Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Sum based on stolz theorem The Sum is as follows:
$$ \lim_{n\to \infty} \frac{1} {\sqrt n} \left( \frac {1} {\sqrt n}+ \frac {1} {\sqrt {n+1}} +...+ \frac{1}{\sqrt{2n}} \right) $$.
I solved as follows using stolz theorem
$$ \lim_{n\to \infty} \frac{x_n-x_{n-1}} {y_n-y_{n-1} } = \lim_{n\to \infty} \frac {\frac {1} {\... | The second step follows from
$$
x_n - x_{n-1} = \frac{1}{\sqrt{2n}} + \frac{1}{\sqrt{2n-1}} - \frac{1}{\sqrt{n-1}},
$$
which you see by carefully noting which terms are present in $x_n$ which are not in $x_{n-1},$ and vice versa. When you go from $x_{n-1}$ to $x_n$ you gain two terms 'at the end', and loose one 'at the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2566676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Combinations for 6-card pattern from 4-deck cards I had programmatically determined all the desired combinations of the 6-card patterns from a 4-deck cards, and now I want to apply combinatorics method to verify my computer generated results.
From my computer simulation I've determined the following patterns and combin... | Five of a kind: Choose which rank is the five of a kind $(13)$, and then which particular five cards of that rank you get ($_{16}C_5$). Then choose the sixth different card from what's left ($192$).
So, $13 \cdot 4368 \cdot 192 = 10902528$ combinations.
Looks like you were successful for that one, in my book.
Let's try... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2566831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Show that a such set of Hahn-Banach extensions is infinite Let $M$ be the subspace of $C([-1,1])$ consisting of all even functions. Let $\psi:M\to \mathbb{R}$ be the linear functional given by $\psi(f)=\int_{-1}^{1}f(t)dt$.
Show that there are infinitely many bounded linear extensions $F$ of $\psi$ to $C([-1,1])$ such... | First calculate $\|\psi\|$. We have:
$$|\psi(f)| = \left|\int_{-1}^1f(t)\,dt\right| \le \int_{-1}^1 |f(t)|\,dt \le \int_{-1}^1 \|f\|_\infty\,dt =2\|f\|_\infty$$
so $\|\psi\| \le 2$.
For the even function $f \equiv 1$ we have $$\|\psi\| \ge \frac{|\psi(f)|}{\|f\|_\infty} = 2
$$
so we conclude $\|\psi\| = 2$.
Consider ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2566946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Limit of $\frac{x^5-1}{x^2-1}$ I need to determine if the following limit exists.
$$\lim_{x\to 1}\frac{x^5-1}{x^2-1}$$
I've already proved using L'Hospital that this limit exists and should equal to $\frac{5}{2}$, but unfortunately I'm not allowed to used anything more than basic analysis for functions, i.e. basic defi... | using Horner's method:
$$x^5 -1 =(x-1)(x^4 +x^3 +x^2 +x+1)$$
$$x^2 -1=(x-1)(x+1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2567028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
} |
Looking for a proof of an interesting identity Working on a problem I have encountered an interesting identity:
$$
\sum_{k=0}^\infty \left(\frac{x}{2}\right)^{n+2k}\binom{n+2k}{k}
=\frac{1}{\sqrt{1-x^2}}\left(\frac{1-\sqrt{1-x^2}}{x}\right)^n,
$$
where $n$ is a non-negative integer number and $x$ is a real number with ... | Using
$$\binom{n}{k}=\frac{1}{2 \pi i}\oint_C\frac{(1+z)^{n}}{z^{k+1}}dz$$ we get (integration contour is the unit cicrle)
$$
2\pi iS_n=\oint dz \sum_{k=0}^{\infty}\frac{(1+z)^{n+2k}x^{n+2k}}{z^{k+1}2^{n+2k}}=\oint dz \frac{(1+z)^n x^n}{z2^n}\sum_{k=0}^{\infty}\frac{(1+z)^{2k}x^{2k}}{2^{2k}z^k}=\\
4\frac{x^n}{2^n}\oi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2567158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
Some version of Hahn-Banach theorem
I understand the first part of the theorem. I can prove the part about $\delta$ if the inf is actually achieved. But I wonder whether this is even true.
| Consider $Z=Y\cup\operatorname{span}\{x\}$. This is obviously a subspace. Now define $f:Z\to \mathbb K$ by
$$f(z)=\lambda\delta,$$
where $z=y+\lambda x$ ,remembering the definition of $Z$. I leave it to you to show that $f$ is bounded and linear, $\|f\|=1$, and $f(x)=\delta$, but feel free to ask hints if you get stuck... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2567260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Maximum likelihood estimator for uniform distribution $U(-\theta, 0)$ Consider $X_1,X_2,...,X_n$ i.i.d $U(-\theta,0)$.
I want to find the maximum likelihood estimator of $\theta$.
I know that $f(x,\theta)=\frac{1}{\theta}$ for $-\theta < x < 0$ and that $L_n(\theta, x)= \frac{1}{\theta^n}$.
If we were looking at $U(0,\... | Note that the likelihood function is a function of $\theta$. In particular,
$$L_n(\theta;\vec X) = \left \{ \begin{matrix}\frac{1}{\theta^n} & \text{if $\theta \ge -X_i$ for $i=1,2,\cdots, n$,} \\ 0 & \text{otherwise.}\end{matrix}\right.$$
Here $\theta \ge -X_i$ comes from $-\theta \le X_i$. Now, $L_n(\theta,\vec X)$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2567387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Product between a column vector and a row vector I know that matrices product is correct when the number of the columns of the first matrix is equal to the number of rows of the second matrix.
Why I can't do the product between a column vector and a row vector? For example:
$$\begin{bmatrix}1 \\ 2 \\ 3 \end{bmatrix} \,... |
I know that matrices product is correct when the number of the columns of the first matrix is equal to the number of rows of the second matrix.
I wouldn't say "is correct", it is only defined in this case. You can invent your own product or way of multiplication, but the standard product of matrices only works, as yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2567679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
proof of roots of characteristic polynomial are eigenvalues How do I prove the 2 directions of this statement?
| A root $r$ of characteristic polynomial provides equation $det(M - r Id)=0$ so $M-r Id$ hasn't a full rank and there exist a nontrivial vector $\tilde{v}$ with $0 =(M-rId)\tilde{v}= M\tilde{v}-r\tilde{v}$ so $M\tilde{v}= r\tilde{v}$ and $r$ is by definition an eigenvalue. Walking backwards alongside my arguments provid... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2567988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Showing two topological spaces are not homeomorphic
Having infinitely many open sets in a topology is a topological property
#1: What are example of two topological spaces, preferably simple ones, which this statement show are not homeomorphic.
But, more importantly why does it show this? Having trouble understanding... | If $X$ and $Y$ are topological spaces with respective topologies $\tau_X$ and $\tau_Y$, and if $f : X \to Y$ is a homeomorphism, then $f$ induces a bijection $\tau_X \mapsto \tau_Y$: each $U \in \tau_X$ is mapped to its image $f(U) \in \tau_Y$ defined as usual by the formula
$$f(U)=\{f(x) \,|\, x \in U\}
$$
The proof... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2568082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Proving restriction of function is continuous The problem is:
If $f : (X, T) \to (Y, S) $ is continuous and $A \subseteq X$, then $f|_A : (A, T_A) \to(Y, S) $ is continuous.
How would I get started on this proof? I understand the restriction of the function $f$ is $f|_A(x) = f(x) $ for all $x\in A $. I jut don't see ... | If $A$ has the subspace topology, then $f^{-1}(U)$ is open for any $U \subset Y$ that is open. But then $f^{-1}(U) \cap A$ is open in $A$, but $f^{-1}(U) \cap A$ is precisely$\dots$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2568165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the spectral decomposition of $A$ $$
A= \begin{pmatrix} -3 & 4\\ 4 & 3
\end{pmatrix}
$$
So i am assuming that i must find the evalues and evectors of this matrix first, and that is exactly what i did.
The evalues are $5$ and $-5$, and the evectors are $(2,1)^T$ and $(1,-2)^T$
Now the spectral decomposition of $A... | The needed computation is
$$\mathsf{A} = \mathsf{Q\Lambda}\mathsf{Q}^{-1}$$
Where $\Lambda$ is the eigenvalues matrix. And your eigenvalues are correct.
Hence you have to compute
$$\mathsf{AQ} = \mathsf{Q\Lambda}$$
Which gives you the solutions
$$a = 2c ~~~~~~~~~~~ d = -\frac{b}{2}$$
You can then choose easy values lik... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2568305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Almost complex structure on $\mathbb{S}^{3} \times \mathbb{S}^{5}$ I would like to check, whether the product space $X = \mathbb{S}^{n} \times \mathbb{S}^{m}$ admits an almost complex structure for odd $m,n$.
For example, if $m=1$ and $n=3$, then $X = \mathbb{S}^{1} \times \mathbb{S}^{3}$ -- in this case one can constr... | Let $n$ and $m$ be odd.
As $\chi(S^n) = 0$, the manifold $S^n$ has a nowhere-vanishing vector field, and hence $TS^n \cong E\oplus\varepsilon^1$ for some rank $n - 1$ vector bundle $E$. Now note that
\begin{align*}
T(S^n\times S^m) &\cong \pi_1^*(TS^n)\oplus\pi_2^*(TS^m)\\
&\cong \pi_1^*(E\oplus\varepsilon^1)\oplus\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2568569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Probability: 2 independent events I am not sure if I answered this question right:
Suppose that A and B are 2 independent events such that the probability that neither occurs is 0.1 and that the probability of B is 0.2. Find the probability of A.
Now I know that 2 events are independent if their intersection = $P(A) ... | You already correctly noted the definition of independent events: $P(A\cap B)=P(A)P(B)$, then note that $A,B$ independent implies $A,B^c$ independent and $A^c,B^c$ independent, etc... We are told $P(A^c\cap B^c)=0.1$ and $P(B)=0.2$.
So, $$0.1=P(A^c\cap B^c)=P(A^c)P(B^c)=(1-P(A))(1-P(B))=(1-P(A))(1-0.2)$$
At this poi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2568691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Let $A$ be an $n \times n$ matrix. If every non-zero vector $v$ is an evector of $A$, prove that $A$ is a diagonal matrix I'll start with the things i already know, I know that for a vector $v$ to be an evector of $A$, then the following must be true
$Av = \lambda v$ this is only true if and only if....
$(A - \lambda ... | Hint:
For $k\in\mathbb{R}$,
$$kI_nv=kv$$
I'm only showing $n=2$. Use the same idea for higher $n$
Suppose $v=\binom{a}{b}$ and $A=\left(\begin{matrix}x_1&y_1\\y_2&x_2\end{matrix}\right)$.
We have
\begin{align}
Av&=\lambda v\\
\left(\begin{matrix}x_1&y_1\\y_2&x_2\end{matrix}\right)\binom{a}{b}&=\lambda \binom{a}{b}\\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2568831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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How can I calculate $\lim_{x \to 0}\frac {\cos x- \sqrt {\cos 2x}×\sqrt[3] {\cos 3x}}{x^2}$ without L'Hôpital's rule? How can I calculate following limit without L'Hôpital's rule
$$\lim_{x \to 0}\frac {\cos x- \sqrt {\cos 2x}×\sqrt[3] {\cos 3x}}{x^2}$$
I tried L'Hôpital's rule and I found the result $2$.
| Hint:
$$\cos x-(\cos2x)^{1/2}(\cos3x)^{1/3}=1-\cos x-[1-(\cos2x)^{1/2}(\cos3x)^{1/3}]$$
Now $\lim_{x\to0}\dfrac{1-\cos x}{x^2}=\cdots=\dfrac12$
On rationalization using $a^6-b^6=(a-b)(\cdots),$
$$1-(\cos2x)^{1/2}(\cos3x)^{1/3}=\dfrac{1-\cos^32x\cos^23x}{\sum_{r=0}^5[(\cos2x)^{1/2}(\cos3x)^{1/3}]^r}$$
$\lim_{x\to0}\sum_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2568920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 4
} |
Is it possible to solve this bitwise equation? I have been googling but I can't get a conclusion. I have this equation:
a = ((b ^ x) >> c) & d | e * ((b ^ x) & f)
Would it be possible to solve this bitwise equation? Assuming all values are known except x.
Thank you in advance.
| If anyone is interested in the answer, finally I solved it with the help of bruteforce.
First, I concluded that the result of ((b ^ x) >> c) & d was always relatively low (in a range of 0-50) and most of times 0 or 1 so I transformed the equation into this:
a = y | e * ((b ^ x) & f)
And defined the value of y with a l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2569024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Good reference for metric topology I would like a "good" book (not really introductory, not too advanced with good theory and exercises) on metric topology covering the following topics:
Metric spaces, open/closed sets, sequences, compactness, completeness, continuous functions and homeomorphisms, connectedness, produ... | You could have a look at Intermediate Mathematical Analysis by R.D. Bhatt. It does not cover all the topics that you mentioned, but provides an excellent treatment (with lots of exercises) for topics upto and including connectedness in your list.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2569151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Relation of convergence in probability and almost sure convergence Let $(Y_n)$ real valued random variables, such that $Y_n\to c \in \mathbb R$ in distribution. It has been shown that $Y_n \to c$ in probability. I want to prove that $Y_n \to c $ a.s. does not hold!
Therefore consider $Y_n \sim Ber_{1/n}\,$, i.e. $\math... | It seems you don't apply Borel-Cantelli correctly. Consider the events {$Y_n=1$} for $n \in \mathbb{N}$. They are independent and
$$\sum_{n=1}^\infty \mathbb{P}(Y_n=1) = \sum_{n=1}^\infty \frac{1}{n} = \infty$$
Thus by Borel-Cantelli, $\limsup_{n \to \infty} Y_n = 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2569276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Improper definite integration with complex bounds I am looking to prove the functional equation for theta function. Source: https://www.youtube.com/watch?v=-GQFljOVZ7I&list=PL32446FDD4DA932C9&index=12
Time about 8.00. We are to integrate:
$$\int_{-\infty+\frac{ik}{x}}^{+\infty+\frac{ik}{x}}e^{-\pi xz^2}dz$$
The argumen... | This is not an answer.
I am just showing that the naïve substitution $u=z-ik/x$ produces the correct answer to this integral with arbitrary real limits $a$ and $b$. Symmetry between the two limits is not leading to error cancellation in this case.
$$I=\int_{a+ik/x}^{b+ik/x} e^{-\pi x z^2}\,dz=\frac{\text{erf}\left(\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2569378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Proof for sum of squares formula ( statistics - related ) I'm new to the domain of statistics and i'm trying to accumulate as much info as i can right now. I've considered that this question should be asked here as it is related to mathematics. The problem is that from the get go most statistics books use the sum of sq... | The sample variance of data $X_1, X_2, \dots, X_n$ is
$$S^2 = \frac{\sum_{i=1}^n(X_i - \bar X)^2}{n-1} = \frac{\sum_i(X_i-\bar X)^2}{n-1} = \frac{\text{SS}_x}{n-1},$$
where
$$\text{SS}_x = \sum_i(X_i - \bar X)^2
= \sum_i(X_i^2 - 2\bar X X_i + \bar X^2)
= \sum_i X_i^2 -2\bar X\sum_i X_i + n\bar X^2\\
= \sum_i X_i^2 - \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2569510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Regression for implicit polynomial functions of n-th order How to do regression using polynomials of the form $a_n x^n + a_{n-1} x^{n-1} y + \cdots+ a_0 y^n+ b_{n-1}x^{n-1}+b_{n-2}x^{n-2}y+\cdots=c$, given some data points $(x_i,y_i)$ (the number of data points is much greater than $2n+2$)?
The sources on the internet ... | Just do a linear regression like you normally would with any basis of functions: calculate the $x^ky^l$, put in the columns of a matrix $\bf \Phi$ in some order, then pose
$${\bf v_o}= \min_{\bf v}\|{\bf \Phi v}-c{\bf 1}\|_2^2$$
If you allow constant term ($k=l=0$), there will exist a trivial solution which you will n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2569663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Order of $2^{36} \pmod{107}$ What is the order of $2^{36}\pmod{ 107}$?
My current thought is $$2^{106} \equiv 1 \pmod{107}$$ according to Euler's Theorem. However, I don't know how to proceed from here or maybe my approach is wrong from the beginning?
| Since $107-1=2\cdot 53$, every quadratic non-residue, with the only exception of $-1$, is a generator of $\mathbb{Z}/(107\mathbb{Z})^*$. $107$ is a prime of the form $8k+3$, hence $\left(\frac{2}{107}\right)=-1$ and $2$ has order $106$ in $\mathbb{Z}/(107\mathbb{Z})^*$. We have $\gcd(36,106)=2$, hence the order of $2^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2569775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
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Eigenvalues of same-row matrices It has previously been discussed here that the eigenvalues of an all-ones $n \times n$ matrix $A$ such as the following are given by $0$ with multiplicity $n - 1$ and $n$ with multiplicity $1$, hence a total multiplicity of $n$ which means that the given matrix is diagonalizable.
$$A =
... | *
*If all the columns are zero, it is the zero matrix which of course must be diagonalizable since the zero matrix is a diagonal matrix. $$0=I\cdot 0\cdot I$$
*If at least one of the column is non-zero, then the rank of the matrix is $1$ and the nullity is $n-1$. We check that the all-$1$ vector is an eigenvector an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2569890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Describe all prime and maximal ideals of $\mathbb{Z}_n$ I know that an ideal P in $\mathbb{Z}_n$ is prime if and only if $\mathbb{Z}_n/P$ is an integral domain and an ideal m in $\mathbb{Z}_n$ is maximal if and only if $\mathbb{Z}_n$/m is a field.
I think I've figured out that $\mathbb{Z}_p$ where p is a prime that di... | Consider the canoncial map $$\pi:\Bbb Z\rightarrow \Bbb Z_n$$ If $\mathfrak m\subset\Bbb Z_n$ is a maximal ideal, then $\pi^{-1}(\mathfrak m)\subset\Bbb Z$ is maximal aswell. This means that $\pi^{-1}(\mathfrak m)=(p)$ for some prime $p\in\Bbb Z$. There are two cases to consider:
*
*If $p\mid n$ then $(p)$ is a maxim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2569986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Derivative of a rotated vector with respect to the quaternion Let us say we have a right-handed unit quaternion, describing the rotation from frame $a$ to frame $b$: $q_a^b$. The rotation matrix formed from this quaternion is $R\left( q_a^b \right)$ and describes a passive rotation. That is, $R\left( q_a^b \right)v$ ... | I may have an answer, taking the derivation straight from "A Primer on the Differential Calculus of 3D Orientations" by Bloesch et al. (Appendix I: Section 3: Derivative of a Coordinate Map)
First, let $\Phi_{BA} \in SO(3)$ be a relative orientation of a coordinate system $B$ w.r.t. a coordinate system $A$. In the pape... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2570065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Trouble understanding pointwise vs uniform boundedness I've read the definition a few times and I am still struggling to understand what these actually mean. As I understand it, a sequence of functions is pointwise bounded if there exists an $M$ for each $x$ such that $f_n(x)<M$, for all $n$, and a function is uniforml... | Yes it is true that pointwise convergence implies pointwise bounded. A proof is similar to a proof that the convergent sequence of numbers $f_i(x)$ in the index $i$ is bounded for a fixed $x$, except you apply the $\forall x$ quantifier.
No uniform convergence does not imply uniform boundedness. Take $f_i(x)=x^2+1/i$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2570218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 0
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derivative of $\frac 2x \sin(x^3)$ by definition The function: $$\frac 2x \sin(x^3)$$ when $x\neq0$, and $0$ while $x=0$.
I need to find if the function is derivation at $x=1$.
First step was to check if the function is continuous. there is just 1 side of limit to check (since its the same function around $x=1$, so I c... | The function is obviously continuous at $1$, because it is a composition of continuous functions.
Your computation is flawed: it is true that $\lim_{x\to0}\frac{\sin x}{x}=1$, but you have $\lim_{h\to0}\frac{\sin((1+h)^3)}{(1+h)^3}$ and the preceding limit doesn't apply.
For the derivative at $1$:
$$
f'(1)=\lim_{h\to0}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2570328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
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Laplace method for $\int_0^1 e^{a(x-1)}\ln(-\ln(x)) \, dx$ The following integral
$$ \int_0^1 e^{a(x-1)}\ln(-\ln(x)) \, dx$$
looks like the one for which the Laplace method is applicable. The function under the integral looks like this:
($a$ increases from red to black), so the leading contribution to the integral sho... | Let $I(a,\epsilon) = \int_\epsilon^{1/e} e^{-a(1-x)}\log\log\frac{1}{x}dx$ and $J(a,\epsilon) = \int_{1/e}^{1-\epsilon} e^{1a(1-x)}\log\log\frac{1}{x}dx$, i.e. we cut the integral at the unique root $1/e$
Then holds for all $\epsilon>0$:
*
*$0\le I(a,\epsilon) \le e^{-a(1-\frac{1}{e})} I(0,\epsilon)$
*$0\ge J(a,\ep... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2570427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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If $B^{1/2}A=AB^{1/2}$. It is true that $AB=BA$? Let $E$ be a complex Hilbert space. Let $A\in \mathcal{L}(\mathcal{H})$ and let $B\in \mathcal{L}(E)^+$.
Assume that $B^{1/2}A=AB^{1/2}$. It is true that $AB=BA$?
Thank you
| $\def\h{^{1/2}}$
$$ AB=AB\h B\h = (B\h A) B\h= B\h (AB\h)=B\h B\h A=BA\,.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2570570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Find all directions such that they decrease the function value after taking a small step towards that direction Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ and all second partial derivatives of $f$ exist and are continuous over the domain of the function, mixed derivatives are continuous and hence $ \forall i, j \frac... | You are correct. By Taylor's theorem, you have
$$f(x_0+tp) =f(x_0) +t\nabla f(x_0)^T p + \frac{t^2}{2}p^TH(x_0)p + o(t^2)= f(x_0) + \frac{t^2}{2}p^TH(x_0)p + o(t^2) $$ because of the fact that $\nabla f(x_0)=0$ by hipothesis. Now take any $p\in \Bbb R^n$ such that $p^TH(x_0)p<0.$ Such a $p$ always exists because of the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2570674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Find floor of sum $\sum_{k=1}^{80} k^{-1/2}$ We have to find the floor $\lfloor S \rfloor$ of the following sum:
$$S = \sum_{k=1}^{80}\frac{1}{\sqrt k}$$
What I did was to find a approximate series that this series is near to. Let that series have general term $T_k$ and original series may have general term $a_k$. We ... | $$\begin{align}
\int_1^{81}\frac 1{\sqrt x}\;\;\text d x
&<\qquad\sum_{k=1}^{80}\frac 1{\sqrt k}
&&<1+\int_1^{80}\frac 1{\sqrt x}\;\;\text d x\\
\bigg[2\sqrt{x}\bigg]_1^{81}
&< \qquad\sum_{k=1}^{80}\frac 1{\sqrt k}
&&<1+\bigg[2\sqrt{x}\bigg]_1^{80}\\
2\big(\sqrt {81}-\sqrt{1}\big)
&< \qquad\sum_{k=1}^{80}\frac 1{\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2570782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
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$\mu,\nu$ ergodic implies $\mu\perp\nu$ Let $T:\Omega\to\Omega$ a measurable function and $\mu,\nu$ $T$-ergodic measures on $\Omega$. I am trying to prove that $\mu\perp\nu$ (this is, they concentrate in disjoint sets).
My attempt was define $w=\mu+\nu$ and use Radon-Nikodým to obtain $f,g\in L^1(\Omega)$ such that $\m... | For a measure $\lambda$ and a measurable function, $f$, let
\begin{align*}
B_{\lambda}^{f}
&= \left\{x: \lim_{n} n^{-1}\sum_{i=1}^{n}{f(T^{i}(x))
=E_{\lambda}[f]}\right\}
\end{align*}
Since $\mu \neq \nu$, there exists a measurable $f^*$ such that $E_{\nu}[f^*] \neq E_{\mu}[f^*]$. Therefore, $B_{\mu}^{f^*} \cap B_{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2570913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Role of binomial coefficient in binomial distribution As I can verify, the binomial distribution $W_N(n) = \binom Nn p^n q^{N-n}$, is indeed a probability distribution:
$$\sum_{n=1}^N W_N(n) = (p+q)^N = 1\ ,$$ with $q = 1 -p$.
In the context of statistical physics, I don't understand why it is essential that we take in... | Expanding on @Falrach's answer; if we, indeed, fix $r$ events who "succeed" amongst $n$ events, where the probability of succeeding resp. failing are $p$ and $q = 1-p$, we have the following argument for $n=5$ and $r=2$.
Denote $P(i)$ for the probability of $i$ succeeding events and $\hat{P}(k,l)$ for the probability o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2571020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 4
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Certain Subset of Sorgenfrey Plane is Closed Note that $L = \{(x,-x) \mid x \in \Bbb{R} \}$ is closed. Then if $A$ is closed in $L$, then it will also be closed in $\Bbb{R}^2_\ell$. According to Munkres, $L-A$ will also be closed, but I am having trouble proving this. The set $L-A$ closed in $\Bbb{R}^2_\ell$ if and onl... | As $\{(x,x)\} = ([x,x+1) \times [x,x+1)) \cap L$, every singleton subset of $L$ is open in $L$ (as an intersection of an open set of the Sorgenfrey plane with $L$).
This means that $L$ is discrete as a subspace: all of its subsets are open (and thus closed) in $L$ and as $L$ is closed in the Sorgenfrey plane, all of it... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2571109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A problem on rank of a matrix over two fields This is a problem from Berkeley problems in mathematics.
If $F$ is a subfield of $K$, and $M$ has entries in $F$, how is the row rank of $M$ over $F$ related to the row rank of $M$ over $K$?
where $M$ is a n by n matrix
The solution says "If a set of rows of $M$ is linearly... | The rank of a matrix is the size of its largest square submatrix with nonzero determinant. Since the determinant is determined only by the matrix's entries, it remains the same during a field extension. Thus the rank remains invariant too.
Alternatively, the rank of a matrix is just the number of nonzero diagonal entri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2571197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Show that the number of solutions of $x^2+y^2 ≡ 1$ (mod $p$) where $0Show that the number of solutions of $x^2+y^2 ≡ 1$ (mod $p$) where $0<x<p$, $0<y<p$, $p$ is odd prime is even iff $p ≡ 3, -3$ (mod $8$).
I learned about quadratic residue and sums of squares.
Let
$S_1$ = {$1^2, 2^2, ... , (p-1)^2$}
$S_2$ = {$1-1^2, 1-... | If $(x,y)$ is a solution, so are $(\pm x, \pm y)$, and since $p$ is odd, all $4$ solutions are distinct, mod $p$.
Hence the number of solutions is a multiple of $4$.
Note:$\;$For this argument, $p$ can be any odd positive integer, not necessarily prime.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2571329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Unsure how to use initial conditions in characteristic value problem $u_{xx} - 7u_{tx} + 12u_{tt} = 0$
$u(t,x) = sinx$ on $t+3x=0$
$u(t,x) = x$ on $t+4x = 0$
I have factorized the above equation, which gives me the result:
$u(t,x) = f(t+3x) + g(t+4x)$
But, I don't know how to proceed. All I have is:
$f(0) + g(t+4x) = ... | $u_{xx} - 7u_{tx} + 12u_{tt} = 0$ ...$(i)$
$u(t,x) = sinx$ on $t+3x=0$ ...$(ii)$
$u(t,x)=x$ on $t+4x=0$ ...$(iii)$
First of all, we observe from equations $(ii)$ and $(iii)$ that $u(t, x)$ is independent of $t$ for all $t$ and $x$, hence the second and third term of equation $(i)$ are zero.
differentiat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2571458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is $f(m,n)=2^m\cdot(2n+1)$ a bijection between $\Bbb{Z_{\geq0}\times Z\to Z}$?
Let $\mathbb Z$ denote the set of integers and $\mathbb Z_{\ge 0}$
denote the set $\{0,1,2,3,...\}$. Consider the map $f:\mathbb Z_{\ge
0}\times \mathbb Z \to \mathbb Z$ given by $f(m,n)=2^m\cdot(2n+1)$. Then
the map $f$ is
(A)injecti... | Hint:
Note that $0$ is not mapped by any pair of $(m,n)$ (as $2^m \geq 1$ for $m \in \mathbb{Z}_{>0}$). So, $\implies \, ?$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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weird brackets in unit interval I have found during looking at the book Linear Algebra and its Applications (K. Nordstrom) some weird (for me) notation for belonging to the unit interval, namely, $\lambda \in \ ] 0,1 [$. Does it mean as always that $\lambda$ belongs to $[0,1]$ or something different?
| The notation $]a,b[$ is used for an open interval, more commonly written as $(a,b)$; meaning:
$$x \in \; ]a,b[ \; \iff a \color{red}{<} x \color{red}{<} b$$
whereas:
$$x \in [a,b] \iff a \color{blue}{\le} x \color{blue}{\le} b$$
So $\lambda \in \; ]0,1[$ would mean values satisfying $0<\lambda<1$, excluding the end po... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2571747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to show it is a rhombus I am trying to solve question 2 (figure 2). I have shown that the diagonals are interesting each other in right angle but I cannot show that AB||GH. Please help.
| Using Alternate Interior Angles
$$\angle GBH=\angle AHB$$ and $$\angle BHG=\angle ABH$$
Again, $$\angle ABH=\angle GBH$$
$$\implies\angle ABH=\angle AHB\implies AB=AH\ \ \ \ (1)$$
Similarly $BG=GH$
In $\triangle ABH, \triangle GBH$ $$\angle ABH=\angle GBH,\angle AHB=\angle GHB$$
and $BH$ being the common side
Using... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2571882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
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Steiner inellipse Hello it's related to my answer for Prove the inequality $\frac{b+c}{a(y+z)}+\frac{c+a}{b(z+x)}+\frac{a+b}{c(x+y)}\geq 3\frac{a+b+c}{ax+by+cz}$
My answer fails but I don't know why ... So I was thinking a generalization of the following formula:
$$\frac{IA^2}{CA\cdot AB}+\frac{IB^2}{BC\cdot AB}+\frac... | Sums $$CA\cdot AB+BC\cdot AB+CA\cdot BC = s^2+(4R+r)r$$ and
$$\frac{1}{CA\cdot AB}+\frac{1}{BC\cdot AB}+\frac{1}{CA\cdot BC}=\frac{1}{2rR}$$
are not related with Steiner inellipse.
Two remaining sums are based on the products of distances from the foci $P$ and $Q$ of the Steiner inellipse of the triangle $\Delta ABC... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2571971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Magnitude of $f_3(n)$ compared to power towers of tens In the fast growing hierarchy , the sequence $f_2(n)$ is defined as $$f_2(n)=n\cdot 2^n$$
The number $f_3(n)$ is defined by $$f_3(n)=f_2^{\ n}(n)$$
For example, to calculate $f_3(5)$, we have to apply the operator $n\cdot 2^n$ five times with start value $5$.
Denot... | We will use $\log$ to mean logarithm base $10$.
Observe that $f_2^k(n) > (2^n \uparrow\uparrow k)n$, as a simple induction shows. Therefore $f_3(n) > 2^n \uparrow\uparrow n$. Since $2^n > 10$ for $n \ge 4$, we thus have
$$\log^{n-2} f_3(n) > 2^{n 2^n}$$
In the other direction, we will show that
$$\log^{n-2} f_3(n) < ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2572111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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$\vec{PP_1}+\vec{PP_2}+\vec{PP_3}=\frac {3}{2}\vec {PO} $. Let $ABC $ an equilateral triangle and $P\in int (ABC) $.
If $O $ is the center of gravity of the triangle and $P_1, P_2, P_3$ are the projections of $P $ on the sides then
$\vec{PP_1}+\vec{PP_2}+\vec{PP_3}=\frac {3}{2}\vec {PO} $.
| Let
$A_1C_2||AC $, $A_2B_1||AB $ and $B_2C_1||BC $ trough $P $
where $A_1,B_2\in [AB] $, $A_2, C_1\in [AC ]$ and $B_1, C_2\in [BC] $.
Then $PA_1AA_2$, $PC_1CC_2$, $PB_1BB_2$ are parallelograms and $A_1PB_2$, $A_2PC_1$, $B_1PC_2$ are equilateral triangles.
Now,
$\vec {PP_1}=\frac {\vec{PB_1}+\vec{PC_2}}{2}$
$\vec {PP... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2572278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Lebesgue measure - natural measure on $S^1$ I know the definition of the Lebesgue measure for subsets of $\mathbb{R}$. I have heard that the Lebesgue measure is the "natural" measure to put on the circle, $S^1$. Why is this so?
How is the Lebesgue measure defined for $S^1$ in $\mathbb{R}^2$ or $\mathbb{C}$? Is the ide... | The Lebesgue measure on $S^1$ can be viewed as the Hausdorff measure of it as a subset of $R^2$ with the 2D Lebesgue measure $m_2$. Alternately it can be viewed as the pushfoward measure $\mu(A)=m_1(f^{-1}(A))$ where $f(t)=(\cos(t),\sin(t)$ with domain $[0,2\pi)$. Either way, the measure of an arc is the length of the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2572341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Understanding Distributions I have been studying distributions, and I am still trying to get the intuition behind the following:
$1)$ Suppose $f\in L^1_{loc}(\mathbb{R})$, and $\phi\in C^{\infty}_c(\mathbb{R})$. A distribution is defined as $\langle T_f, \phi \rangle=\int_{\mathbb{R}}f\phi dx$.
Suppose we are given an... | For 1) there is not a distribution for every $\phi$. The distribution is the linear functional
$$\phi \longmapsto \int_{\mathbb R} f \phi dx$$
This functional is denoted $T_f$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2572452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Proving that if $c^2 = a^2 + b^2$ then $c < a + b$ I am having some trouble in proving a conjecture that occurred to me some time ago, based on the Pythagorean theorem.
If, for a non-degenerate triangle, $$c^2 = a^2 + b^2$$
Then can the following be proven?
$$c < a + b$$
Is this statement always true?
| For all $a,b \in \mathbb{Z}^+$ we have $a^2 + b^2 < (a+b)^2 \implies \sqrt{a^2+b^2} < a+b$. If $c^2$ is defined to be $c^2 = a^2 + b^2$, then we have $c < a+b$. This also explains why the triangle cannot be degenerate, i.e. $a = 0$ or $b = 0$, in order for statement to be true.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2572505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Let A be an n by n matrix with the property that $A^TA = AA^T$ Show that $\left\Vert A^Tx \right\Vert = \left\Vert Ax \right\Vert$ for $x$ in $\mathbb{R}^n$.
So this is what I did. I need someone to check my work.
1) I squared the left hand side of the equation
$$\left\Vert A^Tx \right\Vert^2$$
2) I then applied the ru... | $\Vert A^Tx \Vert^2 = \langle A^T x, A^T x \rangle = \langle x, (A^T)^T A^T x \rangle = \langle x, AA^T x \rangle$
$= \langle x, A^T Ax \rangle = \langle (A^T)^T x, Ax \rangle = \langle Ax, Ax \rangle = \Vert Ax \Vert^2, \tag 1$
whence
$\Vert A^T x \Vert = \Vert Ax \Vert. \tag 2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2572646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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What is the answer to this infamous "Common Core" question? The following question (number 15 of this test) has become infamous as a poor "Common Core" question. What is the correct answer?
Juanita wants to give bags of stickers to her friends. She wants to
give the same number of stickers to each friend. She's not ... | The question is terribly worded. Here is another solution which is consistent with the wording:
The only way Juanita can be unsure whether she needs 4 or 6 bags is if she has exactly 1 or 2 friends; any other number of friends does not divide evenly into both 4 and 6 bags and she would know in those cases that at least... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2572757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 7,
"answer_id": 4
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In showing the convergence of a sequence, does it matter how one chooses $n$? I am doing an exercise in Abbott's Understanding Analysis textbook (question 2.2.1.b).
I want to show that: $\displaystyle\lim\frac{3n+1}{2n+5}=\frac{3}{2}.$
Now, i've begun by first working with the inequality I need: $\displaystyle\left|... | Either one would be fine. As long as you prove that for sufficiently large values of n, $\frac{3n+1}{2n+5}<\epsilon$ then you're done. It doesn't matter how large $n$ needs to be. As long as it is proved for sufficiently large $n$, no matter what the bound is the proof is fine.
However, on re-look there is a mistake in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2572972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculation of Chern number of $U(N)$ principal bundle I am considering $U(N)$ principal bundle on a two-dimensional sphere $S^2$.
(Below the fiber is $N\times N$ unitary matrices for simplicity, and any generalization to general representations will be appreciated.)
The bundle is defined by the transition function $... | Yes. And you can do better.
Observe that the clutching function is determined by its homotopy class $t\in \pi_1(U(N))$. Note that there is a homeomorphism $U(N)\cong U(1)\times SU(N)$. Since $SU(N)$ is simply connected we get $\pi_1(U(N))\cong \pi_1(U(1))$, so we can choose $t$ to factor through the inclusion $U(1)\ho... | {
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Prove that $3^n - 4(2^n) + (-1)^n + 6 \equiv 0 \mod 24 $ Is it possible to prove that $3^n - 4(2^n) + (-1)^n + 6 \equiv 0 \mod 24 $ for $n \geq 1 $ . I know that it is true because $ \frac{3^n - 4(2^n) + (-1)^n + 6}{24}$ represents the number of ways to uniquely $4$-colour an n-cycle , excluding permutations of colours... | Using weak induction,
if $f(n)=3^n - 4(2^n) + (-1)^n + 6,$
$$f(m+2)-f(m)=3^m(3^2-1)-4\cdot2^m(2^2-1)$$ which is clearly divisible by $3\cdot8$ for $m\ge1$
So, $24\mid f(m)\iff24\mid f(m+2)$
Now establish the base cases $f(1),f(2)$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 1
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Estimating distance to travel to each household in a county TL;DR: What's the minimum distance one has to travel to visit each node in a square matrix in which the nodes are D distance apart?
I ask because I'm working on a fun feature on how long it takes Santa to reach every child in the country, using a Travelling Sa... | If you have $N$ nodes in a square matrix it takes $N-1$ times the distance between neighboring nodes to visit them all. You can't get from one to another in less distance and a snaking path will keep you at that minimum. Your grid is $\sqrt N \times \sqrt N$ so the area is $N$ times the square of the spacing, which g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2573350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integral over solid angle in Cartesian coordinates I have an integral that is an average of some (unknown) function $f$ over solid angle:
$$\bar{f} = \frac{1}{4\pi} \iint\limits_\Omega f \sin\theta~\mathrm{d}\theta~\mathrm{d}\phi$$
I use the physics convention where $\theta$ is the polar angle and $\phi$ is the azimuth... | The most obvious choice for the surface is a sphere of radius $1$. For the sphere $\hat n=\hat r$, so $\hat r\cdot\hat n=1$. Also $r=1$, so $1/r^2=1$ on the surface. Note that you still want to use your expression for $\vec F$.
| {
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Calculating $\int_0^\pi \log(1-2a\cos (x)+a^2)\,dx$ I need to calculate this integral using Riemann sum.
$$\int_0 ^\pi \log(1-2\alpha \cos (x) +\alpha^2){\rm d}x$$
a). For $|\alpha|<1$; b). For $|\alpha| > 1$.
I know one way of computing this using substitutions and symmetries, but it is necessary to do with Riemann su... | I don't think it's possible to solve such a hard integral with a Riemann sum.
Anyway, by Cauchy's theorem we have $$I=\oint_{|z|=1} \frac{\log(a-z)}{z}dz = 2\pi i\log(a).$$
(assuming $|a| > 1$ so that the branch point is outside the contour). Now let's parametrise the function using $z= e^{it}$ for $0 \leq t \leq 2\pi$... | {
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Turning sum to integral representation I'd like to turn this sum:
\begin{align}\sum_{n=0}^{\infty} \frac{x^{n+1}}{3^{n+1}(n+1)} \end{align}
into an integral $\displaystyle \int_{a}^{b} g(x) \space dx$.
There seems to be many methods to either change or approximate sums as integrals. So I've become confused which appro... | Well, you could write $$\frac{1}{n+1} = \int_0^1 t^n\; dt$$
so (for $|x| < 3$) your sum becomes
$$
\eqalign{\sum_{n=0}^\infty &\left(\frac{x}{3}\right)^{n+1} \int_0^1 t^n\; dt\cr
= & \frac{x}{3} \int_0^1 \sum_{n=0}^\infty \left(\frac{xt}{3}\right)^n \; dt\cr
= & \frac{x}{3} \int_0^1 \frac{dt}{1-xt/3}\cr
= & \ln\left(... | {
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Show that $f\left(\frac{x_1+x_2}{2}\right)\leq\frac{f(x_1)+f(x_2)}{2}$ Let $f$ be twice differentiable in $(a,b)$ and $f''>0$ in the same interval. If $a<x_1<x_2<b,$ show that $$f\left(\frac{x_1+x_2}{2}\right)\leq\frac{f(x_1)+f(x_2)}{2}.$$
I'm not sure how to start. I know at least that $f'$ is strictly increasing in $... | What you want here is the convexity of $f$. Since you know that $f'$ is monotonically increasing, I will outline a proof for you of how to show $f$ is convex.
Proof outline. Let $a< x < y < b$ and consider the slope $m$ of the line $L$ joining the points $(x,fx)$ and
$(y,fy)$. By the mean value theorem, there exists a ... | {
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Minimize interpolation error for sin$(x)$ on $[0,\pi]$ The interpolation error $R(x) = f(x) - L(x)$ of the interpolation polynom $L$ is given for $x_1 \le x \le x_n$ by
$$R(x) = {f^{n}(\xi) \over n!} \prod_{i=1}^n (x - x_i)$$ where $x_1 < \xi < x_n $ if $x_i$ is sorted in a ascending order. Find (two) sampling points ... | Your idea in general fails because $\xi$ is a function of $x$.
The equioscillation theorem tells you that the minimizer is uniquely characterized by the requirement that there be three points a,b,c where the error is the same value but alternating in sign. By the symmetry this can be achieved by simply taking p to be 1... | {
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Derive Barycentric coordinate distance formula please pardon the poor formatting. (I'll work on learning it in time; I just started this account to see help with this question.)
I've recently started learning about affine geometry and Barycentric coordinates, and I have a question regarding the distance formula for Bar... | This is a nice problem. First, we assume we are in an affine plane over an inner product space such as the Euclidean plane. This means that the (inner) dot product defines a distance measure of line segments by $\;\textrm{dist}(P,Q)^2 = |PQ|^2 = (Q-P)\cdot(Q-P).\;$ Now given a triangle of reference $ABC$ with sides $\... | {
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Show that $\alpha^3 = 10\alpha - 24$ given that $\alpha + \beta = 4$ and $\alpha \beta = 6$ How to show that $\alpha^3 = 10\alpha - 24$ given that $\alpha + \beta = 4$ and $\alpha \beta = 6$
As given in the title. I tried shifting the RHS to the left but I'm not sure if factorization is the way to go. There are earlier... | Sum of roots = $ \alpha + \beta + \gamma =$
$ \frac{-b}{a} = \frac{-0}{1} = 0 $
But, $ \alpha + \beta = 4 $ $\implies \gamma = -4 $
Product of roots $\alpha * \beta * \gamma = $
$ \frac{-d}{a} = -24 $
Sum of roots taken two at a time = $ \alpha\beta + \beta\alpha + \gamma\alpha =$
$ \frac{c}{a} = \frac{10}{1} = 10 $
... | {
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Evaluate perfect $\sqrt[3]{X}$
Is there a method of working out the perfect cube root of a 3 digits number?
Working out the perfect cube root of a 2 digits number which I know.
An example of $$\sqrt[3]{12167}$$
*
*Cross out $16$ (always two digits before the last digit)
*take the cube root of $12$ is bewtween $2$... | Solving equations is equivalent to root finding (as in points where a function equals, not the kind of root in the question).
For any monotonic function a good way to find an integer root if you know the number of digits is to progressively start at the maximal digit and increase one by one until you find it changes si... | {
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In the triangle $ABC$ $R = \frac56 BH = \frac52OH$. Find the angles $ACB$ or $BAC$
In the triangle $ABC$, the height $BH$ is drawn, the point $O$ is the center of the circle circumscribed about it, the length of its radius $R$. Find the smallest of the angles $ACB$ and $BAC$, expressed in radians, if it is known that ... | The hint.
In $\Delta HOB$ we know that $OH=\frac{2}{5}R$, $BH=\frac{6}{5}R$ and $BO=R$.
Thus, by law of cosines we obtain $$\cos\cos\measuredangle HBO HBO=\frac{1+\frac{36}{25}-\frac{4}{25}}{2\cdot\frac{6}{5}},$$
which gives
$$\cos\measuredangle HBO=\frac{19}{20}.$$
In another hand, $\cos\measuredangle HBO=|\alpha-\gam... | {
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Why can we simply substitute the constrain in when maximizing this equation? I am working on some Portfolio Analysis related material at the moment, and am trying to understand why the following approach to solving this maximization problem is correct:
$\max _x \frac{\mu' x-r_f}{(x' \varSigma x)^{0.5}} $ subject to $\m... | There is no need for it to sum up to $1$. The problem is homogeneous in $x$ meaning that if you replace $x$ with $tx$, the objective is the same. Hence, once you have a solution to the problem, you can always scale it suitably afterwards. The only important thing is the relative size of the elements in the allocation.
| {
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Help with trigonometric proof
Show that $\frac {\cos (24)}{\cos {6} }+2\times \sin {24}=\sqrt {3}$
| We need to prove that
$$\cos24^{\circ}+2\sin24^{\circ}\cos6^{\circ}=2\sin60^{\circ}\cos6^{\circ}$$ or
$$\cos24^{\circ}+\sin30^{\circ}+\sin18^{\circ}=\sin66^{\circ}+\sin54^{\circ}$$ or
$$\sin66^{\circ}+\frac{1}{2}+\sin18^{\circ}=\sin66^{\circ}+\sin54^{\circ}$$ or
$$\sin54^{\circ}-\sin18^{\circ}=\frac{1}{2},$$ which is t... | {
"language": "en",
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Is it closed and compact? $A= f(B) \subseteq X$ where $B =\{(x,y)\in \mathbb{R}^2 |1≤ x^2 +y^2 ≤ 2\}$. $X$ is an arbitrary topological space and $f :\mathbb{R}^2 \rightarrow X $ is an arbitrary continuous map.
which of the following is correct.
*
*open
*closed
*compact
*connected
I know that continuous image ... | $B$ is compact $\implies$ $f(B)$ is compact $(\because$ continuous image of compact set is compact.)
| {
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Find $\lim_{x \to 0} (\frac{\tan(x)}{x})^{\frac{1}{x}}$
Find $$\lim_{x \to 0} (\frac{\tan(x)}{x})^{\frac{1}{x}}$$
My first idea to solve this was to try to evaluate it and then apply the L'Hospital's rule. This is what I managed to achieve:
$$\left(\frac{\tan(x)}{x}\right)^\frac{1}{x}=e^{\frac{1}{x}\ln\left(\frac {\t... | If you want to use L'Hospital's rule, I would note that the limit of these two functions are equal:
$\frac{ln(\frac{tanx}{x})}{x}$ and $\frac{(tan^2x+1)x-tanx}{xtanx}$, and then rewrite the second term as $tanx + \frac{x-tanx}{xtanx}$, noting that $tanx$ is eliminated. It remains to find $\frac{x-tanx}{xtanx}$. But by ... | {
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Inverse of sum of nilpotent matrix and identity matrix Suppose $A$ is a $n\times n$ nilpotent matrix of index $m$. i.e. $A^m=0$ but $A^{m-1}\neq0$.
Construct $M=A+\lambda I_n$. It is known that $M^{-1}=\sum_{i=1}^{m}\lambda^{-i}(-A)^{i-1}$
I want to prove $M^{-1}= B+\lambda^{-1}I_{n}$, where B is a nilpotent matrix of ... | Hint Writing out explicitly the first few terms of your summation expression gives $$M^{-1} = B + \lambda^{-1} I_n = \lambda^{-1} I_n - \lambda^{-2} A + p A^2$$
for some expression $p$ polynomial in $A$. So, we can write $B^{m - 1}$ as
$$B^{m - 1} = (- \lambda^{-2} A + p A^2)^{m - 1} .$$
When expanding this expressio... | {
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Invertible 4x4 matrix $$
\begin{pmatrix}
5 & 6 & 6 & 8 \\
2 & 2 & 2 & 8 \\
6 & 6 & 2 & 8 \\
2 & 3 & 6 & 7 \\
\end{pmatrix}
$$
Is this matrix invertible? I would like to show that it is invertible but first I should find the det(Matrix) which should not be equal to zero. To find the determinant, ... | You may find interesting to use Hamilton-Caley Formula, for $4x4$ matrices:
$$\text{det}\mathsf{A} = \frac{1}{24}\left\{(\text{tr}\mathsf{A}^4 - 6\text{tr}(\mathsf{A}^2)(\text{tr}\mathsf{A})^2 + 3(\text{tr}\mathsf{A})^2 + 8\text{tr}(\mathsf{A}^3)\text{tr}\mathsf{A} - 6\text{tr}(\mathsf{A}^4)\right\}$$
| {
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How can I define a span-preserving, linear matrix tansformation? I have a matrix $M \in \mathbb{R}^{n \times m}, n \gt m \geq 3$, where every column of $M$ is mutually orthogonal (so rank($M$)=$m$).
I want to tranform $M$ in such a way that the span of $M$ is preserved. Intuitively, this is a rotation, but when I've i... | First of all, it may interest you to know that $\operatorname{span}(MR) = \operatorname{span}(M)$ as long as $R$ is invertible (of size $m \times m$).
To answer the question that you posed, namely find the family of matrices $R \in \mathbb{R}^{n \times n}$, such that span($RM$)=span($M$) for arbitrary dimension, here i... | {
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related rates sphere volume and area calculus problem I have been given the following problem:
A spherical balloon is expanding at the rate of 60 pie in^3/sec. How fast is the surface area of the balloon expanding when the radius of the balloon is 4 inches?
I don't understand how the way I set the problem up is not ... | We have that $$\frac{\mathrm dV}{\mathrm dt} = 4 \pi r^2\cdot \frac{\mathrm dr}{\mathrm dt} =60$$
which implies $$\frac{\mathrm dr}{\mathrm dt} =\frac{60}{4 \pi r^2}$$
We know that $A = 4 \pi r^2$, and so we also have
$$\frac{\mathrm dA}{\mathrm dt} = 8\pi r\cdot \frac{\mathrm dr}{\mathrm dt} $$
Plug in what we found f... | {
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$3$ mutually tangent circles $\textbf{Problem}$: In $\triangle{ABC}$, $AB=3$, $BC=4$, and $CA=5$. Additionally, we have mutually tangent circles $X$, $Y$, and $Z$ inside the triangle that are tangent to $\{AB, BC\}$, $\{BC, CA\}$, and $\{CA, AB\}$ respectively. Determine the sum of the radii of the circles $X$, $Y$, an... | The hint.
Let $a$, $b$ and $c$ be radius of circles.
Thus, we need to solve the following system.
$$a\sqrt5+2\sqrt{ab}+b=3,$$ $$3c+2\sqrt{bc}+b=4,$$ $$a\sqrt5+2\sqrt{ac}+3c=5.$$
| {
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$(x_n)$ real, $(x_n) \to 0$ and $0So, I'm supposed to show that:
Given $(x_n)$ a sequence of real numbers, with $(x_n) \to 0$, and given $0<c<1$, then $(y_n) \to 0$, where $y_n = c^n x_0 + c^{n-1} x_1 + ... + c^0 x_n$.
Here's my attempt, and I'd appreciate any corrections:
Since $|y_n| < c^n |x_0| + c^{n-1} |x_1| + ...... | Let me suggest a totally different approach.
Suppose that
$$
f(z)=\sum_{n=0}^\infty x_n z^n, \quad g(z)=\sum_{n=0}
^\infty c^nz^n, \quad\text{and then}\quad h(z)=f(z)g(z)=\sum_{n=0}^\infty y_nz^n,
$$
where $\,y_n=x_n+cx_{n-1}+\cdots+ c^n x_0$.
Then, the radius of converge of $f$ and $g$ is at least 1, and hence at leas... | {
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A complicated limit involving floor function Let $f(x) = \lfloor x\lfloor1/x\rfloor \rfloor $ . Find $\lim_{x \to 0^{+} } f(x) $ and $\lim_{x \to 0^{-} } f(x)$ . I think $\lim_{x \to 0^{+} } f(x)$ doesn't exist but I have no idea about $\lim_{x \to 0^{-} } f(x)$ .
| Check the following graph:
You will get the answer automatically.
https://www.desmos.com/calculator/8p190y7wr2
Hint: The limit as a whole is not defined, because negative limit is not equal to positive limit.
| {
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For $\alpha$ a limit ordinal, show $V_{\alpha}=\bigcup_{\beta\lt\alpha}P(V_{\beta})$ For $\alpha$ a limit ordinal, I would like to show
$V_{\alpha}=\bigcup_{\beta\lt\alpha}P(V_{\beta})$
where the $V$'s are members of the cumulative hierarchy and $P$ is the power set.
(This is a continuation of Showing equivalence ... | The main idea for the reverse inclusion, I think, is that $V_\beta \subset P(V_\beta)$ because $V_\beta$ is transitive.
For the first inclusion, what you said was enough, also perhaps you should say more clearly that (with your notations) $\beta' = \beta +1 $, so that $\beta < \alpha \implies \beta' < \alpha$ (as $\al... | {
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Non-split exact sequence of modules
Find a module $M$ and a submodule $N$ such that $|M| = 100$, $|N| = 20$, $M$ is not cyclic, and the exact sequence $0 \rightarrow M \rightarrow N \rightarrow M/N \rightarrow 0$ does not split.
I have tried $M = Z/2 \oplus Z/2 \oplus Z/5 \oplus Z/5$ and $N = Z/2 \oplus Z/2 \oplus Z... | We'll work with $\mathbb Z$-modules, i.e., abelian groups. Note $|M/N| = |M|/|N| = 5$ so $M/N$ is going to be cyclic of order $5$. Thus we can guarentee that the sequence does not split if $\mathbb Z/5$ is not a direct summand of $M$.
One group that has $\mathbb Z/5$ as a quotient but not as a summand is $\mathbb Z/2... | {
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Let $\phi:\Bbb Z_n\rightarrow G$ s.t. $\phi(i)=h^i$ for $0\le i\le n$. Give necessary and sufficient condition for $\phi$ to be homomorphism. The exercise reads
Let $G$ be a group, $h$ and element of $G$, and $n$ a positive integer. Let $\phi : \mathbb{Z}_n\rightarrow G$ be defined by $\phi(i)=h^i$ for $0\leq i\leq n$... | This community wiki answer is to point out that this comment followed by this comment, both posted above by @Timkinsella (who is invited to post his own answer), form an answer to the question.
Summarising them:
1) Prove $\tilde{\phi}: \Bbb Z\to G, i\mapsto h^i$ is always a homomorphism.
2) Think of the condition on $\... | {
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Difference between changing coordinates and changing basis? If we have a vector $v$ then is has some coordinates w.r.t to a basis $B$ say $(a,b)$. What if we transform this into "polar coordinates"? Does this change the basis or just transform the coordinates into some new pair? Giving us new coordinates for the same v... | Changing coordinates linearly allows you to think of this as a change of basis on the whole Euclidean space. If you do some non-linear change of coordinates, like to polar, there is no such way to think about the coordinates themselves.
However, there is a notion of 'tangent space' to a point, which is a vector space o... | {
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Problem with proof of Stone Representation Theorem I am reading Thomas Jech's Axiom of choice in which he gives a concise proof of the Stone Representation theorem using the (Boolean) Prime Ideal theorem. However he states something is trivial, which I am really struggling to see. I shall quote him verbatim:
Stone Repr... | Let $U\in S$ and $u,v\in B$. If $u,v\in U$, then $u\cap v\in U$ since $U$ is a filter. Conversely, if $u\cdot v\in U$, then $u\in U$ and $v\in U$, since $u\cdot v\leq u$ and $u\cdot v\leq v$ and $U$ is a filter. Thus $\pi(u\cdot v)=\pi(u)\cap\pi(v)$. The other equation then follows formally: $$\pi(u+v)=\pi(-(-u\cdo... | {
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Why does $(g^a \bmod n)^b = (g^b \bmod n)^a = g^{ab} \bmod n $? The Diffie–Hellman key exchange protocol relies on the fact that one person, Alice, can perform $(g^a \bmod n)^b $ and another person, Bob, can perform $(g^b \bmod n)^a $ and they will both arrive at the same number: $g^{ab} \bmod n $. This allows Alice an... | It seems that you are thinking of 'mod' as an operation, such as might be found in a programming language. If, instead, you restate the above identity in terms of congruences, it might be clearer:
$$(g^a)^b \equiv g^{ab} \equiv (g^b)^a \pmod n$$
| {
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Take complex roots and apply it to a new polynomial Polynomial $f(x)=x^3-x^2+x+18$ has three distinct complex roots $r_1$,$r_2$, and $r_3$. Denote by $g(x)$ the cubic polynomial with leading coeffecient $1$ such that $g(r_i+\frac{1}{r_i})=0$, for $i=1,2,3$. The value of $g(2)$ can be expressed in the form $\frac{m}{n}$... | You might start with $$g(x) = (x - (r_1 + 1/r_1))(x - (r_2 + 1/r_2))(x - (r_3 + 1/r_3))$$
EDIT:
So $$ \eqalign{g(2) &= \frac{(r_1 - 1)^2 (r_2 - 1)^2 (r_3 - 1)^2}{r_1 r_2 r_3} \cr
&= \frac{f(1)^2}{r_1 r_2 r_3}}$$
| {
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How to calculate the proficiency based on an exam with questions of different difficulty levels? In an examination (multiple choice test), suppose there are n questions each question can have a difficulty level between 1 - 5 (1 for the easiest question and 5 for the difficult one).
If someone answers x questions corre... | Alright, so we have question $i$ with difficult level $X_i$, $i=1,2,3,\ldots,n$.
Say the set $S$ is the collection of the question indices of the questions that are answered correctly.
So the average difficulty level of the questions answered correctly, $P$, is
$$P=\frac{\sum_{i\in S}X_i}{\mathrm{n}(S)}$$
where $\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2576512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
For which complex numbers $\alpha$ and $\beta$ is it true that $\alpha^n+\beta^n$ is always an integer? Possibly a very straightforward question, but:
Question. For which complex numbers $\alpha$ and $\beta$ is it true that $\alpha^n+\beta^n$ is always an integer for all $n=1,2,3\ldots$?
For example, $$\alpha = \frac... | This is certainly true if $\alpha$ and $\beta$ are conjugate quadratic integers because $\alpha^n+\beta^n$ is a symmetric function of $\alpha$ and $\beta$ and so is an integer polynomial expression in $\alpha+\beta$ and $\alpha\beta$.
Conversely, if $\alpha+\beta$ and $\alpha^2+\beta^2$ are integers, so is $2\alpha\bet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2576633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
Find all $x^6 \pmod {17}$
Find all $a$ such that $x^6 \equiv a\pmod {17}$ (not including $0$)
First I thought that we could look at $y^2 \equiv a \pmod {17}$, where $y=x^3$.
Then, by Euler's criterion, it must be that:
$$a^{\frac{17 -1}{2}} \equiv a^8 \equiv 1 \pmod {17}$$
I could develop it to
$$a^8 -1 \equiv 0 \pmo... | The group of nonzero residues modulo $17$ is not merely cyclic, it is cyclic of order $16$, which is relatively prime to $3$. This means that every residue has a unique cube root. As you very perceptively recognized, solving $x^6=a$ is the same as solving $y^2=a$; but you didn’t realize that given such a $y$, there is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2576719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Injectivity and range of $\arctan\left(\sqrt{\frac{1+x}{1-x}}\right)$ To prove that $f(x)=\arctan\left(\sqrt{\frac{1+x}{1-x}}\right)$ is injective I have tried the following:
According to the definition, for a function to be injective $f(a)=f(b) \to a=b$ for all $a,b \in D_f$. Using this I get:
$$\arctan\left(\sqrt{\fr... | The domain is $-1\leq x<1$, $\lim\limits_{x\rightarrow1^-}f(x)=\frac{\pi}{2}$, $f(-1)=0$ and $f$ is a continuous function.
Thus, the range is $\left[0,\frac{\pi}{2}\right).$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2576789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Epsilon delta continuity I know that there are a lot of similar questions like this on this forum but still I can't figure it out one thing of this definition.
Definition: f is continuous at $x_o\in X\subset\mathbb{R}$ if
$$\forall \epsilon>0 \exists \delta>0 |\forall x| |x-x_o|<\delta\Rightarrow|f(x)-f(x_o)|<\epsilon$... | Please try to remember that there is no real number at all that tends to zero. You can't write, in standard analysis, anything like "consider a number $x \to 0$."
The very definition of limit actually gives the piece of notation "$f(x) \to L$ as $x \to x_0$ " a meaning by using quantifiers: for every $\epsilon>0$ ther... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2576902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
find the $\sum\limits_{k=0}^n \frac{(-1)^k}{k! (2k+1)} \frac{1}{(n-k)!}$ I'm stuck on computing the sum of
\begin{align*}
\sum\limits_{k=0}^n \frac{(-1)^k}{k! (2k+1)} \frac{1}{(n-k)!}
\end{align*}
I tried some manipulations which include
\begin{align*}
\frac{1}{n!} \binom{n}{k} = \frac{1}{k! (n-k)!}
\end{align*}
but st... | Start with the binomial theorem:
$$\sum_{k=0}^n \binom{n}{k}x^n=(x+1)^n$$
Substitute $x=y^2$:
$$\sum_{k=0}^n \binom{n}{k}y^{2k}=(y^2+1)^n$$
Integrate both sides:
$$\sum_{k=0}^n \binom{n}{k}\frac{y^{2k+1}}{2k+1}=\int_0^y(t^2+1)^ndt$$
Divide across by $n!$:
$$\sum_{k=0}^n \frac{1}{k!(n-k)!}\frac{y^{2k+1}}{2k+1}=\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2576997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Best way to fill a $8\times 8$ board You have an $8\times 8$ Battleships board and need to place battleships of sizes $1\times 1$, $1\times 2$, $1\times 3$, $1\times 4$, $1\times 5$ on the board to cover as much of the board as possible. The ships cannot touch another ship, even at the corners.
You can place as many of... | EDIT:
It looks as though you can get 32:
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2577095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Existence of a differentiable function satisfying a condition Show that there exists a real number $\epsilon>0$ differentiable function $f:(-\epsilon, \epsilon)\rightarrow \mathbb{R}$ such that
$$e^{x^2+f(x)}=1-\sin(x+f(x)).$$
I have no idea about how to proceed. Any ideas?
| Hint : apply the Implicit function theorem to the function
$$F(x,y)=e^{x^2+y}-1+\sin(x+y).
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2577217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Best way to show that this is a subvariety of the Grassmannian Let $k$ be a field and let $\text{Gr}(r,k^n)$ be the Grassmannian of $r$-dimensional subspaces of $k^n$. Fix a linear map $\phi\colon k^n \rightarrow k^n$. Define $X_{\phi}$ to be the subspace of $\text{Gr}(r,k^n)$ consisting of $r$-dimensional subspaces ... | Let $0 \to U \subset O^{\oplus n} \to Q \to 0$ be the tautological exact sequence of bundles on the Grassmannian. Consider the composition
$$
U \to O^{\oplus n} \stackrel\phi\to O^{\oplus n} \to Q.
$$
Then $X_\phi$ is the zero locus of this morphism, hence is a closed subscheme.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2577428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If H/K is normal in G/K, then is H normal in G? Let $G$ be a group and $H, K$ are two subgroups of it s.t. $K$ is normal in $G$. Now if $H/K$ is a normal subgroup in $G/K$, then can we say $H$ is a normal subgroup of $G$?
| Yes, this is the Lattice or fourth isomorphism theorem. $H$ is normal in $G$ if and only if $H/K$ is normal in $G/K$ for $K$ a normal subgroup of $G$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2577542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Equation involving modulus Solve the equation ,
$|x+1| + |x-2| = |2x-1|$
This was the solution given in my book ,
$|x+1| + |x-2| = |2x-1|$
$(x+1)(x+2) ≥ 0$
Hence , $x≤-1 , x≥2$
However I couldn’t understand the second step. How did they just factorise the equation that was in the modulus ? I initially thought of s... | If you feel yourself lost you can always try cases:
$$\begin{align*}\bullet\;&x<-1&\implies& -(x+1)-(x-2)=-(2x-1)\implies1=1\implies \color{red}{x<-1}\\{}\\
\bullet\;&-1\le x<\frac12&\implies& (x+1)-(x-2)=-(2x-1)\implies3=-2x+1\implies \color{red}{x=-1}\\{}\\
\bullet\;&\frac12\le x<2&\implies&(x+1)-(x-2)=(2x-1)\implies... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2577651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Counting question / solution verification
In a $100$ day period, each of six friends goes swimming on exactly $75$ days. There are $n$ days on which at least $5$ friends swim. Find the largest and smallest possible values of $n$.
My attempt at a solution:
The largest value of $n$ occurs when exactly $5$ friends swim ... | There are $6 \cdot 75 = 450$ swimmer-days, so there is an obvious upper bound to the number of 5-swimmer days of $\frac{450}{5}=90$. Your schedule is a concrete example of such, so 90 days must be the maximum.
A similar argument can be made about the lower bound. There you want to maximize the number of days in which... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2577783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Pivots and linear independence I am doing linear algebra and I am a begginer. A thought struck on my mind that I need to work out that what is the relationship between pivots, pivot columns and linear independence?
As far as I perceived, if a matrix is in a row Echelon form, then the first non-zero entry (not necessar... | Pivot columns are linearly independent with respect to the set consisting of the other pivot columns (you can easily see this after writing it in reduced row echelon form).
This means that if each column is a pivot column, all columns are linearly independent. The converse is also true.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2577895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What are the products of partition numbers? In Neil Sloane's, On Line Encyclopedia of Integer Sequences, A033637 is the sequence whose first few terms are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, \
24, 25, 27, 28, 30, 32, 33, 35, 36, 40, 42, ...
There is no description of what is meant by th... | The sequence is literally what the title says.
First $n$-th partition number (A000041) is the number of integer partitions of $n$. Let $A$ be the set of all partition numbers.
Now let $B$ be the set of all numbers that can be written as the product of elements of $A$. For example since $2, 3 \in A$, $2*3 = 6 \in B$.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2578014",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding the global maxima and minima on closed interval Given a function $f(\theta) = (4\theta-\pi)\sin(\theta)+4\cos(\theta)-2.$ I want to find a value for $\theta$ in interval $\theta \in [0,\frac{\pi}{2}]$ such that for that value of $\theta$ my function has global maximum and minimum.
I took the derivative of the ... | Since you already found the critical points, why not apply the second derivative test ?
$$f'(\theta) = \cos(\theta)(4\theta-\pi)\implies f''(\theta) =4 \cos (\theta)-(4 \theta-\pi ) \sin (\theta)$$ from which
$$f''\left(\frac{\pi }{4}\right)=2 \sqrt{2}\color{red} {>0} \qquad \text{and} \qquad f''\left(\frac{\pi }{2}\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2578264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Intuition behind Algebra When proving, or just straight up solving equations, we often manipulate variables until we get the results we want. For example, the square root of the discriminant is simply an algebraic manipulation to get the result of the distance (or difference) between the roots. However, is there intuit... | An equation like $x+4 = 3x$ returns True for some values of $x$ and False for others.
For example, this particular equation returns false for $x := 1$, because
$$(x+4 = 3x)(x:=1) \iff 1+4= 3\cdot 1 \iff 5 =3 \iff \mathrm{False}.$$
Whereas it returns True for $x:=2$, because
$$(x+4 = 3x)(x:=2) \iff 2+4= 3\cdot 2 \iff 6 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2578390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 7,
"answer_id": 0
} |
Trigonometric equation: $\ln(\sin x + \cos x)^{1+\sin 2x}= 2$
$\ln(\sin x + \cos x)^{1+\sin 2x}= 2$
I am unable to solve it. I tried this way:
$(\sin x + \cos x)^{1+\sin 2x}= e^2$
I know that:
$\sin x + \cos x \le \sqrt2 $
$1+ \sin 2x \le 2 $
I don't know how to utilise this idea in my solution.
| HINT
$$\ln(\sin x + \cos x)^{1+\sin 2x}= 2\iff\ln(\sin x + \cos x)^{1+\sin 2x}= \ln e^2\iff (\sin x + \cos x)^{1+\sin 2x}=e^2$$
but
$$(\sin x + \cos x)^{1+\sin 2x}=(\sin x + \cos x)^{(\sin x + \cos x)^2}\leq\sqrt2^{2}=2$$
Note
$$(\sin x + \cos x)^2=1+\sin 2x\le2\implies \sin x + \cos x \leq\sqrt 2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2578482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
$\int f(x)\,\mathrm{d}x = \left(\int_0^x f(t) \, \mathrm{d}t\right) + C$ If $f(x)$ is a continuous function on $\mathbb{R}$ and I am asked to find
$\int f(x) \, dx$, what is the problem with the following answer:
$$\int f(x)\,\mathrm{d}x = \left(\int_0^x f(t) \, \mathrm{d}t\right) + C$$
| Despite teaching calculus for 20+ years, I have never seen a student do this on an exam, so the novelty of seeing it for the first time would count for something. The first time I would see it, I would accept it and even be somewhat impressed with the student. Far too few students really understand that part of the fun... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2578593",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 5,
"answer_id": 3
} |
Why can we replace a variable with a constant in a limit? We say $\lim_{x \to c} f(x) = L$ that means $f(x)$ may be as close to $L$ as $x$ tends to $c$. "Tends" here means $x$ approaches $c$ but never actually becomes $c$. If it so, then why do we such easily replace $x$ with value $c$, whenever it is appropriate. E.g.... | In order to compute $\lim_{x \to 5} (4 + x)$, we use the fact that
$$\lim_{x \to c} (f(x) + g(x)) = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$$
whenever both limits on the right exist.
Now, $\lim_{x \to 5} 4 = 4$ holds because as $x$ approaches any value (including $5$), the expression $4$ approaches the value $4$, sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2578692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Why can the a combination of an exponential and logarithmus function be used to create a powerlaw distribution I answered a question on StackOverflow where a person wanted a randomized function that produces as many numbers between 0-10, 10-100 and 100-1000.
I offered the following function that does the job:
Math.floo... | One way to generate random numbers according to a particular distribution is to generate random numbers on $[0,1)$ and then use the inverse cumulative distribution function give the distribution.
You want your cumulative distribution function (before rounding to integers) to be of the form $a\log(x)+b$ between two valu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2578780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
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