Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Find the length of a triangle's side I have the following triangle and I'm supposed to find the value of x.
First thought that came to mind is to use the the following tangent equation
$$\tan(y)=\frac{x}{27}$$ and $$\tan(19+y) = \frac{2x}{27}$$ which implies that $$\tan(19+y) =2\tan(y)$$ and solve for $y$ and once I'v... | I think you can just use the sinus rule and pythagorean theorem, then:
$$\frac{x}{\sin(19)} = \frac{\sqrt{x^2+27^2}}{\sin(71-y)}$$
And we see:
$$\sin(71-y)=\frac{27}{\sqrt{4x^2+27^2}}$$
Thus, if we substitute:
$$\frac{x}{\sin(19)} = \frac{\sqrt{x^2+27^2}\sqrt{4x^2+27^2}}{27}$$
Solving for $x$:
$$x=15.15...\lor x=24.05.... | {
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"timestamp": "2023-03-29T00:00:00",
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Why is the inverse of an average of numbers not the same as the average of the inverse of those same numbers? I have a set of numbers (in my case: mean retention time (MRT) in the stomach (h)) of which I want to calculate the average gastric passage rate (/h). Gastric passage rate = 1/MRT.
My question is why 'the avera... | You already got many answers why it doesn't work for a "normal" arithmetic average.
In fact there exists a type of average for which this is true.
It would be the same if you were using a geometric mean instead of an arithmetic mean.
In that case, the algebra becomes:
$$\sqrt[N]{\prod_1^N a_i} = \sqrt[N]{a_1\cdots a_N}... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Cohomology and Eilenberg-MacLane spaces It is well known that for any abelian group $G$, and any CW-complex $X$, the set $[X, K(G,n)]$ of homotopy classes of maps from $X$ to $K(G,n)$ is in natural bijection with the $n^{\mathrm{th}}$ singular cohomology group $H^n(X; G)$ with coefficients in $G$.
My question is, is t... | Assume $X$ connected.
Yes, this is known. For pointed homotopy classes this is $\text{Hom}(\pi_1 X, G)$. This is 1B.9 of Hatcher, usually the first place one sees obstruction theory, and requires less work than the case of $n > 1$.
For unpointed homotopy classes of maps, one quotients by conjugacy of elements of $G$.... | {
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Confirming Axioms of Vector Spaces that rely on modular arithmetic V is a vector space where $$V = \{\mathrm{rotations}\} = \{\theta : θ ~ \text{is a real number and} ~ 0 ≤ θ < 2π\}$$
Addition is defined by $$θ_1 + θ_2 := (θ_1 + θ_2) ~ \mathrm{mod} ~ 2π$$
Scaling by real numbers is defined by $$rθ = rθ ~ \mathrm{mod}... | See both of u/Nick 's comments for the answer to this question. "Mods" can be pulled out of the expression and the additive inverse would be 2$\pi$ - v.
| {
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GCDs for the polynomial ring over a Galois field. You can find many examples of computing the inverse of an element inside a Galois field. (For example here)
What happens if we look at the polynomial ring over a Galois field and would like to compute gcd of two elements? Since this is a euclidean domain the GCD should ... | You ned rules for simplifying expressions in $\mathbf F_8$. With your setting, if you denote $\omega$ the congruence class of $X$ in $\mathbf F_2[X]/(X^3+X+1)$, you know that
$$\omega^3=\omega+1\qquad\text{(we're in characteristic }2),$$
so the last division is written as
$$Y^2=(\omega+1)^{-1}Y^\cdot (\omega+1)+0.$$
N... | {
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Can $x^n-(x-1)^n$ be Prime if $n$ is Not Prime? I'm hoping someone can provide an answer or a link to a proof regarding this question.
Edit:
The question has been put on hold because I did not expound on why the answer to this was of interest to me or the community, so, even though I've received my answer, I will elabo... | Hint: $x^n - y^n$ is divisible by $x^r - y^r$ if $r$ divides $n$.
| {
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Finding the conditional distribution of the uniform distribution
Let $U$ denote a random variable uniformly distributed over $(0,1)$. Compute the conditional distribution of $U$ given that
$$(a)\,\,\,U>a\\
(b)\,\,U<a$$
where $0<a<1.$
For $(a)$ I tried the following,
$$f_{U\mid U>a}(u\mid u>a)=\frac{f(u,u>a)}{f_{... | The conditions do not change the fact that the distribution is uniform, only the interval for which it is defined. Therefore for an interval of $0\le u\le a$, the density function is $\frac{1}{a}$, while for an interval $a\le u\le 1$, the density function is $\frac{1}{1-a}$. Your analysis is correct.
To clarify yo... | {
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How to show there are only two discrete valuation rings with quotient field $k(x)$? I want to show that only discrete valuation rings with quotient field as $k(x)$ containing $k$ are:
$\mathcal{O_{a} (\mathbb{A^{1}})}$ for each $a \in k$ and $\mathcal{O_{\infty}}$;
the former is the set of rational functions on $\math... | Let me start by pointing out the a DVR $R\subset k(x)$ may not contain $k[x]$, and in fact you $\mathscr{O}_\infty$ doesn't. Here's an outline of how to prove the result: (1) using the second exercise you mention (about maximality of DVRs), prove that a DVR R that does contain $k[x]$ must be the localization of $k[x]$ ... | {
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Constrained (and non) extrema of $f(x,y)=2x^2-2xy^3+3y^2$ I need to find the critical points on the boundary,inside $D$,outside $D$, and find the image of this function (constrained on $D$).
$f(x,y)=2x^2-2xy^3+3y^2$
$D=\{2x^2+3y^2\le 9\}$
Critical points non-constrained:
$f_x=4x-2y^3=0$
$f_y=-6xy^2+6y=0$ --> $6y(-xy+1... | Critical points non-constrained:
$$(0,0),\quad \left(\pm2^{-1/4}, \pm2^{1/4}\right)$$
Global minimum:
$$f_{min}=f(0,0)=0$$
Saddle points:
$$f\left(\pm2^{-1/4}, \pm2^{1/4}\right)=2\sqrt2$$
Constrained Critical points: from system
$$\left\{
\begin{array}{c}
4x-2y^3=4x\lambda\\
-6xy^2+6y=6y\lambda\\
2x^2+3y^2-9=0
\end{a... | {
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"timestamp": "2023-03-29T00:00:00",
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How many essentially different strings are there of length $\leq n$ and over an alphabet of size $|\Sigma| = m$? For example, $aaaaaabb \simeq ccccccdd$ essentially, because a smallest grammar algorithm would perform the exact same steps to reduce one as the other. So how can I phrase this in terms of formal strings, ... | The equivalence classes you are trying to count are called "restricted growth strings". The sequence of counts of all RGS of length $n$ are the "Bell numbers", after the mathematician Eric Temple Bell who studied them in the 1930s.
This corresponds to the count of your "essentially different strings" for the case $n=m$... | {
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Let $ X \sim (0,1) $ and $ Y \sim (-1,2)$ be independent. Compute the distribution function of $Z=X+Y$ - how to break into cases?
Let $ X \sim (0,1) $ and $ Y \sim (-1,2)$ be independent. Compute the
distribution function of $Z=X+Y$ - how to break into cases?
I first found the density functions:
$$
f_x(t) =\begin{c... | Assuming that you know that the sum of two independent random variables is their convolution, we have
$$F_z(t)= \int_{-\infty}^{\infty}f_X(t-\tau)f_Y(\tau) \ d \tau = \frac{1}{3} \int_{-1}^{2}f_X(t-\tau) \ d \tau = \frac{1}{3}\int_{t-2}^{t+1}f_X(x) \ dx$$
Now, we have cases according to the integral boundaries and t... | {
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Rational expression $K(t)$ is a transcendental extension of $K$? I know this question was asked before, but none of the previous threads end up answering the question satisfactorily enough for me. So let me try to summarize my problems succinctly:
*
*The notation $\mathbb{Q}(\sqrt{2})$ is commonly used to denote the... | Well, if $t$ is algebraic over $K$, then $K[t]=K(t)$, i.e., ring extension equals field extension.
If $t$ is transcendental over $K$, then $K[t]$ is a polynomial ring and $K(t)$ the field of rational functions of $t$ over $K$. The construction of $K(t)$ from $K[t]$ is the same as the construction of $\Bbb Q$ from $\Bbb... | {
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Calculating $\int_{0}^{\pi}\text{sinh}\left(\sin\left(x\right)\right)\text{d}x$ I was wondering if it exists a beautiful exact value to
$$ \int_{0}^{\pi}\text{sinh}\left(\sin\left(x\right)\right)\text{d}x$$
which has a nice graph over $\left[0,\pi\right]$, but i can't get to compute it.
| If you know the Beta and Gamma functions well, you can see the sequence $$a_n:=\int_0^\pi\sin^{2n+1}x\mathrm{d}x=2\int_0^{\pi/2}\sin^{2n+1}x\mathrm{d}x=\operatorname{B}\left(\frac{1}{2},\,n+1\right)=\frac{n!\sqrt{\pi}}{\Gamma\left(n+\frac{3}{2}\right)}$$satisfies$$a_0=2,\,\frac{a_{n+1}}{a_n}=\frac{2(n+1)}{2n+3}$$so by ... | {
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Proof that $\lim_{x\downarrow 0}x^me^{\frac{-1}{x}} =0,m\in\mathbb{Z}$ with L'Hospital For the case that $m\geq0$ I don't need to apply L'Hospital.
Let $m<0$
We have $x^m=\frac{1}{x^{-m}}$
We also know that $x^{-m}\rightarrow 0$ as $x\rightarrow 0$
We also know that $e^{-\frac{1}{x}}<\epsilon\iff x<-\frac{1}{\ln \epsil... |
Direct application of L'Hospital's Rule does not provide a tractable way forward as mentioned in the OP.
To see this, we begin by writing (for $m<0$, $|m|\in\mathbb{N}$)
$$\begin{align}
\lim_{x\to0^+}\left(x^me^{-1/x}\right)=\lim_{x\to0^+}\left(\frac{e^{-1/x}}{x^{|m|}}\right)\tag1
\end{align}$$
But, differentiating... | {
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Continuous function between topological space mapping to a closed set A function between two topological spaces is continuous if the preimage of every open set is open. I am getting confused with constant functions.
$$f:(X,\tau_X)\to(Y,\tau_Y) \text{ is continuous if } \forall V\in Y \text{ open } f^{-1}(V) \text{ is ... | The statement says that $f$ is continuous if and only if for each open subset $V$ of $\mathbb R$, $F^{-1}(V)$ is also an open set. Note that it says “every open subset $V$ of $\mathbb R$”, not “every subset $V$ of $f(\mathbb R)$ which happens to be an open subset of $\mathbb R$”. So, if, for instance, $f(x)=2$ and $V=(... | {
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How can I find the coefficient of $x^6$ in $(1+x+\frac{x^2}{2})^{10}$ efficiently with combinatorics? To find the coefficient of $x^6$ in $(1+x+\frac{x^2}{2})^{10}$,
I used factorization on $(1+x+\frac{x^2}{2})$ to obtain $\frac{((x+(1+i))(x+(1-i)))}{2}$, then simplified the question to finding the coefficient of $x^6... | Write the expression as $\frac{1}{2^{10}}\left((x+1)^2+1\right)^{10}$.
Hint if you would like to do it using combinatorial arguments : Coefficient of $x^6$ in $(1+x^2)^{n}$ would give you the number of tuples such that $a_1 + a_2 + \cdots a_{n} = 6$ such that $a_i\in \{0,2\}$. This should be easy- just note that any t... | {
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Why is clifford group a group? Let $C(Q)$ denote the clifford algebra of vector space $Q$ with respect to a quadratic form $q:V \rightarrow \Bbb R$. Hence we have the relation $w^2 = Q(w) \cdot 1$ for $w \in V$.
Let $\alpha:C(Q) \rightarrow C(Q)$ be the canonical automoprhism $\alpha^2=id, \alpha=-x$.
The Clifford g... | I will assume, as it is usually defined, that $C(Q)^*$ is the set of invertible elements in $C(Q)$. Then, the existence of an element $x^{-1}\in C(Q)$ such that $xx^{-1}=x^{-1}x=1$ is, by definition, guaranteed for each $x\in C(Q)^*$. What we need to show is that $x^{-1}$ is in fact an element in $\Gamma(Q)$.
First, fo... | {
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Proving "$\forall_{\epsilon>0} \exists_{N \in \mathbb{N}} \forall_{n \geq N}: x^{n} < \epsilon$" for $0I'm trying to prove a rather simple analysis statement but I think I'm overseeing something. For $0<x<1$ I need to prove
$$\forall_{\epsilon>0} \exists_{N \in \mathbb{N}} \forall_{n \geq N}: x^{n} < \epsilon.$$
I no... | Put $y=1/x$, then $y>1$, hence $y=1+z$ for some $z>0$. Then we get, with Bernoulli:
$$\frac{1}{x^n}=y^n=(1+z)^n \ge 1+nz >nz.$$
Hence
$$ x^n < \frac{1}{nz}.$$
Can you proceed ?
| {
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Find $A^5-4A^4+7A^3+11A^2-A-10I$
If$$A= \begin{bmatrix}
1 & 4 \\
2 & 3
\end{bmatrix},$$then find $A^5-4A^4+7A^3+11A^2-A-10I$ where $I$ is Identity matrix of $2^{nd}$ order.
Answer should come in terms of $A$ and $I$.
My approach: I thought it would might end up in a pattern so I found
$$A^2=\begin{bm... | The characteristic polynomial of that matrix is $\lambda^2-4\lambda-5$. So, by the Hamilton-Cayley theorem, $A^2-4A-5\operatorname{Id}=0$. In other words, $A^2=4A+5\operatorname{Id}$. Can you take it from here?
| {
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Show that the function $f(x)=e^{-\frac{1}{x}},\text{ if }x>0\wedge f(x)=0,\text{ if }x\leq 0$ is smooth and all derivatives vanish in $x=0$ I had an idea but I think it is wrong. I have said $e^z$ is smooth because the n-th derivative of $e^z$ is always $e^z$ and then I have substituted $z$ with $-\frac{1}{x}$. But jus... | When $x<0$ then $f^{(n)}(x)=0$ for all $n\geq0$. When $x>0$ then $f^{(n)}(x)=p_n(1/x)e^{-1/x}$ for some polynomial $t\mapsto p_n(t)$. The latter is true for $n=0$ with $p_0(t)\equiv1$, and
$$f^{(n+1)}(x)={d\over dx}\biggl(p_n(1/x)e^{-1/x}\biggr)=\bigg({-1\over x^2}{p_n}'(1/x)+p_n(1/x){1\over x^2}\biggr)e^{-1/x}=:p_{n+1... | {
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Weak Convergence Lemma - Is Banach needed? Lemma. Let $X$ be a normed space.
*
*If $x_n \rightharpoonup x$ in $X$ and $x_n^* \to x^*$ in $X^*$, then $\lim_{n \to \infty} x_n^*(x_n) = x^*(x)$.
*If $X$ is even Banach, then $x_n \to x$ in $X$ and $x_n^* \overset{*}\rightharpoonup x^*$ in $X^*$ implies $\lim_{n \to \i... | Here is a counter-example for the second statement without $X$ being complete. The first statement is valid for non-complete $X$ (by means of embedding into the Banach space $X^{**}$).
Take $X=c_{00}$ provided with the $l^2$-norm. Define $x_n = n^{-1}e_n$ and
$$
x_n^*(y):= n y_n.
$$
Then $x_n^*\rightharpoonup^*0$, $x_... | {
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"url": "https://math.stackexchange.com/questions/3110212",
"timestamp": "2023-03-29T00:00:00",
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probability that at least two part will be defective Question
The probability that a part manufactured by a company will be defective is $0.05$. If $15$ such parts are selected randomly and inspected, then the probability that at least two part will be defective is ________. (round off to two decimal places)
My Appro... | Not quite, $$P=1-(0.95)^{15}-{15\choose 1}((0.95)^{14} \times 0.05)$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Confused by ODE $f''(x)=\lambda f(x).$ So the solutions to the characteristic equation is $r_{1,2}=\pm\sqrt{\lambda}.$ We thus have three cases to consider in order to find the solution of $f''=\lambda f.$ However for this question, the only relevant one is when the roots become complex, that is
Case 1: $\lambda < 0.$
... | 1) First, there's a conlict of notation : the $a$ in $ay''+by'+c=0$ is not the same $a$ appearing in $r_{\pm}=a\pm bi$.
That being said, $f''=\lambda f\Longleftrightarrow f''-\lambda f=0$. Its characteristic equation is $r^2-\lambda=0$, which can be easily solved.
2) She didn't. It's just that in this case, the solutio... | {
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How can I create a seam for an ellipse? I want to create a seam for an ellipse. (Mathematically I think this means that there are constant-length normal lines between the ellipse and the curve that creates the seam, but I'm not 100% sure that definition is accurate.) I know I can't do this by creating another ellipse w... | What you've described is typically called an "offset curve". The offset curve for an ellipse is ... not nice. The points of the ellipse itself satisfy a nice quadratic like
$$
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
$$
while those of the offset curve satisfy...a polynomial of degree 8 perhaps? I can't recall, but it's ce... | {
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prove that : $\dfrac{a^2}{2}+\dfrac{b^3}{3}+\dfrac{c^6}{6} \geq abc$ for $a ,b ,c \in \mathbb{R}^{>0}$ prove that : $\dfrac{a^2}{2}+\dfrac{b^3}{3}+\dfrac{c^6}{6} \geq abc$ for $a ,b ,c \in \mathbb{R}^{>0}$ .
I think that must I use from $\dfrac{a^2}{2}+\dfrac{b^2}{2} \geq ab$ but no result please help me .!
| I'd like to add a calculus approach to the above answers. If we define the function
$$
f(a,b,c) = \dfrac{a^2}{2}+\dfrac{b^3}{3}+\dfrac{c^6}{6} - abc
$$
and look for the positions in $\mathbb{R}_{>0}^3$ where the gradient vanishes, we will find
$$
\overline{\triangledown } f = 0 \Rightarrow a=b=c=1
$$
The value of $f$ a... | {
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"url": "https://math.stackexchange.com/questions/3110691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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If $\cos\theta=\frac{\cos\alpha+\cos \beta}{1+\cos\alpha\cos\beta}$, then prove that one value of $\tan(\theta/2)$ is $\tan(\alpha/2)\tan(\beta/2)$
If $$\cos\theta = \frac{\cos\alpha + \cos \beta}{1 + \cos\alpha\cos\beta}$$ then prove that one of the values of $\tan{\frac{\theta}{2}}$ is $\tan{\frac{\alpha}{2}}\tan{\f... | I think i have found out how to approach it. I hope this proof is satifactory.
Using the half angle formula,
$$\tan{\frac{\theta}{2}} = \pm \sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}} \longrightarrow \text{eq.1}$$
Evaluating $\frac{1 - \cos\theta}{1 + \cos\theta}$ first,
$$\frac{1 - \cos\theta}{1 + \cos\theta} = \fr... | {
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Show that $\lim\limits_{(x,y)\to(0,0)}\frac{x^3y-xy^3}{x^4+2y^4}$ does not exist.
Show that $$\lim_{(x,y)\to(0,0)}\frac{x^3y-xy^3}{x^4+2y^4}$$ does not exist.
I'm not even sure how to approach this. I tried factoring out $xy$ in the numerator to get $xy(x^2 - y^2)$, but I don't think that gets me anywhere with the de... | Let's approach the limit along the line $y=mx.$
$\begin{align}
&\lim_{(x,y)\to (0,0)}\dfrac{x^3y-xy^3}{x^4+2y^4}\\
&=\lim_{x\to 0}\dfrac{x^3mx-x(mx)^3}{x^4+2(mx)^4}\\
&=\lim_{x\to 0}\dfrac{x^4m-m^3x^4}{x^4+2m^4x^4}\\
&=\lim_{x\to 0}\dfrac{m-m^3}{1+2m^4}\\
&=\dfrac{m-m^3}{1+2m^4}\\
\end{align}$
So what can you conclude ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3111000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Evaluating the definite integral of $\cot^2(t)dt$ Given that:
$$\cot(t) \to \infty \text{ when } t=0$$
and:
$$\int \cot^2(t) dt = -\cot(t) -t +C$$
It seems strange that:
$$\left[-\cot(t)-(t)\right]^{\pi+\frac{\pi}{2}}_{\frac{\pi}{2}} = (-\cot^2(\pi+
\left(\frac{\pi}{2}\right))-(\pi+\left(\frac{\pi}{2}\right)))-(-\cot^2... | The fundamental theorem of calculus breaks if there is a singularity in the integration interval.
Check the conditions of application: https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Formal_statements.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3111091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How to find $\lim \limits_{x \to 0} \frac{\sqrt{x^3+4x^2}} {x^2-x}$ when $x\to 0^+$ and when $x\to 0^-$? I'm trying to find:
$$ \lim \limits_{x \to 0} \frac{\sqrt{x^3+4x^2}} {x^2-x} $$
Since there is a discontinuity at $x=0$ I know that I have to take the limits from both sides, $x \to 0^+$ and $x \to 0^-$, and check i... | Limit from right side is
$
\lim \limits_{x \to 0^+} \frac{\sqrt{x^3+4x^2}} {x^2-x} \\
= \lim \limits_{x \to 0^+} \left(\frac{ |x| \sqrt{x+4}} { x(x-1) }\right) \\
= \lim \limits_{\delta \to 0} \left(\frac{ |0+\delta| \sqrt{ (0+\delta) +4}}{ (0+\delta)( (0+\delta) -1 ) }\right) \ [ \ \text{substituting} \ x = 0 + \delta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3111220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 6,
"answer_id": 3
} |
If $\tan(x_1) \cdots\tan(x_n)=1$ for acute $x_i$, then does it follow that $\cos(x_1)+\cdots+\cos(x_n) \leq n\sqrt{2}/2$? It is easily seen that if $x,y\in[0,\pi/2)$ satisfy $\tan(x)\tan(y)=1$, then $$\cos(x)+\cos(y)\le\sqrt 2$$
A much more delicate fact is that if $\tan(x)\tan(y)\tan(z)=1$ (while $0\le x,y,z<\pi/2$), ... | For three variables we ca use C-S:
Let $\tan{x}=\sqrt{\frac{b}{a}},$ $\tan{y}=\sqrt{\frac{c}{b}},$ where $a$, $b$ and $c$ are positives.
Thus, $\tan{z}=\sqrt{\frac{a}{c}}$ and by C-S we obtain:
$$\sum_{cyc}\cos{x}=\sum_{cyc}\frac{1}{\sqrt{1+\tan^2x}}=\sum_{cyc}\sqrt{\frac{a}{a+b}}\leq$$
$$\leq\sqrt{\sum_{cyc}\frac{a}{(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3111299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Is the Quotient Group Cyclic? I'm just wondering how to show that a quotient group $H = (G/N)$ is cyclic?
Let $G= \mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$
Let $N = \left<(2,3)\right>$ , where N is a cyclic subgroup of G
Is it correct to say that $G$ has order $24$ and $N$ has order $2$?
Can I then say th... |
Is it correct to say that G has order 24 and N has order 2?
$G$ has order $24$ because $\mathbb{Z}/4\mathbb{Z}$ has $4$ elements and $\mathbb{Z}/6\mathbb{Z}$ has $6$ elements and $4\cdot 6 = 24$.
In order to see if $N$ has order $2$ you should check what are the elements in $N:\;$ we have the trivial element, of cou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3111408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that the midperpendiculars $MN$ pass through a constant point.
Consider two circles $O_1$ and $O_2$, the intersection of $O_1$ and $O_2$ are $A$ and $B$.
Let $M$ be the point on $O_1$, $N$ on $O_2$, $M,N$ moving clockwise on $O_1,O_2$ and $\angle{AO_1M }=\angle{AO_2N }$.
Show that the midperpendiculars of $MN$... | This has notihng to do with Apollonius.
Step 1: Let line $O_1M$ meet line $O_2N$ at $Q$. Then since $$\angle QO_1A = \angle QO_2A $$ we see that points $O_1, O_2, Q$ and $A$ are concyclic and so $\angle O_1QO_2 = \angle O_1AO_2$ is constatnt.
Step 2: Let $S$ be a midpoint for segment $O_1O_2$. Let us prove that $SP$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3111497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find the indefinite integral of $\int_{} \frac{x}{x^2+4}dx$ I am beginning to question whether the indefinite integral actually exists or I am doing something wrong with my u-substitution.
Let $u = x^2 + 4, du = 2xdx,$
$$
\begin{align}
\int_{} \frac{x}{x^2+4}dx
&= \int_{}x(x^2 + 4)^{-1} \\
&= \frac{1}{2} \int_{} u^{-1... | $$\int \frac{xdx}{x^2+4}=\frac 12 \int \frac{2xdx}{x^2+4}=\frac12 \ln(x^2+4)+C$$ where we have used that
$$\frac{d}{dx}(x^2+4)=2x$$ and $$\int \frac{dt}{t}=\ln|t|+C.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3111626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Relation between eigenvalues of $A^{\top}BB^{\top}A$ and $B^{\top}AA^{\top}B$ I have two real value matrices: $A \in \mathbb{R}^{m \times n}$ and $B \in \mathbb{R}^{m \times p}$.
If I know which are the eigenvalues of $A^{\top}BB^{\top}A$, what can I say about the eigenvalues of $B^{\top}AA^{\top}B$?
I suspect that the... | Fact. The nonzero eigenvalues of $X^\top X$ and $XX^\top$ are the same.
We may apply this fact to your two matrices by taking $X=A^\top B$. Indeed, this gives
\begin{align*}
X^\top X &= (A^\top B)^\top(A^\top B)=B^\top AA^\top B &
XX^\top &= (A^\top B)(A^\top B)^\top = A^\top BB^\top A
\end{align*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3111902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
What is the largest integer value of $n$ for which $8^n$ evenly divides $(100!)$? I know that this may be an unnecessary question, but I am a bit confused. The problem asks for the highest integer $n$ such that $8$ to the power of $n$ is divisible, evenly of course, by $100$. Now, I searched the site, and, in general, ... | It's easiest, I think, to do this with powers of $2:$
$$\left\lfloor{100\over2}\right\rfloor+
\left\lfloor{100\over4}\right\rfloor+
\left\lfloor{100\over8}\right\rfloor+
\left\lfloor{100\over16}\right\rfloor+
\left\lfloor{100\over32}\right\rfloor+
\left\lfloor{100\over64}\right\rfloor=97=32\cdot3+1
$$
so the greatest e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3112037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
Permute columns by pre-multiplying and rows by post-multiplying? I was looking at Gilbert Strang's lectures on Linear Algebra and noticed that in lecture 2, Elimination with Matrices, around the 40nth minute he mentions that you can use the permutation matrix,
$$P=
\begin{bmatrix}
0 & 1 \\
1 & 0
\... | $\newcommand\bm\boldsymbol$
If such $\bm P'$ works for all matrices, then for all $\bm A$,
$$
\bm {AP} = \bm {P'A},
$$
then specifically it works for the identity matrix $\bm I$, i.e.
$$
\bm {IP} = \bm {P'I},
$$
then the only candidate of $\bm P'$ is $\bm P$ again. But clearly
$$
\bm {PA} = \bm {AP}
$$
only holds... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3112210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Proving IID Central Limit Theorem using Lindeberg Conditions. The goal is to prove the IID Central Limit Theorem through Lindeberg's Condition.
Suppose that $X_1,X_2,\ldots\displaystyle\sim\text{i.i.d.}$ with $E[X_i]=\mu$ and $Var[X_i]=\sigma^2<\infty$.
Let $Y_i=X_i-\mu$ and $s_n^2=\sum_{i=1}^{n}Var[Y_i]=n\sigma^2$.
P... | All you need is $EY_1^{2} I_{\{|Y_1| >\epsilon \sigma \sqrt n\}} \to 0$ as $n \to \infty$ and this follows from DCT. [$Y_1^{2} I_{\{|Y_1| >\epsilon \sigma \sqrt n\}} $ is dominated by $Y_1^{2}$ which is integrable. Of course, the events $\{|Y_1| >\epsilon \sigma \sqrt n\}$ decrease to empty set so $Y_1^{2} I_{\{|Y_1| ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3112302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Center is a normal subgroup of G This is a problem from Herstein's Topics in Algebra.
I have already shown the above result using the definition of normal subgroup. But now I want to prove it by constructing a homomorphism such that kernel is center of the group G.
How can I construct such homomorphism?
I was thinkin... | You are very close. You already have a way to transform $g$ into an element of... something ... where $g$ transforms into the identity function if and only if $g$ communtes with everything in $G$. You also know that the something contains some sort of mappings.
Now you need to write that down with correct terms. That i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3112430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
$f$ has the form $f(x) =ax^2+bx+c$. Every differentiable function $f:R \rightarrow R$ with the property that $(2h)f′(x) =f(x+h)−f(x−h)$ for all $x \in R$ and all $h$ has the form $f(x) =ax^2+bx+c$.
I would like to get some help on this one. Taking partial derivatives was an idea, but is that legal?
| We have $f'(x)={1\over2}\bigl(f(x+1)-f(x-1)\bigr)$ for all $x$. Since here
the RHS is differentiable it follows that $f'$ is differentiable as well; in fact $f\in C^\infty$. We now differentiate
$2 h f'(x)=f(x+h)-f(x-h)$ two times with respect to $h$ and obtain
$$0=f''(x+h)-f''(x-h)\qquad\forall\,x\quad\forall\, h\ .... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3112546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Can you keep this raffle fair? So there is a car draw in my area. There are 3221 participants in the draw. The winner is decided by a trustee drawing each digit from a separate drum. So from the first drum there is 0-3, the second 0-9 (this is the same for the third and the fourth drum). So if I draw 0-4-0-7 the winner... | Whether or not the raffle is fair, depends on how you draw the ticket. Let's say I draw, from the four different bins, the numbers $3, 5, 1, 0$. This is an invalid number, so the price cannot be assigned. If I simply restart the whole procedure, each time drawing four numbers until a valid number shows up, then each pa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3112652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How many roots does an exponential polynomial have? Let $s$ be a complex variable and consider two polynomials with real coefficients:
$$A(s) = s^n + a_{n-1}s^{n-1}+\ldots+a_1s+a_0,$$
$$B(s) = s^m + b_{m-1}s^{m-1}+\ldots+b_1s+b_0,$$
where $n \ge m$.
Let $k$ be a real constant. I am looking for roots of the function
$$... | Set $A(s)=-1$, $B(s)=1$, and $k=2\pi$. Then you obtain the equation $$e^{2 \pi s}=1,$$ which has solutions $s = 0, \pm i, \pm 2i, \pm 3i, \ldots$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3112888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Is $Y_n=f(X_n)$ a Markov chain, when $X_n$ is? Let $X_n$ be an independent Markov chain which has values (states) $X_n={0,1,2}$ with its transition matrix. $$\\p=
\begin{pmatrix}
0 & \frac{1}{2} & \frac{1}{2} \\
\frac{1}{2} & \frac{1}{2} & 0 \\
1 & 0 & 0
\end{pmatrix}
$$ Let $Y_n=f(X_n)$ $f(0)=0$ $f(1)=1$ $f(2)... | The concept you're looking for here is called Lumpability. If you aggregate states of a Markov chain and the chain is "lumpable", then the aggreate process that you obtain is again a Markov chain.
Lumpability property (see Theorem 6.3.2 in Finite Markov chain, by Kemeny ans Snell):
A discrete-time Markov chain $\{X_{i}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3112981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Integral representation of Digamma function A similar question was already asked about 2 years ago:
Integral representation of the Digamma function
Someone asked for the derivation of the integral representation of the Digamma-function and it was answered, but I don't see how you get from here:
$$ \psi^{(0)}(x)=\frac{\... | I'll give a more detailed version of Jack D'Aurizio's answer on the other question.Start with the Weierstrass product for the $\Gamma$ function
$$ \Gamma(z+1) = e^{-\gamma z}\prod_{n\geq 1}\left(1+\frac{z}{n}\right)^{-1}e^{z/n}\tag{1}.$$
By definition,
\begin{align*}
\psi(z+1) &= \frac{d}{dz} \log \Gamma(z+1) \\
&= \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3113119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Have similar theories like knot theory been developed in higher dimensions? Well, my question is kind of basic but I hope it would be taken seriously by the community. Also, I'm very new to this topic and I want to study knot theory in future. Knot theory is the study of embedding $S^{1}$ in $\mathbb{R}^3$. Right?
So, ... | A reference to higher-dimensional knot theory:
E. Ogasa, Introduction to higher-dimensional knots.
For classical knot theory, I like the book
D. Rolfsen, "Knots and Links".
It is a bit dated (written by 1970s) but very readable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3113363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Understanding why a limit proof using another limit works Sorry for the title, hopefully I can explain it better. I think the title is about as good as I could get in terms of description.
I have a problem:
Let $x_n \ge 0$ for all $ N \in \mathbb{N}$
If $(x_n) \to x$, show that $(\sqrt{x_n}) \to \sqrt(x)$
Assume that... | You're using the fact that $x_n \rightarrow x$ in the very first step: that's how you know that you can make $|x_n - x|$ arbitrarily small. And you need to be able to make that difference arbitrarily small for the rest of the proof to work.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3113495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Computing the matrix of a Linear Transformation For a matrix $A\in M_n(\mathbb{F})$ consider the
linear transformation $T_A:\mathbb{F}^n\rightarrow
\mathbb{F}^n$. Denote the $B_{st}=\{e_1,...,e_n\}$
the standard basis of $\mathbb{F}^n.$
Compute the matrix $[T_A]_{B_{st}}$.
So I don't really want anyone to solve the pr... | The map $T_A$ is$$\begin{array}{rccc}T_A\colon&\mathbb{F}^n&\longrightarrow&\mathbb{F}^n\\&v&\mapsto&Av.\end{array}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3113617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why $\max\{f,g\}$ is continuous if $f$ and $g$ are continuous? Let $f,g:\mathbb{R}\to \mathbb{R}$ be continuous functions. I need to prove that $x\mapsto \max\{f(x),g(x)\}$ is continuous without using the fact that $\max\{a,b\}=\frac{a+b+|a-b|}{2}$.
Let $\varepsilon >0$ and $a\in\mathbb{R}$. Suppose, without loss of ge... | Rather straightforward:
Let $\epsilon$ be given
For $\epsilon/2$ there are $\delta_{1,2}$ s.t.
$|x-x_0| < \delta = \min \delta_{1,2}$ implies
$|f(x)-f(x_0)| < \epsilon/2$ , and $|g(x)-g(x_0)| < \epsilon/2$.
Then
$|x-x_0| \lt \delta$ implies
$(1/2)|f(x)+g(x)+|f(x)-g(x)|- f(x_0)-g(x_0)-|f(x_0-g(x_0)|| $
$\le (1/2)|f(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3113730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Does $\triangle ABC$ exist such that $\triangle ABC \sim \triangle DEF$, with $D, E, F$ being the incentre, centroid, orthocentre of $\triangle ABC$? Question:
Does $\triangle ABC$ exist such that $\triangle ABC \sim \triangle DEF$, with $D, E, F$ being the incentre, centroid, orthocentre of $\triangle ABC$, resp.?
F... | Yes, this triangle exists. (Found using brute force approximation.)
Coordinates:
$$A\approx(0.182,0.260)\quad B=(0,0)\quad C=(1,0)\\
D\approx(0.229,0.120)\quad E\approx(0.394,0.087)\quad F\approx(0.182,0.571)$$
Angles:
$$a=d\approx107.2957\quad b=e\approx55.0744\quad c=f\approx 17.6299$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3113834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 0
} |
A formula for tan(2x) Help with solving...
Suppose that $\tan^2x=\tan(x-a)·\tan(x-b)$, show that $$\tan(2x)=\frac{2\sin(a)·\sin(b)}{\sin(a+b)}$$
As far as I know, so far the $\tan2x$ can be converted to $\frac{2\tan x}{1-\tan^2x}$ using the double angle formula and the $\tan 2x$ can be further be substituted to the f... | The intial equations can be written
$$t^2=\frac{t-t_a}{1+t\,t_a}\frac{t-t_b}{1+t\,t_b},$$
$$\frac{2t}{1-t^2}=\frac{2t_at_b}{t_a+t_b}.$$
From the second equation, we can draw $t^2$ as a linear function of $t$. Then, replacing $t^2$ several times in the first equation, we should get an identity.
With $c$ denoting the cot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3113894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Find the area of the shaded region of two circle with the radius of $r_1$ and $r_2$ In the given figure , $O$ is the center of the circle and $r_1 =7cm$,$r_2=14cm,$ $\angle AOC =40^{\circ}$. Find the area of the shaded region
My attempt: Area of shaded region $=\pi r^2_2 - \pi r^2_1= \pi( 196-49)= 147\pi$
Is it t... | Use the formula for the area of a sector (the angle is measured in radians) and the formula for the area of a circle:
$$
A=\frac{1}{2}r^2\theta
$$
$$
A=\pi r^2
$$
$40^\circ$ in radians would be:
$$
40^\circ=\frac{40\pi}{180}
$$
The area of the top peice:
$$
A_1=\frac{1}{2}r_2^2\cdot \frac{40\pi}{180} - \frac{1}{2}r_1^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3114006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Determinant of a particular matrix. What is the best way to find determinant of the following matrix?
$$A=\left(\begin{matrix}
1&ax&a^2+x^2\\1&ay&a^2+y^2\\ 1&az&a^2+z^2
\end{matrix}\right)$$
I thought it looks like a Vandermonde matrix, but not exactly. I can't use $|A+B|=|A|+|B|$ to form a Vandermonde matrix. Please ... | Note that
$$
\det\left(\begin{matrix}
1&ax&a^2+x^2\\1&ay&a^2+y^2\\ 1&az&a^2+z^2
\end{matrix}\right) =\det \left(\begin{matrix}
1&ax&x^2\\1&ay&y^2\\ 1&az&z^2
\end{matrix}\right)=a\cdot\det \left(\begin{matrix}
1&x&x^2\\1&y&y^2\\ 1&z&z^2
\end{matrix}\right)
$$ which boils down to Vandermonde determinant
$$
a(x-y)(y-z)(z-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3114122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 2
} |
General closed form for $L(\phi)=\int_0^\phi \log(\sin x)\mathrm dx$ when $\phi\in(0,\pi)$? I would like to know if there is a general closed form for
$$L(\phi)=\int_0^\phi \log(\sin x)\mathrm dx,\qquad \phi \in(0,\pi)$$
Context: (below are also the extent of my search for a closed form.)
I would like to know such a... | Beside Clausen functions, using one integration by parts
$$\int\log (\sin (x)) \,dx=x \log (\sin (x))-\int x \cot(x)\,dx$$ and
$$\int x \cot(x)\,dx=x \log \left(1-e^{2 i x}\right)-\frac{1}{2} i \left(x^2+\text{Li}_2\left(e^{2 i x}\right)\right)$$ making by the end
$$\int_0^\phi\log (\sin (x)) \,dx=\frac{1}{2} i \left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3114200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Let $a,b\in G$ elements of order $5$. If $a^3=b^3$ then $a=b$.
Prove or disprove: let $G$ be a group and $a,b\in G$ elements of order $5$. If $a^3=b^3$ then $a=b$.
I saw the following example which tries to disprove the theorem: $G=\mathbb{Z}_{10}$ and $a=2,b=8$.
I'm not sure about that part, but $o(2)=5$ and $o(2)=8... | Since $$b^5=a^5 =a^3a^2 =b^3a^2\implies a^2=b^2$$
so $$b^3=a^3 =a^2a =b^2a\implies a=b$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3114266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding rational solutions for $3x^2+5y^2 =4$ This question comes from Rational Points on Elliptic Curves (Silverman & Tate), exercise $1.6$ (b).
I want to calculate all rational solutions for $3x^2+5y^2 =4$. However, I think that there are no rational solutions because if we homogenize we get $3X^2+5Y^2 =4Z^2$ and mod... | Your argument isn't quite sufficient. This is because the observation that "mod $3$ the only solution is $Z=Y=0$" applies equally well to the equation $9X^2+5Y^2=4Z^2$, which does have solutions, e.g., $(X,Y,Z)=(2,0,3)$. What you need to say is that "mod $3$ the only solution is $Z=Y=X=0$." (On a side note, it would ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3114412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Simplifying 3rd root of (24 * sqrt(3)) I have problems following a solution towards simplifying a given polynomial.
$$Polynomial: p(x)=x^5+{\sqrt 3}x^4+24{\sqrt 3}x^2+72x$$
the zeros of this function (Polynomial roots? English isn't my native language, so I don't know how to express the point(s) at which the function m... | $24 = 3*8$ so $24 \sqrt {3} = 8*3*\sqrt{3} = 8\sqrt{3}^3$
$-1 = e^{\pi i}$ so $-24\sqrt{3} = 8*(\sqrt 3)^3 e^{\pi i}=2^3(\sqrt 3)^3e^{\pi i}$
And so $\sqrt[3]{-24\sqrt 3}=\sqrt[3]{2^3\sqrt{3}^3e^{\pi i}} = 2\sqrt 3 e^{\frac {(2k + 1)}3\pi i}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $f:X\to X $ be continuous. Does $f $ have a fixed point when $X=[0,1)$ or $X=(0,1) $?
Let $f:X\to X $ be continuous. Show that if $X=[0,1] $, $f $ has a
fixed point(i.e. there exists $x$ such that $f (x)=x$). What happens
if $X $ equals $[0,1) $ or $(0,1) $?
First part of the question is an immediate conseq... | The IVT is not the only ingredient here. The way the theorem works is by setting up this square:
where the line in the middle is $y = x$. A function from $[0, 1]$ to $[0, 1]$ that intersects this line will have a fixed point at the point of intersection.
The IVT kicks in when we have a function whose graph enters the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving an equality using given ones; no need for differentiation Prove that $(\frac ab+\frac bc+\frac ca)(\frac ba+\frac cb+\frac ac)\geqslant9$
The formulas given were
$$\frac{a+b} {2}\geqslant\sqrt {ab}$$
$$a^2+b^2\geqslant2ab$$
$$\frac{a+b+c} {3}\geqslant\root3\of{abc}$$
$$a^3+b^3+c^3\geqslant3abc$$
for all $a\gt0,... | Use that $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq 3\sqrt[3]{\frac{a}{b}\frac{b}{c}\frac{c}{a}}=3$$
and $$\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\geq 3\sqrt[3]{\frac{b}{a}\frac{c}{b}\frac{a}{c}}=3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3114879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Let $N,M$ be normal subgroups of $G$ with $N\cap M=\{e\}$. Prove that $M\subset C_{G}(N)$ and $N\subset C_{G}(M)$. First consider the following definition:
Let $G$ be a group and $H$ a subgroup of $G$. The center:
$$ C_{G}(H)=\{g\in G\,:\,gh=hg,\,\forall h\in H\}$$
Now I'm trying to prove the following theorem:
Let ... | It is useful to see $[M,N] := \langle mnm^{-1}n^{-1} |m\in M$ $ n\in N\rangle $. Shall show this is the trivial subgroup. $mnm^{-1} \in N$ and hence $mnm^{-1}n^{-1}\in N$ but then again $nm^{-1}n^{-1}\in M$ and hence $mnm^{-1} \in M$. So each generator is in $M\cap N=e$. Thus every element of $M$ commutes with every el... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Conditional Independence, Decomposition Is there some set of independence relations between three random variables $X$, $Y$ and $Z$ such that $P(Z \mid X, Y)$ = $P(Z \mid X) \cdot P(Z \mid Y)$? (I feel like there should be, but I can't find it).
| The relationship you are hoping to see:
$$
p(z\mid x,y)=p(z\mid x) p(z\mid y)\tag1$$
isn't true for the graph $x\to z\leftarrow y$ in general. The graph $x\to z\leftarrow y$ represents a joint distribution for $(X,Y,Z)$ that can be factored into the form
$$
p(x,y,z)=p(x)p(y)p(z\mid x,y).\tag2
$$
By summing (2) over $z... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to integrate $\cos^3x$ by parts I've converted $\cos^3(x)$ into $\cos^2(x)\cos(x)$ but still have not gotten the answer.
The answer is $\dfrac{\sin(x)(3\cos^2x + 2\sin^2x)}{3}$
My answer was the same except I did not have a $3$ infront of $x$ and my $2\sin^2x$ was not squared.
Help!
| Since $\cos^3x=(1-\sin^2 x)\cos x$, you can do $\sin x=t$ and $\cos x\,\mathrm dx=\mathrm dt$, thereby getting$$\int1-t^2\,\mathrm dt.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that $n^2-1+n \sqrt{d}$ is always a unit in $\mathbb{Z}[\sqrt{d}]$ We let $n\in \mathbb{Z}$, $n>2$ and $d=n^2-2$. We want to show that $n^2-1+n\sqrt{d}$ is a unit of $\mathbb{Z}[\sqrt{d}]$.
My initial idea was to consider the induced norm $N:R\to\mathbb{Z}$, given by $N(a+b\sqrt{d})=a^2-db^2$. We know that if $R$ ... | Why not prove it directly? Multiply it by its conjugate and see you get $1$.
$$\begin {align} \left(n^2-1+n\sqrt d\right)\left(n^2-1-n\sqrt d\right)&=\left(n^2-1\right)^2-n^2d\\
&=\left(n^2-1\right)^2-n^2\left(n^2-2\right)\\&=1 \end {align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3115250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Poisson Process Conditional Expectation Given $X_t$ a Poisson process such that $\lambda = 1$ find $E[X_1\mid X_2]$ and $E[X_2\mid X_1]$.
The first one is pretty straight forward since we have $E[X_2 - X_1] = E[X_1] = 1$ so then we get $E[X_2 \mid X_1] = E[X_2 - X_1 + X_1 \mid X_1] = E[X_2 - X_1 \mid X_1] + E[X_1\mid ... | By what you calculated, the PMF of $X_1$ conditional on $X_2 =y$ is that of a $\mathrm{Bin}(y,1/2)$ random variable. Therefore, $\mathbb{E}[X_1\mid X_2 = y] = y/2$ and $\mathbb{E}[X_1\mid X_2] = X_2/2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3115416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Should we distinguish the minus sign from the negative sign? In the set $\mathbb{C}$ of complex numbers, the minus sign "-" may be used for following:
*
*As a unary operator $-_u$, given a complex number $a$, $-_ua$ is the unique number (called the negative of a) $c$ such that $a+c=c+a=0$, where $0$ is the additive ... | We do distinguish most of the time, at such a subconscious level that it seems either automatic or unthinking. Heck, we're even capable of resolving meanings that a computer would find contradictory.
Take for instance $$\frac{3}{2} = 1 + \frac{1}{2}.$$ In a cake recipe, you might find something like
1-1/2 cups light b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3115513",
"timestamp": "2023-03-29T00:00:00",
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Is there a proof of $\lnot \forall x, P(x) \iff \exists x, \lnot P(x)$ I am interested in how one would formally prove:
$\lnot \forall x, P(x) \iff \exists x, \lnot P(x)$
I realize that it's basically saying that:
$\lnot(P(x_0) \land P(x_1) \land ... \land P(x_n)) \iff \lnot P(x_0) \lor \lnot P(x_1) \lor ... \lor \lnot... | Fitch style proof:
\begin{array}{lll}
1&\neg \forall x \ P(x) & Assumption\\
2&\quad \neg \exists x \ \neg P(x)&Assumption\\
3&\quad \quad a&\\
4&\quad \quad \quad \neg P(a) & Assumption\\
5&\quad \quad \quad \exists x \ \neg P(x)&\exists \ Intro \ 4\\
6&\quad \quad \quad \bot& \bot \ Intro \ 2,5\\
7&\quad \quad \neg \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3115645",
"timestamp": "2023-03-29T00:00:00",
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Are the projections along orthogonal direction of multivariate normal distribution with diagonal covariance matrix independent? I'm taking a probability class and my prof used the following theorem IIRC.
Let $g\sim\mathcal{N}(\mu,\Sigma)$ where $\Sigma$ is diagonal( I don't know if this condition is necessary) and $\la... | Elaborating on Minus One-Twelfth's comment:
The pair $(g^\top u, g^\top v)$ is jointly normal. (Why?)
Thus independence is equivalent to $\text{Cov}(g^\top u, g^\top v) = 0$.
The covariance is
$$\begin{align}\text{Cov}(g^\top u, g^\top v)
&= E[(g^\top u - E[g^\top u])(g^\top v - E[g^\top v])]
\\
&= E[((g - \mu)^\top u)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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For what values of $a$ and $b$ is the function $\frac{x^ay^b}{x^2+y^2}$ continuous at $(0,0)$? I have the function $$f(x,y)=\begin{cases}\dfrac{x^ay^b}{x^2+y^2} &(x,y)\neq(0,0)\\ 0 &(x,y)=(0,0) \end{cases}$$ I am trying to figure out what constants $a$ and $b$ will make the function continuous at $(0,0)$. I know that t... | $\dfrac{x^ay^b}{x^2+y^2} \overset{POLAR \, \, COORD.}{\Longrightarrow} \dfrac{r^a\cos^a(\theta) r^b\sin^b(\theta)}{r^2} \Longleftrightarrow r^{a+b-2}\cos^a(\theta)\sin^b(\theta)$
What happens for different values of $a+b-2$ in the limit,
$\lim_{r\rightarrow 0}r^{a+b-2}\cos^a(\theta)\sin^b(\theta)=0$ ?
| {
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Linear Algebra, proving subset is a subspace
Let $W$ be a subset of vector space $V$ over $K$. $\forall \mathbf{u}, \mathbf{v} \in W,\alpha \in K, \alpha \mathbf{u} + \mathbf{v} \in W$ , show that $W$ is a subspace over $K$. Hence, show that the set of linear combinations
$$W = \{\alpha_1 \mathbf{v}_1 + \alpha_2\mat... | To check that $W$ is a vector subspace you need to check the 3 following conditions: i) $W$ is non empty (clear if $V$ is non empty), ii)if $\mathbf x \in W$ and $\mathbf y \in W$, then $\mathbf x+\mathbf y \in W$. iii)If $\alpha \in K$, and $\mathbf x \in W$, then $\alpha \mathbf x \in W$
For your second question, you... | {
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Does $\mathbb{F}_9$ contain a 4th root of unity? I realised that I don't know how to construct $\mathbb{F}_9$. I'm guessing that $\mathbb{F}_9 = \mathbb{F_3(\theta)}$, where $\theta$ is the root of some irreducible polynomial over $\mathbb{F}_3[x]$ of degree two?
Must I even construct $\mathbb{F}_9$ in order to determ... | A non-trivial $4$th root of unity is a square root of $-1$, since $(x^4-1)=(x^2-1)(x^2+1)=(x-1)(x+1)(x^2+1)$, and $x^2+1$ is irreducible over $\mathbf F_3$. So the answer is yes:
$$\mathbf F_9\simeq \mathbf F_3[x]/(x^2+1),$$
and if you denote $\omega$ the congruence class of $x$, the non-trivial $4$th roots of unity a... | {
"language": "en",
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How to solve this recursion relation? Suppose:
$$2k(n-k)a_k=n(n-1)+(n-k)(k+1)a_{k+1}+(n-k)(k-1)a_{k-1}$$
where $k=1, 2, ..., n-1$ and $a_n=0$,
how to derive $a_k$?
I tried to find pattern $a_1-a_2=n/2$; $a_2-a3=n(2n-1)/6(n-2)$, it become more and more complicated and I can't find the rule.
| Hint.
Making $b_k = k a_k$ we have
$$
-b_{k-1}+2 b_k - b_{k+1} = \frac{n(n+1)}{n-k}
$$
This can be simplified by making $c_k = b_k-b_{k-1}$ giving
$$
c_k-c_{k+1} = \frac{n(n+1)}{n-k}
$$
After solving for $c_k$ can be then solved for $b_k$
| {
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If $D$ be the differentiation operator on $V$. Find $D^*$.
Let $V$ be the vector space of the polynomials over $R$ of degree less than or
equal to $3$ with the inner product space $(f|g)=\int_{0} ^{1}f(t)g(t) dt$, and let $D$ be the differentiation operator on V. Find $D^*$
Attempt
As I did the calculation was very... | Yes, the idea is correct (I didn't verify the coefficients in the orthonormalization).
Here's an alternative method: Integrating by parts gives
$$\langle f, Dg \rangle = \int_0^1 f g' dx = f(1) g(1) - f(0) g(0) + \int_0^1 - f' g \,dx = f(1) g(1) - f(0) g(0) + \langle -Df, g \rangle.$$
Thus, if we can find an operator $... | {
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Prove that the product of any two numbers between two consecutive squares is never a perfect square In essence, I want to prove that the product of any two distinct elements in the set $\{n^2, n^2+1, ... , (n+1)^2-1\}$ is never a perfect square for a positive integers $n$. I have no idea on how to prove it, but I've al... | For any two numbers $n^2+a,\ n^2+b;\ 0<a<b<(2n+1)$, their product will satisfy $n^4<n^4+(a+b)n^2+ab<(n^2+2n+1)^2$.
All of the squares between $n^4$ and $(n^2+2n+1)^2$ will have the form $(n^2+m)^2=n^4+2mn^2+m^2;\ 1\le m\le 2n$
If $n^4+(a+b)n^2+ab$ is a perfect square, it will be one of the squares between $n^4$ and $... | {
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Partition pairs of numbers such that no partition contain two pairs with the same number. Problem
Integers $1, 2, 3, ..., n$ can form $\binom{n}{2}$ pairs of numbers.
I want to partition these pairs of numbers such that:
*
*The number of partition is as low as possible
*None of the partitions contains two pairs tha... | This is an edge coloring of the complete graph $K_n$. For each color, the edges of that color will form one part of the partition.
Each color can only be used on $\lfloor \frac n2\rfloor$ edges. So we need at least $\binom n2/\lfloor \frac n2\rfloor$ colors: this is $n-1$ when $n$ is even, and $n$ when $n$ is odd.
This... | {
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Show that F vanishes at infinity. Suppose $1 ≤ p < ∞, f ∈ L^p(R)$, and
$F(x) = \int_{x}^{x+1} f(t) dm(t)$
Prove that F vanishes at infinity.
We know that $\int_R |f|^p < \infty$, then, of course, for any $x, F(x)< \int_x^{x+1} |f|^p < \infty$. But I want to show that not only is it finite, but it goes to zero as $x$ g... | Let $g_n(t) = |f(t)|^p\cdot \chi_{[-n,n]}(t)$. Note that $0\leqslant g_n(t)\nearrow |f(t)|^p$ pointwise a.e. as $n\to\infty$. By the Monotone Convergence Theorem,
$$
\int g_n(t)\,dt \to \int |f(t)|^p\,dt \quad\text{as $n\to\infty$}.
$$
Hence the difference
$$
\int|f(t)|^p\,dt - \int_{-n}^n|f(t)|^p\,dt \geqslant \int_n... | {
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Median divisor of even perfect numbers I noticed that when divisors of even perfect numbers are listed in ascending order, the middle divisor (I guess the median), is always of the form $2^n$, some power of 2. If true is there a proof for this, or does it happen all the time? I only checked up to the 8th perfect number... | Every even perfect number is of the form $n=2^{p-1}(2^p-1)$ with $2^p-1$ prime.
Therefore, the factors of $n$ are
$1, 2, 2^2, 2^3, ..., 2^{p-1}, 2^p-1, 2(2^p-1), 2^2(2^p-1), 2^3(2^p-1), ...2^{p-2}(2^p-1),$
and the middle one is $2^{p-1}.$
| {
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If $p^4 | z^2$, then $p^2 | z$ for $p$ a prime and $z$ some positive integer. Problem: Suppose that $p$ is a prime, and $z$ is some positive integer. If $p^4 | z^2$, then $p^2 | z$.
Thoughts:
If for some positive integer $a$, that $p^4 a = z^2$, then necessarily
$p^2 \sqrt{a} = z$, so that if my desired conclusion i... | $\dfrac{z^2}{p^4} = n\in\Bbb Z\ $ so $\ x = \dfrac{z}{p^2}\Rightarrow\,x^2= n\,$ $\Rightarrow\,x\in\Bbb Z\,$ by the Rational Root Test.
| {
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Is it the right way to determine the coefficients? I am having trouble with understanding the logical completeness of a solution of exercise on my textbook (Linear Algebra Done Right).
we have $Tp=(bp(1)p(2),c\sin p(0))$ &$p\in P(R)$. Prove that if T is a linear map, then $b=c=0$. It answers:
consider $f(x)=\pi/2$ and... | A proof is either correct or not. There's no issue of danger.
I dare say mathematics is not dangerous. Nobody ever cut himself on an integral (though metaphorically, yes, to be sure).
One might come up with a simpler, more elegant proof, say; but I don't think it's very likely in this case.
Mathematicians ten... | {
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solve $(3x-1)^{x-1}=1$ Q:How do I solve for $x$? $$(3x-1)^{x-1}=1$$
When I first saw this question I thought it was quadratic equation, $(3x-1)(x-1)=1$. But friend say no! It is raised to the power of $x-1$. What?!!
I could take the log
$$(x-1)\log(3x-1)=0$$
$\log(3x-1)=0$
$3x-1=1$
$x=\frac{2}{3}$.
There is a problem, ... | Hint:$$(x-1)\log(3x-1)=0\implies \begin{array} xx-1=0\\ 3x-1=1\end{array}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3117457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Numbers with exactly 1 square (prime) factor I have recently learned that numbers with no square (prime, assumed in the following) factor are called square-free numbers. I have read that it would asymptotically grows towards
$$\#\{SquareFree\} under\ n = \frac{6n}{\pi^2}$$
I am curious and am wondering if there's a sim... | Here I would proudly present to you the result that I and some friends have actually got after doing more math on it. Would someone read it and check if it's correct?
First, it is known that the number of square-free is
$$\sum_{i=1}^{\sqrt{n}}\mu{(i)}\left\lfloor{\frac{n}{i^2}}\right\rfloor$$
Where $\mu$ is the mobius... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3117573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Hahn-Banach needed to show equality? Let $X$ be a normed space and $x_1, x_2 \in X$. Suppose $x^{\ast}(x_1) = x^{\ast}(x_2)$ for all $x^{\ast} \in X^{\ast}$. Then $x_1 = x_2$.
Do we need Hahn-Banach (hence, equivalently some sort of axiom of choice) to prove this?
| For finite dimensional vector spaces we do not need Hahn Banach. Let $X$ be a finite dimensional space with basis $\{e_1,\dots,e_n\}$, and $\{f_1,\dots,f_n\}$ be the usual basis for $X^*$. Let $x_2=\sum_{i=1}^n\alpha_ie_n$ and $x_1=\sum_{i=1}^n\beta_ie_n$, and suppose $f(x_1)=f(x_2)$ for all $f\in X^*$. In particular t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3117800",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that $\cos^220^\circ-\cos20^\circ\sin10^\circ+\sin^210^\circ=\frac34$ The original exercise is to
Prove that
$$4(\cos^320^\circ+\sin^310^\circ)=3(\cos20^\circ+\sin10^\circ)$$
Dividing both sides by $\cos20^\circ+\sin10^\circ$ leads me to the problem in the question title.
I've tried rewriting the left side in te... | We need $$4\cos^220^\circ-4\cos20^\circ\sin10^\circ+4\sin^210^\circ=3$$
$$\iff2(1+\cos40^\circ)-2(\sin30^\circ-\sin10^\circ)+2(1-\cos20^\circ)=3$$
$$\iff\cos40^\circ-\cos20^\circ=-\sin10^\circ$$
which is evident from Prosthaphaeresis Formulas
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3117979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
How to prove that $\left(\ln(\ln(x)) \right)^2 \lt \ln(x)$ How to prove that $\left(\ln(\ln(x)) \right)^2 \lt \ln(x)$ for sufficiently large $x$
This is what I did. Using L'Hopital's rule we have
$$\lim_{x\to\infty}\frac{\left(\ln(\ln(x)) \right)^2 }{ \ln(x)}=0$$
So this implies that $\left(\ln(\ln(x)) \right)^2 \lt ... | Hint:
Star from an inequality traditionally used in high-school to prove that $\;\lim_{x\to+\infty}\dfrac{\ln x}x=0$:
$$\ln x<\sqrt x\quad\forall x>4,$$
and replace $x$ with $\ln x$: if $\ln x>4$, then
$$\ln(\ln x)<\sqrt{\mkern1mu\ln x\mathstrut}$$
Note that, as $x>4>\mathrm e$, both sides of the inequality are positiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3118145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 1
} |
Notation for "Defined as proportional to" If I say $x$ is defined as $y+z$ then I can say $x := y+z$. If I want to say $x$ is defined as proportional to $y+z$, then how can I say that? Would I say $x :\propto y+z$?
| The symbol $\propto$ is indeed meaning that. It means that there exists $k$ a constant in a $\mathbb{K}$ field so that $x=k\cdot (y+z)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3118241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Tail bound of sum less than sum of tail bounds
Problem: Suppose we have a probability space $(\Omega, \mathcal{M}, P)$, a random variable $X$ on this space, and two nonnegative measurable functions $f,g:\mathbb{R}\to[0, \infty)$. Choose some $\epsilon > 0$. Prove that
$$P(f(X)+g(X)>\epsilon) \le P(f(X)>\epsilon/2) ... | Observe that
$$ \{ \nu \in \Omega \mid f(X)(\nu) + g(X)(\nu) > \epsilon \} \subset \{ \nu \in \Omega \mid f(X) (\nu) > \tfrac \epsilon 2\} \ \cup \ \{ \nu \in \Omega \mid g(X) (\nu) > \tfrac \epsilon 2\}. $$
So
\begin{align*} P\left( \{ \nu \in \Omega \mid f(X)(\nu) + g(X)(\nu) > \epsilon \}\right) & \leq P \left( \{ \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3118302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Probability of an event occurs at least three times before another event occurs cars arrives according to a Poisson process with rate=2 per hour and trucks arrives according to a Poisson process with rate=1 per hour. They are independent.
What is the probability that at least 3 cars arrive before a truck arrives?
My... | Let $M_t$ be the Poisson process which counts the arrival of trucks. Then by the given condition we have $(M_t)_{t>0} \sim PP(1).$ Let $X_1,X_2, X_3, \cdots$ be the time gaps between arrival of cars. Then $X_n \sim \text{iid} \exp (2).$ So the required probability is $P(M_{X_1+X_2+X_3} < 1).$
Now $$\begin{align}
P(M_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3118462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How many ways can n identical balls be distributed into k distinct boxes, such that at least one box is empty? This is a problem in my combinatorics book that uses the principle of inclusion-exclusion. I can follow almost all of what is said, except the book says that if we consider $A_{i}$ to be the set of solutions w... | To begin with, I'd solve the problem differently: By a stars-and-bars, there are $n-1\choose k-1 $ ways to place $n$ balls into $k$ bins such that no bin is empty. Subtract this from the $n+k-1\choose k-1$ ways to place $n$ balls into $k$ bins without restriction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3118564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Lagrange multipliers with $\lambda = 0$ problem We want to maximize the function $f(x_1,x_2,x_3,x_4) = \sum_{i=1}^{4}a_i^2x_i$ over the compact set $\Omega = \{x \in \mathbb R^4: |x| = 1, \langle x,a\rangle = 0\}$ where $a = (a_1,a_2,a_3,a_4)$ is some non zero vector
This is a continuous function over a compact set so ... | Hint If $\lambda_1 = 0$ then $a_i^2 = \lambda_2 a_i$ which implies $\lambda_2 = a_i \Rightarrow a = (\lambda_2, \lambda_2, \lambda_2, \lambda_2)$, or $\lambda_2 = 0$, and you should be able to compute a solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3118893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
$\frac{\int fg dx}{\int g dx}=f(0)$ We already know that
$$\lim_{n \rightarrow +\infty} \int_{-1}^1 (1-x^2)^n dx = 0$$
If we have $f(x) \in C[-1,1]$ then prove
$$\lim_{n \rightarrow +\infty} \frac{\int_{-1}^1 f(x)(1-x^2)^n dx}{\int_{-1}^1 (1-x^2)^n dx } = f(0)$$
My thought is $\lim\limits_{n \rightarrow +\infty} \frac{... | You could observe (by calculus) that $g_n(x) = (1-x^2)^n / \int_{-1}^1 (1-u^2)^ndu$ is the density function for a random variable $X_n$ with expectation $0$ and with variance tending to 0 as $n\to\infty$. (This last by checking $\int_{-1}^1x^2g_n(x)dx\to0$, which is a Beta function calculation.) Hence $X_n\to0$ in pro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3118986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Confusion Dividing A Fraction with a Whole Number... In the lesson I am doing, I divide fractions. Here is my problem:
28/55 / 7
I had to look up how to do this problem. According to Math Is Fun, you divide the denominator by the whole number and then simplify if possible. I did this, and got:
28/385
I couldn't simplif... | For a general purpose, suppose we need to find $y=\frac{\frac{a}{b}}{\frac{c}{d}}$ where $\frac{c}{d}\not=0.$ Then we can do the following -
$$y\frac{c}{d}=\frac{a}{b}\implies \frac{c}{d}=\frac{\frac{a}{b}}{y}\\ \implies \frac{d}{c}=\frac{y}{\frac{a}{b}}\\ \implies y=\frac{d}{c} \times \frac{a}{b}.$$
For your problem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3119074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Dimension of global sections Let $X$ be a hypersurface of degree $d$ in a projective space of dimension $n$. Is there a formula which expresses the dimension of the dimension of the space of global sections of the line bundle $O(k)$ on $X$?
| If $n = 1$, then $X$ is a union of $d$ points, so $\mathcal O_{X}(k) = \mathcal O_X$, which has $d$ linearly-independent global sections.
If $n \geq 2$, then consider the short exact sequence,
$$ 0 \to \mathcal O_{\mathbb P^n} (-d) \to \mathcal O_{\mathbb P^n} \to \mathcal O_X \to 0.$$
Tensoring with $\mathcal O_{\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3119260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Use of the product and quotient rule in differentiation In the process of rounding up the product and quotient rule, I got confused when my textbook said that the product rule should not be used if one of the factors of the product is a constant, and the quotient rule should not be used if the denominator is a single t... | When on of the term is a constant, it is equivalent to using linearity to factor out the constant. Therefore, it is unnecessary use of product and quotient rule, but not false.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3119338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
"A Simple Proof of Zorn's Lemma" article. (explanation of a step) There is a short proof of Zorn's lemma by J.W.Lewin:
A Simple Proof of Zorn's Lemma.pdf
Summary:
I don't understand this step:
"Therefore if z is the least member of
$A\setminus P(B, y)$, we have $P(A, z) = P(B, y)$."
I'm interested in both inclusions f... | Here I want to clarify part of the
@Matematleta's proof which took some of my time.
On the other hand, if $<$, then since $\in (,)$, we have
$\in (,)$, which is a contradiction.
We assume that $<$.
1) From $\in A\setminus (,)$, we obtain $z\in A \land [z\notin B\lor z\geqslant y]$
2) The second OR-case $z\geqslan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3119417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Constructing an isomorphism to show two groups are isomorphic For each $n\geq 2$, consider $C_n = \{ (a,b) \in \mathbb{Z}^2 : a \equiv b \mod \: n\}$. I want to show that $C_n$ is isomorphic to $\mathbb{Z} \times \mathbb{Z}$. To do this, I know I need to construct a bijection that preserves products as in the definitio... | The bijection $f:C_n\to\mathbb Z^2$ may be constructed as follows:
$$f(a,b)=\left(a,\frac{b-a}n\right)$$
Its inverse is
$$f^{-1}(a,b)=(a,bn+a)$$
$f$ is a homomorphism because
$$f(a,b)+f(c,d)=\left(a,\frac{b-a}n\right)+\left(c,\frac{d-c}n\right)=\left(a+c,\frac{(b+d)-(a+c)}n\right)=f(a+c,b+d)$$
$f$ is an injection: supp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3119576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
We have an urn with 6 red balls and 4 green balls. We have an urn with 6 red balls and 4 green balls. We draw
balls from the urn one by one without replacement, noting the order of the
colors, until the urn is empty. Let X be the number of red balls in the first
five draws, and Y the number of red balls in the last fiv... | The simplest approach is to observe that $X + Y = 6$, as there are only $10$ balls in the urn, hence one is guaranteed to have drawn all $6$ red balls. Thus $$\begin{align*}
\operatorname{Cov}[X,Y]
&= \operatorname{Cov}[X, 6-X] \\
&= \operatorname{E}[X(6-X)] - \operatorname{E}[X]\operatorname{E}[6-X] \\
&= \operatorn... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3119677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How do I evaluate this indefinite integral? I am currently working on the following problem:
$\int x (2x+3)^{99}$
I have tried using u-substitution $(u = 2x+3)$ and integration by parts, but have not been able to make any progress that leads me to an answer. I thought about actually computing $(2x+3)^{99}$, but I th... | You have the right idea, but just split the result into $2$ integrations after you make the substitution. In particular,
$$u = 2x + 3 \Rightarrow du = 2dx \Rightarrow dx = \frac{du}{2} \tag{1}\label{eq1}$$
Also,
$$x = \frac{u - 3}{2} \tag{2}\label{eq2}$$
Thus,
$$\int x\left(2x + 3\right)^{99}dx = \int \frac{u - 3}{4}u^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3119798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Finding an explicit formula for a Hamiltonian vector field I've been looking at this question: Existence of vector field given a smooth function
That is: Given a symplectic manifold $M$ of dimension $2n$, with a symplectic form $\omega \in \Omega^2(M)$, do we have for all smooth function $f\in C^\infty(M)$ a vector fie... | My attempt at an answer using the help given by user3257842:
Let $\{ e_1(p),....,e_{2n}(p) \}$ be a basis for $T_p(M)$, orthonormal with respect to the Riemannian metric $\rho$. Define the representing matrix of $\omega_p$ to be $B(p)\in \mathbb{R}_{2n \times 2n}$, by: $ B_{i,j}(p):= \omega_p \Big( e_i, e_j \Big) $
We... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3119919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Injective and surjective homomorphisms between non cyclic group of order $4 $ to $Z_8$
Let $G$ be a non cyclic group of order $4$. Consider the following
statements:
$I:$ There is no one-one map homomorphism from $G$ to $Z_8$
$II:$ There is no onto homomorphism from $Z_8$ to $G$
Then which of these statements are tr... | I think your arguments are correct, but can be shortened.
Part I: Suppose that there is such a homomorphism $\phi$. Then
$\phi(G)\cong G$ is a subgroup of $C_8$, hence cyclic, a contradiction.
Part II: Suppose that $\phi$ is such a homomorphism. Then $\phi(C_8)\cong G$, where $\phi(C_8)$ is cyclic, because $C_8$ is cy... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3120116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Square root of a specific matrix in M(3,Z) over Z Let
$ \begin{bmatrix}
a&b&c \\
d&e&f\\
g&h&i\\ \end{bmatrix}^2 = \begin{bmatrix}
x&0&0\\
0&0&-y\\
0&1&-z\\ \end{bmatrix}
$
Where all of the components are integer.
I am trying to figure out what will be the condition of $a,b,c,d,e,f,g,h,i$ or $x,y z$. so that the equa... | Yeah, so a very normal approach to solve this sort of equation is to generalize to the reals and diagonalize. Suppose your matrix can be factored as $$ M = CDC^{-1}, $$ for $D$ diagonal, then its square can be factored as $$M^2 = C D C^{-1}CDC^{-1} = C D^2 C^{-1},$$so the same coordinate-change matrix $C$ is used for t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3120310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Open balls in $p$-adic numbers. I am new to $p$-adic numbers and was watching an introductory video about it. At 14:51, he says that $r$ only takes values in the form of $p^n$. However, I don't understand why $r$ must be restricted to numbers in the form of $p^n$. Given any r that is not in the form of $p^n$, there wil... | As in any metric space, you can define an open ball of radius any real number. But if the metric only takes values that are powers of $p$, we might as well restrict attention to balls of radius a power of $p$. I suspect that this was precisely the point the video was trying to make.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3120565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why do we divide Permutations to get to Combinations? I'm having a hard time reasoning through the formula for combinations $\frac{n!}{k!\left(n-k\right)!}$. I understand the reason for the permutations formula and I know that for the combinations we divide by $k!$ to adjust for the repeated cases, since order does not... | You divide by k! since that's the number of permutations for the k objects that you've taken. All the permutations/orders of each set of k objects is equivalent since you're dealing with combinations, so you have to divide that out.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3120653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 12,
"answer_id": 9
} |
Wave equation on a disk (circular membrane) Solve wave equation in a disk, axisymmetric case
$$\begin{cases}
\frac{\partial^2u}{\partial t^2}=\frac{c^2}{r}\frac{\partial}{\partial r}(r\frac{\partial u}{\partial r}) \,\,\, \,,0<r<a\quad,t>0\\
u(r,0)=f(r),\quad\frac{\partial u}{\partial t}(r,0)=g(r),\quad u(a,t)=0
\end{c... | The BVP
$$ r^2R'' + rR' - \lambda r^2 R = 0, \quad R(0) < \infty, R(a) = 0 $$
only has a non-trivial solution when $\lambda < 0$.
You can check that $\lambda = 0$ returns a general solution of $A+B\ln r$, and $\lambda > 0$ returns modified Bessel functions, neither of which will satisfy the boundary conditions.
The su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3120756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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