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I understand that $\mathscr{K}(\mathscr{H})$ is a closed subspace of $\mathscr{B}(\mathscr{H})$. Where $\mathscr{K}$ is the compact operators and $\mathscr{H}$ is the bounded operators.
I need to prove that if $S \in \mathscr{H}$ and $T \in \mathscr{K}$ then ST is compact. I started by defining D(ST) = {u| $u \in D... |
As mentioned in Revuz-Yor pg.120, the main reason that we require $H_{s}$ to be predictable, is because we want the integral against an adapted increasing process $A_{s}$
$$H\cdot A:=\int H_{s}dA_{s}$$
to remain adapted and in turn preserve martingale too. So as mentioned in (2.2) Theorem part (a), if we don't h... |
Unless I've missed something here, doesn't $\sum\frac{\beta}{n\alpha + \beta} \ge \sum\frac{\beta}{n\alpha + n}$ already diverge? |
Working in the first order language of set theory.
By $R$-bounded quantifiers its meant those of the form $\forall x \ R \ a \, ( \cdots) $ , or $ \exists x \ R \ a \, ( \cdots) $, and these are defined as $\forall x \, ( x \ R \ a \to \cdots)$ and $\exists x \, (x \ R \ a \land \cdots) $ respectively.
A quanti... |
Can this presentation of reflection be considered foundational? |
To show the question, it is quite straightforward to just use the sequential definition of compact operators.
As for your claims about the domain, if you are given $S,T\in\mathcal{B(H)}$, are you not already assuming the domain to be the whole Hilbert space?
Anyways closed+ bounded = convex is definiely wrong (ta... |
I have read in my textbook that if I have, for example two monomials $A$ e $B$ one or both with rational coefficients,
$$A=\frac 34 x^2y^3, \qquad B=-2xyz$$
for the $\text{lcm}$ or the $\text{gcd}$ we have always $1$ as numerical coefficient. Is it a convention or is there something more complex that I do not kno... |
Working in the first order language of set theory.
By $R$-bounded quantifiers its meant those of the form $\forall x \ R \ a \, ( \cdots) $ , or $ \exists x \ R \ a \, ( \cdots) $, and these are defined as $\forall x \, ( x \ R \ a \to \cdots)$ and $\exists x \, (x \ R \ a \land \cdots) $ respectively.
A quanti... |
Might be a stupid question, but I was thinking about this today and wanted to clear something up. I always thought of f(x) as having different algebraic properties than f*x, and it came as a surprise to me when I noticed that you can modify the equation algebraically, treating $f(x)$ as $f*x$, without butchering the or... |
Might be a stupid question, but I was thinking about this today and wanted to clear something up. I always thought of f(x) as having different algebraic properties than f*x, and it came as a surprise to me when I noticed that you can modify the equation algebraically, treating $f(x)$ as $f*x$, without necessarily butch... |
I'm trying to follow some calculus [lecture notes](https://www.dropbox.com/scl/fi/wmo5g2lfb599hwsulnees/Matrix-Calculus.pdf?rlkey=n9xll4xlref9n6lrr3zsmiv12&e=1&dl=0) and I can reproduce an argument for the sum and product rule. Can I make an argument like this for the chain rule in a similar style, using the definition... |
We introduce the notations:
$$
\begin{aligned}
a &=\sqrt{y^2 +x}\ge 0\ ,\text{ so}&a^2 &=y^2 + x\ ,\\
b &=\sqrt{x^2 +y}\ge 0\ ,\text{ so}&b^2 &=x^2 + y\ ,\\
&\qquad\text{ and there is the relation}\\
y&=(a+x)(b-y)\ .
\end{aligned}
$$
The idea is to eliminate $x,y$, and obtain a single equation connecting the v... |
The answer [here][1] makes sense to me, in the sense that the stochastic integral does not preserve the local martingale property if the integrand is not predictable, but I am confused as to why. I have been taught the following definition of the stochastic integral:
We first define the quadratic variation for a RC... |
Which element in $\Bbb Z_{2024}$ corresponds to the triple of numbers $(1,3,22)$ in the product $\Bbb Z_8 \times \Bbb Z_{11} \times \Bbb Z_{23}$? |
Given circle A and circle B (we got their center points, radiuses and 1 relevant point where they intersect)
[![enter image description here][1]][1]
we want to draw a circle C with a given radius in a way that it is tangential with the other two circles (there are up to 4 possible locations). How to get the possi... |
Find the circle center point that is tangential on 2 other circles that has at least one intersection? |
I'm trying to deepen my understanding of the Fundamental Theorem of Algebra, which asserts that every non-constant single-variable polynomial with complex coefficients has as many roots as its degree, counting multiplicities. Specifically, for a polynomial equation of degree *n*, there should be exactly *n* roots in th... |
We introduce the notations:
$$
\begin{aligned}
a &=\sqrt{y^2 +x}\ge 0\ ,\text{ so}&a^2 &=y^2 + x\ ,\\
b &=\sqrt{x^2 +y}\ge 0\ ,\text{ so}&b^2 &=x^2 + y\ ,\\
&\qquad\text{ and there is the relation}\\
y&=(a+x)(b-y)\ .
\end{aligned}
$$
The idea is to eliminate $x,y$, and obtain a single equation connecting the v... |
Might be a stupid question, but I was thinking about this today and wanted to clear something up. I always thought of f(x) as having different algebraic properties than f×x, and it came as a surprise to me when I noticed that you can modify the equation algebraically, treating $f(x)$ as $f×x$, without necessarily butch... |
It is well known how to parameterize rational solutions to $x^2+y^2=a$ for $a\in \mathbb N^+$: first we know which [$a$ admit integer solutions][1]; our characterization is strong enough to tell us that theses are [exactly the $a$][2] admitting rational solutions; then [if we have one rational solution][3] we can get a... |
When you know two sides and the angle between them, the so-called "side-angle-side" case, older textbooks would recommend using the [law of tangents](https://en.wikipedia.org/wiki/Law_of_tangents) to find the remaining angles, if you don't want to bother finding the remaining side. I think that's what you mean by "dir... |
The scalar product defined for the matrices $A$ and $B$ in $R^{n\times n}$ is
$$ \langle A, B \rangle = trace(A^{T} B) $$.
I want to show that the set of all symmetric matrices ($D$) is closed in $R^{n\times n}$.
My attempt:
I want to show that $\overline{D} \subset D $. Let $A \in \overline{D}$. Then there... |
This question is based on Proposition 1.4.13 in Basic Complex Analysis, by Marsden.
> Definition 1.4.12: A set $C \subset \mathbb{C}$ is **not connected** if there exists open sets $U$ and $V$ such that, a) $\, C \subset U \cup V$, b) $\, C \cap U \not = \emptyset$ and $\, C \cap V \not = \emptyset$, c) $(C \cap U)... |
Suppose $f: [0,1]^2 \mapsto \mathbb R$ is continuous. Let
$$
\underline{y}(x)\equiv \inf \{y\in [0,1]: f(x,y)\geq 0\}.
$$
There is $y_0\in(0,1)$ such that
$$f(0,y_0)= 0,$$
$$f(0,y)< 0 \quad \forall y\neq y_0.$$
Moreover,
$$
f(x,y_0) > 0 \quad \forall x\in (0,1].
$$
*I hope to find out: is it true tha... |
I am attempting to prove (or disprove) that $\mathbb{C}\{z\}$ is Noetherian, where $\mathbb{C}\{z\}$ is the subring of $\mathbb{C}[z]$ consisting of functions holomorphic in a neighborhood of $0 \in \mathbb{C}$.
Firstly, I recognize that in the ring of smooth function $C^{\infty}(\mathbb{R}, \mathbb{R})$ if we consi... |
Is The sub ring $\mathbb{C}\{z\}$ Noetherian? |
Might be a stupid question, but I was thinking about this today and wanted to clear something up. I always thought of f(x) as having different algebraic properties than f×x, and it came as a surprise to me when I noticed that you can modify the equation algebraically, treating $f(x)$ as $f×x$, without _necessarily_ but... |
The well-known identity for complex points forming an equilateral triangle reads
$$z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$$
I have a doubt concerning the uniqueness of this identity
> Is $z_1^2 + z_2^2 + z_3^2 - z_1 z_2 + z_2 z_3 + z_3 z_1$ the only quadratic form in 3 variables that when replacin... |
[This](https://x.com/mayfer/status/1762014984660283717) tweet claims to give an explanation for why one should expect the perimeter of a circle with a rational radius to be irrational. It doesn't strike me as that convincing (although feel free to disagree).
A simple refutation would be to exhibit a smooth curve whi... |
Prove that the division rings $\mathbb{C}((z;\sigma))$ and $\mathbb{H}((x))$ are not isomorphic. The following statement is needed:
\begin{equation}
\text{any ring automorphism $\phi: \mathbb{R}((x)) \rightarrow \mathbb{R}((x))$ has the form:}
\end{equation}
\begin{equation}
\forall a \in \mathbb{R}, \phi(a)=a;
\... |
As I understand it, Peano arithmetic needs the axiom of induction to prevent non-standard models of the natural numbers.
Given $P(0)$ and $P(n)\implies P(n+1)$ I can apply modus ponens to get $P(1)$ and again to get $P(2)$. After 5 iterations, I should have deduced $P(5)$. Is this reasoning correct?
It seems the ... |
Do $P(0)$ and $P(n)\implies P(n+1)$ yield $P(5)$ without an axiom of induction? |
I am attempting to prove (or disprove) that $\mathbb{C}\{z\}$ is Noetherian, where $\mathbb{C}\{z\}$ is the subring of $\mathbb{C}[z]$ consisting of functions holomorphic in a neighborhood of $0 \in \mathbb{C}$.
Firstly, I recognize that in the ring of smooth function $C^{\infty}(\mathbb{R}, \mathbb{R})$ if we consi... |
Eq. 1.4 in [Levinson's paper][1] states $$\dfrac{(-1)^m}{m!} \dfrac{d^{m-1}}{ds^{m-1}} \Big(\dfrac{\zeta'(s)}{\zeta(s)} \Big) = \dfrac{1}{(s-1)^m} - \sum_{n=0}^{\infty} \dfrac{1}{(s+2n)^m} - \sum_{\rho} \dfrac{1}{(s-\rho)^m}.$$ By Eq. 2.12.7 in Titchmarsh's book *The Theory of the Riemann Zeta-Function*, $$\dfrac{\zeta... |
I am attempting to prove (or disprove) that $\mathbb{C}\{z\}$ is Noetherian, where $\mathbb{C}\{z\}$ is the subring of $\mathbb{C}[[z]]$ consisting of functions holomorphic in a neighborhood of $0 \in \mathbb{C}$.
Firstly, I recognize that in the ring of smooth function $C^{\infty}(\mathbb{R}, \mathbb{R})$ if we con... |
As I understand it, Peano arithmetic needs the axiom of induction to prevent non-standard models of the natural numbers.
Given $P(0)$ and $P(n)\implies P(n+1) \forall n\in \mathbb{N}$ I can apply modus ponens to get $P(1)$ and again to get $P(2)$. After 5 iterations, I should have deduced $P(5)$. Is this reasoning c... |
This question is based on Proposition 1.4.13 in Basic Complex Analysis, by Marsden.
> Definition 1.4.12: A set $C \subset \mathbb{C}$ is **not connected** if there exists open sets $U$ and $V$ such that, a) $\, C \subset U \cup V$, b) $\, C \cap U \not = \emptyset$ and $\, C \cap V \not = \emptyset$, c) $(C \cap U)... |
I am attempting to prove (or disprove) that $\mathbb{C}\{z\}$ is Noetherian, where $\mathbb{C}\{z\}$ is the subring of $\mathbb{C}[[z]]$ consisting of functions holomorphic in a neighborhood of $0 \in \mathbb{C}$.
Note : $\mathbb{C}[[z]]$ represents the ring of formal power series with coefficients in the field of c... |
For my bachelors thesis I am working with tubifex worms, and trying to develop a graph theoretical model that can help explain some mechanical and dynamical properties of the worms once they have formed a "worm ball" (if you search up tubifex worms you can see what I mean).
We can define a "linking number" to repre... |
Let $A$ be a discrete time Markov chain, with states $1$ and $2$, and transition probability matrix
$$\begin{pmatrix}
\frac{1}{2}( 1 + \alpha ) & \frac{1}{2}( 1 - \alpha ) \\
\frac{1}{2}( 1 - \alpha ) & \frac{1}{2}( 1 + \alpha )
\end{pmatrix}
$$
where $-1 < \alpha < 1$.
Let $X(1)=R^{(1)}$ and $X(2)=R^{(2)}$... |
Let $p$ be a prime, $d$ a proper divisor of $p-1$, and $\sigma\in S_{p-1}$ of the form (everything is modulo $p$):
$$\sigma=(1,x,\dots,x^{d-1})(i_2,i_2x,\dots,i_2x^{d-1})\dots(i_k,i_kx,\dots,i_kx^{d-1})\tag1$$
(hence $|\sigma|=d$) where $kd=p-1$ and, $\forall i,j=1,\dots,p-1$ such that $i+j\not\equiv_p0$:
$$\sigma(i... |
Let $A$ be a discrete time Markov chain, with states $1$ and $2$, and transition probability matrix
$$\begin{pmatrix}
\frac{1}{2}( 1 + \alpha ) & \frac{1}{2}( 1 - \alpha ) \\
\frac{1}{2}( 1 - \alpha ) & \frac{1}{2}( 1 + \alpha )
\end{pmatrix}
$$
where $-1 < \alpha < 1$.
Let $X(1)=R^{(1)}$ and $X(2)=R^{(2)}$... |
__Hint:__ What is the distance between the centers?
To find (say) $O_3$ (which is internally tangent to both $A$ and $B$), what is $|KO_3|$ and $|O_3M|$?
With that info, determine the possible locations of $O_3$.
Note that there are 2 possible locations. (Why?) |
let $p$ be a prime and $\sigma(n)$ be Euler's totient function and $m,n$ be natural numbers. Then $\sigma(p^k)=\frac{p^{k+1}-1}{p-1}$ and $\sigma(mn)=\sigma(m)(n)$ and $\sigma(p)=p-1$.But this
$$
\sigma(p^k)=\sigma(p)^k=(p-1)^k\neq \frac{p^{k+1}-1}{p-1}=\sigma(p^k)
$$
bothers me. Where is the mistake here ? |
I am attempting to prove (or disprove) that $\mathbb{C}\{z\}$ is Noetherian, where $\mathbb{C}\{z\}$ is the subring of $\mathbb{C}[[z]]$ consisting of functions holomorphic in a neighborhood of $0 \in \mathbb{C}$.
Note : $\mathbb{C}[[z]]$ represents the ring of formal power series with coefficients in the field of c... |
Let $A$ be a $\mathbb Z_{\geq 0}$ graded ring. Then we have that $\operatorname{Proj}A$ is the set of homogenous prime ideals which do not contain the irrelevant ideal $A_+$. We put a topology on this by declaring $V(I)=\{\mathfrak p\in \operatorname{Proj}A:I\subset \mathfrak p\}$ to be the closed sets whenever $I$ is ... |
Let $p$ be a prime and $\sigma(n)$ be Euler's totient function and $m,n$ be natural numbers. Then $\sigma(p^k)=\frac{p^{k+1}-1}{p-1}$ and $\sigma(mn)=\sigma(m)(n)$ and $\sigma(p)=p-1$.But this
$$
\sigma(p^k)=\sigma(p)^k=(p-1)^k\neq \frac{p^{k+1}-1}{p-1}=\sigma(p^k)
$$
bothers me. Where is the mistake here? |
While FTOA tells us that every polynomial has a certain number of roots, it doesn't give us a method for finding them. In general, while the roots of any linear or quadratic polynomial can be found through basic algebra, finding the roots of a cubic or quartic is quite difficult and for quintics and higher there is no ... |
Let $f\in\mathcal{C}^\infty(\mathbb{R})$, what is the form of
$$
\frac{d^n}{dx^n}\left(\frac{f(x)}{x}\right)
$$
for any $n\in\mathbb{N}$?\
I need to pull out $\frac{d^n}{dx^n}f(x)$ if possible. Thank you very much. |
Here I explained how the two-dimensional recurrence can be solved directly without knowing the answer.
Let us first fix $x=1$. Then, we have the following recurrence:
$$P(1,n) = P(1,n-1)\left(\frac{n-2}{n-1}\right).$$
The [solution][1] of this is given by
$$P(1,n) = \frac{c}{n-1},$$
where from $P(1,2)=1... |
Let $A$ be a $\mathbb Z_{\geq 0}$ graded ring. Then we have that $\operatorname{Proj}A$ is the set of homogenous prime ideals which do not contain the irrelevant ideal $A_+$. We put a topology on this by declaring $V(I)=\{\mathfrak p\in \operatorname{Proj}A:I\subset \mathfrak p\}$ to be the closed sets whenever $I$ is ... |
I am following chapter 4 of this [paper (1)](https://pure.manchester.ac.uk/ws/portalfiles/portal/208486475/FULL_TEXT.PDF).
**Background information:** The stochastic differential equations (eq’s 4.2 to 4.5) are given by:
$$
\begin{aligned}
& d S_t=\sqrt{\nu_t} d W_t \\
& d \nu_t=\theta\left(\alpha-\nu_t\right)... |
Let $\mu=\sum_{i=1}^n \lambda_i\delta_{w_i}$, where $\delta$ is the Dirac measure, and $\lambda_i,w_i\in\mathbb{C}$ and $\lambda_i$ is non-zero. Let $\sigma_k$ be Fourier coefficients of $f$:
$$
\sigma_k=\dfrac{1}{2T}\int_{-T/2}^{T/2} f(x)e^{-2\pi i k x}d\mu.
$$
Then why do we have the following equality:
$$
\si... |
The minimal polynomial splits over a field of prime characteristic? |
First of all, it is crucial to point out in the question, that we are interested in writing as the sum of $4$ or more consecutive **positive** integers. Otherwise, almost any positive integer $n$ number will do fine: $$n=(-n+1)+…+(-1)+0+1+…+(n-1)+n.$$
Let the number $n$ be equal to the sum of $m$ consecutive positiv... |
I’ve been following along with the book ”Differential geometry of curves and surfaces” and I came along a question asking to solve for the distance travelled of a point by a rolling circle within one full rotation.
I solved for the arclength integral and got an indefinite integral which looked something like
$-4r... |
I am attempting to prove (or disprove) that $\mathbb{C}\{z\}$ is Noetherian, where $\mathbb{C}\{z\}$ is the subring of $\mathbb{C}[[z]]$ consisting of functions holomorphic in a neighborhood of $0 \in \mathbb{C}$.
Note : $\mathbb{C}[[z]]$ represents the ring of formal power series with coefficients in the field of c... |
Here I explained how the two-dimensional recurrence can be solved directly without knowing the answer.
Let us first fix $x=1$. Then, we have the following recurrence:
$$P(1,n) = P(1,n-1)\left(\frac{n-2}{n-1}\right).$$
The [solution][1] of this is given by
$$P(1,n) = \frac{c}{n-1},$$
where from $P(1,2)=1... |
Unfortunately, I have some difficulties understanding the following
proof from Profinite Groups (https://link.springer.com/book/10.1007/978-3-662-04097-3)
on page 123.
Let $G$ be a profinite group and $x \in G$. Since
profinitely completed integers $\widehat{\mathbb{Z}}=
\prod_{p \in \mathbb{P}} \mathbb{Z}_p$ ... |
Let $p$ be a prime and $\varphi(n)$ be Euler's totient function and $m,n$ be natural numbers. Then $\varphi(p^k)=\frac{p^{k+1}-1}{p-1}$ and $\varphi(mn)=\varphi(m)(n)$ and $\varphi(p)=p-1$.But this
$$
\varphi(p^k)=\varphi(p)^k=(p-1)^k\neq \frac{p^{k+1}-1}{p-1}=\varphi(p^k)
$$
bothers me. Where is the mistake here? |
In my partial differential equations class, my instructor mentioned generalizations of diffusion to non-Euclidean space. As I understand, one may use the Laplace-Beltrami operator on a smooth Riemannian manifold $\mathcal M$ of dimension $n$ to state a local notion of diffusion.
$$\Delta_{\mathcal M} = \frac{1}{\sq... |
Eq. 2.1 in [Levinson's paper][1] states $$\dfrac{1}{(s-1)^m} - \sum_{n=0}^{\infty} \dfrac{1}{(s+2n)^m} - \sum_{\rho} \dfrac{1}{(s-\rho)^m} = - \sum_{|\gamma -t|<1} \dfrac{1}{(s-\rho)^m} + \mathcal{O} (\log t).$$ And it is stated that this is derived by $N(T+1) - N(T) = \mathcal{O} (\log t)$, where $N(T)$ are the number... |
Let $k$ be a commutative ring and $R$ a $k$-algebra. The category $R\text{-Bimod}$ of $R$-bimodules becomes a monoidal category with the tensor product of $R$-bimodules. Denote the category of $k$-modules by $k$-Mod. The category of monoid objects $\text{Mon}(R\text{-Bimod})$ is isomorphic to the undercategory $R/\text... |
I'm solving a coding problem, and I wanted to mathematically prove that the answer exists. Essentially, we are given a string of lowercase letters, and we want to know if it can be rearranged such that no character repeats. Example: `aab` can be written `aba`, but `aaab` can't be rearranged in such a way.
Let $N$ b... |
I am interested in the following problem for research: we are given a graph $G$ and integer $d$. Say that $G$ is "$d$-subset-list colorable" if each vertex $v$ is assigned a subset $s(v)$ of $\{1, ..., d\}$, and for each edge $uv$, the symmetric difference of $s(u)$ and $s(v)$ is at least $d$.
When $d=1$, of course ... |
Assume $f\in \text{BMO}(\mathbb{R}^d)$. Denote $f_Q=\displaystyle\dfrac{1}{|Q|}\int_Q f(x)\,dx$. I want to show for $\alpha>2$ and any cube $Q$ with positive volume, we have $|f_{\alpha Q}-f_Q|\leq C_d \log(\alpha) \|f\|_{\star}$ where $\alpha Q$ is dilation of $Q$ by factor $\alpha$ and $\|f\|_{\star}$ is BMO norm of ... |
The well-known identity for complex points forming an equilateral triangle reads
$$z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$$
I have a doubt concerning the uniqueness of this identity
> Is $z_1^2 + z_2^2 + z_3^2 - z_1 z_2 - z_2 z_3 - z_3 z_1$ the only quadratic form in 3 variables that when replacin... |
It is well known that the primes are distributed such that they occur with an approximate "likelihood" of $1/\log(n)$ around the integer $n$ - or more precisely, the number of primes up to $n$ is $$\pi(n) \sim \int_2^n \frac{dt}{\log(t)}.$$
**Question:** Are there other sequences that have a distribution such that t... |
A methodology using complex numbers representation giving a partial proof.
A first remark is that this "distance" is homogeneous,
$$d(g:k.u,k.v)=k.d(g;u,v)$$
For the verification/non verification of triangular inequality, it can be therefore assumed WLOG that all vectors are with a (euclidean) norm $\le 1$.
... |
For a given function $f(z)$ on an annulus $r<|z|<\frac{1}{r}$, I am trying to define a dilation function $f_t(z)$. That is, for $0< r<s< 1$, how to define a function $f_t(z)$ on annulus such that $f_t(r)=f(s)$ and $f_t(\frac{1}{r})=f(\frac{1}{s})$?
This can be done for a unit disk $\mathbb{D}$, where for fixed $t$, ... |
I am doing a problem that states: If you are throwing a dart at a circular board with radius $R$, what is the expected distance from the centre?
If $x$ is the expected radius, then it would be the distance from the centre such that the area created from the radius $x$ is half that of the total area. Then we have:
... |
I need to compute a solution of the following PDE:
$$\frac{\delta u}{\delta t} + v\cdot \frac{\delta u}{\delta x} = 0$$
For didactic purposes, i need to use an IMEX method. The point is no one ever illustrated such methods to me, so i did some research on the web and on chatgpt and i got to some conclusions. I am... |
A methodology using complex numbers representation giving a partial proof.
A first remark is that this "distance" is homogeneous,
$$d(g:k.u,k.v)=k.d(g;u,v)$$
For the verification/non verification of triangular inequality, it can be therefore assumed WLOG that all vectors are with a (euclidean) norm $\le 1$.
... |
I have found this [very close] to $\phi$:
$\phi \sim \dfrac{e^{e^2}}{\left(\pi^2-\dfrac{10e^2-16e-4}{6e^2-116e+69}\right)^3}$ and [the error is close to] $3.13\times 10^{-10}$
[the error is close to]:
https://www.wolframalpha.com/input/?i=-Golden+ratio%2B%28pi%5E2-%2810e%5E2-16e-4%29%2F%286e%5E2-116e%2B69%29%29%... |
Let $A$ be a $\mathbb Z_{\geq 0}$ graded ring. Then we have that $\operatorname{Proj}A$ is the set of homogenous prime ideals which do not contain the irrelevant ideal $A_+$. We put a topology on this by declaring $V(I)=\{\mathfrak p\in \operatorname{Proj}A:I\subset \mathfrak p\}$ to be the closed sets whenever $I$ is ... |
Is it possible to find a spatial curve $\alpha(s) = (x(s), y(s), z(s))$ such that the lines containing the normal vector $\hat{n}(s)$ at each point on the curve $\alpha(s)$ intersect the $z$-axis at a constant angle $\theta$?
My attempts:
For any point $(r(s) \cos \phi(s), r(s) \sin \phi(s), h(s))$, the unit normal... |
Let $a$ and $b$ in $M$, let $\gamma: [0, 1]\rightarrow M$ a path which is differentiable coordinates by coordinates.
Do we have $\dot{\gamma}(t)=(\frac{d}{dt}\gamma_1(t), ..., \frac{d}{dt}\gamma_n(t))\in T_{\gamma(t)}(M)$?
It seems trivial to me but I can't find this written anywhere explicitly, when I dig into ... |
Is the derivative of a path on a manifold always on the tangent space? |
So I have a set of positive integers S (like {1, 5, 9, 29, 12}). Knowing the set, is it possible to build a function F(x) that can, in constant time (independent of the size of the set), map any element of the set to its position (from smallest to largest).<br/>
I know one could build a polynomial with one point for e... |
In the book [Geometric Analysis on the Heisenberg Group and Its Generalizations][1] proposition 1.3 says: Under the change of coordinates
$$ y_1 = x_1 , \, y_2 = x_2, \, \tau = 4t -2x_1x_2,$$
the vector fields
$$ X = \partial_{x_1}, \, Y = \partial_{x_2} +x_1 \partial_t, \, T = \partial_t $$
are transformed into
... |
In the Metropolis-Hastings algorithm, we choose to accept our sample with probability: $$\rho =min\left\{1, \frac{p(x')g(x|x')}{p(x)g(x'|x)}\right\} $$
Where $x$ refers to the current state of the Markov chain, $x'$ refers to the proposed state, $p(.)$ is the target distribution, and $g(.)$ is the proposal distribut... |
Metropolis_Hastings Algorithm. How do we know that it converges to target distribution? |
Metropolis-Hastings Algorithm. How do we know that it converges to target distribution? |
I am reading "Analysis on Manifolds" by James R. Munkres.
>Definition. Let $S$ be a subset of $\mathbb{R}^k$; let $f:S\to\mathbb{R}^n$. We say that $f$ is of class $C^r$ on $S$ if $f$ may be extended to a function $g:U\to\mathbb{R}^n$ that is of class $C^r$ on an open set $U$ of $\mathbb{R}^k$ containing $S$.
... |
How many positive integers up to 100 can be written as a sum of 4 or more consecutive integers? |
I am reading "Analysis on Manifolds" by James R. Munkres.
>Definition. Let $S$ be a subset of $\mathbb{R}^k$; let $f:S\to\mathbb{R}^n$. We say that $f$ is of class $C^r$ on $S$ if $f$ may be extended to a function $g:U\to\mathbb{R}^n$ that is of class $C^r$ on an open set $U$ of $\mathbb{R}^k$ containing $S$.
... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/CJQ9x.png
This problem is from "An Introduction to Manifolds" by Loring W. Tu.
I think I understand the solution, but I would like to ask if the following idea of approaching it is right.
When I dealt with this problem, I tried... |
I bought the book "The Elements of Statistical Learning" (Hastie, etc.), but struggle with the equations presented. I don't understand how they are derived, there are little explanations to why they are true, the authors only assume that you are on that level. Like this one:
$$EPE(x_0)=E_{y_0|x_0}E_{\mathcal{T}}(y_0... |
The question: Given a random sample $X_1,...,X_n$ show that $\frac{1}{n}\sum_{i=1}^n X_i$ is an unbiased estimator for $E(X_1)$.
My confusion: Given a statistical model $(\Omega,\Sigma,p_{\theta})$, where $p_{\theta}$ is a parameterized collection of probability measures, We define $E_{\theta}(X)=\int_{\Omega}X(\ome... |
where are the poles in the product of zeta functions, and where are their residues? |
I was tutoring a student today and was given a question that stated the following:
Find P(|z| > -0.29).
Seems simple enough. But this doesn't really make sense in concept. Firstly, absolute values always yield a positive result, so it wouldn't really make sense for this question. Would the answer simply be 1? An... |
I'm wondering why sequent calculus doesn't have rules like these (at least in the ones I've come across):
$$
\Gamma \vdash A \rightarrow B, \Pi \qquad \Delta \vdash A, \Sigma \over
\Gamma, \Delta \vdash B, \Pi, \Sigma
$$
And
$$
\Gamma, A \rightarrow B \vdash \Pi \qquad \Delta, A \vdash \Sigma \over
\Gamma, \Del... |
Why is there no conditional inference rule in Sequent Calculus of these forms? |
I'm wondering why sequent calculus doesn't have rules like these (at least in the ones I've come across):
$$
\Gamma \vdash A \rightarrow B, \Pi \qquad \Delta \vdash A, \Sigma \over
\Gamma, \Delta \vdash B, \Pi, \Sigma
$$
And
$$
\Gamma, A \rightarrow B \vdash \Pi \qquad \Delta, A \vdash \Sigma \over
\Gamma, \Del... |
First note that I'm not an expert in set theory, but do have an interest in the topic in that I want to be able, at least in principle, to reduce my mathematics to the underlying axioms of my chosen set theory.
In ["One universe as a foundation for category theory"][1], Saunders Mac Lane proposes that a somewhat goo... |
I was tutoring a student today and was given a question that stated the following:
Find $P(|z| > -0.29)$.
Seems simple enough. But this doesn't really make sense in concept. Firstly, absolute values always yield a positive result, so it wouldn't really make sense for this question. Would the answer simply be $1$... |
If $f(x) =\sin^{1200}(x) \ln(1+x)^{500}\arctan^{300}(x)$ how to find $f^{(2000)}(0)$ without Taylor series? |
I come across this problem while reading hints for **Exercise 3.15 (Brezis)**:
Let $\Omega=(0,1)$ and sequence $f_n$ defined by $f_n(x)=ne^{-nx}$. Prove that:
$$\displaystyle \int_{\Omega} \varphi f_n\rightarrow0,\forall\varphi\in C_c(\Omega)$$
Where $C_c(\Omega)$ is the space of continous function on $\Omega$ wi... |
There is a sequence. The first integer is positive, the second integer is negative. An alternating part is a part that switches between positive and negative (0 is not included)
(a) I have a complete answer to this part
(b) What second term should be chosen to get the longest possible switching between positive a... |
Suppose $f: [0,1]^2 \mapsto \mathbb R$ is continuous. Let
$$
\underline{y}(x)\equiv \inf \{y\in [0,1]: f(x,y)\geq 0\}.
$$
There is $y_0\in(0,1)$ such that
$$f(0,y_0)= 0,$$
$$f(0,y)< 0 \quad \forall y\neq y_0.$$
Moreover,
$$
f(x,y_0) > 0 \quad \forall x\in (0,1].
$$
*I hope to find out: is it true tha... |
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