Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Find $P(X^2+Y^2<1)$ if $X,Y$ are independent standard normal variables Suppose that $X$ and $Y$ are independent $n(0,1)$ random variables.
(a) Find $P(X^2+Y^2<1)$
My solution:
since then are independent, the $F_{X,Y}(x,y)=F_X(x)F_Y(y)=\dfrac{1}{2\pi} e^{\frac{-(x^2+y^2)}{2}}$
then $P[X^2+Y^2]=\int\int F_{X,Y}$
I am not... | If X~N(0,1), then the pdf of $X^2$ is not the square of a normal pdf. Even if you integrate what you have, you'll get the wrong answer. In fact, $X^2$ has a chi-squared distribution with one degree of freedom.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1998691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Let $X$ be a variety such that $\forall x,y\in X$ there exists an open affine $U\in X$ containing $x,y$. Then $X$ is separated I am struggling to prove this. Note that I do not know anything about schemes, so please no schemes.
I know that in order to show $X$ is separated I need to show that $\Delta_{X}=\{(x,x)\in X\... | You want to show that the complement of $\Delta_X$ is open in $X\times X$. Let $x\ne y$, then $(x,y)$ is in the complement of $\Delta_X$, so we want to produce an open neighborhood of $(x,y)$ not intersecting $\Delta_X$. If $x,y\in U$, then it suffices to show that $(U\times U) - \Delta_X = (U\times U) - \Delta_U$ is o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1998787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to identify the pattern of a sequence
Are there some particular methods for identifying the following types of number series?
*
*$6, 10, 19, 44, 93, \cdots$ (Difference being prime no's square starting from 2)
*$1, -2, 15, 52, -512, \cdots $ ( $^*2-4,\ ^*-6+3,\ ^*4-8,\ ^*-10+5$, and so on)
*$4, -2, -7, 25, 9... | You ask
is there any generalized mathematical theorems on these types of
number series?
I'll risk an unsatisfactory answer too long for a comment: essentially, "no".
The sequences school kids work on often come from arithmetic or geometric series, which is probably why you suggest trying them first. But there are... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1998900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Calculate commission percentage by amount In our system, we got a booking of the price 1100, the default commission percentage is 15% hence the raw price is
1100 / 1.15 = 956.52173913
Which means that the current commission amount is
1100 - 956.52173913 = 143.47826087
I need to change the commission percentage of thi... | More general. The following formula can be solved for $x$.
$P\cdot \left(1-\frac{1}{1+x}\right)=C$
with $P=$ gross price, $x$=comission rate and $C$=commission
Multipying out the brackets
$P-\frac{P}{1+x}=C$
$P-C=\frac{P}{1+x}$
Interchanging numerators and denominators
$\frac{1}{P-C}=\frac{1+x}{P}$
$\frac{P}{P-C}=1+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1999024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does $E(X_n)\rightarrow E(X)$ as $n\to\infty$ If there is a sequence of random variables ${X_n}$ converging almost surely to $X$, therm is it true that $E(X_n)\rightarrow E(X)$ as $n\to\infty$ ? Only thing given is that $E(X_n)\le 23$ for all $n$.
I am not getting how to do it. I can't use DCT here, can I?
| No it's not.
Consider: $$X_n(\omega) = \begin{cases}
n & \mbox{ if } \omega\in [0,\frac{1}{n}] \\
0 & \mbox{ otherwise}
\end{cases}$$
Then $X_n \to X$ where $X \equiv 0$, $E[X_n] = 1 \le 23$, so $\lim\limits_{n\to\infty} E[X_n] = 1$ but $E[X] = 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1999156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving this modulo question Let a,b be integers and p is prime
$a^pb^{p^2} \equiv 0\pmod p \Rightarrow a \equiv 0\pmod p$ or $b \equiv 0\pmod p$
and using elemination method
$a^pb^{p^2} \equiv 0\pmod p$ and $a \not\equiv 0\pmod p \Rightarrow b \equiv 0\pmod p$
from here
since $a \not\equiv 0\pmod p \Rightarrow p \... | $Z/p$ is a field so $a^pb^{p^2}=0$ mod $p$ implies that $a^p=0$ mod $p$ or $b^{p^2}=0$ mod $p$.
If $a^p=0$ $mod$ $p$, Little fermat implies that $a^p=a$ mod $p$ done.
If $b^{p^2}=0$ mod $p$, you have $b^{p^2}=(b^p)^p=b$ mod $p$ by little fermat.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1999544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Slopes and lines Given two lines having slope $m_1$ and $m_2$, the angle between them is given by $\tan(\theta)=\frac{m_2- m_1}{1+m_1m_2}$
Does the order of $m_1$ and $m_2$ matter here, and if so what is the significance? Graphical aid would be helpful.
| Note that $m_1=\tan \theta_1$ and $m_1=\tan \theta_2$ are the tangents of the angles $\theta_1$ and $\theta_2$ between the lines and the $x$ axis. So:
$$
\frac{m_2-m_1}{1+m_1m_2}=\frac{\tan \theta_2-\tan \theta_1}{1+\tan \theta_1\tan \theta_2}=\tan (\theta_2-\theta_1)
$$
and:
$$
\frac{m_1-m_2}{1+m_2m_1}=\frac{\tan \the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1999702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
preimages under group homomorphism Let $\varphi : G \rightarrow H$ be a group homomorphism with kernel $K$ and let $a,b \in \varphi(G)$. Let $X = \varphi^{-1}(a)$ and $Y = \varphi^{-1}(b)$. Fix $u \in X$. Let $Z=XY$. Prove that for every $w \in Z$ that there exists $v \in Y$ such that $uv=w$. This is Dummit and Foote e... | Since $w\in Z$, you know that $w=xy$, for some $x\in X$ and $y\in Y$.
By definition, $\varphi(x)=a$ and $\varphi(y)=b$.
Also $\varphi(u)=a$, which implies $u^{-1}x\in\ker\varphi$.
Then
$$
w=xy=u(u^{-1}xy)
$$
Can you finish?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1999834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
} |
Solving $\sin z = i$ I know that
$$\sin z = \frac{e^{iz}-e^{-iz}}{2i}$$
so:
$$\frac{e^{iz}-e^{-iz}}{2i} = i\implies e^{iz}-e^{-iz} = -2$$
but I can't take anything useful from here. How do I solve such equations?
What about $\tan z = 1$? Is there any solutions?
| $\sin z = \cos z $ comes down to ..
$$ e^{iz}-e^{-iz} = i( e^{iz} + e^{-iz} ) $$
$$ (1-i)e^{iz}= (1+i)e^{-iz} $$
$$ e^{2iz}= \frac{ 1+i}{1-i} =i=e^{i( \frac\pi2+2n\pi )}$$
So $$z=\frac\pi4 +n\pi$$
All solutions are real.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1999924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
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Is the Post Office Metric applicable in $\Bbb{R}^n$ for all $n$? I was required to provide a metric space $(X,d)$ with $x,y\in X$ and $0<r<R$ such that $B_R(x)\subsetneq B_r(y)$. After a lot of thinking and reading, I came by a metric function called the "Post Office Metric", always attributed to $\Bbb{R}^2$, in partic... | The idea is fundamentally similar to Mike F's answer, but perhaps simpler or at least more common in analysis.
Consider $X=[0,1]$ with the induced Euclidian metric. Let $x=0$; then $B_R(x)=[0,x)$. Take say, $R=1/2$.
Then take, say, $y= 1/3$ and $r=2/5$, we have that $B_r(y)=[0,11/15)\simeq [0,0.73)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2000058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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If $f$ is differentiable in $B(a)$ and $f(x) \leq f(a)$ for all $x$ in $B(a)$, then $\nabla f(a) = 0$
Assume $f$ is differentiable at each point of an n-ball $B(a)$. Prove that if $f(x) \leq f(a)$ for all $x$ in $B(a)$, then $\nabla {f(a)} = 0.$
I had my proof, but I'm not sure it is correct.
Proof:
Since f is differ... | You never used the fact that $f(x)\leq f(a)$. I may be missing something, but it seems that your proof would imply that all differentiable functions have this property.
My proof is the following: Compute
$$\lim_{h\to 0 } \frac{f(a+hy)-f(a)}{h} = \nabla f(a) \cdot y,$$
and note that $f(a+hy)\leq f(a)$ for all $h$ suffic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2000124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Help needed showing that $f(x,y)=U(x+y)+V(x-y)$ Let $h(u,v)=f(u+v,u-v)$ and $f_{xx}=f_{yy}$ for every $(x,y)\in\mathbb{R}^2$. In addition, $f\in{C^2}.$ Show that $f(x,y)=U(x+y)+V(x-y)$.
I think applying the Taylor theorem could be useful.
$$f(x,y)=f(x+h_1,y+h_2)-\left(\frac{\partial{f(x,y)}}{\partial{x}}h_1+\frac{\part... | This is a d'Alembert form solution for the hyperbolic PDE
$$
f_{xx} - f_{yy} = 0
$$
One changes to variables
$$
\xi = x - y \\
\eta = x + y
$$
and uses the chain rule to get
$$
\frac{\partial f}{\partial x} =
\left(
\frac{\partial f}{\partial \xi}
\right)
\frac{\partial \xi}{\partial x}
+
\left(
\frac{\partial f}{\pa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2000254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Calculate $\lim\limits_{n \to \infty} \frac1n\cdot\log\left(3^\frac{n}{1} + 3^\frac{n}{2} + \dots + 3^\frac{n}{n}\right)$
Calculate $L = \lim\limits_{n \to \infty} \frac1n\cdot\log\left(3^\frac{n}{1} + 3^\frac{n}{2} + \dots + 3^\frac{n}{n}\right)$
I tried putting $\frac1n$ as a power of the logarithm and taking it ou... | Hint. $$0<3^{\frac n 2}+\cdots+3^{\frac n n}\le(n-1)3^{\frac n 2}=\frac{3^n}{3^{\frac n 2}/(n-1)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2000378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
can a Car Registration Number, a combination of prime, be prime? While waiting in my car, I noticed registration number of a car parked in front of my car was 6737. So it was a concatenation of two prime numbers 67 and 37.
Now I know following ways to check whether any number is prime or not
Let $p$ be the number to be... | This is true for the prime numbers 3, 7, 109 and 673 that if you concatenate any two of these numbers in any order , the resulting number will be a prime ; as in this case Concatenating 7 at the end of 673 resulting in 6737 which is a prime . Concatenating 7 in the front of 673, which gives 7673 , is also a prime . So... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2000483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
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How is a morphism different from a function How is a morphism (from category theory) different from a function?
Intuitive explanation + maths would be great
| If $G$ is a group, there is a famous example that constructs a category in which morphisms are not functions: you take it to consist of a single object called "$\bullet$" and state that the morphisms (of $\bullet$ to $\bullet$, because there is no other object available) are the elements of $G$. Check for yourself that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2000595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Show that $f$ is unbounded below
Let $f:\Bbb R\to \Bbb R$ be continuous and satisfies $|f(x)|\ge |x| $ for all $x$.Also $f(x+y)=f(x)+f(y)$ for all $x,y$.Show that $f$ is bijective.
My try:
$f$ is injective ;since $f(x)=f(y)\implies f(x-y)=0\implies |x-y|\le 0\implies x=y$.
To show that $f$ is onto.
Every continuous i... | From $f(x+y)=f(x)+f(y)$ and continuity you get
$f(x)=cx$
https://en.wikipedia.org/wiki/Cauchy's_functional_equation
From $|f(x)|\ge |x|$ you get $c\ne 0$
So $f$ is bijective.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2000660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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For which values of positive integer k is it possible to divide the first 3k positive integers into three groups with the same sum? I'm on a GCSE-a level syllabus currently, and I can't seem to think of any algebraic equation that I could comprise to solve this (with the GCSE/early a level syllabus). The question in fu... | The sum of the first $3k$ numbers is $\frac 12(3k)(3k+1)$, so we want three groups that sum to $\frac 12k(3k+1)$. Clearly $k=1$ doesn't work because the desired sum is $2$ and we have a $3$ which is too big. $k=2$ does work as we have $7=1+6=2+5=3+4$. Intuitively, as we have more numbers we have more freedom, so exp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2000768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Prove that a graph on $n$ vertices with girth $g$ has at most $\frac{g}{g-2}(n-2)$ edges I'm not sure how to go about this. One of my thoughts was to take the number of edges in a complete graph, and subtract all of the edges that would be needed to make a smaller cycle.
If you have $C_5$, there is only one way to mak... | This isn’t true as stated: the Petersen graph has $10$ vertices, $15$ edges, and girth $5$, and
$$15>\frac{40}3=\frac53(10-2)\;.$$
It is true for planar graphs.
HINT: Use Euler’s formula, $v-e+f=2$, where $v,e$, and $f$ are the numbers of vertices, edges, and faces, respectively, of a planar graph when it is embedded ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proof of the disjunction property I am trying to prove the disjunction property "if $\,\vdash\phi\lor\psi$ then $\,\vdash\phi$ or $\,\vdash\psi$" for intuitionistic propositional logic.
So far I thought about choosing two non-tautologies $\phi$ and $\psi$ with two Kripke countermodels:
$$(W, R_W, f_W)~~\mbox{such that}... | Yes, your proof works; you have only to specify that the two countermodels must have disjoint frames $\langle W_1, R_1 \rangle$ and $\langle W_2, R_2 \rangle$.
The new model will have $W = \{ w_0 \} ∪ W_1 ∪ W_2$ and the "extended" accessibility relation will be :
$xRy$ iff $x=w_0$ or $xR_1y$ or $xR_2y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2000978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Rolles Theorem - problem with interval I am trying to show that there is only one solution for the equation $x-\cos x = 1$ in the interval $]0,\frac{\pi}{2}[$.
Using $f(x) = x-\cos x -1 = 0$, I took the derivative $1 + \sin x$.
Now I would expect to find solutions for $1 + \sin x = 0$ within the interval, but the next... | Rolles theorem does not apply here : For Rolles theorem you need two distinct real numbers $a$ and $b$ with $\ f(a)=f(b)\ $. Here, no such pair within the interval $\ [0,\frac{\pi}{2}]\ $ exists.
The correct way is using $\ f'(x)=1+\sin(x)>0\ $ to show that there is at most one solution and looking at the signs of $f(0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2001114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Multivariable Calculus Help with Laplacian in Polar coordinates I am trying to see why
$\big(\partial_{xx} + \partial_{yy}\big) u(r, \theta) = u_{rr} + \frac{1}{r}u_r + \frac{1}{r^2}u_{\theta\theta}$
I first use the chain rule to say that:
$\frac{\partial u}{\partial x} = u_r r_x + u_{\theta} \theta_x$
And then I calc... | Let's start with
$$
\begin{cases}
x=r\cos(\theta)\\
y=r\sin(\theta).
\end{cases}
$$
We compute first $u_r:$
$$
u_r=u_xx_r+u_yy_r=\cos\theta u_x+\sin\theta u_y.
$$
$$
u_{rr}=\cos\theta u_{xr}+\sin\theta u_{yr}=\cos\theta (u_{xx}x_r+u_{xy}y_r)+sin\theta(u_{xy}x_r +u_{yy}y_r)=\\
=\cos^2\theta u_{xx}+2\cos\theta\sin\theta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2001216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Coefficients of a formal power series satisfying $\exp(f(z)) = 1 + f(q\,z)/q$ Let $(q;\,q)_n$ denote the $q$-Pochhammer symbol:
$$(q;\,q)_n = \prod_{k=1}^n (1 - q^k), \quad(q;\,q)_0 = 1.\tag1$$
Consider a formal power series in $z$:
$$f(z) = \sum_{n=1}^\infty \frac{(-1)^{n+1}P_n(q)}{n!\,(q;\,q)_{n-1}}z^n,\tag2$$
where ... | Using the first recurrence relation here, we can find a recurrence for the polynomials $P_n(q)$:
$$P_1(q) = 1, \quad P_n(q) = \sum_{k=1}^{n-1} {{n-1} \choose {k-1}} {{n-2} \brack {k-1}}_q P_k(q) \, P_{n-k}(q) \, q^{n-k-1},$$
where $n \choose k$ is the binomial coefficient, and ${n \brack k}_q$ is the $q$-binomial coeff... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2001325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
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Is the axiom of choice needed to prove $|G/H| \times |H| = |G|$ (Lagrange's Theorem)? Consider the following sequence of assertions, each of which implies the next. Nothing below has any topology, we just have sets and discrete groups.
*
*If $F \hookrightarrow E \twoheadrightarrow B$ is fibre bundle of sets (this j... | Yes, the axiom of choice is needed, to some extent.
As bof mentions in the comments, it is always the case that $\Bbb{ Q\hookrightarrow R\twoheadrightarrow R/Q}$. However, as explained by Andrés E. Caicedo on MathOverflow, it is consistent that $\Bbb{R/Q}$ cannot be linearly ordered, as a set. In that case, it is impos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2001438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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How to construct the set E invoving an almost constant function? Assume that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$, and for all $h\in \mathbb{R}$, the set $E_h=\{x:f(x+h)-f(x)\neq 0,x\in \mathbb{R}\}$ is a finite set which has no more than 2016 elements. Prove that there exists a set $E$ which has no mor... | Let $x_m=min(E_1)+1$ and $x_M=max(E_1)$.
We claim that $f(x)=f(x_m-1)$ for all $x<x_m$. In fact, assume by contradiction that $x<x_m$ and $f(x)\ne f(x_m-1)$. Then $f(x)=f(x-n)$ for all $n\in\Bbb{N}$, since otherwise $f(x-j)\ne f(x-j-1)$ for some $j\in\Bbb{N}_0$ and then $x-j-1\in E_1$, but $x-j-1<x_m-1=min(E_1)$. Simil... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $d \geq b+f$. Let $a, b, c, d, e, f$ be positive integers such that:
$$\dfrac{a}{b}<\dfrac{c}{d}<\dfrac{e}{f}$$
Suppose $af - be = -1$. Show that $d \geq b+f$.
Looked quite simple at first sight...but havent been able to solve this inequality. Have no idea where to start. Need help. Thanks!!
| Hint: Try to derive that $bf<d$. What can you conclude from there?
(Hint 2: $bf = (b-1)(f-1) + (b+f) - 1$.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2001685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $\mathbb{E}\left(\bar{X}_{n}\mid X_{(1)},X_{(n)}\right) = \frac{X_{(1)}+X_{(n)}}{2}$
Let $X_{1},\ldots,X_{n}$ be i.i.d. $U[\alpha,\beta]$ r.v.s., and let $X_{(1)}$ denote the $\min$, and $X_{(n)}$ the $\max$. Show that
$$
\mathbb{E}\left(\overline{X}_{n}\mid X_{(1)},X_{(n)}\right) = \frac{X_{(1)}+X_{(n)}}{... | After having used a translation and rescaling, we can assume that $\alpha=0$ and $\beta=1$. Assume that $1\lt i\lt n$. Denote $X_{(i)}$ the $i$th greater element among $X_1,\dots,X_n$ (which is almost surely well-defined, as the vector $\left(X_1,\dots,X_n\right)$ has a continuous distribution). Then for each Borel sub... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2001780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Every Real number is expressible in terms of differences of two transcendentals Is it true that for every real number $x$ there exist transcendental numbers $\alpha$ and $\beta$ such that $x=\alpha-\beta$?
(it is true if $x$ is an algebraic number).
| Suppose there is some real number x that can't be written as the difference of transcendental numbers. Then for every transcendental number y there exists a unique algebraic number z = y+x. But this means there is an injective function f(a) = a+x from the transcendental numbers to the algebraic numbers, which implies... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2001922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 5,
"answer_id": 2
} |
Discrete Math composition functions Give examples of sets $X$, $Y$, $Z$ and functions $f: X \to Y$, $g: Y \to Z$, so that
the composition $g\circ f: X \to Z$ is a bijection, although neither $f$ or $g$ it is.
I have no idea to begin thinking on amounts.
| WLOG we can take $Z = X$ and $g \circ f = Id$
If you can apply $f$ without "losing information", it's because $f$ is bijective on its image, i.e. injective.
Thus take Y bigger than X.
Then you just have to left invert $f$.
If you want to take $Z \neq X$ and $g \circ f \neq Id$, just left compose with a bijection from ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2002063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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What kind of pde is and how can I solve it? I could not find a category of PDE where I can classify this equation and I do not know how to begin to solve it.
$\Delta u - u = 1$ in $\Omega = [0,\pi]\times [0,\pi]$
with the boundary conditions:
$u = 0$ on $[0,\pi]\times\{0\}\cup[0,\pi]\times\{\pi\}$ and
$\frac{\partial u... | The conditions given result in a function $u(x,y)$ that is constant in $x$. That is, $u(x,y)=U(y)$. And,
$$
U''(y)-U(y)=1,\;\;\; U(0)=0,\; U(\pi)=0.
$$
That can be solved with ODE methods and the annihilator method
$$
D(D-1)(D+1)U=0.
$$
The general solution is
$$
U(y)=-1+Be^{y}+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2002140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Correct formulation of axiom of choice In a paper (Wayback Machine), Asaf Karagila writes:
Definition 1 (The Axiom of Choice). If $\{A_i
\mid i ∈ I\}$ is a set of non-empty sets, then there exists
a function $f$ with domain $I$ such that $f(i) ∈ A_i$
for all $i ∈ I$.
Does this formally make sense? Shouldn't it sa... | It does make sense subject to a generous interpretation of the role of the indexing set $I$ and the indexing function $i \mapsto A_i$. It would be much better style (in my opinion) either to write it as you suggested, stating explicitly that the indexing function $i \mapsto A_i$ is part of the data or to write it witho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2002278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Steady State Differential Equation I am trying to find the steady-state solution of the following ODE:
$$(D^2+D+4.25I)y=22.1\cos(4.5t)$$
The answer from the back of my textbook is:
$$y_p = -1.28\cos(4.5t)+0.36\sin(4.5t)$$
I found
$$\begin{align}
y_p&=k_1\sin(4.5t)+k_2\cos(4.5)t\\
y_p'&=4.5k_1\cos(4.5t)-4.5k_2\sin(4.5t)... | You are almost there, we choose the particular solution:
$$y_p(x) = a \cos(4.5 t) + b \sin(4.5 t)$$
We take:
$$(D^2 + D + 4.25 I)y_p = y_p'' + y_p' + 4.25 y_p = 22.1 \cos(4.5t)$$
Using your derivatives and adding all these terms and simplifying, we get:
$$(-4.5 a-16 b) \sin (4.5 t)+(-16 a+4.5 b) \cos (4.5 t) = 22.1 \co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2002380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Fibonacci Identity with Binomial Coefficients A friend showed me this cool trick: Take any row of Pascal's triangle (say, $n = 7$):
$$1, 7, 21, 35, 35, 21, 7, 1$$
Leave out every other number, starting with the first one:
$$7, 35, 21, 1$$
Then these are backwards base-5 "digits", so calculate:
$$7 + 35 \cdot 5 + 21 \cd... | To cancel out the binomial coefficients with even bottom indices, the sum can be written as
$$
\begin{align}
\frac1{2^{n-1}}\sum_{k=0}^n\binom{n}{k}\frac{\sqrt5^k-\left(-\sqrt5\right)^k}{2\sqrt5}
&=\frac{\left(\frac{1+\sqrt5}2\right)^n-\left(\frac{1-\sqrt5}2\right)^n}{\sqrt5}\\
&=F_n
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2002702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 4,
"answer_id": 3
} |
What is the general formula for a convergent infinite geometric series? This question is related, but different, to one of my previous questions (Does this infinite geometric series diverge or converge?). To avoid the previous question getting off-topic, I have created a separate question.
I'm looking for the general f... | In my opinion, the simplest way to memorize the formula is
$$ \frac{\text{first}}{1 - \text{ratio}} $$
So whether you're computing
$$ \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \ldots $$
or
$$ \sum_{n=3}^{\infty} 2^{-n} $$
or
$$ \sum_{n=0}^{\infty} \frac{1}{8} 2^{-n} $$
you can quickly identify the sum as
$$ \frac{ \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2002780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
When does the tangent line to the sine curve pass through the origin? I am trying to find values $a$ and $w$ for which the line $y=ax$ is tangent to the curve $y=\sin(x)$ at $x=w$.
One immediate solution is $a=1$ and $w=0$, but I would like $a<0$ and $w\in (\pi,3\pi/2)$, and I am having a hard time finding such a solut... | I guess the equation you need to solve is
$$\frac{\sin(x)}{x} = \cos(x).$$
The left hand side is the slope of the line through the points $(x,\sin(x))$ and the right hand side is the slope of the line tangent to the graph of $y=\sin(x)$ at the point $x$. You want these to be equal.
The reason you're getting the complai... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2002921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
What is the smallest n for which n! has more than 10 digits? I saw this exercise from the open sourced "Book of Proof", which I'm using in an abstract math class. I could only find out the answer by manually calculating different factorials. But I'm also curious if there is better method for it. Any hint and answers ar... | A way to reduce complexity is by keeping only n (say n=3) significant digits when doing your computations (since you only care about the number of digits). But honestly speaking, what question asks is not that much computationally complex anyway.
If you would have for example $10^{100}$ digits, than it may be useful to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2003015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove $f(x)\le0$ if $f(0)=0$ and $\int_0^xf(t)\mathbb dt\ge xf(x)$
$f(x)$ is a differentiable real valued function that satisfies following conditions:
$$f(0)=0$$
$$\int_0^xf(t)\mathbb dt\ge xf(x)\quad$$
Prove that for all $x>0$
$$f(x)\le0$$
I tried but couldn't derive the conclusion from the given conditions... | Hint: write is as $$\int_0^x \big(f(t)-f(x)\big) dt \ge 0$$
[ EDIT ] The above implies that for a given $x$, there exists a $c \in (0,x)$ such that $f(c) \ge f(x)$. Therefore the set $C_1 = \{ c \in (0,x) \mid f(c) \ge f(x)\}$ is not empty and, being obviously bounded, it has an infimum $0 \le c_1 = \inf C_1 \lt x$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2003158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
What is $(-1)^{\frac{1}{7}} + (-1)^{\frac{3}{7}} + (-1)^{\frac{5}{7}} + (-1)^{\frac{9}{7}} + (-1)^{\frac{11}{7}}+ (-1)^{\frac{13}{7}}$? The question is as given in the title. According to WolframAlpha, the answer is 1, but I am curious as to how one gets that. I tried simplifying the above into $$6(-1)^{\frac{1}{7}}$$ ... | We have
$$x^7+1=(x+1)(x^6-x^5+x^4-x^3+x^2-x+1).$$ Let $x=(-1)^{1/7}$ and get the result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2003317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Prove that the matrix $I_n - A$ is invertible Let $A$ be an $n \times n$ matrix with the property that $A^3 = O_n$, where $O_n$ denotes the $n \times n$ matrix which
has all the entries equal to $0$. Let $I_n$ be the $n \times n$ identity matrix. Prove that the matrix $I_n - A$
is invertible, and indicate how you would... | I think that it should read $I_n -A$ instead of $I_n \rightarrow A$.
Observe that
$ I_n=I_n-A^3=(I_n-A)(A^2+A+I_n)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2003429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Regarding the Tricomi confluent hypergeometric function Is the following equation true for Tricomi confluent Hypergeometric function? $$\phi(1,0,ax)=1-ax\phi(1,1,ax)$$ here $\phi(.,.,.)$ is the Tricomi confluent hypergeometric function. Thanks in advance.
| As usual examine carefully :)
Using DLMF 13.2.11, 13.6.6
$U(1,0,z)=z\cdot U\left(2,2,z\right)=e^{z}\cdot E_{2}\left(z\right)$
$z\cdot U(1,1,z)=z\cdot e^{z}E_{1}\left(z\right)$
So your question is
$E_{2}\left(z\right)+z\cdot E_{1}\left(z\right)=e^{-z}$
Which matches 8.19.12
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2003541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Find the largest possible integer n such that $\sqrt{n}+\sqrt{n+60}=m$ for some non-square integer m. The solution to this problem says that:
Squaring both sides gives us that $n(n+60)$ is a perfect square
I do not understand this, can someone explain me why this is true.
squareing gives:
$2n+60+2\sqrt{n}\sqrt{n+60}=m... | As you have found out it is necessary that $n(n+60)$ is a perfect square. Therefore we want $n(n+60)=(n+r)^2$, or $(60-2r)n=r^2$, for some integer $r\geq0$. It follows that $r$ has to be even, and writing $r=2s$ we arrive at the condition $$(15-s)n=s^2\tag{1}$$ for integer $s$ and $n\geq0$. Considering the cases $0\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2003624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
What is the limit $\lim_{n\to\infty}\sqrt[n]{n^2+3n+1}$? Can you try to solve it? I tried to do something but I do not know how to continue it:
\begin{align}
& \lim_{n\to\infty}\sqrt[n]{n^2+3n+1}=\lim_{n\to\infty}(n^2+3n+1)^{1/n} = e^{\lim_{n\to\infty} \frac{1}{n}\ln(n^2+3n+1)} \\[10pt]
= {} & e^{\lim_{n\to\infty}\frac... |
PRIMER
In THIS ANSWER, I showed using only the limit definition of the exponential function along with Bernoulli's Inequality that the logarithm function satisfies the inequalities
$$\frac{x-1}{x}\le \log(x) \le x-1 \tag 1$$
for $x>0$.
We will now use $(1)$ to show that $\lim_{n\to \infty}\frac1n \log(n)=0$. To do ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2003708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Difference of Squares functional equation Find all twice differentiable functions mapping the reals to the reals such that $$f(x)^2-f(y)^2=f(x+y)f(x-y)$$.
From plugging in $x=y=0$, I got $f(0)=0$. Then from plugging in $x=0$, we get that $f(x)=-x$.
However, I can't continue further. Any help?
| Hint (suggested by the "twice differentiable" condition): take the derivative in $y$.
$$-2f(y)f'(y) = f'(x+y)f(x-y) - f(x+y)f'(x-y)$$
Now take the derivative in $x$.
$$
\begin{align}
0 & = f''(x+y)f(x-y) + f'(x+y)f'(x-y) - f'(x+y)f'(x-y) - f(x+y)f''(x-y) \\
& = f''(x+y)f(x-y) - f(x+y)f''(x-y)
\end{align}
$$
With $x+y ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2003825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluate the summation of $(-1)^k$ from $k=0$ to $k=n$ $$\sum_{k=0}^n(-1)^k$$
I know that the answer will be either -1 or 0 depending on whether there are an odd or an even number of sums in total, but how can I determine this if $k$ goes to infinity (which I am thinking means there is neither an even nor odd amount of... | Using the general form derived as:
\begin{align}
\sum_{k=0}^{n} x^{k} &= \sum_{k=0}^{\infty} x^{k} - \sum_{k=n+1}^{\infty} x^{k} = \frac{1}{1-x} - \frac{x^{n+1}}{1-x} = \frac{1 - x^{n+1}}{1-x}
\end{align}
then for $x = -1$ the result becomes
$$ \sum_{k=0}^{n} (-1)^{k} = \frac{1 + (-1)^{n}}{2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2003955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Power set of Cartesian product I'm trying to see the differences between a Power set of a cartisian product and the cartisian product of two power sets. I have these 2 examples:
$P(\{1, 2\} \times \{3, 4\})$
$P(\{1, 2\}) \times P(\{3, 4\})$
With the first one I can arrive here, but I'm not sure how to continue to compu... | Well, on a practical and obvious level . $A \times B$ = a set of ordered pairs so $P(A\times B)$ = a set of sets of ordered pairs. $P(A)$ = a set of set of elements. So $P(A) \times P(B)$ = a set of ordered pairs of sets.
It might seem abstract but a set of ordered pairs of sets = {({..},{....})}, is completely di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2004106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
} |
Parametrization of the intersection between a sphere and a plane I can't find a way to get the parametric equation $\gamma(t)=(x(t),y(t),z(t))$ of a curve that is the intersection of a sphere and a plane (not parallel to any coordinate planes). That is
$$\begin{cases} x^2+y^2+z^2=r^2 \\ ax+by+cz=d \end{cases}$$
I don'... | Just to present a vectorial approach to the problem
Consider the plane equation written as:
$$
\frac{{a\,x + b\,y + c\,z}}{{\sqrt {a^{\,2} + b^{\,2} + c^{\,2} } }} = \frac{d}{{\sqrt {a^{\,2} + b^{\,2} + c^{\,2} } }}
$$
That means that the points ${\bf p} = \left( {x,y,z} \right)$ on the plane
shall project onto th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2004224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Define equivalence relation on set of real numbers so distinct equivalence classes are $[2k,2k+2)$ Define an equivalence relation $\sim$ on $\mathbb{R}$ such that the distinct equivalence classes of $\sim$ are $[2k,2k+2)$, where $k$ is an integer (Hint: find an appropriate function $f$ with all real numbers as its doma... | When they say that $A $ is an equivalence class they mean that $a, b \in A \rightarrow a \sim b $. So they are asking you to create an equivalence relation $R $ for which the classes are the intervals of the form $[2k, 2k + 2) $.
Let $f(x) = \left\lfloor{\frac{x }{2}}\right\rfloor$
Now define the relation $R = \{(x, y)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2004320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Analog of Gauss-Bonnet formula for principal bundles over manifolds with boundary The Gauss-Bonnet formula gives a topological invariant as an integral over a local density on the given manifold. In particular, when there is a boundary, GB formula has to be supplemented by a boundary term, for example the extrinsic cur... | What you are looking for is the theory of characteristic classes of principal bundles. You can take any connection on the principal bundle, plug it in the appropriate $G$-invariant polynomial and what you get is an element of cohomology that descends to the base manifold. For a certain choice you would get the Chern cl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2004431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
equivalent expressions for trace norm I understand the trace norm (or nuclear norm) of a matrix $X\in\mathbf{R}^{n\times m}$is usually defined as
$$\|X\|_{tr}=\sum_{i=1}^{\min\{m,n\}}\sigma_i$$
where $\sigma_i$'s are the singular values of $X$.
However, some papers uses an alternative definition:
$$\|X\|_{tr}=\min_{X=A... | Note that, since $X^*X$ is positive-semidefinite (and square), the square roots of its eigenvalues are the eigenvalues of the square root. Thus
$$
\|X\|_{\rm tr}=\sum_j\lambda_j(X^*X)^{1/2}=\sum_j\lambda_j((X^*X)^{1/2})=\sum_j\lambda_j(|X|)=\text{Tr}\,(|X|)
$$
(hence the name of the norm). Now, if $X=AB$, then with $AB... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2004542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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If $\dfrac x{x^{1.5}}=(8x)^{-1}$ and $x>0$, then $x=\;?$
If $\dfrac x{x^{1.5}}=(8x)^{-1}$ and $x>0$, then $x=\;?$
I solved this many different ways and every time I get a different answer...
Attempt #1:
$$\frac1{\sqrt x}=\frac1{8x}$$ $$8x=\sqrt x$$ $$64x^2=x$$ $$x(64x-1)=0$$ $$x=0,\;\frac1{64}$$ $$x=\frac1{64}$$
At... | ʜᴇʟʟᴏ ᴅᴇᴀʀ!
$$\frac{x}{x^{1.5}} = (8x)^{-1}$$
$$x^{(1-1.5)} = \frac{1}{8x}$$
$$ x^{-0.5}= \frac{1}{8x}$$
$$\frac{1}{\sqrt{x}} = \frac{1}{8x}$$
$$x = \frac{\sqrt{x}}{8}$$
On squaring both sides,
$$x^2=\frac{x}{64}$$
Given $x>0$:
$$x = \frac{1}{64}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2004639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Determining convergence with taking the limit of a function greater than it Why does the Sandwhich Theorem have to be used here? Why couldn't we just test for the result of the right side of the inequality and say that $a_n$ converges, since the function greater than itself converges?
| You may have confused with this correct theorem (called comparison test):
If $\sum b_n$ converges and $0\leq a_n\leq b_n$ then $\sum a_n$ also converges.
What you mentioned: " $a_n$ converges, if the function greater than itself converges" is not correct:
Example: $a_n=(-1)^n\leq 2$, and 2 being a constant sequence su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2004796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Does $f\colon x\mapsto 2x+3$ mean the same thing as $f(x)=2x+3$? In my textbook there is a question like below:
If $$f:x \mapsto 2x-3,$$ then $$f^{-1}(7) = $$
As a multiple choice question, it allows for the answers:
A. $11$
B. $5$
C. $\frac{1}{11}$
D. $9$
If what I think is correct and I read the equation as:
$$f(x)... | As a matter of not merely style but of writing in complete sentences you should write $x=f^{-1}(7)\iff f(x)=7\iff 2x-3=7\iff (2x-3)+3=7+3\iff 2x=10\iff x=5.$ This reduces errors and shows the flow of the reasoning. In this case the sequence of "$\iff$" shows that the implications go in both directions. The importance ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2004895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 6,
"answer_id": 4
} |
Find all prime numbers p such that 2p+1 is a perfect cube. $$2p+1=n^3$$
$$ 2p = n^3 - 1$$
$$ 2p = (n-1)(n^2+n+1)$$
The only number that fits the criteria is 13, how can I prove this?
| HINT: The factorization of $2p$ shows that $n-1$ must be either $2$ or $p$. (The other two cases, $n-1=1$ and $n^2+n+1=1$, are trivially eliminated.) You already know what happens when $n-1=2$. Otherwise, $n-1$ must be an odd prime $p$. In that case $n$ is even. What does that tell you about $n^3-1$?
| {
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$m>2$ and $n > 2$ are relatively prime $\Rightarrow$ no primitive root of $mn$
Show that if $m>2$ and $n > 2$ are relatively prime, there is no primitive root of $mn$
I know that $mn > 4$, and thus $\varphi(mn)$ is an even number so that I might write $\varphi(mn) = 2x$ for an integer $x$. If I could prove that $x = ... | Carmichael's reduced totient $\lambda(m)$ and $\lambda(n)$ are both even, so $\lambda(mn) < \phi(mn)$ and there is no primitive root.
| {
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Will the resulting Euclidean space be Hilbertian? In the linear space of sequences $x=( x_{1}, x_{2}, ... ), (x_{k}\in \mathbb{R})$ such that $\sum_{k=1}^{\infty }x_{k}^{2}<\infty$.
Let $(x,y)=\sum_{k=1}^{\infty }\lambda_{k}x_{k}y_{k}$, where $\lambda_{k}\in \mathbb{R}, 0<\lambda_{k}<1$. Will the resulting Euclidean s... | The form $(x,y)$ is bilinear symmetric i.e. a scalar product in
$$
H=\{( x_{1}, x_{2}, ... )\in\mathbb R^\infty\;|\,\sum_{k=1}^{\infty }x_{k}^{2}<\infty\}
$$
Whether $H$ is Hilbertian or not depends on $\lambda_k$. For eample, if
$$
\lambda_k=2^{-k},
$$
then $H$ is not hilbertian. In fact, the sequence
$$
x^{(k)}=(1,... | {
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Find all numbers such that "Product of all divisors=cube of number". While solving some old olympiad problems I came across this one. As I m stuck at it, so I m here.
The problem is: Find all positive integers $N$ such that the product of all the positive divisors of N is equal to $N^3$.
Since I was not able to solve t... | Hint: The way to go is by prime factorisation. You need only consider the case up to four distinct prime factors (the rest should follow easily from your discussion). If you realise, all your found examples follow the pattern $pq^2$, where $p,q$ are primes.
| {
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Jacobian and local invertibility of function Following is a question in the text book:
Show by an example that the condition that the Jacobian vanishes at a point is not necessary for a function to be locally invertible at that point..... and the answer given is f : R$^2$- R$^2$ f(x,y) = (x$^3$, y$^3$)
I am a bit conf... | It looks like a typo in the text. It should say "...the condition that the Jacobian does not vanish...".
$f(x,y)=(x^3,y^3)$ is invertible. Its inverse is $f^{-1}(x,y)=(\sqrt[3]{x},\sqrt[3]{y})$. What happens is that the inverse is not differentiable.
| {
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CCRT = Constant case CRT: $\,x\equiv a\pmod{\! 2},\ x\equiv a\pmod{\! 5}\iff x\equiv a\pmod{\!10}$ Problem: Find the units digit of $3^{100}$ using Fermat's Little Theorem (FLT).
My Attempt: By FLT we have $$3^1\equiv 1\pmod2\Rightarrow 3^4\equiv1\pmod 2$$ and $$3^4\equiv 1\pmod 5.$$ Since $\gcd(2,5)=1$ we can multiply... | Yours is a valid, clean argument. It is based on this:
If $m$ and $n$ divide $a$, then $lcm(m,n)$ divides $a$.
In your case, you have that $2$ and $5$ divide $3^4-1$, and so $10=lcm(2,5)$ divides $3^4-1$.
| {
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What is $E(X^2)$ mean in literal terms? In my probability and statistics class we learned about expected value, or $E(X)$. We also did some work about finding expected values of functions and such, like $E(g(x))$. And in the case of finding the variance, one of the steps involve finding $E(X^2)$. Does this mean anythin... | Here's an example. Suppose $X$ is a random variable that represents the outcome of a roll of a die numbered $1$ to $6$ inclusive. No assumption is made about the fairness of the die. Then $X^2$ is a random variable that represents the outcome of the square of the roll; whereas $$X \in \{1, 2, 3, 4, 5, 6\},$$ we have... | {
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Deriving Taylor series without applying Taylor's theorem. First, a neat little 'proof' of the Taylor series of $e^x$.
Start by proving with L'Hospital's rule or similar that
$$e^x=\lim_{n\to\infty}\left(1+\frac xn\right)^n$$
and then binomial expand into
$$e^x=\lim_{n\to\infty}1+x+\frac{n-1}n\frac{x^2}2+\dots+\frac{(n-... | Alan Turing, at a young age, derived the series expansion of $\arctan$ without using (and, purportedly without knowing) calculus whatsoever.
Using the identity
$$\tan 2x = \frac{2 \tan x}{1 - \tan^2 x},$$
he obtained
$$\tan(2 \arctan x) = \frac{2x}{1-x^2},$$
and
$$2 \arctan x = \arctan\left( \frac{2x}{1-x^2}\right).$$
... | {
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Prove that if P(x) is a quartic polynomial, then there can only be at most one line that is tangent to it at two points. Prove that if P(x) is a quartic polynomial, then there can only be at most one line that is tangent to it at two points.
From what I have seen, the only case in which I have found a tangent line at 2... | Suppose $P(x)$ is quartic and $T(x)=Ax+B$ is a line that is tangent to $P(x)$ at $\xi_1,\xi_2$. Then you can factor $P-T$ as
$$P(x)-T(x) = (x-\xi_1)^2(x-\xi_2)^2$$
Then $$P''(x)= 2((x-\xi_1)^2+4(x-\xi_1)(x-\xi_2)+ (x-\xi_2)^2))$$
So that $P''(\xi_1) = P''(\xi_2)= 2(\xi_1-\xi_2)^2$. This determines $\xi_1,\xi_2$ uniquel... | {
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Whispering wall function After visiting the Whispering wall where a whisper can be clearly transmitted between two locations on the surface of a dam wall over more than 100 metres, I was puzzled as to how this occurs. I decided to try to solve the following problem: What must be the shape of a curved wall such that any... | Following @Djura Marinkov's remark, the correct equation to study is
\begin{equation}
y'^2 + \frac{x^2 - y^2 - a^2}{x y} y' -1 = 0. \tag{1}
\end{equation}
Using the substitution $y(x)^2 = z(\eta(x))$ where $\eta(x) = x^2$, we obtain
\begin{equation}
z'^2 + \left(1-\frac{z+a^2}{\eta}\right) z' - \frac{z}{\eta} = 0, \tag... | {
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Why is the supremum this? $M=\{y\in \mathbb R |y=3x+10:x\in(9,14)\}$
From what I've learned, this means
$9<x<14$?
DEF: Supremum
A figure $u\in \mathbb R$ so
*
*$a\leq u$ for all $a\in A$
*for all $\epsilon>0$ there exists such $a\in A$ so $u-\epsilon<a$
How come 52 is said to be the supremum..? I would like to sa... | Remember that $x$ doesn't have to be an integer! For example, take $x=13.9999$. Then $y=3x+10$ will be very very close to $52$ (specifically, $51.9997$).
| {
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Is it possible to get an example of two matrices Is it possible to get an example of two matrices $A,B\in M_4(\mathbb{R})$ both having $rank<2$ but $\det(A-\lambda B)\ne 0$ i.e it is not identically a zero polynomial. where $\lambda$ is indeterminate, I mean a variable. I want to say $(A-\lambda B)$ is of full rank mat... | Note that $rank\ A, B < 2$ means their ranks are 1 (If any of them is 0 your question has a trivial answer). That means that all columns of $A$ are $v, \alpha_v v, \beta_v v, \delta_v v$ and the columns of $B$ are $u, \alpha_u u, \beta_u u, \delta_u u$. Now note that the columns of $(A - \lambda B)$ will be sums of tho... | {
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Solve for $x$ $2\log_4(x+1) \le 1+\log_4x$ $2\log_4(x+1) \le 1+\log_4x$
I did:
$$2\log_4(x+1) \le 1+\log_4x \Leftrightarrow \log_4(x^2+1) \le 1+\log_4(x) \Leftrightarrow \log_4(\frac{x^2+1}{x}) \le 1 \Leftrightarrow 4 \ge \frac{x^2+1}{x} \Leftrightarrow 4x \ge n^2 +1 \Leftrightarrow 0\ge x^2 -4x +1$$
Using the quadrati... | I think you have made mistake in the first step. It should be
$$2\log_4(x+1)=\log_4(x+1)^2=\log_4(x^2+2x+1).$$
Correcting it should give you the solution in your book.
| {
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$\lim \limits_{x \to e} (1-\log x)\log (x-e)$ without l'hopital's rule $\lim \limits_{x \to e}\space\space(1-\log x)\log (x-e)$
How can is it possible to eliminate indeterminate without using L'Hospital's Rule ? I tried to manipulate this formula but still the same problem.
| Let put $$x=te$$ and compute the limit
$$\lim_{t\to1^-}(1-log(te))log(e-te)$$
$$=-\lim_{t\to1^-}log(t)(1+log(1-t))$$
$$=-\lim_{t\to1^-}log(1+(t-1))(1+log(1-t))$$
$$=-\lim_{t\to1^-} \frac{log(1+(t-1))}{t-1}(t-1+(t-1)log(1-t))$$
$=-1(0-0)=0$.
| {
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How do i integrate this? Hyperbolic substitution? I'm trying to integrate
$\int_0^\pi \sqrt{2+t^2}\; dt$
I know there it involves with something along the lines of either hyperbolic substitution or trigonometric.
I tried $t= \sqrt {2}\tan(u)$ and $u=\arctan(\frac{x}{\sqrt{2}})$. But then, I get lost and confused. Any... | Try out $t=\sqrt{2} \sinh(z)$. This leads to an easier calculation compared to the gonimetric substitution. More precisely, $t=\sqrt{2} \sinh(z)$, $dt=\sqrt{2} \cosh(z) dz$ and our integral becomes:
$$ I= \sqrt{2} \int \sqrt{1+\sinh^2(z)} \cosh(z) \, dz = \sqrt{2} \int \cosh^2(z) \, dz = \sqrt{2} \int \frac{1+\cosh(2z)... | {
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Finding Polynomial Equations for $1^4 + 2^4 + 3^4 + \ldots n^4$ Find a polynomial expression for
$$ 1^4 + 2^4 + 3^4 + ... + n^4 $$
I know you have to use the big theorem but I can't figure out how you would start to compute the differences. Suggestions?
| Even though this might be too much for simple task you want (you can assume the polynomial of fifth degree and calculate coefficients) I think it is worth mentioning that there is a general formula for finding polynomial of sum of $p$-th powers $1^p+2^p+\dots+n^p$, and it is called Faulhaber's formula.
In short
$$1^p+... | {
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Is $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\ldots}}}$ rational, algebraic irrational, or transcendental? Let
$$
x=\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\ldots}}}.
$$
Assume $x$ is algebraic irrational. By the Gelfond-Schneider Theorem, $x^x$, which is also $x$, is transcendental, a contradiction.
But I have no idea how to do the res... | The only reasonable meaning of $\sqrt 2^{\sqrt2^{\sqrt2^{\cdots}}}$ would seem to be the limit of the sequence
$$ \sqrt2, \sqrt2^{\sqrt2}, \sqrt2^{\sqrt2^{\sqrt2}}, \ldots $$
which can also be defined recursively as
$$ x_0 = \sqrt 2 \\
x_{n+1} = \sqrt2^{x_n} $$
This sequence converges to the number $2$, which is ration... | {
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Analysis problem: Show that$ f(x)$ is less or equal than$ g(x)$ Analysis problem:
Let $f$ and $g$ be differentiable on $ \mathbb R$. Suppose that $f(0)=g(0)$ and that $f' (x)$ is less or equal than $g' (x)$ for all $x$ greater or equal than $0$ Show that $f(x)$ is less or equal than$g(x)$ for all $x$ greater or equal t... | Let $h(x)=f(x)-g(x);x\in [0,\infty)$
$h^{'}(x)=f^{'}(x)-g^{'}(x)\le 0\implies h$ is decreasing on $[0,\infty)\implies h(x)\le h(0)\forall x\in [0,\infty)\implies f(x)\le g(x)$
| {
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How can I check my solution for this ODE? I have $x''(t) - x(t) = e^{t}$ with boundary conditions $x(0) - x(1) = 0$ and $x'(0) - x'(1) = 0$.
I find the solution to be $x(t) = e^t \frac{4t-2}{8} + c_{1} e^{t} + c_{2}e^{-t}$, but finding $c_{1},c_{2}$ is tedious to find. I find $c_{1} = \frac{-\frac{1}{4} ((1-e) + e + 1)... | To solve:
$$\text{x}''\left(t\right)-\text{x}\left(t\right)=e^t$$
Use Laplace transform:
$$\text{s}^2\text{X}\left(\text{s}\right)-\text{s}\text{x}\left(0\right)-\text{x}'\left(0\right)-\text{X}\left(\text{s}\right)=\frac{1}{\text{s}-1}$$
Solving $\text{X}\left(\text{s}\right)$:
$$\text{X}\left(\text{s}\right)=\frac{\f... | {
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If we are given $3$ positive integers $a,b,c$ such that $a>b>c$ , and $91b>92c>90a$ . What is the minimal value of $a+b+c$? If we are given $3$ positive integers $a,b,c$ such that $a>b>c$ , and $91b>92c>90a$ . What is the minimal value of $a+b+c$?
I am getting the bounds of the fractions $\frac{a}{b},\frac{b}{c},\frac{... | This follows up on my former comment and proves $413=139+138+136$ is the minimal sum.
$92 c \gt 90 a \implies 2c > 90(a-c) \ge 90 \cdot 2 = 180$ therefore $c \gt 90 \iff c \ge 91$.
$91 b \gt 92 c \implies 91(b-c) \gt c \ge 91$ therefore $b-c \gt \frac{91}{91} = 1 \iff b-c \ge 2 \iff b \ge c+ 2$.
Since $b \ge c+2$ and $... | {
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Sum of the digits of $N=5^{2012}$ The sum of the digits of $N=5^{2012}$ is computed.
The sum of the digits of the resulting sum is then computed.
The process of computing the sum is repeated until a single digit number is obtained.
What is this single digit number?
| Hint : You can prove that if $d(n)$ denotes sum of digits of $n$, then $n$ is congruent to $d(n)$ mod 9. Also, $N<10^{2012}$ implies $d(N)<9*2012<20000$.
| {
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Finding matrix BA given AB
Given a matrix $$AB = \begin{bmatrix}-2&-14&14\\5&15&-10\\4&8&-3\end{bmatrix},$$ where $A$ is a $3\times 2$ matrix, and $B$ a $2\times 3$ matrix, how do I find the matrix $BA$?
I was told to find the basis for the rowspace of $AB$, and that $(AB)^2 = 5AB$. However, I do not see how these 2 ... | I will use isomorphisms of matrix algebras with corresponding linear operator spaces a lot. Thus, $A\colon\Bbb R^2\to\Bbb R^3$ and $B\colon\Bbb R^3\to \Bbb R^2$.
If you find basis for rowspace of $AB$ you will find that $r(AB) = 2$, where $r$ denotes rank. That also means that $n(AB) = 1$, by rank-nullity theorem (wher... | {
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How do you actually calculate numbers like $2^{\pi}$ or $\sqrt[\leftroot{-2}\uproot{2}160]{2}$ My question is very simple, but I started to wonder how does one calculate numbers like $2^{\pi}$ or $\sqrt[\leftroot{-2}\uproot{2}160]{2}$?
For example I know that:
$$2^{3/2} = \sqrt{2^3}=\sqrt{8}\approx2.83,$$
which is eas... | How I would think about $2^{\pi}$ is the following:
Consider an infinite sequence $(a_{0},a_{1},a_{2}\ldots)$ such that:
$$\sum_{n=0}^{\infty}a_{n}=\pi$$
Then
$$2^{\pi}=2^{\sum_{n=0}^{\infty}a_{n}}=2^{a_0+a_1+a_2+\cdots}=2^{a_{0}}2^{a_{1}}2^{a_{2}}\cdots$$
Now $2^{\pi}$ can be represented as a infinite product:
$$2^{\p... | {
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How to calculate the actual distance flown by a golf ball? I am interested in finding out what the actual distance is flown by a golf ball on a drive. You can assume no wind, 10 degree launch angle, goes straight down the middle of the fairway (no draw, no slice...) and no erratic backspin on the ball (so it wont "bal... | This is a non mathematical answer but still mildly interesting and worth mentioning. For someone that really wanted to know this but didn't have the math skills to solve it (like me), I suppose someone could record the flight of the ball on their smartphone such that the entire ballflight could be recorded. Then they... | {
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Analytical continuation of complete elliptic integral of the first kind I am dealing with a problem involving the complete elliptical function of the first kind, which is defined as:
$K(k)=\int_0^{\pi/2} d\theta \frac{1}{\sqrt{1-k^2\sin^2(\theta)}}=\int_0^1 dt \frac{1}{\sqrt{1-t^2}\sqrt{1-k^2 t^2}} $
for $k^2<1$. I am ... | On an abstract level, the elliptic integral $K$ is an inverse of the elliptic functions which are doubly periodic (in the complex plane) with periods $K$ and $iK'$ (this in a sense answers your question).
To be concrete, one can answer your question given the integral representation you have given (setting $m=k^2$) for... | {
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Inequivalent cusps of $\Gamma_0(4)$ I have been trying to do this by finding the $\Gamma_0(4)$ orbit of $\infty$, then finding an element not in here (namely $0$) and considering the orbit of this. But I feel like there is a slight error in my computation as I suspect there are in fact 3 non-equivalent cusps.
Here is w... | I guess you know how to compute the number of inequivalent cusps of $\Gamma_{0}(N)$: $\sum_{d\mid N}\phi(\gcd(d,N/d))$. So the number of cusps of $\Gamma_{0}(4)$ has three inequivalent cusps. We know that the cusps $\infty$ and 0 are inequivalent. I claim that $1/2$ is not equivalent to neither $\infty$ nor $0$.
If $1/... | {
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Isomorphism with Directional Graphs I was told isomorphism was when two graphs are the same but have different forms. In order for it to be the same two vertices must be adjacent across the graphs. So if vertex1 is adjacent to vertex2 and vertex3 in one graph, than it must do so in the other. Also that isomorphism can ... | In the original graph, vertices have following degrees.
$1$ — $2$ outs, $1$ in.
$2$ — $1$ out, $2$ ins.
$3$ — $2$ outs, $1$ in.
$4$ — $1$ out, $2$ ins.
Let's give the names to the vertices of the second graph:
Here's their degrees:
$A$ — $1$ out, $2$ ins.
$B$ — $2$ outs, $1$ in.
$C$ — $2$ outs, $1$ in.
$D$ — $1... | {
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$\sum_\limits{n=0}^{\infty} a_n$ converges $\implies \sum_\limits{n=0}^{\infty} a_n^2$ converges
Let $(a_n)$ be a sequence of positive terms and suppose that $\sum_\limits{n=0}^{\infty} a_n$ converges. Show that $\sum_\limits{n=0}^{\infty} a_n^2$ converges.
This is in the section on the Comparison Test so that must b... | For any $\epsilon>0$, there must be an $N$ such that $a_n<\epsilon$ for all $n>N$. If this were not true, you'd be adding an infinite amount of terms which didn't approach zero, meaning the sum would diverge, which, according to the statement, is false.
Let $\epsilon=1$ so that we have some $N$ such that $a_n<1$ for a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2008442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 2
} |
Solving ${\log_{\frac{2}{\sqrt{2-\sqrt{3}}}}(x^2-4x-2)=\log_{\frac{1}{2-\sqrt{3}}}(x^2-4x-3)}$ This was given to me by my Math Teacher almost a year ago and I've not been able to make much progress on it. I am hoping to see it resolved by our community members. $$\large{\log_{\frac{2}{\sqrt{2-\sqrt{3}}}}(x^2-4x-2)=\log... | Hint
If $x^2-4x-2<0$ or $x^2-4x-3<0$ then the corresponding $x$ cannot be a solution. But forget about that for a second and explain the meaning of the equation below:
$$\log_{\frac{2}{\sqrt{2-\sqrt{3}}}}(x^2-4x-2)=\log_{\frac{1}{2-\sqrt{3}}}(x^2-4x-3).$$
If for an $x$ the equation holds then there must exist some $A$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2008534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How do you prove the series $\sum_\limits{n=0}^{\infty}(2n+1)(-x)^n$ is convergent? The Wikipedia article about Formal Power Series states that
$$
S(x)=\sum_{n=0}^{\infty}(-1)^n(2n+1)x^n,
$$
if considered as a normal power series, has radius of convergence 1. How do we prove this?
I understand that for positive $x$ the... | The root test for me,
since I know that
$\lim_{n \to \infty} a^{1/n}
=\lim_{n \to \infty} n^{1/n}
=1$
for any $a > 0$.
Then
$\lim_{n \to \infty} (2n+1)^{1/n}
\le \lim_{n \to \infty} (3n)^{1/n}
\le \lim_{n \to \infty} 3^{1/n}\lim_{n \to \infty} n^{1/n}
=1
$
and
$\lim_{n \to \infty} (2n+1)^{1/n}
\ge 1$
so the radius of c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2008633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Find $1/\alpha$ when basis is {1, $\alpha$, $\alpha^2$, $\alpha^3$} I am trying to calculate a sub field and in the process, I need to state $1/\alpha$ in terms of $\alpha$.
Now, my $\alpha$ = $\sqrt{3+\sqrt{20}}$.
I can't for the life of me do this simple calculation!!
The minimal polynomial is $x^4-6x^2-11$ and I kno... | Note that $\alpha^4-6\alpha^2=11$. Divide on both sides by $11\alpha$, and you're done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2008736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Can you determine the remainder when divided by 6?
An integer $x$ gives the same remainder when divided by both $3$ and
$6$. It also gives a remainder of $2$ when divided by $4$, can you
determine an unique remainder when $x$ is divided by $6$?
I feel like you can't since $x=4q+2$ for integer $q$. Listing out som... | The only value possible is 2. Because x gives also a remainder of 2 when divided by 4. It means that we can exclude 0 as common remainder.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2008851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Find all real solutions of this equation $x^2=2y-1$,$x^4+y^4=2$. Find all real solutions of this equation $x^2=2y-1$,$x^4+y^4=2$.
My attempt:I put the value of $x^2$ in the second equation.I get:
$(2y-1)^2+y^4=2 \Rightarrow [(2y-1)^2-1^2]+(y^4-1^4)=0 \Rightarrow 4y(y-1)+(y-1)(y+1)(y^2+1)=0 \Rightarrow (y-1)(y^3+y^2+5y+... | There are no other real solutions.
From $2y-1=x^2\ge 0,$ $y$ has to satisfy $y\ge \frac 12$.
However, if $y\gt 0$, then $y^3+y^2+5y+1\gt 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2009001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
(Geometry) Circle, angles and tangents problem Let P be an external point of a circle with center in O and also the intersection of two lines r and s that are tangent to the circle. If PAB is a triangle such that AB is also also tangent to the circle, find AÔB knowing that P = 40°.
I draw the problem:
Then I tried to ... | First of all, note that $\angle PAB + \angle PBA = 140^\circ$. That means that $\angle MAB + \angle NBA = 220^\circ$.
Then we see that $AO$ bisects $\angle MAB$, and $BO$ bisects $\angle NBA$, so $\angle OAB + \angle OBA = 110^\circ$.
Lastly, looking at the quadrilateral $AOBP$, we see that $x = 360^\circ - 40^\circ - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2009098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Finding sum of infinite series $1+\frac{x^3}{3!}+\frac{x^6}{6!}+\frac{x^9}{9!}+\ldots $ So the question is 'express the power series $$1+\frac{x^3}{3!}+\frac{x^6}{6!}+\frac{x^9}{9!}+\ldots $$
in closed form'.
Now we are allowed to assume the power series of $e^x$ and also we derived the power series for $\cosh x$ usin... | Let $1, w, w^2$ be the cube roots of unity. Let $f(x) = (e^x + e^{xw} + e^{xw^2})/3$. Expand $f$ in a Taylor series. Because the cube roots of unity sum to 0, all terms vanish except where the exponents are multiples of 3, in which case they give the coefficients of your series. You can convert f(x) to $\frac{e^x}{3} +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2009189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Are these two scenarios equivalent ? (random walks on chessboard) The random walks start at $(x,y)=(0,0)$ on an infinite chessboard which covers the whole upper plane. Let's say $(0,0)$ is white.
Random walk 1: At every step, I always go up one square, and either one square to the left or to the right with probability ... | This is a partial answer. In RW1 the end point will be $(X,t)$ where $X=\sum_1^t Y_k$ and the $Y_k$'s are independent random variables with $P(Y_k=-1)=P(Y_k=1)=1/2$. The variance of each of the $Y_k$'s is $1$ and therefore
$$
\sigma_1=\sqrt{t}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2009309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$\lim_{ n\to \infty} \frac{n^{100}}{1.01^n} = ?$ And how to prove it？ There are many such similar limits. But it seems that each of the proof is isolated, there are any good ways to solve it?
| In all generality,
$$\lim_{n\to\infty}n^ab^n=\lim_{n\to\infty}\left(nb^{n/a}\right)^a=\lim_{n\to\infty}\left(nc^n\right)^a=\left(\lim_{n\to\infty}nc^n\right)^a,$$ if it exists, with $c:=\sqrt[a]b$.
Then every time you increment $n$, the expression between the parenthesis is multiplied by $$\frac{n+1}nc=\left(1+\frac1n\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2009397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Weyl Tensor undefined or vanishing? So we have that the Weyl tensor in component form satisfies in dimension $n$
\begin{equation}
C_{abcd}=R_{abcd} -\frac{2}{n-2}\left(R_{a[c}g_{b]d}+R_{b[d}g_{c]a} \right)+\frac{2}{(n-1)(n-2)}Rg_{a[c}g_{b]d}
\end{equation}
The confusion is simple really. Many texts say that when $n=2$,... | Your formula for the Weyl tensor is wrong. I haven't double-checked my computation, but I think the first term in parentheses should be $R_{a[c}g_{d]b}$, not $R_{a[c}g_{b]d}$, and the last term should be a multiple of $Rg_{a[c}g_{d]b}$.
In any case, there has to be something wrong with your formula, because the curvatu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2009500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Cancellation in quotient of fractional ideals When reading about fractional ideals of rings of integers, I came upon the following footnote:
For fractional ideals $\mathfrak{a}$, $\mathfrak{b}$ and $\mathfrak{c}$ with $\mathfrak{a} \supset \mathfrak{b}$, $$\displaystyle ^{\mathfrak{a}\mathfrak{c}}/_{\mathfrak{b}\mathf... | Here is the main statement:
Proposition: Let $R$ be a Dedekind domain (for instance the ring of integers $R=\mathcal O_K$ of some finite extension $K$ of $\Bbb Q$).
Let $\mathfrak{a}$, $\mathfrak{b}$ and $\mathfrak{c}$ be non-zero fractional ideals of $R$ with with $\mathfrak{a} \supset \mathfrak{b}$.
Then there i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2009591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Can an equivalence relation $C$ on $A$ relate two non-equal elements of a set $A$? I was working through one of the exercises in Topology: A First Course by Munkres, and I came across this:
Let $f : A \to B$ be a surjective function. Let us define a relation on $A$ by setting $a_0 C a_1$ if $f(a_0) = f(a_1)$. Show tha... | Yes, an equivalent relation can, and usually does, relate elements that are not originally equal to each other, generalizing what we know as equality.
Let us take, for instance, $\mathbb{Z}$ with the following equivalence relation: $x \sim y$ iff $x - y$ is even. With the above relation, we generalize the idea of "equa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2009734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Sum of all possible combinations Guys I just discovered something amazing. Can someone please confirm this? The sum of all possible ways to form a number with $n$ digits, using its digits, without repetition, is equal to $11\ldots1\cdot m(n-1)!$, where $m$ is the sum of the digits of the number, and the amount of $1$'s... | Each digit has a "chance" to "occupy" every column.
Hence the sum is that digit multiplied by $\underbrace{111\cdots 1}_n$.
Each column will be occupied by every digit, which when summed, gives $m$ for that column.
...Multiply by $m$ to give $\underbrace{111\cdots 1}_n\cdot m$
When a given column occupies a give... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2009888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Prove $\gcd(a,b)=\gcd(a+b,\operatorname{lcm}(a,b))$ Any ideas on how to prove this equality?
I tried various methods, using properties of gcd and lcd, but I can't prove it.
| You may want to try and prove first that
$\operatorname{gcd}(a, b) = \operatorname{gcd}(b, a - b)$
This identity should be enough to get you rolling. You will just have to be able to use it correctly.
Alternatively, try this approach.
Suppose $d $ is the gcd of $a $ and $b $. Show it divides $a+b $ and $\operatorname {... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2009979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Two men, each placing one ball in five out of ten boxes There are 10 boxes and each box can hold any number of balls. A man having 5 balls randomly puts one ball in each of the arbitrary chosen five boxes. Then another man having five balls again puts one ball in each of the arbitrary chosen five boxes. The probability... | Each of the three conditions you present represents a different way to fill the ten boxes with a different amount of balls so that at least 8 boxes have one ball.
In the first case, you have two boxes with two balls, two boxes with zero balls, and six boxes with one ball. If you labelled the ten boxes, you are then lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2010094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Finding elliptic curve with exactly p+1 number points over F_p Hi all I am beginner in Elliptic Curves. I want to design an elliptic curve with exactly $p+1$ points over $\mathbb{F}_p$. Any approach towards starting to solve this problem or recent progress or any references would be really helpful.
Thanks
| The quantity $1+p - |E(\mathbb F_p)|$ is usually denoted $a_p$, and so you are asking for curves for which $a_p = 0$. When $p \geq 5$, this condition is equivalent to the elliptic curve being supersingular.
In general, there is a finite positive number of elliptic curves over $\overline{\mathbb F}_p$, and they can alw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2010212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Rewrite rational function $f(x)$ as a series if the quadratic expression in the denominator has no roots A function of the type
$$f(x)=\frac{ex+f}{ax^2+bx+c}$$
with $b^2-4 a c \geq 0$ can be written as a series using partial fraction decomposition and geometric series.
But if one has the same function with $b^2-4 a c ... | More than likely, too simplistic !
You wrote the function $$F(x)=\frac{ex+f}{ax^2+bx+c}$$ Since you look for an expansion around $x=0$, let us rewrite it as $$F(x)=\frac{f+ex}{c+bx+ax^2}$$ and now use the long division to get $$F(x)=\frac{f}{c}+\frac{ c e-b f}{c^2}x+\frac{ b^2f-a c f-b c
e}{c^3}x^2+O\left(x^3\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2010337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Doolittle transformation is non-unique for singular matrices Decomposing the singular matrix $$A = \begin{bmatrix}
1 & 2 \\
1 & 2
\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}\begin{bmatrix}1 & 2 \\ 0 & 0\end{bmatrix}=LU$$
by Doolittle decomposition seems to be unique for this case. But how to proove tha... | The row of zeroes in your $U $ allows you to play with the second column of $L $. You have
$$
\begin{bmatrix}
1 & 2 \\
1 & 2
\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 1 & x\end{bmatrix}\begin{bmatrix}1 & 2 \\ 0 & 0\end{bmatrix}
$$ for any choice of $x $.
If you need $x=1$, there is no other choice and the decompositi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2010470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to calculate wedge product of differential forms Consider the differential forms on $\mathbb{R}^3$,
$\omega_1 = xy \space dx + z \space dy + dz$ , $\omega_2 = x \space dy + z \space dz$.
I need to determine $\omega_1 \wedge \omega_2$.
However, I do not know how to find such wedge products. Any help is apprecia... | $$ \sum_I a_I dx^I \wedge \sum_J b_J dx^J: = \sum_{I, J} (a_I b_J)\ dx^I \wedge dx^J$$
$\textbf{Example}$:
$$(x dx + y dy) \wedge (2 dx - dy) = 2x \ dx \wedge dx- x \ dx \wedge dy + 2y \ dy \wedge dx- y \ dy \wedge dy\\ \hspace{-.41in}= (-x-2y)\ dx \wedge dy$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2010563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Equivalent of the integral of a function sequence I gave an exercise to a student last wednesday, but I had some trouble finding a correct solution to the last question !
For $n\ge2$, let $f_n$ be the function
$$f_n:[0,+\infty[\to\mathbb R,\,t\mapsto \frac{t^n}{1+t+t^{n-1}}$$
The first question asks the punctual limit ... | HINT:
The substitution $t\to t^{1/n}$ yields
$$\int_0^1 \frac{t^n}{1+t+t^{n-1}}\,dt=\frac1n\int_0^1 \frac{t^{1/n}}{1+t^{1/n}+t^{(n-1)/n}}\,dt\le \frac1n$$
Alternatively, simply note that $\left|\int_0^1 \frac{t^n}{1+t+t^{n-1}}\,dt\right|\le \int_0^1 t^n\,dt=\frac{1}{n+1}$.
EDIT: The OP is requesting the first-order ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2010686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding the integral: $\int_{0}^{\large\frac{\pi}{4}}\frac{\cos(x)\:dx}{a\cos(x)+b \sin(x)}$ What is
$$\int_{0}^{\large\frac{\pi}{4}}\frac{\cos(x)\:dx}{a\cos(x)+b \sin(x)}?$$
$a,b \in \mathbb{R}$ appropriate fixed numbers.
| $$\mathcal{I}\left(\text{a},\text{b}\right)=\int_0^{\frac{\pi}{4}}\frac{\cos\left(\text{x}\right)}{\text{a}\cos\left(\text{x}\right)+\text{b}\sin\left(\text{x}\right)}\space\text{d}\text{x}=$$
$$\frac{\text{a}}{\text{a}^2+\text{b}^2}\int_0^{\frac{\pi}{4}}1\space\text{d}\text{x}-\frac{\text{b}}{\text{a}^2+\text{b}^2}\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2010770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
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