instruction stringlengths 12 30k |
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I've been reading through Farb and Margalit's book on the action of the mapping class group on Teichmuller space to get a moduli space. This is a very topological/geometric construction, looking at hyperbolic surfaces and homeomorphisms. Specifically, $T(S)=(X,h)/\sim$ for $S$ closed orientable of genus $g\geq 2$, $X$ ... |
Hunter mentiones in his [notes][1] (p.44) that the function $f(x) := x^{-1}\sin(x^{-1})+\cos(x^{-1})$ lacks a defined Lebesgue integral, which must because both $f^+$ and $f^-$ have infinite integrals. Could someone provide some help as to how to show that
$$\int f^+$$
is indeed infinite?
[1]: https://www.math... |
Let $f(x) = x^{-1}\sin(x^{-1})+\cos(x^{-1})$. How can one prove that $\int f^+ = \infty$? |
Let $f(s) = \mathcal{L} \{ x(t) \}(s).$
Then
$$\mathcal{L}\{tx^{(3)}(t)\}(s)=-1\frac{d}{dx}\mathcal{L}\{x^{(3)}(t)\}(s)=-\frac{d}{ds}[s^2f(s)]=-[2sf(s)+s^2f'(s)]$$
using the identities
$$
f^{\prime}(t) \quad s F(s)-f(0)
$$
and
$$
t^n f(t), \quad n=1,2,3, \ldots \quad(-1)^n F^{(n)}(s).
$$
Should ... |
Off by a negative sign in Laplace transform $\mathcal{L}\{tx^{(3)}(t)\}(s)$? |
Let $S:\mathbb{N}\rightarrow\mathbb{Z}^+$ satisfy
$$S(n)=S(A^n\%n)+S(B^n\%n)$$
$$S(0)=1$$
($\%$ denotes modulo if it wasn't clear) My question is, is this function always surjective for all values of $A,B≥2$ with $A≠B$?
|
Is this recursive function always surjective? |
I am reading the articles:
1) *Optimal control for systems described by difference systems*, Hubert Halkin, Advances in Control Systems, Vol 1, Academic Press, New York-London, 1964, Pages 173-196, [MR183564](https://mathscinet.ams.org/mathscinet-getitem?mr=183564).
and the follow up work:
2) *Directional co... |
I was reading McGuire's [paper][1] on why the minimum number of clues in a Sudoku puzzle is 17 when I came across a curious comment:
> In 2008, a 17-year-old girl submitted a proof of the nonexistence of a 16-clue sudoku puzzle as
an entry to Jugend forscht (the German national science competition for high-school s... |
This is stemming from a programming problem, and I am trying to figure out if there is an easier way to go about the issue.
I need to get $\log_{a/b}(n)$, with $0 < a/b < 1$.
Is is possible to rewrite $\log_{a/b}(n)$ as something along the lines of $\frac{\log_{a}(n_1)}{\log_b(n_2)}$?
Yes I know that $\log_{... |
Sotomayor's Theorem for Discrete Dynamical Systems? |
Hunter mentiones in his [notes][1] (p.44) that the function $f(x) := x^{-1}\sin(x^{-1})+\cos(x^{-1})$ lacks a defined Lebesgue integral, which must because both $f^+$ and $f^-$ have infinite integrals. Could someone provide some help as to how to show that
$$\int_0^1 f^+$$
is indeed infinite?
[1]: https://www.... |
If we know our strategy is to go from $\frac {x+1}{x-1} < \frac 1x$ to multiply both sides by $x$ and both sides by $x-1$ to get $(x+1)x=x^2 +x ??? x-1$ and then subtract $x$ from both sides to get $x^2 ??? -1$, then we know the key points we have to consider is when $x-1$ and $x$ change signs and how often the $<$... |
[Green's theorem](https://en.wikipedia.org/wiki/Green%27s_theorem) says that:
$$
\int_C L \ dx + \int_C M \ dy = \iint_D \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \ dx \ dy
$$
If the M and N statisfy $\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} = 1$, then the formula can be use... |
Consider a population $\mathcal{C}$ of $N$ real numbers, possibly with multiplicities. For an integer $n\leq N$, let $A_n$ be the random variable denoting the mean of $n$ random samples of $\mathcal{C}$ **without** replacement. Let $B_n$ denote the mean of $n$ random samples **with** replacement. Let $\mu$ denote the m... |
Mean of samples without replacement is no less concentrated than with replacement? |
It [is well-known](https://en.wikipedia.org/wiki/Idempotent_matrix#Trace) and [can easily be proven](https://math.stackexchange.com/a/2345818/480910) that if a matrix $A$ is idempotent, then its trace equals its rank:
$$
A^2 = A \Rightarrow \mathrm{tr}(A) = \mathrm{rk}(A)
$$
Does the inverse also hold? If yes, ... |
If the trace of a matrix equals its rank, is it idempotent? |
I need help to solve this integral, can you tell me in what order to perform the operations? I have tried to first make the integral of each squared function, then subtract them, but I can't get to the result in the book friends, it gives $0.0000134621$
$\displaystyle \int_{-1}^{1}(\frac{1}{2}\cos(x)+\frac{1}{3}sen(... |
solve this integral, can you tell me in what order to perform the operations? |
I have to write a function which solves the diophantine equation $p(s)x^2 = q(s)$ (in $x$) where $p,q$ are integers polynomials in $s.$ This is doable since $p(s) \mid q(s)$ has only finite solutions (in my context at least). I also know $\deg q=4, \deg p = 2.$
For example $(45s^2 + 264s + 376)^2=x^2(21s^2 + 120s ... |
> Let $S:\mathbb{N}\to\mathbb{Z}^+$ satisfy
$$
\left\{
\begin{aligned}
S(n) &= S(A^n \bmod n) + S(B^n \bmod n) \\
S(0) &= 1
\end{aligned}
\right.
$$
My question is, is this function always surjective for all values of $A, B \geq 2$ with $A \neq B$? |
I arrived at the answer CP = 12/5 units. Since ∆APB — ∆DPC. So there ratio is BP/CP = AB/DC, 2/CP = 5/6, 1/CP = 5/12, CP = 12/5.
Is this correct way to solve? |
Let the set $S = \{0,1,2,3,4,5,6,7,8,9\}$.
Let T be a family of sets of S. Every element of T has 5 elements.
If two distinct elements A and B of T always satisfy
$|A \cap B| \leq 3 $
what is the maximum number of elements in T?
**Additional context**
The program has shown that the number of elements of... |
I asked a question recently on here and got very insightful answers. Now that I understand the proof of the variational method to obtain the EL equation, I have a new question.
The proof that I went through started with the supposition that there is a special path $y(t)$ which minimized the action integral $S$. We ... |
If a path satisfies Euler-Lagrange, does that necessarily mean that it has to minimize the action? |
I was doing a calculation about Nuclear Physics and in one step of the calculation I obtained
$$
\varepsilon_{kbc} n_k n_b \tau_c
$$
where $\tau_c$ is $2\times2$ Pauli matrix and $n_k$ is the k-component of a normalized vector (i.e. $n_a n_a =1$). To obtain the correct result is needed that
$$
\varepsilon_{kbc} ... |
Is this expression with the Levi-Civita tensor correct? |
I am a beginner of functional analysis. I have a simple question when I study this subject.
Let $L(X)$ denote the Banach algebra of all bounded linear operators on Banach space $X$, $T\in X$ is invertible, then $||T^{-1}||=||T||^{-1}$? Is this result correct?
|
This may be simple, but I want to know if my reasoning is ok. I came across a problem whose essential set up is:
let $f_k$ be a sequence of functions in $L^1(\mathbb{R})$ (Lebesgue integrable functions on $\mathbb{R}$). Suppose that
$$\displaystyle \lim_{t \to \infty} f_k(t) = 0 \text{ for all } k \in \mathbb{N}... |
For $u,v\in \mathcal{S}(\mathbb R^d)$, $s,s_1,s_2\in\mathbb R$,
$$
\|uv\|_{H^s}\lesssim_{d,s,s
_1}\|u\|_{H^{s_1}}\|v\|_{H^{s_2}}.
$$
where $s_1+s_2=s+d/2>0$.
**My attempts:** We work in a direct estimate in Fourier side except introducing Bony's decomposition. Indeed, $$
|(\hat{u}\ast \hat{v})(\xi)|\le \left... |
Given two circles:
$\color{red}{\Gamma_1: x^2+y^2=1}$
$\color{yellow}{\Gamma_2: x^2+(y+\frac{1}{2})^2=\frac{1}{4}}$
$\color{green}{\Gamma_3: \dots ?}$
where $\color{green}{\Gamma_3}$ touchs $\color{red}{\Gamma_1}$ and $\color{yellow}{\Gamma_2}$, and lies in the $3^\text{rd}$ quadrant.
Can we, without tri... |
There is an absolutely amazing and very mathematically non-rigorous book by [Stanley Farlow Partial Differential Equations for Scientists and Engineers](https://www.amazon.com/Differential-Equations-Scientists-Engineers-Mathematics/dp/048667620X/ref=sr_1_1?crid=37FOYN5JOHHN0&dib=eyJ2IjoiMSJ9.-MBTFS4ROYNp9IYPtsz0gBwdVS-... |
This may be simple, but I want to know if my reasoning is ok. I came across a problem whose essential set up is:
let $f_k$ be a sequence of functions in $L^1(\mathbb{R})$ (Lebesgue integrable functions on $\mathbb{R}$). Suppose that
$$\displaystyle \lim_{t \to \infty} f_k(t) = 0 \text{ for all } k \in \mathbb{N}... |
I am curious if there is a closed form that represents the coefficients of the inverse of the modified Bessel function of the first kind $I_{0}(x)$. I can find the series representation using InverseSeries[Series[BesselI[0,x],{x,0,20}]] in Mathematica.
$$I^{-1}_{0}(x)=\sum^{\infty}_{n=1}{(-1)^{n+1}a_{n}(n-1)^{n-\fra... |
Given two circles:
$\color{red}{\Gamma_1: x^2+y^2=1}$
$\color{blue}{\Gamma_2: x^2+(y+\frac{1}{2})^2=\frac{1}{4}}$
$\color{green}{\Gamma_3: \dots ?}$
where $\color{green}{\Gamma_3}$ touchs $\color{red}{\Gamma_1}$ and $\color{blue}{\Gamma_2}$, and lies in the $3^\text{rd}$ quadrant.
Can we, without trigono... |
In a finite simple graph $X$, for any $t\in\mathbb{R}$, a **vector $t$-coloring** of $G$ is a mapping $\phi_t: V(X)\longrightarrow S^m$ for some $m\in\mathbb{N}$ (where $S^m$ is the $m$-sphere in $\mathbb{R}^{m+1}$) such that for any $x, y\in V(X)$, $\langle\, \phi_t(x) \,,\, \phi_t(y) \,\rangle \leq -\dfrac{1}{t-1}$ w... |
This may be simple, but I want to know if my reasoning is ok. I came across a problem whose essential set up is:
let $f_k$ be a sequence of functions in $L^1(\mathbb{R})$ (Lebesgue integrable functions on $\mathbb{R}$). Suppose that
$$\displaystyle \lim_{t \to \infty} f_k(t) = 0 \text{ for all } k \in \mathbb{N}... |
Define $P_\phi := e^{-iP\phi}$, where $P$ is a [Pauli matrix][1] with some overall phase factor and $\phi\in[0,2\pi)$. It is claimed (see Page 1 of [this paper](https://arxiv.org/pdf/1808.02892.pdf)) that if $P'P = -PP'$ i.e. we have two anticommuting Pauli matrices $P, P'$, then
$$P_{\frac{\pi}{4}}P'_{\phi} = (iPP'... |
Define $P_\phi := e^{-iP\phi}$, where $P$ is a [Pauli matrix][1] with some overall phase factor and $\phi\in[0,2\pi)$. It is claimed (see Page 1 of [this paper](https://arxiv.org/pdf/1808.02892.pdf)) that if $P'P = -PP'$ i.e. we have two anticommuting Pauli matrices $P, P'$, then
$$P_{\frac{\pi}{4}}P'_{\phi} = (iPP'... |
I am trying to simplify ${}_1F_1(b,1/2,x)$ to see if I can bring it to the form ${}_1F_1(a,1,x)$ for some $a$. I have tried to expand the series representation as follows:
\begin{equation}
\begin{split}
{}_1F_1 \left(b,\frac12,x\right) & = \frac 1{\Gamma(b)} \sum_{k=0}^{\infty} \frac{\Gamma(b+k)}{\Gamma\left(k+\fr... |
I have prices p_i, sizes s_i, with average weighted price A=sum(p_i*s_i)/sum(s_i)
I want to calculate s_hat_i, such that
sum(s_hat_i) = sum(s_i)
to give a desired new average weighted price
B = sum(p_i * sum_s_hat_i) / sum(sum_s_hat_i)
What is the best way to do this?
|
I'm curious about solving a differential equation for the displacement vs. time on the vertical movement of a slinky when it is suspended from rest and starts to oscillate.
I have gotten to these steps, but how do you solve the whole differential equation?
$mg$ - $cv^2$ - $kx$ = m$\frac{dv}{dt}$ , where m = mass... |
Let $k$ be a local field, and consider the local ring of the formal power series $k[[x_1, \cdots, x_n]]$.
Consider the field of fraction $k((x_1,\cdots, x_n)):=\text{Frac}(K[[x_1, \cdots, x_n]])$.
It is not a local field, but we can define a discrete valuation for it.
I am interested in automorphisms of $k((x... |
I have prices $p_i$, sizes $s_i$, with average weighted price $A=\frac{\sum(p_i s_i)}{\sum(s_i)}$
I want to calculate $s'_i$, such that
$\sum(s'_i) = \sum(s_i)$
to give a desired new average weighted price
$B = \frac{\sum(p_i s'_i)} { \sum(s'_i)}$
What is the best way to do this?
Would it be possible... |
I am curious if there is a closed form that represents the coefficients of the inverse of the modified Bessel function of the first kind $I_{0}(x)$. I can find the series representation using InverseSeries[Series[BesselI[0,x],{x,0,20}]] in Mathematica.
$$I^{-1}_{0}(x)=\sum^{\infty}_{n=1}{(-1)^{n+1}a_{n}(x-1)^{n-\fra... |
I am trying to simplify ${}_1F_1(b,1/2,x)$ to see if I can bring it to the form ${}_1F_1(a,1,x)$ for some $a$. I have tried to expand the series representation as follows:
\begin{equation}
\begin{split}
{}_1F_1 \left(b,\frac12,x\right) & = \frac 1{\Gamma(b)} \sum_{k=0}^{\infty} \frac{\Gamma(b+k)}{\Gamma\left(k+\fr... |
I am trying to find a reference to the following "obvious facts" (not sure if they are true or not, but should have some comparable similar results) regarding a **non-commutative** $C^\ast$ algebra $A$.
1. For $a\in A,$ let $\Phi_A$ be the set of all multiplicative linear functionals $A \to \mathbb C.$ Then the spe... |
So far I realized that any polynomial with complex roots has the even number of complex roots.Because for every $(x-(a-b*i))$ there is $(x-(a+b*i)) $ in order for coefficiants to be real.That means that odd degree polynomials have at least one real root.
Because of symmetry of complex numbers roots e.g. $a-b*i$ ... |
Is the angle on on Cartesian coordinate system between dots of all complex roots of polynomial with real coefficients the same? |
I am looking for an example of a concave function whose derivative is bounded over $\mathbb{R}$. Could someone provide such an example? |
I tried to calculate the following:
$$
\int_0^{1/2}\displaystyle\left[\sup_{y\neq x, \ y\in[0, 1/2]}\dfrac{1}{y-x}\int_x^yt^{-1}\ln^{-2}(t)dt\right](-x\ln(x))dx
$$
and obtained $\dfrac{6-\pi^2}{24}$. Is that correct? |
This is a natural follow-up to my previous question, here: https://math.stackexchange.com/questions/4888763/examples-of-two-finite-magmas-which-satisfy-the-same-equations-but-not-the-same. In the answer to that question, Keith Kearnes said that any two magmas on $\{0,1\}$ that satisfy the same equations are isomorphic.... |
Smallest possible cardinality of finite set with two non-elementarily equivalent magmas which satisfy the same quasi-equations? |
Assume that $u(x)$ is the classical solution solving
$$a_{ij}(x)\partial_{ij}u(x)+b_i(x)\partial_iu(x)+c(x)u(x)=f(x)$$
on $\mathbb{R}^n$ for some smooth enough coefficients and uniformly elliptic $a_{ij}$. I am looking for the gradient bound of $u$ explicitly on the behavior of the coefficients.
I found that in Gi... |
>In a given grocery store, apples have an average weight of $194$g and standard deviation of $40$g. Suppose that their weights are independent and follow a normal distribution.
If a customer requests $1$ kg of apples and given that the employer chooses randomly each apple and continues until the total weight is the... |
For a video game, I am trying to figure out whether the ship is headed to the right direction, that is, forward relative to the race direction as it's a racing game in 3 dimensions.
Basically, the game would show up a ***wrong way*** indicator when the ship is driving in the opposite direction.
This is pretty eas... |
Figuring out if an object is headed in the right direction at a crossing? |
$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}}$
> I'm assuming I'm using negation elimination on to get the conclusion, and Existential Elimination once I have transformed the premise into ∃x¬(Fx→Gx)
Yes, and no. A *reduction to absurdity* proof is the way to go. But what exist... |
$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}}$
> I'm assuming I'm using negation elimination on to get the conclusion, and Existential Elimination once I have transformed the premise into ∃x¬(Fx→Gx)
Yes, and no. A *reduction to absurdity* proof is the way to go. But what exist... |
Can a number be positive and negative at the same time? |
In a robotics application (3d Cartesian Space), I need to transform my Point position and orientation in a particular frame to the world frame. The Frame Position and Orientation is known and I was able to get the position by calculating the Rotation Matrix and concatenating it inside the Homogeneous Transformation Mat... |
I have a question about the application of the definition of a random measure on the empirical spectral distribution.
Let $(X , \mathcal{B})$ be some measurable space and let $(\Omega, \mathcal{F}, \mathbb{E})$ be a probability space. A random measure is a mapping $M : \Omega \times \mathcal{B} \to \mathbb{R}$ such ... |
I'm working on graph theory and I've encountered a challenge in determining whether a given graph contains a subdivision of $K_5$ or $K_{3,3}$, which would imply that the graph is non-planar by Kuratowski's theorem.
I have a graph (attached below) with 11 vertices and a number of edges, and I am unsure how to identi... |
Does this graph contain a subdivision of $K_5$ or $K_{3,3}$? |
In a robotics application (3d Cartesian Space), I need to transform my Point position and orientation in a particular frame to the world frame. The Frame Position and Orientation is known and I was able to get the position by calculating the Rotation Matrix and concatenating it inside the Homogeneous Transformation Mat... |
In a robotics application (3d Cartesian Space), I need to transform my Point position and orientation in a particular frame to the world frame. The Frame Position and Orientation is known and I was able to get the position by calculating the Rotation Matrix and concatenating it inside the Homogeneous Transformation Mat... |
**Problem:** True or False: Let $E \subset [0, 1] \subset \mathbb{R}$ be a countable subset. Then, for any $\epsilon> 0$, there is a finite cover of E by open intervals $\{I_k\}_{k=1}^{n}$ such that
$$
\sum_{k=1}^{n} m(I_k) < \epsilon
$$
This sounds like a quite easy problem, but I don't how to solve ... |
It would not be easy to produce simple formulas. Roughly speaking, the connection between arithmetic and bitwise operations is complicated as seen from computer science. Had there been, math education in elementary school can be significantly simplified, since the formula must be able to re-produce the results for natu... |
> Three fair 6−sided dice are rolled and their upfaces are recorded.
> Find the probability that the values showing upon rolling all three
> dice again is the same as the original three values recorded.
Let $X_1$, $X_2$ and $X_3$ be the three first rolls. We observe $X_1=x$, $X_2=y$ and $X_3=z$ and we are interest... |
All I really know is the quotient rule and the chain rule, but this problem blew up and took several pages of my notebook and is still wrong. Is there a way to solve this in less than a page?
$$
z=arctan(\frac{x+y}{1-xy})
$$
$$
\text{find: } z_{xx}
$$ |
How do you find the double partial derivative of a quotient inside of a trigonometric function? |
All I really know is the quotient rule and the chain rule, but this problem blew up and took several pages of my notebook and is still wrong. Is there a way to solve this in less than a page?
$$
z=\arctan\left(\frac{x+y}{1-xy}\right)
$$
$$
\text{find: } z_{xx}
$$ |
There probably is no simple expression, but here is a recurrence relation that will let you compute this number using dynamic programming.
Imagine that you are observing the rolls, as they arrive, one at a time, and keeping track of just enough information to determine whether it contains at least $n$ consecutive nu... |
It seems that some authors use this without proving it since it is sort of intuitively obvious, but I would like to show that in a space $X$, if there exists a path from $x$ to $y$ then there exists a path from $y$ to $x$.
Here is what I have so far. Let $f:[0,1] \rightarrow X$ be a path from $x$ to $y$ and conside... |
In a robotics application (3d Cartesian Space), I need to transform my Point position and orientation in a particular frame to the world frame. The Frame Position and Orientation is known and I was able to get the position by calculating the Rotation Matrix and concatenating it inside the Homogeneous Transformation Mat... |
The polynomial $24x^{2}+14x+14a+4$ is multiplied by 2a−x and divided by x−a to give a remainder of −14/3. What is the value of a?
I havent really tried anything except subbing a into the polynomial, which gives $24a^{2}+28a+4$.
from there im not sure what to do |
A plane flies 1.4 hours at 120 mph on a bearing of 10degrees. It then turns and flies 7.9 hours at the same speed on a bearing of 100degrees. How far is the plane from its starting point? |
[enter image description here][1]
[enter image description here][2]
[1]: https://i.stack.imgur.com/COEIr.png
[2]: https://i.stack.imgur.com/vkJSa.png
How was the state transition probability matrix obtained for this example question? I hope you can provide help, thank you!
|
[enter image description here][1]
[enter image description here][2]
[1]: https://i.stack.imgur.com/COEIr.png
[2]: https://i.stack.imgur.com/vkJSa.png
How was the state transition probability matrix obtained for this example question? I hope you can provide help, thank you!
|
**Problem:** For $a > 0$, let $(S_a f)(x) = f(x/a)$ for Lebesgue measurable functions $f$ on $\mathbb{R}$. Then for any $f\in L_1(\mathbb{R},m)$, $S_a f\rightarrow f$ in $L_1$ as $a\rightarrow 1$.
Can anyone check if my proof below is correct or not? Or is there a simple way? I think the difficulty here is that $f... |
This is the question I got:
We've seen that if the eigenvalues of a 2 by 2 matrix are purely imaginary, then the flow induced by the equation dY/dt = AY acts by rotation.
A. What condition on the eigenvalues and eigenvectors can you check to determine whether the rotation is clockwise or counterclockwise?
B. Does t... |
[enter image description here][1]
[enter image description here][2]
How was the state transition probability matrix obtained for this example question? I hope you can provide help, thank you!
[1]: https://i.stack.imgur.com/KXzLs.jpg
[2]: https://i.stack.imgur.com/FSdhL.jpg |
[enter image description here][1]
[enter image description here][2]
[1]: https://i.stack.imgur.com/KXzLs.jpg
[2]: https://i.stack.imgur.com/FSdhL.jpg
How was the state transition probability matrix obtained for this example question? I hope you can provide help, thank you! |
This is the question I got:
We've seen that if the eigenvalues of a 2 by 2 matrix are purely imaginary, then the flow induced by the equation dY/dt = AY acts by rotation.
A. What condition on the eigenvalues and eigenvectors can you check to determine whether the rotation is clockwise or counterclockwise?
B. Does... |
How to determine the direction of rotation for differential equation by the conditions of their purely imaginary eigenvalues? |
How to find both of the degrees and length of the hypotenuse in a circle if were stretched? |
I'm aware that every topological group is uniformizable: given a neighborhood $U\in\mathcal N(e)$ of the identity, the set $D_U=\{\langle x,y\rangle:x^{-1}y\in U\text{ and }xy^{-1}\in U\}$ is an entourage of the diagonal and $\mathbb D=\{D_U:U\in\mathcal N(e)\}$ is a base for a uniformity compatible with the topology.
... |
What is the universal/fine uniformity on a topological group? |
I am trying to understand parts of the authors solution given to the following question:
>The generators of $\mathrm{SO}(3)$ can be chosen as
>$$t^1=\begin{pmatrix}0 & 0 & 0\\ 0 & 0 & -i \\\ 0 & i & 0 \\ \end{pmatrix},\ t^2=\begin{pmatrix}0 & 0 & i\\ 0 & 0 & 0 \\ -i & 0 & 0 \\ \end{pmatrix},\ t^3=\begin{pmatrix}0 & ... |
This is the question I got:
We've seen that if the eigenvalues of a 2 by 2 matrix are purely imaginary, then the flow induced by the equation $\frac{dY}{dt} = AY$ acts by rotation.
A. What condition on the eigenvalues and eigenvectors can you check to determine whether the rotation is clockwise or counterclockwise... |
When is $\frac{5^n - 1}{4}$ squarefree? |
Is the following correct?
The maximal ideal of $\mathbb{Q}[x,y]$ corresponding to $(\sqrt{2}, \sqrt{2})$ and $(-\sqrt{2}, -\sqrt{2})$ (which are two Galois-conjugate points of $\mathbb{A}^2_{\overline{\mathbb{Q}}}$) should be $(x^2-2, x-y)$. Whereas, that corresponding to the pair $(-\sqrt{2}, \sqrt{2})$ and $(\sqrt... |
It is well-known that the cdf (just 'distribution function' from now on) of a discrete random variable is a "staircase function" going from 0 at minus infinity to 1 at plus infinity. Intuitively this is quite clear. I would like to make the notion of 'staircase function' precise, and also to show a "characterization th... |
**Question:**
There is a game that involves $n$ ordered boxes each with a hidden value associated with it. The value is sampled from a probability distribution density function $P(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2}$. You observe each box’s value in order from box 1 to $n$. After observing any given box’... |
The polynomial $24x^{2}+14x+14a+4$ is multiplied by $2a−x$ and divided by $x−a$ to give a remainder of $\frac{−14}{3}$. What is the value of $a$?
I haven't really tried anything except subbing $a$ into the polynomial, which gives $24a^{2}+28a+4$.
from there I'm not sure what to do. |
I am trying to understand parts of the authors solution given to the following question:
>The generators of $\mathrm{SO}(3)$ can be chosen as
>$$t^1=\begin{pmatrix}0 & 0 & 0\\ 0 & 0 & -i \\\ 0 & i & 0 \\ \end{pmatrix},\ t^2=\begin{pmatrix}0 & 0 & i\\ 0 & 0 & 0 \\ -i & 0 & 0 \\ \end{pmatrix},\ t^3=\begin{pmatrix}0 & ... |
In a course I am studying on Stochastic Processes, I encountered the following exercise:
> Let $X_t = B_t + ct$ for some $c \in \mathbb{R}$ and where $B$ is a standard Brownian Motion. Now define $T_x$ to be the first hitting time of $x$.More formally: $$ T_x = \inf \{ t > 0 : X_t = x \}$$ Calculate $\mathbb{E}(\ex... |
In a course I am studying on Stochastic Processes, I encountered the following exercise:
> Let $X_t = B_t + ct$ for some $c \in \mathbb{R}$ and where $B$ is a standard Brownian Motion. Now define $T_x$ to be the first hitting time of $x$. More formally: $$ T_x = \inf \{ t > 0 : X_t = x \}$$ Calculate $\mathbb{E}(\e... |
I have $y = Ax$ where x, y are vectors and A is a matrix.
I want to get the best K such that $||y|| \leq K||x||$. Clearly K is a matrix norm. Especually, K is a subordinate matrix norm. I tried with Induced norm and Frobenius norms and they provide a very loose bound for K. Are there studies im this direction?
If... |
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