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**How to determine the convergence/divergence of an improper integral:**
$$
\int_{0}^{\infty} \frac{\arctan(ax) - \arctan(bx)}{x}dx
$$
I know that this integral can be written into the sum of two integrals by the additivity property:
$$
\int_{0}^{1} \frac{\arctan(ax) - \arctan(bx)}{x}dx + \int_{1}^{\infty} \frac{... |
Find all values of $x$ such that $(x^2-\frac{1}{x^2})^2+2(x+\frac{1}{x})^2=2024$.
I arrived at the solution $x=\pm \sqrt{22 \pm \sqrt{483}}$, $\pm \sqrt{-23 \pm 4 \sqrt{33}}$, by simplifying to a point where there is no more linear term, and then substituting $y=x^2$. It took many steps and was quite messy. Is there... |
I am wondering if the below equations are equivalent?
$$P(\{\omega: \forall \varepsilon > 0 \ \exists N\in \mathbb{N} \ \forall n\ge N \ |X_n(\omega) - X(\omega)| < \varepsilon \}) = 1$$
$$\forall \varepsilon > 0 \ P(\{\omega: \ \exists N\in \mathbb{N} \ \forall n\ge N \ |X_n(\omega) - X(\omega)| < \varepsilon \}) ... |
This question was answered to some satisfaction in the comment by [Jyrki Lahtonen](https://math.stackexchange.com/users/11619/jyrki-lahtonen), as written by the poster in another comment.
From the comments:
"$[27;20;8]_{27}$ (Reed-Solomon) and a single ternary parity check?" |
Let $f\colon (X,\mathcal{O}_X)\rightarrow (Y,\mathcal{O}_Y)$ be a map of ringed spaces.
In the Stacks Project, in the proof of Tag 09U8, they basically claim at some point, that for any flat $\mathcal{O}_Y$-module $\mathcal{F}$ its pullback $f^*\mathcal{F}$ is also flat.
Now they do link to a proof of this result, bu... |
I started with the following inequality:
$$
\Big|\frac{x^2-9}{x^2-3x}-2\Big|<\epsilon.
$$
Then I could not progress after this:
$$
\Big|\frac{x+3}{x}-2\Big|<\epsilon.
$$
Could anyone help me to find $\delta=\delta(\epsilon)?$ |
This is pretty clear geometrically. Why do you need an algebraic proof?
The projections of those three vertices of a cube will always form a triangle unless two of them are collinear in the $xy$-plane. Your "looking upward" hypothesis prevents that since if two of the projections were collinear the third vertex woul... |
>Prove with delta-epsilon definition: $\displaystyle\lim_{x\to 3} \frac{x^2-9}{x^2-3x}=2$
I started with the following inequality:
$$
\Big|\frac{x^2-9}{x^2-3x}-2\Big|<\epsilon.
$$
Then I could not progress after this:
$$
\Big|\frac{x+3}{x}-2\Big|<\epsilon.
$$
Could anyone help me to find $\delta=\delta(\epsi... |
The Interpretation Theorem is the following excerpt from Kunens old Set Theory
[![the theorem][1]][1]
[1]: https://i.stack.imgur.com/IOLa5.png
The name is from Shoenfields Mathematical Logic. Shoenfield proves essentially the same theorem for a very small Hilbert-style proof-stystem that is very far removed fr... |
Viewing integration as the accumulation of *something* can be a reasonable way to think about it sometimes. But it's not possible actually to *define* integration just using a few words like that. All you can do with a description like "accumulation of change" is to hint at an idea that motivates the actual definition ... |
Given an algebraic field $K$ and denote by $\mathfrak O_K$ its ring of integers. Suppose that $\mathfrak p$ is a prime ideal and $p$ is a prime contained in $\mathfrak p$. Under what conditions of $\mathfrak p$ can $\mathfrak O_K/\mathfrak p^m$ be isomorphic to $\mathbb Z/p^m\mathbb Z$ as $\mathbb Z$-modules?
Note ... |
Let $F$ be Stieltjes measure function (nondecreasing and right continuous) and $\mu$ be corresponding measure on $(\mathbb{R}, \mathcal{R})$. Assume that $F$ is continuous. Show that
$$
\int_\mathbb{R} 2F(x)dF(x)=1
$$
---
I found that this question is a corollary of Durrett's textbook exercise 1.7.3:
> Let... |
If a plane contains a vector, is the vector still considered parallel to the plane? If the answer is no, how can I determine when a vector is parallel using the general equation of planes? |
When is a vector considered parallel to a plane? |
If a plane contains a vector, is the vector still considered parallel to the plane? How can I determine when a vector is parallel using the general equation of planes? |
If a plane contains a vector, is the vector still considered parallel to the plane? How can I determine when a vector is parallel to the plane using the general equation of planes? |
I don't understand the solution my textbook gives for this problem:
$ \int \frac{x^3}{(x+1)^5} \mathrm{d}x $
I thought it had to be done with partial fractions, but I couldn't get it right, and when I checked the solution, it said "through Taylor's formula, you get:"
$\frac{x^3}{(x+1)^5} = \frac{-1 +3(x+1) ... |
Textbook says to integrate a fraction using 'Taylor's formula'? |
When is a vector considered parallel to a plane? (3D) |
Let $F$ be Stieltjes measure function (nondecreasing and right continuous) and $\mu$ be corresponding measure on $(\mathbb{R}, \mathcal{R})$. Assume that $F$ is continuous. Show that
$$
\int_\mathbb{R} 2F(x)dF(x)=1
$$
---
I found that this question is a corollary of Durrett's textbook exercise 1.7.3:
> Let... |
The Batista-Costa surface is a triply periodic minimal surface. Two photos of part of the same surface are below:
[![This is the Batista-Costa surface with many units][1]][1]
[![This is the unit surface of the Batista-Costa surface.][2]][2]
which were both taken form the research paper: [The New Boundaries of 3D... |
What is the Equation for the Batista-Costa Minimal Surface? |
[Hint](https://math.stackexchange.com/a/52497/242) $\, \ \begin{vmatrix} \rm \color{#0af}{x\!+\!y} & \rm u\!+\!v \\ \rm \color{#0af}{x\!-\!y} & \rm u\!-\!\,v\end{vmatrix}\,$ $=\, \color{#0a0}{\begin{vmatrix} 1 & 1 \\ -1 & 1\end{vmatrix}}$ $\begin{vmatrix}\rm y &\rm v \\ \rm x &\rm u\end{vmatrix}\ $ and $\rm \ \overbr... |
Considering $k,x\in \mathbb{R}$
$$k^x=1+\sum_{n=1}^{\infty}\frac{x^n(\ln(k))^n}{n!}$$
Does it have an elementary proof? |
How to prove that? |
Let $F$ be Stieltjes measure function (nondecreasing and right continuous) and $\mu$ be corresponding measure on $(\mathbb{R}, \mathcal{R})$. Assume that $F: \mathbb{R}\to [0,1]$ is continuous. Show that
$$
\int_\mathbb{R} 2F(x)dF(x)=1
$$
---
I found that this question is a corollary of Durrett's textbook exe... |
Let $(e_n)$ be an orthonormal sequence in $H$ is parable. Prove that there exists an orthonormal basis of $H$ that contains $\bigcup_{n\geq 1} \{e_n\}$
My try:
Define the set $\Omega$ as follows:
$$\Omega = \{X \subseteq H : \{e_1\} \subseteq X \text{ and } X \text{ is an orthonormal sequence of } H\}$$
No... |
Let $U \subset \mathbb R^2$ be an open, regular (meaning $U$ is the interior of its closure), bounded and simply connected set. Assume there exists a simple curve $\gamma : [0,1] \rightarrow \partial U$, $\gamma(0) \neq \gamma(1)$.
Does $\partial U \setminus \gamma((0,1))$ stays connected ?
I think this should be... |
How to prove the constant to the x power series? |
I'm currently a high school freshman trying to find some resources for studying higher math. I've read through some Intro to Proof books, Stewart's Calculus (and other supplementary calculus materials), linear alg by W. Curtis, and diff eq by Simmons.
I read through the first two chapters and did the problems in th... |
Let $U \subset \mathbb R^2$ be an open, regular (meaning $U$ is the interior of its closure), bounded and simply connected set. Let $\gamma : [0,1] \rightarrow \partial U$ be a simple curve such that $\gamma(0) \neq \gamma(1)$.
Is $\partial U \setminus \gamma((0,1))$ still connected ?
I think this should be true.... |
[Hint](https://math.stackexchange.com/a/52497/242) $\, \ \begin{vmatrix} \rm \color{#0af}{x\!+\!y} & \rm u\!+\!v \\ \rm \color{#0af}{x\!-\!y} & \rm u\!-\!\,v\end{vmatrix}\,$ $=\, \color{#0a0}{\begin{vmatrix} 1 & 1 \\ -1 & 1\end{vmatrix}}$ $\begin{vmatrix}\rm y &\rm v \\ \rm x &\rm u\end{vmatrix}\ $ and $\rm \ \overbr... |
How to prove the constant's power x series? |
$\mu$ is a measure on $(\mathbb{R}, B(\mathbb{R}))$ which is invariant by translation ($\forall a,b,h\in\mathbb{R}:\,\mu(]a+h, b+h]) = \mu]a, b]$) and such that $\mu(]0,1])=1$.
What I need to show is that $\mu(]0,x])=x$, for $x\in \mathbb{N}$, then for $x\in \mathbb{Q}_+$, then for $x\in \mathbb{R}_+$, and finally, ... |
I was thinking about the trivial topology, in which the only open sets are $X$ and $\emptyset$, where $X$ is the entire space we are working in.
My doubt is: are all the subset of $X$ compact and connected?
I understand that they are all closed, since the only open sets are $X$ and $\emptyset$. But compact means ... |
> This means that a change of coordinates cannot be generally thought of as a change of basis (invertible linear transformation). Is that correct?
Yes, a change of coordinates is not always simply a change of basis: take $\mathbb R ^2$ as $\mathbb R$-vector space and consider a simple translation by $b\in \mathbb R ... |
Yesterday night I watched the [video][1] on youtube and tried my chance :) I was able to rule out some small number of cases.
> **If $b^{2a}-4a^b$ is a perfect square, then any $a>1$ must be even.**
*Proof.* Assume that $a > 1$ is an odd number and let $b^{2a} - 4a^b = t^2$ for some positive integer $t$. F... |
I want to show that $\Bbb R \Bbb P^1 \simeq \Bbb S^1$, So for this I consider the quotient $\Bbb R^2 / \{(0, 0) \}$ induced by the equivalence relation $x \sim y$ iff $x= \lambda y$ with $\lambda \in \Bbb R -\{0\}$
We will say that $\Bbb R \Bbb P^1$ is the map quotient of $\Bbb R^2 / \{(0, 0) \}$ induced by $\pi:\B... |
Let $f\colon (X,\mathcal{O}_X)\rightarrow (Y,\mathcal{O}_Y)$ be a map of ringed spaces.
In the Stacks Project, in the proof of [Tag 09U8](https://stacks.math.columbia.edu/tag/09u8), they basically claim at some point, that for any flat $\mathcal{O}_Y$-module $\mathcal{F}$ its pullback $f^*\mathcal{F}$ is also flat.
N... |
Using the three-point central difference formula,
$$f'(x) \approx \frac{f(x+h)-f(x-h)}{2h}$$
to approximate the $f'(x)$ of $f(x)= e^{-x} sin(x)$ for the interval $[0,9]$ using different step sizes ($h$) produces the following error vs $x$ [graph][1].
I wanted to ask why there are these uniform 'bumps' in the erro... |
**TASK:**
Suppose $X_i (i=1,2,3,…,n)$ are i.i.d random variables with PMF: $f(x, \theta ) = exp(\theta-x), x>\theta$.
Is the estimate unbiased: $\mu = \frac{1}{n} + $min$(X_i)$?
**Answer:**
First, we find the CDF for $\mu$: $F(x) = P(\mu < x) = P($min$(X_i) < x - \frac{1}{n}) = 1 - P($min$(X_i)\geq x - \frac{... |
This is an argument in the proof of complete reducibility of representation of semisimple Lie Algebras.
We have the following SES of a complex Lie Algebra representation,
$$0\to \mathbb{C} \to W \to \mathbb{C} \to 0$$ Now the easy claim would be: $(\rho,W)$ is a two dimension representation and $\rho(x)$ is stri... |
I am reading [The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces](https://arxiv.org/abs/1811.02086), and I came across a usage of the words "covariant" and "contravariant" that I did not understand.
Here is the relevant passage:
> Generalizing the construction of $C... |
I don't understand the solution my textbook gives for this problem:
$$
\int \! \frac{x^3}{(x+1)^5} \, \mathrm{d}x
$$
I thought it had to be done with partial fractions, but I couldn't get it right, and when I checked the solution, it said "through Taylor's formula, you get:"
$$
\frac{x^3}{(x+1)^5}
= \fra... |
In [Sets, Logic, Computation](https://slc.openlogicproject.org/), the compactness theorem is stated as follows:
> For any sentences $\Gamma$ and $A$, the following holds:
>
> 1. $\Gamma\models A$ iff there is some finite $\Gamma_0\subseteq\Gamma$ s.t. $\Gamma_0\models A$.
>
> 2. $\Gamma$ is satisfiable iff for a... |
The rate of change of position with respect to time is velocity. But we measure two common velocities, average velocity over an interval of time and instantaneous velocity.
Now, I understand average velocity as the rate of change. I believe it is the slope of the secant line whose x values are the values across whic... |
I am working with an infinite series of the form $f(x) = \sum_{i=0}^{\infty} a_i x^i$, and I am interested in understanding how to transform this series into a new series where each coefficient $a_i$ is replaced by its reciprocal, yielding a series of the form $\sum_{i=0}^{\infty} \frac{1}{a_i} x^i$.
Given the ini... |
Problem:
A manufactures of times wants to advertise a mileage interval that excludes
no more than $10\%$ of the milage on tired he sells. All he knows is that for
a large number of tires tested, the mean milage with $25,000$ miles, with a
standard deviation of $4,000$. What interval would you suggest?
Answer... |
The Interpretation Theorem is the following excerpt from Kunens old Set Theory
> **8. Appendix $1$: More on relativization**
>
> We sketch here a more formal treatment than that in $\S 2$. There is a general notion of relativization in logic, but we shall discuss only the special case of interest for set theory.
... |
I have a series of independent Binomial random variables, $X_1, X_2, \dots, X_n$. Here, $X_i$ has parameters $(n_i, p_i)$. I want to calculate the probability
$$P\left(\sum_{i=1}^n c_i X_i < c\right)$$
for some constants $c$ and $c_i$. I'll have to calculate this sum repeatedly with the $p_i$'s remaining the sam... |
I'm not sure if this is a math probablem, but I've searched for an question to this question for a very long time and still can't find an answer. Turning to the math community for help is my last hope.
The 3-dimension tensor I need satisfies:
$A_{ijk} = A_{ikj} = A_{jik} = A_{jki} = A_{kij} = A_{kji}$
Certain... |
How to store 3-dimension tensor efficiently with certain symmetries? |
I'm not sure if this is a math probablem, but I've searched for an question to this question for a very long time and still can't find an answer. Turning to the math community for help is my last hope.
The 3-dimension tensor I need satisfies:
$A_{ijk} = A_{ikj} = A_{jik} = A_{jki} = A_{kij} = A_{kji}$
Certain... |
I have a problem connected to Markowitz model. Through some calculations, my task comes down to finding extreme points. I have vector $a \in \mathbb{R}^4$, some $S$ and $V$, where $S$ - expected profit, $V$ - covariation matrix.
So, my problem is: how many extreme points can there be? I have example with $4$ extrem... |
I have a smooth function $f : P \rightarrow R$, where $P$ is the positive orthant of $R^3$. I know that $\partial f / \partial x,\; \partial f / \partial y$ are both $> 0$ everywhere on $P$ and that $\partial f / \partial z < 0$, also everywhere on $P$. Finally, I know that all three partials are functions only of $z$... |
Is a level set of a real valued function f(x,y,z) with non-vanishing gradient and partials that depend only on z connected? |
Using the three-point central difference formula,
$$f'(x) \approx \frac{f(x+h)-f(x-h)}{2h}$$
to approximate the $f'(x)$ of $f(x)= e^{-x} sin(x)$ for the interval $[0,15]$ using different step sizes ($h$) produces the following error vs $x$ [graph][1].
I wanted to ask why there are these uniform 'bumps' in the err... |
I am looking for some basic properties of weighted Sobolev space, like the embedding, the weak topology, their dual space, etc. Is there any good book or note for this? Adam's Sobolev space book seems don't discuss this. |
The rate of change of position with respect to time is velocity. But we measure two common velocities, average velocity over an interval of time and instantaneous velocity.
Now, I understand average velocity as the rate of change. I believe it is the slope of the secant line whose x values are the values across whic... |
Let $(X,\Sigma)$ be a measurable space and $f:X\to \mathbb{R}^d$ be a measurable function. Suppose that $\Vert \cdot \Vert $ is a norm of $\mathbb{R}^d$. How can I show that there's a sequence $(\varphi_n)_{n\in\mathbb{N}}$ of simple functions such that $\lim_{n\to\infty}\varphi_n(x)=f(x)$ and $\Vert \varphi _n(x)\Vert... |
I am looking for some basic properties of weighted Sobolev spaces, like the embeddings, the weak topology, their dual spaces, etc. For example, if $\|u_n\|_{W_g^{1,2}}$ is bounded, is that true $u_n \to u_\infty$ weakly in $W_g^{1,2}$ and $u_n \to u_\infty$ strongly in $L_g^{2}$? Is $W_g^{1,2}$ self-dual?
Is there ... |
Let $(X,\Sigma)$ be a measurable space and $f:X\to \mathbb{R}^d$ be a measurable function. Suppose that $\Vert \cdot \Vert $ is a norm of $\mathbb{R}^d$. How can I show that there's a sequence $(\varphi_n)_{n\in\mathbb{N}}$ of simple functions such that $\lim_{n\to\infty}\varphi_n(x)=f(x)$ and $\Vert \varphi _n(x)\Vert... |
Given an indexed collection of functions $\{f_i : X_i \to Y_i\}_{i\in I}$, there is a product map $\prod_{i\in I}f_i:\prod_{i\in I}X_i \to \prod_{i\in I}Y_i$ defined by $(x_i)_{i\in I} \mapsto (f_i(x_i))_{i\in I}$. This is, I believe, the standard notion of a product map, which makes the Cartesian product into a multif... |
Problem:
A manufactures of times wants to advertise a mileage interval that excludes
no more than $10\%$ of the milage on tired he sells. All he knows is that for
a large number of tires tested, the mean milage with $25,000$ miles, with a
standard deviation of $4,000$. What interval would you suggest?
Answer... |
Problem:
A manufactures of times wants to advertise a mileage interval that excludes
no more than $10\%$ of the milage on tired he sells. All he knows is that for
a large number of tires tested, the mean milage with $25,000$ miles, with a
standard deviation of $4,000$. What interval would you suggest?
Answer... |
I have an observable denoted by $C$, related to a complex potential $B$ by :
$$ C= \bar{B}B ,$$
where $B$ is a complex potential. I know that $ \left. C \right|_0 =C_0 $, a known constant, where the evaluation at $_0$ denotes an equilibrium \ reference state. From this, I can not make any conclusions on $... |
I'm not sure if this is a math probablem, but I've searched for an answer to this question for a very long time and still can't find an answer. Turning to the math community for help is my last hope.
The 3-dimension tensor I need satisfies:
$A_{ijk} = A_{ikj} = A_{jik} = A_{jki} = A_{kij} = A_{kji}$
Certainly... |
I'm not sure if this is a math probablem, but I've searched for this question for a very long time and still can't find an answer. Turning to the math community for help is my last hope.
The 3-dimension tensor I need satisfies:
$A_{ijk} = A_{ikj} = A_{jik} = A_{jki} = A_{kij} = A_{kji}$
Certainly there is a w... |
Consider the family $$C_{\alpha}:\big({2(\alpha-\cos{t})\,\cos{t},\,2(\alpha-\cos{t})\,\sin{t}}\big)\,,\;t\in[0,2\pi]\,,\;\alpha\in\mathbb{R},$$ of closed parametric curves.
1. For which values of $\alpha$ these curves are homotopy-equivalent to curve $C_{0.5}$?
2. For those curves which are homotopy-equivalen... |
I'm not sure if this is a math probablem, but I've searched for this question for a very long time and still can't find an answer. Turning to the math community for help is my last hope.
The 3-dimension tensor I need satisfies:
$A_{ijk} = A_{ikj} = A_{jik} = A_{jki} = A_{kij} = A_{kji}$
Certainly there is a w... |
I'm not sure if this is a math probablem, but I've searched for this question for a very long time and still can't find an answer. Turning to the math community for help is my last hope.
The 3-dimension tensor I need satisfies:
$A_{ijk} = A_{ikj} = A_{jik} = A_{jki} = A_{kij} = A_{kji}$
Certainly there is a w... |
I'm not sure if this is a math probablem, but I've searched for this question for a very long time and still can't find an answer. Turning to the math community for help is my last hope.
The 3-dimension tensor I need satisfies:
$A_{ijk} = A_{ikj} = A_{jik} = A_{jki} = A_{kij} = A_{kji}$
Certainly there is a w... |
I'm not sure if this is a math probablem, but I've searched for this question for a very long time and still can't find an answer. Turning to the math community for help is my last hope.
The 3-dimension tensor I need satisfies:
$A_{ijk} = A_{ikj} = A_{jik} = A_{jki} = A_{kij} = A_{kji}$
Certainly there is a w... |
I would try to estimate this statistically in Excel. A square with each side 1 unit can have its lower left corner at the origin. Randomly select 3 x and y coordinate pairs where one coordinate in the pair is 0 or 1 and the other can be determined with =rand(), use the distance formula for each pair to find the sides... |
I'm not sure if this is a math probablem, but I've searched for this question for a very long time and still can't find an answer. Turning to the math community for help is my last hope.
The 3-dimension tensor I need satisfies:
$A_{ijk} = A_{ikj} = A_{jik} = A_{jki} = A_{kij} = A_{kji}$
Certainly there is a w... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/Xpbhx.png
I am having trouble understanding where these last two expressions come from, namely:
$ P(S_n = i) = p^{\frac{n-j}{2} + \frac{i}{2}} q^{\frac{n-j}{2} - \frac{i}{2}} $
$ P(S_n = -i) = p^{\frac{n-j}{2} - \frac{i}{2}} q^{\fr... |
I'm not sure if this is a math problem, but I've searched for this question for a very long time and still can't find an answer. Turning to the math community for help is my last hope.
The 3-dimension tensor I need satisfies:
$A_{ijk} = A_{ikj} = A_{jik} = A_{jki} = A_{kij} = A_{kji}$
Certainly there is a way... |
This is the approximated version of average case of quicksort. With the iteration method we end up with T(n) = nT(1)+nlogn. We can say that this is easily O(nlogn),but the tight bound is requested. Is nc+nlogn enough to show that it is Θ(nlogn) too, and how? |
In my lecture notes on growth of groups, we are given two definitions for a $c$-step nilpotent group $G$. It goes as follows:
A group $G$ is called $c$-step nilpotent if its lower central series, $\gamma_i(G)$ stabilizes at step $c$, i.e. $\gamma_c(G) = 1$ and $\gamma_{c-1}(G) \ne 1$. Equivalently, $G$ is $c$-step ... |
I'm not sure if this is a math problem, but I've searched this question for a very long time and still can't find an answer. Turning to the math community for help is my last hope.
The 3-dimension tensor I need satisfies:
$A_{ijk} = A_{ikj} = A_{jik} = A_{jki} = A_{kij} = A_{kji}$
Certainly there is a way to ... |
Consider a laser line position estimation by fitting using the Least Square Method (LSM) and prove (or disprove) that it can be considered as a convolution with some function and finding the center by looking for the maximum (zero‐crossing by the derivative). What is the smoothing function?
The Least Square Method (... |
Can the Least Squares Method be expressed as a convolution with a kernel? |
When the cdf $F_{X_1+\dots+X_n}(c)$ of the sum of iid zero mean random variables is decreasing in $n$? |
Let $X$ be a topological space, $Y$ be an ordered topological space
(see definition below) and $f,g: X\to Y$ be continuous maps. Prove
that:\
The map $h: X\to Y$ defined by $$h(x)=\min\{f(x),g(x)\}$$ is continuous on $X$.
PS: Assume $(X,\leq)$ is the set with linear order (if $x,y$ in $X$ then $x\leq y$ or $y\l... |
Given $(n_{11}, n_{12},n_{21},n_{22}) \sim \text{Mult }(n; \pi_{11}, \pi_{12},\pi_{21},\pi_{22})$, consider the problem of testing the null hypothesis $H_0: \pi_{11} \pi_{22}=\pi_{12}\pi_{21} $ against its one- or two-sided alternative.
a. Formulate the underlying testing problem as that of testing one of the parame... |
I am looking for some basic properties of weighted Sobolev spaces, like the embeddings, the weak topology, their dual spaces, etc. For example, if $\|u_n\|_{W_g^{1,2}}$ is bounded, is that true $u_n \to u_\infty$ weakly in $W_g^{1,2}$ and $u_n \to u_\infty$ strongly in $L_g^{2}$ for some subsequence? Is $W_g^{1,2}$ sel... |
Let $I=\langle x-y,xy \rangle \subset \mathbb{C}[x,y]$ be the ideal generated by $u=x-y$ and $v=xy$.
Notice that $x^2 \in I$, since $I \ni xu+v=x(x-y)+xy=x^2-xy+xy=x^2$.
$I$ is not a radical ideal, sicne $x^2 \in I$ but $x \notin I$, hence $I$ is not a maximal ideal.
Notice that the generators of $I$, $u$ and ... |
When $\langle u,v \rangle$ is a maximal ideal in $\mathbb{C}[x,y]$? |
As in the original post, suppose:
$$a=\overline{a_1a_2a_3...a_n}$$
$$b=\overline{b_1b_2b_3...b_n}$$
It goes to show that $\sum_{i=0}^{n} a_i=\sum_{i=0}^n b_i=28$, which is given. If the sum of digits of a number is $27$, that would mean the number is divisible by $9$. Hence, if the sum of digits of a number is $28... |
As in the original post, suppose:
$$a=\overline{a_1a_2a_3...a_n}$$
$$b=\overline{b_1b_2b_3...b_n}$$
It goes to show that $\sum_{i=0}^{n} a_i=\sum_{i=0}^n b_i=28$, which is given. If the sum of digits of a number is $27$, that would mean the number is divisible by $9$. Hence, if the sum of digits of a number is $28... |
In the proof of Sard's Theorem presented in John M. Lee's Introduction to Smooth Manifolds i.e. when trying to show that the set of critical values of a smooth map $F : M \rightarrow N$ between smooth manifolds has measure zero, we identify $M$ with a subset $U \subseteq \mathbb{R}^n$ and let $C \subseteq U$ denote the... |
Sard's Theorem proof in Introduction to Smooth Manifolds -- Why are $C$ and $C_{k}$'s closed in $U$? |
Is homology dual to cohomology or is homotopy dual to cohomology? |
As in the original post, suppose:
$$a=\overline{a_1a_2a_3...a_n}$$
$$b=\overline{b_1b_2b_3...b_n}$$
It goes to show that $\sum_{i=0}^{n} a_i=\sum_{i=0}^n b_i=28$, which is given. If the sum of digits of a number is $27$, that would mean the number is divisible by $9$. Hence, if the sum of digits of a number is $28... |
At the end of a long chapter in which they introduce the basic definitions of linear independence, basis, and dimension (Chapter 2.3 of the second edition), Hoffman and Kunze (HK) provide the following discussion where they seem to define linear independence now for $n$-tuples (rather than, as before, for sets).
I'... |
In their book *A Biography of the World's Most Mysterious Number*, Posamentier and Lehmann mentioned that "Conway has indicated that if you separate the decimal value of $\pi$ into groups of ten places, the probability of each of the ten digits appearing in any of these blocks is about $\frac{1}{40000}$."
I wonder ... |
This is the approximated version of average case of quicksort. With the iteration method we end up with $T(n) = n T(1) + n \log n$. We can say that this is easily $O(n \log n)$, but the tight bound is requested. Is $nc + n \log n$ enough to show that it is $\Theta(n \log n)$ too, and how? |
As in the original post, suppose:
$$a=\overline{a_1a_2a_3...a_n}$$
$$b=\overline{b_1b_2b_3...b_n}$$
It goes to show that $\sum_{i=1}^{n} a_i=\sum_{i=1}^n b_i=28$, which is given. If the sum of digits of a number is $27$, that would mean the number is divisible by $9$. Hence, if the sum of digits of a number is $28... |
I'm trying to understand Example 10.11.1 on page 225 of the book *"An introduction to quiver representations"* by Harm Derksen and Jerzy Weyman (see the attached screenshot below)
[![enter image description here][1]][1]
I want to understand the line underlined in red.
>How does the choice of vectors $(\lambd... |
Given:
- \( \Omega \subseteq \mathbb{R}^n \).
- \( f: \Omega \rightarrow \mathbb{R}^n \) and \( g: \Omega \rightarrow \mathcal{L}(\mathbb{R}^n, \mathbb{R}^n) \) are two \( C^1 \)-class functions on \( \Omega \).
- Define \( F: \Omega \times \Omega \rightarrow \mathbb{R}^n \) as \( F(x, y) = g(x)(f(y)) + f(x) \).
... |
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