instruction stringlengths 12 30k |
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> Let $\mu = $ Lebesgue measure on $[0,1]$ and $f: \mathbb [0,1] \to \mathbb C$ defined as:
$$f(x) =
\begin{cases}
1 & \text{$x \in \mathbb Q \cap [0,1]$} \\
0 & \text{$x \in [0,1] - \mathbb Q$}
\end{cases}$$
> 1. Prove $f \in L^{\infty}(\mu)$.
> 2. Using the definition of $\lVert f \rVert_{\infty}=\inf\{a\in \... |
I tried to solve an olympiad-level problem with complex numbers by myself. Can you please look at my solution and tell me if it is correct and complete, and how many points I would have gotten out of 7? I hope I didn't miss anything.
> Prove that for any complex numbers $z_1 , z_2$ the following inequality holds $$|... |
Given a $n*n$ matrix M, set a $m*n$ (m>n) matrix A as argument. I want to optimize the condition number $cond(AM)$, so I should write the computation of condition number $cond(AM)$ in functional form like $f(A)$, so that I can perform operations (such as derivation) on matrix $A$. My question is that how should I wri... |
Experimental data show that in any triangle with sides $(a,b,c)$ and circumradius $R$, if $x > 2$ then,
$$
\frac{1}{x} - \frac{4}{x^2 - 4} < \frac{1}{a-b+Rx} + \frac{1}{b-c+Rx} - \frac{1}{c-a+Rx}
< \frac{1}{x} + \frac{4}{x^2 - 4} \tag 1
$$
Can this be proved or disproved?
**Corollary**: In the interval $(... |
Does every triangle satisfy $\frac{1}{a-b+5R} + \frac{1}{b-c+5R} > \frac{1}{c-a+5R}$? |
Let $(a,b,c)$ be the sides of a triangle inscribed inside a unit circle such that the vertices of the triangle are distributed uniformly on the circumference. Let $x$ be a real and let
$$
f(x) = P\left(\frac{1}{a-b+x} + \frac{1}{b-c+x} \ge \frac{1}{c-a+x}\right)
$$
The plot of $x$ vs $f(x)$ is show below along ... |
I am familiar with the proof that shows the powerset of the naturals is of the same cardinality as the reals using binary representation. Here's a quick rundown of the proof:
1. Showing that $f:(0, 1) \rightarrow \mathbb R$ where $f(x) = \frac {1}{1-x^2}$ (or any other suitable function) is bijective.
2. Showing ... |
I'm curious about the properties of the space of smooth functions with compact support, $C^\infty_c(\mathbb{R})$. Specifically, I'm interested in whether this space is closed under the operation of differentiation. If we take a function $f$ that is in $C^\infty_c(\mathbb{R})$ and differentiate it, is the resulting func... |
I am puzzed that whether the product manifold $S^{1} \times \text{Bryant soliton on } R^{n-1}$ is non-collapsed, as my understanding of the definition non-collapsed and computation, I believe it is ture, Right? |
Is the product manifold $S^{1} \times \text{Bryant soliton on } R^{n-1}$ is non-collapsed? |
Assume a non-linear relation between the random variables $\mathbf{Y} = f(\mathbf{X})$, where $\mathbf{Y}\sim p_Y$ takes values $\mathbf{y} \in \mathbb{R}^M$ and $\mathbf{X}\sim p_X$ takes values $\mathbf{x} \in \mathbb{R}^N$, with $M\leq N$.
**Direct problem** - If we know the PDF $p_X$, then the PDF $p_Y$ is for... |
Say $K$ is the compact support of a smooth $F$. Let $f$ be the gradient of $F$. Is $f$ also supported on $K$ and therefore compactly supported? Yes. Simply because on $K^c$ $F$ is identically zero, by definition, and the derivative at points there may be computed by only working on $K^c$ - it being open! - and the der... |
What is the value of the integral F(a)? |
How to prove that solution of recurrence relation $ T(n) = n + \sum_{i=1}^{\sqrt n - 1} T(i) $ is O(n) using substitution method for n>= 1? |
Given $A(x)$, with $A:\mathbb{R^n}\rightarrow\mathbb{R^{m\times n}}$ and $\mathbf{f}:\mathbb{R^n}\rightarrow \mathbb{R^n}$, with $\mathbf{f}$ Lipschitz, is it possible to derive a Lipschitz bound of $A(x)\mathbf{f}(x): \mathbb{R}^n \rightarrow \mathbb{R}^m$?
I tought of proceeding as follows,
$$
\begin{align}
\|... |
Given an $n \times n$ matrix M, I want to optimize the condition number $\text{cond}(AM)$, where $A$ is a $m\times n$ ($m>n$) matrix.
I should write the computation of condition number $\text{cond}(AM)$ in functional form like $f(A)$, so that I can perform operations (such as differentiation) on the matrix $A$. My ... |
Try directly construct a constant-valued sequence, e.g., ${x_n}^{y_n} = 1/2$ for all $n$.
If we set $y_n = 1 / n$, then solve for $x_n$, we will get $x_n = \sqrt[1/n]{1/2} = \dfrac{1}{2^n}$.
Then is it correct for $(x_n, y_n) \to (0, 0)$? |
I have been looking through my textbook and have noticed subgroup lattices. I understand the premise of them under addition, but this one example, in particular, I can't figure out.
Let $G = Z^*_{26}$. I know the order of $G$ is $12$ with $7$ being a generator of $G$.
Am I just testing the length of every element... |
I'm studying Algebraic geometry and have been stuck at processing the concept of Transcendental degree of a polynomial and came across the following argument online.
> The transcendence degree of the field of rational functions \(F(x)\) over a field \(F\) is one. This means that the field \(F(x)\) can be thought of ... |
Given $A(x)$, with $A:\mathbb{R^n}\rightarrow\mathbb{R^{m\times n}}$ and $\mathbf{f}:\mathbb{R^n}\rightarrow \mathbb{R^n}$, with $\mathbf{f}$ Lipschitz, is it possible to derive a Lipschitz bound of $A(x)\mathbf{f}(x): \mathbb{R}^n \rightarrow \mathbb{R}^m$?
I tought of proceeding as follows,
$$
\begin{align}
\|... |
I'm studying Algebraic geometry and have been stuck at processing the concept of Transcendental degree of a polynomial and came across the following argument online.
> The transcendence degree of the field of rational functions $F(x)$ over a field $F$ is one. This means that the field $F(x)$ can be thought of as a o... |
This was confusing me when learning about curvature and smoothness. The condition for smoothness on interval $I$ is given as:
1. $\mathbf r'$ is continuous;
2. $\mathbf r'(t) \neq \mathbf 0$.
In this particular example, the space curve of this parametrization is just a straight line passing through the origin, bu... |
Matrix multiplication is associative.
However, I am confused by the following issue. Consider the following matrices and vectors:
$$W=\left(\begin{array}{ccc}
1&2&3\\
4&5&6\\
7&8&9
\end{array}\right),
D=\left(\begin{array}{ccc}
1&0&0\\
0&2&0\\
0&0&3
\end{array}\right),
v= \left(\begin{array}{ccc}
4\\
5... |
when H is order r subgroup of G, $gHg^{-1}$ is yet anther order r subgroup of G.
suppose H is non-normal, mark all subgroups produced by H as Set: {$gHg^{(-1)}$ | for all g in G}
I tried dihedral d_4 with order 2 non-normal subgroup, there are total 4 of them,
H1,H2,H3,H4, where H1={e,f}, H2={e,rf}, H3={e,$r^2f$} et... |
How do we compute the projective resolution of a representation of a quiver with relations.
For example consider the Beilinson quiver $B_4$
[![enter image description here][1]][1]$.
[1]: https://i.stack.imgur.com/RO8iB.png
with the relations $\{\alpha_j^k\alpha_i^{k-1}=\alpha_i^k\alpha_j^{k-1}:1\leq i,... |
If we model $Y = \beta X$, we can estimate $\beta$ to minimize
$$\sum (Y_i - \beta X_i)^2$$
Taking derivatives and solving for 0, we get $\sum 2\beta X_i^2 - 2Y_1X_i = 0 \implies \beta = \frac{\sum Y_i X_i}{\sum X_i^2}.$
However (assume $Y$ and $X$ are normalized) $$\sigma^2(\beta X) = \left(\sum Y_i X_i\righ... |
Why doesn't the linear regression preserve the standard deviation? |
If we model $Y = \beta X$, we can estimate $\beta$ to minimize
$$\sum (Y_i - \beta X_i)^2$$
Taking derivatives and solving for 0, we get $\sum 2\beta X_i^2 - 2Y_1X_i = 0 \implies \beta = \frac{\sum Y_i X_i}{\sum X_i^2}.$
Why does our best fit for $Y = \beta X$ not even satisfy that it has the same variance? |
For the following equation, I don't understand how the terms $df(X_t)$ and $f'(X_t)$ differ.
$$df(X_t) = f'(X_t)dX_t + 1/2 f''(X_t)(dX_t)^2$$
As I understand, both terms refer to the change in the function $f$ with respect to $X_t$. Where am I going wrong?
Thanks in advance. |
The Borel-Cantelli Lemma states that:
- **Lemma 1:** If $\sum_{n=1}^{\infty} P(E_n) < \infty$ then $P(\limsup E_n) = 0$
- **Lemma 2:** If $\{E_n\}$ are independent and $\sum_{n=1}^{\infty} P(E_n) = \infty$ then $P(\limsup E_n) = 1$
As I understand, "lim sup" can be understood in the following ways:
- For ... |
With some intermediate derivation results I found the two integrals are exactly the same. But why?
\begin{align}
I=\int_0^\infty\frac{\sin x}{x + x^2}\mathrm dx=\int_0^\infty\frac{\pi-2\tan^{-1}x}{2 e^x}\mathrm dx
\end{align}
Both are equal to
\begin{align}
I&=\operatorname{Si}(1)\cos(1)-\operatorname{Ci}(1)\si... |
$y = 0$. For what range of $ x$, will there be a unique, strictly positive $r$?
$x > 0$
$0 = -3x + \frac{5}{(1+r)} + \frac{X}{(1+r)^2}$
let $p = 1+r $
using the quadratic equation:
$p = \frac{-5 \pm \sqrt{5^2 - 4 \times (-3x) (x) }}{2 \times -3x}$
$1+r = \frac{-5 \pm \sqrt{5^2 - 4 \times (-3x) (x) }}... |
when H is order r subgroup of G, $gHg^{-1}$ is yet anther order r subgroup of G.
suppose H is non-normal, mark all subgroups produced by H as Set: {$gHg^{(-1)}$ | for all g in G}
I tried dihedral d_4 with order 2 non-normal subgroup, there are total 4 of them,
$H_1$,$H_2$,$H_3$,$H_4$, where $H_1$={e,f}, $H_2$={e,rf}... |
I'm studying Algebraic geometry and have been stuck at processing the concept of Transcendental degree of a polynomial and came across the following argument online.
> The transcendence degree of the field of rational functions $F(x)$ over a field $F$ is one. This means that the field $F(x)$ can be thought of as a o... |
Is my logic correct? A bit string of n with more 0s than 1s |
compute $\sum_{k=0}^{n} \frac{{(-1)^k}}{k+1}{n \choose k}$.
I have done it in the following way:
Notice first that $\sum_{k=0}^{n} \frac{{(-1)^k}}{k+1}{n \choose k} = \sum_{k=0}^{\infty}\frac{(-1)^k}{k+1}{n \choose k}$ Because for $k > n$ we have $n \choose k$ = 0.
Now:
$$(1+x)^n =\sum_{k=0}^{\infty}{n \cho... |
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $A_1,\dots,A_n$ be some specific events in $\Omega$. I am currently reading a quite technical paper which at one points states essentially that for a specific given event $A_n$ we have
$$\mathbb{P}(A_n)=\mathbb{P}(A_n\mid A_{n-1})\cdots\mathbb{P}(A_2\m... |
Is there a name for an equality of the like $\mathbb{P}(A_n)=\mathbb{P}(A_n\mid A_{n-1})\cdots\mathbb{P}(A_2\mid A_{1})\mathbb{P}(A_1)$? |
Find $\sum_{k=0}^{n} \frac{{(-1)^k}}{k+1}{n \choose k}$ as a function of n.
I have done it in the following way:
Notice first that $\sum_{k=0}^{n} \frac{{(-1)^k}}{k+1}{n \choose k} = \sum_{k=0}^{\infty}\frac{(-1)^k}{k+1}{n \choose k}$ Because for $k > n$ we have $n \choose k$ = 0.
Now, using the binomial theor... |
Given $A(x)$, with $A:\mathbb{R^n}\rightarrow\mathbb{R^{m\times n}}$ and $\mathbf{f}:\mathbb{R^n}\rightarrow \mathbb{R^n}$, with $\mathbf{f}$ Lipschitz, is it possible to derive a Lipschitz bound of $A(x)\mathbf{f}(x): \mathbb{R}^n \rightarrow \mathbb{R}^m$?
I tought of proceeding as follows,
$$
\begin{align}
\|... |
The posterior mean prediction of a Gaussian Process is given by
$$\mu(x_*) = \sum_{i=1}^n\alpha_i k(\mathbf{x}_i,\mathbf{x}_*) $$ where
$$\alpha = (K + \sigma_n^2I)^{-1} \mathbf{y}$$
Assuming I want to compute $\alpha$ with more data. i.e. train the GP model with more data. Is it possible to show that $\alpha$ ... |
Let $ M $ be a manifold and let $ X $ be a vector field defined on $ M $. Given some $ m\in M $, my book (R. Abraham, J. E. Marsden, _Foundations of Mechanics_) defines a _flow box_ at $ m $ as a triple $ (U,\epsilon,\Phi) $ where:
1. $ m\in U\subset M $ is an open set and $ \epsilon > 0 $ or $ \epsilon = +\infty $... |
Find $\sum_{k=0}^{n} \frac{{(-1)^k}}{k+1}{n \choose k}$ as a function of n.
I have done it in the following way:
Notice first that $\sum_{k=0}^{n} \frac{{(-1)^k}}{k+1}{n \choose k} = \sum_{k=0}^{\infty}\frac{(-1)^k}{k+1}{n \choose k}$ Because for $k > n$ we have $n \choose k$ = 0.
Now, using the binomial theor... |
Given $A(x)$, with $A:\mathbb{R^n}\rightarrow\mathbb{R^{m\times n}}$ and $\mathbf{f}:\mathbb{R^n}\rightarrow \mathbb{R^n}$, with $\mathbf{f}$ Lipschitz, is it possible to derive a Lipschitz bound of $A(x)\mathbf{f}(x): \mathbb{R}^n \rightarrow \mathbb{R}^m$?
I tought of proceeding as follows,
$$
\begin{align}
\|... |
I'm somewhat confused as to how the general version of Cauchy's Theorem does not follow (almost) immediately from its version in a disk. At least Ahlfors as well as Stewart & Tall prove the general theorem with arguments more complicated than the one I have in mind, which leads me to ponder: is the proof below wrong?
... |
Would the following not be a correct proof of Cauchy's Integral Theorem? |
I am trying to build an example of a topology on $\mathbf{R}^2$ in which addition is discontinuous and multiplication by scalar is continuous. I think that such a topology is generated by the collection of all intervals on all straight lines passing through the origin $(0,0)$. Then, for example, adding points $(1,1)$ a... |
I'm somewhat confused as to how the general version of Cauchy's Theorem does not follow (almost) immediately from its version in a disk. At least Ahlfors as well as Stewart & Tall prove the general theorem with arguments more complicated than the one I have in mind, which leads me to ponder: is the proof below wrong?
... |
This is very difficult to explain, but what I'm looking for, is kind of like sigma but instead I want the middle to do something like this "x=n+1" instead of doing (n+1)+(n+1)+(n+... etc. etc.
So, as an explanation, I am in short trying to recreate the exponent symbol function following these requirements;
- All... |
I'm solving the following ODE:
$$(1 + y^2) = (\arctan(y) - x)\frac{dy}{dx},$$
with $y(x)$ being the unknown function. The Solutions sheet recommends substituting $u(x) = \arctan(y(x))$, so that $\frac{du}{dx} = \frac{1}{y^2 + 1}\frac{dy}{dx}$. After making the substitution, the sheet reports the equation becoming... |
In Wikipedia's [proof of Bertrand's Postulate](https://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate), in the second lemma, it is concluded that:
$$R = R(p,{{2n}\choose{n}}) \le \log_p 2n$$
where $R(p,n)$ is the [p-adic order](https://en.wikipedia.org/wiki/P-adic_valuation) of ${2n}\choose{n}$
Later in... |
(ii) "When p=1 the argument is parallel with the above but simpler. Here we take $f_n(x)=(sign g(x))X_{E_n}(x)$,where $E_n$ is an increasing sequence of sets of finite measure whose union is X." I can not figure out how to do that.I am not smart. Can anyone know how to do that? |
Some pointers:
- In *The Topology of Roman Mosaic Mazes* (1992), author Anthony Philips discusses the topology of the 46 mazes that were 'well-preserved enough to be intelligible'. He finds that that these fall into 25 distinct topological types, which in turn can be built up out of seven elementary submazes. <a h... |
Given matrices $\pmb{A}\in\mathbb{R}^{p\times n}$ and $\pmb{B}\in\mathbb{R}^{p\times n}$ with $p>n$, I need to solve the following linear system in symmetric matrix $\pmb{X}\in\mathbb{R}^{p\times p}$
$$\pmb{X}\pmb{A}=\pmb{B}.$$
Based on [this post][1], it seems that $\pmb{X}$ has an explicit formula.
My issue is... |
if i have a function like f(x) = e^sqrt(x+(2/x)).
why is the result of using [this formula][1], where x is 2 and h is 0.1, greater than the exact derivative of f(x)?
[1]: https://i.stack.imgur.com/J3AAf.png |
why can the result of the four point finite difference formula of a function, be greater than the exact derivative of the function? |
if i have a function like $f(x) = e^{\sqrt{x+\frac{2}{x}}}$.
why is the result of using the formula below, where $x$ is $2$ and $h$ is $0.1$, greater than the exact derivative of $f(x)$?
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/J3AAf.png |
I want to know if the function $f(x) = \frac{\sin(x)}{x}$ is integrable and if it is, then what's its integral?
My high school book says its a non-integrable function while WolframAlpha says its integral is Si$(x) +$constant.
Please shed some light on this topic and explain from basic level like what is Si$(x)$,... |
I have the following question that starts from this expression:
$$
u(x)=\int_{\partial B}\left(-\frac{\partial u}{\partial n}(y) \Gamma(y-x)+u(y) \frac{\partial \Gamma}{\partial n}(y-x)\right) d \sigma(y) .
$$
If $h \in C^2(\bar{B})$ is harmonic and Green function $G(x, y)=\Gamma(y-x)-h(y)$, the above integral le... |
Say we have cubic curve $\vec{C}(t)_ = (C_x(t), C_y(t), C_z(t))$ which approximates some parametric function $\vec{F}(t)$ within error less than $\epsilon$. The cubic curve is $C^2$ continuous and is constructed by the usual method of least squares fit to some set of points $(\vec{P}_i, t_i), \quad i<n$ with $\vec{P}_i... |
I am learning about combinatorial and bit strings.
I decided to use combinatorial reasoning and wanted to see if my logic made sense.
**The question: How many bit strings of length n contain more 0’s than 1’s?**
**Note: the length is unknown**
----
Let there be a bit string of length \( n \)
with \(k\... |
Say we have cubic curve $\vec{C}(t)_ = (C_x(t), C_y(t), C_z(t))$ which approximates some parametric function $\vec{F}(t)$ within error less than $\epsilon$. The cubic curve is $C^2$ continuous and is constructed by the usual method of least squares fit to some set of points $(\vec{P}_i, t_i), \quad i<n$ with $\vec{P}_i... |
We are given $f(0)\neq0$. The equation $f(x+f(y))=f(x)+f(y)$ seems to be entirely dependent on the values of $f$. It seems impossible to have anything to say about $f(x+y)$. Clearly, functions of the form $x+b$ (with $b\neq 0$) satisfy this equation. So, it suggests the function in the question has some properties of t... |
Does $f(x+f(y))=f(x)+f(y)$ imply $f(x+y)=f(x)+f(y)-f(0)$, given $f(0)\neq0$? |
I need to random numbers of the form $\frac 1 {(1+x) \log 2}$ over the interval $[0, 1]$. I know that I can generate this using inverse transform method. Are there any other methods beside it? Is there a way to generate this distribution using acceptance-rejection method or transformation method? |
We say, that a graph $G$ is packable, if it is isomorphic to a subgraph of its complement.
In more formal terms:
A graph $G$ is packable, if there is a permutation $\sigma : V(G) \to V(G)$ such that
$$\forall_{u,v \in V(G)} \quad : \quad uv \in E(G) \quad \Rightarrow \quad \sigma(u)\sigma(v) \notin E(G)$$
... |
What are other graphs of order $n$ than the star $K_{1, n-1}$ which are not packable? |
Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a continuous function. We assume that $f$ is periodic: for every $x \in \mathbb{R}^n$ and every $a \in \mathbb{Z}^n$ we find $f(x) = f(x + a)$. Is it true that $f$ must have fixed point? |
Does a periodic function on $\mathbb{R}^n$ have fixed point? |
Let $R \subset S$ an injective map of commutative rings, $M \subset N $ an inclusion of $R$-modules. We identify $M, N$ respectively as $M \otimes_S 1_R \subset N \otimes_R 1_S \subset N \otimes S$.
In [this answer][1] Haran elaborated that if $R \subset S$ a faithfully flat extension, then the intersection of ... |
By definition, the basis of the product topology on the ∏Xα(α∈J) is the set of the form "∏Uα(α∈J)" where the Uα≠Xα only finitely many values of α, then could there be infinitely many sets between Uα1≠Xα1 and Uα2≠Xα2?
The answer depends on the cardinality of the set J and it`s easy to see there are no such basis fo... |
For a basis of product topology on the X×X ...X×X...，could there be infinitely many sets between two sets that are not the X? |
For each $n \geq 1$, denote $C_n= \int_0^\pi x^3cos(nx)dx$.
Calculate $\sum_{n=1}^{\infty} {C_n}^2$
My suggestion:
The given sequence is proportional to Fourier coefficients of $x^3$ expansion to an even function in $[-\pi,\pi]$.
The sum that I get is: $\frac{9\pi^8}{224}$.
Am I in the right direction, how do ... |
Given $A(x)$, with $A:\mathbb{R^n}\rightarrow\mathbb{R^{m\times n}}$ and $\mathbf{f}:\mathbb{R^n}\rightarrow \mathbb{R^n}$, with $\mathbf{f}$ Lipschitz, is it possible to derive a Lipschitz bound of $A(x)\mathbf{f}(x): \mathbb{R}^n \rightarrow \mathbb{R}^m$?
I thought of proceeding as follows,
$$
\begin{align}
\... |
I am trying to prove that the set of equivalence classes of measurable functions with the metric $\rho(f,g)=\int_{0}^{1}\frac{|f(x)-g(x)|}{1+|f(x)-g(x)|} \, d\mu$ is a vector topological space.
By definition, a vector topological space is a linear space, with a topology in which addition and multiplication by a sca... |
Consider the following initial value problem for $\dot{x} =F(t,x)$:
\begin{equation}
\dot{x} = x^{1/3}, \quad x(0)=0
\end{equation}
Now, the general solution reads
\begin{equation}
x(t)= \left[ \frac{2}{3} (t+ C) \right]^{3/2}
\end{equation}
with $C$ being an arbitrary constant.
If $C=0$ the initial conditio... |
I am struggeling to find the derivative of
$f(x) = a^T(x) b(x) c(x)$, where $a,b,c,x$ are all $\in R^{3x1}$.
Using single variable chain rule I get:
$f'(x) = \frac{d}{dx}\left(a^T(x)\right) * b(x) * c(x) + a(x) ...)$
However, already here I am hold back by a dimension mismatch:
A vector derived by a vector... |
So lets say we are trying to figure out if:
> **P(x, y) is the predicate of $x^2 < y$ as defined for x, y ∈ Z.**
This means using quantifiers: ∀ and ∃, I can logically say:
**1. ∀x∃y of P(x,y) is possible!**
This is because we can say $ y = x^2 + 1 $ which means $ x < x^2 $. This means for every x there is... |
Consider the heat kernel on the 2D-torus
$$
K(t,z-z_1):= \sum_{m \in \mathbb{Z}^2} \exp\Big(-(1-4\pi^2|m|^2)t +2\pi i m \cdot (z-z_1)\Big)
$$
It satisfies the semigroup property
$$
K(t_1+t_2,z_1-z_2)=\int_{\mathbb{T}^2} K(t_1,z_1-z) K(t_2,z_2-z) dz
$$
Let now $\rho_\epsilon$ a mollifier and consider the reg... |
I am struggeling to find the derivative of
$f(x) = a^T(x) b(x) c(x)$, where $a,b,c,x$ are all $\in R^{3x1}$.
Using single variable chain rule I get:
$f'(x) = \frac{d}{dx}\left(a^T(x)\right) * b(x) * c(x) + ...$
However, already here I am hold back by a dimension mismatch:
A vector derived by a vector shoul... |
Let $R$ be discrete valuation ring with field of fractions $K$, and $\phi:R \to A$ a *finite, local* (ie $A$ dominates $R$ in the sense that it's maximal ideal contains the image of maximal ideal of $R$) map of local rings.
What are the weakest conditions to assure that $A$ is already discrete valuation ring too?
... |
Finite map of local rings $R \to A$ with $R$ DVR. When is Target DVR too? |
By definition, a basis of the product topology on
$$
\prod_{\alpha \in J} X_\alpha
$$
consists of sets of the form
$$
\prod_{\alpha \in J} U_\alpha,
$$
where the $U_\alpha \neq X_\alpha$ only finitely many values of $\alpha$.
> Then, could there be infinitely many sets between $U_{\alpha_1} \neq X_{\alp... |
For a basis of product topology on the $X \times X \times \cdots$，could there be infinitely many sets between two sets that are not the $X$? |
Consider the following family of integrals for $\varepsilon > 0$:
$$I(\varepsilon) := \int_{|z| = 1 + \varepsilon} \frac{|\mathrm{d}z|}{|z - 1|} = \int_0^{2\pi} \frac{(1 + \varepsilon)\,\mathrm{d}\theta}{|(1+\varepsilon)e^{i\theta} - 1|}$$
> **Question**: The objective would be to show the following behaviour:
$$\... |
A group $G$ is called an extension of a group $Q$ by $N$ if we have the following short exact sequence: $1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$. If there is a homomorphism $f: G \rightarrow N$ that is the identity on $N$, then such extensions give quite a simpler structure of $G$, namely, $G \cong N... |
I'm reading *Complex Analytic and Differential Geometry* by *J-P. Demailly*. In section 2.4.2, the author gives the following proposition:
Suppose $A=V(\mathcal{J})$, where $J$ is a prime ideal of $\mathcal{O}_n=\mathcal{O}_{\mathbb{C}^n,0}$. Set $\mathcal{J}_k=\mathcal{J}\cap \mathbb{C}\{z_1,\cdots,z_k\}$.
****P... |
Let $R$ be discrete valuation ring with field of fractions $K$, and $\phi:R \to A$ a *finite, local* (ie $A$ dominates $R$ in the sense that it's maximal ideal contains the image of maximal ideal of $R$) map of local rings.
What are the weakest conditions to assure that $A$ is already discrete valuation ring too?
... |
Finite map of local rings $R \to A$ with $R$ DVR. When is target $A$ is DVR too? |
[Khan Academy][1] used the approach I thought of in the question above.
Though they have done it in a much more intuitive way.
The most important thing is that it shows that the approach was valid.
[1]: https://www.youtube.com/watch?v=7br_-emGNec |
The answer is no, and we give some counter-examples below: two in $2$ and one in $3$ and more dimensions. The latter example also appears in a slightly different form in [these old lecture notes of M. Wolf][1] (Example 8.1). However, in my opinion, this specific result is rather difficult to find if you're not already ... |
A group $G$ is called an extension of a group $Q$ by $N$ if we have the following short exact sequence: $1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$. If there is a homomorphism $f: G \rightarrow N$ that is the identity on $N$, then such extensions give quite a simpler structure of $G$, namely, $G \cong N... |
Let $R$ be discrete valuation ring with field of fractions $K$, and $\phi:R \to A$ a *finite, local* (ie $A$ dominates $R$ in the sense that it's maximal ideal contains the image of maximal ideal of $R$) map of local rings.
What are the weakest conditions to assure that $A$ is already discrete valuation ring too?
... |
What is $\sqrt{-1}$? circular reasoning defining $i$ (How to define the square root of negatives ) |
Let $P_{m}^{\left(a,b\right)}\left(s\right)$ the $m$-th [Jacobi polinomial][1]. An old result of Bateman shows that $$
\left(1-x\right)^{\alpha+\mu}\frac{P_{m}^{\left(\alpha+\mu,\beta-\mu\right)}\left(x\right)}{P_{m}^{\left(\alpha+\mu,\beta-\mu\right)}\left(1\right)}=\frac{\Gamma\left(\alpha+\mu+1\right)}{\Gamma\left(... |
For the beginning I want to clarify the quantifiers (I am assuming that we are always working in $\mathbb{Z}$):
Let's take a look at $\forall x. A$. Where $A$ is a statement which can be either true or false.
The $\forall x.$ quantifier, says that the following statement has to be true **for every** possible $x$.... |
A group $G$ is called an extension of a group $Q$ by $N$ if we have the following short exact sequence: $1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$. If there is a homomorphism $f: G \rightarrow N$ that is the identity on $N$, then such extensions give quite a simpler structure of $G$, namely, $G \cong N... |
For the beginning I want to clarify the quantifiers (I am assuming that we are always working in $\mathbb{Z}$):
Let's take a look at $\forall x. A$. Where $A$ is a statement which can be either true or false.
The $\forall x.$ quantifier, says that the following statement has to be true **for every** possible $x$.... |
In my calculus class we were solving: integral{0 to x}(sin(x^4)) via power series. My professor told us we can't solve it by any other means, so I went to try to brute force it anyways using Desmos.
I ended up finding that 2x*cos((x^4)/2)*sin((x^4)/2) fit the graph of the integral seemingly perfectly. Even when test... |
Why does 2xcos((x^4)/2)sin((x^4)/2 seem to match the integral[0 to x](sin(x^4))? |
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