Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How do I compute the following complex number? This was the problem I was given:
Compute the complex number for $\frac{(18-7i)}{(12-5i)}$.
I was told to write this in the form of $a+bi$.
So please give me a hint of how to do this. :)
| You can "rationalize" or more accurately "real-ize" the denominator by multiplying the numerator and denominator by denominator's conjugate. It is just like with radicals. You will get:
$$\frac{18-7i}{12-5i}=\frac{(18-7i)(12+5i)}{(12-5i)(12+5i)}=\frac{216-84i+90i-35i^2}{144-60i+60i-25i^2}=\frac{216+6i+35}{144+25}=\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1757021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Conditional probability involving coin flips A coin has an unknown head probability $p$. Flip $n$ times, and observe $X=k$ heads. Assuming an uniform prior for $p$, then the posterior distribution of $p$ is $B(\alpha = k + 1, \beta = n - k + 1)$. Consider $Y$ = number of additional flips required until the first head a... | $\mathsf P(Y=j\mid p=\theta)$ is the conditional probability of $j$ aditional flips until another head shows for a given bias for the coin (after $X=k$ in $n$ flips). What is the (conditional) distribution used for this? (Hint: you do not need to resort to Bayes' for this; just identify the model.)
$\mathsf P(p=\th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1757142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that there is a step function $g$ over $[a,b]$
Assume $f$ is integrable over $[a,b]$ and $\epsilon > 0$. Show that there is a step function $g$ over $[a,b]$ for which $g(x) \leq f(x)$ for all $x \in [a,b]$ and $\displaystyle \int_{a}^b (f(x)-g(x))dx < \epsilon$.
I am having trouble coming up with a step functio... | This follows from the definition of Riemann integral:
For given $\epsilon>0$ there exists $\delta>0$ such that for every partition of $[a,b]$ that is finer than $\delta$, the lower and upper Riemann sum for that partition differ by less than $\epsilon$ from $\int_a^bf(x)\,\mathrm dx$, which is between them. Let $g$ be ... | {
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"url": "https://math.stackexchange.com/questions/1757238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Understanding the proof of Möbius inversion formula I am trying to understand one step in the proof of the Möbius inversion formula.
The theorem is
Let $f(n)$ and $g(n)$ be functions defined for every positive integer $n$ satisfying $$f(n) = \sum_{d|n}g(d)$$
Then, g satisfies $$g(n)=\sum_{d|n}\mu(d) f(\frac{n}{d})$$... | First, considering the sum
\begin{align*}
\sum_{d|n}\mu(\frac{n}{d})\sum_{m|d}g(m),
\end{align*}
let's take a look into the indices
$$
n=\frac{n}{d}d=\frac{n}{d}\frac{d}{m}m=khm,
$$
with
$$
\frac{n}{d}=k, \frac{d}{m}=h.
$$
Thus, we have
\begin{align*}
\sum_{d|n}\mu(\frac{n}{d})\sum_{m|d}g(m)
&=\sum_{dk=n}\mu(k)\sum_{hm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1757352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 1
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number of strings of $5$ lower case letters $a\cdots z$ that do not contain any letter twice or more What are the number of strings of $5$ lower case letters $a\cdots z$ that do not contain any letter twice or more?
I think it would be $26*25*24*23*22$ because the first position can be filled in $26$ ways because the... | community wiki answer so the question can be closed
Your answer is correct.
| {
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"url": "https://math.stackexchange.com/questions/1757477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
} |
The $\cos(\alpha-\beta)$ formula always need $\alpha > \beta$ or not? I'm a beginner student study the proof of sum and difference trigonometry formula.
There is a formula that:
$$\cos(\alpha-\beta) = \cos(\alpha)\cos(\beta)+\sin(\alpha)\sin(\beta)$$
In the tutorial it shows only the case when $\alpha > \beta$
Can this... | The formula holds for all $\alpha,\beta$. If you only know it for the case that $\alpha-\beta>0$ note that for $\alpha-\beta<0$,
$$ \cos(\alpha-\beta)=\cos(\beta-\alpha)=\cos(\beta)\cos(\alpha)+\sin(\beta)\sin(\alpha)$$
(and of course for $\alpha=\beta$, $1=\cos 0=\cos^2(\alpha)+\sin^2(\alpha)$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Then there exists a unique natural number $b$ less than $p$ such that $ab \equiv 1 \pmod{p}$. Full question: Let $p$ be a prime and let $a$ be an integer such that $1 \leq a < p$. Then there exists a unique natural number $b$ less than $p$ such that $ab \equiv 1 \pmod{p}$.
Looking for the proof. Is Fermat's little theo... | Hint: When does a linear congruence equation $ax\equiv b($mod $m)$ have a solution?
EDIT: If you know the rule regarding division in modular arithmetic, you can find uniqueness.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1758111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 4
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If $f(x-f(y))=f(-x)+(f(y)-2x)\cdot f(-y)$ what is $f(x)$
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x-f(y))=f(-x)+(f(y)-2x)\cdot f(-y), \quad \forall x,y \in \mathbb{R}$$
It's easy to see that $f(x)=x^2$ is a function satisfying the above equation. Thus I thought it would be wise to fir... | This is a loose derivation.
Let $x = 0$, to have:
$$
f(-f(y))=f(0)+f(y)\cdot f(-y)
$$
Let $ y = -y$:
$$
f(-f(-y))=f(0)+f(-y)\cdot f(y)
$$
So $f(-(f(y)) = f(-f(-y))$, I think this is sufficient to conclude that $f$ is even, by apply $f^{-1}$ on both sides and multiply $-1$.
Now with $f$ even,
$$
f(-f(y))=f(0)+f(y)\cdot ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1758348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Tridiagonal matrix inner product inequality I want to show that there is a $c>0$ such that
$$
\left<Lx,x\right>\ge c\|x\|^2,
$$
for alle $x\in \ell(\mathbb{Z})$ where
$$
L=
\begin{pmatrix}
\ddots & \ddots & & & \\
\ddots & 17 & -4 & 0 & \\
\ddots & -4 & 17 & -4 & \ddots \\
& 0 & -4 & 17 & \ddots \\
... | You can also finish your proof by noting that $k \le 2 \, \|x\|^2$ (by applying Hölder's inequality). Hence,
$$\langle L \, x , x \rangle \ge -4 \, k + 17 \, \|x\|^2 \ge 9 \, \|x\|^2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1758470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Solution of $x^2e^x = y$ The other day, I came across the problem (or something that reduced to the problem):
Solve for $x$ in terms of $y$ and $e$: $$x^2e^x=y$$
I tried for a while to solve it with logarithms, roots, and the like, but simply couldn't get $x$ onto one side by itself without having $x$ on the other side... | Solution with Lambert W:
$$
x^2 e^x=y
\\
x e^{x/2} = \sqrt{y}
\\
\frac{x}{2}\;e^{x/2} = \frac{\sqrt{y}}{2}
\\
\frac{x}{2} = W\left(\frac{\sqrt{y}}{2}\right)
\\
x = 2\;W\left(\frac{\sqrt{y}}{2}\right)
$$
One solution for each branch of the W function.
Other solutions by taking the other square-root:
$$
x = 2\;W\left(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1758587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Limit of a complex sequence So I wanted to calculate $$\lim_{n\rightarrow\infty}\frac{n^2}{(4+5i)n^2+(3+i)^n}$$
I thought that I could do it easier if I calculate $\lim_{n\rightarrow\infty}\frac{(3+i)^n}{n^2}$. First I write $\phi=\arctan(\frac{1}{3})$ so that $3+i=\sqrt{10}(\cos\phi+i\cdot\sin\phi)$. Now we have $\lim... | The correct way of doing this is to show that
$$\lim_{n \to \infty} \left| \frac{n^2}{(4+5i)n^2 + (3+i)^n} \right| =0$$
Now, write
$$\frac{n^2}{(4+5i)n^2 + (3+i)^n} = \frac{1}{(4+5i) + (3+i)^n/n^2}$$
and using triangular inequality, $$|(4+5i) + (3+i)^n/n^2| \ge |(3+i)^n/n^2| - |4+5i| =$$ $$ =|3+i|^n/n^2 - |4+5i| = \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1758708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show the Cauchy-Riemann equations hold but f is not differentiable
Let $$f(z)={x^{4/3} y^{5/3}+i\,x^{5/3}y^{4/3}\over x^2+y^2}\text{ if }z\neq0
\text{, and }f(0)=0$$
Show that the Cauchy-Riemann equations hold at $z=0$ but $f$ is not differentiable at $z=0$
Here's what I've done so far:
$\quad$As noted above, ther... | HINT: Are the limits as $x \to 0$ and $y \to 0$ the same? Because if the limit does not exist and equal the same value for EVERY direction of approach to the origin, then the limit does not exist there.
EDIT: The same argument would work with directional derivatives at the origin; if any two are not equal, then the fun... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Show that doesn't exist a group $(G,*)$ such that $\mathbb{R}$ is closed under $*$ and the restriction to $\mathbb{R}$ is the usual multiplication
Show that doesn't exist a group $(G,*)$ such that $\mathbb{R}\subset G$ such that $\mathbb{R}$ is closed under $*$ and the restriction to $\mathbb{R}$ is the usual multipli... | In a group, every element that is not the identity has a inverse element. $1$ is the identity element of the group under multiplication. Now, does zero have an inverse in $\mathbb{R}$? That will answer your question.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1758911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Derivative of unknown compound function The problem says:
What is $f'(0)$, given that
$f\left(\sin x −\frac{\sqrt 3}{2}\right) = f(3x − \pi) + 3x − \pi$, $x \in [−\pi/2, \pi/2]$.
So I called $g(x) = \sin x −\dfrac{\sqrt 3}{2}$ and $h(x)=3x − \pi$.
Since $f(g(x)-h(x))=3x − \pi$, I called $g(x)-h(x) = j(x) = \sin x −\fr... | Given: $$f(\sin x - \frac{\sqrt 3}{2}) = f(3x-\pi) + 3x - \pi$$
Taking derivatives on both sides w.r.t. $x$, we get,
$$f'(\sin x - \frac{\sqrt 3}{2}).\cos x = 3.f'(3x-\pi) + 3$$
Put $x=\frac{\pi}{3}$, we get,
$$f'(\sin \frac{\pi}{3} - \frac{\sqrt 3}{2}).\cos \frac{\pi}{3} = 3.f'(3.\frac{\pi}{3}-\pi) + 3$$
$$f'(0).\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1759027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Finding the Limit of a Sequence (No L'Hopital's Rule) Okay, I feel almost silly for asking this, but I've been on it for a good hour and a half. I need to find:
$$\lim_{n \to\infty}\left(1-\frac{1}{n^{2}}\right)^{n}$$
But I just can't seem to figure it out. I know its pretty easy using L'Hopital's rule, and I can "see"... | $(1 - \frac1{n^2} )^{n^2} \to e^{-1}$ as $n \to \infty$.
Thus $(1 - \frac1{n^2} )^{n^2} \in [\frac12,1]$ as $n \to \infty$.
$\def\wi{\subseteq}$
Now $(1 - \frac1{n^2} )^n \in \left( (1 - \frac1{n^2} )^{n^2} \right)^\frac1n \wi [\frac12,1]^\frac1n \to {1}$ as $n \to \infty$.
Thus by the squeeze theorem $(1 - \frac1{n^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1759124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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"answer_id": 3
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proving every nonempty open set is the disjoint union of a countable collection of open intervals i'm studying real analysis with royden, and i've looked up similar qeustions and answers but i couldn't get the exact answer that i need.
i don't need the whole process of proof , and i confused with certain phrase.
Firs... | Each $I_x$ contains a rational number say $q_x$.
Since the $I_x$ are disjoint, each $q_x$ belongs to exactly one $I_x$.
This produces a bijection between the sets $\{I_x\}$ and $\{q_x\}$, which is a subset of the rationals and hence countable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1759246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What's the value of $x$ in the following equation?
So this is how I approached this question, the above equations could be simplified to :
$$a = \frac{4(b+c)}{b+c+4}\tag{1!}$$
$$b = \frac{10(a+c)}{a+c+10}\tag{2}$$
$$c=\frac{56(a+b)}{a+b+56}\tag{3}$$
From above, we can deduce that $4 > a$ since $\frac{(b+c)}{b+c+4} < 1... | Your deductions are wrong and that is what is misleading you. Integers can be both positive and negative.
If you solve equations (1), (2) and (3) simultaneously you can find a, b and c.
I did this to find $$a=3$$
$$b=5$$
$$c=7$$
You can then plug this into the forth equation given in the problem to solve for x.
$$x = \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1759327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Closed form for binomial sum with absolute value Do you know whether the following expression has a (nice) closed form or a close enough approximation?
$$\frac{1}{2^n}\sum_{k=0}^{n} \binom{n}{k}|n-2k|$$
Thanks a lot :)
Cheers,
M.
| If we assume that $n$ is even, $n=2m$, our sum times $2^n$ equals:
$$ \sum_{j=0}^{m-1}\binom{2m}{j}(2m-2j)+\sum_{j=m+1}^{2m}\binom{2m}{j}(2j-2m) =\sum_{j=0}^{m-1}\binom{2m}{j}(4m-4j)$$
where:
$$\sum_{j=0}^{m-1}\binom{2m}{j} = \frac{4^m-\binom{2m}{m}}{2}$$
and:
$$\sum_{j=0}^{m-1}\binom{2m}{j}j = 2m\sum_{j=0}^{m-2}\binom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1759420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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limit to infinity : trouble with l'hopital Given the following limit for s positive constant
$\lim_{x\to \infty} xe^{-sx}(\sin x-s\cos x) $
how can I prove that the above is equal to $0$ ?
I re-write the limit as $ \frac{x(\sin x-s\cos x)}{e^{sx}} $ and then I use de l'Hopital theorem but it seems that I only go ro... | See $0\leq sin(x),cos(x)\leq 1$. So numerator is just oscillating between $x,0$ so now if you separate out you will see that denominator is growing so rapidly that at large x the value is almost $0$. So the limit as $x->\infty$ is $0$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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These two spaces are not homeomorphic...right? why is $\Bbb R\times[0,1]\not \cong \Bbb R^2$? we can't use the popular argument of deleting a point and finding that one has more path components than the other here.
So my idea is to delete a strip $\{0\}\times[0,1]$ from $\Bbb R\times[0,1]$.
But is $\Bbb R^2-f(\{0\}\tim... | The property of simple connectivity will distinguish between $\Bbb{R} \times[0,1]$ and $\Bbb{R}^2$. When we remove one point from $\Bbb{R} \times[0,1]$ then it is simply connected but removing one point from $\Bbb{R}^2$ then it is not simply connected.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1759660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$
Let $f : [0,1] \to \mathbb{R}$ be a continuous function such that
$$\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$$
Determi... | Calculate the difference of first and second integral then second and third.
$$\int_0^1f^2(x)-f^3(x)dx=0$$
$$\int_0^1f^3(x)-f^4(x)dx=0$$
Subtract both equations:
$$\int_0^1f^2(x)-2f^3(x)+f^4(x)dx=0$$
$$\int_0^1f^2(x)\left[1-2f(x)+f^2(x)\right]dx=0$$
$$\int_0^1f^2(x)(1-f(x))^2dx=0$$
Look at the integrand it consists of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1759736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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"answer_id": 2
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How do I calculate the number of unique permutations in a list with repeated elements? I know that I can get the number of permutations of items in a list without repetition using
(n!)
How would I calculate the number of unique permutations when a given number of elements in n are repeated.
For example
ABCABD
I want ... | Do you want the number of combinations of a fixed size? Or all sizes up to 6? If your problem is not too big, you can compute the number of combinations of each size separately and then add them up.
Also, do you care about order? For example, are AAB and ABA considered unique combinations? If these are not considered ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1759845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Understanding a proof by induction In the following proof by induction:
Problem: Prove by induction that $1+3+ \ldots+ \ (2n-1)=n^2$
Answer:
a) $P(1)$ is true since $1^2=1$
b)Adding $2n+1$ to both sides we obtain:
$$
1+3+..+(2n-1)+(2n+1)=n^2+2n+1=(n+1)^2
$$
Why $2n+1$ ? Where does this come from?
And how does knowing t... | $P(n)$ is the statement $1+3+\dots+(2n-1)=n^2$. To carry out a proof by induction, you must establish the base case $P(1)$, and then show that if $P(n)$ is true then $P(n+1)$ is also true.
In this problem, $P(n+1)$ is the statement $1+3+\dots+(2n-1)+(2n+1)=(n+1)^2$, because $2(n+1)-1=2n+1$. So by starting with $P(n)$ a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Is a ball always connected in a connected metric space? If I have a connected metric space $X$, is any ball around a point $x\in X$ also connected?
| There are even complete path-connected metric spaces that contain a point $x$ such that no ball around $x$ is connected, for example
$$ \{\langle x,0\rangle \mid x\ge 1\} \cup
\{\langle 1,y\rangle \mid 0\le y\le 1 \} \cup
\{\langle x,\tfrac1x \rangle \mid x\ge 1 \} \cup
\bigcup_{n=3}^\infty \{ \langle x,\tfrac1n \rangl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1760148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Picking two random points on a disk I try to solve the following:
Pick two arbitrary points $M$ and $N$ independently on a disk $\{(x,y)\in\mathbb{R}^2:x^2+y^2 \leq 1\}$ that is unformily inside. Let $P$ be the distance between those points $P=d(M,N)$. What is the probabilty of $P$ being smaller than the radius of the ... | Let $R$ be the distance between $O$, the origin, and $M$. The probability that $R$ is less than or equal to a value $r$ is
$$P(R\le r) = \begin{cases}
\frac{\pi r^2}{\pi\cdot 1^2} = r^2, & 0\le r\le 1\\
1, &r>1\\
0, &\text{otherwise}
\end{cases}$$
The probability density function of $R$ is
$$f_R(r) = \frac{d}{dr}P(R\le... | {
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"url": "https://math.stackexchange.com/questions/1760230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is a map that preserves the hyperbolic distance biholomorphic? Let $\lVert z \rVert_w = \frac{|z|}{1 - |w|^2}$ be the hyperbolic distance in $\mathbb{D}$, and let the hyperbolic metric be $d(z, w) = \inf_\gamma \int_0^1 \lVert \gamma'(t) \rVert_{\gamma(t)} \, dt$. Automorphisms of $\mathbb{D}$ preserve this metric. I'... | Yes, this is true. The proof is not specific to hyperbolic metric: one can argue the same way about the Euclidean metric on the plane, or in higher dimensions.
Step 1: Compose $f$ with a Möbius transformation that sends $f(0)$ to $0$. This reduces the problem to the case $f(0)=0$.
Step 2: Since $f$ is an isometry, ... | {
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"answer_id": 0
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Type 2 Error Question - How to calculate for a two tailed? The modulus of rupture (MOR) for a particular grade of pencil lead is known to have
a standard deviation of 250 psi. Process standards call for a target value of 6500 psi
for the true mean MOR. For each batch, an inspector tests a random sample of 16
leads. Man... | You have not said what significance level you are using, but it seems that what you did for one tail used $5\%$ and
*
*standard error of the mean $\dfrac{250}{\sqrt{16}} = 62.5$
*If the true population mean is $6500$ then there is a $5\%$ probability that the the sample mean will be below $6500 + 62.5 \Phi^{-1}(0.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1760432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Is the unit circle "stretchy" with respect to its norm? Suppose we have a collection of metric spaces on $\mathbb{R}^n$, each of which has a different p-norm, $1\leq p \leq \infty$. ($p=2$ is Euclidean distance, $p=1$ is taxicab distance, etc.) Then, suppose we have a point $x$ in $\mathbb{R}^2 $ that's within the uni... | Yes, this is true by a simple continuity argument. Note that for any fixed $x\in\mathbb{R}^2$, the map $p\mapsto \|x\|_p$ is a continuous function $[1,\infty]\to\mathbb{R}$ (it is obvious that this is continuous for $p<\infty$; continuity as $p\to \infty$ requires a little work but is not hard). So by the intermediat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1760556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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inequality between median length and perimeter Is there an inequality between the sum of median lengths and the perimeter? If there is, can you specify a proof as well? I need to use this to solve a question.
I tried using Apollonius theorem.
$$m_a=\sqrt{\frac{2b^2+2c^2-a^2}{4}}$$
| Yes, there is :
$$\frac 34(\text{the perimeter})\lt \text{(the sum of median lengths)}\lt \text{(the perimeter)}$$
Proof :
Let $G$ be the centroid of $\triangle{ABC}$, and let $X,Y,Z$ be the midpoint of the side $BC,CA,AB$ respectively.
First of all,
$$GY+GZ\gt YZ,\quad GZ+GX\gt ZX,\quad GX+GY\gt XY$$
and so
$$2(GX+G... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1760826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Prove: $\arcsin\left(\frac 35\right) - \arccos\left(\frac {12}{13}\right) = \arcsin\left(\frac {16}{65}\right)$ This is not a homework question, its from sl loney I'm just practicing.
To prove :
$$\arcsin\left(\frac 35\right) - \arccos\left(\frac {12}{13}\right) = \arcsin\left(\frac {16}{65}\right)$$
So I changed all t... | How exactly did you convert to arctan? Careful:
$$\arccos\left(\frac {12}{13}\right) = \arctan\left(\frac {5}{12}\right)
\ne \arctan\left(\frac {12}{5}\right)$$
Draw a right triangle with hypotenuse of length 13, adjacent side (from an angle $\alpha$) with length 12 and opposite side with length 5; then $\cos\alpha = 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1760940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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finding sup and inf of $\{\frac{n+1}{n}, n\in \mathbb{N}\}$ Please just don't present a proof, see my reasoning below
I need to find the sup and inf of this set:
$$A = \{\frac{n+1}{n}, n\in \mathbb{N}\} = \{2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \cdots\}$$
Well, we can see that:
$$\frac{n+1}{n} = 1+\frac{1}{n} > 1$$... | Answer on "how do I prove it?"
If $a\in A$ is an upper bound of $A$ then any $c$ with $c<a$ is not an upper bound of $A$ since $a$ is an element of $A$ that does not satisfy $a\leq c$.
We conclude that $a$ must be the least upper bound of $A$.
*
*$1\leq1+\frac{1}{n}$ for each $n$ so $1$ is a lower bound of $A$.
*... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1761130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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In MATLAB, $\pi$ value is given as 355/113. why? $\pi$ is an irrational number. MATLAB shows it equal to 355/113 in fractional format. Is there no better fractional representation than 355/113 within the limits of the finite precision the computers use? How is the value arrived at?
| The irreducible fraction $\frac{355}{113}$ of MATLAB gives certainly a good approximation, as @almagest comment. One could argue that the irreducible $\frac{208341}{66317}$ ($\frac{22}{17}+\frac{37}{47}+\frac{88}{83}$ indeed) is another rational approximation with an accuracy of nine decimal digits of exactness. Howeve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1761234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Finding Laplace Transform of $te^{-t}$ I started with this integral:
$$ \int_{0}^{\infty} e^{-st}\cdot te^{-t}dt$$
= $$\int_{0}^{\infty} te^{-(s+1)t}dt$$
let $dv=e^{-(s+1)t}dt, u=t$ and thus $v=-\frac{1}{s+1}e^{-(s+1)t}dt, du=dt$
$\rightarrow$
$-\frac{t}{s+1}e^{-(s+1)t}|_0^\infty + \frac{1}{s+1}\int_{0}^{\infty}e^{-(s... | If you see an integral of the form $$\int t f(t) \,dt$$ then try partial integration!
Assuming $s\neq 1$
\begin{align}\int_0^{\infty} t f(t) \,dt &= \left[t\frac{-1}{s+1}e^{-(s+1)t} \right]_{t=0}^\infty-\int_0^{\infty} \frac{-1}{s+1}e^{-(s+1)t} \,dt \\&= 0 - 0 + \left[ \frac{-1}{(s+1)^2}e^{-(s+1)t} \right]_{t=0}^\inft... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1761367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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A variant of the exponential integral Consider the following integral (for $x,y\in \mathbb{R}_{>0}$)
$$E(x,y) = \int_0^1 \frac{\mathrm{e}^{-x/s-ys}}{s}\,\mathrm{d}s,$$
which is a variant of the usual exponential integral $E_1(x)$ to which it reduces when $y=0$.
I am interested in the efficient evaluation of $E(x,y)$, ... | $\int_0^1\dfrac{e^{-\frac{x}{s}-ys}}{s}~ds$
$=\int_\infty^1se^{-xs-\frac{y}{s}}~d\left(\dfrac{1}{s}\right)$
$=\int_1^\infty\dfrac{e^{-xs-\frac{y}{s}}}{s}~ds$
$=K_0(x,y)$ (according to https://core.ac.uk/download/pdf/81935301.pdf)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1761491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Prove: $\frac{a+c}{b+d}$ lies between $\frac{a}{b}$ and $\frac{c}{d}$ (for positive $a$, $b$, $c$, $d$) I am looking for proof that, if you take any two different fractions and add the numerators together then the denominators together, the answer will always be a fraction that lies between the two original fractions.
... | Here's one way to look at it:
You're taking a class. Suppose you get $a$ points out of $b$ possible on Quiz 1, and $c$ points out of $d$ possible on Quiz 2.
Your overall points are $a+c$ out of $b+d$ possible. And your overall percentage should be between your lower quiz score and your higher quiz score.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1761557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 4
} |
Linear Algebra with functions Basically my question is - How to check for linear independence between functions ?!
Let the group $\mathcal{F}(\mathbb{R},\mathbb{R})$
Be a group of real valued fnctions.
i.e $\mathcal{F}(\mathbb{R},\mathbb{R})=\left\{ f:\mathbb{R}\rightarrow\mathbb{R}\right\} $
Let 3 functions $f_{1},f... | Hint: Use Wronskian and show that the Wronskian-Determinant does not vansish.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1761628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 7,
"answer_id": 2
} |
A human way to simplify $ \frac{((\sqrt{a^2 - 1} - a)^2 - 1)^2}{(\sqrt{a^2 - 1} - a)^22 \sqrt{a^2 - 1}} - 2 a $ I end up with simplifying the following fraction when I tried to calculate an integral(*) with the residue theory in complex analysis:
$$
\frac{((\sqrt{a^2 - 1} - a)^2 - 1)^2}{(\sqrt{a^2 - 1} - a)^22 \sqrt{a^... | Start with
$$\begin{align}\left(\sqrt{a^2-1}-a\right)^2-1&=a^2-1-2a\sqrt{a^2-1}+a^2-1\\
&=\sqrt{a^2-1}\left(2\sqrt{a^2-1}-2a\right)\\
&=2\sqrt{a^2-1}\left(\sqrt{a^2-1}-a\right)\end{align}$$
So you are now down to
$$\frac{\left(2\sqrt{a^2-1}\left(\sqrt{a^2-1}-a\right)\right)^2}{\left(\sqrt{a^2-1}-a\right)^2\cdot2\sqrt{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1761694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to show that $X^p-t\in\mathbb{F}_p(t)[x]$ is irreducible? This question is previously asked here, but there is no complete solution of it.
I understand that the root $\alpha$ exist in the algebraic closure of $\mathbb{F}_p(t)[x]$, and it is the only root because $f(X)=(X-\alpha)^p$, but how do we proceed to show th... | Because it only has one root, and none of them are in $\Bbb F_p(t)$. Recall for any root $\zeta$ we have $\zeta^p=t$, but then since $\Bbb F_p(t)[x]$ is a UFD, it means that $(x-\zeta)^p=x^p-t$ has just the one root. So if it is reducible, it reduces all the way, and in fact there is an element of $\Bbb F_p(t)$ such th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1761815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluation of $ \int_0^\infty\frac{x^{1/3}\log x}{x^2+1}\ dx $ The following is an exercisein complex analysis:
Use contour integrals with $-\pi/2<\operatorname{arg} z<3\pi/2$ to compute
$$
I:=\int_0^\infty\frac{x^{1/3}\log x}{x^2+1}\ dx.
$$
I don't see why the branch $-\pi/2<\operatorname{arg} z<3\pi/2$ would work... | Sketch of a possible argument. Use the branch with the argument from
zero to $2\pi$ of the logarithm, the function
$$f(z) = \frac{\exp(1/3\log z)\log z}{z^2+1}$$
and a keyhole contour with the slot aligned with the positive real
axis.
The sum of the residues at $z=\pm i$ is (take care to use the chosen branch of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1761922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
$|f| $ is Lebesgue integrable , does it implies $f$ is also? If $ f $ is Lebesgue integrable then $|f|$ is Lebesgue integrable but does the converse of the result is also true?
| Think of |f| as a division of f into two functions: $f_+$ and $f_-$.
$f_+$ we define as equal to f on the domain {x: f(x) is non-negative}, and 0 on all other x.
$f_-$ we define as equal to -f on the domain {x: f(x) is negative} and 0 elsewhere. $|f|=f_+ + f_-$.
If |f| is finite, then necessarily both $f_+$ and $f_-$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1762006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
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Right Triangle's Proof A right triangle has all three sides integer lengths. One side has length 12. What are the possibilities for the lengths of the other two sides? Give a proof to show that you have found all possibilities.
EDIT: I figured out that there are a total of 4 combinations for a side with length 12.
$$a^... | Wolfram MathWorld gives the number of ways in which a number $n$ can be a leg (other than the hypotenuse) of a primitive or non-primitive right triangle. For $n=12$, the number is $4$. It also gives the number of ways in which a number $n$ can be the hypotenuse of a primitive or non-primitive right triangle. For $n=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1762101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Can we find a non central element of order 2 in a specific 2-group? Let $G$ be a non-abelian group of order $2^5$ and center $Z(G)$ is non cyclic. Can we always find an element $x\not\in Z(G)$ of order $2$ if for any pair of elements $a$ and $b$ of $Z(G)$ of order $2$, the factor groups $G/\langle a\rangle$ and $G/\lan... | This is not complete answer; but a partial information, which in addition to Holt's comment may simplify your job. (I will try to write complete proof as I get some directions on it)
Suppose $G$ is a group of order $2^5$ satisfying conditions in your question. We show that $Z(G)=C_2\times C_2$.
For this, by hypothesis... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $a \sqrt{b + c} + b \sqrt{c + a} + c \sqrt{a + b} \le \sqrt{2(a+b+c)(bc + ac + ab)}$ for $a, b, c > 0$ Prove for $a, b, c > 0$ that
$$a \sqrt{b + c} + b \sqrt{c + a} + c \sqrt{a + b} \le \sqrt{2(a+b+c)(bc + ac + ab)}$$
Could you give me some hints on this?
I thought that Jensen's inequality might be of use ... | Use the Cauchy-Schwarz inequality on the two vectors $(\sqrt a, \sqrt b, \sqrt c)$ and $(\sqrt{a(b+c)}, \sqrt{b(a+c)}, \sqrt{c(a+b)})$ (then take the square root on both sides or not, depending on which version of the CS inequality you use), and lastly note that we have:
$$
a(b+c) + b(a + c) + c(a + b) = 2(bc + ac + a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1762452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Example of a function $u \in W^{1,p}(\Omega)$ whose extension $\hat{u}$ to be $0$ outside $\Omega$ $\hat{u} \notin W^{1,p}(\mathbb{R}^n)$ Let's consider a function $u \in W^{1,p}(\Omega)$, where $W^{1,p}(\Omega)$ is the Sobolev Space and $\Omega$ is an open set. When we extend $u$ to $\hat{u}$ like this:
$$\hat{u}(x)=\... | If $\Omega$ is bounded, it suffices to take $u(x)\equiv 1$. Then $\hat u$ cannot be in $W^{1,p}(\mathbb R^n)$ for $p>n$ since it is discontinuous. (Sobolev embedding)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1762543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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generalization of the Pythagorean theorem In school, students learn that in a triangle ABC, ACB is a right angle if and only if AB^2=AC^2+BC^2. This deep relation between geometry and numbers is actually only a partial result as one can say much better : the angle in C is
*
*acute if and only if AB^2 < AC^2+BC^2,
... | The relation can actually be thought of in terms of the cosine rule:
$a^2 = b^2 + c^2 - 2bc \cos(A)$, where $a, b, c$ are the sides of the triangle and $A$ is the angle opposite to side $a$.
Clearly, if $A = 90^\circ$, then $a^2 = b^2 + c^2$
If $A < 90^\circ$, then $\cos(A) > 0$, hence $a^2 < b^2 + c^2$ and vice versa ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1762687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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GCH is preserved when forcing with $Fn(\lambda,\kappa)$. Given a countable transitive model $M$ where $GCH$ holds it is an exercise from Kunen's book to show that GCH also holds in $M[G]$ when $G$ is a $P-$generic filter over $M$, and $P=Fn(\lambda,\kappa)$ ($\aleph_0\leq\kappa<\lambda$ in $M$). Recall that $Fn(\kappa,... | Regarding CH:
Working in $M$ we have $\mid Fn(\kappa,\lambda)\mid=\kappa^\lambda{=}\lambda^+$ and also $(\lambda^+)^{\aleph_0}=\lambda^+$ because $cf(\lambda^+)=\lambda^+>\aleph_0$ and $GCH$ holds in $M$. So using the well-known argument with nice names and $\lambda^+-cc$-ness, we conclude $(2^{\aleph_0}\leq (\lambda^+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1762782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the points on the sphere $x^2 + y^2 + z^2 = 4$ that are closest to, and farthest from the point (3, 1, -1). Find the points on the sphere $x^2 + y^2 + z^2 = 4$ that are closest to, and farthest from the point $(3, 1, -1)$.
I identified that this is a constrained optimisation problem which I will solve using an aux... | $(2/\sqrt{11})(3,1,-1)$ is the closest point and $(-2/\sqrt{11})(3,1,-1)$ is the most distant point.
If you have to use Lagrange multipliiers....
minimize/maximize: $(x-3)^2 + (y-1)^2 + (z+1)^2$ (this is the distance squared from x,y,z to your point.)
constrained by: $x^2 + y^2 + z^2 = 4$
$F(x,y,z,\lambda) = (x-3)^2 +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1762885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Function that is second differential continuous Let $f:[0,1]\rightarrow\mathbb{R}$ be a function whose second derivative $f''(x)$ is continuous on $[0,1]$. Suppose that f(0)=f(1)=0 and that $|f''(x)|<1$ for any $x\in [0,1]$. Then $$|f'(\frac{1}{2})|\leq\frac{1}{4}.$$
I tried to use mean value theorem to prove it, but I... | A bit more tricky than I thought at first. The idea is easy, the calculation may look complicated. The idea is to find a second order polynomial with second derivate $=1$ which has the same values as $f$ for $x=0 $ and $x= \frac{1}{2}$, and then to show that this function $-f$ is convex, which allows to get an estimate... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1763014",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Using contour integration to solve $ \int ^\infty _0 \frac {\ln x} {(x^2+1)} dx$ Question:
Find the value of contour integration $$ \int ^{\infty}_0 \frac {\ln x} {(x^2+1)} dx$$
Attempt:
I just calculate
$$\text{Res}(f,z=i) = 2\pi i\lim_{z\to i}(z - i)\frac{\ln z}{z^2+i} = \frac{\pi^2 i}2$$
Im not too sure how to m... | Consider the branch $f(z) = \frac{\ln z}{z^2 +1}$ where $|z| > 0 , -\frac{\pi}{2}< \arg z < \frac{3\pi}{2}$. Take the path $C = L_2 + L_1 + C_{\rho} + C_R$ where $\rho < 1 < R$ and $C_R$ and $C_{\rho}$ are the semi-circles with radius $R$ and $\rho$ respectively. See the figure below.
$\hskip.75in$
By Cauchy's Theorem ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1763174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Show that Set in $M:=\{x\in \Bbb R^3 : x_1^2\ge2(x_2^3+x_3^3) \}$ is closed I have to show this regarding the Euclidean metric.
I've already shown that it isn't bounded by showing that the $d(x,y)\:\forall x,y \in M$ isn't bounded.
I know that in order to show the closedness i have to show that the complement is open, ... | Let $f(x_1,x_2,x_3)= x_1^2-2(x_2^3+x_3^3)$, which is a polynomial, hence continuous. Then the set in question is $f^{-1}([0,\infty))$ which is closed, hence by the definition of continuity, the inverse image of this closed set is closed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1763255",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Continuous function injective over a compact set, and locally injective on each point of the set Suppose we have a function $F: \mathbb R^n \rightarrow \mathbb R^k$ continuous over some open set $U \in \mathbb R^n$, and let compact set $K \subset U$. $F$ satisfies the following properties:
1) F is injective over K
2) F... | This is a very nice problem indeed. Thank you.
Let's call $B(K,\epsilon)$ the set $\{x\in \mathbb{R} : \mathrm{dist}(x,K)
<\epsilon\}$. It is easy to see that for any $\epsilon$,
$\overline{B(K,\epsilon)}$ is compact, and that there exists an
$n\in\mathbb{N}$ such that $B(K,\frac{1}{n})\subseteq U$. We may assume witho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1763351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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(Elegant) proof of : $x \log_2\frac{1}{x}+(1-x) \log_2\frac{1}{1-x} \geq 1- (1-\frac{x}{1-x})^2$ I am looking for the most concise and elegant proof of the following inequality:
$$
h(x) \geq 1- \left(1-\frac{x}{1-x}\right)^2, \qquad \forall x\in(0,1)
$$
where $h(x) = x \log_2\frac{1}{x}+(1-x) \log_2\frac{1}{1-x}$ is th... | Hopefully, this is right!
Note that from the weighted AM-GM inequality, We have that $$h(x)=\log_2{\frac{1}{x^x(1-x)^{1-x}}} \ge \log_2\frac{1}{x^2+(1-x)^2}$$
Thus we have to show $$\left(1-\frac{x}{1-x}\right)^2 \ge 1-\log_2\frac{1}{2x^2-2x+1}=\log_2{(4x^2-4x+2)}$$ Substitute $x=\frac{a+1}{a+2}$, and we have $$f(a)=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1763483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 9,
"answer_id": 1
} |
How many draws to have a 90% chance of getting the ace of spades? You have a standard 52-card deck, and you want to take the minimum number of draws from a random/shuffled deck such that you have a 90% chance of drawing the ace of spades. How would you find the minimum number of draws to achieve this 90% probability of... | Without replacement is (for once) easier. The Ace of Spades is equally likely to be in any of the $52$ positions, so we need to draw $n$ cards, where $n$ is the smallest integer $\ge (0.90)(52)$.
For with replacement, the probability we don't see the Ace of Spades in $n$ draws is $\left(\frac{51}{52}\right)^n$. We want... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1763559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Show that $\lim_{n \rightarrow \infty} n \int_{0}^{1}e^{-nx}f(x)dx=f(0)$ when $f:[0,1] \rightarrow \mathbb{R}$ is continuous. Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function.
Show that $\lim_{n \rightarrow \infty} n \int_{0}^{1}e^{-nx}f(x)dx=f(0)$
I am making a claim,
$\lim_{n \rightarrow \infty} n \int_... | Use integration by parts:
$$ \lim_{n\to\infty} n\left[\left.{\frac{f(x)e^{-nx}}{n}}\right|^0_1-\frac{1}{n}\int_{0}^{1}f'(x)e^{-nx}\text{d}x\right] $$
$$ = f(0) - \lim_{n\to\infty}f(1)e^{-n} - \int_{0}^{1}\lim_{n\to\infty}f'(x)e^{-nx}\text{d}x $$
$$ = f(0) $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1763652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Accurate summation of mixed-sign floating-point values Due to the finite representation, floating-point addition loses significant bits. This is particularly noticeable when there is catastrophic cancellation, such that all the significant bits can disappear (e.g. $10^{-10}+10^{10}-10^{10}$).
Reordering the terms of th... | I would use compensated summation. Basically you can recover most of the error
from a single floating point addition. This was noticed long time ago by e.g. Dekker, Knuth, and others.
There are a lot of references, e.g.
T. Ogita, S.M. Rump, and S. Oishi, Accurate sum and dot product,
SIAM J. Sci. Comput., 26 (2005), pp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1763764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Existence of how many sets is asserted by the axiom of choice in this case? Applying the axiom of choice to $\{\{1,2\}, \{3,4\}, \{5,6\},\ldots\}$, does only one choice set necessarily exist, or all of the $2^{\aleph_0}$ I "could have" chosen? Or something in between? It seems if only one, then I didn't really have m... | In this case you don’t need the axiom of choice to get a choice function: just pick the smaller member of each pair. Various other choice functions are explicitly definable: we could just as well pick the larger member of each pair, for instance. Or we could pick the member that is divisible by $3$ if there is one, and... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can you use both sides of an equation to prove equality? For example:
$\color{red}{\text{Show that}}$$$\color{red}{\frac{4\cos(2x)}{1+\cos(2x)}=4-2\sec^2(x)}$$
In high school my maths teacher told me
To prove equality of an equation; you start on one side and manipulate it algebraically until it is equal to the othe... | It is enough.. Consider this example:
To prove:
$a=b$
Proof:
$$a=c$$
$$b=c$$
Since $a$ and $b$ are equal to the same thing, $a=b$.
That is the exact technique you are using and it sure can be used.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1763978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
"answer_count": 7,
"answer_id": 6
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How to find the radius of convergence of $\sum_{n=0}^{\infty}[2^{n}z^{n!}]$ I tried with $$1/R = \lim_{n\to\infty}{\sup({\sqrt[n]{2^n}})} = \lim_{n\to\infty}{2} = 2$$
But that don't seem correct.
Thank you for your help!
| We have
$$\sqrt[n]{2^n\left|z^{n!}\right|}= 2|z|^{(n-1)!} \to \begin{cases}0&,|z|<1\\\\2&,|z|=1\\\\\infty&,|z|>1\end{cases}$$
and the series converges when $|z|<1$ and diverges otherwise.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1764098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Explain why a gamma random variable with parameters $(t, \lambda)$ has an approximately normal distribution when $t$ is large. Explain why a gamma random variable with parameters $(t, \lambda)$ has an approximately normal distribution when $t$ is large.
What I have come up with so far is:
Let $X=$ the sum of all $X_i... | That's pretty much it except that you assumed $t$ is an integer and you used "$i$" rather then "$t$" at one point. One way of dealing with non-integer values of $t$ is to go back to the proof of the CLT that uses characteristic functions and make a minor modification in the argument to accomodate non-integers.
(BTW, o... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Is the set of periodic functions from $\mathbb{R}$ to $\mathbb{R}$ a subspace of $\mathbb{R}^\mathbb{R}$? A function $f: \mathbb{R} \to \mathbb{R}$ is called periodic if there exists a positive number
$p$ such that $f(x) = f(x + p)$ for all $x \in \mathbb{R}$. Is the set of periodic
functions from $\mathbb{R}$ to $\mat... | The idea is that if $f$ is $a$-periodic and $g$ is $b$-periodic, and $\frac{a}{b}\in \mathbb{Q}$, then it's easy to see that $f+g$ is periodic : if $\frac{a}{b} = \frac{p}{q}$ with $p,q\in \mathbb{N}$ then put $c = qa=pb$.
Since $f$ is $a$-periodic, it's also $c$-periodic. Likewise, $g$ is $c$-periodic since it's $b$-p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1764349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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What are those weighed graphs called? Let $G = (V, E)$ be a directed graph, and define the weight function $f : V \sqcup E \to \mathbb{R}^+$ as follows:
*
*sum of weights of vertices is 1,
*if a vertex has edges coming out of it, their weights sum to 1, and
*if a vertex has edges coming into it, their weights sum ... | For undirected graphs, constant weight of neighboring vertices has been called "weighted-regular", generalizing the concept of regular graph (when all the vertex weights are equal).
That definition is not used often enough to have become a standard term.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1764424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Can$A \cap (B' \cap C')$ be $(A \cap B') \cap (A \cap C')$? If I use the above statement, provided that it is right, in a question, would I have to prove it as well?
| In this case, the proof is rather trivial: since the $\cap$ operator is both associative and commutative, all the following are equivalent:
$$ (A \cap B) \cap (A \cap C) $$
$$ A \cap B \cap A \cap C $$
$$ A \cap A \cap B \cap C $$
$$ A \cap B \cap C $$
$$ A \cap (B \cap C) $$
As others said, whether or not you actually... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1764511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Maximum concyclic points Given n points, find an algorithm to get a circle having maximum points.
| The method of choosing
all sets of 3 points,
finding the circle that
passes through that set,
and seeing which other points
lie on that circle
has one big problem:
roundoff error.
If you try to use
any method that involves
taking square roots,
roundoff can cause problems.
Here is a method,
based on some previous work o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1764722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Shortest path tree different than minimum spanning tree My professor brought up in class that a shortest path tree can be different than minimum spanning tree for an undirected graph. However, I have not been able to find a case where this is possible.
| Remember that the shortest path is between two points, while the minimum spanning tree is the tree that spans the entire graph, and not just two points.
If you consider a triangle with side lengths of 1, can you see the MSP and the minimum path between all pairs of vertices in your head? do they differ for one pair of ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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multiplication of finite sum (inner product space) I am having difficulty to understand the first line of the proof of theorem 3.22 below. (taken from a linear analysis book)
Why need to be different index, i.e. $m,n$ when multiplying the two sums? This is very basic, but I really need help for explanations.
| Orthornormality refers to the basis $e_i$. When a basis is orthonormal it means the inner product between any two elements of the basis $e_i,e_j$ is $\langle e_i, e_j \rangle = \delta_{ij}$ (see kronecker delta). More generally, two vectors $u,v$ are orthogonal if $\langle u, v \rangle = 0$. The normality part comes fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1764927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 2
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Mistake while evaluating the gaussian integral with imaginary term in exponent I am trying to evaluate the integral $I=\int_0^\infty e^{-ix^2}\,dx$ as one component of evaluating a contour integral but I am dropping a factor of $1/2$ and after checking my work many times, I worry that I am making a conceptual mistake i... | The mistake is in my eyes that you allow the polar angle to be in $\phi\in[0,\pi]$ although just integrating over a quadrant integration domain in cartesian coordinates,$(x,y)\in[0,R]^2$. I'd suggest putting $\phi\in[0,\pi/2]$ in order to account for the integration in the first quadrant. This modification of your calc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1765002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Triangle group $(\beta,\beta,\gamma)$ is a subgroup of the triangle group $(2,\beta, 2\gamma)$. Let $1/\alpha+1/\beta+1/\gamma<1$, and let us consider the triangle group $(\alpha,\beta,\gamma)$, i.e. the subgroup of $\mathbb{P}\mathrm{SL}(2,\mathbb{R})$ induced by the hyperbolic triangle with angles
$\pi/\alpha,\pi/\... | It's the subgroup $\langle xyx,y \rangle$ of $\langle x,y \mid x^2,y^\beta,(xy)^{2\gamma} \rangle$.
The index of this subgroup $2$, so checking that the subgroup has the presentation $\langle z,y \mid z^\beta,y^\beta,(xy)^\gamma \rangle$ (with $z=xyx$) is routine.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1765144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculate $\int_D \rvert x-y^2 \rvert dx \ dy $ $$\int_D \rvert x-y^2 \rvert dx \ dy $$
$D$ is the shape that is delimited from the lines:
$$
y=x \\
y=0 \\
x=1 \\$$
$$D=\{ (x,y) \in \mathbb{R}^2: 0 \le x \le 1 \ , \ 0 \le y \le x \}$$
$$\rvert x-y^2 \rvert=x-y^2 \qquad \forall (x,y) \in D $$
$$\int_0^1 \Big( \int_0^x... | This is correct, you are correct that the absolute value integrand is equal to x-y^2 for all x and y in your region of integration.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1765260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Multiplying two logarithms (Solved) I was wondering how one would multiply two logarithms together?
Say, for example, that I had:
$$\log x·\log 2x < 0$$
How would one solve this? And if it weren't possible, what would its domain be?
Thank you!
(I've uselessly tried to sum the logs together but that obviously wouldn't ... | There is no particular rule for the product of logarithms, unlike for the sum.
Applying the latter, you can rewrite
$$\log(x)\log(2x)=\log(x)(\log(x)+\log(2))=t(t+\log(2))$$
and proceed as usual to find the domain of $t$. Then $x=e^t$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1765403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Maximum of $xy+y^2$ subject to right-semicircle $x\ge 0,x^2+y^2\le 1$ Maximum of:
$$
xy+y^2
$$
Domain:
$$
x \ge 0, x^2+y^2 \le1
$$
I know that the result is:
$$
\frac{1}{2}+\frac{1}{\sqrt{2}}
$$
for
$$
(x,y)=\left(\frac{1}{\sqrt{2(2+\sqrt{2})}},\frac{\sqrt{2+\sqrt{2}}}{2}\right)
$$
But I don't know how to get this resu... | Hint$$x^2+y^2=x^2+(3-2\sqrt{2})y^2+(2\sqrt{2}-2)y^2 $$
Now notice $$x^2+(3-2\sqrt{2})y^2+(2\sqrt{2}-2)y^2 \ge 2(3-2\sqrt{2})^{\frac{1}{2}}xy+(2\sqrt{2}-2)y^2 (\because \text{AM-GM})$$Now note $(\sqrt{2}-1)^2=3-2\sqrt{2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1765612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Given $\tan a = -7/24$ in $2$nd quadrant and $\cot b = 3/4$ in $3$rd quadrant find $\sin (a + b)$. Say $\tan a = -7/24$ (second quadrant) and $\cot b = 3/4$ (third quadrant), how would I find $\sin (a + b)$?
I figured I could solve for the $\sin/\cos$ of $a$ & $b$, and use the add/sub identities, but I got massive unwi... | $\tan(a) = -7/24$
Opposite side $= 7$ and adjacent side $= 24$
Pythagorean theorem
$\Rightarrow$ hypotenuse $= \sqrt{49+576} = 25$
$\sin(a) = 7/25$ (sin is positive in second quadrant)
$\cos(a) = - 24/25$ (cos is negative in second quadrant)
$\cot(b) = 3/4 \Rightarrow \tan(b) = 4/3$
Opposite side $= 4$ and adjace... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Showing a C* Algebra contains a compact operator In my functional analysis class we are currently dealing with C* Algebras, and I just met this problem:
Let $ \mathbb{H} $ be a separable Hilbert space, and suppose we have $ A \subset B(\mathbb{H}) $ a C* Algebra of bounded operators of bounded operators on H. Now we s... | Here is an answer in terms of a standard result of C*-algebra theory.
Theorem: Every injective $*$-homomorphism from one C*-algebra to another isometric.
In addition to this result, we will need to know that the quotient of a C*-algebra by a closed ideal makes sense, and that the result is a C*-algebra. In particular... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Does there exist a non-parallelisable manifold with exactly $k$ linearly independent vector fields?
For any positive integer $k$, is there a smooth, closed, non-parallelisable manifold $M$ such that the maximum number of linearly independent vector fields on $M$ is $k$?
Note that any such $M$, for any $k$, must have... | Consider $M$, the product of the Klein bottle $K^2$ with $k-1$-torus $T^{k−1}$. This manifold is nonorientable, hence, nonparallelizable. On the other hand, it is the total space of a circle bundle over $T^k$, since $K^2$ is a circle bundle over the circle. Let $H$ be a (smooth) connection on this bundle. Take $k$ inde... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1765967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Sum of sum of binomial coefficients $\sum_{j=1}^{n}{\sum_{k=0}^{j}{{n}\choose{k}}}$ I know there is no simple way to solve the sum:
$$\sum_{k=0}^{j}{{n}\choose{k}}$$
But what about the equation:
$$\sum_{j=1}^{n}{\sum_{k=0}^{j}{{n}\choose{k}}}$$
Are there any simplifications or good approximations for this equation?
| $$\begin{align}
\sum_{j=\color{red}0}^{n}{\sum_{k=0}^{j}{{n}\choose{k}}}
&=\sum_{k=0}^n\sum_{j=k}^n\binom nk\\
&=\sum_{k=0}^n (n-k+1)\binom n{n-k}\\
&=\sum_{k=0}^n (j+1)\binom nj
&&\text{putting }j=n-k\\
&=\sum_{k=0}^n j\binom nj+\sum_{k=0}^n \binom nj\\
&=n 2^{n-1}+2^n\\
\sum_{j=\color{red}1}^n\sum_{k=0}^n\binom nk&=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1766025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
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Cosine Inequality Show that given three angles $A,B,C\ge0$ with $A+B+C=2\pi$ and any positive numbers $a,b,c$ we have $$bc\cos A + ca \cos B + ab \cos C \ge -\frac {a^2+b^2+c^2}{2}$$
This problem was given in the course notes for a complex analysis course, so I anticipate using $$bc\cos A + ca \cos B + ab \cos C=\math... | I applied the Cosine Rule three times (from each side to the opposite angle) and added to arrive at the inequality (actually an equality in a triangle?)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1766099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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Distribution of $aXa^T$ for normal distributed vector $a$ Let $a$ be $1\times n$ random vector with entries chosen independently from normal distribution with zero mean and unit variance. What is the distribution of $aXa^T$ for a given $n\times n$ matrix $X$.
If $X$ is symmetric matrix, then the above is a Wishart dis... | $X$ can be written as a sum $X_{s} + X_a$, where $X_{s}$ is symmetric and $X_a$ is antisymmetric. But $a X_a a^\text{T} = 0$ for any $a$ and any antisymmetric $X_a$, so WLOG, suppose $X = X_s$. Then your penultimate sentence applies.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1766203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is $\langle A,B\rangle =\operatorname{trace}(AB^T)$ an inner product in $\mathbb R^{n\times m}$? I don't understand why one should take transpose of $\operatorname{tr}(AB^T)$ and why we use the fact that $\operatorname{tr}(M)=\operatorname{tr}(M^T)$ for any $M$ that is a square matrix to solve the problem.
| If you grind through the details, you will see that
$\operatorname{tr} (A B^T) = \sum_{i,j} [A]_{ij} [B_{ij}]$, hence
this is the 'standard' inner product if you view the matrices
$A,B$ as giant columns.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1766303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find how many solutions the congruence $x^2 \equiv 121 \mod 1800$ has I want to find how many solutions the congruence $x^2 \equiv 121 \mod 1800$ has.
What is the method to find it without calculating all the solutions?
I can't use euler criterion here because 1800 is not a primitive root. thanks!!!
| Here is a different approach to find the number of solutions to the congruence
$$ x^2 \equiv a \space(\bmod n)$$
If
$$ n = 2^kp_1^{k_1}\cdots p_r^{k_r}$$
where $p_1, \dots ,p_r$ are odd different primes, $k \ge 0$ and $k_1,\dots,k_r \ge 1$.
If the congurence has solutions, the number of the solutions will be equal to $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1766505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Show that $3P_{\lceil n \rceil}-2=\sum_{k=1}^{A}\left(4-\left\lceil \frac{\pi(k)}{n}\right\rceil^2\right) $ We proposed a formula for calculating nth prime number using the prime counting function.
Where $\lfloor x\rfloor$ is the floor function and $\lceil x\rceil$ is a ceiling function.
$\pi(k)$ is prime counting func... | Dusart showed that $\pi(n) \ge n (\log n + \log \log n - 1)$ for $n\ge 2$. From it's not too hard to calculate that for any $n\ge 2$,
$$\pi(2n) \ge 2n (\log 2n + \log \log 2n - 1) \ge 2n( \log n + \log \log 2n - 0.307) > A.$$
In particular (ignoring very small values of $n$), for all values of $k$ in the sum, $\pi(k) ... | {
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"timestamp": "2023-03-29T00:00:00",
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Sign of composition of transpositions Let $\sigma \in S_n$.
Definition: Suppose that $\text{sign}\sigma=(-1)^N$, where $N$ - number of inversions in permutation $\sigma$.
Suppose that $\tau_1$ and $\tau_2$ transpositions. How to prove that $\text {sign}(\tau_1\circ \tau_2)=\text {sign}\tau_1\cdot \text {sign}\tau_2?$
I... | During a long drive this evening I realized my other answer, involving determinants, was just a long trip around Robin Hood's Barn. Here's the short proof.
Suppose that
$\tau_1$ is a composition of $k$ transpositions $X_1, \ldots, X_k$,
$$
\tau_1 = X_k X_{k-1} \cdots X_2 X_1,
$$
and that
$\tau_2$ is a composition of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1766735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to calculate the radius of a circle inscribed in a regular hexagon? If I know how long one side of a regular hexagon is, what's the formula to calculate the radius of a circle inscribed inside it?
Illustration:
| Draw the six isosceles triangles.
Divide each of these triangles into two right angled triangles.
Then you have
$s = 2x = 2 (r \sin \theta)$
where $r$ is the radius of the circle, $\theta$ is the top angle in the right angled triangles and there are in total $12$ of these triangles so its easy to figure out $\theta$. $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1766870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Which are the connected components of $K=\{1/n\mid n\in\mathbb{N}\}\cup\{0\}$? Which are the connected components of the topological subspace $K=\left\{\left.\dfrac{1}{n}\ \right|\ n\in\mathbb{N}\right\}\cup\{0\}$ of $\mathbb{R}$?
I think they are every single point, but I can't find open set of $\mathbb{R}$ separating... | You cannot find such an open set. What you need to check is that all points of the form $\frac{1}{n}$ are isolated points.
Any subset $C$ with more than 1 point and at least one isolated point $p$ is disconnected (use $\{p\}$ and its complement in $C$, both of which are open and closed in $C$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1767072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Prove that the sum of the squares of two odd integers cannot be the square of an integer. Prove that the sum of the squares of two odd integers cannot be the square of an integer.
My method:
Assume to the contrary that the sum of the squares of two odd integers can be the square of an integer. Suppose that $x, y, z \in... | Let $a=2n+1$, $b=2m+1$. Then $a^2 + b^2=4n^2 + 4n +4m^2 +4m+2$. This is divisible by $2$, a prime number, but not by $4=2^2$. Hence it cannot be the square of an integer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1767200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Find all prime p such that Legendre symbol of $\left(\frac{10}{p}\right)$ =1 In the given question I have been able to break down $\left(\frac{10}{p}\right)$=
$\left(\frac{5}{p}\right)$ $\left(\frac{2}{p}\right)$. But what needs to be done further to obtain the answer.
| Hint:
By the second supplementary law of quadratic reciprocity,$\biggl(\dfrac 2p\biggr)=1$ if and only if $p\equiv \pm 1\mod 8$.
On the other hand, $\biggl(\dfrac 5p\biggr)=\biggl(\dfrac p5\biggr)=1\;$ if and only if $p\equiv \pm 1\mod 5$.
So you have to solve the systems of congruences:
\begin{align*}
\begin{cases}
p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1767306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Help solving the inequality $2^n \leq (n+1)!$, n is integer I need help solving the following inequality I encountered in the middle of a proof in my calculus I textbook:
$2^n \leq (n+1)!$
Where $\mathbf{n}$ in an integer.
I've tried applying lg to both members, but got stuck at:
$n \leq \lg(n+1) + \lg(n) + \lg(n-1) + ... | $(n+1)!=2\cdot 3\cdot \dots\cdot(n+1)$ here a product of $n$ numbers all are at least 2 so the result follows...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1767427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Problem based on area of triangle In the figure, E,C and F are the mid points of AB, BD and ED respectively. Prove that: $8\triangle CEF=\triangle ABD$
From the given,
$ED$ is the median of $\triangle ABD$
So, $\triangle AED=\triangle BED$
Also, by mid point Theorem
$EC||AD$ and $CF||AB$.
Now, what should I do next?
| Well $\triangle$AED=$\frac{1}{2}\triangle ABD$.
So the problem reduces to showing that
$$\triangle CEF = \frac{1}{4} \triangle BED$$
Since F is the midpoint of ED, we have by similarity that:
$$|EB| = 2|FB|$$
in other words:
$$\triangle CFD = \frac{1}{4} \triangle BED$$
Likewise, also by similarity,
$$\triangle BEC = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1767501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Multiplicative inverse of $x+f(x)$ in $\Bbb Q[x]/(f(x))$ So I have $f(x) = x^3-2$ and I have to find the multiplicative inverse of $x + f(x)$ in $\mathbb{Q}[x]/(f(x))$. I'm slightly confused as to how to represent $x + (f(x))$ in $\mathbb{Q}[x]/(f(x))$. Would I be just finding the inverse of $x+1$? How would I do that?... | $1=(x+1)(\frac{x^2-x+1}{3})-\frac{x^3-2}{3}$, then $\frac{x^2-x+1}{3}$ is the multiplicative inverse of $x+1+\langle f\rangle$ in $\frac{\mathbb{Q}[x]}{\langle f\rangle}.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1767591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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A different notion of convergence for this sequence? I was thinking about sequences, and my mind came to one defined like this:
-1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, ...
Where the first term is -1, and after the nth occurrence of -1 in the sequence, the next n terms of the sequence are 1, followed by -1, and so... | As André Nicolas wrote,
the Cesaro mean,
for which the $n$-th term
is the average of the
first $n$ terms
will do what you want.
In both your cases,
for large $n$,
if $(a_n)$ is your sequence,
if
$b_n = \frac1{n}\sum_{k=1}^n a_k
$,
then
$b_n \to 1$
since the number of $-1$'s
gets arbitrarily small
compared to the numb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1767682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Determine the derivative $\frac{dy}{dx}$ of the integral
Determine the derivative of the integral $$ \,\int_{\sqrt x}^{0}\sin (t^2)dt $$
What does this question mean.
I do not understand it and I think you can't integrate $\sin t^2\,$.
| You can take the derivative of an integral, even though you can't directly integrate it.
let
$I = \int \sin t^2 dt$
The value that you care about is $$R = I(0) - I(\sqrt x)$$
since we are asked to evaluate the integral from $\sqrt x$ to $0$.
Now, differentiating $R$ by $dx$, we get the expression
$$\frac{dR}{dx} = \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1767804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Legendre polynomial with argument zero $P_n ( 0 )$ I want to find the expression for $P_n(x)$ with $x = 0$, ie $P_n(0)$ for any $n$. The first few non-zero legendre polynomials with $x=0$ are
$P_0(0) = 1$, $P_2(0) = -\frac{1}{2}$, $P_4(0) = \frac{3}{8}$, $P_6(0) = -\frac{5}{16}$, $P_8(0) = \frac{35}{128}$ but I can't ... | One way to define Legendre polynomials is through its generating function:
$$\frac{1}{\sqrt{1-2xt+t^2}} = \sum_{n=0}^\infty P_n(x)t^n$$
Together with following sort of well known expansion:
$\displaystyle\;\frac{1}{\sqrt{1-4z}} = \sum_{k=0}^\infty \binom{2k}{k} z^k$,
we have
$$\sum_{n=0}^\infty P_n(0) t^n = \frac{1}{\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1767969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Does every invertible matrix A has a matrix B such that A=Adj(B)? I'm trying to understand if it's always true, always true over $\mathbb C$ or never true.
I know that if $A$ is invertible, than there exists $A^{-1}$.
$$A=\frac{1}{det (A^{-1})}Adj(A^{-1})$$
So I have an adjoint matrix multiplied by a scalar, but how do... | There is no solution over $\mathbb R$ if $n \ge 3$ is odd and $\det(A) < 0$.
$\det(\text{adj}(B))= \det(\det(B) B^{-1}) = \det(B)^{n-1}$, which can't be negative in this case.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1768076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to show $\frac{19}{7}How can I show $\dfrac{19}{7}<e$ without using a calculator and without knowing any digits of $e$?
Using a calculator, it is easy to see that $\frac{19}{7}=2.7142857...$ and $e=2.71828...$
However, how could this be shown in a testing environment where one does not have access to a calculator?... | $$ \int_{0}^{1} x^2 (1-x)^2 e^{-x}\,dx = 14-\frac{38}{e},$$
but the LHS is the integral of a positive function on $(0,1)$.
Another chance is given by exploiting the great regularity of the continued fraction of $\coth(1)$:
$$\coth(1)=[1;3,5,7,9,11,13,\ldots] =\frac{e^2+1}{e^2-1}$$
gives the stronger inequality $e>\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1768195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 5,
"answer_id": 1
} |
Evaluating integral with $e^{\sin x}$ I had this integral $ \int e^{\sin(x)} {\sin(2x)} dx$
I tried to split it up using integration by parts but I can't evaluate integral of $e^{\sin x}$
| By parts works perfectly,
$$\int2\sin(x)\left(\cos(x)e^{\sin(x)}\right)dx=2\sin(x)e^{\sin(x)}-2\int\cos(x)e^{\sin(x)}dx\\
=2\sin(x)e^{\sin(x)}-2e^{\sin(x)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1768333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
If $A,B,C$ are $3\times 3$ matrices such that $A,(A-B)$ are invertible and if $(A-B)C=BA^{-1}$, show that $C(A-B)=A^{-1}B$.
If $A,B,C$ are $3\times 3$ matrices such that $A,(A-B)$ are invertible and if $(A-B)C=BA^{-1}$, show that $C(A-B)=A^{-1}B$.
Usually, $AB$ may not be equal to $BA$.
I tried starting from the ans... | If you have $AB=I$, then it follows $ABA=A$ , and therefore $BA=I$.
So, a matrix always commutes with its inverse. This is exactly what you need to prove your claim.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1768412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Using Euclid's algorithm to find Multiplicative Inverse 71 mod 53 I begin by writing out the recursion until a mod b == 0
53 -> 71-> 53-> 18-> 17 ->1 -> 0
to get in the form $sa+tn$
starting with $1 = 18-17$ I then substitute $17 = 53-(18\cdot2)$
this gives me $18\cdot3-53$
I then substitute $18 = (71-53)$ which gives... | You have done almost all the work yourself. You just need to interpret what you already have.
Your arrangement in the second last line gives you $71\cdot3-53\cdot4=1$ which on rearrangement is $71\cdot3=53\cdot4 + 1$ which exactly implies by modular property that $3\cdot71=1 \pmod{53}$ i.e. in modulo group $\mathbb{Z}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1768773",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Show that $\lim_{x \to 0}\sqrt{1-x^2} = 1$ with the help of the definition of a limit The original problem is to calculate $\lim_{x \to 0}\dfrac{1-\sqrt{1-x^2}}{x}$
I simplified the expression to $\lim_{x\to 0}\dfrac{x}{1 + \sqrt{1-x^2}}$
The only definitions and theorems I can use are the definition of a limit and the... | Let $f(x)$ be our function. We want to show that for any given $\epsilon\gt 0$, there is a $\delta$ such that if $0\lt |x-0|\lt\delta$, then $|f(x)-0|\lt \epsilon$.
Note that $1+\sqrt{1-x^2}\ge 1$, at least when $|x|\le 1$. (When $|x|\gt 1$, it is not defined.) It follows that for such $x$ we have
$$\left|\frac{x}{1+\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1768873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to prove that $\lim_{x \to 0^{+}} \frac{e^{-1/x}}{x} = 0$ Today someone asked me how to calculate $\lim_{x \to 0^{+}} \frac{e^{-1/x}}{x}$. At first sight that limit is $0$, because the exponential decreases faster than the lineal term in the denominator. However, I didn't know how to prove it formally.
I thought of... | Putting $t = 1/x$ we see that $t \to \infty$ as $x \to 0^{+}$. Also the function is transformed into $t/e^{t}$. Next we put $e^{t} = y$ so that $y \to \infty$ as $t \to \infty$. Thus the function is transformed into $(\log y)/y$. Since $y \to \infty$ we can assume $y > 1$ so that $\sqrt{y} > 1$. We have the inequality ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1769000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 5
} |
Prove that $\mathscr{F}[f] \in L^2(\mathbb{R})$ Let $f \in L^2(\mathbb{R})$ (square integrable functions), I'm trying to prove that his Fourier transform also does: $\mathscr{F}[f] \in L^2(\mathbb{R})$.
I have tried to bound it
\begin{align}
\int_{-\infty}^{+\infty}|\ \hat{f}(\omega)\ |^2 d\omega &= \int_{-\infty}^{+\i... | Let $f$ be absolutely and absolutely square integrable on $\mathbb{R}$. Then,
$$
\overline{\hat{f}(s)}=\overline{\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)e^{-ist}dt}=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\overline{f(t)}e^{ist}dt = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\overline{f(-t')}e^{-ist... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1769101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is the sum of all the Fibonacci numbers from 1 to infinity. Today I believe I had found the sum of all the Fibonacci numbers are from $1$ to infinity, meaning I had found $F$ for the equation
$F = 1+1+2+3+5+8+13+21+\cdots$
I believe the answer is $-3$, however, I don't know if I am correct or not. So I'm asking if... | None of these answers adhere to analytic continuation, which is clearly what you are looking for. A fairly non-rigorous method for doing this is to use the generating function for the Fibonacci series, namely
$$\frac{1}{1-x-x^2} = 1 + 1x + 2x^2 + 3x^3 + 5x^4 + 8x^5 +\cdots$$
Clearly we get the answer $-1$ if we plug th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1769145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 4
} |
Trying to show that $\ln(x) = \lim_{n\to\infty} n(x^{1/n} -1)$ How do I show that $\ln(x) = \lim_{n\to\infty} n (x^{1/n} - 1)$?
I ran into this identity on this stackoverflow question. I haven't been able to find any proof online and my efforts to get from $\ln(x) := \int_1^x \frac{\mathrm dt}t$ to that limit have been... | You can even do a bit more using Taylor series $$x^{\frac 1n}=e^{\frac 1 n \log(x)}=1+\frac{\log (x)}{n}+\frac{\log ^2(x)}{2
n^2}+O\left(\frac{1}{n^3}\right)$$ which makes $$n(x^{\frac 1n} -1)=\log (x)+\frac{\log ^2(x)}{2 n}+O\left(\frac{1}{n^2}\right)$$ which shows the limit and also how it is approached.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1769256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 8,
"answer_id": 2
} |
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