Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Apollonian gasket Okay , is there a way to find the radius of the nth circle in a apollonian gasket ..
Something like this
Its like simple case of apollonian gasket ..
I found from descartes' theorem
$R_n = 2\cdot\sqrt{ R_{n-1}\cdot R_1 + R_n-1\cdot R_2 + R_1\cdot R_2} +R_{n-1} + R_1 +R_2 $
I have the values of $R_1,... | Sure, there is a way.
To simplify the formula, I will relabel your circles as follows.
*
*Let $S_a$ and $S_b$ be your circle $C_1$ and $C_2$.
*Let $S_0, S_1, S_2, \ldots$ be your circle $C_3, C_4, C_5, \ldots$.
*Given $C_1, C_2, C_3$, there are two possible configurations of $C_4$.
Let $S_{-1}$ be the other ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1349032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find radius of Circle There is a circle C1 of Radius R1 and another circle C2 of radius R2 (R2 ≤ R1) such that it touches circle C1 internally
There is another circle C3 with radius R3 such that it touches the circle C1 from inside and the circle C2 from outside (it is given that R3 + R2 ≤ R1)
Now there is a circle C... | You have just to apply Descartes' theorem.
Assuming that $R_i$ is the radius of $C_i$ and $\kappa_i=\frac{1}{R_i}$, $\kappa_4$ and $\kappa_5$ are given by:
$$ \kappa_1+\kappa_2+\kappa_3\pm 2\sqrt{\kappa_1\kappa_2+\kappa_1\kappa_3+\kappa_2\kappa_3},$$
so:
$$ \kappa_4+\kappa_5 = 2(\kappa_1+\kappa_2+\kappa_3). $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1349108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How do I prove that for any two points in $\mathbb{C}$, there exists a $C^1$-curve adjoining them? Let $G$ be an open-connected subset of $\mathbb{C}$.
Let $a,b$ be two distinct points in $G$.
How do I prove that there exists a $C^1$-curve $\alpha:[0,1]\rightarrow G$ such that $\alpha(0)=a$ and $\alpha(1)=b$?
Here's ho... | Another approach: A corollary of the Weierstrass approximation theorem says that if $f:[0,1]\to \mathbb {R}$ is continuous, then there is a sequence of polynomials $p_n$ converging uniformly to $f$ on $[0,1]$ such that $p_n(0) = f(0),p_n(1) = f(1)$ for all $n.$
Now let $\alpha :[0,1]\to G$ be a continuous map (like you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1349199",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Does the equation $2\cos^2 (x/2) \sin^2 (x/2) = x^2+\frac{1}{x^2}$ have real solution?
Do the equation
$$2\cos^2 (x/2) \sin^2 (x/2) = x^2+\frac{1}{x^2}$$
have any real solutions?
Please help. This is an IITJEE question.
Here $x$ is an acute angle.
I cannot even start to attempt this question. I cannot understand.... | Observe $x^2+\frac{1}{x^2} \geq 2$, and simplify left hand side using $\sin(2\theta)=2\sin(\theta)\cos(\theta)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1349276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 1
} |
For what values of $a$ does $\int_{0}^{1}(-\ln x)^adx$ converge? For what values of $a$ does $\int_{0}^{1}(-\ln x)^adx$ converge? I have seen a duplicate of this question but the answer there, though very good and creative, isn't clear about negative values. When $a=0$ it is trivial. I actually did arrive at something ... | By replacing $x$ with $e^{-t}$ we have:
$$ \int_{0}^{1}(-\log x)^{\alpha}\,dx = \int_{0}^{+\infty}t^{\alpha}e^{-t}\,dt = \Gamma(\alpha+1) $$
so the integral is converging for any $\alpha>-1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1349382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculating $\sum_{k=0}^{n}\sin(k\theta)$ I'm given the task of calculating the sum $\sum_{i=0}^{n}\sin(i\theta)$.
So far, I've tried converting each $\sin(i\theta)$ in the sum into its taylor series form to get:
$\sin(\theta)=\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\frac{\theta^7}{7!}...$
$\sin(2\theta)=2\theta... | If you know complex variables,
$$
e^{ik\theta}=\cos(k\theta)+i\sin(k\theta).
$$
Then
$$
\sum_{k=0}^n\sin(k\theta)=\Im\left(\sum_{k=0}^ne^{ik\theta}\right)=\Im\left(\sum_{k=0}^n\left(e^{i\theta}\right)^k\right).
$$
This, however, is a geometric series (provided $\theta$ is not a multiple of $2\pi$), so we know its for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1349466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Probability that an integer is divisible by $8$ If $n$ is an integer from $1$ to $96$ (inclusive), what is the probability that $n(n+1)(n+2)$ is divisible by 8?
| Hint: the product is divisible by $8$ if:
*
*One factor is divisible by 8, or
*One factor is divisible by 4 and another is divisible by 2, or
*All three factors are divisible by 2.
The last one is impossible in your situation (why?) What are the possible remainders when $n$ is divided by $8$ such that the first ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1349566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How do i evaluate this integral $ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $? Is there some one show me how do i evaluate this integral :$$ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $$
Note :By mathematica,the result is :
$\frac{Gamma\left(\frac1 4\right)Gamma\left(\frac5 4\right)}{\sqrt{\pi}}-\s... | $$
\begin{align}
\int_{\pi/4}^{\pi/2}\frac{\sqrt{\tan(x)}}{\sin(x)}\,\mathrm{d}x
&=\int_{\pi/4}^{\pi/2}\frac{\sqrt{\tan(x)}}{\sin(x)}\frac{\mathrm{d}\tan(x)}{\sec^2(x)}\tag{1}\\
&=\int_1^\infty\frac{\sqrt{u}}{\frac{u}{\sqrt{1+u^2}}}\frac{\mathrm{d}u}{1+u^2}\tag{2}\\
&=\frac12\int_0^\infty\frac{\mathrm{d}u}{\sqrt{u(1+u^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1349654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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Taking limits on each term in inequality invalid? So this inequality came up in a proof I was going through.
$$c - 1/n < f(x_n) \leq c$$
Where $c$ is a real number, $f(x_n)$ is the image sequence of some arbitrary sequence being passed through a function and $n$ is a natural number. At this point the author simply conc... | Its a pretty standard theorem which was originally called "Sandwich Theorem" but nowadays more popularly known as "Squeeze Theorem".
Squeeze Theorem: If $\{a_{n}\}, \{b_{n}\}, \{c_{n}\}$ are sequences such that $a_{n} \leq b_{n} \leq c_{n}$ for all $n$ greater than some specific positive integer $N$ and $\lim_{n \to \i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1349773",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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"answer_id": 3
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Confused by certain interpretation of expected value... I read the following in Stein / Shakarchi's Fourier Analysis book, where they discussed the notion of expectation of a probility density.
"Consider the simpler (idealized) situation where we are given that the particle can be found at only finitely many different ... | The whole purpose of calculating the arithmetic mean of a distribution is to minimize the expectation of the square of the error. It all makes sense if you are penalized one dollar per square of the error. So if you know that tossing 3 fair coins gives you the usual distribution of outcomes, your best guess for the num... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1349845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Fermat's little theorem This is a very interesting word problem that I came across in an old textbook of mine. So I mused over this problem for a while and tried to look at the different ways to approach it but unfortunately I was confused by the problem and I don't really understand how to do it either, hence I am una... | In problem (a), use Fermat's little theorem, which says (or a rather, a very slightly different version says) that for any prime number $p$, and any integer $n$ that's not divisible by $p$, we have
$$n^{p-1}\equiv 1\bmod p$$
In particular, use $n=2$ and $p=17$. Keep in mind that $2017=(126\times 16)+1$.
In problem (b),... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1349923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Two questions on the Grothendieck ring of varieties 1) In the definition of the Grothendieck ring of varieties over a field $k$, which definition of the various notions of "variety" is chosen? Finite type and separated, or maybe more?
2) If $\mathbb{L}$ is the class of the affine line in the Grothendieck ring, does the... | From what I've read, it seems like separated and finite type are fairly standard. Poonen, in his paper "The Grothendieck Ring of Varieties is not a Domain" adds geometrically reduced- I haven't seen this other places, but I haven't looked too hard.
For the second, there does exist a fibration relation- seeing it is dif... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1350018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Necessary condition for local maximum Let $\Omega\subset \mathbb{R}^n$ open, bounded and let $f:\Omega\to\mathbb{R}$ be a $C^2$-function.
I want to prove: Necessary for a interior maximum $x_0\in\Omega$ is that $D^2f(x_0)$ is negative semidefinite.
I'm stuck, I want to know how to finish my proof.
First case: Suppose ... | It might be that there is no contradiction in the third case.
Namely, it can happen that $D^2f(x_0)$ is zero and therefore positive semidefinite and positive semidefinite.
You don't really have to work by cases.
A matrix $A$ being negative semidefinite means that for all $v$ you have $v^TAv\leq0$.
If this is not the ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1350106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Find the dimension of a vector subspace I'm doing a problem on finding the dimension of a linear subspace, more specifically
if $\:$ {$f \in \mathcal P_n(\mathbf F): f(1)=0, f'(2)=0$} is a subspace of $P_n$, what is this dimension of this subspace? Here $\mathcal P_n(\mathbf F)$ denotes a vector space of Polynomials o... | Intuitively, dimension is the number of degrees of freedom. The elements of $\mathcal{P}_n(\mathbb R)$ are polynomials of degree $n$ (more precisely, at most $n$), so they look like
$$ a_0 + a_1 x + a_2 x^2 + \dotsb + a_{n-1} x^{n-1} + a_n x^n $$
To specify such a polynomial, you have to specify $n+1$ numbers, the coe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1350215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
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How can $f(x)=x^4$ have a global minimum at $x=0$ but $f''(0)=0$? $f(x)=x^4$ has a global minimum in $\Bbb R$ at the point $x=0$, but $f''(0)=0$.
This case confuses me. For every $0\neq x\in I$, $f(x)>f(0)$. So how can it be that $f''(0)=0$, following $f'(x)$ doesn't change its sign at $x=0$?
I could accept it if there... | $f'(x)$ does change it sign at $0$ though. Calculate $f'(x)$ and try point either side of $0$ and you will see this.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1350296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that in any triangle, we have $\frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right),$ Show that in any triangle, we have $$\frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right),$$
where $R$ is the circumradius of the triangle.
Here is... | Since the sine theorem implies:
$$\sum_\text{cyc}a\sin A = \frac{1}{2R}\sum_\text{cyc}a^2 \tag{1}$$
we just need to prove:
$$ \sum_\text{cyc} a \cos A = \frac{abc}{2R^2}=\frac{2\Delta}{R}\tag 2$$
that is trivial since twice the (signed) area of the triangle made by $B,C$ and the circumcenter $O$ is exactly $aR\cos A$:
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1350412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Prove that the graph dual to Eulerian planar graph is bipartite. How would I go about doing this proof I am not very knowledgeable about graph theory I know the definitions of planar and bipartite and dual but how do you make these connection
| So $G$ is planar and eulerian. We must prove $G'$ is bipartite. Asume $G'$ is not bipartite. Now I want you to forget about the fact that $G'$ is the dual of $G$. Just think of $G'$ as a normal graph in which the vertices of $G'$ are drawn as vertices and not as the faces of $G$.
Since $G'$ is not bipartite it has an o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1350552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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When do I use a z-score vs a t-score for confidence intervals? I have a set of 1000 data points. I would like to estimate their mean using a confidence interval. I read somewhere that if the sample size, $n$, is bigger than 30 you should use a t-score, and else use a z-score.
Is that true?
| If you don't know the variance of the population, then you should formally always use the $t$-distribution. If you do know the population variance, you can use the standard normal distribution.
However, as $n \to \infty$, the $t$-distribution becomes the same as the standard normal distribution. Even for relatively sma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1350635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
If a unit ball is compact then why a ball of radius 5 has to be compact too? So if I use the definition of compactness that every open cover has a finite sub-cover, then as the unit ball is compact , there exists a finite subcover. But if I increase the radius of the ball, why does it still need to be compact. Intuitiv... | Strictly positive scaling is a homeomorphism on closed balls. So either none is compact or all.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1350679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
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Latin square property sufficient? So I know that for any group table, Every row must contain distinct group elements and the same holds for every column for a group table. And this property is called the Latin Square property. However, Every time I read a book about Abstract Algebra, They say that the latin square prop... | Taken from http://science.kennesaw.edu/~sellerme/sfehtml/classes/math4361/chapter4section1outline.pdf
Let $G=\{1,2,3,4,5\}$ with multiplication table
\begin{array}{|c||c|c|c|c|c|}
\hline
*&1&2&3&4&5\\
\hline
\hline
1&1&2&3&4&5\\
\hline
2&2&1&4&5&3\\
\hline
3&3&4&5&2&1\\
\hline
4&4&5&1&3&2\\
\hline
5&5&3&2&1&4\\
\hline... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1350756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Interchanging sum and differentiation, almost everywhere Let $\{F_i\}$ be a sequence of nonnegative increasing real functions on $[a,b]$ with $a<b$ such that $F(x):=\displaystyle\sum_{i=1}^\infty F_i(x)<\infty$ for all $x\in [a,b],$ then show $F'(x)=\displaystyle\sum_{i=1}^\infty F'_i(x)$ a.e. on $[a,b]$.
At first note... | Consider the derivarive of $\mu_{F_i}$, $\mu_{F_i}[a,x] = \int_a^x F'_i(s)\, ds$.
Now note that $F'_i \geq 0$.
Compute define $G_N(x):=\left(\sum_{i=1}^N F_i(x)\right)$
$$G_N'(x) = \sum_{i=1}^N F'_i(x) = g_N(x) $$
Note that $g_N(x) \uparrow g_\infty(x)$ as $N \to \infty$.
This implies that $F'(x)= g_\infty(x)= \sum_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1350827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Square roots equations I had to solve this problem:
$$\sqrt{x} + \sqrt{x-36} = 2$$
So I rearranged the equation this way:
$$\sqrt{x-36} = 2 - \sqrt{x}$$
Then I squared both sides to get:
$$x-36 = 4 - 4\sqrt{x} + x$$
Then I did my simple algebra:
$$4\sqrt{x} = 40$$
$$\sqrt{x} = 10$$
$$x = 100$$
The problem is that when... | Method $\#1:$
As for real $a,\sqrt a\ge0\ \ \ \ (1)$
$(\sqrt x+\sqrt{36-x})^2=36+2\sqrt{x(36-x)}\ge36$
$\implies\sqrt x+\sqrt{36-x}\ge6\ \ \ \ (2)$ or $\sqrt x+\sqrt{36-x}\le-6\ \ \ \ (3)$
Finally $(1)$ nullifies $(3)$
Method $\#2:$
WLOG let $\sqrt x=6\csc2y$ where $0<2y\le\dfrac\pi2\implies\sqrt{x-36}=+6\cot2y$
$\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1350897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Should I use the comparison test for the following series? Given the following series
$$\sum_{k=0}^\infty \frac{\sin 2k}{1+2^k}$$
I'm supposed to determine whether it converges or diverges. Am I supposed to use the comparison test for this? My guess would be to compare it to $\frac{1}{2^k}$ and since that is a geometr... | You have the right idea, but you need to do a little more, since some of the terms are negative. Use your idea and the fact that $|\sin x|\le 1$ for all $x$ to show that
$$\sum_{k\ge 0}\frac{\sin 2k}{1+2^k}$$
is absolutely convergent, i.e., that
$$\sum_{k\ge 0}\left|\frac{\sin 2k}{1+2^k}\right|$$
converges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1350969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Finding a Lyapunov Function for a system involving a trigonometric function I'm dealing with determining if $(0,0)$ is stable or not for the following system via constructing a Lyapunov function. The system is
$$ \begin{cases}
x'(t)=(1-x)y+x^2\sin{(x)}& \\
y'(t)=-(1-x)x+y^2\sin{(y)}&
\end{cases}
$$
My... | I just want to point out that you can not use Lyapunov's second methods (Lyapunov functions) to show instability. The notion of Lyapunov's second method is not strong enough to do so. You can ONLY SHOW STABILITY.
In a less ambiguous way: if you can show stability its fine and you are save that the system is stable. If... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1351028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Understanding the proof that $\lim_{n\to\infty}x^{1/n}=1$ ($x>0$) in Tao's Analysis I was reading sequences in Terence Tao Analysis book and I came across the question:
Prove that $\forall x>0$, $\lim\limits_{n\to\infty}x^{1/n}=1$
In the hint it says that
... you may need to treat the cases $x\geq1$ and $x<1$ separa... | If $0 < x < 1$, then put $y = \dfrac{1}{x}$, we can consider only the case $x > 1$. By AM-GM inequality: $1 \leq \sqrt[n]{x}=\sqrt[n]{1\cdot 1\cdot 1\cdots \sqrt{x}\cdot\sqrt{x}}\leq \dfrac{(n-2)+2\sqrt{x}}{n}=1+\dfrac{2(\sqrt{x}-1)}{n}$.Next apply squeeze theorem to conclude.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1351067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
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Why does $\sum a_i \exp(b_i)$ always have root? Let $z$ be complex.
Let $a_i,b_i$ be polynomials of $z$ with real coefficients.
Also the $a_i$ are non-zero and the non-constant parts of the polynomials $b_i$ are distinct.
Let $j > 1$.
$$f(z) = \sum_{i=1}^j a_i \exp(b_i)$$
It seems there always exists a complex value $s... | This follows from the theory of entire functions of finite order
in complex analysis. Specifically, we have:
Proposition:
Suppose $f(z) = \sum_{i=1}^j a_i \exp b_i$ for some polynomials $a_i,b_i$
(which may have complex coefficients, though the question specifies
real polynomials). Then if $f\,$ has no complex zeros ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1351157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Projections: Ordering Given a unital C*-algebra $1\in\mathcal{A}$.
Consider projections:
$$P^2=P=P^*\quad P'^2=P'=P'^*$$
Order them by:
$$P\leq P':\iff\sigma(\Delta P)\geq0\quad(\Delta P:=P'-P)$$
Then equivalently:
$$P\leq P'\iff P=PP'=P'P\iff\Delta P^2=\Delta P=\Delta P^*$$
How can I check this?
(Operator algebrai... | The assertion $P\leq Q$ means $Q-P\geq0$. Then
$$
0\leq P(Q-P)P=PQP-P\leq P^2-P=P-P=0.
$$
So $P=PQP$. Now we can write this equality as $$0=P-PQP=P(I-Q)P=[(I-Q)P]^*[(I-Q)P],$$
so $(I-Q)P=0$, i.e. $P=QP$. Taking adjoints, $P=PQ$.
The converse also holds: if $P=PQ=QP $, then $$Q-P=Q^2-QPQ=Q (I-P)Q\geq0. $$
Edit: for the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1351252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Product of cosines: $ \prod_{r=1}^{7} \cos \left(\frac{r\pi}{15}\right) $
Evaluate
$$ \prod_{r=1}^{7} \cos \left({\dfrac{r\pi}{15}}\right) $$
I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ \cos(A+B)+\cos(A-B)] $$
but I couldn't find the product.
Any help w... | Since an elegant solution has already been provided, I will go for an overkill.
From the Fourier cosine series of $\log\cos x$ we have:
$$ \log\cos x = -\log 2-\sum_{n\geq 1}^{+\infty}\frac{(-1)^n\cos(2n x)}{n}\tag{1} $$
but for any $n\geq 1$ we have:
$$ 15\nmid n\rightarrow\sum_{k=1}^{7}\cos\left(\frac{2n k \pi}{15}\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1351337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "36",
"answer_count": 5,
"answer_id": 2
} |
this inequality $\prod_{cyc} (x^2+x+1)\ge 9\sum_{cyc} xy$ Let $x,y,z\in R$,and $x+y+z=3$
show that:
$$(x^2+x+1)(y^2+y+1)(z^2+z+1)\ge 9(xy+yz+xz)$$
Things I have tried so far:$$9(xy+yz+xz)\le 3(x+y+z)^2=27$$
so it suffices to prove that
$$(x^2+x+1)(y^2+y+1)(z^2+z+1)\ge 27$$
then the problem is solved. I stuck in here
| with loss of generality, assume $x\ge y $
consider $f(x, y, z) = (x^2 + x + 1)(y^2 + y + 1) (z^2 + z + 1) - 9 (z(x+y) + xy)$
*
*we first show that
$$f(x, y, z) \ge f((x+y)/2, (x+y)/2, z)$$
since
$$f(x,y,z) - f((x+y)/2, (x+y)/2, z) = (x-y)((z^2 + z + 1)(z -2 - 2xy) + 9)$$
we simply show that
$$(z^2 + z + 1)(z -2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1351422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Show $ \lim_{n\rightarrow \infty} 2^{-1/\sqrt{n}}=1$ I am tasked with proving the following limit:
$$ \lim_{n\rightarrow \infty} 2^{-1/\sqrt{n}}=1$$
using the definition of the limit. I think I have done so correctly. I was hoping to have someone confirm my proof. Here is my reasoning:
We want
$$ \left| 2^{-1/\sqrt{n}... | The minus sign is not essential, we can remove it by inversion. We can assume $n$ to be a perfect square (as $2^{1/\sqrt n}<2^{1/\lfloor\sqrt n\rfloor}$) and study
$$\lim_{n\to\infty}2^{1/n}.$$
Then,
$$\frac{2^{1/n}-1}{2-1}=\frac{2^{1/n}-1}{(2^{1/n})^n-1}=\frac1{(2^{1/n})^{n-1}+(2^{1/n})^{n-2}+\cdots1}<\frac1n.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1351488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Maximum determinant of $3 \times 3$ matrix Good one guys!
I'm studying to the maths olympiads in my college and I ran to the following problem:
What is the possible matrix $3 \times 3$, that you can write using digits from $0 $ to $9$, (you can repeat them), that gives you the maximum determinant?
I got by brute force... | More generally, consider the $n \times n$ case.
Since the determinant is a linear function of the entries in any given row or column, it's clear that there is an optimal solution with all entries $0$ or $9$. Dividing by $9$, you have an $n \times n$ $0-1$ matrix with maximum determinant, and that corresponds to a norm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1351586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Analytic Functions on Upper Half Plane Satisfying Inequality
Original Problem. Let $\mathbb{H}=\left\{z\in\mathbb{C} : \Im(z)>0\right\}$ be the open upper half plane. Determine all analytic functions $f:\mathbb{H}\rightarrow\mathbb{C}$ that satisfy the inequality
$$\left|f(z)\right|\leq\Im(z) \tag{1}$$
This is an old... | Your solution is fine. (Answered to keep the question off the unanswered list.)
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do I solve $2^x + x = n$ equation for $x$? I need to solve the equation $$2^x + x = n$$ for $x$ through a programming-based method. Is this possible? If not, then what would be the most efficient way to approximate it?
| You could either use numerical methods such as interval bisection, Newtons Method of finding roots applied to $f(x) = 2^x + x - n$, as in Winther's solution.
Secondly, you could solve this equation manually to get the solutions as $x=0$ for $n=1$ and $$x=n-\frac{W_{c}\left(2^n \ln 2\right)}{\ln 2}$$ where we let $c$ ra... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding an explicit formula from a recursive formula. I have the recurrence relation:
$$g(k, 0, x) = k,$$
$$g(k, n, x) = \dfrac{1}{2} \log_{k}{\left(\dfrac{k^{g(k, n - 1, x)}x}{g(k, n - 1, x)}\right)},$$
and I would like to solve it, if it is possible.
By the way, $\lim_{n \to \infty}{g(k, n, x)} = f^{-1}(k, x), f(k, x... | I understand simplify the $n$-th term not solve the recurrence relation.
Noting $u_n=g(k,n,x)$ one has
$$2u_n = \log_{k}{\left(\dfrac{k^{g(k, n - 1, x)}x}{g(k, n - 1, x)}\right)}$$
i.e.$$2u_n=\log_{k}\frac {k^{u_{n-1}}x}{u_{n-1}}\iff k^{2u_n}=\frac {k^{u_{n-1}}x}{u_{n-1}}$$
Hence $$u_{n-1}k^{(2u_n-u_{n-1})}=x$$
►"An... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1351842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Identities For Generalized Harmonic Number I have been searching for identities involving generalized harmonic numbers
\begin{equation*}H_n^{(p)}=\sum_{k=1}^{n}\frac{1}{k^p}\end{equation*}
I found several identities in terms of $H_n^{(1)}$, but I am looking for some interesting identities for $H_n^{(2)}$. Does anyone k... | There is a nice list and a set of references at mathworld. Additionally, I discovered this one while writing a thesis on the Riemann Zeta function.
$$\sum_{n=1}^\infty \frac{H_n^{(s)}}
{n^s}=\frac{\zeta(s)^2+\zeta(2s)}{2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1351906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Find a point so that the triangle is equilateral We have O(0,0), A(3,4) and B(x,y). Find $x,y\in{R}$ so that the OAB triangle is equilateral.
I tried using the fact that the median is also the altitude(height) of the equilateral triangle. I calculated that distance from O to A(it's 5), meaning all the sides of the tria... | Using rotation transformation rotate the vector on x-axis, $ 5\,i, $ by angle
$$ \pi/3 + tan^{-1} \frac 43 $$
for vector tip to get to required position $(x,y).$ It simplifies easily.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1351957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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Proof that If G = (V , E ) is a graph, then S is an independent set ⇐⇒ V − S is a vertex cover. I have the following :
Proof:
Proof. ⇒ Suppose $S$ is an independent set, and let $ e = (u, v )$
be some edge. Only one of $u, v$ can be in S . Hence, at least one of
$u, v$ is in $V − S$ . So, $V − S$ is a vertex cover.
... | It's correct, but perhaps the wording is a bit vague/imprecise. A better way to say it is:
At most one of $u,v$ can be in $S$.
Then when we negate both sides, we see where the "at least one" part came from.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1352044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determine whether or not the series $\sum\limits_{n=2}^\infty (\frac{n+4}{n+8})^n$ converges or diverges Does the series converge or diverge?
The series $\sum\limits_{n=2}^\infty (\frac{n+4}{n+8})^n$
I tried the root test to get rid of the nth power but the limit equaled $1$ so the test is inconclusive. How else can I ... | You can rewrite the expression for a term in this series using long division:
$$\frac{n+4}{n+8}=1-\frac{4}{n+8}$$
We recognize the limit:
$$\begin{align}\lim_{n\to\infty} \left(1-\frac{4}{n+8}\right)^n &= \lim_{n\to\infty} \left(\left(1-\frac{4}{n+8}\right)^{n+8}\right)^{\frac{n}{n+8}}\\
&=\left(e^{-4}\right)^1\\
&\neq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1352141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
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Finding the equation of a function
Find the function $y=f(x)$ whose graph is the curve $C$ passing through the point $(2,1)$ and satisfying the following property: each point $(x,y)$ of $C$ is the midpoint of $L(x,y)$ where $L(x,y)$ denotes the segment of the tangent line to $C$ at $(x,y)$ which lies in the first quad... | At any point of your curve you have information about the slope of the curve at your point. This means that you have to solve a differential equation.
Suppose $(x_0,y_0) \in C$ is a point of your curve. Then (I skip boring computations) you have that the slope of the curve at $x_0$ is
$$f'(x_0)=-\frac{y_0}{x_0}$$
so y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1352276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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how do they calculate these following columns I have these data:
I am sorry the data is in Portuguese, and it is an image so I can't convert it to a table but the translate "probably" ( i am not a native speakers for Portuguese language) is:
*
*The first column is the minute that the cars have entered to my garage.... | I would say:
forth column = third column / 170 * 100 (e.g. 2*20 / 170 * 100 = 23.53)
fifth column:
5.88 + 0 = 5.88
5.88 + 6.47 = 12.35
12.35 + 14.12 = 26.47
etc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1352360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is a polynomial $f$ zero at $(a_1,\ldots,a_n)$ iff $f$ lies in the ideal $(X_1-a_1,\ldots,X_n-a_n)$? This is probably a very silly question:
If $R$ is an arbitrary commutative ring with unit and $f\in R[X]$ a polynomial, then for any element $a\in R$ we have
$$f(a)=0 \Longleftrightarrow X-a ~\mbox{ divides }~ f \Longle... | Yes. Another way to see this is by the following:
We can assume without loss of generality that $X_n$ appears in $f$. Then $$f=\varphi_0+\varphi_1X_n+\cdots + \varphi_kX_n^k$$
for some $\varphi\in R[X_1,...,X_{n-1}]$ with $\varphi_j(a_1,..,a_{n-1})\neq 0$ for some $j\leq k$. This reduces to the case you are okay with!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1352606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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Finding directional derivative in direction of tangent of curve just something small I couldnt get. $C$ is my curve that described by intersection of two planes: $$2x^2-y^2\:=1 ,\:2y-z=0 $$
The point $(1,1,2)$ is on the curve. $n$ is the vector whos direction is the tangent to $C$ in the point (in a way that its create... | You have to take the positive square root since your point is $(1,1,2)$ where $x$ value is positive.
Based on the information of the angle, the direction of the tangent has a negative $z$ value. So once you find the tangent direction, change the signs if necessary.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1352699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determine whether $\sum \frac{2^n + n^2 3^n}{6^n}$ converges For the series $$\sum_{n=1}^{\infty}\dfrac{2^n+n^23^n}{6^n},$$ I was thinking of using the root test? so then I would get $(2+n^2/n+3)/6$ but how do I find the limit of this?
| We might compare and then apply either the root or ratio test, if these are your preferred tests. That is
$$
\sum_{n=1}^{\infty}\dfrac{2^n+n^23^n}{6^n} < \sum_{n = 1}^\infty \frac{n^23^n + n^23^n}{6^n} = 2\sum_{n = 1}^\infty \frac{n^2 3^n}{6^n} = 2\sum_{n = 1}^\infty \frac{n^2}{3^n},
$$
and you can now use either the r... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Picking and replacing balls from a bag until you are relatively certain you have picked each one at least once Suppose I have an unknown number of balls ($N$), each of a different color, hidden in a bag.
How many times must I draw a single ball, make a note the color and return it to the bag in order to be sure (within... | If there are a reasonable number of balls you can model this as a Poisson process for each ball. Each time you draw the ball has $\frac 1N$ chance to be picked, so the mean after $k$ tries is $\frac kN$ and the chance the ball has not been seen is $e^{-k/N}$. The chance that all the balls have been seen is then $(1-e... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Mechanics: Projectiles involving a ball shot out of a cannon, moving in the opposite direction of the shot
A child is playing with a toy cannon on the floor of a long railway carriage. The carriage is moving horizontally in a northerly direction with acceleration $a$. The child points the cannon southward at an angle ... | Let the point at which the child fires the toy cannon from be $P$. Let us consider the situation relative to point $P$. Let $d$ be the distance the shell travels horizontally. Let $T$ be the time for the shell to land after being fired.
Vertical motion:
$s = ut + \frac{1}{2} at^2$, we get $$ 0 = VT\sin \theta - \frac{g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1353169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding the first three terms of a geometric sequence, without the first term or common ratio.
Given a geometric sequence where the $5$th term $= 162$ and the $8$th term $= -4374$, determine the first three terms of the sequence.
I am unclear how to do this without being given the first term or the common ratio. ple... | I had a similar question and tried to solve it alone i used this way
since the initial value is f(1) so u use this equation f(n) = ar^n-1
since u have f(5) as initial value or actually (known value) then we have to change it to this
f(n) = a5 * r (n-5) , so we know that the 8 th term f(8) = -4734
so : f(8) = a5 * ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1353281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Use the Laplace transform to solve the initial value problem. $$
y''-3y'+2y=e^{-t}; \quad\text{where}~ ~ y(2)=1, y'(2)=0
$$
Hint given: consider a translation of $y(x)$.
I am stuck on this problem on our homework. I don't understand what they mean by a "translation". Do they just mean a substitution?
Update - after spe... | Firstly I can't see where you get the $\delta$-function from, please consider showing your working when you post a question here.
You want to use the following properties of the Laplace transform:
$$\mathcal{L}[f'(t)]=sF(s)-f(0)$$
$$\mathcal{L}[f''(t)]=s^2F(s)-sf(0)-f'(0)$$
where $F(s)$ denotes the Laplace transform of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1353355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Euler-Mascheroni constant [strategic proof] I know two proofs about the approximation of Euler-Mascheroni constant $\gamma$ that are very technical. So I would like to know if someone has a strategic proof to show that $0.5<\gamma< 0.6$.
Let be $\gamma\in \mathbb{R}$ such that
$$\large\gamma= \lim_{n\to +\infty}\left... | Let $f(x)=\frac{1}{x}$ and $H_n=\sum_{k=1}^nf(k)$. For every $k$ take the segments $\overline{P_kP_{k+1/2}}$ and $\overline{P_{k+1/2}P_{k+1}}$, where $P_k=(k,f(k))$. Note that for every $k$ the sum of area of trapezes $Q_kP_kP_{k+1/2}Q_{k+1/2}$, and $Q_{k+1/2}P_{k+1/2}P_{k+1}Q_{k+1}$, where $Q_k=(0,k)$, is greater tha... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
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Showing a Group $G$ is a Semidirect Product of $S_n$ and the Group of Diagonal Matrices with Entries $±1$. Consider $G$ to be the set of $n$ $\times$ $n$ matrices with entries in $\{\pm1\}$ that have exactly one nonzero entry in each row and column. These are called signed permutation matrices.
Show that $G$ is a grou... | Define $|(a_{i,j}):=(|a_{i,j}|)$. Show that for matrices with only one nonzero entry per row and column we have $|AB|=|A||B|$. Not only does this show us the existence of inverses in $G$ (for $A\in G$, $|A|$ is a permutation matrix, has an inverse, hence $B:=A|A|^{-1}$ is a matrix with $|B|=I$, hence a diagonal matrix ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Distribution of decimal repunit primes The prime number theorem describes the distribution of prime numbers in positive integers. Is there a similar theorem describing the distribution of primes among positive integers of the form $\frac{10^n - 1}{9}$?
If so, what techniques are used to arrive at the theorem?
| Before the Prime Number Theorem was proved, mathematicians used calculations and heuristics to determine that there 'should' be about one prime every $\log x$ numbers around $x$. We can work similarly: suppose a repunit is just like other numbers of its size which are relatively prime to 10. Even numbers are twice as l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1353753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Separation of variables Calculus The given differential equation I need to solve is $dy/dx=1/x$ with the initial conditon of $x=1$ and $y=10$
My attempt:
$dy=\frac 1x dx$
Integrating yields
\begin{align*}
y&=\log x+C\\
x=1 & y=10\\
10 &=\log 1 + C\\
10=C\\
\leadsto y&=\log x+10
\end{align*}
Something seems ... | We have $$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{x}$$ Hence, separating the variables gives us $$\int \mathrm{d}y = \ln x + c \implies y=\ln x + c.$$ Using the initial condition of $x=1, y=10$ gives us $$10 = 0 + c \iff c = 10.$$ So the solution is $$\bbox[10px, border: solid lightblue 2px]{y = \ln x + 10}$$ which ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1353934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $\frac{a}{b}<\frac{c}{d}$ and $\frac{e}{f}<\frac{g}{h}$, then $\frac{a+e}{b+f} < \frac{c+g}{d+h}$.
If $a, b, c, d, e, f, g, h$ are positive numbers satisfying $\frac{a}{b}<\frac{c}{d}$ and $\frac{e}{f}<\frac{g}{h}$ and $b+f>d+h$, then $\frac{a+e}{b+f} < \frac{c+g}{d+h}$.
I thought it is easy to prove. But I could ... | This is false.
For example,
$$\frac{1}{3} < \frac{5}{12}, \quad \frac{52}{5} < \frac{11}{1}, \quad \frac{53}{8} > \frac{16}{13}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1354015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Independence of intersections Let $(\Omega,\mathfrak{A},P)$ be a probability space and $A,B,C\in\mathfrak{A}$ some events where $A$ and $B$ are independent. I'm a bit confused now as I intuitively think that this also implies independence of $A\cap C$ and $B\cap C$. Am I right? Unfortunately, I immediately get stuck wh... | Consider $A = \{ 1,2 \}$, $B = \{1,4\}$ and $C = \{1\}$ on $\Omega = \{1,2,3,4\}$, with $P( X ) = \frac{|X|}{|\Omega|}$
$$P(A \cap B ) = 1/4$$
$$P(A) P(B) = 1/4$$
So A and B are independent, yet
$$P( (A \cap C) \cap (B\cap C) ) = 1/4$$
But, note that
$$P(A \cap C) P (B\cap C) = 1/16.$$
Therefore, these events are no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1354063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Are there more transcendental numbers or irrational numbers that are not transcendental? This is not a question of counting (obviously), but more of a question of bigger vs. smaller infinities. I really don't know where to even start with this one whatsoever. Any help? Or is it unsolvable?
| Hint: the set of algebraic numbers is Countable.
For proving this, you can show that the polynomials with integer coefficients form a countable set, and that the set of their roots is also countable. The latter is just the Set of Algebraic Numbers (over $\mathbb{C}$). As the Algebraic Numbers over $\mathbb{R}$ form a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1354153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
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Conservative field? Let a vector field $F$ what it is defined by $F(x,y)=(\frac{-y}{(x-1)^2+y^2},\frac{x^2+y^2-x}{(x-1)^2+y^2})\ \forall \ (x,y)\epsilon\mathbb{R}^2$\ {$(1,0)$} then... is the vector field $F$ a conservative field in $\mathbb{R}^2$\ { $(1,0)$}? Why? I dont know how I can prove that, I know that if I ca... | If you shift $x$ by $1$ to get $G(x,y)=F(x+1,y)=(\frac{-y}{x^2+y^2},1+\frac{x}{x^2+y^2})$ then it is easy to see that $G$ is not conservative, e.g. calculate the curve integral along the unit circle to see it is not zero. Neglecting the constant component $(0,1)$ it is the electromagnetic field around $z$-axis.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Greatest common divisors equal? Let $a,b$ be natural numbers. Show that $gcd(a^n,b^n)$ = ($gcd(a,b)^n)$ for any integer $n$.
How I started was first proof by contradiction, and then tried to do an inductive proof when that didn't work, but neither of them worked out for me. I think I'm confused as to the notation I sho... | If you're just looking for a nudge in the right direction try looking at the prime factorization of $a$ and $b$ vs $a^n$ and $b^n$. (You could also try thinking of $a$ as $gcd(a,b)*p_1p_2....$ and $b$ as $gcd(a,b)*q_1q_2....$ where $p_i$ and $q_j$ are all prime). Further "nudge": Could it possibly be the case that $p_i... | {
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"source": "stackexchange",
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Representing any number in $R_+$ by means of two numbers I have the following question:
Let $R_+$ denote the nonnegative reals. Let $0<a<1$ and $b>0$, and set $p = a^n b^m$. Is it possible to find a $n,m\in N$ such that $p$ can approximate any number in $R_+$ arbitrarily close?
| This cannot be true in general, because a simple counter-example:
Set $a=\frac{1}{2}$ and $b=2$. Now, the only number that can be arbitrarily approximate is $0$, as is easily seen.
But I believe that it could be true when $\frac{a}{b}$ is irrational.
| {
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Does every projection operator satisfy $\|Px\| \leq \|x\|\,$? It's well known that an orthogonal projection satisfies $\|Px\|\leq\|x\|$.
Does this property hold for any general projection operator $P$, which is defined by $P^2=P$?
| The property does not hold. For example, consider
$$
P = \pmatrix{1&0\\\alpha &0}
$$
for some $\alpha \neq 0$. Note that $P^2 = P$, but
$$
\left\|
\pmatrix{1&0\\\alpha &0} \pmatrix{1\\0}
\right\| =
\left\| \pmatrix{1\\ \alpha} \right\| \geq
\left\| \pmatrix{1\\0} \right\|
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A proof regarding Lebesgue measure and a differentiable function Could anyone kindly provide a hint on the following problem? My guess is to do some change of variables? Thank you!
Let $f:\Bbb{R}\rightarrow\Bbb{R}$ be a continuously differentiable function. Define $\phi:\Bbb{R}^2\rightarrow\Bbb{R}^2$ by $\phi(x,y)=(x+... | Clearly, the function $\phi$ has continuous partial derivatives. Its Jacobian is given as follows: $$\mathbf J_{\phi}(x,y)=\left[\begin{array}{rr}1+f'(x+y)&f'(x+y)\\-f'(x+y)&1-f'(x+y)\end{array}\right]$$ and $\det\mathbf J_{\phi}(x,y)=1$ for all $(x,y)\in\mathbb R^2$.
It can be shown also that $\phi$ is injective. To s... | {
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Seven points in the plane such that, among any three, two are a distance $1$ apart Is there a set of seven points in the plane such that, among any three of these points, there are two, $P, R$, which are distance $1$ apart?
| Yes. Normally I like to give some more motivation for an answer, but in this case it's probably hard to say anything suggestive that the illustration below does not, besides perhaps that after experimenting one might guess that the four-point "diamond" configurations are helpful (and probably necessary) in constructing... | {
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How to rewrite $\pi - \arccos(x)$ as $2\arctan(y)$? I get the following results after solving the equation $\sqrt[4]{1 - \frac{4}{3}\cos(2x) - \sin^4(x)} = -\,\cos(x)$, :
$$
x_{1} = \pi - \arccos(\frac{\sqrt{6}}{3}) + 2\pi n: n \in \mathbb{Z}\\
x_{2} = \pi + \arccos(\frac{\sqrt{6}}{3}) + 2\pi n: n \in \mathbb{Z}
$$
Wo... | We define $y = \arccos \left( \dfrac{\sqrt{6}}{3} \right)$. Since $0 < \dfrac{\sqrt{6}}{3} < 1$, we have $0 < y < \dfrac{\pi}{2}$, and $\dfrac{\pi}{2} < \pi - y < \pi$. Therefore,
\begin{equation*}
\tan \left( \frac{\pi - y}{2} \right) = \sqrt{\frac{1 - \cos (\pi - y)}{1 + \cos (\pi + y)}} = \sqrt{\frac{1 + \cos y}{1 -... | {
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Incompleteness in other areas of mathematics I read in "Apostolos Doxiadis:Uncle Petros and Goldbach's conjecture" that SPOILER ALERT Uncle Petros practically stopped working on Goldbach's Conjecture when he learnt about the Incompleteness Theorem. He asked Godel if it could be decided if a particular statement is true... | What about questions asking whether some big, multi-variable polynomial equation (with integer coefficients) has an integer solution? There are equations of this sort for which there is no integer solution but the usual Zermelo-Fraenkel axioms of set theory can't prove that. (If you add to ZFC the assumption that ther... | {
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Spectrum: Boundary Problem
Given a Hilbert space $\mathcal{H}$.
Denote for readability:
$$\Omega\subseteq\mathbb{C}:\quad\|\Omega\|:=\|\omega\|_{\omega\in\Omega}:=\sup_{\omega\in\Omega}|\omega|$$
Denote for shorthand:
$$\Omega\subseteq\mathbb{R}:\quad\Omega_+:=\sup_{\omega\in\Omega}\omega\quad \Omega_-:=\inf_{\omega\in... | Define the value:
$$\omega:=\omega_-:=\mathcal{W}(H)_-$$
By linearity of range:
$$\mathcal{W}(H-\omega)=\mathcal{W}(H)-\omega\geq0$$
For bounded operators:
$$\mathcal{W}(H-\omega)\geq0\implies\sigma(H-\omega)\geq0$$
For normal operators:
$$\sigma(H-\omega)_+=\|\sigma(H-\omega)\|=\|\mathcal{W}(H-\omega)\|=\mathcal{W}(H-... | {
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Example of self-adjoint linear operator with pure point spectrum on an infinite-dimensional separable Hilbert space Let $\mathcal{H}$ be an infinite-dimensional, complex, separable Hilbert space.
Besides the well-known one-dimensional Harmonic oscillator on $\mathcal{H}=(\mathcal{L}^{2}(\mathbb{R})\,;d\mathit{l})$, doe... | Take any orthonormal basis $\{e_{n} \}_{n=1}^{\infty}$ of a Hilbert space $H$ and define $Lf = \sum_{n=1}^{\infty}n(f,e_{n})e_{n}$ on the domain $\mathcal{D}(L)$ consisting of all $f\in H$ for which $\sum_{n}n^{2}|(f,e_n)|^{2} < \infty$. The operator $L : \mathcal{D}(L)\subset H \rightarrow H$ is a densely-defined self... | {
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Behavior of $x^2+y^2+\frac{y}{x}=1$. In my mathematical travels, I've stumbled upon the implicit formula $y^2+x^2+\frac{y}{x}=1$ and found that every graphing program I've plugged it in to seems to believe that there is large set of points which satisfy the equation $(y^2+x^2+\frac{y}{x})^{-1}=1$ which do not satisfy t... | Here is a color plot (aka. scalar field) of $$x^2+y^2+{y\over x}$$
White is around zero, gray is positive and red is negative. The thick line is $0$ and the thin line is $1$, your original curve.
It's easy to see that by taking the inverse, the thin line will be preserved, but you will get a discontinuity at the thick... | {
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Conditional probability for classification In Book "Machine Learning: A probabilistic Perspective" Page 30, it tries to give a generative classifiers solution using conditional probability as
$$p(y=c|x,\theta)=\frac{p(y=c|\theta)p(x|y=c,\theta)}{\sum_{c'}p(y=c'|\theta)p(x|y=c',\theta)}$$
I understand the numerator but ... | Starting from the definition of a conditional probability using Bayes' rule
$$p(y=c|x,\theta)=\frac{\color{blue}{p(y=c,x,\theta)}}{p(x,\theta)}=\frac{\color{blue}{p(x|y=c,\theta)p(y=c|\theta)p(\theta)}}{p(x,\theta)}$$
where we use the chain rule in the numerator (parts highlighted in blue), to express the joint density... | {
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upper bound on sum of exponential functions Let $$\sum_{k=1}^{N} |a_k| = A$$ where $A$ is some constant. I am looking for an upper bound on the sum
$$\sum_{k=1}^{N} e^{|a_k|},$$ independent of $N$, but may depend on $A$. That is, I am looking for something like
$$\sum_{k=1}^{N} e^{|a_k|} \le f(A).$$
We can find the u... | There is no upper bound independent of $N$, as every term in the sum is at least $1$, so the sum is at least $N$.
| {
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When is the union of $\sigma$-algebras atomless? Suppose that we are given a probability space $(\Omega, \mathcal{F}, \mathsf P)$ and an increasing sequence of $$\mathcal{F}_1\subset \ldots \subset\mathcal{F}_n\subset \mathcal{F}_{n+1} \subset \ldots \subset \mathcal{F}$$
of $\sigma$-algebras. Assume that $\mathsf P|_{... | An atom is an indivisible set in the sigma algebra with positive measure.
Consider $\mathcal{F}_n$ a sigma algebra of subsets of the space $[0,1]^2$.
The construction is a bit artificial. so I will divide it into steps.
1) consider a the sets $[0,1] \cup (n,n+1)\backslash E_n$ as the basic sets of your sigma algebra $\... | {
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"source": "stackexchange",
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What does the quotient group $(A+B)/B$ actually mean? I understand that $A+B$ is the set containing all elements of the form $a+b$, wit $a\in A, b\in B$.
When you do the quotient group, that's like forming equivalence classes modulo $B$. All elements in $B$ should be congruent to 0 modulo $B$, as far as I understand. ... | The error is if $B\not\subset A$ then we cannot take the quotient. This theorem essentially says if this is the case, then we can either expand $A$ to the smallest group containing $A$ and $B$ and then quotient by $B$, or we can restrict to $A$ and quotient by $A\cap B$ and, they give the same result.
Note that if $B\s... | {
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Sum of Two Continuous Random Variables Consider two independent random variables $X$ and $Y$. Let $$f_X(x) =
\begin{cases}
1 − x/2, & \text{if $0\le x\le 2$} \\
0, & \text{otherwise}
\end{cases}$$.Let $$f_Y(y) =
\begin{cases}
2-2y, & \text{if $0\le y\le 1$} \\
0, & \text{otherwise}
\end{cases}$$. Find the probability... | Let $Z=X+Y$. We know $0\leq Z\leq 3$ but for the density of $Z$ we have three cases to consider due to the ranges of $X$ and $Y$.
If $0\leq z\leq 1$:
\begin{eqnarray*}
f_Z(z) &=& \int_{x=0}^{z} f_X(x)f_Y(z-x)\;dx \\
&=& \int_{x=0}^{z} (1-x/2)(2-2y)\;dx \\
&=& \left[ 2x-2zx+zx^2/2+x^2/2-x^3/3 \right]_{x=0}^{z} \\
&=& 2z... | {
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Show that if n+1 integers are choosen from set $\{1,2,3,...,2n\}$ ,then there are always two which differ by 1 Considering n=5 i have
$\{1,2,3,...,10\}$ .Making pairs such as $\{1,2\}$ ,$\{2,3\}$ ... total of $9$ pairs which are my holes and $6$ numbers are to be choosen which are pigeons .So one hole must have two p... | Well, in your case when $n=5$, you shouldn't make the pairs $\lbrace1,2\rbrace,\lbrace2,3\rbrace,...,\lbrace8,9\rbrace,\lbrace9,10\rbrace$, but rather $\lbrace1,2\rbrace,\lbrace3,4\rbrace,...,\lbrace7,8\rbrace,\lbrace9,10\rbrace$, as in the first case the same integer appears in two different pairs.
When we have the co... | {
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Functions that are their own inverse. What are the functions that are their own inverse?
(thus functions where $ f(f(x)) = x $ for a large domain)
I always thought there were only 4:
$f(x) = x , f(x) = -x , f(x) = \frac {1}{x} $ and $ f(x) = \frac {-1}{x} $
Later I heard about a fifth one
$$f(x) = \ln\left(\frac {e... | Technically Any functions satisfying $$f(x)=-x+a$$ for real numbers $a$ are their own inverses
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1356095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do we decide the direction of vector, that is orthogonal to some other vector? assume that two vector given with relationship below: $$\vec{n}\cdot \vec{u} = 0 $$ Then $\vec{n}$ and $\vec{u}$ are orthogonal vectors. Assume that we now have the direction of $\vec{n}$
Then there are infinitely many possible directio... | There is a special case for twice-differentiable curves acting in $R^3$ For any point of such a curve you can explicitly define an orthonormal basis in $R^3$ that includes the normalized direction and two orthonormal vectors.
The trick is to realize that if $f''$ exists and is and not zero
$(f'(t) \cdot f'(t))' = 0 = ... | {
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How to generate the icosahedral groups $I$ and $I_h$? The icosahedral groups $I$ with 60 elements and $I_h = I \times Z_2$ are also three dimensional point groups. However, ever unlike other point groups, it seems there is rarely reference to give their representation (e.g., matrix representation for each elements). C... | The explicit matrix representations of all 5 irreducible representations of $I$ are given in Hu, Yong, Zhao, and Shu, "The irreducible representation matrices of the icosahedral point groups I and Ih", Superlattices and Microstructures, Volume 3, Issue 4, 1987, pages 391-398, DOI:10.1016/0749-6036(87)90212-6. You can ... | {
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Find conformal mapping that maps set $\{(x,y)\in\mathbb{R}^2 : \operatorname{y\le 0}\}$ to unit ball. Find conformal mapping that maps set $A=\{(x,y)\in\mathbb{R}^2 : \operatorname{y\le 0}\}$ to unit disk. I know that such a mapping exists from Riemann Theorem.
Note: I don't want full answer. I expect only some starti... | A map from the complex half-plane of numbers with positive real part to the unit disc is given by $$ z \mapsto \frac{z-1}{z+1}$$
For details on this see an earlier question Mapping half-plane to unit disk?
You should be able to modify this for you purpose; note that your set corresponds to the half-plane with negative... | {
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Differentiate the Function: $y=\frac{e^x}{1-e^x}$ $y=\frac{e^x}{1-e^x}$
Utilize the quotient rule:
$y'=\frac{(1-e^x)\cdot\ e^x\ - [e^x(-e^x) ]}{(1-e^x)^2}$
$y'=\frac{(1-e^x)\cdot\ e^x\ - e^x(+e^x) }{(1-e^x)^2}$
I am confused how to solve the problem from this point. Can I factor out the negative sign in the numerator?... | $y=-1+(1-e^x)^{-1} \implies y'=e^x(1-e^x)^{-2}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Multivariable Functions - Second Derivative Problem So here is the problem:
Calculate the second class derivative on $(1,1)$ of the equation $x^4+y^4=2$
I found this problem on my proffesor's notes. However it doesn't state whether a partial or a total derivative must be calculated. My guess would be the total.
So my... | we have $x^4+y^4=2$ we assume the the derivative exists and we get $$4x^3+4y^3y'=0$$ divided by $4$ we get $$x^3+y^3y'=0$$ and the same thing again:
$3x^2+3y^2(y')^2+y^3y''=0$ with $$y'=-\frac{x^3}{y^3}$$ you will get an equation for $$y''$$ allone.
| {
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"timestamp": "2023-03-29T00:00:00",
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Solving set of 2 equations with 3 variables I'm working through an example and my answer is not coming out right.
Two equations are given and then the solution is shown.
Equations:
$$\begin{aligned}20q_{1}+15q_{2}+7.5q_{3}&=10\\
q_{1}+q_{2}+q_{3}&=1\end{aligned}$$
Solution Given:
$(X, (1/3)(1-5X), (1/3)(2+2X))$ for a... | Did you set up the matrix equation correctly?
$$\pmatrix{20 & 15 &7.5\\ 1 & 1 & 1}\pmatrix{q_1 \\ q_2\\ q_3} = \pmatrix{10\\1}$$
When I solve it, I get
$q=\pmatrix{-1+\frac{3}{2}X\\2-\frac{5}{2}X\\X}$
Note that you can confirm you did your math wrong somewhere because in your answer $q_1+q_2+q_3 \neq 1$ for every $X$. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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finding the partial bell polynomial of $e^x$ $$
\left(e^{x+z} - e^x\right) = \sum_{n=1}^\infty \frac{z^n}{n!} \frac{d^n}{dx^n}[e^x]
$$
$$
\left(e^{x+z}-e^x\right)^k = \sum_{n \geq k} Y^{\Delta}_{e^x}(n,k,x)z^n
$$
Where:
$$
Y^{\Delta}(n,k,x) = \frac{k!}{n!}B_{n,k}^{e^x}(x)
$$
$$
[z^n]\left(e^{x+z}-e^x\right)^k = e^{kx}[... | I think your calculation is correct.
Observe that
\begin{align*}
B_{n,k}^{e^x}(0)=\frac{1}{k!}\sum_{j=0}^k\binom{k}{j}j^n(-1)^{k-j}=\left\{n \atop k\right\}
\end{align*}
yield the Stirling Numbers of the second kind.
| {
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Is this formula true for $n\geq 1$:$4^n+2 \equiv 0 \mod 6 $? Is this formula true for $n\geq 1$:$$4^n+2 \equiv 0 \mod 6 $$.
Note :I have tried for some values of $n\geq 1$ i think it's true such that
:I used the sum digits of this number:$N=114$,$$1+1+4\equiv 0 \mod 6,1²+1²+4²\equiv 0 \mod 6,1^3+1^3+4^3\equiv 0 \mod 6,... | The answer is yes, because $4^2\equiv 4\pmod{6}$, and hence $4^n\equiv 4\pmod{6}$ for all $n\ge 1$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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a college entrance examination problem! how to find a geometric progression A Chinese college entrance examination math problem:
Given an arithmetic progression sequence $a_{1},a_{2},a_{3},a_{4}$, that are all positive real numbers, the common difference of the arithmetic progression is set as $d\;(d\in \mathbb{R}\set... |
*Let the numbers be $b-d,b,b+d,b+2d$. Then $b-d,b^2,(b+d)^3,(b+2d)^4$ are to form a geometric series.
We need $(b-d)(b+d)^3=(b^2)^2$. Let $e=b/d$, then it simplifies to $2e^3-2e-1=0$.
We need $(b^2)(b+2d)^4=(b+d)^6$. This simplifies to $2e^5+9e^4+12e^3+e^2-6e-1=0$.
Do long division to check that the cubic and the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1356982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Root space $L_\alpha$ is completely contained in simple ideal? I'm having trouble understanding a section in Humphrey's Lie algebras on page 74.
Suppose $L$ is a semisimple Lie algebra which decomposes as a direct sum of simple ideals $L_1\oplus\cdots\oplus L_t$. Let $H$ be a maximal toral subalgebra, so $H=H_1\oplus\... | We have $0\neq [H_i,L_\alpha] = \{\alpha(h)x\,\big|\, h\in H_i, x\in L_\alpha\}$. In particular there exists $h\in H_i$ such that $\alpha(h) \neq 0$. But this means $[H_i,L_\alpha]= L_\alpha$. Since $L_i\subseteq L$ is a simple ideal, we also have $[L_i,L_\alpha]\subseteq L_i$. Putting things together, we arrive at
$$L... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1357074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
The limit of the integral when the set is decreasing in probability to zero. This is an exercise problem(#2 in section 3.2) from 'A course in probability theory'.
If $E(\vert X \vert ) < \infty$ and $\lim_{n \to \infty} P(A_n) = 0,$ then $\lim_{n \to \infty} \int_{A_n} X \ dP = 0.$
I tried using the modulus inequality ... | The answer given by @d.k.o. is almost correct: Let us assume that $\int_{A_n} X \, dP \not \to 0$. Then there is some $\epsilon > 0$ and a subsequence $A_{n_k}$ with $|\int_{A_{n_k}} X \, dP | \geq \epsilon$ for all $k$.
Now, since $P(A_n) \to 0$, we can choose a further subsequence $A_{n_{k_\ell}}$ (which we call $B_\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1357254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Why does the equation of a circle have to have the same $x^2,y^2$ coefficients? In one of my geometry texts, it tells me they should be the same but not why. I am unsatisfied with this.
Suppose that:
$$ax^2+by^2 + cx + dy + f = 0 \text{ such that } a \neq b$$
is the equation of some circle. Upon completing the square a... | ****According to the definition of a circle , it's a locus of a points from a fixed point at a constant distant ****
Now take a fixed point C( h,k)which we call it Center of circle and constant distance is 'r' which is it's radius
Let $P(x,y)$ be any point in locus
By the definition we have
$PA=r$
The distance betwe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1357350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
What is "an increasing sequence of step functions"? I'm reading Alan Weir's "Lebesgue Integration and Measure".
In exercise 8 on page 30 he talks about "...an increasing sequence of step functions $\{\phi_n\}$..." and "...an increasing sequence of sets $\{S_n\}$..." However, he doesn't seem to define what these are.
Ca... | The first one means a pointwise increasing sequence of step functions.
He is probably approximating a function from below by a pointwise increasing sequence of step functions so that the limit is almost everywhere the desired function.
An increasing sequence of sets means a nested sequence of sets: every set contains t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1357468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How do you show $ \lim_{n\to \infty} {\frac{\ln(1)+\ln(2)+\ldots+\ln(n)}{n}} = \infty$? $$ \lim_{n\to \infty} {\frac{\ln(1)+\ln(2)+\ldots+\ln(n)}{n}} = \infty$$
I know I should show that it is greater then something that approaches to $\infty$ but I don't see what.
| Suppose $n$ is even. Then in the numerator, at lease $n/2$ of the terms are $>\ln (n/2).$ Thus
$$\frac{\ln 1 + \cdots + \ln n}{n} > \frac{(n/2)\ln (n/2)}{n} = (1/2)\ln (n/2) \to \infty$$
as $n\to \infty$ through even $n.$ This gives the idea; handling odd $n$ is a small step from the above.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1357540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 6
} |
Example for an ideal which is not flat (and explicit witness for this fact)
I'm looking for an ideal $\mathfrak{a}$ of an commutative (possibly nice) ring $A$ together with an injective $A$-module homomorphism $M\hookrightarrow N$ such that the induced map $\mathfrak{a}\otimes_A M\to\mathfrak{a}\otimes_A N$ is not inj... | First of all we are looking for an ideal $\mathfrak a$ which isn't flat. (A simple example is $\mathfrak a=(X,Y)$ in $A=K[X,Y]$.)
It would be nice if $\mathfrak a$ contains an $A$-sequence $a,b$, that is, $a$ is a non-zero divisor on $A$ and $b$ is a non-zero divisor on $A/(a)$. (For instance, in the previous example ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1357714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Properties of $L^2(-1,1)$ functions I want to show that there is no function $v \in L^2(-1,1)$ with $\int_{-1}^{1} v(x)\phi(x) dx = 2\phi(0)$ for all $\phi \in C^\infty_0(-1, 1)$ ($\phi$ is $0$ everywhere but $[-1,1] $).
I know about the delta distribution or the dirac measure, but I'd like to solve this without using ... | Suppose the assertion is true; We may then define smooth functions that are $0$ near $0$, i.e., for a small interval $(-a,a)$ we have that $\phi_{a,r,s} (x) \equiv 0$ on it, but $\phi_{a,r,s} \equiv 1$ on the intersection of the complement of $(-2a,2a)$ and some interval $(r,s)$.
Then we have that $\langle v, \phi_{a,r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1357821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 1
} |
Finding period of $f$ from the functional equation $f(x)+f(x+4)=f(x+2)+f(x+6)$ How can I find the period of real valued function satisfying $f(x)+f(x+4)=f(x+2)+f(x+6)$?
Note: Use of recurrence relations not allowed. Use of elementary algebraic manipulations is better!
| Observe, since
$$
f(x)+f(x+4)=f(x+2)+f(x+6),
$$
we can substitute $x+2$ for $x$ to get
$$
f(x+2)+f(x+6)=f(x+4)+f(x+8).
$$
Equating these, we know that $f(x)=f(x+8)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1357922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Cauchy-Schwarz inequality proof (but not the usual one) Before you downvote/vote-to-close, I am not asking for a proof of:
$$\sum^n_{i=1}a_ib_i\le\sqrt{\sum_{i=1}^na_i^2}\sqrt{\sum^n_{i=1}b_i^2 } $$
Which is what EVERY link I've found assumes is the inequality (for a proof of that see: http://www.maths.kisogo.com/index... | Since $\langle tx+y,tx+y\rangle\ge0$ for all $t$, $\;\;\langle x,x\rangle t^2+2\langle x,y\rangle t+\langle y, y\rangle\ge0$ for all $t$, so
$(2\langle x,y\rangle)^2-4\langle x,x\rangle\langle y,y\rangle\le0\implies \langle x,y\rangle^2\le\langle x,x\rangle\langle y,y\rangle=\lvert\lvert x\rvert\rvert^2 \lvert\lvert y\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1357968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 8,
"answer_id": 6
} |
convergence of $ \sum_{k=1}^\infty \sin^2(\frac 1 k)$
convergence of $ \sum_{k=1}^\infty \sin^2(\frac 1 k)$
How can I do this? Should I use the Ratio Test (I tried this but it started getting complicated so I stopped)? Or the Comparison test(what should I compare it to?)?
| Comparison: $ \displaystyle \left(\sin \frac 1 k \right)^2 < \left( \frac 1 k \right)^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1358037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Is there a necessary form of consecutive composites? For every $n \geq 3$ there is a tuple of $n-1$ consecutive composites, namely the composites of the form $n! + 2, \dots, n!+n$. However, must a tuple of $n$ consecutive composites take the form? It seems plausible to me, and I have not yet seen a counterexample.
| No, we can find consecutive composites that are not of this form. The point of $n!$ is just that it is a "very divisible number". We can obtain a lost of many other examples just like this one.
For example the numbers $n!^2+2,n!^2+4\dots +n!^2+n$ or $n!^3+2,n!^3+3\dots n!^3+2$.
Also $kn!+2,kn!+3\dots k!n+n$ works for a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1358250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to prove that the Fibonacci sequence is periodic mod 5 without using induction? The sequence $(F_{n})$ of Fibonacci numbers is defined by the recurrence relation
$$F_{n}=F_{n-1}+F_{n-2}$$
for all $n \geq 2$
with $F_{0} := 0$
and
$F_{1} :=1$.
Without mathematical induction,
how can I show that
$$F_{n}\equiv F_{n+2... | (This is @mathlove 's solution slightly streamlined.)
The sequence
$$G_n:=F_{n+5}-3F_n\qquad({\rm mod} \ 5)$$
satisfies the same recursion as the $F_n$. Furthermore one easily checks that $G_0=G_1=0$. This implies $G_n=0$ for all $n$, so that $$F_{n+5}=3F_n \qquad({\rm mod} \ 5)$$
for all $n$. It follows that $$F_{n+2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1358310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "42",
"answer_count": 13,
"answer_id": 11
} |
Grasping the concept of equivalence classes more concretely I have read about equivalence classes a lot but unable to understand exactly what they are. I know about equivalence relations, but I am having a hard time understanding what equivalence classes are exactly.
Can anyone explain the concept of equivalence classe... | Just in case you wanted a nice example: Consider the following shopping list made into a mathematical set $S$
\begin{equation}
S = \{a=\text{apple}, b=\text{banana}, c=\text{chicken}, d=\text{date}, e=\text{elk}\}.
\end{equation}
Define the equivalence relation to be
\begin{equation}
aRb\text{ if and only if a is the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1358420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 1
} |
What does it mean when two functions are "orthogonal", why is it important? I have often come across the concept of orthogonality and orthogonal functions e.g in fourier series the basis functions are cos and sine, and they are orthogonal. For vectors being orthogonal means that they are actually perpendicular such tha... | The concept of orthogonality with regards to functions is like a more general way of talking about orthogonality with regards to vectors. Orthogonal vectors are geometrically perpendicular because their dot product is equal to zero. When you take the dot product of two vectors you multiply their entries and add them to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1358485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "52",
"answer_count": 7,
"answer_id": 2
} |
Find the roots of the summed polynomial
Find the roots of: $$x^7 + x^5 + x^4 + x^3 + x^2 + 1 = 0$$
I got that:
$$\frac{1 - x^8}{1-x} - x^6 - x = 0$$
But that doesnt make it any easier.
| $$x^7 + x^5 + x^4 + x^3 + x^2 + 1 = 0$$
$$(x^7 + x^4) + (x^5 + x^3) + x^2 + 1 = 0$$
$$x^4(x^3 + 1) + x^3(x^2 + 1) + x^2 + 1 = 0$$
$$x^4(x^3 + 1) + (x^2 + 1)(x^3 + 1) = 0$$
$$(x^3 + 1)(x^4+x^2 + 1) = 0$$
from there we can continue with factoring
$$x^3+1=x^3+1^3=(x+1)(x^2-x+1)$$
and
$$x^4+x^2 + 1 =(x^2)^2+2x^2+1-x^2=(x^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1358551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Measurability of $t \mapsto \int_{\Omega(t)}f(t)g(t)h(t)$ given measurability of $t \mapsto \int_{\Omega(t)}f(t)g(t)$? Suppose I know that, given $f(t), g(t) \in L^2(\Omega(t))$,
$$t \mapsto \int_{\Omega(t)}f(t)g(t)$$
is measurable on $([0,T], Lebesgue) \to (\mathbb{R}, Borel)$. Suppose $h(t) \in L^\infty(\Omega(t))$ i... | Let $\Omega(t) = [-1,1]$ and $f(t) = 1$. Let $A$ be a non-measurable set in $[0,1]$,
$$g(t)(x) = \cases{1 & if $x \ge 0$ and $t \in A$\cr
-1 & if $x \le 0$ and $t \in A$\cr
0 & otherwise\cr}$$
Let $h(t) = 1$ on $[0,1]$ and $0$ on $[-1,0)$. Then $\int_{\Omega(t)} f(t) g(t) = 0$, but $\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1358626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the limit $\lim_{x\to 0}\frac{1-\cos 2x}{x^2}$
Find the limit $$\lim_{x\to 0}\frac{1-\cos 2x}{x^2}$$
This is what I did: $$\lim_{x\to 0}\frac{1-\cos 2x}{x^2} = \frac{0}{0}$$
Then, if we apply L'hopital's, we get: $$\lim_{x\to 0}\frac{2\sin 2x}{2x} =\frac{0}{0}.$$
Once again, using L'hopital's rule, we get:$$\li... | Without L'Hopital: Multiply top and bottom by $1+\cos 2x$ to get
$$\frac{1-\cos^2 2x}{x^2(1+\cos 2x)}= \frac{\sin^2 2x}{x^2(1+\cos 2x)}.$$
It's easily seen that $(\sin^2 2x)/x^2 \to 4.$ Thus the limit is
$4\cdot [1/(1+1)] = 2.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1358696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 1
} |
Let $Y=1/X$. Find the pdf $f_Y(y)$ for $Y$. The Statement of the Problem:
Let $X$ have pdf
$$f_X(x) =
\begin{cases}
\frac{1}{4} & 0<x<1 \\
\frac{3}{8} & 3<x<5 \\
0 & \text{otherwise}
\end{cases}$$
(a) Find the cumulative distribution function of $X.$
(b) Let $Y=1/X$. Find the pdf $f_Y(y)$ for $Y$. Hint: Co... | Crude sketch of CDF of Y based on a simulation. Perhaps helpful as a check on your work.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1358786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Differential Equations- Wronskian Fails? I was doing a problem where the goal was to find whether two functions: $$f(x) = \sin(2x) ,~~~~~ \text{and}~~~~~~ g(x) = \cos(2x)$$ are linearly independent or not using the wronskian.
The problem is simple enough, and after evaluating the wronskian I got a result of $~-2~$. Th... | In order for $f(x)$ and $g(x)$ to be linearly dependent functions, there have to be constants $a$ and $b$ such that
$$
af(x)+bg(x)=0
$$
for all $x$. You have found constants that work for one specific value of $x$, but they won't work for other values.
Notice that I haven't said anything about the Wronskian yet. There ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1358854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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