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We know that if $f(z)$ is analytic at $a \in \mathbb{C}$, then by Taylor,
$$f(z) =\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(z-a)^n$$
By Cauchy, we also know that
$$f^{(n)}(a) = \frac{n!}{2 \pi i } \int_C \frac{f(s)}{(s-a)^{n+1}} \, ds$$
Combining these two, I'm imagining that it would then follow that
... |
> Is there a simple example such that $m^*(A\cup B)<m^*A+m^*B$?
Here $m^*$ is the Lebesgue outer measure on $\Bbb R$.
Of course, we can show the existence by contradition, and prove it by Berstein set...
Let $x,y\in [0,1],x\sim y\leftrightarrows x-y\in\Bbb Q$, then let $F$ be a set which intersets each equiva... |
Is there a simple example such that $m^*(A\cup B)<m^*A+m^*B$? |
I'm pretty sure "separatrix" is older than "fence" and "funnel". For example,
Lefschetz "[Lectures on Differential Equations](https://www.google.ca/books/edition/Lectures_on_Differential_Equations/1rbkRZZQ5IAC?hl=en)" from 1946 contains "separatrix". |
Is there a fast, reasonable way to compute the approximate value of $\log(- \log x)$? |
> Is there a simple example such that $A\cap B=\emptyset, m^*(A\cup B)<m^*A+m^*B$?
Here $m^*$ is the Lebesgue outer measure on $\Bbb R$.
Of course, we can show the existence by contradition, and prove it by Berstein set...
Let $x,y\in [0,1],x\sim y\leftrightarrows x-y\in\Bbb Q$, then let $F$ be a set which in... |
Is there a simple example such that $A\cap B=\emptyset, m^*(A\cup B)<m^*A+m^*B$? |
Let us consider here the continuous elements of $L^2$.
It is often stated that the family $e_k(x) = e^{-2 \pi i k x}$ is an orthonormal basis of $L^2[0, 1]$, in that a function can be written as
$$
f(x) = \sum_{k \in \mathbb{Z}} c_k e_k(x),
$$
for an appropriate complex sequence $c_k$.
However, my confusi... |
Confusion regarding orthonormal basis of $L^2[0, 1]$, in requiring $f(0) = f(1)$? |
If I have a proposition $P(n), n \in \{ 1,2,\ldots, m\}$ and I want to prove it by induction, can I proceed like this:
1) Show that $P(1)$ is true
2) Suppose that $P(n)$ is true for all $n \in \{1,2,\ldots,t\}$, where $t \in \mathbb{N}^*, t \le m-1$ and show that $P(t+1)$ is true.
This is something like complete... |
Compute $$\int \frac{\sin(x)\cos(x)}{x^2+1}dx$$
**Attempt**
$u=\sin(x)\Rightarrow du=\cos(x)dx$
$$\int\frac{u}{x^2+1}du$$
But, now I have both $u$ and $x$ in the function. How do I resolve?
Second attempt:
using Maclaurin series for $\sin(x)$ and $\cos(x)$,
$$\sin(x)=1-x^3/3!+x^5/5!...$$
$$\cos(x)=1-x^2/2!... |
According to this answer:
https://math.stackexchange.com/a/2582334/1098426
We know $\forall x \ \exists y \ \forall z$ differs from $\forall x \forall z\ \exists y$ insofar as $y$ depends on $x$ versus $y$ depends on $x$ and $z$.
I traditionally translate the definition of uniform continuity on some set $A\sub... |
Let $Y$ be any scheme, $S$ a set and $\underline{S}_Y$ the constant sheaf on any reasonble site over $X$, say to something concrete in mind the small etale site $X_{\text{et}}$. Let $f: X \to Y$ any morphism of schemes.
It is well known that the pullback sheaf $f^*\underline{S}_Y$ is a constant sheaf too isomorphic to... |
I am asked to find all singularities for the function $f(z) = sin(cot(1/z))$.
Here is my attempt, would appreciate any feedback if this is correct or otherwise where I am wrong / not rigorous enough:
At $z = 0$, the function is undefined, however this is a "limit point of the poles" (or "non isolated singularity"... |
I am asked to find all singularities for the function $f(z) = \sin(\cot(1/z))$.
Here is my attempt, would appreciate any feedback if this is correct or otherwise where I am wrong / not rigorous enough:
At $z = 0$, the function is undefined, however this is a "limit point of the poles" (or "non isolated singularit... |
Let $Y$ be any scheme, $S$ a set and $\underline{S}_Y$ the constant sheaf on any reasonble site over $X$, say to something concrete in mind the small etale site $X_{\text{et}}$. Let $f: X \to Y$ any morphism of schemes.
It is well known that the pullback sheaf $f^*\underline{S}_Y$ is a constant sheaf too isomorphic to... |
I am self-studying number theory and came across the question asked by another user in this post [here][1]. Specifically part (b).
My initial solution is to use the fact that the Legendre Symbol is completely multiplicative to split $(\frac{-5}{p})$ = $(\frac{-1}{p})*(\frac{5}{p})$, then follow the same steps to com... |
Let $G$ be a finite group. Let $A<G$.
When is it true that $G$ has only a single maximal subgroup containing (or equal to) $A$?
The necessary condition is that there must be $x \in G$ such that $\left<A \cup \{x\}\right>=G$ and the order of $\left<x\right>$ is a power of a prime. In many cases, this is also suff... |
Let $Y$ be any scheme, $S$ a set and $\underline{S}_Y$ the constant sheaf on any reasonble site over $X$, say to something concrete in mind the small etale site $X_{\text{et}}$. Let $f: X \to Y$ any morphism of schemes.
It is well known that the pullback sheaf $f^*\underline{S}_Y$ is a constant sheaf too isomorphic to... |
Let us consider here the continuous elements of $L^2$.
It is often stated that the family $e_k(x) = e^{-2 \pi i k x}$ is an orthonormal basis of $L^2[0, 1]$, in that a function can be written as
$$
f(x) = \sum_{k \in \mathbb{Z}} c_k e_k(x),
$$
for an appropriate complex sequence $c_k$.
However, my confusi... |
Let $Y$ be any scheme, $S$ a set and $\underline{S}_Y$ the constant sheaf on any reasonble site over $X$, say to have something concrete in mind the small etale site $X_{\text{et}}$. Let $f: X \to Y$ any morphism of schemes.
It is well known that the pullback sheaf $f^*\underline{S}_Y$ is a constant sheaf too isomorph... |
I have a question that goes:
Find the number of functions $f: X \to X$, such that the range of the $m$ times composed function $f^{(n)}$ has only one element. Given that $n(X) = n$
It is obvious that when $n = 1, 2$, the number of those functions are $0, 2$. I observed by python that if $n \le 8 $, the number of ... |
I am studying measure theory, and especially Lebesgue differentiation theorem. Let $f\: \mathbb{R}^n \to \mathbb{R}$ be a Lipschitz function, and let $N=f^{-1}(0)$. Show that for almost
every point $x\in N$ the derivative $Df(x)\: \mathbb{R}^n\to \mathbb{R}$ of $f$ at $x$ exists and $Df(x)=0$.
Following a hint f... |
How to calculate the area of a projected path onto a plane? |
This question is motivated by a simple exercise in Peter Cameron's *Combinatorics: Topics, Techniques, Algorithms*:
> A restaurant near Vancouver offered Dutch pancakes with ‘a thousand and
one combinations’ of toppings. What do you conclude?
The intended solution (according to [Cameron's website](https://webspa... |
How to find all positive integers $n,k$ such that ${n\choose k}=m$ for a given $m$? |
Consider $B\in \mathbb{R}^{m\times h}$ and $A \in \mathbb{R}^{h\times n}$. We can make sense of
\begin{align*}
\lVert BA\rVert_{1,\infty} = \sup_{\{x \in \mathbb{R}^n:\lVert x\rVert_1 = 1\}}\lVert BAx\rVert_{\infty}
\end{align*}
and we know
\begin{align*}
\lVert BA\rVert_{1,\infty} \leq \lVert B\rVe... |
I have a question that goes:
Find the number of functions $f: X \to X$, such that the range of the $m$ times composed function $f^{(m)}$ has only one element. Given that $n(X) = n$, where $n(X)$ is the number of the elements in $X$.
It is obvious that when $n = 1, 2$, the number of those functions are $0, 2$. I o... |
Thank Mhenni Benghorbal for the hint! I have worked out (i) so far:
The singular points for (i) are $\frac{k\pi}{2}, k \in \mathbb{Z}$,
The case $k=0$ was justified by Mhenni,
For $k \neq 0$ the singular points are simple poles, since $$\lim_{z \to \frac{k\pi}{2}}\frac{z-\frac{k\pi}{2}}{z\sin(2z)}=\lim_{z \to \f... |
According to this answer:
https://math.stackexchange.com/a/2582334/1098426
We know $\forall x \ \exists y \ \forall z$ differs from $\forall x \forall z\ \exists y$ insofar as $y$ depends on $x$ versus $y$ depends on $x$ and $z$.
I traditionally translate the definition of uniform continuity on some set $A\sub... |
Let $\mathcal{C}$ be a family of subsets of some finite set $X$, and let $\Omega$ be the $\sigma$-algebra generated from $\mathcal{C}$. Let $\mathcal{A}$ be the set of subsets of $S \subseteq X$ such that
1) For any set $C \in \Omega$, either $S \cap C = \emptyset$ or $S \cap C = C$.
2) There is no proper superset ... |
I have the following problem:
There are the Proth numbers $h\cdot 2^k+1$ with $h<2^k$ odd.
Let $p$ be an odd prime and $r := \operatorname{ord}_p(2)$ be the multiplicative order of $2$ modulo $p$.
Now I want to show that if $l$ is a natural number and $l\equiv k \pmod r$, then $h\cdot 2^k+1 \equiv 2^l+1\pmod r$.
... |
From the textbook: Introduction to Differential Topology by TH. Brocker and K. Janich.
> Defintion 1.3: An atlas of a manifold is called differentiable if all its chart transformations are differentiable.
Then a few lines below:
> If $\mathfrak{U}$ is a differentiable atlas on the manifold $M$, then the atlas ... |
I'm trying to compute the following limit:
$$\lim_{x\to 0} \frac{e^{ax}-e^{bx}}{x}$$
And I have to use some of the following limits for it:
$$\lim_{x\to 0}(1+x)^{\frac{1}{x}}=e=\lim_{x\to \infty}\left(1+\frac{1}{x}\right)^{x}$$
I tried some substitutions, specially the first limit but I got only to:
$$\l... |
How to compute $\lim_{x\to 0} \frac{e^{ax}-e^{bx}}{x}$? |
Assume there is a functor $L$ from a category $C$ to a category $D$ which satisfies the following property: for any objects $X,Y,Z$ from $C$ and morphisms $f\colon X\to Y, g\colon X\to Z$ such that $L(g)=\varphi\circ L(f)$ for some morphism $\varphi\colon L(Y)\to L(Z)$ there is a morphism $h\colon Y\to Z$ such that $\v... |
What is the name of this functor's property? |
Does Hennessy-Milner Theorem hold when I weaken its condition (image-finite) a little? |
Consider $B\in \mathbb{R}^{m\times h}$ and $A \in \mathbb{R}^{h\times n}$. We can make sense of
\begin{align*}
\lVert BA\rVert_{1,\infty} = \sup_{\{x \in \mathbb{R}^n:\lVert x\rVert_1 = 1\}}\lVert BAx\rVert_{\infty}
\end{align*}
and we know
\begin{align*}
\lVert BA\rVert_{1,\infty} \leq \lVert B\rVe... |
More precisely, if anyone has found it, could they provide links, please? I have been trying to find such a solution and have not seen one.
For context, when I try to solve the strict partition function for n = 10 billion, but only for triplets, my understanding is that the brute-force method would have me find all ... |
Has anyone found a closed-form expression for the strict partition function? |
Consider $B\in \mathbb{R}^{m\times h}$ and $A \in \mathbb{R}^{h\times n}$. We can make sense of
\begin{align*}
\lVert BA\rVert_{1,\infty} = \sup_{\{x \in \mathbb{R}^n:\lVert x\rVert_1 = 1\}}\lVert BAx\rVert_{\infty}
\end{align*}
and we know
\begin{align*}
\lVert BA\rVert_{1,\infty} \leq \lVert B\rVe... |
I am trying to apply Assumed Density Filtering (ADF) according to the paper [Lightweight Probabilistic Deep Networks](https://arxiv.org/abs/1805.11327) to my own model, and I need to implement the variational approximation layer of Sigmoid and SiLU function.
I tried to look for the equations for Sigmoid layer. In th... |
How can I use the change of basis theorem to find the standard matrix of a linear transformation? |
I want to prove that solution to Laplace's equation do not support local maximum or minimum using Hessian.
Suppose $f(x,y,z)$ is a real-valued function of three real variables that satisfies the Laplace equation, $\nabla^2f=0$. Consider the Hessian matrix
$$
H(x,y,z)=
\begin{bmatrix}
f_{xx} & f_{xy} & f_{xz}\\
... |
I'm trying to compute the following limit:
$$L=\lim_{x\to 0} \frac{e^{ax}-e^{bx}}{x} \tag{1}$$
And I have to use some of the following limits for it:
$$\lim_{x\to 0}(1+x)^{\frac{1}{x}}=e=\lim_{x\to \infty}\left(1+\frac{1}{x}\right)^{x}$$
I tried some substitutions, specially the first limit but I got only t... |
**Edited** I want to prove that solution to Laplace's equation do not support local maximum or minimum using Hessian.
Suppose $f(x,y,z)$ is a real-valued function of three real variables that satisfies the Laplace equation, $\nabla^2f=0$. Consider the Hessian matrix
$$
H(x,y,z)=
\begin{bmatrix}
f_{xx} & f_{xy} ... |
Consider $B\in \mathbb{R}^{m\times h}$ and $A \in \mathbb{R}^{h\times n}$. We can make sense of
\begin{align*}
\lVert BA\rVert_{1,\infty} = \sup_{\{x \in \mathbb{R}^n:\lVert x\rVert_1 = 1\}}\lVert BAx\rVert_{\infty}
\end{align*}
and we know
\begin{align*}
\lVert BA\rVert_{1,\infty} \leq \lVert B\rVe... |
**Edited** I want to prove that solution to Laplace's equation do not support local maximum or minimum using Hessian.
Suppose $f(x,y,z)$ is a real-valued function of three real variables that satisfies the Laplace equation, $\nabla^2f=0$. Consider the Hessian matrix
$$
H(x,y,z)=
\begin{bmatrix}
f_{xx} & f_{xy} ... |
I try to solve following exersize (7.24 from *3264 & All That Intersection Theory in Algebraic Geometry* by Eisenbud and Harris)
> Let $S \subset \mathbb P^3$ be a smooth cubic surface and $L \subset S$
> a line. Let $C_t$ be the pencil of conics on
> S cut out by the pencil of planes $H_t$ containing L. How many ... |
I saw this problem:If $x^2-16\sqrt x =12$ what is the value of $x-2\sqrt x$?
To make notations simpler Let $x=t^2$ and $t^4-16t=12$ and $l= t^2-2t$.
I tried to use the fact the $\frac{12} l =\frac{t^4-16t}{t^2-2t}= t^2+2t +4 + \frac{8}{t-2}= (t-2)^2 + 6t+ \frac{t}{8l}= \frac{l^2}{t^2}+6\frac{l}{t-2}+ \frac{t}{8l... |
If $x^2-16\sqrt x =12$ what is the value of $x-2\sqrt x$? |
I have a question that goes:
Find the number of functions $f: X \to X$, such that the range of the $n$ times composed function $f^{(n)}$ has only one element. Given that $|X| = n$, where $|X|$ is the number of the elements in $X$.
It is obvious that when $n = 1, 2$, the number of those functions are $1, 2$. I obs... |
This is question will be about the theorem behind an efficient algorithm, which has been [used][1] in LeetCode problem 3086. 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 pick o... |
Find minimum sum of distance to all points in a vector, using an accumulator? |
When investigated the Wilf function
$$W(z)=\frac{\arctan\sqrt{2\operatorname{e}^{-z}-1}\,}{\sqrt{2\operatorname{e}^{-z}-1}\,},$$
see the preprint [1] below, I proposed a guess which reads that the sequences
\begin{equation}\label{approx2pi-guess-qi}\tag{QG}
\frac{1}{2^k}\sum_{j=0}^{k-1}\frac{(-1)^j}{2j+1} \sum_{\el... |
Question:
**Evaluate in terms of the Gamma function the integral** $$\int_{0}^{\infty}\frac{x^k}{k^x} dx$$
And state the condition on k such that the integral converges.
**Attempt**
Let $$I = \int_{0}^{\infty}\frac{x^k}{k^x} = \int_{0}^{\infty}e^{-x\ln(k)}*x^k dx$$
So $k>0$ and $k$ can not be negative.
Let... |
Question:
**Evaluate in terms of the Gamma function the integral** $$\int_{0}^{\infty}\frac{x^k}{k^x} dx$$
And state the condition on $k$ such that the integral converges.
**Attempt**
Let $$I = \int_{0}^{\infty}\frac{x^k}{k^x}dx = \int_{0}^{\infty}e^{-x\ln(k)}\cdot x^k dx$$
So $k>0$ and $k$ can not be negativ... |
I've read in Jech's chapter on Borel and analytic sets that there is a universal $\Sigma_{\alpha}^0$ set $U \in \mathcal{N} \times \mathcal{N}$ such that for every $\Sigma_{\alpha}^0$ set $A$ in $\mathcal{N}$, there exists some $a \in \mathcal{N}$ such that:
$$
A = \{x : (x,a) \in U \}
$$
Having constructed $U$... |
Borel and analytic sets - Why did Jech do this? |
Let $k$ be a field of characteristic $0$ and $A=k\langle\langle x_1,\dotsc,x_n\rangle\rangle$ be a free completed associative algebra. The space of continuous derivations $\mathrm{Der}(A)$ is isomorphic, as a vector space, to $A^{\oplus n}$. $\mathrm{Der}(A)$ is a non-commutative analogue of the space of formal vector ... |
I've read in Jech's chapter on Borel and analytic sets that there is a universal $\Sigma_{\alpha}^0$ set $U \in \mathcal{N} \times \mathcal{N}$ such that for every $\Sigma_{\alpha}^0$ set $A$ in $\mathcal{N}$, there exists some $a \in \mathcal{N}$ such that:
$$
A = \{x : (x,a) \in U \}
$$
Having constructed $U$... |
I've read in Jech's chapter on Borel and analytic sets that there is a universal $\Sigma_{\alpha}^0$ set $U \in \mathcal{N} \times \mathcal{N}$ such that for every $\Sigma_{\alpha}^0$ set $A$ in $\mathcal{N}$, there exists some $a \in \mathcal{N}$ such that:
$$
A = \{x : (x,a) \in U \}
$$
Having constructed $U$... |
Let A = ($a_{ij}$) be an $n×n$ matrix and $A_{ij}$ be the co-factor of $a_{ij}$.
<br>
Show that: <br> $\sum$ $a_{ik}$$A_{jk}$ = det($A$) if i=j, else 0
I couldn't think of any way to do this other than trying to open the co-factor, but I think that would be vigorous.
EDIT- I've tried expanding, bu... |
How many total elements do you get from sets $A_i$'s if you do not consider the equality of elements? The answer is $30\times 5=150$. Now, note that $$\bigcup_{i=1}^{30}A_i=S$$
and every element in $S$ occurs in exactly $10$ different $A_i$'s.
We are interested in the size of $S$, so we look at the total elements w... |
I've read in Jech's chapter on Borel and analytic sets that there is a universal $\Sigma_{\alpha}^0$ set $U \in \mathcal{N} \times \mathcal{N}$ such that for every $\Sigma_{\alpha}^0$ set $A$ in $\mathcal{N}$, there exists some $a \in \mathcal{N}$ such that:
$$
A = \{x : (x,a) \in U \}
$$
Having constructed $U$... |
Suppose $V$ is a $3$-dimensional vector space, suppose I have $2$ planes $S_1$ and $S_2$ going through the origin, then they must at least intersect at a line.
Suppose $V$ is a $4$-dimensional vector space, suppose I have $2$ planes $S_1$ and $S_2$ going through the origin, then they must at least intersect at the o... |
$$f(s) = \zeta(s)\zeta(s+1)\Gamma(s) $$
This has a double pole at $s=0$ , and I got -1/2 , and then to find the other
$$ \lim_{s\to0} s(f(s) - \frac{-1/2}{s^2}) $$
(Actually, it is $-log\sqrt{2\pi} = \zeta'(0)$ from my book)
In my calcuation, It comes down to
$$ \lim_{s\to0} (s\zeta(s+1)\Gamma(s+1))' = ... |
We are performing fft on N samples of a function f(x) that is periodic on the interval [0,L] where x is in meters. What would be the most common definition of the units of the domain of the transformed signal ? Will it be cycles per meter or radian per meter ?
|
I'm pretty sure "separatrix" is older than "fence" and "funnel". For example,
Lefschetz "[Lectures on Differential Equations](https://www.google.ca/books/edition/Lectures_on_Differential_Equations/1rbkRZZQ5IAC?hl=en)" from 1946 contains "separatrix".
[EDIT]
It seems "fence" and "funnel" were introduced in 1991 in... |
$$f(s) = \zeta(s)\zeta(s+1)\Gamma(s) $$
This has a double pole at $s=0$ , and I got -1/2 , and then to find the other
$$ \lim_{s\to0} s(f(s) - \frac{-1/2}{s^2}) $$
(Actually, it is $-log\sqrt{2\pi} = \zeta'(0)$ from my book)
In my calcuation, It comes down to
$$ \lim_{s\to0} (s\zeta(s+1)\Gamma(s+1))' = ... |
I’ve seen work before on relevance logic and set theory, but is there some connection too between relevance logic and mereology? I’ve heard that mereological relations are akin to the inclusion relation in set theory. Could there be a “relevant mereology”? |
Relation between mereology and relevant logic? |
this comes from a comment of [this][1]
If
and
are ×
and ×
matrices,
and
have the same nonzero eigenvalues:
is an eigenvector of
iff
is an eigenvector of
, with the same nonzero eigenvalue.
I try to proof this
if
is an eigenvector of =>
is an eigenvector of
$\begin{align}
... |
What is an example of non isomorphic field extensions? I would like such a non example as I think it would explain the significance of the commutativity of the square condition in an isomorphism of field extensions $L/K\cong M/N$ where $\mu$, $\sigma$ are isomorphisms:
[![][1]][1]
[1]: https://i.stack.imgur.com/A... |
I've read in Jech's chapter on Borel and analytic sets that there is a universal $\Sigma_{\alpha}^0$ set $U \subseteq \mathcal{N} \times \mathcal{N}$ such that for every $\Sigma_{\alpha}^0$ set $A$ in $\mathcal{N}$, there exists some $a \in \mathcal{N}$ such that:
$$
A = \{x : (x,a) \in U \}
$$
Having construct... |
Let $E/\Bbb C : y^2=x^3+ax+b$ be an elliptic curve over $\Bbb C$. Let
$$E(\Bbb C)=\{(x,y)\in\Bbb C^2:y^2=x^3+ax+b\}$$
be the affine variety generated by $E$. Finally, let
$$I(E(\Bbb C))=\{f(x,y)\in\Bbb C[x,y]: f(p)=0,\, \forall p\in E(\Bbb C)\}.$$
I am trying to show that $I(E(\Bbb C))=I_E$ is the ideal generat... |
I was looking at the proof of Euler product at Mathworld https://mathworld.wolfram.com/EulerProduct.html
First we expand the product, this I understand, then “we write each term as a geometric series.” But I see that we don’t write “each term” as a geometric series. The whole term is $\frac{1}{1-\frac{1}{p_k^s}}$ bu... |
We are performing fft on N samples of a function f(x) that is periodic on the interval [0,L] where x is in meters. What would be the most common definition of the units of the domain of the transformed signal ? Will it be cycles per meter or radians per meter ?
|
This question was asked in my assignment of linear algebra and I am not able to make any significant progress on this question.
> Question :Let V be the space of all linear transformations from $\mathbb{R}^2 \to \mathbb{R}^2$ under usual addition and scalar multiplication. Then show that V is a vector space of dime... |
How can I find the dimension of this vector space? |
I have been working on this question for two days,
does $\sum_{n=0}^\infty\arcsin(3^{-n})$ converges
?
----------
I did that.
$$arcsinx<\frac{x}{\sqrt{1-x^2}}$$ then i put $3^{-n}$ as x , $3^{-n}∈(0,1]$
$$arcsin(3^{-n})<\frac{3^{-n}}{\sqrt{1-(3^{-n})^2}}$$
$$arcsin(3^{-n})<\frac{1}{\sqrt{(3^{... |
does the $\sum_{n=0}^\infty\arcsin(3^{-n})$ converge? |
I have been working on this question for two days,
does $\sum_{n=0}^\infty\arcsin(3^{-n})$ converge?
----------
I did that.
$$\arcsin x<\frac{x}{\sqrt{1-x^2}}$$ then i put $3^{-n}$ as $x$ , $3^{-n}\in (0,1]$
$$\arcsin(3^{-n})<\frac{3^{-n}}{\sqrt{1-(3^{-n})^2}}$$
$$\arcsin(3^{-n})<\frac{1}{\sqrt{(3^{2n})-1... |
Does the series $\sum_{n=0}^\infty\arcsin(3^{-n})$ converge? |
Define $f:A \to A$ where $A$ contains $n$ distinct elements. How many functions exist such that $ \forall x \in A, f^m(x)=x$, $(m<n)$ (and $m$ is prime to avoid the mistake pointed out in the comments) where $f^m$ represents the $m^{th}$ composition of $f(x)$?
The key idea I recognized here was that there are only... |
How can I find nicely behaving functions which are easy to compute and which fit well to the upper limits of the (2-logarithm) of the nCr function?
What I am interested in in practice is to be able to model "how many binary digits (bits) will a digital representation of nCr(a,b)" have? A guess of 1 or 2 too many bit... |
> Let $f: \Omega\to\mathbb{C}$ be defined and continuous on $\Omega$,
> $\Omega\subset\mathbb{C}$ is open. Assume that $f$ is holomorphic on
> $\Omega\setminus A$, $A$ is a discrete set that has no limit points on
> $\Omega$. Prove that $f$ can be extended to a holomorphic function $g$
> on $\Omega$, i.e there exis... |
I have equations such as
$\frac{dx}{dt} = y \\
\frac{dy}{dt} = \mu y + x - x^{2} + xy$
This system is known to be homoclinic bifurcation at the origin. To find the critical value of $\mu_{c}$, one found $\mu_{c}$ by the numerical calculation. In this case, how can I find critical value of $\mu_{c}$ where the bif... |
This question was asked in my assignment of linear algebra and I am not able to make any significant progress on this question.
> Question: Let $V$ be the space of all linear transformations from $\mathbb{R}^2 \to \mathbb{R}^2$ under usual addition and scalar multiplication. Then show that $V$ is a vector space of ... |
The problem occurs when reading [Bombardini et al., 2023, "Did US Politicians Expect the China Shock?", *American Economic Review*, Vol.1, PP174-209](https://bcf.princeton.edu/wp-content/uploads/2021/03/BLT25jan2021.pdf).
The authors define $\xi_{it}$ to be a Gaussian term obeying $N(0,2\sigma_{\xi}^2)$ (*see line 7... |
suppose V is 4-dimensional vector space,
$v_1$ and $v_2$ are linear independent in V,
now for arbitrary plane S that goes through origin,
can I find two new linear independent vector
in plane S that
condition:
\begin{align}
v_1 \cdot v_3 >=0 \ \& \ v_2 \cdot v_3 >=0,
& v_1 \cdot v_4 >=0 \ \& \ v_2 \cdot v_4... |
Why are all pairs of order-12 subgroups of $A_5$ conjugate? |
Let $X_1,\cdots,X_n$ are i.i.d. r.v.s with expectation $0$ and variance $1$, then according to central limit theorem we have
$$\frac{S_n}{\sqrt{n}}\sim N(0,1)$$ where $S_n=\sum_{i=1}^nX_i$.
Thus $\mathbb P(\frac{S_n}{\sqrt{n}}>a)\leq \exp(-\frac{a^2}{2})$, i.e., $\mathbb P(S_n>a\sqrt{n})\leq \exp(-\frac{a^2}{2})$
... |
why non-gaussian case we do not have $S_n>an$ being rare? |
Let $F$ be an ordered field and give $F$ the order-topology. Then is $F$ a topological field (that is, are the operations
\begin{equation}
\begin{split}
+&:F\times F\to F\\
-&:F\to F\\
\times&:F\times F\to F\\
^{-1}&:F\setminus\{0\}\to F
\end{split}
\end{equation}
continuous)?
I checked that $+,-$ and $^{... |
What are the maximal $p$-subgroups of $A_5$? |
Why is a Sylow $p$-subgroup normal iff there's only one of them? |
I'm currently starting to learn K-homology and something bothers me about the definition of Fredholm modules. I looked in 5/6 papers and each time a different definition is given... For example in [1] and [2]:
**Definition (Fredholm module)** : Let $\mathcal{A}$ be a separable $C^*$-algebra. A **Fredholm module** ov... |
suppose V is 4-dimensional vector space,
$v_1$ and $v_2$ are linear independent in V,
now for arbitrary plane S that goes through origin,
can I find two new linear independent vector
in plane S that
condition:
\begin{align}
v_1 \cdot v_3 >=0 \ \& \ v_2 \cdot v_3 >=0,
& v_1 \cdot v_4 >=0 \ \& \ v_2 \cdot v_4... |
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