Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
$Q \cap N_G(P)=Q\cap P$ where $P,Q$ are both Sylow $p-$groups We just need to prove that $Q \cap N_G(P) \subset P$. This problem becomes to show $gPg^{-1}=P, g \notin P, g\in Q \implies g \in P$. So how exactly $g$ being a member of another $p$-Sylow group helps him to be a member of $P$?
| Apply Sylow Theory in $N_G(P)$: $Q \cap N_G(P)$ is a $p$-group in $N_G(P)$, and hence must be contained in some Sylow $p$-subgroup of $N_G(P)$. Since $P \unlhd N_G(P)$, $P$ is the unique one and so $Q \cap N_G(P) \subseteq P$. (Note that $Q$ does not even have to be a Sylow $p$-group here, just being a $p$-subgroup suf... | {
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"timestamp": "2023-03-29T00:00:00",
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An expression of scalar curvature In the book "Hamilton's Ricci flow" by B. Chow, P. Lu and L. Ni, at the page- 99, Exercise 2.8, the statement says:
If $(M,h)$ is a Riemannian surface and $g=uh$ for some function $u$ on $M$, then
$$R_g=u^{-1}(R_h-\Delta_h\log u),$$
where $R_h$ and $R_g$ are scalar curvature of the met... | One place to find a proof is in my Introduction to Riemannian Manifolds (2nd ed.), Theorem 7.30. (The notation is a little different from yours, but it should be easy to translate between the two notations.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3332561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Showing that the limit of non-eigenvector goes to infinity Let $A$ be a $3$ by $3$ real matrix with the triple eigenvalue $1$. Also, further suppose its eigenspace corresponding to $1$ is only of dimension $1$. Thus, we can find a basis of $\mathbb{R}^3$, denoted by $v$. $w_1$. $w_2$ where $v$ is an eigenvector of $A$.... | The theorem is correct. Without loss of generality and by using Schur decomposition, we can write the most general form of matrix $A$ as$$A=\begin{bmatrix}1&a&b\\0&1&c\\0&0&1\end{bmatrix}$$Since the eigenspace of $A$ has dimension $1$, and the eigenvector is $v=\begin{bmatrix}v_1\\v_2\\v_3\end{bmatrix}$ therefore the f... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find $(x_1,\dots,x_n) \in (\mathbb{R}_+^*)^n$ to minimize $\sum_{k=1}^{n}x_k\prod_{i=1}^{k}{(1 + x_i)}$ such that $\sum_{k=1}^{n}{x_k} = 1$ I want to find $(x_1,\dots,x_n) \in (\mathbb{R}_+^*)^n$ to minimize
$$ \sum_{k=1}^{n}{x_k \prod_{i=1}^{k}{(1 + x_i)}} $$
with the following constraint $\displaystyle\sum_{k=1}^{n}{... | Not an answer.
We could try Lagrange multipliers. Take the Lagrangian
$$\mathcal L (\textbf{x},\lambda) = \sum_{k=1}^n x_k \prod _{j=1}^k (1+x_j) + \lambda \left (\sum _{k=1}^n x_k -1\right ) =: f(\textbf{x}) + \lambda g(\textbf{x}). $$
Find the candidate solution(s) by finding stationary point(s) i.e $\nabla\mathcal ... | {
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"timestamp": "2023-03-29T00:00:00",
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Milnor Number for holomorphic map germ Definition: Let $f : (\mathbb{C}^{n}, p) \longrightarrow (\mathbb{C}^{n}, q)$ be a holomorphic map germ. The multiplicity of $f$ at $p$, or Milnor number de $f$ at $p$, noted $\mu_{p}(f)$, is the dimension of the $\mathbb{C}$-linear space $\mathcal{Q}_{f}$.
Here, we have:
1) $\ma... | Expand the function as a power series around $(i,0,0)$
$$
f_1 = z_2 +z_1z_3 =z_2 +(z_1-i+i)z_3 = z_2 +iz_3 + (z_1-i)z_3\\
f_2 = -z_2z_3 \\
f_3 = z_1^2+z_2^2+1 = (z_1-i+i)^2+z_2^2+1 = 2i(z_1-i) + (z_1-i)^2+z_2^2
$$
$\mathcal{O}_{(i,0,0)}= \mathbb{C}\{z_1-i, z_2, z_3\}$ is the ring of convergent power series in the varia... | {
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Why are there $2^n-1$ terms in the inclusion-exclusion formula of $n$ sets? Why are there $2^n-1$ terms in the inclusion-exclusion formula of $n$ sets?
An example of what I mean by inclusion-exclusion formula is this:
There are three sets (i.e. $n$ $=$ $3$): $A, B,$ and $C$.
$A \cup B \cup C = |A| +|B|+|C|-|A\cap B| -... | This is because you have to take these sets $1$ by $1$, then $2$ by $2$, &c. and ultimately $n$ by $n$, which makes all nonempty subsets of the set $\{A_1,A_2,\dots, A_n\}$, and because there are $2^n$ subsets of a set with $n$ elements (including the empty subset).
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve the equation: $y^3 + 3y^2 + 3y = x^3 + 5x^2 - 19x + 20$ A question asks
Solve the equation: $y^3 + 3y^2 + 3y = x^3 + 5x^2 - 19x + 20$ for positive integers $x$ and $y$.
I tried factoring the LHS by adding $1$ to both sides so we get $(y+1)^3$ in the LHS. But I couldn't get any factorisation for the RHS, neither... | You are asking for
$$ x^3 + 5 x^2 - 19 x + 21 = (y+1)^3 $$
For large enough positive $x,$ (you need to find out explicit lower bound for $x$),
$$ (x+1)^3 < x^3 + 5 x^2 - 19 x + 21 < (x+2)^3 $$
and cannot be a cube. Then check the small values of $x$ remaining.
so, $x^3 + 3 x^2 + 3x + 1 < x^3 + 5 x^2 - 19 x + 21,$ or... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3333099",
"timestamp": "2023-03-29T00:00:00",
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Calculate $P(X>Y^2)$ Calculate $P(X>Y^2)$ given that $$f(x,y)=6(x-y)$$ for all $(x,y)$ such as $$0\le y\le x\le1$$
My solution: $$6\cdot\int_{0}^{y^2}\int_{0}^{x}(x-y)dydx$$
which in turn results in : $y^6$. However I have no way of knowing whether or not I have actually solved this correctly and hence arrived at the ... | Refer to the graph:
$\hspace{3cm}$
The total probability is:
$$P(\underbrace{0\le Y\le X\le 1}_{\text{the gray region}})=P(0\le Y\le X \ \cap \ 0\le X\le 1)=\int_0^1\int_0^x6(x-y)dydx=1.$$
The required probability is:
$$P(X>Y^2)=P(Y^2<X)=P(0\le Y^2<X\le 1)=P(\underbrace{0\le Y<\sqrt{X}\le 1}_{\text{the gray and orange ... | {
"language": "en",
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Why is every homogeneous space with $G$ action isomorphic to $G/H$ for some closed subgroup $H$? This is a statement in Lang, Real and Functional Analysis, Chpt XII, sec 4. The space below means homogeneous space equipped with $G$ action where $G$ is some topological group.
"Such a space is isomorphic to $G/H$ with som... | Given a continuous, transitive action $\alpha:G\times X\to X$ of a topological group $G$ on a Hausdorff (or merely $T_1$) space $X$, the $H$ that's used in the statement you quoted is just the stabilizer of any chosen point $x_0\in X$. That is, $H=\{g\in G:\alpha(g,x_0)=x_0\}$. This $H$ is closed because it is the pre-... | {
"language": "en",
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Almost sure convergence to 0 implies probability convergence to 0 I've seen proof of almost sure convergence implying probability convergence but I want to ask whether or not the "proof" for the 0 case is correct:
$X_n$ converges almost surely to $X$ if $\mathbb{P}(\lim_{n\to\infty}X_n = X) = 1$.
Given that $\lim_{n\t... | Here's a way to salvage your proof. Note that $P(|X_n|>\epsilon) = E(1_{|X_n|>\epsilon})$. Since $X_n \to 0$ a.s, we have $1_{|X_n|>\epsilon} \to 0$ a.s.
Since $1_{|X_n|>\epsilon}\leq 1$, the dominated convergence theorem applies and yields $\lim_n E(1_{|X_n|>\epsilon}) = E(0) = 0$.
| {
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"timestamp": "2023-03-29T00:00:00",
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How to prove $\sum_{n=-\infty}^ \infty {\rm sinc}\bigl( \pi(t-n)\bigr) = 1$? Thank you by avance for your help.
So, I found on this website, that $\sum_{n=-\infty}^{\infty} {\rm sinc}( \pi n)= 1$.
But I could not find any way to prove it. I know it’s about fourrier, but I don’t know how to do so...
Does anyone know how... | After getting help of @reuns, here the full demonstration :
$f(x)=e^{2i\pi tx}$
$f(0)=e^0=1$
We take a period of 1.
So with $C_{n}(f)=\frac{1}{T}\int_{-T/2}^{T/2}f(t)e^{-i2\pi\frac{n}{T}t}dt$
We get : $C_{n}(f)=\int_{-1/2}^{1/2}f(t)e^{-i2n\pi t}dt$
We use $f(x)$ from above :
$= C_{n}(f)=\int_{-1/2}^{1/2}f(t)e^{-i2n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find the greatest common divisor of $2^m+1$ and $2^n+1$ that $m,n$ are positive integers. I am confused of a question that needs to know the greatest common divisor of $2^m+1$ and $2^n+1$ ($m,n$ are positive integers), but I don't really know. I am pretty sure that the greatest common divisor of $2^m-1$ and $2^n-1$ ($m... | Your conjectured formula is correct; here is the proof.
For integer $m,n\ge 0$, let $d(m,n):=\gcd(2^m+1,2^n+1)$. Assuming for definiteness $m\ge n$, we have
\begin{align*}
d(m,n) &= \gcd(2^m-2^n,2^n+1) \\
&= \gcd(2^n(2^{m-n}-1),2^n+1) \\
&= \gcd(2^{m-n}-1,2^n+1) \\
&= \gcd(2^{m-n}+2^n,2^n+1... | {
"language": "en",
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Conditional probability against sum of two normal random variables I am trying to solve the following problem:
Let $X$, $Y$ be two variables with the distribution $\mathcal{N}(0,\lambda^2)$. Find the distribution of $X$ under the condtion $X+Y=t$.
So what we are looking for is, for $B$ a Borel set, the following thing:... | Here's another way. I'm assuming $X$and $Y$ are independent.
Since $\begin{pmatrix}X\\X+Y\end{pmatrix} = \begin{pmatrix}1 & 0\\ 1 & 1\end{pmatrix} \begin{pmatrix}X\\Y\end{pmatrix}$, the joint distribution of $(X,X+Y)$ is $\mathcal N_2\left(0,\lambda^2 \begin{pmatrix}1 & 1\\ 1 & 2\end{pmatrix}\right)$.
Note also that... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What will be the value of $\cos (n\pi x /l)$ for $x=0$? $\cos0 = 1$ but the value of the above expression is $1$. How it is equal to $1$? If $\cos0 =1$ then the value should be $n\pi/l$.
| Here we assume $l\neq 0$. If $x=0$, we have that
$$
\cos(n\pi\times 0)/l=\cos(0)/l=1/l.
$$
If you mean $\cos(n\pi x/l)$, then, at $x=0$, we have
$$
\cos(n\pi\times 0/l)=\cos(0)=1.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3333905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is $\mathbb{Z}[{ \sqrt 8 } ] $ a Euclidean domain?
Is $\mathbb{Z}[{ \sqrt 8 } ] $ a Euclidean domain ?
I have some confusion that is what is difference between euclidean domain and euclidean Norms ?
My attempt : I thinks yes
i know that $d( a+b \sqrt 8) = |a^2 - 8b^2 | $ as i can show it is euclidean domai... | It is more straight forward to give a counter example. Since $4=2 \cdot 2 = (\sqrt{8}+2)(\sqrt{8}-2)$, $\mathbb{Z}[\sqrt{8}]$ is not a unique factorisation domain (UFD), hence not a Euclidean domain. Note that those factors are irreducible.
Suppose to the contrary that $2$ is not irreducible. There exists $a,b \in \mat... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Unique factorization of an element in an UFD By definition:
An integral domain $R$ is a unique factorization domain if the following conditions are satisfied:
*
*Every element $a \in R$, $a \neq 0$ that is not a unit can be factored into a product $a = c_1 \cdots c_n$ where $c_1,\dots,c_n \in R$ are irreducib... | The action you take is identical to the following situation where you consider words that are monomials in several variables: for example $$(\frac{3}{4}x)(5y)(x)(\frac{5}{3}x)(\frac{2}{5}z)(\frac{1}{4}y) $$
First group all associates: $$(\frac{3}{4}x)(x)(\frac{5}{3}x).(5y)(\frac{1}{4}y). (\frac{2}{5}z)$$
Then for each ... | {
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Finding the sum of the first term and common difference when given sum of first 5 and sum of first 10 the sum of the first 5 terms of an arithmetic series is 110 and the sum of the first 10 terms is 320. How do i go about finding the first term and common difference.
Sn = n/2 [2a+d(n−1)] is the equation for working ou... | Sn = n/2 (2a + (n-1) d)
110 = 5/2 (2a+(5–1)d) (Eq. 1)
320=10/2(2a+(10–1)d (Eq.2)
110=2.5(2a+5d-1d) - 110=2.5(2a+4d) (Eq.3)
320=5(2a+10d-1d) - 320=5(2a+9d) (Eq.4)
64=(2a+9d) - Divided both sides 5 from equation 4 (eq.5)
44=2a+4d - Divided both sides 2.5 from equation 3 (eq.6)
20=5d - Simulataneous Equations - just minu... | {
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Theorem 5.13 in "Principles of Mathematical Analysis" by Walter Rudin L'Hospital's Rule L'Hopital's Rule I am reading "Principles of Mathematical Analysis" by Walter Rudin.
Thank you Saaqib Mahmood.
I copied and pasted your text
Theorem 5.13 on p.109:
Suppose $f$ and $g$ are real and differentiable in $(a, b)$, a... | If $g(x)=g(y)$ then by Theorem 5.10 (Lagrange), $0=g(x)-g(y)=(x-y)g'(t)$ for some $t\in (x,y)$. It follows that $g'(t)=0$, contradicts with the assumption $g'(x)\neq 0$ for all $x\in (a,b)$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Conditions to exploit Polar coordinates in limits.
Evaluate, $$\lim_{(x,y)\rightarrow(0,0)}f(x,y)=\lim_{(x,y)\rightarrow(0,0)}\dfrac{2x^2y}{x^4+y^2}$$
When I used polar coordinates with $x=r\cos\theta, y=r\sin\theta$,
$$\lim_{r\rightarrow0}\dfrac{r\cos\theta\sin2\theta}{r^2\cos^4\theta+\sin^2\theta}=0$$
But when I us... | If in polar coordinates the function takes the form
$$
g(r) \, h(r,\theta)
$$
where $g(r) \to 0$ as $r \to 0^+$ (standard single-variable limit) and the function $h$ is bounded for all $\theta$ and all $r$ in some region $0 < r < R$, then you can draw the conclusion that the two-variable limit is zero.
But that's not w... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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on solving $\frac{d^2y}{dx^2} = cy$ in matlab The differential equation $\frac{d^2y}{dx^2} = my$ has two solutions
$y = e^{\sqrt{m}x}$ and $y = e^{-\sqrt{m}x}$.
When I use ode45 (or any other IVP solver) in matlab, it always picks up $y = e^{\sqrt{m}x}$.
How do I make matlab pick the other solution, namely $y = e^{-\... | MATLAB does not pick $y = e^{\sqrt{m} x}$. Since $y_1 = e^{\sqrt{m} x}$ and $y_2 = e^{-\sqrt{m} x}$ are both solutions, then the solution is the linear combination of both namely
$$y(x) = c_1 y_1(x) + c_2y_2(x)$$
Now, depending on the initial conditions you've passed ode45, $c_1,c_2$ will be adjusted accordingly.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the directional derivatives of a function
Consider the function $f : R^2 → R$ given by
$$
f(x, y) =
\begin{cases}
\begin{matrix}
\frac{x^2y}{x^4+y^2} & \mathrm{if}\ (x, y) \ne(0, 0)\\
0 & \mathrm{if}\ (x, y) = (0, 0) \\
\end{matrix}
\end{cases}
$$
Using the definition, compute the directional derivative $\partia... | The way I have edited the question now, the problem lies at the end of the line in case 2:
$$\lim_{h\to0}\left(\frac{u_1^2u_2}{h^2 u_1^4+u_2^2}\right) = \frac{u_1^2u_2}{u_2^2} = \frac{u_1^2}{u_2}$$
Hence $\partial_u f(0,0) = \frac{u_1^2}{u_2}$.
Have a look at these plots to get an inuitive understanding of the behavio... | {
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Why the rotation of two equal surfaces (with different shape) do not give the same volume I will explain my question with an example:
Let's take two surfaces that have the same area:
SURFACE A
$$ f(x) = x, x \in [0,6] $$
SURFACE B
$$ f(x) = 3, x \in [0,6] $$
Both surfaces are equal:
Surface A, Sa = 18
Surface B, Sb =... | There is a result that says that equal areas rotated about axes an equal distance away from the center-of-mass (or centroid) of the areas, produce equal volumes. If the area is not distributed equally with respect to the distance from the axis of rotation, you will get different volumes.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Expected number of tosses to get 3 consecutive heads, what's wrong with my solution? Apparently the answer is 14 Expected number of tosses to get 3 consecutive Heads, but I got 11. Can someone pinpoint the error to my solution?
Let $X_i$ = Expected number of tosses to ith head. Hence,
$$X_1 = 1 + 1/2*X_1$$
$$X_2 = X_1 ... | Your argument is almost correct.
Consider $X_2$
Your type of reasoning means that half the time this will be $X_1+1$ and half the time this will be $(X_1+1)+X_2$. So your second line should have "+1+" instead of "+1/2+".
Ditto in line 3.
Then your method gives $X_1=2, X_2=6,X_3=14.$
| {
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Problems in $\sqrt{5x+4}=x-2$ So when I solve $\sqrt{5x+4}=x-2$, I end up with $x(x-9)=0$. Yet only when $x=9$ is the original inequality satisfied. Can somebody give me some details on what exactly goes wrong here?
| At $x=0$, the LHS is $2$, while the RHS is $-2$. So the new solution come from squaring both sides, because $(2)^2=(-2)^2$.
| {
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Showing a complex inequality Let $z,w \in \mathbb C$ s.t $\bar z w \neq 1$ and $|z|\leq 1, |w| \leq 1$ then $ |\dfrac{z-w}{1-\bar z w}| \leq 1$
The hint is to show that $ |z-w|^2\leq |1-\bar z w|^2$ but i cant relate those 2
| $$|1-\overline zw|^2=(1-\overline zw)(1-\overline wz)=1+|wz|^2-\overline z w-\overline wz,$$
$$|z-w|^2=(z-w)(\overline z-\overline w)=|z|^2+|w|^2-z\overline w-w\overline z.$$
The difference is
$$1+|wz|^2-|w|^2-|z|^2.$$
| {
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"url": "https://math.stackexchange.com/questions/3335197",
"timestamp": "2023-03-29T00:00:00",
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Uniform bound on derivatives and uniform convergence If $f_n:[0,1]\to \mathbb{R}$ are differentiable, $|f_n'(x)|\leq C$ for all $n\in\mathbb{N}$ and $x\in [0,1]$, $f_n\to f$ uniformly, $f_n'(x)\to g(x)$ pointwise and $f$ is differentiable, can we conclude that $f'=g$? Equivalently, can we conclude that
$$
\lim_{n\to \... | No. For a counterexample on the whole $\Bbb R$ (but the difference is unessential), consider the $C^\infty$ bump $$\Phi(x)=\begin{cases}\exp\frac1{x^2-1}&\text{if }-1<x<1\\ 0&\text{if }x\le-1\lor x\ge 1\end{cases}$$ and $g_n(x)=\Phi(nx)$, $f_n(x)=\int_{-\infty}^x g_n(y)\,dy$. Then: $$\begin{align} \lvert g_n(x)\rvert&\... | {
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Finding solution to a double integral Well, here it is
$$\int\int_{[0,1]^{2}}\sqrt{1+x^2+y^2}\mathrm dx\mathrm dy\tag*{}$$
Maybe there's a really nice way of doing it, hopefully someone knows. Good luck!
| We can do this integral in polar coordinates by recognizing a symmetry - divide the square in half by the line $y=x$. The integral on the top half will equal the integral on the bottom half, so we will do one of the integrals and multiply its value by 2.
$$\iint_{[0,1]^2}\sqrt{1+x^2+y^2}dA = \int_0^{\pi/4} \int_0^{\sec... | {
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Find $f(x)$ for a function $f: R \to R$, which satisfies condition $f(x+y^{3}) = f(x) + [f(y)]^{3}$ for all $x,y \in R$ and $f'(0)≥0$. Find $f(x)$ for a function $f: R \to R$, which satisfies condition $f(x+y^{3}) = f(x) + [f(y)]^{3}$ for all $x,y \in R$ and $f'(0)≥0$
My attempt:
Replacing $x$ and $y$ by $0$, $f(0)=0$
... | What you've shown so far is that $f'(x)$ is a constant, since $f'(x) = f'(0)$. And we know that $f(0) = 0$. This means that your solution is going to be something in the form $f(x) = ax$ for a non-negative constant $a$ (since you've specified that $f'(x) \geq 0$).
So what constants work? Well, we know that $f(x^3) = ax... | {
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Derivation of alternative form of Chernoff bound I have encountered an alternative form for the Chernoff bound for the sum of $n$ coins which I have not been able to derive.
Specifically, let $X_1,...,X_n$ be independent Poisson trials, let $X = \sum_{i=1}^{n} X_i$ and let $\mu = \mathbb{E}(X)$. Then
$$
\forall t > 0 .... | For each $n$, let $X_{n}$ be a random variable bounded between $a_{n}$ and $b_{n}$.
Let $X\equiv X_{1}+\cdots+X_{n}$ and $\mu \equiv \mathbb{E}X$.
Hoeffding's inequality states that
$$
\mathbb{P}(X \geq \mu + t)\leq\exp\left(-\frac{2t^{2}}{\sum_{i=1}^{n}\left(b_{i}-a_{i}\right)^{2}}\right).
$$
In your case, $b_{i}=1$ ... | {
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Use $x\ge 0 \implies y = 1$ and $x<0 \implies y = 0$ into a linear programming solver For a binary variable $y$ and another decision variable $x$, $x$ being integer, I want to be able to use the following two non-linear constraints into a linear solver:
\begin{align}
x\ge 0 \implies y = 1\\
x< 0 \implie... | $$
\begin{align}
x\ge 0 \implies y = 1\\
x< 0 \implies y = 0
\end{align}
$$
I assume your variable $y\in \{0,1\}$ because there is no another possibility. Let $M$ be a big positive number.
$$
\begin{align}
x \geq (y-1)M\\
x < yM\\
-M \leq x < M
\end{align}
$$
Using the intervals $[-M... | {
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Cardinality of a p-Sylow Let $G$ be a group with cardinal $n=p^{\alpha}m$, (prime $p$ dividing $n$ and $\gcd(m,p)=1$).
Let $E$ be the set of subsets of $G$ containing $p^{\alpha}$ elements.
I'm trying to understand why $p$ does not divide $\vert E \vert = \binom{p^{\alpha}m}{p^{\alpha}}$
Writing
\begin{equation}
\disp... | Recall that the $p$-adic valuation of $N$ is $v_p(N)=r$ when $p^r$ is the highest power of $p$ dividing $N$, so that $(p,N)=1$ if and only if $v_p(N)=0$.
By counting powers of $p$ smaller than $n$ we get the formula
$$
v_p(n!)=\lfloor \frac np\rfloor+\lfloor\frac n{p^2}\rfloor+\lfloor\frac n{p^3}\rfloor+\cdots.
$$
Now ... | {
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Is group homomorphism $F: SO(3) \to \mathbb R-\{0\}$ unique? If we assume that $F$ is smooth, then such $F$ is unique. Under this assumption, the question is equivalent to find all 1-dimensional representation of $SO(3)$, and by considering Lie algebra there is only the trivial one.
But what if we remove the smoothness... | Yes, it's unique: indeed, the group $\mathrm{SO}(3)$ is perfect: actually every element is a commutator.
Indeed, consider any element $q$: this is a rotation; hence square of another rotation (with same axis) $r$, namely $q=r^2$. Then $r$ and $r^{-1}$ being rotations of the same angle, are conjugate (by any element rev... | {
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Does every odd integer $m$ satisfy $3^x(m)-2^y=1$ for some integer values of $x$ and $y$?
Does the equation $$3^x(m)-2^y=1$$ have positive integer solutions $x, y$ for for every positive odd number $m$?
For example, for $m = 1$, we have $x = 1, y = 1$: $3^1(1)-2^1=1$. For $m=3$, the (only) solution is $x=1,y=3$. But ... | No, take $m$ to be a power of $3$, your question reduces to:
Distance between powers of 2 and 3
Hope it helps:)
| {
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Poisson paradigm: Why is $\lambda$, the rate of occurrence of events, equal to the sum of the probabilities of all the events that occur? My notes say the following about the Poisson paradigm:
Let $A_1, A_2, \dots, A_n$ be events with $p_j = P(A_j)$, where $n$ is large, the $p_j$ are small, and the $A_j$ are independe... | If $A_i$ occurs with probability $p_i$, then the expected number of $A_i$ occurring is also $p_i$. Since expected values are additive under all circumstances, $E[X]=\sum p_i$. So if this is a Poisson distribution, then necessarily one with the correct expected value, which is $\lambda$.
| {
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Intersection of nested sequence of non-empty compact sets is non-empty (using sequential compactness) Let $(X,d)$ be a metric space, and let $K_1, K_2, K_3, \ldots$ be a sequence of non-empty compact sets in this metric space such that $$K_1 \supseteq K_2 \supseteq K_3 \supseteq \cdots$$ Then the intersection $\bigcap_... | Your proof is indeed correct. As you noticed, it requires the axiom of choice, and that's perhaps the reason why it is avoided.
| {
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Confusing Limit With Logarithms So I am probably below the knowledge level of the average mathematician here- computer science student here studying algorithms at the graduate level and came across this peculiarity that I would appreciate some context to. Hopefully this is a softball question for you folks.
Why is
$$\l... | We can convert this limit of logs by making the substitution $x=e^{e^{y}}$ (in the spirit of Yuriy S's comment). This will of course not really change the structure of the limit, but I at least find it easier to think about the size of exponentials than the size of logarithms.
Noting that $\ln(e^{e^y})=e^y$ and $\ln(\l... | {
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Impossible hyperbolic integral I'm quite good with integral calculation, very rarely happens that I'm not able to solve an indefinite integral. I've tried for 7 days to solve this monster but in the end, I surrendered to this beast.
I'm sure that it has a solution because it was a challenge from my calculus professor... | First make the substitution $y=\cosh x,$ then simplify the radicals by removing radicals from all denominators, to obtain the simpler $$\int\frac{\sqrt {2y+2}}{2y+2}\frac{\sqrt{(y-7)(y-3-\sqrt{2y+2})}}{\left({(y-7)^2(y-3-\sqrt{2y+2})}\right)^{1/3}}\mathrm d y.$$ Then make the substitution $2y+2=z^2.$ The integral becom... | {
"language": "en",
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Why if $\lim_{x \to a}f(x)=b$ then $\lim_{n \to \infty}f(x_n)=b$? Why is that if $$\lim_{x \to a}f(x)=b \Rightarrow \lim_{n \to \infty}f(x_n)=b?$$
This is often used to prove that the same properties that apply to sequences, also apply to functions, but I don't know from where this implication is coming from.
It is giv... | If $(x_{n})_{n\in \mathbb{N}}$ is a sequence such that $\lim_{n \to \infty} x_{n} = a$
Then if $\lim_{x \to a} f(x) = b$ we have that for all $\varepsilon >0$ there exist $\delta >0$ such that if $|x-a|<\delta \implies |f(x)-b| < \varepsilon$
So, if we take a fixed $\varepsilon>0$, there exist a $\delta>0$ then by defi... | {
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$3$ times a number, plus $4$, is equal to $10$ I was helping my brother with his math homework and there was this question:
$3$ times a number, plus $4$, is equal to $10$. What is that number?
My first thought was that $3x+4 = 10$ and then, solve for $x$. But then, my brother told me maybe it’s $3(x+4)=10$. Now I’m con... | Almost certainly the problem is intended to be $3x+4=10$.
But your brother has made an important discovery. Until the sixteenth century, all mathematical textbooks would write out such problems in words, like "3 times a number, plus 4, is equal to 10". As your and your brother have discovered, this can be difficult to ... | {
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Proving that $\rho(x,y) = \frac{d(x,y)}{1+d(x,y)}$ is a metric and that $\rho(x,y)$ and $d(x,y)$ are equivalent metrics As you can see the proof is divided into 2, the first part consists on proving that $\rho(x,y)$ is a metric
My attempt
i) $\rho(x,y) \geq 0$, which is clear since $d(x,y)$ is a metric, and it is $0$ i... | In my functional analysis class, we were allowed to prove metric equivalence as follows. Plot the function $f(x)=\dfrac{1}{1+x}$ to obtain a graph that is decreasing from $y=1$ to $y=0.$ Then note that inside any open ball on the $x$-axis, you can pull back those values to the $y$ axis and fit an open ball inside the r... | {
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Number of roots in the first quadrant I want to find how many roots of the equation $z^4+z^3+1=0$ lies in the first quadrant.
Using Rouche's Theorem how to find ?
| Look at the family of polynomials $z^4+tz^3+1$. For $t=0$ we know the solutions $z=\sqrt{\frac12}(\pm1\pm i)$ which has one root per quadrant. We additionally know that the set of roots is continuous in the coefficients of the polynomial.
Now if changing $t$ from $0$ to $1$ were to change the number of roots in the fir... | {
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second derivative of a function equals the function squared Can someone solve the following differential equation for me please?
The second derivative of a function equals the function squared.
Find $y(x)$ if
$$
\frac{d^2 y}{dx^2} = y^2
$$
| How familiar are up with Weierstrass elliptic functions?
Let $\mathcal{P}(x;a,b)$ be that value of $z$ which makes
$$
\int_{-\infty}^z \frac{1}{\sqrt{4t^3 - at - b}}dt = x
$$
This is the Weierstrass $\mathcal{P}$ function.
The general solution to
$$
\frac{d^2 y}{dx^2} = y^2
$$
is
$$
y = \sqrt[3]{6} \,\mathcal{P} \l... | {
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Logic: How to prove this argument is not valid Good day to all,
I need calcification on how to show this argument is not valid by finding a counterexample, but without using a truth table. Since there are 5 propositions I would need a 32 row truth table. It would be too time consuming to construct.
... | The argument is valid.
For the four premises to be all true, we must evaluate (1) $p$ as true, (2) $q$ as either true or false (we cannot infer which), (3) $r$ as false or $s$ as true (since $r\to s$ is an inference from evaluating $p$ being true), and (4) $t$ as false or $r$ as true.
In short, we must resolve to have ... | {
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Linear ordering isomorphic to an initial segment Question: Suppose $\left( L, \prec \right)$ is a linear ordering such that for every $X \subseteq L, \left( X, \prec \right)$ is isomorphic to an initial segment of $\left( L, \prec \right)$. Show that $\left( L, \prec \right)$ is a well ordering.
This is what i have go... | We don't have an isomorphism $f:(X,\prec) \, \to\, (L,\prec)$, only an order-preserving embedding such that the range of $f$ is an initial segment of $L$.
Now it's very easy: if we show that $L$ has a smallest element, then every initial segment of $L$ will have a smallest element, thus by the embedding being order-pre... | {
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Regarding perfect squares Is there any positive integer $n > 2$ such that $(n - 1)(5n - 1)$ is a perfect square? It is observed that $(n - 1)(5n - 1)$ is of the form $4k$ or $4k+ 1$.
Affirmative answers were given by Pspl and Mindlack (by providing some examples).
Now my question is the following:
Is there any charact... | If $m^2=(n-1)(5n-1)$, then $5m^2=5(n-1)(5n-1)=(5n-3)^2-4$.
Write this as $x^2-5y^2=4$, for $x=5n-3$ and $y=m$.
Write this as $N\left(\frac{x+y\sqrt5}{2}\right)=1$. Then $\frac{x+y\sqrt5}{2}=\left(\frac{1+\sqrt5}{2}\right)^{2k}$, since $\frac{1+\sqrt5}{2}$ is a fundamental unit of norm $-1$.
Thus, $\frac{x+y\sqrt5}{2}=\... | {
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How To Use The Steps on Page 149 in Calculus Made Easy To Solve Chapter XIV Example 12 Please how do I take the derivative of
$ y = \left(\frac{1}{a^x}\right)^{ax} $
from Calculus Made Easy Chapter XIV Example 12 using the steps used
to solve
$y=a^x$
on page 149.
The steps are
\begin{align*}
y & = a^x\\
\log_ey & =... | A good place to start is to consider that
$$\bigg(\dfrac{1}{a^x} \bigg)^{ax} = (a^{-x}) ^{ax} = a^{-ax^2}.$$
Once you get there, apply logarithmic differentiation to get
\begin{array}[rcl]
$y & =&a^{-ax^2}\\
\ln y& =& -ax^2 \ln a\\
\dfrac{1}{y}\cdot\dfrac{dy}{dx}&=&-2a\ln a \cdot x\\
\dfrac{dy}{dx}&=& -2a\ln a \cdot a... | {
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Elements and Subsets This is my first discrete math course so this questions might seem simple but I would like clarification.
Say I have a set $A=\{1,2,3\}$ and a set $D=\{1,2,3\}$. Is it true that set $A$ is both an element and a subset of set $D$?
Now say $D=\{\{1,2,3\}\}$. Is it true that set $A$ is no longer an el... | In the first case, $A = D$, which is different from $A \subseteq D$ (true) or $A \in D$ (false). In the second case, $A \in D$, while $A \ne D$ and $A \not\subseteq D$.
| {
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Continuous convolution between two functions I'm studing for an exam and I'm stuck on a simple exercise about convolution between two functions. It says:
A system has a triangular impulse response (LSF) centered at the origin of the plane $h(x)=Λ(x)$. We input an image with two impulses, defined as $f(x)=δ(x-x_0)+δ(x-2... | Since response of system to $input=\delta(x)$ is impulse response and equal to
$$h(x)=Λ(x)$$
Input to the system is a summation of two shifted impulses
$$f(x)=\delta(x-x_0)+\delta(x-2x_0)$$
What's the Output of system ?
$$Output=f(x)\star h(x)=\int_{-\infty }^{+\infty} \Big(\delta(x-x_0)+\delta(x-2x_0)\Big) Λ(x)dx$$
B... | {
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Rudin exercise: If $f$ is a diferentiable mapping on an open connected set $E$ and $f'(x) = 0$ for all $x \in E$, then $f$ is constant. Suppose $f: E \subseteq \mathbb{R}^n \to \mathbb{R}^m$ is a differentiable map with $E$ open and connected. If $f'(x)=0$ for all $x \in E$, prove that $f$ is constant.
My attempt:
For... | For your attempt, note that $E$ is actually path connected, and the range of a path is always compact. Can you see how to proceed?
Here’s a different method altogether:
Pick any value $c$ taken by the function. Consider the set
$$\{x\in E : f(x)=c\}$$
By continuity, this is closed. By the lemma you stated, it is open.... | {
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Algorithm to decompose a number into the product of an integer and a base two exponencial So i`ve been asked to code an algorithm that decomposes an integer into the product of a base two exponencial and some integer.Something like number = k.(2^n) , k and n being random integers with k restricted to being odd.
My que... | One simple way is to start with the $k$ you are given and $n=0$. Then divide $k$ by $2$ as many times as possible, incrementing $n$ each time.
while $k$ is even
$\quad k=k/2 $
$\quad n=n+1$
| {
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Possible Error in Elementary Analysis by Ross I am having some serious trouble understanding example 5 from chapter 8 of Elementary Analysis by Ross. The example is to do with proving sequence are convergent, and here is the definition of convergence that the example uses:
A sequence ($s_n$) is said to converge to the... | What I am confused about is as to where he got $|_−|<\sqrt{s} \epsilon$.
First, what he should have said was that for any $\epsilon' > 0$, we can define $\epsilon = \sqrt{s}\epsilon'$. Now the assumption in your first highlighted box shows that there's some number $N$ such that $n > N$ implies
$$
|s_n - s | < \epsil... | {
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Limit of Lebesgue integrals
Let $a>0$ and $f,g:[0,+\infty) \to \Bbb{R}$ where $f$ is a Lebesgue integrable function and $g$ has the property: $$|\frac{g(t)}{t}| \leq a, \forall t \geq 1$$.
Prove that $\lim_{t \to +\infty}\frac{1}{t}\int_1^tf(x)g(x)dx = 0$
Here is my proof:
$$|\frac{1}{t}\int_1^tf(x)g(x)dx| \leq \f... | Your proof is correct. But you can also get this as an immediate consequence of DCT: $\frac 1 t I_{(1,t)} f(t)g(t) \to 0$ for each $x$ and this function is dominated in absolute value by $a|f|$ which is integrable.
| {
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What does it mean to take the ratio of two equations?
The line joining the origin and the point of intersection of the curves $ax^2+2hxy+by^2+2gx=0$ and $a_1x^2+2h_1xy+b_1y^2+2g_1x=0$ will be mutally perpendicular if $g(a_1+b_1)=g_1(a+b)$
This is solved in my reference as
$$
ax^2+2hxy+by^2=-2gx\\
a_1x^2+2h_1xy+b_1y^2... | The step you are concerned with is justified by the simple fact that division is a well-defined binary operation on the real numbers, assuming that the denominator is nonzero. To be precise:
Given $r,s,t,u \in \mathbb R$, if $r=t$ and if $s=u \ne 0$ then $\frac{r}{s} = \frac{t}{u}$.
In your problem, you have two equa... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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divisibility by 21 It's a simple problem but I am stuck.
The multiple of 21 between 700 and 950 are 714, 735, 756, 777, 798, 819, 840,861,882,903,924,945.
So, there are 12 multiples of 21 between 700 and 950 but 21 *12=252.
So 12 multiples of 21 should take an interval of 252 , and there are only 250 numbers between 7... | Let's look at multiples of $3$ between $2$ and $13$. There are $3, 6, 9, 12$, which is four of them. But $4 \times 3 = 12$, and between $2$ and $13$ is only $11$ numbers.
The problem is that the number $4 \times 3$ doesn't represent the length of the interval containing the four multiples. There are only three gaps of... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the Fourier transform of $1$? For sure, $g(x)=1$ has no fourier transform, but it has a Fourier transform in distribution sense. We have for $\varphi \in \mathcal C_0^\infty (\mathbb R)$, $$\left<\hat 1,\varphi \right>:=\left<1,\hat \varphi \right>=\int_{\mathbb R}\hat \varphi (x)dx=\int_{\mathbb R}\hat \varphi... | $g$ is a locally integrable function, hence a tempered distribution. Fourier transform of a tempered distribution $u$ is defined by $\hat {u} (\phi)=u(\hat {\phi})$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3339067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the sum of the second numbers in the first $100$ rows of Pascal's triangle (excluding the first row)?
What is the sum of the second numbers in the first $100$ rows of Pascal's triangle (excluding the first row, the row containing a single $1$)? The sum should be from the second to the hundredth row.
Startin... | To make the computations more transparent, I'll start indexing with 0. Therefore we want to sum up the elements $a_{k,1}$ ($k\geq 1$).
I'll also assume, that the elements, that are 'outside' of the triangle are all equal 0 (in particular $a_{k,k+1}=0$ and $a_{0,1}=a_{0,2}=a_{1,2}=0$)
Note, that for $n\geq 1$ we have:
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3339223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Fréchet derivative of non-coercive energy functional $\frac{1}{2}\int_\Omega |\nabla u|^2 - \frac{1}{p}\int_\Omega |u|^p$ Let $\Omega \subseteq \subseteq \mathbb{R}^n$, $n \geq 3$, and for $2 \leq p \leq 2^* := 2n/(n - 2)$ define $E \colon H^1_0(\Omega) \to \mathbb{R}$ by
$$E(u) := \frac{1}{2}\int_\Omega |\nabla u|^2 ... | It should first be noted that $E$ is well-defined thanks to the Sobolev embedding $$W^{k,p}(U)\subset L^{\frac{np}{n-kp}}(U)$$ for bounded, open $U\subset\mathbb R^n$ and $1\le k < \frac np$.
Now, using Hölder's inequality and the Sobolev inequality $$\lVert u\rVert_{L^{\frac{np}{n-kp}}(U)}\lesssim \lVert u\rVert_{W^{k... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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roots of holomorphic functions I've got to check out if there is a holomorphic function $f$ such that $f(z)^3=z^3-1$ for all $a)$ $z \in B_1(0)$ and $b)$ $z\in B_1(1)$, where $B_r(z_0)$ is the open ball around $z_0$ with radius $r$.
I think the easiest way would be to take the Taylor series $f(z)= \sum_{n=0}^{\infty... | In $B_1(0)$, yes, there is such a function. If $z\in B_1(0)$, then $\lvert z\rvert<1$. Therefore, $\lvert z\rvert^3<1$ and so $z^3-1$ belongs to the halfplane $\{z\in\mathbb C\,|\,\operatorname{Re}z<0\}$. So, your idea is fine: you can work with an appropriate branch $\log$ of the logarithm and define $f(z)=\exp\left(\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Limit of a greatest integer function (sided limit)
What is the value of $\lim\limits_{x\to 0^+} \dfrac{b}{x}\left\lfloor\dfrac{x}{a}\right\rfloor$ for $a>0$ and $b>0$. Note that $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.
I know that $\left\lfloor\dfrac xa\right\rfloor=0$. But when it c... | Since $\frac{b}{x}\left \lfloor{\frac{x}{a}}\right \rfloor=0,\forall x\in (0,a)$, it follows that $\lim_{x\rightarrow 0^+}\frac{b}{x}\left \lfloor{\frac{x}{a}}\right \rfloor=0$ .
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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verification of convergence of random variable For $n \in \mathbb{N}$, let $X_n$ be a random variable such that $\mathbb{P} [X_n = \frac{1}{n}] = 1 − \frac{1}{n^2}$ and $\mathbb{P}[X_n = n] = \frac{1}{n^2}$. Does $X_n$ converge in probability? In $L^2$?
My attempt:
To converge in probability we must have
$$
P(|X_n - X|... | Not really. Here $X$ is chosen to let $X_n\to X$ in probability as $n\to\infty$, so we can not let "$n$" appear in $X$. Actually, we can choose $X=0$.Let $\epsilon>0$, for $n>\frac1\epsilon$, we have
$$P(|X_n-0|>\epsilon)=P(X_n=n)=\frac1{n^2}\to0,$$
so $X_n\to 0$ in probability.
For the $L^2$ convergence, since $E|X_n-... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the notation $\int_{\Bbb R}$ equivalent to $\int_{-\infty}^\infty$? The question is pretty straightforward. In class, I have seen $\int_{\Bbb R}$ and $\int_{-\infty}^\infty$ being used interchangeably. However $\int_{\Bbb R}$ contains no implicit sense of direction, so technically isn't $\int_\infty^{-\infty}$ also ... | The notation $\int_a^b$ (or, in an extended sense, $\int_{-\infty}^{\infty}$) is meant to be suggestive of Riemann integration, which, as you said, entails an orientation in your Riemann sums and hence, your integrals.
The notation $\int_{\mathbb{R}}$ is a more general notation from the theory of Lesbegue integration, ... | {
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A truncated alternating sum of product of binomial terms While solving a question I came across the following alternating sum; $C(j,n): = \sum\limits_{i=j}^{n} (-1)^{i}\binom{n+1}{i+1} \binom{i}{j}$ where $j$ and $n$ are integers with $n \geq j \geq 0$.
By hand I computed that $C(j, j+r) = (-1)^{j}$ for small positive... | \begin{align}
\sum_{i=j}^{n}(-1)^i\binom{n+1}{i+1}\binom{i}{j}&=(-1)^j\sum_{i=0}^{n}(-1)^i\binom{n+1}{i+1}[x^j](1-x)^i\\&=(-1)^j[x^j]\sum_{i=0}^{n}\binom{n+1}{i+1}(x-1)^i\\&=(-1)^j[x^j]\frac{x^{n+1}-1}{x-1}=(-1)^j.
\end{align}
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Inverse of a skew-symmetric matrix For $a, x, y, z \in \mathbb R$, let $$M= \left(
\begin{array}{cccc}
\cos(a) & \sin(a) \, x & \sin(a)\, y & \sin(a) \, z \\
-\sin(a) \, x & \cos(a) & \sin(a) \,z & -\sin(a)\, y \\
-\sin(a) \, y & -\sin(a) \, z & \cos(a) & \sin(a) \, x \\
-\sin(a) \, z & \sin(a) \, y & ... | Here is a proof of the first part (inversibility of $M-I_4$)
and a computation of the inverse (though I do not get a result in the form of yours).
One obtains an efficient simplification by using half-angle formulas (https://en.wikipedia.org/wiki/Tangent_half-angle_substitution) :
$$\cos(a)=\dfrac{1-t^2}{1+t^2} \ \ \t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Intersection of closed sets is closed proof without Morgan's theorem I wish to prove that the intersection of closed sets is closed. However, all proofs that I have come across use Morgan's Theorem, which we have not seen in class (and therefore cannot use).
I was thinking that maybe I could use something like the fac... | Suppose $C_i, i \in I$ is a family of closed subsets, and let $C$ be their intersection. Suppose $x \in C'$ (a limit point of $C$). Then as $C \subseteq C_i$ for each $i$, $x$ is in $C'_i$ for all $i$, and as each $C_i$ is closed, we know that for each $i$, $C'_i \subseteq C_i$ and thus $x \in \bigcap_i C_i = C$. As $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3340169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What does the math symbol $\propto$ mean? I came across this symbol in my engineering class and I have never seen it before. Anyone know this?
| It typically means proportional to. Such that
If $$y=cx$$ for some constant $c$ we say
$$y\propto x$$ so that when x grows, y grows proportionally by the ratio $c$
Alternatively inverse proportionality is when
$$y=c\frac{1}{x}$$ so that when x gets smaller, y gets bigger proportionally by $c$
$$y\propto \frac{1}{x}$$... | {
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"timestamp": "2023-03-29T00:00:00",
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How can we generalize the linear equation $Ax=b$ from finite to uncountable dimensions? Starting with the equation $Af=g$ where $f,g \in$ $\mathbb{R}^2$ and $A$ is a $2$x$2$ matrix, suppose that we generalize the two indexes of $f=$ ($f$$1$, $f$$2$) to a continuum, e.g all real numbers, so that $f$ becomes a function $... | You would need a basis. And it's really a matter of convention whether one exists in general (you need the Axiom of Choice, in the form of Zorn's lemma). At any rate, it's impossible to actually construct / write down a concrete basis for spaces like the functions $\Bbb R\to\Bbb R$. And since you can't write down a bas... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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proving that $a + b \sqrt {2} + c \sqrt{3} + d \sqrt{6} $ is a subfield of $\mathbb{R}$ The question is given below:
My questions are:
1- How can I find the general form of the multiplicative inverse of each element?
2-How can I find the multiplicative identity?
3-Is the only difference between the field and the sub... | Check the field axioms for such expressions. You really need only that the sum and product has the same form, identify the additive and multiplicative inverses; the others follow as you are operating on real numbers. They hold, and the set is clearly a (proper) subset of $\mathbb{R}$. Done.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Autonomous equilibrium points Given $$\frac{dx}{dt}= 3x-x^2$$
I don't understand how $$x=0$$ is not semistable.
I get the following 0 points: $$x = 0, x = 3$$
Here are the values of $\frac{dx}{dt}$ I get when plugging in and my reasoning:
$-3 \to -18$
$-2 \to -10$
$-1 \to -4$
It would seem to me that clearly the slope... | The zeroes--critical points--of the equation
$\dot x = 3x - x^2 \tag 1$
occur where
$\dot x = 0, \tag 2$
that is, where
$3x - x^2 = 0; \tag 3$
it is easy to see that the values of $x$ satisfying this quadratic are
$x = 0, 3; \tag 4$
the stability of these critical points is, in accord with the well-known theory, determ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Are conic sections obtained from a cone or a double cone? According to Wikipedia,
In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.
However most of the images actually show a double cone instead of a cone and it makes sense to me since a h... | Cone/Plane Intersection scenario
A full cone consists of two nappes/sheets.
Only when the inclinations of cone generator semi-vertical angle $ \alpha$ is more than sectioning plane angle $\beta$ to the cone axis we get the hyperbola as a real disjuncted double curve.
A parabola or ellipse is produced by intersection... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Stem and leaf diagrams I have the following data:
$2.6$ $ $ $3.3$ $ $ $2.4$ $ $ $1.1$ $ $ $0.8$ $ $ $3.5$ $ $ $3.9$ $ $ $1.6$ $ $ $2.8$ $ $ $2.6$ $ $ $3.4$ $ $ $4.1$ $ $ $2.0$ $ $ $1.7$ $ $ $2.9$ $ $ $1.9$ $ $ $2.9$ $ $ $2.5$ $ $ $4.5$ $ $ $5.0$
Built stem and leaf plot:
$Stem$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $Leaf$ $ ... | I created a different diagram than yours. Please see below. It is clear that the probability is as your instructor suggested by the probability definition, namely:
P(X)=Number of occurrences of Elements in Interval / Total Number of Elements.
If this is not clear let me know.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Prove that if $\mathbf A$ is an invertible matrix then $\mathbf A^{-1}$ is invertible and $\mathbf (\mathbf A^{-1})^{-1} = \mathbf A$ I am asked to prove following proposition:
If $\mathbf A$ is an invertible matrix then $\mathbf A^{-1}$ is invertible and $\mathbf (\mathbf A^{-1})^{-1} = \mathbf A$
My attempt:
Let $\... | You can use the definition of an inverse matrix to do it. We know that, if A is a matrix of order $ n $ and it's inverse, then exists B such that:
$$ AB = BA = I_n $$
And we know that $ B = A^{-1} $, just by notation.
Then it follows that $ A $ is the inverse of $ B $. Just like before, $ A = B^{-1} $, so $
A = (A^{-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3341143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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About the fact that $\mathbb Q$ has gaps In my book there are two theorems:
*
*$1.$ The number $\sqrt 2$ is irrational because if we put $\sqrt 2 = \frac pq$ for some integers $p, q$ where $p, q$ have no common factors, then we get a contradiction.
*$2.$ The set $A = \mathbb Q \cap (0, \sqrt 2)$ has no largest num... | Yeah, it would be more noteworthy for the book to say that $ \mathbb Q \cap (0, \sqrt 2]$ has no largest number and $\mathbb Q \cap [\sqrt 2, \infty)$ has no smallest number. Or, to say it without using the real numbers, that the set $D=\{x\in\mathbb Q^+\mid x^2<2\}$ has neither a largest member nor a least upper boun... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why are direct proofs often considered better than indirect proofs? As the title indicates, I'm curious why direct proofs are often more preferable than indirect proofs.
I can see the appeal of a direct proof, for it often provides more insight into why and how the relationship between the premises and conclusions wor... | Direct proof is kind of proof which don't depend on number of values your logic can take - in 2-value logic contradiction is just shortcut to take all the option at once.
Such proof will need certain modifications before using it to 2< value logic which means you need new proof for new environment.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3341323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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Why do these integration steps hold true? Can someone explain the first three steps of the solution to this integral to me? I have searched but not found a lot:
$$\begin{align}\int e^{ax}\cos(bx)dx &= \frac{1}{a}\int \cos(bx)de^{ax} \\
&= \frac{1}{a}e^{ax}\cos(bx)+\frac{b}{a}∫e^{ax}\sin(bx)dx\\
& = \frac{1}{a}e^{ax}\c... | The first step involves a shorthand way of your usual $u$-sub.
Let $u=e^{ax}$. Then, $du=ae^{ax}\ dx$ and $e^{ax}\ dx = \dfrac{1}{a} du$.
The second step involves integration by parts.
Let $v=\cos(bx)$ and $du=du$. Then, $dv=-b\sin(bx)\ dx$ and $u=u$.
\begin{align*}
& \int e^{ax} \cos(bx) dx \\
=\ & \dfrac{1}{a}\int ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Determining convergence of $\sum_{n=1}^\infty \frac{3n i^n}{(n+2i)^3}$ I've tried to solve this problem about convergence:
$\sum_{n=1}^\infty \frac{3n i^n}{(n+2i)^3}$
it's supposed to be solved using ratio, root tests or by testing the limit of the sumand. Anyways, I've tried both 3 and I had no success:
I get to a poi... | Hint:
$$\left| \frac{3n i^n}{(n+2i)^3} \right| =\frac{3n}{\sqrt{(n^2+4)^3}}<\frac{3n}{\sqrt{(n^2)^3}} = \frac{3n}{n^3}=\frac{3}{n^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3341578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How should we characterize a continuous family of continuous mappings? Let $S$ and $S'$ be two topological spaces. It is known that two continuous mappings $f,g:S\to S'$ are homotopic if there is a continous mapping
$$F:[0,1]\times S\to S'$$
with
$$F(0,\cdot)=f(\cdot)$$
$$F(1,\cdot)=g(\cdot)$$
The continuity of $F$ is ... | You can topologize $C(S, S')$ in a convenient way by taking the subsets $\mathcal{K}(C,V) = \{f \in C(S, S'): f[C] \subseteq V\} \subseteq C(S,S')$, where $C \subseteq X$ is a compact subspace and $V \subseteq Y$ is open, as a subbasis. The subsequently generated topology is the so-called compact-open topology.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3341727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find solution for $|x| + |y| \frac{dy}{dx}=0$ I want to solve following differential equation
$|x| + |y| \frac{dy}{dx}=0$
with initial condition $y(2)=-1$.
@Robert Z, since the it pass through $(2,-1)$
\begin{align}
x - y \frac{dy}{dx}=0
\end{align}
\begin{align}
x dx = y dy
\end{align}
with the initial c... | Hint. Start by solving the Cauchy problem in the quadrant which contains the initial point $(2,-1)$ where
$$x -y(x) y'(x)=0.$$
Edit. The solution that you obtained
$$y(x)= - \sqrt{x^2-3}$$
is valid for $x \in [\sqrt{3},+\infty)$ (where $x\geq 0$ and $y\leq 0$). Now extend the solution in $[0,\sqrt{3}]$ and then in $(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A particle starts its motion from rest and moves with constant acceleration for time $t_1$ and then it retards with constant rate for $t_2$. And comes to rest. Then the ratio of maximum speed and average speed during the complete motion will be
MY SOLUTION
Let acceleration be a.
Max speed
$$v=at_1$$
Also distance cover... | You have assumed that the magnitude of the acceleration and deceleration is equal, which is not given.
For the acceleration phase, $v_{max}=a_1t_1$ and distance covered is $\frac12a_1t_1^2$.
For the deceleration phase, $v^2-u^2=-a_1^2t_1^2=-2a_2s$ giving the distance covered as $\frac{a_1^2t_1^2}{2a_2}$. You also have... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is $T$ (defined below) a distribution?
How to show that
$$
\langle T, \varphi\rangle = \int\limits_{0}^{+\infty} \frac{\varphi(x)-\varphi(0)}{x^{3/2}} dx,\quad \varphi \in C_0^\infty(\mathbb{R}),
$$
is a distribution?
I would first rewrite the definition of $T$, using $ \varphi(x)-\varphi(0) = \int_0^x \varph... | Two inequalities:
*
*$|\varphi (x)- \varphi (0)|\le \|\varphi'\|_\infty |x|.$
*$|\varphi (x)- \varphi (0)|\le 2\|\varphi|_\infty.$
Thus
$$\int_0^\infty\left |\frac{\varphi (x)- \varphi (0)}{x^{3/2}}\right |\,dx$$ $$ \le \left (\int_0^1x^{-1/2}\,dx \right)\|\varphi'\|_\infty + \left (\int_1^\infty x^{-3/2}\,dx\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3342062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that $\mathbb R^n$ is not the union of finitely many its proper subspaces. I am reading An Introduction to Algebraic Topology by Rotman. After proving
Theorem 2.7: For every $k\geq 0$, euclidean space$\mathbb R^n$ contains $k$ points in general position,
the book remarked: There are other proofs of this theorem ... | Every affine proper subspace of $\mathbb{R}^n$ i contained in an affine hyperplane.
Now, let $C=\{(t,t^2,t^3,\ldots,t^n),\,t \in \mathbb{R}^n\}$.
Since every nonzero polynomial with degree at most $n$ has at most $n$ roots, an affine hyperplane of $\mathbb{R}^n$ contains at most $n$ points in $C$.
So if you have a c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3342105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Stationarity of AR(2) process I am new to time series modeling and currently struggling with stationarity.
Can someone please explain why the roots of the following AR polynomial are $ - 1 $ and $1/2$?
The AR(2) process is $X_t = X_{t-1} + 2 X_{t-2} + Z_t$
To my best knowledge so far, I could use backward shift opera... | The shift operator $1-B-2B^2$ factorizes as such:
$$
1-B-2B^2=(1+B)(1-2B).
$$
Note that in your question you had a sign error when writing down the shift operator - the $2X_{t-2}$ moves from the right to the left side of the equation and thus becomes a $-2B^2$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Linear Algebra intuition behind subspaces Hello I am trying to understand better the intuition behind subspaces. I am aware that the subspace of a vector space must have zero(go through the origin), has to be closed by addition, and has to be closed by multiplication. The aspect I fail to understand is why must it equa... | A subspace has to be a vector space and therefore must be closed under scalar multiplication. There is always a $0$ value in the field of scalars, so $0\cdot\vec v$, which is the origin $\vec0$, has to be in the subspace.
You might be familiar with thinking of Euclidean vector spaces like $\mathbb R^3$, where the nontr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3342394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Given the function $f(x) = e^x - ax$ solve for $a$ such that $f(x) \geq 1$. So I have the function $f: \mathbb{R} \to \mathbb{R}$, $f(x) = e^x - ax$ and it is known that $a > 0$. I need to find $a$ such that $f(x) \geq 1$, $\forall x\in \mathbb{R}$.
What I have done so far is to set the derivative equal to $0$ in hope... | No matter what $a$ is, you'll have $f(0)=e^0-a\cdot 0=1$.
So your only hope of getting $f(x)\ge 1$ everywhere is if you have $f'(0)=0$. It turns out that there is exactly one $a$ that achieves this, namely $a=1$.
Since $e^x-ax$ is easily seen to be convex no matter what $a$ is, $f'(0)=0$ will also guarantee that it has... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3342528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
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What method to use for differential equations I'm having doubts as to what method to use in the following ODE:
$$2t+3x+(x+2)x'=0$$
As this can be changed into:
$$x'=\frac{-2t-3x}{x+2}$$
I'm thinking it can be solved by using homogeneous equations method, but I'm not sure this applies because for the $x+2$ in the deno... | Compressing the d'Alembert treatment: Insert $p=x'$, then
$$
2t+3x+(x+2)p=0.
$$
If $p$ is constant, then the $t$ derivative of this equation gives
$$
2+3p+p^2=0\implies p=-1\text{ or } p=-2.
$$
In the other cases, locally use $p$ as parameter and use $T(p)$, $X(p)$ as the dependent functions. Then from the chain rul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3342652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the pointwise limit and determine if convergence is uniform
Find the pointwise limit of $f_n(x)=nx^n(1-x^n)$ and determine if the convergence is uniform.
Solution
For $x=1$
$$\lim_{n\to\infty}(nx^n(1-x^n))\to 0$$
For $0\le x<1$
$$\lim_{n\to\infty}(nx^n(1-x^n))\to0$$
as $x^n\to0$ as $n\to\infty$.
Thus the pointwi... | It is always interesting to plot graphical representations concretizing algebraic/analytic proofs. Here it is for the 8 first curves with coordinates of maximum (red stars)
$$(\sqrt[n]{\frac12}, \ \frac{n}{4}).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3342771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
divisibility of big powers of 71 Since
$$ 71 \equiv7\equiv-1 \pmod 8 $$
$\implies71^2 \equiv 1 \pmod 8$
$\implies $ any power of $71$ would leave either $1$ or $-1$ mod $8$.
Is this logic ok ?
| You are absolutelly right. It's good to understand and always keep in mind the basic rules of modular arythmetic, so that you never have doubts about your reasonings. For instance, if:
$$a \equiv b \pmod k$$
$$c \equiv d \pmod k$$
Then
$$ac \equiv bd \pmod k$$
And, as a consequence (set $a=c$, $b=d$ and apply the abov... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3342920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Inverse of nth power of a linear transformation Using matrices it is easy to show that doing a linear transformation n times and then taking inverse is same as inverting the linear transformation and then doing it n times:
$$(A^n)^{-1}=(AAAA\ldots)^{-1} = \ldots A^{-1}A^{-1}A^{-1}A^{-1}=(A^{-1})^n~~\blacksquare$$
I'm w... | You can prove it by induction: for $n=1$ it’s banal. Suppose thesis true for a certain $n$. Then $(f^{n+1})^{-1}=(f\circ f^n)^{-1}=(f^n)^{-1}\circ f^{-1}=(f^{-1})^n\circ f^{-1}=(f^{-1})^{n+1}$
Did you asked for a similar demonstration?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
$P$ and $A$ are square matrices and $P$ is invertible. Prove that $(P^{-1} AP)^{n} = P^{-1}A^{n}P $ The proposition I would like to prove is following:
Proposition. Let $\mathbf{P}$ and $\mathbf{A}$ be square matrices and a matrix $\mathbf{P}$ be invertible. Prove that $({\mathbf{P}^{-1}} \mathbf {AP})^{n} = \mathbf{P... | Your proof looks good.
When you're starting your journey towards becoming a mathematician, it can seem natural to give an extra example like your $n=2$ case to provide extra illustration to the reader. But if you have confidence in your ability to write clear proofs, you'll come to see it as dead weight that is just m... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that $A^5 \neq I$ $A \in M_{5}(\mathbb{C})$, $\operatorname{trace}(A) = 0$, $I-A$ is invertible. Prove that $A^5 \neq I$
I think this problem has something to do with eigenvalue and the fact that trace of a matrix = sum of the eigenvalues. But, I have no idea how to proceed!
Thanks!
| Assume, by the way of contradiction, that $A^5 = I$ holds. In this case, as $A^5 - I = 0$, we can factorize this expression in $(A - I)(A^4 + A^3 + A^2 + A + I) =0 $.
As $A - I $ is invertible, the first factor is nonzero, hence the last one is and the minimal polynomial of $A$ divides it, forcing $A$ to have all eig... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Induction for $(A_1 \cap A_2 \cap ... \cap A_n)^c = A^c_1 \cup A^c_2 \cup ... \cup A^c_n$. I need to do induction on this problem:
$(A_1 \cap A_2 \cap ... \cap A_n)^c = A^c_1 \cup A^c_2 \cup ... \cup A^c_n$.
Induction is new to me and this problem is hard to understand. The base case here I'm guessing is just that $A^c... | The base case is $(A_1\cap A_2)^c = A_1^c\cup A_2^c$. To see this, we have
\begin{align}
x \in (A_1\cap A_2)^c &\iff x\notin A_1\cap A_2\\
&\iff x\notin A_1 \vee x\notin A_2\\
&\iff x\in A_1^c \vee x\in A_2^c\\
&\iff x\in A_1^c\cup A_2^c,
\end{align}
where $\vee$ denotes logical disjunction (OR).
For the induction ste... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3343413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Let $G$ be an open set in $\mathbb{C}$ and $a \in G$ with $B_X(a,r)\cap \delta G = \emptyset$. Then $B_X(a,r) \subseteq G$. I came up with the following and wanted to be sure if my proof is correct:
Theorem: Let $G$ be an open set in $\mathbb{C}$ and $a \in G$ with $B_X(a,r)\cap \delta G = \emptyset$. ($\delta G$ is th... | If $B_X(a,r) \cap \partial G= \emptyset$, $B_X(a,r)$ cannot intersect $G^\complement$ or the ball would be disconnected (by the disjoint cover $G$ and $\Bbb C\setminus \overline{G}$, intersected by $B_X(a,r)$.
This argument works for any connected neighbourhood of $a$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3343514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Trouble proving $3^2 + 3^3 + ... 3^n = 9 \cdot \frac{3^{n-1} - 1}2$ by induction So i'm supposed to prove by mathematical induction that this formula: $3^2 + 3^3 + ... 3^n = 9 \cdot \dfrac{3^{n-1} - 1}2$ holds true for all numbers greater than 2.
I started with the base case and just plugged in 2, it worked.
Then I ass... | Suppose $t=3^{n-1}$
Then:
$$\frac 92 (t-1)+9t=\frac 92t+9t-\frac92=\frac{27}{2}t-\frac92$$
$$=\frac92(3t-1)$$
Then sub back in, we get: $\frac 92(3^n-1)$ as required.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3343635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A matrix of order 8 over $\mathbb{F}_3$ What is an example of an invertible matrix of size 2x2 with coefficients in $\mathbb{F}_3$ that has exact order 8?
I have found by computation that the condition that the 8th power of a matrix
$\begin{bmatrix}a & b\\c & d\end{bmatrix}$ is the identity is
$$
b c (a + d)^2 (a^2 +... | We can use the same approach but reduce drastically the complexity of the system in the entries $a, b, c, d$ if we instead look for a square root of some matrix with order $4$.
The matrix $A = \pmatrix{0&-1\\1&0}$ satisfies $A^2 = -I$ and so has order $4$ over any field of characteristic not $2$. In particular, if we c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3343777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 1
} |
X be a set, $\sum$ be any $\sigma$- algebra on X, is there a measure on X such that every element of $\sigma$- algebra is measurable? In particular, on $\mathbb{R}$, $2^{\mathbb{R}}$ be a $\sigma$- algebra, is there a measure on $\mathbb{R}$ such that every element of $2^{\mathbb{R}}$ is measurable?
| Pick any $x \in X$ and define $\mu(A)=1$ if $x \in A$, $\mu(A)=0$ for $x \notin A$. This is a measure and every set is measurable. This works for any set and any sigma algebra on it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3343874",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
An upper bound on the expected value of the square of random variable dominated by a geometric random variable Let $X$ and $Y$ be two random variables such that:
*
*$0 \leq X \leq Y$.
*$Y$ is a geometric random variable with the success probability $p$ (the expected value of $Y$ is $1/p$).
I would be grateful for... | If $0 \leq X \leq Y$ almost surely, then $0 \leq X^2 \leq Y^2$ almost surely, as $x \mapsto x^2$ is monotonous on $\mathbb{R}_+$. We also know, that if $X \leq Y$ almost surely, then $EX \leq EY$. Thus $0 \leq E(X^2) \leq E(Y^2) = (EY)^2 + Var(Y) = \frac{1}{p} + \frac{1 - p}{p^2} = \frac{1}{p^2}$. Thus $E(X^2) \in [0; ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3344006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Let d1 and d2 be two metrices on X . Then $d(x,y)= d1(x,y)*d2(x,y)$ $x, y\in X$ is also metric on X? I'm able to solve three properties of metric such as
*
*$d(x,y)\geq 0$ for all $x, y \in X $
*$d(x,y)=0$ iff $x=y$
*$d(x,y)= d(y,x)$ for all $x, y \in X $
But facing problem to solve triangle inequality.
Please he... | $|0-\frac12|^2 + |\frac12 - 1|^2 = \frac12 < 1 = |0-1|^2$ so $d_1 = d_2$ equal to the standard distance on $\Bbb R$ already gives a counterexample to the triangle inequality.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3344124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find n in sum that results in a number $aaa$ Lets say that we have the sum $1+2+3+\ldots+n$ where $n$ is a positive natural number and that this sum should equal a three digit number in which all the digits are the same, for example $111, 222,$ and so on.
What would be the best way to find the $n$ that would result in ... | $$\begin{align}
\frac {n(n+1)}2&=111m\qquad (m=1,2,3,\cdots,9)\\
n^2+n-222m&=0\\
n&=\frac {-1\pm \sqrt{1+888m}}2\end{align}$$
Check for values of $m$ where $(1+888m)$ is a perfect square of an odd number.
The only solution is $m=6$, where $\sqrt{1+888m}=73$. This gives
$$n=\frac {-1\pm 73}2=36\qquad (n>0)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3344200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How can I say a function $F:\mathbb{R}^n \longrightarrow \mathbb{R}^m$ $nMy doubts are about an application of the Implicit Function Theorem: I do not understand how a function $F: \mathbb{R}^n \longrightarrow \mathbb{R}^m$, with $n<m$ (!!!) can locally be an homeomorphism between a neighbourhood of $x$ and a neighbour... | The proposed solution is nonsense, you should complain to the people who wrote it. (In the 3rd edition of ``Berkeley Problems in Mathematics" that I have, this is a "solution" of Problem 2.2.10.) A correct solution is to consider the gradient function $G(x)=\nabla F(x), x\in {\mathbb R}^n$. This function is $C^1$ and h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3344318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Combinatorics problems that can be solved more easily using probability I'm looking for examples of combinatorics problems which would be very difficult to solve by direct enumeration, but can be easily solved using ideas from probability, like independence, commuting sums and expectations, etc. I know I have seen suc... | Here are some problems:
*
*Find the sum of the number of all continuous runs of all possible sequences with $2019$ ones and $2019$ zeros
*https://artofproblemsolving.com/community/c6h366278p2018435
*https://artofproblemsolving.com/community/c6h60752p366512
*https://artofproblemsolving.com/community/q2h1151650p545... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3344468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Why is this proof invalid? I don't understand why this theorem is false.
Suppose that $A \subseteq C$, $B \subseteq C$, and $x \in A$. Then $x \in B$.
Invalid Proof:
Suppose that $x \notin B$. Since $x \in A$ and $A \subseteq C$, $x \in C$. Since $x \notin B$ and $B \subset C$, $x \notin C$. But now we have proven b... |
Since $x \not\in B$ and $B \subseteq C$, $x \not\in C$.
Actually, if $x \not\in B$ and $B \subseteq \color{blue}{C}$, both $\color{red}{x \in C}$ and $\color{magenta}{x \not\in C}$ are possible. In a picture:
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3344671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
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