Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How about the converse of the property that a product of manifolds is a manifold? We know that the Cartesian product of two manifolds is a manifold, but is the converse true? Let us assume that we have $A$ and $B$ two second countable Hausdorff topological spaces, and $M = A \times B$. If we assume that $M$ is a $n$-ma... | The dogbone space is not a manifold but its product with $\mathbb{R}$ is homeomorphic to $\mathbb{R}^4$. This was proved in this paper by Bing.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Logarithm power law discrepancy According to the power law:-
$$\log_a (x^k) =k\log_a x $$
So take the following example:-
$$\log_2 ((-2)^2) $$
On solving $\log_2 4=2$
However, if we use the power law, then on simplifying, $2\log_2 (-2)$ is not defined.
So how do I justify this?
| This dicrepancy is due to the fact that
*
*$\log x^2$ is defined for $x\neq 0$ but
*$2\log x$ is defined for $x>0$
then the two expression are equal $\iff x>0$.
What is true $\forall x\neq 0$ is that
*
*$\log x^2=2\log |x|$
where we have used that $\sqrt {x^2}=|x|$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2694848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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general equation to find cubic polynomial from two minimums? I tried researching and found that I can use a system of linear equations and solve by an inverse matrix to find the cubic equation given 4 points which satisfy the function f(x) of the general form $f(x)=ax^3+bx^2+cx+d$
I can also find a cubic of the form $a... | Well, since the extrema are the roots of the derivate, it would be nice to have the minimum and the maximum (and an initial condition too.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2694934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find a 1-dimensional sufficient statistic for theta. $X_1, X_2,\ldots,X_n$ is a sample from random variable $X$.
$$f(x\mid\theta)=\frac 1 {\sqrt{2\pi}}\exp{(-\frac{1}{2}(x-\theta)^2)}$$
I have to find a 1-dimensional sufficient statistic for $\theta$ and I'm not sure how to go about doing this. Any help would be apprec... | Use the factorization theorem.
Hint #$1$:
$$
\sum(x_i - \theta)^2 =\sum x_i^2-2\theta n\bar{x}_n+n\theta^2,
$$
Hint #$2$:
The MLE is always a function of the MSS. What is the MLE of $\theta$?
| {
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How to decide the neighbourhood when proving a function is locally Lipschitz-continuous. I have come across a problem in one of my textbooks that I understand but could not replicate.
Show that the function $f : \mathbb{R}^+ → \mathbb{R}$ where $f(x) = {1\over x}$ is locally Lipschitz-continuous in $\mathbb{R}^+ = (0, ... | Fact: If the derivative of a function is bounded over an interval, then the function is Lip on that interval with bound equal to the supremum of the derivatie there.
Derivative of $1/x$ is $1/x^2$ which is bounded from above as long as we're away from zero. So, in fact $1/x$ is Lipschitz over any $[\delta, +\infty)$, ... | {
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Orthonormal basis. Linear Algebra. Hint.
Let $\lbrace u_{1}, ..., u_{n} \rbrace$ an orthonormal basis of $\mathbb{R}^{n}$, then $\displaystyle x = \sum_{i=1}^{n}\langle x, u_{i} \rangle u_{i}$ for all $x \in \mathbb{R}^{n}$.
I know that $\langle u_{i}, u_{i} \rangle = 1$ and $\langle u_{i}, u_{j} \rangle = 0$ if $i \... | Hint: Write $x=x_1u_1+\dots+x_nu_n$ and take its inner product with $u_i$.
| {
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How to find the image under a Möbius transformation. Find the image of the set $D_1\cap D_2$, where $$D_1 = \{z : |z| < 1\}$$ and
$$D_2 = \{z : |z + 1/2| > 1/2\}$$ under the transformation
$$f(z) = \frac{z − i}{z + 1}$$
I have done the picture of $D_1\cap D_2$, but I don't know how to do it. If anybody could help me, p... | We have $f^{-1}(z) = \frac{z + i}{1 - z}$.
Let $w \in f(D_1)$. Then $f^{-1}(w) \in D_1$ and by definition we have $|f^{-1}(w)| = \left|\frac{w+i}{1-w}\right| < 1$, that is
$$|w+i| < |w-1|$$
So we can conclude that $f(D_1)$ contains all points $w$ that are closer to $-i$ than they are to $1$, i.e. the "half plane" $\{x+... | {
"language": "en",
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prove that $\{a_n\}_{n=1}^{\infty}$ is bounded and monotone increasing. I am trying to prove the sequence $\{a_n\}_{n=1}^{\infty}$, which is defined by $a_{n+1} = \frac {2(a_n+1)}{a_n+2}$, and $a_1=1$.
(1) prove that it is monotone increasing. ($a_{n+1} \ge a_n$)
proof by induction.
$P(1)$ : $a_2= \frac {2(1+1)}{2+1} ... | I'll play around
and see what happens.
$a_{n+1} = \frac {2(a_n+1)}{a_n+2},
a_1=1
$
$\begin{array}\\
a_{n+1}^2-2
&= \dfrac {4(a_n+1)^2}{(a_n+2)^2}-2\\
&= \dfrac {4a_n^2+8a_n+4-2(a_n^2+4a_n+4)}{(a_n+2)^2}\\
&= \dfrac {4a_n^2+8a_n+4-2a_n^2-8a_n-8}{(a_n+2)^2}\\
&= \dfrac {2a_n^2-4}{(a_n+2)^2}\\
&= 2\dfrac {a_n^2-2}{(a_n+2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Whats the probability you lose money? Suppose you are playing a game that costs $ $8 to play. You flip 10 coins and, for every head, you win $2. Whats the probability you lose money ?
$$
\begin{array}{c|cccccccccc}
x& 1 & 2 & 3 & 4 & 5 & 6 & 7 &8 &9&10 \\ \hline
p(x)
& 5/512 ... | Don't forget about the possibility of a super bad luck event, that is you might get all tails.
probability of losing money is equal to
$$\frac{1}{2^{10}}\sum_{i=0}^3 \binom{10}{i}= \frac{1+10+45+120}{1024}\approx 0.1719$$
| {
"language": "en",
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Name of the numerical method I came across one numerical method, however I have lost the link or is not able to find it on my history.
It used the following variables:
$$G=\frac{p'(x_k)}{p(x_k)} \ \& \ \ H=G^2-\frac{p''(x_k)}{p(x_k)}$$
Or something to that effect, not entirely sure. Thanks in advance!
| That looks like part of the Laguerre method for polynomial roots.
$$
x_{k+1}=x_k-\frac{n}{G\pm\sqrt{(n-1)(nH-G^2)}}.
$$
| {
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How to show that $f$ is monotone?
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be defined by
$$f(x) = \int_{e^{x^3 +x}}^{1+e^{x^3+x}}e^{r^2} dr$$
for all $x\in\mathbb{R}$. Prove that $f$ is monotone.
I was thinking about Leibniz rule. Actually my main problem is that how to show that f is continuous...
I am struck at... | For the first derivative of your integral we get $$e^{x^3+e^{2 \left(x^3+x\right)}+x} \left(e^{2
e^{x^3+x}+1}-1\right) \left(3 x^2+1\right)$$ this derivative is positive.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What is the probability of drawing a "conditionally" specific set of cards in a hand of 7? With the following specifics:
*
*Deck of 60 cards:
4 copies of $\mathcal A$,
3 copies of $\mathcal B$,
16 copies of $\mathcal C$ and
37 copies of $\mathcal Z$
*Hand of 7 cards drawn from the deck of 60, shuffled beforehand
*... | Sketch:
Let the $4-$ tuple $(a,b,c,z)$ represent the number of cards of each type in your hand. These are the patterns you wish to count. Of course we must have $a+b+c+z=7$ but there are other restrictions as well.
The good hands satisfy: $$4≥a≥1\quad 3≥b≥1\quad c≥2$$
We remark that we can drop the $4≥a$ restriction... | {
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I don't understand how the diameters of $ \bar A$ and $A$ are equal I'm given the assignment to prove that $\operatorname{diam}\bar A = \operatorname{diam}A$, where $\operatorname{diam}A=\sup\{\rho(a,b): a,b \in A\}.$
How can they be equal if $\bar A= \partial A \cup A$:
Say $x\in \partial A$, then does it not follow t... | We don't necessarily have any ordering among the points of the metric space (consider e.g. $\Bbb R^n$), so your statements $x>a$ and $x>b$ are meaningless.
To prove the claim, take any $a, b\in\bar A$ and consider sequences $a_n, b_n\in A$ with $a_n\to a, \ b_n\to b$, and note that $\rho(a_n, b_n) \to\rho(a, b) $.
| {
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"timestamp": "2023-03-29T00:00:00",
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Solving $2^x \cdot 5^y = 0,128$ $x,y \in \mathbb Z$
$2^x \cdot 5^y = 0,128$
$x+y = ?$
My attempt:
I know that
$$0,128 = \frac{128}{1000}$$
$$5^3 = 125$$
$$2^{-3} = \frac{1}{8}$$
EDIT:
$2^7 = 128$
Then we need to get
$0,128$
| You can write 5 in terms of 10 and 2... so you can write the problem as:
$2^x \cdot (\frac{10}{2})^y=.128$
$2^{x-y} \cdot 10^y=.128$
Find $y$ first and then $x $
| {
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"timestamp": "2023-03-29T00:00:00",
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A pattern appearing in a dynamical sequence of polynomials This question came up while I was looking at the dynamics of the map in this question
Let $f : \Bbb C[x,y,z] \to \Bbb C[x,y,z]$ given by $f(P)(x,y,z) = P(y(x+y),2z^2,z(x+y))$.
Consider the sequence of polynomials obtained by iterating $f$ on $z$ :
$P_0(x,y,z) ... | The results for this problem are typical. Define the homogeneous quadratic polynomial:
$\;L(x,y,z)=(y(x+y),2z^2,z(x+y)),\,$ and using initial value $L_0(x,y,z) = (x,y,z),\,$ define sequence $\,L_n\,$ by recursion $\,L_{n+1}=L(L_n).$
Now name the three components: $\;(r_n,q_n,p_n):=L_n(x,y,z).\;$ For example, $\,r_2=2z... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluating the lambda expression $(.)(.)(.)$ In the book of Haskell Programming by C. Allen, at page 39, it is given the following lambda expression
$$(.)(.)(.)$$
According to me, this equals to by applying the left two expression as an input for the rightmost expression
$$(.)(.)(.) = (.)(.)y = (.)yz = yzy,$$
and if ... | It appears that the book first presents one example and then just moves to another one, as $(λxy.xxy)(λx.xy)(λx.xz)$ is not equivalent to $(λxyz.xz(yz))(λmn.m)(λp.p)$.
In order to not get confused with clashing variable names I prefer to use De Bruijn indices; the initial expression would then be:
$(λλ221)(λ12)(λ12)$
a... | {
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How do I verify if $\phi : \mathbb{Z}_{6} \rightarrow \mathbb{Z}_{36}$ is well-defined? Given $\phi : \mathbb{Z}_{6} \rightarrow \mathbb{Z}_{36}$ and $\phi ([a]_{6})=[a]_{36}$, verify that $\phi$ is a well-defined function.
My understanding is that well-defined is the converse of injective. So given a function $f:A\rig... | To prove that a statement is false, you just need to find a counter example. You cannot say "this statement is false because I cannot prove it". For here, you just need to find some $[a]_{6}=[b]_{6}$ such that $\phi([a]_{6})\neq \phi([b]_{6})$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2697837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Convex sets. Real Analysis.
Let $C \subset \mathbb{R^{n}}$ a convex set. Fixed $p \in \mathbb{R}^{n}$, let $\varphi: C \longrightarrow \mathbb{R}$ the function defined by $\varphi(x) = |x-p|=\sqrt{\langle x-p,x-p\rangle}$. Theres exists as most one point $a \in C$ such that $\varphi(a) = \inf \lbrace \varphi(x) | x \i... | As mentioned, we need $C$ to be closed for this. Assume $a, b \in C$ are both distance minimizers to $p$. Let $\varphi(a) = \varphi(b) = d$ be the minimum value.
Using the paralellogram identity, we obtain:
\begin{align}
4d^2 &= 2\|a - p\|^2 + 2\|b-p\|^2 \\
&= \|(a-p)+(b-p)\|^2 + \|(a-p)-(b-p)\|^2 \\
&= 4\underbrace{\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2698041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Inequality: $0<\alpha<1$, $(\sum_{i=1}^na_i)^{\alpha} \leq \sum_{i=1}^na_i^\alpha$ I would like to know if the following if true for $0<\alpha<1$, $a_i>0$ for all $i$.$$(\sum_{i=1}^na_i)^{\alpha} \leq \sum_{i=1}^na_i^\alpha$$
This looks like Jensen's inequality with counting measure. But $x^\alpha$ is
a concave functi... | For $n=2$, and $a>0$
\begin{align*}
(1+a)^{\alpha}-a^{\alpha}&=\int_{0}^{1}\dfrac{d}{dt}((1-t)a+t(a+1))^{\alpha}dt\\
&=\int_{0}^{1}\alpha((1-t)a+t(a+1))^{\alpha-1}dt\\
&=\int_{0}^{1}\alpha(a+t)^{\alpha-1}dt\\
&\leq\int_{0}^{1}\alpha t^{\alpha-1}dt\\
&=1,
\end{align*}
so $(1+a)^{\alpha}\leq 1+a^{\alpha}$.
So $(a_{1}+a_{... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove linear transformation is one to one The linear operator T: R2→R2 defined by the equations
w1 = 4x1 - 6x2
w2 = -2x1 + 3x2
is not one-to-one. Using the methods in class, show why this is true. Once you have done this, provide a simple, specific, numerical example, where the output vector is not the zero vector,... | You are suppose to find $x_1, x_2, y_1, y_2$ such that $(x_1, x_2) \neq (y_1, y_2), (w_1, w_2) \ne (0,0),$ and
$$4x_1-6x_2=w_1 = 4y_1-6y_2$$
$$-2x_1+3x_2=w_2 = -2y_1+3y_2$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Showing that $(a,b)=1$ where $d=(m,n)$ and $m=ad$ and $n=bd$ If $d=(m,n)$, then $d|m$ and $d|n$ so there must exist integers $a$ and $b$ such that $m=ad$ and $n=bd$. Now $d=(m,n)=(ad,bd)=d(a,b)$ and so $(a,b)=1$. Is this correct?
| Yes, the proof is flawless.You have proved that$$ d=(m,n)=(ad,bd)=d(a,b)\implies (a,b)=1$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Exponentials inequality: $|e^x + e^y| \leq |x - y|$ for $x,y<0$. I am trying to prove $|e^z - e^\omega| \leq |z - \omega|$ for $z,\omega\in\{z\in\mathbb{C}\;:\;\Re(z)<0\}$ and I get stuck in an inequality of the stated in the title : $|e^x + e^y| \leq |x - y|$ for $x,y<0$.
This is what I have tried:
\begin{equation}\le... | Um, take $x=y=-1$;
$$|e^{-1}+e^{-1}|>|x-y|=0$$
seems like your theorem isn't true.
| {
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"timestamp": "2023-03-29T00:00:00",
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For what non trivial values of $a$ and $k$, $a^k + 1$ will be prime?
For what non trivial values of $a$ and $k$, $a^k + 1$ will be prime?
If $a=1$, then $1^k+1=1+1=2$, where 2 is prime.
If $k=1$ then we have$a^k+1=a+1$, but if some prime $p$ is equal to $a+1$, then only we can say $a+1$ is prime.
But for the rest p... | consider the case $k=3$ then we get $$a^3+1=(a+1)(1-a+a^2)$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Showing that a linear map does not achieve its norm If we take the Banach spaces $(X, \|\cdot \|_X )$ and $(Y, \|\cdot \|_Y )$ and the bounded linear map $F: X \rightarrow Y$, then the norm of $F$ is $$\|F\| = \underset{\|x\|_X = 1}{\textrm{sup}} \|F(x)\|_Y.$$ My question is how we can determine, generally, whether or ... | Firstly, when considering $F$ as a map from $C([0,1])$ we have $\|F\|=\frac{1}{2}$. One way to see this is that
$$\left|\int_0^1 xf(x)\ dx\right|\leq\int_0^1|x||f(x)|\ dx\leq\|f\|\int_0^1x\ dx=\frac{1}{2}\|f\|,$$
and the constant function $f(x)=1$ gives us this upper bound.
Now let's write $\tilde X=\{f\in C([0,1])... | {
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"timestamp": "2023-03-29T00:00:00",
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Find all positives integers $n$ such that $n^3+1$ is a perfect square A solution as follows:
$n^3+1=x^2$
$n^3=x^2-1$
$n^3=(x-1)(x+1)$
$x-1=(x+1)^2~~or~~x+1=(x-1)^2$
$x^2+x+2=0~~or~~x^2-3x=0$
$x(x-3)=0$
$x=0~~or~~x=3~~\Longrightarrow~~n=2$
Does it cover all possible solutions? How to prove that 2 is the only which solve... | Hint: see that $m^2=n^3+1$ gives $(m-1)(m+1)=n^3$. What factors can $m-1$ and $m+1$ have in common? How can their product be a perfect cube?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Why does this integral not depend on the parameter? While working with stable distributions, we for $\alpha \in (1,2) $ have encountered the following integral:
$$\lim_{\varepsilon \to 0} \int_{(\varepsilon, \pi/2 - \varepsilon) \cup (\pi/2 + \varepsilon, \pi-\varepsilon)} \frac{\log \left(|\cos \theta|^\alpha + |\sin ... | A partial answer, designed for simplifying further attempts. Now an answer, with some detours not really needed but left here for documentation purposes.
$\cos\left(\theta+\frac{\pi}{2}\right)=-\sin\theta$ and $\sin\left(\theta+\frac{\pi}{2}\right)=\cos\theta$, hence the given integral can be written as
$$ \int_{0}^{\p... | {
"language": "en",
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Maximal ideals in a polynomial ring I am having a lot of trouble figuring out which ideals are maximal in rings that aren't fields. I read that maximal ideals of $\mathbb{Z}[x]$ are of the form $(p,f(x))$ where $p$ is a prime number and $f(x)$ is a polynomial in $\mathbb{Z}[x]$ which is irreducible modulo $p$.
$\\$I a... | We show that $(x+1)$ is not a maximal ideal in $\mathbb{Z}[x]$. Since $(x+1)\subset (2,x+1)\subset \mathbb{Z}[x]$, we just need to show that $(x+1)\neq (2,x+1)$ and $(2,x+1)\neq \mathbb{Z}[x]$. The first statement holds because $2\notin (x+1)$. The second statement is true by the result you said in the first paragraph.... | {
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Inequality $\frac{a^3+b^3+c^3}{3}\geq\sqrt{\frac{a^4+b^4+c^4}{3}}$ Let $a$, $b$ and $c$ be non-negative numbers such that $(a+b)(a+c)(b+c)=8$. Prove that:
$$\frac{a^3+b^3+c^3}{3}\geq\sqrt{\frac{a^4+b^4+c^4}{3}}$$
Some attempts:
*
*From the condition follows $a^3+b^3+c^3 = (a+b+c)^3 -24$
*It is known (see here)
$$\... | I repeat the above hints:
Setting $2x=a+b$, $2y = b+c$, $2z = a+c$, we can express $a =x+z-y$ etc. Further, we have $a^3+b^3+c^3 = (a+b+c)^3 -24 = (x+y+z)^3 -24$.
The condition then becomes $xyz = 1$ which can be parametrized with free variables $0\leq q \leq 2 \pi /3 $ and arbitrary $r$ by
$$
x = \exp(r \cos q) \; ;... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2699175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Showing $P(S_mLet $X_1, X_2,\ldots$ be a sequence of iid random variables such that for each $i$, $X_i$ takes value as nonnegative integer and is in $L^1$. Let $ S_n = \sum_{i=1}^n X_i$. How to show that
\begin{equation}
P(S_m<m, \forall\ 1\leq m\leq n | S_n)=\max\{0, 1-S_n/n\} ?
\end{equation}
I think that there is s... | Let $E_n = \bigcap_{i=1}^n \{S_i<i\}$. We will prove $P(E_n|S_n=k)=(1-k/n)^+$ by induction on $n$, where $x^+=\max(0,x)$.
Given a list of numbers $X = (X_1,\dots,X_n)$, let $Y = (Y_1,\dots,Y_n)$ be the same list rearranged in weakly increasing order. We will prove the stronger fact that for any deterministic $y=(y_1,\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2699295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How does WolframAlpha get an exact answer here? ${}{}$ I had a simple thing to compute with a calculator:
$$\sin\left(2\cos^{-1}\left(\frac{15}{17}\right)\right)$$
I got the decimal answer of about $0.83044983$, but when I typed it in WolframAlpha, it also gave an exact answer of $\frac{240}{289}$. How in the world wou... | $ (8,15,17)$ are lengths of a Pythagorean triple right triangle. A narrow right triangle of these side lengths can be drawn if needed.
$$\sin(2\cos^{-1}\frac{15}{17}) = \sin(2\sin^{-1}\frac{8}{17}) = 2 \cdot \frac{8}{17}\cdot \frac{15}{17} =\frac{240}{289}.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 7,
"answer_id": 1
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What groups have the property: All nontrivial isomorphic subgroups have a nontrivial intersection? This is a generalization of my previous question: What are all finite groups such that all isomorphic subgroups are identical?
Specifically, what finite groups $G$ have the following property: For any two subgroups $1 < H... | The question has an answer here.
[In such a group] every Sylow $p$-subgroup is either cyclic or generalized quaternion (because these are the only $p$-groups that have a unique subgroup of
order $p$). Apparently, Zassenhaus (in the 1930s) classified all
solvable groups where all the Sylow $p$-subgroups for odd pri... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Minkowski's triangle inequality for $0I have a trouble in the following theorem introduced "Minkowski's inequality for $0<p<1$":
Let $0<p<1$ and let $x,y\ge0$. Then, $|x^{p}-y^{p}|\le|x-y|^{p}$.
I have proved the case for $p=\frac{1}{2}$.
How do I prove it in the general case ?
Give some advice! Thank you!
| A concave function $f:[0,\infty)\to\Bbb R$ such that $f(0)\ge0$ must be subadditive: if by any chance $f(a+b)>f(a)+f(b)$, then $\frac{f(a+b)-f(a)}{b}>\frac{f(b)-f(0)}{b}$, against the hypothesis of concavity.
So, for $p\in(0,1)$ and $x,y\ge 0$, we have that $$(\max\{x,y\}-\min\{x,y\})^p+(\min\{x,y\})^p\ge (\max\{x,y\})... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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In general, what techniques can be used to show that 2 groups are not isomorphic? Say I have 2 groups $G$ and $H$, what techniques can be used to show that they are not isomorphic?
A simple one I can think of is proving that their order is different, thus showing there cannot be a bijection in between the 2. However I ... | Look for any difference in the groups such as
*
*Order of the groups.
*One group has an element of order $n$, and the other does not have an element of order $n$.
*One group has a subgroup of order $n$, and the other does not.
*The orders of the centers of the groups are different.
*One group is Abelian, the oth... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2699708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Minimizing $f(x,y,z)=y$ In the problem of minimizing $f(x,y,z)=y$ over the constraint set $z=y^3 - x^2$ and $z=x^2$, I have managed to solve the problem directly and obtain that the minimum occurs at $x=0, y=0, z=0$, yielding a value of $f(0,0,0)=0$, but when I write the Lagrangian and try to solve it, I realize that i... | Let $g(x,y,z):=(z-y^3+x^2, z-x^2)$, $h(x,y,z, \lambda, \mu):=f(x,y,z)+\lambda(z-y^3+x^2)+ \mu(z-x^2)$ and $\phi:= \nabla h$.
Lagrange says: if(!) $f$ has in $(x_0,y_0,z_0) $ a local extremum under the constraints $g(x,y,z)=(0,0)$ and if(!) $rank g'(x_0,y_0,z_0)=2$, then there are $\lambda_0, \mu_0$ such that $\phi(x_0,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $\cos(x)$ is continuous I'm stuck at a particular step and could use some help.
By definition, a function is continuous at $x=a$ iff $\lim_{x \to a} f(x) = f(a)$.
So I assume to prove $\cos(x)$ is continuous we must use the definition of a limit to show that:
$$\lim_{x \to a} \cos(x) = \cos(a) \iff \forall \epsi... | Use that $|\sin u|\leq\min\{|u|,1\}$, then $2\left|\sin\left(\dfrac{x+a}{2}\right)\right|\cdot\left|\sin\left(\dfrac{x-a}{2}\right)\right|\leq 2\left|\dfrac{x-a}{2}\right|=|x-a|<\epsilon$ if we take $\delta=\epsilon$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Can I combine two functions (express them as a single function) where they meet a certain criteria at different times. Say I have a function $F(x) = 7x$. That meets my criteria when $0 \leq x \leq 10$.
And another function $G(x) = x^2$. That meets my criteria when $x > 10$.
Is there a way to combine these two into on... | One way to do this is to define a piecewise function like N. F. Taussig said in the comments above. However I have my own alternative way of defining such funcion, using the properties of floor and ceiling functions.
First notice one special function that I found:
$$y=\Bigg\lceil\frac{\lfloor x\rfloor}{x}\Bigg\rceil$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2700065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Help me understand how we define closed sets in real analysis I'm currently reading the second version of Understanding Analysis by Stephen Abbott and on page $90$ theorem $3.2.8.$ states that "A set $F \subseteq \mathbb{R}$ is closed if and only if every Cauchy sequence contained in $F$ has a limit that is also an ele... | You are correct when you say that a singleton set does not contain any limit points.
However, think of the proposition in the following fashion : $F$ is closed, if the set of limit points of $F$ is a subset of $F$.
Then, the set of limit points of a singleton set, is the empty set, and is therefore contained in $F$. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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unexplained modulo, rs ≡ 1 (mod m) where r = 3, s= 59 and m = 176 I'm trying to work through the math behind public key encryption, I'm a computer programmer, but not a mathematician. I came across this wonderful example, but I'm confused about the use of (mod x) my understanding of modulus is that it is the remainder ... | You should know that $a \equiv b (\text{mod $m$)}$ means that $a-b$ is divisible by $m$.
For this modulo, you should note that $0$ is divisible by all positive natural numbers (result would be $0$), so $1$ when divided by any positive natural numbers (except $1$) will have $1$ as a remainder $\Rightarrow 1 \equiv 1 (\t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to evaluate $\int_a^b(b-x)(x-a)dx$ in a smart way? It holds that $\int_a^b(b-x)(x-a)dx=1/6(b-a)^3$. I was wondering if there is a smart way of seeing this, instead of brute-force solving the intergral (it's not that tedious, but if there is a quick way about it, I would like to know it)
If it's not possible to do i... | You may use the fact that for quadratic polynomials the Simpson $1-4-1$ rule is an exact quadrature formula. In particular
$$ \int_{a}^{b}(b-x)(x-a)\,dx = \frac{b-a}{6}\cdot 4\left(\frac{a-b}{2}\right)^2 =\frac{1}{6}(b-a)^3.$$
As an alternative, you may substitute $x=a+\lambda(b-a)$ and exploit
$$ \int_{0}^{1}\lambda(1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Counting 3-cycles in $S_6$ I've bolded the incorrect part.
Find the number of order $3$ elements in $S_6$
My "Solution":
There are two ways to obtain these, either a 3-cycle or product of 2 disjoint 3-cycles.
Counting 3-cycles:
Choose 3 of 6 elements, this will determine the first 3 elements. We don't worry about the... | Two elements of $S_{n}$ are conjugate if and only if they have the same cycle structure. There are two possible cycle structures for an element of order $3$ in $S_{6}$, as you noted yourself. Hence we need to count the conjugates of $x = (123)$ and of $y = (123)(456)$ in $S_{6}$ ( and add these numbers to get the numbe... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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The right adjoint of a functor is unique up to a natural isomorphism I'm having trouble showing that the right adjoint of a functor $F \colon \mathcal{C} \to \mathcal{D}$ is unique up to isomorphism. (My definition of adjoint functors is the one using units and counits.)
Let $G_1$ and $G_2$ be two right adjoints of $F$... | Hint 1: Find the arrows in the following diagram which make its squares commutative, using naturality.
$$\matrix{G_1 & \to & G_2FG_1 & \to & G_2\\
\downarrow && \downarrow && \downarrow \\
G_1FG_1 & \to & G_1FG_2FG_1 & \to & G_1FG_2 \\
&&\downarrow && \downarrow \\
&&G_1FG_1 & \to & G_1}$$
Hint 2: Alternatively, we c... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the equation of the normal at the point $(1,2)$ to the curve $y=x+\frac{1}{x}$. The answer in the textbook is listed as $x=1$, however I don't understand why this is.
My working:
$$y=x+\frac{1}{x}$$
$$f'(x)=1-x^{-2}$$
$$f'(x)=1-\frac{1}{x^{2}}$$
$$f'(1)=1-\frac{1}{1}=0$$
Therefore the gradient of the tangent to th... | Perhaps a picture will clarify the situation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2700938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Can this statement about supremum be proven directly? Theorem (I.32, pg 27, Apostol's Calculus Vol.1): Let $\varepsilon \in \mathbf{R}^+$ and let $S \subset \mathbf{R}$. If $\sup S$ exists, then there exists $x\in S$ such that $$x > \sup S -\varepsilon.$$
(Reminder: The real number $B = \sup S$ is the least upper bou... | For a direct prove:
Let $\epsilon >0$ be given.
Note that $ (\sup S - \epsilon) < \sup S $
Therefore $ (\sup S - \epsilon)$ is not an upper bound of S, because $\sup S$ is the least upper bound of S.
Thus there is an element of S, say $x$ such that $x> \sup S - \epsilon$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2701008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding value of $\int\frac{\ln(x)}{1+x^2}dx$
Finding value of $\displaystyle \int\frac{\ln(x)}{1+x^2}dx$
Try: let $$I=\int\frac{\ln(x)}{1+x^2}dx=\frac{1}{2}\int\bigg[\frac{\ln x}{1-ix}+\frac{\ln x}{1+ix}\bigg]dx$$
$$I=\frac{1}{2}\int\frac{\ln x}{1-ix}dx+\frac{1}{2}\int\frac{\ln x}{1+ix}dx$$
Put $1+ix =t$ Then $dx=-i... | Note that
$$\ln(x)+C=\int\frac1x{\rm~d}x$$
and
$${\rm Li}_2(x)+C=\int\frac{\ln(1-x)}x{\rm~d}x$$
where ${\rm Li}$ is the polylogarithm, since
$${\rm Li}_1(x)=\ln(1-x)$$
and
$${\rm Li}_{s+1}(x)=\int_0^x\frac{{\rm Li}_s(t)}t{\rm~d}t$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2701123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is Cartesian product of Graph and Composition of Graph the same thing? As you must have seen from my previous questions, I am mostly dealing with probabilistic maths so Graph is completely new to me. Still, I am trying to understand a paper that utilizes Graph theory.
I understand the Cartesian product of Graphs, my qu... | No, the composition of two graphs $G_1$ and $G_2$ is their Cartesian product plus an edge between every pair of vertices $(u_1,v_1)$ and $(u_2,v_2)$ where $u_1$ and $u_2$ are adjacent in $G_1$ and $v_1\neq v_2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2701222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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The equality of two conditional probability Suppose that a company has had $70\%$ female applicants and $30\%$ male applicants since it was founded. We also assume that $70\%$ of all staff are female.
Let $F$ denote female and $M$ denote male and $S$ denote the event that an applicant is successful.
The question is: I... | You're saying that given an applicant is female, they are guaranteed to get the job, and similarly for males. That is probably not true. $P(S\cap F)$ does not necessarily equal $0.7$. If there were $70$ female applicants of $100$ total applicants and $7$ females got the job then $P(S\cap F)=0.7\cdot0.1=0.07.$
We have
$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Gradient of a function with respect to a matrix How can I compute the gradient of the following function with respect to $X$,
$$g(X) = \frac{1}{2}\|y-AX\|^2$$
where $X\in\mathbb{R}^{n\times n}$, $y\in\mathbb{R}^m$, and $A:\mathbb{R}^{n\times n}\to \mathbb{R}^m$ is linear. We can assume that $A$ is of the form,
$$A = \b... | Write the cost function as
$\phi
= \frac{1}{2} \sum_i z_i^2
$
with $z_i=
\mathrm{tr}
\left( \mathbf{A}_i^T \mathbf{X} \right)
-y_i
$.
Then taking a differential approach,
we obtain
$$
d\phi
= \sum_i z_i
\mathrm{tr}
\left( \mathbf{A}_i^T d\mathbf{X} \right)
=
\mathrm{tr}
\left(
\sum_i z_i
\mathbf{A}_i^T d\mathbf{X}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2701503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Analyzing a function with abstract algebra For an $n$-tuple $S$ of decreasing positive integers, we can define $f(S)$ as subtracting $1$ from every element of $S$, prepending $n$, and then removing $0$s and re-ordering in decreasing order if neccecary. For example, $f((4,2,1))=(3,3,1)$
We are just learning group theory... | You just have to prove that if $x=(x_1,\dots,x_n)$ is your tuple then you only have a finite number of elements in the sequence $$x,f(x),f(f(x)),\dots $$
Hint: look at the sum of the elements constituting the tuple in $x$ and in $f(x)$.
If you do this then you will necessarily reach twice the same element in the seq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2701610",
"timestamp": "2023-03-29T00:00:00",
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Given a straight line with some distance, a fixed rate of acceleration/deceleration, and a time, find the velocity needed I have a path which is a straight line with some distance d.
I have a particle which is at position 0 in the line, with a resting velocity of 0 m/s. The particle has a fixed rate of acceleration at ... | You can draw a line AB, then draw points AC and DB, where AC = DB, which is the distance between the point where the particle starts accelerating to the point where it reaches max speed (same thing for deceleration). So, CD would be the distance it travels at maximum speed.
AC and DB would be:
$t(t-1)$ Since the distan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2701719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Topological space , containing a point whose closure is the whole space, is contractible? Let $X$ be a topological space containing a point whose closure is the whole space $X$. Then is $X$ contractible ?
I feel it is, but I am unable to come up with a proof. Please help.
| If $\eta$ is the dense point and $I=[0,1]$ the map $f:X\times I\to X$ defined by $$f(x,0)=x \quad \operatorname {and} \quad f(x,t)=\eta \quad \operatorname {for} t\gt0$$ is the required contraction of $X$ to $\eta$ .
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2701847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to find two monic polynomials with the same number of roots mod p for each prime p Given an arbitrary monic polynomial $f_1$, is there an elegant way of identifying another $f_2$ such that both have the same number of roots (counting repeated roots as distinct) of $f_i \equiv 0 \bmod p$ for each prime $p$, excludin... | Note that the better question here is why $f_1$ and $f_2$ have the same number of roots over $\mathbb{F}_p$ for every prime $p$, although $f_1$ and $f_2$ are not related by a transformation $x\mapsto x+a$. Indeed,
$$
f_1(x+a)=x^3 + 3x^2a + 3x(a^2 + 252) + a^3 + 756a + 1302,
$$
so that $f_2=f_1(x+a)$ would imply $a=0$, ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prox Operator of a First Order Perturbation (Adding Linear Term to the Function) Given a function $f$ we can describe its proximal operator as,
$$\mbox{prox}_{\frac{1}{\rho}f}(x) = \arg\min\limits_{u} f(u) + \frac{\rho}{2}\|x-u\|^2$$
How does this change if we introduce a linear perturbation, i.e. we replace $f$ with $... | [Corrected via comments] Sure. The optimality condition for your original prox function is
$$0 \in \partial f(u) - \rho ( x - u)$$
For the perturbation, it is
$$0 \in \partial f(u) + \mu - \rho ( x - u ) = \partial f(u) - \rho ( x - \rho^{-1} \mu - u)$$
So basically, your perturbation is solved by $$\textstyle\mathop{\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show probability bound. Let $Y$ be the number of successes in $n$ trials of a Bernoulli experiment with success probability $p$. Show that:
$$ Pr(|\frac{Y}{n} - p |<e) \geq 1 - \frac{1}{4ne^2}$$
I tried starting with Chebychev and work backwards but I got stuck:
I used $np(1-p)$ for variance since binomial.
$$
\begin{... | Check out Hoeffding's inequality on Wikipedia. It won't give you exactly the answer that you're looking for: instead, it will give you a far stronger one (asymptotically, as $n$ gets large)! Regarding the precise formlation that you have, you're almost there: all you need to do is note that
$$ p(1-p) \le \tfrac14 \quad... | {
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$G$ is a group of order $pq$ and $P_q$ and $P_p$ are Sylow subgroups... If $(G,*)$ is a group of order $pq$, then it is clear that there are Sylow subgroups $P_q$ and $P_p$ of order $q$ and $p$ in $G$. If $q>p$ then $P_q$ is normal.
I want to find a decomposition for all $g \in G$, there exists $(q',p') \in P_q \times... | Since $p$ and $q$ are different primes, Bézout's Lemma guarantees the existence of $m,n \in \mathbb{Z}$, with $1=mp+nq$. Hence $g=(g^p)^m \cdot (g^q)^n$. Now prove that $g^p \in P_q$ and $g^q \in P_p$. Hence powers of these elements are in the same respective subgroups.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2702241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Solving following System of ordinary differential equations. How to solve system of ODEs which contains independent variable like t in equation as in this particular case :
$$\frac{dx}{dt}= -x + ty $$ $$ \frac{dy}{dt} = tx-y$$ can we solve them or additional information may be needed (may be about t) ?
| Hint...you can subtract the equations and write $u=y-x$ to obtain a seperable variable DE in $u$ and $t$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2702348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How to show that the normalizer is the largest subgroup of a group in which a set is normal? Assume we have $H \trianglelefteq K\subseteq G$
We want to prove the $K \subseteq N_G(H)$
However all I have so far is:
Assume $K \subseteq G$ such that $H$ is normal in $K$. Then it must be that $kHk^{-1} \subseteq H$
If $kHk... | As $H$ is a normal subgroup of $K$, for $k \in K$, you have that $kHk^{-1} \subseteq H$ and $k^{-1}Hk \subseteq H$. So
$$H = (kk^{-1})H(kk^{-1}) = k(k^{-1}Hk)k^{-1} \subseteq kHk^{-1} \subseteq H.$$
This gives you that $kHk^{-1} = H$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2702476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Orthogonal vector in a plane I have two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$. Now I want to find a vector $\boldsymbol{v}$ orthogonal to $\boldsymbol{a}$ that is in the plane spanned by $\boldsymbol{a}$ and $\boldsymbol{b}$. Is it ok if I do $\boldsymbol{v} = \boldsymbol{a} \times (\boldsymbol{a} \times \bolds... | Yes. $\boldsymbol a\times (\boldsymbol a\times \boldsymbol b)$ is perpendicular to both $\boldsymbol a$ and $\boldsymbol a\times \boldsymbol b$. Being perpendicular to $\boldsymbol a\times \boldsymbol b$ means being on the plane generated by $\boldsymbol a$ and $\boldsymbol b$.
See the other answer for a less expensive... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2702604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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How to find the general solution of $x^2y'' - x(x+2)y' + (x+2)y = 0$ $$x^2y'' - x(x+2)y' + (x+2)y = 0$$
where a particular solution is $y_1(x) = x$
So, can I start off by subbing in $x$ for $y$ since I have a particular solution?
Which would give me:
$\frac{x}{x+2}y'' - y' + y = 0$
If this is a valid approach, then so... | Here $y_1(x)=x$ is your known integral. For finding complete solution in terms of known integral you put: $y=vx$, where $v$ is also function of $x$; then the above ODE becomes $\dfrac{d^2v}{dx^2}-\dfrac{dv}{dx}=0$, which can be easily solve for $v$. Then substitute $v$ in $y=vx$ you got general solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2702723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
When every minimal subgroup is contained in the center
Let $G$ be a finite group and $(*)$ be the property:
$(*)$: Every minimal normal subgroup is contained in the center.
$(a)$ Let $N$ and $M$ be normal subgroups of $G$, both of which satisfy $(*)$. Then prove:
$NM$ satisfies $(*)$.
$(b)$ If $G$ satisfies $(*)$, t... | I have an argument for (a) in the case $N \cap M = 1$, which implies $NM = N \times M$. Let $K$ be some minimal normal subgroup of $NM$. Then by considering the quotient $KN / N \unlhd NM / N \cong M$ either $KN \le N$ or $KN / N$ is minimal normal in $MN / N$, in the last case $[x,y] \in N$ for each $x \in K, y \in N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2702817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Simplex Solver - is it possible to enforce maximum number of non-zero weights? I have a problem which I am trying to solve using a simplex solver (but I am happy to use any approach that works).
$$
a_1 x_1 + a_2 x_2 + ... + a_n x_n= TotalCost
$$
Where $a_i$ are constant inputs and $x_i$ are the weights that I am optim... | You could add extra binary variables $y_i$ that take value $0$ when $x_i=0$ and penalize $1-y_i$ in the objective function:
Add the following constraint, where $\varepsilon$ is a small constant $<1$:
$$
x_i\ge \varepsilon \;y_i\\
y_i \in \{0,1\}
$$
Then minimize the following term in the cost function, where $p$ is ano... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2703133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Discontinuity set of an increasing right continuous function Let $f:[0,\infty) \rightarrow [0,\infty] $ be both right-continuous and increasing. Is it possible that $f$ is discontinuous at dense subset of $\mathbb R$? I already know that if we drop right-continuity, then $f$ can be discontinuous at every rational.
Any ... | This should work:
$f(x)=\sum_{q_n \leq x} 2^{-n},$ where $\{q_n\}$ is an enumeration of the rationals $\mathbb{Q} \cap [0,+\infty)$.
Tell me if you have some doubts.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2703268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Shouldn't $2^x \sin\frac{180}{2^x}$ approach $\pi$ as $x$ gets large? Take a circle with diameter $1$. Obviously its circumference is $\pi$.
Draw a square inside this circle (biggest such). Since the circle's diameter is 1, we can work out that each side of the square would be $\frac{1}{\sqrt{2}}$. Its perimeter, ther... | The limit should be $180$ if you are taking the sine in radians. Yes, it should be $\pi$ if you are taking the sine in degrees. You should be graphing for larger values of $x$ as even $\frac {180}{2^8} \gt \frac 12$ is not so small. Below is a plot from Alpha that shows nice convergence to $180$ using radians
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
how many fish are needed to have probability 2/3 of at least two fish in a bucket?
Lets say we have $f$ fish and $b = 10$ buckets. You toss the fish independetly and uniformly into the buckets.
Define $A_{f} =$ "There exists a bucket that has at least two fish"
Let $p_{f} = Pr(A_{f})$
What is the lowest value of $f$ s... | This is sometimes called the birthday problem.
In general if there are $f$ fish and $b$ buckets then the probability that no bucket has two fish is:
$(1-\frac{1}{b})(1-\frac{2}{b})\dots (1-\frac{f-1}{b})$
So the probability that there is at least one bucket is:
$1-(1-\frac{1}{b})(1-\frac{2}{b})\dots (1-\frac{f-1}{b})$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2703490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Question on projection operator and direct sum Here is question 10,section 6.6,Hoffman and Kunze:
Let $F$ be a field of characteristic 0.Let $V$ be a finite dimensional vector space over $F$.Suppose that $E_1,..,E_k$ are projections of V such that $E_1+..+E_k=I$.Prove that $E_i E_j$=0.
My attempt:
Let $W_i=E_i(V)$.
The... | Hint:
Assume that for some $\alpha\in V$, $\ E_iE_j\alpha\neq 0$, and then conclude that $E_j\alpha\in W_i\cap W_j$ which contradicts $V$ being the direct sum of $W_1,..., W_k$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2703657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Intuition behind the Banach fixed-point theorem The theorem appeared as an exercise in my real analysis book and only considered functions in $\mathbb{R}$ but the proof of the general theorem seems to be almost identical after looking up the wikipedia article. Is there a somewhat intuitive way of thinking about this th... | Generally, a function between metric spaces can be very wild with respect to the distances. But if we demand that the function actually decreases the distance between any two points, then it becomes difficult to actually construct such functions. Looking at a few examples of such functions one sees that one easy way to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2703770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Linear mapping of a binary vector to a decimal-based binary vector Given a binary vector $\mathbf{v}$, where $\mathbf{v} \in \{0,1\}^N$ and the binary-to-decimal conversion of $\mathbf{v}$ is equal to $j$, is there a way to linearly map the vector $\mathbf{v}$ to a binary vector $\mathbf{e}_j$ with $\mathbf{e} \in \... | If by "linearly map" you mean a real matrix $H$ such that $\mathbf{e} = H \mathbf{v}$ yields what you want, the answer is no. If $$\left(\begin{array}{c} 0\\ 0\\ 1\\ 0 \end{array}\right)=H \left( \begin{array}{c} 0\\ 1 \end{array}\right)$$ (encoding of 2) and $$\left(\begin{array}{c} 0\\ 1\\ 0\\ 0 \end{array}\right)=H ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2703879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the equilibrium position of an object which satisfies the equation of motion Find the equilibrium position of an object which satisfies the equation of motion. I know how that there is a formula $$ d^{2}x/dt^{2} = - \omega^{2}x $$ but I can't see the usage of it when there are so many "things" on the right side of... | Recall Newton's law
$$
\frac{{\rm d}^2 x}{{\rm d}t^2} = -\frac{{\rm d}V}{{\rm d}x}
$$
In you case, it is pretty straightforward to find the potential $V$
$$
V(x) = \frac{x^4}{16} - \frac{x^3}{12} + \frac{x^2}{8} - \frac{x}{4}
$$
You see that the dominant term is $\sim x^4$, this means that the potential goes to infinit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2703982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the value of $\frac{\tan A}{\tan B}$, given $\frac{\sin A}{\sin B}=5$
If $\displaystyle \frac{\sin A}{\sin B}=5$, then find the value of $\displaystyle \frac{\tan A}{\tan B}$
Try using the Componendo and Dividendo formula:
$$\frac{\sin A+\sin B}{\sin A-\sin B}=\frac{3}{2}$$
$$\frac{\tan(A+B)/2}{\tan(A-B)/2}=\fra... | suppose $\dfrac{SinA}{SinB}=\dfrac{x}{y}=5$
so from right tringle we know $cosA=\sqrt{1-x^{2}}$ and so $$tanA=\dfrac{x}{\sqrt{1-x^{2}}}.$$ the same for y so :
$$\dfrac{tanA}{tanB}=\dfrac{\frac{x}{\sqrt{1-x^{2}}}}{\frac{y}{\sqrt{1-y^{2}}}}$$
and we have $x=5y$ so $x^{2}=25y^{2}$ , $1-x^{2}=1-25y^{2}$
$$\dfrac{tanA}{tanB... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2704183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
An urn has 4 balls of 4 different colours Red,Blue,Green,Yellow.
An urn has $4$ balls of $4$ different colours; red, blue, green, and
yellow. I pick one ball at random at first and if it is red, I paint
it blue and return it to the urn. If it is blue, I paint it green. If
it is green, I paint it yellow. If it is... | This problem is equivalent to one in which there are four people in four rooms that are joined cyclically by corridors. Initially, each room has one of the four people, and at each turn, one person (not one room) is chosen at random, and this person moves counterclockwise. How long before they end up in the same room... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2704389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 0
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Is $\frac{cos^4(x)}{4}-\frac{cos^2(x)}{2}+C$ a correct evalution of $\int sin^3(x)cos(x)\ dx$ The answer on Khan Academy states that the integral evaluates to $1 \over 4$$\sin^4(x)+C$
However, I performed a u-substitution that I cannot find a mistake in (maybe I am blind).
Here's the working:
$I=\int \sin^3(x)\cos(x)dx... | Your solution is correct (with minor error at the end) and it is equivalent to the one given in Khan Academy:
$$\begin{align}I&=\frac{\cos^4(x)}{4}\overbrace{\require{cancel}\cancel{+}}^{-}\frac{\cos^2(x)}{2}+C=I= \\
&=\frac{(1-\sin^2 x)^2}{4}-\frac{\cos^2(x)}{2}+C=\\
&=\frac14-\frac{\sin^2 x}{2}+\frac{\sin^4 x}{4}-\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2704467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is wrong in $(\frac{1}{8})^{-\frac{1}{3}}$? I tried this:
$$\biggl (\frac{1}{8}\biggr)^{-\frac{1}{3}}= \frac{1^{-\frac{1}{3}}}{8^{-{\frac{1}{3}}}}=\frac{-\sqrt[3]{1}}{-\sqrt[3]{8}}=\frac{-1}{-2}$$
Also, is it possible to see what I don't understand here or are there maybe several things..
| $$(1/8)^{-1/3}=((1/8)^{1/3})^{-1}=(1/2)^{-1}=2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2704575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $4^n-1$ is divisible by $3$, for all $n\in\Bbb N$? Prove $4^n-1$ is divisible by $3$, for all $n\in\Bbb N$?
I started by assuming there exists some $k\in\Bbb N$ s.t. $4^n-1=3k\iff \dfrac{4^n}3-\dfrac 13=k$, so for $k$ to be a natural number, $4^n\equiv 1\mod 3$ must be true, but this tells us no new information, ... | With only middle school tools:
$$4^n-1=4^n-1^n=(\underbrace{4-1}_{\textstyle 3})(4^{n-1}+4^{n-2}+\dots +4+1).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2704691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 1
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Solve $\sqrt {x^2-3}=x-3$ in $\mathbb R$
Solve $\sqrt {x^2-3}=x-3$ in $\mathbb R$
My attempt:
$|x^2-3|=(x-3)^2$
So $-(x^2-3)=(x-3)^2$ or $(x^2-3)=(x-3)^2$
If $-(x^2-3)=(x-3)^2=x^2+9-6x$
So no solutions in $\mathbb R$
And if $(x^2-3)=(x-3)^2$
So $x^2-3=x^2+9-6x$
Now, can I delete $x^2$ with $x^2$ ? Like this
$x^2-x... | I was taught to always find the domain of possible solutions first. We have
\begin{cases}
x-3 \ge 0, \\
x^2-3 \ge0
\end{cases} or \begin{cases}
x \ge 3, \\
-\sqrt{3} \le x \le \sqrt{3}
\end{cases}
which has no solutions
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2704837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Find $\oint\limits_{|z-\frac{1}{3}|=3} z \text{Im}(z)\text{d}z$ In my test on complex analysis I encountered following problem:
Find $\oint\limits_{|z-\frac{1}{3}|=3} z \text{Im}(z)\text{d}z$
So first I observed that function $z\text{Im}(z)$ is not holomorphic at least on real axis. Therefore we have to intgrate usin... | Note that since $\text{Im}(z)=\frac1{2i}(z-\bar z)$, that
$$z\text{Im}(z)=\frac1{2i}(z^2-|z|^2)$$
Since $z^2$ is analytic, we have
$$\begin{align}
\oint_{|z-\frac13 |=3}z\text{Im}(z)\,dz&=\frac i2\oint_{|z-\frac13 |=3}|z|^2\,dz\\\\
&=-\frac {3}2 \int_0^{2\pi} \left|\frac13 +3e^{i\phi}\right|^2 e^{i\phi}\,d\phi\\\\
&=-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2704955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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perturbation of differential equation I am struggling with the following problem, which on first site looks easy, but I can't see it.
Given the DE:
$$\frac {d^2y}{dx^2} +y =\frac{\cos 2x}{a+ \epsilon y}$$
with initial conditions: $y(-\pi/4) =y(\pi/4) = 0$, $a>0$ and $|\epsilon| \ll1$
By using the scaling: $y=\alpha z$... | Let's substitute $y=\alpha z$ into your equation:
$$\alpha\left(\frac {d^2z}{dx^2} +z\right) =\frac{\cos 2x}{a+ \alpha\epsilon z}$$
$$\frac {d^2z}{dx^2} +z =\frac{\cos 2x}{\alpha a+ \alpha^2\epsilon z}$$
Let $\alpha = 1/a$. Then
$$\frac {d^2z}{dx^2} +z =\frac{\cos 2x}{1+ \frac{\epsilon}{a^2} z}$$
Introduce $\delta = \e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2705343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Evaluating $\int\limits_0^\infty{\frac{1}{1+x^2+x^\alpha}dx}$ I'm trying to evaluate$$f(\alpha)=\int\limits_0^\infty{\frac{1}{1+x^2+x^\alpha}dx}$$
I proved:
$f(\alpha)$ converges when $\alpha\in\mathbb{R}$
$f(2-\alpha)=f(\alpha)$
$f(0)=f(2)=\frac{\pi}{2\sqrt{2}}$
$f(1)=\frac{2\pi}{3\sqrt{3}}$
$f(-\infty)=f(\infty)=\fra... | Not a full answer, some considerations.
First, we make a substitution:
$$x=\tan t$$
The integral becomes (I use $a$ instead of $\alpha$ for convenience):
$$f(a)=\int_0^{\pi/2} \frac{dt}{1+\cos^{2-a} t~ \sin^a t}$$
Which by the way, makes the functional equation from the OP very clear.
For the values of $a$ satisfying $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2705423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 1,
"answer_id": 0
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How to prove divergent sequences? For this question, I know that the sequence diverges to infinity, but I'm not sure if I am doing it right. Here is what I have so far. Can anyone please help me out?
Determine whether the following sequence is convergent or divergent
$a_n = \{8n^3 + n^2 -2\}$
$\lim_{n \to \infty} a_n =... | We fix $M>2$ and we look for $N$ such that $\forall n>N$
$$a_n = 8n^3 + n^2 -2>M$$
then choose $n=M$ and check that
$$8M^3 + M^2 -2>M\iff 8M^3 + M^2>M+2$$
which is true.
Then it suffice to choose $N\ge M$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2705509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Finding the locus of $z=at+\frac bt$ I have to find the locus of$$
z=at+\frac{b}{t},
$$
where $a, b \in \mathbb{C}$ are constants. I took $a=a_1+ia_2$ and $b=b_1+ib_2$, but could not get the solution.
| In a real setting, the curve given by parametric equations
$$\tag{1}x=t, \ \ \ y=\dfrac{1}{t},$$
i.e., with cartesian equation $y=\dfrac{1}{x}$, is a (equilateral) hyperbola.
Of course, (1) can be written
$$\tag{2}x+iy=t\color{red}{1}+\dfrac{1}{t}\color{red}{i}$$
Thus $z=t\color{red}{a}+\dfrac{1}{t}\color{red}{b}$ (s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2705686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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A Problem on Theory Of Equations Let $f(x) = x^2 + x$, for all real $x$. There exist positive integers $m$ and $n$, and distinct nonzero real
numbers $y$ and $z$, such that $f(y) = f(z) = m + \sqrt{n}$ and $f(1/y) + f(1/z) = 1/10$ . Compute $100m + n$.
| Hint. Let $a=m+\sqrt{n}>0$. Note that $y$ and $z$ are the two solutions of the quadratic equation $x^2+x-a=0$. Therefore $y+z=-1$ and $yz=-a$. Hence
$$\frac{1}{y}+\frac{1}{z}=\frac{y+z}{yz}=\frac{-1}{-a}=\frac{1}{a}.$$
Moreover
$$\frac{1}{y^2}+\frac{1}{z^2}=\frac{y^2+z^2}{y^2z^2}=\frac{(y+z)^2-2yz}{(yz)^2}=\frac{(-1)^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2705758",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Expressing $\tan 20°$ in terms of $\tan 35°$ If $\tan 35^\circ = a$, we are required to express $\left(\frac{\tan 145^\circ - \tan 125^\circ}{1 + \tan 145^\circ\tan 125^\circ}\right)$ in terms of $a$.
Here's one way to solve this: $$\frac{\tan 145^\circ - \tan 125^\circ}{1 + \tan 145^\circ\tan 125^\circ} = \tan (145^\... | $\tan145^\circ=\tan(180^\circ-35^\circ) =-\tan35^\circ $ and $\tan125^\circ=\tan(90^\circ+35^\circ) =-\frac{1}{\tan35^\circ} $
For your two answers, have you find their values with a calculator?
Actually, you have proven an equality in $a$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Using differentiation for computing $\int_{0}^{1}\frac{\arctan(ax)}{x\sqrt{1-x^2}}\,dx$ How can I prove that I can differentiate the integral:
$$ \int_{0}^{1} \frac{\arctan(ax)}{x\sqrt{1-x^2}}\,dx $$
First I have to prove this integral converges.
Next I have to prove the integral $$ \int_{0}^{1} f_{a}'(x, a)\,dx $$ con... | The given hints contain pretty much everything. $\frac{d}{da}\arctan(ax)=\frac{x}{1+a^2 x^2}$ and
$$ \int_{0}^{1}\frac{dx}{(1+a^2 x^2)\sqrt{1-x^2}} \stackrel{x\mapsto\sin\theta}{=} \int_{0}^{\pi/2}\frac{d\theta}{1+a^2\sin^2\theta}=\int_{0}^{\pi/2}\frac{d\theta}{1+a^2\cos^2\theta}\\\stackrel{\theta\mapsto\arctan u}{=}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2706052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Graph of $\log(3-x)$ So if $y=\log(3-x) = \log(-x+3)$ then you reflect $\log(x)$ in the $y$ axis to get $\log(-x)$.
Then because it is $+3$ inside brackets you then shift to the left by $3$ giving an asymptote of $x=-3$ and the graph crossing the $x$ axis at $(-4,0)$.
However this does not work. The answer shows the $... | Start with $y=\log(x)$. To shift this left three units, replace "$x$" with "$x+3$". Now you have $y=\log(x+3)$.
Now reflect over the $y$-axis. To do this, replace "$x$" with "$-x$". Now you have $y=\log(-x+3)$.
The order that the horizontal graph transformations happen is opposite from what you might think by the order... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2706165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Why is $M_n(A)$ a von Neumann algebra I'm trying to verify that for each von Neumann algebra $A$ the algebra of matrices with entries in $A$ is again von Neumann.
I do already know, that those kind of matrix algebras are again C-*-algebras.
I'd prefer a basic argument involving only weak or strong closedness over other... | Strong convergence of a net in $M_n(A)$ is equivalent to entrywise strong convergence.
Assume $x_\lambda$ is a net in $M_n(A)$ converging strongly to $x \in B(H^n)$. Let $h,k \in H$ and $1 \leq i,j \leq n$. Let $\xi \in H^n$ be the vector which has $h$ in the $i$-th component and $0$ elesewhere, and let $\eta \in H^n$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2706252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What values of $a$ make matrix $A$ diagonalisable? I have the following question in an assignment paper.
Let $$A=\begin{bmatrix} 0 & a & 0\\ 1 & 0 & a\\ a & 1 & 0\end{bmatrix}$$ For what values of $a$ is $A$ diagonalisable?
Simply put, I don't know how to do it. In the $2 \times 2$ case we were asked, I completed the... | Taking $a^3 = \frac{32}{27} \; , $ this includes complex $a:$
$$
\frac{1}{864a}
\left(
\begin{array}{rrr}
40 & 36 a^2 & 24 a \\
-100 & -90 a^2 & 156 a \\
-9 a^2 & -24 a & 32
\end{array}
\right)
\left(
\begin{array}{rrr}
0 & a & 0 \\
1 & 0 & a \\
a & 1 & 0
\end{array}
\right)
\left(
\begin{array}{rrr}
6 a & -12 a & 54 a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2706368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Spivak's Calculus: Chapter 1, Problem 18b (Quadratic determinant less than zero) The problem in question is as follows:
18b Suppose that $b^2 -4c \lt 0$. Show that there are no numbers $x$ that satisfy $x^2 + bx + c = 0$; in fact, $x^2 + bx + c \gt 0$ for all $x$. Hint: complete the square.
Trying to apply the hint, I ... | What you are missing is that$$x^2+bx+c=\left(x+\frac b2\right)^2-\frac{b^2-4c}4.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2706487",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 7,
"answer_id": 4
} |
probability - distinguishable vs indistinguishable objects/persons I am solving one simple problem just for fun.
The problem is as follows.
$n$ people have to be seated randomly in a cinema hall which has $(n+k)$ seats.
What is the probability that in this process, some fixed $m$ places ($m \leq n$) get occupied/tak... | The term $\binom{n}{m}$ counts the number of ways $m$ of the $n$ people can occupy the specified $m$ seats. That leaves $n + k - m$ seats available. The remaining people can occupy $n - m$ of them. Hence, the number of favorable cases is
$$\binom{n}{m}\binom{n + k - m}{n - m}$$
as you suspected. The denominator $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2706573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
is $(1+\pi)/(1-\pi)$ a transcendental number? I know it's a open problem to show if $\frac{\pi}{e}$ is a trans. number or not. But what about quotient between numbers in function only of $\pi$, which is trans, such as $\frac{1+\pi}{1-\pi}$ or $\frac{1+i\pi}{1-i\pi}$. Wolfram says all theses numbers are trans.
$\frac{... | $1-{2\over{\pi-1}}$ is algebraic implies that ${1\over{\pi-1}}=a$ where $a$ is algebraic, this implies that $\pi-1=1/a$ and $\pi=1+1/a$ contradiction
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2706688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Riemann surface $y=(1-x^3)^{1/3}$ Let $X\subset \mathbb C^2$ be the Riemann surface given by （the multivalued function) $y=(1-x^3)^{1/3}$, and let $\phi:X \to \mathbb C $ be the induced map. Let $X'\subset P(\mathbb C^2)$ be the complex curve $x^3+y^3=z^3$ (in homogeneous coordinates). I want to define a complex manifo... | If I got well, $X$ is the zero locus $x^3+y^3=1$ in $\mathbb{C}^2$ and your $\phi $ is the projection on one of the coordinate (I will assume is the $y$ just to be clear).
Now you take $X'$ that is ,as you can easily prove, the projective closure of $X$ in $\mathbb{P}^2(\mathbb{C})$.
As you correctly said , you need th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2706771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$(p \wedge q) \wedge p$ convert to CNF I was doing some exercises to convert formulas to CNF by means of the axioms and I imagined the following exercise $(p \wedge q) \wedge p$, but for this exercise, I do not find an axiom that can take me to CNF. However, I know that if I use the truth table I could solve it. I woul... | $\begin{align}(p\wedge q)\wedge p &= p\wedge(p\wedge q) &&\text{commutivity}\\ &= (p\wedge p)\wedge q &&\text{associativity}\\ &=p\wedge q && \text{idempotence}\end{align}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2706877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What is wrong with my proof of Law of Large Numbers? Need help in deriving variance for LLN, but am getting something wrong here, can you help point out where I'm going wrong?
Thanks a ton!
For i.i.d random variables $X_1,X_2,\ldots,X_n$ with mean $\mu$ and variance $\sigma^2$.
Let $X=\frac{1}{n}\displaystyle\sum_{i=1}... | You say "from LLN", but you are not using LLN anywhere.
Moreover, it's much easier to do this:
$$
\text{Var}\left(\frac{1}{n}\sum X_{i}\right)=\frac{1}{n^{2}}\sum\text{Var}\left(X_{i}\right)=\frac{n\sigma^{2}}{n^{2}}=\frac{\sigma^{2}}{n}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2706954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
On which topological spaces, can we give a group structure to make it a topological group? Let $X$ be a non-empty set. It is known that we can give a group structure on $X$. Now let $X$ be a non-empty topological space. Then can we give a group structure on $X$ so that it becomes a topological group w.r.t. its original... | No, you cannot do that for all spaces $X$.
If $X$ has the structure of a topological group, it implies a lot of extra facts about it, and those give necessary conditions that $X$ should fulfill.
Some examples of such properties:
*
*If $X$ is $T_0$ it must also be $T_{3\frac{1}{2}}$ (Tychonoff). (it's uniformisable)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2707066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Create Approximately Follows Distribution Symbol Using MathJax. I want to use MathJax to create an "approximately follows distribution" symbol. It is a tilde with a dot above and a dot below. The closest I get is $\overset{\cdot}{\underset{\cdot}{\sim}}$ but the distance from the bottom dot to the tilde is greater than... | Using the lower and raise command should give you a desirable result like this:
$\overset{\lower{0.5ex}{\cdot}}{\underset{\raise{1ex}{\cdot}}{\sim}}$.
Hope you find this useful, thanks!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2707198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Distributing $2$ identical balls to $8$ distinct boxes Find the number of ways to distribute $2$ identical balls to $8$ distinct boxes.
This is how I reasoned it:
As the boxes are distinct, the only way by which we could get different configurations would be by considering the number of ways in which $2$ distinct pair... | It depends on what you mean by "distribute":
a) if you mean "(randomly) throw the balls into the boxes" ,
then it means that you consider equi-probable that at each launch you can
choose one of the $8$ boxes, so a total of $8^2=64$ ways to do that;
b) if instead you mean "(randomly) pour the balls into the boxes... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2707314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A problem about the order of element in a group Assume $a \in G$, where $G$ is a group and element $a$'s order is $mn$ with $(m,n) = 1$, the problem is to prove there exist elements $b$ and $c$ in the group where $a = bc = cb$ , $b$'s order is $m$ and $c$'s order is $n$ and $b,c$ are unique.
I have proven that the exi... | Hint: $\mathbb Z_{mn}\cong \mathbb Z_m\times \mathbb Z_n$ where $(m,n)=1$. The uniqnes comes from the propery of "direct product".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2707415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Cancellation law for multiplication of natural numbers I'm trying to prove the following cancellation law for multiplication of natural numbers: if $xz=yz$ for natural numbers $x,y$ and $z$, where $z$ is non-zero, then $x=y$.
I'm working with the peano-axioms and I've already proven elementary properties of multiplicat... | 1) Prove If $x \ne y$ then eithere there exists a $c \ne 0$ so that $x + c =y$ or that $x = y + c$.
2) Prove that if $c \ne 0$ then $z(c + x) \ne z(0 + x)$
=====
Actually You could use $1$ to define subtraction. If $a \le b$ then, by definition there is a $c$ so that $a + c = b$. If $a \le b$ then define $b - a= c$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2707555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What is the maximum value of $a + b + c$, given $ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{5}$ What is the maximum value of $a + b + c$, where $a, b, c\in \mathbb{Z}$, and
$$
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{5}
$$
Note: I could solve the question if the question asks "minimum" instead of "m... | Lacking any insight, what follows is a purely mechanical approach. We'll show that there are only finitely many possibilities for $a,b,c$. We will not assume that they are all positive.
Taking any solution, sort it so that $|a|≤|b|≤|c|$. We remark that $$\frac 15=\big \vert \frac 1a+\frac 1b+\frac 1c\big \vert≤ \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2707707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Calculate angle from accelerometer on a moving axis For the purpose of this question, let's say that I have an accelerometer that is on a foot of a human. ( red square on by the heel in the below photo )
I'd calculate the angle the following way, as the leg goes from step 1 to step 2.
$\theta = tan^{-1}(\frac{x}{y})$
... | In the plane $y-z$ the rotation would be
$$\alpha = \tan^{-1}\left(\frac{z}{y}\right)$$
with positive values for $\alpha$ when moving back.
A general expression for the absolute angle between the leg and y axis is
$$\phi = \tan^{-1}\left(\frac{\sqrt{x^2+z^2}}{|y|}\right)$$
For a general description you could use spheri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2707845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is κ(G) and κ′(G) and δ(G) for graph G?
Am I correct to say that G is 4-connected?
So κ(G) = 4 but then κ′(G)=2 but that cannot happen since κ(G)<=κ′(G)<=δ(G)
I know δ(G)=4 so wouldn't κ′(G)=4 then? However, I don't see how that would happen.
Could someone please explain this?
| I hope it can help you
Vertex Connectivity: $\kappa(G)$ is the minimum size of a vertex set S s.t. G\S is disconnected.
Edge Connectivity: $\lambda(G) $ or $\kappa'(G)$ is the minimum size of edge set F s.t. G\F has more than one component.
in your graph $\kappa(G)=4$: for example $S=\{f,l,i,c\}$ and
$\lambda(G)=4 $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2707934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Generating function of a sequence with a sum I am new to generating functions and I know some of the basics how to create them. But now I am not sure how to create a generating function of this sequence:
$$a_0 = 0$$
$$ a_n = n + c (\sum_{k=0}^{n-1} a_k)$$ for n >=1
where c is a real non-zero constant.
I guess I need to... | Note that if
$$
A(x)=\sum_{n=0}^\infty a_n x^n;\quad
B(x)=\sum_{n=0}^\infty b_n x^n
$$
are two formal power series then (by definition)
$$
A(x)B(x)=\sum_{n=0}^\infty
\left(
\sum_{k=0}^na_kb_{n-k}
\right)
x^n.\tag{1}
$$
In particular taking $B(x)=(1-x)^{-1}$, we have that
$$
\frac{A(x)}{1-x}=\sum_{n=0}^\infty
\left(
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2708045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Probability question regarding coin flips. There are 20 coins in a jar. Of these, 8 are quarters, 5 are dimes, 3 are nickels, and 4 are pennies. 8 coins are drawn at random, without replacement from the jar.
What is the chance that the fourth coin is a quarter and the eighth coin is a dime?
For this question I used com... | What is the chance that the fourth coin is a quarter and the eighth coin is a dime?
There is actually a simpler way but let’s push forward with your combinatorics method. For the first 4, the cases are that there may be 1, 2, 3, or 4 quarters by the time the fourth coin is picked so we need to take care of that. We don... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2708392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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