Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How do you parameterize a circle? I need some help understanding how to parameterize a circle.
Suppose the line integral problem requires you to parameterize the circle, $x^2+y^2=1$. My question is, if I parameterize it, would it just be: $r(t)=($cos $t)i+($sin $t)j$? And how would that change if say my radius became $... | Your parametrization is correct. Once you have a parameterization of the unit circle, it's pretty easy to parameterize any circle (or ellipse for that matter): What's a circle of radius $4$? Well, it's four times bigger than a circle of radius $1$!
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Solving $2\cos^2 x-2\sin^2 x-2\cos x=0$ $$f(x) = 2\cos^2 x-2\sin^2 x-2\cos x$$
Need values of x that which make $f(x) = 0$
Tried $a^2-b^2 = (a+b)(a-b)$ with no luck
Really just need a hint that could bring me in the right direction
Thanks
EDIT: Solution thanks to everyones help! :D
$$f(x) = 2\cos^2 x-2\sin^2 x-2\cos x... | Hint:
Use $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\sin^2x=1-\cos^2x$
Update:
$$\cos x=-\frac{1}{2},1$$
$$\cos x=-\dfrac12=\cos\left(\frac{2\pi}3\right)\implies x=2k\pi\pm\dfrac{2\pi}3$$
$$\cos x=1\implies x=2k\pi$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Why are integers subset of reals? In most programming languages, integer and real (or float, rational, whatever) types are usually disjoint; 2 is not the same as 2.0 (although most languages do an automatic conversion when necessary). In addition to technical reasons, this separation makes sense -- you use them for qui... | There are very many reasons, but first of all, if the real numbers did not contain the integers then it would be very very very difficult to do elementary arithmetic.
Computers don't have any way to represent "real numbers", they can represent only integers and rationals.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1055501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "47",
"answer_count": 11,
"answer_id": 7
} |
y' = y^3 - y stable points So I have this differential equation:
$$y' = y^3 - y$$
And I need to find out which of the points (0, 1, -1) are stable.
So here we go:
$$y = \pm \frac{1}{\sqrt{e^{2x+c}+1}}$$ (solution to differential equation without y=0)
Assume that
$y = y^*(x, y_0^*)$ is a solution which satisfies $y(x_0)... | Let me show you a way I was taught
$$
y = y_0 + \delta y
$$
where $y_0 = \lbrace{0,\pm1\rbrace}$
Linearise the original equation given by
$$
y' = F(y)
$$
as
$$
\delta y' = \frac{\partial F(y)}{\partial y}\lvert_{y_{0}}\delta y = \left(3y_0^2-1\right)\delta y
$$
solutions for $\delta y$ i.e the perturbations
$$
\delta y... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Simplifying $\frac{x^6-1}{x-1}$ I have this:
$$\frac{x^6-1}{x-1}$$
I know it can be simplified to $1 + x + x^2 + x^3 + x^4 + x^5$
Edit : I was wondering how to do this if I didn't know that it was the same as that.
| Ill show an "tricky" method.
$\displaystyle \frac{x^6 - 1}{x-1}$
$= \displaystyle \frac{x^6 -x + x - 1}{x-1} = \frac{x^6 - x}{x-1} + 1 = \frac{x^6 - x^5 + x^5 - x}{x-1} + 1 = x^5 + 1 + \frac{x^5 - x}{x-1} = \frac{x^5 - x^4 + x^4 - x}{x-1} + x^5 + 1 = \frac{x^4(x - 1) + x^4 - x}{(x-1)}$
Do you see the pattern?
This i... | {
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"answer_id": 5
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Let R be an integral domain. Show that if the only ideals in R are {0} and R itself, R must be a field I know that if (x)={0} then the if 0=r0 such that r belongs to R therefor it's a field.
Most likely I'm wrong but I need help with the second part if the ideal is R
| If $R$ with unit.
Let $0\neq x\in R$ then $ 0\neq ( x)$ is an ideal of $R$ , hence $(x)=R$ so there exists $a$ in $R$ sush that $ax=1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Double substitution when integrating. I need to integrate
$$f(x) = \cos(\sin x)$$
my first thought was substituting so that $u = \cos(x)$, but that doesn't seem to do the trick. Is there any way to do a double substitution on this?
Any ways on how to proceed would be appreciated.
| Although the indefinite integral $($or anti-derivative$)$ cannot be expressed in terms of elementary functions, its definite counterpart does have a closed form in terms of the special Bessel function:
$$\int_0^\tfrac\pi2\cos(\sin x)~dx~=~\int_0^\tfrac\pi2\cos(\cos x)~dx~=~\frac\pi2J_0(1).$$
Also, $$\int_0^\tfrac\pi2\s... | {
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Miklos Schweitzer 2014 Problem 8: polynomial inequality Look at problem 8 :
Let $n\geq 1$ be a fixed integer. Calculate the distance:
$$\inf_{p,f}\max_{x\in[0,1]}|f(x)-p(x)|$$ where $p$ runs over
polynomials with degree less than $n$ with real coefficients and $f$
runs over functions $$ f(x)=\sum_{k=n}^{+\infty... | Your inequality does not hold since $x^{n-1}$ is not the best approximation polynomial of $x^n$ with respect to the uniform norm. By Chebyshev's theorem we have that if $p(x)$ is the best approximation polynomial for $f(x)$, then $f(x)=p(x)$ holds for $\partial p+1$ points in $[0,1]$.
For instance, if $f(x)=x^n$ and $... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Riemann integrals real analysis How would I do this question?
$$
\int_{-1}^{2}(4x - x^2)dx
$$
I am using a similar method that I did for a different question but I think my problem is I don't know how to tackle the $x^2$.
I am getting an answer of $15$, but $15$ is not the answer.
| When using the Riemann definition, we first have to define the partition and the interpolation points. In the region $[-1,0]$ we use $x_i=-\frac{i}{N}$, and every interval has length $\frac{1}{N}$. In the region $[0,2]$ we use $x_i=2\frac{i}{N}$ and every interval has length $\frac{2}{N}$. It follows that the integral ... | {
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"timestamp": "2023-03-29T00:00:00",
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Simplifying the sum $\sum\limits_{i=1}^n\sum\limits_{j=1}^n x_i\cdot x_j$ How can I simplify the expression $\sum\limits_{i=1}^n\sum\limits_{j=1}^n x_i\cdot x_j$?
$x$ is a vector of numbers of length $n$, and I am trying to prove that the result of the expression above is positive for any $x$ vector. Is it equal to $\... | Yes,
$$\sum_{i=1}^n\sum_{j=1}^nx_i x_j=\left(\sum_{i=1}^ nx_i\right)^2\;.$$
To see this, let $a=\sum_{i=1}^ nx_i$; then
$$\sum_{i=1}^n\sum_{j=1}^nx_i x_j=\sum_{i=1}^n\left(x_i\sum_{j=1}^nx_j\right)=\sum_{i=1}^na x_i=a^2\;.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1056337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Finding a basis for Kernel and Image of a linear transformation using Gaussian elimination. This is a very fast method for computing the kernel/nullspace and image/column space of a matrix. I learned this from my linear algebra teacher but I haven't seen it mentioned online apart from this reference in wikipedia.
The ... | The image of $A$ is precisely the column space of $A$, that is, the vector space spanned by the columns of $A$. Performing column operations does not change the column space of $A$ (see below), so the nonzero columns in the echelon form are a basis for the column space of $A$ (and, hence, for the image of $A$).
A vect... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1056478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Find the value of the sum $$\arctan\left(\dfrac{1}{2}\right)+\arctan\left(\dfrac{1}{3}\right)$$
We were also given a hint of using the trigonometric identity of $\tan(x + y)$
Hint
$$\tan\left(x+y\right)\:=\:\dfrac{\tan x\:+\tan y}{1-\left(\tan x\right)\left(\tan y\right)}$$
| Let $$\arctan\left(\dfrac{1}{2}\right)+\arctan\left(\dfrac{1}{3}\right)=u $$
Take tangents on both sides using hint given.
$$ \dfrac{1/2 +1/3}{1- {\dfrac{1} {6}}} = tan(u), $$
$$ tan(u) =1$$
$$ u = \pi/4 $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1056578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Prove that the sequence $\cos(n\pi/3)$ does not converge EDIT: Using a rigorous formal proof, I need to prove that this sequence does not diverge. I of course understand why it doesn't converge...
$n=1$ to infinity of course.
So, I have a bit of trouble understanding the definition. Here's what I have so far:
I know it... | If the sequence converges, then all its subsequences have the same limit. However, the subsequence $\big( \cos \big( \frac{(6k+3) \pi}{3} \big) \big)_{k \in \mathbb{N}}$ converges to $-1$ and the subsequence $\big( \cos \big( \frac{(6k) \pi}{3} \big) \big)_{k \in \mathbb{N}}$ converges to $1$. Hence, the limit does not... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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If matrix $\sum_0^\infty C^k$ is convergent, how can I prove that $A(\sum_0^\infty C^k)B$ is convergent? For an $n \times n$ matrix $C$ and If $\sum_0^\infty C^k$ is convergent, how can I prove that for two matrices $A$ and $B$, $A(\sum_0^\infty C^k)B$ is convergent?
It seems quite obvious that you just multiply the co... | One approach is to note that the map
$$
X \mapsto AXB
$$
is continuous. As Marc van Leeuwen notes below,
$$
A(\sum_{k=0}^N C^k)B = \sum_{k=0}^N AC^kB
$$
so that the limit must be $\sum_{k=0}^\infty AC^kB$, which converges (by our statement of continuity).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Evaluate the integral by using Gauss divergence theorem. Evaluate $\int\int_SF.dS$ where $F=(xz,yz,x^2+y^2)$
by using the Gauss divergence theorem.
Where $S$ is the closed surface obtained from the surfaces $x^2+y^2\leq 4,z=2,x^2+y^2\leq 16,z=0$ on the top and the bottom and $z=4-\sqrt{x^2+y^2}$ on the side.
My calcula... | Anyway, this is what I get:
$$2 \int\int\int z dV = 2\int_0^2 \int_0^{4-z} \int_0^{2\pi} z r\ d\theta dr dz = 4\pi \int_0^2 \int_0^{4-z} zr\ dr dz$$
$$ = 2\pi \int_0^2 z(4-z)^2 dz = 2\pi \int_0^2 \bigg(z^3 - 8z^2 + 16z\bigg)dz$$
$$ = 2\pi \bigg( \frac{2^4}{4} - \frac{16(2^3)}{3} + \frac{16(2^2)}{2}\bigg) = \frac{88\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1056940",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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How to evaluate $\lim\limits_{n\to \infty}\prod\limits_{r=2}^{n}\cos\left(\frac{\pi}{2^{r}}\right)$ How do I evaluate this limit ?
$$\lim_{n\to \infty}\cos\left(\frac{\pi}{2^{2}}\right)\cos\left(\frac{\pi}{2^{3}}\right)\cdots\cos\left(\frac{\pi}{2^{n}}\right)$$
I assumed it is using this formaula $\displaystyle \cos(A)... | Hint: $\cos x = \dfrac{\sin (2x)}{2\sin x}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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"answer_id": 1
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A finite sum over $\pm 1$ vectors $$\mbox{What is a nice way to show}\quad
\sum_{\vphantom{\Large A}u\ \in\ \left\{-1,+1\right\}^{N}}
\left\vert\,\sum_{i\ =\ 1}^{N}u_{i}\,\right\vert
= N{N \choose N/2}
\quad\mbox{when}\ N\ \mbox{is even ?.}
$$
Could there be a short inductive proof ?.
| If there are $r$ -1s in $u$, then $|\sum u_i| = |N-2r|$
Let $M = \frac{N}{2} -1$. Thus we are looking at
$$ \sum_{r=0}^{N} \binom{N}{r} |N - 2r| = 2\sum_{r=0}^{M} \binom{N}{r} (N-2r) $$
(using $\binom{N}{r} = \binom{N}{N-r}$)
Now if $P(x) = \sum_{k=0}^{N} a_k x^k$, then the partial sum $a_o + a_1 + \dots + a_i$ is give... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1057107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
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Construct quadrangle with given angles and perpendicular diagonals The following came up when I worked on the answer for a different question (though it was ultimately not used in this form):
Proposition.
Given positive angles $\alpha,\beta,\gamma,\delta$ with $\alpha+\beta+\gamma+\delta=360^\circ$, $\beta<180^\circ$, ... | Consider a hypothetical solution $\square ABCD$ with diagonals meeting at $X$, and with angle measures and segment lengths as shown:
Then
$$\tan \alpha = \tan A = \tan(\alpha_1 + \alpha_2) = \frac{\tan\alpha_1+\tan\alpha_2}{1-\tan\alpha_1\tan\alpha_2} = \frac{\frac{d}{a}+\frac{b}{a}}{1-\frac{d}{a}\frac{b}{a}} = \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1057310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 0
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c(n,k) equals subdivisions To compare n files, the total comparison count is:
$$
{{n}\choose{k}} = C^k_n = \dfrac{n!}{k! ( n - k )!}
$$
with k = 2. Input space is composed by all pairs of files to compare. I want to split input space by the number of processor available to parallelize comparison. Dividing the input s... | There's an amusing solution in the case of $4$ processors.
Consider the space of all $(i,j)$ file indexes you're going to compare. This is a triangle, the half of a square with side $n$.
You can split this triangle into four identical ones by means of the other diagonal and the medians.
This generalizes to all powers ... | {
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Problem of Partial Differential Equations
For this question, I get stuck when I apply the second initial equation. My answer is $θ= Ae^-(kλ^2 t)\cos λx$, where $A$ is a constant. Would anyone mind telling me how to solve it?
| Notice that the general solution to the heat equation with Neumann boundary conditions is given by
$$ \theta(x,t) = \sum_{n=0}^{\infty} A_n\cos\bigg(\frac{nx}{2}\bigg)\exp\bigg(-k\bigg(\frac{n}{2}\bigg)^{2}t\bigg) $$
We are given
$$ \theta(x,0) = 2\pi x - x^{2} $$
But our series solution at $ t = 0$ is
$$ \theta(x,0) =... | {
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"timestamp": "2023-03-29T00:00:00",
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Determine the number of zeros for $4z^3-12z^2+2z+10$ in $\frac{1}{2}<|z-1|<2$. I'm faced with the problem in the title
Determine the number of zeros for $4z^3-12z^2+2z+10$ in the annulus $\frac{1}{2}<|z-1|<2$.
Clearly this requires a nifty application of Rouche's Theorem. Why this isn't so easy for me is because th... | Well I just figured out what to do while writing this, so I'm going to write the trick out here for anybody that might run across the same problem I had.
Just write the polynomial $4z^3-12z^2+2z+10$ as a Taylor Polynomial centered at $z=1$. You don't actually need to use Taylor's theorem for this - using algebra could ... | {
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Another way to prove that if n$^{th}$-degree polynomial $r(z)$ is zero at $n+1$ points in the plane, $r(z)\equiv 0$? The original problem is as follows
Let $p$ and $q$ be polynomials of degree $n$. If $p(z)=q(z)$ at $n+1$ distinct points of the plane, the $p(z)=q(z)$ for all $z\in \mathbb{C}$.
I attempted showing thi... | When people say “Fundamental Theorem of Algebra”, they usually mean the result that every complex polynomial has a complex root. This is far simpler than that, since it’s true over any field at all, not even of characteristic zero.
For a proof, let $\rho_1,\dots,\rho_n,\rho_{n+1}$ be the roots. Then by Euclidean divisi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1057794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Colouring graph's edges. Let $G$ be a graph in which each vertex except one has degree $d$. Show that if $G$ can be edge-coloured in $d$ colours then
(1) $G$ has an odd number of vertices,
(2) $G$ has a vertex of degree zero
Please help me with it.
| Hint:
*
*Let $u$ be the vertex of non-$d$ degree. We have that $\deg(u) < d$, so there is a color $c$ not used by that vertex.
*Remove all the edges of color $c$, either
*
*$\deg(u)$ is still smaller and we reduced the problem, or
*all the degrees are equal, but the sum of degrees we removed is an even number.
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove $x^2=-1$ has a solution in $\mathbb{Q}_p$ iff $p=1\mod 4$ Let $p$ be prime and let $\mathbb{Q}_p$ denote the field of $p$-adic numbers.
Is there an elementary way to prove $x^2=-1$ has a solution in $\mathbb{Q}_p$ iff $p=1\mod 4$?
I need this result, but I cannot find a reference. Can some recommend a good... | Are you familiar about number theory? This is equivalent to saying that $x^2 \equiv -1$ in $\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$ iff $p \equiv 1$ (mod $4$). But this is a basic fact on number theory, since
\begin{equation*}
\Big( \frac{-1}{p} \Big) = (-1)^{\frac{p-1}{2}} = 1 \text{ iff } p \equiv 1 (\text{ mod }4)
\e... | {
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How to Find the Matrix of a Linear Map If a map $f$ from the set of polynomials of degree 3 to the real numbers is given by $f(u) = u'(-2)$, how do I find the matrix that represents $f$ with respect to the bases $[1, t, t^2, t^3]$ and $[1]$?
I have worked out $f(1) = 0$, $f(t) = 1$, $f(t^2) = -4$ and $f(t^3) = 12$ and ... | Actually you are done already:
$$f \equiv \pmatrix{0&1&-4&12}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1058075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Derangement formula; proof by induction Proof by induction that
$
d_{n}=nd_{n-1}+(-1)^{n}
$
where $d_{n}$ is number of $n$-element derangements.
| Hints
The number of derangements is given by
$$d_n = n! \sum_{k=0}^n \frac{(-1)^k}{k!}.$$
To prove your relationship by induction, show that
*
*it is correct for the base case $n=1$, i.e. that $d_1 = 1 d_0 + (-1)^1$ by plugging in.
*it is correct in the inductive step; i.e., assuming that this is true for all $k=1,... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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asymptotics of sum I wanna find asymptotic of sum below
$$\sum\limits_{k=1}^{[\sqrt{n}]}\frac{1}{k}(1 - \frac{1}{n})^k$$
assume I know asymptotic of this sum (I can be wrong):
$$\sum\limits_{k=1}^{n}\frac{1}{k}(1 - \frac{1}{n^2})^k \sim c\ln{n}$$
So I use Stolz–Cesàro theorem and wanna show that
$$\sum\limits_{k=1}^{[\... | Another method you could try is rewriting your sum as
$$
\sum_{k=1}^{\lfloor\sqrt{n}\rfloor} \frac{1}{k} - \sum_{k=1}^{\lfloor\sqrt{n}\rfloor} \frac{1}{k} \left[1-\left(1-\frac{1}{n}\right)^k\right],
$$
then using Bernoulli's inequality to show that the second sum is $O(n^{-1/2})$.
| {
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"url": "https://math.stackexchange.com/questions/1058259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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There is a function which is continuous but not differentiable I have a function which is a convergent series:
$$f(x) = \sin(x) + \frac{1}{10}\sin(10x) + \frac{1}{100}\sin(100x) + \cdots \frac{1}{10^n}\sin(10^nx)$$
This function is convergent because for any E you care to specify, the function has a term which is small... | Notice that if you plug $x=k\pi$, where $k$ is any integer, into the derivative of the that function, you get $\infty$, thereby making the derivative discontinuous. This, in other words, means that the function is not differentiable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1058367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Why Rational Numbers do not include pairs $(a,b)$ with $b=0$? Let $X=Z\times Z$
If we have the relation $R$ on $X$ defined by $(a,b)R(c,d)$ if and only if $ad=bc$. Then, what is the problem if $b=0$?
Obviously, I'm not looking for the answer that we cannot divide by 0, but rather something more fundamental. I thought ... | As anorton said, this fails to be an equivalence relation when you include $(0,0)$.
You don't need to throw away all ordered pairs with $b=0$, though; you could just remove $(0,0)$, then get a valid equivalence relation on $\mathbb Z\times\mathbb Z\setminus\{(0,0)\}$. You could think of these as the "extended rational... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1058488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Do the columns of an invertible $n \times n$ matrix form a basis for $\mathbb R^n$? The columns of an invertible $n \times n$ matrix form a basis for $\mathbb R^n$.
I follow the definition from the text book, then I guess because the matrix is invertible, each vector in the matrix is linearly independent, thus the basi... | If you have an $n \times n$ invertible matrix, then the columns (of which there are $n$) must be independent. How do we know that these vectors -- the columns -- span $\mathbb{R}^n$? We know this because $\mathbb{R}^n$ is an $n$-dimensional space, so any independent set of $n$ or more vectors will span the space. Hence... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1058555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What are the double union ($\Cup$) and double intersection ($\Cap$) Operators? Finale of THIS.
Unicode says that $\Cup$ and $\Cap$ are double union and intersection, respectively. I was wondering if there was an actual operation that went with these symbols. If not, would these definitions make sense for these operator... | I think a more useful definition would be $A\Cup B:=\{a\cup b\mid a\in A, b\in B\}$, respectively $A\Cap B:=\{a\cap b\mid a\in A, b\in B\}$.
I've never seen the symbol before, but I could think of situations where that might be useful. I'm sure there are situations where the diagonal of $(A\cup B)^2$ (which you're usin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1058656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Unit sphere weakly dense in unit ball I'm studying for an exam and came across a problem: I want to prove that the unit sphere in a Hilbert space $\mathcal{H}$ is weakly dense in the unit ball.
I already had to prove that the unit ball is weakly closed, so the weak closure of the unit sphere is contained in the unit ba... | A basic neighbourhood of a point $p$ in the weak topology can be written in the form
$$U(y_1, \ldots, y_n; p) = \{x: \left|\langle y_j, x - p \rangle\right| < 1, j = 1 \ldots n\}$$
where $\{y_1, \ldots, y_n\}$ is any finite set of vectors in $\mathcal H$.
If $\|p \| < 1$, find $v$ so $\langle y_j, v\rangle = 0$ for $j ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1058727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Arithmetic Progressions in slowly oscillating sequences An infinite sequence ($a_0$, $a_1$, ...) is such that the absolute value of the difference between any 2 consecutive terms is equal to $1$. Is there a length-8 subsequence such that the terms are equally spaced on the original sequence and the terms form an arithm... | Far from a full answer, but I have some (hopefully) new information. Length $4$ equally spaced, AP subsequences can be found from all finite $(a_n)_{n=1}^N$ with $N>10$ and $\forall n(|a_{n+1}-a_n|=1)$. This can very easily be brute forced, as there exist only $2^N$ distinct length $N$ sequences which obey the absolute... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1058858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
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expression of the 4-tensor $f \otimes g$ in given basis Let $f$ and $g$ be bilinear functions on $\mathbb{R}^n$ with matrices $a = \{a_{ij}\}$ and $B = \{b_{ij}\}$, respectively. How would I go about finding the expression of the $4$-tensor $f \otimes g$ in the basis$$x_{i_1} \otimes x_{i_2} \otimes x_{i_3} \otimes x_{... | Let $\{e_1, \dots, e_n\}$ be the usual basis for $\mathbb{R}^n$, dual to $x_1, \dots, x_n$. We have $$f\otimes g = \sum_{1 \le i_1, i_2, i_3, i_4 \le n} f \otimes g(e_{i_1}, e_{i_2}, e_{i_3}, e_{i_4})x_{i_1} \otimes x_{i_2} \otimes x_{i_3} \otimes x_{i_4}.$$$($Proof: Let$$H = \sum_{1 \le i_1, i_2, i_3, i_4 \le n} f \ot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1058989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Lattice Paths problem I was assigned to determine the number of "lattice paths" that are in a 11 x 11 square.
Recalling that I can only go upwards and rightwards, here is my approach:
Note: The red square is the restricted area I cannot go through.
I computed the result in WolframAlpha, and I got the following:
(Corr... | Your solution is okay.
$$2\times\left[\binom{11}{0}\binom{11}{11}+\binom{11}{1}\binom{11}{10}+\binom{11}{2}\binom{11}{9}+\binom{11}{3}\binom{11}{8}\right]$$
If you go 'right-under' then it is for
certain that you will arrive $\left(11,0\right)$, $\left(10,1\right)$,
$\left(9,2\right)$ or $\left(8,3\right)$.
It cannot ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1059091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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evaluating a contour integral where c is $4x^2+y^2=2$ Consider the integral
$$\oint_C \frac{\cot(\pi z)}{(z-i)^2} dz,$$ where $C$ is the contour of $4x^2+y^2=2$.
The answer seems to be $$2 \pi i\left(\frac{\pi}{\sinh^2 \pi} - \frac{1}{\pi}\right)$$ but I do not know how to proceed.
I would be grateful if an answer con... | Hints.
(A) $\cot(\pi z)$ is a meromorphic function having residue equal to $\frac{1}{\pi}$ for every $z\in\mathbb{Z}$;
(B) $\frac{1}{(z-i)^2}$ is a meromorphic function with a double pole in $z=i$;
(C) If $\gamma$ is a simple closed curve in the complex plane and $f(z)$ is a meromorphic function with no singularities o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1059169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Great contributions to mathematics by older mathematicians It is often said that mathematicians hit their prime in their twenties, and some even say that no great mathematics is created after that age, or that older mathematicians have their best days behind them.
I don't think this is true. Gauss discovered his Theore... | Louis de Branges proved Bieberbach's Conjecture at age 53.
LINK wikipedia
(I found the picture when researching his birthdate)
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
"answer_count": 11,
"answer_id": 7
} |
proving gradient of a function is always perpendicular to the contour lines Can someone give an explanation of how such a proof would go, given a function
example: $y = f(x)$
| I have an intuitive (not formal, just to get a good image) answer for you: What is a gradient? It is the direction of fastest increase for the function $g$. What is a contour line? It is a line on which $g$ has the same value everywhere. But what direction must the gradient have with respect to the contour line to pre... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Integrate a periodic absolute value function \begin{equation}
\int_{0}^t \left|\cos(t)\right|dt = \sin\left(t-\pi\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor\right)+2\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor
\end{equation}
I got the above integral from https://www.physicsforums.com/threads/closed-for... | A quick experimentation gave me that
$$
\int_0^t\sin\left(\frac{s-t}{2}\right)\left|\sin\left(\frac{s}{2}\right)\right|\,\mathrm{d}s=
\mathrm{sign}\left(-\sin \left(\frac{t}{2}\right)\right) \left(2 \pi \cos \left(\frac{t}{2}\right) \left\lceil \frac{\left\lfloor \frac{t}{2 \pi }\right\rfloor }{2}\right\rceil +\sin \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1059400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
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How many ways are there to arrange k out of n elements in a circle with repetition? If you a set of the n elements, in how many ways $Q(n,k)$ can you take some of them and arrange them on a $k$-gon, when repetition of one element is allowed but rotations of one arrangement are not counted twice?
If I am not mistaken, t... | Ok, if I understand correctly what you want is basically the number of ways to color the vertices of a $k$-gon having $n$ colors available, but taking two colorings to be the same if you can rotate one of them so that they look the same.
To do this we need a little bit of group theory. The intuition is the following, f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Eigenvalue problem $y'' + \lambda y = 0,$ $y'(0) = 0$, $y(1) = 0$
Find the eigenvalues of
$$y'' + \lambda y = 0, \; y'(0) = 0, y(1) = 0$$
For $\lambda >0$,
$$y(x) = c_1 \cos(\sqrt{\lambda} x) + c_2 \sin(\sqrt{\lambda}x)$$
We get that $y'(0) = 0 \implies c_2 = 0$, but when I try to solve for $\lambda$ when doing $y(... | Apply your boundary condition at $x=1$:
$$
y(1)=c_1\cos(\sqrt{\lambda})=0\implies \sqrt{\lambda}={(2n-1)\pi\over 2},\ n=1,2,\dots
$$
(recall that the zeros of the cosine function occur at odd multiples of $\pi/2$), but then squaring both sides,
$$
\lambda=\lambda_n=\left({(2n-1)\pi\over 2}\right)^2,\ n=1,2,\dots
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1059571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Difficult general integral definite 0 to 1 $$\int_{0}^{1} \log^2(x)\cdot x^{k+1} dx$$
I tried integration by parts but it leads to an extremely complicated computation, which didnt lead me anywhere.
Then
I tried differentiating the beta function. That was partly successful. But the problem was when I substituted $k+2$ ... | The two answers are excellent, but there is a third giving something interesting, i.e. a relation to the Gamma function:
$$\int_0^1 \log(x)^2 x^{k+1}\ dx=$$
Substituting $u=-\log(x)$:
$$\int_0^{\infty} u^2 \exp(-(k+1)u)\exp(-u)\ du=$$
Substituting $v=(k+2)u$:
$$(k+2)^{-3}\int_0^{\infty} v^2 \exp(-v)\ dv= (k+2)^{-3} \Ga... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1059674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
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Tensor products- balanced maps versus bilinear When defining tensor products $M\otimes_R N$ over a commutative ring $R$ one can use a universal property with respect to bilinear maps $M\times N\rightarrow P$.
On the other hand, in the general case, for noncommutative rings one has to use balanced maps $M \times N \righ... | The right bilinear maps would yield a tensor product of right modules if we construct the corresponding universal object, where $mr\otimes n=m\otimes nr$. The tensor product of two right modules is again a right module. The corresponding construction can also be done for left modules, which is a left module, and we can... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Understanding proof of Peano's existence theorem I'm studying the proof of Peano's existence theorem on this paper.
At page 5 it is said that the problem
$$\begin{cases}
y(t) = y_0 & \forall t ∈ [t_0, t_0 + c/k] \\
y'(t) = f(t − c/k, y(t − c/k)) & \forall t ∈ (t_0 + c/k, t_0 + c]
\end{cases}
$$
has a unique solution.... | The proof proceeds stepwise. At the outset, the solution is given for $t\in[t_0,t_0+c/k]$. Note carefully that then the right hand side of the second equation $y'(t)=f(t-c/k,y(t-c/k))$ is known for $t\in[t_0+c/k,t_0+2c/k]$, so you can integrate:
$$y(t)=y(t_0+c/k)+\int_{t_0+c/k}^\tau f(\tau-c/k,y(\tau-c/k))\,d\tau,\qqua... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Open vs Closed Sets for studying topology Topologies can be defined either in terms of the closed sets or the open sets. Yet most proofs, examples, problems, etc. in standard texts concern the open sets.
I would think closed sets are easier and more intuitive for most people. So, is there a particular reason it is b... | Perhaps the reason is that many proofs and definitios rely on open neighbourhoods. I think that an open set that contains a point is more intuitive that a closed set that doesn't.
Take, for example, this definition of limit:
$\lim_{x\to a}f(x)=L$ if for every $\epsilon>0$ there exists some $\delta>0$ such that $|a-x|\g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1059987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find the field of intersection Let $\mathbb{F}_{p^{m}}$ and $\mathbb{F}_{p^{n}}$ be subfields of $\overline{Z}_p$ with $p^{m}$ and $p^{n}$ elements respectively. Find the field $\mathbb{F}_{p^{m}} \cap \mathbb{F}_{p^{n}}$.
Could you give me some hints how I could show that??
| Hints:
For all prime $\;p\;$ and $\;n\in\Bbb N\;$ :
$$\begin{align}&\bullet\;\; \Bbb F_{p^n}=\left\{\;\alpha\in\overline{\Bbb F_p}\;:\;\;\alpha^{p^n}-\alpha=0\;\right\}\\{}\\
&\bullet\;\; \Bbb F_{p^n}\le\Bbb F_{p^m}\iff n\mid m\;\;\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1060072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Help with a recurrence with even and odd terms I have the following recurrence that I've been pounding on:
$$
a(0)=1\\
a(1)=1\\
a(2)=2\\
a(2n)=a(n)+a(n+1)+n\ \ \ (\forall n>1)\\
a(2n+1)=a(n)+a(n-1)+1\ \ \ (\forall n\ge 1)
$$
I don't have much background in solving these things, so I've been googling around looking for ... | This isn't a full answer, but might make a solution more tractable.
Define $S(n) = a(n+2) - a(n)$. (FYI I wasn't able to find $S(n)$ sequence in OEIS either)
Then, we get that
$S(2n) = a(2n+2) - a(2n) = a(n+2) + a(n+1) + (n+1) - a(n+1) - a(n) - n = a(n+2) - a(n) + 1 = S(n) + 1$
and
$S(2n+1) = a(2n+3) - a(2n+1) = a(n+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1060154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Basic set theory proof about cardinality of cartesian product of two finite sets I'm really lost on how to do this proof:
If $S$ and $T$ are finite sets, show that $|S\times T| = |S|\times |T|$.
(where $|S|$ denotes the number of elements in the set)
I understand why it is true, I just can't seem to figure out how to p... | OK here is my definition of multiplication:
$$m\cdot 0=0$$
$$m\cdot (n+1)=m\cdot n +m$$
(you need some such definition to prove something so basic.)
Now let $|T|=m$ and $|S|=n$.
If $n=0$ then $S=\emptyset $ and so $T\times S=\emptyset$ and we are done by the first case.
If $n=k+1$ let $s \in S$ be any element and let $... | {
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Prove that if $f$ is integrable on $[0,1]$, then $\lim_{n→∞}\int_{0}^{1} x^{n}f(x)dx = 0$.
Prove that if $f$ is integrable on $[0,1]$, then $\lim_{n→∞}\int_{0}^{1} x^{n}f(x)dx = 0$.
Attempt: this exercise has two parts. I did part a) already. From part a) we know that $g_{n} \geq 0$ is a sequence on integrable functi... | If you know that $f$ is bounded, so that $-M \le f \le M$, then you have $\int_0^1 x^n f(x)\,dx \le M \int_0^1 x^n\,dx$. But you can compute $\int_0^1 x^n\,dx$ directly and show that it converges to 0. Likewise, $\int_0^1 x^n f(x)\,dx \ge -M \int_0^1 x^n\,dx$. Use the squeeze theorem.
Alternatively, use the fact tha... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Evaluating $\frac{d}{dx} \int_{1}^{3x} \left(5\sin (t)-7e^{4t}+1\right)\,\mathrm dt$
$$\dfrac{d}{dx} \int_{1}^{3x} \left(5\sin (t)-7e^{4t}+1\right)\,\mathrm dt$$
The answer I come up with is: $5\sin(3x)(3)-7e^{4(3x)}(3)$, however this was not on the answer choice. What is the correct way to do this? Thanks.
| Let $$\begin{align}
f(x)&=\dfrac{d}{dx} \int_{1}^{3x} \left(5\sin (t)-7e^{4t}+1\right)\,\mathrm dt\\
&=\dfrac{d}{dx} \int_{0}^{3x}\left( 5\sin (t)-7e^{4t}+1\right)\,\mathrm dt-\dfrac{d}{dx} \int_0^{1}\left(5\sin (t)-7e^{4t}+1\right)\,\mathrm dt\\
&=\dfrac{d}{dx} \int_{0}^{3x}\left( 5\sin (t)-7e^{4t}+1\right)\,\mathrm d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1060435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Calculating how many elements are in the product of Cartesian multiplication Let $A = \{1,3\}, B = \{1, 2\}, C = \{1, 2,3\}$.
How many elements are there in the set
$\{(x,y,z) \in A \times B \times C | x + y = z \} $ ?
Two things I'm not familiar with here,
First, how do I do Cartesian multiplication between 3 sets... | The Cartesian product of the three sets $A$, $B$, and $C$ is just the triplet $(a,b,c)$ such that $a∈ A$, $b∈B$, and $c∈C$. We wish to find any and all $(a,b,c)$ that satisfies the property $a+b=c$, where $a∈ A$, $b∈B$, and $c∈C$.
So $1+1=2$, and $1∈A$, $1∈B$, and $2∈C$, so we found one element, namely $(1,1,2)$.
$1+2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1060647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $(E\cup Z_1)\setminus Z_2$ has the form $E\cup Z$ I'm trying to do an exercise as follows:
Let $(X, {\mathbf X}, \mu)$ be a measure space and let ${\mathbf Z}=\{E\in {\mathbf X}:\mu(E)=0\}$. Let $\mathbf X'$ be the family of all subsets of $X$ of the form $(E\cup Z_1)\setminus Z_2, E\in \mathbf X$, where $Z_... | Clearly $E \cup Z = (E \cup Z) \setminus \emptyset \in \mathbf X'$.
On the other hand, suppose that $E \in \mathbf X$, $N_1,N_2 \in \mathbf Z$, $Z_1 \subset N_1$, and $Z_2 \subset N_2$.
Now work out that $$(E \cup Z_1) \setminus Z_2 = (E \setminus N_2) \cup \bigg[(E \cap (N_2 \setminus Z_2)) \cup (Z_1 \setminus Z_2)\b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Probability of guessing the colors of a deck of cards correctly 10 years ago when I was about 15 I sat down with a deck of shuffled cards and tried to guess if the next card in the deck would be red or black. In sequence I guessed 36 cards correctly as red or black, at that point my older bother came in and took told m... | If you're aiming to guess the first 36 cards right with as high a probability as possible, then the best you can do is to choose some sequence of guesses with 18 reds and 18 blacks, and all those sequences are equally likely.
Each of them has probability
$\frac{(26\times 25\times 24\times\dots\times9)^2}{52\times51\t... | {
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"url": "https://math.stackexchange.com/questions/1060782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How find $a,b$ if $\int_{0}^{1}\frac{x^{n-1}}{1+x}dx=\frac{a}{n}+\frac{b}{n^2}+o(\frac{1}{n^2}),n\to \infty$ let
$$\int_{0}^{1}\dfrac{x^{n-1}}{1+x}dx=\dfrac{a}{n}+\dfrac{b}{n^2}+o(\dfrac{1}{n^2}),n\to \infty$$
Find the $a,b$
$$\dfrac{x^{n-1}}{1+x}=x^{n-1}(1-x+x^2-x^3+\cdots)=x^{n-1}-x^n+\cdots$$
so
$$\int_{0}^{1}\dfrac... | $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
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$[1-x_n]^{n}\to 1$ implies that $n x_n\to 0$ as $n\to\infty$? As the title says $[1-x_n]^{n}\to 1$ implies that $n x_n\to 0$ as $n\to\infty$?
We know that $\left(1+\frac{x}{n} \right)^n \to e^x$ as $n\to\infty$. This implies (not sure why) that $\left(1+\frac{x}{n} +o(1/n) \right)^n \to e^x$.
So the result obtains sin... | For $n$ be sufficiently large, we have $(1-x_n)^n >0$ (since it converges to 1).
We can then say that $n \log(1-x_n)$ tends to zero. Necessarily $\log(1-x_n)$ tends to zero.
So $x_n$ tends to zero and we have that : $ \quad \log(1+x) \underset{0}{\sim} x$
Then $nx_n \to 0$.
| {
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How many invertible matrices are in $M_{2}(\mathbb{Z}_{11})$? I tried to solve this question but without a success.
How many invertible matrices are in $M_{2}(\mathbb{Z}_{11})$?
Thanks
| The answer is $ (11^2 - 1)(11^2 - 11) $
The first term is how many non-zero vectors there are in $ (Z_{11})^2 $, the other one is how many there are that are linearly independent of the first one chosen.
You can easily generalize that fact: the power of all invertible $ n\times n$ matrices over a finite field $\mathbb{... | {
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Evaluate $\lim_{x→0}\left(\frac{1+\tan x}{1+\sin x}\right)^{1/x^2} $ I have the following limit to evaluate:
$$ \displaystyle \lim_{x→0}\left(\frac{1+\tan x}{1+\sin x}\right)^{1/x^2} $$
What's the trick here?
| $$\lim_{x\to 0}\left(\frac{1+\tan x}{1+\sin x}\right)^{\frac{1}{x^2}}$$ $$=\exp\left(\lim_{x\to 0}{\frac{1}{x^2}}\left(\frac{1+\tan x}{1+\sin x}-1\right)\right)$$
$$=\exp\left(\lim_{x\to 0}{\frac{1}{x^2}}\left(\frac{\tan x-\sin x}{1+\sin x}\right)\right)$$
$$=\exp\left(\lim_{x\to 0}{\frac{1}{x^2}}\left(\frac{\frac{2tan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1061142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Closed-form of sums from Fourier series of $\sqrt{1-k^2 \sin^2 x}$ Consider the even $\pi$-periodic function $f(x,k)=\sqrt{1-k^2 \sin^2 x}$ with Fourier cosine series $$f(x,k)=\frac{1}{2}a_0+\sum_{n=1}^\infty a_n \cos2nx,\quad a_n=\frac{2}{\pi}\int_0^{\pi} \sqrt{1-k^2 \sin^2 x}\cos 2nx \,dx.$$ This was considered in th... | We have:
$$ \frac{\pi}{2}\,a_n=\int_{0}^{\pi}\sqrt{1-k^2\sin^2\theta}\cos(2n\theta)\,d\theta =\frac{k^2}{4n}\int_{0}^{\pi}\frac{\sin(2\theta)\sin(2n\theta)}{\sqrt{1-k^2\sin^2\theta}}\,d\theta.\tag{1}$$
If we set:
$$ b_m = \int_{0}^{\pi}\frac{\cos(2m\theta)}{\sqrt{1-k^2\sin^2\theta}}\,d\theta,\qquad c_m = \int_{0}^{\pi}... | {
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Finding the pdf of $(X+Y)^2/(X^2+Y^2)$ where $X$ and $Y$ are independent and normal
$X$ and $Y$ are iid standard normal random variables. What is the pdf of $(X+Y)^2/(X^2+Y^2)$?
I am guessing you would transform into polar coordinates and go from there, but I am getting lost. Do we need two variable transformations... | $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
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Evaluate the Cauchy Principal Value of $\int_{-\infty}^{\infty} \frac{\sin x}{x(x^2-2x+2)}dx$ Evaluate the Cauchy Principal Value of $\int_{-\infty}^\infty \frac{\sin x}{x(x^2-2x+2)}dx$
so far, i have deduced that there are poles at $z=0$ and $z=1+i$ if using the upper half plane. I am considering the contour integral... | Write $\sin{x} = (e^{i x}-e^{-i x})/(2 i)$. Then consider the integral
$$PV \oint_{C_{\pm}} dx \frac{e^{\pm i z}}{z (z^2-2 z+2)} $$
where $C_{\pm}$ is a semicircular contour of radius $R$ in the upper/lower half plane with a semicircular detour into the upper/lower half plane of radius $\epsilon$. For $C_{+}$, we hav... | {
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Prove by induction that an expression is divisible by 11
Prove, by induction that $2^{3n-1}+5\cdot3^n$ is divisible by $11$ for any even number $n\in\Bbb N$.
I am rather confused by this question. This is my attempt so far:
For $n = 2$
$2^5 + 5\cdot 9 = 77$
$77/11 = 7$
We assume that there is a value $n = k$ such th... | Keep going!
$64\cdot 2^{3k-1} + 45\cdot 3^k = 9(2^{3k-1} + 5\cdot3^k) + 55\cdot2^{3k-1}$
| {
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Why is the grade of the wedge product of two arbitrary blades the sum of the two blades' grades independently? I'm reading Geometric Algebra For Computer Science, An Object Oriented Approach to Geometry and it says that this is true of any two arbitrary blades.
$\ grade( \textbf{ A} \wedge \textbf{B})=grade( \textbf{ A... | I don't think there's any issue here. $(e_1 \wedge e_2) \wedge (e_2 \wedge e_3)$ is the zero 4-vector. It should not be confused with the zero scalar, although in geometric algebra, we can and often do use the same symbol (0) to denote any zero $k$-vector, or whole linear combinations of these zero $k$-vectors.
I wou... | {
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Why do we focus so much in math on functions (as a subclass of relations)? Why is it that math so focuses on the subclass of relations known as functions? I.e. why is it so useful for us in nearly all branches of mathematics to focus on relations which are left-total and left-unique? Left- (or even right-) totality see... | A function models a deterministic computation: if you put in $x$, you always get out the same result, $f(x)$, hence the left-uniqueness.
The asymmetry of the definition (left uniqueness rather than right uniqueness) is because the left side models the input and the right side models the output, and the input is logic... | {
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Conjugation map on a complex vector space Let $V$ be a complex vector space and $J:V\to V$ a map with the following properties:
(i) $J(v+w)=J(v)+J(w)$
(ii) $J(cv)=\overline{c}J(v)$
(iii) $J(J(v))=v$
for all $v,w\in V$ and $c\in\mathbb{C}$. Put $W=\{v\in V\,|\, J(v)=v\}$; this is a real vector space with respect to the ... | Given $z \in \mathbb{C}$, we have $\operatorname{Re}(z) = \frac{1}{2}(z + \bar{z})$ and $\operatorname{Im}(z) = \frac{1}{2i}(z - \bar{z})$. Let's try the analogous thing, where in place of $\mathbb{C}$ we have $V$, and instead of conjugation, we have $J$.
Let $w = \frac{1}{2}(v + J(v))$ and $u = \frac{1}{2i}(v - J(v))$... | {
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When to use Binomial Distribution vs. Poisson Distribution?
A bike has probability of breaking down $p$, on any given day.
In this case, to determine the number of times that a bike breaks down in a year, I have been told that it would be best modelled with a Poisson distribution, with $\lambda = 365\,p$.
I am wonder... | For Binomial, we assume the bike breaks in a given day and it doesn't break again that day (one Bernoulli trial). For Poisson, bike might get fixed and break yet again in the same day (as said by @JMoravitz). Still, if the chosen time interval (day is an arbitrary choice) is narrowed down to so small that likelihood of... | {
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What is a counting number? The definition of natural number is given as The counting numbers {1, 2, 3, ...}, are called natural numbers. They include all the counting numbers i.e. from 1 to infinity. at the link http://en.wikipedia.org/wiki/List_of_types_of_numbers.
What does counting number mean, are there any number... | The counting numbers are $\mathbb{Z}^{+} = \big\{1, 2,3....,∞\big\} $ also known as the natural numbers.
Even though the counting numbers go on on forever, they can be counted.
Given a set $X$,
$X$ is denumerable or enumerate if there is a bijection $\mathbb{Z}^{+} → X$.
A set is countable if either it is finite or ... | {
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Is that true that the normalizer of $H=A \times A$ in symmetric group cannot be doubly transitive? Is that true that the normalizer of $H=A \times A$ in the symmetric group cannot be doubly transitive where $H$ is non-regular, $A \subseteq H$ and $A$ is a regular permutation group?
| Let the group $A$ act regularly on a set $X$. Then the centralizer of $A$ in the symmetric group $S_X$ is a regular group $A'$ isomorphic to $A$. (The centralizer of the (image of the) left regular representation of $A$ is the right regular representation.) Note that $A \cap A' = Z(A)$, so we only get $A \times A$ acti... | {
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Reading the coefficients with Maple If I have
$$
a+(b-c \cdot a) \cdot x = 5+7x
$$
then I know that
$$
a = 5
$$
and
$$
b-c \cdot a = b-5c = 7
$$
Can I get this result with Maple?
If I just write solve(a+(b-c*a)*x = 5+7*x) it will solve it instead of just 'reading' the coefficient.
| You could do as follows
lhs:=a+(b-c*a)*x
rhs:=5+7x
solve(coeff(lhs,x,0)=coeff(rhs,x,0),a)
This will output $a=5$. You are solving the equation that the coefficients of the 0 degree terms on each side are equal. To solve for $a$ and say $b$ simultaneously you could try
solve({coeff(lhs,x,0)=coeff(rhs,x,0),coeff(lhs... | {
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Evaluate $\int_0^{\pi/2}x\cot{(x)}\ln^4\cot\frac{x}{2}\,\mathrm dx$ How to evaluate the following integral ?:
$$
\int_{0}^{\pi/2}x\cot\left(\, x\,\right)\ln^{4}\left[\,\cot\left(\,{x \over 2}\,\right)\,\right]\,{\rm d}x
$$
It seems that evaluate to
$$
{\pi \over 16}\left[\,
5\pi^{4}\ln\left(\, 2\,\right) - 6\pi^{2}\zet... | Let $$J = \int_0^1 {\frac{{\arctan x{{\ln }^4}x}}{x}dx} \qquad K = \int_0^1 {\frac{x{\arctan x{{\ln }^4}x}}{{1 + {x^2}}}dx}$$
Then by M.N.C.E.'s comment, $$\tag{1}I = \int_0^{\pi /2} {x\cot x{{\ln }^4}\left( {\cot \frac{x}{2}} \right)dx} = 2J - 4K$$
Here is a symmetry of the integrand that we can exploit:
$$\begin{al... | {
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I am working on proving or disproving $\cos^5(x)-\sin^5(x)=\cos(5x)$ True or false? $$\cos^5(x)-\sin^5(x)=\cos(5x)$$ for all real x.
I have no idea how to prove or disprove this. I tried to expand $\cos(5x)$ using double angle formula but I wasn't sure how to go from that to $$\cos^5(x)-\sin^5(x)$$
| $x=\dfrac{\pi}{4} \implies \cos^5 x-\sin^5 x=0 \neq\cos \dfrac{5\pi}{4}$.
| {
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How does Wolfram Alpha come up with the substitution $x = 2\sin u$? Integration/Analysis I have to integrate
$$
\int_0^2 \sqrt{4-x^2} \, dx
$$
I looked at the Wolfram Alpha step by step solution, and first thing it does is it substitutes
$x = 2\sin(u)\text{ and } \,dx = 2\cos(u)\,du$
How does it know to substitute $2\... | This is a common substitution technique known as Trig substitution, you substitute $x$ with a trigonometric function.
Typically, when something is in the form $\sqrt{a-x^2}$, you substitute $x=\sqrt a\sin u$
| {
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Number of winning tern in a deck of cards and other 3 related questions There is a deck made of $81$ different card. On each card there are $4$ seeds and each seeds can have $3$ different colors, hence generating the $ 3\cdot3\cdot3\cdot3 = 81 $ card in the deck.
A tern is a winning one if,for every seed, the correspon... | For question one, ask a simpler question: Look at just the first seeds, how many ways can you color them?
Two possibilities are: Card 1: Red, Card 2: Red, Card 3: Red
and Card1:Red, Card2: Blue, Card3: Yellow.
Now for four seeds, you make the choice of colors four times.
You should check that you haven't chosen the sam... | {
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Is "being harmonic conjugate" a symmetric relation? The question is:
Prove or disprove the following: If $u,v:\mathbb{R}^2 \to \mathbb{R}$
are functions and $v$ is a harmonic
conjugate of $u$, then $u$ is a harmonic conjugate of $v$ (in other words, show whether or
not the relation of being a harmonic conjugate is symm... | Here's the simplest example I can think of. $f(z)=z$ is clearly holomorphic, but $g(z)=Im(z)+iRe(z)$ is not, because when you look at the second Cauchy-Riemann equation you get $1 \neq -1$.
| {
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Limit of the geometric sequence $\{r^n\}$, with $|r| < 1$, is $0$?
Prove that the $\lim_{n\to \infty} r^n = 0$ for $|r|\lt 1$.
I can't think of a sequence to compare this to that'll work. L'Hopital's rule doesn't apply. I know there's some simple way of doing this, but it just isn't coming to me. :(
| If $r=0$ it's trivial, so we can skip this case. Now assume $r\neq0$.
$$\lim_{n\to\infty}r^n=0 \Longleftrightarrow(\forall \epsilon \in \mathbb{R}^+)(\exists \delta\in\mathbb{R})(\forall n_{> \delta})(|r^n| < \epsilon)$$
Now we can note, that both sides of above inequality are positive. We can use logarithms.
$$|r^n|<... | {
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For a $C^1$ function, the difference $|{g'(c)} - {{g(d)-g(c)} \over {d-c}} |$ is small when $|d-c|$ is small
Suppose $g\in C^1 [a,b]$. Prove that for all $\epsilon > 0$, there is $\delta > 0$ such that $|{g'(c)} - {{g(d)-g(c)} \over {d-c}} |{< \epsilon }$ for all points $c,d \in [a,b]$ with $0 <|d-c|< \delta$
First, ... | This problem has nothing to do with the mean value theorem. This problem is designed to test your understanding of uniform continuity and/or compactness. It seems likely to me that you have just learned (in the context of whatever book or class this is from) the compactness argument that a continuous function on a comp... | {
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"Novel" proofs of "old" calculus theorems Every once in a while some mathematicians publish (mostly on the American Mathematical Monthly) a new proof of an old (nowadays considered "basic") result in analysis (calculus).
This article is an example.
I would like to collect a "big list" of such novel proofs of old result... | Andrew Bruckner's survey paper Current trends in differentiation theory includes a lot of examples of new simple proofs of results previously having difficult proofs, but most of the examples are probably past the level you want. Probably more appropriate would be the use of full covers in real analysis.
(ADDED NEXT DA... | {
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If a function is convergent and periodic, then it is the constant function. I have to prove that if a function f is convergent : $$ \lim\limits_{x\to +\infty} f(x) \in \mathbb{R}$$ and f is a periodic function :
$$\exists T \in \mathbb{R}^{*+}, \forall x \in \mathbb{R}, f(x + T) = f(x)$$
Then f is a constant function.
... | Argue by contradiction. Suppose $f$ is not constant. Then there exist $a,b \in \mathbb{R}$ with $f(a) \neq f(b)$. Now suppose $c = \lim_{x \to \infty} f(x)$. Then for all $
\epsilon > 0$ there exists an $N > 0$ such that $|c - f(x)| < \epsilon$ for all $x > N$. Since $f$ is periodic, there exists $x > N$ with $f(x)=f(a... | {
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Probability generating function for urn problem without replacement, not using hypergeometric distribution UPDATE: Thanks to those who replied saying I have to calculate the probabilities explicitly. Could someone clarify if this is the form I should end up with:
$G_X$($x$) = P(X=0) + P(X=1)($x$) + P(X=2) ($x^2$) + P(X... | You don't need to use the formula for a hypergeometric distribution. Simply observe that the most number of balls you can draw before obtaining the first red ball is $3$, so the support of $X$ is $X \in \{0, 1, 2, 3\}$. This is small enough to very easily compute explicitly $\Pr[X = k]$ for $k = 0, 1, 2, 3$.
| {
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How to sum $\sum_{n=1}^{\infty} \frac{1}{n^2 + a^2}$? Does anyone know the general strategy for summing a series of the form:
$$\sum_{n=1}^{\infty} \frac{1}{n^2 + a^2},$$
where $a$ is a positive integer?
Any hints or ideas would be great!
| Use the fractional expansion of $\cot z$, you can get:
$$\frac{1}{e^t-1} -\frac{1}{t} +\frac{1}{2} =\sum_{n=1}^{\infty}\frac{2t}{t^2 +4n^2\pi^2} $$
| {
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Why can a matrix whose kth power is I be diagonalized? Say A is an n by n matrix over the complex numbers so that A raised to the kth power is the identity I. How do we show A can be diagonalized?
Also, if alpha is an element of a field of characteristic p, how do we show that the matrix A=[1, alpha; 0, 1] satisfies A ... | If you have the machinery of Jordan forms and/or minimal polynomials you can settle the questions with those. When working over $\Bbb{C}$ user152558's answer points at a useful direction, and Pedro's answer shows that this won't work over the reals.
Lacking such technology I proffer the following argument.
Remember tha... | {
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if $m, n \in \mathbb{N}$, $m < n$, then $S_m$ isomorphic to subgroup of $S_n$ How do show that if $m, n \in \mathbb{N}$ and $m < n$, then $S_m$ is isomorphic to a subgroup of $S_n$, without using any "overpowered" results?
| We first prove a lemma.
Lemma. Let $X$ be a set and let $G = \{f: X \to X\text{ }|\text{ }f\text{ is a bijection}\}$. This is a group under composition. Let $A \subseteq X$ be a subset, and let $H_A = \{f \in S_X\text{ }|\text{ }f(a) = a\text{ for all }a \in A\}$. Then $H_A$ is a subgroup of $S_X$.
Proof. We first show... | {
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"answer_id": 2
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Compact sets closed in Hausdorff spaces without choice? An elementary proof that compact sets are closed in Hausdorff spaces involves making arbitrary choices based on the Hausdorff property. Is there a way to avoid invoking choice?
| Yes, there is.
Let $\langle X,\tau\rangle$ be a Hausdorff space, and let $K$ be a compact subset of $X$. Suppose that $K$ is not closed. Then we can pick $p\in(\operatorname{cl}K)\setminus K$. (Note that this does not require $\mathsf{AC}$: it’s a single choice.) Let
$$\mathscr{U}=\{U\in\tau:p\notin\operatorname{cl}U\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1064394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Probability that Amy, Bill and Poor pete win money 6.042 MIT I was doing the following problem:
Problem 2. [12 points] Amy, Bill, and Poor Pete play a game:
*
*Each player puts \$2 on the table.
*Each player secretly writes a number between 1 and 4.
*They roll a fair, four-sided die with faces numbered ... | You've understood Pete's behavior well, as well as the independence of Pete's behavior and that of Amy & Bill's. But you haven't explored Amy & Bill's behavior.
The key is to understand what a pair of distinct numbers at random means.
Amy and Bill, as a pair, select one of:
$\begin{array}{cc}
\text{Amy} & \text{Bill} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1064475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Diagonalization and find matrix that corresponds to the given condition Diagonalize the matrix
$$
A=
\begin{pmatrix}
1 & 2\\
0 & 3
\end{pmatrix}
$$
and find $B^3=A$.
I derived $A \sim \text{diag}(1,3)$ but I have problem finding any $B$. I tried to solve it by writing $B= \begin{pmatrix} 1 & x\\ 0 & 3\end{pmatrix}$, bu... | The eigen values are $1,3 $ clearly. So it is diagonalizabe(distinct eigen values).
And so, there exits $P$ such that
$$A=P\left(\begin{array}{cc}1& 0\\ 0& 3\end{array}\right)P^{-1}.$$
Now we need $B$ such that $B$ such that $B^3=A$
Supose there exists such a $B$ then, $$B^3=A=P\left(\begin{array}{cc}1& 0\\ 0& 3\end{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1064546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
$x\in \{\{\{x\}\}\}$ or not? I wonder if we can we say $x\in \{\{\{x\}\}\}$?
In one viewpoint the only element of $\{\{\{x\}\}\}$ is $\{\{x\}\}$. In the other viewpoint $x$ is in $\{\{\{x\}\}\}$, for example all people in Madrid are in Spain.
| The answer to your question is no. Note that the set $ \{ \{ \{ x \} \} \} $ has one element, which is $ \{ \{ x \} \} $, not $x$. To say $ x \in \{ \{ \{ x \} \} \} $ means that $x$ is an element of $ \{ \{ \{ x \} \} \} $, implying that $ x = \{ \{ x \} \} $, which is not true if $x$ is a number.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1064663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 8,
"answer_id": 6
} |
Estimation of a sum independent of $n$ Suppose $f$ is differentiable on $[0,1]$, $f(0)=f(1)$, $\int_0^1 f(x)dx=0$, $f'(x)\neq 1$. Furthermore, let $g(x)=f(x)-x$, $n\geq 2$ is an integer.
Show that $$\left|\sum_{i=1}^{n-1}g\left(\frac{i}{n}\right)\right|<\frac12.$$
I do not know how to prove it, only can prove $$-\frac{... | let $$f(x)\equiv 0 \Longrightarrow g(x)=-x$$, and such all condition
But
$$\sum_{i}^{n-1}g\left(\dfrac{i}{n}\right)=-\sum_{i=1}^{n-1}\dfrac{i}{n}=-\dfrac{n-1}{2}\to -\infty,n\to \infty$$
In fact, I think your reslut is true
we have $f'(c)=0,c\in(0,1)$ since $f(0)=f(1)$
since $f'(x)\neq 1$,so $f'(x)<1$.
let $$g(x)=f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1064745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Rotate a line by a given angle about a point Given the coefficients of a line $A$ , $B$ and $C$.
$$ Ax + By + C = 0$$
I wish to rotate the line by angle say $\theta$ about a point $x_0$ and $y_0$ in clockwise direction. How can I achieve this so that I get new coefficients then?
| Take a point on this line, say $$A=\binom{x}{\frac{-C-Ax}{B}}$$ multiply the coordinates of this point by the rotation-matrix defined by $$R_\theta=\bigg(\matrix{\cos\theta &&-\sin \theta \\\sin\theta && \cos\theta}\bigg)$$ Immediately you get $$A^\prime=\binom{x\cos\theta+\frac{C+Ax}{B}\sin\theta}{x\sin\theta-\frac{C+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1064832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
If $(f\circ g)(x)=x$ does $(g\circ f)(x)=x$? Given $$(f◦g)(x)=x$$ (from R to R for any x in R)
does it mean that also $$(g◦f)(x)=x$$
I feel like its not true but I can't find counter example :(
I tried numerous ways for several hours but I cant counter it though I almost know for sure that this will only be true if ... | Let $f(x)=\begin{cases}\tan(x),&\text{if }x \in (-\pi/2,\pi/2)\\0,&\text{otherwise}\end{cases}$
and $g(x)=\tan^{-1}(x)$ with image $(-\pi/2,\pi/2)$.
Then $f(g(x))=x$ but $g(f(2\pi))=0\ne 2\pi$.
The hypothesis is indeed true if both $f$ and $g$ are continuous. $g(\mathbb{R})$ is an interval, and $f|_{g(\mathbb{R})}$ has... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1064935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
Dimension of a vector space below I have to prove that the dimension of the vector space of real numbers over Q (rational numbers) is infinity. How can I prove? I have no idea.
| Hint: Show that all $\log(p)$ for $p$ prime are linearly independent over $\mathbb{Q}$.
Apply the exponential function to the equation $\lambda_1\log(p_1)+\cdots +\lambda_n\log(p_n)=0$, and conclude that all $\lambda_i$ are zero. It follows that $\dim_{\mathbb{Q}}\mathbb{R}\ge n$ for all $n\ge 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1065041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
bounds of Riemann $\zeta(s)$ function on the critical line? I vaguely remembered that
$$0\leq|\zeta(1/2+i t)|\leq C t^{\epsilon},\qquad t>>1,\epsilon>0$$.
Is this bound correct?
Thanks-
mike
| In: http://www.math.tifr.res.in/~publ/ln/tifr01.pdf pp.97-99 , it is proven that:
$$\zeta(s) < A(d)t^{1-d}, \text{for } \sigma=\mathrm{Re}(s) \geq d, 0 < d < 1 ; t=\mathrm{Im}(s) \geq 1 .$$
with $A(d) = (1/(1-e) + 1 + 3/e)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1065157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Probability of selecting a red ball first An urn contains 3 red and 7 black balls. Players A
and B withdraw balls from the urn consecutively
until a red ball is selected. Find the probability that
A selects the red ball. (A draws the first ball, then
B, and so on. There is no replacement of the balls
drawn.)
How do I c... | Here's what I was thinking.
$$\color{RED}R+BB\color{RED}R+BBBB\color{RED}R+BBBBBB\color{RED}R$$
$$P(A)=\frac{3}{10}+\frac{7}{10}\frac{6}{9}\frac{3}{8}+\frac{7}{10}\frac{6}{9}\frac{5}{8}\frac{4}{7}\frac{3}{6}+\frac{7}{10}\frac{6}{9}\frac{5}{8}\frac{4}{7}\frac{3}{6}\frac{2}{5}\frac{3}{4}=\frac{7}{12}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1065249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Points in general position I'm really confused by the definition of general position at wikipedia.
I understand that the set of points/vectors in $\mathbb R^d$ is in general position iff every $(d+1)$ points are not in any possible hyperplane of dimension $d$.
However I found that this definition is equivalent to affi... | $\def\R{\mathbb{R}}$General position is not equivalent to affine independence. For instance, in $\R^2$, you can have arbitrarily many points in general position, so long as no three of them are collinear. But you can have at most 3 affinely independent points.
Any random set of points in $\R^d$ is almost certain (with ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1065337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Prove that $G$ is abelian iff $\varphi(g) = g^2$ is a homomorphism I'm working on the following problem:
Let $G$ be a group. Prove that $G$ is abelian if and only if
$\varphi(g) = g^2$ is a homomorphism.
My solution: First assume that $G$ is an abelian group and let $g, h \in G$. Observe that $\varphi(gh) = (gh)^2 ... | Assume $\varphi(g) = g^2$ is a homomorphism and $G$ is non-abelian; we will show a contradiction.
Since $G$ is non-abelian, we can choose $g, h \in G : gh \neq hg$.
Then since $\varphi$ is a homomorphism,
$$ \varphi(gh) = \varphi(g) \varphi(h) = gghh $$
So $$ ghgh = gghh $$
Left multiply be $g^{-1}$ and right multiply... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1065396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Prove that $T,S$ are simultaneously diagonalizable iff $TS=ST$. Definition: We say that $S,T$ are simultaneously diagonalizable if there's a basis, $B$ which composed by eigen-vectores of both $T$ and $S$
Show that $S,T$ are simultaneously diagonalizable iff $ST=TS$.
I tried both directions, but couldn't get much fur... | hint: If $S = P^{-1} D_S P$ and $T = P^{-1} D_T P$ with $D_{S,T}$ diagonal
then what are the values of $ST$ and $TS$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1065505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Equivalence of geometric and algebraic definitions of conic sections I have not been able to find a proof that the following definitions are equivalent anywhere, thought maybe someone could give me an idea:
*
*A parabola is defined geometrically as the intersection of a cone and a plane passing under the vertex of a... | All of these can be proved by using Dandelin spheres. And Dandelin spheres can also be used to prove that the intersection between a plane and a cylinder is an ellipse.
Both spheres in this picture touch but do not cross the cone, and both touch but do not cross the cutting plane. The points at which the spheres tou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1065655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Find the number of pathways from A to B if you can only travel to the right and down. I would like to solve the following, using Pascal's Triangle. Since there are shapes withing shapes, I am unsure as to where I should place the values.
EDIT 1:
Where do I go from here? How do I get the value for the next vertex?
EDI... | This problem can be solved by labeling every vertex with the number of ways you can get to $B$ by traveling only right and down. Then the label at vertex $A$ is the desired number.
As to how you can do this labeling, work backwards from $B$. Starting at the vertex directly above $B$, there is only one way to get to $B$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1065811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Probability: Random Sample Problem I need some help on the following problem:
Let $X_1$ and $X_2$ be random sample from the pdf
\begin{equation}
f(x) = \begin{cases}
4x^3,&0<x<1\\
0, & \text{otherwise}
\end{cases}
\end{equation}
Obtain $P(X_1X_2\geq 1/4)$.
So here is what I did:
$P(X_1X_2\geq 1/4)=1-P(X_1X_2< 1/4)$
Nex... | The distribution of $X_1X_2$ is $f(x,y)=16x^3y^3$, where I've used $x$ for $X_1$ and $y$ for $X_2$. The limits are just $0<x,y<1$ as before.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1065903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Evaluate $\int_0^4 \frac{\ln x}{\sqrt{4x-x^2}} \,\mathrm dx$
Evaluate
$$\displaystyle\int_0^4 \frac{\ln x}{\sqrt{4x-x^2}} \,\mathrm dx$$
How do I evaluate this integral? I know that the result is $0$, but I don't know how to obtain this. Wolfram|Alpha yields a non-elementary antiderivative for the indefinite inte... | First let $t = x-2$ this way $4x-x^2 = 4 - (x-2)^2 = 4-t^2$. Substitute,
$$ \int_{-2}^2 \frac{\log(t+2)}{\sqrt{4-t^2}} ~ dt $$
Now let, $\theta = \sin^{-1}\tfrac{t}{2}$ so that $2\sin \theta = t$ and hence, after substitute,
$$ \int_{-\pi/2}^{\pi/2} \frac{\log [2(1+\sin \theta)]}{2\cos \theta} 2\cos \theta ~ d\theta ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1066006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 2
} |
Show the intersection of a nonidentity normal subgroup and the center of P is not trivial P is p-group and M is a nontrivial normal subgroup of P. Show the intersection of M and the center of P is nontrivial.
By the class equation, I proved that Z(P)is not 1. Then, how do prove I the intersection of M and the center o... | The Class Equation for normal subgroups reads (for details, see here):
$$|M|=|M \cap Z(P)|+\sum_{m \in \{Orbits \space rep's\}}\frac{|P|}{|C_P(m)|} \tag 1$$
where:
*
*$C_P(m)$ is the centralizer of $m$ in $P$;
*"$Orbits$" (capital "O") are the conjugacy orbits in $M$ of size greater than $1$.
Now, by hyphothesis,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1066120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Hypothetical contradiction to Bolzano-Weierstrass we've learned about the Bolzano-Weierstrass theorem that states that if a sequence is bounded, then it has a subsequence that converges to a finite limit.
Let's define $a_n$ as the digits of $\pi$, i.e. $a_1$ = 3, $a_2$ = 1, $a_3$ = 4, and so on infinitely. Certainly t... | Sure, take the subsequence of every occurrence of $1$. If there aren't infinitely many $1$s, then $2$s; if not $2$s then $3$s, etc. As $\pi$ is irrational, at least one non-zero digit in the decimal expansion is repeated infinitely many times.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1066220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
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