a rule.
| Lemma
Given any matrix $A$ of order at least $2$, there exists an invertible matrix $E$ such that $(E^{-1}AE)_{11} = 0$.
Proof: Let $A = [a_{ij}]_{n \times n}$, $n \ge 2$. If $a_{11} = 0$, the result holds with $E = I$, the identity matrix of order $n$. If $a_{12} = 0$, then let $E$ be the permutation matrix obtained b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1805500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A curious approximation to $\cos (\alpha/3)$ The following curious approximation
$\cos\left ( \frac{\alpha}{3} \right ) \approx \frac{1}{2}\sqrt{\frac{2\cos\alpha}{\sqrt{\cos\alpha+3}}+3}$
is accurate for an angle $\alpha$ between $0^\circ$ and $120^\circ$
In fact, for $\alpha = 90^\circ$, the result is exact.
How can ... | Close but not exact even with regard to the linear term. With $x$ and $y$ as per Paramanand Singh's answer, if $x=1-\delta$, then, ignoring $o(\delta)$ throughout,
$$\begin{align}
y&=(1-\delta)[4(1-2\delta)-3]\\
&=(1-\delta)(4-8\delta-3)\\
&=(1-\delta)(1-8\delta)\\
&=1-9\delta.
\end{align}$$
By Paramanand Singh's equat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1805643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Find all the natural numbers $n$ such that $\sigma(n)=15$
Find all the natural numbers $n$ such that $\sigma(n)=15$
Where $\sigma (n)$ is the sum-of-divisors function
My attempt:
$$n=p_1^{\alpha_1}\cdots p_s^{\alpha_s}$$
$$\sigma(n)=\frac{p_1^{\alpha_1+1}-1}{p_1-1}\cdots \frac{p_s^{\alpha_s+1}-1}{p_1-1}=15$$
$1.\qua... | Here is a way to finish the s = 1 case in the way you started it and continuing from your last line:
$(p^\alpha - 15)p = 14$.
Since p is prime and divides 14 we must have either $p = \pm 2$ or $p = \pm 7$. So we are left with four cases: $2^\alpha - 15 = 7$, $2^\alpha - 15 = -7$, $7^\alpha - 15 = -2$ and $7^\alpha -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1805743",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find the range of the function $f(x) = 4x + 8$ for the given domain $D = \{-5, -1, 0, 6, 10\}$ The question is to find the range of each function for the given domain $f(x)=4x+8$, $D=\{-5, -1, 0,6, 10\}$. Is the range just $R= \{-12,4,8,32,48\}$ or am I mistaken? Could you elaborate why my answers are correct?
| Sure. The range (or image) of a function is just the set of all possible values of the function that you can get by plugging in values in the domain. If the domain is a finite list of numbers, you can find the range just by plugging in every number in the list, and removing duplicates.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1805823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Discontinuity and Dirac's Delta Function
Can someone help me understand how he came up with Dirac's function to differentiate that discontinuous periodic function? I am familiar with Dirac's function, but I don't understand where it came from in this case. Thanks
| The function in the figure can be constructed by the sum of $i(t)=-t/T$ {which is a straight line at -45 degree slope through the origin} and $j(t)=\sum{} h(t-nT)$ {which is a 'staircase' function comprised of the sum of Heaviside step functions} .
The derivative of $i(t)+j(t)$ yields your results. So the Dirac distr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1806052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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about non normality of Sorgenfrey plane There is a classical theorem saying that a regular space with a countable basis is normal. The Sorgenfrey plane is regular since it is the product of two regular spaces (which in fact are normal) and it has a countable basis since it is separable which would say that it should b... | Countable basis $\Longrightarrow$ separable | in general topological spaces.
Separable $\Longrightarrow$ countable basis | only in metrizable topological spaces.
Sorgenfrey’s plane is not metrizable. It is separable, but does not admit a countable basis.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1806171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving $\int_{0}^{\infty}\frac{4x^2}{(x^4+2x^2+2)^2}dx\stackrel?=\frac{\pi}{4}\sqrt{5\sqrt2-7}$ $$\int_{0}^{\infty}\frac{4x^2}{(x^4+2x^2+2)^2}dx=\frac{\pi}{15}$$
$$\int_{0}^{\infty}\frac{4x^2}{[(1+(1+x^2)^2]^2}dx=\frac{\pi}{15}$$
$u=\tan(z)$ $\rightarrow$ $du=\sec^2(z)$
$u$ $\rightarrow \infty$, $\tan(z)=\frac{\pi}{2}... | Hint. One may write
$$
\begin{align}
\int_{0}^{\infty}\frac{4x^2}{(x^4+2x^2+2)^2}dx&=4\int_0^{\infty}\frac1{\left(\left(x-\frac{\sqrt{2}}x\right)^2+2+2\sqrt{2}\right)^2}\:\frac{dx}{x^2}
\\\\&=2\sqrt{2}\int_0^{\infty}\frac1{\left(\left(x-\frac{\sqrt{2}}x\right)^2+2+2\sqrt{2}\right)^2}\:dx \quad (x \to \sqrt{2}/x)
\\\\&=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1806291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
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Prove $\ \sin(x) < x \ \ \ \forall x \in(0, 2\pi)$
Problem : Prove $\sin(x) < x \ \ \ \forall x \in(0, 2\pi)$
Now I have a possible solution for this, using limits and the first derivatives of $\sin(x)$ and $x$, but I don't feel it's a very rigorous or succinct way to prove this. Can any of you find better ways to p... | If you know that $\cos x < 1$ for $0 < x < 2\pi$, then
$$
0 < \int_0^{x} 1-\cos(t) \, dt = x -\sin(x)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1806387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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How to show that $\sup(\mathbb{Q} \cap (a,b)) = b$?
Let $a<b$ be two real numbers. Show that $\sup(\mathbb{Q} \cap (a,b)) = b$ and $\inf(\mathbb{Q} \cap (a,b)) = a$.
This intuitively makes sense. Since a sequence of rationals will infinitely approach $a$ and $b$, it makes sense that the question is true. How do I pro... | Let $A=\Bbb{Q}\cap (a,b)$. Of course, $b\geq x$ for all $x\in A$. Suppose that the supremum of $A$ is $m<b$. By density, there exists $r\in\Bbb{Q}\cap(m,b)$. Hence, $r\in A$ and $r>m$, which contradicts the maximality of $m$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1806463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Normal subgroups of matrices
Let $G=\begin{bmatrix}1&a\\0&b\end{bmatrix}$ so that $a,b\in\mathbb C$ and $b\ne0$. I need to prove that $G$ has infinitely many normal subgroups.
I attempt to do this by constructing some family of normal subgroups but I keep failing, as most of the things I try aren't even subgroups.
| Let $g$ and $h$ be two elements in $G$. Calculate $ghg^{-1}$.
You will immediately see under what conditions $ghg^{-1}$ is "of the same form" as $h$. That is, $ghg^{-1} \in H$ where $H$ is a subgroup of $G$.
This condition $ghg^{-1} \in H$ means that $H$ is normal. This method will give you infinitely many subgroups... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1806603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to verify $(1+\frac{1}{n})^2(1-\frac{1}{n^2})^{n-1}\geq \exp(\frac{1}{n})$ How to verify this inequality? Assuming that $n\in \mathbb{N}^+$, we have:
$$\left(1+\frac{1}{n}\right)^2\left(1-\frac{1}{n^2}\right)^{n-1}\geq \exp\left(\frac{1}{n}\right).$$
| Consider $$A_n=\left(1+\frac{1}{n}\right)^2\times\left(1-\frac{1}{n^2}\right)^{n-1}$$ Take logarithms $$\log(A_n)=2\log\left(1+\frac{1}{n}\right)+(n-1)\log\left(1-\frac{1}{n^2}\right)$$ Now, use the Taylor series $$\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}+O\left(x^6\right)$$ and replace $x$ b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1806723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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$\exp(x)$ as defined by a net Motivation:
So, I had an idle thought last week, and I thought I would ask it here before I forget about it. It is well known that we can define
$$
e^x = \lim_{n \to \infty} \left(1 + \frac x{n} \right)^{n}
$$
Where $x$ here can be taken as either a number or a linear operator.
This is of... | Having come across one of my old questions, I've decided to leave an answer (following Daniel's hint in the comment).
Fix $x \in \Bbb C$. The Taylor expansion for $\log(1 + z)$ (with the Peano form of the remainder) tells us that
$$
\log(1 + \mu x) - \mu x = [1 + \zeta(\mu x)]\frac {\mu^2 x^2}2
$$
where $\zeta(z) \to 0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1806855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Which elements of $\mathbb{R}$ make sense as representatives for cosets of $\mathbb{Q}$ in the group $\mathbb{R/Q}$ I am trying to better understand the group $\mathbb{R/Q}$. It's unclear to me when two irrational numbers will give the same coset of $\mathbb{Q}$, but I know that this must happen since, for example $\pi... | $\mathbb{R/Z}$ is isomorphic to the unit circle $S^1$, the set of all complex numbers of absolute value $1$, seen as a multiplicative group.
$\mathbb{Q/Z}$ is isomorphic to the torsion subgroup of $S^1$, formed by the elements of finite order, that is, all roots of unity.
$\mathbb{R/Q}$ is thus isomorphic to $S^1/tor(S... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1806926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Why can we choose a greatest ordinal $\beta$, such that $\omega^\beta\leq \alpha$? I am reading a proof of Cantor's normal form theorem. In it, I read:
for arbitrary $\alpha>0$ let $\beta $ be the greatest ordinal such that $\omega^\beta \leq \alpha$.
Why should such an ordinal exist?
| There are ordinals $\varepsilon$ such that $\omega^{\varepsilon} > \alpha$. By the well-ordering of the ordinals, there is hence a smallest ordinal $\gamma$ with $\omega^{\gamma} > \alpha$.
For limit ordinals $\lambda$, we have
$$\omega^{\lambda} = \bigcup_{\delta\in \lambda} \omega^{\delta} = \sup \{ \omega^{\delta} :... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1807017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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coordinate geometry in polar coordinate Let $G= \{(x, f(x)) \mid x \text{ lies between } 0 \text{ and } 1 \}$
Let $(1,0)$ belong to $G$.
It is given that tangent vector to $G$ at any point is perpendicular to radius vector at that point.
Is $G$ parabola or ellipse?
| Well, you are given that $(1,f'(x)) \perp (x,f(x))$ and $f(1) = 0$. Thus, $x + f'(x)f(x) = 0$. We can rewrite this as
$$ \frac{d}{dx} f^2(x) = -2x. $$
Integrating, we get $f^2(x) = -x^2 + C$ and by plugging in $f(1) = 0$ we see that $C = 1$. Thus, $f(x) = \pm \sqrt{1 - x^2}$ and thus $G$ is (part of) a circle.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1807116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Set of marginals is convex Let $[n] = \{1,2,\cdot,n\}$ and $[m] = \{1,2,\cdot,m\}$. Let $Z_{1,2}$ denote the set of all probability distribution on the Cartesian product $[m]\times [n]$. Let
$S_{1,2}$ denote a convex closed subset of $Z_{1,2}$. Let $S_1$ denote the set of probability distributions on the set $[n]$ obt... | Think of this as a linear map from $\phi: Z_{1,2} \subset \mathbb{R}^{mn} \to \mathbb{R}^n$ where $Z_{1,2}$ is your compact set and the map is the marginalizing map. Since $\phi$ is linear, it is continuous and since and $S_{1,2}$ is compact, the image $\phi (S_{1,2})$ is compact (and therefore closed).
On the other ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1807185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that the $7$-th cyclotomic extension $\mathbb{Q}(\zeta_7)$ contains $\sqrt{-7}$
Prove that the $7$-th cyclotomic extension $\mathbb{Q}(\zeta_7)$ contains $\sqrt{-7}$
I thought that the definition of the $n$-th cyclotomic extension was: $\mathbb{Q}(\zeta_n)=\{\mathbb{Q}, \sqrt{-n}\}$. Is this correct?
How could ... | I'm sure there are more elegant ways to observe this, but here's one possible way:
*
*First, show that the extension is Galois (not too hard) and that $Gal(\mathbb{Q}(\zeta_7)/\mathbb{Q})\cong Z_6$. This has subgroups of order 2 and 3.
*Now find the fixed field of the subgroup of order 3. This is a degree 2 extens... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1807278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 1
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Irrationality of the concatenation of the rightmost nonzero digits in $n!$ Surfing the internet I bumped into a very interesting problem, which I tried to solve, but got no results.
The problem is following: let $h_n$ be the most right non-zero digit of $n!$, for example, $10!=3628800,$ so $h_{10}=8$. The task is to pr... | To show that $0,h_1h_2\ldots$ is irrational, it's enought to show that the sequence $h_n$ is not eventually periodic.
Assume on contrary that $h_n$ is eventually periodic.
Then $h_n$ is eventually periodic as a sequence modulo $5$.
Let $\mathbb Q^+$ denote the multiplicative group of positive rationals.
The set of prim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1807380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove that random variables are independent $X_1,X_2,X_3$ are independent random variables with with exponential distribution with parameter $\lambda=1$. I'd like to prove that variables $\frac{X_1}{X_2 +X_1}, \frac{X_1 +X_2}{X_1 + X_2 +X_3},X_1 + X_2 +X_3$ are independent too. I can calculate the distribution od those... | Use the Jacobian formula. If $X, Y, Z$ are iid exponential($\lambda$), then we obtain $U,V,W$ by applying the transformation $(x,y,z)\mapsto(u,v,w)$ defined by
$$\begin{align}
u&=\frac x{x+y}\\
v&=\frac{x+y}{x+y+z}\\
w&=x+y+z.\end{align}$$
Invert this mapping to obtain
$$\begin{align}x&=uvw\\y&=(1-u)vw\\ z&=(1-v)w.\end... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1807438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show that $(a,b)+(c,d) = (a+c,b+d)$
Let $a<b$ and $c<d$ be real numbers. Show that $(a,b)+(c,d) = (a+c,b+d)$.
I don't understand the question. Since $(a,b)$ and $(c,d)$ are intervals, what does it mean to add them?
| In terms of sets and set notation:
$(a,b)$ = all the points of R that are between a and b exclusively =$\{x\in \mathbb R| a < x < y\}$
If $A$ and $B$ are sets, than we say $A + B =\{x+y|x \in A; y \in B \}$.
So the statement $(a,b)+(c,d) = (a+b,c+d)$ means that $(a,b) + (c,d)=\{x+y|a <x <b;c <y<d\}$ is the same as $(a+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1807527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $T^{-1}:Y \to X$ exists and is bounded.
Let $T$ be a bounded linear operator from a normed space $X$ onto a normed space $Y$. If there is a positive $b$ such that $$||Tx||\ge b||x||,$$ for all $x \in X$, show that $T^{-1}:Y \to X$ exists and is bounded.
My attempt:
Suppose $Tx =0$ then clearly $||Tx~||=0 \i... | As I said in a comment, you should also check that $T^{-1}$ is linear but the rest seems fine. Here is the proof.
Let $y_1, y_2 \in Y$ and $\lambda, \mu \in \mathbb{C}$. Using the linearity of $T$ you've got
$$ \lambda y_1 + \mu y_2 = \lambda TT^{-1}(y_1)+ \mu TT^{-1}(y_2) \\= T\Big(\lambda T^{-1}(y_1) + \mu T^{-1}(y_2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1807612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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I'm stuck in a logarithm question: $4^{y+3x} = 64$ and $\log_x(x+12)- 3 \log_x4= -1$ If $4^{y+3x} = 64$ and $\log_x(x+12)- 3 \log_x4= -1$ so $x + 2y= ?$
I've tried this far, and I'm stuck
$$\begin{align}4^{y+3x}&= 64 \\
4^{y+3x} &= 4^3 \\
y+3x &= 3 \end{align}$$
$$\begin{align}\log_x (x+12)- 3 \log_x 4 &= -1 \\
\log_x ... | You're right up to $y+3x=3$.
Now consider the other statement $\log_x(x+12)-3\log_x 4=-1$
$\log_x{x+12 \over 64 }=-1$
${x+12 \over 64 }={1 \over x}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1807717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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proof that $\int_{a}^{x} = \int_{x}^{b}$ I want to show that if $f$ is a continuous function on the interval $[a,b]$ then there must exist some $x \in [a,b]$ such that: $$\int_{a}^{x} = \int_{x}^{b}$$
Intuitively this seems very easy and I can see why it is true, its just the structure of the proof that I'm confused ab... | We know that there exists a continuous function on the interval $x \in [a, b]$, $F(x)$, such that:
$$
\int_a^bf(t)dt = F(a) - F(b) \\
\int_a^xf(t)dt = F(x) - F(a) \\
\int_x^bf(t)dt = F(b) - F(x) \\
$$
Set the last two equations to be true:
$$
F(x) - F(a) = F(b) - F(x) \longrightarrow
F(x) = \frac{F(a)+ F(b)}{2}
$$
Usin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1807798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 2
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If $f(x)$ has a minimum at (3,2) what does $y = \frac{5}{3+f(x)}$ have at x = 3? No calculus allowed.
I can get the value by substituting 2 for $f(3)$, and that it should be a maximum turning point as $f(x+\delta h)$ and $f(x -\delta h) > f(x)$ but am not sure how to proceed further.
| Let $g(x)=y=5/(3+f(x))$. For any $x$, we have
$$
3+f(x)\geq3+f(3)=3+2=5>0\implies g(x)=\frac{5}{3+f(x)}\leq\frac{5}{3+f(3)}=g(3).
$$
And so $y=g(x)$ is everywhere defined (because $3+f(x)>0$ always; see above) and has a global maximum at $x=3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1807915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Calculate the normal cone of a convex set at a point Let $C$ be a convex set in $\mathbb{R}^d$ and $\overline{x}\in C$. We define the normal cone of $C$ at $\overline{x}$ by
\begin{equation}
N_C(\overline{x}) = \{ y \in \mathbb{R}^d \ <y ,c-\overline{x}> \leq 0 \forall c \in C \}.
\end{equation}
I found in a book of ... | In your case
\begin{align}
N_C(0,0) &= \{ (y_1,y_2)\in \mathbb{R}^2|y_1c_1+y_2c_2\le 0, \forall (c_1,c_2)\in C\}\\
&=\{ (y_1,y_2)\in \mathbb{R}^2|y_1c_1\le 0, \forall\, c_1\in [0,1]\}\\
&= \{ (y_1,y_2)\in \mathbb{R}^2 | y_1\le 0\}
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1808136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How work out the length of this side?
This is probably so basic, but I just cannot see it. If you do not know that the left side is $5x$ and are only given $3x$ and $4x$, how do you deduce $5x$?
| As Emilio Novati implies, all right triangles, with given angle $\theta$, are "similar triangles". So since the legs have ratio $\frac{3x}{4x}$, the hypotenuse must be a multiple of the hypotenuse with legs of length 3 and 4 which is, of course, 5. The hypotenuse must have length 5x.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1808234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Uniform bounded of Riemann-like sum and improper integral For any $h>0$, suppose $\{(y_i,y_{i+1}]\mid i\in \mathbb{Z}\}$ be a uniform partition of $\mathbb{R}$ with mesh size $h$. I am considering under what condition for a continuous transition density function $p(h,x,y)$ of a stochastic process, we can have
$$\sum_i ... | Looks not that complicated, if I understood the statements correctly.
For simplicity let me set $x=0$, $y_0=0$.
$i\geq 0\,\, \sum_{i\geq 0} p(h,0,\xi _i)h\leq \frac{h^\frac{1}{2}}{2\pi}+ \sum_{i\geq 0} q(h,0,\xi_i)h\leq\frac{h^\frac{1}{2}}{2\pi}+\int _{y\geq0}p(h,0,y)dy\leq \frac{h^\frac{1}{2}}{2\pi}+\frac{1}{2}\,\,\, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1808414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What are some examples of when Mathematics 'accidentally' discovered something about the world? I do not remember precisely what the equations or who the relevant mathematicians and physicists were, but I recall being told the following story. I apologise in advance if I have misunderstood anything, or just have it pla... | Science News article accidental astrophysicists 13 June 2008 explains how a math proof became a physics proof of gravitational lensing.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1808523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "243",
"answer_count": 26,
"answer_id": 4
} |
Simple rocket model I have a problem creating a model for a horizontal rocket flight.
I want to model a rocket with constant force, drag constant and gravity. I also have to account for a changing mass and drag. I know I could calculate movement of this rocket by something like Runge-Kutta algorithm but I want to know ... | Assuming that you rocket is still not fast enough. By Newton's second law (like gt6989 said):
$m\ddot{x}+\dot{m}\dot{x}=F(t)-gm-c\dot{x}^2$ or $$m\ddot{x}+\dot{m}\dot{x}+c\dot{x}^2=F(t)-gm$$
You still need to give models for the thrust and also for the mass which is lost. An explicit solution will not be that easy.
You... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1808546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Let $n \in \mathbb{N}$. Proving that $13$ divides $(4^{2n+1} + 3^{n+2})$ Let $n \in \mathbb{N}$. Prove that $13 \mid (4^{2n+1} + 3^{n+2} ). $
Attempt: I wanted to show that $(4^{2n+1} + 3^{n+2} ) \mod 13 = 0. $ For the first term, I have $4^{2n+1} \mod 13 = (4^{2n} \cdot 4) \mod 13 = \bigg( ( 4^{2n} \mod 13) \cdot ( 4... | By the binomial theorem,
$$
4^{2n+1} + 3^{n+2}
=4\cdot 16^n+9\cdot 3^n
=4\cdot (13+3)^n+9\cdot 3^n
=4(13a+3^n)+9\cdot 3^n
=13(4a+3^n)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1808629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
2-norm of matrix How to prove that for a symmetric matrix A with eigenvalues $\lambda_1 \leq \lambda_ 2, ... \leq \lambda_n$ it holds that
$$\vert\vert A \vert\vert = \text{max}(-\lambda_1, \lambda_n)$$
where $\vert\vert \cdot\vert\vert$ denotes the 2-norm? I am familiar with the usual matrix norm, i.e. $\text{max}(\l... | If $v_i$ are the corresponding orthonormal eigenvectors of $A$ then you have
$$ ||Av||^2 = \left< A\left( \sum_{i=1}^n \left< v, v_i \right>v_i \right), A \left( \sum_{i=1}^n \left< v, v_j \right> v_j\right) \right> = \left< \sum_{i=1}^n \left<v, v_i \right> \lambda_i v_i, \sum_{j=1}^n \left< v, v_j \right> \lambda_j v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1808712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How to find the range of the given function?
Find the range of $$f(x)=\dfrac{x^2+14x+9}{x^2+2x+3}$$ where $x\in \mathbb R$
I thought of finding derivative but this will get too complicated so i am completely blank.
Thanks in advance!
| I think you'd have to find the derivative for this problem. But don't worry, finding the derivative is not as cumbersome as you think. Just use the quotient rule and you'll get something reasonable.
You should get $-\frac{12(x^2+x-2)}{(x^2+2x+3)^2}$ and setting that equal to 0 and solving for $x$ should be straightforw... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How did we derive this general term for the series? We have this series of numbers:
$1, 3, 6, 10, 15$
The general term can be described wit: $\frac{r(r + 1)}{2}$
Apparently the following series:
$1, 4, 10, 20, 35$
Can be described with $\frac{r(r + 1)(r + 2)}{6}$ based on the first series.
But I am not clear how this i... | Note: That from the given series you are not able to derive a unique expression for the general term. There are infinitly many solutions. There are some which are more obvious (see JMoravitz), but there is no mathematical definition for more obvious as far as I know.
E.G. a polynomial of 6th degree can fit all points, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$\sin\alpha + \sin\beta + \sin\gamma = 4\cos{\frac{\alpha}{2}}\cos{\frac{\beta}{2}}\cos{\frac{\gamma}{2}}$ when $\alpha + \beta + \gamma = \pi$ Assume: $\alpha + \beta + \gamma = \pi$ (Say, angles of a triangle)
Prove: $\sin\alpha + \sin\beta + \sin\gamma = 4\cos{\frac{\alpha}{2}}\cos{\frac{\beta}{2}}\cos{\frac{\gamma}... | You may go the other way around:
$$
\cos\frac{\gamma}{2}=\cos\frac{\pi-\alpha-\beta}{2}=
\sin\frac{\alpha+\beta}{2}=
\sin\frac{\alpha}{2}\cos\frac{\beta}{2}+
\cos\frac{\alpha}{2}\sin\frac{\beta}{2}
$$
so the right hand side becomes
$$
4\cos\frac{\alpha}{2}\cos\frac{\beta}{2}
\sin\frac{\alpha}{2}\cos\frac{\beta}{2}+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1808948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Least Squares Alternates- approximating functions I was given this least squares problem to solve:
Find a linear function $\ell(x)$ such that
$\displaystyle\int_0^1(e^x-\ell(x))^2{\rm d}x$ is minimized.
As an answer, I got $\ell(x)=0.5876+0.5519x$, which I am pretty sure but not positive that it is right. I am sup... | About the main problem, since
$$ \int_{0}^{1} e^{x}\,dx = (e-1), \qquad \int_{0}^{1} e^{x} P_1(2x-1)\,dx = (3-e) $$
the best $L^2$ approximation is given by $\color{red}{(4e-10)+(18-6e)x}$ as already pointed by Winther in the comments. About $(a)$ and $(b)$, given the convexity of $e^x$ it is quite trivial that the bes... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1809035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Show every irreducible subset of a topological space $X$ is contained in a maximal irreducible subset
Let $X$ be a topological space. A subset $A$ is irreducible if for every open $U,V\subseteq A$, we have $U\cap V\neq\varnothing$. Show that any irreducible subset $A\subseteq X$ is contained in a maximal irreducible s... | First, let me note that there might be more than one maximal irreducible set containing $A$. For instance, let $X=\{a,b,c\}$ with $\{b\}$ and $\{c\}$ as the only nontrivial open sets. Then $A=\{a\}$ is irreducible, but $\{a,b\}$ and $\{a,c\}$ are two different maximal irreducible sets containing it.
In particular, in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1809159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Strange PDE solution Given the linear equation $$u_t -xt u_x = x$$ $x\in\mathbb{R}$, $t>0$, with IVP $u(x,0)=u_0(x)$, my solution comes to $u(x,t) = u_0(xe^{t^2/2})+xt$, but Maple gives a much more complicated solution to this IVP. I would appreciate if someone could please point out what I might not be doing right.
As... | Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:
$\dfrac{dt}{ds}=1$ , letting $t(0)=0$ , we have $t=s$
$\dfrac{dx}{ds}=-xt=-xs$ , letting $x(0)=x_0$ , we have $x=x_0e^{-\frac{s^2}{2}}=x_0e^{-\frac{t^2}{2}}$
$\dfrac{du}{ds}=x=x_0e^{-\frac{s^2}{2}}$ , letting $u(0)=f(x_0)$ , we have $u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1809316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Linearising shallow-water wave equations We are given the equations
$$\frac{\partial{u}}{\partial{t}}+u\frac{\partial{u}}{\partial{x}}+g\frac{\partial{h}}{\partial{x}}=0$$
and
$$\frac{\partial{h}}{\partial{t}}+\frac{\partial{(hu)}}{\partial{x}} = 0$$
We are then asked
By linearising these equations about a uniform me... | Following a suggestion from Semiclassical on the chat and in comment. We can write the systems in the following form:
$$\begin{align*}\frac{\partial u^{\prime}}{\partial t} &= -u_{0}\frac{\partial u^{\prime}}{\partial x}-g\frac{\partial h^{\prime}}{\partial x} \\ \frac{\partial h^{\prime}}{\partial t}&= -u_{0}\frac{\pa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1809452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Support Vector Machines: Hype or Hallelujah? - what is alfa? I at the moment trying to understand how SVM works with the help of this paper
The paper itself explains things pretty well, but there is an alfa term, which doesn't seem to be documented anywhere? could any of elaborate on what it means? and what effect it ... | $c$ (resp. $d$) is a point on the convex hull of the points in Class 1 (resp. Class 2). Therefore, $c$ (resp. $d$) can be represented as a convex combination of the points in Class 1 (resp. Class 2). More specifically,
$$
c = \sum_{y_i \in \text{ Class 1}} \alpha_i x_i\quad \text{ for some }\quad \sum_{y_i \in \text{ C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1809567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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100 pieces of paper in a box, one of which has a black dot on it. Probability Question. There are $100$ pieces of paper in a box, one of which has a black dot on it. If $100$ people go up one by one and pick a paper from the box, which one has the lowest probability of getting the black dot, and which one has the highe... | The probability $p_k$ that the $k$-th person, $1\le k\le 100$ has the black dot is equal to $$p_k=\frac{99}{100}\cdot\frac{98}{99}\cdot\;\cdots\;\cdot\frac{100-(k-1)}{100-(k-1)+1}\cdot\frac{1}{100-k+1}=\frac1{100}$$ so it is independent of $k$ and equal for every one.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1809698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Why L'hopital's rule only proved for indetermined forms? In this proof of L'hopital's rule, $\lim\limits_{x\to a}\, f(x)=0$ and $g(x)=0$ seems have no role to paly.
So what goes wrong when the limits mentioned before are not equal to $0$?
(My guess is that you can't assume that "f(a)=g(a)=$0$" without making the functi... | The proof requires an application of Rolle's theorem to
$$h(x) = f(x) - \frac{f(b)}{g(b)}g(x).$$
Since $h(b) = 0$, this requires $h(a) = 0$ for all $b >a$. This is satisfied if $f(a) = g(a) = 0$ either outright or by continuous extension using $f(x),g(x) \to 0$ as $x \to a+$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1809776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Solve Poisson Equation Using FFT I am trying to solve Poisson equation using FFT. The issue appears at wavenumber $k = 0$ when I want to get inverse Laplacian which means division by zero.
We have
${\nabla ^2}\phi = f$
Taking FFT from both side we get:
$-k^2\hat\phi = \hat f $
or
$\hat\phi = \frac{\hat f}{-k^2} $
... | This is conceptually similar the integrating constants that show up when you are solving a differential equation by other methods. Usually these integrating constants are decided by your boundary conditions.
Instead of doing any division you can simply rewrite it as a least-norm problem: $$\|-k^2\hat \phi +\hat f\|_2^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1809871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
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Find $\lim_{n \to +\infty} \frac{1}{n}\sqrt[n]{\frac{(2n)!}{n!}} $ using Riemann integral Wonder how to determine this limit by the use of Riemann integral. The limit is as follows:
$$\lim_{n \to +\infty} \frac{1}{n}\sqrt[n]{\frac{(2n)!}{n!}} $$
My instructor told me that the usage of Riemann integral gives spectacular... | I will repeat basically the same approach as in this answer: Find the value of : $\lim\limits_{n\to \infty} \sqrt [n]{\frac{(3n)!}{n!(2n+1)!}} $ It is also the same approach as suggested in Paramanand Singh's comment. (I see that the OP asks specifically about a proof using Riemann's integral, but this seems interestin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1809965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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Using the digits of $\pi$ to generate random numbers. Let's say I've been captured by Russian operatives and am locked in a room with only one object: a book listing the digits of $\pi$.
I'm told to generate a sequence of binary digits. If this sequence is random, they will cut off one of my arms and let me free; if th... | There are long lists of digits for $\pi$ online. You could write a computer program that calculates different types of statistics to test your hypothesis.
One interesting statistic could be to measure $$P(X_{i+1}=1|X_{i},\cdots,X_{i-n})$$That is: the conditioned probability of one bit being $1$ given that we know the ... | {
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"timestamp": "2023-03-29T00:00:00",
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Difference between viewer and camera in a 3d projection I have been programming a 3D graphics library for myself and I have used the following wikipedia page to help me.
https://en.wikipedia.org/wiki/3D_projection#Perspective_projection
The article references both a camera position and a viewers position. I assume aft... | It should be roughly like this:
*
*The camera projects the 3D world on its 2D display.
*The viewer (observer) takes a look at that display.
*The field of view field is a property of the camera, describing what part of a sphere around the camera is visible to it
| {
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"url": "https://math.stackexchange.com/questions/1810135",
"timestamp": "2023-03-29T00:00:00",
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Prove or disprove: there exists a basis $p_0, p_1, p_2, p_3 \in P_3(F)$ such that none of the polynomials $p_0, p_1, p_2, p_3$ has degree 2 Prove or disprove: there exists a basis $p_0, p_1, p_2, p_3 \in P_3(F)$ such that none of the polynomials $p_0, p_1, p_2, p_3$ has degree 2
This is a repeat of Does there exist a... | Suppose we have a list of vectors $(v_1,v_2,v_3,v_4)$, and that this list forms a basis for the space $V$. Our task is to prove that $$(v_1+v_4,v_2+v_4,v_3+v_4,v_4)$$ also forms a basis of $V$.
Our first step is to prove that $(v_1+v_4,v_2+v_4,v_3+v_4,v_4)$ is linearly independent. To do so, we'll examine the followi... | {
"language": "en",
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What is the classifying space G/Top? I simply can't find the definition(except in one book on surgery where a definition was not actually given but instead they alluded to what the definition is) and I have spent an hour and half looking. My professor used the notation when talking me today. I would be grateful for a... | A reference available online is Rudyak's survey Piecewise Linear Structures on Topological Manifolds. Beware that what you are calling $G$ he calls $F$ (I think both notations are common --and awful).
$G/TOP$ can be defined as the homotopy fiber of the canonical map $BTOP \to BG$, where $BTOP$ is the classifying space ... | {
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Isomorphism of two graphs using adjacency matrix How can I show that the following two graphs are isomorphic:
Steps:
The given graphs can be written as:
| I am keeping this answer as simple as I can, so kindly pardon the layman's language.
Observe the to graphs,
From G1, G2 can be obtained if the first line and second line of the graph G1 are interchanged. So, this graph is definitely iso-"morphic". (bijective and satisfies the edge adjacency property). And the mapping ... | {
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Sums involving binomial coefficients in a finite field Consider the field $\mathbb{F}_q$ where $q=p^k$ for some prime $p$. I have some identities related to binomial coefficients over such a field, which I wish to prove. So, can someone tell me a source where I could read up on these?
An example of the identities that ... | I found the solution to the above problem. A general term of the above sum looks like ${(q-t)(q-1) \choose (q-a)(q-1)}$ which is the coefficient of $x^{(q-a)(q-1)}$ in $(1+x)^{(q-t)(q-1)}$. Hence, the sum is the coefficient of $x^{(q-a)(q-1)}$ in $(1+x)^{q(q-1)}+(1+x)^{(q-1)^2}+...+(1+x)^{(q-a)(q-1)}$. We can continue ... | {
"language": "en",
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Is there a rigorous proof of this combinatorial identity?
Theorem: For any pair of positive integers $n$ and $k$, the number of $k$-tuples of positive integers whose sum is $n$ is equal to
the number of $(k − 1)$-element subsets of a set with $n − 1$
elements.
Does anyone know of a rigorous mathematical proof to ... | Stars and bars will explain it, but suppose we go for an inductive proof. So let $S(n,k)$ mean that the number of solutions to
$$x_1+x_2+ \cdots +x_k=n \tag{1}$$ with positive $x_j$ (order mattering) is given by $\binom{n-1}{k-1}.$ A few base cases are easily established, so we turn to breaking up the solutions of (1) ... | {
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Group generated by $x,y$. With relations $x^3=y^2=(xy)^2=1$.
Let $G$ be a group generated by $x,y$ with the relations $x^3=y^2=(xy)^2=1$. Then show that the order of $G$ is 6.
My attempt: So writing down the elements of $G$ we have $\{1,x,x^2,y,\}$. Other elements include $\{xy, xy^2, x^2y\}$ it seems I am counting m... | One group presentation for the dihedral group $D_n$ is $\langle x,y|x^2=1,y^n=1,(xy)^2=1\rangle $. Hence the group is indeed isomorphic to $D_3$. Here $x$ with $x^2=1$ corresponds to a reflection, and $y$ with $y^3=1$ to a rotation of $60$ degrees. Finally we have $xyx^{-1}=y^{-1}=y^2$, which is how rotation and reflec... | {
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Clarification about injectivity of a function Let $f:{X\to Y}$ be a function.
Show:
$f$ is injective $\Leftrightarrow \exists \space h:{Y\to X} \ni h\circ f=id_X$.
If we have $f(x)=f(y)$ then $x=y$. My problem is how can we have the function $h$? f is injective but not surjective this means that there might be values... | You can define how $h$ works with those values $\alpha \in Y $ for which does not exists a $x \in X$ such that $f(x)=\alpha$ in this way: $h(\alpha)=x$ if $f(x)=\alpha$ and $h(\beta)=z \in X$ if $\not\exists x \in X \space ,f(x)=\beta.$
This function do what we want.
| {
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Why $h$ has zero topological degree? I am trying to prove that $f,g : M^n \to S^n$, both $C^1$ (indeed just $C^0$ is enough) with the same topological degree are homotopic.
I saw on a book that the trick is as follows:
Take $W = M\times [0,1]$ and define $h(x,0) = f(x),$ and $h(x,1) = g(x).$ Then the map $h : \partial ... | The boundary of $M\times [0,1]$ is the disjoint union of two copies $M_1=M\times\{0\}$ and $M_2=M\times\{1\}$ of $M$ with opposite orientation. Thus the degree of $f$ and $g$ are integers with opposite signs.
| {
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Help with sum and product of roots. I'm having trouble with a question from my textbook relating to roots of an equation. This is it:
Let a and b be the roots of the equation:
$x^2-x-5=0$
Find the value of $(a^2+4b-1)(b^2+4a-1)$, without calculating values of $a$ and $b$.
What I do know however, is that the book has hi... | HInt: You have $a+b=1$ and $ab=-5$. Now expand the product $(a^2+4b-1)(b^2+4a-1)=(ab)^2+4(a^3+b^3)+16ab-4(a+b)-(a^2+b^2)+1$, then try to express the terms $a^3+b^3$ and $a^2+b^2$ in terms of $ab$ and $a+b$.
| {
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Maximum likelihood estimators for gamma distribution I'm having trouble with an exercise about maximum likelihood estimators.
Specifically, the exercise gives me values of a protein which was found in 50 adults. We assumed that the data follow a gamma distribution:
$X \sim \Gamma(r,\lambda)= \frac {\lambda^{r}}{\Gamm... | We know that $\Gamma(r,\lambda)= \frac {1}{\Gamma(r)}\lambda^{r}x^{r-1}e^{-\lambda x} $ if $x\ge0$.
In this case the likelihood function $L$ is $$\prod_i \Gamma(r,\lambda)_{x_i}=\frac{1}{\Gamma(r)^{n}}\lambda^{nr}x_1^{r-1}x_2^{r-1}...x_n^{r-1}e^{-\lambda T}$$
where $T=x_1+...+x_n$;
By apllying the logaritmic function t... | {
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If $\varphi \in E'$ and $A$ is convex and open then $\varphi (A)$ is an open interval
Let $E$ be a real normed space and $\varphi \in E'$, $\varphi \neq 0$.
Suppose that $A \subset E$ is an open convex not empty subset.
Show that $\varphi(A)$ is an open interval.
Since $A$ is connected and $\varphi$ is continuou... | Since $\varphi(A)$ is an $\mathbb{R}$-interval, it is of the form $(a,b)$, $[a,b]$, $(a,b]$, $[a,b)$, with $-\infty\le a,b\le\infty$. Supose $\varphi(A)$ is not open. Then, WLG we can assume $\varphi(A)=[a,b)$. Then, there is $x\in A$ such that $\varphi(x)=a$.
Since $A$ is open, there is $r>0$ such that if $\|x-y\|<r$... | {
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Showing that $\int_{0}^{1}{x(x-1)(x+2)\over (x+1)^3}\cdot{1\over \ln(x)}dx={\ln{\pi\over 2}-{7\zeta(3)\over 4\pi^2}}$ Showing that
$$\int_{0}^{1}{x(x-1)(x+2)\over (x+1)^3}\cdot{1\over \ln(x)}dx=\color{brown}{\ln{\pi\over 2}-{7\zeta(3)\over 4\pi^2}}$$
Applying substitution
$u=\ln(x)\rightarrow du={1\over x}dx$ and $x=... | $$\int_{0}^{1}\frac{1-x}{(1+x)\log x}\,dx =\log\frac{\pi}{2}\tag{1}$$
is a straightforward consequence of Frullani's theorem and Wallis' product.
So it is enough to compute:
$$\begin{eqnarray*} \int_{0}^{1}\frac{1-x}{(1+x)^3\log x}\,dx &=& \int_{0}^{+\infty}\frac{e^{-x}-e^{-2x}}{x(1+e^{-x})^3}\,dx\\&=&-7\cdot\zeta'(-2... | {
"language": "en",
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Why does $\mathrm{d}l=v\,\mathrm{d}t$ imply that $\frac{\mathrm{d}}{\mathrm{d}t}=v\frac{\mathrm{d}}{\mathrm{d}l}$? If we have a lengthening pendulum and the length $l$ of the pendulum at time $t$ is $$l=l_0+vt$$ where $l_0$ is the initial length of the pendulum and $v$ is the velocity for which the pendulum's length is... | The chain rule gives
$$\begin{align}
\left(\frac{d}{dt}\right)f(l(t))&=\frac{df(l)}{dl}\frac{dl}{dt}\\\\
&=\left(v\frac{d}{dl}\right)f(l)
\end{align}$$
| {
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Is span $\{[1,0],[0,1]\}$ a vector space? I can't figure this out. I would think that it is a vector space because it has the zero vector, and it seems to me to be closed under addition and scalar multiplication. But $[1,0]+[0,1] = [1,1]$ which is definitely not in the set. Can someone clarify? Is the span a vector spa... | Yes, the (linear) span is a vector space. By definition it is the smallest vector space that contains all the elements in the set. In particular it will contain all linear combinations of those elements (and will in fact contain exactly all linear combinations that can be formed with those elements).
So in your example... | {
"language": "en",
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Locally symmetric spaces and the curvature tensor
Let $M$ be a Riemannian manifold. Suppose $\nabla R=0$ where $R$ is the curvature tensor (we then say $M$ is locally symmetric). Then if $\gamma$ is a geodesic of $M$ and $X,Y,Z$ are parallel vector fields along $\gamma$ then $R(X,Y)Z$ is a parallel field along $\gamma... | The covariant differential $\nabla T$ of a tensor $T$ of order $r$ is defined as:
$$\nabla T (Y_1, \ldots, Y_r, Z) = Z(T(Y_1, \ldots, Y_r)) - T(\nabla_Z Y_1, \ldots, Y_r) - \cdots - T(Y_1, \ldots, Y_{r-1}, \nabla_Z Y_r)$$
In this case $T = R$ and $r = 4$ because the Riemann tensor takes 4 arguments $R(X,Y,Z,W)$. So we ... | {
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Books on graph/network theory with linear algebra focus I am interested on getting feed back on books that are graph theory with focusing on linear algebra(have taken several courses on Linear Algebra)
I have gone through
*
*Introductory Graph Theory by Gary Chartrand
*Graph Theory and Complex Networks: An Introd... | This question is very similar to the question
Textbook on Graph Theory using Linear Algebra
except the stress on the networks so I could find
Book recommendation for network theory
but I could not find a question on network theory and linear algebra, except that what is the difference between network theory and gr... | {
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Prove that if $ p | x^p + y^p $ where $p$ is a prime number greater than $2$, then $p^2 | x^p + y^p$ I was trying to solve the following problem recently:
Prove that if $ p | x^p + y^p $ where $p$ is a prime number greater than $2$, then $p^2 | x^p + y^p$. Here $x$ and $y$ are both integers.
$a|b$ reads $a$ divides $b$... | $p|x^p+y^p$ given
since $p$ is a prime number $> 2$, it is an odd prime number
For any odd power $p$, $x^p+y^p$ has a factor $(x+y)$
Using Fermat's little theorem
$$x^p \equiv x \mod p\\
y^p \equiv y\mod p\\
x^p+y^p \equiv (x+y) \equiv 0\mod p$$ since $p|x^p+y^p$
Therefore $p|(x+y)$, so $(x+y) = m*p$ where $m$ is an ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Optimization inside integral I want maximize the integral
$$\int_a^b \left( 2 cx y(x) - e y(x)^2 \right) \, \mathrm{d}x$$
with respect to to $y(x)$.
If I discretize the problem, I get
$$ \frac{b-a}{n}\sum_{i=1}^n 2c(i/n(b-a)+a)y_i-eyi^2$$
If I take the derivative with respect to each $y_i$, I find in undiscretized vers... | No fancy calculus is required. Since there are no constraints and no derivatives of $y$ occur in the integral, only $y$ itself, the integral is directly maximised by maximising the integrand at each point separately. Setting the derivative of $2cxy-ey^2$ with respect to $y$ to $0$ yields $2cx-2ey=0$ and thus $y=\frac{c... | {
"language": "en",
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Complex numbers inequalities and optimisation I'm now aware that you can't definitely with ease say that one complex number is greater than another.
Though what about imaginary numbers? Is $5i > 3i$? Is $i>-i$?
Is it possible to optimise (find the minimum or maximum) of a complex number function? I assume if it's not ... | I Think we can't say that 5i>3i.
But if the modulus is greatest then we say the complex number is greater.|5i|=5 &|3i|=3. So we may say that 5i>3i. But it is true that we can't say that 5i>3i
| {
"language": "en",
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Evaluating $\int_{0}^\infty \frac{\log x \, dx}{\sqrt x(x^2+a^2)^2}$ using contour integration I need your help with this integral:
$$\int_{0}^\infty \frac{\log x \, dx}{\sqrt x(x^2+a^2)^2}$$
where $a>0$. I have tried some complex integration methods, but none seems adequate for this particular one.
Is there a specific... | $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
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\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
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Area of a triangle whose vertices lie on a parallelogram In the parallelogram $ABCD$, $X$ and $Y$ are the midpoints of $BC$ and $CD$. Then prove that $$Ar(\triangle AXY) = \frac {3}{8} Ar(ABCD)$$
My Attempt :
Construction; Joining $BY$ and $AC$, I got
$$\triangle AYC=\triangle BCY$$.
But I couldn't move further from ... | Use $$S_{ABC}=\frac12 a\cdot h_a$$ and $$S_{ABCD}= a\cdot h_a$$
Let $Ar(ABCD)=S$, then $Ar(\triangle ABX)=\frac14S$, $Ar(\triangle ADY)=\frac14S$, $Ar(\triangle CXY)=\frac18S$
Then $Ar(\triangle AXY)=S-\frac14S-\frac14S-\frac18S=\frac38S$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1812190",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Convergence under a Hilbert space Let $\{\varphi_n\}_{n=1}^\infty$ be an orthonormal sequence (not necessarily a basis) in a Hilbert space. Let $\{\lambda_n\}_{n=1}^\infty$ be a sequence of numbers
Define $T:H\to H$ by $Tx=
\sum_{n=1}^{\infty} \lambda_n\langle x, \varphi_n\rangle \varphi_n$
Show that if $sup_n|λ_n| < ... | Hint:
Since $(\varphi_n)_{n=1}^{\infty}$ is an orthogonal sequence, the expression
$$
\sum_{n=1}^{\infty}|\langle x, \varphi_n\rangle |^2
$$
always converges. This follows from Bessel's inequality.
Using the fact that $\quad\sup_n|\lambda_n|<\infty\quad$, there's an $M>0$ such that $|\lambda_n|<M$ for all $n\in\B... | {
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Calculating $\lim \limits_{x \to \infty} \frac{x+\frac12\cos x}{x-\frac12\sin x}$ using the sandwich theorem Calculating $\lim \limits_{x \to \infty} \dfrac{x+\frac12\cos x}{x-\frac12\sin x}$
Correct me if I'm wrong:
$\cos x$ and $\sin x$ are bounded so that
$$|\cos x|\le 1,\qquad |\sin x|\le1$$
Therefore I can say:
$... | Yes, your solution is correct.
Indeed : $|\cos x| \leq 1 \Leftrightarrow -1 \leq \cos x \leq 1 $ and $|\sin x| \leq 1 \Leftrightarrow -1 \leq \sin x \leq 1$
This means that your fraction is bounded between the values that it gets for the upper and lower bounds of $\cos x, \sin x$. $x$ approaches $\inf$ which makes the ... | {
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Integrate $\int^a_0x^3 - x \mathop{\mathrm{d}x}$ to find the area
(a) Solve the equation: $$\int^a_0x^3 - x \mathop{\mathrm{d}x} = 0, a > 0$$
(b) For this value of $a$, find the total area enclosed between the $x$-axis and the curve $y=x^3 - x$ for $0 \leq x \leq a$.
I can quite easily solve the first part but for p... | Actually problem part(a) is not different from part (b), as it is asking to just check your work.. so getting zero between $ 0, \sqrt 2 $ shows your result is in fact correct.
| {
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Are standalone statements conventionally considered to imply truth? From what I understand, the statement $\exists x(p(x) \vee q(x))$ in the English language sounds something like this: "There exists $x$ such that $p(x)$ or $q(x)$". But this sounds like an incomplete claim; "There exists $x$ such that $p(x)$ or $q(x)$ ... | Logical formulae like this are always meant to be true, and if what they express is false, so is the formula.
That is, the formula you gave essentially says "It is true that there is an x which is p or q", but since the "It is true that..."-part is already part of the semantics of logical formulae anyway, you wouldn't ... | {
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Uniqueness of cyclic groups up to isomorphism. I am reading something about abstract algebra. Can anyone please tell me if the following statements are saying: a cyclic group can only be isomorphic to either $\mathbb{Z}/n\mathbb{Z}$ if it is finite or $\mathbb{Z}$ if it is infinite?
Thanks a lot.
The group $\mathbb{Z}$... | Yes, that is correct. If a group $G$ is cyclic, then $G$ is generated by (the positive and negative powers of) a single element $a$, i.e there exists an $a \in G$ such that $G=\{a^n: n \in \mathbb{Z} \}$. If $G$ is finite, then there exists an isomorphism from $G$ to $C_n$, and if $G$ is infinite, then there exists an... | {
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"timestamp": "2023-03-29T00:00:00",
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Finding all pairs of integers that satisfy a bilinear Diophantine equation The problem asks to "find all pairs of integers $(x,y)$ that satisfy the equation $xy - 2x + 7y = 49$.
So far, I've got
\begin{align}
xy - 2x + 7y &= 49 \\
x\left(y - 2\right) + 7 &= 49
\\
y &\leq 49
\end{align}
I can't get any further. Any hel... | hint: $xy+7y = 2x+49 \implies (x+7)y = 2x+49 \implies y = \dfrac{2x+49}{x+7}= 2 + \dfrac{35}{x+7}\implies (x+7) \mid 35\implies x+7 = \pm 1, \pm 5, \pm 7, \pm 35$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the function an entire function? Is the following an entire function? (Here $z\in \mathbb{C}$)
$$\sum_{n=0}^\infty \frac{2^n}{n!}z^{3n}$$
($***$) So, here I first note that the function is a sum of powers of $z$. Now if I show that the sum converges for all $z\in \mathbb{C}$, the problem will be solved, right?
Usin... | The function defined by the sum is entire, yes. For this, it suffices to check that the radius of convergence is infinite (which you have done); the result then follows as this implies that the power series is uniformly convergent everywhere.
| {
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Condition on $a$ for $(x^2+x)^2+a(x^2+x)+4=0$ Find the set of values of $a$ if $$(x^2+x)^2+a(x^2+x)+4=0$$ has
$(i)$ All four real and distinct roots
$(ii)$ Four roots in which only two roots are real and distinct.
$(iii)$ All four imaginary roots
$(iv)$ Four real roots in which only two are equal.
Now if I set $x^2+x=t... | Alternatively, graphing it:
Let $a=y$, then $(x^2+x)^2+y(x^2+x)+4=0$
$\displaystyle y=\frac{1}{4}-\left( x+\frac{1}{2} \right)^{2}+
\frac{4}{\frac{1}{4}-\left( x+\frac{1}{2} \right)^{2}}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1814099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Prove the polynomial $P_a=X^5 + a$ is reducible over a field
Let $(K, +, \cdot)$ a finite field so that the polynomial $P=X^2-5$ is
irreducible.
Prove that:
a) $1+1 \ne 0$
b) The polynomial $P_a=X^5 + a$ is reducible $\forall a \in K$
a) This is the easy part.
Suppose $1+1=0$. Then $5=1$ therefore $P=(X-1)(X+1)$ con... | If $a=0$,we're done. So, suppose $a\in F^\times$.
Let $p=$char$ F$. Then, $F=\Bbb{F}_{p^k}$. If $k$ is even, then, $\Bbb{F}_{p^2}\subseteq F$. However, $X^2-5$ splits over $\Bbb{F}_{p^2}$, as splitting field of a degree two polynomial is of degree two, and $\Bbb{F}_{p^2}$ is the only degree two extension of $\Bbb{F}_p$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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why cannot the limits be $-1$ and $-2$ I came across a problem in definite integral as : Evaluate
$$I=\int_{0}^{3} x\sqrt{1+x}\:dx$$
By the substitution $1+x=t^2$
so book has given lower and upper limits as $t=1$ and $t=2$ which is obtained as
$t^2=1$ $\implies$ $t=1$ and
$t^2=4$ $\implies$ $t=2$
we get $$I=\int_{1}^{... | The book does a bad service to students, in my opinion. The substitution should be $t=\sqrt{1+x}$, making it clear that $t\ge0$ and also providing for automatic substitution of the bounds.
Of course you can also do the substitution $t=-\sqrt{1+x}$, so the bounds are $-1$ and $-2$, but the integral becomes
$$
\int_{-1}^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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uniform rod revolving around a vertical axis with given angular velocity and given length of rod A uniform rod of given length and given angular velocity is revolving around a vertical axis.
Clearly it can do so in a horizontal plane with respect to vertical axis.
At what other angle can it do so?
That is, what is t... | The torque due to gravity and centripetal force must be equal. Let's call the coordinate along the rod $x$, varying from 0 to $L$. A small piece, of length $dx$ has a mass of $\frac{m}{L} dx$. The torque due to gravity of this small piece is $$d\tau_g=\frac{m}{L} dx g x \sin \theta$$ The total torque due to gravity is ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convergence in probability
Can anyone tell me how they got the regions $0<\epsilon<\theta$ and $\epsilon >0 $.
Also to clarify, is the last step where it says $\lim_{n \to \infty} P(|Y_n-\theta|>\epsilon)=0$
| \begin{align}
\mathbb{P}(|Y_n - \theta | \geq \epsilon) &= 1- \mathbb{P}(-\epsilon \leq Y_n - \theta \leq \epsilon) \\
&= 1 - \mathbb{P}(Y_n \leq \theta + \epsilon) + \mathbb{P}(Y_n \leq \theta - \epsilon) \\
&= 1 - F_{Y_n}(\theta + \epsilon) + F_{Y_n}(\theta-\epsilon). \\
&= F_{Y_n}(\theta- \epsilon).
\end{align}
Not... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How many solutions has the equation $\sin x= \frac{x}{100}$ ? How many solutions has the equation $\sin x= \frac{x}{100}$ ?
Usually when I was asked to solve this type of problem, I would solve it graphically but this one seems to be trickier. It doesn't seem wise to put $f(x)=\sin x$ and $g(x)=\frac{x}{100}$ in the s... | First, we may suppose $x\ge 0$ since both sides are odd functions.
Using the Intermediate value theorem, there'll be two non-negative solutions on each interval $]2k\pi,2(k+1)\pi[$ as long as $\frac x{100}\le 1$, i.e. $x\le 100$. There results the number of non-negative solutions is equal to $2\times \biggl\lfloor \dfr... | {
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proof that diagonals of a quadrilateral are perpendicular if $AB^2+CD^2=BC^2+AD^2$ proof that diagonals of a quadrilateral are perpendicular if $AB^2+CD^2=BC^2+AD^2$.
My Attempt:we know that if diagonals of a quadrilateral are perpendicular then we have $AB^2+CD^2=BC^2+AD^2$.But have to proof opposite of it?
| Assume the intersection of the to diagonals is $O$. Let $|OA|=a,|OB|=b,|OC|=c,|OD|=d$. Assume $\angle AOB=\gamma$. Then
$$|AB|^2=a^2+b^2-2ab \cos\gamma,$$
$$|CD|^2=c^2+d^2-2cd \cos\gamma,$$
$$|BC|^2=b^2+c^2-2bc \cos(\pi-\gamma)=b^2+c^2+2bc \cos\gamma,$$
$$|AB|^2=a^2+d^2-2ad \cos(\pi-\gamma)=a^2+d^2+2ad \cos\gamma.$$
F... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Expression for Euler characteristic in differential geometry Is the Euler-Poincare$ ^{\prime}$ characteristic $\chi$ a bending invariant? If so, how should it be isometrically expressed in terms of first fundamental form coefficients?
Is not stretching invariant better as terminology? If so how can it be expressed in t... | The Euler characteristic in a homotopy invariant, and that is a more precise statement that the one you want.
The Gauss-Bonnet theorem states that the Euler characteristic of a closed orientable surface is equal, up to normalization, to the integral of the gaussian curvature.
More generally the Chern-Gauss-Bonnet theo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1814943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do I calculate the gradient of a function in a $n$-dimensional space? $q(x)=x^TAx+b^Tx+c$
$A$ is matrix. $x,b\in \mathbb{R}^n$ and $c\in \mathbb{R}$
I really don't know how to calculate it for this function.
| $$q(x+h)=(x+h)^TA(x+h)+b^T(x+h)+c=\\=x^TAx+b^Tx+c+\color{blue}{h^TAx}+x^TAh+b^Th+h^TAh=\\ =\color{red}{x^TAx+b^Tx+c}+\color{blue}{x^TA^Th}+x^TAh+b^Th+\color{brown}{h^TAh}=\\=\color{red}{q(x)}+x^T(A^T+A)h+b^Th+\color{brown}{O(\lVert h\rVert^2)}$$
Hence, $$\nabla_xq=x^T(A^T+A)+b^T$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1815026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 2
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Prove that $\lambda(f) = o(2^n)$ for almost all boolean functions How to prove that $\lambda(f) = o(2^n)$ for almost all boolean functions $f$ of $n$ variables?
Here $\lambda(f)$ denotes minimal length (i.e. count of terms) of all possible disjunctive normal forms (DNFs) of $f$.
| This is the Korshunov–Kuznetsov Theorem. Quoting [1] which restates the theorem:
Theorem. (Korshunov–Kuznetsov, 1983) The optimal DNF size for a random Boolean function is $(K + o(1))\frac{2n}{\log n \log\log n}$, where $1\leq K\leq 1.54169$ (and the $\log$ is in base $2$)
They refer to the article of Pippenger [2], ... | {
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"timestamp": "2023-03-29T00:00:00",
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Disprove: $f\circ g = f \circ h \implies g=h$ for a surjective function $f$ I tried using a very specific counterexample here where I select a surjective function for which the compositions are equal but the functions within are not.
This is probably off-base, but it's what I've got so far.
Assume $f \circ g = f \cir... | Let $f: \mathbb{R} \rightarrow [-1,1]$, $f(x)=\sin(x)$. Then let $g(x)=x$, $h(x)=x+2 \pi$. $f$ is surjective and $f \circ h= f \circ g$, but we clearly don't have $h=g$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Expected value of $\max\{X_1,\ldots,X_n\}$ where $X_i$ are iid uniform. Let $X_i\sim\mathrm{Uniform}(0,\theta)$ be iid, what is $E[\max\{X_1,\ldots,X_n\}]$? Apparently the answer is $$\frac{n}{n+1}\theta,$$
but I do not see why?
It seems intuitive in that you would "expect" them to be spaced out evenly, hence the maxim... | Your intuition is correct.
To see this mathematically, suppose $X_1, \ldots, X_n$ are independent and uniformly distributed and $M_n = \max(X_1,X_2,\ldots,X_n).$
The distribution function of the maximum is the joint probability that $X_k \leq x$ for all $k.$ This is a product of marginal probabilities since the varia... | {
"language": "en",
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Given $a,b_0,\ldots,b_n$, there exists a polynomial of degree $\le n$ s.t. the derivatives $f^{(i)}(a)=b_i$ Just exploring some maths problems from a book until I came across this question.
Let $a, b_0, . . . , b_n ∈ R$. Show that there exists a polynomial $f(x)$ of degree at most n such that
$$f(a) = b_0, f'
(a) = b_... | Consider the general polynomial of degree $n$
$$
f(x) = a_0 + a_1x + \ldots a_nx^n
$$
Since we know that a polynomial is uniquely determined by its Taylor expansion
, consider the taylor expansion of $f(x)$ about $a$, which is
$$
f(x) = f(a) + \frac{(x - a)f'(a)}{1!} + \frac{(x - a)^2f''(a)}{2!} + \ldots \frac{(x - a)^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find the center of the circle through the points $(-1,0,0),(0,2,0),(0,0,3).$ Find the center of the circle through the points $(-1,0,0),(0,2,0),(0,0,3).$
Let the circle passes through the sphere $x^2+y^2+z^2+2ux+2vy+2wz+d=0$ and the plane $Ax+By+Cz+D=0$
So the equation of the circle is $x^2+y^2+z^2+2ux+2vy+2wz+d+\lamb... | follow this process
*
*find the plane passing through the three points (as the 3 points are points on axis it is very simple)
*take normal vector to the plane
*consider any two sides of the triangle made by the three points and take their mid-point
*now take cross product between side's vector and plane's norma... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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"answer_id": 1
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Prove that $X^4+X^3+X^2+X+1$ is irreducible in $\mathbb{Q}[X]$, but that it has two different irreducible factors in $\mathbb{R}[X]$
Prove that $X^4+X^3+X^2+X+1$ is irreducible in $\mathbb{Q}[X]$, but it has two different irreducible factors in $\mathbb{R}[X]$.
I've tried to use the cyclotomic polynomial as:
$$X^5-1=... | let $$P(x)=x^4+x^3+x^2+x^1+1$$
We know if $x=\frac{a}{b}$ is root of $P(x)$ then $b|1\,$ , $\,a|1$. In the other words $a=\pm 1 $ and $b=\pm 1 $ but $P(1)=5$ and $P(-1)=1$, thus we let
$$P(x)=(x^2+ax+b)(x^2+cx+d)$$
as a result
\begin{align}
& bd=1 \\
& ad+bc=1 \\
& b+d+ac=1 \\
& a+c=1 \\
\end{align}
This sys... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1815662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Ten digit numbers divisible by 3. I came across an interesting property of 10-digit numbers that are constructed using each digit only once: e.g. $9867534210$ or $352147890$. These numbers are exactly divisible by $3$. Each and every of the $10!$ combinations are also divisible by $3$.
But why is this property emerging... | You have $10$ digits $[0,1,2,3,4,5,6,7,8,9]$ and if you construct any possible number by taking each digits once,you'll get $10!$ numbers.
A number is divided by $3$,if sum of the digits of the number is divided by $3$
For all these numbers (Sum of digits)=$(0+1++2+3+4+5+6+7+8+9)=45$ is divided by $3$.
So,all these n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1815742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove that the condition is sufficient? I was solving a question at a programming website.
The question specifies that a person is standing at point $(a, b)$ in an infinite 2D grid. He wants to know if he can reach point $(x, y)$ or not.
The only operation allowed is to move to point $(a, a+b)$ or $(a+b, b)$ or ... | Let $d = gcd(a, b) = gcd(x, y)$
For simplicity, I am assuming that $a, b, x, y$ are nonnegative. Otherwise, you can easily modify them until they are.
You can use the euclidean algorithm to get from $(a, b)$ to $(x, y)$:
The algorithm looks like this (quoting Wikipedia):
function gcd(a, b)
while a ≠ b
if a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Dual space with $W=\ker T$ Let $f \in V^*$ with $V$ a vector space and $W=ker f $. If $v_0 \in V$ is a vector such that $f(v_0)\ne 0$ then for every $v \in V$ there exist unique $w \in W$ and scalar $c$ such that $v=cv_0+w$.
How can I prove this I don't understand it, please if someone can help me. Thanks for you time... | Since there exists $v_0 \in V$ such that $f(v_0) \neq 0$, we know $f$ is not trivial. Since $V$ is a vector space over a field $F$, and $f:V \rightarrow F$ is linear, we must then have $f$ surjective, since $F$ has no nontrivial proper subspaces (this is a property of fields). Then we have by the first isomorphism theo... | {
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