instruction stringlengths 12 30k |
|---|
I have to find the splitting field of the following in $\mathbb{Q}$:
a) $f(x) = x^6 + 1 $
b) $f(x) = (x^2-3)(x^3+1) $
For a), finding the 6th roots of -1, I concluded that $\pm i, \frac{\pm \sqrt{3} \pm i}{2}$ should be in the splitting field. The minimum adjoined splitting field is thus $\mathbb{Q}[\sqrt{3},i... |
Problem :
Let :
$$\frac{\Gamma(x+1)}{(x+x^2)}=f(x),x>0$$
Then let $A=\sqrt{\frac{\pi}{4}}-1$ And $y$ be the global minimum over $x\in (0,\infty)$ of $f(x)$ then it seems we have :
$$2+A-A^4-30A^6-103A^8-76A^{10}-23A^{12}-22A^{14}-3A^{16}-\cdots=y$$
I cannot pursue it because of the floating point in comp... |
Your sum has a slight mistake. It should be
$$w_n = \sum_{i=0}^n \sum_{j=0}^i\sum_{k=0}^j \binom{n}{i}\binom{i}{j}\binom{j}{k}$$
Can you see why?
----------
Solving:
$$\begin{align}
w_n &= \sum_{i=0}^n \sum_{j=0}^i\sum_{k=0}^j \binom{n}{i}\binom{i}{j}\binom{j}{k} \\
&= \sum_{i=0}^n \sum_{j=0}^i \binom{n}{... |
To prove this i was thinking to start by supposing that $f(x_1)=f(x_2)$ this implies that $(sin(x_1),cos(x_1),x_1^2 ) = (sin(x_2),cos(x_2),x_2^2)$.This gives us that,$ sin(x_1)=sin(x_2) \Rightleftarrow x_1=x_2 .$
$ cos(x_1)=cos(x_2) \Rightleftarrow x_1=x_2 $ and
$ x_1^2=x_2^2 \Rightleftarrow x_1=x_2.$
Is this corre... |
In Boyd and Vandenberghe (https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf), section 4.1.3, they say:
> We call two problems equivalent if from a solution of one, a solution
> of the other is readily found, and vice versa. (It is possible, but
> complicated, to give a formal definition of equivalence.)
I'm... |
I would like to prove that $X_n \rightarrow X$ almost surely iff $\forall \varepsilon >0 \ \lim_{N \to \infty} P \left( \bigcap_{n=N}^{\infty} \{ \omega: |X_n(\omega)-X(\omega)| \le \varepsilon\} \right)=1$.
I know how to prove that but one technical thing I cannot understand.
$X_n \rightarrow X a.s. \iff P (\{ \... |
I would like to prove that $X_n \rightarrow X$ almost surely iff $\forall \varepsilon >0 \ \lim_{N \to \infty} P \left( \bigcap_{n=N}^{\infty} \{ \omega: |X_n(\omega)-X(\omega)| \le \varepsilon\} \right)=1$.
I know how to prove that but one technical thing I cannot understand.
$X_n \rightarrow X a.s. \iff P (\{ \... |
Here's a question I'm sturglling with:
> Show that for an increasing sequence of events $$A_1\subset A_2\subset
A_3\subset ...$$ the next equation holds
> $$P\big(\bigcup_nA_n\big)=\lim_{n\rightarrow\infty}P(A_n)$$
Here's what I have so far: let $(B_n)_{n=1}^{\infty}$ be a sequence of events such that
$$B_{1}... |
I looked at the proof by induction of the multinomial theorem on [Wikipedia](https://en.wikipedia.org/wiki/Multinomial_theorem#Proof "Wikipedia proof by induction of the multinomial theorem") and do not understand how to get the last step. Specifically, I do not know why this equality is true:
$$\sum_{k_1 + k_2 + \cd... |
I am self-studying some basic linear algebra. I encountered the following theorem:
> **Theorem**$\quad$ Let $A$ be an $n \times n$ matrix.
>
> 1. $A$ is diagonalizable if and only if it has eigenvectors $\mathbf{x_1},\mathbf{x_2},\dots,\mathbf{x_n}$ such that the matrix $P=\begin{pmatrix}\mathbf{x_1} & \mathbf{x_2} ... |
When we study quantum mechanics, teacher often says that $H = -\Delta$ is a Hermitian operator (more precisely, a self-adjoint operator), and then reference linear algebra to note that its eigenvectors can form a complete basis.
The function $e^{i \mathbf{p}\cdot \mathbf{x}}$ is such an eigenvector. This aligns with... |
Why harmonic functions does not belong to basis while it's eigenvector of Laplacian operator? |
To prove this i was thinking to start by supposing that $f(x_1)=f(x_2)$ this implies that $(\sin(x_1),\cos(x_1),x_1^2 ) = (\sin(x_2),\cos(x_2),x_2^2)$.This gives us that, $ \sin(x_1)=\sin(x_2) \longleftrightarrow x_1=x_2.$
$ \cos(x_1)=\cos(x_2) \longleftrightarrow x_1=x_2 $ and
$ x_1^2=x_2^2 \longleftrightarrow x_1=... |
Let $f,g:\mathbb R^n\to\mathbb C$ be two Lebesgue measurable functions. Thus $(x,y)\mapsto f(x-y)g(y)$ is also a mesaurable function. To be rigorous, we define let
$$f\ast g(x)=\int_{\mathbb R^n}f(x-y)g(y)dy.$$
We say *the convolution is defined*, if for almost every $x\in\mathbb R^n$, $f(x-\cdot)g$ is a Lebesgue i... |
I am self-studying some basic linear algebra. I encountered the following theorem:
> **Theorem**$\quad$ Let $A$ be an $n \times n$ matrix.
>
> 1. $A$ is diagonalizable if and only if it has eigenvectors $\mathbf{x_1},\mathbf{x_2},\dots,\mathbf{x_n}$ such that the matrix $P=\begin{pmatrix}\mathbf{x_1} & \mathbf{x_2} ... |
To prove this, I was thinking to start by supposing that $f(x_1)=f(x_2)$; this implies that $(\sin(x_1),\cos(x_1),x_1^2 ) = (\sin(x_2),\cos(x_2),x_2^2)$. This gives us $\sin(x_1)=\sin(x_2) \Longleftrightarrow x_1=x_2$,
$ \cos(x_1)=\cos(x_2) \Longleftrightarrow x_1=x_2 $ and
$ x_1^2=x_2^2 \Longleftrightarrow x_1=x_2$... |
How to how that the function: $ f: \mathbb{I} \to \mathbb{R}^3, x \mapsto (\sin(x),\cos(x),x^2 )$ is injective? |
I am going through a maths book and unable to understand the following:
> Given that $x$ and $y$ are rational numbers, then if the equation
$$2\sqrt{3}-3=(x+y)\sqrt{3}-2\sqrt{3xy}$$ is true, then we must have $x+y=2$ and $2\sqrt{3xy}=3$.
I don't understand this. We cannot reach such a conclusion with equations ... |
Why must rational $x$ and $y$ satisfy $x+y=2$ and $2\sqrt{3xy}=3$ in order for $2\sqrt{3}-3=(x+y)\sqrt{3}-2\sqrt{3xy}$ to be true? |
this question is about a proof that a particular group representation of $\mathcal{S}_{3}$ on vector space $\mathbb{C}^{3}$ is indeed correct.
We define the following arrow :
$$
\theta_{4} : \left \{ \begin{cases} \mathcal{S}_{3} & \rightarrow GL(\mathcal{C}^{3}) \\
\sigma & \mapsto g_{\sigma} \end{cases}
$$... |
I'm reading through Stein and Shakarchi's Complex Analysis textbook, but I'm a bit confused by their proof that
$$ \int_{0}^{\infty} \frac{1-\cos(x)}{x^2} dx = \frac{\pi}{2}$$
They consider the function $f(z) = \frac{1-e^{iz}}{z^2}$ and an indented semicircle as their contour[![enter image description here][1]][1]... |
this question is about a proof that a particular group representation of $\mathcal{S}_{3}$ on vector space $\mathbb{C}^{3}$ is indeed correct.
We define the following arrow :
$$
\theta_{4} : \left{ \begin{cases} \mathcal{S}_{3} & \rightarrow GL(\mathbb{C}^{3}) \\
\sigma & \mapsto g_{\sigma} \end{cases} \right.... |
How to show that the function: $ f: \mathbb{I} \to \mathbb{R}^3, x \mapsto (\sin(x),\cos(x),x^2 )$ is injective? |
How to show that the function: $ f: \mathbb{I} \to \mathbb{R}^3, x \mapsto (\sin(x),\cos(x),x^2 )$ is injective?
---
To prove this, I was thinking to start by supposing that $f(x_1)=f(x_2)$; this implies that $(\sin(x_1),\cos(x_1),x_1^2 ) = (\sin(x_2),\cos(x_2),x_2^2)$. This gives us $\sin(x_1)=\sin(x_2) \Longlef... |
How to show that the function: $ f: \mathbb{I} \to \mathbb{R}^3, x \mapsto (\sin(x),\cos(x),x^2 )$ is injective? |
$f:\varepsilon \rightarrow R$, where $\varepsilon$ is a linear space.
$Df(x): \varepsilon \rightarrow R$ is the differential of $f$ at $x$, that is, it is the linear map defined by:
$$Df(x)[v]=\lim\limits_{t\rightarrow 0} \frac{f(x+tv)-f(x)}{t}.$$
How to prove $Df(x)$ is a linear map using the definition above? I ... |
How to understand the differential is a linear map? |
this question is about a proof that a particular group representation of $\mathcal{S}_{3}$ on vector space $\mathbb{C}^{3}$ is indeed correct.
We define the following arrow :
$\theta_{4} \left.: \begin{cases} \mathcal{S}_{3} & \rightarrow GL(\mathbb{C}^{3}) \\
\sigma & \mapsto g_{\sigma} \end{cases} \right.
... |
this question is about a proof that a particular group representation of $\mathcal{S}_{3}$ on vector space $\mathbb{C}^{3}$ is indeed correct.
We define the following arrow :
$\theta_{4} \left.: \begin{cases} \mathcal{S}_{3} & \rightarrow GL(\mathbb{C}^{3}) \\
\sigma & \mapsto g_{\sigma} \end{cases} \right.
... |
This is a nice discovery! But why does it work?
Let $n^2$ and $(n+1)^2$ be two consecutive squares.
If you factor $(n+1)^2$, you will get $n^2+2n+1$.
$(n^2+2n+1)-(n^2)=2n+1$.
This means the difference between consecutive squares is always the odd numbers.
Now, we can say that $((n+2)^2-(n+1)^2)-((n+1)^2-n^2)=(2(n+... |
this question is about a proof that a particular group representation of $\mathcal{S}_{3}$ on vector space $\mathbb{C}^{3}$ is indeed correct.
We define the following arrow :
$\theta_{4} \left.: \begin{cases} \mathcal{S}_{3} & \rightarrow GL(\mathbb{C}^{3}) \\
\sigma & \mapsto g_{\sigma} \end{cases} \right.
... |
this question is about a proof that a particular group representation of $\mathcal{S}_{3}$ on vector space $\mathbb{C}^{3}$ is indeed correct.
We define the following arrow :
$\theta_{4} \left.: \begin{cases} \mathcal{S}_{3} & \rightarrow GL(\mathbb{C}^{3}) \\
\sigma & \mapsto g_{\sigma} \end{cases} \right.
... |
Hmm... it's not as obvious as I thought
See, my first thought was this as a case of if $W$ is irrational and $a,b,c,d$ are rational and $a+bW = c+dW$ then $b=d$ and $a=c$.
Why? Because $a-c = (d-b)W$. The left hand side is rational so the right hand side must be rational. But $W$ is irrational and the *only* wa... |
We want to know if a smooth surface under a $C^{\infty}$ diffeomorphism (infinitely differentiable with a infinitely differentiable inverse) map to a smooth surface too?
Further, it would be nice to understand what happens if the diffeomorphism was $C^k$ for some $k\geq 1$.
So far I have looked at these books:
... |
I was trying to solve this question:
Sole in R the Equation
$7^x-5^x=[x^2]+1$
Where [t] is the floor function
It is known that $[x^2]\leq x^2 $
so
$7^x-5^x<x^2+1 $
but i would prove
$7^x-5^x-x^2>1$
for $x>1 $
But it is hard with derivatives
Other aproaches? |
How to show that the function: $ f: (0,1) \to \mathbb{R}^3, x \mapsto (\sin(x),\cos(x),x^2 )$ is injective?
---
To prove this, I was thinking to start by supposing that $f(x_1)=f(x_2)$; this implies that $(\sin(x_1),\cos(x_1),x_1^2 ) = (\sin(x_2),\cos(x_2),x_2^2)$. This gives us $\sin(x_1)=\sin(x_2) \Longleftrigh... |
How to show that the function: $ f: (0,1) \to \mathbb{R}^3, x \mapsto (\sin(x),\cos(x),x^2 )$ is injective? |
I was trying to prove $L^\infty$ is complete with $(X,\mathcal{A},\mu)$ given and $\mu$ is $\sigma$-finite.
Outline of my attempted proof:
- (i) Assume Cauchy in norm - let $\{f_n\}$ be a Cauchy sequence which converges in $L^\infty$ norm. That is, let $\epsilon>0$, and assume $\left\Vert f_n-f_m \right\Vert_\inf... |
When we study quantum mechanics, teacher often says that $H = -\Delta$ is a Hermitian operator (more precisely, a self-adjoint operator), and then reference linear algebra to note that its eigenvectors can form a complete basis.
The function $e^{i \mathbf{p}\cdot \mathbf{x}}$ is such an eigenvector. This aligns with... |
A Stochastic Block Model (SBM) is a random graph model with a block structure. Given parameters n ∈ Z+, k ∈ Z+, α ∈ [0, 1], and β ∈ [0, 1], the model
works as follows. Create k groups of vertices with n members each. Connect
pairs of vertices within the same group uniformly at random with probability α
and pairs a... |
Am came across this problem on varsity tutors. A part of the answer walk through states that sin^2 can equal cos^2 - 1.
This is stated more than once on the site. I do not see how this is possible.
I can see that sine^2 = 1 - cos^2
cos^2 = 1 - sin^2
if cos^2-1 = were to be the equation. then it ... |
I've been dabbling with matrix computations in finite fields, and I've stumbled upon a pattern that I can't understand. Perhaps someone here could shed some light on it?
So, here's what's happening: In my application, after performing certain computations, I end up with a vector (like this one from $\mathbb{Z}_5$: $... |
\begin{align*}
\begin{array}{c|cc}
0 &0 &0 \\
1 & \frac{1}{2} & \frac{1}{2} \\
\hline
& \frac{1}{2} & \frac{1}{2}
\end{array}
\end{align*}
How is the trapezoidal method derived from the implicit RK tableau above?
For implicit RK methods, I have that:
\begi... |
For my question (2), instead of using Munroe's approach to establish the limit, we can use problem 19.7 in Aliprantis and Burkinshaw 2nd edition, or Theorem 2.30 in Folland:
Assume that $\{f_n\}$ is Cauchy in measure, if and only if there exists a measurable function $f$ such that $\lim_{n}f_n=f$ in measure, i.e. for ... |
I came across linear programming while trying to find a solution to a problem. I didn't use LP before so pardon if it is naive question. I hope this forum can help.
There are n tasks (x1, x2 .., xn) with current assigned values (r1, r2 .. rn) and new target values (t1, t2 ... tn). Each of the tasks has certain know... |
For a smooth function $f:\mathcal{E} \rightarrow R$, where $\mathcal{E}$ is a linear space.
$Df(x): \mathcal{E}\rightarrow R$ is the differential of $f$ at $x$, that is, it is the linear map defined by:
$$Df(x)[v]=\lim\limits_{t\rightarrow 0} \frac{f(x+tv)-f(x)}{t}.$$
How to prove $Df(x)$ is a linear map using the... |
For my question (2), instead of using Munroe's approach to establish the limit, we can use problem 19.7 in Aliprantis and Burkinshaw 2nd edition, or Theorem 2.30 in Folland:
Assume that $\{f_n\}$ is Cauchy in measure, if and only if there exists a measurable function $f$ such that $\lim_{n}f_n=f$ in measure, i.e. for ... |
This is exercise $1.2$ in Nicholas Loehr's book "Combinatorics".
**Which is larger: the number of binary strings of length $56$, or the number of permutations of the English alphabet ($26$ letters)?**
Letting $D$ and $P$ denote these sets, respectively, it is obvious that $|D|=2^{56}$ and $|P|=26!$. I checked us... |
Am came across this problem on varsity tutors. A part of the answer walk through states that $sin^2(x)$ can equal $cos^2(x) - 1.$
This is stated more than once on the site. I do not see how this is possible.
I can see that $$sin^2(x) = 1 - cos^2(x) $$
$$ cos^2(x) = 1 - sin^2(x)$$
if $cos^2(x)-1$ ... |
Am came across this problem on varsity tutors. A part of the answer walk through states that $\sin^2(x)$ can equal $\cos^2(x) - 1%.
This is stated more than once on the site. I do not see how this is possible.
I can see that $$\sin^2(x) = 1 - \cos^2(x) $$
$$\cos^2(x) = 1 - \sin^2(x)$$
if $\cos^2(... |
This is exercise $1.2$ in Nicholas Loehr's book "Combinatorics".
**Which is larger: the number of binary strings of length $56$, or the number of permutations of the English alphabet ($26$ letters)?**
Letting $D$ and $P$ denote these sets, respectively, it is obvious that $|D|=2^{56}$ and $|P|=26!$. I checked us... |
I came across linear programming while trying to find a solution to a problem. I didn't use LP before so pardon if it is naive question. I hope this forum can help.
There are n tasks (x1, x2 .., xn) with current assigned values (r1, r2 .. rn) and new target values (t1, t2 ... tn). Each of the tasks has certain know... |
Thank you for reading my questions.
It is known that the stationary phase method has this form:
$$
\int_a^bg(t)e^{jf(t)dt}\approx\sum\limits_{t_0\in\Sigma}g(t_0)e^{jf(t_0)+j*\text{sign}(f''(t_0))\frac{\pi}{4}}\frac{\sqrt{2\pi}}{\sqrt{|f''(t_0)|}} \tag{1}
$$
You can check the wiki for this:
https://en.wikipedia.... |
How to use stationary phase method when the zero point is at infinity? |
Let $f,g:\mathbb R^n\to\mathbb C$ be two Lebesgue measurable functions. Thus $(x,y)\mapsto f(x-y)g(y)$ is also a mesaurable function. To be rigorous, we define let
$$f\ast g(x)=\int_{\mathbb R^n}f(x-y)g(y)dy.$$
We say *the convolution is defined*, if for almost every $x\in\mathbb R^n$, $f(x-\cdot)g$ is a Lebesgue i... |
I was trying to solve this question:
Solve in $\mathbb{R}^+$ the Equation
$7^x-5^x=\lfloor x^2 \rfloor+1$
Where $\lfloor t\rfloor$ is the floor function
It is known that $\lfloor x^2\rfloor\leq x^2 $
so
$$7^x-5^x<x^2+1 $$
but i would prove
$$7^x-5^x-x^2>1$$
for $x>1 $
But it is hard with... |
I came across this problem on varsity tutors. A part of the answer walk through states that $\sin^2(x)$ can equal $\cos^2(x) - 1$.
This is stated more than once on the site. I do not see how this is possible.
I can see that $$\sin^2(x) = 1 - \cos^2(x) $$
$$\cos^2(x) = 1 - \sin^2(x)$$
if $\cos^2(x... |
I am writing a boid simulation in a toroidal space. The boids wrap around from the left to right edge and the top to bottom edge. To calculate what boids are visible, I need to find the minimum distance between those two boids' positions. I have been doing the following
1. $\Delta x = \lvert(x_2 - x_1)\rvert,\Delta... |
I came across this problem on varsity tutors. A part of the answer walk through states that $\sin^2(x)$ can equal $\cos^2(x) - 1$.
This is stated more than once on the site. I do not see how this is possible.
I can see that
\begin{align}
\sin^2(x) &= 1 - \cos^2(x) \\
\cos^2(x) &= 1 - \sin^2(x)
\end{align}
if... |
Let $x_1\leq x_2\leq ... \leq x_n$ and $y_1<y_2< ... <y_n$ be positive integers such that $x_1\geq 2$, $x_i < y_i$ and $y_i + 1<y_{i+1}$. Do we have that $$x_ny_n (\sum_{i=1}^n x_i)^2 \leq (\sum_{i=1}^n x_iy_i)^2$$
The proposed inequality has a certain "Cauchy-Schwarz" flavour, but from the lower bound side of the s... |
Are you solving for a falling object ?
Its easier to multiply both sides by $e^{\frac kmt}$ so
$$y''+y'{\frac km}=-g\hat{j}\\ \times e^{\frac kmt} \\
e^{\frac kmt}y''+e^{\frac kmt}y'{\frac km}=-ge^{\frac kmt}\hat{j}\\ (e^{\frac kmt}y')'=-ge^{\frac kmt}\hat{j}$$ no apply integral both sides
$$\int =\int \\e^{\f... |
I have to evaluate the following quantity
**$P\left(C|AB\right)$**
Where A, B, and C are events.
Since I know the following conditional rule
**$P\left(A|B\right)=\frac{P\left(AB\right)}{P\left(B\right)}$**
I thought I could write
**$P\left(C|AB\right)=\frac{P\left(CAB\right)}{P\left(AB\right)}$**
... |
For a Brownian motion $B_t$ there are many results about its $t \rightarrow + \infty$ or $t \rightarrow 0+$ behavior, such as
$$\limsup_{t\rightarrow \infty} \frac{B_t}{(2t\log\log t)^{1/2}} = 1 \quad a.s.\\
\liminf_{t\rightarrow \infty} \frac{B_t}{(2t\log\log t)^{1/2}} = -1 \quad a.s.$$
I interpret the above res... |
I've stumbled across this integral: $$\int_{0}^{\pi/2} \left(\frac{\sin5x}{\sin x}\right)^2 \,dx $$
I was on a time limit and my intuition told me: $$\int_{0}^{\pi/2} \left(\frac{\sin5x}{\sin x}\right)^2 \,dx = \int_{0}^{\pi/2}5dx= 5x\Big|_0^{\pi/2}=\frac{5\pi}{2}$$
which turned out to be the right answer.
I... |
In a casino a slot machine displays pairs of symbols which are either a **club** or a **spade**.
A player may obtain the following options: double club, double spade, club-spade, or spade-club.
Each entry (club or spade) of the pair is obtained independently.
The club appears with probability $\theta\in(0,1)$ and ... |
Are variables of this casino game dependent? |
First, we observe that among first countable spaces, every compact is sequentially compact: https://topology.pi-base.org/spaces/?q=First+Countable%2BCompact%2B%7ESequentially+Compact
But the converse fails: https://topology.pi-base.org/spaces/?q=First+Countable%2B%7ECompact%2BSequentially+Compact The particular coun... |
Can you give me a hint for the following? |
In a casino a slot machine displays pairs of symbols which are either a **club** or a **spade**.
A player may obtain the following options: double club, double spade, club-spade, or spade-club.
Each entry (club or spade) of the pair is obtained independently.
The club appears with probability $\theta\in(0,1)$ and ... |
>Let $b_n$ be a sequence of positive real numbers such that the limit from the Raabe-Duhamel's test is positive, i.e. $$\lim_{n \to \infty} n\left( \frac{b_n}{b_{n+1}}-1\right)=p > 0$$
the limit exists and it has a positive value. Then $b_n$ must approach $0$.
My attempt:
Let $\varepsilon > 0$. Then there is $n_... |
I recently solved the following integral:
$$
\int _1^{\sqrt{3}}\frac{\sqrt{1+x^2}}{x^2}dx
$$
After integrating, I obtained the result:
$$
\frac{1}{2}\ln\frac{2-\sqrt2}{2-\sqrt3}+\frac{1}{2}\ln\frac{2+\sqrt3}{2+\sqrt2}+\sqrt2-\frac{2}{\sqrt3}
$$
However, the answer provided in my textbook differs slightl... |
We know that [the solution of a linear SDE is a Gaussian process](https://ethz.ch/content/dam/ethz/special-interest/mavt/dynamic-systems-n-control/idsc-dam/Lectures/Stochastic-Systems/SDE.pdf#page=40). I wonder whether the same is true for a SPDE casted to an infinite-dimensional SDE on a Hilbert space $H$ like $${\rm ... |
Is the solution of the stochastic heat equation a Gaussian process? |
With no correlation we can show sample variance is unbiased:
$$E[s^2] = E\left(\frac{\sum^n_{i=1}(X_i - \bar{X})^2}{n-1}\right) = \sigma^2$$
Proof
$$E\left(\sum^n_{i=1}(X_i - \bar{X})^2\right) = E\left(\sum^n_{i=1}X_i^2 -2\bar{X}\sum^n_{i=1}X_i + n \bar{X}^2 \right ) = E\left(\sum^n_{i=1}X_i^2 -2\bar{X}n\bar{X} + ... |
> Let $f:[0,\infty)\to\mathbb{R}$ be continuous on $[0,\infty)$ and differentiable on $(0,\infty)$
suppose: $$\lim_{x\to\infty}{f(x)+f'(x)}=5$$ Show that $f$ is uniformly continuous.
I've thought of proving that $\lim_{x\to\infty}{f(x)}$ is finite but I'm not even sure if $f$ converges at infinity.
Another idea ... |
I am writing a boid simulation in a toroidal space. The boids wrap around from the left to right edge and the top to bottom edge. To calculate what boids are visible, I need to find the minimum distance between those two boids' positions. I have been doing the following
Given: $$P_1 = (x_1, y_1), P_2 = (x_2, y_2)$$
... |
It is often claimed that the following equation holds for the [modular discriminant](https://en.wikipedia.org/wiki/Weierstrass_elliptic_function#Modular_discriminant) $\Delta=g_2^3-27g_3^2$ of the Weierstrass elliptic funtion and the [Dedekind eta ($\eta$) function](https://en.wikipedia.org/wiki/Dedekind_eta_function) ... |
For which of the Weierstrass elliptic function periods do this equation of the modular discriminant and the Dedekind eta function apply? |
I am writing a boid simulation in a toroidal space. The boids wrap around from the left to right edge and the top to bottom edge. To calculate what boids are visible, I need to find the minimum distance between those two boids' positions. I have been doing the following
Given: $$P_1 = (x_1, y_1), P_2 = (x_2, y_2)$$
... |
Recently for a bonus homework assignment in my algebra class, I was asked to review the literature and write up a proof that $\pi$ is transcendental. Essentially every source I found ("The Transcendence of $\pi$" by Steve Mayer for example) presents the classic proof of Lindemann, which heavily relies on symmetric func... |
**Question:** Does a finite constant $C>0$ exist such that for $\forall\ p\in\mathbb{C}[z_1,z_2]$ we have:
$$\sup_{z\in r \mathbb D^2}|p(z_1,z_2)|\le C\sup_{z\in \mathbb D^2}|p(z_1,z_2)|$$
where $r>0$?
I define as $p\in\mathbb{C}[z_1,z_2]$ a finite function of the following formula:
$$p(z_1, z_2) := \sum_{... |
Does a constant $C>0$ exist such that for $\forall\ p\in\mathbb{C}[z_1,z_2]$ we have: $\sup_{z\in rD^2}|p(z_1,z_2)|\le C\sup_{z\in D^2}|p(z_1,z_2)|$? |
How Can I Improve the Construction of this Embedding? |
Trying to find answers for this year french intern agregation (Algebra exam), i found this question about a proof that a particular group representation of $\mathcal{S}_{3}$ on vector space $\mathbb{C}^{3}$ is indeed correct.
We define the following arrow :
$\theta_{4} \left.: \begin{cases} \mathcal{S}_{3} & \r... |
First, we observe that among first countable spaces, every compact is sequentially compact: https://topology.pi-base.org/spaces/?q=First+Countable%2BCompact%2B%7ESequentially+Compact
But the converse fails: https://topology.pi-base.org/spaces/?q=First+Countable%2B%7ECompact%2BSequentially+Compact The particular coun... |
So I'm currently learning continuum mechanics and I am a bit confused about my course notes introducing first order transformation rules.
[![enter image description here][1]][1]
[![enter image description here][2]][2]
[![enter image description here][3]][3]
When it says we project $\hat{i}_{i}^{'}$ onto the $... |
I tried to imagine 2 more scenarios and answer them to get a good intuition on the matter , I may be wrong and I would love nothing more than to be corrected ,
case 1:
if $$T(x)^2=0$$ does $$T(x)=0$$ have to be zero as well ,
I went and started with the thinking if there is a case where this is not true could not th... |
So I'm currently learning continuum mechanics and I am a bit confused about my course notes introducing first order transformation rules.
[![enter image description here][1]][1]
[![enter image description here][2]][2]
[![enter image description here][3]][3]
When it says we project $\hat{i}_{i}^{'}$ onto the $... |
First, we observe that among first countable spaces, every compact is sequentially compact: https://topology.pi-base.org/spaces/?q=First+Countable%2BCompact%2B%7ESequentially+Compact
But the converse fails: https://topology.pi-base.org/spaces/?q=First+Countable%2B%7ECompact%2BSequentially+Compact The particular coun... |
I tried to imagine 2 more scenarios and answer them to get a good intuition on the matter , I may be wrong and I would love nothing more than to be corrected ,
case 1: if $T(x)^2=0$ does $T(x)=0$ have to be zero as well ?,
I went and started with the thinking if there is a case where this is not true could not ... |
Be wary not to confuse this post with [this one](https://math.stackexchange.com/questions/3046393/is-a-monotone-class-which-is-closed-under-complements-and-finite-disjoint-unions). In this case, $M$ is simply a subset of the power set of $X$.
>Let $X$ be a set and $M$ be a nonempty collection of subsets of $X$. Is ... |
Subset of $X$ closed under countable disjoint unions and complements a $\sigma$-algebra? |
Let $\{a_i\}_{i\in \mathbb{N}}$ be a sequence of positive real numbers, whose series converges i.e. $\sum\limits_{i\in \mathbb{N}}a_i = a < \infty.$ Does this imply the convergence of the double series $\sum\limits_{i,j \in \mathbb{N}}a_i a_j?$
If yes, how can this be proven? If not, what are some counterexamples?
... |
First, we observe that among first countable spaces, every compact is sequentially compact: https://topology.pi-base.org/spaces/?q=First+Countable%2BCompact%2B%7ESequentially+Compact
But the converse fails: https://topology.pi-base.org/spaces/?q=First+Countable%2B%7ECompact%2BSequentially+Compact The particular coun... |
Locally $p-$integrable function on an open set that satisfies additional condition must be $0$ almost everywhere on the open set? |
I'm reading Category Theory in context and I got a bit confused by problem 2.1.ii. The problem is as follows:
> Prove that if $F: C \rightarrow \text{Set}$ is representable, then $F$ preserves monomorphisms, i.e., sends every monomorphism in $C$ to an injective function. Use the contrapositive to find a covariant s... |
Covariant representable functors preserve monomorphisms. What about contravariant representable functors?? |
While reading about Sylvester–Gallai theorem, I got the simple proof of the theorem using Induction.
Statement:
*Given a finite set of points in Euclidian space such that any line passing through two of the points passes through at least one more of them, all points have to be collinear.*
Proof
Let n be... |
Let $L$ be the (holomorphic) line bundle on $\mathbb{P}^1$ for which the glueing function from $\mathbb{P}^1\setminus \{\infty\}$ to $\mathbb{P}^1\setminus \{0\}$ on the overlap $\mathbb{C}^\times = \mathbb{P}^1\setminus \{0,\infty\}$ is given by multiplication with $g(z) = \exp(1/z)$.
A global (holomorphic) sectio... |
I'm going through problems in chapter one of Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler.
I need some help with question 39 in Section 1.5 problems.
$\epsilon$, $\delta$ are positive infinitesimal, is the following expression infinitesimal, finite, or infite?
The expression is this:
... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.