instruction stringlengths 12 30k |
|---|
What alternative proof can be used on this theorem? |
This is problem `6.3.8` from *Trigonometry: A Complete Introduction*, by Hugh Neill (ISBN 1473678498). It's in the surveying chapter, which is a practical continuation of the sine and cosine rules of the chapter before that. So I'm assuming you don't need much more than those two rules. It's a simple problem, but I'm ... |
Suppose I have two spheres of arbitrary size $S_1$ and $S_2$ and I intersect them so they overlapped some extent.
What kind of shape would their intersection create? In other words, what does $S_1 \cap S_2$ look like? |
Is the intersection of two spheres and ellipse (or circle)? |
I was playing around with the following series
$$S(x) = \frac{1}{x^2}+\frac{1}{x^3}+\frac{1}{x^5}+\frac{1}{x^7}+\frac{1}{x^{11}}+...=\sum_{p\in primes}\frac{1}{x^p}$$
for $x\in\mathbb{R}$ and $|x|>1$.
And it seems, that for all $x\ge2$ the value of this sum is pretty close numerically to the following expression:
... |
What shape does the intersection of two spheres yield? |
This is problem `6.3.8` from *Trigonometry: A Complete Introduction*, by Hugh Neill (ISBN 1473678498). It's in the surveying chapter, which is a practical continuation of the sine and cosine rules of the chapter before that. So I'm assuming you don't need much more than those two rules. It's a simple problem, but I'm ... |
The random arrival of k phone calls within a time interval of length t is described by the following pdf
$$f(t) = \frac{\lambda^k}{(k-1)!} t^{k-1} e^{-\lambda t}$$
where parameter $$\lambda$$ represent arrival rate and the arrivals are assumed to be independent of each other
(a) Assume that N time interval $$t... |
I believe that the stability of vector bundles or coherent sheaves (defined as the inequality of 'slopes' of its subsheaves) comes naturally from GIT. However, in any literature I can find, it is introduced in a way that consider the family of semi-stable bundles first and take the GIT quotient. I see this is technical... |
This is problem `6.3.8` from *Trigonometry: A Complete Introduction*, by Hugh Neill (ISBN 1473678498). It's in the surveying chapter, which is a practical continuation of the sine and cosine rules of the chapter before that. So I'm assuming you don't need much more than those two rules. It's a simple problem, but I'm ... |
I actually asked a version of this question before, here: https://math.stackexchange.com/questions/4328632/what-is-the-formal-rigorous-definition-of-a-differential-equation. However, I should have asked what an equation is, first, which I did here: https://math.stackexchange.com/questions/4329160/what-is-the-formal-def... |
I'd like to determine the ellipse that is tangent to the $x$ axis at $r_1 = (x_1, 0), x_1 \gt 0$ and the $y$ axis at $r_2 = (0, y_2), y_2 \gt 0$, and also tangent to the line $n \cdot (r - r_0) = 0 $ at unknown points $r_3$, where $r = [x, y]^T$ and $n$ and $r_0$ are given 2-vectors. This is the task.
**My attempt:... |
The random arrival of $k$ phone calls within a time interval of length $t$ is described by the following pdf
$$f(t) = \frac{\lambda^k}{(k-1)!} t^{k-1} e^{-\lambda t}$$
where parameter $\lambda$ represents the arrival rate and the arrivals are assumed to be independent of each other.
(a) Assume that $n$ time in... |
I'd like to determine the ellipse that is tangent to the $x$ axis at $r_1 = (a, 0), a \gt 0$ and the $y$ axis at $r_2 = (0, b), b \gt 0$, and also tangent to the line $\mathbf{n}\cdot (\mathbf{r} - \mathbf{r_0}) = 0 $ at unknown points $r_3$, where $\mathbf{r} = [x, y]^T$ and $\mathbf{n}$ and $\mathbf{r_0}$ are given 2... |
I was trying to prove the following theorem: If $f(x, y)$ is continuous on some region $R \subseteq \mathbb R^2$ then any solution of IVP $$y'(x)=f(x,y(x)), y(x_0)=y_0 \tag{1}$$ is also a solution of integral equation $$y(x)=y_0+\int_{x_0}^x f(t,y(t)) \, dx \tag{2}$$ Converse also holds, meaning that, any continuous so... |
I'm confused why I end up with two matrices that are transposes of each other when I take a tensor inner product of a third order tensor with a vector, when I use two different Kronecker product matrix tensor representations that I believe are equal.
Let $A$ be a matrix and $x$ and $y$ be vectors. Using index notat... |
Does the derivative with respect to a matrix have a Kronecker product matrix representation? |
I'd like to determine the ellipse that is tangent to the $x$ axis at $r_1 = (a, 0), a \gt 0$ and the $y$ axis at $r_2 = (0, b), b \gt 0$, and also tangent to the line $\mathbf{n}\cdot (\mathbf{r} - \mathbf{r_0}) = 0 $ at unknown points $r_3$, where $\mathbf{r} = [x, y]^T$ and $\mathbf{n}$ and $\mathbf{r_0}$ are given 2... |
Given an equalatteral triangle with vertices $A,B,C$, find the perimeter of the triangle if the height of this triangle is $8\sqrt{6}$.
I do not have a visual image for this problem, because I cannot see. However, this shape is not complex, and I think it is really not needed here.
When I first attempted this pro... |
I got this integral in one quantum mechanics problem: $$\int_{-\infty}^{\infty} e^{-i\cdot p_y y/\hbar}e^{-\beta y^2}dy$$ and the solution was apparently $\sqrt{\frac{\pi}{\beta}} e^{-\frac{p_y^2}{4\hbar^2 \beta}}$. Like how? I can see that $\sqrt{\frac{\pi}{\beta}}$ factor since it is a solution to the integral $\int_... |
________________________
1.
Take any separable Hilbert space ($\ell^2$, for example) and a maximal orthonormal set (which must be countable, because our space is separable). This orthonormal set is a Schauder basis. Furthermore we notice that the following series is convergent:
$$S=\sum_{j=1}^\infty \frac{... |
I’ve been trying to understand the details of orthogonal projections within the context of linear regression models, and I’ve come across the following scenario:
> Given a linear regression model $Y=\beta_0+\beta_1X_1+\beta_2X_2+\epsilon$ where $X=\begin{pmatrix} 1 & x_{1,1} & x_{1,2} \\ 1 & x_{2,1} & x_{2,2} \\ 1 ... |
Given an equalatteral triangle with vertices $A,B,C$, find the perimeter of the triangle if the height of this triangle is $8\sqrt{6}$.
I do not have a visual image for this problem, because I cannot see. However, this shape is not complex, and I think it is really not needed here.
When I first attempted this pro... |
Given an equilateral triangle with vertices $A,B,C$, find the perimeter of the triangle if the height of this triangle is $8\sqrt{6}$.
I do not have a visual image for this problem, because I cannot see. However, this shape is not complex, and I think it is really not needed here.
When I first attempted this prob... |
Approaching an infinite natural x value for the expression $5^{2^x}$, one is left with the following p-adic number: $$...212890625$$ Which is equal to its own square. With basic algebra one can see that, since this value is equal to itself squared, it's equal to 1. Therefore, $$\lim_{x \in \mathbb{N} \to \infty} 5^{2^x... |
Proof using p-adic numbers that $\lim_{x \in \mathbb{N} \to \infty} 2^x = 0$? What am I doing wrong? |
What I mean is that the number 3 comes up a lot in these fields as sort of a boundary between decidability and undecidability, or chaos and order.
Examples:
1. Quadratic Diophantine equations are always decidable, but cubic ones are not.
2. Time independent second order ODEs cannot admit chaotic solutions, but ... |
Suppose I have the dynamical system $$x_{i+1} = A_{i+1} x_{i}$$ with state vector $x \in \mathbb{R}^{n}$. If its given that $\lim_{i \to \infty} \Vert x_{i} \Vert = 0$, does this imply $\Vert x_{i} \Vert \leq M$? How do I prove this? |
Is the state norm of an asymptotically stable linear system always bounded? |
I was wondering if you could help me settle a discussion with a co-author. Neither of us is a hard-core topologist but our current paper forced us into this, so we figured this forum was a good place to settle the discussion!
**Setting**
We are thinking of the space $\mathbb{R}^X$ where X is an arbitrary set (so, ... |
Let $A_0$ be a finite set, and consider the sets $A_i = A_{i-1} \cup \mathcal{P}(A_{i-1})$ for all $i > 0$, where $\mathcal{P}$ denotes powerset. Does $\bigcup_{j=0}^{\infty} A_j$ exist as a set?
Intuitively, it seems true because each $A_i$ is a finite set, so the class corresponding to the union must be countabl... |
Proof that the union of iterated powersets of a finite set, is a set? |
I'm trying to calculate the probability that a certain scenario will be dealt in Blackjack (known as "21" to some), after a very random shuffle of a pack of 52 cards. Let me describe the "rules" that should be kept in mind for this question:
* There are 6 players and one dealer.
* The dealer deals each of the six p... |
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 following is my proof.
Since $\lim_{\xi \rightarrow 0} ... |
Given two (1-)categories $\mathcal{C}, \mathcal{D}$, and given the 0-category (class) of funtors $\mathcal{C} \to \mathcal{D}$, denoted $Func(\mathcal{C} \to \mathcal{D})$, let's say we want to make **some** choice of (1-)category denoted $[[\mathcal{C}, \mathcal{D}]]$ such that $Ob([[\mathcal{C}, \mathcal{D}]]) = Func... |
Among morphisms of morphisms, what makes commutative squares special? |
**Guess 1:** We need the morphisms of $Arr(\mathcal{D})$/$[[\mathbb{2}, \mathcal{D}]]$ to in some way be "expressible" in terms of the objects, so that we can define "whiskerings". (Compositions of "1-morphisms" with "2-morphisms" to create new "2-morphisms".)
Commutative squares obviously allow this. But are they a... |
I'm trying to prove the propositions which are in the comments of [Willie Wong's answer](https://mathoverflow.net/a/463341/94522). Let $S = \{ (-2)^k : k \in \mathbb{N} \}, 0\lt \epsilon \lt \frac14$ and $$\chi(t) = \begin{cases}
1 & \exists s\in S, \quad |t-s| < \epsilon \\
0 & \text{otherwise} \end{cases}$$ Also th... |
I'm trying to show that
$$e^{-\mu}\left(\left(\frac{e}{1+\delta}\right)^{(1-\delta)\mu} + \left(\frac{e}{1-\delta}\right)^{(1-\delta)\mu}\right) \le 2e^{-C\mu\delta^2},$$
for an absolute constant $C > 0$, where $\delta \in (0,1]$ and $\mu > 0$.
Using the idea in [this answer][1], I wrote
$$\log(1+\delta) = 0 + \... |
Is the Euclidean group $E(3)$ on $\mathbb{R}^3$ amenable? |
let $ p $ be a prime number.
Consider $ \Bbb Z_p[x] $. Is possible that for a certain polynomial degree bigger or equal to one ; the only irriducibile polynomials of that degree were non monic? |
The following are excerpt from Folland:
Proposition 2.12:
>2.12 Proposition. Let $(X, \mathcal{M}, \mu)$ be a measure space and let $(X, \bar{\mathcal{M}}, \bar{\mu})$ be its completion. If $f$ is an $\mathcal{M}$-measurable function on $X$, there is an $\bar{\mathcal{M}}$-measurable function $g$ such that $f = g$ $\... |
I would like to evaluate the following sum in terms of $n$. The sum is essentially a weighted sum of a binomial distribution with N = 2n and p = $\frac{1}{n}$. I can't figure out how to do it, despite looking through a lot of posts on here and a few papers. Could anyone help?
$\sum_{t=1}^{2n} {2n\choose{t}} (\frac{1... |
If $\sum b_n$ diverges, and $\sum a_n$ converges, and $b_n$ is monotonic does $\lim_{n\to\infty} \frac{a_n}{b_n}$ exist? |
Below is proof outline. Several places switch the order of expectation $f(E(x))=f(E(x))$ which was confirmed to hold in numeric simulations, but I'm curious how to show it more rigorously:
Given $x\sim \mathcal{N}(0, 1)$, define $y=Ax$ for invertible $d\times d$ matrix $A$ and consider 2 solutions of this system:
... |
The wedge product of a $k$-form $\alpha \in \Omega^k(M)$ and a $1$-form $\beta \in \Omega^1(M)$ is given by
$$
\alpha \wedge \beta(X_0,\dots,X_k) = \sum \text{sgn}(\sigma)\alpha(X_{\sigma(0)},\dots,X_{\sigma(k-1)})\beta(X_{\sigma(k)})
$$
where $X_i$'s are vector fields and $\sigma$ runs over all $(k+1)$-shuffl... |
*Note: This is not a duplicate of https://math.stackexchange.com/questions/2947201/understanding-a-proof-that-17-mid-2x3y-iff-17-mid9x-5y or https://math.stackexchange.com/questions/1968750/understanding-a-proof-that-2x-3y-is-divisible-by-17-iff-9x-5y-is-divis*.
I was reading through Naoki Sato's notes on number the... |
How do you find the equation for a best fit line that passes a specific point? |
Below is proof outline. Several places switch the order of expectation $f(E(x))=f(E(x))$ which was confirmed to hold in numeric simulations, but I'm curious how to show it more rigorously:
Given $x\sim \mathcal{N}(0, 1)$, define $y=Ax$ for invertible $d\times d$ matrix $A$ and consider 2 solutions of this system:
... |
I was wondering if you could help me settle a discussion with a co-author. Neither of us is a hard-core topologist but our current paper forced us into this, so we figured this forum was a good place to settle the discussion!
**Setting**
We are thinking of the space $\mathbb{R}^X$ where $X$ is an arbitrary set (so... |
Following this question:https://math.stackexchange.com/questions/1563669/show-that-every-lebesgue-integrable-function-can-be-approximated-in-norm-and-alm
Let $f\in L^1(\mathbb{R})$. Show that there exists a sequence of continuous functions $\{g_n\}$ so that $g_n\to f$ as $n\to \infty$.
In that answer, we find a... |
I am working with the Exercise II.5.14 in Hartshorne's *Algebraic Geometry*. And its problem (a) is stated roughly as follows.
> Let $X$ be a closed subscheme of $\mathbb{P}^r_k$, a projective $r$-space over an algebraically closed field $k$, let $S$ be the homogeneous coordinate ring of $X$, i.e. $S=k[x_0,\dots,x_r... |
$\color{brown}{\textbf{What do we prove?}}$
Really, our goal is to prove the inequality
$$\ln(\ln(500(\sqrt5+1))<2,$$
With $LHS\approx 1.999\,987\,955\,779.$
We can get the intermediate level in table
$e^3\approx 20.085\,536\,923\,19,$
$e^6\approx 403.428\,793\, 432\,7,$
$500ж(\sqrt5+1)\approx 1618.033\,98... |
*Note: This is not a duplicate of https://math.stackexchange.com/questions/2947201/understanding-a-proof-that-17-mid-2x3y-iff-17-mid9x-5y or https://math.stackexchange.com/questions/1968750/understanding-a-proof-that-2x-3y-is-divisible-by-17-iff-9x-5y-is-divis*.
I was reading through Naoki Sato's notes on number the... |
$\color{brown}{\textbf{What do we prove?}}$
Really, our goal is to prove the inequality
$$\ln(\ln(500(\sqrt5+1))<2,$$
We can get the intermediate level in table
$e^3\approx 20.085\,536\,923\,19,$
$e^6\approx 403.428\,793\, 432\,7,$
$500ж(\sqrt5+1)\approx 1618.033\,988\,75,$
$\ln(500(\sqrt5+1))\approx 7.3... |
I tried to solve this problem but got stuck. Here is my work:
Since there are six cafes left to visit, knowing that Cafes R and M are the start and end cafes, I have two choices for each cafe. Hence, the answer is 2^6 or **64**.
After some testing and writing down some possibilities, I saw that this theory was wron... |
Three cafes,R S,and M take one step to go to one of these cafes to another. How many unique ways are there to start at R and end at M in seven steps? |
How to prove two functions cannot be solutions of a same ODE? |
Consider
[![enter image description here][1]][1]
I believe that there is something wrong with this text. In particular, how is
$$\Delta=0 \quad \Longrightarrow \quad V(\Delta)=0$$
completely non trivial by linearity of operators?
Moreover, I do not understand how this is used to find lie point symmetries.
... |
Multiply your series by $$\Gamma(N)\Gamma(a)/ \Gamma(a+N)$$ and your series becomes $$\sum_{n=1}^{\infty}\frac{\Gamma(a)\Gamma(N)\prod_{k=0}^{n-1} (N+k)}{\Gamma(a+N)\prod_{k=0}^{n-1} (a+N+k)}=\sum_{n=1}^{\infty} B(N+n, a) =\int_0^1 (1-x)^{a-1}\sum_{n=1}^\infty x^{N+n-1} \mathrm d x = \int_{0}^1(1-x)^{a-2} x^{N} \mathrm... |
>We have three distinct balls (representing the people here) and we have an unlimited supply of four different types of stickers - S, M, T, W (representing the different days / birthdays). In how many ways can each ball be labeled with a sticker such that exactly two balls have the same type of sticker. Will the answer... |
>Calculate the following limit
$$\lim_{n \to \infty} \int_0^\pi \frac{2(2\sin{x}-\sin{((2n+1)x)}+\sin{((2n-1)x)}}{5-4\cos{(2nx)}-\cos{(4nx)}}dx$$
After some trigonometry manipulation (writing the numerator in terms of $\sin{(nx)}$ and the denominator in terms of $\sin^2{(nx)}$ by double angle cosine formulas) we ge... |
Let $X$ and $Y$ be complete and separable metric spaces with Borel probability measures $\mu$ and $\nu$, respectively. Now take any distribution $p$ on $X\times Y$ so that the marginal of $p$ on $X$ is $\mu$ and the marginal on $Y$ is $\nu$. That is, we have
$\begin{align} p(A\times Y) &= \mu(A) \textrm{ for any me... |
Let $p$ be a probability dist. on $X\times Y$ s.t. marginal of $p=\mu$. Is $p\otimes p(A\times (Y\times Y)) = \mu\otimes \mu(A)$ true? |
I was thinking about math and came up with the following idea. It seems very obvious but I would like to see a concrete rigorous proof (albeit possibly a one-liner) for the following.
Consider a regular $n-$gon. Take one arbitrary point on each the sides of the $n-$gon (these points cannot be vertices). Prove that i... |
In the literature of symmetry groups, Sophus Lie define the symmetry of a pde or an ode by a vector field defined in the tangent space of the submanifold (defined by solutions of the pde or ode) and search when this pde or ode is invariant under the action of the one parameter group of transformations generated by the ... |
How we define the Lie symmetry of a stochastic differential equation? |
I'm not sure if Folland did assume (either implicitly or explicitly) that the measure space it complete, but based on the presence of Proposition $2.12$, I'm guessing he did (although he may not have said to explicitly). Other things you're quoting here seem to confirm that. However, let's cover why, regardless of what... |
Regarding the last section **B**, the gamma 1/3 cube:
$$\,_2F_1\left(\frac{1}{12},\frac{7}{12};1;\frac{3^2}{253^2}\right) = \frac{253^{1/6}\,\Gamma^3\big(\tfrac{1}{3}\big)}{2\pi^2\sqrt{6}} $$
where $253 = 11\times23.\,$ [Testing it ][1] in Wolfram. Source: https://iamned.com/math.
[1]: https://www.wolfram... |
In reading the following article: https://www.researchgate.net/publication/262736434_The_Chetaev_Theorem_for_Ordinary_Difference_Equations
Theorem 1 seems to prove a discrete-time analog of Chetaev theorem. The proof is quite elementary, but I believe it is wrong. In particular, I do not think the quantity $\gamma$ ... |
---
Context:
---
You and 4 other people are sitting in a circle. You are given a ball to start the game. Every second of this game, the person with the ball has three choices they can make. They can either pass the ball to the left, pass the ball to the right, or keep the ball (all with equal probability). This ga... |
I have a 4 by 4 matrice:
$$
A = \begin{bmatrix}
2 & -2 & 0 & 0\\
2 & -4 & 2 & 0\\
0 & -2 & 4 & -2\\
0 & 0 & 2 & -4
\end{bmatrix}
$$
How would I use Gaussian elimination to find the LU factorisation of the matrix?
Please could someone explain how to do this!? I have an exam where a similar question will come... |
I'm not sure if Folland did assume (either implicitly or explicitly) that the measure space it complete, but based on the presence of Proposition $2.12$, I'm guessing he did (although he may not have said to explicitly). Other things you're quoting here seem to confirm that. However, let's cover why, regardless of what... |
I would like to evaluate the following sum in terms of $n$. The sum is essentially a weighted sum of a binomial distribution with N = 2n and p = $\frac{1}{n}$. I can't figure out how to do it, despite looking through a lot of posts on here and a few papers. Could anyone help?
$$\sum_{t=1}^{2n} {2n\choose{t}} \left(\... |
Let $\gamma$ be a real constant with $\gamma^2\neq 1$ for the parametric functional $S[x, y]=\int_{0}^{1}[\sqrt{\dot{x}^2+2\gamma\dot{x}\dot{y}+\dot{y}^2}-\lambda(x\dot{y}-\dot{x}y)]dt, \lambda>0,$ with the boundary conditions $x(0)=y(0)=0, x(1)=R>0$ and $y(1)=0$. The stationary paths of this parametric functional are ... |
How should I show that the equations for $x$ and $y$ can be written in the form $(1-\gamma^2)x'=D-X$? |
The following are excerpt from Folland:
Proposition 2.12:
>2.12 Proposition. Let $(X, \mathcal{M}, \mu)$ be a measure space and let $(X, \bar{\mathcal{M}}, \bar{\mu})$ be its completion. If $f$ is an $\mathcal{M}$-measurable function on $X$, there is an $\bar{\mathcal{M}}$-measurable function $g$ such that $f = g$ $\... |
What is the definition of a vector field generating a Lie group of symmetries which is not necesserally a 1-parameter group? |
I am aware that in a first countable space (and thus any metric space) is completely determined by its convergent sequences *and* their limits, *i.e.*,
> If $\tau_1$ and $\tau_2$ are two first countable topologies on a set $X$ such that $x_i\to c$ in $\tau_1$ iff $x_i\to c$ in $\tau_2$, then $\tau_1 = \tau_2$.
Ho... |
Is the metric topology determined by its convergent sequences? |
Multiply your series by $$\Gamma(N)\Gamma(a)/ \Gamma(a+N)$$ and your series becomes $$\sum_{n=1}^{\infty}\frac{\Gamma(a)\Gamma(N)\prod_{k=0}^{n-1} (N+k)}{\Gamma(a+N)\prod_{k=0}^{n-1} (a+N+k)}=\sum_{n=1}^{\infty} B(N+n, a) =\int_0^1 (1-x)^{a-1}\sum_{n=1}^\infty x^{N+n-1} \mathrm d x = \int_{0}^1(1-x)^{a-2} x^{N} \mathrm... |
For some unknown $A \in \mathbb{R}^{n\times n}$, if $A$ is symmetric, I think the function $f: \mathbb{R}^n \to \mathbb{R}$, $x \mapsto x^{\top}Ax$ determines what $A$ is. I think this is true, and I should've known how to prove this, but right now I'm not very sure on how to proceed.
One always have $x^{\top}Ax = \... |
Multiply your series by $$\Gamma(N)\Gamma(a)/ \Gamma(a+N)$$ and your series becomes $$\sum_{n=1}^{\infty}\frac{\Gamma(a)\Gamma(N)\prod_{k=0}^{n-1} (N+k)}{\Gamma(a+N)\prod_{k=0}^{n-1} (a+N+k)}=\sum_{n=1}^{\infty} B(N+n, a) =\int_0^1 (1-x)^{a-1}\sum_{n=1}^\infty x^{N+n-1} \mathrm d x = \int_{0}^1(1-x)^{a-2} x^{N} \mathrm... |
When I put this function in GeoGebra calculator suite, I am getting the minimum value 4 but when I tried to solve, the minimum value I'm getting is different:
**My attempt:**
${27^6}$ +$\frac{1}{(^2−^2−^2+2)}$ can be written as ${27a^6}$+$\frac{1}{a^2-(b-c)^2}$
**i.e. ${27a^6}$+$\frac{1}{(a+b-c)(a-b+c)}$**
... |
Which distribution $p$ is most similar to a Gaussian but with $[0,\infty)$ support (i.e., $p(0)=0$)? |
Which distribution is most similar to a Gaussian but with $[0,\infty)$ support? |
Is the only continuous outer automorphism of $\text{GL}(n, \mathbb{R})$, the transpose inverse map:
$$
GL(n,\mathbb{R}) \ni g \mapsto (g^\intercal)^{-1} ?
$$
If not, what other possible continuous outer automorphisms are there? |
I want to find $\varphi: \mathbb{S}^2 \to \mathbb{C}_{\infty}:=\mathbb{C} \cup \{\infty \}$, so let us start from the following fact
When considering the equation of the line passing from the North Pole to a point in the complex extended plane $l(t)=((1-t)x, (1-t)y, t)$ for $0 \leq t \leq 1$ we find the point of int... |
Let $X$ and $Y$ be complete and separable metric spaces with Borel probability measures $\mu$ and $\nu$, respectively. Now take any distribution $p$ on $X\times Y$ so that the marginal of $p$ on $X$ is $\mu$ and the marginal on $Y$ is $\nu$. That is, we have
$\begin{align} p(A\times Y) &= \mu(A) \textrm{ for any me... |
The famous Ramanujan constant $ e^{\pi \sqrt{163}} $ is a near-integer.
see the link [here.][1]
I tried to calculate this number with matlab and failed. Matlab cannot even deliver the first 9 apparently because of the double float precision.
It seems that we need a customized algorithm. Could anyone give a ... |
I have studied Hypothesis testing from [thinkstats2][1] which does simulation to estimate p value.
For e.g. In coin toss example, we run the simulation, to generate the test statistic i.e. Difference between the number of Heads and Tails. p value is the proportion of test statistics greater than observed. So intuiti... |
The famous Ramanujan constant $ e^{\pi \sqrt{163}} $ is a near-integer.
see the link [here.][1]
I tried to calculate this number with matlab and failed. Matlab cannot even deliver the first 9 apparently because of the double float precision.
```
2.625374126407683e+17
```
It seems that we need a ... |
Background:
For the [dual numbers](https://en.wikipedia.org/wiki/Dual_number), we extend the reals with an additional unit vector $\epsilon$ subject to the constraint that $\epsilon^2 = 0$. We can write dual numbers as $x_0 + x_1 \epsilon$ for $x_0,x_1 \in \mathbb{R}$.
We have a rule for multiplication,
$$
(x_0... |
I am learning about the St Petersburg Paradox https://en.wikipedia.org/wiki/St._Petersburg_paradox - here is my attempt to summarize it:
- A fair coin is tossed at each stage.
- The initial stake begins at 2 dollars and is doubled every time tails appears.
- The first time heads appears, the game ends and the play... |
I am learning about the St Petersburg Paradox https://en.wikipedia.org/wiki/St._Petersburg_paradox - here is my attempt to summarize it:
- A fair coin is tossed at each stage.
- The initial stake begins at 2 dollars and is doubled every time tails appears.
- The first time heads appears, the game ends and the play... |
I am learning about the St Petersburg Paradox https://en.wikipedia.org/wiki/St._Petersburg_paradox - here is my attempt to summarize it:
- A fair coin is tossed at each stage.
- The initial stake begins at 2 dollars and is doubled every time tails appears.
- The first time heads appears, the game ends and the play... |
I am learning about the St Petersburg Paradox https://en.wikipedia.org/wiki/St._Petersburg_paradox - here is my attempt to summarize it:
- A fair coin is tossed at each stage.
- The initial stake begins at 2 dollars and is doubled every time tails appears.
- The first time heads appears, the game ends and the play... |
I am learning about the St Petersburg Paradox https://en.wikipedia.org/wiki/St._Petersburg_paradox - here is my attempt to summarize it:
- A fair coin is tossed at each stage.
- The initial stake begins at 2 dollars and is doubled every time tails appears.
- The first time heads appears, the game ends and the play... |
I am learning about the St Petersburg Paradox https://en.wikipedia.org/wiki/St._Petersburg_paradox - here is my attempt to summarize it:
- A fair coin is tossed at each stage.
- The initial stake begins at 2 dollars and is doubled every time tails appears.
- The first time heads appears, the game ends and the play... |
I am learning about the St Petersburg Paradox https://en.wikipedia.org/wiki/St._Petersburg_paradox - here is my attempt to summarize it:
- A fair coin is tossed at each stage.
- The initial stake begins at 2 dollars and is doubled every time tails appears.
- The first time heads appears, the game ends and the play... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.