Newest Questions
1,689,782 questions
-2
votes
1
answer
85
views
Trying to understand $C([0,1])$ is separable. [closed]
I'm trying to understand this proof A proof that $C[0,1]$ is separable but I can't understand why $||f-g||_{\infty}\leq \epsilon$. I know that in the comments they were discussing about this, but I ...
1
vote
1
answer
25
views
Examples of connected graphs that are hard to break (at least 3-connected) and have the following property
So I was trying to find examples of connected, simple graphs $G$ that have the following two properties:
$G$ is at least 3-connected (you need to remove at least $3$ vertices to disconnect it)
There ...
- 142
-3
votes
0
answers
90
views
Try to prove $e\pi$ is irrational
(My motivation for this try is from the idea in Irrationality of $\pi\ln(2)$)
Try to prove $e\pi$ is irrational
Let $e\pi=\frac{a}{b}$ where $a,b \in \mathbb{N}=\{1,2,3,...\}$ and $gcd(a,b)=1$.
Then $...
- 1,679
7
votes
2
answers
83
views
What [is the minimal] differential equation is satisfied by $y = \sum_{n=0}^\infty {x^n}(n!)^{-k}$?
Specific Question
What [is the minimal] differential equation is satisfied by $y = \sum_{n=0}^\infty {x^n}(n!)^{-k}$?
That is, we are hunting for a monic differential operator of minimal order which ...
- 4,210
1
vote
0
answers
57
views
Minimal polynomial of an algebraic integer
The definition I learned for the minimal polynomial is: the minimal polynomial of an algebraic integer $a$ is the monic integer polynomial $f$ with minimal degree such that $f(a)$ = 0.
Could there be ...
- 31
0
votes
0
answers
28
views
Recurrence relation between $n^{-1}$ and $(n+1)^{-1}$ in GF(p)
In a prime field $GF(p)$ is there a recurrence relation between $n^{-1}$ and $(n+1)^{-1}$?
Both answers to this post explain how to obtain $(n+1)^{-1} \bmod n^2$, which made me wonder whether a ...
- 101
0
votes
1
answer
19
views
Conditional expectation of a mixture distribution with measure zero event
Suppose $\theta \sim U[0,1]$. If $\theta = x$, you observe a signal $m = x$. If $\theta < t < x$, you also observe the signal $m = x$. How do you compute the conditional expectation of $m$? I ...
1
vote
0
answers
35
views
Question about an application of Hopf-Rinow Theorem
Let $\varphi: \Bbb{R}^2 \rightarrow \Bbb{R}^3, (x,y) \mapsto (x,2x,x^2+y)$ and $S= \varphi(\Bbb{R}^2)$. If we consider the Riemannian metric for $S$ induced by the usual product of $\Bbb{R}^3$, prove ...
- 585
0
votes
0
answers
16
views
Trees isomorphic [closed]
Let $T_1$ and $T_2$ be trees of order $n$ with degree sequences $D = (d_1, d_2, \ldots, d_n)$ and $X = (x_1, x_2, \ldots, x_n)$, where $d_n \geq \dots \geq d_1$ and $x_n \geq \dots \geq x_1$. We say ...
2
votes
0
answers
20
views
Constructing the Hecke Theta Function for a Number Field
I'm working through writing down a proof for the analytic continuation of the Dedekind zeta function of a number field $K$, of degree $d$, signature $(r_{1},r_{2})$ and rank $r_{K} = r_{1}+r_{2}-1$, ...
- 305
1
vote
0
answers
22
views
Justifying the interchange of limit and integral in a sequence involving a diffeomorphism with non-uniform determinant bounds
Let $f: \mathbb{R}^m \to \mathbb{R}^m$ be a diffeomorphism such that $f(B) \subseteq B$, where $B$ is the closed unit ball in $\mathbb{R}^m$. Suppose that for all $x \in B$, we have $|\det Df(x)| < ...
- 21
0
votes
0
answers
83
views
Show that $\sqrt{a}+\sqrt{2\sqrt{a}}+\ldots \leq \frac{3}{2}\sqrt{a}$?
Let $(a_k)_k$ be the real sequence defined recursively by $a_0 = \sqrt{a}$ and $a_{k+1} = \sqrt{2a_k}$ for $k>1$. Show that $$\sum_{k=0}^\infty a_k \leq \frac{3}{2}\sqrt{a}$$
I tried to manipulate ...
- 1,119
1
vote
0
answers
32
views
Intuition on when and why algebraic manipulations work with operator equations
I have written the following proof for the Euler-Maclaurin formula, taking references from many sources.
Let $f$ be an analytic function which has a Taylor series $f(y)=\sum_{n=0}^\infty \frac{f^{(n)}(...
- 855
-3
votes
1
answer
36
views
compute Cantor normal form of $(\omega k )^n$ [closed]
Recall the,Cantor normal form(CNF) in base $\omega$: Every ordinal $\alpha$ can be uniquely written in “base $\omega$” as
$$\alpha=\omega^{\alpha_1}n_1+\omega^{\alpha_2}n_2\cdots\omega^{\alpha_k}n_k,$$...
- 91
1
vote
0
answers
28
views
Existence of a function in L1(G) whose Fourier transform never vanish
I am reading Wiener Tauberian theorem on locally compact abelian group setting and there I have read that:
The closed linear span of the translates of $ f $ is $ L^1(G) $ if and only if $ \hat{f}$ ...
- 11