instruction stringlengths 12 30k |
|---|
Why does the simplified version of $x^{\frac{6}{8}}$ have a domain of $x≥0$ while the unsimplified has a domain of $x\in{\Bbb{R}}$? Shouldn't they have the same domain being that they are the same expression? If a negative number were to be plugged into $x^{\frac{3}{4}}$ then it would take the 4th root of a negative nu... |
Why does $x^{\frac{3}{4}}$ have a domain of $x≥0$, while $x^{\frac{6}{8}}$ has a domain of $x\in{\Bbb{R}}$? |
Usually, when we have a complete noncompact manifold, we can look at its universal cover so that many geometric property can be lifted to a simply-connected manifold. If we take a large compact subset of this complete noncompact manifold such that the complement has only one component with compact boundary (probably ... |
I am quite new to the concept of optimization on manifolds, however in my research I have stumbled upon a problem which I believe is amenable to this type of analysis.
Specifically, I am concerned with the following convex optimization problem
\begin{align*}
&\min_{K_k, \Sigma_{k}} \ &&J_{\Sigma} = \sum_{k=0}^{N-1} ... |
Find all the values of r so that the equation dx/dt=cos(rx) defines a vector field on the circle.
My answer is that ;
By the definition of a vector field on the circle, dx/dt=cos(rx) must be real valued and 2π periodic function. Since cos x has all real numbers in its domain and -1≤ cos x≤ 1, hence cos(rx) is a ... |
I am quite new to the concept of optimization on manifolds, however in my research I have stumbled upon a problem which I believe is amenable to this type of analysis.
Specifically, I am concerned with the following convex optimization problem
\begin{align*}
&\min_{K_k, \Sigma_{k}} \ &&J_{\Sigma} = \sum_{k=0}^{N-1} ... |
I recently came across [this][1] paper by Friedgut that shows low sensitivity boolean functions are close to juntas. This was an unintuitive result to me, as I thought I could easily imagine a function on the boolean hypercube that is a linear combination of many low-degree monomials, and therefore has low sensitivity,... |
I am quite new to the concept of optimization on manifolds, however in my research I have stumbled upon a problem which I believe is amenable to this type of analysis.
Specifically, I am concerned with the following convex optimization problem
\begin{align*}
&\min_{K_k, \Sigma_{k}} \ &&J_{\Sigma} = \sum_{k=0}^{N-1} ... |
If $p_1<p_2<...<p_n$ are $n$ different prime numbers, and the sum of any three numbers is still a prime number, what is the maximum value of $n$?
I only can find $3$ numbers $3$, $5$, $11$ that the sum is $19$ which is still prime... |
If $p_1<p_2<...<p_n$ are $n$ different prime numbers, and the sum of any three numbers is still a prime number, what is the maximum value of $n$? |
I noticed that using the integration reduction formula to reduce any nth term of sinx results in a different answer than reframing a power of sin^n x as (sin^2 x)^n/2.
The reduction formula:
integral sin^n x dx = (-1/n)sin^(n-1)(x)*cos(x) + ((n-1)/n)*integral sin^(n-2)(x) dx
Using the reduction formula on... |
$$f(x)= \lim_{n\rightarrow\infty} \left(\dfrac{n^n(x+n)(x+\dfrac{n}{2})\cdots(x+\dfrac{n}{n})}{n!(x^2+n^2)(x^2+\dfrac{n^2}{2^2})\cdots(x^2+\dfrac{n^2}{n^2})}\right)^{x/n}
, \quad x>0$$
How can I represent this limit in a simple form?
I tried that
above fomula $\left(\dfrac{\prod\limits_{k=1} \left(\dfrac{k... |
How to represent this limit:$\lim\limits_{n\to\infty}\left(\frac{n^n}{n!}\prod_{k=1}^n\frac{x+\frac{n}{k}}{x^2+\frac{n^2}{k^2}}\right)^{\frac{x}{n}}$? |
If I have one child half my chromosomes are passed on.
If I have two children somewhere between half and all my chromosomes will be passed on.
From a statistical perspective what is the average number of chromosomes inherited with N children? N<10 |
If I have N children, what percentage of my chromosomes will be passed on? |
Choose positive integers $D$ and $N$. Roll a fair $D$-sided die $N$ times, recording the number of times each of the $D$ outcomes are rolled, say $r_1, r_2, \ldots, r_D.$ What are the asymptotics of $\min(r_1, r_2, \ldots, r_D)$?
They're not independent, so I can't directly apply the <a href="https://en.wikipedia.or... |
I am working through a book to learn differential equations. There is a step of solution, and I don't get two things:
1) How is possible to substitute $\frac{1}{16}x^4$ with $\color{red}{\frac1{16}(4\cdot x^3)}$
2) Why did they substitute the $y$ function on the left side where the $\mathrm dy/\mathrm dx$ was. I wo... |
I've proven Thomae's function
$$
f(x)=\begin{cases}\frac{1}{q},&\text{if }x=\frac{p}{q}\in\mathbb{Q}\text{ in lowest terms, }x\neq0;\\0,&\text{ otherwise}.\end{cases}
$$
is Riemann integrable and integrates to zero. Consider the slight modification
$$
f(x)=\begin{cases}\frac{1}{p},&\text{if }x=\frac{p}{q}\in\math... |
A more generalized integral:
For $q,p\in\mathbb{Z}_{\ge1}$ with $q+p$ is even number, we have
\begin{gather}
\int_0^{\frac{\pi}{4}}\ln^{q-1}(\tan(x))\operatorname{Li}_p(\tan^4(x))\mathrm{d}x=(1-(-1)^q)2^{2p-3}(1-2^{-p})|E_{q-1}|\left(\frac{\pi}{2}\right)^{q}\eta(p)\\
-(q-1)!2^{2p-1}\sum_{k=0}^{\lfloor{\f... |
I got a question below with answer from the professor, although I think the answer may be wrong, but is there something I missed?
1. Determine the form of a particular solution to the ODE below. Do not solve for the coefficient.
$y'' + 4y' + 5y = e^{2t} + t^2 \sin(t) - e^{2t} \cos(t)$
I know the solution for t... |
Let $(X, \mathcal{T})$ be a topological space, $P(\mathcal{T})$ be the set of all compact-and-Hausdorff subsets of $X$, and
$$\mathcal{T} = \{U \subset X : \forall C \in P(\mathcal{T}): U \cap C \in \mathcal{T}|C\},$$
where $\mathcal{T}|C$ denotes the subspace topology in $C$.
Are the sets in $P(\mathcal{T})$ ... |
Are compact and Hausdorff subsets closed if the topology is determined by them? |
$$\sum_{n=0}^{\infty} x^{n q} \prod_{k=1}^{t} \left(\sum _{m=0}^n \frac{1}{(x^{a_k})^m}\right)$$
For $t=1$:
\begin{align*}
\sum_{n=0}^{\infty} x^{n q} \left(\sum _{m=0}^n \frac{1}{x^{a_1 m}}\right) &= \sum_{n=0}^{\infty} x^{n q} \left( \frac{\frac{1}{x^{(n+1)a_1}}-1}{\frac{1}{x^{a_1}}-1}\right)\\
&= \frac{1}{ \le... |
Show that the equation of a straight line passing through the point with position vector $\vec{b}$ and perpendicular to the line $\vec{r}=\vec{a}+\mu \vec{c}$ is of the form $\vec{r}=\vec{b}+\beta \vec{c}×\{(\vec{a}-\vec{b})×\vec{c}\}$.
How to derive the vector parallel to the required line? I get that this vector m... |
Suppose that I have a a circle of radius, $R$ with $n$-many non-overlapping circles in it where each circle has a radius, $r$.
The first block of code below generates random circles of the same radius inside a larger circle but not allowing overlapping. It has a set amount of tries to get a user input defined amount... |
What is the normalized frequency distribution for the shortest tour length of identical circles in a larger circle? |
Let $(X, \mathcal{T})$ be a topological space such that
$$\mathcal{T} = \{U \subset X : \forall C \in P(\mathcal{T}): U \cap C \in \mathcal{T}|C\},$$
where $P(\mathcal{T})$ be the set of all compact-and-Hausdorff subsets of $X$ and $\mathcal{T}|C$ denotes the subspace topology in $C$.
Are the sets in $P(\mathc... |
In terms of unknown, it may be abstract. Let's recall in our first multivariable class, with numerical example. Like find a line passing through $b=(\pi,1,2)$ such that it is perpendicular to the line $$\ell:(1,2,3)t+(1,1,1)$$
How do you solve it? You first identify that the set of vectors perpendicular to $\ell$ must... |
Let
$$S=\sum_{0\le i}\sum_{<j\le n} {n \choose i}~~~~~~~~(*)$$
Consider the summation:
$$\sum_{j=0}^{n}\sum_{i=0}^{n} A_i B_j=\sum_{i=0}^n A_i B_i+\sum_{0\le i<j\le n} A_i B_j+ \sum_{n\ge i>j\ge 0} A_i B_j$$
If $B_j=1$, then the second and the third sums in RHS above are equal, then
$$\sum_{0\le i<j\le n}A_i=\fra... |
If $f,g$ are degree-1 monotone maps of the circle, **why** do we generally have $\rho(f\circ g)\neq\rho(f)+\rho(g)$?
I mean, you might say that we have no right to expect an equality. After all, it's not hard to come up with counterexamples. For example, if $f$ is the piecewise-affine map that interpolates $[0,\frac... |
Why are rotation numbers not homomorphic? |
Let $M,\,N$ be smooth manifolds and $f:\,M\longrightarrow N$ be a smooth map. The *rank* of $f$ at a point $p\in M$ is defined by the rank of the differential map $df_p$, that is the dimension of $\,df_p(TM_p)\leq TN_{f(p)}$. If the rank of $f$ is constant at every point of $M$, then we say $f$ *has constant rank* on $... |
Let $(X, \mathcal{T})$ be a topological space such that
$$\mathcal{T} = \{U \subset X : \forall C \in P(\mathcal{T}): U \cap C \in \mathcal{T}|C\},$$
where $P(\mathcal{T})$ is the set of all compact-and-Hausdorff subsets of $X$ and $\mathcal{T}|C$ denotes the subspace topology in $C$.
Are the sets in $P(\mathc... |
In Royden's Real Analysis, the problem given on page 43 as problem 18 as follows:
Let $E$ have finite outer measure. Show that there exists an $F_\sigma$ set $F$ and a $G_\delta$ set $G$ such that $F \subseteq E \subseteq G$ and $m^*(F) = m^*(E) = m^*(G)$.
But according to the errata, the problem has been revised... |
This construction taken from J. Kollar's book *Singularities of MMP*. We just consider varieties over $\mathbb{C}$:
> **[Def]** Let $X$ be a projective variety with a line bundle $\scr{L}$, we define
$$
\mathsf{AC}(X,\mathscr{L}):=\mathrm{Spec}\bigoplus_{m\geq0}H^0(X,\mathscr{L}^{\otimes m}).
$$
>
>Also we cons... |
Higher push-forward of structure sheaf of partial resolution of Affine Cones associated to ample line bundles? |
In Royden's Real Analysis, the problem given on page 43 as problem 18 as follows:
"Let $E$ have finite outer measure. Show that there exists an $F_\sigma$ set $F$ and a $G_\delta$ set $G$ such that $F \subseteq E \subseteq G$ and $m^*(F) = m^*(E) = m^*(G)$."
But according to the errata, the problem has been revis... |
$$f(x)= \lim_{n\rightarrow\infty} \left(\dfrac{n^n(x+n)(x+\dfrac{n}{2})\cdots(x+\dfrac{n}{n})}{n!(x^2+n^2)(x^2+\dfrac{n^2}{2^2})\cdots(x^2+\dfrac{n^2}{n^2})}\right)^{x/n}
, \quad x>0$$
How can I represent this limit in a simple form?
I tried that
above fomula $\left(\dfrac{\prod\limits_{k=1}^n \left(\dfrac... |
I am reading an interesting paper [One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational](https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=427&option_lang=eng) by V.V. Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive integer parameters satis... |
Let $\mathbb{F}_{p^n}$ be a finite field of characteristic $p$, then we can define discrete valuation $v_p$ in $\mathbb{F}_{p^n}((x))$ by
$$ v_p(f):=\min \{n~|~a_n \neq 0\}, $$
where $f(x)=\sum_{n \geq i} a_n x^n \in \mathbb{F}_{p^n}((x))$.
This may be called a one-dimensional local field known as the field of for... |
Is the $\mathbb F_{p^n}((x,y))$ a local field? |
Let $\mathbb{F}_{p^n}$ be a finite field of characteristic $p$, then we can define discrete valuation $v_p$ in $\mathbb{F}_{p^n}((x))$ by
$$ v_p(f):=\min \{n~|~a_n \neq 0\}, $$
where $f(x)=\sum_{n \geq i} a_n x^n \in \mathbb{F}_{p^n}((x))$.
This may be called a one-dimensional local field known as the field of for... |
In Royden's Real Analysis, the problem given on page 43 as problem 18 used to be as follows:
"Let $E$ have finite outer measure. Show that there exists an $F_\sigma$ set $F$ and a $G_\delta$ set $G$ such that $F \subseteq E \subseteq G$ and $m^*(F) = m^*(E) = m^*(G)$."
But according to the errata, the problem has... |
Given a convex optimization problem $$\min f(x), x \in D$$ $f, D$ convex.
The **first-order optimality condition** says $x$ is the minimizer if and only if $\nabla f(x)^T (x-y) \geq 0, \forall y\in D.$ For unconstrained problems, this is $\nabla f(x) = 0$.
This seems to be perfectly suited for finding minimize... |
What's the purpose of the KKT condition when first-order optimality condition exists? |
Let
$$S=\sum_{0\le i}\sum_{<j\le n} {n \choose i}\tag{*}$$
Consider the summation:
$$\sum_{j=0}^{n}\sum_{i=0}^{n} A_i B_j=\sum_{i=0}^n A_i B_i+\sum_{0\le i<j\le n} A_i B_j+ \sum_{n\ge i>j\ge 0} A_i B_j$$
If $B_j=1$, then the second and the third sums in RHS above are equal, then
$$\sum_{0\le i<j\le n}A_i=\frac{1}... |
I have the following PDE (with small parameter $0<\epsilon\ll 1$):\
$$
\begin{cases}
\dfrac{\partial u(x,t)}{\partial t}+\dfrac{\partial ^2u(x,t)}{\partial x^2}-\epsilon u(x,t)^2=0, & x\in (0,1), t>0\\
\\
u(x,0)=u_0(x), &x\in (0,1)\\
\\
u(0,t)=u(1,t)=0, &t>0
\end{cases}
$$
I now have to find the equations f... |
Let $\mathcal{E}$ be an edge measure, and function $f$ be edgewise linear, i.e., it is continuous and its restriction to edge interior is a linear function. Then, at the bottom of the page 7 of [this handout][1], the author claims that, for an edge $e=(u,v)$, we have
$$\int_{e}f\mathrm{d}\mathcal{E} = [f(u)+f(v)]\fr... |
I saw [this question][1]and I was able to prove that $f(x)= e^x \cdot \left(\frac{x+1}{x^2+1}\right)^x \frac{(x+1)}{e^{2\arctan(x)}} $ but a new question came to my mind : does the sequence $f_n(x):=\left(\frac{n^n}{n!}\prod_{k=1}^n\frac{x+\frac{n}{k}}{x^2+\frac{n^2}{k^2}}\right)^{\frac{x}{n}}$ converge uniformly to... |
Do $f_n(x):=\left(\frac{n^n}{n!}\prod_{k=1}^n\frac{x+\frac{n}{k}}{x^2+\frac{n^2}{k^2}}\right)^{\frac{x}{n}}$ converge uniformly? |
If $p_1<p_2<...<p_n$ are $n$ different prime numbers, and the sum of any three numbers is still a prime number, what is the maximum value of $n$?
I only can find $3$ numbers $3$, $5$, $11$ that the sum is $19$ which is still prime...
[Update] I think it is feasible to consider the remainders.
There are ... |
Does $f_n(x):=\left(\frac{n^n}{n!}\prod_{k=1}^n\frac{x+\frac{n}{k}}{x^2+\frac{n^2}{k^2}}\right)^{\frac{x}{n}}$ converge uniformly? |
Let $E$ be a globally generated vector bundle bundle of rank $r$ on $\mathbb P^n$. Then there exists a S.E.S: $0 \to K \to \mathcal O_{\mathbb P^n}^{\oplus N} \to E \to 0$. Then is it necessarily true:
$(i)$ $c_1(E) \geq 0$?
We have $0=c_1(K) +c_1(E)$, why can't we have $c_1(K)$ to be positive?
$(ii)$ Is there... |
I saw [this question][1] and I was able to prove that $f(x)= e^x \cdot \left(\frac{x+1}{x^2+1}\right)^x \frac{(x+1)}{e^{2\arctan(x)}} $ but a new question came to my mind : does the sequence $f_n(x):=\left(\frac{n^n}{n!}\prod_{k=1}^n\frac{x+\frac{n}{k}}{x^2+\frac{n^2}{k^2}}\right)^{\frac{x}{n}}$ converge uniformly t... |
Find all natural numbers n for which: in a triangle with sides |AB| = 33, |AC| = 21 and |BC| = n we can place point D on AB and point E on AC such that |AD| = |DE| = |EC| = m, where m is also a natural number.
Attempt: I saw that m=21 satisfied the condition, so n can be any natural number between 12 and 53 by Trian... |
Find all natural numbers n for which: in a triangle with sides $|AB| = 33$, $|AC| = 21$ and $|BC| = n$ we can place point $D$ on $AB$ and point $E$ on $AC$ such that $|AD| = |DE| = |EC| = m$, where $m$ is also a natural number.
Attempt: I saw that $m=21$ satisfied the condition, so $n$ can be any natural number betw... |
> Any suggestions on where I made an error in my logic?
The problem with your logic is that you are introducing an extra restriction. You are allowing only the last ball to be paired with another ball. In other words you are forcing the last ball to necessarily be in the bin with $2$ balls. This is a restriction tha... |
Let $E$ be a globally generated vector bundle bundle of rank $r$ on $\mathbb P^n$. Then there exists a S.E.S: $0 \to K \to \mathcal O_{\mathbb P^n}^{\oplus N} \to E \to 0$. Then is it necessarily true:
$(i)$ $c_1(E) \geq 0$?
Is this because restricting to a line $0 \to K|_L \to \mathcal O_{L}^{\oplus N} \to E|_L ... |
I wanted to invert the following Laplacian expression which was not possible with simple manipulation.
$$ \frac{s}{s^2 + \omega^2}\exp\bigg(-\sqrt{\frac{s}{\alpha}}x\bigg) $$
Thus, I used the Taylor Series on each part and simplified it as shown:
$$ \implies \frac{1}{s} \bigg( 1 + \frac{\omega^2}{s^2}\bigg)^{... |
Let $U$ be an open set in $\mathbb{R}^k$ for some positive integer $k$. Consider the following set (for some positive integer $n$):
$$\bigcup_{x_1,\cdots,x_n \in U} \left( (U \setminus\{x_1,\cdots,x_n\}) \times \{x_1\} \times \{x_2\} \times \cdots \times \{x_n\} \right).$$
Is this an open set in $\mathbb{R}^{k(n+... |
Is this exotic set an open set? |
### Problem
> Let $a>0$ and $B$ be a standard $\mathbb{R}$-valued Brownian motion. Define the stopping time $S_a:=\inf\{t\geq 0\ \vert \left\lvert B_t\right\rvert = a\}$. Compute $\mathbb{E}\left[S_a^2\right]$.
### Flawed attempt at showing that $\mathbb{E}\left[S_a\right] = a^2$
I am following the solution fo... |
Suppose we have some financial data, e.g., stock return time series. The theoretical distribution is unknown, while we can construct the empirical distribution through historical data. The problem is to compute the probability that the random process will exceed (i) some $x$ value, and (ii) some $x$ value conditional t... |
This is an equation from Rau's book "Quantum theory-an Information Processing Approach". It's leading to "lossless steering" of a particle if you insert an infinite number of filters between the initial state and the last filter and for the last step you have to substitute x for $2^n+1$ (n: number of filters) an then... |
This is an equation from Rau's book "Quantum theory-an Information Processing Approach". It's leading to "lossless steering" of a particle if you insert an infinite number of filters between the initial state and the last filter and for the last step you have to substitute x for $2^n+1$ (n: number of filters) an then... |
For $p(s^{\prime} | s, a) = (1 - \alpha)\rho(s^{\prime}) + \alpha p(s^{\prime} | s, a)$ for some $\alpha \in (0, 1)$, clearly, $\tilde{p}(\cdot | s, a)$ is a probability for every $(s, a)$. For every arbitrary policy $\pi$, define the transition kernel:
$\tilde{p}_{\pi}(s^{\prime}, a^{\prime} | s, a) $ $=$ $\pi(a^{\pr... |
I am quite new to the concept of optimization on manifolds, however in my research I have stumbled upon a problem which I believe is amenable to this type of analysis.
Specifically, I am concerned with the following convex optimization problem
\begin{align*}
&\min_{K_k, \Sigma_{k}} \ &&J_{\Sigma} = \sum_{k=0}^{N-1} ... |
### Problem
> Let $a>0$ and $B$ be a standard $\mathbb{R}$-valued Brownian motion. Define the stopping time $S_a:=\inf\{t\geq 0\ \vert \left\lvert B_t\right\rvert = a\}$. Compute $\mathbb{E}\left[S_a^2\right]$.
### Flawed attempt at showing that $\mathbb{E}\left[S_a\right] = a^2$
I am following the solution fo... |
A more generalized integral:
For $q,p\in\mathbb{Z}_{\ge1}$ with $q+p$ is even number, we have
\begin{gather}
\int_0^{\frac{\pi}{4}}\ln^{q-1}(\tan(x))\operatorname{Li}_p(\tan^4(x))\mathrm{d}x=-(1-(-1)^q)2^{2p-3}(1+2^{-p})|E_{q-1}|\left(\frac{\pi}{2}\right)^{q}\eta(p)\\
-(q-1)!2^{2p-1}\sum_{k=0}^{\lfloor{\... |
### Problem
> Let $a>0$ and $B$ be a standard $\mathbb{R}$-valued Brownian motion. Define the stopping time $S_a:=\inf\{t\geq 0\ \vert \left\lvert B_t\right\rvert = a\}$. Compute $\mathbb{E}\left[e^{-\frac{\lambda^2}{2}S_a}\right]$ for $\lambda\in\mathbb{R}$.
### My (edited) attempt
I noticed that the expressi... |
### Problem
> Let $a>0$ and $B$ be a standard $\mathbb{R}$-valued Brownian motion. Define the stopping time $S_a:=\inf\{t\geq 0\ \vert \left\lvert B_t\right\rvert = a\}$. Compute $\mathbb{E}\left[S_a^2\right]$.
### Flawed attempt at showing that $\mathbb{E}\left[S_a\right] = a^2$
I am following the solution fo... |
For $p(s^{\prime} | s, a) = (1 - \alpha)\rho(s^{\prime}) + \alpha p(s^{\prime} | s, a)$ for some $\alpha \in (0, 1)$, clearly, $\tilde{p}(\cdot | s, a)$ is a probability for every $(s, a)$. For every arbitrary policy $\pi$, define the transition kernel:
$\tilde{p}_{\pi}(s^{\prime}, a^{\prime} | s, a) $ $=$ $\pi(a^{\pr... |
Suppose we have some financial data, e.g., stock return time series. The theoretical distribution is unknown, while we can construct the empirical distribution through historical data. The problem is to compute the probability that the random process will exceed (i) some $x$ value, and (ii) some $x$ value conditional t... |
I have the following PDE (with small parameter $0<\epsilon\ll 1$):\
$$
\begin{cases}
\dfrac{\partial u(x,t)}{\partial t}+\dfrac{\partial ^2u(x,t)}{\partial x^2}-\epsilon u(x,t)^2=0, & x\in (0,1), t>0\\
\\
u(x,0)=u_0(x), &x\in (0,1)\\
\\
u(0,t)=u(1,t)=0, &t>0
\end{cases}
$$
I now have to find the equations f... |
Let $A$ be a $m \times r$ matrix and $B$ be a $r \times n$ matrix, I wonder if there exists an inequality like the following:
$$
\left \| AB \right \|_F \leq f(m,r,n)g(A,B) ,
$$
or
$$
\left \| AB \right \|_F \ge f(m,r,n)g(A,B) ,
$$
where $f(m,r,n)$ is a constant only depending on the dimension of matrix $m,r,n... |
Is there any inequality involving the Frobenius norm and the dimension of matrix? |
Are these assertion true? 1) Let X be a random variable. If X^3 admit an expected value, then X admit an expected value. 2) Let X be a real random variable. If e^X admit an expected value, then X admit an expected value. |
The Frobenius norm is sub-multiplicative since it is an operator norm. Meaning:
$$\|AB\|_F \leq \|A\|_F\|B\|_F.$$
Is this what you are looking for? I think [this wikipedia article][1] contains good explanations for this.
[1]: https://en.wikipedia.org/wiki/Matrix_norm |
Let $\Gamma \le \operatorname{PSL}(2,\mathbb{R})$ be a discrete subgroup and $\Sigma:=\mathbb{D}^2/\Gamma$ be the quotient. Then $\Sigma$ is a hyperbolic surface whose universal cover is $\mathbb{D}^2$ and the fundamental group $\pi_1(\Sigma,x)$ ($x\in\Sigma$) is isomorphic to $\Gamma$. Fix a point $\tilde{x}\in\mathbb... |
Your computations are correct but the conclusions ar wrong.If two events cannot occur simultaneously they are *not* independent but mutually exclusive, i.e. they are the null event. That event has zero probability of occuring.
You can think it like this: "I'm currently at 5. What is the probability of being at 7?" S... |
I know a statistic $T$ is complete if for any measurable function $g$, $E(g(T))=0$ implies $g(T)=0$ a.s. But what if I take $g(T)=T-E(T)$?
The expectation of $g(T)$ is 0, but not necessarily also $g$ is. Following this, the sample mean in a normal model, for example, would not be complete.
Where am I getting this w... |
I am a beginner in koopman operators，and I want to apply it to the robotics. But I think the koopman operator approximated by EDMD is more like a fitting tool. Because we always choose some dictionary functions $\Phi=\left \{ \phi_1(x),\phi_2(x),...,\phi_n(x) \right \} $, and the approximation of koopman operator is ... |
Is Koopman operator just like curve fitting? |
Problem: Find a nice simple closed curve which passes through the points $(0,0)$ and $(1,0)$ on the Cartesian plane and whose length is $L\pi$.
If the given condition is not the loop length but the loop area, it is easy to find a nice curve. The ellipse
$$(2x-1)^2+\frac{y^2}{4A^2}=1$$
has the area $A\pi .$
Can ... |
I have looked everywhere and I can't find a clear definition of hamiltonian isotopy. I have these definitions in my lecture notes but they are rather confusing
[![enter image description here][1]][1]
Acording to the last definition a Hamiltonian isotopy $\phi_t$ (the one they given in the definition of Hamiltonia... |
There is an exercise in András Frank’s book *Connections in Combinatorial Optimization*,
> An easy exercise shows that every affine matroid can be represented as a linear matroid, and vice versa.
I think I need to prove that given a set of vectors $S$ in $\mathbb{R}^n$, there exists a transformation $m$ from $S$ t... |
There is an exercise in András Frank’s book *Connections in Combinatorial Optimization*,
> An easy exercise shows that every affine matroid can be represented as a linear matroid, and vice versa.
I think I need to prove that given a set of vectors $S$ in $\mathbb{R}^n$, there exists a transformation $m$ from $S$ t... |
I saw [this question][1] and I was able to prove that $f(x)= e^x \left(\frac{x+1}{x^2+1}\right)^x \frac{(x+1)}{e^{2\arctan(x)}} $ but a new question came to my mind : does the sequence $f_n(x):=\left(\frac{n^n}{n!}\prod_{k=1}^n\frac{x+\frac{n}{k}}{x^2+\frac{n^2}{k^2}}\right)^{\frac{x}{n}}$ converge uniformly to $f(x... |
I came across this post https://math.stackexchange.com/questions/2562148/distances-defined-in-manifold-of-symmetric-positive-definite-matrices/4885473#4885473 because I had the same questions. I did not understand the answers provided, so I wanted to try to answer it myself. Originally, I posted my justifications and i... |
I am trying to find bounds for the following quantity. Take two functions $f,h \in L^{1} \cap L^2$ but $\|h \|_{\infty} = \infty$. Is there a way to obtain a bound of the following type:
$$ | (h * f^2) (x)| \leq \| f\|^2_2 g(h)$$
where $g$ is some function of $h$? This would be the case if we could take the sup n... |
Let $E/K$ be an elliptic curve over a field $K$.
Let $Aut(E)$ be a group of all automorphism of $E$ as algebraic curves.
Then, does $AutE\cong E$ holds ?
There is a natural injection $E\to \text{Aut}(E)$ given by $P\mapsto \{E\to E, Q\mapsto Q+P\}$.
To prove this is surjective, we should prove every automor... |
Does $\text{Aut}(E)\cong E$ hołd for elliptic curves? |
Suppose we have a $l\times l$ real matrix $M=(X'X)^{-1}X'\Sigma X (X'X)^{-1}$, where $X$ is a $n\times l$ real matrix and $\Sigma$ is a $l\times l$ real symmetric and positive definite matrix. $X'X$ is invertible and its minimum eigenvalue $\lambda_{min}(X'X)$ diverges to $\infty$ as $n\rightarrow \infty$. How to show ... |
How to show the quadratic form of the following matrix converges to zero? |
Is there any way to simplify this fomula below:
$\sqrt[3]{\tan x+1}$ or $\sqrt[3]{\left(\tan x\right)^{3}+\left(\tan x\right)^{2}-\tan x-1}$? |
anyway to simplify $\sqrt[3]{\tan x+1}$ or $\sqrt[3]{\left(\tan x\right)^{3}+\left(\tan x\right)^{2}-\tan x-1}$? |
Need help solving a system of three trigonometric equations in three unknowns. These equations are used to find the coordinates φs, θs and time ts at the signal source.
(φ1, θ1); (φ2, θ2); (φ3, θ3) - coordinates of the three points to which the signal arrived.
t1, t2, t3 - the time at which the signal arrived at ea... |
I was reading the second answer of the following question here https://math.stackexchange.com/questions/122274/why-xpn-x1-is-irreducible-in-mathbbf-p-only-when-n-1-or-n-p-2/4175356#4175356 :
Prove that $f(X) = X^{p^n} - X + 1$ is irreducible over $\mathbb F_{p}$ if and only if either $n = 1$ or $n = p = 2.$
An... |
Why $[\mathbb{F}_{p}(\alpha): \mathbb{F}_{p^n}] = p$? |
Let H be a Hilbert space. Working on locally compact quantum groups, I have met with operator of the form $\iota \otimes \omega_{\xi, \eta}$ with $\xi,\eta \in H$ and $\omega_{\xi,\eta}(T) = (T\xi,\eta)$. These operators should be defined on $B(H \otimes H)$ but I can't find their definitions.
First, I thought we c... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.