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Let $A\subset \mathbb{P}^2$ be the set defined by the equation $X^3=Y^2Z$. I have the map $f:\mathbb{P}^1\rightarrow V$, $[S,T]\rightarrow [S^2T, S^3, T^3]$. I want to find another rational map $g:V\rightarrow \mathbb{P}^1$ (not necessarily a morphism) such that $f\circ g$ and $g\circ f$ are the identity wherever they ... |
>Doesn't it just show that $g(n)=100n^3$ is bounded by $C∗n^3$?
Well that is what it means to be $O(n^3)$ by definition. Recall that $g$ is $O(f)$ if there exists a positive constant $C$ and some $x_0$ such that $|g(x)|\leq C\cdot f(x)$ for all $x\geq x_0$. In other words, $g$ is $O(f)$ if it is *eventually* dominat... |
This is not an answer... Just to give a simple example why it is not always solvable. Imagine a function $\mathbb R^2 \to \mathbb R$.
Let us prescribe any non-closed (importante!) contour of gradient orthogonal to the direction of the contour with a prescribed $\nabla g = \|\nabla g\|\hat n$. Now prescribe every ot... |
Question from an old exam:
>The King of Squares sets out to patrol the City of Squares. The City of Squares is an infinite ladder, i.e., a graph $(V, E)$ where $V = \mathbb{N} \times \{0,1\}$ and the vertex $(x, y)$ is connected to $(x-1, y)$ for $x>0,(x+1, y)$ and $(x, 1-y)$. Unfortunately, due to the revolt, some pa... |
My question is motivated by this question:
https://math.stackexchange.com/questions/78445/does-law-of-large-numbers-converge-in-l1
that asks about the the convergence in $L^1$-norm of the sample mean $\bar{X_n}$ to the population mean $\mu.$
I looked at the answers and comments and I understood that the answer... |
I am trying to understand $\mathbb{C}P^2$. Since I understand the Hopf fibration quite well, I like the following construction:
- Attach a point $\mathbb{D}^0$ (0-cell) to $\mathbb{D}^2$ (2-cell) to get $S^2$.
- Now attach $\mathbb{D}^4$ to $S^2$ by gluing the boundary $\partial \mathbb{D}^4=S^3$ to $S^2$ using th... |
### Question:
When $f \in C^2[a;b]$ we define $ \|f\|_2=\sqrt{\int_{[a;b]}|f|^{2}}$ and when $f$ is bounded on $[a;b]$ we define $\|f\|_{\infty} = \sup_{x \in [a;b]} |f(x)| $.
Is it possible that it exists a $C>0$ that verify for all continuous function on $[a;b]$; $||f||_{\infty} \leq C \|f\|_2 $ ?
##... |
Is it possible that it exists a $C>0$ that verify for all continuous function on $[a;b]$; $\|f\|_{\infty} \leq C \|f\|_2 $? |
Consider a finite group $G$ with subgroups $N$ and $M$, where $N$ is normal in $G$, $N \cap M = 1$, and $[M, N] \leq \Phi(N)$, where $[M, N]$ denotes the commutator subgroup of $M$ and $N$, and $\Phi(N)$ represents the Frattini subgroup of $N$. The question arises: is $M$ necessarily normal in $G$?
To explore this q... |
This is not an answer... Just to give a simple example why it is not always solvable. Imagine a function $\mathbb R^2 \to \mathbb R$.
Let us prescribe any non-closed (importante!) contour of gradient orthogonal to the direction of the contour with a prescribed $\nabla g = \|\nabla g\|\hat n$. Now prescribe every ot... |
Is there an "elementary irrational number" without a certain digit in its decimal presentation? |
Let $\mathscr{C}$ be a pointed $\infty$-category. Then, $\Sigma : \mathscr{C} \rightleftarrows \mathscr{C} : \Omega$ can be defined through $\Sigma = \operatorname{colim}( * \leftarrow X \to *)$ and $\Omega = \lim ( * \to X \leftarrow *)$.
There seems to be some part in the theory of (co-)limits and adjunctions for ... |
My question is motivated by this question:
https://math.stackexchange.com/questions/78445/does-law-of-large-numbers-converge-in-l1
that asks about the the convergence in $L^1$-norm of the sample mean $\bar{X_n}$ to the population mean $\mu.$
I looked at the answers and comments and I understood that the answer... |
I am trying to understand the definition of spectra in stable homotopy theory.
One definition is as the colimit $\lim_{\rightarrow} (Space^* \overset{\Sigma^{red}} \rightarrow Space^* \overset{\Sigma^{red}} \rightarrow \cdots )$, where $Space^*$ are based spaces and $\Sigma^{red}$ is the reduced suspension. My underst... |
I am trying to find integer solutions for a set of equations and would appreciate any help or insights on methods to determine if solutions exist for certain cases or generally. The equations are as follows:
1. For the case where $\frac{3^n - 1}{2}$ is odd:
$$
x = \frac{\left(\frac{3^n - 1}{2}\right) - 2^y}{2 \t... |
I am trying to understand the definition of spectra in stable homotopy theory.
One definition is as the colimit $\lim_{\rightarrow} (Space^* \overset{\Sigma^{red}} \rightarrow Space^* \overset{\Sigma^{red}} \rightarrow \cdots )$, where $Space^*$ are based spaces and $\Sigma^{red}$ is the reduced suspension. My underst... |
If a function $G(x)$ is defined by
$$G(x)=\int_a^xf(t)dt$$
Then by the [Fundamental Theorem][1] I can see that $G(x)$ is an antiderivative of $f(x)$ and thus
$$G'(x)=f(x)=\frac{dG(x)}{dx}$$
My confusion starts when I look at the integrand $f(t)$ and its variable $t$. The integrand $f(t)$ must have an antiderivative... |
Hint: let $$f(x_1, \ldots, x_{n-1}) = \sum_{0\le i< j\le n} \frac1{x_j - x_i}$$
and $\Omega:=\left\{(x_1, \ldots, x_{n-1}): 0=x_0 < x_1 < \cdots < x_{n-1} < x_n =1\right\}$.
Can you prove that $f$ is convex? What is the limit of $f$ on the boundary of $\Omega$? Is $\Omega$ open? What is the gradient of $f$? |
Say I have a $\sigma$-algebra $\mathcal{F}$ and its completion $\mathcal{F}^{\ast}$. If a set $E\in \mathcal{F}^{\ast}$ but $E\notin \mathcal{F}$, does that mean that I definitely have sets $A,B\in\mathcal{F}$ s.t. $A\subset E\subset B$ and $\mu\left(B\backslash A\right)=0$?
That is how we defined the $\sigma$-algeb... |
Hint: let $$f(x_1, \ldots, x_{n-1}) = \sum_{0\le i< j\le n} \frac1{x_j - x_i}$$
defined on $\Omega:=\left\{(x_1, \ldots, x_{n-1}): 0=x_0 < x_1 < \cdots < x_{n-1} < x_n =1\right\}$.
Can you prove that $f$ is convex? What is the limit of $f$ on the boundary of $\Omega$? Is $\Omega$ open? What is the gradient of $f... |
If I have two complex numbers $ a,b \in \mathbb{C} $，and $ |a|^2 + |b|^2 = 1 $，I need to get the max of
\begin{aligned}
(aa^*-bb^*)^2 - [(ab^*)^2 - (a^*b)^2]^2
\end{aligned}
In the answer book, the auther let
\begin{aligned}
a = \cos \frac{\beta}{2}
\\
b = \sin \frac{\beta}{2} e^{i\alpha}
\end{aligned}
the... |
I would like to show that $$A = x^T G(x) \tanh(x) \geq 0$$ (with equality only for $x=0$), for $x\in\mathbb{R}^n$. It is known that $G(x)\in\mathbb{R}^{n\times n}$ is a positive definite matrix.
Or come up with a counter-example if it does not hold.
I tried transforming to the associated quadratic forms $$A = \fra... |
### Question:
Let $f \in C^1([a;b])$, prove that $ \forall x,y \in [a;b]$ we have $|f(x)-f(y)| < || f' ||_2 \sqrt{|x-y|} $
### Answer:
1- $f \in C^1([a;b]) \Rightarrow f'(\chi) $ exists and is defined: $ |f(x)-f(y)|= |\int_x ^y f'(\chi) d \chi | \leq \int_x ^y |f'(\chi)| d \chi$ by triangle inequality.... |
According to [OEIS][1], the formula for getting the number of arrangements of any subset of `n` distinct objects is:
$$\sum_{k=0}^n{n \choose k}k! = \sum_{k=0}^n\frac{n!}{k!(n-k)!}k!\\ = n!\sum_{k=0}^n \frac{1}{(n-k)!} = n! \sum_{k=0}^n \frac{1}{k!}$$
From the first expression to the second expression is the form... |
Why $n! \sum_{k=0}^n \frac{1}{(n-k)!} = n! \sum_{k=0}^n \frac{1}{k!}$? |
Claim: If for any $\epsilon >0$, there exists a $\delta >0$, such that for any $|x-x_0| < \delta$, it holds
$$
(L-\epsilon)^k \leq f(x) \leq (L+\epsilon)^k,
$$
where $k$ and $L$ are positive constants. Then we have
$$
\lim_{x \rightarrow x_0} f(x) =L^k.
$$
-------------------------------------
The fol... |
with the help of the comment of https://math.stackexchange.com/users/4829/jose27
3- We continue by writing $(\int_x ^y |f'(\chi)| d \chi)^2 = (\int_x ^y |f'(\chi)| \cdot 1 d \chi)^2 $ thus we can identify the following inner product $\int_x ^y |f'(\chi)| \cdot 1 d \chi = <|f'(\chi)| | 1 > $
4- By CS inequality... |
I just found this kind of 'conjecture' recently. Define the set of positive prime numbers ≥ 3. Suppose $$ n \ \epsilon \ P $$ Define S = {1; $p_1$; ... $p_{n-1}$} that are primes and 1 which are smaller than $n$.
1. Can you find $a, b \ \epsilon \ S$ that $a + b + c = n$ and $c = p_{n-1}$? Does this apply for all p... |
Are primes made of smaller primes and one? |
How would we decide increasing or decreasing function if the function is not differentiable? Can a discontinuous function be increasing or decreasing?
And can it be monotonic? |
I have a random variable which is distribuited as a multinomial standard gaussian $z\sim N^d(0, I)$, and another r.v. that instead is distribuited as a bernoulli $\alpha \sim Bern(\pi)$. Is it possible to compute the mutual information analytically? If yes, can you show me all the steps to the results? |
I have this graph here picture[![enter image description here][1]][1]. And i tried to calculate this myself with removing an edge and contracting an edge here.
Below [![enter image description here][2]][2]
[1]: https://i.stack.imgur.com/Km5iF.png
[2]: https://i.stack.imgur.com/R51iT.png
And i got my Po... |
How to find this polynomial chromatic function for this graph? |
There are $15$ treadmills at my gym each person uses them for $40$ minutes. The possible times spent on the treadmill, for example $5$ minutes or $20.23$ minutes, are equally likely. I arrive at my gym, I see every treadmill is occupied.
What is the probability that at least one person gets off within $10$ minutes ... |
I just found this kind of 'conjecture' recently. Define the set of positive prime numbers. Suppose $$ n \ \epsilon \ P, n ≥ 3 $$ Define S = {1; $p_1$; ... $p_{n-1}$} that are primes and 1 which are smaller than $n$.
1. Can you find $a, b \ \epsilon \ S$ that $a + b + c = n$ and $c = p_{n-1}$? Does this apply for al... |
I am trying to understand $\mathbb{C}P^2$. Since I understand the Hopf fibration quite well, I like the following construction:
- Attach a $\mathbb{D}^2$ (2-cell) to a point $\mathbb{D}^0$ (0-cell) to get $S^2$ (thanks to @Leo Mosher for suggesting the more sensible order of attaching)
- Now attach $\mathbb{D}^4$ t... |
I just found this kind of 'conjecture' recently. Define the set of positive prime numbers. Suppose $$ n \ \epsilon \ P, n ≥ 3 $$ Define S = {1; $p_1$; ... $p_{x-1}$} that are primes and 1 which are smaller than $n$.
1. Can you find $a, b \ \epsilon \ S$ that $a + b + c = n$ and $c = p_{x-1}$? Does this apply for al... |
I just found this kind of “conjecture” recently. Define the set of positive prime numbers $\geq 3$. Suppose $$ n \in P. $$ Define $S := \{1,p_1,\dots,p_{n-1}\}$ that are primes and $1$ which are smaller than $n$.
1. Can you find $a, b \in S$ that $a + b + c = n$ and $c = p_{n-1}$? Does this apply for all primes?
2.... |
More detailed hints than in the comments.
Why there is no Hamiltonian path in this graph.
For convenience, let the vertices of degree 2 be red. Let the other vertices be blue.
First, note that the red vertices in each path cannot be neighbors, since they form an independent set.
It follows that if a Hamilto... |
I just found this kind of “conjecture” recently. Define the set of positive prime numbers $P$. Suppose $$ n \in P, n\geq 3. $$ Define $S := \{1,p_1,\dots,p_{n-1}\}$ that are primes and $1$ which are smaller than $n$.
1. Can you find $a, b \in S$ that $a + b + c = n$ and $c = p_{n-1}$? Does this apply for all primes?... |
Consider a bookstore that has an inventory of used books $U = \{\text{'To Kill a Mockingbird', '1984', \ldots}\}$, and an inventory of new books $N = \{\text{'Project Hail Mary', 'The Midnight Library', \ldots}\}$. The book selected by a customer buying a used book is a random variable $U$ taking values in the set $\ma... |
When I am plotting two quadratic surfaces $y^2/2 + z^2/2 - 16.7 z = -30$ and $-(x^2/2) + 16.7 z = 200$, then its intersecting orbit is found to be a single orbit (see attached image).
[![enter image description here][1]][1]
But, when I am changing the RHS value of both the expressions, then the shape of the i... |
I just found this kind of “conjecture” recently. Define the set of positive prime numbers $P$. Suppose $$ n \in P, n\geq 3. $$ Define $S := \{1,p_1,\dots,p_{n-1}\}$ that are primes and $1$ which are smaller than $n$.
1. Can you find distinct $a, b \in S$ that $a + b + c = n$ and $c = p_{n-1}$? Does this apply for al... |
Consider a bookstore that has an inventory of used books $U = \{\text{'To Kill a Mockingbird', '1984'}, \ldots\}$, and an inventory of new books $N = \{\text{'Project Hail Mary', 'The Midnight Library'}, \ldots\}$. The book selected by a customer buying a used book is a random variable $U$ taking values in the set $\ma... |
I just found this kind of “conjecture” recently. Define the set of positive prime numbers $P$. Suppose $$ n \in P, n\geq 3. $$ Define $S := \{1,p_1,\dots,p_{n-1}\}$ that are primes and $1$ which are smaller than $n$.
1. Can you find at most 2 distinct $a, b \in S$ (it's like $n$ can be $c$ + $a$ whereas $a$ = 2 and ... |
In my scenario, I have a map of 2d points and a set of disjoint circles of radius 5 that are on the map. I'm shooting a laser beam from position (0,0) at some angle until I shoot a circle. The probability of observing the circle is 1 along its whole area.
The goal is to find the probability distribution of the circle ... |
I want to test Proth's numbers for primality.
These are numbers of the form $h\cdot 2^k+1$ for a natural $k$ and an odd $h<2^k$.
I am wondering whether I can find a faster method than the "normal" trialdivision.
So my thought is:
Let $p$ be an odd prime. Then we have
$$
h\cdot 2^k+1\equiv 0 \pmod p
\Leftrightarr... |
I was reading Ozawa's book and in Chapter 5 they discuss $C^* $-algebras of locally compact Hausforff étale groupoids. I have a question regarding the reduced $C^* $-algebra construction.
It is first defined by considering the $* $-algebra $C_c(G)$ and endow it with a $C_0(G^{(0)})$-valued inner product defined for ... |
Mathematica gives the following asymptotic expansion:
$$
\int_0^\Lambda \frac{1-\cos(qx)}{q} \mathrm{d}q \overset{\Lambda\to\infty}{\sim}\gamma+\log\left(x\Lambda\right) +\mathcal{O}\left(\frac{1}{x\Lambda}\right)
$$
where $\gamma\approx 0.577$ is the Euler-Mascheroni constant. How can this expansion be obtained a... |
I just found this kind of “conjecture” recently. Define the set of positive prime numbers $P$. Suppose $$ n \in P, n\geq 3. $$ Define $S := \{1,p_1,\dots,p_{x-1}\}$ that are primes and $1$ which are smaller than $n$.
(in which $p_{x-1}$ is the prime directly before n)
1. Can you find at most 2 distinct $a, b \in S$ (... |
I was reading Ozawa's book and in Chapter 5 they discuss $C^* $-algebras of locally compact Hausforff étale groupoids. I have a question regarding the reduced $C^* $-algebra construction.
It is first defined by considering the $* $-algebra $C_c(G)$ and endow it with a $C_0(G^{(0)})$-valued inner product defined for ... |
I was reading Ozawa's book and in Chapter 5 they discuss $C^* $-algebras of locally compact Hausforff étale groupoids. I have a question regarding the reduced $C^* $-algebra construction.
It is first defined by considering the $* $-algebra $C_c(G)$ and endow it with a $C_0(G^{(0)})$-valued inner product defined for ... |
Let $K = K(x,y) \in L^{2}(\mathbb{R}^{d}\times \mathbb{R}^{d})$. This function defines an integral operator on $L^{2}(\mathbb{R}^{d})$, which I denote by $K$, as follows:
$$(Kf)(x) = \int_{\mathbb{R}^{d}}K(x,y)f(y)dy$$
This operator $K$ is proven to be a Hilbert-Schmidt operator, with Hilbert-Schmidt norm $\|K\|_{\ma... |
*comment*
Here is the graph of $\sin(xe^x)$
[![graph][1]][1]
Of course the integral $\int_{-\infty}^0$ converves. But how do we know that $\int_0^\infty$ converges?
[1]: https://i.stack.imgur.com/hkDDZ.jpg |
Consider a set $A$ in $\mathbb{R}^3$. Let $(x,y,z)$ denote a representative element of $\mathbb{R}^3$. Let
$$
A(x=3)\equiv\{(x,y,z)\in A: x=3\}.
$$
Does this set has a specific technical name? For instance, something related to the notion of projection? |
In an academic research paper in computer science, I have a formula using the set builder notation in order to define a set $A$, as follows:
$A = \{ y(b_i)=a_i | b_i \in B \}$
In the particular example, I want to convey in a concise way that each element $a_i \in A$ results from the transformation given by the f... |
I have a few questions about solutions that arise from differential equations where Picard-Lindeloef can be applied:
1. In the problem $y'=f(t,y)=-y^2$, solutions have the form $\frac{1}{x-c}$ and always have a discontinuity point, what is causing this? Is it that $f(t,y)$ has a zero for some value of $y$? Will this... |
The problem occurs in the [Convex Optimization II lecture by Stephen Boyd][1]. There we want to choose $\alpha_i$'s to minimize $$\frac{R^2+G^2 \sum_{i=1}^k \alpha_i^2}{2\sum_{i=1}^k \alpha_i}$$
where $\alpha_i>0,R>0,G>0$. The professor argued this in the [video 20:00][2] with the statement that a quadratic over a p... |
How to show that a quadratic over linear function is convex? |
In an academic research paper in computer science, I have a formula using the set builder notation in order to define a set $A$, as follows:
$$A = \{ y(b_i)=a_i | b_i \in B \}.$$
In the particular example, I want to convey in a concise way that each element $a_i \in A$ results from the transformation given by th... |
I am trying out different FFT solvers, with two of them being FFTW and FAST FFT ( https://www.kurims.kyoto-u.ac.jp/~ooura/fft.html).
I am using DCT/inverse DCT to accomplish the following:
(1) **solve a discrete Poisson equation on an NxN map** => result is same between FFTW, FFT
(2) I multiply the DCT result ... |
I am trying out different FFT solvers, with two of them being FFTW (fftw version 3.3.3.8, https://www.fftw.org/index.html) and FAST FFT ( https://www.kurims.kyoto-u.ac.jp/~ooura/fft.html).
I am using DCT/inverse DCT to accomplish the following:
(1) **solve a discrete Poisson equation on an NxN map** => result is ... |
$u,v\in\mathbb{R}^2$, $p(v)$ is bounded and non-negative, and $\int_{\mathbb{R}^2}p(v)dv=1$, prove $f(u,t)=\int_{\mathbb{R}^2}\left[\nabla^\perp\ln||u-v||\right]p(v)dv$ (Here $\nabla$ is gradient operator ,the vector $\mathbf{a}^\perp$ is obtained by rotating vector $\mathbf{a}$ by 90 degrees) satisfies the condition
... |
I am having issues computing Riemann tensor of 2D manifold. It should be very simple, but I cannot see the mistake I am making.
Assume a 2D manifold with metric in very simple form $$\tilde{g}=\frac{1}{f(x)^2}(dx^2+dy^2).$$
Since this is conformally flat, I wanted to use this to my advantage. For example from htt... |
Question from the test:
>Let $X_n$ (for $n=1,2,3,\ldots$) be a random variable with a binomial distribution with parameters $n$ and $p$, and let $\alpha$ be such that $P(Z \geq \alpha)=\frac{1}{2024}$, for $Z \sim N(0,1)$. Find a sequence $a_n$ such that $P(X_n > a_n)$ tends to $\frac{1}{2024}$. (In the formula, you c... |
Given a linear system of equations $A\vec x = \vec b$, if the determinant $\det(A)$ is $0$, then how do we know if the system has no solutions or infinitely many solutions?
Two sub-questions:
a) Using Cramer's Rule, the determinant of $A$ being zero means that a situation of "Division by zero" arises. Then, there... |
So I know that I should have some matrix $A$ such that $\langle x,y\rangle = x^T A y$ for the $L^2 (D)$ inner product. I would like to find this $A$. My domain $D$ is a $3 \times 3$ square grid in $\mathbb{R}^2$. I am assuming that the points making up this grid are $x_i = x_0 + i$ for $i = 0,1,2,3$ and $y_j = y_0 + j$... |
I have a finite set size $N$. If $Count$ is defined as the count of all subsets, $Count(P(N))$ = $2^N$.
My questions, what is $$\sum_{a \in P(N)} Count(P(a))$$?
I imagine there is a basic combinatorial identity that I - an humble programmer - am unaware of. |
Let $K = K(x,y) \in L^{2}(\mathbb{R}^{d}\times \mathbb{R}^{d})$. This function defines an integral operator on $L^{2}(\mathbb{R}^{d})$, which I denote by $K$, as follows:
$$(Kf)(x) = \int_{\mathbb{R}^{d}}K(x,y)f(y)dy$$
This operator $K$ is proven to be a Hilbert-Schmidt operator, with Hilbert-Schmidt norm $\|K\|_{\ma... |
I am reading a paper titled [Transformer can do Bayesian inference][1].
I am lost in proof of insight 1, in which they derive (equation 3):
$$-\int_{D,x,y}p(x,y,D)\log q_{\theta}(y|x,D) = -\int_{D,x}p(x,D)\int_{y}p(y|x,D)\log q_{\theta}(y|x,D)$$
How do they arrive with the term on the right-hand side? is this f... |
I would like to show that $$A = x^T G(x) \tanh(x) \geq 0$$ (with equality only for $x=0$), for $x\in\mathbb{R}^n$. It is known that $G(x)\in\mathbb{R}^{n\times n}$ is a symmetric and positive definite matrix.
Or come up with a counter-example if it does not hold.
I tried transforming to the associated quadratic fo... |
The complementary slackness condition within the KKT optimality conditions states that $\lambda_i^* f_i(x^*) = 0$ for all inequality constraints $f_i(x) \leq 0$ and for optimal primal variable $x^*$ and dual variables $\lambda_i^* \geq 0$. It follows that $\lambda_i^* > 0 \implies f_i(x^*) = 0$ and $f_i(x^*) < 0 \impli... |
Say I have box of 100 balls. 2 are black and 98 white.
Chance to draw 1 black is $2/100=\frac{2\choose1}{{100}\choose{1}}$
Now whats the chance to draw 1 black without return with 2 tries? This must be a bigger chance right?
I know I don't have to use combinatorics, but its just for practice:
$\frac{2\choose... |
Let $X$ be a finite set. Recall that the power set of $X$ is the set of all subsets of $X$. We denote the power set of $X$ by $\mathcal{P}(X)$. It can be proved that, if $X$ has $n$ elements $(n \in \mathbb{N}_0)$, then $\lvert \mathcal{P}(X) \rvert = 2^n$.
**Question.** What is the value of
$ \sum_{A \in \mathcal{... |
I was reading Ozawa's book and in Chapter 5 they discuss $C^* $-algebras of locally compact Hausforff étale groupoids. I have a question regarding the reduced $C^* $-algebra construction.
It is first defined by considering the $* $-algebra $C_c(G)$ and endow it with a $C_0(G^{(0)})$-valued inner product defined for ... |
Say I have box of 100 balls. 2 are black and 98 white.
Chance to draw 1 black is $2/100=\frac{2\choose1}{{100}\choose{1}}$
Now whats the chance to draw at least 1 black without return with 2 tries? This must be a bigger chance right?
I know I don't have to use combinatorics, but its just for practice:
$\frac... |
I am having issues computing Riemann tensor of 2D manifold. It should be very simple, but I cannot see the mistake I am making.
Assume a 2D manifold with metric in very simple form $$\tilde{g}=\frac{1}{f(x)^2}(dx^2+dy^2).$$
Since this is evidently conformally flat (indeed all 2D manifolds are, but here one can se... |
I am trying to get better at doing these formal limit proofs, but I think that I'm kind of unsure how the AP integrates into these.
I split it into two cases; let's first consider $x=0$. Then, let $\epsilon \gt 0$ be arbitrary. By AP, $\exists N \in \mathbb{N}$ s.t. if $n \in \mathbb{N} \gt N$, then $\vert x_n-0 \ve... |
Consider the following statements in $\newcommand{\ZF}{\sf (ZF)}\ZF$:
- (Inj) If $A, B$ are sets, then there is an injection $\iota:A\to B$, or an injection $\iota:B\to A$, or both.
- (Surj) If $A, B$ are non-empty sets, then there is a surjection $s:A\to B$, or a surjection $s:B\to A$, or both.
Clearly, (Inj) i... |
Is "existence of injections / surjections" weaker than AC? |
there are 4 boys and 4 girls, how many ways they are arranged to sit in circular table if the 3 boys always together?
i found someone's video and the answer is just 5!3!, there are 4 boys why he didn't choose 3 boys from 4 boys? let say the 4 boys are A B C D and the girls E F G H why just count cyclic from ABC D E ... |
Cyclic permutation with restriction, am i wrong? |
The question is very straightfoward: It's true that given real vector spaces $V$, $W$, $X$, $Y$ such that $V\cong X$ and $W\cong Y$, then $V\otimes W\cong X\otimes Y$? And if it's true, what is the prove of the unicity, exaclty? I could prove that exists an isomorphisms betwen them, but I couldn't prove the unicity.
... |
If $V\cong X$ and $W\cong Y$, then $V\otimes W\cong X\otimes Y$? |
So I have the PDE given by $u_t+x^2u_x=0$ with the auxiliary condition of $u=\sin(x)$ for $t=0$ and $x>0$. Applying the method of characteristics to solve for $u$ we get the path
$$\frac{dx}{dt}=x^2, x(\epsilon,0)=\epsilon$$
since $\frac{du}{dt}=0$, the PDE reduces to an ODE. From this we get that $u(\epsilon,t)=... |
Question from the test:
>Let $X_n$ (for $n=1,2,3,\ldots$) be a random variable with a binomial distribution with parameters $n$ and $p$, and let $\alpha$ be such that $P(Z \geq \alpha)=\frac{1}{2024}$, for $Z \sim N(0,1)$. Find a sequence $a_n$ such that $P(X_n > a_n)$ tends to $\frac{1}{2024}$. (In the formula, you c... |
I was reading Ozawa's book and in Chapter 5 they discuss $C^* $-algebras of locally compact Hausforff étale groupoids. I have a question regarding the reduced $C^* $-algebra construction.
It is first defined by considering the $* $-algebra $C_c(G)$ and endow it with a $C_0(G^{(0)})$-valued inner product defined for ... |
For example, consider the function $\phi:\mathbb{Q} \rightarrow \mathbb{Z}\times\mathbb{N}$ defined by $\phi(\frac{p}{q})=(p,q)$ for any $\frac{p}{q}\in\mathbb{Q}$. The way this function is defined relies on how the set $\mathbb{Q}$ is defined, that is, $\mathbb{Q}=\{\frac{p}{q}:p\in\mathbb{Z},q\in\mathbb{N}\}$.
I ... |
What is the proper notation when defining certain functions? |
The Lah numbers satisfy the following rising-falling factorials relation
\begin{equation}
x^{\overline{n}} = \sum_{k=0}^{n} L(n, k) x^{\underline{k}},
\end{equation}
where $x^{\overline{n}}$= $x(x+1)\times ... \times (x+n-1)$ and $x^{\underline{n}}$ = $x(x-1) \times ... \times (x-n+1)$.
What would be the p... |
In Statistical Inference, we were taught this theorem,
Consider an estimator $T_n$ of population parameter $\theta$, using $n$ samples. $T_n$ is a Consistent Estimator of $\theta$ if $$E[T_n] \to \theta \text{ and $Var(T_n) \to 0$}$$
However the converse of the above is not true. To show this, I made the fol... |
This is essentially very similar to Sangchul's solution, but I think the approach might be easier to understand/motivate.
Hints/Observations towards a solution. If you're stuck, explain what you've tried.
1. $a_n$ is a decreasing sequence.
- OP mentioned this, but did not explicitly show this. ... |
Let $D$ be a set of lines in the real projective plane such that for all $d\in D$, $\bigcap (D\setminus \left\{d\right\})=\emptyset$ and let $Dot(D)$ be the set of all dots that belong to at least two members of $D$.
$D$ is *complete* if for any pair of dot in $Dot(D)$ the line that contain both dots of the pair is... |
Here is the problem:
The time it takes for a baker to bake a loaf of bread without it being underbaked is normally distributed with a mean of µ minutes and a standard deviation of σ minutes. Bread is considered to be slightly baked if it is baked for longer than (µ + 1.5σ) minutes. What is the probability that bread r... |
This is essentially very similar to Sangchul's solution, but I think the approach might be easier to understand/motivate.
Hints/Observations towards a solution. If you're stuck, explain what you've tried.
1. Show that $a_n$ is a decreasing sequence.
- OP mentioned this, but did not explicitly sho... |
Consider an homogeneous PPP in $\mathbb{R}^2$ with intensity parameter $\lambda > 0$, and denote the set of _generator seeds_ in the spatial domain as $P = \{p_1, p_2, \dots, p_N\}$. At the same time, envision a Voronoi tessellation of the planar point set such that the space is sparated into regions based on the prox... |
For example, consider the function $\phi:\mathbb{Q} \rightarrow \mathbb{Z}\times\mathbb{N}$ defined by $\phi(\frac{p}{q})=(p,q)$ for any $\frac{p}{q}\in\mathbb{Q}$ such that p and q have no common divisor except 1. The way this function is defined relies on how the set $\mathbb{Q}$ is defined, that is, $\mathbb{Q}=\{\f... |
Consider an homogeneous PPP in $\mathbb{R}^2$ with intensity parameter $\lambda > 0$, and denote the set of _generator seeds_ in the spatial domain as $P = \{p_1, p_2, \dots, p_N\}$. At the same time, envision a Voronoi tessellation of the planar point set such that the space is sparated into regions based on the prox... |
# Background
I am trying to understand the paper ["On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg][1]. However, despite my efforts to decipher the paper, I can't follow what it is describing. This paper is written around complex differential geometry, and appears either to p... |
Can you explain Section 2.2 of this paper to someone who hasn't learned Differential Geometry? |
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