Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
What is going on with the symmetry group of the cube? I've been studying the symmetry groups of the platonic solids, and the cube has be confused. I've been considering the four diagonals on the cube. By writing out all permutation in $S_4$, it's clear that almost all permutations of the four diagonals can be reached b... | The action of the orientation preserving symmetry group of the group induced on the diagonals is indeed an isomorphism between two groups isomorphic to $S_4$. The orientation preserving symmetry group is of index $2$ in the full symmetry group of the cube (a group of order $48$). Note that the reflection at the center ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1676176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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If $a^2+b^2=c^2$ then $a$ or $b$ is even. I am having trouble proving this directly. I know that it is easy to prove by contradiction by assuming both $a$ and $b$ to be odd, but how should I start to try to prove this? This is a homework problem and I am just looking for help getting started, not the full answer.
| It's a bit of a fine line between proving directly and by contradiction. Try breaking it into cases, such as
Suppose $c^2$ is even/odd and WLOG $b$ is odd, then show that $a$ must be even.
Since the conclusion isn't a contradiction, but rather that $a$ is even, this is a direct proof. You should end with something like... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1676265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Picking Multiples of 4 I recently came up with and tried to solve the following problem: If you are randomly picking integers in the range $[1,30]$ out of a hat without replacement, on average, how many integers will you have to pick until you have picked all of the multiples of $4$?
There are $7$ multiples of $4$ that... | The probability of getting all multiples of $4$ precisely on draw $k$ is
$$
\frac{\binom{k-1}{6}}{\binom{30}{7}}
$$
Thus, the expected number of draws would be
$$
\begin{align}
\frac1{\binom{30}{7}}\sum_{k=7}^{30}k\binom{k-1}{6}
&=\frac7{\binom{30}{7}}\sum_{k=7}^{30}\binom{k}{7}\\
&=\frac7{\binom{30}{7}}\binom{31}{8}\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1676374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 1
} |
Proving density of a set I want to show that $A=\{(x_n)\in c_{00}:\sum\limits_{n=1}^{\infty}x_n=0\}$ is dense in $(c_0,\|.\|_{\infty})$.
Let $x=(x_n)\in c$ and let $\epsilon >0$. Thus there exists $n_0\in \mathbb N$ such that $|x_n|<\epsilon $ for all $n\geq n_0$. Now the question is how to choose an element from $A$?... | We must show that $$\forall a_n\in c_0\quad,\quad\forall\epsilon>0\quad,\quad\exists b_n\in A\quad,\quad||a_n-b_n||_\infty<\epsilon$$also according to definition $$||a_n-b_n||_\infty<\epsilon\leftarrow\rightarrow \sup_{n}|a_n-b_n|<\epsilon\leftarrow\rightarrow|a_n-b_n|<\epsilon\quad,\quad\forall n$$we try to make such ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1676489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What's so special about a prime ideal? An ideal is defined something like follows:
Let $R$ be a ring, and $J$ an ideal in $R$. For all $a\in R$ and $b\in J$, $ab\in J$ and $ba\in J$.
Now, $J$ would be considered a prime ideal if
For $a,b\in R$, if $ab\in J$ then $a\in J$ or $b\in J$.
To my (admittedly naive) eyes,... | The easiest way to answer this, I think, is with an example. Let $R=\mathbb{Z}$, and let's consider the ideal $I=(6)$. This is the set of all integers that are multiples of $6$. You can see that it's an ideal because if you take any multiple of $6$ and multiply it by any other integer, the result is still a multiple... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1676578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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How to show that this cone doesn't have a generating set that forms a basis of $\mathbb Z^2$? Consider the cone $\sigma=\mathbb R_{\ge0} e_1 +\mathbb R_{\ge0}(e_1+2e_2)$ in $\mathbb R^2$. I am trying to show that $\sigma$ is not a smooth cone. That is I want to show that $\sigma$ doesn't have a generating set that form... | Here's a concrete way to explain it: show that there is no linear combination (over $\Bbb Z$) of $e_1$ and $e_1 + 2e_2$ that produces the vector $e_2 \in \Bbb Z^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1676700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Splitting Lemma where $C=\mathbb{Z}.$ Given a short exact sequence
$$ 0 \xrightarrow{\theta_3} A \xrightarrow{\theta_2} B \xrightarrow{\theta_1} \mathbb{Z} \xrightarrow{\theta_0} 0 $$
show that $B \cong A \oplus \mathbb{Z}.$
So far I have that $\theta_2$ is injective and as $0 \to \operatorname{Im}(\theta_3) \to A \to ... | This is the elementary way to proof it:
Let $b \in B$ with $\theta_1(b)=1$ and define a homomorphism $s: \mathbb Z \to B, 1 \mapsto b$.
Then you can show that
$$A \oplus \mathbb Z \to B, (a,z) \mapsto \theta_2(a) + s(z)$$
is an isomorphism.
Using results from homological algebra, one would just say that $\mathbb Z$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1676870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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integrating this infinite gaussian integral How does one integrate
$\int_{-\infty}^{+\infty}x e^{-\lambda ( x-a )^2 }dx $
where $\lambda$ is a positive constant.
My integral tables are not returning anything useable. The best it return is non-definite gaussian integrals. Useless!
Help please
| $$
\begin{align}
\int_{-\infty}^{+\infty}x e^{-\lambda(x-a)^2}\,\mathrm{d}x
&=\int_{-\infty}^{+\infty}(x+a) e^{-\lambda x^2}\,\mathrm{d}x\tag{1}\\
&=\frac{a}{\sqrt{\lambda}}\int_{-\infty}^{+\infty}e^{-x^2}\,\mathrm{d}x\tag{2}\\
&=a\sqrt{\frac\pi\lambda}\tag{3}
\end{align}
$$
Explanation:
$(1)$: substitute $x\mapsto x+a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1676958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$L^p-L^q$ estimates for heat equation - regularizing effect Where can I find a proof of the following estimate
$$\|S(t)v\|_{L^p(\Omega)}\leq C t^{-\frac{N}{2}\left(\frac{1}{q}-\frac{1}{p}\right)}\|v\|_{L^q(\Omega)}, $$
where $1\leq p<q<+\infty$, $\Omega\subset \mathbb{R}^N$ is an open bounded set and $\{S(t)\}_{t\geq 0... | Let $N_t:\mathbb{R}^N\to \mathbb{R}$, $t>0$, be the function defined by
$$N_t(x)=(4\pi t)^{-N/2}e^{-|x|^2/4t}.$$
Since
$$\int_{\mathbb{R}^N} e^{-a|x|^2}dx=\left(\frac{\pi}{a}\right)^{N/2},\tag{1}\label{1}$$
we can see that $N_t\in L^1(\mathbb{R}^N)$ and $\|N_t\|_{ L^1(\mathbb{R}^N)}=1$.
We know that $S(t)v=N_t\ast v$.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proof that $\sum_{i=0}^n 2^i = 2^{n+1} - 1$ $\sum_{i=0}^n 2^i = 2^{n+1} - 1$
I can't seem to find the proof of this. I think it has something to do with combinations and Pascal's triangle. Could someone show me the proof? Thanks
| Mathematical induction will also help you.
*
*(Base step) When $n=0$, $\sum_{i=0}^0 2^i = 2^0 = 1= 2^{0+1}-1$.
*(Induction step) Suppose that there exists $n$ such that $\sum_{i=0}^n 2^i = 2^{n+1}-1$. Then $\sum_{i=0}^{n+1}2^i=\sum_{i=0}^n 2^i + 2^{n+1}= (2^{n+1}-1)+2^{n+1}=2^{n+2}-1.$
Therefore given identity ho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1677359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
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Are there order statistics for a Gaussian variable raised to a power? Let $X$ be a random variable with a standard normal distribution. Let $Y = |X|^{2p}$. I am trying to find the distribution for $Y_{(n)}$, i.e., the largest value of $Y$ out of $n$ samples.
I have derived the pdf to be:
$$f_{Y_{(n)}} = n \left(\frac{... | In this answer I will try to derive an analytic formula for you. I think my answer is the same as yours. Anyway, you can use it to compare your result and check what might have gone wrong.
Let $\Phi$ denote the cumulative distribution function of the standard normal distribution. That is
$$
\Phi(x) = \int_{-\infty}^{x}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1677449",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
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If $R$ is an equivalence relation, does $R^2$ too? I think that yes,
$I_A \subseteq R$
$R = R^{-1}$
$R^2 \subseteq R$
And now we can show.
Reflex: $I_A = I_A^2 \subseteq R \subseteq R^2$
A lil bit struggling with symm. And trans. Can you show me how will you prove it?
| The key is that $R_1\subseteq R_2$ implies $R_1^2\subseteq R_2^2$. Together with $I_A^2=I_A$ and $(R^{-1})^2=(R^2)^{-1}$ the three properties for $ 2$ follow.
But actually the situation is much simpler: While transitivity (alone) of $R$ in fact just means $R^2\subseteq R$, transitivity plust reflexivity means that $$R^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1677529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Getting the sides of unit circle Im very confused on how to get the sides of the unit circle. By that i mean like sin/cos of 0,90,180,360... I can get the others by this logic for example:
Image:
For example if it asks me
*
*Sin 150: I would first see the quadrant.. Its Quadrant II so it
means Sin is the only one po... | In the following triangles, use the Pythagorean theorem to find the missing lengths, then find the trigonometric ratios for $0,\pi/2, \pi/3,\pi/4,$ and $\pi/6$. Don't even bother memorizing, just draw the appropriate triangle.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1677643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Defining term with $3$-vectors and $3 \times 3$ matrices I don't normally ask questions, but my professors isn't responding back to me. I do not want an answer or anything, just want a point in the right way so I can figure this set of question out.
The question is:
$3$-vectors with Cartesian coordinates and $3 \time... | Let $M$ be a $3 \times 3$ matrix given by columns $M_1, M_2, M_3$ (so each of $M_1, M_2, M_3$ is a three vector in column format). Then we have $$M = [M_1|M_2|M_3].$$
Let Let $W$ be a $3 \times 3$ matrix given by rows $W_1, W_2, W_3$ (so each of $W_1, W_2, W_3$ is a three vector in row format). Then we have $$\pmatrix{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1677753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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What is the probability that three are males and two are female? The human sex ratio at birth is commonly thought to be 107 boys to 100 girls. Suppose five infants are chosen at random.
(A) What is the probability that three are males and two are female?
(B) What is the probability that at least one of them is a male?... | 107:100 implies that the chance of being male is $p = \frac{107}{207}$.
Notice that we have $n = 5$ (presumably independent) trials (children) with probability $p = 107/207$ of success (if I consider selecting a boy being as a success). Then the number of boys selected $N$ follows a binomial distribution with $n = 5, p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1677894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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What is the probability that Andrea, Melissa and Carol end up being on the same team? I am stuck on a word problem about picking teams. I thought it would be very simple, but to my surprise, I could not solve it. So here's the problem..
Andrea, Melissa, and Carol are in a class of 27 girls. The teacher chooses
studen... | Consider that there are $27$ slots available.
$\Large\boxed{.}\Large\boxed{.}\Large\boxed{.}\;\;\Large\boxed{.}\Large\boxed{.}\Large\boxed{.}\;\;\Large\boxed{.}\Large\boxed{.}\Large\boxed{.}\;\;\Large\boxed{.}\Large\boxed{.}\Large\boxed{.}\;\;\Large\boxed{.}\Large\boxed{.}\Large\boxed{.}\;\;\Large\boxed{.}\Large\boxed{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1678004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that the "theta-space" is not homeomorphic to $S^{1}$. Let $X$ be the "$\theta$-space":
\begin{equation*}
X = \{ (x,y) \in \mathbb{R}^{2} \colon x^{2} + y^{2} = 1 \} \cup \{ (x,0) \colon -1\leq x \leq 1 \}.
\end{equation*}
Prove that $X$ is not homeomorphic to $S^{1}$.
My initial thought was to show that $X$ is n... | You can also give an alternative proof using $\pi_1$. Since the theta-space is homotopic to the figure eight space i.e $\mathbb{S}^1 \vee \mathbb{S}^1$, it has fundamental group $\mathbb{Z} * \mathbb{Z}$. However, the fundament group of $\mathbb{S}^1$ is $\mathbb{Z}$. Since homeomorphic spaces have isomorphic fundament... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1678110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Proving a nash equilibria does not exist At a certain warehouse, the price of tobacco per pound in dollars, $p$, is related
to the supply of tobacco in pounds, $q$, by the formula
$p=10−(q/100000)$
Thus the more tobacco farmers bring to the warehouse, the lower the price. However,
a price support program ensures that t... | Without loss of generality, let $q_1'=600000$ and let $q_2'+q_3'=Q<375000$, $q_2',q_3'\in(0,600000)$.
To show that a NE does not exist, you need to demonstrate that at least one player $i$ can earn more by not playing $q_i'$. In other words, you'd show that $q_1',q_2',q_3'$ are not mutual best responses.
Obviously one ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1678213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove a useful formula for computing expectations Suppose $X$ is a non-negative random variable, and $h$ is a non-decreasing function on $\mathbb{R}_+$ such that $h(0)=0$ and $h$ is absolutely continuous on each bounded interval. ($h(a) = \int_0^a h'(s) ds$ for all $0\leq a <\infty$) Then,
\begin{align}
\mathbb{E}[h(X)... | Since $X\geqslant0$ a.s. and $h$ is increasing, we have
$$\mathbb E[h(X)] = \int_0^\infty (1-F_{h(X)}(x)\ \mathsf dx = \int_0^\infty (1-F_X(h^{-1}(y))\ \mathsf dy, $$
where $$h^{-1}(y)=\min\{x: h(x)=y\}. $$
Applying the change of variables $s=g^{-1}(x)$, we have
$$\mathbb E[h(X)]=\int_0^\infty(1-F_X(s))h'(s)\ \mathsf d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1678328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$|.|$ and $d'$ are not equivalent (in metric sense)? Please i need a counter example why $|.|$ and $d'$ are not metricly equivalent where:
$$d'(x,y)=\left\{
\begin{array}{cc}
0, & x=y, \\
|x|+|y|, & x\neq y.
\end{array}\right.$$
Thank you very much .
| Hint: This can also help, since metric equivalence implies topological equivalence:
Given any $x \in \mathbb{R}\setminus\{0\}$, let $\varepsilon =\frac{|x|}{2}$, then the epsilon ball $B_{d'}(x,\varepsilon)$ consists only of $\{x\}$. For, if $y\not= x$ and $y \in B_{d'}(x,\varepsilon)$, we must have $\varepsilon < |x|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1678407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Limit of sum of the series What would be the sum of following ?
$$\lim_{n\to\infty} \left[\frac{1}{(n+1)^{2}} + \frac{1}{(n+2)^{2}} + \frac{1}{(n+3)^{2}} + \cdots + \frac{1}{(n+n)^{2}}\right]$$
I tried to turn it into integral :
$\displaystyle\int \frac{1}{(1+\frac{r}{n})^{2}}\frac{1}{n^{2}} $ but
I can't figure out ... | $$\lim_{n\to\infty}0\le\lim_{n\to\infty} [\frac{1}{(n+1)^{2}} + \frac{1}{(n+2)^{2}} + \frac{1}{(n+3)^{2}} + ... + \frac{1}{(n+n)^{2}}]\le\lim_{n\to\infty}\frac{n}{(n+1)^2}$$
$$\lim_{n\to\infty}0\le\lim_{n\to\infty} [\frac{1}{(n+1)^{2}} + \frac{1}{(n+2)^{2}} + \frac{1}{(n+3)^{2}} + ... + \frac{1}{(n+n)^{2}}]\le\lim_{n\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1678539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
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Show that $\int\cdots\int{p_1^{x_1}\cdots p_m^{x_m} \, dp_1\cdots dp_m}=\frac{x_{1}!\cdots x_{m}!}{\left(m-1+\sum_{i=1}^{m}{x_i}\right)!}$
How to prove that,$$\int\int\cdots\int p_1^{x_1} p_2^{x_2} \cdots p_m^{x_m} \, dp_1 \, dp_2 \cdots dp_m = \frac{x_1!\cdots x_m!}{\left(m-1+\sum_{i=1}^m x_i\right)!}$$
where $0\le... | *I know this question has existed for a long time. It happens that I also came across this formula learning Bose-Einstein distribution and was confused for a moment. However, it turns out that the step-by-step calculation wasn't that daunting for some lower dimension case. And indeed the whole formula can be derived by... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1678656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Reflection of a Light Ray I found this problem to be very hard while studying for the exam:
Let
$$L: \vec r(t)=<1,-2,3>+t<-5,4,1>, \qquad t \in \mathbb{R}$$
be a line. Light is traveling along the line $L$ in the direction of increasing $t$ value. Let $\Omega: x=0$ (the $yz$ plane) be a mirror. Find the vector eq... | Perhaps I’ve misunderstood something in the problem statement, but this seems rather straightforward. The problem talks about lines, not rays, so the point at which the given line intersects the mirror isn’t directly relevant.
Forget about the line for the moment. What is the image of an arbitrary vector $\langle x,y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1678735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Uniform convergence of $\frac{n^2\sin(x)}{1+n^2x}$ My goal is to show that
$$\frac{n^2\sin(x)}{1+n^2x}$$
does not converge uniformly on $S=(0,\infty)$ but does so on any compact subset of $S$. First, we find the limit function
$$\frac{n^2\sin(x)}{1+n^2x} \to \frac{\sin(x)}{x} \qquad \text{ as } n \to \infty$$
Now, to s... | For $0 < a \leqslant x$ we have
$$\left|\frac{n^2\sin(x)}{1+n^2x}-\frac{\sin(x)}{x} \right| = \left|\frac{\sin (x)}{n^2x^2 + x}\right| \leqslant \frac{1}{n^2a^2 + a} \to 0
$$
and convergence is uniform on $[a, \infty)$.
For $x > 0$
we have
$$\left|\frac{n^2\sin(x)}{1+n^2x}-\frac{\sin(x)}{x} \right| = \left|\frac{\sin (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1678843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove $\gcd(a^3,b^3)$ = $\gcd(a,b)^3$ $\gcd(a^3,b^3)$ = $\gcd(a,b)^3$
Let there be integers $s,t,x,y$
$a^3s + b^3t = (ax + by)^3 $
Should I start like from the above?
| $\displaystyle ~ \gcd(a^3, b^3) = \prod_{1 \le k \le n} p_k^{\min(3r_k, 3s_k)} = \prod_{1 \le k \le n} p_i^{3\min(r_k, s_k)} = \bigg(\prod_{1 \le k \le n} p_i^{\min(r_k, s_k)} \bigg)^3 = (\gcd(a, b))^3.$
| {
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"url": "https://math.stackexchange.com/questions/1678959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the proper way to set up this integral to find the area bounded by the curves?
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle. Then find the area.
$$y = 2x^2, y = 8x^2, 3x + y = 5, x ≥ 0$$
Here is my drawing: http://w... | You can use double integral,
the intersection points:
between $y=8x^2$ and $y=5-3x$ are at $x=-1$ and $x=5/8$
between $y=2x^2$ and $y=5-3x$ are at $x=-5/2$ and $x=1$
let's assume that we want $\text{d$y$d$x$}$
for $I_1$: when $y$ goes from $y=2x^2$ to $y=8x^2$ then $x$ goes from $0$ to $5/8$
for $I_2$: when $y$ goes... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1679044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Why is it important to find both solutions to a second order linear differential equation? Given the equation $$y'' + y=0$$
A solution is $y=\sin(t)$
Why can't we stop there since we know a way to solve the system? Why should we consider all of the ways to solve the system?
I would really like to see a real world exam... | Consider a point mass $m= 1 \ \mathrm{kg}$ attached to one end of a spring with spring constant $k = 1 \ \mathrm{N/m}$. Suppose that the spring is suspended vertically from an immovable support. The oscillations of the mass around its equilibrium position can then be described by the equation
$$y''+y = 0$$
where $'$ de... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1679146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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How canonical is Gauss's law of composition of forms Gauss defined the composition of binary quadratic forms $f$ and $g$ to be another binary quadratic form $F$ such that there exist integral quadratic forms
$$ \begin{align} r(x_0,x_1,y_0,y_1) &= p_0 x_0y_0 + p_1 x_0y_1 + p_2 x_1y_0 + p_3 x_1y_1\\
s(x_0,x_1,y_0,y_1) ... | The composition of binary quadratic forms that has always been used is a version developed by Dirichlet. This can be found on page 49 in the first edition of Cox, with a correction in the second edition.
It is quite recent that Bhargava showed that there were actually 14 distinct ways to, say, implement the conditions... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1679228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Exist unique $g_0 \in H^1(0, 1)$ such that $f(0) = \int_0^1 (f'g_0' + fg_0) \text{ for all }f \in H^1(0, 1)$? The mapping $f \mapsto f(0)$ from $H^1(0, 1)$ into $\mathbb{R}$ is a continuous linear functional on $H^1(0, 1)$. Does there exist a unique $g_0 \in H^1(0, 1)$ such that$$f(0) = \int_0^1 (f'g_0' + fg_0) \text{ ... | Hint:
$H^1(0,1)$ is a Hilbert space. If you can show that
$$L : H^1(0,1) \to \mathbb R, \ \ \ L(f) = f(0)$$
is a bounded linear functional, the Riesz representation theorem tells you that
$$L(f) = \langle f, g_0\rangle\ \ \ \forall f\in H^1(0,1),$$
which is exactly what you want.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1679342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Prove that $\sup{(A+B)} = \sup{A}+\sup{B}$
Let $A,B$ be subsets of $\mathbb{R}$. Prove that $\sup{(A+B)} = \sup{A}+\sup{B}$.
I think in order to solve this we are going to have to use the mathematical definition of supremum. Maybe we can break this up into $4$ cases: $A$ is finite, $B$ infinite; etc. This would allo... | $A \leq \sup A$, $B \leq \sup B$, so $\sup A + \sup B \geq \sup(A+B)$
Also, $A + B \leq \sup (A+B)$, so $A \leq \sup(A+B) - B$, so $\sup A \leq \sup (A+B) -B$, then $\sup B \leq \sup (A+B) - \sup A$. Done
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Correct set notation for "all integers which are not multiples of 7"? What is correct set notation for "all integers which are not multiples of $7$"? My best guess is:
$$ \{ x : (\forall k \in \mathbb{Z})(\neg(7k = x)) \}$$
Or
$$ \{ x : \neg(\exists k \in \mathbb{Z})(7k = x) \}$$
However this seems unlike other example... | The most accurate translation of the English condition would be
$$
\left\{n\in\mathbb{Z}:\frac n7\not\in\mathbb{Z}\right\}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1679543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "42",
"answer_count": 9,
"answer_id": 5
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Why is one relation transitive but the other is not? From what I have read about a transitive relation is that if xRy and yRz are both true then xRz has to be true.
I'm doing some practice problems and I'm a little confused with identifying a transitive relation.
My first example is a "equivalence relation"
$S=\{1,2,... | A relation $R$ of the set $S$ is transitive if:
$$\forall a{\in} S~\forall b{\in}S~\forall c{\in}S: \Big(\big((a,b){\in}R\wedge(b,c){\in}R\big)\to (a,c){\in}R\Big)$$
That definition is equivalent to:
$$\neg \exists a{\in}S~\exists b{\in}S~\exists c{\in}S: \Big(\big((a,b){\in}R\wedge(b,c){\in}R\big)\wedge (a,c){\notin}R... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1679615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Solving an IVP with Laplace Transforms I'm trying to solve the following IVP (differential equations) with the Laplace Transform method:
\begin{cases}
y''+9y=36t\sin(3t)\\
y(0) = 0\\
y'(0) = 3
\end{cases}
After taking the Laplace Transform of both sides, I obtain
$$s^2Y(s) - 3 + 9sY(s) = \frac{216}{(s^2+9)^... | Notice:
*
*$$\mathcal{L}_{t}\left[y'(t)\right]_{(s)}=sy(s)-y(0)$$
*$$\mathcal{L}_{t}\left[y''(t)\right]_{(s)}=s^2y(s)-sy(0)-y'(0)$$
*$$\mathcal{L}_{t}\left[t\right]_{(s)}=\frac{1}{s^2}$$
*$$\mathcal{L}_{t}\left[\sin(t)\right]_{(s)}=\frac{1}{1+s^2}$$
*$$\mathcal{L}_{t}\left[\sin(at)\right]_{(s)}=\frac{a}{a^2+s^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1679771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove $\forall x:P(x) \implies \exists x:P(x)$ without using UI? In standard FOL, can we prove $\forall x: P(x) \implies \exists x:P(x)$ without introducing a new free variable by universal instantiation, i.e without using $\forall x: P(x) \vdash P(y)$ where $y$ is not does not occur in $P(x)$?
I have tried dir... | If you use the following system from A Primer for Logic and Proof, Holly P. Hirst and Jeffry L. Hirst,
Axioms
Axiom 1: $A \implies (B \implies A)$
Axiom 2: $(A \implies (B \implies C)) \implies ((A \implies B) \implies (A \implies C))$
Axiom 3: $(\lnot B \implies \lnot A) \implies ((\lnot B \implies A) \impl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1679909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Is inversion ever a field automorphism? Is the function sending every element of a field to its multiplicative inverse (and 0 to 0) ever a field automorphism?
| Proposition. Inversion is an automorphism of a field $F$ if and and only if $F$ is one of $\mathbb F_2$, $\mathbb F_3$, or $\mathbb F_4$.
Proof:
Let $F$ be a field and suppose that inversion is a field automorphism of $F$. If $x \in F \setminus \{0,1\}$, then $x$ and $1-x$ are both invertible, and hence, by our assumpt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1679965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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Integrating $\int\frac{x^2-1}{(x^2+1)\sqrt{x^4+1}}\,dx$ I came across a question today...
Find $$\displaystyle\int\dfrac{x^2-1}{(x^2+1)\sqrt{x^4+1}}\,dx$$
How to do this? I tried to take $x^4+1=u^2$ but no result. Then I tried to take $x^2+1=\frac{1}{u}$, but even that didn't work. Then I manipulated it to $\int \dfr... | Hint Divide the numerator and denominator by $x^2$, to get:
$$\int \frac{(1-\frac{1}{x^2})dx}{(x+\frac{1}{x})(\sqrt {(x+\frac{1}{x})^2-2} )}$$
Then put $x+\frac{1}{x}=t$
$$\int \frac{dt}{t(\sqrt{t^2-2})}$$
Which is easily taken care of by putting $t=\sqrt2 \sec\theta$,
$$\int \frac{(\sqrt2 \sec\theta\tan \theta)d\thet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1680049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Solving two specific limits without L'Hôpital's Rule Greetings to any and all that might read this. I am a 12-th grade student from Portugal and, here, it is not allowed for us to solve limits through L'Hôpital's Rule at this level (in fact, it isn't even taught, though we learn derivatives as well...), which places me... | The first one is elementary if we set $f(x) = e^{x+2}, g(x) = \ln (7x+15).$ The expression then equals
$$\frac{(f(x)-f(-2))/(x-(-2))}{(g(x)-g(-2))/(x-(-2))}.$$
By definition of the derivative, the above $\to f'(-2)/g'(-2),$ which is easy to compute.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1680242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do I show that if $f$ is entire and $\{\lvert f(z)\rvert < M\}$ is connected for all $M$, then $f$ is a power function?
Let $f$ be a non constant entire function satisfying the following conditions :
*
*$f(0)=0$
*for every positive real $M$, the set $\{z: \left|f(z)\right|<M\}$ is connected.
Prove that $f(x)=c... | Let $D_M=\{z:|f(z)|<M\}$ open connected set; since $f$ is non-constant $D_M$ is not the plane for any $M$; given any Jordan curve $J \subset D_M$, the interior of $J$ is contained in $D_M$ by maximum modulus, hence $D_M$ is simply connected.
Let now $B_{2r}$ a small disc of radius $2r$ around $0$ where $f$ vanishes onl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1680346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
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Finding the value of $\sum_{k=1}^{\infty} \frac{x^{4k-3}}{4k-3}$
This is my power series: $$\sum_{k=1}^{\infty} \frac{x^{4k-3}}{4k-3}$$
I need to find the sum of it. Unfortunately, I have kind of no idea how to do it. I think I need to substitute with something. One of the series that I do know of is $\sum_{k=0}^{\in... | $$ \frac{x^{4k-3}}{4k-3} = \int_{0}^{x} z^{4k-4}\,dz $$
hence:
$$ \sum_{k\geq 1}\frac{x^{4k-3}}{4k-3} = \int_{0}^{x}\sum_{k\geq 1} z^{4k-4}\,dz = \int_{0}^{x}\frac{dz}{1-z^4} $$
but:
$$ \frac{1}{1-z^4} = \frac{1}{2}\left(\frac{1}{1+z^2}+\frac{1}{1-z^2}\right) $$
so:
$$ \sum_{k\geq 1}\frac{x^{4k-3}}{4k-3} = \frac{1}{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1680445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Random Variable - Probability and Statistic Suppose that $f(x)$ is a continuous and symmetric pdf, where symmetry is the property that $f(x) = f(-x)$ for all $x$. Show that $P(-a ≤ X ≤ a) = 2F(a) - 1$
Does anyone have any idea what this is even asking? I have honestly no idea what to do.
| $$
P(-a\le X \le a) = F_X(a) - F_X(-a) = F_X(a) - (1-F_X(a)) = 2F_X(a) -1
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1680574",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Why does the Axiom of Selection solve Russell's Paradox in Set Theory? I am a beginner in mathematics and I was reading a text on Set Theory that talked about how Zermelo's Axiom of Selection "solves" Russel's Paradox.
I understand that the the axiom does not allow constructions of the form
$$\{x \:: \text S(x) \}$$ an... | The way that the Axiom of Selection prevent's Russell's Paradox is by preventing you from selecting from all sets. Rather you are selecting from the elements of A. Since $ A \in A $ is forbidden by the Axiom of regularity the paradox can't arise.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1680720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 2
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Calculating the expected value and variance of $n$ independent observations of $X$ I am attempting to find the expected value and variance of the random variable $X$ analytically (in addition to a decimal answer). $X$ is the random variable expression(100)[-1] where expression is defined by:
def meander(n):
x =... | So, your random variable is $$
X = 3X_1+\dots+3X_{100} = \sum_{k=1}^n 3X_k
$$
with $n=100$, where $X_1,\dots, X_n$ are independent, identically distributed random variables that are uniform in $[0,1)$. In particular, $\mathbb{E}\left[ X_k \right] = \frac{1}{2}$ and $\operatorname{var} X_k = \frac{1}{12}$ for every $1\l... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability of sum to be divisible by 7 6 fair dice are thrown simultaneously. What is the probability that the sum of the numbers appeared on dice is divisible by 7 ?
| One way is to add the coefficients of $x^7+x^{14}+x^{21}+x^{28}+x^{35}$ in the expression $(x+x^2+x^3+x^4+x^5+x^6)^6$ which will be symmetrical around the middle.
Another way is to use stars and bars, and apply inclusion-exclusion by preplacing $6$ in one or more of the $6$ cells, e.g. for a sum of $21$,
$\binom{20}{5}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1680976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Showing that a group with transposition acting transitively on ${1, \dots, n}$ is $S_n$ I am trying to prove that a group is $S_n$ where $n$ is any integer.
Right now I have gotten that it acts transitively on $\{1, \dots, n\}$, and contains a transposition.
Can I conclude from this that it is $S_n$? I think I'm missin... | Like MooS said, it is not enough. I sum up some conditions that imply that a permutation group $H$ is $S_n$.
*
*If $n$ is prime, $H$ contains a transposition and acts transitively on $\{1,\dots,n\}$.
*If $n$ is an integer, $H$ contains a transposition and acts doubly transitively on $\{1,\dots,n\}$.
*If $n$ is an ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1681091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Definition of connectedness? Connectedness is defined as: "A metric space $E$ is connected if the only subsets of $E$ which are both open and closed are $E$ and $\varnothing$. A subset $S$ of a metric space is a connected subset if the subspace $S$ is connected."
Can someone provide me with a more trivial/simple defini... | Topological space is connected if and only if the only subsets in it, simultaneously open and closed, are the space itself and the empty subset.
In case of a metric space, the topology is induced by the metric.
As a counter-example, consider say a pair of parallel lines or planes as a single topological space - it is n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1681188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
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CDF of absolute value of difference in random variables Let $X$ and $Y$ be independent random variables, uniformly distributed in the interval $[0,1]$. Find the CDF and the PDF of $|X - Y|$?
Attempt
Let $Z = |X - Y|$, so for $z \geq 0$, the CDF $F_{Z}(z) = \mathbf{P}(Z \leq z) = \mathbf{P}(|X - Y| \leq z) = \mathbf{P}(... | For fixed $z\in\mathbb{R}$ we have:$$F_Z(z)=P\left(\left|X-Y\right|\leq z\right)=\int\int1_{\left(-\infty,z\right]}\left(\left|x-y\right|\right)f_{X,Y}(x,y)dxdy$$
where $f_{X,Y}(x,y)$ denotes the density of $\langle X,Y\rangle$.
Substituting this density we arrive at:
$$F_Z(z)=P\left(\left|X-Y\right|\leq z\right)=\int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1681363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
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Prove or disprove $ab\mid ac\Longrightarrow b\mid c$ I need to prove or to give a counter-example:
$$ab\mid ac\Longrightarrow b\mid c$$
My attempt:
Yes, this is correct, let's divide $ab$ and $ac$ by $a$ (assuming that $a\neq 0$) we get $$b\mid c$$ and that's it.
Is my attempt correct?
| Yes. Your attempt is correct. Let me simplify it.
Proof:
Assuming $a$ is not 0.
$$ab|ac \implies kab = ac \implies kb = c \implies b|c$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1681452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Square root of $\sqrt{1-4\sqrt{3}i}$ How can we find square root of the complex number
$$\sqrt{1-4\sqrt{3}i}?$$
Now here if I assume square root to be $a+ib$ i.e.
$a+ib=\sqrt{\sqrt{1-4\sqrt{3}i}}$, then after squaring both sides, how to compare real and imaginary part?
Edit: I observed
$\sqrt{1-4\sqrt{3}i}=\sqrt{4-3-... | You need to first find the square root of $1 - 4 \sqrt{3}i$ using the same method: let $(c+di)^2 = 1 - 4 \sqrt{3} i$ and then compare real and imaginary parts to find $c$ and $d$ explicitly. This will give you two different answers. Then assume $(a+bi)^2 = c + di$, and compare real and imaginary parts to find $a$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1681555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
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Elementary question about the limit $\big( 1 - \frac 1 {\sqrt n}\big )^n$, $n\to\infty$. When calculating the limit $L=\big( 1 - \frac 1 {\sqrt n}\big)^n$, $n\to\infty$, what allows me to do the following:
$$
L=\lim \left(\left(1-\frac {1}{\sqrt n}\right)^\sqrt{n} \right)^\sqrt{n}
$$
As the term inside the outer parent... | You are looking for the following statement.
Let $(a_n)$ be a converging sequence of non-negative real numbers with $\lvert \lim_{n → ∞} a_n \rvert < 1$. Furthermore, let $(e_n)$ be an unbounded, increasing sequence. Then $({a_n}^{e_n})$ converges to zero.
Proof idea. Let $a = \lim_{n → ∞} a_n$. Let $K > 1$ be a real n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1681772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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Any countable free group is embeddable into a free group of rank $2$ I have found this proof of the fact that any countable free group is embedable in a free group of rank $2$ (see the last page, Proposition 2). But isn't this proof incorrect?
First off, it says $w=b^{-i_1}a^{\epsilon_1}b^{i_1}\dots$. Shouldnt it be $w... | Note that $$x_{i_1}^{\epsilon_1}=b^{-i_1}a^{\epsilon_1}b^{i_1}$$
Thee is no collapsing when going from $a^{\epsilon_j}a^{\epsilon_{j+1}}$ to $a^{\epsilon_j+\epsilon_{j+1}}$. Note that we speak about words over the alphabet $\{a,b,a^{-1},b^{-1}\}$ and that $a^n$ is just a notational shorthand for $\underbrace{aaa\ldots ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1681851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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How would one solve the following equation? This equation is giving me a hard time.
$$e^x(x^2+2x+1)=2$$
Can you show me how to solve this problem algebraically or exactly? I managed to solve it using my calculator with one of its graph functions. But I would like to know how one would solve this without using the calcu... | The answer given by Desmos for intersection of the two curves $y=e^x$ and $y=\frac {2}{(x+1)^2}$ is $\color{red}{x=0.249}$. Now we have
$$(x+1)^2=2e^{-1}\iff x^2+2x+1=2(1-x+\frac{x^2}{2}-\frac{x^3}{6}+\frac{x^4}{24}-\frac{x^5}{60}+O(x^6))$$ hence $$1-4x-\frac{x^3}{3}+\frac{x^4}{12}-\frac{x^5}{60}+20\cdot O(x^6)=0$$
Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1681953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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If $f:\mathbb{R}\mapsto\mathbb{R}$ is a non-constant periodic function than $\lim_{x \to \infty} f(x)$ does not exist I am trying to prove if $f:\mathbb{R} \mapsto \mathbb{R}$ is any non-constant periodic function than $\lim_{x \to \infty} f(x)$ does not exist. What I have so far is this:
Suppose $f$ is periodic with ... | Your reasoning implies that $|f(x)-L|\le|f(x)-L|$, which it is not a contradiction. In fact, as comments suggest, the supossition that $f$ is periodic and has limit at infinity should imply that $f$ is constant.
So, let's suppose that $f$ is periodic and not constant. Then, there exist two real numbers $x,y$ such that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1682005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
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Weird Inequality that seems to be true Is it true that:
$$\left (3x+\frac{4}{x+1}+\frac{16}{y^2+3}\right )\left (3y+\frac{4}{y+1}+\frac{16}{x^2+3}\right )\geq 81,\ \forall x,y\geq 0$$
I have proved that $3x+\frac{4}{x+1}+\frac{16}{x^2+3}= 9 +\frac{(x-1)^2 (3x^2+1)}{(x+1)(x^2+3)}, \ \forall x\geq 0$, but I did not succe... | This inequality is false, e.g. $x=2.5$, $y=0.5$. You will obtain a value of $79.9901<81$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1682088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Bayesian Statistics: Finding Sufficient Statistic for Uniform Distribution The example: let $y_1,\dots,y_n \overset{\text{i.i.d.}}\sim U([0,\theta])$, where $\theta >0$ is unknown. Find a sufficient statistic for $\theta$.
Solution attempt:
$$g(y_1,\dots,y_n) = c\quad \text{(constant)}$$
$$P(y_i\mid\theta) = \frac{1}{... | I find it at best irritating to use the same symbol to refer both to the random variable and to the argument to the density function. We can understand such things as $\Pr(Y\le y) = (\text{a certain function of } y)$ because capital $Y$ and lower-case $y$ mean two different things.
Write the density like this and see ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to define a sequence with an infinite amount of each natural number? I was trying to create a sequence such that for every $x \in \mathbb{N}$, that sequence has a subsequence that converges to $x$.
Basically I came up with the sequence that has each natural number an infinite amount of times. Now I am having troub... | How about just:
$$ a_n=1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,\ldots$$
Clearly each natural number appears an infinite amount of times, so you can extract a subsequence converging to whichever natural number you want.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to show that a map without fix point from annular region to annular region is homotopic to antipodal map $\Omega=\{x\in R^3: 1\le||x||\le2\}$
If $L:\Omega\rightarrow \Omega $ is continuous and without fix point , how to show $L$ is homotopic with antipodal map $x\rightarrow -x$?
| The homology of $\Omega$ is $H_0(\Omega,\Bbb{Q})=H_2(\Omega,\Bbb{Q})=\Bbb{Q}$ and $H_i(\Omega,\Bbb{Q})=0$ if $i\ne 0,2$.
The Lefschetz formula tells you that a map f without fixpoints necessarily
$0=Tr(H_0(f))+Tr(H_2(f)=H_0(f)+H_2(f)$. But $H_0(f)=H_0(id)=1$ for all $f$,
hence in our case $H_2(f)=-1=H_2(-id)$. By th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1682411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Number of integer solutions to the equation $x_1+x_2+x_3+x_4=100$ $x_1+x_2+x_3+x_4 = 100$ with
$1 \le x_1\le10$
$2\le x_2\le15$
$x_3\ge5$
$0\le x_4\le10$
Apparently this is the same as
$y_1 + y_2 + y_3 + y_4 = 92$ with
$y_1 \le 9$
$y_2 \le13$
$y_4 \le10$
I understand the $3$ conditions, but what has happened to $x_3... | We are defining $y_1=x_1-1, y_2=x_2-2,y_3=x_3-5,y_4=x_4$ and demanding that all the $y$'s be $\ge 0$. Once we defined $y_3=x_3-5$, there is no constraint on $y_3$ except that it be nonnegative, so it is not listed in the constraints.
| {
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"timestamp": "2023-03-29T00:00:00",
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What is the difference between ($\tan x \sec^2x$) and ($\sin x/\cos^3x$)? Why is the answer to the integration different? $$\int \:\frac{\left(\sin x+\tan x\right)}{3\cos^2x}dx$$
I know I have to split the equation into
$$\frac{1}{3}\int \:\left(\:\frac{\sin x}{\cos x}\right)\left(\frac{1}{\cos x}\right)dx+\frac{1}{3}... | Both the answers differ by a constant(1/6). So both the answers and both the methods are correct. The constant of integration takes care of the constant(1/6).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1682610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Finding the zero-state output The input and output of a stable network are related via the following equation.
$$\frac{d^2y(t)}{d(t)} + \frac{2*dy(t)}{d(t)} + 10y(t) = \frac{dx(t)}{d(t)} + x(t)$$
x(t) = input, y(t) = output, u(t) = unit function.
The input is $$\frac{3u(t)}{e^t}$$
I want to find the zero-state output.... | First find the solutions of $y''(t) + 2y'(t) + 10y = 0$. Passing to the characteristic polynomial, it has roots $\lambda_{1,2} = -1\pm 3i$, so every function of the form
$$
y_0(t) = Ae^{-t}e^{i3t} + Be^{-t}e^{-i3t}
$$
solves the homogenous equation.
Now consider the non-homogenous one. If $u$ is differentiable you get
... | {
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Is limit of function -1/0 ok? A quick question, i'm determining the limit of this function:
$$\lim_{x→1}\frac{x^2 - 2x}{x^2 -2x +1}$$
When I divide numerator and denominator by $x^2$ and fill in $1$, I get $-1/0$. This is an illegal form right? Or does it indicate it is going to $∞$ or $-∞$?
| Notice, $$\lim_{x\to 1}\frac{x^2-2x}{x^2-2x+1}$$
$$\lim_{x\to 1}\frac{(x^2-2x+1)-1}{x^2-2x+1}$$
$$=\lim_{x\to 1}\left(1-\frac{1}{(x-1)^2}\right)\longrightarrow \color{red}{-\infty}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1682838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Proof on set equality I was making a few exercises on set proofs but I met an exercise on which I don't know how to start:
If $A \cap C = B \cap C $ and $ A-C=B-C $ then $A = B$
Where should I start? Should I start from $ A \subseteq B $ or should I start from this $ ((A\cap C = B\cap C) \land (A-C = B-C)) \Rightarr... | You show $A\subseteq B$ and $B \subseteq A$, as one usually would when showing that two sets are equal. Since the conditions are symmetric in $A$ and $B$, the two proofs are completely analoguous, so I will only do one of them.
To show $A \subseteq B$, take an $a \in A$, and note that either $a \in C$ or $a \notin C$. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Are any of these in the kernel? Let $T:P_2 \rightarrow P_3$ be the linear transformation with rule $T(p)(t) = tp(t)$. Which of the following (if any) are in the kernel of $T$?
*
*$p_1(t) = t^2$
*$p_2(t) = 0$
*$p_3(t) = 1+ t$
Here is my question. To be in the Kernel of $T$, $T(u) = 0$. Would that mean only the se... | The equation $T(u)=0$ is to be read in a way that the right hand side refers to the zero polynomial function, i.e. $f:\mathbb R \rightarrow \mathbb R, t\mapsto 0$, which is equal to zero for all $t$.
Thus, the first interpretation is correct, the result has to be equal to zero for all $t$, not just a particular choice.... | {
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Prove that the inequality $\sin^8(x) + \cos^8(x) \geq \frac{1}{8}$ is true for every real number. Prove that the inequality $\sin^8(x) + \cos^8(x) \geq \frac{1}{8}$ is true for every real number.
| $$\sin^8 x+\cos^8x \ge \frac 18;$$
$$ \left (\frac{1-\cos2x}{2} \right )^4+ \left (\frac{1+\cos2x}{2} \right )^4\ge \frac 18;$$
$$(1-\cos2x)^4+(1+\cos2x)^4 \ge2$$
$$1-4\cos2x+6\cos^22x-4\cos^32x+\cos^42x+$$
$$+1-4\cos2x+6\cos^22x-4\cos^32x+\cos^42x\ge 2$$
$$12\cos^22x+2\cos^42x \ge 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1683097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Evalute $\lim\limits_{n\to\infty}\sin^2(\pi\sqrt{ n^2+n})$. I like to find the following limit:
$\lim\limits_{n\to\infty}\sin^2(\pi\sqrt{ n^2+n})$.
Any ideas or insight would be greatly appreciated.
| Hint: Note that
$$\sqrt{n^2+n}=\sqrt{n^2+n}-n+n=\frac{n}{\sqrt{n^2+n}+n}+n=\frac{1}{\sqrt{1+1/n}+1}+n.$$
| {
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Find a $2 \times 2$ matrix $A$ with each main diagonal entries $0$, and with $A^2 = -I.$ I'm not sure how to tackle this problem.
I'm not certain what is meant by "each main diagonal entries 0". Does this mean:
$$A=\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}$$
I'm also not sure what it means by $A^2 = -I$?
From W... | The main diagonal runs from the top left to the bottom right, so you're looking for a matrix like
$$A = \left[ \begin{array}{cc} 0 & a \\ b & 0 \end{array}\right]$$
such that $A^2 = -I$. This can be done by trial and error, or by writing a system of a few equations after finding what $A^2$ is.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1683351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why would the category of topological spaces be a balanced category (i.e. monic epimorphisms are isomorphisms)? I've just read on this page that
For example, $\mathsf {Set}$ (the cateogry of sets), $\mathsf {Grp}$ (the category of groups), and $\mathsf {Top}$ (the category of topological spaces) are all balanced.
(B... | As it was pointed out in the comments (by Pedro Sánchez Terraf and Rob Arthan), the PlanetMath page is wrong. It is not true that every monic epimorphism in $\sf Top$ is an isomorphism.
Other examples of such morphisms can be found in the category of Hausdorff spaces $\sf Haus$ (looking at the inclusion $\Bbb Q \hookri... | {
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Finding the mean time to second failure? So I can find the mean time to failure by using the equation $∑kpk$ which is the expected number of trails. I then used the formula $(1−x)^{−2}=1+2x+3x^2+⋯$ to simplify the above formula to $1/p$ which is the mean time to failure. So what about the mean time to second failure? W... | If we are counting the number of trials, then notice that the 2nd success occurs on the $k$th trial. Hence, there is one success in the previous $k-1$ trials. there are $\binom{k-1}{1}$ ways to choose where the success happens, there are 2 successes with probability $p = 1/2$, and $k-2$ failures with probability $1-p$.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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prove or disprove if $\|AB - I\|<1$ then $\|BA - I\|<1$ can we say if $\|AB - I\|<1$ then $\|BA - I\|<1$ for some arbitrary norm.
I am trying to make counter example but I stuck please help me.
| For given matrices $A$ and $B$, it depends on the norm you use (see this for examples of matrix norms), so it is important to specify your norm. We have already seen in other answers that for some norms this is false. For the Frobenius norm, namely $\|A\|_F \overset{def}= \sqrt{\mathrm{tr}(A^{\top}A)}$, this is true fo... | {
"language": "en",
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Establish the identity $\frac{\cot\theta + \sec\theta}{\cos\theta + \tan\theta} = \sec\theta \cot\theta\$ Establish the identity:
$$\dfrac{\cot\theta + \sec\theta}{\cos\theta + \tan\theta} = \sec\theta \cot\theta$$
The first step I got was:
$$\sec\theta \cot\theta = \dfrac{\sec\theta \cot\theta\,\big(\cos\theta + \tan\... | Continuing from what you got:
$$\sec\theta \cot\theta = \dfrac{\sec\theta \cot\theta\,\big(\cos\theta + \tan\theta\big)}{\cos\theta + \tan\theta}$$
and since $\sec \theta \cos \theta = 1, \cot \theta \tan \theta = 1$, expand the brackets:
$$\sec\theta \cot\theta = \frac{\cot \theta + \sec \theta}{\cos \theta + \tan \th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How can trigonometric functions be negative? I cannot understand why $\cos(180-\theta)$ say is $-\cos\theta$. This is probably because my teacher first introduced trigonometry in triangles. I do not understand it for obtuse angles because I cannot think of them in a right triangle.
I realised that I couldn't feel wha... | The cosine of an obtuse angle simply does not come from ratios of the lengths of the sides of an obtuse angle. It's defined to be the $x$ coordinate of the intersection of the terminal side of the angle with the unit circle. That's all. There's nothing forcing us to make this definition, except that it's immensely usef... | {
"language": "en",
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Maximum minimum values in trigonometry
Find minimum value of $2\sin^2a+3\cos^2a$
Solving it we get $2+ \cos^2a$
Answer: $3$ (taking $\cos a$ as $-1$)
Why are we using the minimum cosine value as $-1$ instead of using the cosine as $0$?
This can make the minimum value as $2$.
| $$2 \sin^2a+3 \cos^2 a= 3\cos^2a+2-2\cos^2 a=\cos^2a+2$$
$$0\le \cos^2a \le 1 \Rightarrow 2\le \cos^2a+2 \le 3$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Size issues in 2-categories I was playing a bit the 2-category Cat trying to have a better understanding of the notion of a 2-category (strict I guess). The usual definition of a category that I use assumes that $Hom(A,B)$ is a set.
What is an analogue of that condition in 2-categories? I guess you need to have some si... | Paul Blain Levy has a short (1-page) note on the topic.
| {
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proving that $\frac{(n^2)!}{(n!)^n}$ is an integer How to prove that $$\frac{(n^2)!}{(n!)^n}$$ is always a positive integer when n is also a positive integer. NOTE i want to prove it without induction. I just cancelled $n!$ and split term which are $n^2-(a^2)=(n-a)(n+a)$ where a is a perfect square. nothing more i coul... | $S_n \times S_n \times \cdots \times S_n$ is a subgroup of $S_{n^2}$.
The number in question is the index of that subgroup and thus an integer by Lagrange's theorem.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 4
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Prove the following tautology: $\big[(p\leftrightarrow q) \land (\lnot q \to r) \land (p \to r)\big]\to r $
Prove the following tautology:
$$\big[(p\leftrightarrow q) \land (\lnot q \to r) \land (p \to
r)\big]\to r $$
My effort
I am trying to prove this with a direct reasoning,i.e without using truth tables.
Now sin... | Here is a natural deduction proof in a Fitch-style proof checker:
The law of the excluded middle (LEM) on $Q \lor \neg Q$ is referenced at the end.
Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/
| {
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Proof by induction: Show that $9^n-2^n$ for any $n$ natural number is divisible by $7$. Can someone please solve following problem.
Show that $9^n-2^n$ for any $n$ natural number is divisible by $7$. ($9 ^ n$ = $9$ to the power of $n$).
I know the principle of induction but am stuck with setting up a formula for this... | Assume $7\mid(9^n-2^n)$; then you can write $9^n-2^n=7m$, for some integer $m$, or $9^n=7m+2^n$ as well. Then
$$
9^{n+1}-2^{n+1}=
9\cdot 9^n-2^{n+1}=
9(7m+2^n)-2^{n+1}=
63m+(9-2)2^n=
7(9m+2^n)
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 3
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Find a non-trivial solution to $5x^2+7y^2=3z^2$. Find a non-trivial solution to $5x^2+7y^2=3z^2$. Note: $x,y,z$ are integers.
My attempt: By a theorem I have $x^2 \equiv 1 \pmod 5$, $x^2 \equiv 1 \pmod 7$ and $x^2 \equiv 1 \pmod 3$. I just guessed $x=1$. So I have $5+7y^2=3z^2$. I now guessed $y=1$ and $z=2$.
Is there ... | The existence of integer solutions of $ax^2+by^2+cz^2=0$ or other ternary quadratic forms can be decided by using Legendre's theorem.
Theorem (Legendre): Let $a,b,c$ coprime positive integers, then $ax^2 + by^2 = cz^2$ has a nontrivial solution in integers (or rational numbers) $x,y,z$ if and only if $$\left(\frac{-bc... | {
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Notation for double-integrals - partial or full differentials? When you are trying to find a volume for a function z = f(x,y), the common notation is to find:
$$\int\biggr(\int f(x,y)dx\biggr)dy$$
However, when you do this, you are actually keeping the $y$ constant on the first integral. To me, for this to be the way ... | Apparently, you are correct, because when you use implicit differentation to solve partial derivatives of multivariable functions in the form $z = f(x, y)$, you will take the partial derivative of $z$ with respect to $x$, which means you will take the derivative of the $x$-terms and leave all $y$ terms as constants. It... | {
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Four statements, One statement is false math problem When trying to recall some facts about the ages of his three aunts, Josh made the
following claims:
*
*Alice is fifteen years younger than twice Catherine’s age.
*Beatrice is twelve years older than half of Alice’s age.
*Catherine is eight years younger than Bea... | The approach you suggest would work - as long as you find that only 1 of the situations is possible, and the other 3 aren't (like for example if someone has a non-integer/negative age, or you reach another contradiction of some sort).
On the other hand, you may find that there are several possibilities for which Josh's... | {
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Show $\frac{3997}{4001}>\frac{4996}{5001}$ I wish to show that $$\frac{3997}{4001}>\frac{4996}{5001}.$$
Of course, with a calculator, this is incredibly simple. But is there anyway of showing this through pure analysis? So far, I just rewrote the fractions:
$$\frac{4000-3}{4000+1}>\frac{5000-4}{5000+1}.$$
| Using long multiplication we get
$$3997\times5001=19988997>19988996=4996\times4001$$
which implies the desired result (because positive multiplication preserves order).
| {
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"url": "https://math.stackexchange.com/questions/1684883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Some Trouble Understanding set theory I'm currently in a discrete mathematics class and we've recently been discussing set theory. I feel like I have basic understanding of how to actually prove set relations when a question asks to do so. However I am having a lot of trouble when initially presented with questions, wh... | Part (1):
The light gray area represents $A - B$. If you combine the dark and light gray areas it is $A - C$. Evidently, $A - B \subset A - C$ but $C \not\subset B$. How did I come up with this diagram? I purposely drew $B$ and $C$ so that $C \not\subset B$, and then I drew $A$ so that $B$ overlapped more with it than... | {
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Proving set equivalence Let $S =\{[12],[3]^{-1},[13][4]\}$ and $T= \{[6^{16}],[24]+[67],[-158]\}$ be subsets of $\mathbb Z_{17}$. I am trying to prove $S=T$
So far I have $S =\{[12],[6],[1]\}$ since $[a^{-1}]=[b] \iff ab \equiv 1(mod\,x)$ and $T= \{[1],[6],[12]\}$
But I am not sure if this is sufficient. Does anyone kn... | While you arrived at the right conclusion, you may want to be a bit more verbose in your presentation.
*
*Since $6 \cdot 3 \equiv 18 \equiv 1 \mod 17$, we have $[3]^{-1} = [6]$.
*Since $13 \cdot 4 \equiv 52 \equiv 1 \mod 17$, we have $[3]\cdot [14] = [1]$.
This yields $S = \{[12],[6],[1]\}$ and we may also note t... | {
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"answer_id": 0
} |
Can you apply the fundamental theorem of calculus with the variable inside the integrand? I was wondering if I can apply the FTOC: $\frac { d }{ dx } \left( \int _{ a }^{ x }{ f(t) } dt \right) =f(x)$ to an implicit function with the variable being differentiated implicit in the function.
For example: $\frac { d }{ dx... | By definition
$$\frac{d}{dx}\int_a^xf(x,t)dt=\lim_{\Delta x\to 0}\frac{1}{\Delta x}\left[\int_a^{x+\Delta x}f(x+\Delta x,t)dt-\int_a^x f(x,t)dt\right]$$
$$=\lim_{\Delta x\to 0}\frac{1}{\Delta x}\left[\int_a^{x+\Delta x}\left(f(x,t)+\frac{\partial f}{\partial x}\Delta x\right)dt-\int_a^x f(x,t)dt\right]$$
$$=\lim_{\Delt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1685283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Expansion of an expression. I want to know how to expand expressions like $(x+y+z)(a+b+c)$, I currently have a problem I want to solve but I know FOIL if it is $(x+y)(a+b)$ but what do I do when it is $(x+y+z)(a+b+c)$?
| Use the distributive property.
We have that
$$(x+y+z)(a+b+c)=x(a+b+c)+y(a+b+c)+z(a+b+c)$$
Can you continue?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1685416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What is the domain of $f(x)=x^x$? What is the domain of $f(x)=x^x$ ?
I used Wolfram alpha where it is said that the domain is all positive real numbers. Isn't $(-1)^{(-1)} = -1$ ? Why does the domain not include negative real numbers as well?
I also checked graph and its visible for only $x>0$ . Can someone help me cl... | The expression $x^y$ can be assigned a reasonable meaning for all real $x$ and all rational numbers of the form $y=m/n$, where $m$ is even and $n$ is odd and positive. Thus $x^y=(x^m)^{1/n}$, interpreted as the unique real $n$th root of $x^m$ (define $0^0$ to be $1$). Since every real number can be arbitrarily well app... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1685523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Coverings of connected sum of four copies of $\mathbb{R}P^2$ G.Baumslag in one of his papers asserts that a group $G = \langle a,b,c,d | a^2b^2c^2d^2 = 1 \rangle$ contains all fundamental groups of closed compact orientable surfaces of genus $g\geq 2$? I think it can be proved using covering theory. The group $G$ is th... | $M_2$ cannot cover your space, since they both have the same (non-zero) Euler characteristic.
EDIT: Just to expound a bit, this means $\pi_1(M_2)$ is not a subgroup of your group.
DOUBLE EDIT: This also rules out all even-genus closed orientable surfaces, since they would have to go through the orientable double-cover ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1685716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Does there exist an analytic function such that $|f(z)|=x$ for $z=x+iy \in D$? Does there exist an analytic function $f=u+iv$ in $D=\{z:|z|<1\}$ such that $|f(z)|=x$ for $z=x+iy \in D$? Prove your response.
I am pretty stuck on this.
I know that $f(z)=u(z)+iv(z)$ and
$$u_x=v_y \quad u_y=-v_x$$
Further, I want to know i... | Consider $f$, restricted to the imaginary axis. However, if you mean $|f(z)| = |x|$ and only want to work with the Cauchy-Riemann differential equations (CRDE), derive $u^2 + v^2 = x^2$ with respect to $y$ and use CRDE to see that $u = cv$ for some $c$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1685794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find a value of $n$ such that the coefficients of $x^7$ and $x^8$ are in the expansion of $\displaystyle \left(2+\frac{x}{3}\right)^{n}$ are equal
Question: Find a value of $n$ such that the coefficients of $x^7$ and $x^8$ are in the expansion of $\displaystyle \left(2+\frac{x}{3}\right)^{n}$ are equal.
My attempt... | The coefficient of $x^7$ is
$$\binom{n}{7}\frac{2^{n-7}}{3^7}$$
And the coefficient of $x^8$ is
$$\binom{n}{8}\frac{2^{n-8}}{3^8}$$
Comparing them we get:
$$\binom{n}{8}=\binom{n}{7}\frac{3}{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1685895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
A random invertible matrix I work on a project, for these project i need to generate a square random invertible matrix.
I found out how to generate a square random matrix, still i want to be sure that this is an invertible one, without having to compute the determinant or to generate this matrix multiple times, can you... | A mean to be sure that a matrix has nonzero determinant is to take it as diagonally dominant (say for example on each column $j$, $|a_{jj}|> \sum_{i=1...n, i \neq j}|a_{ij}|$) https://en.wikipedia.org/wiki/Diagonally_dominant_matrix It can be also done on rows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1686116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Solving an infinite series $$\sum_{c=1}^\infty \frac{1}{(2c+1)^2(2c-1)^2} = \frac{1}{16}(\pi^2 -8)$$
I got the result using wolfram alpha but I don't know how to calculate it. I tried breaking it into telescopic sums but it can't be separated like that. Any hints?
| $$\frac1{(2n+1)^2(2n-1)^2}=\frac14\left(\frac1{2n-1}-\frac1{2n+1}\right)^2=\frac14\left(\frac1{(2n-1)^2}-\frac2{4n^2-1}+\frac1{(2n+1)^2}\right)$$
Now, we get
$$\frac{\pi^2}6=\sum_{n=1}^\infty\frac1{n^2}=\frac14\sum_{n=1}^\infty\frac1{n^2}+\sum_{n=1}^\infty\frac1{(2n-1)^2}\implies$$
$$\sum_{n=1}^\infty\frac1{(2n-1)^2}=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1686187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that (p ∧ q) → (p ∨ q) is a tautology? I am having a little trouble understanding proofs without truth tables particularly when it comes to →
Here is a problem I am confused with:
Show that (p ∧ q) → (p ∨ q) is a tautology
The first step shows: (p ∧ q) → (p ∨ q) ≡ ¬(p ∧ q) ∨ (p ∨ q)
I've been reading my text boo... | As $\lnot p\lor p$ is trivial and $p→q$ means $p$ necessitates $q$. This gives $\lnot p\lor q$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1686369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 8,
"answer_id": 5
} |
Program to create graph with modified bessel function $$e =\frac{1}{x}\frac{I_{1}(2x)}{I_{0}(2x)}$$
$$e =\frac{2}{1+\sqrt{1+x²\frac{8}{3}}}$$
$$e =\frac{1}{\sqrt{x\frac{4}{3}}} \frac{I_{\frac{2}{3}}(x\frac{4}{3}^{3/2})}{I_{\frac{-1}{3}}(x\frac{4}{3}^{3/2})}$$
I wanted to plot those functions but I'm having a hard time ... | Have you tried Wolfram Alpha ? Just type Plot[BesselI(1,2x)/BesselI(0,2x)/x,{x,-5,5}]
into the dialog box, and press Enter. If you want to plot more than one function at once,
the syntax is Plot[{... , ... , ...},{x, ... , ...}], where the dots are to be replaced
by the definitions of the functions, and the values o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1686476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Possible to determine position of number based on base 3 order of magnitude? I struggled with a good title for this - sorry if it ended up being confusing. I am attempting to try to partition a series of decimal numbers (starting at 1) by base 3 orders of magnitude. Essentially breaking them into ordered groups of 1,... | Let's focus on the last number $a_n$ of the $n$th partition (where $n$ starts from $0$):
$$
1, 4, 13, 40, \ldots
$$
Observe that it is the partial sum of a geometric series:
$$
a_n = 1 + 3 + \cdots + 3^n = \frac{3^{n + 1} - 1}{3 - 1}
$$
Taking the inverse function and using a ceiling function, we conclude that the $m$t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1686595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is a function satisfying these premises, ae. positive or zero? Let g be defined on $[0,1]$ such that the integral between $t_0$ and $t_1$, for all $t_0 < t_1$, is positive or zero. Does g satisfy $g(x) \ge 0$ ae. ?
If not, what if we add the continuity ?
I thought about saying that if it was negative on some measure-po... | Suppose $g$ is integrable and $\int_a^b g \ge 0$ for all $0 \le a \le b \le 1$. Then $g(x) \ge 0$ ae. $x \in [0,1]$.
Let $\phi(t) = \int_0^t g$, then the Lebesgue differentiation theorem shows that
$\phi$ is ae. differentiable with $\phi'(t) = g(t)$ ae.
Suppose $x<y$, then $\phi(y) = \phi(x)+ \int_x^y g \ge \phi(x)$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1686690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving $T(n) = T(n-2) + \log_2 n$ to be $\Omega(n\log_2 n)$ As title, for this recursive function $T(n) = T(n-2) + \log_2 n$, I worked out how to prove that it belongs to $O(n\log n)$; however I'm having trouble proving it to be also $\Omega(n\log n)$, i.e. lower-bounded by $n\log_2 n$ asymptotically.
Following the st... | The recurrence relation is given as $T(n) = T(n-2) + \lg n$.
Let's unroll the formula half the way, that is, we will do $k = \lfloor n/4\rfloor$ steps
\begin{align}
T(n) &= T(n-2) + \lg n \\
&= T(n-4) + \lg (n-2) + \lg n \\
&\ \ \vdots \\
&= T(n-2k-2) + \underbrace{\lg (n-2k) + \lg \big(n-2(k-1)\big) + \ldots + \lg ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1686792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Non-separated quotient of separated scheme I am reading Mumford's GIT book. I found the following claim there.
Let $X$ be an algebraic variety. Let $G$ be an algebraic group acting on $X$. Then the categorical quotient of $X$ by $G$ may be non-separated.
Question: Could you construct an example?
| This answer is given by Takumi Murayama. I am just writing down the details.
Consider the quotient of $X = \mathbb{C}^2 \backslash \{ 0 \}$ by $\mathbb{C}^*$ action $\lambda (x, y) = ( \lambda x, \lambda^{-1} y )$.
We can consider two charts
$U_1 = \{ (x,y) \in \mathbb{C}^2 \backslash \{ 0 \}$ such that $x \neq 0 \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1686872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How to use Borel-Cantelli specifically to show that the probability of a simple random walk returning to the origin in finite time is 1? Suppose we have that $X_i$ are iid random variables with $P(X_i =1) = P(X_i = -1) = 1/2$ and that $X_0 = 0$. Then, we define the simple symmetric random walk to be $S_n = \sum_{i=1}^n... | As pointed out in comments, there could be good ways of doing this using not Borel-Cantelli, but the properties of either Markov Chains or (I would recommend) martingales, since your random walk is both a Markov chain and a martingale.
However, there is a neat argument which uses the Borel-Cantelli lemma. Suppose that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1686977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Show that a regularizing operator $K : C_c(\Omega) \to \mathcal{D}'(\Omega)$ has kernel $k \in C^\infty(\Omega \times \Omega)$. I am reading Francois Treves' Introduction to pseudodifferential and Fourier integral operators, vol. I.
Let $\Omega \subseteq \mathbb{R}^n$ be open. On page 11, Treves defines what it means ... | This result can be found in the Hormander's book, The Analysis of Linear Partial Differential Operators I. More precisely, Theorem 5.2.6., page 132.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1687088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Identity having to do with weak derivative For $a<b \in \mathbb{R}$, let $(a,b) = G \subset \mathbb{R}$ be a bounded interval in the real numbers.
Show that there exists no $v \in L^2(G)$ and no $y \in G$ such that $$ \int_G v \varphi \text{d}x = \varphi(y) $$
holds for all test functions $\varphi \in C^{\infty}_0(G)... | HINT: Argue by contradiction that such $v,y$ exist. By Cauchy-Schwarz, the set $$A=\{ \varphi (y) : \varphi \in C^{\infty}_0 (G), ||\varphi||_2 \le 1\}$$ is bounded (it is contained in the interval $(-||v||_2, ||v||_2)$).
Now, try to construct some sequence $\{ \varphi_n \}_{n \ge 1}$ of bell-shaped smooth functions wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1687177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is a good notation for an “even falling factorial”? It has been suggested to me that I use this notation:
$$
\lfloor n \rfloor_2 = 2 \left\lfloor \frac n 2 \right\rfloor = \text{“even floor of $n$''} = \text{largest even integer}\le n.
$$
I also want to write about an “even falling factorial” that, for example, g... | Could you just define it as a two-variable function $f(n,k)?$
You could avoid introducing new notation if you wrote it
$$f(n,k)=\displaystyle\prod_{i=0}^{k-1} \left(2\left\lfloor\frac{n}{2}\right\rfloor - 2i\right)$$
Another way would be
$$f(n,k) = k! \cdot 2^k \cdot \binom{\left\lfloor\frac{n}{2}\right\rfloor}{k}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1687282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Show that if $ \lim_{x \to a} f(x) = -\infty$ then $\lim_{x \to a} (f(x))^2 = \infty$
Let $a \in \mathbb{R}$. Use the $\epsilon-\delta$ definition of the limit to show that if $\displaystyle \lim_{x \to a}f(x) = -\infty$ then $\displaystyle \lim_{x \to a} (f(x))^2 = \infty$.
We are given that $$\forall M < 0, \exists... | The two $M$s can be distinct provided they can be arbitrarily large. Here is how I would write it.
$\forall\,M > 0, \exists\,\delta > 0, |x - a| < \delta \Rightarrow f(x) < -M \iff -f(x) > M \Rightarrow (f(x))^{2} > M^{2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1687386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
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