Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Is the equation $\phi(\pi(\phi^\pi)) = 1$ true? And if so, how? $\phi(\pi(\phi^\pi)) = 1$
I saw it on an expired flier for a lecture at the university. I don't know what $\phi$ is, so I tried asking Wolfram Alpha to solve $x \pi x^\pi = 1$ and it gave me a bunch of results with $i$, and I don't know what that is either... | It's a joke based on the use of the $\phi$ function (Euler's totient function), the $\pi$ function (the prime counting function), the constant $\phi$ (the golden ratio), and the constant $\pi$. Note $\phi^\pi\approx 4.5$, so there are two primes less than $\phi^\pi$ (they are $2$ and $3$), so $\pi(\phi^\pi)=2$. There i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/861618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 1,
"answer_id": 0
} |
The difference between a fiber and a section of a vector bundle If $ E_x := \pi^{-1}(x) $ is the fiber over $x$ where $(E,\pi,M)$ is the vector bundle. And the section is $s: M \to E $ with $\pi \circ s = id_M $. This implies that $\pi^{-1} = s $ on $M$. So then whats the difference between a fiber over $x$ and the se... | A section is any function s that assigns to every point p in the base an element in its fiber $E_p$ The restriction s(p) assigns to p a point in $E_p$. As an example, a section of the tangent bundle is a vector field, i.e., an assignment of a tangent vector to each tangent space $T_pM$; its restriction to p assigns to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/861697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Cholesky factorization and non-positive definite matrices When Cholesky factorization fails, is there an alternative method to obtain the $\mathbf{L}$ matrix in:
$\mathbf{A}=\mathbf{L}\mathbf{L}^{*}$
I'm dealing with a matrix not guaranteed to be positive-definite; I'm wondering if there is a sure-fire way to find $\ma... | In order that such a decomposition exists, $A$ must be Hermitian ($A=A^H$) and positive semi-definite. Then we can diagonalize $A$ with an unitary matrix $U$ ($UU^H=I$):
$$
A = U^HDU.
$$
$D$ is a diagonal matrix with non-negative diagonal entries. Then we can write
$$
A = (U^HD^{1/2}U)(U^HD^{1/2}U),
$$
where $D^{1/2}$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/861810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Limit of $a(k)$ in $ \sum_{k=1}^n \frac{a_k}{(n+1-k)!} = 1 $ For n = 1, 2, 3 ... (natural number)
$ \sum_{k=1}^n \frac{a_k}{(n+1-k)!} = 1 $
$ a_1 = 1, \ a_2 = \frac{1}{2}, \ a_3 = \frac{7}{12} \cdots $
What is the limit of {$ a_k $}
$ \lim_{k \to \infty} a_k $ = ?
I have no idea where to start.
| Note that
$$
\begin{align}
\frac{x}{1-x}
&=\sum_{n=1}^\infty x^n\\
&=\sum_{n=1}^\infty\sum_{k=1}^n\frac{a_k}{(n-k+1)!}x^n\\
&=\sum_{k=1}^\infty\sum_{n=k}^\infty\frac{a_k}{(n-k+1)!}x^n\\
&=\sum_{k=1}^\infty\sum_{n=0}^\infty\frac{a_k}{(n+1)!}x^{n+k}\\
&=\frac{e^x-1}{x}\sum_{k=1}^\infty a_kx^k\tag{1}
\end{align}
$$
Theref... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/862914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Closed-form term for $\sum_{i=1}^{\infty} i q^i$ I am interested in the following sum
$$\sum_{i=1}^{\infty} i q^i$$
for some $q<1$.
Is there a closed-form-term for this? If yes, how does one derive this?
I am also interested in
$\sum_{i=x}^{\infty} i q^i$
for some $x>1$.
| $$\sum_{i=1}^{\infty} i q^i=q\sum_{i=1}^{\infty} i q^{i-1}=q\frac{d}{dq}\sum_{i=1}^{\infty} q^{i}=q\frac{d}{dq}\sum_{i=0}^{\infty} q^{i}=q\frac{d}{dq}\frac{1}{1-q}=\frac{q}{(1-q)^2}$$
$$\sum_{i=x}^{\infty} i q^i=q\sum_{i=x}^{\infty} i q^{i-1}=q\frac{d}{dq}\sum_{i=x}^{\infty} q^{i}=q\frac{d}{dq}\left[\sum_{i=0}^{\infty}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/862966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
A single-segment Newton polygon implies henselian? I have a question about Newton polygons and henselian fields.
In p149 of Neukirch’s book(algebraic number theory:the beginning of Proposition 6.7), he says that
“We have just seen that the property of $K$ to be henselian follows from the condition that the Newton polyg... | Where do you see a problem?
Theorem 6.6 says that a field is henselian (that is Hensel's Lemma holds) if and only if the valuation extends uniquely to every algebraic extension.
The proof of the implication $\Rightarrow$ is already given in Theorem 6.2.
For the proof of the implication $\Leftarrow$ one assumes uniquene... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/863076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Easy way to compute $Pr[\sum_{i=1}^t X_i \geq z]$ We have a set of $t$ independent random variables $X_i \sim \mathrm{Bin}(n_i, p_i)$.
We know that $$\mathrm{Pr}[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$
But is there an easy way to compute:
$$\mathrm{Pr}\left[\sum_{i=1}^t X_i \geq z\... | I'll give you a general idea. If you want the details, look up an article by Tomas Woersch (here).
So you have
$$
S_m(n) = \sum_{k=0}^{m} \binom{n}{k}x^k = 1 + \binom{n}{1}x +\ldots + \binom{n}{m}x^m\\
\frac{S_m(n)}{\binom{n}{m}x^m} = 1 + \ldots + \frac{\binom{n}{1}x}{\binom{n}{m}x^m} + \frac{1}{\binom{n}{m}x^m} = 1 +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/863145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Deriving marginal effects in multinomial logit model For the multinomial logit model, it holds that:
$$P[y_i=j]=\frac{\exp{\beta_{0,j} + \beta_1 x_{ij}}}{\sum_h \exp(\beta_{0,h} + \beta_1 x_{ih})}$$.
Now my book states that the marginal effect is as follows:
$$\dfrac{\partial \operatorname{P}[y_i = j]}{\partial x_{ij}}... | Cross multiply the equation to obtain:
$$\operatorname{P}(y_i=j)\sum_h \exp(\beta_{0,h} + \beta_1 x_{ih})=\exp(\beta_{0,j} + \beta_1 x_{ij})$$
Then deriving with respect to $x_{ij}$ on both sides of the equality gives the following:
$$\dfrac{\partial \operatorname{P}(y_i = j)}{\partial x_{ij}}\sum_h \exp(\beta_{0,h} + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/863258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Definite integral $\int_0^{2\pi}(\cos^2(x)+a^2)^{-1}dx$ How do I prove the following?
$$ I(a)=\int_0^{2\pi} \frac{\mathrm{d}x}{\cos^2(x)+a^2}=\frac{2\pi}{a\sqrt{a^2+1}}$$
| If anyone wants to see a complex analysis solution.
Let $\gamma$ be the unit$\require{autoload-all}$ circle. This proof holds for all complex $a$ such that the integral exists.
$$ I(a)=\int_0^{2\pi}\!\!\! \frac{\mathrm{d}t}{\cos^2(t)+a^2}$$
$$
\toggle{
\text{Set} \; x = e^{it}\quad\enclose{roundedbox}{\text{ Click for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/863339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Proving a lemma for the irrationality of $e$ Define
$$I_n = \int_0^1 e^tt^ndt$$ where $n$ is a non-negative integer.
In a related question here, I asked how $e$ can be proven based on the following definitions and results:
$$I_{n+1} = e - (n + 1)I_n$$
$$I_n = (-1)^{n + 1}n! + e\sum_{r = 0}^n (-1)^r\frac{n!}{(n-r)!}$$
$... | Notice that if $n \geq 1$, then $e^tt^n \leq e^1t^n$ for all $t \in [0, 1]$. This implies that:
$$\int_0^1 e^tt^ndt \leq e\int_0^1 t^ndt = \frac{e}{n+1} < \frac{e}{n}$$
Hence, $\displaystyle I_n < \frac{e}{n}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/863414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Two functions whose order can't be equated - big O notation Our teacher talked today in the class about big O notation, and about order relations.
she mentioned that the set of order of magnitude, is not linear
Meaning, there are function $f,g$ such that $f$ is not $O(g)$ and $g$ is not $O(f)$, but she gave no such exa... | $f(x)=x^2\sin x$ and $g(x)=x.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/863622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Proof about AM-GM inequality generalized Note: I'm not sure this type of questions are welcome on the site. In case tell me.
Let's define the $p$ mean as
$$M_p(x_1, \dots, x_n) = \sqrt[p] { \frac 1n \sum_{i = 1}^n x_i^p}$$
for $x_1, \dots, x_n > 0$.
Your goal is to prove that
$$\dots \le M_{-2} \le M_{-1} = \mathcal{... | The magic word is convexity, and there is really not much more to it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/863720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
About Trigonometry Is there anything cool about trigonometry? I was just curious. I'm learning trig right now and I often find myself asking myself, "What's the point?" I feel if I knew what I was working on and why, I'd be more successful and goal-oriented.
| *
*The beautiful rhythmic patterns of trigonometric identities.
*The fact that things like Fourier series exist. Look at what it led to: http://en.wikipedia.org/wiki/List_of_Fourier_analysis_topics
*And look at this list of cycles
*The uses of trigonomety.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/863824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 0
} |
Find the speed S by using radical equations There are two word problems that I cannot write as radical equations.
1.A formula that is used for finding the speed s, in mph, that a car was going from the length L, in feet, of its skid marks can be written as s=2 square root of 5L. In 1964, a jet powered car left skid mar... | For the first question, we can see that the speed if a function of the length of the skid mark. Therefore,
$$
S = f(L)
$$
We also know that the relation between $ S $ and $ L $ is that $ S $ is $2$ times square root of $5L$
$$
S = 2 \times \sqrt{5L}
$$
Similarly, for the next question,
$$
S = 6.75 \times \sqrt[7]{P}
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/863924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Show that f is measurable Let $U$ be a open Set of $\mathbb{R} \times [0,\infty]$ and let f be defined as
$$f: \mathbb{R}\mapsto [0,\infty], \quad f(x) := \max\{0,\sup\{y| (x,y) \in U\}\} $$
How can I show that $f$ is measurable?
| Define $g:\mathbb{R}\rightarrow\left[0,\infty\right]$ by $x\mapsto\sup\left\{ y\mid\left(x,y\right)\in U\right\} $ and let $c\in\mathbb R$ be a constant.
If $x\in\left\{ g>c\right\} $ then set $U$ contains an element $\left(x,y\right)$
with $y>c$.
Find some $\epsilon>0$ s.t. $y-\epsilon>c$ and $\left(x-\epsilon,x+\eps... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/864013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Use a proof by cases to show that $\lfloor n/2 \rfloor$ * $\lceil n/2 \rceil$ = $\lfloor \frac{n^2}{4} \rfloor$ for all integers $n$. Question
Use a proof by cases to show that $\lfloor n/2 \rfloor$ * $\lceil n/2 \rceil$ = $\lfloor \frac{n^2}{4} \rfloor$ for all integers $n$.
My Attempt:
I can only think of two cases, ... | So you've got the case when $n$ is even. When $n$ is odd, then $\lfloor n/2 \rfloor * \lceil n/2 \rceil = \frac{n - 1}{2} * \frac{n + 1}{2} = \frac{n^2 -1}{4}$.
We want to show that this equals the right hand side. Since $n^2/4 = (n/2)^2$, we can rewrite $n^2/4$ as $(m + 1/2)^2$ where $m$ is an integer, and $m = \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/864140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Is there a finite non-solvable group which is $p$-solvable for every odd $p\in\pi(G)$? Let $G$ be a finite non-solvable group and let $\pi(G)$ be the set of prime divisors of order of $G$. Can we say that there is $r \in \pi(G)-\{2\}$ such that $G$ is not a $r$-solvable group?
| Yes. A finite group $G$ is $p$-solvable if every nonabelian composition factor of $G$ has order coprime to $p$.
If $G$ is not solvable, then it has a nonabelian simple group as a composition factor. This nonabelian simple group must have order divisible by some odd prime $r$ (finite groups of order $2^k$ are solvable).... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/864227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Lines in $\mathbb{CP}^n$ and in $\mathbb{CP}^2$ The question is, in how many points does a line in $\mathbb{CP}^n$ intersect $\mathbb{CP}^2$?
By a line in $\mathbb{CP}^n$, I mean a copy of $\mathbb{CP}^1$. I have tried with a system of equations, (because a line in $\mathbb{CP}^n$ is the zero locus of a polynomial in $... | It depends on $n$, and on the line. $\def\C{\mathbb C}\def\P{\mathbb P}\def\CP{\C\P}$We consider the embedding:
$$ \CP^2 = \{[x_0:x_1:x_2: 0 :\cdots : 0] \mid [x_0:x_1:x_2] \in \CP^2\} \subseteq \CP^n $$
Lifting this to $\C^{n+1}$, we have $\C^3 \subseteq \C^{n+1}$ embedded as the first three coordinates. A line in $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/864294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Constant in Sobolev-Poincare inequality on compact manifold $M$; how does it depend on $M$? Let $M$ be a smooth compact Riemannian manifold of dimension $n$. Let $p$ and $q$ be related by $\frac 1p = \frac 1q - \frac 1n$. There is a constant $C$ such that for all $u \in W^{1,q}(M)$
$$\left(\int_M |u-\overline{u}|^p\ri... | No, it's not nearly that simple. The constant $C$ quantifies the connectivity of the manifold. It can be imagined as the severity of traffic jams that occur when all inhabitants of the manifold decide to drive to a random place at the same time. For example, let $M$ be two unit spheres $S^2$ joined by a thin cylinder o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/864393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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How to find know if function is onto? How do you figue out whether this function is onto?
$\mathbb{Z}_3\rightarrow \mathbb{Z}_6:f(x)=2x$
Onto is of course is for all the element b in the codomain there exist an element a in the domain such that $f(a)=b$
Here the co domain is mod 6
So let $k\in\mathbb{Z}_6$
But I am not... | To put it a bit differently from Bananarama, the map gives you the values {$f(0),f(1),f(2)$}$(Mod 6)$. Even if these values are a different, can the map be onto?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/864485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Decomposing a square matrix into two non-square matrices I have a matrix $A$ with dimensions $(m \times m)$ and it is positive definite. I want to find the matrix $B$ with dimensions $(n \times m), (n \ll m)$, which follows the following expression: $$A = B'B$$ Here $B'$ is the transpose of $B$.
I was also thinking of... | If $A$ is positive definite (all eigen-values positive) then you will not ever be able to find such $B$ that gives you exactly $A = B'B$ because $A$ is invertible but $B'B$ is not (it will have rank at most $n < m$). However, you can use the SVD to come up with an approximation as $A = B'B$ with $B = \sqrt{D_n} Q'$, wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/864552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Constructing a vector space of dimension $\beth_\omega$ I'm trying to solve Exercise I.13.34 of Kunen's Set Theory, which goes as follows (paraphrased):
Let $F$ be a field with $|F| < \beth_\omega$, and $W_0$ a vector space over $F$ with $\aleph_0 \le \dim W_0 < \beth_\omega$. Recursively let $W_{n+1} = W_n^{**}$ so ... | Hint: First prove the case when $F$ is countable. For the case $|F|<\beth_\omega$, consider the prime field $K$ of $F$. Let $W'_\omega$ be the space obtained using the construction above considering $W_0$ as a $K$-vector space, then $|W_\omega'|=\beth_\omega$. As there is a copy of $W_\omega'$ in $W_\omega$, we obtain ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/864640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Numerical integration of $\sin(p_{m})$ and $\cos(p_{m})$ for a polynomial $p_{m}$ I was wondering if anyone knew about any numerical methods specifically designed for integrating functions of the form $\sin(p_{m})$ and $\cos(p_{m})$ where $p_{m}$ is a polynomial of degree $m$. I don't think $m$ will be too large in the... | I think you can use Gaussian quadrature with weight functions $w(x)=\sin(p_m)$ and $w(x)=\cos(p_m)$. For any weight function there is a family of orthogonal polynomials related to that function in $[a,b]$. This family can be obtained by Gram-Schmidt orthogonalization process on $1,x,x^2,\ldots,x^n$.
Let $w(x)$ be a wei... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/864714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Why is a random variable called so despite being a function? According to my knowledge, its a function $P(X)$ which includes all the possible outcomes a random event.
| IMO, the random variable X can be considered as a variable resulted from a function, and it's often being used as a variable in $P(X)$. For example, $y=mx+c$ is a function, but it can be used as a variable to $f(y)=y^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/864830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
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Is $-|x|\le\sin x\le|x|$ for all $x$ true? I have seen in Thomas' Calculus that says to prove $\lim_{x\rightarrow0}\sin x=0$, use the Sandwich Theorem and the inequality $-|x|\le\sin x\le|x|$ for all $x$.
My question is how could the inequality be true? If we derive from $-1\le\sin x\le1$, we could only get $-|x|\le|x|... | The unit circle is parametrized by $\theta\mapsto(\cos\theta,\sin\theta)$:
$\hskip 1.5in$
Consider an angle $0\le\theta\le\frac{\pi}{2}$, as depicted above. The red line's length is $\theta$, and $\sin\theta$ is the purple length. Since they both travel the same vertical distance but the red one also travels horizonta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/864925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Completing the following equation by the suitable method i got this linear equation two variable problems for my school.
I understand the basics of the normal linear equation but this seems different instead having a pure number after the "=" they got a ration, here is the problem.
$$X:2Y = 5:14$$
$$(X+4) : (3Y-21) = 2... | We can start by solving both equations for the same variable.
$$X:2Y = 5:14\implies Y = \frac{7 X}{5}$$
$$(X+4) : (3Y-21) = 2:3 \implies Y = \frac{X + 18}{2}$$
We now equate the two "solutions" of $Y$.
$$\frac{7 X}{5}=\frac{X + 18}{2}\\
\implies \frac{7 X}{5} = \frac{X}{2} + 9\\\implies 14X-5X =90\\
\implies X=10\qqua... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/864999",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Greatest value of f If $f'(x)=6-x$ then which of the following has the greatest value?
*
*$f(2.01)-f(2)$
*$f(3.01)-f(3)$
*$f(4.01)-f(4)$
*$f(5.01)-f(5)$
*$f(6.01)-f(6)$
I know the answer is $f(2.01)-f(2)$ but how to prove?
| Hint:Use the
$$f'(x)\approx \frac{f(x+h)-f(x)}{h}$$ with $h=0.01$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/865081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 0
} |
Big $O$, little $o$ Notation So I've gone through the typical undergraduate math sequence (two semesters of real analysis, two semesters of abstract algebra, some measure theory, but I haven't taken discrete math) and in various posts online, I keep on seeing things such as
$$O(x^2) $$
which I've never encountered in ... | Big $O$ notation:
$f(n)=O(g(n))$:
$\exists \text{ constant } c>0 \text{ and } n_0 \geq 1 \text{ such that, for } n \geq n_0: \\ 0 \leq f(n) \leq c g(n)$
$$$$
Little $o$ notation:
$f(n)=o(g(n))$:
$\text{ if for each constant } c>0 \text{ there is } n_0 \geq 1 \text{ such that } \\ 0 \leq f(n) < c g(n) \ \forall n \geq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/865174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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How to find the limit of the following function? I found this task in test $$\lim_{x \to 0+} \frac{1}{x} \int_0^{2x} (\sin{t})^t\,\mathrm{d}t$$. The answer is 2. But I can't find out the algorithm of solving of such an improper integral.
| Using the L' Hospital rule we have: $$\lim_{x\to 0_+ } \frac{1}{x}\int_0^{2x} (\sin t)^t dt =\lim_{x\to 0_+ } 2(\sin 2x )^{2x} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/865232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Showing $\ln(\sin(x))$ is in $L_1$
Prove $\ln[\sin(x)] \in L_1 [0,1].$
Since the problem does not require actually solving for the value, my strategy is to bound the integral somehow. I thought I was out of this one free since for $\epsilon > 0$ small enough, $$\lim_{\epsilon \to 0}\int_\epsilon^1 e^{\left|\ln(\sin(x... | A simpler approach would be to observe that the function $x^{1/2}\ln \sin x$ is bounded on $(0,1]$, because it has a finite limit as $x\to 0$ -- by L'Hôpital's rule applied to $\dfrac{\ln \sin x}{x^{-1/2}}$. This gives $|\ln \sin x|\le Mx^{-1/2}$.
As Byron Schmuland noted, $e^{|\ln \sin x|} = 1/\sin x$, which is no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/865293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
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Function is defined on the whole real line and $|f(x) -f(y)| \leq |x-y|^\alpha$, then.... Given: $f(x)$ is defined on $\mathbb{R}$ and $|f(x) -f(y)| \le |x-y|^\alpha$. Which of the following statements are true?
I. If $\alpha > 1$, then $f(x)$ is constant.
II. If $\alpha = 1$, then $f(x)$ is differentiable.
III. $0... | Hints: For (I) $0\le\vert\frac{f(x)-f(y)}{x-y}\rvert\le\vert x-y\rvert^{\alpha-1}$
For (II) Think about the function $x\mapsto\lvert x\rvert$
For (III) $0\le\lvert f(x)-f(y)\rvert\le\lvert x-y\rvert^{\alpha}$ let $x$ tend to $y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/865365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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What exactly is a number? We've just been learning about complex numbers in class, and I don't really see why they're called numbers.
Originally, a number used to be a means of counting (natural numbers).
Then we extend these numbers to instances of owing other people money for instance (integers).
After that, we consi... | There is no concrete meaning to the word number. If you don't think about it, then number has no "concrete meaning", and if you ask around people in the street what is a number, they are likely to come up with either example or unclear definitions.
Number is a mathematical notion which represents quantity. And as all q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/865409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "64",
"answer_count": 19,
"answer_id": 11
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Finding a matrix projecting vectors onto column space I can't find $P$, for vectors you can do $P = A(A^{T}A)^{-1}A^T$. But here its not working because matrices have dimensions that can't multiply or divide. help
| The dimensions of the matrices do match.
Matrix $A$ is 3x2, which matches with $(A^TA)^{-1}$, which is 2x2.
The result $A(A^TA)^{-1}$ is again 3x2.
When multiplying it with $A^T$, which is 2x3, you get a 3x3 matrix for $P$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/865478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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I need help finding the critical values of this function. So $h(t)=t^{\frac{3}{4}}-7t^{\frac{1}{4}}$. So I need to set $h'(t)=0$. So for $h'(t)$ the fattest I've gotten to simplifying os $h'(t)=\frac{3}{4 \sqrt[4]{t}}-\frac{7}{4\sqrt[4]{t^3}}$ and that is as farthest as I can simplify. So i'm having a had time having $... | Hint
To make your life easier, just define $x=t^{\frac{1}{4}}$. So $h(t)=x^3-7x$ and then $$\frac{dh}{dt}=\frac{dh}{dx}\frac{dx}{dt}=(3x^2-7)\frac{dx}{dt}$$ So,$h'(t)=0$ if $x^2=\frac{7}{3}$ that is to say $\sqrt t=\frac{7}{3}$ then $t=\frac{49}{9}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/865552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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limit of a sum of powers of integers I ran across the following problem in my Advanced Calculus class:
For a fixed positive number $\beta$, find
$$\lim_{n\to \infty} \left[\frac {1^\beta + 2^\beta + \cdots + n^\beta} {n^{\beta + 1}}\right]$$
I tried manipulating the expression inside the limit but didn't come up with a... | $$\frac{1^{\beta}+\cdots+n^{\beta}}{n^{\beta+1}}=\frac{1}{n}\Big(\Big(\frac{1}{n}\Big)^{\beta}+\cdots+\Big(\frac{n}{n}\Big)^{\beta}\Big)\to\int_0^1x^{\beta}dx=\frac{1}{\beta+1}.$$
If You did not have integrals yet use Stolz theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/865619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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why is it that the conjugate of a+bi is a-bi? if a+bi is an element of a group, then its conjugate is a-bi,
how can we prove this by using the fact that the conjugate of an element g of a group is h if there is an x in the group such that h=xgx^(-1)?
| Given a mathematical structure $S$ and a group of automorphisms $G \subseteq \mathrm{Aut}(S)$, we often say that the elements $\sigma(x)$ for $\sigma \in G$ are the conjugates of $x$. And if $\sigma$ is any particular automorphism, we might call $\sigma(x)$ the conjugate of $x$ by $\sigma$.
For the complex numbers, $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/865723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\forall$ At the beginning or at the end? I have a set of real numbers $x_1, x_2, \ldots, x_n$ and two functions $f:\mathbb{R} \rightarrow \mathbb{R}$ and $g:\mathbb{R} \rightarrow \mathbb{R}$.
What are the differences between the following statements?
*
*$\forall v \in \{1, \ldots, n\} ~f(x_v) = 0 \vee g(x_v) = 0$
... | The second one is simply wrong, that's not how things should be written.
It's unambiguous here because there is only one quantifier, but if had been something like $\exists yP(x,y)\forall x$ you wouldn't know whether $\forall x\exists yP(x,y)$ or $\exists y\forall xP(x,y)$ was meant. In my experience, the second optio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/865871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Radius of $\sum a_n b_n x^n$ via radii of $\sum a_n x^n$ and $\sum b_n x^n$
Series $\sum a_n x^n$ and $\sum b_n x^n$ have radii of convergence of
1 and 2, respectively. Then radius of convergence R of $\sum a_n b_n x^n$ is
*
*2
*1
*$\geq 1$
*$ \leq 2$
My attempt was to apply ratio test:
$$\lim... | The problem in the attempt is that we cannot use in general the ratio test because we are not sure that the terms are different from $0$.
Since the radius of convergence of $\sum_n b_nx^n$ is $2$, the series converges at $1$ hence $b_n\to 0$. This implies that the series $\sum_n |a_nx^n|$ is convergent if $|x|<1$.
Con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/865944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Quaternion Group as Permutation Group I was recently, for the sake of it, trying to represent Q8, the group of quaternions, as a permutation group. I couldn't figure out how to do it.
So I googled to see if somebody else had put the permutation group on the web, and I came across this:
http://mathworld.wolfram.com/Perm... | Follow Cayley's embedding: write down the elements of $Q_8=\{1,-1,i,-i,j,-j,k,-k\}$ as an ordered set, and left-multiply each element with successively with each element of this set - this yields a permutation, e.g. multiplication from the left with $i$, gives you that the ordered set $(1,-1,i,-i,j,-j,k,-k)$ goes to $(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/866026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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What are the properties of the set of the Real Numbers without the Integers? This question came up in a lunchtime discussion with coworkers. None of us are professional mathematicians or teachers of math. I apologize for any incorrect math or sloppy terminology.
We were discussing getting from one number to another a... | The space you describe
$...(-(n+1),-n)...(-2,-1)(-1,0)(0,1)(1,2)...(n,n+1)...$
is homeomorphic to countably infinite ("$\omega$") many copies of $(0,1)$. The Long Line you refer is made of uncountably infinite many copies of $[0,1)$. It can be thought of as a generalization of the nonnegative reals $[0,\infty)$, becau... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/866154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove or disapprove the statement: $f(n)=\Theta(f(\frac{n}{2}))$ Prove or disapprove the statement:
$$f(n)=\Theta(f(\frac{n}{2}))$$
where $f$ is an asymptotically positive function.
I have thought the following:
Let $f(n)=\Theta(f(\frac{n}{2}))$.Then $\exists c_1,c_2>0 \text{ and } n_0 \geq 1 \text{ such that } \forall... | Just set $f=f(n) = 2^{\frac{n}{2}}, \ g=g(n) = 2^n$ and take $\lim_n \frac{f}{g} = 0$, so $f <c_1 g \ \forall \ c_1>0 $ and $n>n_0$ or simply $f=o(g)$. Clearly $\lim_n \frac{g}{f} = \infty$, so LHS of the inequality is never fulfilled: $\not \exists c_2 \ \text{s.t.} f>c_2g \ \forall n>n_0$ or simply $g=\omega(f)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/866220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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At what extent I can use trigonometric functions and properties with parametric curves? I have a know-how and a library about trigonometry and trigonometric operations, I would like to know if I can possibly rely on trigonometry for parametric curves too and how the trigonometry from the circle with $\mathbb{ \mathcal{... | A few examples ...
If the vector $U = (u_x, u_y)$ is the derivative vector of a Bézier curve (or any other curve, actually), then $\text{atan2}(u_x,u_y)$ is the angle between the curve tangent and the $x$-axis.
Once you know to calculate this angle, you can find places where the tangent is horizontal or vertical, which... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Putnam Exam question Prove or disprove: if $x$ and $y$ are real numbers with $y\ge 0$ and $y(y+1)\le (x+1)^2$, then $y(y-1)\le x^2$.
How should I approach this proof? The solution starts with assuming $y\ge 0$ and $y\le 1$, but I'm not sure how to arrive at that second assumption or go from there. Thank you in advanc... | Draw a diagram. For $y\ge0$, the curve
$$y(y+1)=(x+1)^2$$
is the upper half of a hyperbola with turning point at $(-1,0)$. One of the asymptotes of this hyperbola is $y=x+\frac{1}{2}$. The inequality
$$y(y+1)\le (x+1)^2$$
defines the region below this hyperbola. The hyperbola
$$y(y-1)=x^2$$
has turning point $(0,1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/866415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Are the real numbers really uncountable? Consider the following statement
Every real number must have a definition in order to be discussed.
What this statement doesn't specify is how that loose-specific that definition is.
Some examples of definitions include:
"the smallest number that takes minimally 100 syllables t... | The countably infinite set $c{\mathbb R}$ of computable real numbers is difficult to define and to handle. But it is embedded in the uncountable set ${\mathbb R}$ which is easy to handle and is characterised by a small set of reasonable axioms. Beginning with data from $c{\mathbb R}$ we freehandedly argue in the enviro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/866447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 6
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Is $\tan^2\theta+1=\large\frac{1}{\sin^2\theta}$ a Fundamental Identity Wrote this down during class, and I am wondering if I incorrectly transcribed from the board. Is this identity true? And if so, how?
| It is not true. For example, let $\theta=\frac{\pi}{6}$ ($30$ degrees). Then $\tan^2\theta+1=\frac{1}{3}+1=\frac{4}{3}$ while $\frac{1}{\sin^2\theta}=4$.
But $\tan^2\theta+1=\frac{1}{\cos^2\theta}$ is true.
To show that the identity $\tan^2\theta+1=\frac{1}{\cos^2\theta}$ holds, recall that $\sin^2\theta+\cos^2\the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/866542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why can one take the power of $e$ directly? The definition of Euler's constant to the power $x$, $e^x$, is
$$e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + {...}$$
And of course, we have the number $e$ defined as
$$e = \sum_{n=0}^\infty \frac{1}{n!} = 1 + \frac{1}{1!} + \f... | When you say calculate the value of $e$ and then take the power $e^x$, what does taking a power $a^x$ mean? By definition, we let
$$
a^x = \exp(x\log(a)),
$$
where $\exp$ is defined as the power series you mentioned and $\log$ is its left inverse. Thus,
$$
e^x = \exp(x\log(e)) = \exp(x\log(\exp(1)) = \exp(x),
$$
since ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/866657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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The definition of the right regular representation I'm having difficulties understanding the definition of the right regular representation as it appears in Dummit & Foote's Abstract Algebra text. On page 132 it says
Let $\pi:G \to S_G$ be the left regular representation afforded by the action of $G$ on itself by left... | You seem to worry about these sentences:
"Let $\lambda : G \to S_G$ be the permutation representation afforded by the corresponding
right action of $G$ on itself, and for each $h \in G$ denote the permutation $\lambda(h)$ by
$\tau_h$. Thus $\tau_h(x)=xh^{−1}$ for all $x \in G$ ($\lambda$ is called the right regul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/866728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Find y-coordinate on a line between two (known) points Iam a littlebit stuck with a simple task and hope to find some help here, since my days in school are now long time over and to be honest i can’t remember so well how to do it.
I have a straight line between two points, lets say (8,20) (300,50) and i want to figur... | Compute the gradient,
$$ m= \frac{y_2-y_1}{x_2-x_1} $$
Then the y-intercept is
$$ c=y_1 - mx_1 $$
or you could do $c=y_2-mx_2$ which would give the same answer.
Then to find $y$ for a particular $x$ (e.g. $x=200$), you just do
$$y = mx+c$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/866905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Simplify [1/(x-1) + 1/(x²-1)] / [x-2/(x+1)] Simplify: $$\frac{\frac{1}{x-1} + \frac{1}{x^2-1}}{x-\frac 2{x + 1}}$$
This is what I did.
Step 1: I expanded $x^2-1$ into: $(x-1)(x+1)$. And got: $\frac{x+1}{(x-1)(x+1)} + \frac{1}{(x-1)(x+1)}$
Step 2: I calculated it into: $\frac{x+2}{(x-1)(x+1)}$
Step 3: I multiplied $x-\... | It simplifies things a lot if you just multiply the numerator and denominator by $(x+1)(x-1)$
$$\frac{\frac{1}{x-1} + \frac{1}{x^2-1}}{x-\frac 2{x + 1}}\cdot\frac{\frac{(x+1)(x-1)}{1}}{\frac{(x+1)(x-1)}{1}} = \frac{(x+1)+1}{x(x+1)(x-1)-2(x-1)}=\frac{x+2}{(x-1)(x(x+1)-2)}$$
$$=\frac{x+2}{(x-1)(x^2+x-2)}=\frac{x+2}{(x-1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/866935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find $\lambda$ if $\int^{\infty}_0 \frac{\log(1+x^2)}{(1+x^2)}dx = \lambda \int^1_0 \frac{\log(1+x)}{(1+x^2)}dx$ Problem : If $\displaystyle\int^\infty_0 \frac{\log(1+x^2)}{(1+x^2)}\,dx = \lambda \int^1_0 \frac{\log(1+x)}{(1+x^2)}\,dx$ then find the value of $\lambda$.
I am not getting any clue how to proceed as if I ... | Setting $x=\tan y,$
$$I=\int_0^\infty\frac{\ln(1+x^2)}{1+x^2}\ dx=\int_0^{\dfrac\pi2}\ln(\sec^2y)\ dy=-2\int_0^{\dfrac\pi2}\ln(\cos y)\ dy (\text{ as } \cos y\ge0 \text{ here})$$
which is available here : Evaluate $\int_0^{\pi/2}\log\cos(x)\,\mathrm{d}x$
The Right Hand Side can be found here : Evaluate the integral: ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/867024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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$\lim_{n \rightarrow \infty} n ((n^5 +5n^4)^{1/5} - (n^2 +2n)^{1/2})$ $$\lim_{n \rightarrow \infty} n ((n^5 +5n^4)^{1/5} - (n^2 +2n)^{1/2})$$
Please, help me to find the limit.
| Use the general fact
$$n(n+a-\sqrt[k]{n^k +akn^{k-1}})\rightarrow \frac{k+1}{2}a^2$$ as $n\to \infty$
to get a limit of
$$\frac{2+1}{2}-\frac{5+1}{2}=-\frac{3}{2}$$
For proof of the above fact.
If we let $A=\sqrt[k]{n^k +akn^{k-1}}$
then $$n((n+a)-A)=n\frac{(n+a)^k-A^k}{(n+a)^{k-1}+\cdots +A^{k-1}}
=\frac{\binom{k}{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/867115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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finite dimensional integral domain containg $\mathbb C$ Let $ R$ be an integral domain containing $\mathbb C$.
Suppose that $R$ is a finite dimensional $\mathbb C$-vector space . Show that $R=\mathbb C$.
One side $\mathbb C \subset R$ is obvious. What about the other?
Show me right way.Thanks in advance.
| Hint: If it's finite dimensional, over $\mathbb{C}$, then it must be a field. This follows from integrality, or more simply just write down a polynomial killing any non-zero element of $R$, and show how you can make an inverse from this equation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/867178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\frac{1}{\sqrt{2\pi}}\int_\frac {1}{2}^0\exp(-x^2/2)dx$ How do we analytically evaluate $J=\frac{1}{\sqrt{2\pi}}\int_\frac {-1}{2}^0\exp(-x^2/2)dx$?
This is what I tried:
$$ J^2=\frac{1}{{2\pi}}\int_\frac {-1}{2}^0\int_\frac {-1}{2}^0\exp(-(x^2+y^2)/2)dxdy \\
=\frac{1}{{2\pi}}\int_\pi ^\frac {3\pi}{2}\int_0 ^\frac {1}... | You could use
$$e^x=\sum_{i=1}^{\infty}\frac{x^i}{i!}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/867262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Let $p$ be a prime of the form $3k+2$ that divides $a^2+ab+b^2$ for some integers $a,b$. Prove that $a,b$ are both divisible by $p$. Let $p$ be a prime of the form $3k+2$ that divides $a^2+ab+b^2$ for some integers $a,b$. Prove that $a,b$ are both divisible by $p$.
My attempt:
$p\mid a^2+ab+b^2 \implies p\mid (a-b)(a... | Suppose that $p=3k+2$ is prime and
$$
\left.p\ \middle|\ a^2+ab+b^2\right.\tag1
$$
then, because $a^3-b^2=(a-b)\left(a^2+ab+b^2\right)$, we have
$$
\left.p\ \middle|\ a^3-b^3\right.\tag2
$$
Case $\boldsymbol{p\nmid a}$
Suppose that $p\nmid a$, then $(2)$ says $p\nmid b$. Furthermore,
$$
\begin{align}
a^3&\equiv b^3&\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/867413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
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Number theory proofs regarding perfect squares How do you prove that $3n^2-1$ is never a perfect square
| Let $3n^2-1=b^2, \text{ for a } b \in \mathbb{Z}$
$$3n^2-1 \equiv -1 \pmod 3 \equiv 2 \pmod 3$$
$$b=3k \text{ or } b=3k+1 \text{ or } b=3k+2$$
Then:
$$b^2=9k \equiv 0 \pmod 3 \text{ or } b^2=3n+1 \equiv 1 \pmod 3 \text{ or } b=3n+1 \equiv 1 \pmod 3$$
We see that it cannot be $b^2 \equiv 2 \pmod 3$,so the equality $3n^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/867476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the equation of a line tangent at a specific point I have to find an equation for the line tangent to the graph of
$\large\frac {\sqrt{x}}{6x+5}$
at the point $(4,f(4))$, and write it out in the form of $y=mx+b$
Using the quotient rule I get..
$(6x+5)\frac12 x^{-{\frac12}} - \large\frac{(6\sqrt{x})}{(6x+5)^2}$
I t... | First you can simplify your derivative to: $$\dfrac{\mathrm d}{\mathrm dx}\left\{\dfrac{\sqrt{x}}{6x+5}\right\}=\dfrac{5-6x}{2\sqrt{x}(6x+5)^2},$$ which would make your calculations a little bit simpler. To find $b$ you just use the fact that $(4,f(4))$ lies in your tangent line, and you use the point-slope formula: $$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/867567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Prove a statement with elements for Set Theory I am stuck on this proofing question and I would like some clarification.
Q: $A\subseteq B \iff A\cap B^{\prime} = \emptyset$
I already proved that LHS goes to RHS, but I am confused for the other way around because the textbook answer key gives a weird answer.
It says tha... | The point is that there's a contradiction here; this is how a standard proof by contradiction goes: Start by assuming something (that is hopefully false), and use it to get something you know is false. Then the original statement must be false too.
So to clarify the proof, I'll expand it a bit:
Suppose, intending a co... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Geometric series of matrices I am currently reading 'Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach' by J. Hubbard and B. Hubbard. In the first chapter, there is the proposition:
Let A be a square matrix. If $|A|<1$, the series $$S=I+A+A^2+\cdots$$
converges to $(I-A)^{-1}$.
The proof ... | Since $|A|<1$, and since you state that $\left|A^k\right|\le|A|^k$, you clearly have $\lim_{k\to\infty}\left|A^k\right|\to0$.
I don't know which norm you are using, but for every norm this also means $\lim_{k\to\infty}A^k\to0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/867768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 0
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Double integral of a rational function Consider the region $D$ given by $1\leq x^2+y^2\leq2\land0\leq y\leq x$. Compute $$\iint_D\frac{xy(x-y)}{x^3+y^3}dxdy$$
Attempt: The region $D$ is part of a ring in the first quadrant below the line $y=x$
Any hints are wellcome.
| Changing to polar coordinates, $x=\rho \cos\theta$, $y=\rho \sin\theta$, and the Jacobian of the transformation is $J=\rho$. Then:
$$\int_1^\sqrt2 \rho d\rho\int_0^\frac{\pi}{4}\frac{\sin\theta\cos\theta(\cos\theta-\sin\theta)}{\cos^3\theta+\sin^3\theta}d\theta$$
The first integral is immediate and yields $\frac{1}{2}$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Surface area of sphere $x^2 + y^2 + z^2 = a^2$ cut by cylinder $x^2 + y^2 = ay$, $a>0$ The cylinder is given by the equation $x^2 + (y-\frac{a}{2})^2 = (\frac{a}{2})^2$.
The region of the cylinder is given by the limits $0 \le \theta \le \pi$, $0 \le r \le a\sin \theta$ in polar coordinates.
We need to only calculate t... | Given the equations
$$
x^2+y^2+z^2=a^2,
$$
and
$$
x^2+y^2 = ay,
$$
we obtain
$$
ay + z^2 = a^2.
$$
Using
$$
\begin{eqnarray}
x &=& a \sin(\theta) \cos(\phi),\\
y &=& a \sin(\theta) \sin(\phi),\\
z &=& a \cos(\theta),\\
\end{eqnarray}
$$
we obtain
$$
a^2 \sin(\theta) \sin(\phi) + a^2 \cos^2(\theta) = a^2
\Rightarrow \s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/867961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Ramanujan type sum Let $$f_b(x)=\sum\limits_{a=1 , (a,b)=1}^{b}\frac{1}{1-e^{2\pi i \frac{a}{b}}x}$$
For example:
$$f_6(x) = \frac{1}{1-e^{2\pi i \frac{1}{6}}x}+\frac{1}{1-e^{2\pi i \frac{5}{6}}x}$$
I'm wondering if there is a simple closed for for my function. For instance, if we get rid of the $(a,b)=1$ condition, we... | Define
$$g_b(x) = \sum\limits_{a=1 }^{b}\frac{1}{1-e^{2\pi i \frac{a}{b}}x}.
$$
Then it's clear that
$$
g_b(x) = \sum_{d\mid b} f_b(x)
$$
(where the sum is over all positive integers $d$ dividing $b$). By Mobius inversion, we conclude that
$$
f_b(x) = \sum_{d\mid b} \mu(b/d) g_d(x) = \sum_{d\mid b} \mu(b/d) \frac d{1-x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/868153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is this Goldbach-type problem easy to solve? Problem: Given an odd prime number $p$, are there odd prime numbers $q$, $p'$, $q'$ such that $\{p,q\} \neq \{ p',q'\}$ and $p+q = p'+q'$ ?
This comment informs that it's an obvious corollary of the Polignac's conjecture.
This conjecture is still open, and my proble... | It seems likely that your result can be proven using methods like those used to bound the number of exceptions to the Goldbach conjecture. Let $E(x)$ be the number of even integers $\le x$ that cannot be written as a sum of two primes. It is known that $E(x) \in O(x^{1-\delta})$ for some $\delta>0$ (for instance, see... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/868224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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How to call two subsets that can be deformed into each other? Given a topological space $X$, is there a canonical name for the equivalence relation generated by the following relation on the subsets of $X$?
$A \sim B :\Leftrightarrow \exists \text{ continuous } h:[0,1]\times X \to X,\; h(0, •) = id_X,\; h(1, A) = B$
th... | This very closely matches the notion of homotopy. However, homotopy is slightly different, as it represents the notion of a map $h\colon [0,1] \times A \rightarrow X$ such that $h(0,-)$ is the identity and $h(1,-) = B$. This is basically a deformation of maps into $X$, but doesn't require that all of $X$ be mapped at... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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about the intersection of nested intervals Consider a sequence $\{a_n\}$ (we have not informations about its convergence) and moreover consider a sequence of semi-open intervals of $\mathbb R$:
$$\left[\frac{a_0}{2^0},\frac{a_0+1}{2^0}\right[\supset \left[\frac{a_1}{2^1},\frac{a_1+1}{2^1}\right[\supset\cdots\supset\lef... | As Daniel Fischer noted, the length of the intervals shrinks to $0$ so the intersection is either empty or contains exactly one point.
If you choose $a_n = 0$ for each $n$, the intervals will be nested and their intersection is $\{0\}$. On the other hand, if you choose $a_n$ so that $\{\frac{a_n + 1}{2^n}\}$ is constan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/868485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Help with composite functions? Suppose that $u$ and $w$ are defined as follows:
$u(x) = x^2 + 9$
$w(x) = \sqrt{x + 8}$
What is:
$(u \circ w)(8) = $
$(w \circ u)(8) = $
I missed this in math class. Any help?
| When the function isn’t too complicated, it may help to express it in words. So, your $u$ is “square the input, and then add $9$, to get your final output”. And your function $w$ is “add $8$ to your input, and then take the square root to get your final output”. And I’m sure you know that $u\circ w$ means to perform th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integration by substitution, why do we change the limits? I've highlighted the part I don't understand in red. Why do we change the limits of integration here? What difference does it make?
Source of Quotation: Calculus: Early Transcendentals, 7th Edition, James Stewart
| The original limits is for variable $x$, and the new limits is for the new variable $u$. If you can get a primitive function
$$\frac{8}{27}(1+\frac{9}{4}x)^\frac{3}{2}$$
by observation, it is unnecessary.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/868720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 3
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Calculating the central point with minimal average distance to other points I work at an office with colleagues coming from all over the country. Our office is quite centrally located, but some colleagues have to travel quite a lot further than others. I often wondered how I could calculate a central point which minimi... | The average distance is the sum of distances divided by the number of colleagues. Since the latter is fixed, you can as well ask for the point which minimizes the sum. Which, by the way, indicates that for two employees the situation would be not as simple as you make it to be, since any point on the connecting line wi... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
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Can two distinct formulae (or series of formulae) have the same Gödel number? As I am studying Gödel's incompleteness theorem I am wondering if two distinct formulae or series of formulae can have the same Gödel number? Or the function mapping each formula or series of formulae to a Gödel number is not invertible?
| Two different formulas, or strings of formulas, cannot have the same index (Gödel number).
In the usual indexing schemes, not every natural number is an index. So the function is one to one but not onto.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/868870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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2 dimensional (graphical) topological representation of a sphere One page 37 of this pdf - Surfaces - it gives a graphical representation of a sphere in 2 dimensional topological format. I don't see how the image for a sphere here actually describes a sphere. Does anyone know how this image describes a sphere?
| It might be easier to think about it the other way round. Imagine you have a sphere and you cut from the north pole down to the south pole and then pull it apart and try to lay it flat. If the material was suitably stretchy then it would be able to lay flat on a table and would be (topologically) a disk because there a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is it possible to describe $Q(x)$ as the extension field of $R$ freely generated by $\{x\}$? Given an integral domain $R$, the polynomial ring $R[x]$ can be defined as the commutative $R$-algebra freely generated by $\{x\}$. Also, let $Q$ denote the field of fractions associated to $R$. Then we can describe $Q(x)$ as t... | In the category of extension fields of $R$, the morphisms $Q(x) \rightarrow S$ correspond bijectively to the elements of $S$ that are transcendental (over $Q$). So if you define $\mathcal{R}$ to be the functor that sends an extension field of $R$ to the set of elements that are transcendental (over $Q$), then its left ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/869066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A function $f$ such that $f(x)$ increases from $0$ to $1$ when $x$ increases from $0$ to infinity? I am looking for a function f(x) with a value range of [0,1].
f(x) should increase from 0 to 1 while its parameter x increases from 0 to +infinity.
f(x) increases very fast when x is small, and then very slow and eventual... |
Simple version:
$$f(x)=1-\mathrm e^{-a\sqrt{x}}\qquad (a\gt0)$$
Slightly more elaborate version:$$f(x)=1-\mathrm e^{-a\sqrt{x}-bx}\qquad (a\gt0,\ b\geqslant0)$$
Every such function fits every requisite in the question, including the infinite slope at $0$. The parameter $a$ can help to tune the increase near $0$. Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/869150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
"answer_count": 8,
"answer_id": 3
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Composition of an analytic function with a continuous function that is analytic If $f$ is a continuous function such that $g(z)=\sin{f(z)}$ is analytic, then is $f$ analytic?
I know we can take $f(z)=\bar{z}$ then $f$ is continuous but $g$ is not analytic. Same holds if we take $f(x+iy)=x$.
I tried letting $f(z)=u+iv$... | For points where $f(z) \neq \left(k+\frac{1}{2}\right)\pi$, the sine is locally biholomorphic, and
$$f(z) = \arcsin \left(\sin f(z)\right)$$
is holomorphic in a neighbourhood of $z$ as a composition of two holomorphic functions.
It remains to deal with the points where $f(z) = \left(k+\frac{1}{2}\right)\pi$ for some $k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/869250",
"timestamp": "2023-03-29T00:00:00",
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How to solve this elementary induction proof: $\frac{1}{1^2}+ \cdots+\frac{1}{n^2}\le\ 2-\frac{1}{n}$? This is a seemingly simple induction question that has me confused about perhaps my understanding of how to apply induction
the question;
$$\frac{1}{1^2}+ \cdots+\frac{1}{n^2}\ \le\ 2-\frac{1}{n},\ \forall\ n \ge1.$$
... | What you really need is $2 − \frac{1}{k} + \frac{1}{(k+1)^2} \leq 2 − \frac{1}{(k+1)}$,
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/869358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $\lim_{a\to 0^+} \int \frac{e^{-|x|/a}}{2a}f(x)dx=f(0)$ I'm trying to show $\displaystyle \lim_{a\to 0^+} \int \frac{e^{-|x|/a}}{2a}f(x)dx=f(0)$ where $f$ is continuous with compact support.
I've already shown that for any $a>0, \displaystyle\int\frac{e^{-|x|/a}}{2a}dx=1$ and that for any fixed $\delta>0$, $... | Do you know about mollifiers? If you do, the last thing to see is that
$$\frac{e^{-|x|/a}}{2a} \ge 0 \qquad \forall a > 0$$
And then notice that
$$\int \underbrace{\frac{e^{-|x|/a}}{2a}}_{=: \phi_a} f(x) \mathrm dx = (\phi_a \ast f)(0)$$
Then use that $(\phi_a)_{a>0}$ is a mollifier famliy and use
$$\lim_{a\searrow 0} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/869419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the GCD of two Gaussian integers How do you calculate the GCD of $6-17i$ and $18+i$ in $\Bbb Z [i]$?
| You can use the Euclidean algorithm in $\Bbb Z[i]$: divide one number by the other to obtain a quotient and remainder, then repeat with the previous divisor and remainder, and so on. The quotient must of course be an element of $\Bbb Z[i]$, and the remainder must have norm less than that of the divisor: to ensure this... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/869514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to prove that some set is a Borel set If $B$ is a borel set, is $B+c$ a borel set for some constant $c$ ? I know that it is not possible to characterize a Borel set.
| Yes. You know that the set $B+c$ is the inverse image of the measurable set, $B$, under the continuous function: $f(x)=x-c$, and since continuous functions are measurable, and the inverse image of a measurable set is measurable, $B+c$ is also measurable.
Note: there is nothing particularly special about Lebesgue meas... | {
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"timestamp": "2023-03-29T00:00:00",
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How find the maximum of the $x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}$ Let $$0\le x_{i}\le i,\, i=1,2,3$$ be real numbers. Find the maximum of the expression
$$x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}$$
My idea: I guess
$$x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}\le 0^3+2^3+3^3-0\cdot2\cdot 3=35$$
But I can't prove it. C... | Hint:
As the objective function is convex and continuous, and the domain of interest is compact and convex, so we must have maximum when each $x_i \in \{0, i\}$. This cuts down the problem (even the general $n$ variable case) to checking a few extreme cases, many of which are trivial...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/869687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How prove $A^2=0$,if $AB-BA=A$
let $A_{2\times 2}$ matrix, and The matrix $B$ is order square,such
$$AB-BA=A$$ show that
$$A^2=0$$
My idea: since $$Tr(AB)=Tr(BA)$$ so
$$Tr(A)=Tr(AB-BA)=Tr(AB)-Tr(BA)=0$$
Question:2
if $A_{n\times n}$ matrix,and the matrix $B$ is order square,such
$$AB-BA=A$$
then we also have
... | An alternative geometric approach:
*
*We have $A\in \mathfrak{sl}_2$ and it can be assumed that $B\in \mathfrak{sl}_2$ as well. Hence $A$, $B$ can be seen as vectors $\vec{a},\vec{b}\in \mathbb{C}^3$. In this picture, $[A,B]\sim \vec{a}\wedge \vec{b}$ and $\operatorname{Tr}AB\sim \vec{a}\cdot\vec{b}.$
*Now since $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/869777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is it possible to find square root using only rational numbers and elementary arithmetic operators Suppose I have a number a
How can I find it's square root using only +, -, /, * and rational numbers?
If it is impossible how to prove it?
| If you allow infinite number of operations, then you can use some algorithm.
One easy example is root searching via Newton's method. Here we do the iteration
$$x_{n+1} = \frac{a + x_n^2}{2x_n},$$
which eventually converges to $\sqrt{a}$ if $a$ and $x_0$ are positive.
See https://en.wikipedia.org/wiki/Methods_of_computi... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove that $\lim_{x \to -1^-}\frac{5}{(x+1)^3} = -\infty $ using the $\delta M$ definition of infinite limits I am posting this for you guys to let me know whether it's wrong and/or give me any advice regarding the proof.
Thank you.
Given $ M < 0 $ we need $\delta > 0$ such that $ -1 -\delta< x < -1 \Rightarrow \frac{... | Edit: The proof now starts out correctly. There are still issues.
For example, the line
$ \frac{5}{(x+1)^3} < M \Rightarrow$$x + 1 > \sqrt[3]{\frac{5}{M}}$
though correct, has the implication running in the wrong direction. We want to show that if $\delta$ is chosen appropriately, then $\frac{5}{(1+x)^3}$ is $\lt M$.... | {
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"timestamp": "2023-03-29T00:00:00",
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An integral inequality with inverse Let $f:[0,1]\to [0,1]$ be a non-decreasing concave function, such that $f(0)=0,f(1)=1$. Prove or disprove that :
$$ \int_{0}^{1}(f(x)f^{-1}(x))^2\,\mathrm{d}x\ge \frac{1}{12}$$
A friend posed this to me. He hopes to have solved it, but he is not sure. Can someone help? Thanks.
| I am not so sure this works. Take $f(x)=nx$ for $x \in [0;\frac{1}{n+1}]$ and $f(x)=\frac{n}{n+1} + (x-\frac{1}{n+1})\frac{1-\frac{n}{n+1}}{1-\frac{1}{n+1}} = \frac{n}{n+1} + \frac{1}{n}(x-\frac{1}{n+1})$ on $]\frac{1}{n+1},1]$.
Then $f^{-1}(y)= \frac{y}{n}$ if $y \in [0,\frac{n}{n+1}]$ and $f^{-1}(y)= n(y-\frac{n}{n+1... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Find $\delta >0$ such that $\int_E |f| d\mu < \epsilon$ whenever $\mu(E)<\delta$ I am studying for a qualifying exam, and I am struggling with this problem since $f$ is not necessarily integrable.
Let $(X,\Sigma, \mu)$ be a measure space and let
$$\mathcal{L}(\mu) = \{ \text{ measurable } f \quad| \quad \chi_Ef \in L... | Suppose not: there is $\varepsilon_0\gt 0$ such that for each positive $\delta$, there is a measurable set $A$ such that $$\mu(A)\lt \delta\quad \mbox{ and }\quad \int_A|f|\mathrm d\mu\gt\varepsilon_0.$$
In particular, for each integer $k$ and $\delta:=2^{-k}$, there exists $A_k$ of measure smaller than $2^{-k}$ for wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/870142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Proving that $nb \text{ (mod m)}$ reaches all values $\{0 \dots (m-1)\}$ if $n$ and $m$ are relatively prime I am trying to prove the frobenius coin problem which requires me to prove the following lemma:
If $n$ and $m$ are relatively prime and $b$ is any integer, then the set of all possible values of $$nb \text{ (mod... | By Euclid, $\,(m,n)=1,\ m\mid nx\,\Rightarrow\, m\mid x,\,$ i.e. $\,nx\equiv 0\,\Rightarrow x\equiv 0\,$ in $\,\Bbb Z/m,$
i.e. the linear map $\,x\mapsto nx\,$ has trivial kernel, thus it is $\,\color{#c00}1$-$\color{#c00}1\,$ on $\,\Bbb Z/m.$
But $\,\Bbb Z/m\,$ is finite, thus, by $\rm\color{#c00}{pigeonhole}$, the ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/870298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Proof by contradiction using counterexample Why can't we use one counterexample as the contradiction to the contradicting statement?
Example:
Let a statement be A where a-->b.
We can prove A is not true by finding a counter example.
Now, in another space and time, Let a new statement be B where it is the same as a-->no... | You can disprove the statement $a \implies \sim b$ using a counterexample if you can find one. However, this does not prove the statement $a \implies b$. This is because the negation of $a \implies \sim b$ isn't $a \implies b$.
Recall that the negation of an implication is not an implication. $\sim (a \implies b) \e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/870381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Showing that Gaussians are eigenfunctions of the Fourier transform I'm having a bit of trouble on this problem:
I've tried to evaluate the integral directly (using the trick from multivariable calculus where you "square" the integral and convert to polar coordinates), but that hasn't gotten me anywhere. Does anyone ha... | You have $e^{-x^2/2} e^{-itx}$. The exponent is
$$
\begin{align}
-\frac{x^2}{2} - itx & = -\frac 1 2 (x^2 + 2itx) \\[10pt]
& = -\frac{1}2 \left((x^2+2itx+ (it)^2) - (it)^2\right) \tag{completing the square} \\[10pt]
& = -\frac 1 2 \left( (x+it)^2 - t^2\right)
\end{align}
$$
So you're integrating
$$
e^{-(1/2)(x+it)^2} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/870484",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Parametrizing curve for complex analysis integral I'm trying to show that
$$
\int_{|z-z_0| = R} (z-z_0)^m \, dz = \begin{cases}0, & m \neq -1 \\ 2\pi i, & m =- 1. \end{cases}
$$
Here's my attempt at a solution:
We parametrize the curve at $z(\theta) = z_0 + Re^{i\theta}$ and therefore $dz = iRe^{i\theta} \, d\theta$. S... | For $m\geqslant 0, f(z) = (z - z_{0})^{m}$ is analytic on $\mathbb{C}$. Thus the line integral is 0 ( by FTOC ).
For $m = -1, f(z) = 1$ is analytic on the closed disc $\rvert z - z_{0}\rvert\leqslant R$, so by Cauchy's integral formula this implies that $$1 = \frac{1}{2\pi i}\int_{\rvert z - z_{0}\rvert = R}\frac{1}{z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/870566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Images of Regions Under Cayley's Transformation I'm working on the following problem for my complex analysis course:
Problem $\bf 1$: Find the images of the followings under the Cayley's transformation: $a)$ imaginary axis $b)$ real axis $c)$ upper half plane $d)$ horizontal line through $i$
I can't seem to find ... | For the image of the imaginary axis under the Cayley's transformation: $$0 + yi\mapsto\frac{yi - i}{yi + i} = \frac{y - 1}{y + 1}\:\text{ ( the real-axis ) }$$
The map for the real-axis: $$x + 0i\mapsto\frac{x - i}{x + i}\:\text{ ( Note $\rvert\frac{x - i}{x + i}\rvert = \rvert\frac{\overline{x + i}}{x + i}\rvert = 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/870639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
A simple question in combinatorics. A university bus stops at some terminal where one professor,one student and one clerk has to ride on bus.There are six empty seats.How many possible combinations of seating?
My problem:I know that if there are six people to fill the 3 seats then there are $6\times5\times4=120= ^6P_3$... | Let's see how you arrived at $3^6$.
Each seat can be occupied by $3$ different people, and there are $6$ seats, so the number of possibilities is $3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^6$.
So one of these $3^6$ possibilities is: First seat is occupied by Professor. Second seat has $3$ choices: Professor, S... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/870715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Singular vector of random Gaussian matrix Suppose $\Omega$ is a Gaussian matrix with entries distributed i.i.d. according to normal distribution $\mathcal{N}(0,1)$. Let $U \Sigma V^{\mathsf T}$ be its singular value decomposition. What would be the distribution of the column (or row) vectors of $U$ and $V$? Would it be... | The original question suggested IID entries, which means that in the limit that the matrix gets big, the singular values follow a Marchenko-Pastur distribution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/870816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Is finiteness of rational points preserved by duality? Sorry if this is obvious. I don't know much about Abelian varieties.
Let $A/k$ be an abelian variety. Let's say $k$ has characteristic zero.
Let $\widehat{A}$ be the dual abelian variety.
Suppose that the set $A(k)$ of $k$-rational points is finite.
Is $\widehat{A}... | Since any abelian variety is isogenous to its dual, the answer is yes.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/870904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How many ordered pairs of integers $(a, b)$ are needed to guarantee that there are two ordered pairs $(a_1, b_1)$ and $(a_2, b_2)$ such that $\dots$ Question:How many ordered pairs of integers $(a, b)$ are needed to guarantee that there are two ordered pairs $(a_1, b_1)$ and $(a_2, b_2)$ such that $a_1 \bmod 5 = a_2 \b... | Yes, the answer is correct. There are $5$ options for the congruence of $a$ and $5$ options for the congruence of $b$ modulo $5$ so there are $25$ options for the congruence of the pair $(a,b)$. Clearly if you have $26$ then at least two will be in the same "congruence pair class". However with $25$ it could be we had ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/870984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Evaluate: $\lim_{x \to \infty} \,\, \sqrt[3]{x^3-1} - x - 2$ Find the following limit $$\lim_{x \to \infty} \,\, \sqrt[3]{x^3-1} - x - 2$$
How do I find this limit? If I had to guess I'd say it converges to $-2$ but the usual things like L'Hôpital or clever factorisation don't seem to work in this case.
| Hint: It is best to use series. However, we can do it with algebraic manipulation. Let $a=(x^3-1)^{1/3}$ and let $b=x+2$. Multiply top and (missing) bottom by $a^2+ab+b^2$.
Another way: Make the substitution $x=1/t$. We end up wanting
$$\lim_{t\to 0^+} \frac{\sqrt[3]{1-t^3} -1-2t}{t}.$$
Now one Hospital round does i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/871073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 8,
"answer_id": 3
} |
How do I reduce radian fractions? For example, I need to know $\sin (19π/12)$.
I need to use the subtraction formula. How do I get $(\text{what}) - (\text{what}) = 19π/12$? I am stuck at what are the radians
Do I divde it by something? What is the process?
| The denominator is $3\cdot4$ and you (should) know the values for denominators $3$ and $4$, so just decompose in two fractions
$$\frac{4\pi}3+\frac\pi4.$$
Now apply the addition formula.
This can be obtained by solving
$$\frac a3+\frac b4=\frac{19}{12}$$ or
$$4a+3b=19.$$
You can work this out by trial and error.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/871177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 5
} |
Probability Help! (X,Y) ~ f(x,y) = 8xy $I_D(x,y)$ a) $f_X (x) =$ ?
b) $P( X + Y < \frac{1}{2}) =$ ?
c) $f_Y(y \,| \, X = \frac{3}{4}) =$ ?
d) $P( Y < \frac{1}{2} \, | \, X = \frac{3}{4}) = $ ?
Any help is greatly appreciated! Thanks!!
Here is my work so far...
a) To get marginal density of $x$, I need to integrate $f(x... | $\operatorname{\bf I}_D(x,y)$ is an indicator function; a characteristic equation that has the value of $1$ when the argument exists within the domain, and a value if $0$ when it does not. Here the argument is $(x,y)$ and the domain is $D$.
This is sometimes written as: ${\large\bf 1}_D(x,y)$.
It is compact notation f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/871279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Fresnel Integral multiplied with cosine term. $$I=\int_a^b \sin(\alpha-\beta x^2)\cos(x)\, dx.$$
Can anybody tell me, how to solve this integral ?
I know that this is related to Fresnel Integral if the $\cos(x)$ term is absent.
| If $\beta>0$
$c+d=2\alpha-2\beta x^{2}$
$c-d=2x$
so $c=\alpha+x-\beta x^{2}=(\alpha+\frac{1}{4\beta})-(\frac{1}{2\sqrt{\beta}}-\sqrt{\beta}x)^{2}$ and $d=\alpha-x-\beta x^{2}=(\alpha+\frac{1}{4\beta})-(\frac{1}{2\sqrt{\beta}}+\sqrt{\beta}x)^{2}$
then using that $\sin(c)+\sin(d)=2\sin(\frac{1}{2}(c+d))\cos(\frac{1}{2}(c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/871412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Frenet-Serret formula proof Prove that $$\textbf{r}''' = [s'''-\kappa^2(s')^3]\textbf{ T } + [3\kappa s's''+\kappa'(s')^2]\textbf{ N }+\kappa \hspace{1mm}\tau (s')^3\textbf{B}.$$
What is $\tau$, I can't figure that part out.
All ideas are welcome.
| Take into account that, for a generic function $f$,
$$
f'=\frac{df}{dt}=\frac{df}{ds}\frac{ds}{dt}=s'\frac{df}{ds}
$$
so that
$$
\mathbf{N}'=s'\frac{d\mathbf{N}}{ds}
$$
and
$$
\frac{d\mathbf{N}}{ds}=-\kappa\mathbf{T}+\tau\mathbf{B}
$$
see Frenet-Serret formulas.
Also, your $\kappa'$ is indeed $\frac{d\kappa}{dt}$, whi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/871526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Evaluate $\int \frac{1}{(2x+1)\sqrt {x^2+7}}\,\text{d}x$. How to do this indefinite integral (anti-derivative)?
$$I=\displaystyle\int \dfrac{1}{(2x+1)\sqrt {x^2+7}}\,\text{d}x.$$
I tried doing some substitutions ($x^2+7=t^2$, $2x+1=t$, etc.) but it didn't work out.
| Using Euler substitution by setting $t-x=\sqrt{x^2+7}$, we will obtain
$x=\dfrac{t^2-7}{2t}$ and $dx=\dfrac{t^2+7}{2t^2}\ dt$, then the integral turns out to be
\begin{align}
\int \dfrac{1}{(2x+1)\sqrt {x^2+7}}\ dx&=\int\frac{1}{t^2+t-7}\ dt\\
&=\int\frac{1}{\left(t+\dfrac{\sqrt{29}+1}{2}\right)\left(t-\dfrac{\sqrt{29... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/871595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
finding right quotient of languages Can someone enumerate in detail, the steps to find right quotient of languages, i.e. $L_1/L_2$. Using an example will be great.
| The right quotient of $L_1$ with $L_2$ is the set of all strings $x$ where you can pick some element $y$ from $L_2$ and append it to $x$ to get something from $L_1$. That is, $x$ is in the quotient if there is $y$ in $L_2$ for which $xy$ is in $L_1$.
Let's agree to write the quotient of $L_1$ by $L_2$ as $\def\Q{\oper... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/871662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Matrices: $AB=0 \implies A=0 \text{ or } \ B=0$ When A and B are square matrices of the same order, and O is the zero square matrix of the same order, prove or disprove:-
$$AB=0 \implies A=0 \text{ or } \ B=0$$
I proved it as follows:-
Assume $A \neq O$ and $ B \neq O$:
then, $$ |A||B| \neq 0 $$
$$ |AB| \neq 0 $$
$$ A... | You are saying that if $A \neq O$ then, $det(A) \neq O$, which is false in general. Consider any diagonal matrix different from $O$ which has at least one zero in the diagonal.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/871744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Reduce distance computation overhead between a point and several rectangles We are given several rectangles in the plane, without loss of generality, assume there are three of them, namely $R_1$, $R_2$ and $R_3$.
For a point $P$, we can compute three distances $d_1$, $d_2$ and $d_3$ between $P$ and each rectangle respe... | If the rectangles are not too many, you can store the Voronoi diagram of your configuration in a convenient way (as a set of linear and quadratic inequalities, for instance), then find, for each point $P$, the index $i$ of the region where it belongs, in order to compute only $d(P,R_i)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/871831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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