instruction stringlengths 12 30k |
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So the above statement is a question from a course on Theory of Computation. I have already proven it using the pumping lemma, but am looking for proofs using the Myhill-Nerode theorem, or more specifically, it's corollary, which states that a language, $L$, has a finite number of equivalence classes under the relation... |
From what I understand we say that a function is increasing on an interval $I$ if
$$
x_1 < x_2 \quad\Rightarrow\quad f(x_1) < f(x_2).
$$
for all $x_1,x_2\in I$. I understand that some might call this *strictly increasing* and allow for equality when saying *increasing*.
We know that when a function is differenti... |
This is a projection excercise I'm stuck on.
Lets say we have two planes in R3, $A$ and $B$, which both go through the origin. We also have two orthogonal projections, $ProjA$ and $ProjB$, projecting on plane $A$ and $B$ respectively. $ProjA$ is given by the projection matrix $P1$, so that $$ProjA(v) = P1v$$ $ProjB$ i... |
Let $A\in\mathbb R^{n\times n}$ be a symmetric and positive definite matrix. Consider the function
$$f: u\in\mathbb R^n\mapsto f(u)= Au\cdot u\in\mathbb R.$$
Let $||\cdot||_1$ denote a norm and set
$$n(u):= \big(||u||^2_1 + f(u)\big)^{1/2}.$$
As an exercise, I have to prove that
$$n(u+v)\le n(u) +n(v)\quad\for... |
Given a sequence of functionals $(\omega_n)_{n \in \mathcal{N}}$ on a Von Neumann algebra $\mathcal{W}$ converging to $\omega$, I have the following doubts:
1. Is it true that $\omega_n$ is pure $\forall n \in \mathcal{N} \iff \omega$ is pure?
2. Given G group, is it true that $\omega_n$ is G-invariant $\forall n... |
### problem statement
Zagreb and Yerevan play a restricted version of nim as follows: They start with 2 possibly empty (but not both empty) piles of rocks, and take turns Z-Y-Z-Y-Z... until the last player who has a move wins. On a turn, each must take at least one rock out of exactly one pile, so that the total numb... |
I came across the following problem while studying series of functions:
> Prove that $S(x)=\sum_{n=1}^\infty \frac{\sin(2^n x)}{2^n}$ is not differentiable at integers.
I am aware that this series is commonly known as the [Weierstrass function](https://en.wikipedia.org/wiki/Weierstrass_function), and according to... |
In this exercise find and sketch the image of $|z-2|<1$ under $w=\frac{2}{z-1}$ my professor said that is sufficient to just look what happens at the boundary and where is one point inside mapped. But why does the boundary of my set goes to the boundary of the image ? And why the internal part of the set goes to intern... |
I encountered the phrase "sheaf of germs of continuous real-valued functions on $\mathbb{R}$" on Page 5, Example 1.4 of the book "Sheaf Theory" by Bredon.
I am trying to figure out the explicit description of this sheaf without using presheaf. Is there a way to do it? I have tried to formulate something (which I am... |
I'm having difficulty grasping how the zeros of Riemann zeta function is calculated. I thought studying a simpler function such as $f(z)=z^2$ may help.
I'm reading this [LibreText document (PDF)][1]. I understand how the function $f(z)=z^2$ maps the lines $x=1$ and $y=1$ in the z-plane into parabolas in the w-plane.... |
The group $SL_2(\mathbb{Z})$ act on the projective spaces $P(\mathbb{Z}_p)$ and the upper half of the complex plane $\mathbb{H}$ by linear fractional transformations. I am wondering whether there is a direct connection between these actions.
For example: Is there a natural projection $\mathbb{H}\rightarrow P(\mathb... |
Let $A\in\mathbb R^{n\times n}$ be a symmetric and positive definite matrix. Consider the function
$$f: u\in\mathbb R^n\mapsto f(u)= (Au\cdot u)^{1/2}\in\mathbb R.$$
Let $||\cdot||_1$ denote a norm and set
$$n(u):= \big(||u||^2_1 + (f(u))^2\big)^{1/2}.$$
As an exercise, I have to prove that
$$n(u+v)\le n(u) +n... |
I'm having difficulties in understanding the proof for the following theorem which is stated in section 1.8 of Paul Halmos' *Finite-Dimensional Vector Spaces*:
**Theorem.** The number of elements in any basis of a finite-dimensional vector space $V$ is the same as in any other basis.
**Proof.** For the proof, he ... |
> Claim 1: {convex and non-decreasing func.} ◦ {convex func.} is convex
> Claim 2: Norms are convex (The triangle inequality is the same thing as convexity of the norm)
> Claim 3: {convex and non-decreasing func.} ◦ {metric} is probably not a metric (or, {**concave** and non-decreasing func.} ◦ {metric} is still... |
Let $\{f_n(x)\}_{n=1}^\infty$ be a sequence of bounded functions that converge pointwise on $[0,1]$ to some bounded function $f(x)$.
I am trying to determine whether it is true that $\sup_{x\in[0,1]}f_n(x)\to\sup_{x\in[0,1]}f(x)$ pointwise on $[0,1]$. I think it is, and here is my proof:
Let $x\in[0,1]$. Fix $\ep... |
If $f_n\to f$ pointwise on $[0,1]$, then $\sup_{[0,1]}f_n\to\sup_{[0,1]}f$ pointwise? |
I came across the following problem while studying series of functions:
> Prove that $S(x)=\sum_{n=1}^\infty \frac{\sin(2^n x)}{2^n}$ is not differentiable at integers.
I am aware that this series is commonly known as the [Weierstrass function](https://en.wikipedia.org/wiki/Weierstrass_function), and according to... |
Rectangle $ABCD$ is tilted such that its base $AB$ makes an angle of $\theta$ with the horizontal floor. It has its vertex $A$ in contact with the horizontal floor, and the rectangle is released from this position, to fall freely and the vertex at $A$ is free to slide along the floor.
What will be the rectangle's ... |
Let $\mathcal{H}$ be a separable Hilbert space. Let $(S_\alpha)_{\alpha\in\Lambda}$ be a decreasing net of closed subspaces of $\mathcal{H}$, i.e. such that for each $\alpha,\beta\in\Lambda$, there exist $\gamma\ge\alpha,\beta$ such that $S_\gamma\subseteq S_\alpha\cap S_\beta$.
Using separability of $\mathcal{H}$,... |
Whether $\sum_{k=1}^{K} a_k^2 / \sum_{k=1}^{K} a_k$ can converge to $0$ as $K\rightarrow \infty$ if $a_k > 0$, $a_k\rightarrow 0$ but $\sum_{k=1}^{\infty} a_k \rightarrow +\infty$? If take $a_k = \frac{1}{k}$ it's right, but if take $a_k = \frac{1}{\sqrt k}$ it's not sure (I don't know whether it is true since it's $\f... |
Convergence of $\sum_{k=1}^{\infty} a_k^2 / \sum_{k=1}^{\infty} a_k$? |
There are <strike>31 known</strike> primitive solutions to $a^4+b^4+c^4 = d^4$ with $d<10^{28}.$ (*Update*: As of Feb. 21, there are now **93**. See [this MSE table][1].) Old statistics are,
\begin{array}{|c|c|}
\hline
\text{Range} & \text{# of sol} \\
\hline
10^5-10^7 & \;3 \\
\hline
10^7-10^9 & 12 \\
\hl... |
Can anyone help me with this homework question of mine? I'm actually new to proofs.
Here's the question, "Prove, by contradiction, that no chord of a circle is longer than a diameter."
My only knowledge on proof by contradiction is on conditional statements, for instance, given $P\rightarrow Q$, then the proof ... |
If geometric brownian motion is given by:
$$S_t = S_0 e^{\mu t - \frac{1}{2}\sigma^2t + \sigma dW_t}$$
Then what would $\frac{\partial S_t}{\partial t}$ be?
Mainly because for:
$$
\begin{aligned}
& d S_t=\sqrt{\nu_t} d W_t, \\
& d \nu_t=\theta\left(\alpha-\nu_t\right) d t+\xi \sqrt{\nu_t} d B_t, \\
& ... |
partial derivative of geometric brownian motion wrt time? $S_t = S_0 e^{\mu t - \frac{1}{2}\sigma^2t + \sigma dW_t}$ |
If geometric brownian motion is given by:
$$S_t = S_0 e^{\mu t - \frac{1}{2}\sigma^2t + \sigma W_t}$$
Then what would $\frac{\partial S_t}{\partial t}$ be?
Mainly because for:
$$
\begin{aligned}
& d S_t=\sqrt{\nu_t} d W_t, \\
& d \nu_t=\theta\left(\alpha-\nu_t\right) d t+\xi \sqrt{\nu_t} d B_t, \\
& d... |
partial derivative of geometric brownian motion wrt time? $S_t = S_0 e^{\mu t - \frac{1}{2}\sigma^2t + \sigma W_t}$ |
In section $14.14$, Tom Apostol is deriving curvature vector as follows:
$$
\frac{dT}{ds} = \frac{dt}{ds} \frac{dT}{dt} = \frac{1}{s'(t)} T'(t) = \frac{1}{v(t)}T'(t)
$$
As far as I'm aware (theorem $6.7$ in his book), $\frac{dt}{ds} = \frac{1}{s'(t)}$ only if $s(t)$ has inverse, and in general speed $v(t)$ does... |
If geometric brownian motion is given by:
$$S_t = S_0 e^{\mu t - \frac{1}{2}\sigma^2t + \sigma W_t}$$
Then what would $\frac{\partial S_t}{\partial t}$ be? |
In section $14.14$, Tom Apostol is deriving curvature vector as follows:
$$
\frac{dT}{ds} = \frac{dt}{ds} \frac{dT}{dt} = \frac{1}{s'(t)} T'(t) = \frac{1}{v(t)}T'(t)
$$
As far as I'm aware (theorem $6.7$ in his book), $\frac{dt}{ds} = \frac{1}{s'(t)}$ only if $s(t)$ has inverse, and in general speed $v(t)$ does... |
Let $S\subset \mathbb{R}^n$ a (path-)connected subspace in the usual topology with the property that, for all $x\in S$, $S-\{x\}$ has two connected components.
Is $S$ homeomorphic to $R$ with the usual topology?
My attempts at (dis)proving this have been unfruitful, with the tools that I now have.
I kindly ask... |
A version of the [disintegration theorem][1] states:
> Let $(X,\mathcal{A})$ be a [standard Borel space][2]. Let $p$ be a probability measure on $(X,\mathcal{A})$ $\mathcal{B}\subseteq\mathcal{A}$ be a sub-sigma-algebra.
There exists a $\mathcal{B}$-measurably indexed family $y\mapsto p_y$ of probability measures ... |
Presumably the intended question was
<p>
<ul>
<li>If $f_n\to f$ pointwise on $[0,1]$, must $\sup_{[0,1]}f_n\to\sup_{[0,1]}f$?
</ul>
<p>
If so, the answer is "no".
<p>
For a counterexample, if we let
$$
f_n(x)=
\begin{cases}
1&\text{if}\;x={\large{\frac{1}{n}}}\\[4pt]
0&\text{otherwise}\\
\end{cases}
... |
The wiki article https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula
says
$$ \pi = \sum_{k=0}^{\infty} \left[ \frac{1}{16^k} \left( \frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15} \right) \right] $$
In trying to implement an algorithm for the actual computation, I found this... |
Let $\mathcal{H}$ be a separable Hilbert space. Let $(\mathcal{S}_\alpha)_{\alpha\in\Lambda}$ be a decreasing net of closed subspaces of $\mathcal{H}$, i.e. such that for each $\alpha,\beta\in\Lambda$, there exist $\gamma\ge\alpha,\beta$ such that $\mathcal{S}_\gamma\subseteq \mathcal{S}_\alpha\cap \mathcal{S}_\beta$. ... |
I am learning alone and I have just start studying about manifolds and below a question. I want to be sure that my answer and so my understanding is correct.
------------------------
### Question:
Let $ \lambda \in \mathbb{R}$ and let $S = \left \{ (x_1,x_2,x_3) \in \mathbb{R}^3 : x_1^2+x_2^2-x_3^2 = \lambd... |
After $ ( A_\mu B_\nu )_{; \sigma ; \rho} - ( A_\mu B_\nu )_{; \rho ; \sigma} = A_\alpha B_\nu R^\alpha_{\mu \rho \sigma} + A_\mu B_\alpha R^\alpha_{\nu \rho \sigma} $ where $ A_\mu B_\nu $ is outer product of two vector $ A_\mu $ and $ B_\nu $, I understand until substituting $ A_\mu B_\nu $ to $ V_{\mu \nu} $ so it b... |
I am currently working on a project where we are administering different treatments to subjects, but with some potential side effects. I am trying to measure the effect each method has on the well being of the subjects based on some welfare parameters. One problem is that, due to reasons, I am not able to test the same... |
Suppose $V\subset \mathbb{D}^2$ ($V$ is a subset of the unit bidisc) is a set. Also, suppose that the following conditions hold:
1. If $\hat{V}$ is the polynomial convex hull of $V$, then $\hat{V}\cap \mathbb{D}^2 =V$. So $V$ is a closed subset of $\mathbb{D}^2$.
2. For any $\lambda\in \mathbb{D}^2-V$, there is a b... |
Why can we substitute $ V_{\mu \nu} $ to $ V_{\mu ; \nu} $ while inducing contracted Bianchi identity? |
Suppose $V\subset \mathbb{D}^2$ ($V$ is a subset of the unit bidisc) is a set. Also, suppose that the following conditions hold:
1. If $\hat{V}$ is the polynomial convex hull of $V$, then $\hat{V}\cap \mathbb{D}^2 =V$. So $V$ is a closed subset of $\mathbb{D}^2$.
2. For any $\lambda\in \mathbb{D}^2-V$, there is a b... |
Conjecture :
Let $x<0,n>2$ ,$n$ an integer such that $|x|<(n!+1)(n^2+1)$ then it seems that :
$$1/(x/(n!+1))!-1/(x/(n^2+1))!=0$$
Have no integer solution in $x$ where :
$$x!=\Gamma(x+1)$$
Or :
$$\Gamma(x/(n!+1)+1)=\Gamma(x/(n^2+1)+1)$$
Based on empirical trial it seems true for low $n$ .It generates ... |
I am learning alone and I have just start studying about manifolds and below a question. I want to be sure that my answer and so my understanding is correct.
------------------------
### Question:
Let $ \lambda \in \mathbb{R}$ and let $S = \left \{ (x_1,x_2,x_3) \in \mathbb{R}^3 : x_1^2+x_2^2-x_3^2 = \lambd... |
Suppose $V\subset \mathbb{D}^2$ ($V$ is a subset of the unit bidisc) is a set. Also, suppose that the following conditions hold:
1. If $\hat{V}$ is the polynomial convex hull of $V$, then $\hat{V}\cap \mathbb{D}^2 =V$. So $V$ is a closed subset of $\mathbb{D}^2$.
2. For any $\lambda\in \mathbb{D}^2-V$, there is a b... |
Let $S\subset \mathbb{R}^n$ a (path-)connected subspace in the usual topology with the property that, for all $x\in S$, $S-\{x\}$ is disconnected and has two connected components.
Is $S$ homeomorphic to $R$ with the usual topology?
My attempts at (dis)proving this have been unfruitful, with the tools that I now h... |
Identify $t\in\mathbb{R}$ such that (basically it is a determinant of a block matrix of size $n$):
$$
\left| \begin{array}{cc}
\mathbf{x}+t\mathbf{y}
&
\mathbf{B} \\
\mathbf{c}
&
\mathbf{D}
\end{array} \right| = 0
$$
where $\mathbf{x}\in\mathbb{R}^{m\times1}$, $\mathbf{y}\in\mathbb{R}^{m\times1}$, $\math... |
My question is taking place more specifically in a tensor category $(\mathcal{C},\otimes,\phi,\psi)$ (I will denote this category with $\mathcal{C}$ going forward) where $\otimes:\mathcal{C} \times \mathcal{C} \to \mathcal{C}$ is a *bifunctor*, $\phi$ is the *associator* and $\psi$ is the *commutator*.
Now, let $U$ ... |
**Background**
For a certain project, I collected a considerable number of definite integrals whose evaluations amounted to numbers of the form $P :=a+b \pi^{2} $, where $a, b \in \mathbb{Q}$. I won't list them all here (the document comprises 34 pages, including references), but state the ones that stood out to me ... |
Convergence of $\lim_{K\rightarrow +\infty}\sum_{k=1}^{K} a_k^2 / \sum_{k=1}^{K} a_k$? |
>A natural number $a$ is selected from the first $100$ natural numbers. The probability that $a=[\frac a2]+[\frac a3]+[\frac a5]$, where [.] represents greatest integer function, is $\frac mn$ where $m,n$ are coprime then $(m+n)$ is equal to
My Attempt:
Let $a=30n+\gamma$, where $0\le\gamma\lt30$
Putting this... |
How many natural numbers $a\le100$ are there such that $a=[\frac a2]+[\frac a3]+[\frac a5]$, where [.] represents the greatest integer function? |
Whether $(\sum_{k=1}^{K} a_k^2) / (\sum_{k=1}^{K} a_k)$ can converge to $0$ as $K\rightarrow \infty$ if $a_k > 0$, $a_k\rightarrow 0$ but $\sum_{k=1}^{\infty} a_k \rightarrow +\infty$? If take $a_k = \frac{1}{k}$ it's right, but if take $a_k = \frac{1}{\sqrt k}$ it's not sure (I don't know whether it is true since it's... |
Convergence of $\lim_{K\rightarrow +\infty}(\sum_{k=1}^{K} a_k^2) / (\sum_{k=1}^{K} a_k)$? |
Identify $t\in\mathbb{R}$ such that (basically it is a determinant of a block matrix of size $n$):
$$
\left| \begin{array}{cc}
\mathbf{x}+t\mathbf{y}
&
\mathbf{B} \\
\mathbf{c}
&
\mathbf{D}
\end{array} \right| = 0
$$
where $\mathbf{x}\in\mathbb{R}^{m\times1}$, $\mathbf{y}\in\mathbb{R}^{m\times1}$, $\math... |
In introduction I would like to say that I just begin to study (alone) what is are submanifold so I need your help in order to be sure that my understanding of those concept is correct.
### Question:
Let $S = \left \{ (x;y;z) \in \mathbb{R} \times \mathbb{R}^{+*} \times \mathbb{R} : x^2 + z^2 = 1/y \right... |
I have the coordinates (3D) of the vertices of an icosahedron (not a regular one) and I want to find all dihedral angles. I am fine with just equations to which they need to fulfill as I would think that you need information of many/all vertices before you can calculate one of the dihedral angles (so not expecting a cl... |
Take standard 2d-Heisenberg group over finite ring Z/p. Choose standard generators $x_i, y_i$.
Consider generating polynomial for growth: $ g(t) = \sum_i g_i t^i $ , where $g_i$ are the ball sizes.
What is the explicit expression for g(t) ?
It should be not so complicated and quite well-known. But I cannot G... |
I'm supposed to do a variety of matlab experiments for different factorizations and two of those are QR and LQ factorizations, which for matrix $A$ ${m \times n}$ yield $$A=Q[R_1\;R_2]$$
where $R_1$ is upper triangular of order m, $R_2$ is $m\times(n-m)$ and Q is orthogonal of order m, and
$$A=[L\;0]Q,$$
where $L$ i... |
I'm currently going over the proof of the uniqueness of the infinite open cluster in Grimmet (Probability on Graphs, http://www.statslab.cam.ac.uk/~grg/books/USpgs-rev3.pdf, p. 92).
There he defines the quantities $S= \{x \in \mathbb{Z}^d: dist(0,x) \leq n\}$ and $N_S(0),N_S(1)$ when we say for an event that all eg... |
Consider $T, g: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ which are bounded on any bounded interval, satisfy $T(x)=2 T(x / 2)+g(x)$ for every real number $x \geqslant 1$ and $g(x)=\mathcal{O}(x)$.
I would like to show $T(x)=\mathcal{O}(x \log (x))$.
For this I want to show $lim_{x\rightarrow \infty} \dfrac{T(x)... |
Consider $\mathbb{C}^3$ equipped with product $\bullet$ given as
$$(x_1,x_2,x_3)\bullet(y_1,y_2,y_3)~=~ (x_1y_1,x_1y_2+x_2y_1,x_1y_3+x_2y_2+x_3y_1)$$
and norm $\|(x_1,x_2,x_3)\|=|x_1|+|x_2|+|x_3|$. $A:=(\mathbb{C}^3,\bullet,\|\!\cdot\!\|)$ is then a unital abelian Banach algebra with unit $(1,0,0)$.
How would one ... |
I'm currently going over the proof of the uniqueness of the infinite open cluster in Grimmet (Probability on Graphs, http://www.statslab.cam.ac.uk/~grg/books/USpgs-rev3.pdf, p. 92).
There he defines the quantities $S= \{x \in \mathbb{Z}^d: dist(0,x) \leq n\}$ and $N_S(0),N_S(1)$ when we say for an event that all eg... |
Yes, the answer is $\frac{29}{100}$. There are a few details that need to be taken care of.\
You are setting $\gamma$ to be one of $\{1,2,\dots,29\}$ and each value of $\gamma$ generates a value for $n$, which then determines $a=30n+\gamma$.
Why is $\gamma\ge \left\lfloor \frac{\gamma}{2} \right\rfloor + \left\lflo... |
Consider $T, g: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ which are bounded on any bounded interval, satisfy $T(x)=2 T(x / 2)+g(x)$ for every real number $x \geqslant 1$ and $g(x)=\mathcal{O}(x)$.
I would like to show $T(x)=\mathcal{O}(x \log (x))$.
For this I want to show $\lim_{x\rightarrow \infty} \dfrac{T(x... |
This is a multiple choice question which had four options and in the answer booklet it gives out of many others one particular answer regarding which I have some doubts. The question is as follows:
"Let $\mathbb{R}$ be the set of real numbers, $\mathbf{}$ be a subspace of $\mathbb{R^3}$ and $\mathbf{M} \in \mathbb{ℝ^{... |
I came across the following problem while studying series of functions:
> Prove that $S(x)=\sum_{n=1}^\infty \frac{\sin(2^n x)}{2^n}$ is not differentiable at integers.
I am aware that this series is commonly known as the [Weierstrass function](https://en.wikipedia.org/wiki/Weierstrass_function), and according to... |
Suppose we have a finite list of $n$ triples of positive integer numbers, as:
$$
\mathcal{L}=\{(a_{i1},a_{i2},a_{i3}):a_{ij}\in \mathbf{N}\setminus\{0\}, \text{ for } j=1,2,3\}_{i=1,\dots,n}.\
$$
Is there a software, where I can give $\mathcal{L}$ as in-put and which gives as out-put a possible formula, denot... |
Let $D$ be the open unit disc in $\mathbb{C}$ and let $D^* = D\setminus\{0\}$. Now, Picard's Great Theorem implies the following fact.
>> If $f\colon D^\ast \to \mathbb{C}$ is a holomorphic function with finite fibres, then $f$ does not have an essential singularity at $0$.
Here is the simple argument (using Pica... |
If $a,b$ are norms then $n=(a^2+b^2)^{1/2}$ is as well. For this question we will only bother to check triangle inequality. Since $(x^2+y^2)^{1/2}$ is the standard norm on $\mathbb R^2$, it satisfies the triangle inequality: for all $x_1,x_2,y_1,y_2 \in \mathbb R$, we have $((x_1+x_2)^2+(y_1+y_2)^2)^{1/2}\le (x_1^2+y_1... |
Let $A\in\mathbb R^{n\times n}$ be a symmetric and positive definite matrix. Consider the function
$$f: u\in\mathbb R^n\mapsto f(u)= (Au\cdot u)^{1/2}\in\mathbb R.$$
Let $\|\cdot\|_1$ denote a norm and set
$$n(u):= \big(\|u\|^2_1 + (f(u))^2\big)^{1/2}.$$
As an exercise, I have to prove that
$$n(u+v)\le n(u) +n... |
Let $A = [0,1]$ and $F_A$ be the $\sigma$ field generated by $A$. I believe that this means that $F_A$ is the sigma field generated by all open intervals of $A$. Is this correct?
Also, $B = [2,3]$ and $F_B$ be the $\sigma$ field generated by $B$.
I need to describe $F_A \cup F_B$. Would this just be the set gener... |
I am studying alone and I would like to have a feedback on my two first answers in "a)" and "b)" and get help on "c)".
### Question:
We have the following parametric surface in $\mathbb{R}^3$, $ \vec{\phi}(r ; \theta ) = \begin{pmatrix}
r \cdot cos(\theta)\\
1/r^2 \\
r \cdot sin(\theta)
\end{pmatrix}$... |
Let X and Y be iid normal distribution with mean $\mu = 0$, let $Z=\frac{|X|}{|Y|}$, what is the distribution of Z?
I'm thinking of using the distribution method but I'm stuck by the integration. I'm also wondering if change of variable technique can be used. |
Suppose X,Y are iid normal distribution with $\mu = 0$, what is the distribution of ratio of absolute value of X to Y? |
Let $\Omega$ $= \mathbb R$, $A = [0,1]$ and $F_A$ be the $\sigma$ field generated by $A$. I believe that this means that $F_A$ is the sigma field generated by all open intervals of $A$. Is this correct?
Also, $B = [2,3]$ and $F_B$ be the $\sigma$ field generated by $B$.
I need to describe $F_A \cup F_B$. Would th... |
[This](https://math.stackexchange.com/questions/775328/proof-that-infinitely-many-f-exist-if-ffx-fx2013) was a question that appeared in CMI BSc (Hons) Mathematics exam. It asks about finding a real function $f(x)$ such that $$f(f(x))=f(x)^{2013}$$
Along with the answer already given in the question linked to above,... |
I want to learn Algebraic geometry. My main focus will be on **Varieties**.
Can you recommend a good textbook for Algebraic geometry for a beginner? |
Can you recommend a good textbook for Algebraic geometry for a beginner? |
I wanted to confirm whether or not I am correctly identifying a mistake in the author-provided solution to the following problem (located within Spivak's *Calculus* 4th Ed.):
> Let $f$ be defined by: $\begin{cases} \frac{e^x-1}{x}\quad &x \neq 0\\ 1\quad & x=0\end{cases}\quad$ Find the Taylor polynomial of degree $n... |
The title pretty much sums it up. For a 2-manifold with metric $\gamma_{AB}$, a vector field $Y^A$ is said to be a conformal Killing vector field if $\nabla_A Y_B + \nabla_B Y_A = \nabla_C Y^C \gamma_{AB}$. In the case of a two-sphere with the round metric, this means $\partial_{\bar{z}} Y^z = \partial_z Y^{\bar{z}} = ... |
Why are there only six linearly independent global conformal Killing vector fields on the two-sphere with the round metric? |
Consider a number in its decimal representation that begins with an consecutive digits of 3, followed by 2. My question is: Can such a number ever be a **perfect square**?
To clarify, the numbers we're considering take the form
Here are the specific points I'm curious about:
1. Is there a mathematical approa... |
Can $3\cdots32$ be a Square Number? |
I think I ve found a solution for "c)" in all the cases I will be happy to have your feedback on what I ve writen.
c)
To prove that $\vec{\phi} (r ; \theta) $ is a bijection from $r > 0 , 0 \leq \theta < 2\pi $ to $S$ I need to demonstrate that $\vec{\phi} (r ; \theta) $ is injective and surjective.
1- *Inj... |
> Any matrix $A$ is similar to a matrix of form $\begin{pmatrix}B&\\&0\end{pmatrix}$, where $B$ is a matrix with rank $r$?
Is this right? that is, is there exists an invertible matrix $P$, such that $P^{-1}AP=\begin{pmatrix}B&\\&0\end{pmatrix}$, with $rank B=r$.
$B$ may not be a $r\times r$ matrix$.
As is well... |
In tao's book "Nonlinear dispersive equations Appendix A",a Bernstein inequality puzzled me a long time:
Suppose that $P_{\geq N}$ is smoothed out projection to region $|\xi|\geq N$, where $N\in 2^{\mathbb{Z}}$ is a dyadic number, $\hat{\varphi}(\xi)$($|\hat{\varphi}(\xi)|\leq1$) is a real valued raidally symmetric ... |
I think I ve found a solution for "c)" in all the cases I will be happy to have your feedback on what I ve writen.
c)
To prove that $\vec{\phi} (r ; \theta) $ is a bijection from $r > 0 , 0 \leq \theta < 2\pi $ to $S$ I need to demonstrate that $\vec{\phi} (r ; \theta) $ is injective and surjective.
1- *Inj... |
> Any singular matrix $A$ is similar to a matrix of form $\begin{pmatrix}B&\\&0\end{pmatrix}$, where $B$ is a matrix with rank $r$?
Is this right? that is, is there exists an invertible matrix $P$, such that $P^{-1}AP=\begin{pmatrix}B&\\&0\end{pmatrix}$, with $rank B=r$.
$B$ may not be a $r\times r$ matrix$? Or m... |
When I was an elementary student, I'd suffered from understanding basic things like multiplication table and other simple things and I had to memorized them. Last hours I was searching for genesis of multiplication table for the first time (I was curious, If that wasn't folded by experience or observation, So how was m... |
why learning math in elementary school was harder for me rather than upper grades? |
When I was an elementary student, I'd suffered from understanding basic things like multiplication table and other simple things and I had to memorized them. Last hours I was searching for genesis of multiplication table for the first time (I was curious, If that wasn't folded by experience or observation, So how was m... |
How do I get $c$ and $d$ parameters? |
When I was an elementary student, I'd suffered from understanding basic things like multiplication table and other simple things and I had to memorized them. Last hours I was searching for genesis of multiplication table for the first time (I was curious, If that wasn't folded by experience or observation, So how was m... |
Distributing the error in a frequency table so that the $\chi^2$ statistics are distributed according to the actual distribution? |
> Any singular matrix $A$ is similar to a matrix of form $\begin{pmatrix}B&\\&0\end{pmatrix}$, where $B$ is a matrix with rank $r$?
Is this right? that is, is there exists an invertible matrix $P$, such that $P^{-1}AP=\begin{pmatrix}B&\\&0\end{pmatrix}$, with $rank B=r$.
$B$ may not be a $r\times r$ matrix? Or ma... |
I am trying to find an identity for $\arctan(ab)$ in terms of any trigonometric function involving $a$ and $b$ being separate.
I have attempted doing so using $(\arctan(a)+\arctan(b))^2$ but that got me nowhere. I have not tried any other ideas because I can't think of anything that is worth putting down on paper an... |
**Problem**: a set of integers has a coprime cycle if it can be listed in a cyclic list of the form $(a_1, a_2, a_3, \ldots, a_n)$, where
* each element of the set appears exactly once and $\gcd(a_i, a_{i+1}) = 1$ for $i = 1, \ldots, n$ and
* $a_{n+1} = a_1$.
For example, $(7, 8, 9, 10, 11, 14, 13, 12)$ is a co... |
The Lah numbers satisfy the following rising-falling factorials relation
\begin{equation}
x^{\overline{n}} = \sum_{k=0}^{n} L(n, k) x^{\underline{k}},
\end{equation}
where $x^{\overline{n}}$= $x(x+1)\times ... \times (x+n-1)$ and $x^{\underline{n}}$ = $x(x-1) \times ... \times (x-n+1)$.
What would be the ... |
Is this proof of the Lah Numbers Polynomial Identity With Induction on $n$ correct? |
Hint: Set $z=e^x$. This gives
$$z^4+8z^3+13z^2-8z+1 \, = \, \left(z^2 + 3 z - 1\right) \left(z^2 + 5 z - 1\right) \, =\, 0$$
Each quadratic corresponds to two zeroes, for a total of four $z$ values. Finally, you can take their natural log, to find the four corresponding $x$ values. [All four $z$ values](https://... |
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