Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Proof by induction that $3^n - 1$ is an even number How to demonstrate that $3^n - 1$ is an even number using the principle of induction?
I tried taking that $3^k - 1$ is an even number and as a thesis I must demonstrate that $3^{k+1} - 1$ is an even number, but I can't make a logical argument.
So I think it's wrong... | Hint: try substituting back in $3^k = 2x+1$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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First-order linear ordinary differential equation with piecewise constant source term Find a continuous solution satisfying the DE: $\frac{dy}{dx} + 2y = f(x)$
$$\begin{align}
f(x) &= \begin{cases}
1, & 0 \leq x \leq 3 \\
0, & x>3\text{.}\end{cases}\\
y(0)&=0\end{align}
$$
I don't get this problem at all. Can anyone ... | This ODE is first order linear. You've probably seen it as follows: Consider
$$ \left \{ \begin{array}{cc} y'(x) + p(x) y(x) = g(x) \\y(0) =0 \end{array} \right.$$
The solution is found using an integrating factor
$$ \mu (x) = \exp \left [ \int_{0}^x p(s) ds \right ]$$
and "factoring" the ODE to obtain
$$ y(x) = \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1319227",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Convolution of binomial coefficients As part of a (SE) problem I've been working on, I came up with this expression:
$$
\sum_{i=0}^M\binom{M-1+i}{i}\binom{M+i}{i}
$$
I'd like to get a closed form for this, but after a considerable amount of time searching my references and online sources (not to mention the time I've s... | Here is an alternate representation of the sum.
Suppose we seek to evaluate
$$S_M =
\sum_{q=0}^M {q+M-1\choose M-1} {q+M\choose M}.$$
Introduce
$${q+M\choose M} = \frac{1}{2\pi i}
\int_{|w|=\epsilon}
\frac{(1+w)^{q+M}}{w^{M+1}} \; dw.$$
and furthermore introduce
$$[[0\le q \le M]]
= \frac{1}{2\pi i}
\int_{|z|=\epsilon... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1319317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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confidence intervals with a mean How many people must be surveyed to estimate $90\%$ confidence for the mean, if the standard deviation is known to be $21.0$ and the allowable margin of error is $5$?
What I have so far is:
$\frac{1}{2}(1-0.90)=0.05$,
$Z= 1.645$
$(1.645\cdot 21.0/5)^2= 47.73$
Is that right?
| Yes, what you have is correct. Confidence intervals are represented as
$$\bar{X}\pm E$$
Where $E$ is the maximum error of estimate and
$$E=z_{\alpha/2}\left(\frac{\sigma}{\sqrt{n}}\right)$$
Solving the last equation for $n$ yields
$$n=\left(\frac{z_{\alpha/2}\cdot\sigma}{E}\right)^2$$
So for your values
$$n=\left(\f... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Monotone convergence theorem in a special case
Suppose $C^{+}[0,1]$ be the set of all continuous functions with
domain $[0,1]$ taking non-negative values only. Let $\lambda : C^{+}[0,1] \to [0,\infty)$ be a map that satisfies
$$\lambda(f+g)=\lambda(f)+\lambda(g)$$ Now suppose $\{g_k\}$ be a
sequence of functions... | We consider $f_n = g-g_n \in C^+([0,1])$. Then $f_n\ge f_{n+1}\ge \cdots$ and $\{f_n\}$ converges uniformly to the zero function. Consider the constant function $I(x) = 1$ and let $C = \lambda (I)$. One can show by induction that
$$\lambda \left(\frac 1k I\right) = \frac{C}{k}.$$
Now for each $k$, there is $n_k$ so th... | {
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An inequality, comprising of liminf and Sup of a set The following was an example in a book called Applied analysis by Hunter. I am not exactly sure of what it really means or to how to approach it
{ $ x_{n,\alpha}\in \mathbb R $ |$ n \in \mathbb N $ and $\alpha \in A $} is a set of real numbers indexed by the natural ... | Being indexed by two sets $X$ and $Y$ is effectively (or literally, depending on your definitions) the same as being indexed by the product space $X \times Y$. Your intuition, however is dead on. Just think of it that for each $a\in A$, you have a sequence.
Actually solving the problem more or less involves writing out... | {
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On the existence of field morphisms Let $K$ and $L$ be two fields, does the existence of two field morphisms $f\colon K\rightarrow L,\ g\colon L\rightarrow K$ imply that, as abstract fields, $K\cong L$ (not necessarily via $f$ or $g$)?
| I think I found a counterexample:
Let $F:=\overline{\Bbb Q(x_1,x_2,\dots)}$, i.e. the algebraic closure of the extension of $\Bbb Q$ by infinitely many independent transcendent elements.
Let $K:=F(x_0)$ a simple transcendent extension. This field is not algebraically closed.
Finally, let $L:=\overline K$, its algebraic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1319708",
"timestamp": "2023-03-29T00:00:00",
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Compute $I=\int e^{it}\cos t\,dt$ Compute $I:=\int e^{it}\cos t\,dt$, where $i\in\mathbb{C}$.
Attempt: Let $u=e^{it}$, $dv=\cos t\,dt$. Then $du=ie^{it}\,dt$ and $v=\sin t$
Using integration by parts, we have $I=u=e^{it}\sin t-i\int e^{it}\sin t\,dt$. Say $I_1:=\int e^{it}\sin t\,dt$. By the same method, we have $I_1=... | Your work is correct and means that, for a imaginary exponent, the classical double integration by part (*), that I suppose you well know for real exponent, does not work. Simply you have found that your primitive function $I$ is defined up to constant $i= e^{it \sin t}+i e^{it}\cos t$.
(*)
For a real exponent $a$ we... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Proof that ${{2n}\choose{n}} > 2^n$ and ${{2n + 1}\choose{n}} > 2^{n+1}$, with $n > 1$ i'm trying to proof these two terms. I started with an induction, but I got stuck...
Can anybody help?
| Induction works pretty well, since:
$$\frac{\binom{2n+2}{n+1}}{\binom{2n}{n}}=\frac{4n+2}{n+1}=2\cdot\frac{2n+1}{n+1}\geq 3 $$
and:
$$\frac{\binom{2n+3}{n+1}}{\binom{2n+1}{n}}=2\cdot\frac{2n+3}{n+2}\geq 3$$
as soon as $n\geq 1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Definition of exponential function, single-valued or multi-valued? If we define
$$e^z=1+z+\frac{z^2}{2!}+\cdots$$
then it is single-valued.
However, if we write
$$e^z=e^{z\ln e}$$
then it is multi-valued.
Besides, $a^z$ is multi-valued in general. It is kind of strange if only when the base is $e$ that it is single-val... | It's true that the complex map $\ln$ is multi-valued, but in the case of your second definition it doesn't matter, because $\exp$'s periodicity kills the extra arguments of $\ln$, so the definition becomes single-valued and perfectly ok.
For example, $\arg(e)=\Theta=0$ and $\ln|e|=1$, so your definition becomes:
$$e^{z... | {
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"timestamp": "2023-03-29T00:00:00",
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Pulling balls out of a box A box has $b$ blue balls and $r$ red balls. We pull the balls without returning them. What is the probability that the $k$th pull is the first red ball and what is the probability that the last ball is red?
Also: do you know a site which fully explains these kinds of problems and all the vari... | Answering the first question:
$\left(\prod\limits_{n=0}^{k-2}\dfrac{b-n}{b+r-n}\right)\cdot\left(\dfrac{r}{b+r-(k-1)}\right)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1320056",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "5",
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Jordan-homomorphism; equivalent properties Let $\phi:A\to B$ a linear map between $C^*$-algebras $A$ and $B$. I want to know, why the following properties are equivalent: $(i) \phi(ab+ba)=\phi(a)\phi(b)+\phi(b)\phi(a)$ and $(ii) \phi(a^2)=\phi(a)^2$ for all $a,b\in A$.
The direction => is ok: Let $a\in A$, it is $\phi(... | If we assume that $\varphi(a^2)=\varphi(a)^2$ for all $a$, then for $a,b$, we have, using linearity and the square-preserving property,
$\varphi((a+b))^2=(\varphi(a)+\varphi(b))^2=\varphi(a)^2+\varphi(a)\varphi(b)+\varphi(b)\varphi(a)+\varphi(b)^2.$
Now, observe that
$\varphi((a+b)^2)=\varphi(a^2+ab+ba+b^2)=\varphi(a^2... | {
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"timestamp": "2023-03-29T00:00:00",
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What would a base $\pi$ number system look like? Imagine if we used a base $\pi$ number system, what would it look like? Wouldn't it make certain problems more intuitive (eg: area and volume calculations simpler in some way)?
This may seem like a stupid question but I do not remember this concept ever being explored in... | The picture below may answer your queries.
My opinion is that it would help in some problems.
*
*Wikipedia: Non-integer representation
| {
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"url": "https://math.stackexchange.com/questions/1320248",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "16",
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Prove $ \lim\limits_{x\to\infty}y_{n}=\sqrt{x}$ if $y_{n}=\frac{1}{2}\left(y_{n-1}+\frac{x}{y_{n-1}}\right),n\in \mathbb{N},x>0,y_{0}>0$ Can someone say how to solve this problem? In solution, it says that it stars with $$\frac{y_{n}-\sqrt{x}}{y_{n}+\sqrt{x}}=\left(\frac{y_{n-1}-\sqrt{x}}{y_{n-1}+\sqrt{x}}\right)^2,n\g... | We have that
$$y_n=\frac12\left(y_{n-1}+\frac{x}{y_{n-1}}\right)\implies y_{n-1}^2-2y_ny_{n-1}+2x=0 \tag 1$$
Now, multiplying $(1)$ by $x^{1/2}$ yields the following obvious identity
$$y_{n-1}^2\sqrt{x}-2y_ny_{n-1}\sqrt{x}+2x\sqrt{x}=-(y_{n-1}^2\sqrt{x}-2y_ny_{n-1}\sqrt{x}+2x\sqrt{x}) \tag 2$$
Next, adding $y_ny_{n-1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1320326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Find $\lim\limits_{x\to \infty} \frac{x\sin x}{1+x^2}$
$$\lim_{x\to \infty} \frac{x\sin x}{1+x^2}$$
Using L'hopital I get:
$$\lim_{x\to \infty} \frac{x\cos x + \sin x}{2x}=\lim_{x \to \infty}\frac{\cos x}{2}$$
However, how is it possible to evaluate this limit?
| Some answers are saying L'Hopital is not applicable here because the numerator does not $\to \pm \infty.$ Actually L'Hopital is valid if we assume only the denominator $\to \pm \infty:$ If $f,g$ are differentiable on $(a,\infty), \lim_{x\to \infty} g(x) = \pm \infty,$ and $\lim_{x\to\infty} \frac{f'(x)}{g'(x)} = L,$ th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 4
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Show that $30 \mid (n^9 - n)$ I am trying to show that $30 \mid (n^9 - n)$. I thought about using induction but I'm stuck at the induction step.
Base Case: $n = 1 \implies 1^ 9 - 1 = 0$ and $30 \mid 0$.
Induction Step: Assuming $30 \mid k^9 - k$ for some $k \in \mathbb{N}$, we have $(k+1)^9 - (k+1) = [9k^8 + 36k^7 + 84... | And here are the congruences requested to complement the answer by Simon S
$$\begin{array}{c|ccccc}
n&n-1&n+1&n^2+1&n^4+1\\
\hline
0&4&1&1&1\\
1&0&2&2&2\\
2&1&3&0&4\\
3&2&4&0&2\\
4&3&0&2&2
\end{array}$$
And one can see that one of these factors is always $\equiv 0\pmod{5}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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Show that a zero of $f$ is a fixed point of $g$ I want to show that a solution of the equation $x^2+cos(x)-10x=0$ is a fixed point of $g(x)=(x^2+cos(x))/10$. I tried using the quadratic equation but my solution doesn't simplify nicely in $g$. I'm not sure how to deal with the $cos(x)$ term- any ideas?
| If $x$ solves $x^2+\cos(x)-10x=0$, then $x^2+\cos(x)=10x$. So
$$
g(x)=(x^2+\cos(x))/10=(10x)/10=x
$$
No need for the quadradic equation. I have absolutely no idea what values of $x$ even look like here. but certainly if you bothered to find one it would be a fixed point of $g$.
| {
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"timestamp": "2023-03-29T00:00:00",
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If $U=\bigcup_{j=1}^{\infty}K_j$ then $U=\bigcup_{j=1}^{\infty}\mathring{K}_j$ Let $U\subset\mathbb{R}^n$ be a domain (connected open set), such that $U=\bigcup_{j=1}^{\infty}K_j$ where each $K_j$ is a compact set, and $K_j\subset K_{j+1}$ for all $j\ge 1$.
Is the following proposition true?
$U=\bigcup_{j=1}^{\infty}\m... | No, consider $n=1$ with $U=(-1,1)$ and $K_n= [-1+1/n,-1/n]|\cup \{0\} \cup [1/n,1-1/n].$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Creating a CFL - based off unknown Regular language Suppose $A\subseteq\Sigma^{\ast}$ is a regular language.
Let $B=\{xy^R:x,y\in\Sigma^* , |x|=|y|, x \ XOR \ y \in A\}$
Prove that B is context free.
I am struggling with understanding B.
My only thought on how to start, is to assume A is in Chomsky Normal Form then use... | In your example, you say that $100$ and $001$ are in $B$, but that's not quite correct. I think you may be confusing $x$ and $y$ with the elements of $B$.
Let's let $A = \{001\}$. Then an $x$ and a $y$ such that $x \text{ XOR } y \in A$ could be $x = 111$, $y = 110$. So the string $xy^R = 111011$ would be in $B$.
Like ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Confused with $x$ and $a$ of Taylor Series Given general formula of Taylor Series:
$$T_{n}(x)=\sum_{n=0}^{\infty}{\frac{f^{(n)}(a)}{n!}(x-a)^{n}}$$
Why Taylor Series for $f(x)=\sin{x}$ is always evaluated at $x=0$ (the evalution means $a=0$) so we can get desired $\sin{x}$ value like $\sin{\pi}?$
| The derivatives of increasing order at $a$ are
$$\sin(a),\cos(a),-\sin(a),-\cos(a),\sin(a),\cos(a),-\sin(a),-\cos(a),\cdots$$periodically, and the development is
$$T_a(x)=\sin(a)+\cos(a)(x-a)-\sin(a)\frac{(x-a)^2}2-\cos(a)\frac{(x-a)^3}{3!}\cdots.$$
If you evaluate it at $x=a$, all terms but the first vanish:
$$T_a(a)=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1321148",
"timestamp": "2023-03-29T00:00:00",
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Finding strict Liapunov function Find a strict Liapunov function for the equilibrium point (0, 0) of
$$x' = −2x − y^ 2$$
$$y' = −y − x^2 .$$
Find δ > 0 as large as possible so that the open disk of radius δ and center (0, 0) is contained in the basin of (0, 0).
My attempt was $V = \frac{ax^2+by^2}{2}$, which gives
$... | Write $\dot{V}$ in this form:
$$\dot{V}=-(2a+by)x^2-(ax+b)y^2$$
For it to be less than $0$, we need
$$2a+by>0\implies y>-\frac{2a}{b}\\
ax+b>0\implies x>-\frac{b}{a}$$
So the disc has radius the minimum of $\frac{2a}{b}$ and $\frac{b}{a}$, or $2r$ and $\frac{1}{r}$, with $r=\frac{a}{b}$. Can you continue from here?
| {
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"timestamp": "2023-03-29T00:00:00",
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Improper integral: $\int_0^\infty \frac{\sin^4x}{x^2}dx$ I have been trying to determine whether the following improper integral converges or diverges:
$$\int_0^\infty \frac{\sin^4x}{x^2}dx$$
I have parted it into two terms. The first term: $$\int_1^\infty \frac{\sin^4x}{x^2}dx$$ converges (proven easily using the comp... | For the second term, you can re-write your function as $\frac{\sin x}{x} \cdot \frac{\sin x}{x} \cdot \sin^2(x)$. Note that on $[0,1]$, this function is continuous (by the Fundamental Trig Limit you can extend the function to be defined as $f(0)=1$ at $x=0$). But then any continuous function on a closed interval is i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Rooks Attacking Every Square on a Chess Board 8 rooks are randomly placed on different squares of a chessboard. A rook is said to attack all of the squares in its row and its column. Compute the probability that every square is occupied or attacked by at least 1 rook.
The first step I took was to state that there are ... | Consider e.g. In how many different ways can we place $8$ identical rooks on a chess board so that no two of them attack each other? for the number of ways that 8 rooks can be placed so that all squares are attacked or occupied.
Then, how many ways in total can you place 8 rooks regardless of attacked or occupied squar... | {
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Finding proper subfields Let $\omega$ denote the cube root of unity such that $\omega\neq 1$. I want to find the subfields properly contained in $\mathbb Q(\sqrt[3]{2},\omega)$ and containing $\mathbb Q$ properly.
Two of then certainly $\mathbb Q(\sqrt[3]{2})$ and $\mathbb Q(\omega)$. I have been told that the correct... | *
*$\mathbb Q(\omega + \sqrt[3]2)$ is actually equal to $\mathbb Q(\omega,\sqrt[3]2)$. (I can't say this with 100% certainty since I didn't work it out by hand, but it should be.)
*$\mathbb Q(\omega\sqrt[3]2)$ is a third subfield.
*$\mathbb Q(\omega^2\sqrt[3]2)$ is the fourth subfield.
You're probably wondering wh... | {
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Problem solving $2\times 2$ equation system $$\left\{\begin{align}
2x + 3y &= 10 \\
4x - y &= - 1
\end{align}\right.
$$
1) I don't get why I have to multiply first equation by $-2$ and not $+2$
2) How does $x=\frac{1}{2}$
I'm only $15$ so try and explain as simple as possible. Thanks.
| it don't matter whether you multiply by the first equation by $2$ or $-2.$ what you want to do is to go from a set of simultaneous equations of two variables $x, y$ to an equation with just one variable. in this example you are trying to eliminate $x$ so that you end up with an equation in only $y.$
if we multiply the ... | {
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Which Coefficient will Make a 2-Variable Linear System Solvable? This is most likely a pretty simple problem although my textbook doesn't quite explain how to solve it. I have a linear system with two equations and two variables (x and y) below:
2x - 5y = 9
4x + ay = 5 ('a' being the unknown coefficient of y)
The que... | The only thing that would keep the system from yielding a single value is if $2a+20=0$ or $a=-10$. In that case the loci of the two equations would be parallel. No intersections means no solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1321669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Coupled differential equation arising in flow line. So, I ran (certainly not literally) across these two coupled differential equation given by:
$$x'(t)=\left(x(t)\right)^2-\left(y(t)\right)^2 $$
$$ y'(t)=2x(t)y(t)$$
These equation occurred when I was trying to figure flow lines, to a vector field given by.
$\v... | what happens if we make a change of variables $$ x = r \cos \theta, y = r \sin \theta. $$ we have $$\begin{align} r'\cos \theta-r \sin \theta \, \theta' &= r^2 \cos (2\theta)\\
r'\sin \theta+r \cos \theta\, \theta' &= r^2 \sin 2 (\theta)\\\end{align} $$
from these we get $$r' = r^2\cos \theta, \theta' = r\sin \theta... | {
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"timestamp": "2023-03-29T00:00:00",
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} |
proof of existence While I was studying for my finals i found the following problem:
Let $f:[a,b]\rightarrow \mathbb{R}$ be a Riemann integrable real function, also let $\displaystyle \int\limits_a^bf(t)\,\mathrm{d}t=3$
Prove that there exist $\displaystyle t_1,t_2 \!\in \! (a,b):\int\limits_{t_1}^{t_2}f(t)\,\mathrm{d}... | Hint:
The function $F\colon [a,b]\to \mathbb{R}$ defined by $F(x) = \int_a^x f(t)dt$ is continuous, and $F(a)=0$, $F(b)=3$. Use the Intermediate Value Theorem on it.
Then:
You can find $c\in(a,b)$ such that $F(c)=2$. Can you continue?
And then:
Now, do a similar step with $G\colon [a,c]\to \mathbb{R}$ defined by ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1321819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Single variable improper integral Say I have an integral of $x/(1+x^2)$ that goes from negative infinity to infinity, and then part it into two integrals $A + B$ (let $A + B = I_\text{tot}$) where $A$ and $B$ have the limits from R to Infinity and negative infinity to R, respectively, and where R is any R-number. Now, ... | $$
\int_{-\infty}^r \frac x {1+x^2} \,dx = -\infty \text{ and }\int_r^\infty \frac x {1+x^2}\,dx = +\infty.
$$
One can say that one of these diverges to $-\infty$ and the other diverges to $+\infty$. However, notice also that
$$
\lim_{r\to\infty} \int_{-r}^r \frac x {1+x^2}\,dx = 0\text{ and } \lim_{r\to\infty} \int_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1321883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Calculate the limit?
The answer sheet says:
I understand all of this apaprt from the last bit where it says
I don't understand how/why they have put logs and exponentials in there.
| Since $exp$ and $log$ are reciprocals, then
\begin{equation*}
a^{b}=e^{\ln (a^{b})}=e^{b\ln a}.
\end{equation*}
So,
\begin{equation*}
\left( \frac{3}{4}\right) ^{x}=e^{x\ln \left( \frac{3}{4}\right) }=\frac{1}{%
e^{-x\ln \left( \frac{3}{4}\right) }}=\frac{1}{e^{x\ln \left( \frac{4}{3}%
\right) }}.
\end{equation*}
On th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1322081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Evaluating a sum (begginer) I am stuck on this evalutaion number.
So the sum is $$\sum_{k=1}^n{1\over (k+1)^2}-{1\over k^2}$$
I can't find a way :(. Do we use Comparasion or partial sums?
Thanks for the help!
| $$\sum_{k=1}^n\ \left(\frac{1}{(k+1)^2} - \frac{1}{k^2}\right)$$
$S_1 = \frac{1}{2^2} - \frac{1}{1^2}$
$S_2 =\left(\frac{1}{2^2} - \frac{1}{1^2}\right)+\left(\frac{1}{3^2}-\frac{1}{2^2}\right)$
$S_3 = \left(\frac{1}{2^2} - \frac{1}{1^2}\right)+\left(\frac{1}{3^2}-\frac{1}{2^2}\right)+\left(\frac{1}{4^2}-\frac{1}{3^2}\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1322148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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$n$-words from the alphabet $A=\{0, 1, 2, 3\}$. How many of them have an even number of zeros and ones? Consider all $n$-words from the alphabet $A=\{0, 1, 2, 3\}$. How many of them have an even number of zeros and ones?
I showed that the number of $n$-words from $\{0, 1, 2, 3\}$ with an even number of zeros is $\displ... | For a word $w$, let $w(k)$ be the number of times that $k$ appears in $w$, and $l(w)$ the length of $w$.
Let $A_n=\{w:l(w)=n,w(0)\text{ even and }w(1)\text{ even}\}$.
Let $B_n=\{w:l(w)=n,w(0)\text{ odd and }w(1)\text{ even}\}$.
Let $C_n=\{w:l(w)=n,w(0)\text{ even and }w(1)\text{ odd}\}$.
Let $D_n=\{w:l(w)=n,w(0)\text{ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1322232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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How do I solve $y''+y'+7y=t$? How do I solve $y''+y'+7y=t$ where $y(0)=0$ and $y'(0)=0$ $(t\geq 0)$?
I tried to solve this by Laplace transformation, but I couldn't find the inverse of $1/(s^2(s^2+s+7))$.
How would I solve this?
| You have the Laplace transform, so lets do a partial fraction decomposition. We want to find $a,b,c,d$ such that $$\frac{1}{s^2(s^2 + s +7)} = \frac{a}{s} + \frac{b}{s^2} + \frac{cs + d}{s^2 + s + 7}.$$
So we have
$$as(s^2 + s +7) + b(s^2 + s +7) + cs^3 + ds^2 = 1.$$
Collecting coefficients we have
$$ (a+c)s^3 + (a+b+d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1322380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Mathematical function for weighting football results Let's assume team A has scored x goals, and team B has scored y goals in a match.
I'm trying to find a function that returns a number between 0 and 1 based on the goals from the match.
Ideally, the function would reach 1 when y is 0 and x approaches infinity.
I have ... | One possibility is
$$f(x,y)=\frac{x+1}{x+y+2}$$
This never equals $1$ or $0$, but the more lopsided the win for $x$ the closer it gets to $1$. This also is symmetric in $x$ and $y$, as you said you wanted in a comment.
The score $1-0$ gives a value of $\frac 23$, while $10-0$ gives a value of $\frac{11}{12}$. This seem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1322464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$ Let $X=C[0,1]$ be the set of all continuous functions on $[0,1]$. For any two functions $f,g\in X$, set $$d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|.$$
Prove $(X,d)$ is a metric space.
My attempt:
This q... | For all $x \in [0,1]$ we have $$ |f(x)-g(x)|\leq |f(x)-h(x)|+|h(x)-g(x)|\leq \| f-h\|+\|h-g\|$$ and the last bound doesn't depend on $x$ so $\sup _{x\in [0,1]}|f(x)-g(x)|\leq \| f-h\|+\|h-g\|$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1322571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Bounding $\sum_{k=1}^N \frac{1}{1-\frac{1}{2^k}}$ I'm looking for a bound depending on $N$ of $\displaystyle \sum_{k=1}^N \frac{1}{1-\frac{1}{2^k}}$.
The following holds $\displaystyle \sum_{k=1}^N \frac{1}{1-\frac{1}{2^k}} = \sum_{k=1}^N \sum_{i=0}^\infty \frac{1}{2^{ki}}=\sum_{n=0}^\infty \sum_{\substack{q\geq 0 \\ 1... | We have $$\sum_{k\leq N}\frac{2^{k}}{2^{k}-1}=N+\sum_{k\leq N}\frac{1}{2^{k}-1}
$$ and recalling that holds the identity $$\sum_{k=1}^{\infty}\frac{1}{1-a^{k}}=\frac{\psi_{\frac{1}{a}}\left(1\right)+\log\left(a-1\right)+\log\left(\frac{1}{a}\right)}{\log\left(a\right)}
$$ where $\psi_{q}\left(z\right)
$ is the $q
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1322644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Finding a polynomial by divisibility
Let $f(x)$ be a polynomial with integer coefficients. If $f(x)$ is divisible by $x^2+1$ and $f(x)+1$ by $x^3+x^2+1$, what is $f(x)$?
My guess is that the only answer is $f(x)=-x^4-x^3-x-1$, but how can I prove it?
| One way you could express this idea is through functions $g,h$ where $$f(x) = g(x)(x^2+1) \\ f(x)+1 = h(x)(x^3+x^2+1)$$ This means that $$g(x)(x^2+1)+1 =h(x)(x^3+x^2+1)$$ or that $$h(x) = \frac{g(x)(x^2+1)+1}{x^3+x^2+1}$$ So you could begin by choosing any polynomial $g$ you want so long as it forces $h$ to be a polyno... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1322741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Trigonometric Equation involving tangent - How do I solve it? I would like to know how to solve this and most importantly make intuitive sense of it. Thanks in advance!
Equation: $\tan(x) = -1.5$
(using a calculator with $\arctan(-1.5)$)
My Results: $x = -0.9820 + 2 \pi n$
Book says the result is: $x = 2.1588 + 2\pi n... | When you have an equation $$\tan x = \tan \alpha$$ then there are an infinte number of solutions of x. The solutions such that $x\in(-\pi,\pi]$ are called the principle solutions. The general solution of the above equation is $$x=n\pi+\alpha\,;\;\;\;\;n\in\mathbb{Z}.$$ Here $\alpha = \arctan -1.5$
So the solutions are ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1322844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is there proof show that $\log x$ is undefined and make no sense at $ x=0$? I was asked by someone: why $\log x$ is undefined at $x=0 $?
Is there proof show that $\log x$ is undefined at $x=0$?
Note(01):: log is the inverse function of the exponential function.
note(02): I edited my question as I meant why it's not ma... | The natural logarithm is defined as
$$
\log x = \int_{1}^{x} \frac{1}{t} \, dt
$$
Naively plugging in $0$, we get
$$
\log 0 = \int_{1}^{0} \frac{1}{t} \, dt = - \int_{0}^{1} \frac{1}{t} \, dt,
$$
However, the area under $y = 1 / t$ from $0$ to $1$ diverges to $\infty$, which would imply
$$
\log 0 = - \infty
$$
Intuiti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1322925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 4
} |
Confusion with the formula to sum the terms of a finite geometric series The formula to sum the terms of a finite geometric series is the following:
$$\frac{a_1(1 - r^{n+1})}{1 - r}$$
where $a_1$ is the first term, $r$ is the common ratio, and $n + 1$ is the number we want to sum up to.
Now, my problem is really in thi... | In general, for $n\geq m$ and $r\neq 1$,
$$
\sum_{k=m}^n r^k = r^m \sum_{k=m}^n r^{k-m}
= r^m \sum_{\ell=0}^{n-m} r^{\ell} = r^m\cdot \frac{1-r^{n-m+1}}{1-r}
$$
To remember it: the exponent in the denominator is the number of terms in the sum.
To check (very basic check): when $n=m$, you only have one term, and it's $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1323037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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If $x,y$ is a 2-edge cut of a graph $G$; every cycle of G that contains $x$ must also contain $y$ $x,y$ are cut edges; if I understand the definition, it means that if we delete both $x$ and $y$ our resulting graph is disconnected.
I'm very confused because I started like this:
Let $C$ be a cycle that contains $x$ but ... | You missunderstood the definition of a cut. I hope is clear now thanks to @Gregory J. Puleo.
That being said here is my answer:
Let $C$ be a cycle that contains $x$ and that does not contain $y$. Since $x\in [S,\overline{S}]$ and $x \in C$ it must exist another edge $z\in[S,\overline{S}]$ to complete the cycle between... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Need a hint to evaluate $\lim_{x \to 0} {\sin(x)+\sin(3x)+\sin(5x) \over \tan(2x)+\tan(4x)+\tan(6x)}$ I know that $\sin A + \sin B + \sin C = 4\cos({A \over 2})\cos({B \over 2})\cos({C \over 2})$ when $A+B+C=\pi$. If ${x \to 0}$ then I have a half circle, right?
If it is right then I have
$\tan(2x) + \tan(4x) + \tan(6x... | In what follows I will provide full details of the idea of Abhishek Parab.
First divide top and bottom by $x,$ then
\begin{eqnarray*}
\frac{\sin x+\sin 3x+\sin 5x}{\tan 2x+\tan 4x+\tan 6x} &=&\frac{\left(
\dfrac{\sin x+\sin 3x+\sin 5x}{x}\right) }{\left( \dfrac{\tan 2x+\tan
4x+\tan 6x}{x}\right) } \\
&& \\
&=&\frac{\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1323196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
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Fourier transform of a Green's function I was studying for an exam and I found this question which has caused me a bit of trouble:
Given the Green's function that satisfies the equation
$$\Box G(\mathbf{r}-\mathbf{r}',t-t')=-4\pi\delta(\mathbf{r}-\mathbf{r}',t-t'),$$ where $\Box$ is the d'Alembertian operator
$$\Box=\D... | Generally you would Fourier transform the differential equation. The elements of the D'Alembertian transform to $\vec{k}\cdot\vec{k}$ (the Laplacian) and $-\omega^2$ (the 2nd time derivative), while the delta function transforms to $1$.
Then you get something like
$$ (2\pi)^4 \left(\left\lVert \vec{k}\right\lVert^2 - \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1323291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Experimental Units I am doing a paper airplane project for school.
I am doing a two-factor experiment, each with three levels, the factors are design of the plane and angle of launch. I am also having three people make a copy each of the airplane designs to account for variation when manufacturing the planes.
Am I corr... | If your factors are design of the plane and angle of launch you have three of each so $9$ cases. You have decided that the manufacturer and the repetition do not matter systematically, so repeating them will reduce the random variability. You have $81$ data points that you have decided to bin into $9$ bins of $9$ poi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1323373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Differentiability of $\cos |x-5|+\sin |x-3|+|x+10|^3-(|x|+4)$. Question : Test the differentiability of the function $$\cos |x-5|+\sin |x-3|+|x+10|^3-(|x|+4)$$ at the points $5, 3, -10, 0$.
Solution : Now $\cos |x|$ is differentiable everywhere. So is $\sin |x|$ as well as $|x|^3$. Therefore the only issue is the funct... | No, it is not correct. First, $\cos |x-5|$ does not vanish at $5$. Remember $\cos 0=1\neq 0$.
$\cos |x|$ is differentiable everywhere.
$\sin |x|$ is differentiable everywhere except at $x=0$.
More generally, $|x|^n$ is differentiable everywhere for any $n\in\Bbb Z_{\ge 2}$.
$|x|$ is differentiable everywhere but ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1323655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Combinatorics of given alphabet I'm looking for the formula to determine the number of possible words that can be formed with a fixed set of letters and some repeated letters. For instance take the 8-letter word SEASIDES and find out how many possible anagrams can be made: there are 3x S, 2x E, and 1 each of A, I and D... | You can use $$\displaystyle \sum_{s=0}^3 \sum_{e=0}^2 \sum_{a=0}^1 \sum_{i=0}^1 \sum_{d=0}^1 \dfrac{(s+e+a+i+d)!}{s!\,e!\,a!\,i!\,d!}$$ and if you like ignore the $ a!\,i!\,d!$ which is always $1$. This will give $9859$ possibilities. You may want to subtract $1$ if you want to exclude $0$-letter anagrams, leaving $98... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1323772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Partition Generating Function a)
Let
$$P(x)=\sum_{n=0}^{\infty} p_nx^n=1+x+2x^2+3x^3+5x^4+7x^5+11x^6+\cdots$$
be the partition generating function, and let $Q(x)=\sum_{n=0}^{\infty} q_nx^n$, where $q_n$ is the number of partitions of $n$ containing no $1$s.
Then $\displaystyle\frac{Q(x)}{P(x)}$ is a polynomial. What p... | Hint: Try to understand why
$$P(x)=\frac{1}{\prod_{n=1}^{\infty}\left(1-x^n\right)},$$
and what are the corresponding expressions for $Q(x)$ and $R(x)$.
Illustration:
\begin{align*} \frac{1}{1-x}\cdot \frac{1}{1-x^2} \cdot \frac{1}{1-x^3}\cdot\ldots =\,\Bigl(&1+x^1+x^{1+1}+x^{1+1+1}+\ldots\Bigr)\\ \times\,&\Bigl(1+x^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1323848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Simplifying sum equation. (Solving max integer encoded by n bits) Probably a lack of understanding of basic algebra. I can't get my head around why this sum to N equation simplifies to this much simpler form.
$$
\sum_{i=0}^{n-2} 2^{-i+n-2} + 2^i = 2^n - 2
$$
Background
To give you some background I am trying to derive... | The thing you've asked to show isn't too hard:
\begin{align}
\sum_{i=0}^{n-2} \left(2^{-i+n-2} + 2^i\right)
&= \sum_{i=0}^{n-2} \left(2^{-i+n-2}\right) + \sum_{i=0}^{n-2}\left(2^i\right) \\
&= \sum_{i=0}^{n-2} \left(2^{-i+n-2}\right) + \left(1 + 2 + \ldots + 2^{n-2}\right) \\
&= \sum_{i=0}^{n-2} \left(2^{-i+n-2}\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1323931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
finding out if two vectors are perpendicular or parallel I'm not sure if I quite get this. For example,
(1, -1) and (-3, 3)
take the cross product, you will end up with
-3 + (-3)
This doesn't equal 0, so it's not perpendicular. So that leaves me with it being parallel. When are two vectors parallel?
| They are parallel if and only if they are different by a factor i.e. (1,3) and (-2,-6).
The dot product will be 0 for perpendicular vectors i.e. they cross at exactly 90 degrees.
When you calculate the dot product and your answer is non-zero it just means the two vectors are not perpendicular.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1324005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
A collection $\{f_\alpha\}_{\alpha \in A}$ so that $\sup_{\alpha \in A} f_{\alpha}(x)$ is finite and non-measurable Background
Give an example of a collection of measurable non-negative functions $\{f_\alpha\}_{\alpha \in A}$ such that if $g$ is defined by $g(x)=\sup_{\alpha \in A} f_{\alpha}(x)$, then $g$ is finite fo... | Your proof is correct except specifying in more detail about
$$\{x:f_{\alpha}(x)>\beta\}\in \{\varnothing, \{\alpha\},\mathbb{R}\}$$
like
$$
\{x:f_{\alpha}(x)>\beta\}=\begin{cases}
\varnothing & \text{ if } \beta\geqslant1
\\
\{\alpha\} & \text{ if } 0\leqslant\beta<1
\\
\mathbb{R} & \text{ if } \beta<0
\end{cases}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1324125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Sum of weighted chi square distributions Let $X_1 \sim \chi_{k}^2$ and $X_2 \sim \chi_{k}^2$ be i.i.d and both $a_1$ and $a_2$ positive real values.
How can be expressed the PDF of $Y = a_1X_1 + a_2X_2$? Is it also a chi-square distribution?
thanks
| PDF of weighted sum of TWO iid chi-square distributions:
Please check page 5 Corollary 1
https://arxiv.org/pdf/1208.2691.pdf
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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What conditions must be checked for that $c$ is Cohen over $V$. $\textbf{Hechler forcing} $
Let $\mathbb{D}=\{(s,f): s \in \omega^{<\omega},f\in \omega^{\omega} \text{and} s \subseteq f\}$, ordered by $(t,g)\leq (s,f)$ if $s \subseteq t$, $g$ dominates $f$ everywhere, and $f(i) \leq t(i)$ for all $i \in |t|\setminus|s|... | A real $c\in2^\omega$ is Cohen over $V$ if and only if, for all $D\subseteq2^{<\omega}$, if $D$ is dense in $2^{<\omega}$ (meaning that every $s\in2^{<\omega}$ has an extension in $D$) and $D\in V$, then $D$ contains an initial segment $c\upharpoonright n$ of $c$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1324459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Prove that $K$ is finite Galois over $\mathbb{Q}$ I just need a bit of quick help in understanding some solutions to a problem set.
The question is this:
(a) Let $K=\mathbb{Q}(\alpha)$ with $\alpha$ a zero of $f(x) = x^3-3x+1$. Prove that $K$ is Galois over $\mathbb{Q}$.
I easily showed, $f(x)$ is irreducible, and det... | Have you heard of the discriminant? This is one way to show that K is Galois over the field of rational numbers.
| {
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"timestamp": "2023-03-29T00:00:00",
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Plotting a part of curves (with possible solution as an attempt) I may mess up with this question!
I plotted them using http://www.desmos.com/calculator
How to plot only a part of some curves?
For example:
Plot of $y=x^2$:
But if i want to plot the curve from $x=1$ to $x=2$ which will look something like:
How to a... | Rory's suggestion is quite good, and it should work for Desmos. But apparently what you want is to stealthily include domain information; that's what it will boil down to, as we'll see.
To this end, you can cook up auxiliary "characteristic functions" that are $1$ at least on the domain you want, and $0$ certain places... | {
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Strange logic question, truth of predictions 1: Half of my predictions come true;
2: I predict A;
3: I predict B.
Now, suppose A come true, so that the prediction 2 is true; and B come false.
So, half of my predictions came true and 1 is also true; BUT in this way, 2/3 of my predictions came true so 1 is false. Contrad... | This doesn't seem like a contradiction to me: (1) and (3) are false, and (2) is true.
Alternatively, we can make things more comlicated by introducing time (https://en.wikipedia.org/wiki/Temporal_logic) into the equation, so that there is a moment when (1) is true (immediately after (2) has been verified and (3) dispr... | {
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When does equality hold in this inequality? The following inequality can be proven as follows:
Let $n\geq3$ and $0=a_0<a_1<\dots<a_{n+1}$ such that $a_1a_2+a_2a_3+\dots+a_{n-1}a_n=a_na_{n+1}$. Show that
\begin{equation*}
\frac{1}{{a_3}^2-{a_0}^2}+\frac{1}{{a_4}^2-{a_1}^2}+\dots+\frac{1}{{a_{n+1}}^2-{a_{n-2}}^2}\g... | Deduced from the equality condition of Cauchy-Schwarz inequality, we have:
$$ \forall 0 \leqslant k \leqslant n-1, \, \, a_{k+3}^2 = a_k^2 + c$$
where $c > 0$ is a constant. So the equation has 3 possiblities that depends on $n$.
For example, if $n = 3K + 1$:
$$\sum_{k=0}^{K-1} (a_0 + kc)(a_1 + kc) + (a_1 + kc)(a_2 +... | {
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If $\phi: M_1 \to M_2$ a diffeomorphism between diff. manifolds, prove that if $M_2$ is orientable then so is $M_1$ Let $\phi: M_1 \to M_2$ a local diffeomorphism between two differentiable manifolds $M_1,M_2$. I want to prove that if $M_2$ is orientable so is $M_1$.
Attempt: In order a manifold to be orientable the d... | Equivalent condition for manifold $M$ to be orentable is that there exists non-vanishing form $\Omega$ of degree equal to the dimension of manifold.
So let $M_2$ be orientable of dimension $n$. Then there exists non-vanishing $\Omega_2\in\Omega^n(M_2).$ Let $\phi:M_1\rightarrow M_2$ be a diffeomorphism. Set $\Omega_1:=... | {
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Integration of $\frac{f'(x)}{f(x)}$? $\frac{f'(x)}{f(x)}$ integrated must be
$$\int\frac{f'(x)}{f(x)}dx=\ln\rvert f(x)\rvert+c.$$
But when trying to show this with partial integration I get another result:
$$
\begin{align*}
&\int\frac{\frac{d}{dx}f(x)}{f(x)}dx \\
=\quad&\int\frac{d}{dx}(f(x))\cdot (f(x))^{-1}dx \\
=\qu... | Yes, you are correct. In fact you can always summerize constants.
Another example:
Let's take two approaches to finding
for $x\in (-1,1)$ $$\int -\frac{1}{\sqrt {1-x^2}}dx$$
Approach 1:
Since
$$(\arccos x)'=-\frac{1}{\sqrt{1-x^2}}$$
we have
$$\int -\frac{1}{\sqrt {1-x^2}}dx=\arccos x+C \tag{1}$$
Approach 2:
Since
$$... | {
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"timestamp": "2023-03-29T00:00:00",
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What does $\sum_{k=0}^\infty \frac{k}{2^k}$ converge to? This problem comes from another equation on another question (this one).
I tried to split it in half but I found out that
$$\sum_{k=0}^\infty \frac{k}{2^k}$$
can't be divided.
Knowing that $$\sum_{k=0}^\infty x^k=\frac{1}{1-x}$$
I wrote that
$$\sum_{k=0}^\infty \... | Start with:
$$\frac{1}{1-x}=\sum_{k=0}^{\infty}x^k.$$
Then take derivative with respect to $x$.
$$\frac{1}{(1-x)^2}=\sum_{k=1}^{\infty}kx^{k-1}.$$
Multiply by $x$.
$$\frac{x}{(1-x)^2}=\sum_{k=1}^{\infty}kx^{k}.$$
Now substitute $x=\frac{1}{2}$.
| {
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Geometric justification of the trigonometric identity $\arctan x + \arctan \frac{1-x}{1+x} = \frac \pi 4$ The trigonometic identity
$$
\arctan x + \arctan \frac 1 x = \frac\pi 2\quad\text{for }x>0
$$
can be seen to be true by observing that if the lengths of the legs of a right triangle are $x$ and $1$, then the two ac... | Here's another geometrical picture of what the identity means.
The tangent of an angle can also be interpreted as the slope of a line. That is, take a straight line through the origin of an XY cartesian coordinate system, then its equation in cartesian coordinates is
$$y=\tan{\theta} \; x$$
where $\theta$ is the angle... | {
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"timestamp": "2023-03-29T00:00:00",
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Why $\lim_{R\to\infty}\int_{0}^{\pi}\sin(R^{2}e^{2i\theta})iRe^{i\theta}\:\mathrm{d}\theta = -\sqrt{\frac{\pi}{2}}$ This is a short question, but I'm simply not sure where to start, I know by Jordan's Lemma that the integral is not $0$, but I only know the below result due to Mathematica.
$$\lim_{R\to\infty}\int_{0}^{... | Just in order to add something to the other answers, in order to compute the Fresnel integrals you may also use the Laplace transform. Since:
$$\mathcal{L}^{-1}\left(\frac{1}{\sqrt{x}}\right) = \sqrt{\frac{1}{\pi s}},\qquad\mathcal{L}(\sin x)=\frac{1}{1+s^2},\qquad\mathcal{L}(\cos x)=\frac{s}{1+s^2} $$
we have:
$$ \int... | {
"language": "en",
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Maximal ideals of polynomial ring We know that if $k$ is algebraically closed, then each maximal ideals of $k[x_1, x_2, \ldots , x_n]$ are of the form $(x_1 - a_1, x_2 - a_2, \ldots, x_n - a_n),$ where $a_1, a_2, \ldots , a_n \in k$ (Hilbert's Nullstellensatz Theorem). In the case when $k$ is not algebraically closed ... | Yes. Set $R=k[x_1...x_n]$ and suppose $R/m\cong k$. Then we have a homomorphism $R\to k$ given by composing $R\to R/m$ with this isomorphism, and each $x_i$ is sent to some element $a_i$ of $k$. This completely determines the kernel of $R\to k$, for instance since $R$ is a free algebra, and so realizes $m$ in the desir... | {
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Proving continuity for all $x$. I am having difficulty in proving the following problem. Any hints would be greatly appreciated.
Let $f:[0,1]\times [0,1] \rightarrow \mathbb{R}$ be such that for each $y \in [0,1]$, $f(\cdot,y)$ is continuous and for each $ x \in [0,1]$, $f(x,\cdot)$ is measurable. Assume that $f$ is ... | This is simply the Dominated Convergence Theorem.
Write $f_x(y) := f(x,y)$, then $F(x) = \int_0^1 f_x(y)dy$. For any $x_0 \in [0,1]$, as $x \to x_0$, $f_x \to f_{x_0}$ pointwise by continuity along the horizontal lines. Since $|f_x|$ is uniformly bounded for all $x$, we have by the DCT that
$$
\lim_{x \to x_0}F(x) = \l... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is $2|x|/(1+x^{2})<1$ true if $|x|<1$? [Solved] I was asked to show that the series
$$\sum_{k\geq 0}a_{k}\left ( \frac{2x}{1+x^{2}} \right )^{k}$$
is convergent for $x\in (-1,1)$, where $\{ a_{k}\}_{k\geq 0}$ is a real bounded sequence.
My main question is, how do I know that $|2x/(1+x^{2})|<1$, that is, $2|x|/(1+x^{... | The claim is true if $2|x| < 1+x^2$ which is true by AM-GM inequality, because equality occurs only when $x^2=1$. Because $|x|<1 \implies x\neq 1$ the inequality is strict. The AM-GM inequality holds for the domain of positive reals, but all terms involved are positive even when $x$ is negative so the above inequality ... | {
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Is there a theory of induced representations for semigroups? Given a semigroup $G$, a subgroup $H\subseteq G$ (not merely a subsemigroup) and a representation $\rho: H\rightarrow GL(V)$ for some vector space $V$, is there a canonical definition of an induced representation on some vector space $V^\prime$ containing $V$... | I am writing a book on the representation theory of monoids. The latest version is http://www.sci.ccny.cuny.edu/~benjamin/monoidrep.pdf
You will see that there is a more subtle way to induce representations of a monoid from its maximal subgroups than the way suggested by Phil which is essential for constructing the sim... | {
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How prove $\frac{(a-b)^4+(b-c)^4+(c-a)^4}{(a+b+c)^4}=2$ Let $a,b,c\in R$,and such $ab+bc+ac=0,a+b+c\neq 0$
show that
$$\dfrac{(a-b)^4+(b-c)^4+(c-a)^4}{(a+b+c)^4}=2$$
| Hint: Express the numerator in terms of elementary symmetric polynomials, as the denominator and constraint already are. You get
$$(a-b)^4+(b-c)^4+(c-a)^4 \\= 2(a+b+c)^4 - 12(ab + bc + ca) (a+b+c)^2 + 18(ab+bc+ca)^2$$
| {
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Calculating the radius of convergence of a series.
Let $d_n$ denote the number of divisors of $n^{50}$ then determine the radius of convergence of the series $\sum\limits_{n=1}^{\infty}d_nx^n$.
So obviously we need to calculate the limit of $\frac{d_{n+1}}{d_n}$. I am guessing I need some information about the asympt... | Note that $d_1=1$, and for all $n \ge 2$, $d_n \le n^{50}$. So $$\frac{1}{R}=\limsup_{n \to \infty}{d_n}^{\frac{1}{n}}$$ and $$1 \le \limsup_{n \to \infty}{d_n}^{\frac{1}{n}} \le \lim_{n \to \infty}(n^{\frac{1}{n}})^{50}=1$$
| {
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Find two finite abelian groups of same order $G$ and $H$ such that $G \ncong H.$ Find two finite abelian groups of same order $G$ and $H$ such that $G \ncong H.$
I found groups $\mathbb{Z}_4$ and $V$ (Klein's group) that satisfy it, but would like more examples. I'm trying to use the result that $p$ is a prime number t... | The three abelian groups of order $8$, namely $G_1$, $G_2$ and $G_3$, belong to three different isotopy classes and hence there can't be two of them being isomorphic.
| {
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How are basis elements also elements of the topology? I read the following definitions in Munkres' Topology (2nd Edition):
If $X$ is a set, a basis for a topology on $X$ is a collection
$\mathcal B$ of subsets of $X$ (called basis elements) such that
*
*For each $x\in X$, there is at least one basis element... | Just expanding on Daniel Fischer's comment a bit:
From the definition of an open set from a basis, we have:
A subset $U$ of $X$ is said to be open iff for each $x\in U$, there is a basis element $B\in\mathcal B$ such that $x\in B\subseteq U$.
So consider this:
Take a $B \in \mathcal{B}$ and further let $x\in B$.
Ther... | {
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Bounded sequence with subsequences all converging to the same limit means that the sequence itself converges to the same limit. I have a question regarding one exercise in Stephen Abbotts' Understanding Analysis. The question is: Assume $(a_n)$ is a bounded sequence with the property that every convergent subsequence o... | Here is how I would approach the problem - since $(a_n)$ is a bounded sequence, there exist subsequences $(a_{n_k})$ and $(a_{n_l})$ such that \begin{align}\lim_{k\to\infty} a_{n_k} &= \limsup_{n\to\infty} a_n\\
\lim_{l\to\infty} a_{n_l} &= \liminf_{n\to\infty} a_n.
\end{align}
(It is a good exercise to prove the above... | {
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Solving $\int_0^{\infty}\,dk\,\exp{(-\delta^2k^2)}\,\frac{J_1(kR)}{k^2}$ I would like to understand if there is a closed formula for this integral:
$$\int_0^{\infty}\,dk\,\exp{(-\delta^2k^2)}\,\frac{J_1(kR)}{k^2}$$
where $R,\delta>0$ and $J_1(\cdot)$ is the bessel function of first kind of order 1.
By using the series ... | With the help of Mathematica I got:
$$\mathcal{L}\left(\frac{J_1(R\sqrt{x})}{x^{3/2}}\right)=\frac{|R|}{2}\left(1-2\gamma-\frac{4s}{R^2}\left(1-e^{-\frac{R^2}{4s}}\right)+\log\frac{4}{R^2}-\Gamma\left(0,\frac{R^2}{4s}\right)\right)$$
and by expanding the RHS as a series it is not difficult to check it matches your seri... | {
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Integral of Absolute Value of $\sin(x)$ For the Integral: $\int |\sin (ax)|$, it is fairly simple to take the Laplace transform of the absolute value of sine, treating it as a periodic function.
$$\mathcal L(|\sin (ax)|) = \frac{\int_0^\frac{\pi} {a} e^{-st} \sin (ax)} {1 - e^{-\frac{ \pi s} {a}}} = \frac{a (1 + e^{-\f... | Without loss of generality, let us assume $a=\pi$ and compute
$$\int_0^X|\sin(\pi x)|dx.$$
The function has period $1$ so that when integrating from $0$ to $X$, there are $\lfloor X\rfloor$ whole periods and a partial one from $X-\lfloor X\rfloor$ to $X$, i.e. by shifting, from $0$ to $X-\lfloor X\rfloor$.
Hence
$$\int... | {
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Working with $i$ and finding roots How do you find the cubic polynomial $f(x)$ with integer coefficients, where $1-3i$ and $-2$ are the roots of $f(x)$?
It has been too long since I have worked with $i$ and I have been unable to generate it.
| The other solutions correctly point out that the third root must be $1+3i$, and therefore the polynomial must be $$(x-(1-3i))(x-(1+3i))(x-2)$$
or a multiple of it (notice that multiplying through the entire thing by a constant does not change the zeroes).
I will add to this one tip for simplifying the expression: Noti... | {
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High School Trigonometry ( Law of cosine and sine) I am preparing for faculty entrance exam and this was the question for which I couldn't find the way to solve (answer is 0). I guess they ask me to solve this by using the rule of sine and cosine:
Let $\alpha$, $\beta$ and $\gamma$ be the angles of arbitrary triangle... | First lets consider
$$\frac{b-2a\cos\gamma}{a\sin\gamma}$$
The numerator looks similar to the RHS of the cosine law, but not quite. It would be nice to see $-2ab\cos\gamma$ instead of $-2a\cos\gamma$. So lets just multiply by $b$. Then the numerator would look like this
$$b^2-2ab\cos\gamma$$
Now all that is missing is ... | {
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Are projections surjective? Suppose $A,B$ are non empty and consider projection $P: A \times B \to A $ given by $P( (a,b) ) = a $. Show $P$ is surjective.
Attempt:
Let $y \in A $ be arbitrary. We know that for every $x \in B $, it follows that $P ( ( y,x) ) = y $. So, for every $y \in A $, we can always find an element... | Pretty much yes. You need to cite non-emptiness of $B$ explicitly... but I would still give full credit.
| {
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Visual approach to abstract algebra I'm currently finding abstract algebra to be very fascinating. However, one of the things that pulls me back is that I sometimes find it hard to understand something visually.
For example, one could visualise the First Isomorphism Theorem as being a circle with a smaller circle insi... | For group theory see Nathan Carter's Visual Group Theory and its accompanying software, Group Explorer.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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suggest an elementary text in analysis I have just completed my undergraduate course in mathematics but I don't feel better in analysis,
there is a mugup of books in my book collection But don't know what to choose who will help me to teach me analysis I have the options of Battle, Apostol and Rudin and some other loc... | Well I have quite a collection of analysis books myself. Here are my favorites.
"Introduction to real analysis - Bartle and Sherbert" Is my favorite book on the subject. Everything is well laid out and easy to understand. Unfortunately the treatment very elementary in nature. For example there is no treatment of improp... | {
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Can ANY 2 or 3 dimensional shape be reversed engineered to give an equation (formula) for its shape?? Can ANY 2 or 3 dimensional shape be reversed engineered to give an equation (formula) for its shape?
In other words given ANY 2 or 3 dimensional shape that ones draws on a graph can one reverse engineer it to find a fo... | It depends. If you do not know the shape, then the answer is simply no, because there is no way to observe everything about the shape in most reasonable models of the world. For example, if a shape is infinitely divisible and you can only observe one point at a time, then even if its boundary is a continuous surface, y... | {
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System of Equations: any solutions at all? I am looking for any complex number solutions to the system of equations:
$$\begin{align}
|a|^2+|b|^2+|c|^2&=\frac13
\\ \bar{a}b+a\bar{c}+\bar{b}c&=\frac16 (2+\sqrt{3}i).
\end{align}$$
Note I put inequality in the tags as I imagine it is an inequality that shows that this has ... | By Cauchy-Schwarz, we must have$$\left(|a|^2+|b|^2+|c|^2 \right) \left(|\overline{c}|^2+|\overline{a}|^2+|\overline{b}|^2 \right) \ge \left| a\overline c + b\overline a + c\overline b\right|^2$$
$$\implies \frac19 \ge \frac1{36}|2+\sqrt3 \; i|^2 = \frac7{4\cdot9}$$
which is obviously not possible...
| {
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System of equations $x + xy + y = 11$ and $yx^2 + xy^2 = 30$ I have problem with solving this one.
Total number of solutions from system of equations?
\begin{cases}
x + xy + y = 11 \\
x^2y + y^2x = 30
\end{cases}
There is a system of equation and I have tried to get some normal solutions, but I always get the fourth... | Hint: Use the substitution a=x+y
and b=xy.
Thus your two equations become a+b=11, ab=30.
Can you solve the above equation, and from that solve for x and y?
| {
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} |
Does there exist a connected metric space, with more than one point and without any isolated point, in which at least one open ball is countable? Does there exist a connected metric space, with more than one point and without any isolated point, in which at least one open ball is countable?
| Assume a ball $B(x_0,r)\subseteq X$ is countable, $x_0\in X$, $r>0$.
As $X$ is not a one-point space, there exists $x_1\ne x_0$ and we may assume $r\le d(x_0,x_1)$.
Then by pigeon-hole there exists $0<\rho <r$ such that $d(x_0,x)\ne \rho$ for all $x\in B(x_0,r)$. By definition of $B(x_0,r)$, in fact $d(x_0,x)\ne\rho$ f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1327483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Understanding a matrix bound/inequality I came across the following statements;
For a positive matrix $A$, that is bounded $0 \leq A \leq I,$ where $I$ is the identity matrix and a statement like $Y \leq X$ means that $X-Y$ is positive semidefinite. The following inequality is true:
$$A(I-A)A \leq \frac{4}{27} I \tag{1... | If $A$ is positive semidefinite, $A$ is unitarily similar to a diagonal matrix with nonnegative diagonal entries ($0\leq A\leq I$ implying that these diagonal entries are in the interval $[0,1]$). So $A=UDU^*$, where $D$ is this diagonal matrix and $U$ is unitary. Then
$$
A(I-A)A=UDU^*(I-UDU^*)UDU^*=U[D(I-D)D]U^*.
$$
N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1327557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Disjoint refinement of a set of sets Say I have some set $F$ of sets. This is obviously a cover of $\cup F$. Is there a general algorithm that makes "this" cover disjoint? That is, a set $F'$ of pairwise disjoint sets with the properties $\cup F = \cup F'$ and $(\forall A \in F')(\exists B \in F)(A \subset B)$?
This qu... | I can tell you how to prove that such a refinement exists. It's not entirely elementary, and it probably doesn't count as an "algorithm". I don't know how much set theory you know - you can find the required background for what's below by searching Wikipedia for "ordinal", "Well ordering theorem", and "transfinite recu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1327631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Sums $a_k=[\frac{2+(-1)^k}{3^k}, (\frac{1}{n}-\frac{1}{n+2}), \frac{1}{4k^2-1},\sum_{l=0}^k{k \choose l}\frac{1}{2^{k+l}}]$
Determien the sums of the following series'.
1:$\sum_{k=0}^\infty \frac{2+(-1)^k}{3^k}$
2:$\sum_{k=0}^\infty (\frac{1}{n}-\frac{1}{n+2})$
3:$\sum_{k=0}^\infty \frac{1}{4k^2-1}$
4:$\sum_{k=0}^\inf... | $(1)$ Looks good to me.
$(2)$ You are correct that this is a telescoping series and your answer is correct (but the index should start at $k=1$, not $k=0$) You could also rewrite the series as $$\begin{align}\sum_{k=1}^\infty \frac{1}{k}-\frac{1}{k+2} = \sum_{k=1}^\infty \frac{1}{k}-\sum_{k=1}^\infty \frac{1}{k+2} \\ =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1327732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Galois extension and Galois group Let $x$ be a real root of the polynomial $X^3-X+1$, and $y,\overline{y}$ two other roots in $\mathbb{C}$, and $K$ be the cubic field $\mathbb{Q}[x]$.
Show that $y+\overline{y}=-x$ , $y\overline{y}=-1/x$, and $[(y-x)(\overline{y}-x)(y-\overline{y})]^2=-23$.
Show that $L=K[\sqrt{-23}]=... | Hint: Every extension that contains a complex number has even order, since if it contains $i$ it can be written as $\mathbb Q (\alpha, i) / \mathbb Q (i) / \mathbb Q$. By the tower law, the order of this field over $\mathbb Q$ is $[\mathbb Q (\alpha, i) : \mathbb Q(i)] [\mathbb Q (i) : \mathbb Q]$. Since $[\mathbb Q(i)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1327809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
The fundamental vector fields of a principal bundle are vertical. Let $p:P\to M$ be a principal $G$-bundle. To each $A$ in the Lie algebra of $G$ corresponds a fundamental vector field $A^*$ on $M$ defined by
$$A^*_u=\frac{d}{dt}|_{t=0} u(exp(tA))$$
How can we see that $A_u^*$ is a vertical vector?
Idea: We need to ver... | First of all a fundamental vector field is a vector field on $P$ not on $M$ as you wrote in your question. As you say the action of $G$ on $P$ preserves the fibers i.e. $p(u) = p(u g)$ for all $u \in P , g \in G$. If $A$ is in the Lie algebra of $G$ then $exp(tA)$ is a monoparametric group. For $u \in P$ we have a curv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1327911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Maple Help format output collect terms with same coefficients and multiple variables I realize this may have been asked before, however I have no idea what to search for specifically, since the topic is so generic it returns so many results that I cannot find what I am looking for.
eqn3:=3.9024*x+3.9024*y;
How can I ge... | Note that automatic simplification distributes numeric coefficients, e.g.
if you enter
> expr:= 3.9024*(x+y);
the result will be
$$ expr := 3.9024 x + 3.9024 y $$
To defeat this, one way is to use the `` function.
> E:= map(t -> (t = ``(t)), indets(expr,float)):
> collect(subs(E, expr),``);
$$ (x + y) (3.9024) $$
Un... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1327952",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Computing $\sum_{n\geq 0}n\frac{1}{4^n}$ Can I compute the sum
$$
\sum_{n\geq 0}n\frac{1}{4^n}
$$
by use of some trick?
First I thought of a geometrical series?
| The idea is to take the derivative of the geometric series. For $|x|<1$,
$$\frac1{(1-x)^2}=\sum_{n=1}^\infty nx^{n-1}$$
So,
$$\frac x{(1-x)^2}=\sum_{n=1}^\infty nx^n.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1328050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
Characterising a rotation matrix We have the operator $T: \mathbb{R}^3 \to \mathbb{R}^3$ given by $T(\vec{x}) = A\vec{x}$
$$A = \frac{1}{3}\begin{pmatrix} -2 & 1 & 1 \\ 1 & -2 & 1 \\ 1 & 1 & -2 \end{pmatrix}$$
I have orthogonally diagonalized A and found that
$$Q = \begin{pmatrix} \frac{1}{\sqrt{3}} & \frac{-1}{\sqrt{... | After orthogonally diagonalizing your matrix, you should find that $A = QDQ^T$, where $Q$ is as you've given it and
$$
D = \pmatrix{
0&0&0\\
0&-1&0\\
0&0&-1}
$$
We can easily see what happens to a vector in the eigenvector basis. That is, if we define
$$
(a,b,c)_B = Q \pmatrix{a\\b\\c}
$$
Then you can see that
$$
A(a,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1328140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Every subset has first and last element -> set is finite Let $X$ be a partially ordered set, so that every non empty subset of $X$ has a first and a last element.
Show that $X$ is a finite set.
And what if every subset only has a first element?
Well, I proved it is a linear order, but now I'm stuck.
Anyone ready to ... | One may also give a more "direct" proof. Assume $X$ is infinite. We shall construct an strictly increasing/decreasing sequence which will contradict the existence of a last/first element.
Start with some element $x_1\in X$. At least one of the sets $\left\{ x>x_1 \right\}, \left\{ x<x_1 \right\}$ is infinite. Assume wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1328215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
A bit confused about definition of set of mappings in Herstein's Algebra I have been just trying to start working with the books Topics in Algebra by I.N. Herstein, and I am having a bit of trouble understanding a definition.
It is,
Definition: If $S$ is a nonempty set then $A(S)$ is the set of all one-to-one mappings... | Yes, by Herstein's definition, $A(X)=\{\text{bijective functions }f:X\to X\}$.
Just for emphasis, since this seems to be something you're confused on, the elements of $A(X)$ are precisely the bijective functions from $X$ to $X$. There's nothing else in $A(X)$ other than that, and every such bijective function is inclu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1328297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
evaluate $\int_0^{2\pi} \frac{1}{\cos x + \sin x +2}\, dx $ This is supposed to be a very easy integral, however I cannot get around.
Evaluate:
$$\int_0^{2\pi} \frac{1}{\cos x + \sin x +2}\, dx$$
What I did is:
$$\int_{0}^{2\pi}\frac{dx}{\cos x + \sin x +2} = \int_{0}^{2\pi} \frac{dx}{\left ( \frac{e^{ix}-e^{-ix}}{2i... | Via corindo's rearrangement: we have
$$
I = \int_0^{2 \pi} \frac{1}{\sqrt{2}\cos(x - \pi/4) + 2}\,dx =\\
\frac 1{\sqrt{2}} \int_0^{2 \pi} \frac{1}{\cos(x - \pi/4) + \sqrt{2}}\,dx = \\
\frac 1{\sqrt{2}} \int_0^{2 \pi} \frac{1}{\cos(x) + \sqrt{2}}\,dx =\\
\frac 1{\sqrt{2}} \oint_{|z| = 1} \frac{1}{iz((z + z^{-1})/2 + \sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1328371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Fundamental polygon square $abab$ What is the most convenient description of the space with fundamental polygon a square, with all vertices identified, glued by $abab$? If we were to identify only opposite vertices, we would get $\mathbb{R}P^2$. So I believe one description is $\mathbb{R}P^2$ with two points identified... | I'm putting what I have wrote in the comments together in this answer :
Claim The polygon $abab$ with all the vertices identified is homotopy equivalent $\Bbb RP^2 \vee S^1$.
First consider the polygon $abab$ with diagonally opposite pair of vertices identified, which is homeomorphic to $\Bbb RP^2$. Since all of the ve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1328490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Slow decreasing function that exhibits asymptotic behaviour I am currently doing some work on modelling the effects of treated nets usage on mosquito populations. Nets do not retain their maximum efficacy forever. They lose their chemical efficacy after about three years and all that is left is the physical protection ... | How about something like
$$
y=1-0.8\,\frac{e^{ax}-1}{e^{1095a}-1}?
$$
The sign of $a$ determines concavity or convexity; $a=-0.005$ gives a nice graph.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1328566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Regular Quadratic Space - isotrope vector I am currently trying to solve the following exercise:
Show that every regular quadratic space of finite dimension $E$ that contains at least one isotrope vector, has a basis consisting only of isotrope vectors.
I think my problem with this exercise is that I don't really h... | You don't need the full Witt decomposition, just split off a hyperbolic plane, so
$$V = W\oplus \langle u, u' \rangle$$ with $f(u,u')=1$, ...
Take any basis $w_1$, $\ldots$, $w_m$ of the subspace $W$. We get the basis of $V$
\begin{eqnarray}
&w_i& - \frac{q(w_i)}{2}(u+u') \text{ for } 1 \le i \le m, \ \text{and}\
u, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1328709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
When I can reverse the logical operators? I heard say that is logically equivalent to say it:
$$\neg (p \vee q) = p \land q$$
So every time you have a negation operator in front can make a "distributive" altering the operator from within? And if $\neg(p \vee \neg q)$ then result is $p \land \neg q$? Can anyone give som... | Example, as requested...
Suppose I want to know what is the domain of the function
$$
\frac{1}{x^2-4x+3}
$$
That is, I want to find all the $x$ where that denominator is not zero.
So I factor the denominator. $x^2-4x+3=(x-1)(x-3)$.
When is it zero? Either $x=1$ or $x=3$. So when is it nonzero?
$$
\neg\;\big(x=1\;\lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1328790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
The rows of an orthogonal matrix form an orthonormal basis
A matrix $A \in \operatorname{Mat}(n \times n, \Bbb R)$ is said to be orthogonal if its columns are orthonormal relative to the dot product on $\Bbb R^n$.
*
*By considering $A^TA$, show that $A$ is an orthogonal matrix if and only if $A^T = A^{−1}$.
... | We are given this definition: matrix $A \in \mathbb{R}^{n \times n}$ is "orthogonal" if and only if its columns are orthonormal.
We want to show that the rows of $A$ form an orthonormal basis for $\mathbb{R}^n$ (technically an orthonormal basis for $V=$"the space of $n$-component row vectors over $\mathbb{R}$", but the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1328874",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
} |
How big is the chance that a arbitrary man is taller than a arbitrary woman? I'm a first year mathematics student, and I'm having trouble with computing the following:
Assume that in a country the height $X$ of men is normally distributed, with $\mu_X = 180$ (the expected value), $\sigma_X = 6$ (the standard deviation)... | The probability density function of a $N(\mu,\sigma)$ random variable $X$ is given by:
$$ f_X(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$
hence our probability is given by the integral:
$$ \mathbb{P}[X\geq Y]=\frac{1}{60\pi}\iint_{x>y}\exp\left(-\frac{(x-180)^2}{72}-\frac{(y-173)^2}{50}\right)\,d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1328959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If $f:\mathbb{R}\rightarrow\mathbb{R}$, $f\in C^{\infty}(\mathbb{R})$ and $f(0)=0$ then $\frac{f(x)}{x}\in C^{\infty}(\mathbb{R})$ If $f:\mathbb{R}\rightarrow\mathbb{R}$, $f\in C^{\infty}(\mathbb{R})$ and $f(0)=0$ then $\frac{f(x)}{x}\in C^{\infty}(\mathbb{R})$.
Following ther is what i did:
the Maclaurin series for $f... | We can write $f(x) = \int_0^x f'(t)\,dt = x\int_0^1 f'(xs)\,ds.$ So
$$g(x) = \int_0^1f'(xs)\,ds.$$
for all $x.$ You can differentiate all day through the integral sign, showing $g\in C^\infty.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1329076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Rotation of a line by a matrix
Give the equation of the line $\ell'$ that is obtained by rotating $\ell$: $x+2y=5$ by an angle of $\theta=\frac{1}{2}\pi$ with center point $O(0,0)$.
The rotation matrix is $\left.\begin{pmatrix}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{pmatrix}\right|_{\alpha=\frac{1... | I notice that the point $(1,2)$ is on the original line. After a rotation by $\pi/2$, this point becomes $(-2,1)$. I also notice that the original slope was $-1/2$. A rotation by $\pi/2$ has the effect of finding a line perpendicular to our starting line, so it will have slope $2$.
The equation of a line with slope $2$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1329193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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