Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Why does $n \choose k$ get you the $k^{th}$ (starting from 0) coefficient of $(a+b)^n?$ I'm aware of the connection between Pascal's triangle and the binomial theorem, and how each edge, left and right if we consider the triangle to be a graph, represents multiplying by $a$ or $b.$ But how do we relate this to combinat... | $$(a+b)^n=\underbrace{(a+b)(a+b)\cdots(a+b)}_{\text{n times}}$$ To get a term of the form $a^kb^{n-k}$ we need to choose an $a$ out of the $n$ factors $k$ times and a $b$ out of the remaining $(n-k)$ factors $(n-k)$ times. Hence the coefficient of $a^kb^{n-k}$ in the expansion is
$$\binom{n}{k}\binom{n-k}{n-k}=\binom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1903442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to formalize "It doesn't matter how places the brackets" Let $X$ be a set and $X\times X\to X: (a, b)\mapsto a\cdot b$ an associative operation on $X$. Now one can prove by induction that it doesn't matter how one places the brackets in a product $x_1\cdot x_2\cdot\text{ }\dots\text{ } \cdot x_n$, the product alway... | This is called the associative law which says precisely: $$\forall x,y,z [(x*(y*z))=((x*y)*z))] $$
That it extends to arbitrary finite products can then proven using induction.
Some more details: you will prove that every expresention is equivalent to the one with brackets on left e.g. to $(((x*y)*z)*w))$ (every expres... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1903507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 2
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If $f\not\in L^{\infty}$ then $\lim_{p\rightarrow\infty}||f||_p=\infty $. I want to prove that
If $f\not\in L^{\infty}$ then $\lim_{p\rightarrow\infty}||f||_p=\infty $.
I'd like any hint that approaches to solution.
I now that for every real number $r$ the set $\{x:|f(x)|>r\}$ has positive measure $f$ but I do not kn... | If $f \not\in L^{\infty}$, then given any $M > 0$, we have $|f| > M$ on a set $E$ of positive measure. Therefore, $|f|^p > M^p$ on $E$, so
$$\int |f|^p \geq \int_E |f|^p \geq M^p \mu(E)$$
If $\mu(E) = \infty$ then this shows that $\|f\|_p = \infty$ for all $p<\infty$, so the result certainly holds.
Otherwise,
$$\|f\|_p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1903601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Second order derivative of the inverse matrix operator
Let $f : Gl_{n}(\mathbb{R}) \to Gl_{n}(\mathbb{R})$ defined by $f(X)=X^{-1}$. Compute $f''(X)(H,K)$.
I calculated $f'(X).H=-X^{-1}HX^{-1}$ so I tried to use some composition of linear functions but did not find the appropriate functions. Can anyone help me in th... | Using the Taylor formula, (As Rodrigo de Azevedo and user1952009 did) you can calculate $f''(X)(H,H)$. If you want the gneral formula $f''(X)(H,K)$, then you can calculate the derivative (with respect to $X$, considering $H$ as a fixed vector) of $f'(X)(H)=-X^{-1}HX^{-1}$.
We obtain $f''(X)(H,K)=X^{-1}KX^{-1}HX^{-1}+X... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1903677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If $a$, $b$, and $c$ are sides of a triangle, then $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$.
Let $a,b,c$ be the lengths of the sides of a triangle. Prove that
$$\sum_{\text{cyc}}\frac{a}{b+c}=\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2\,.$$
Attempt. By clearing the denominators, the required inequality is equiv... | \begin{align*}
\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} & = \frac{2a}{2(b+c)}+\frac{2b}{2(c+a)}+\frac{2c}{2(a+b)} \\
&< \frac{2a}{a+b+c} + \frac{2b}{c+a+b} + \frac{2c}{a+b+c} \\
&= 2
\end{align*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1903775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Zero Vectors for Vector Spaces other than $R^n$ I understand what a zero vectors is in $R^n$ but I need some help visualising other zero vectors:
For example, the vector space of all functions $${ y : \mathbb R \rightarrow \mathbb R \ \ | \ y''+xy'+e^xy=0 } $$
Is the zero vector just $z(x)=0$ ? Explicit examples of le... | A vector space is among other things a group with respect to sum. So the zero vector is exactly the zero of the sum, the unique element that can be added to any other element without changing it.
In both your examples the zero is the constant $z(x) = 0$. In the second example, all the functions $z(x)=ax+b$ are in fact ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1903888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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To test convergence of infinite series $x^2(\log 2)^p + x^3(\log 3)^p + x^4(\log 4)^p +\dots$
To test convergence of infinite series $x^2(\log 2)^p + x^3(\log 3)^p + x^4(\log 4)^p +\dots$
My approach to the above problem:
let $u_n = x^{(n+1)}(\log(n+1))^p$. the $u_{(n+1)}= x^{(n+2)}(\log(n+2))^p$
now, $n \log\frac{u_... | There are a lot of possibilties to check convergence.
E.g. the ratio test here for $p\ge 0$ is:
$$|\frac{x^{n+1}(\ln(n+1))^p}{x^n(\ln n)^p}|=|x|(\frac{\ln(n+1)}{\ln n})^p<1$$
This means $|x|<(\frac{\ln n}{\ln(n+1)})^p<1$: Convergence for $|x|<1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1904052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Is it possible to have a $f(\vec{r})$ satisfy this relation? It is known that:
$$
(\nabla^2+k^2)(-\frac{e^{ikr}}{4\pi r})=\delta(\vec{r})
$$
where $k>0$ and $\delta(\vec{r})$ is the three dimensional Dirac delta function.
My question is, is it possible to find a function $f(\vec{r})$ that satisfies the following relati... | Fourier transforming your equation, you can check that the function $f({\bf r})$ is given by
$$f({\bf r}) = \int \frac{d^3 q}{(2\pi)^3} \frac{e^{i {\bf q} \cdot {\bf r}}}{(q_3+\alpha)^2 - q_1^2 -q_2^2 -k^2 }$$
with ${\bf q} = (q_1,q_2,q_3)$. You can still impose boundary conditions on $f({\bf r})$ which corresponds to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1904135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Conflicting definitions of continuity (strict or non-strict inequality)? On page 97 of Kreyzig's functional analysis book he provides a proof that a linear operator $T$ is continuous if and only if it is bounded.
When proving that $T$ is continuous implies that $T$ is bounded he says that if we assume $T$ is continuous... | If you know that you can show this with $\le$ and need $<$ just apply the $\le$ case to $\frac{\varepsilon}{2}$ in order to get the $<$ for $\varepsilon$ (and vice versa)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1904220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Sequence converging to definite integral Let's define
\begin{equation*}
I_0 := \log\frac{6}{5}
\end{equation*}
and for $k = 1, 2, \ldots, n$
\begin{equation*}
I_k := \frac{1}{k} - 5 I_{k-1}.
\end{equation*}
How the value $I_n$ is linked with the value of $$\int_0^1\frac{x^n}{x+5} \mathrm{d}x \ ?$$
| If we set
$$ I_n = \int_{0}^{1}\frac{x^n}{x+5}\,dx \tag{1}$$
we clearly have $I_0=\log\frac{6}{5}$ and
$$ I_n+ 5I_{n-1} = \int_{0}^{1}\frac{x^n+5 x^{n-1}}{x+5}\,dx = \int_{0}^{1}x^{n-1}\,dx = \frac{1}{n}.\tag{2} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1904317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Why can't an improper transfer function be realized? A major result in control system theory is that a transfer function,
$$G\left( s \right) = \frac{{Y\left( s \right)}}{{U\left( s \right)}}$$
has a state space realization if and only if the degree of $Y(s)$ is less than or equal to the degree of $U(s)$. I cannot find... | To realize an improper transfer function, derivatives of the input would be needed. The answer above by Rodrigo de Azevedo helps make clear why. The problem is that it is not possible to realize perfect derivatives. A number of arguments are helpful in understanding why.
The modulus of the frequency response of a diff... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1904469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
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Can the limit $\lim_{x\to0}\left(\frac{1}{x^5}\int_0^xe^{-t^2}\,dt-\frac{1}{x^4}+\frac{1}{3x^2}\right)$ be calculated?
$$\displaystyle\lim_{x\to0}\left(\frac{1}{x^5}\int_0^xe^{-t^2}\,dt-\frac{1}{x^4}+\frac{1}{3x^2}\right)$$
I have this limit to be calculated. Since the first term takes the form $\frac 00$, I apply th... | By bringing the fractions to the same denominator, start by writing the limit as
$$\displaystyle\lim_{x\to0}\frac{3\int_0^xe^{-t^2}\,dt -3x+x^3 }{3x^5}$$
Now, since this is of the form $0/0$ by L'H and FTC you get
$$\displaystyle\lim_{x\to0}\frac{3e^{-x^2}-3+x^2 }{15x^4}$$
From here it is easy.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1904553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Tips on identifying pigeon and pigeonhole I always have trouble trying to exactly identifying the exact pigeon and the pigeonholes for questions with slightly more integers. For example, questions like this.
Eleven integer are chosen from 1 to 20 inclusive. Use pigeonhole principle to prove the selection include integ... | For the first problem you want to divide the integers from $1$ through $20$ into $10$ pairs of consecutive integers: $\{1,2\}$, $\{3,4\}$, and so on up through $\{19,20\}$. These $10$ pairs are your pigeonholes, and the $11$ numbers that you choose are your pigeons. Since you have $11$ pigeons and only $10$ pigeonholes... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1904643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do I calculate $u(w)=\int_0^\infty \frac{1-\cos(wt)}{t}\,e^{-t}\,dt$? How do I calculate
$$u(w)=\int_0^\infty \frac{1-\cos(wt)}{t}\,e^{-t}\,dt$$
I tried to do it, I use partial integration but I get lost. Is there any nice simple way to calculate it?
| $\int \frac {(1−\cos\omega t)e^{−t}}{t}dt$ cannot be evaluated into elementary functions. You need to get tricky.
$F(s) = \int_0^{\infty} \frac {(1−\cos\omega t)e^{−st}}{t}dt$
and if we can find $F(1)$ we are done.
$\frac {dF}{ds} = $$\int_0^{\infty} -(1−\cos\omega t)e^{−st}dt\\
\frac 1s e^{-st} + \frac {-s\cos\omeg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1904748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Unconventional Hardy-Ramanujan number $a^3 + b^3 = c^3 + c^3$ - can it's existence / non-existence be proven?
Can it be proven that a number exists such that
$$\text{number} = a^3 + b^3 = c^3 + c^3,$$ where $a,b$ and $c$ are $3$ distinct positive integers?
If it cannot be proven, can it be proven that such a numbe... | If such a triple existed with $a<b$, then $a,c,b$ would be an arithmetic progression of cubes. But it is known that there cannot be three $n$-th powers in arithmetic progression if $n\geq 3$, see for example the paper of Darmon and Morel here.
There is probably an elementary proof of this when $n=3$, possibly in the p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1904825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Which matrix satisfies the following condition? $P$ is a real (symmetric) positive definite matrix. Let $P_i$ and $P_j$ represent the $i$'th and $j$'th columns of $P$, respectively. Further, let $P_{ki}$ represent the element situated at the $k$'th row of the column vector $P_i$.
I want to find additional conditions on... | Let $A$ be the following matrix $a_{ik} = \log_2 P_{ik}$. Then
$$
a_{ij} + a_{kj} \leq 1 + a_{ik} + a_{jj} \tag{*}
$$
is sufficient for
$$
\frac{P_{ij} P_{kj}}{c + P_{jj}} < \frac{P_{ij} P_{kj}}{P_{jj}} \leq 2P_{ik}.
$$
If $a_{ik} > -1, a_{jj} > 0$ and the matrix $A$ is diagonally dominant $|a_{jj}| \geq \sum_{k=1}^n ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1904934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is A276175 integer-only? The terms of the sequence A276123, defined by $a_0=a_1=a_2=1$ and $$a_n=\dfrac{(a_{n-1}+1)(a_{n-2}+1)}{a_{n-3}}\;,$$ are all integers (it's easy to prove that for all $n\geq2$, $a_n=\frac{9-3(-1)^n}{2}a_{n-1}-a_{n-2}-1$).
But is it also true for the sequence A276175 defined by $a_0=a_1=a_2=a_3=... | Yes, $(a_n)$ is a sequence of integers.
To prove this we first need to study some auxiliary sequences that satisfy a polynomial recurrence relation (unlike $(a_n)$ which has a rational fraction as its recurrence).
Consider the sequences $(b_n)$ of positive reals satisfying the recurrence relation $b_nb_{n+4} = b_{n+1}b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1905063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
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Let $R$ be a commutative Noetherian ring (with unity), and let $I$ be an ideal of $R$ such that $R/I \cong R$. Then is $I=(0)$?
Let $R$ be a commutative Noetherian ring (with unity), and let $I$ be an ideal of $R$ such that $R/I \cong R$. Then is it true that $I=(0)$ ?
I know that a surjective ring endomorphism of a ... | Assume $a$ is a proper ideal. Suppose they were isomorphic. Then $\varphi: A \to A/a$ is some arbitrary isomorphism, and correspondingly $\varphi(a) := I_{1} \subset A/a$ is an ideal (proper inclusion as $a \subset A$ is a proper inclusion). By the correspondence principle, $I_{1} \subset A/a$ pulls back to an ideal $a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Trouble understanding proof of the inequality - $(\frac{1}{a}+1)(\frac{1}{b}+1)(\frac{1}{c}+1) \ge 64 $, for $a,b,c > 0$ and $a+b+c = 1$ I was looking into this problem in a book discussing inequalities, However I found the proof quite hard to understand.The problem is as follows:
Let $a,b,c$ be positive numbers with ... | using for $$\frac{1}{a},\frac{1}{b},\frac{1}{c}$$ the AM-GM inequality we obtain
$$\frac{1}{3}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq \sqrt[3]{\frac{1}{abc}}$$ and for $$\frac{1}{ab},\frac{1}{bc},\frac{1}{ca}$$ the same we get $(3)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1905278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $\int_0^\infty \frac{dx}{\sqrt{(x^4+a^4)(x^4+b^4)}}=\frac{\pi}{2 \sqrt2 a b} ( \text{agm} (\frac{a+b}{2},\sqrt{\frac{a^2+b^2}{2}} ))^{-1}$ The following definite integral turns out to be expressible as the Arithmetic-Geometric Mean: $$I_4(a,b)=\int_0^\infty \frac{dx}{\sqrt{(x^4+a^4)(x^4+b^4)}}=\frac{\pi}{2 \sqrt2... | Substitition is sufficient.
Let $$\displaystyle z=x-\frac1x,w=x+\frac1x$$
then $$\displaystyle
\frac{\mathrm dx}{\sqrt{x^8+p x^4+1}}=\frac12\left(
\frac{\mathrm dz}{\sqrt{z^4+4z^2+2+p}}+\frac{\mathrm dw}{\sqrt{w^4-4w^2+2+p}}\right)
$$
So $$f(p)=\displaystyle
\int_0^\infty
\frac{\mathrm dx}{\sqrt{x^8+p x^4+1}}
=\frac12\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1905349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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Why does $\frac{1}{x} < 4$ have two answers? Solving $\frac{1}{x} < 4$ gives me $x > \frac{1}{4}$. The book however states the answer is: $x < 0$ or $x > \frac{1}{4}$.
My questions are:
Why does this inequality has two answers (preferably the intuition behind it)?
When using Wolfram Alpha it gives me two answers, but w... | Here is an important aspect which should be always considered. If someone asks me:
Problem: Find the solution of
\begin{align*}
\frac{1}{x}<4
\end{align*}
I would not answer the problem, but instead ask: What is the domain of $x$?
Please note the problem is not fully specified if the domain of $x$, the range of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1905422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "60",
"answer_count": 13,
"answer_id": 8
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Problem 3.2 in Gilbarg-Trudinger: Elliptic PDEs of second order I am strugging on problem 3.2 in Gilbarg-Trudinger, which says that
if $L=a^{ij}(x)D_{ij}+b^{i}(x)D_{i}+c(x)$ is an elliptic operator in a
bounded domain $\Omega \subset \mathbb{R}^{n}$ with $c<0$, and $u\in C^{2}(\Omega)\cap
> C^(\overline\Omega)$ sati... | There is a version of the maximum principle where you use the zeroth order term instead of uniform ellipticity to get the estimate. I'll sketch the proof below, and I'll take $c<0$ to be a constant (don't have the book in front of me right now). The argument should work just as well if $c(x)$ is negative and bounded aw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1905498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Discontinuous at infinitely many points While doing a worksheet on real analysis I came across the following problem.
$Q$. Let $f$ be a function defined on $[0,1]$ with the following property.
For every $y \in R$, either there is no $x$ in $[0,1]$ for which $f(x)=y$ or there are exactly two values of $x$ in $[0,1]$ for... | To construct a function satisfying your condition, we first construct such a function $f$ on $\Bbb R$.
Decompose $\Bbb R$ into union of $[n,n+1)$, construct a function $f$ whose restriction $f_n:[n,n+1)\to[n,n+1)$ is defined by
$$\begin{align}
f_n(x) &= x\quad\text{;}\quad n\le x<n+\frac 12 \\
&= x-\frac 12\quad\te... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1905560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 1
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When is $\sin(x) = \cos(x)$? How do I solve the following equation?
$$\cos(x) - \sin(x) = 0$$
I need to find the minimum and maximum of this function:
$$f(x) = \frac{\tan(x)}{(1+\tan(x)^2}$$
I differentiated it, and in order to find the stationary points I need to put the numerator equal to zero. But I can't find a w... | Approach $1$ (Squaring):
$$(\sin x-\cos x)^2=0$$
$$(\sin^2x+\cos^2x)-2\sin x\cos x=0$$
$$1-\sin2x=0$$
$$\sin2x=1$$
$$2x=\frac{\pi}2+2n\pi,n\in\Bbb{N}$$
$$x=\frac{\pi}4+n\pi,n\in\Bbb{N}$$
Approach $2$ (By definition of $\sin x$ and $\cos x$):
$$\cos t=\frac{e^{it}+e^{-it}}2=\frac{e^{it}-e^{-it}}{2i}=\sin t$$
$$(1+i)e^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1905640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 8,
"answer_id": 5
} |
find position of a point rotate about an arbitrary axis I have the axis $u=[1,1,0]$
Follow by $u$ is a vector $v$ which perpendicular to $u$ (i.e. $z \times u=v$ which $z=[0,0,1]$)
I want to find vector $p = u+v$ when rotate $v$ about $u$ by angle $\theta$
Basically, it's about finding the transformation matrix to find... | I will use $\hat{u}=u/|u|$. The vector $w=v\times\hat{u}$ is perpendicular to $w$ and $u$, and $v\times w$ is along $u$. $v'$, which we obtain by rotating $v$ around $u$ by an angle $\theta$ is therefore in the plane of $v$ and $w$. It has a component $|v| \cos\theta$ along $v$, and a component $|v|\sin\theta$ along $w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1905736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Problem in finding limit function of $f_n(x)=1-(1-x^2)^{n}$. Its answer is $f(x) =
\begin{cases}
0, & \text{when $x=0$ } \\
1, & \text{when $0<\vert x \vert< \sqrt 2$}
\end{cases}$.
I'm not getting how second line of $f(x)$ came?
Apologies if the post is too basic, but I've invested a great time in understanding this ... | Assuming you want the limit as $n\to\infty$, then you need $|1-x^2|<1$. This is the open interval $(0,\sqrt 2)$. So
$$
\lim_{n\to\infty}f_n(x) =
\begin{cases}
0, & \text{when $x=0$ } \\
1, & \text{when } 0<x< \sqrt 2.
\end{cases}
$$
For $x\geq \sqrt 2$ or $x<0$ the limit does not exist, since $|1-x^2|>1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1905825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How do you show that two sets are disjoint? Here's a problem I am trying to solve for recreation.
$$
A\cap B\subset C' \text{ and } A\cup C\subset B. \hspace{2 mm}\text{ Show that $A$ and $C$ are disjoint.}
$$
I can clearly see how A and C would be disjoint. Essentially, if my understanding is correct, A and C are ... | Let's try to capture your argument algebraically.
I think your core idea is that if you take $B$ as the universe, then $A$ and $C$ are still subsets of the universe, and $A \cap C$ within $B$. The question, now, is how to translate that back to the actual universe you're working in.
The main thing you want here is that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1905935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $a+b+c=0$ then the roots of $ax^2+bx+c=0$ are rational? If $a+b+c=0$ then the roots of $ax^2+bx+c=0$ are rational ?
Is it a "If and only if " statement or "only if " statement ?
For $a,b,c \in \mathbb Q$ , I think it is a "if and only if" statement . Am I correct ?
I can prove that if $a+b+c=0$ and $a,b,c \in \mathb... | If $c = -a-b$, then the discriminant is $$b^2-4ac = b^2 +4a(a+b) = b^2 +4ab +4a^2 = (b+2a)^2.$$
Since the discriminant is a perfect square, then the roots are always rational.
Their values are
$$\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}
=\dfrac{-b \pm (b+2a)}{2a}
\in \left\{ 1, -\dfrac{a+b}{a} \right\}$$
Of course, now ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1906065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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When not to treat dy/dx as a fraction in single-variable calculus? While I do know that $\frac{dy}{dx}$ isn't a fraction and shouldn't be treated as such, in many situations, doing things like multiplying both sides by $dx$ and integrating, cancelling terms, doing things like $\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}$ w... | There are places where it is "obvious" that we should not blindly
apply the laws of arithmetic to $\frac{dy}{dx}$ as if it were
a ratio of real numbers $dy$ and $dx$.
An example from another question is
$$
\frac{dy}{dx}+\frac{du}{dv} \overset ?= \frac{dy\,dv+dx\,du}{dx\, dv},
$$
where the left-hand side has a clear int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1906241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "77",
"answer_count": 8,
"answer_id": 4
} |
Solving quadratic inequalities I have a quadratic inequality I am halfway through solving but cannot quite figure out the concept.
$$3+\frac{4-x}x>0$$
I so far understand that without knowing whether x is positive, I would need to square it, but I do not know what comes next.
| We have
$$3+\frac{4-x}{x} >0$$
$$\frac{3x+4-x}{x} >0$$
$$\frac{2x+4}{x} >0$$
Divide both sides by $2$
$$\frac{x+2}{x} >0$$
Now use wavy curve method
$-\infty+++++ (-2)-----(0)++++++ \infty$
As we want L.H.S. to be positive, hence we get
$x \in (-\infty, -2) \cup (0, \infty)$
Edit 1
Let me explain you how I decided the... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
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How can I argue that Lie derivative is not a connection? I am reading Lee's book of riemannian geometry and he asks to show that Lie derivative of two vector fields on a riemannian manifold is not a connection.
How can I argue that this is true?
He also asks to show that there is a vector field $V$ on $\mathbb{R}^2$ su... | Your example for the second problem would be OK, except that I suppose your $V$ vanishes on the $y$-axis instead of the $x$-axis.
Your example is also the sort of thing you should think about to solve your first problem. The axiom $\nabla_{fX} Y = f \nabla_X Y$ for a connection is actually equivalent to saying that, f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1906479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Invert Cubic Bezier Curve I am trying to 'invert' a path.
Lines are fairly easy as they just need to be multiplied by -1, however, I am struggling to convert the bezier curve path into its inverse.
The relevant code is:
c0,0,1.628,34.086-32.059,34.086
c-33.688,0-32.059-34.086-32.059-34.086
where c denotes the start of... | Your clarification does not clarify (for me, at least). I still don't know what you mean by "invert".
If you want to "flip" the curve (mirror it about a vertical line), then negate the x-coordinates of all the control points.
If you want to reverse the direction of the curve (trace out the same curve, but in the opposi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1906602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A concrete example of an orthonormal basis of a Hilbert module over $K(H)$, the algebra of compact operators Suppose $H$ be a Hilbert space, $K(H)$ be the set of compact operators on $H$, $E$ be Hilbert module over $K(H)$ and $(e_{\lambda})_{\lambda \in I}$ be a orthonormal bass for $E$,
Can you mention example for $H... | Why don't you take any finite-dimensional Hilbert space $H$ and $E=K(H)$. Then $E$ is just the algebra of matrices $M_n$ with $n=\dim H$. The standard matrix units in $M_n$ certainly form an orthonormal basis.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1906687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Given two equivalent norms, convergence in one implies convergences in other. Suppose we have two equivalent norms, $\|\cdot \| \sim \left\vert\!\left\vert\!\left\vert \cdot\right\vert\!\right\vert\!\right\vert$.
We assume that $x_n \rightarrow x $ in $(X,\|\cdot \|)$. Show that $x_n \rightarrow x $ in $(X,\left\vert\!... | I think you mean $J(\epsilon) = K(\epsilon/\beta)$.
For all $n \ge J(\epsilon)=K(\epsilon/\beta)$, we have $\|x_n-x\| \le \epsilon/\beta$, so
$$||| x_n-x||| \le \beta \|x_n-x\| \le \beta \epsilon/\beta=\epsilon.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1906782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Standard deviation is diverging... I've made some mistake. I'm supposed to find the standard deviation of the Fourier transformed function $f(t) = e^{-|t|/a}$. Just a note: to make it easier on you guys I'll be leaving off the normalization constant. Let me know for some reason you think I should add it back in.
The ... | Comment: This is a Laplace distribution with median 0 and scale parameter $\alpha.$
The Wikipedia article on this distribution states that its variance is $2\alpha^2.$ It also givee the correct characteristic function ('CF').
I don't think it is difficult to get the variance directly
from the density function.
If your... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1906893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How many ways can a natural number n be expressed as a sum of one or more positive integers, taking order into account? Q: The number 4 can be expressed as a sum of one or more positive integers, taking order into account, in 8 ways:
\begin{array}{l} 4&=1+3&=3+1&=2+2&=1+1+2\\
&=1+2+1&=2+1+1&=1+1+1+1. \end{array}
In ge... | Since your title carefully distinguishes between natural numbers (which include $0$) and positive integers (which do not, at least not for the English sense of "positive"), I think a formula should be given that gives the proper value (namely $0$, since at least one summand was required) for $n=0$, and $2^{n-1}$ does ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1907123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 3,
"answer_id": 2
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Why is the distance between two circles/spheres that don't intersect minimised at points that are in the line formed by their centers? From GRE 0568
From MathematicsGRE.Com:
*
*I'm guessing the idea applies to circles also?
*Is there a way to prove this besides the following non-elegant way?
*
*Form a line... | Basically the question in your title is answered by the fact that a straight line is the shortest path between two points. If $P,Q$ are the points on your two spheres (or circles, if you are in a plane), and $C_i$ and $r_i$ are their respective centres and radii, for $i=1,2$, then $C_1-P-Q-C_2$ is a path from $C_1$ to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1907243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
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Rationalizing denominator with cube roots Rationalize the denominator of $$\frac{6}{\sqrt[3]{4}+\sqrt[3]{16}+\sqrt[3]{64}}$$ and simplify.
I already have an answer. I just want to compare answers with others. Maybe someone has different solutions? Also, I really disagree with the answer found at the back of the questio... | $$\sqrt[3]{64}=4\;,\;\;\sqrt[3]{16}=4^{2/3}\;\implies$$
$$\sqrt[3]4+\sqrt[3]{16}+\sqrt[3]{64}=4^{1/3}+4^{2/3}+4=4^{1/3}\left(1+4^{1/3}+4^{2/3}\right)=$$
$$=4^{1/3}\frac{1-4}{1-4^{1/3}}=3\cdot4^{1/3}\frac1{4^{1/3}-1}\implies$$
$$\frac6{\sqrt[3]4+\sqrt[3]{16}+\sqrt[3]{64}}=\frac{2(4^{1/3}-1)}{4^{1/3}}=2^{1/3}(4^{1/3}-1)=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1907342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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In a triangle $ABC$, $AB = a-b$ and $BC = 2\sqrt{ab}$, then find $\angle B$? Is this question solvable?
In $\Delta ABC$, $AB = a-b$ and $BC = 2\sqrt{ab}$, then $\angle B$ is
(a) $\: 60^{\circ}$
(b) $\: 30^{\circ}$
(c) $\: 90^{\circ}$
(d) $\: 45^{\circ}$
| Using @JanEerland ‘s suggestion, $BC = k \sin A$, $AC = k \sin B$, and $AB = k \sin C$; for some $k \ne 0$.
Substituting in the cosine law (wrt B), we get
$$\sin^2 B = \sin^2 A + \sin^2 C – 2 \sin A \sin C \cos B$$
(There might be others) but one solution of it is $B = 90^0$ and $A$ is then complement to $C$.
Note that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1907436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Evaluate the integral $\int_0^\pi \sin{(x \cos{t}})\cos{t}\; dt$ How to evaluate:
$\int \sin{(x \cos{t}})\cos{t}\; dt$
or:
$\int_0^\pi \sin{(x \cos{t}})\cos{t}\; dt$
| $\int\sin(x\cos t)\cos t~dt=\int\sum\limits_{n=0}^\infty\dfrac{(-1)^nx^{2n+1}\cos^{2n+2}t}{(2n+1)!}~dt$
For $n$ is any non-negative integer,
$\int\cos^{2n+2}t~dt=\dfrac{(2n+2)!t}{4^{n+1}((n+1)!)^2}+\sum\limits_{k=0}^n\dfrac{(2n+2)!(k!)^2\sin t\cos^{2k+1}t}{4^{n-k+1}((n+1)!)^2(2k+1)!}+C$
This result can be done by succe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1907526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Minimize the number of points in a piecewise linear approximation I have $m$ data points $(x_i,y_i)$ in a given interval. I would like to find a piecewise linear function $f(x)$ that approximate these $m$ points with a minimum number of points $n$ so that my approximation error is below a tolerance $\epsilon$.
My $m$ p... | Here is the way that looks obvious to me; maybe someone wiser will point out how it's inefficient, or fails on perverse input.
Consider the $(a,b)$ plane in which each point represents a function $y=ax+b$. Each of your inputs, with its tolerances, defines a band in that plane. An intersection of such bands is a conve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1907587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 1
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How to solve this complex equation for the modulus of z? The question is as follows:
All the roots of the equation $11z^{10}+10iz^9+10iz-11=0$ lie:
$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (i=\sqrt{-1})$
(a) inside $|z|=1$
(b) on $|z|=1$
(c) outside $|z... | Substitution $z=e^{it}$ gives the trigonometrical equation
$$11\sin5t+10\cos4t=0,\qquad(1)$$
or
$$\cos 4t=-1.1\sin 5t.$$
Easy to see that
$$RHS\left(\dfrac{2k+1}{10}\pi\right)=1.1(-1)^{k+1}$$
for $k=-3,-2,-1,0,1,2$, so LHS and RHS have at least five intersections for $t\in\left(-\dfrac\pi2,\dfrac\pi2\right)$.
This ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1907706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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For any $q\in [0,1]$ there exist $p>0$ such that $q(1+p)=1$ I was reading what is mathematics and to prove that if $q$ is a number between $0$ and $1$ then $q^n$ tends to $0$, they use that $q$ can be written as $q=\frac{1}{1+p}$ with $p>0$.
It's equivalent to:
for any $q\in (0,1)$ there exist $p>0$ such that $q(1... | How about just solving the equation?
$$q(1+p)=1$$
$$1+p = \frac{1}{q}$$
$$ p=\frac{1}{q}-1$$
Of course this doesn't work for $q=0$ (division by zero) or $q=1$ (then $p =0 \not \gt 0$) , but in those cases, the statement is clearly false.
Thus, for $q \in (0,1)$, we know that $\frac{1}{q}>1$, so $p=\frac{1}{q}-1>0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1907781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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What logic rule is used to show $(p \lor q) \land (p)=p~$?
What logic rule is used to show: $(p \vee q) \wedge (p)=p~$?
This is obvious, because if p is true the whole expression is true, and if p is false, the whole expression is false, but I'm not sure which logic rule is used to draw this conclusion.
| One can show this with Boolean algebra as follows:
$$
(p \vee q)\wedge p = (p \vee q)\wedge (p\vee F) = p \vee (q \wedge F) = p \vee F = p
$$
If we have $\cdot$ for $\wedge$ and $+$ for $\vee$, then here's what this looks like:
$$
(p+q)p = (p+q)(p+0) = p+q0 = p+0 = p
$$
To prove this, I am using the "distributive laws"... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1907918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Rolling an $n$-sided die until repeat is found Problem: We are rolling an $n$-sided die. You roll until you reach a number which you have rolled previously. I need to calculate the probability $p_m$ that we have rolled $m$ times for such a repeat.
My first thought was to try some inputs. I took $n=6$. I noticed that wh... | for a die with $n$ sides, if you haven't already seen a duplicate, the probability of getting a repeat on the $k$th roll is
$$\frac{k-1}{n}$$
So, to get a repeat exactly on the $j$ you must first succeed at getting to the $j$th roll without any repeats, and then roll a repeat:
$$\begin{align}P(j) &= \frac{j-1}{n}\cdot\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1907979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the sixth side of hexagon. You are given a hexagon inscribed in a circle. If the lengths of $5$ sides taken in order are $3,4,6,8$ and $7$ units, find the length of $6^\text{th}$ side.
Not got the slightest of idea how to proceed, so I can't show my attempts.
| If the radius of the circle is $r$, the sixth side length is $s$, and you label the central angles $\theta_1,\ldots,\theta_6$, then using the Law of Cosines
$$\begin{align}
2\pi&=\theta_1+\ldots+\theta_5+\theta_6\\
2\pi&=\arccos\left(1-\frac{3^2}{2r^2}\right)+\ldots+\arccos\left(1-\frac{7^2}{2r^2}\right)+\arccos\left(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1908096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to integrate $\int_1^\infty e^{-\alpha x}J_0(\beta\sqrt{x^2-1})\mathrm{d}x \,$? This integral is from (6.616.2) in Gradshteyn and Ryzhik.
$$
\int_1^\infty e^{-\alpha x}J_0(\beta\sqrt{x^2-1})\mathrm{d}x \,=\frac{1}{\sqrt{\alpha^2+\beta^2}}e^{-\sqrt{\alpha^2+\beta^2}}
$$
I want to know how to do this integral and th... | The two expressions are equal by analytical continuation whenever the left hand side exists. The only problem for the convergence of the integral is at $x\to \infty$. We have the asymptotic expansion $(|\arg z| < \pi)$
$$ J_0(z) \sim \sqrt{\frac{2}{\pi z}} \cos(z-\pi/4).$$
Thus, for $x\to \infty$, we have that the inte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1908181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Exact probabilities in a $M/M/1$ queue Suppose we have a $M/M/1$ queue with arrival rate $\lambda$, service rate $\mu$, and $\lambda<\mu$. Suppose also that there are initially $k$ people in the queue.
I want to find the exact probability that there are $n$ people in the queue at time $t$, for arbitrary $n$ and $t$. Ho... | This is covered in Wikipedia's M/M/1 queue article, it is the transient solution of the model. You're looking for
$$p_n(t)=e^{-(\lambda+\mu)t} \left[ \rho^{\frac{n-k}{2}} I_{n-k}(at) + \rho^{\frac{n-k-1}{2}} I_{n+k+1}(at) + (1-\rho) \rho^{n} \sum_{j=n+k+2}^{\infty} \rho^{-j/2}I_j(at) \right]$$
where $p_n(t)$ is the pro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1908275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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All simple modules over a PID. The exercise is to classify all simple modules over a PID. I tried the following:
If $M$ is a simple module over $R$ (PID) then for a $m \in M$ with $m \neq 0$ we have $M = (m)$ then i can define $f:R \rightarrow M$ by $f(r) = rm$. The homomorphism theorem give that $R/\ker(f) \simeq M$ ... | Let $S$ be a simple module over a commutative ring $R$. Then $S\ne\{0\}$ by definition and, if $x\in S$, $x\ne0$, we have $Rx=S$ because $S$ is simple.
Then the map $\varphi\colon R\to S$ defined by $r\mapsto rx$ is surjective and so
$$
S\cong R/\ker\varphi
$$
Since $S$ is simple, it follows from the homomorphism theor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1908463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Implication in Zassenhaus Lemma I need to proof Zassenhaus Lemma using the First Isomorphism Theorem and I have a problem with the following implication:
$H' \vartriangleleft H < G, K' \vartriangleleft K < G \Longrightarrow H'(H \cap K') \vartriangleleft H'(H \cap K) < H.$
I have shown that $H \cap K' \vartrianglelef... | Perhaps it is more clear if you prove the following:
Claim: If $C \trianglelefteq H$ and $A \trianglelefteq B \leq H$, then $CA \trianglelefteq CB \leq H$.
Then your claim follows with $C = H'$ and $A = H \cap K'$, $B = H \cap K$. Note that in this situation $A \trianglelefteq B$ since $K' \trianglelefteq K$.
Maybe ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1908534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What is $\Pr(Y\in[\pi,X+\pi]\mid X)$ if $X \sim U(0,\pi)$ and $Y \sim U(0,2\pi)$? Let $X \sim U(0,\pi)$ and $Y \sim U(0,2\pi)$ be two uniform independent distributions. What is $\Pr(\left.Y\in[\pi,X+\pi]\right|X)$? Intuitively I know that the result is $\frac{1}{4}$ but how can I formally derive the density function i... | It depends on the joint distribution of $X$ and $Y$. If almost surely, $Y=2X$, then $\Pr(Y\in [\pi, \pi+X]|X) = \mathbb I(X>\pi/2)$.
If $X$ and $Y$ are independent, $\Pr(Y\in [\pi, \pi+X]|X) = X/(2\pi)$. Indeed, the unconditional probability in this last case is 1/4. I can't find, off the top of my head, a situation w... | {
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"timestamp": "2023-03-29T00:00:00",
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Proving that $GH$ is parallel to $AD$ in the given figure
In the above figure, $ABCD$ and $AECF$ are two parallelograms such that $EF$ is parallel to $AB$. $DHF$ and $BGE$ are straight lines intersecting $EC$ and $AF$ at $H$ and $G$ respectively. We need to prove that $GH \parallel AD$.
One approach may be by using Ba... | Notice first of all that all lines are symmetric around the common center $O$ of parallelograms $ABCD$ and $AECF$. It follows that $EHFG$ is a parallelogram and $EO=FO$.
Produce $GH$ to meet $AB$ at $M$. By similar triangles one has:
$$
EO:BM=OG:GM=FO:AM.
$$
Hence $AM=BM$ and $GH$ belongs to line $OM$, connecting the m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1908733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Confused by proof of the irrationality of root 2: if $p^2$ is divisible by $2$, then so is $p$. In typical proofs of the irrationality of $\sqrt{2}$, I have seen the following logic:
If $p^2$ is divisible by $2$, then $p$ is divisible by $2$.
Perhaps I am being over-analytical, but how do we know this to be true? IE.... | The quickest proof of that fact is to note that every whole number $n$ is either even or odd.
If $n$ is even, $n=2k$ for some whole number $k$: $n^2 = 4k^2 = 2(2k^2)$ is even.
If $n$ is odd, $n=2k+1$ for some whole number $k$: $n^2 = (2k+1)^2 = 4k^2 +4k + 1 = 2(2k^2 +2k) +1$ is odd.
Therefore the square of a whole numb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1908946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 12,
"answer_id": 4
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Confusion about derivative notation I have always been confused about Leibniz notation. Not the notation itself, but the fact that it treats the differential operators ($d$, $\partial$) as being multipliable. The most famous example would probably be the Schrödinger equation, which if often denoted something like this:... | What is happening is that the physicists writing out the equation are using an operator notation, where $\frac{\partial}{\partial x}$ is a shorthand for the operator of taking the partial derivative in the $x$ direction of whatever appears on the right of the operator.
The reason this appears like multiplication is tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1909019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find all numbers $z \in \mathbb{C}$ such that $(z−i)^5 = \sqrt{3} +i$ This is a follow-up question to finding all solutions for $z \in \mathbb{C}$ such that $z^5 = \sqrt{3} +i$ but I have no idea how to approach this question (might just be having a brain fart)
The only way I can think of solving it would be to expand ... | Hint:
Write $\sqrt 3+i$ in exponential form., and you'll find out it's a problem of finding the $5$th roots of a complex number with modulus $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1909116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that any integer $n>7$ can be written as the sum of $3$s and $5$s exclusively
Show that any integer $n>7$ can be written as the sum of $3$s and $5$s exclusively,
i.e.,
$$
8= 5+3 \\
9=3+3+3 \\
10 = 5+5 \\
11 = 5+3+3 \\
12 = 3+3+3+3
$$
So I've started in a couple directions without progress. I think it makes ... | $2*3 - 5 = 1$
$2n*3 - 5n = n$.
$3(2n - 5k) + 5(3k - n)= n$.
To assure that $2n - 5k \ge 0$ and $3k - n > 0$...
If $n = 3m - r; r = 0, 1,2$ then $k$ can be anything equal or greater than $m$ so long as $2n - 5k \ge 0$ i.e. $6m - 2r - 5k \ge 0\implies k \le 6m/5 - 2r/5= m + \frac{m-2r}5$.
So long as $m \ge 4$ we will a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1909194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Intuition for the epsilon-delta definition of continuity This is my first question so I hope this sort of thing is OK to ask. I'm working my way through Rudin's Principles of Mathematical analysis, and I'm up to chapter 4, which is on continuity in the context of functions between metric spaces. It introduces what I un... | That is an almost correct intuitive formulation of what continuity is. Somehow you also need to get across that the actual size of the allowable deviations does not have anything to do with it. You could do that by saying "for any interpretation of the word 'small'", or something like that.
It does definitely show the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1909351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Locally Path Connected Definition Why are the following two statements equivalent for any topological space $X$?
1) $X$ is locally path connected (meaning, it has a basis of path connected sets).
2) Every point of $X$ has a path connected neighborhood.
Is it simply that a path connected neighborhood is an open set in ... | They aren't equivalent. Indeed, any path-connected space satisfies (2), since you can take the neighborhood to just be $X$ itself. But not every path-connected space is locally path-connected (see https://math.stackexchange.com/a/135483/86856, for instance).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1909443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Counting using permutation and combination How many solutions are there to the equation
$x_1+x_2+x_3+x_4+x_5=21$,
where $x_i,i=1,2,3,4,5$, is a nonnegative integer such
that
$ 0 ≤ x_1 ≤ 3$, $1 ≤ x_2 < 4$, and $x_3 ≥ 15$?
I tried it .My Approach-:
$ x_3=x_3'+15 \implies x_1+x_2+x_3'+15+x_4+x_5=21 \implies x_1+x_2+x_3... | Let $y_3 = x_3 -15$. We need the number of solutions to
$x_1+x_2+y_3+x_4+x_5 = 21 - 15$ with $y_3 \geq 0$, $0 \leq x_1 \leq 3$ and $1 \leq x_2 <4$. The number of solutions is the coefficient of $x^6$ in
\begin{align*}
(1+x+x^2+x^3)&(x+x^2+x^3)(1+x+x^2+\cdots)(1+x+x^2+\cdots)(1+x+x^2+\cdots)\\
&= x(1+x+x^2+x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1909525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Empty Set $\{\}$ is the Only Basis of the Zero Vector Space $\{0\}$ Question
Suppose we want to find a basis for the vector space $\{0\}$.
I know that the answer is that the only basis is the empty set.
Is this answer a definition itself or it is a result of the definitions for linearly independent/dependent sets and ... | The standard definition of basis in vector spaces is:
$\mathcal B$ is a basis of a space $X$ if:
*
*$\mathcal B$ is linearly independent.
*The span of $\mathcal B$ is $X$.
You can easily show both of these statements are true when $X=\{0\}$ and $\mathcal B= \{\}$. Again, you have to look at the definitions:
*... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show $A^T$ has an eigenvector with all components rational
Matrix $A$ is a $5 \times 5$ matrix with rational entries such that $(1, \sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5})^T$ is an eigenvector of A. Show that $A^T$ has eigenvector with all components rational.
My idea is: let the eigenvalue associated with the above ... | Let $v = (1, \sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5})^T$ and assume $A v = \lambda v$. From the first row we get
$$a_{1,1} + 2 a_{1,4} + a_{1,2} \sqrt{2} + a_{1,3} \sqrt{3} + a_{1,5} \sqrt{5} = \lambda$$
From the second row we get:
$$a_{2,1} + 2 a_{2,4} + a_{2,2} \sqrt{2} + a_{2,3} \sqrt{3} + a_{2,5} \sqrt{5} = \lambda ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to find tangents to curves at points with undefined derivatives I will explain my question with the help of an example. We need to find the tangent at origin to the curve $$x^3 + y^3 =3axy$$
The derivative at origin is $0/0$ or indeterminate, found after implicit differentiation. But the tangents exist (via Wolfram... | For algebraic curves, you can use the notion of tangent cone.
Consider the curve defined by $P(x, y) = 0$, where $P(x, y)$ is a polynomial. Write
$$P(x, y) = P_m(x, y) + P_{m+1}(x, y) + \dotsb + P_{m+k}(x, y)$$
where each $P_i(x, y)$ is a polynomial of degree $i$, and $P_m(x, y) \neq 0$, i.e. $P_m(x, y)$ is the homogen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1909877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving $|z-1|<|z-i|$ is an open set Consider this set:
$$|z-1|<|z-i|$$
Suppose $z=x+iy$, then:
$$\sqrt{(x-1)^2+y^2}<\sqrt{x^2+(y-1)^2}\implies$$
$$(x-1)^2+y^2<x^2+(y-1)^2\implies$$
$$x^2-2x+1+y^2<x^2+y^2-2y+1\implies$$
$$-2x<-2y\implies y<x$$
First of all, am I right?
Now, in order to prove that the set $O = \{(x,y); ... | If you draw the picture, you see that your set is just $\mathcal O =\left \{ (x,y):y<x \right \}$ so pick a point $(x_0,y_0)\in \mathcal O$ and observe that the ball centered at $(x_0,y_0)$ of radius $\frac{\vert x_0-y_0\vert }{2\sqrt{2}}$ lies entirely in $\mathcal O$.
Or if you know that $f(z)=|z-1|-|z-i|$ is contin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that $4^n+ 1$ is not divisible by $3$ For all integers $n \ge 0$, prove that the value $4^n + 1$ is not divisible by 3.
I need to use Proof by Induction to solve this problem. The base case is obviously 0, so I solved $4^0 + 1 = 2$. 2 is not divisible by 3.
I just need help proving the inductive step. I was tryin... | I think that if you need to use induction, instead of proving "$4^n+1$ is not divisible by $3$", you should prove the more specific "$4^n+1$ has remainder $2$ when divide by $3$".
$$4^n+1=3k+2\implies4^n=3k+1\implies4^{n+1}=12k+4$$
$$\implies4^{n+1}+1=12k+5\implies4^{n+1}+1=3(4k+1)+2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1910085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 9,
"answer_id": 1
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Lateral limits of an endpoint of the interval. Imagine we have a the domain $D=[d_1,d_2]$ of a continuous function $f$.
The definition of right limit I'm using is the following:
$$\lim_{x\rightarrow a^+}f(x)=b \Leftrightarrow \forall_{\epsilon}\exists_{\delta}\forall_{x}(x\in D \ \cap \ ]a,a+\delta[\ \Rightarrow \ f(x)... | I think I get what Fujisaki is talking about. My definition of right limit is incomplete. I should have demanded, right at the begining of the definition, that $a$ be an adherent point to the set $D \cap ]a,+\infty[$, otherwise we get this problem, since $[b,a] \cap ]a,a+\delta[=\emptyset$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What does $f|A$ mean? Let $X$ be some space, $A$ a subspace of $X$, $f:X \rightarrow X$ a function. My first guess for what $f|A$ is would be that the domain of $f$ is restricted to $A$, but I can't find any confirmation that this is actually the case. The only notation that I'm aware of is $f|_A$.
For reference, the a... | You are correct, this is indeed just a restriction.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Discriminant of splitting field Let K number field, $O_K$ be its integer domain. We all know $O_K$ is a free $\mathbb{Z}$-module. If L is a finite (or galois) extension of K, whether $O_L$ is a free $O_K$-module?
In addition, let $f$ be a irreducible polynomial in $\mathbb{Q}$, α is a root of $f$, $K$ is the splitting... | Answer to your 2nd question:
In the relative situation $K/k$, one defines the discriminant ideal $\Delta (K/k)$ as being the ideal of $O_k$ generated by all the discriminants of all the $k$-bases of $K$ consiting of integral elements. A finer invariant is the different $\mathfrak D(K/k)$, which is an ideal of $O_K$ def... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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A nice identity involving urns and balls problem Prove the identity: $$\frac{\displaystyle\sum_{k=0}^{a} {n+a-k-2\choose n-2}}{\displaystyle {n+a-1\choose a}} = 1$$ where $C_{i}^{j}$ is defined as the number of ways to simultaneously choose $j$ objects from $i$ objects.
My attempt: I was trying to use a combinatorial a... | ${n+a-1 \choose n-1}={n+a-1 \choose a}$ is the number of non-negative integer solutions to:
$x_1+x_2+\ldots+x_n=a$
Let $x_1=k$ with $k \in \{0,1,\ldots,a\}$. Then the corresponding number of solutions is ${n+a-k-2 \choose n-2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1910602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Lower bound on the sum of divisor function It there a lower bound on the sum of divisors function? More specifically is there a "simple" function $f(n)$ such that $\sigma(n) \ge f(n)$ for all $n$ large enough?
| The best function is actually $f(n)=n+1$ which holds iff $n$ is prime.
Since there are infinitely many primes you can't expect anything better for $n$ large enough.
Ramanujan under the assumption of Riemann hypothesis has shown that $\sigma(n)<e^{\gamma}n{\log\log}n$ for $n$ sufficiently large.
Unfortunately, $\sigma(n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1910739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Can we take derivative of $x$ over $f(x)$ ? I have, probably the most dumb question ever:
if we have $f = f(x)$
can we take derivative:
$$\frac{dx}{df} = \frac{dx}{df(x)} ?$$
I was thinking of that it may be possible, if we find some inverse, or smth like substitution. So if $f= cx^2$, then:
$$\frac{dx}{df} = \frac{dx}... | Suppose that $f$ is a function $D\to\Bbb R$. To make sense of what you write, we need that $f$ has an inverse function, that is, that there exists $g$ such that for all $x\in D$, $g(f(x)) = x$. Then using the chain rule, we have that $\frac{d(g\circ f)}{dx}(x) = \frac{dg}{df}(f(x))\times \frac{df}{dx}(x)$ for all $x\in... | {
"language": "en",
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Finding the smallest index $l$ with $x_l$ = $x_{2l}$ in a seqeunce
The sequence $(x_i)_{\displaystyle i \geq 0}$ has the preperiod
$2,3,5,7,11,13,17,19,23,29$ and the periodic part $31,37,41,43$. Find
the smallest index $l \in \mathbb{N}$ with $x_l = x_{2l}$.
In other words, the sequence is
$$2,3,5,7,11,13,17,19... | The sequence verifies $a_{10+k} = a_{10 + (k \; {\rm mod}\; 4)}$ for $k\geq 0$ (and there are no other relations). So look for $\ell=10+k$, $k\geq 0$ so that
$a_{10+k}=a_{20+2k}=a_{10+(10+2k)}$. And this is equivalent to $k\geq 0$ and
$$ k \equiv 10+2 k \ {\rm mod} \ 4 $$
or $k \equiv 2 \ {\rm mod} \ 4$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1910957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proof that lim sup of union equals union of lim sup My home work is: let $ A_{n},B_{n}$ be subsets of the sample space. Prove that
$$
\limsup_{n\to\infty} (A_{n}\cup B_{n}) = \limsup_{n\to\infty} A_{n}\cup\limsup_{n\to\infty} B_{n}
$$
I managed to get to this:
$$
\bigcap_{1}^{n}\bigcup_{n\geq m}^{ } A_{m}\cup \bigcap_{... | I think you can prove this using the distributivity laws of sets. $x \in \text{limsup} A_n \cup B_n \iff x \in \cap_n \cup_{ k \geq n} (A_k \cup B_k) \iff x \in \cap_n [ (\cup_{k \geq n} A_k) \cup (\cup_{k \geq n} B_k)] \iff x \in [\cap_n \cup_{k \geq n} A_k] \cup [\cap_n \cup_{k \geq n} B_k] \iff x \in \text{limsup} A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1911034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove the product of a polynomial function of the roots of another polynomial is an integer. I noticed this while solving another problem on this site.
Let $P(x)$ be a polynomial in $x$ with integer coefficients, and let the roots of $P(x)=0$ be $r_1, r_2 \ldots ,r_n$, where multiple $r_i$ might be equal if there are r... | I agree with Bill that symmetric polynomials should somehow be the standard solution, but I wanted to point out that Galois theory makes this straightforward.
It is easy to show that $s = \prod_i Q(r_i)$ is an algebraic integer, so it is enough to show that $s\in\mathbb{Q}$. By the Galois correspondence, this is the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1911132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that if $\sum |a_n|^2$ converges then $ \sum \frac{a_n}{n}$ converges Let $ \{a_n\}\in \mathbb{C}$. Prove that if $\sum |a_n|^2$ converges than $\sum \frac{a_n}{n}$ converges.
Note that this problem is taken from the first chapter on series of a calculus book, so it should be solvable with very basic tools (e.g. ... | With Cauchy-Schwarz inequality and the fact that $\sum_n 1/n^2 < \infty$, the result follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1911211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Permutations of $\{1,2,3,...,n\}$ where first k elements precede each other. Number of permutations of $\{1,2,3,...,n\}$ where first $k$ elements have the property that element $1$ precedes element $2$ which precedes element $3$ ...... which precedes $k-1$ which precedes element $k$ (not necessarily immediately).
E.g:... | Your argument is correct. A slightly different way to say it is that there are $\binom{n}k$ ways to choose which positions in the permutation contain the numbers $1,\ldots,k$, whose order within those positions is fixed, and there are then $(n-k)!$ ways to arrange the remaining numbers in their positions. The total num... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1911276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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integral solutions to a prime number?
Find the number of non-negative integer solutions to:
$$(x_1+x_2+\cdots + x_n)(y_1+y_2+\cdots + y_p)=P$$
where $n\in\mathbb{N}$ and $P$ is a prime number.
I know the answer is $2n\binom{p+n-1}{p}$ but I don't understand why?
| Hint: $ P $ being prime means that its factors are $ 1 $ and $ P $, so the integer solutions to that equation satisfy either
$$ x_1 + \dots + x_n = 1 \text{ and } y_1 + \dots + y_p = P$$
or
$$ x_1 + \dots + x_n = P \text{ and } y_1 + \dots + y_p = 1 $$
So your answer should be the sum of the numbers of solutions to the... | {
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"timestamp": "2023-03-29T00:00:00",
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Does this matrix have a name? (Maybe in combinatorics?) Or rather, is there a name for the class of matrices that resemble this one:
\begin{bmatrix}
1 & 1 & 1 \\
1 & 1 & 0 \\
1 & 0 & 1\\
1 & 0 & 0\\
0 & 1 & 1\\
0 & 1 & 0\\
0 & 0 & 1\\
0 & 0 & 0
\end{bmatrix}
I made the matrix because I wanted to be abl... | I don't think there's a standard name for this matrix, but the idea that it represents is well known and useful. Good for you if you invented it yourself.
Each row of your matrix describes one of the $2^n$ subsets of an $n$ element set (you have $8$ rows since $n=3$) using a string of $n$ bits, each either $1$ or $0$, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$n \geq 4$ men, among whom are A, B and C stand in a row.then what is the probability that C stands somewhere between A and B $n \geq 4$ men, among whom are A, B and C stand in a row. Assume that all possible orderings of the $n$ men are equally likely, then what is the probability that C stands somewhere (not necessar... | Your explanation is correct. To make it a bit more explicit, consider the positions 3 occupied by $A$, $B$, and $C$, regardless of how the three are arranged within those positions. there are $\binom n3$ such positions, and for each there are $6$ orderings of $A$, $B$, and $C$ from leftmost to rightmost. By symmetry, e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1911617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Number of ways to arrange n people of increasing height The task is to arrange $n$ people in a single line such that exactly $x$ of them are visible from left and $y$ of them visible from right (Since some taller people block the view of the shorter ones).
For example, if the people were arranged in line with heights 3... | This is building on @Fimpellizieri's answer. As @Fimpellizieri showed, we can reduce this to the problem of calculating $l(n,k)$, the number of ways of ordering $n$ people so that exactly $k$ are visible from the left. (Note that $r(n,k) = l(n,k)$, since you can reverse to order to switch left-visibility into right-v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1911769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
If $X$ and $Y$ are independent exponential then $\{\min(X,Y)>z\}$ and $\{X
Given that $X$ and $Y$ are exponential random variables, possibly with different rates. Also, $Z = \min(X,Y)$. Show that the event $\{Z>z\}$ is independent of $\{X<Y\}$.
So, I've found the probabilities for both $P(Z>z)$ and $P(X<Y)$. But I don'... | I assume that $X$ and $Y$ are independent. Let $\lambda_X$ and $\lambda_Y$ be the rates of $X$ and $Y$, respectively.
We want to find the following probability:
$$P(Z > z \wedge X < Y) = P(\min(X,Y) > z \wedge X <Y).$$
The set $\mathcal{S}_z = \{(X,Y) : \min(X,Y) > z \wedge X <Y\}$ is defined by inequalities $X < Y$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1911852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Limit of $\sum\limits_{k=0}^n \frac{{n\choose k}}{n^k(k+3)}$ when $n\to\infty$ What is $$ \lim_{n \to \infty} \sum_{k=0}^n \frac{{n\choose k}}{n^k(k+3)}\ ?$$ I know the way by integration and that the answer is $e-2$ but I am more interested in use of sandwich theorem which provides a maxima or a closed form to it. Ex... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
Linear subspaces dimensions (2) I have the following question:
$U$ and $V$ are two linear subspaces of $\mathbb{R}^7$, with $\dim(U) = \dim(V) = 5$.
What is the minimum dimension of $U ∩ V$?
Using the Grassmann Formula I have:
$\dim(U ∩ V) + \dim(V + U) = \dim (U) + \dim(V)$
$\dim(U ∩ V) + \dim(V + U) = 5 + 5 = 10$... | You're correct. By Grassmann's formula,
$$
\dim(U\cap V)=\dim U+\dim V-\dim(U+V)
$$
so it is minimal when $\dim(U+V)$ is maximal. Since it is possible that $\dim(U+V)=7$ (an example would be needed), but not more, the minimum value for $\dim(U\cap V)$ is $5+5-7=3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1912139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How are matrices related to vectors? I know it's a silly question, but there's something I feel like I'm missing in my understanding of matrices.
I'm studying linear algebra, and much of what we covered in the first few topics related to vectors (vector spaces, linear independence, etc.), but then all of a sudden we s... | A vector is a linear array of quantities.
A matrix is a 2-dimensional array of quantities.
Three dimensional and higher dimensional arrays also exist, they are called Tensors.
A matrix can be thought of a sequence of column vectors, but also as a sequence of row vectors, both interpretations are useful.
An example of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1912236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 2
} |
Prove Without Induction: $\sum\limits_{k=2}^{n} \frac{1}{k(k-1)} = 1 - \frac{1}{n}$ everybody.
I'm suppose to prove this without induction:
Prove Without Induction: $\sum\limits_{k=2}^{n} \frac{1}{k(k-1)} = 1 - \frac{1}{n}$
I'm not sure how to do it. I tried a bit of algebraic manipulation, but I'm not sure how to do i... | That's a telescoping series. Use partial fraction techniques to do the following split: $$\frac{1}{k(k-1)} = \frac{1}{k-1} - \frac{1}{k},$$ and proceed from there.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1912321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
parentheses vs brackets for indexed family of sets I understand the definition of parentheses and brackets. However this problem involving indexed family of sets has me questioning the definitions.
Q: Let $\mathscr A = \{[-x,0]: x \in \mathbb R \text{ and } 0 < x < 1\}$.
Find the union over $\mathscr A$ and the interse... | $(-1,0]$ is correct.
The number $-1$ is not an element of any set in the family $\mathcal{A}$ and therefore cannot be in the union.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1912435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
prove that $R$ is an equivalence relation $$\forall a,b \in \mathbb{Q} \quad aRb \Leftrightarrow \quad \exists k \in \mathbb{Z}: \quad b=2^ka$$
1) Reflexivity:
$\forall a \in \mathbb{Q}\quad aRa \Leftrightarrow \quad \exists k \in \mathbb{Z}: \quad a=2^ka $
choosing $k=0 \quad \Rightarrow a=2^0a=a \Rightarrow aRa \Ri... | Your proofs for parts (1) and (3) are correct. For symmetry, suppose $aRb$, so that
\begin{equation}
b = 2^ka
\end{equation}
for some $k\in\mathbf{Z}$. Can you think of an integer $l$ so that
\begin{equation}
a = 2^lb?
\end{equation}
(Hint: Remember that negative integers are integers too!) Once you have such an int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1912548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Example of a countable compact set in the real numbers Can someone give me an example or a hint to come up with a countable compact set in the real line with infinitely many accumulation points?
Thank you in advance!
| What about if we define $H = \{ \frac{1}{n} : n\in \mathbb{N}\} \cup \{0\}$, a sort of standard countable compact set with 0 at its sole limit point, then define your countable compact set to be:
$$S = \{ x + y \mid x, y \in H\}.$$
To unpack the thought behind this definition:
*
*This set $S$ is countable.
*This se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1912628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 3
} |
A proof of the identity $ \sum_{k = 0}^{n} \frac{(-1)^{k} \binom{n}{k}}{x + k} = \frac{n!}{(x + 0) (x + 1) \cdots (x + n)} $. I have to prove that
$$
\forall n \in \mathbb{N}_{0}, ~ \forall x \in \mathbb{R} \setminus \mathbb{N}_{0}:
\qquad
\sum_{k = 0}^{n} \frac{(-1)^{k} \binom{n}{k}}{x + k}
= \frac{n!}{(x + 0) (x + ... | I thought it might be instructive to present a proof by induction.
First, we establish a base case. For $n=0$, it is straightforward to show that the expression holds.
Second, we assume that for some $n\geq 0$, we have
$$\sum_{k=0}^n \binom{n}{k}\frac{(-1)^k}{x+k}=n!\prod_{k=0}^n\frac{1}{x+k}$$
Third, we analyze t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1912720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 3
} |
Payoff of a dice game I came across this question today:
"A fair die is tossed. If 2,3 or 5 occurs, the player wins that number of rupees, but if 1, 4 or 6 occurs, the player loses that number if rupees. Then find the possible payoffs for the player".
My textbook has then proceeded to solve it like this:
What is ... | For such simple excercise you can imagine to throw the die a big number of times, obtaining perfect statistics.
In your case let's throw the die 600 times, obtaining 100 ones, 100 twos and exactly 100 polls for all the figures.
Our earnings would have been $$100*(-1)+100*2+100*(3)+100*(-4)+100*(5)+100*(-6)=-100$$ so, f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1912822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Galois Basic Question (Cyclic Order 4 Extension of Q cannot contain i) Suppose $K$ is an extension of $\mathbb{Q}$ in $\mathbb{C}$, where $Gal(K/\mathbb{Q})$ is cyclic of order 4. Show that $i\notin K$.
($i$ is the imaginary number $i^2=-1$.)
My Galois theory is quite weak, hope someone can check if my attempt is corr... | Here is a slick solution, I think. Consider generally a cyclic extension $F/k$ and try to embed it in an over-extension $K/F/k$ such that $K/k$ is cyclic. For simplification, suppose that $F/k$ has degree $p^n$, $K/F$ has degree $p$ and $k$ contains a primitive $p$-th root $\zeta$ of unity ($p$ a prime). Then, using Ku... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1912901",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
} |
Enumerative combinatorics applications in Computer Science I am interested in specific examples and applications of enumerative combinatorics in Computer Science -- concrete problems in this field that make explicit use of the concepts and ideas from combinatorics. Are there any good references that you can point me to... | The emphasis is on enumeration rather than counting if I understand
the question correctly. The perfect match would be the combstruct
package that is included with Maple. This software is a companion to
the book Analytical Combinatorics by Flajolet and Sedgewick, which
is the canonical text and basically provi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1912995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 4
} |
$A$ is a $3\times3$ matrix and $A^2+4 A−12 I =0 $. If det$(A+2I )>0$, what is $det(A+2I )$? I need to rearrange the following $A^2+4A-12I=0$, where $A$ is an unknown $3\times3$ matrix so that I can find det of $(A+2I)$
| A has -6 and 2 and 0 as Eigen values. It can be a matrix with all elements zero except diagonal matrix with -6,2,0 in the diagonal. Now the determinant of A+2I should be greater than zero and it will be if 2 is at 1,1 and -6 at 3,3.
This matrix satisfies all the conditions u asked. And determinant of A+2I is 4. Not the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1913072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Metrizability of the weak topology $\sigma(E,E')$ of a normed space $E$ Let $(E , \left \| \cdot \right \|_E)$ a normed space with topology $\mathcal{T}_E$ induced by norm. We have that
(1) If $\mathrm{dim}(E) < \infty$ then $\sigma(E,E') = \mathcal{T}_E$ and weak topology $\sigma(E,E')$ is metrizable.
(2) If $\mathrm... | It boils down to this:
Metrizable topological vector spaces (and hence normed spaces) are locally bounded, and what this argument shows is that the $\sigma(E,E')$ topology on $E$ is not locally bounded.
To see this, pick $x_0\in N(u)$ with $x_0\neq0$. The Hahn-Banach theorem furnishes some $v\in E'$ with $v(x_0)=\|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1913169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Serre's Trick for flatness of a morphism of schemes I'm reading some exercises on abelian varieties and I came across the following claim:
Claim (Serre's Trick): Let $X,Y,S$ be schemes and suppose that $X \times_S Y$ is flat over $S$. If $X(S) \neq \emptyset$ then $Y$ is flat over $S$.
Evidently to prove the claim ... | Let $f:C\rightarrow A$ be a map (which is non zero, I assume you have units in the ring involved), you have a non zero $A$-bilinear map
$h:C\times B\rightarrow B$ defined by $h(c,b)=f(c)b$, by the universal property of the tensor product, $h$ factors by a map $\bar h:C\otimes_AB\rightarrow B$, henceforth, $C\otimes_AB... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1913276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
How to prove that the rank of a matrix is a lower semi-continuous function? I need to prove that rank($\mathrm{A}$) is not continuous everywhere but is lower semi-continuous everywhere, where $\mathrm{A}\in \mathbb{C}^{n\times m} $
| Let $A\in{\mathbb C}^{n×m}$; we use the symbol $m$ to denote a square minor (a selection of $k$ rows and $k$ columns). For any minor $m$ of the matrix define a function $f_m:{\mathbb C}^{n×m}\to {\mathbb R}$ in this way: if the minor is invertible then $f_m(A) = k$ otherwise $f_m(A) = 0$. Obviously $f_m$ is lower sem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1913394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 2,
"answer_id": 1
} |
How would you solve this polynomial? Is there a way to find the roots of equations such as $x^3-9\sqrt[3]{2}+9=0$?
I've just been using Wolfram Alpha to factor it into $(x-\sqrt[3]{4}+\sqrt[3]{2}-1)(x^2+(1-\sqrt[3]{2}+\sqrt[3]{4})x+3\sqrt[3]{4}-3)$. But for harder equations such as $$x^3-63\sqrt[3]{20}+9=0$$, Wolfram A... | We can find $\sqrt[3]{9(\sqrt[3]2-1)}$ by the following way without WA.
Indeed, let $\sqrt[3]2=x$.
Hence, $$x^3=2$$ or $$9(x^3-1)=9$$ or $$9(x-1)=\frac{9}{1+x+x^2}$$ or $$9(x-1)=\frac{27}{x^3+3x^2+3x+1}$$ or
$$\sqrt[3]{9(\sqrt[3]2-1)}=\frac{3}{\sqrt[3]2+1}$$ or
$$\sqrt[3]{9(\sqrt[3]2-1)}=\sqrt[3]4-\sqrt[3]2+1$$
and we ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1913452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
limit of product of $(a_1a_2.\dots a_n)^{\frac{1}{n}}$ How to calculate the following limit
$$ \lim_{n\rightarrow \infty} \left[ \left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)\cdots \left(1+\frac{n}{n}\right) \right]^\frac{1}{n} .$$
I was trying this by taking the $\log $ of the product and then limit but I am ... | Answering your second question as the previous answers all answer your first.
Select a sequence $\{b_n\}$ such that $a_n = e^{b_n}$
Now, instead, we have ${\left({e^{\left(\sum\limits_{i=1}^n b_i \right)}}\right)^{\frac 1 n}} = \left(\prod\limits_{i=1}^n a_i \right)^{\frac 1 n}$
This may allow you to leverage identiti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1913595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 9,
"answer_id": 5
} |
Let $F,G$ to be distribution. Is the product $FG$ also a distribution?
Let $F,G$ to be distribution functions of a random variable, say $X$. Is the product $FG$ also a distribution?
We define $\psi(x) = F(x)G(x)$. We first have:
$$\lim_{x \to \infty } \psi(x) = \lim_{x \to \infty} F(x)G(x) = 1,$$
$$\lim_{x \to -\inft... | Your reasonment is correct, your function $FG$ matches all criteria of a CDF:
*
*increasing
*is $0$ in $-\infty$ and $1$ in $\infty$
*defined on $\mathbb{R}$
Interestingly, $FG$ is the cumulative distribution function of $max(X_F,X_G)$ if $(X_F,X_G)$ are independent random variables with F and G as CDF :
$\forall ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1913678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Show that a particular mapping of an open ball in $\mathbb{R}^k$ onto $\mathbb{R}^k$ is a diffeomorphism. This is a problem from Guilleman and Pollack: Page 5, question 4.
Let $B_a$ be the open ball $\{ x: |x|^2 < a \}$ in $\mathbb{R}^k$, where $|x|^2 = \sum_i x_i^2$.
Show that the map
\begin{equation}
x \mapsto \frac... | This is listed as the first typo of the book by Ted Shifrin here, so at least Ted agree with you.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1913765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove that the following spaces are complete/not How to show if the following facts are true/false:
*
*$(0,1)$ with the usual topology admits a metric which is complete.
*$ [0,1]$ with the usual topology admits a metric which is not complete.
Since usual metric gives usual topology and we know that $(0,1)$ is ... | The important thing to remember is that it is possible that a metrizable topological space $(X,\tau)$ supports two different metrics $d_1,d_2$ that induce the topology $\tau$ for which $(X,d_1)$ is complete but $(X,d_2)$ isn't.
For the first question, note that $(0,1)$ is homeomorphic to $\mathbb{R}$. Choose some home... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1913923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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