Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Two examples of modules: quotienting and direct sum, need some clarification.
*
*Any ring $A$ has the natural structure of a left, resp. right, module over itself. A submodule $J \subset A$ is just the same thing as a left, resp. right, ideal of $A$. Therefore, the set $A/J$ also has the natural structure of a left,... | *
*In $A/J$ the elements are of the from $a+J$ for some $a\in A$. Now what is the natural choice to multiply this by $a'\in A$? Well the easiest thing to do is to take $a'a+J$ which does the job.
For the direct sum remember that you can define them as functions from $\{1, \dots , n\}$ to $A$, here the natural choice f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1925385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find $\int\limits^{\infty}_{0}\frac{1}{(x^8+5x^6+14x^4+5x^2+1)^4}dx$ I was asked to prove that
$$\int\limits^{\infty}_{0}\frac{1}{(x^8+5x^6+14x^4+5x^2+1)^{4}}dx=\pi\frac{14325195794+(2815367209\sqrt{26})}{14623232(9+2\sqrt{26})^\frac{7}{2}}$$
I checked the result numerically and the first digits correct using W|F
$$\in... | Hint. A route.
One may recall the following result, which goes back at least to G. Boole (1857).
Proposition. Let $f \in L^1(\mathbb{R})$ and let $f$ be an even function. Then
$$
\int_{-\infty}^{+\infty}x^{2n}f\left(x-\frac1x\right) dx=\sum_{k=0}^n \frac{(n+k)!}{(2k)!(n-k)!}\int_{-\infty}^{+\infty} x^{2k}f(x)\: dx. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1925458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Proving $\angle QAP=45^\circ$ if $ABCD$ is a square with points $P$ in $BC$, $Q$ in $CD$ satisfying $\overline{BP}+\overline{DQ}=\overline{PQ}$ Here is the problem:
Let $ABCD$ be a square with points $P$ in $BC$, $Q$ in $CD$ satisfying $\overline{BP}+\overline{DQ}=\overline{PQ}$. Prove that $\angle QAP=45^\circ$.
So ... | Using the cosine rule:$$|AP|^2+|AQ|^2-2|AP||AQ|cos\alpha=|PQ|^2$$Replacing $|AP|,|AQ|$ and $|PQ|$:$$|AP|=\sqrt{a^2+r^2}$$$$|PQ|=\sqrt{b^2+r^2}$$$$|PQ|=a+b$$
Here is $a=|BP|, b=|DQ|$:$$a^2+r^2+b^2+r^2-2\sqrt{a^2+r^2}\sqrt{b^2+r^2}cos\alpha=(a+b)^2$$$$\implies2r^2-2\sqrt{a^2+r^2}\sqrt{b^2+r^2}cos\alpha=2ab$$Note that $b ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1925585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that if a|c, b|c and gcd(a,b)=1, then ab|c Let $a,b,c \in \mathbb{Z}$ such that $a|c$, $b|c$, gcd$(a,b)=1$. Prove that $ab|c$.
My thoughts so far: By the Unique Factorization Theorem, we can rewrite $a$ as $p_1^{\alpha_1}p_2^{\alpha_2}...p_k^{\alpha_k}$ where $p_i$'s are primes that make up $a$. Similarly, we can ... | Any time you see $gcd(a,b)=1$, it can often be helpful to think of Bezout's identity.
$$
gcd(a,b)=1\Rightarrow \exists \; x,y\in\mathbb{Z}\;s.t\; 1=ax+by
$$
Then we can scale to get
$$
c=cax+cby
$$
And now using that $a\vert c,\;b\vert c \Rightarrow c=ak_1=bk_2$ our identity becomes
$$
c=cax+cby=axbk_2+aybk_1\Rightarr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Inverse of series $z+a_2z^2+...$ Let $p$ be a power series with integer coefficients of the special form $p(z)=z+a_2z^2+a_3z^3+..$. I wonder if the inverse (composition not $1/p$) series has again integer coefficients.
I have calculated some of such series so I guess yes. What do you think?
| Given
$$
f(x)=x+a_2x^2+a_3x^3+\dots\tag{1}
$$
and
$$
g(x)=x+b_2x^2+b_3x^3+\dots\tag{2}
$$
formally, for $n\ge2$, the coefficient of $x^n$ in $g\circ f$ is
$$
0=a_n+b_n+\left[x^n\right]\sum_{k=2}^{n-1}b_k\left(x+a_2x^2+a_3x^3+\dots+a_{n-1}x^{n-1}\right)^k\tag{3}
$$
For $n=2$, $(3)$ says that $b_2=-a_2$. Then inductively... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove by induction that $(n!)^2\geq n^n$ How does one prove by induction that
$(n!)^2\geq n^n$
for all $n \geq 1$
Hint:$(1+x)^r\geq 1+rx$
, for $r\geq0$ and $x\geq-1$
Step 1
For $n=1$, the LHS=$1!^2=1$ and RHS=$1^1=1$. So LHS$\geq$ RHS.
Step 2
Suppose the result be true for $n=k$ i.e.,
$(k!)^2 \geq k^k$
Step 3
For... | Start by showing that it works for 3 (I'm starting with 3 because 1 and 2 are trivially true):
Base Case:
We have: $(3!)^2 = 6^2 = 36$
We also have: $(3)^3 = 27$. So, $36 \geq 27$. Done.
Inductive Step: Assume that it works for all n.
Prove it works for all (n+1):
Let's set it up as follows:
$\begin{align} ((n+1)!) ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to evaluate $\lim_{n\to\infty}n!/n^{k}$ I do not know how to go about finding the limit of this sequence. I know it diverges to infinity yet I can't find terms to use the squeeze lemma effectively.
$a_n = \frac{n!}{n^{1000}}$
$\lim_{n\rightarrow\infty} \frac{n!}{n^{1000}}$
I know that when $n>1000$, the numerator ... | $$\log\frac{n!}{n^k} = \log n!-\log n^k = \sum_2^n \log i - k\log n$$
Let's apply the intergral test. Note that:
$$\int_2^{n} \log x\; dx-k \log n = x\log x\Bigg|_{2}^{n} - k\log n$$
So, for $n>k$, we get:
$$x\log x\Bigg|_{2}^{n} - k\log n = (n-k)\log - 2\log 2 n>0$$
And
$$\lim_{n\to \infty} (n-k)\log n - 2\log 2 = \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1926106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is max($ab^3$) if $a+3b=4$? This question was asked by my professor,i think in the set $\mathbb Z$,its answer is 1.But,he advised me for the further exploration in $\mathbb R$.But,what i think(i may be wrong) that in $\mathbb R
$ its answer is not unique,but it should be unique.
Any suggestions are heartly welc... | If $a,b>0$, then $$a+3b=4 \ge 4\sqrt[3]{ab^3}$$From the mean inequality. Thus $ab^3 \le 1$.
If $ab<0$, then $ab^3<0$. Thus the maximum of $ab$ cannot be find here.
If $a,b<0$, then $a+3b<4$. Thus a contradiction.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Spherical Cake and the egg slicer Recently I baked a spherical cake (3cm radius) and invited over a few friends, 6 of them, for dinner. When done with main course, I thought of serving this spherical cake and to avoid uninvited disagreements over the size of the shares, I took my egg slicer with equally spaced wedges(... | I assume you mean something like this (although this is intended for five persons). If so, then that is obviously not a fair way of dividing the cake (the other slices will be much smaller than the inner slices).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1926312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Distributing 4 distinct balls between 3 people
In how many ways can you distribute 4 distinct balls between 3 people such that none of them gets exactly 2 balls?
This is what I did (by the inclusion–exclusion principle) and I'm not sure, would appreciate your feedback:
$$3^4-\binom{3}{1}\binom{4}{2}\binom{2}{1}^2+\bi... | One person gets $4$ balls: $3$ ways
One person gets $3$ balls, one person gets $1$: $\dbinom41\times3!=24$
$27$ in total.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1926503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 2
} |
$P:\mathbb{R}^2 \to \mathbb{R}, P(x,y) = x.y$ is continuous. I need to prove $P:\mathbb{R}^2 \to \mathbb{R}, P(x,y) = x.y$ is continuous.
I just proved that the sum is continuous, but I'm lost at how to manipulate the inequalities in this case.
My attempt:
Let $\epsilon>0$.
$d((x,y),(a,b)) = \sqrt{(x-a)^2 + (y-b)^2} $.... | Since
$$
\underbrace{(x_1^2+x_2^2)}_{\|x\|^2}\underbrace{(y_1^2+y_2^2)}_{\|y\|^2}=\underbrace{(x_1y_1+x_2y_2)^2}_{(x\cdot y)^2}+(x_1y_2-x_2y_1)^2
$$
We have
$$
|x\cdot y|\le\|x\|\,\|y\|
$$
Thus,
$$
\begin{align}
\left|\,x\cdot y-a\cdot b\,\right|
&=\left|\,(x-a)\cdot b+x\cdot(y-b)\,\right|\\
&\le\|x-a\|\,\|b\|+\|y-b\|\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1926624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is calculus considered to be difficult? I am going to take a calculus course soon and everyone tells me that it is very difficult. What is considered difficult about it and why do so many people fail in calculus courses?I am asking these questions so that I can prepare myself beforehand and do not face any difficul... | One of the difficulties with calculus is the amount of time spent solving one problem. Most of that time is simplifying, evaluating, and the like, before applying the calculus-level concept at the end. If you can still do precalculus-level work fluently and without many mistakes, then the exercises won't take too long;... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1926739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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What is needed to justify taking the derivative of a geometric series? Of course, if $|p| < 1$,
$$\sum_{n=0}^{\infty}p^n=\dfrac{1}{1-p}\text{.}$$
One fact that is particularly useful in probability and financial mathematics is taking the derivative of both sides leads to
$$\sum_{n=1}^{\infty}np^{n-1}=\dfrac{1}{(1-p)^2... | You can fix $k$ and consider series of the form
$$f_k(p)=\sum_{n=0}^kp^n=\frac{1-p^{k+1}}{1-p}$$
Then $f_k$ is differentiable and
$$f_k'(p)=\sum_{n=1}^knp^{n-1}\tag{$*$}$$
The series $\sum_{n=1}^\infty np^{n-1}$ converges uniformly on sets of the form $\left\{p:|p|\leq r\right\}$, where $r<1$ by Weierstrass' test, or y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1926814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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A Neat Identity Involving Zeta Zeroes While playing around, I encountered the following very curious and cool identity. Consider the exponential integral $\text{Ei}(x)$ and the $n$th nontrivial zero of the Riemann Zeta function $p_n$.
Now, look at the first few imaginary parts of the following function:
$$f(x)=\sum_{n=... | The result found by OP turns out to be quite generic; it holds for a wide range of sequences $\rho_n$ and not just the zeros of the $\zeta$-function. A precise formulation of what is observed is the following:
$$\lim_{N\to\infty} \frac{1}{N}\sum_{n=1}^N \text{Ei}(\rho_n) = \pi i \tag{1}$$
The sum above is the Cesaro me... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1926964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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Help in evaluating $\displaystyle\int\ \cos^2\Big(\arctan\big(\sin\left(\text{arccot}(x)\right)\big)\Big)\ \text{d}x$ Is there an easy way to prove this result?
$$\int\ \cos^2\Big(\arctan\big(\sin\left( \text{arccot}(x)\right)\big)\Big)\ \text{d}x = x - \frac{1}{\sqrt{2}}\arctan\left(\frac{x}{\sqrt{2}}\right)$$
I tri... | As mentioned in comments, draw the triangles. You have
$$
\frac x 1 = x = \cot\theta = \frac{\text{adjacent}}{\text{opposite}}
$$
so if you have a triangle in which $\text{opposite}=1$ and $\text{adjacent} = x$ then you have $\text{hypotenuse} = \sqrt{x^2+1}$ and so
$$
\sin\theta = \frac{\text{opposite}}{\text{hypoten... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1927025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 1
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Is $(x+1)/(x^2-1)$ defined for $x=-1$? It might sound like a silly question, but I can't come up with a clear answer.
By looking at the expression, the answer should be "no", since $(-1)^2=1$ and we're in trouble.
However, if I factorize: $(x^2-1) =(x+1)(x-1)$, $x=-1$ is still illegal. But now the terms cancel out and... | This is what's called a removable singularity. In this case, we have that
$$\lim_{x\to -1}\frac{x+1}{x^2-1}$$
exists and is equal to $\lim_{x\to -1}\frac{1}{x-1}=-\frac12$, so setting the value of that function as $-\frac12$ yields a new function that is equal to the old one wherever the old one is defined $(\mathbb{R}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1927096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Greatest common divisor relatively prime integers proof Let $gcd(c,m)=g.$
Show that if $kc+lm=g$, then $gcd(k,l)=1$
I can see that its true with this example
Let $c=5, m=15$. We have $gcd(5,15)=5$, then let $k=-2$ and $l=1$ $gcd(-2,1)=1$ but I'm not sure how to generalize it.
| We have
$$
\left\{ \begin{gathered}
\gcd (c,m) = g \hfill \\
kc + lm = g \hfill \\
\end{gathered} \right.\quad \Rightarrow \quad \left\{ \begin{gathered}
\gcd (c',m') = 1 \hfill \\
kc' + lm' = 1 \hfill \\
\end{gathered} \right.
$$
Suppose $ \gcd (c,m) \ne 1$ , then you would get the contradiction
$$
\begin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1927189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Existence of $\xi$ and $\eta$ such that $f'(\xi)+f'(\eta)=\xi+\eta$ Let $f$ be continuous on $[0,1]$, differentiable in $(0,1)$. Assume futher that $f(0)=0$, $f(1)=1/2$. Show that there exist $\xi,\eta\in (0,1)$ such that $f(\xi)+f'(\eta)=\xi+\eta$.
I saw this problem in a draft. I do not know whether it is true. Up t... | For $f(\xi)+f'(\eta)=\xi+\eta$ there is a counterexample given in
Prove $\exists \xi, \eta \in (0,1)$, such that $f(\xi)+f'(\eta)=\xi+\eta$.?
On the other hand the statement with $f'(\xi)+f'(\eta)=\xi+\eta$ is true.
Let $F(x)=f(x)-x^2/2$.
Now by the MVT there is $\xi\in (0,1/2)$ such that
$$\frac{F(1/2)-F(0)}{1/2-0}=F... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1927339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Prove inequality $2e^x>x^3+x^2$ If $x \in \Bbb R$, show that
$$2e^x>x^3+x^2$$
This inequality is right, see
Own ideas: If $x\in \Bbb R$,
$$f(x)=2e^x-x^3-x^2$$
$$f'(x)=2e^x-3x^2-2x$$
$$f''(x)=2e^x-6x-2$$
$$f'''(x)=2e^x-6$$
$$f''''(x)=2e^x>0$$
Like the symbol cannot judge $f$ sign.
So how can we show this $$f(x)>0 \text... | Consider that for all x in $\mathbb R$, $e^{x}$ > 0, so $2e^{x}$ > 0 for all $\mathbb R$.So clearly for $x\leq 0$ in $\mathbb R$, the statement is true since $x^2 \geq 0$ and $x^3 \leq 0$. Since $|x^3| \geq x^2$ for x in $\mathbb R$, $x^3 + x^2 \leq 0$ for $x\leq 0$ in $\mathbb R$.
Now consider where $x>0$ in $\mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1927471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Can completing the square apply to more higher degree? I have some trouble about completing the square lessons.
As we know , minimum value of $\ \ 3x^{2} + 4x + 1 $ is $-\frac{1}{3}$ when $x=-\frac{2}{3}$ by completing the square.
Then , how about minimum value of $\ \ x^5 +4x^4 + 3x^3 +2x^2 + x + 1$ when $(-2<x<2... | Yes we have something same. But if we look to the topic like following.
$3x^2+4x+1$ has a complete square $3x^2+4x+\frac43$ corresponding to it in which their difference, $-\frac13$, is of degree maximum two degrees less than the original.
For any polynomial in one variable we have a unique multiple of a power of a $(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1927629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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In gambling, how much should I bet to win a specific amount of money? I have a seemingly simple problem but I can't work it out. Say I start with a bank of £20 - I would like to work out how much I should bet take my bank to £25 (assuming a win).
In this case we'll assume odds of 2.00. Since the amount I want to win is... | Your mistake is that $£5:2.00=£5$ profit, not $£2.50$ as you say, so your bank would be at $£25.00$, as you wanted.
If the odds are $d$ and you make a stake of $s$, then your profit $p$ is:
$$p=s(d-1)$$
So for a profit of $p$, you should stake;
$$s=\frac p{d-1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1927741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$1$ heap of sand $+\ 1$ heap of sand $= 1$ heap of sand? My uncle, who barely passed elementary school math (which leads me to believe he read this in some kind of joke magazine), once told me this when I was very young.
$$1 \text{ heap of sand } + 1\text{ heap of sand } = 1\text{ heap of sand}
.$$
It does sound like ... | One thing you must notice here is that:
$$1\, \text{heap of sand} + 1\, \text{heap of sand} = 1\, \text{heap
of sand} \not \large\Rightarrow 1+1=1$$
The reason is that $1$ heap is not well-defined or quantitative or mathematical enough for being used to perform mathematical operations like addition,subtraction and m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1927856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 0
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The series $\sum\limits_ {k=1}^{\infty} \frac{1}{(1+kx)^2}$ converges for $x>0$ $$ \sum_ {k=1}^{\infty} \dfrac{1}{(1+kx)^2}$$ $$x\in (0,\infty) $$
This series converges on given interval but how exactly can I show this is true?
| $$x>0\implies(1+kx)^2\ge k^2x^2\implies\frac1{(1+kx)^2}\le\frac1{k^2x^2}$$
and now use the comparison test.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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My solution of this integral does not match. Where am I doing wrong? I've been trying to calculate the following integral. But I always get the wrong result....
$$
S(a, b) = \int_0^{2\pi}\frac{d\theta}{\left(a + b\cos\theta\right)^2},
\quad\quad\quad\mbox{for}\quad a > b > 0
$$
Assume substitution: $z = e^{i\theta}$. ... | I noted that while computing the residues you used $2az + b z^2 + b =(z-z_+)(z-z_-)$ instead of $2az + b z^2 + b =b(z-z_+)(z-z_-)$.
Therefore I think that a factor $1/b^2$ is missing.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1928174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convergence/divergence of $\sum_{k=1}^\infty\frac{2\times 4\times 6\times\cdots\times(2k)}{1\times 3\times 5\times\cdots\times(2k-1)}$ A problem asks me to determine if the series
$$\sum_{k=1}^\infty \frac{2 \times 4 \times 6 \times \cdots \times (2k)}{1 \times 3 \times 5 \times \cdots \times (2k-1)}$$
converges or div... | $$\sum_{k=1}^\infty \frac{2 \times 4 \times 6 \times \cdots \times (2k)}{1 \times 3 \times 5 \times \cdots \times (2k-1)}=$$
$$\sum_{k=1}^\infty \frac{\prod_{j=1}^k(2j)}{\prod_{j=1}^k(2j-1)}=$$
$$\sum_{k=1}^\infty \prod_{j=1}^k\frac{(2j)}{(2j-1)}$$
The product $a_k = \prod_{j=1}^k\frac{(2j)}{(2j-1)}$ is
$$\left(1+\dfra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1928286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Homeomorphisms between circles and rectangles In our topology class we learn that in $\mathbb{R}^2,$ circles and rectangles are homeomorphic to each others.
I can understand the underline idea intuitively.
But can we find an explicit homeomorphic between them?
If so how?
Also our professor said that, "we can describe ... | consider $S^1$ parametrized by $(\cos(t),\sin(t))$. Extend this vector from the origin to its first intersection with the square. You can explicate the first quadrant and then just argue by symmetry. For $0 \leq t \leq \pi/4$, you're going to intersect with the $x=1$ line segment, and for $\pi/4 \leq t \leq \pi/2$, the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1928386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Characteristic equation of a recurrence relation? I am trying to find the general term of the following recurrence relation:
$$a_{n + 1} = \frac{1}{2}(a_{n} + \frac{1}{a_{n}})$$
where $a_1 = 3$. I'm failing to write the characteristic equation.
| Here is what I came upon:
If we have: $a_{n} = \frac{1}{2}(a_{n} + \frac{\lambda^2}{a_{n}})$, then it holds that: $\frac{a_{n} - \lambda}{a_{n} + \lambda} = (\frac{a_{1} - \lambda}{a_{1} + \lambda})^{2^{n-1}}$, solving for $a_{n}$ from the above example we have:
$$\frac{a_{n} - 2}{a_{n} + 2} = (\frac{3 - 2}{3 + 2})^{2^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1928503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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proving a map is a covariant functor between modules I'm trying to show that $(\cdot)\otimes_A N$ is a covariant functor from $MOD_A\to MOD_A$. Can somebody ensure I'm not oversimplifying the task (which I feel to be the case). It is clear that this functor does indeed take objects in $MOD_A$ to objects in $MOD_A$, and... | Yeah, that's it. In principle, of course, you need to check that your formula actually defines a homomorphism, although that's hardly a problem here. Another approach just uses the universal property and the Yoneda lemma: if $f:P\to Q$, then for any $R$ and bilinear map $a:Q\times N\to R$ we get a bilinear map $af:P\ti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1928587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Limit of ratio of two Gamma functions with negative integer arguments When using the hypergeometric representation for a Legendre polynomial, I encounter, for integer n and l, the following ratio: $$\frac{\Gamma(n-l)}{\Gamma(-l)}$$
Where $n \leq l$ (the quantity is definitely zero for $n > l$, as it should be in the de... | We have for $k\in\mathbf{N}$ and $x\rightarrow 0$
$$\Gamma(-k+x) \sim \frac{1}{k!x} + O(1)$$
and therefore
$$\lim_{x\rightarrow 0}\frac{\Gamma(n-l+x)}{\Gamma(-l+x)} = \frac{1}{(l-n)!}/\frac{1}{l!} = \frac{l!}{(l-n)!}$$
Of course this is only a very special limit. It is finite for $n=l$ but I do not know what you expec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1928697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving that the set $\{f \in \mathcal{F}(A,\mathbb{F}): f(a_0) = 0\}$ contains the zero vector. Suppose that $A$ is a nonempty set and $\mathbb{F}$ is a field. I would like to show that for any $a_0 \in A$, the set $\{f \in \mathcal{F}(A,\mathbb{F}): f(a_0) = 0\}$ is a subspace of $\mathcal{F}(A,\mathbb{F})$, where $\... | The set $\{f \in \mathcal{F}(A,\mathbb{F}): f(a_0) = 0\}$ is non-empty IFF set $A$ is not empty. In this case the function $f\equiv0$ is always exist and is contained in aforementioned subspace. Thus the subset is (mnemonically trivial) nonempty.
Don't worry :)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1928864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Question regarding a PDE I can see where to start on a $f(x,t)$ that solves a PDE $f_t+f*f_x+f_{xxx}=0$, then to show that it will also be true for $f(x-ct,t)+c$ for any number $c$
I can't see where to start on a problem. If a function $f(x,t)$ solves the PDE
$$f_t + f f_x + f_{xxx} = 0$$
I want to show that $f(x-ct,t... | I have started with putting the whole expression in like this
$\frac{\partial (f(x-ct,t)+c)}{\partial t}+f(x,t) \frac{\partial (f(x-ct,t)+c)}{\partial x}+\frac{\partial (f(x-ct,t)+c)^3}{\partial^3 x}$ and then get $(-c,0)+(f(x-ct)+c)+(1,0) +0$ but it doesn't seme right?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1928972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Prove $\dfrac{ax+b}{cx+d}=\dfrac{b}{d}$ if $ad=bc$. Well obviously if $\dfrac{ax+b}{cx+d}=\dfrac{b}{d}$ holds then $cbx+bd=bd+adx$ and it holds for any $x$ if $ad=bc$.
However, my question is to 'algebraically' or 'directly' calculate the $\dfrac{ax+b}{cx+d}$ if $ad=bc$. By multiplying both denominator any numerator by... | Multiplying
$$\frac{ax+b}{cx+d}$$
by $\frac{bd}{bd}\ (=1)$ gives
$$\frac{b(adx+bd)}{d(bcx+bd)}=\frac{b(bcx+bd)}{d(bcx+bd)}\tag1$$
since $ad=bc$.
Now $(1)$ equals $\frac bd$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1929090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Prove the inequality $\sqrt{a^2+b^2}+\sqrt{c^2+b^2}+\sqrt{a^2+c^2}<1+\frac{\sqrt2}{2}$ Let $a,b,c -$ triangle side and $a+b+c=1$. Prove the inequality
$$\sqrt{a^2+b^2}+\sqrt{c^2+b^2}+\sqrt{a^2+c^2}<1+\frac{\sqrt2}{2}$$
My work so far:
1) $a^2+b^2=c^2-2ab\cos \gamma \ge c^2-2ab$
2) $$\sqrt{a^2+b^2}+\sqrt{c^2+b^2}+\sqrt{... | Try to use Lagrange multipliers to solve this problem.
max $[\sqrt{a^2+b^2}+\sqrt{c^2+b^2}+\sqrt{a^2+c^2}+\lambda(a+b+c-1)]$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1929245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Polar to cartesian form of r=cos(2θ) This is possible for $r=\sin(2θ)$: Polar to cartesian form of $ r = \sin(2\theta)$
Surely there is some trig identity that may substitute for $cos(2θ)$ and allow for a similar coordinates transfer. What is the cartesian form of $\cos(2\theta)$?
I found something remotely similar:
$... | All polar to Cartesian / Cartesian to polar transformations derive from these simple rules
$r^2 = x^2 + y^2\\
x = r \cos \theta\\
y = r \sin \theta$
$r = cos 2\theta$ the 4 petaled rose. If it had an elegant form in Cartesian we would teach it. It will likely be a cubic or quartic equation.
$r = \cos^2 \theta - si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1929330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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"The following are equivalent" What does it mean for several statements to be equivalent? And why does it suffice to prove a "cyclic" chain
$$A_1\implies A_2\implies \cdots\implies A_n\implies A_1$$
in order to show that the conditions $A_1, \dots, A_n$ are equivalent?
| If you have shown $$A_1\Longrightarrow A_2 \Longrightarrow\ldots\Longrightarrow A_n $$
To show equivalence of all of these statements it suffices to finally show $A_n \Longrightarrow A_1$, because the former implications all hold and so you get equivalence between any two statements.
For instance, $A_2\iff A_5$ becaus... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1929453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 4,
"answer_id": 1
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limit of $a_n=\sqrt{n^2+2} - \sqrt{n^2+1}$ as $n$→∞ $a_n=\sqrt{n^2+2} - \sqrt{n^2+1}$ as $n$→∞
Both limits tend to infinity, but +∞ −(+∞) doesn't make sense. How would I get around to solving this?
| Multiply by the conjugate over the conjugate, then go from there. In your case, you should multiply your expression by $\frac{\sqrt{n^2+2}+\sqrt{n^2+1}}{\sqrt{n^2+2}+\sqrt{n^2+1}}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Let $f(a) = \frac{13+a}{3a+7}$ where $a$ is restricted to positive integers. What is the maximum value of $f(a)$? Let $f(a) = \frac{13+a}{3a+7}$ where $a$ is restricted to positive integers. What is the maximum value of $f(a)$?
I tried graphing but it didn't help. Could anyone answer? Thanks!
| $f'(a)=-\frac{32}{(3a+7)^2}<0$ so $f$ is strictly decreasing for $a>-\frac73$. So $f(1)>f(2)>f(3)>...$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1929660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Does $\sum_{n = 2}^{\infty} [\zeta(n) - 1]$ converge? Question in the title. Does $\sum_{n = 2}^{\infty} [\zeta(n) - 1]$ converge? If not, how about $\sum_{n = 1}^{\infty} [\zeta(2n) - 1]$?
| Figured out the solution:$\sum_{n=2}^{\infty} [\zeta(n) - 1] = \sum_{n=2}^{\infty}\sum_{k=2}^{\infty} n^{-k} = \sum_{n=2}^{\infty}\frac{1}{1-\frac{1}{n}} - 1 - \frac{1}{n} = \sum_{n=2}^{\infty}\frac{1}{n(n-1)}$, which converges by comparison with $\zeta(2)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1929731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
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How to compute that $\int_{-\infty}^{\infty}x\exp(-\vert x\vert) \sin(ax)\,dx$ $$
\int_{-\infty}^{\infty}x\exp(-\vert x\vert)
\sin(ax)\,dx\quad\mbox{where}\
a\ \mbox{is a positive constant.}
$$
My idea is to use integration by parts. But I have been not handle three terms..
Please, help me solve that.
| or note
$$
2\int_{0}^{\infty}x\exp(-x)
\sin(ax)\,dx
=
2\;\mathrm{Im}\int_{0}^{\infty}x\exp((-1+ia)x)\;dx
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1929818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $|f(x)|^p\ln|f(x)|$ is a bounded function (with some conditions) I wish to prove that $g(x,p)=|f(x)|^p\ln|f(x)|$ is a bounded function of $(x,p)$, where $0<|f(x)|\leq M$ for all $x$, and $p\in[p_1,p_2]$, where $0<p_1<p_2<\infty$.
$f$ is measurable but not necessary continuous.
My attempt:
Let $(x,p)$ be an ... | If $t<1$, then
$$|t^p \ln t| \le |t|^{p_1} |\ln t|.$$
Since $|t|^{p_1} |\ln t| \to 0$ as $t\to 0$, $|t|^{p_1} |\ln t|$ can be extended to a continuous function defined on $[0,1]$ and thus is bounded on $[0,1]$. Thus $t^p \ln t$ is also bounded on $[0,1] \times [p_1\times p_2]$. Together with case one you have that $t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1929970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the equation of two ...
Find the single equation of two straight lines that pass through the point $(2,3)$ and parallel to the line $x^2 - 6xy + 8y^2 = 0$.
My Attempt:
Let, $a_1x+b_1y=0$ and $a_2x+b_2y=0$ be the two lines represented by $x^2-6xy+8y^2=0$.
then,
$$(a_1x+b_1y)(a_2x+b_2y)=0$$
$$(a_1a_2)x^2+(a_1b_2+b... | The lines $$(x-2)^2 - 6(x-2)(y-3) + 8(y-3)^2 = 0$$ on transfer of origin to $(2,3)$, using the transformations $X = x-2, Y= y-2$ becomes $$X^2 - 6XY + 8Y^2 = 0$$ Hence the required equation is $$(x-2)^2 - 6(x-2)(y-3) + 8(y-3)^2 = 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1930128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Solve $\iint_D\sqrt{9-x^2-y^2}$ Where $D$ is the positive side of a circle of radius 3
Solve $\displaystyle\iint_D\sqrt{9-x^2-y^2}$ Where $D$ is the positive
side of a circle of radius 3 ($x^2+y^2=9,x\ge0,y\ge0$)
I tried to subsitute variables to $r$ & $\theta$:
$$x = r\cos\theta$$
$$y = r\sin\theta$$
$$E = \{0\le ... | It just is the half of the capacity of the ball: ($x^2+y^2+z^2=9)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1930244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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If $\ker f^k = \ker f^{k+1}$, then $\ker f^k = \ker f^{k+m}$ for all $m\in\mathbb N$ If $f: V\rightarrow V$ is a linear transformation, with $V$ being a finite-dimensional vector space over $\mathbb F$, such that $\ker f^k = \ker f^{k+1}$, then $\ker f^k = \ker f^{k+m}$ for all $m\in\mathbb N$.
We prove by induction o... | Here's the inductive step:
Suppose $\ker f^{k+n}=\ker f^k$ for some $n>0$.
$\ker f^k\subset \ker f^{k+n+1}$ is trivial. To prove the reverse inclusion, let $x\in\ker f^{k+n+1}$. So $f(x)\in\ker f^{k+n}$. By the inductive hypothesis, $f(x)\in\ker f^k$, i.e. $x\in\ker f^{k+1}=\ker f^k$ (initial step). The inductive step... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1930330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Equation with Vieta Jumping: $(x+y+z)^2=nxyz$.
Find all $n\in\mathbb{N}$ so that there exist $x,y,z\in \mathbb{N}$ that solve:
$$(x+y+z)^2=nxyz$$
I tried to attack it finding solutions, but all the solutions doesn't seem to have something in common. For example:
$$ (x,y,z,n)=(1,1,1,9)$$
$$ (x,y,z,n)=(1,2,3,6)$$
$$ ... | Besides Vieta jumping one can successfully use Pell's equation. For $n=1$ this is demonstrated here, giving infinite families of solutions. This may work for other $n$, too. The author, Titu Andreescu, has written several notes on such Diophantine equations. If you search his articles, in particular on quadratic Diopha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1930438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
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Continuity and limits at end point of interval I am bad a calculus and I have question about continuity. If I have a polynomial, then the function is continuous on $\mathbb{R}$ because $\lim_{x\to a} f(x) = f(a)$ for all $a\in \mathbb{R}$.
My question is if $f(x) = \sqrt{x}$ is continuous at $0$. My text book doesn't s... | We say a function $f$ is continuous at some interior point $a$ $\iff$
$$\forall \epsilon >0, \exists \ \delta>0, \ s.t. \ |x-a|<\delta\implies|f(x)-f(a)|<\epsilon$$
Or more simply notated as:
$$\lim_{x\to a^+}f(x)=\lim_{x\to a^-}f(x)=f(a)$$
Considering one sided limits, we say $\lim{_{x\to a^+}f(x)}$ exists and equals ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1930526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 3
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What is the amount of different four-digit numbers that can be created from the digits $0, 1, 2, 3, 4$ and $5$?
What is the amount of different four-digit numbers that can be created from the digits $0, 1, 2, 3, 4$ and $5$?
The solution is $431,$ but I have no idea how this solution was found. How can I solve this pr... | As stated in some of the comments, the problem is in the description of the question. We can distinguish the following cases:
*
*Using the digits to form a four-digit number. The first digit cannot equal $0,$ while all digits can be used for the second, third and fourth digit. As such, the number of possibilities eq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1930649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Show that a function $f:P(X)\to P(X)$ preserving the subset relation has a fixed point We have a map $f:P(X)\to P(X)$, where $P(X)$ means the part of $X$ and the function is monotone (by considering inclusion "$\subseteq$"). So $\forall \space A\subseteq B $ we have $f(A)\subseteq f(B)$.
Show that this map has a fixed... | HINT: Consider the set $\bigcup\{A\subseteq X:A\subseteq f(A)\}$. (Be sure to show that there is at least one $A\subseteq X$ such that $A\subseteq f(A)$.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1930743",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Can the elasticity of a concave function be strictly increasing If a real valued function $f$ defined on the positive orthant is strictly concave, can its elasticity, defined as:
$$E(f(x))=\frac{xf'(x)}{f(x)}$$
be strictly increasing in $x$?
| Let $f(x) = \sqrt{x+1}$. Then $f$ is strictly concave on $(0,\infty)$. Furthermore,
$$(Ef)(x) = \frac{xf'(x)}{f(x)} = \frac{x\frac{1}{2\sqrt{x+1}}}{\sqrt{x+1}} = \frac{x}{2(x+1)} = \frac{1}{2}\left(1-\frac{1}{x+1}\right)$$
is strictly increasing on $(0,\infty)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1930891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Give an example of two distinct sets $A$ and $B$ such that $A \times B = B \times A$ This is a question from my textbook. The book gives the answer $A = \varnothing$ and $B = \{1\}$.
The definition of $A \times B$ is $\{(a,b): a \in A \land b \in B\}$.
But if $A = \varnothing$, what could be in it?
Even if it's $(\varn... | You are on the right track as if one of the two sets is empty so is the cartesian product. Formally
$$
\emptyset\in \{A,B\}\Longrightarrow A\times B=\emptyset
$$
then you can choose $A=\emptyset,B=\{1\}$.
One can, moreover, show that all the solutions are of the form $(A=\emptyset,B \not=\emptyset)$ or $(A\not=\empty... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1930953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Closed-form condition for concyclicity $A_i(x_i, y_i)$, where $i=0,1,2,3$, are four points such that no three of them are collinear.
Is there a closed form on the condition that they are concyclic?
Answers with $x_0 = y_0 = x_1 (\text { or } y_1) = 0$ are also welcome.
| *
*Construct the circumcircle of $A_0$, $A_1$ and $A_2$. This circle will be unique and well-defined, as it has been stipulated that no three points are collinear. This circle can be found in closed form in a few ways – by the method I described here, for example. Let the centre of this circle be $(R_x,R_y)$ and its r... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Cross product (high school level) I know that the cross product has this property: $v \times u = u \times -v$
Yet I still am struggling getting the order right. I am also not sure if I fully understand the concept of normal vectors. For instance, this question in my maths book:
Line $l$ passes through point $A (-1, 1,... | For (b), the normal of a plane is perpendicular to all vectors that lie in that plane, by definition. Since $\Pi$ contains $l$, it contains $\mathbf d$; since it contains $A$ and $B$ it contains the vector $AB$. Then $AB$ and $\mathbf d$ are both perpendicular to $\mathbf n$.
Both $AB\times\mathbf d$ and $\mathbf d\tim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1931301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding the gradient of a given function Given the function $\phi=B_0/r=B_0/\vert \mathbf{x}\vert$ in a spherical axisymmetric geometry, where $B_0$ is a constant. Find $\nabla\phi$.
The given answer is
$$\phi=\frac{B_0}{r}=\frac{B_0}{\vert \mathbf{x}\vert}\implies \nabla\phi = -\frac{B_0\mathbf{x}}{\vert \mathbf{x}\... | 1º Note that r=$\sqrt{x^2+y^2+z^2}$, then ∇ϕ=($\frac{∂}{∂x}\frac{B_0}{\sqrt{x^2+y^2+z^2}}$,$\frac{∂}{∂x}\frac{B_0}{\sqrt{x^2+y^2+z^2}}$,$\frac{∂}{∂x}\frac{B_0}{\sqrt{x^2+y^2+z^2}}$).
2º$\frac{∂}{∂x}\frac{B_0}{\sqrt{x^2+y^2+z^2}}$=$-\frac{B_0}{2(\sqrt{x^2+y^2+z^2})^3} 2x$.
3º ∇ϕ=($-\frac{B_0}{\sqrt{x^2+y^2+z^2})^3} x,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1931395",
"timestamp": "2023-03-29T00:00:00",
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"answer_count": 3,
"answer_id": 2
} |
How to find area under sines without calculus? In the section establishing that integrals and derivatives are inverse to each other, James Stewart's Calculus textbook says (pp325--pp326, Sec.4.3, 8Ed):
When the French mathematician Gilles de Roberval first found the area under the sine and cosine curves in 1635, this ... | Let us show that
$$ I(a,b)=\int_{a}^{b}\cos(x)\,dx = \sin(b)-\sin(a) \tag{1}$$
through Riemann sums. We have to compute:
$$ \lim_{n\to +\infty}\frac{b-a}{n}\sum_{k=1}^{n}\cos\left(a+\frac{(b-a)k}{n}\right) \tag{2}$$
but the RHS of $(2)$ is a telescopic sum in disguise, hence $(1)$ boils down to proving
$$ \lim_{n\to +\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1931509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "48",
"answer_count": 4,
"answer_id": 2
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Is it possible to swap sums like that? Say that I have two sums like this :
$$\sum_{a=0}^n\sum_{b=0}^m f_{ab}$$
Would it be true to say that this expression can be considered as equal to :
$$\sum_{a=0}^m\sum_{b=0}^n f_{ab}$$
As long as the expression that comes after the sums is the same is both cases ? If it is true, ... | Not at all, look:
$$\sum_{a=1}^2\sum_{b=1}^3 \frac ab=\sum_{a=1}^2 \left(\frac a1+\frac a2+\frac a3\right)=\frac 11+\frac 12+\frac 13+\frac 21+\frac 22+\frac 23=\frac {11}2,$$
and
$$\sum_{a=1}^3\sum_{b=1}^2 \frac ab=\sum_{a=1}^3 \left(\frac a1+\frac a2\right)=\frac 11+\frac 12+\frac 21+\frac 22+\frac 31+\frac 32=9.$$
Y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1931602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 2
} |
Solve $\cot^2x=\csc x$ in degrees $$\cot^2x = \csc x$$
I have to solve for $x$ in degrees. Here's what I did:
$$\cot^2x=\csc^2x - 1$$
$$\csc^2x -\csc x - 1 = 0$$
If $y=\csc x$:
$$y^2-y-1=0$$
and now I cannot proceed to the given answers of 38.2° and 141.2°.
| since $$\frac{\cos(x)^2}{\sin(x)^2}=\frac{1}{\sin(x)}$$ you will get $$\cos(x)^2=\sin(x)$$ this means $$1-\sin(x)^2-\sin(x)=0$$
and so you will get a quadratic equation in $$\sin(x)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1931705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Find the volume of the solid obtained by rotating the region between the curves around the line $y = -1$ I'm stuck on this volume problem - everytime I calculate the volume I get zero. Here is the problem:
Find the volume of the solid obtained by rotating the region between the curves around the line $y = -1$.
$$
\be... | Your set up is not completely correct. You do have $\pi \int _{\pi / 4}^{5\pi/4} R^2-r^2 dx$ where $R=1+\sin (x)$ and $r=1+\cos (x)$ but then you claim that $R=\cos (x) $ and $r= \sin (x)$ and this is incorrect. To see why (I believe you confused a trigonometric identity because you were correct until that simplificati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1931787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Calculate $\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^T\frac{\zeta(\frac{3}{2}+it)}{\zeta(\frac{3}{2}-it)}dt$ as $\sum_{n=1}^\infty\frac{\mu(n)}{n^3}$ Using a well known theorem for Dirichlet series can be justified that (for $T>0$) $$\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^T\frac{\zeta(\frac{3}{2}+it)}{\zeta(\frac{3}{2}-i... | $$\frac{\zeta(3/2+it)}{\zeta(3/2-it)} = \sum_{n=1}^\infty n^{-3/2} \sum_{d | n} \mu(d) (d^2/n)^{it}$$ Note it converges absolutely when $t$ is real.
If $x \ne 1$ : $$\lim_{T \to \infty} \frac{1}{2T}\int_{-T}^T x^{it}dt = \lim_{T \to \infty} \frac{1}{2T} \frac{x^{iT}-x^{-iT}}{i\ln x} = 0$$
(with $x=1,x^{it} =1$ you hav... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1932081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Verification: Give a definition of a $G$-Set so that it may be viewed as an algebra A $G$-Set for those who may not be familiar with the terminology is as follows:
Let $G$ be a group and $S$ be a set. $G$ acts on $S$ define by the map $\star : G \times S \to S $ where $e \star s = s$ and $g\star (h \star s) = gh \star ... | Your definition is fine, although using $\star$ for what is (as you point out) a unary operation maybe isn't ideal.
For a group $G$ acting on a set $X$, define the $G$-set $\langle X, \overline{G}\rangle$ to be the algebra with universe $X$ and operations $\overline{G} = \{\bar{g} \mid g \in G\}$, one (unary) operation... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1932200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
5 odd-numbered taxis out of 9 to 3 airports
A fleet of 9 taxis must be dispatched to 3 airports: three to airport A, five to B and one to C. If the cabs are numbered 1 to 9, what is the probability that all odd-numbered cabs are sent to airport B?
What I have have come up with so far:
*
*Probability the cabs are o... | There is only $1$ way to send all $5$ odd numbered cabs to $B$,
against an unrestricted distribution of $\binom95 = 126$
Thus $Pr = \dfrac1{126}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1932330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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How to prove this relation between the laplacian of the logarithm and the dirac delta function? Why is this true in two dimensions?
$$\nabla^2\bigg(\ln(r)\bigg)=2\pi\delta^{(2)}(\mathbf{r}),$$
where $\delta^{(2)}$ denotes the two-dimensional $\delta$-function and $r=\sqrt{x^2+y^2}$ in Cartesian coordinates.
I understan... | Hint: Let $\phi \in C^\infty_0(\mathbb{R}^2)$. You need to show
\begin{align}
\int_{\mathbb{R}^2} \log|x| \Delta \varphi(x)\ dx = 2\pi \varphi(0).
\end{align}
This can be done by splitting the left-hand side into
\begin{align}
\int_{B(0, \epsilon)} \log|x| \Delta \varphi(x)\ dx + \int_{\mathbb{R}^2\backslash B(0, \ep... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1932451",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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How to add IEE754 half precision numbers I'm getting stuck with an exercise on adding two IEE754 half precision numbers, the numbers are:
$1110001000000001$
$0000001100001111$
I have tried to solve it using this procedure:
Half precision is:
$1$ sign bit, $5$ bits exponent and $10$ bits mantissa
So I rewrote the n... | The information you need is all in the Wikipedia pages on floating point arithmetic
and IEEE 754.
I agree with the comment suggesting that you may do better on a computer stackexchange site, but let me sketch a few hints about the mathematics that may help you understand the Wikipedia pages.
The basic mathematical ide... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1932521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If the radius of a sphere is increased by $10\%$, by what percentage is its volume increased?
Question: If the radius of a sphere is increased by $10\%$, by what percentage is its volume increased? Use Calculus.
Answer: It increases by approximately $33.1\%$.
I did the above question using calculus but my answer cam... | we have $$V_1=\frac{4}{3}\pi \cdot r^3$$ then we get $$V_2=\frac{4}{3}\pi\cdot r^3\left(\frac{11}{10}\right)^3$$ and we get $$\frac{V_2}{V_1}=\left(\frac{11}{10}\right)^3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1932597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Showing Euclidean metric and metric in $R^{2}$ produces same topology
Question:
Prove that the Euclidean metric and the metric $d_\infty \left ( x,y \right ):=\max\left \{ \left | x_{1}-y_{1} \right |,\left | x_{2}-y_{2} \right | \right \}$ defines the same topology in $\mathbb{R}^{2}$.
The Euclidean metric is defi... | To show that $d_1(x,y) = \sqrt{(x_1 - y_1)^2 + (x_2-y_2)^2}$ and the distance $d_2(x,y)=\max\{|x_1-$ $y_1|, |x_2-y_2|\}$ produce the same topology, it is enough to see the following inequality:
There exist constants $c_1$ and $c_2$ such that for all $x,y$, $c_1 d_2(x,y) \leq d_1(x,y) \leq c_2 d_2(x,y)$.
Proof: On one... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1932683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Series Expansion for $\ln(x)$ I have a mathematics assignment, which requires me to proof that $$\ln\frac{2}{3} = \sum_{n=1}^{\infty}\frac{(-1)^{n}}{2^{n}n}$$.
I know, I can solve this by proving $\ln x$ = $\sum_{n=1}^{\infty }\frac{1}{n}\left ( \frac{x-1}{x} \right )^{n}$, but I don't know how to prove this, so can an... | Using the definition of geometric series
$$\frac{1}{1-x}=\sum_{k=0}^{\infty} x^{k} $$
Integrating both sides
$$\int\frac{1}{1-x}=\sum_{k=0}^{\infty}\frac{(-1)^k x^{k+1}}{k+1} $$
So $$\ln (1-x)=-\sum_{k=1}^{\infty} \frac { x^k}{k} $$
$$\ln (y)= \sum_{k=1}^{\infty} \frac{1}{k} \left( \frac{y-1}{y} \right) ^{k} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1932795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Prove $\lim_{(x,y)\to(1,1)} x^2+xy+y=3$
Prove that $$\lim_{(x,y)\to(1,1)} x^2 + xy + y = 3$$ using the epsilon-delta definition.
What I have tried:
Let $\epsilon > 0$ be arbitrary. We must show that for every $\epsilon$ we can find $\delta>0$ such that
$$0 < \|(x,y) - (1,1)\| < \delta \implies \|f(x,y) - 3\| < \epsi... | \begin{align}
x^2+xy+y-3 &= (x-1)^2+2x-1+(x-1)(y-1)+x+y-1+(y-1)+1-3 \\
&=(x-1)^2+(x-1)(y-1)+(y-1)+3x+y-4 \\
&=(x-1)^2+(x-1)(y-1)+(y-1)+3(x-1)+(y-1)\\
&=(x-1)^2+(x-1)(y-1)+2(y-1)+3(x-1)\\
\end{align}
Let $\delta= \min(1, \frac{\epsilon}7),$
Then
\begin{align}
|x^2+xy+y-3| &\leq |x-1|^2+|x-1||y-1|+2|y-1|+3|x-1| \\
&\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1932891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Multiplication of series Suppose $a_n\geq0$ and $\sum a_n$ is convergent. Show that $ \sum 1/(n^2\cdot a_n)$ is divergent.
I haven't been able to get any result from any of my approaches (which include the general tests for positive series). However if I were to create $\sum b_n$ such that the terms are in the same or... | Assuming $a_n\ne 0$:
If $\sum\limits_{n=1}^{\infty}{\frac{1}{n^2a_n}}<\infty$, then by Cauchy-Schwarz we would have
$$\left(\sum\limits_{n=1}^{\infty}{\frac{1}{n}}\right)^2 = \left(\sum\limits_{n=1}^{\infty}{\sqrt{a_n}\frac{1}{n\sqrt{a_n}}}\right)^2 \le \sum\limits_{n=1}^{\infty}{a_n}\sum\limits_{n=1}^{\infty}{\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1933001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Integral representation of $\sum_{k=0}^{n} \frac{x^k}{(k!)^2}$? I would like to know if there is any integral representation of the following sum : $\displaystyle{S_n(x) := \sum_{k=0}^{n} \frac{x^k}{(k!)^2}}$.
I'm willing to have an idea of how fast this sums goes to infinity when $x$ is a sequence $(x_n )$ such that $... | Since by De Moivre's formula
$$\binom{2k}{k}= \frac{4^k}{\pi}\int_{-\pi/2}^{\pi/2}\cos(\theta)^{2k}\,d\theta \tag{1}$$
we have
$$f(x)=\sum_{k\geq 0}\frac{x^k}{k!^2}=\int_{-\pi/2}^{\pi/2}\sum_{k\geq 0}\frac{(4x)^k\cos(\theta)^{2k}}{\pi(2k)!}\,d\theta \tag{2}$$
hence
$$ \boxed{\,f(x)=\color{red}{\frac{1}{\pi}\int_{-\pi/2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1933129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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show $h : \underline{A} \to \underline{B}$ is a homomorphism iff $h$ is a subuniverse of $A \times B$ show $h : \underline{A} \to \underline{B}$ is a homomorphism iff $h$ is a subuniverse of $\underline{A} \times \underline{B}$ where $\underline{A}$ and $\underline{B}$ are similar algebras
$\Rightarrow$
Assume $h$ is a... | If $h : \mathbf{A} \to \mathbf{B}$ is a homomorphism, and $f$ is a $n$-ary operation of the type of the algebras, then take $n$ elements of $h$ as a subset of $A \times B$:
$$(a_1,b_1), \ldots, (a_n,b_n) \in h,$$
which means they have the form (as you say)
$$(a_1,h(a_1)), \ldots, (a_n,h(a_n)).$$
Now we want to prove th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Create a logical expression that has this truth table The following table has 3 input value.
I need to make a logical expression that has this truth table.
X Y Z OUTPUT
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 1
I att... | Consider rows, where output equals to $1$, all we need to do is to write combinations which produce each of these rows and then make union of them
$$
Output = (\neg X\land \neg Y \land Z) \lor (X \land \neg Y \land \neg Z) \lor (X \land \neg Y \land Z) \lor (X\land Y \land Z)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1933360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Easier way to calculate the derivative of $\ln(\frac{x}{\sqrt{x^2+1}})$? For the function $f$ given by
$$
\large \mathbb{R^+} \to \mathbb{R} \quad x \mapsto \ln \left (\frac{x}{\sqrt{x^2+1}} \right)
$$
I had to find $f'$ and $f''$.
Below, I have calculated them.
But, isn't there a better and more convenient way to do t... | By implicit differentiation:
Let
$$
y(x) = \log\left[\frac{x}{\sqrt{x^2 + 1}}\right].
$$
Then
$$
(x^2 + 1)e^{2 y(x)} = x^2.
$$
Differentiating both sides,
$$
(x^2 + 1)e^{2y(x)}y'(x) + x e^{2y(x)} = x.
$$
Solving for $y'(x)$,
$$
y'(x) = \frac{x(e^{-2y(x)}-1)}{(x^2 + 1)} = \frac{1}{x(x^2+1)}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1933460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 5
} |
Sequence that is neither increasing, nor decreasing, yet converges to 1 Give an example of a sequence which is neither increasing after a while, nor
decreasing after a while, yet which converges to 1.
My solution: $1.01,\ .99,\ 1.001,\ .999,\ 1.0001,\ .9999,\ \text{etc}\dots$
Does that satisfy all the conditions? Also... | Yes, your sequence satisfies all required conditions.
Another example sequence:
$$a_n = \begin{cases}
1+\frac 1n & \text{if } \log_{10}n \in \mathbb N \cup \{0\} \\
1 & \text{otherwise}
\end{cases}$$
has terms:
$$\begin{align}
a_1 & = 2 \\
a_{10} & = 1.1 \\
a_{100} & = 1.01 \\
a_{1000} & = 1.001 \\
a_{10000} & = 1.0001... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1933596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
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"answer_id": 0
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How to find Laurent series for this real valued function $1/(1-x)$? http://www.wolframalpha.com/input/?i=1%2F(1-x)+taylor
This page says that
$$\dfrac{1}{1-x}=\sum\limits_{n=0}^\infty x^n = 1+x+x^2+x^3 +...
$$
when $|x|<1$,
and
$$\dfrac{1}{1-x}=\sum\limits_{n=0}^\infty -x^{-(n+1)} = -\dfrac{1}{x}-\dfrac{1}{x^2}-\dfr... | If $\left|x\right| > 1$, then $1/\left|x\right| < 1$. Hence,
$$
\frac{1}{1 - x} = \frac{1/x}{1/x - 1} = \frac{-1}{x}\cdot\left(\frac{1}{1 - 1/x}\right).
$$
Now use the series for $\frac{1}{1 - u}$ for $\left|u\right| < 1$ with $u$ replaced by $1/x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1933729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Probability of never getting k tails in infinite flips I am currently trying to solve a problem but am at a standstill. The question is:
If a coin is flipped infinitely many times, what is the probability that there will never be j successive tails?
I have a recursive sequence of a_n, where a_n is the number of ways of... | Suppose we flip a coin $j$ times, and the chance that any flip lands tails is $p \in (0,1)$. Then, assuming that individual flips are independent, the probability that all flips land tails is simply $p^j$. Let's call such a group of flips a single trial. Consequently, if we repeat this experiment $n$ times--i.e., we... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Example of space $T_3$ which is not regular Does there exist example of a topological space in which any closed set and a point can be seperated by open sets i.e. space is $T_3$.
But there exist a pair of points which can't be seperated by points(i.e. points not closed) i.e. space not $T_1$. Hence space not regular bec... | Take the indiscrete topology on any set with more than one point. Then it's not $T_1$, but it's $T_3$ because any closed set which does not contain a point is empty.
(In fact, any example is basically the same: a $T_3$ space is regular iff it is $T_0$, and a space is $T_3$ iff its $T_0$ quotient is regular. So the on... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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QR algorithm for finding eigenvalues and eigenvectors of a matrix Let $A$ be symmetric and diagonalizable, and let $\{\lambda_1, \cdots, \lambda_n\} $ be its spectrum. A consequence of the Spectral Theorem assures that $\exists Q$ orthogonal s.t.
$\begin{pmatrix}
\lambda_1 & 0 &\cdots & 0\\
0 & \lambda_2 & \cdots &... | *
*eigenvalues are distinct: the more relevant property would be that all eigenvalues are simple. This is guaranteed for symmetric or more generally normal matrices. This only has to do with convergence results, and has no influence in the considered case of symmetric matrices.
*$A^{(k)}$ converges to a triangular ma... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Dirichlet hyperbola methods : estimate functions # of ordered pairs Let $f$ be an arithmetic function defined by $$f(n) = |A_n|$$ where $A_n = \{(a, b) : n = ab^2\}$.Estimate $$\sum_{n \leq x} f(n)$$ where $x \in \mathbb{R}^+$, using Dirichlet hyperbola method. The error term should be $O(x^{1/3})$.
Dirichlet hyperbola... | For each $n$ we are computing $f(n) = \displaystyle\sum_{ab^2=n} 1$. Now the indicator function of the square is just $\sum_{d|n}\lambda(d)$ where here $\lambda(d)$ is the Liouville function, $\lambda(d) = (-1)^{\Omega(d)}$. Writing this out we have
$$f(n) = \sum_{d|n}\sum_{e|d}(-1)^{\Omega(e)}$$
Because this will have... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1934192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that $(X\times Y)\setminus (A\times B)$ is connected
Problem.
Let $\emptyset \subset A\subset X$ and $\emptyset \subset B\subset Y$. If $X$ and $Y$ are connected, show that $(X\times Y)\setminus (A\times B)$ is also connected by using the criteria of connectedness that if for any continuous function $f$ such that... | The proof is essentially the same as the one linked to. Suppose we have a function $f:(X\times Y)\setminus(A\times B)\to\{\pm1\}$. Begin by choosing $a\in X\setminus A$ and $b\in Y\setminus B$ (which is possible because both are proper subsets). Now, let $(x,y)\in(X\times Y)\setminus(A\times B)$ be arbitrary. We will s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1934259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Show that there exists $c\in[a,b]$ such that $f(c)=\frac{1}{n} ( f(x_1)+f(x_2)+...f(x_n))$
Given $f$ a continuous function on $(a,b)$ such that $x_1,x_2,...,x_n$ $n$ elements of $(a,b)$, show that it exists $c\in[a,b]$ such that $f(c)=\frac{1}{n} ( f(x_1)+f(x_2)+...f(x_n))$
That equals to show that : $f(c)+f(c)+...+f... | Find $l,h \in \{1,2,\ldots ,n \}$ such that $f(x_l)\leq f(x_i)$ for all $i \in \{1,2,\ldots ,n \}$ and $f(x_h)\geq f(x_i)$ for all $i \in \{1,2,\ldots ,n \}$. Clearly such $l$ and $h$ exist. Now if $f(x_l)=f(x_h)$, then all $f(x_i)$ are equal so there is nothing to prove. So $f(x_l)<f(x_h)$ and so now assume WLOG $x_l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1934345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
A geometric locus in a equilateral triangle Find the locus of the points $P$ in the plane of an equilateral triangle $ABC$ that satisfy :
$$\max\{PA,PB,PC\} = \frac{PA+PB+PC}{2}.$$
I have never dealt with locus problems like these. So any help would be appreciated. (And please mention the intuition behind the answer to... |
Tricky question. I will give you just a substantial hint. Let we consider a point $P$ on the minor $BC$-arc of the circumcircle of $ABC$. By applying Ptolemy's theorem to the cyclic quadrilateral $PBAC$ we get that $PA=PB+PC$, from which
$$ PA = \frac{PA+PB+PC}{2}.$$
Can you guess now what the wanted locus is? Conside... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1934450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Mathematical notation of set with $n+3$ members One of the math problems I have describes a set of numbers this way:
let there be a set A such that $A=\{{1,2,3,...,n+3}\}$.
I don't understand what the $n+3$ means and how the set actually looks like.
| The $n$ is arbitrary and finite. Basically, choose an $n$ so
$$ A_n = \{ 1, 2, \ldots, n, n+1, n+2, n+3 \}.$$
For example, $A_4 = \{1, 2, \ldots, 5, 6, 7 \}.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1934524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Prove that $a/(p-a) + b/(p-b) + c/(p-c) \ge 6$ Prove that $a/(p-a) + b/(p-b) + c/(p-c) \ge 6$ , where $a,b,c$ are the sides of a triangle and $p$ is the semi-perimeter .
| By C-S $\sum\limits_{cyc}\frac{a}{p-a}\geq\frac{(a+b+c)^2}{\sum\limits_{cyc}(pa-a^2)}=\frac{2(a+b+c)^2}{(a+b+c)^2-2(a^2+b^2+c^2)}\geq\frac{2(a+b+c)^2}{(a+b+c)^2-\frac{2}{3}(a+b+c)^2}=6$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1934649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How to find $f(x)$ in order to find $f^{(10)}(3)$?
(a) Find the radius of convergence of $\sum_{n=1}^\infty (-1)^n \frac{(x-3)^n}{(2n+1)}$ and its derivative.
(b) Denote by $f(x)$ the function represented by the above power series within its region of convergence. Find $f^{(10)}(3)$, i.e., its 10th derivative at $x = ... | You don't need to find $f(x)$. To do part (b), you only need to think about the coefficients of the Taylor series of $f(x)$ at $x=3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1934770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Solving system of $9$ linear equations in $9$ variables I have a system of $9$ linear equations in $9$ variables:
\begin{array}{rl}
-c_{1}x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + x_{9} &= 0 \\
x_{1} - c_{2}x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + x_{9} &= 0 \\
x_{1} + x_{2} - c_{3}... | Let
$$\mathrm A := 1_n 1_n^T - \mbox{diag} (1 + c_1, \dots, 1 + c_n)$$
where $c_i \neq -1$ for all $i \in \{1,2,\dots,n\}$. Using the matrix determinant lemma,
$$\det (\mathrm A) = \left( 1 - \sum_{i=1}^n \frac{1}{1 + c_i} \right) (-1)^n \left( \prod_{i=1}^n (1+c_i)\right)$$
We want the homogeneous linear system $\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1934857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Convergence of Complex Sequence Let $z_n = \left(\frac{1+i}{3}\right)^n$ be a complex sequence. Show that $(z_n)$ converges.
I'm unsure how to do this because I've only just started learning about complex sequences. If this were real and, say we replaced $i$ with just $1$, I would note that $2^n < 3^n$ for all $n \geq ... | Yes, the key here is that
$$
\left|
{1 + i \over 3}
\right|
\leq {2 \over 3} < 1.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1935007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Determinant involving function of $x$ If $f(x)$ is a polynomial satisfying $$f(x)=\frac{1}{2}
\begin{vmatrix} f(x) & f(\frac{1}{x})-f(x) \\ 1 & f(\frac{1}{x}) \end{vmatrix} $$ and $f(3)=244$ then $f(2)$ is what?
My attempt—
Replacing $x$ by $\frac{1}{x}$ we get $$f\left(\frac{1}{x}\right)=\frac12\begin{vmatrix} f\left(... | Your starting idea is indeed great. Using $f(x)$ and $f(1/x)$ both, and adding them up, we have
$$
f(x)+f(1/x) = f(x)f(1/x).
$$
This gives
$$
f(1/x)-1=\frac{1}{f(x)-1}. \ \ \ \ (*)
$$
As $x\rightarrow\infty$, we have $f(0)=1$.
By assuming that $n=\textrm{deg}(f)$, we have by multiplying $x^n$ both sides,
$$
\frac{x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1935122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$L^p_0(\Omega)\cap L^p(\Omega)$ dense in $L^p(\Omega)$ when $m(\Omega)=\infty$? Let $1\leq p < \infty$ and $\Omega\subset\mathbb{R}^n$ be a measurable (Lebesgue) set. I know that $L_0^p(\Omega)\cap L^p(\Omega)$ is dense in $L^p(\Omega)$ when $m(\Omega)$ is finite. For the proof I used the absolute continuity of the int... | Sure it is. Since $\mathbb{R}^n$ is $\sigma$-finite. Intersect your $\Omega$ with sets of the form $\{n<|x|<n+1\}$ and approximate on each annulus with error at most $\epsilon/2^n$, then you will see the total error will be less than $\epsilon$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1935278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to solve $x^3 = x^{x^2-2x}$? Well i guess this is somehow pretty easy but there is something i don't understand.
I know that if $\;x>0\;$ then I can compare the exponents: $\; 3=x^2-2x$ , and from here I get that $\;x=3\; or \;x=-1\;$, but because $\;x>0\;$ that leaves me only with $\;x=3$ .
Second thing is that if... | $x^a=x^b\implies a=b$ for all $x$,whether $x>0$ or $x<0$. You got $3=x^2-2x$, and solved this quadratic equation to get $x\in\{3,-1\}$. So $-1$ is indeed a solution. It depends on the equation as to which methods will you follow, there is no general algorithm. But yes, keep using exponent rules, and you will eventually... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1935367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Elementary number theory and quadratic Find the value(s) of $a$ for which the equation
$ax^2-4x+9=0$
has integral roots(i.e. $x$ is an integer).
My attempt:
By hit and trial I am getting answer as $a=\frac{1}{3}$. No information about nature of $a$ is given.
| I propose a graphical understanding of this question:
Consider it as looking for the intersection points of these two curves
$$\cases{y=4x-9\\y=ax^2}$$
*
*The first one is a fixed straight line on which we define points with integer abscissas $P_k(k,4k-9) \ \ (k \neq 0).$
*the second one is a variable parabola wit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1935515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Proof that $ \|x+y\|^2 - \|x\|^2 \geq b(1 - 2^{-n})\|y\|^2 + 2^n( \|x+2^{-n}y\|^2 - \|x\|^2), \quad \forall x,y \in E, \forall n \in \mathbb N $ Suppose that $E$ is a Banach space over $\mathbb R$ satisfying the following inequality, for some $b > 0$
$$ \|x+y\|^2 + b\|x-y\|^2 \leq 2\|x\|^2 + 2 \|y\|^2, \quad \forall x,... | I think I might have solved the problem. Let's re-write (I) as
$$ \|x+y\|^2 - \|x\|^2 \geq b(\frac{2^n - 1}{2^n})\|y\|^2 + 2^n\|x+\frac{1}{2^n}y\|^2 - 2^n\|x\|^2 $$
$$ 2^n\|x+y\|^2 - 2^n\|x\|^2 \geq b(2^n - 1)\|y\|^2 + (2^n)^2\|x+\frac{1}{2^n}y\|^2 - (2^n)^2\|x\|^2 $$
$$ 2^n\|x+y\|^2 - 2^n\|x\|^2 \geq b(2^n - 1)\|y\|^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1935587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Example of a left compatible relation on a semigroup that is not right compatible. Definition : Let $S$ be a semigroup. let $R$ be a relation on $S$.
*
*Left compatibility: $R$ is left compatible if
$$ (\forall a , s ,t \in S) \ \ (s,t) \in R \ \ \Rightarrow (as , at) \in R $$
*Right compatibility: $R$ is is ri... | The Green's relation $\mathcal{R}$ that you mentioned in this question is left compatible (but in general not right compatible). Dually the Green's relation $\mathcal{L}$ is right compatible (but in general not left compatible)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1935681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
limit of a function of three variables I would like to ask you how to solve the limit at the origin of the following function:
$$f(x,y,z)=\frac{x^3y^3z^2}{x^6+y^8+z^{10}}$$
I am quite sure that it is $0$, but I cannot find a function majorizing $f$ and going to $0$ at the origin (in order to use the sandwich thm).
Tha... | Let $X=x^3$, $Y=y^4$, $Z=z^{5}$. Then $r^2=X^2+Y^2+Z^2=x^6+y^8+z^{10}$ and
$$|x^3y^3z^2|=|X||Y|^{3/4}|Z|^{2/5}\leq r^{1+3/4+2/5}=r^{43/20}$$
where
$$X=r\cos\theta\cos\phi,\quad Y=r\cos\theta\sin\phi,\quad Z=r\sin\theta\ .$$
Therefore, as $(x,y,z)\to(0,0,0)$, it follows that $r\to 0$ and
$$\left|{x^3y^3z^2\over x^6+y^8+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1935761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
The smallest topology in $\mathbb C$ in which every singleton set is closed
Let $\tau$ be the smallest topology on $\mathbb C$ such that every singleton set under $\tau$ is closed. Then which of the following is true ?
$1.(\mathbb C,\tau) \text{ is not Hausdorff}.$
$2.(\mathbb C,\tau) \text{ compact }$
$3.(\mathbb ... | First of all, notice that nothing about this problem is particular to $\Bbb C$; you can replace it with any infinite set whatsoever (and then $\Bbb Z$ is replaced by your favorite infinite subset).
It does turn out to be true that the smallest suitable topology is the co-finite topology (so awesome that you know the an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1935857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Codes and Codewords I'm sorry for the very dumb question, but I just can't grasp the concept of codes.
If C is an [n, k] linear code, what are n and k exactly? I know k is the dimension of C and that n is the number of tuples of the field where C is a subset of. Is n the dimension of the field? If so, why should it be ... | An $[n,k]$ linear code over some field $F$ is a $k$-dimensional subspace of the vectorspace $F^n$. The codewords are elements of $F^n$, that is they are $n$-tuples of elements of $F$. Thus yes the $a_i$ are from the field.
The $n$ is the length of the code, i.e., the codeword. The $k$ is the dimension of the code, whi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1935979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Double Integral $\iint\limits_D\frac{dx\,dy}{(x^2+y^2)^2}$ where $D=\{(x,y): x^2+y^2\le1,\space x+y\ge1\}$ Let $D=\{(x,y)\in \Bbb R^2 : x^2+y^2\le1,\space x+y\ge1\}$. The integral to be calculated over $D$ is the following:
\begin{equation}
\iint_D \frac{dx\,dy}{(x^2+y^2)^2}
\end{equation}
I do not know how to approac... | Transforming to polar coordinates $(\rho,\phi)$, we find that on $x+y=1$ we have $\rho =1/(\cos(\phi)+\sin(\phi))$. Therefore, we can write
$$\begin{align}
\int_D \frac{1}{(x^2+y^2)^2}\,dx\,dy&=\int_0^{\pi/2}\int_{1/(\sin(\phi)+\cos(\phi))}^1 \frac{1}{\rho^3}\,d\rho\,d\phi\\\\
&=\frac12\int_0^{\pi/2}\sin(2\phi)\,d\phi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1936076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Help with Epsilon-Delta proof for limits I tried to teach myself these types of proofs. I understand the reasoning behind it very well, but I have trouble understanding specific parts when simplifying inequalities. Let me give an example:
Say I wanted to prove the following:
$$\lim_{x\to1}(x^2+3)=4$$
I start by supp... | We get to choose our $\delta$, hence to get the conclusion that
$$3|x-1|<\epsilon$$
I can choose my $\delta$ to be $$\delta=\min(1, \epsilon/3)$$
Hence, For $|x-1| <\delta$,
$$|(x+1)(x-1)|<3|x-1|<3\delta\leq 3(\epsilon/3)=\epsilon$$
$|x+1|<3$ is due to $\delta \leq 1$, $3\delta \leq \epsilon$ is due to we choose $\del... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1936157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Coordinate Systems I have two axes: X,Y,Z and x',y',z'. I am given 4 relations: X is 60 degrees from x', Y is 90 degrees from x', Y is 120 degrees from z', Y is 30 degrees from y'. Knowing all of this, how do I find the rotation matrix relating the XYZ axis to the x'y'z' axis?
| If you are asking for a rotation matrix, we are speaking of two sets of orthogonal systems.
First step, determine $Y$. Its position is clear, it is on the $y,z$ plane, and with the given angles it cannot be but$(0,\sqrt {3} /2, -1/2)$.
Second step, determine $X$, by imposing that its dot product with $x=(1,0,0)$ be $1/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1936240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Linear combination of columns In the following question I am trying to determine if vector $b$ is a linear combination of the columns of $A$. If this is false I need to explain why, but if it is true I need to write down the linear combination.
Matrix A:
$$ \begin{bmatrix}
1& -1& 2& 1\\
2& -3& 2& 0\\
-1& 1& 2& 3 \... | If you perform row reduction on the augmented matrix
$$
\begin{bmatrix}
1& -1& 2& 1&|&2\\
2& -3& 2& 0&|&3\\
-1& 1& 2& 3 &|&6\\
-3& 2& 0& 3 &|&9\end{bmatrix},
$$
you'll obtain
$$
\begin{bmatrix}
1&0&0&-1&|&-5\\
0&1&0&0&|&-3\\
0&0&1&1&|&2\\
0&0&0&0&|&0
\end{bmatrix},
$$
which shows that the system $Ax=b$ is consi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1936359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Given u and v are harmonic in some region R prove the following If u and v are harmonic in some region R prove the following
$ ( \frac {\partial u}{ \partial y} - \frac {\partial v}{ \partial x}) + i(\frac {\partial u}{ \partial x} + \frac {\partial v}{ \partial y})$
is analytic in R. Does this mean if u and v are har... | Define the function $\hat{u}(x,y) := \frac{\partial u}{\partial y}(x,y) - \frac{\partial v}{\partial x}(x,y)$ and $\hat{v}(x,y) := \frac{\partial u}{\partial x}(x,y) + \frac{\partial v}{\partial y}(x,y)$. Then consider the function $f(x,y) := \hat{u}(x,y) + i \cdot \hat{v}(x,y)$, this is your function in question. Now ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1936495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Conditional probability that the first toss resulted in heads
A fair coin is tossed until two heads have appeared.
*
*Given that exactly $k$ tosses were required, what is the conditional probability that the first toss resulted in heads?
*If $p_k$ is the probability that at least $k$ tosses are required, f... | A = First toss Was a head
B = k tosses required
$P(A|B) = P(A \cap B)/P(B)$
for $P(A \cap B)$ (probability that first toss was a head then k tosses were reuired to get a second head) we need to consider that in this case, toss 1 was a head, and toss k was a head, all tosses in between were tails - therefore it is a a s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1936726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Numbers $1,2,3, ...., 2016$ arranged in a circle A student wrote numbers $1,2,3, ...., n$ arranged in a circle, then began with erasing number $1$ then he Leaves $2$ then he erased $3$ and leaves $4$ ......
exemple: if $n=10$ :
in the first round he erased $1,3,5,7,9$ , and leaves $2,4,6,8,10$
in the second round he er... | Let it be that $a_{n}$ is the remaining number if we are dealing
with the numbers $1,2,3,4,5\dots,n$.
If $n$ is sufficiently large then after erasing $1$ and leaving
$2$ we have the numbers $3,4\dots,n,2$ ahead of us.
Comparing this with the situation in which we have the numbers $1,2,3,\dots,n-2,n-1$
ahead of us we co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1936853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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