Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Is $1-\alpha x\le (1-x)^{\alpha}\le 1-\alpha x+\frac{\alpha(\alpha-1)}{2}x^2,$ an inequality about generalized binomial coeffients true? Is the following inequality true?
$$1-\alpha x\le (1-x)^{\alpha}\le 1-\alpha x+\frac{\alpha(\alpha-1)}{2}x^2$$ for real numbers $x,\alpha.$ We may assume $0\le x\le 1$ and put some re... | For the upper bound, use
\begin{align}
(1-x)^r\leq e^{-rx}= 1-rx+ \frac{r^2x^2}{2!}-\frac{r^3x^3}{3!}+\ldots
\end{align}
for all $x<1$ and $r>0$.
Next, consider the following fact: for $x<0$, we have
\begin{align}
e^x \leq T_{2n}(x)
\end{align}
for all $n$ where
\begin{align}
T_n(x) = 1+x+\frac{x}{2!}+\ldots +\frac{x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2252384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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continuity of $e^{-1/z}$ in $\mathbb C$ Let f be the function defined by $f(z)=e^{-1/z}$ in $\mathbb C$, prove that $f$ is continuous in the set $0< \vert z \vert < 1$ and $\vert arg(z) \vert <\pi/2$ but it's not uniformly continuous on it.
I think an easy way to prove it is firstly to show that $f$ is analytic, theref... | For continuity, note that $-1/z$ maps $\{0<|z|<1\}$ into $\{1<|z|<\infty\}$ continuously, $e^z$ is continuous everywhere, and the composition of continuous maps is continuous. As for uniform continuity, consider the behavior of $f(x+i\sqrt x) - f(x)$ as $x\to 0^+.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2252482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Relationship between the number of edges in the dual of graph with the degree of the original graph? Is the number of edges in the dual of a graph (not necessarily a true dual) related to the degree of the nodes of the (original) graph? If so, is there a generalized formula for this relationship?
| To avoid bad cases like stars, I’ll consider only finite simple 3-connected planar graphs. (In particular, all triangulations are 3-connected). Due to Steinitz theorem such graphs are exactly the 1-skeletons of convex polyhedra. By Whitney's theorem, all plane embeddings of a polyhedral graph $G$ are equivalent, that i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2252566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is uniform convergence for one function and how is that equivalent to continuity? My teacher wanted to prove that function
$$\frac{1}{x} - \sum_{n=1}^\infty \frac{2x}{n^2-x^2}$$
is continuous. He said that it is equivalent to proving that
$$\sum_{n=1}^\infty \frac{1}{n^2-x^2}$$
uniformly converges. However uniform... | Theorem: the uniform limit of a sequence of continuous functions on an interval is a continuous function on that interval.
If you examine the teacher's proof that $\sum_{n=1}^\infty \frac{1}{n^2 - x^2}$ converges (presumably on some interval that doesn't contain any positive integer $n$), it may have in it some estim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2252672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to show that $\{(x_1,x_2) \in \mathbb{R}^2 | \exp(x_1) + \exp(x_2) \leq c\}$ is unbounded if $c > 0$ Define $f(x) := \exp(x_1) + \exp(x_2) $
Let the sublevel set be given by $\{x = (x_1,x_2) \in \mathbb{R}^2 | \exp(x_1) + \exp(x_2) \leq c\}$
A plot of this function along with its contour is given as:
Clearly, the ... | Since $\lim_{x\to -\infty} e^x =0$ it follows that for any $\frac{c}{2} \in \mathbb {R}^+$ there is some $a$ such that $e^x\leq \frac{c}{2}$ for all $x<a$ so $e^x + e^y \leq c$ for $x<a$ and $y<a$
This implies that the sublevel set $f^{-1}((-\infty,c])$ contains the set $\{(x,y) \in \mathbb {R}^2|x<a , y<a\}$ which is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2252788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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In which cases do multiple hyperbola branches have two intersection points? I am researching on hyperbolic localization techniques. In these techniques there are usually three anchor nodes $a_1, a_2$ and $a_3$ trying to position a blind node $b$. To do this, hyperbola branches are estimated which pass through the blind... |
Without loosing in generality we can place two anchor nodes ($A_m$ and $A_p$) symmetrically on the $x$ axis , and
place the third ($C$) in the upper half-plane as shown in the scheme above.
Let's consider the localization effectuated by node $C$ respectively with nodes $A_m$ and $A_p$, by crossing
the red and blue ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2252889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Show that there exists a step function Suppose $f:[a,b] \rightarrow X$ is a continuous map. By an argument based on uniform continuity, show that for any $\epsilon>0$ there exists a step-function $u: [a,b] \rightarrow X$ such that for all $x \in [a,b]$, $||f(x)-u(x)||<\epsilon$.
Okay, so I have looked back to the unifo... | We know that $x-\lfloor{x}\rfloor\leq1$. Using this fact,
$$u(x)=\epsilon\cdot\lfloor \frac{f(x)}{\epsilon}\rfloor$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2253029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Substitution rule for dirac measure I am trying to apply a substitution rule to a Lebesgue integral and get very strange results.
Substitution rule (an instantiation of Fremlin, D.H. (2010), Measure Theory, Volume 2, Theorem 263D): Let $\phi: \mathbb{R} \to \mathbb{R}$ be an injective function with derivative $\phi': \... | What you wrote doesn't match Theorem 263D in my copy of Fremlin. I think you missed an important sentence at the beginning of Section 263.
Throughout this section, as in the rest of the chapter, $\mu$ will denote Lebesgue measure on $\mathbb R^r$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2253160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do I simplify $\tan(\alpha-\beta)$ into $\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$? How do I simplify $\tan(\alpha-\beta)$ into $\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}$?
I tried:
$$\tan(\alpha-\beta) = \\\frac{\sin(\alpha-\beta)}{\cos(\alpha-\beta)}=\\\frac{\sin(\alpha)\cos(\beta)-\cos(\alpha)... | From
$$ \frac{\sin \alpha \cos \beta - \cos \alpha \sin \beta}{\cos \alpha \cos \beta + \sin \alpha \sin \beta} $$
divide the numerator and denominator by $\cos \alpha \cos \beta$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2253320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Does rotating a matrix change its determinant? For a $2 \times 2$, it is easy to see the determinant only changes sign.
\begin{align*}
\left(
\begin{array}{cc}
a & b \\ c & d
\end{array}
\right) \mapsto
\left(
\begin{array}{cc}
c & a \\ d & b
\end{array}
\right)
\end{align*}
We can see that $\det(A) = -\det(A')$, whe... | With a $4\times 4$ matrix, rotating preserves the determinant.
In general rotating means transposing (determinant-preserving) followed by turning matrix upside down (multiplies determinant by $(-1)^{\lfloor n/2\rfloor}$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2253433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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How many ways are there to go from the point $(0,0)$ to $(m,n)$ using,right up and down moves that we don't pass a point more than once? How many ways are there to go from the point $(0,0)$ to $(m,n)$ using,right up and down moves that we don't pass a point more than once?
I tried using to calculate every case(dependin... | Under the assumption that movement is only allowed in the grid formed by corner points $(0,0), (m, 0), (0, n), (m,n)$:
For each column, there are $n+1$ ways to pick where the horizontal movement occurs. After placing the horizontal movements for each column, there's only one way to connect them all (and these connectio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2253520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A more modern textbook on Axiomatic Set Theory, at the same level of rigor as Suppes? I'm currently using Suppes textbook to learn axiomatic set theory. Is there a more modern textbook that is just as well-written? I'm thinking of a textbook that still has a treatment of urelements (for example), but is modern enough t... | My personal preference is Set Theory: An Introduction To Independence Proofs by K.Kunen. Although it does not cover ur-elements. It's almost entirely on ZF and ZFC. And I get lost in the details in the def'n of Godel's constructable class L. For an easy and useful def'n of L, I suggest the essay in the book Lectures In... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2253640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Equation with a permutation composition Is there any method to solving such an equation:
$$f_1\circ f = f_2$$
Where $f_1, f, f_2 \in S_7$ and:
$f_1 = (1234)(5)(6)(7)$
$f_2 = (172536)(4)$
| So to my mind that would be the solution:
1) first we are to find $f_1^{-1}$:
$$f_1^{-1}\circ f = id \rightarrow f_1^{-1} = (13421)(5)(6)(7)$$
2) now we are to solve:
$$f_1^{-1} \circ f_2 = (17)(254)(36)$$
And that is our permutation $f$ we were looking for.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2253744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find parameter so that the equation has roots in arithmetic progression Find the parameter $m$ so that the equation
$$x^8 - mx^4 + m^4 = 0$$
has four distinct real roots in arithmetic progression.
I tried the substitution $x^4 = y$, so the equation becomes
$$y^2 -my + m^4 = 0$$
I don't know what condition should I put... | Zero can't be a root, else $m=0$, in which case all the roots would be zero.
If $r$ is any root, so is $ri$, hence there must be exactly $4$ real roots, and $4$ pure imaginary roots.
Also, if $r$ is a root, so is $-r$, hence the real roots sum to zero.
Ordering the real roots in ascending order, let $d > 0$ be the c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2253825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $ab$ is a square number and $\gcd(a,b) = 1$, then $a$ and $b$ are square numbers.
Let $n, a$ and $b$ be positive integers such that $ab = n^2$. If $\gcd(a, b) = 1$, prove that there exist positive integers $c$ and $d$ such that $a = c^2$ and $b = d^2$
So far I have tried this:
Since $n^2 = ab$ we have that $n = \s... | Let $ab=c^2$ for some $c\in N$ then the result will hold if any one of the integers is 1 as 1^2=1.
So let us take a>1,b>1 and c>1. We can use prime factorization and represent the integers as follows:
a=$p_1^{d_1} * p_2^{d_2}$
b= $q_1^{e_1} * q_2^{e_2}$
and c=$k_1^{l_1} *k_2^{l_2}$
thus $ab=c^2$ becomes
$p_1^{d_1}... | {
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"timestamp": "2023-03-29T00:00:00",
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Given the product is measurable, is each factor measurable? Given a random variable $M$ on $(\Omega,\mathscr F, \Bbb P)$ and $M=X\cdot Y$, can we proof that $X$ and $Y$ are also measurable?
To be more specific, I was thinking about if a process $M_t=X_t\cdot Y_t$ is adapted to its natural filtration $\mathscr F_t^M$. I... | No; not necessarily.
Witness a sample space, $(\Omega,\mathcal F) =\big( \{(1,1),(1,2),(2,1)\},\{\emptyset, \{(1,1)\}, \{(1,2),(2,1)\}, \Omega\}\big)$.
We define the random variables $X:(x,y)\mapsto x$, $Y:(x,y)\mapsto y, M:(x,y)\mapsto xy$.
Then $M(\omega)=[X\cdot Y] (\omega)$, and $M$ is $\mathcal F$-measureable, but... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2254071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Question on Limits - Asymptote As $x$ gets larger, $(x^3-8)/(x^2-4)$ approaches
a. 0
b. 1
c. 2
d. 3
e. infinity.
The answer is 3 but I do not think it is correct. Shouldn't it be infinity as we will have a slanted asymptote?
| We have an $x^3$ in the numerator and an $x^2$ in the denominator. $x^3$ increases much faster than $x^2$, so as $x$ gets large, the fraction will approach infinity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2254139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove that $\int_{0}^{\infty}{\sin^4(x)\ln(x)}\cdot{\mathrm dx\over x^2}={\pi\over 4}\cdot(1-\gamma)?$
How to prove that
$$\int_{0}^{\infty}{\sin^4(x)\ln(x)}\cdot{\mathrm dx\over x^2}={\pi\over 4}\cdot(1-\gamma).\tag1$$
Here is my attempt:
$$I(a)=\int_{0}^{\infty}{\ln(x)\sin^4(x)\over x^a}\,\mathrm dx\tag2$$... | I will outline a self-contained approach, too. By differentiating the integral definition of the $\Gamma$ function, we get the following Lemma:
$$ \mathcal{L}(\log x) = -\frac{\gamma+\log(s)}{s}\tag{1} $$
and it is not difficult to compute from $(1)$ the Laplace transform of $\sin^4(x)\log(x)$.
By Euler/De Moivre's for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2254266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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On generating set for abelian $p$-group Let $G$ be a finite abelian $p$-group. Let $\{x_1,\cdots,x_r\}$ be a subset of $G$ with following property:
(1) $\langle x_1, \cdots, x_r\rangle=\langle x_1\rangle \oplus \cdots \oplus \langle x_r\rangle$.
(2) No $x_i$ is a $p$-th power in $G$. [In other words, all $x_i$'s are ou... | No. Let $G = \langle a \rangle \oplus \langle b \rangle$ where $a$ and $b$ have orders $p$ and $p^3$, respectively, and let $H = \langle x \rangle$ with $x = ab^p$. So $H \cong C_{p^2}$ and is not a direct summand of $G$.
(I once made a mistake myself in a proof and this was essentially the counterexample.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2254375",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Square and multiply algorithm I'm trying to understand the square and multiply algorithm:
If I understand it correctly, whenever the exponent is even, we divide it by 2 but square the base, and whenever it is odd, we take an x out and subtract 1 off the exponent.
So, when running the algorithm on $2^{10}$, I was expec... | Error: in second step, As @Kenny mentioned, $x$ should be $2^2$ instead of $2$. We are talking about $(x^2)^{y/2}$ and here, $x$ happens to be $2^2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2254489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $f$ is integrable on $[a,b] $ Suppose that $f(x)=0$ for all $x$ in $[a,b]$ except for some $ z $ in $[a,b]$ Prove that $f$ is integrable on $[a,b] $.
My try:If we can show that $f$ is continuous in $[a,b]$,then the result will follow.Thank you.
| Continuity would give integrability, but this function cannot be continuous since $\displaystyle \lim_{x \rightarrow z} f(x) = 0 \neq f(z)$, and this condition must be true for every $y \in [a,b]$ if $f$ is indeed continuous on this interval. However, it isn't too difficult to tackle the problem directly from the defi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2254602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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If $\lim_{n \rightarrow \infty} (a_{n+1}-\frac{a_n}{2})=0$ then show $a_n$ converges to $0$. I have been stuck on this question for a while now. I have tried many attempts. Here are two that I thought looked promising but lead to a dead end:
Attempt 1:
Write out the terms of $b_n$:
$$b_1=a_{2}-\frac{a_{1}}{2}$$
$$b_2=a... | Let $\epsilon > 0$.
Since $a_{n+1}-a_n/2$ converges to $0$, there is an integer $m$ such that
for any $n \ge m$, $|a_{n+1}-a_n/2| \le \epsilon/4$.
For such an $n$, you have
$|a_{n+1}| - \epsilon/2 \\
\le |a_{n+1} - a_n/2| + |a_n/2| - \epsilon/2 \\
\le \epsilon/4 + |a_n|/2 - \epsilon/2 \\
= |a_n|/2 - \epsilon/4 \\
= (|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2254694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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What's the magnitude of a real number? As a student of mathematics (first year master degree) I have to admit that I'm somewhat ashamed to ask this.
We know that if $z=x+iy$ is a complex number then we can identify it as $z=r\cdot\exp(i\theta)$. But what if $z$ is real - in other words its $y$ equals 0? Then $z=r\exp(i... | Since it has not been pointed out so far, it is crucial to realize that while cartesian coordinates are unique, polar coordinates are not unique, because the pair $(r\cos(t),r\sin(t))$ is exactly the same as $(r\cos(t+2πn),r\sin(t+2πn))$ for any integer $n$. Furthermore, $(0\cos(t),0\sin(t)) = (0,0)$ for any real $t$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2254792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 6
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One-step transition probabilities for a markov chain? Imagine m balls being exchanged between two adjacent chambers (left and right) according to the following rules. At each time step, one of the m balls is randomly selected and moved to the opposite chamber, i.e., if the selected ball is currently in the right chambe... | $\{0,1,2,3\}$ is the set of states, that is, the set of the possible numbers of balls in the left chamber.
Assume that the system is in state $i$ ($i=0,1,2,3$). The probability that the sytem goes to state $i-1$ is $\frac i3$ because this is the probability that one selects a ball from the left box. The probability tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2254873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How can we prove that there are $2^{\mathfrak{c}}$ Hamel bases? I know that there are $2^{\mathfrak{c}}$ distinct Hamel bases for $\mathbb{R}$ over $\mathbb{Q}$ but what is the demonstration for that?
| There are many ways to show this; here's one simple way. Let $B$ be any Hamel basis for $\mathbb{R}$ over $\mathbb{Q}$, which must have cardinality $\mathfrak{c}$, and partition $B$ into two subsets $C$ and $D$ with a bijection $f:C\to D$ (so $|C|=|D|=\mathfrak{c}$). For any subset $S\subseteq C$, the following set i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2255021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does Yoneda embedding reflect equivalent categories? Let $\mathsf{Cat}$ denote the category of small categories. For categories $\mathcal A$ and $\mathcal B$ in $\mathsf{Cat}$, let $[\mathcal A,\mathcal B]$ denote the category whose objects are functors form $\mathcal A$ to $\mathcal B$ and morphisms are natural transf... | Yes. This is a special case of the $2$-categorical Yoneda Lemma. Here is a direct proof.
Assume that $F^*$ is an equivalence for all categories $C$. In particular, $F^* : [B,A] \to [A,A]$ is essentially surjective. Choose some $G : B \to A$ with $GF \cong \mathrm{id}_A$. We have $FG \cong \mathrm{id}_B$ since $FGF \con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2255283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find a non-zero integer matrix $X$ such that $XA=0$ where $X,A,0$ are all $4 \times 4$ Let $A$ be the following $4 \times 4$ matrix.
\begin{bmatrix}1&2&1&3\\1&3&2&4\\2&5&3&7\\1&4&1&5\end{bmatrix}
How can we find a non-zero integer matrix $C$ such that $CA = 0_{4 \times 4}$
Note that $0$ is a $4 \times 4$ matrix.
| Hint : The rank of $A$ is $3$ so : $$\exists P,Q \in GL_n(\mathbb R),A=P\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&0\end{bmatrix}Q$$
$$CA=0 \Rightarrow CP\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&0\end{bmatrix}Q=0\Rightarrow C\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&0\end{bmatrix}=Q^{-1}P^{-1}$$
Calcu... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
Quotient group is complete, so is the group Let $G$ be a topological metrizable group and $K$ a normal subgroup of $G.$ Consider the homogeneous space $G/K$ and assume that both $K$ and $G/K$ are complete. I need to prove $G$ is complete.
More specifically, assume there is a right-invariant metric on $G$ such that the ... | I can finish your proof as follows. For each $n$ pick an element $x’_n\in \dot x$ such that $d(x_n, x’_n)< d(x_n, \dot x)+1/n$. Since the sequence $(\dot x_n)$ converges to $\dot x$ and is fundamental, the sequence $(x’_n)$ is fundamental too. Since the space $\dot x\supset (x’_n)$ is complete in the induced metric, t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2255476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Integration of f(x) where f(x) is x in binary, used as a decimal
Define $f(x)$ when $x$ $\in [0,1]$ as $x_2$ ($x$ base 2) considered as a decimal value.
Therefore, $f(0.75) = 0.11$, and $f(0.25) = 0.01$.
Compute $\int_0^1f(x)dx$.
To do this, I figured that the answer might be $0$ because integrals aren't defined by ... | Here's why zero can't be right. You can see $f(x) \geq 0.1$ on $[0.5,1.0)$.
So $\int_0^1 f(x)\,dx \geq (0.1)(0.5) = 0.05$.
Also, $f(x) \geq 0.01$ on $[0.25,0.5)$, $f(x) \geq 0.1$ on $[0.5,0.75)$, and $f(x) \geq 0.11$ on $[0.75,1)$. So
$$
\int_0^1 f(x)\,dx \geq (0.01)(0.25) + (0.1)(0.25) + (0.11)(0.25) = (0.22)(0.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2255564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Symmetries of Tetrahedral Dice I am trying to find the number of distinguishable tetrahedral dices, where the sides are numbered 1,2,3,4. I found this webpage (http://mathworld.wolfram.com/PolyhedronColoring.html) that claims that there are 2 distinct ways to do so, but I don't see how they came to that answer. I have ... | We can answer the questions about the number of colorings of the faces
of a tetrahedron using some maximum number of colors or some specific
number of colors using the cycle index of the symmetries acting on the
faces of the tetrahedron and applying Burnside. The cycle index is
quite simple here, we now show how... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2255661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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I don't really understand this calc question What do you even do here? Take the other two variable to RHS?
| HINT:
Start by considering,
$$z^z = \dfrac{c}{x^xy^y}$$
Next, apply logarithm laws to obtain,
$$z\log (z) = \log (c) - x\log (x) + y\log (y)$$
Hopefully, from here you can apply ideas of partial differentiation to obtain your answer (the product rule will come in handy).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2255783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving $8yy'^2 - 2xy' + y = 0$ I'm solving the differential equation $8yy'^2 - 2xy' + y = 0$
My attempt:
We divide both sides by $x$, obtaining:
$$8\frac{y}{x}y'^2 - 2y' + \frac{y}{x} = 0$$
Then, we introduce $t = y'$, hence the differential equation becomes:
$$8\frac{y}{x}t^2 - 2t + \frac{y}{x} = 0$$
from which follo... | You lost the factor $x$ in the last term while differentiating the parametric equation.
It is even easier to multiply with $y$ and then substitute $u=y^2$ to get
$$
2u'^2-xu'+u=0\iff u=xu'-2u'^2
$$
which is a Clairaut differential equation. This has the lines
$$
u=cx-2c^2
$$
as solutions and their envelope which is th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2255914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Concurrency of the heights of a tetrahedron with opposite edges perpendicular. Can anyone give me a vectorial solution to the following problem:
Prove that if each pair of opposite edges of the tetrahedron $ABCD$ is perpendicular (that is, $AB \perp CD$ and $AC \perp BD$ and $AD \perp BC$), then the heights of the tet... | That is a simple exercise in visualization. Imagine that $A,B,C$ are embedded in the $xy$ plane (the screen) and $D$ lies on the $z$-axis (orthogonal to the screen), so that the origin $O$ is the projection of $D$ on the plane through $A,B,C$. Since $DB\perp AC$ (in $3$D) we have $OB\perp AC$ (in $2$D). Similarly we ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2256055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Find the maximum and minimum value of $P =x+y+z+xy+yz+zx$ Let $x^2+y^2+z^2\leq27$ and $P = x+y+z+xy+yz+zx$. Find the value of $x, y, z$ such that $P$ is the maximum value and minimum value.
My attempt :
$$(x-y)^2 + (y-z)^2 + (z-x)^2 \geq 0$$
$$27 \geq x^2+y^2+z^2 \geq xy+yz+zx\tag{1}$$
$$(x+y+z)^2 \leq 3(x^2+y^2+z^2) \... | You may use the same method to find the minimum. First, we obtain a lower bound for $P$:
$$
\begin{align}
P&=x+y+z+xy+yz+zx\\
&=\frac12 [ (x+y+z+1)^2 - (x^2+y^2+z^2) - 1 ]\\
&\ge\frac12 (0 - 27 - 1)\tag{1}\\
&= -14.
\end{align}
$$
Next, note that at $\left(\frac{\sqrt{53}-1}2,-\frac{\sqrt{53}+1}2,0\right)$, we have $x+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2256277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Finding the preimage I want to find the preimage of $]-2,4]$ for the function $f(x)=x^2-x$
This is what I have done so far:
We have $0=x^2-x-y$ and therefore the inverse is:
$$f^{-1}(y)=\frac{-1\pm \sqrt{1+4y}}{2}$$
And how do I find out the boundaries of the preimage? If I put the boundaries $-2$ and $4$ into the func... | $$f(x) = x^2-x=x^2-x+\frac14-\frac14 = (x-1)^2 - \frac14 \geq -\frac14 $$
you are asked to find $x$ such that $-2\leq f(x) \leq 4$, but $f \geq -\frac14$ so the first inequality doesn't restrict us at any way.
$$ f(x) \leq 4 \Rightarrow (x-1)^2 -\frac14 \leq 4 \Rightarrow (x-1)^2 \leq \frac{17}4\Rightarrow$$
$$ -\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2256389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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How to find modulo using Euler theorem? I don't know how that's possible using phi, the question starts with this one:
a) Decompose 870 in prime factors and compute, ϕ(870)
I know how to resolve this, first 870 = 2*3*5*29 and ϕ(870)= 224
Now this is the question I don't know how to resolve:
b) Compute 77^225 modulo ... | Use euler's theorem. If $(a,n)=1$, then
$$
a^{\varphi(n)}\equiv1\pmod{n}
$$
Since $77$ and $870$ are coprime (their prime factorizations have no prime in common)
$$
77^{225}
\equiv
77^{224}\times77\equiv1\times77\equiv77\pmod{870}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2256561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Heat equation with different boundary conditions Consider the heat equation
$$
u_t=u_{xx}
$$
on an interval $[-L,L]$
with Dirichlet, Neuman and periodic boundary conditions.
Am I Right that with Dirichlet b.c. all solutions are exponentially decaying in $L_2$-Norm (and that this corresponds to a spectrum in the left h... | The spectrum is determined by the $x$ equation and associated endpoint conditions after performing separation of variables. The equation in $X$ is
$$
-X''(x)=\lambda X(x)
$$
and there are two general types of conditions:
*
*Separated Conditions, which as described as a two-parameter family in real $\a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2256651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Height of hill when angle of elevation for each vertex is same
The angle of elevation of the top of a hill from each of the vertices $A, B$ and $C$ of a horizontal triangle is $\alpha$. Prove that the height of the hill is $\frac{a}{2} \tan\alpha \csc(A)$.
Could someone help me to approach this question. I am not g... | radius of circumcentre R= a/2sinA =b/2sinB= c/2sinC IN WHICH AH = BH =CH= R
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2256794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Doubt over the proof of Cayley- Hamilton heorem I am having some doubt in the proof of Cayley Hamilton theorem. This theorem says that every matrix is a root if its characteristic polynomial.
Proof goes as follows:
Let us assume that matrix $A$ is of order $n\times n$. If $P(\lambda)$ be its characteristic polynomial... | If $Q(\lambda)$ is the adjoint of $P-\lambda I$, by definition $Q(\lambda)(P-\lambda I)=det(P-\lambda I) I$.
Compare the maximum degree element of each matrix of the equality. If an element $q_{i,j}$ of $Q(\lambda)$ had degree $k$, then element in each diagonal element of the matrix would have max degree element would ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2256914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Show that if $E$ is not measurable, then there is an open set $O$ containing $E$ that has finite outer measure and for which $m^*(O-E)>m^*(O)-m^*(E)$ Let $E$ have finite (Lebesgue) outer measure. Now we need to show that if $E$ is not measurable, then there is an open set $O$ containing $E$ that has finite outer measur... | How do you define $m^*(U)$ for $U$ open? In order to say that for any $\epsilon$, $\exists U \supseteq E$ open such that $m^*(U) - m^*(E) < \epsilon$, it seems like you need to make precise how you could find this $U$ with outer measure between $m^*(E)$ and $m^*(E) + \epsilon$.
Other than this point, the proof looks co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2257097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Help with finding expectancy So I have $x\sim U(\{1,2,..., 20\})$
and I need to find $E(x^2)$.
I have tried searching our textbook but could not really understand the logic behind the steps they showed.
Where am i supposed to start solving something like this?
| $x$ is uniformly distributed on the set $\{1, 2, \dots, 20\} $, so there are the same number of $1$'s as $2$'s as $3$'s, etc. Now, the variable we are investigating to find the expected value is $x^2$. So, the set $\{1, 4, 9, \dots, 400\}$ of squares will be uniformly distributed. The expected value is the average of t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2257196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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absolute values of algebraic numbers under Galois automorphism This could be very easy question, but I have no idea about it in depth.
Q. Let $\alpha$ be an algebraic integer. Let $\sigma$ be an automorphism of Galois group of $\mathbb{Q}(\alpha)$ (over $\mathbb{Q}$). If, as complex number, $|\alpha|<1$, is it necessa... | In general, no.
For example, let $\alpha = 2-\sqrt{2},\,$ and consider the automorphism $\sigma$ of $\mathbb{Q}(a)$ such that $\sigma(\sqrt{2}) = -\sqrt{2}$.
Then
$$|a| = |2-\sqrt{2}| < 1$$
but
$$|\sigma(a)| = |2+\sqrt{2}| > 1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2257309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is $f(x + dx) -f(x) = f'(x) \,\mathrm dx$ a valid equation? $\def\d{\mathrm{d}}$We know that it is true that$$\lim_{\Delta x \to 0} \frac{f(x + \Delta x) -f(x)}{\Delta x} = \frac{f(x + \d x) -f(x)}{\d x} = f'(x),$$
where $\d x$ is define to be an infinitesimal.
Then we could rearrange the equation and say that$$f(x + \... | Yes its valid we use this result in cases where we want to find an approximate value without using calculator. For eg say we want the value of $\sqrt {64.1} $ so we define $f (x)=\sqrt {x}$ then using your equation its $\sqrt {64.1} \approx \sqrt {64}+0.1 \frac {1}{2\sqrt {64}}=8+0.1/16=8.00625$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2257417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
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Applying Fundamental theorem of calculus problem In this link provided is a question about the Fundemental Theorem of Calculus since I don't know how to use LaTeX. I somehow can't get the right answer, I'm using the chain rule and everything but still getting it wrong
link
$$F(x)=\int_0^{x^3} 4\sin \pi t^2dt$$
Find $F(... | The interesting thing is that what you put for your answer is right. There must be some formatting issue with your homework software. Have you tried something like changing the "pi" to a "Pi"?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2257549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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How to find all the positive integers n of 4 digits such that all its digits are perfect squares and n is a multiple of 2, 3, 5 and 7? I was trying to use the divisibility rules of 2,3,5 and 7 but I becomes very tedious and couldn's solve the problem. I think there could be a faster way to solve it or to apply those ru... | Hint: The last digit has to be $0$ to be divisible by $2$ and $5$ simultaneously. Thus, The number you are seeking is $ABC0$ where $A,B,C\in\{0,1,4,9\}$ but $A \ne 0$.
Also, $A+B+C=3k$ for divisibility via $3$-Equation $(1)$
And $A+B+C =7 \lambda$-Equation$(2)$
Although I'd recommend checking divisibility rules for $7$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2257667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does $x_n$ converge, given $\lim(3x_{n+1} - x_{n})=1 $ I want to prove that $x_n$ converges, given that $\lim (3x_{n+1} - x_n ) = 1$
Attempt: Since $\lim (3x_{n+1} - x_n ) = 1$, set $\epsilon > 0, $ such that $\forall n > N, |3x_{n+1} - x_n -1 | < \epsilon. $ Then,
$$|x_{n+1} - x_n|< \min(\frac{\epsilon + 1 - 2x_N}{3}... | The limit, if it exists, is $\frac{1}{2}$, so I think it is easier to work with $u_n:=x_n-\frac{1}{2}$. The hypothesis then is $3 u_{n+1} - u_n \to 0$. We want to prove that $u_n\to 0$.
Let $\epsilon>0$. For big enough $n$ we can get $3 |u_{n+1}|\leqslant |u_n| +\epsilon$, and then
$$
|u_{n+k}|\leqslant \frac{1}{3^k}|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2257757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is generating function proof of Fibonacci formula correct? The proof goes as follows:-
Let $F = 1 + x + 2x^2 + 3x^3 + 5x^4 + 8x^5 + ...$
Then
$$\begin{align} 1 + Fx + Fx^2 &= 1 + (x + x^2 + 2x^3 + 3x^4 + ...) + (x^2 + x^3 + 2x^4 + 3x^5) \\
1 + Fx + Fx^2 &= 1 + x + (x^2+x^2 + 2x^3+x^3 + 3x^4+2x^4 + ...) \\
1 + Fx... | One way to do this is to first prove that the series converges and then do these calculations, as Foobaz John outlines. But in fact, all these manipulations are perfectly valid without even taking convergence into consideration if you frame them in the right way - that is, in terms of formal power series.
A formal p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2257866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 3,
"answer_id": 0
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Closed subset of $C([0,1],\mathbb R$)
Let $B=\{f\in C^1[0,1]:\Vert f\Vert _\infty \le A\}$. Is $B$ a closed subset of $C([0,1],\mathbb R)$?
Here's what I tried to do:
Let $\{f_n\}$ be a sequences of functions of $B$ such that $f_n \to f$, with $f\in C([0,1],\mathbb R)$.
If I prove that $f\in B$, then I finish the pr... | Ahh, another simple example of sequence of functions is $$ f_{n} = \sqrt{x + \frac{1}{n}} $$
which uniformly convergences to $f(x)= \sqrt x $ which is not differentiable at $x=0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2257978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Show $f(x,y)=x^2\log(x^4+y^2)$ is differentiable at $\vec 0$ I have to show that $f$ is differentiable at $\vec 0$, where
$$
f(x,y)=x^2\log(x^4+y^2),
$$
and $f(0,0)=0$.
I’ve already shown that $f$ is continuous at $\vec 0$. I started off by calculation the first partial derivative:
$$
D_1f(\vec 0)=\lim_{t\to 0}t\log t^... | $t\log t^4=4t\log t\to 0$ as $t\to 0_+$ is a standard limit from high school.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2258077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
smallest number of socks to guarantee that the selection contains at least $10$ pairs A drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks and $40$ black socks.
A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn.
What is the smal... | Suppose you put $n$ socks into $4$ color boxes such that there are a total of exactly $k$ pairs of socks in the $4$ color boxes. Then, $n$ must be at least $2k$ because that many socks are required for the $k$ pairs. The maximum possible value of $n$ is $2k+4$ because each of the $4$ color boxes can have an odd numbe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2258175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
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Definite Integral in the study of Prophet Inequalities I wish to integrate the function $\frac{1}{a + x - x \cdot \ln x}$ for $x$ going from $0$ to $1$, where $0<a<1$ is a constant. It's easy to see that when $a=0$ this function has a simple indefinite integral $-\ln(\ln x - 1)$. However, for non-zero $a$, solvers like... | The bad news is that the integral actually diverges at $0,$ despite the nice indefinite integral. The good news is that if you evaluate it (say, numerically) at one point ($a=1$ is good), you have a rapidly converging power series expansion. Namely, by differentiating Brevan's expression under the integral sign with re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2258303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Help needed with modulus addition and multiplication proof We have recently started working with modular arithmetic in my discrete mathematics course, and I found two problems in my textbook that I am having trouble with. What are these kinds of proofs called, and what is the usual approach that is undertaken? Lastly, ... | The idea here is to write $a\mod m$ as $a+k_1m$, where $k_1$ is some constant. Then we can construct a proof quite easily.
For 1, we have $(a+k_1m) + (b + k_2m) = a+b+(k_1+k_2)m \equiv a+b\mod m$
And for 2, we have $(a+k_1m)(b + k_2m) = ab+(bk_1+ak_2+k_1k_2m)m \equiv ab \mod m$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2258459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Why the equation of an arbitrary straight line in complex plane is $zz_o + \bar z \bar z_0 = D$ Why the equation of an arbitrary straight line in complex plane is $zz_o + \bar z \bar z_0 = D$ where D $\in R$
I understand that a vertical straight line can be defined by the equation $z+\bar z= D$ because suppose $z =x+yi... | Hint: given any two points $z_1, z_2 \in \mathbb{C}\,$, then $z$ is collinear with $z_1, z_2$ iff there exists $\lambda \in \mathbb{R}$ such that $z-z_1 = \lambda(z-z_2)$. Eliminate $\lambda$ between the following, then define $z_0, D$ appropriately:
$$
\begin{cases}
\begin{align}
z-z_1 &= \lambda(z-z_2) \\
\bar z- \ba... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2258557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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If $\int_{a}^{b}f(x)g^n(x)dx=0, \quad \forall n \in \mathbb{N}, \; $ then $\; f \equiv 0$ Let g be continuous, not negative, and strictly increasing in [a, b]. Prove that if $f$ is continuous and
$$\int_{a}^{b}f(x)g^n(x)dx=0, \quad \forall n \in \mathbb{N},$$
then $f\equiv 0$.
With a change of variable I have arrived h... | Hagen von Eitzen above has indeed given an elegant solution valid when the given g is strictly increasing but no assumption that g be non-negative is needed or used. However it is not necessary assume the integral (1) I[a,b]f(x)(g(x))^n dx =0 for all non negative integers n= 0.1,2,3... . It suffices to assume this fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2258659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is meaning of same thing is different in mathematical logic and english? Is meaning of "if and only if" is different in mathematical aspects and English aspect?
let's take an example:
Example:I will go home if and only if it is not raining.
Now according to me in english aspect ,I cannot comment anything about rai... | There is indeed a mismatch between the way we typically treat the 'if and only if' statement in English (or any other natural language) and the way we treat the logical $\leftrightarrow$. Take the following example:
'Mary lives in France if and only if Mary lives in Germany'
In any natural language would immediately sa... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find how many complex roots the equation has How many complex roots has each of the equations:
$$z^3 = \overline{z}$$
$$z^{n-1} = i \overline{z}$$
Where $\overline{z}$ is the complex conjugate of $z$.
For the first one I tried giving the form $z = a + bi,\:a, b \in \mathbb R$ and find the roots but I think there's some... | For the first equation, we have $$|z|^3=|\bar z|\implies |z|^3 = |z|\implies |z| \in\{0,1\}.$$
If $z\neq 0$, we can multiply the first equation by $z$ to get $z^4 = z\bar z = |z|^2 = 1$ and we have four different solutions $\{\pm 1,\pm i\}$. Since $z=0$ is solution of the original equation as well, this gives total of ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Triangles within a Parallelogram ABCD is a parallelogram.
E is the point where the diagonals AC and BD meet.
Prove that triangle ABE is congruent to triangle CDE.
| AE and EC are on the same line, so have the same gradients. Same goes for BE and ED. Because it is a parallelogram, AB=CD, as it has to be so that the shape holds. Therefore, you have proved this by the Angle-Side-Angle rule, where two triangles with two identical angles and a side are congruent, even if they are refle... | {
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"source": "stackexchange",
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Is it possible to compute Right Triangle's Legs starting from another Right Triangle with the same Hypotenuse? In an application of the Manhattan Distance trough the haversine formula, I was stuck in a problem that doesn't allow me to compute the right distance among two points in a space.
Despite the scope, it could b... | To know all sides of a triangle you must know either:
*
*One side and two angles, or
*Two sides and one angle, or
*Three sides
You only know one side and one angle. Ergo, you cannot compute any further sides in the blue triangle.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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A question about the equivalence relation on the localization of a ring. Let $A$ be a ring and $S$ a multiplicative closed set. Then the localization of $A$ with respect to $S$ is defined as the set $S^{-1}A$ consisting of equivalence classes of pairs $(a, s)$ where to such pairs $(a,s), (b,t)$ are said to be equivalen... | For a more geometric example, let $A = \mathbb{R}[x,y] / (xy)$ and $S = \{ 1, x, x^2, x^3, \ldots \}$. (In algebraic geometry, $A$ represents a ring of functions on the union of the two axes in $\mathbb{R}^2$.) Then in $S^{-1} A$, $y/1 = 0/1$ even though $y \cdot 1 - 0 \cdot 1 \ne 0$ in $A$ - but in fact, $x (y \cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2259166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Mental division of two fractions? I've got a non-calc paper coming up, and when going through a test, this fraction came up:
$$
\frac{8}{-0.4} \equiv \frac{8}{\big(\frac{-2}{5}\big)}
$$
Going through the answers he says: $$8/2=4$$ I then assume he did -(4*5)
so: $$\frac{8}{\big(\frac{-2}{5}\big)} = -20$$
I can see wha... | What he is using is that $\frac{a}{\frac{1}{b}} = a*b$. That way he could transform a division involving decimals into simple integer multiplication.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2259290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Line Integral and Residue Theorem
I know applying the residue theorem to some integral gives that when you differentiate $(1+z)^{2n} $ n times and evaluate at 0 you get 2n choose n but I don't understand how the highlighted step is taken as automatic~
| HINT:
Apply the binomial theorem to $\left(\frac{z+z^{-1}}{2}\right)^n$ and exploit
$$\oint_{|z|=1}z^m\,dz=\begin{cases}0&,m \ne -1\\\\2\pi i&,m=-1\end{cases}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can I find the area between the graphs of a function and its inverse? I have the following function $$f(x)=x\cdot e^{x^2-1} $$ and I want to find the are between this function and its inverse. I'm not sure how to calculate the integral because I know that for this type of problem I need to find where the two functi... | HINT: The functions intersect where $f(x)=x$.
HINT2: This is at $-1, 0, 1$
HINT3: The inverse function is the function reflected around the line $y=x$, so you find the intersections of $f(x)$ and $f^{-1}(x)$ when $f(x)$ intersects with $y=x$. You use this to integrate and find the area between $f(x)$ and $x$, then mult... | {
"language": "en",
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"source": "stackexchange",
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Decomposition of the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}[i]$ into a product of fields I am trying to solve the following problem:
For each rational prime $p$, describe the decomposition of the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}[i]$ into a product of fields, where $\ma... | In general, tensoring a number field with $ \mathbf Q_p $ gives you an appropriate direct product of the completions of it at the different primes lying over $ p $. The key isomorphism is
$$ \mathbf Q_p \otimes \mathbf Q(i) \cong \mathbf Q_p \otimes \mathbf Q[x]/(x^2 + 1) \cong \mathbf Q_p[x]/(x^2 + 1) $$
Now, we have ... | {
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"source": "stackexchange",
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Galois Theory Quadratic Subfield Let $ζ_7=e^{i2\pi/7}$ be a 7th root of unity. The field $\Bbb{Q}(ζ_7)$ contains a quadratic subfield that can be expressed in the form of $\Bbb{Q}(\sqrt{D})$ where D is an integer. What is D?
I understand that there is a field extension of order 6 and therefore there will be a quadrati... | Not every field extension of degree $ 6 $ has a quadratic subfield. For example, $ \mathbf Q(\sqrt{1 + \sqrt[3]{2}}) $ has no subfield that is quadratic over $ \mathbf Q $. It is, however, true that every Galois extension of degree $ 6 $ has a unique quadratic subfield. It has been pointed out in another answer how to ... | {
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"source": "stackexchange",
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Markov Chains - understand proof that if x and y communicate then if x is recurrent then y must also be recurrent I am trying to understand the proof that if two states that communicate, then if one state is recurrent the other must also be recurrent.
The book I'm looking at has this proof:
Suppose $x$ is recurrent, an... | Regarding $p_{k+n+l}(y,y) \ge p_l(y,x) p_n(x,x) p_k(x,y)$ can be intuitively explained as follows:
*
*The left-hand side is the probability of starting at $y$ and after $k+n+l$ steps landing at $y$ again.
*The right-hand side is the probability of starting at $y$, then after $l$ steps landing at $x$, then after $n$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Maps to a Subobject Classifier Let $\Omega$ be the subobject classifier of a category $C$. Choose some map $m: X \rightarrow \Omega$. Is there necessarily a subobject $p: P \rightarrow X$ for which $m$ is the characteristic map?
It seems that in the definition of subobject classifier you can only go the other way aro... | Usually you only speak of a subobject classifier in a category that has all pullbacks. In that case, the pullback of the universal map $1\to\Omega$ along $m$ is a subobject $p:P\to X$ for which $m$ is the characteristic map. Really, the existence of at least all pullbacks of this form should be part of the definition... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the real and imaginary parts of $ln(z)$ This is on my homework on differentials and partial differentiation, so I'm not sure what application these could have on the natural log of z
| The answer of Iti Shree is correct, but under tacit assumptions, which I would
like to clarify here. In particular, $\Im(\ln(z)) = \arctan(\frac{y}{x})$ is
not defined if $x = 0$ and does not distinguish between opposite complex
numbers.
First, $\ln(z)$ needs to be defined properly. Notice that it cannot be defined
con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2260028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Why are random walks in dimensions 3 or higher transient? I watched this PBS video a while ago (relevant part here) and have been trying to get my head around the idea of transient walks. The video says that a recurrent random walk is one that is guaranteed to return to it's starting position - all 1D and 2D walks - an... | In 1 dimension, although the expected number of times that you will return to the origin before a give time approaches infinity as time approaches infinity, it varies sublinearly with time. When you have 3 independent 1-dimensional random walks all starting at the origin, although you can expect that each of them indiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2260163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Am I supposed to use generating functions or combinatorics or something for this question? If there are 201 seats in the Parliamentary chamber, how many different ways are there for the numbers
of seats to be allocated amongst three political parties, subject to no one party having an overall
majority?
I could figure i... | We seek the number of integer solutions to the equation
$$
x_1+x_2+x_3=201\tag{1}
$$
where $ 0\leq x _i \leq 100$. You can solve this using generating functions by noting that this is equivalent to finding the coefficient of $x^{201}$ in the generating function of
$$
(1+x+\dotsb x^{100})^3
=\frac{(1-x^{101})^3}{(1-x)^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Ratio of sum of squares of Normal Distributions Find $$E[\frac{\sum_{i=1}^{51} X_i^2}{\sum_{i=51}^{101} X_i^2}], $$if $X_1,X_2,...,X_{51}$ are independent and are distributed as N(0,1). This question appeared in my final exam of probability and stochastic processes, that occurred today .I couldn't answer this in the ex... | You can write the ratio as $\frac{Y}{X+Z} + \frac{X}{X+Z}$. Then $\frac{Y}{X+Z}$ has an F-distribution with $d_1 = 50$ and $d_2 = 51$ degrees of freedom. And $\frac{X}{X+Z}$ has a Beta-distribution with parameters $\alpha = \frac{1}{2}, \beta = \frac{50}{2} = 25$ .
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2260437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Identity theorem for $\mathbb{R}^n$ This question is follow up to an interesting question I found here.
The results of this question states the following:
If $U$ is a domain, and $f,g$ are two real-analytic functions defined
on $U$, and if $V\subset U$ is a nonempty open set with $f\lvert_V
\equiv g\lvert_V$, then... | $M$ being open is not a necessary condition for $M$ to be a uniqueness set in higher dimensions. For example, if $f,g$ are real analytic on $\mathbb R^2$ and $f = g$ on $\cup_{k=1}^\infty \{(x,y):x=1/k\},$ then $f\equiv g.$ Also any set of positive measure, in any dimension, will give the result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2260532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Which fallacy is this? Assume that all people are either right-handed or left-handed, and likewise either right-footed or left-footed. 90% of people are right-handed. 90% of right-handed people are right-footed, but only 50% of left-handed people are left-footed as well. Which is more common: left-handedness or left-fo... | https://en.wikipedia.org/wiki/Prosecutor's_fallacy
People will see that half of the the left handed people are left footed and assume that left handed people are more common, but they don't realize that a small set of a large population can be comparable to a large set of a small population.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2260623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find the exact value of $A(\beta)=8\pi-16\sin(2\beta)$ with $\tan(\beta)= \frac{1}{2}$
The picture below represents a semi-circumference of diameter [AB] and
center C. Point D belongs to the semi-circumference and it's one of
the vertices of the triangle $ABC$. Consider that BÂD = $\beta (\beta
\in ]0,\frac{\pi}{... | Given: $\tan \beta = \frac 12$, so $2\sin\beta = \cos \beta$
$\sin 2\beta = 2\sin\beta\cos\beta = \cos^2\beta = \frac 1{\sec^2 \beta} = \frac 1{1 + \tan^2 \beta} = \frac 1{ 1 + \frac 14} = \frac 45$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2260722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show $\sum_{l=0}^\infty \sum_{i=0}^\infty x^{{2^l}(2i+1)}=\sum_{j=1}^\infty x^j$
Show $\sum_{l=0}^\infty \sum_{i=0}^\infty x^{{2^l}(2i+1)}=\sum_{j=1}^\infty x^j$ where $|x|<1$
Thus show it is the geometric series.
I know, I could just write out the sum and it would make sense, but is there a more formal proof for thi... | If $|x| < 1$,
$$\sum_{i=0}^\infty x^{2^l(2i+1)} = x^{2^l} \sum_{i=0}^\infty x^{2^{l+1} i} = \frac{x^{2^l}}{1 - x^{2^{l+1}}}.$$
Now, intuitively, if you sum the first $L$ terms you get the geometric series except for terms where the power of $x$ is divisible by $2^{L+1}$. So, you would expect the $L$th partial sum of t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Implication without if statement In logic, can we have an implication if there is no "if"?
Ex, John shall do $x$ regardless
Then is this an implication?
| The word "regardless" in your example doesn't modify the meaning of the sentence. Its job is merely to explicitly point out that there is no implication or condition present.
It is up to the reader/listener to figure out from context which potential implication it is he should note isn't there.
In mathematics, where ou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2260965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Characterization for basis of generated topology
Definition: Given a set $X$ and $\mathcal{B}$ a collection of subsets of $X$, the topology generated by $\mathcal{B}$ is the set:
$$[\mathcal{B}]=\bigcap_{\tau\in T}\tau$$
where $T=\{\tau \in \mathscr{P}(X):\tau$ is a topology on $X$ and $\mathcal{B}\subset\tau\}$
Quest... | The set $\tau$ of arbitrary unions of elements of $\mathcal{B}$ is a topology.
It is clear that $X\in\tau$, because of property 1; also $\emptyset\in\tau$ (union of the empty family).
It is also clear that $\tau$ is closed under finite intersections. Indeed, suppose
$$
U=\bigcup_{i\in I}B_i,
\qquad
V=\bigcup_{j\in J}C_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2261030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to interpret limit notation $\lim\limits_{x \to a} f(x)= L$ is by most; intuitively thought of "as $x$ gets close to $a$, $f(x)$ gets close to $L$", however my lecturer said this is not correct. She told me to go away and somehow find out why, by formal definition, the intuition "$f(x)$ is close to $L$, for all $x$... | The epsilon-delta definition is pretty straight-forward:
$$\lim _{{x\to c}}f(x)=L\iff (\forall \varepsilon >0)(\exists \ \delta >0)(\forall x\in D)(0<|x-c|<\delta \ \Rightarrow \ |f(x)-L|<\varepsilon )$$
What does this mean? Well, we break it down, part by part:
*
*$(\forall\varepsilon>0)\dots(\dots|f(x)-L|<\varep... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2261123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How does one evaluate $\lim _{n\to \infty }\left(\sqrt[n]{\int _0^1\:\left(1+x^n\right)^ndx}\right)$? I tried using Lebesgue's dominated convergence theorem and I'm getting $\lim _{n\to \infty }\left(1+\int _0^1x^ndx\:\right)$ which is $1$. But the answer should be 2.
| Let $I_n$ be given by
$$I_n=\int_0^1(1+x^n)^n\,dx \tag1$$
From the binomial theorem, we can write
$$(1+x^n)^n=\sum_{k=0}^n\binom{n}{k}x^{nk}\tag 2$$
Using $(2)$ in $(1)$ reveals
$$\begin{align}
I_n&=\int_0^1 \sum_{k=0}^n\binom{n}{k}x^{nk}\,dx\\\\
&=\sum_{k=0}^n\binom{n}{k}\frac{1}{1+nk}\tag3
\end{align}$$
Clearly fr... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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A function is differentiable $n$ times. Assume there are $n+1$ distinct points. Prove that $\exists$ one point $y$ such that $f^{(n)}(y)=0$ So I'm stuck on this question, I have an idea on the question but I missed the lecture which it pertained to. So I'm unsure of the theory behind it so, it'd be appreciated if someo... | By Rolle's Theorem, there exist $c_i$ such that $x_i < c_i < x_{i+1}$ and $f'(c_i) = 0$. Now apply Rolle's Theorem in this way to the $c_i$'s. Keep doing this.
| {
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"timestamp": "2023-03-29T00:00:00",
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Evaluate $\int_0^\infty \frac{\ln^2(z)}{1+z^2}$dz by contour integration Background: This is part b of problem 12.4.3 from Arfken, Weber, Harris Math Methods for Physicists to show that $\int_0^\infty \frac{\ln^2(z)}{1+z^2}$dz$=4(1-\frac{1}{3^3}+\frac{1}{5^3}-\frac{1}{7^3}+\dots)$.
Part b of the question asks to show ... | $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
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\newc... | {
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Convert statement from English to logic: "to pass philosophy it is not necessary to make notes every week" I saw this on a previous thread,
To pass philosophy it is not necessary to make notes every week.
Let $p = \text{Pass phil}$ and $m = \text{make notes}$,
Then basically what the sentence is is,
if you take notes... | You might use: $(m\land p) \lor (\neg m \land p)$ which is equivalent to $p$.
See truth table at: http://www.wolframalpha.com/input/?i=truth+table+((m%26p)or(~m%26p))
| {
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"url": "https://math.stackexchange.com/questions/2261592",
"timestamp": "2023-03-29T00:00:00",
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The group corresponding to the Rubik's cube Why is this group never studied in a group theory course at university? Is it too complicated or is it just not useful in connecting ides to other systems like vector spaces, creating modules, etc?
I would like to study this group but I did not find much useful info. Can ... | The Rubik's cube group is studied in some universities:
W.D.Joyner's Lecture notes on the mathematics of the Rubik's cube
The Mathematics of the Rubik's cube
Group Theory and the Rubik’s Cube
Mathematics of the Rubik's Cube
Rubik’s Magic Cube
$\cdots$
| {
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"url": "https://math.stackexchange.com/questions/2261755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Intuitive understanding of maximum value of quadratic function In trying to understand why the maximum area of a rectangle with a fixed perimeter occurs when the base is equal to the height, I got as far as this expression:
$A = (p/2)x - x^2$
from
$p = 2x + 2y,x + y = p/2, y = p/2 -x, A = x(p/2 - x)$
I know that I nee... | A quadractic equation
$$
q(x) = a x^2 + bx + c
$$
with non-zero coefficient $a$ has either a minimum or a maximum, depending on the sign of $a$.
If you plot the graph you will see the typical parabola shape.
One way to determine the extremum is to bring $q$ into the form
$$
q(x) = a (x - S)^2 + T
$$
where $(S, T)$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2261884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
Functional equations $f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}$ and $f(x)=\frac{f\left(\frac{2}{3}x\right)+f\left(\frac{4}{3}x\right)}{2}$. Suppose $f$ is continuous and $f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}$. Can we claim that $f(x)=kx$?
What if $f$ only satisfy $f(x)=\frac{f\left(\frac{2}{3}x\right)... | For the first problem, fix $x, y \in \Bbb{R}$ and define the set $S_{x,y}$ by
$$S_{x,y} = \{\lambda \in [0, 1] : f(\lambda x + (1-\lambda) y) = \lambda f(x) + (1-\lambda)f(y) \}. $$
We easily check that $0, 1 \in S_{x,y}$ and if $\alpha, \beta \in S$ then $\frac{\alpha+\beta}{2} \in S_{x,y}$. It immediately follows tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2262104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Basic question on countable intersection and union of sets I am just a beginner at measure theory, and I have a very basic question on the following fact found in Robert Ash, Probability and Measure theory , page 7:
Now this strikes me as a little asymmetric.
Very informally speaking, and I know this makes no sense, b... | *
*Take $F_n=(a−1/n,b)$.
*I may be very ignorant of this, but I think uncountable intersection is not a thing. I have only encountered finite or infinite intersection, which compile the arbitrary intersection. I don't think you can define $F_n$ if the intersection is uncountable, since the notation implies that it is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2262311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
RSA Cryptography: Given $n$ and $\varphi(n)$, find $e$ such that $e=d$ The modulus, $n=8633$, is given (it's simple to find $p$ and $q$ such that $n=pq$, i.e. $p=89$ and $q=97$) and the task is to find an encoding exponent, $e$, such that the correpsonding decoding exponent, $d$, is equal to $e$.
As $n=89 \times 97$, $... | So $e^2\equiv 1(\textrm{mod}\quad 89)$ and $e^2\equiv 1(\textrm{mod}\quad 97)$. This tell us that $e\equiv\pm 1(\textrm{mod}\quad 89)$ and similarly $e\equiv\pm 1(\textrm{mod}\quad 97)$. Take a nontrivial pair, (for example, $e\equiv-1(\textrm{mod}\quad 89)$ and $e\equiv 1(\textrm{mod}\quad 97)$) and use Chinese remai... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2262400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to get the inverse of this function, when we have qudratic? $$f(x)=-x^2+6x-5$$
How do I find the inverse? I tried by making it
$$y=-x^2+6x-5$$
Then swapping $y$ with $x$, and then solve it for y, but I got $y^2$.
The domain is $x$ greater or equal to $m$, and in this case $m=5$.
After that we need to find the dom... | The simplest one is by completing the square
$f(x)=y=-((x-3)^2-4)$,
$4-y=(x-3)^2$,
$x=3\pm\sqrt{4-y}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2262519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Weak * lower semicontinuity I was asking myself the following question: let $X$ be a Banach space, dual of a separable Banach space, if $I: X \to R$ is a convex weak lower semicontinous functional, does it follow that $I$ is weak-* lower semicontinous?
If not, does the claim is true if we consider sequential lower sem... | The proof of the "general theorem" uses the following arguments:
*
*Since $I$ is convex, its epigraph is convex.
*Since $I$ is l.s.c., the epigraph is closed.
*Closed and convex sets are weakly closed.
*Hence, the epigraph is weakly closed and $I$ is weakly l.s.c.
The third step does not work with the weak-* to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2262668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Exterior derivative of 2-forms and divergence I'm working through A Geometric Approach to Differential Forms. The deriative of a $2$-form $\omega$ in $\mathbb{R}^3$, denoted $d\omega$ and operating on vectors $U, V, W \in T_p \mathbb{R}^3$, is defined as
$$d\omega(U, V, W) = \nabla_U \omega(V, W) - \nabla_V \omega(U, W... | You are using $dx\wedge dz$ and the wiki page is using $dz\wedge dx$. So there is no difference.
remark: In general for an $n-1$ form, one usually insert some $(-1)$'s to deal with this: a general $n-1$ form $\alpha$ is written as
$$\alpha = \sum_{i=1}^n (-1)^{i-1} \alpha_i dx^1 \wedge \cdots \wedge\widehat{dx^i} \we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2262808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Artin–Schreier polynomial Suppose we have the finite field $K=\Bbb{F}_{p^n}$ ($p$ prime and $n>0$) and an Artin–Schreier polynomial $f=x^p-x+\gamma \in K[x]$.
Suppose that $f$ is irreducible. How do we prove that $tr_K(\gamma)=\gamma+\gamma^p+\cdots +\gamma^{p^{n-1}} \neq 0$ ?
I think it helps to see that $tr_K(\gamma... | This follows from Hilbert's Satz 90 for finite cyclic extensions, i.e if $ L/K $ is a finite cyclic extension with Galois group generated by $ \sigma $, then for an $ x \in L $, we have that $ \textrm{Tr}_{L/K}(x) = 0 $ if and only if $ x = \sigma(y) - y $ for some $ y \in L $. The extension $ \mathbb F_{p^n}/\mathbb F... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2262937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Multivariable Optimization - Distance Formula, Use of Square Root I'm working through a problem in which I am trying to find the point $P(x,y,z)$ closest to the surface $f(x,y)$. I am not concerned with the actual distance, I just want to find the closest point $P$.
To do this I am minimizing the distance between the ... | You are correct in that you don't need the square root:
If $\sqrt{h(x,y,f)}$ is at a minimum, then $h(x,y,f)$ is at a minimum, for $h\geq 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2263081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
If $H/\Gamma$ is a compact Riemann surface , the generator of $\Gamma$ is not commutative In the book Compact Riemann Surfaces by Jurgen Jost, the Exercises for 2.4 is asking to prove that:
Let $H/\Gamma$ be a compact Riemann surface. Show that each nontrivial abelian subgroup of $\Gamma$ is infinite cyclic group. Whe... | Here's a (long) hint. The automorphism group of the upper-half plane is isomorphic with $\operatorname{PSL}(2,\mathbb{R})$, which contains three types of elements: elliptics, parabolics, and hyperbolics, depending on whether the absolute value of the trace is less than, equal to, or greater than $2$. If $\Gamma$ contai... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2263148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Index of subgroup is 2 then for any $g$, $g^2$ belongs to subgroup If index of a subgroup $H$ is 2, then $g^2 \in H$ for every $g$ in G.
Proof:
Since index is 2, there are only two distinct cosets. Now if $g \in H$ then it trivially holds because $H$ is a subgroup. Let $H$ and $gH$ be cosets where $g \notin H$ theref... | Typically one shows that if $|G:H|=2$ then $H$ is normal in $G$. This follows directly since for $g\in G$ and $g\notin H$ we have $G = H \cup gH = H \cup Hg$ which implies $gH = Hg$.
Therefore $G/H \cong \mathbb{Z}_2$ since there is only one group of order 2. Hence $g^2H = H$ for any $g$ which implies $g^2 \in H$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2263264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Eigenvalues of a $2\times2$ matrix Let $a,b$ be distinct eigen values of a $2\times2$ matrix $A.$Then which of the following statement is true?
*
*$A^2$ has distinct eigen values.
*$A^3=\frac{a^3-b^3}{a-b}A-ab(a+b)I$
*Trace of $A^n$ is $a^n+b^n$ for every positive integer n.
*$A^n$ is not a scalar multip... | Hint: For option $4$, construct a matrix $A$ with the diagonal elements $1$ and $-1$. Then check $A^2$ = identity matrix.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2263404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How can we integrate integral(s) of this type? So I was able to free $dx$ from the power. Now only wolfram can solve this Integral. How can I do this on my own?
$$r=\int_0^1\left(\frac{x^{12}}{(1-x^4)^3}+1\right)^{1/4}~dx$$
| Hint:
$\int_0^1\left(\dfrac{x^{12}}{(1-x^4)^3}+1\right)^\frac{1}{4}~dx$
$=\int_0^1\left(\dfrac{x^3}{(1-x)^3}+1\right)^\frac{1}{4}~d\left(x^\frac{1}{4}\right)$
$=\dfrac{1}{4}\int_0^1x^{-\frac{3}{4}}\left(\dfrac{x^3}{(1-x)^3}+1\right)^\frac{1}{4}~dx$
$=\dfrac{1}{4}\int_1^0(1-x)^{-\frac{3}{4}}\left(\dfrac{(1-x)^3}{x^3}+1\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2263535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If the connected sum $A\#B$ is homeomorphic to $S^2$ then $A\cong B \cong S^2$ I was looking for this, but I can't find anything.
Problem. Let $A, B$ two compact surfaces such that $A\#B \cong S^2$ then $A\cong B \cong S^2$.
I considerd infinite connected sum $A\#B\#A\dotsc$ and $B\#A\#B\dotsc$ These are homeomorphic ... | As you have mentioned, $A\#(B\#A\#B\#\cdots)\simeq \mathbb{R}^2$ and $B\#A\#B\#\cdots\simeq \mathbb{R}^2$.
But observe that if $X$ is a compact surface, $X\#\mathbb{R}^2 \simeq X\setminus\{x_0\}$ (=$X$ minus a point) Therefore, it means that $A$ minus a point is homeomorphic to $\mathbb{R}^2$, and then it is easy to c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2263751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Parametrization of two surfaces $\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$ and $\frac{x^2}{p}+\frac{y^2}{q}=2z$. Can someone please help me to parametrize the following surfaces in terms of hyperbolic(for second it might not be possible but i need some more convenient set of parametric equation than mine ) an... | For the first: $$\frac{x^2}{a^2}\color{red}{-}\left(\frac{y^2}{b^2}+\frac{z^2}{c^2}\right)=1$$
\begin{eqnarray}
&x&=a\cosh\theta,\\
&y&=b\cos\phi\sinh\theta,\\
&z&=c\sin\phi\sinh\theta.\\
\end{eqnarray}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2263853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
series solutions to 2nd order ODEs i am very confused about how to tell whether a point is ordinary or regular singular. i know the definitions but think that I am physically doing it wrong. Do you sub the point into p @ q and see if you get zero, or see if they are equal? can you please explain in the most basic manne... | Let's look at the definitions: given a differential equation
$$ y'' + p(x) y' + q(x) = 0, $$
(the leading coefficient must be $1$ to do this: if it isn't, divide by it!) the point $x=a$ is
*
*An ordinary point if $p(x)$ and $q(x)$ are regular at $x=a$ (continuous or bounded in a neighbourhood is good enough).
*A re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2263964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Longest, First Attempt, Chain of Heads on a Coin Flip So taking the question of :
If everyone in the world flipped a coin until they got tails what is the most likely longest chain of heads?
Assuming population is 7.347*10^9
And everyone has an ideal coin and has the ability to flip a coin "randomly"
What would the ans... | Suppose the population of the world is $k$.
Then the probability they all have at least one tail in up to $n$ flips is $$\left(1-\frac1{2^n}\right)^k \approx \exp\left(- \dfrac{k}{2^n}\right)$$ so the probability that the longest string of heads is exactly $n$ $$\left(1-\frac1{2^{n+1}}\right)^k - \left(1-\frac1{2^{n}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2264120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proving the nth derivative of a polynomial of degree n-1 is zero using linear Algebra. I want to prove by using linear algebra only that the nth derivative of a polynomial of degree n-1 is zero. My idea is using proving first that every square matrix $A$ such the only not zero entries are those that $j=i+1$ then $A^{n}... | Note that $[D]_{\beta}$ is an $(n+1)\times (n+1)$ matrix. What we actually have is that $([D]_{\beta})^{n+1} = 0$, not $([D]_{\beta})^n=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2264233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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