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This is question will be about the theorem behind an efficient algorithm, which has been [used][1] in LeetCode problem [3086][2]. Let me start by exposing the problem in a friendly way:
You are a bus driver who will pick rockers from the Rock'n Rio festival and take them to a car parking lot, were each rocker will p... |
I don't have much background in MDPs but I was reading a text in which I came across the following which I am finding difficult to understand. Consider a Markov decision process with state space $S$ where the only action possible for state $s=0$ is $a=0$ and the reward on completing the action is $r_a$ and the task tim... |
Peano arithmetic is the following list of axioms (along with the usual axioms of equality) plus induction schema.
1. $\forall x \ (0 \neq S ( x ))$
2. $\forall x, y \ (S( x ) = S( y ) \Rightarrow x = y)$
3. $\forall x \ (x + 0 = x )$
4. $\forall x, y \ (x + S( y ) = S( x + y ))$
5. $\forall x \ (x \cdot... |
> There is a committee of 11 people where 3 people are from Group A, 6
> from Group B, and 2 from Group C. The first part is to find the number
> of ways they can sit, if people in the same group are
> indistinguishable. Now the second part is to find the probability that
> the first and last people in the row are ... |
First, it should be noted that computers store [floating-point numbers](https://en.wikipedia.org/wiki/IEEE_754) in what's essentially scientific notation, with a sign bit ($+$ or $-$), fixed-width, “significand” or “mantissa”, and exponent indicating a power of two:
$$x = (-1)^sm2^p, s \in \{0, 1\}, m \in [1, 2), p ... |
I know that [all trees are planar][2], and I know that 3-connected graphs of degree $n \geq 4$ are planar (this is given as Exercise 5.14 in [[1]]). With this, I couldn't help but wonder if 2-connected graphs were planar. I would imagine that this is true given that all 2-connected graphs have an ear decomposition (The... |
Are all 2-connected graphs planar? |
I'm trying to **construct a totally ramified covering (of order $d$) of the complex projective line $\mathbb{P}^1$ with exactly 3 branch points $0,1,\infty$**.
Here is what I'm trying: consider the smooth projective curve obtained as the normalization of the affine curve $$y^d=x^a(x-1)^b,$$ where $a$, $b$, and $a+b... |
Given the open endedness of this question, I hope you don't mind if I answer with an anecdote.
When I was an undergraduate at MIT, I took abstract algebra. The professor was the legendary Michael Artin. At the time I was reading *Adventures in Group Theory* by David Joyner, and in the book, it stated that if you t... |
Let $K$ be a finite extension of $\mathbb{Q}$. Is it possible for non-trivial elements in $\text{Gal}(K/\mathbb{Q})$ to be continuous everywhere on $K$? Yes -- for instance, if $K$ is *not* a real number field, then complex conjugation is a non-trivial automorphism that is continuous.
But if $K$ is real, then non-tr... |
If it weren't for the square-root in the last step of the Laguerre root-finding algorithm, the algorithm itself could operate over the p-adics (or just about anything that has a useful concept of derivative).
Is there an alternative operation that could be substituted for the square-root, even if that meant slower c... |
p-adic Laguerre root-finding algorithm? |
I just started taking my first statistics course and I seem to be having trouble understanding this concept and answering this question.
If the radius of a circle is an exponential random variable, how do you find the density function of the area?
I am wondering if anyone can point me to any resources or help me ... |
Say that one polygon $P$ is _inscribed_ in another one $Q$ if $P$ is contained entirely in (the interior and boundary of) $Q$ and every vertex of $P$ lies on an edge of $Q$. It's clear that a regular $m$-gon can be inscribed in a regular $n$-gon if $m\mid n$ or $m=2n$. Well-known results (e.g. the [Square Peg problem][... |
**Question**
Prove by contradiction the following statement:
One cannot place the numbers 1, 2, 3, 4, 5, 6, and 7 to make the sum of the numbers on the hexagon equal to 7 times the number in the center.
[![Image of hexagon][1]][1]
Let a, b, c, d, e, f be the numbers on the hexagon, and g the number in the c... |
Please, how do I decompose a signed permutation into simple transpositions? I need a concrete example to do this. I represented the signed permutation 3 -4 -1 2 in its cycle form (1 3 - 1 3) (2 -4 -2 4) but I cannot decompose this permutation into simple transpositions. |
Consider $x=0$. Using Stirling's approximation you find
$$P(S_{2m}=0)=\binom{2m}{m}2^{-2m}\sim \frac{1}{\sqrt{\pi m}},$$
which yields
$$\sum_{m=0}^\infty P(S_{m}=0)= \infty.$$
Define
$$\tau_{0}^{(m)}:=\inf\{m>\tau_{0}^{(m-1)}: S_m=0\}$$
and use the Markov property to find $P(\tau_{0}^{(m)}<\infty)= P(... |
I got stuck in a problem that in the end I needed to know what is the normal cone to a ball $B = \left\{ x \in \mathbb{R}^n : \lVert x \rVert \leq 1 \right\}$ (here $\lVert \cdot \rVert$ is an arbirtrary norm), at a point at the boundary $\partial B$. Recalling that the normal cone to a closed convex set $C$ at a point... |
Let $x \in [0,1]$ be an *irrational algebraic number*, and $y \in [0,1]$ algebraic. Does there exist an effective procedure for determining whether there exists some integer $k \geq 1$ such that
$$
kx \equiv y \mod 1?
$$
This can be thought of a question about circle rotations from dynamics: marking a point at $0$... |
Let $K$ be a finite extension of $\mathbb{Q}$. Is it possible for non-trivial elements in $\text{Gal}(K/\mathbb{Q})$ to be continuous everywhere on $K$? Yes -- for instance, if $K$ is *not* a real number field, then complex conjugation is a non-trivial automorphism that is continuous.
But if $K$ is real, then non-tr... |
I need to prove this:
Let X,Y metric compact spaces, f:X -> Y continuous and bijective, then f is an homeomorfism.
But i have been investigating, and there's no such theorem, cause it says that necessary "Y" needs to be a Hausdorff space.
Can someone explain me why "Y" needs to be a Hausdorff space?
Tha... |
¿Does it needs to be a Hausdorff space? |
I need to prove that:
> Let $X$, $Y$ be metric compact spaces, $f:X \to Y$ be continuous and bijective. Then $f$ is an homeomorphism.
But I have been investigating, and there's no such theorem, because it says that $Y$ needs to be a Hausdorff space.
Can someone explain me why $Y$ needs to be a Hausdorff space?... |
Does a metric space need to be a Hausdorff space? |
Is it true that a field automorphism must fix the elements of any proper subfield?
Ie, since $\mathbb{Q}\subset\mathbb{R}$, all automorphisms from R to itself must preserve the map from $I:\mathbb{Q}\to\mathbb{Q}$?
note: I know that there is only one automorphism of $\mathbb{R}$ --- I am more curious to see if t... |
**Definitions**
Consider a generic smooth closed curve $c(t):[0,1]\mapsto \mathbb{R}^2$ and define its *barycenter* $g$ as
\begin{equation*}
g \triangleq \frac{\int_0^1 c(t)\,\lVert \dot{c}(t)\rVert \text{ d}t}{L}
\end{equation*}
where $L\triangleq\int_0^1 \lVert \dot{c}(t)\rVert \text{ d}t$ and
\begin{equation... |
[enter image description here][1]
I was trying to understand this lemma in the book of Rotman, but I am confused while understanding the well-defined part, mainly how they defined the map. Can anyone please explain how the map is well-defined?
[enter image description here][2]
To avoid this, I did it another... |
I want to solve the Forward Kolmogorov equation by Finite element method:
$Lf = (\nabla f)^T b + \frac{1}{2}tr((\nabla^2f)D)$
where $D = \begin{pmatrix}
D_{11} & D_{12}\\
D_{21} & D_{22}
\end{pmatrix}$ is the diffusion matrix.
For standard laplacian problem:
$\Delta f = 0, x\in \Omega$, I know the weak fo... |
I know that [all trees are planar][2], and I know that 3-connected graphs of degree $n \geq 4$ are planar (this is given as Exercise 5.14 in [[1]]). With this, I couldn't help but wonder if 2-connected graphs were planar. I would imagine that this is true given that all 2-connected graphs have an ear decomposition (The... |
I would like to ask Python to compute the determinant of a large symmetric matrix where all off diagonal entries are known. The diagonal entries could vary. Since I need to compute the determinant many times with different diagonal entries, it seems a waste of time when the computation involves the multiplications of a... |
How to efficiently compute the determinant of a matrix with unknown diagonal entries? |
So in metric spaces, basic open sets are open balls, right?
Now pick two points $a\neq b$ from, there you can define distance between them in metric space, say it is $\delta (>0)$, and now construct the open balls $B(a;\frac{\delta}{4})$ (centered at $a$) and $B(b;\frac{\delta}{4})$, now is there anything inside int... |
So in metric spaces, basic open sets are open balls, right?
Now pick two points $a\neq b$ from $Y$, there you can define distance between them in metric space, say it is $\delta (>0)$, and now construct the open balls $B(a;\frac{\delta}{4})$ (centered at $a$) and $B(b;\frac{\delta}{4})$, now is there anything inside... |
I know that [all trees are planar][2], and so now I'm wondering whether 2-connected graphs are necessarily planar. I would imagine that this is true given that all 2-connected graphs have an ear decomposition (Theorem 4.10 in [[1]]), and that every ear decomposition I have seen is planar, like the following example (Fi... |
I'm currently studying multiple linear regression on my own, and I've come here with three questions.
Let **y** $=(y_1, y_2, ... , y_n)^{T}$ the response variable and **X** be the n $\times$ p matrix. Using the parameter vector $\beta = (\beta_1, \beta_2, ... , \beta_p)$, the following equation can be yielded: **y**... |
Let $A,B$ be real $n\times n$ matrices (if we want complex entries, replace transpose with hermitian conjugate). Write them as an array of columns so that $A=\begin{pmatrix} A_1~|&\cdots& |~A_n \end{pmatrix}$, and $B=\begin{pmatrix} B_1~|&\cdots& |~B_n \end{pmatrix}$. By definition, the matrix product $AB$ is an array ... |
Let $u(x, t)$ be the solution to the initial-boundary value problem of the
wave equation for x, t > 0:
$u_{tt} = 4u_{xx}, u_x(0, t) = 0, u(x, 0) = φ(x) = 2x − 1$ and
$u_t(x, 0) = ψ(x) = 2x$ if $x ∈ (2, 10)$ and $0$ if $x ∈ (0, 2] ∪ [10,∞)$.
So I know how to set this equation up using $u(x, t) = 1/2 [φ_{odd}(x + 2... |
*Too long for comments.*
Using q-Pochhammer symbols amke some inexpected factors to appear in the problem.
$$f_n(a) = \prod_{i=0}^n ( \sin(i) +a)$$
$$f_n(a) = i\,\frac {(-1)^n}{2^{n+1} }\,e^{-\frac{i}{2} n (n+\pi +1)}\left(i \left(a-\sqrt{a^2-1}\right);e^i\right){}_{n+1} \left(i
\left(a+\sqrt{a^2-1}\right... |
I know that $\forall x [R(x) \rightarrow S(x)]$ is NOT equivalent to $\forall x R(x) \rightarrow \forall x S(x)$, because universal quantification does not distribute over disjunction. However, it seems to me that $$(*) \forall x [R(x) \rightarrow S(x)] \rightarrow [\forall x R(x) \rightarrow \forall x S(x)]$$ just int... |
why does $\mathbb{Q}(3^{1/8},\zeta)=\mathbb{Q}(3^{1/8},2^{1/2}, i)$?
I know that $\mathbb{Q}(3^{1/8},\zeta)=\mathbb{Q}(3^{1/8},3^{1/8}e^{2\pi/8},3^{1/8}e^{4\pi/8},3^{1/8}e^{6\pi/8},3^{1/8}e^{8\pi/8},3^{1/8}e^{10\pi/8},3^{1/8}e^{12\pi/8},3^{1/8}e^{14\pi/8})$
$=\mathbb{Q}(3^{1/8},3^{1/8}e^{\pi/4},3^{1/8}e^{\pi/2},3^{... |
I'm currently studying multiple linear regression on my own, and I've come here with three questions. Photos are included, but I'll try my best to explain in LaTeX code as much as I possible. Please consider the photos for references.
Let $Y=(y_1, y_2, ... , y_n)^{T}$ the response variable and $X$ be the n $\times$ ... |
Assume
$$
F\left( x \right) := \sum_{n=-\infty}^{+\infty}{x^{\left( n+\frac{1}{2} \right) ^2}}, G\left( x \right) := \sum_{n=-\infty}^{+\infty}{x^{n^2}}, H\left( x \right) := \sum_{n=-\infty}^{+\infty}{\left( -1 \right) ^nx^{n^2}},
$$
I have to prove
$$
\left( G\left( x \right) \right) ^2+\left( H\left( x \right)... |
[![enter image description here][1]][1]
I was trying to understand this lemma in the book of Rotman, but I am confused while understanding the well-defined part, mainly how they defined the map. Can anyone please explain how the map is well-defined?
[![enter image description here][2]][2]
To avoid this, I di... |
> **Lemma 10.26.** Let $G$ be a group acting transitively on a set $Y$, and let $y_0 \in Y$. Then $$\mathrm{Aut}(Y) \cong N_G(G_0)/G_0,$$ where $G_0$ is the stabilizer of $y_0$.
I was trying to understand this lemma in the book of Rotman, but I am confused while understanding the well-defined part, mainly how they d... |
>The natural question which now arises is whether the Fourier series of $f$
converges to $f$, or, more generally, whether $f$ is determined by its Fourier series.
That is to say, if we know the Fourier coefficients of a function, can we find
the function, and if so, how?
This was on page 186 and then on page 18... |
> **Lemma 10.26.** Let $G$ be a group acting transitively on a set $Y$, and let $y_0 \in Y$. Then $$\mathrm{Aut}(Y) \cong N_G(G_0)/G_0,$$ where $G_0$ is the stabilizer of $y_0$ and $N_G(G_0)$ is the normalizer of $G_0$.
I was trying to understand this lemma in the book of Rotman, but I am confused while understandin... |
**Idea**
There are associative, commutative, and idempotent algebraic structures. This gives eight categories, an "eight-fold way". What is the ideal terminology for such a categorization as it relates to discrete mathematics, graph theory, and theoretical computer science? More formally:
**Introduction To Que... |
> **Lemma 10.26.** Let $G$ be a group acting transitively on a set $Y$, and let $y_0 \in Y$. Then $$\mathrm{Aut}(Y) \cong N_G(G_0)/G_0,$$ where $G_0$ is the stabilizer of $y_0$ and $N_G(G_0)$ is the normalizer of $G_0$.
I was trying to understand this lemma in the book of Rotman, but I am confused while understandin... |
Let $(P, \le)$ be a partially ordered set. A subset $A\subseteq P$ is a *chain* if any two elements of its elements are comparable. A subset $A\subseteq P$ is *cofinal* if every element of $P$ is less than or equal to some element of $A$.
> **Question:** Suppose $P$ admits some cofinal chain. Does every cofinal s... |
In a poset with a cofinal chain, does every cofinal subset admit a cofinal chain? |
Let us consider the dynamical system
\begin{aligned}
\dot{x}&=\frac{(a-x)}{b} -z-u, \\
\dot{y}&=x-\min (\beta u, c y), \\
\dot{z}&=x-y, \\
\dot{u}&=x-1
\end{aligned},
where $a,b,c$ are parameters
We can observe that the equilibrium point of the system is of the form $(x, y,z, u) = (1,1,\frac{a-1}{b}-t,t$, wh... |
> **Lemma 10.26.** Let $G$ be a group acting transitively on a set $Y$, and let $y_0 \in Y$. Then $$\mathrm{Aut}(Y) \cong N_G(G_0)/G_0,$$ where $G_0$ is the stabilizer of $y_0$.
$N_G(G_0)$ is the normalizer of $G_0$ and $\mathrm{Aut}(Y)$ is set of all $G-$ isomorphisms, $Y\rightarrow Y$ (bijective $G-$ maps $f(g\... |
Suppose a sample of a certain substance decayed to 65.2% of its original amount after 300 days. (Round your answers to two decimal places.)
(a)
What is the half-life (in days) of this substance?
i just cant process how to get all the values for the Decay Rate Formula:
M(t) = M(0)e^kt |
how do you find the initial amount of a decay problem when you dont have one in the problem? |
Let $A,B$ be real $n\times n$ matrices (if we want complex entries, replace transpose with hermitian conjugate). Write them as an array of columns so that $A=\begin{pmatrix} A_1~|&\cdots& |~A_n \end{pmatrix}$, and $B=\begin{pmatrix} B_1~|&\cdots& |~B_n \end{pmatrix}$. By definition, the matrix product $AB$ is an array ... |
Let us consider the dynamical system
\begin{aligned}
\dot{x}&=\frac{(a-x)}{b} -z-u, \\
\dot{y}&=x-\min (\beta u, c y), \\
\dot{z}&=x-y, \\
\dot{u}&=x-1
\end{aligned},
where $a,b,c$ are parameters
We can observe that the equilibrium point of the system is of the form $(x, y,z, u) = (1,1,\frac{a-1}{b}-t,t)$, w... |
Suppose a sample of a certain substance decayed to $65.2\%$ of its original amount after $300$ days.
What is the half-life (in days) of this substance? (Round your answers to two decimal places.)
I just cant process how to get all the values for the Decay Rate Formula:
$M(t) = M(0)e^{kt}$
|
Let us consider the dynamical system
\begin{aligned}
\dot{x}&=\frac{(a-x)}{b} -z-u, \\
\dot{y}&=x-c y, \\
\dot{z}&=x-y, \\
\dot{u}&=x-1
\end{aligned},
where $a,b,c$ are parameters
We can observe that the equilibrium point of the system is of the form $(x, y,z, u) = (1,1,\frac{a-1}{b}-t,t)$, where $t \in \Bbb... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/HIR5N.png
The first part of this exercise seems straightforward to me, but I'm a bit confused about the second part. In the case $\alpha$ is transcendental we know that $k(\alpha)$ is actually isomorphic to $k(t)$ so it must be referring t... |
The following is from [Rasmussen and Williams (2006)][1]:
> A simple example of a Gaussian process can be obtained from our Bayesian linear regression model $f(x)=\phi(x)^Tw$ with prior $w\sim\mathcal N(0,\Sigma_p)$. We have for the mean and covariance
$$
\begin{aligned}
\mathbb E[f(x)]&=\phi(x)^T\mathbb E[w]=0\\... |
Can a Gaussian process have a degenerate subset? |
I am trying to prove the following.
**Theorem?** (Seifert-Van Kampen theorem for Lie-groupoids, a partial statement). Let $G_1\substack{\to \\ \to}G_0$ (denoted simpy by $G$) be a Lie-groupoid. Let $x_0\in G_0$. If $G_0$ is the union of open sets $A_\lambda$ each containing $x_0$ and if each restriction $G|_{A_\lamb... |
Seifert-Van Kampen theorem for Lie-groupoids? |
Let $f(x)$ and $g(x)$ be two functions with finite bandwidth, i.e., the support of their Fourier transform has length $a$ and $b$, respectively. How can we show that the bandwidth of $f(x)g(x)$ is upper bounded by $a+b$? It's not direct to see this result by simply doing the Fourier transform $\frac{1}{2\pi}\int_{-\inf... |
Bandwidth of the product of two functions? |
Seifert-Van Kampen theorem for Lie-groupoids (or orbifolds)? |
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 ma... |
If I have a equation
$$\int_0^{t} \int_{s(\tau)}^D f(x, \tau) d x d \tau=0$$
and I want to solve $s(t)$ for any given $t$. Here $D$ is a constant and $f(x,\tau)$ is a known function, is there a numerical method for solving $s(t)$? I was thinking a method like Runge–Kutta method for ODE, but I'm not sure. Any help is... |
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 ma... |
I have always wanted to solve geometry problems more simply using algebraic methods (for example, point geometry is one way, which uses one variable instead of two in 2D plane geometry). Still, the problem down below is so frustrating (and common, too) that I cannot find a good way to solve it.
I have four points $A... |
Fix $y_0$. Given $\varphi\in\mathrm{Aut}_G(Y)$, $\varphi(y_0)\in Y$. Since the action is transitive, there exists a $g\in G$ such that $gy_0=\varphi(y_0)$. Rotman wants to map it to a coset of $G_0$ in $N_G(G_0)$, namely to the coset $g^{-1}G_0$. But $g$ need not be unique. So he needs to verify two things for this f... |
let
$$ C_0= { f | f \in C[0,1] \land f(\frac{1}{2})=0 } $$
then $ C_0$ is a subspace of C[0,1] and $ C_0/C[0,1] $ is isomorphic to R.
I have proved that $ C_0 $ is a vector subspace.
I tried to prove that $ C_0/C[0,1] $ is isomorphic to R
I know that
$ C_0/C[0,1] = \{ f+C_0 : f \in C[0,1] \} $
I tried th... |
I think the lemma 2.3 in this [paper](https://arxiv.org/pdf/1401.7158.pdf) will help a lot in my homework, but I am confused about some steps of the proof. The lemma and its proof are as follows:(The proof in the paper begins at page 9.)
Lemma: Let $h \not\equiv 0$ be a holomorphic function on the disc $\Delta r$ in... |
$$\sin(\pi x)=\pi x \prod_{n \ge 1}\left(1-\frac{x^2}{n^2}\right)$$
$$\pi = \frac{\sin(\pi x)}{x\prod_{n \ge 1}\left(1-\frac{x^2}{n^2}\right)}$$
Let $x=\frac{1}{2}$
$$\pi = \frac{2}{\prod_{n \ge 1}\left(1-\frac{1}{4 n^2}\right)}$$
The question is Why no one uses this formula to calculate $\pi$? Maybe I am ... |
Why no one uses the product formula for sine function to calculate $\pi$? |
let
$$ C_0= \{ f | f \in C[0,1] \land f(\frac{1}{2})=0 \} $$
then $ C_0$ is a subspace of C[0,1] and $ C_0/C[0,1] $ is isomorphic to R.
I have proved that $ C_0 $ is a vector subspace.
I tried to prove that $ C_0/C[0,1] $ is isomorphic to R.
I know that
$ C_0/C[0,1] = \{ f+C_0 : f \in C[0,1] \} $
I t... |
>Find all entire functions such that $|f(z+z')|\leq |f(z)| + |f(z')|$, for all $z,z'\in\mathbb{C}$
In particular, let $z=z'$ yields $|f(2z)|\leq2|f(z)|$. This gives that $\frac{f(z)}{f(2z)}=c, $ for some $c\in\mathbb{C}$. By considering continuity at 0, we have $f(0)=\lim_{n\to\infty}c^n f(1)$. Then $c\in\mathbb{R}$... |
Fix $y_0$. Given $\varphi\in\mathrm{Aut}_G(Y)$, $\varphi(y_0)\in Y$. Since the action is transitive, there exists a $g\in G$ such that $gy_0=\varphi(y_0)$. Rotman wants to map it to a coset of $G_0$ in $N_G(G_0)$, namely to the coset $g^{-1}G_0$. But $g$ need not be unique. So he needs to verify two things for this f... |
I know how to construct $\mathbb{Z}, \mathbb{Q}, \mathbb{R}$ from $\mathbb{N}$ in set theory. For example, the construction of $\mathbb{Z}$ is,
$$\mathbb{Z}=\mathbb{N}^2/\sim$$
$$(a, b)\sim(c, d)\Leftrightarrow a+d=b+c$$
However, I do not know how to construct tuples and quotients in PA.
Is it possible to con... |
Is it possible to construct a real number theory on Peano arithmetic? |
For fixed real numbers $a,b,c,d$ define the map $f : \mathbb{R}^2 \to \mathbb{R}^2$ by
$$f : (x, y) \longmapsto (\phi(ax) + b \cdot \phi(ay),\hspace{1em}
\phi(cx) + d \cdot \phi(cy)),
\qquad \phi(z) := \sin(z).$$
Given a point $(x_0, y_0) \in \mathbb{R^2}$ we consider its orbits under t... |
Symmetry of King's dream fractal (orbit of a 2-dimensional iterated function system is a parallelogram?) |
Let $K$ be a Galois extension of $\mathbb{Q}$. Is it possible for non-trivial elements in $\text{Gal}(K/\mathbb{Q})$ to be continuous everywhere on $K$? Yes -- for instance, if $K$ is *not* a real number field, then complex conjugation is a non-trivial automorphism that is continuous.
But if $K$ is real, then non-tr... |
I face the same question (for case 2) now. Could you please tell me whether you have addressed this problem? |
I was reading **Complex Geometry** by Daniel Huybrechts. On page 68, section 2.2 we have a proposition of holomorphic line bundle over $\mathbb P^n$,
> Proposition 2.2.6 The projection $\pi:\mathcal O(-1)\rightarrow\mathbb P^n$ is given by projecting to the first factor. Let $\{U_i\}_{i=0}^n$ be an open covering of ... |
*Conceptually* $\, \dfrac{m}{18} = a = \dfrac{n}{25}\,$ so it's least denominator divides *coprimes* $18,25$ so is $1,\,$ so $\,a\in\Bbb Z$
**Remark** $\ $ This is an additive analog of this (multiplicative) group result
$$ a^{\large 18} = 1 = a^{\large 25}\,\Rightarrow\, {\rm ord}(a)\mid 18,25\,\Rightarrow\, {\... |
> But how slowly it converge to ?
Are you not able to run numerical calculations yourself?
Let $x_N = 2/\prod_{n=1}^N (1 - 1/(4n^2))$. We have
$$
x_{100} = 3.133\ldots, \
x_{10^3} = 3.1408\ldots, \
x_{10^4} = 3.1415141\ldots, \
$$
$$
x_{10^5} = 3.1415847\ldots, \
x_{10^6} = 3.1415918\ldots, \
x_{10^... |
I am reading the Evan's PDE, theorem 1 ( p.23 ) and some questions arise. I found assoicated question but I think that I don't get satisfactory answer yet.
Now assume $f\in C^{2}_C(\mathbb{R}^n)$.
> Theorem 1. ( Solving Poisson's euqation ) Define $u$ by $(8)$ ;
$$ u(x)=\int_{\mathbb{R}^n} \Phi(x-y)f(y)dy = ... |
let
$$ C_0= \{ f | f \in C[0,1] \land f(\frac{1}{2})=0 \} $$
then $ C_0$ is a subspace of C[0,1] and $ C_0/C[0,1] $ is isomorphic to R.
I have proved that $ C_0 $ is a vector subspace.
I tried to prove that $ C_0/C[0,1] $ is isomorphic to R.
I know that
$ C[0,1]/C_0 = \{ f+C_0 : f \in C[0,1] \} $
I t... |
*Conceptually* $\, \dfrac{m}{18} \!=\! a \!=\! \dfrac{n}{25}\,$ [so](https://math.stackexchange.com/a/825871/242) its least denominator divides *coprimes* $18,25$ so is $1,\,$ so $\,a\in\Bbb Z$
**Remark** $\ $ This is an additive analog of this better-known (multiplicative) group result
$$ a^{\large 18} = 1 = a^... |
I am reading the Evan's PDE, theorem 1 ( p.23 ) and some questions arise. I found assoicated question but I think that I don't get satisfactory answer yet.
Now assume $f\in C^{2}_C(\mathbb{R}^n)$.
> Theorem 1. ( Solving Poisson's euqation ) Define $u$ by $(8)$ ;
$$ u(x)=\int_{\mathbb{R}^n} \Phi(x-y)f(y)dy = ... |
A prime is not a product of primes, so why is “every positive integer, except 1, […] a product of primes”? |
I was reading **Complex Geometry** by Daniel Huybrechts. On page 68, section 2.2 we have a proposition of holomorphic line bundle over $\mathbb P^n$,
> Proposition 2.2.6 The projection $\pi:\mathcal O(-1)\rightarrow\mathbb P^n$ is given by projecting to the first factor. Let $\{U_i\}_{i=0}^n$ be an open covering of ... |
Let
$$ C_0= \{f \in C[0,1] \mid f(1/2)=0 \} $$
then $ C_0$ is a subspace of $C[0,1]$ and $C[0,1]/C_0$ is isomorphic to $\Bbb R$.
I have proved that $ C_0 $ is a vector subspace.
I tried to prove that $C[0,1]/C_0$ is isomorphic to $\Bbb R$.
I know that
$ C[0,1]/C_0 = \{ f+C_0 : f \in C[0,1] \} $
I trie... |
This is a follow up question based on this post: https://math.stackexchange.com/questions/4120560/bounding-rows-of-random-matrices/4123276#4123276.
I can not see why we can use Bernstein's inequality as mentioned in the answer. In particular, it seems like the variables $(\delta_{ij} - p)^2$ are not centered: $\math... |
Why is Bernstein's Inequality legal here? |
I am reading the Evan's PDE, theorem 1 ( p.23 ) and some questions arise. I found assoicated question but I think that I don't get satisfactory answer yet.
Now assume $f\in C^{2}_C(\mathbb{R}^n)$.
> Theorem 1. ( Solving Poisson's euqation ) Define $u$ by $(8)$ ;
$$ u(x)=\int_{\mathbb{R}^n} \Phi(x-y)f(y)dy = ... |
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... |
I realized that the function $p(y;b)$ cannot be a log-concave for all fixed $y$. Let me show this for $y=0$. In this case,
$$p(0;b)=\int_{0}^{\infty}\dfrac{1}{\sqrt{b+\frac 5u}} u^{1/2}\exp\left(-\dfrac{2u}{3}\right)du.$$
is **both convex and log-convex** over $b>0$.
Considering that the function $$\frac{1}{\sq... |
> **Lemma 10.26.** Let $G$ be a group acting transitively on a set $Y$, and let $y_0 \in Y$. Then $$\mathrm{Aut}(Y) \cong N_G(G_0)/G_0,$$ where $G_0$ is the stabilizer of $y_0$.
$N_G(G_0)$ is the normalizer of $G_0$ and $\mathrm{Aut}(Y)$ is set of all $G-$ isomorphisms, $Y\rightarrow Y$ (bijective $G-$ maps $f(g\... |
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