text stringlengths 83 79.5k |
|---|
H: can i have open sets in closed interval $[0,1]$?
I have to show that if $A=[0,1]$ then $(a,1]$ and $[0,b)$ are open in $A$ for $0\le a<1$ and $0<b\le 1$. when i finish my proof a friend tell me that is not true because when $x=1$ in $(a,1]$ i can't find a $r>0$ such that the open ball is complete contain in $(a,1]$... |
H: Non-trivial normal subgroup of $G$ intersects the center $Z(G)$ non-trivially
I've seen a lot of information about this problem when $G$ is a $p$-group. But that need not be the case here.
Let $G$ be a group such that $G/Z(G)$ is abelian. Let $H$ be a non-trivial normal subgroup of $G$. Show $H\cap Z(G)$ is a non-... |
H: Find the arclength of the curve defined by $r(t)=i+9t^2j+t^3k$ for $0 \leq t \leq \sqrt28$.
First I found $r'(t)=\langle 1,18t^2,3t^2\rangle$ and so the magnitude of $r'(t)= \sqrt{1+(18t)^2+(3t^2)^2}$ thus the integral from $0$ to $\sqrt{28}$ of $\sqrt{1+324t^2+9t^4} dt$. When I plugged $\sqrt{28}$ in, I get $\sqrt... |
H: Is this function lebesgue integrable or not?
I'm trying to see if this function is lebesgue integrable.
$$\int_0^1 \frac{(-1)^{\lfloor 1/x \rfloor}}{x^2} dx.$$
How can I prove it?
I try the following:
Let $f(x)=\frac{(-1)^{\lfloor 1/x \rfloor}}{x^2}$.
\begin{align*}
\int_0^1 |f(x)| dx&=\sum_{n=1}^{\infty} \int_{1/(... |
H: Show the following defines a topology on $\mathbb{R}$
For the open sets of $\mathbb{R}$ take those sets that are open in the
usual sense and periodic with period $1$ ( $t \in U \iff t+1 \in U \;
, \forall U \; \; open $) PROVE this is a topology on R
attempt:
I am having some trouble proving this.Trivially $\varn... |
H: How can we conclude that $\mathbb{P}(A)\leq \mathbb{P}(B)?$
Given two events $A$ and $B$, if the event $A$ happens, then $B$ happens. How can we conclude that
$$\mathbb{P}(A)\leq \mathbb{P}(B)?$$
AI: "if A happens then B happens" means that
$A \subseteq B$, that imply also $\mathbb{P}[A] \leq \mathbb{P}[B]$ |
H: Topological difference between the compact interval $I$ and the Cantor set
There is an homeomorphism between the Cantor set $X = 2^\omega$ (with the product topology) and the Cantor ternary set $\mathcal{C}=[0,1] \smallsetminus \bigcup_{n=0}^\infty \bigcup_{k=0}^{3^n-1} \left(\frac{3k+1}{3^{n+1}},\frac{3k+2}{3^{n+1... |
H: A question on a sequence problem
The question:
Given 6 beads which consist of red, blue, yellow, green, white and black. How many ways can a rings of beads be formed?
I'm confused by the question, would this mean this is asking for the combination or permutation of the colors above?
AI: It is hopefully asking for p... |
H: $\text{Q Find the area enclosed by the curves "}y^2+x^2=9\text{" and "}\left|\left(x^2-y\left|x\right|\right)\right|=1$
$\text{Q Find the area enclosed by the curves "}y^2+x^2=9\text{" and "}\left|\left(x^2-y\left|x\right|\right)\right|=1\text{" which contains the origin}.$
I tried to plot the graph on desmos. and ... |
H: express a cosine product as a single cosine
Problem
I have this expression $c\cos(\theta_1)\cos(\theta_2),$
where $c$ is a constant term that eats away any constant factor e.g the following can be considered true:
$$c\cos(\theta_1)\cos(\theta_2) = c\cos(\theta_1)\cos(\theta_2) / \pi$$
Because both expressions diffe... |
H: While using the method of proof by contradiction, are we "assuming" consistency?
I am aware of threads here and here which asks something similar. However, I had something very specific to ask under the same context.
I have a very elementary question about the connection between Godel's second incompleteness thorem... |
H: If $A,B,C$ form a partition of $\Omega$ then describe the smallest $\sigma$-algebra containing the sets $A,B$ and $C$.
Question:
If $A,B,C$ form a partition of $\Omega$ then describe the smallest $\sigma$-algebra containing the sets $A,B$ and $C$.
This seems like a straightforward question but it is giving me a r... |
H: If 2 dice are rolled what is probability that "product of given values of dices is > 10" | "given the result is a double"
I am new in the field of probabilities. I came accross this problem.
Is my solution correct?
If 2 dices are rolled, what is the probability of P("product of given values of dices is > 10" | "gi... |
H: finding value of $p,q$ such that function $f(x)$ is continuous at $x=-1$
If $$f(x)=\left\{\begin{matrix}
\sin\bigg(\pi(x+p)\bigg)\;\; , &x<-1 \\
q\bigg(\lfloor x \rfloor^2+\lfloor x \rfloor\bigg)+1\;\;,\;\;&
x\geq -1 \end{matrix}\right.$$ where $\lfloor x \rfloor $ represent the integer part of $x,$ . Then wha... |
H: Prove that if $E[X_1^p]<\infty$, then $\frac{\max_{1\le i\le n} X_i}{n^{1/p}} \rightarrow 0$ in probability where $\{X_n\}$ is i.i.d and non-negative
Suppose $\{X_n\}_{n\geq 1}$ are iid and non negative. Define $M_n=\max \limits_{i=1,\ldots,n}\{X_i\}.$
Prove if $E[X_1^p]<\infty$, then $\frac{M_n}{n^{1/p}}\rightarr... |
H: The difference between the statements for sequences of function $f_n(x)$
Let I be an interval and c ∈ I.
Statement A: For all $\epsilon$ > 0, there is $\delta$ > 0 such that,for all $n ∈ \mathbb{N}$ and for all $x ∈ I$ satisfying $|x−c|≤\delta$, $|f_n(x)−f_n(c)| ≤ \epsilon$.
Statement B: For all $n ∈ \mathbb{N}$ an... |
H: Is there a way to add positives and negatives with the same algorithm?
It is taught to us in grade school that you can add (positive) numbers like this:
5 2 3
4 5 6
-------
5 7 9
But if we change it to 523 + (-456), we cannot use the same algorithm
5 2 3
-4 -5 -6
---------
1 -3 -3 ... |
H: Point on line closest to origin given the line's parametric equations
So I'm given the line described by:
$$\begin{cases} x = \frac{2}{3} + \lambda \\ y = \frac{1}{3} + \lambda \\ z = \lambda \end{cases}$$
And I'm asked to determine the point that is closest to the origin. I have the following formula for the dista... |
H: Does a reduced row echolon only have one solution?
In general, when finding the reduced row echolon for a matrix, is there only one solution, even if there is no solution?
$$\begin{pmatrix}2&3&5\\ \:-2&-3&-3\end{pmatrix}$$
I got the following answer by hand (no solution because $1$ does not $= -2$):
$$\begin{pmatr... |
H: A Doubt regarding Cycloid
If we consider a cycloid made by a wheel. Then will the cycloid intersect the wheel when the wheel touches the topmost point of the cycloid? Thus will the radius of curvature be same to that of the wheel at that point?
AI: From the parametric equations
$$\begin{align}x&=R(\theta-\sin\theta... |
H: Kolmogorov's three-series theorem- what can be said about distribution of $X_1$?
We assume that $X_n$ is i.i.d. and we know that $\sum_{n=1}^{\infty} X_n$ converges almost surely.
The question is what can be said about $X_1$ distribution.
I wanted to use Kolmogorov's three-series theorem. Then I know that:
$\su... |
H: A statistics problem (normal distribution)
If $X\sim N(0, \sigma^2)$, how can I compute $\operatorname{Var}(X^2)$? Here is my idea... but I cannot get there.
$$\operatorname{Var}(X^2) = E(X^4) - (E(X^2))^2$$
AI: There are many ways to do this, but here's one. The moment-generating function of $X$ is $\exp\frac{\si... |
H: Is $dX/dt=X(t)$ the correct ODE for $X(t)=e^t$?
For a school project for chemistry I use systems of ODEs to calculate the concentrations of specific chemicals over time. Now I am wondering if
$$ \frac{dX}{dt} =X(t) $$
the same is as
$$ X(t)=e^t . $$
As far as I know, this should be correct, because the derivativ... |
H: why $(3/4,3/4)\notin B(0,1)$?
I have an excerpt from my textbook which explain that elements of the product topology are not all of the from $U_1\times U_2$ for $U_i\in\mathcal{T}_i$ , where the profuct topology is defined as :
Suppose that $(T_1,\mathcal{T}_1)$ and $(T_2,\mathcal{T}_2)$ are two topological spaces... |
H: Is this proof correct? [$\lim_{x\to-\infty}f=\lim_{x\to+\infty}f=+\infty\implies\ f$ has a global minimum]
I'm trying to prove that if $f : \Bbb R\to\Bbb R$ is a continuous function that verifies:
$$\lim\limits_{x\to-\infty}f=\lim_{x\to +\infty}f+\infty$$
Then $f$ has a global minimum
So, since:
$\lim\limits_{x\to-... |
H: What is the intuition behind the formula $p \leftrightarrow (q \leftrightarrow (r \leftrightarrow ...))$?
I cooked up the formula $p \leftrightarrow (q \leftrightarrow (r \leftrightarrow ...))$ and naively thought it is a sort of "equivalence" relation. It turns out I am wrong. Suppose you have four variables in to... |
H: Find coordinates of the center of the mass - line integral
Find the coordinates of the center of the mass of the curve
$$ x^2+y^2=1, x+2y+3z=12 $$
I find calculating line integrals in 3D problematic and really don't know how to approach this one. I think that the curve we get is an elipse but how to find its param... |
H: A question about regularity of parameterizations of a surface
Can someone give an example of two permetrazations (1-1) of a surface that satisfy: At a same point (on the surface) one permetrazation is regular but the other is not regular
AI: Consider the $xy$-plane $S$. Then we can give a 1-1 parametrization $\over... |
H: Set notation: $U\otimes V\simeq V\otimes U$ with $V,U$ vector spaces
Let $U,V$ be finite-dimensional vector spaces over a common field and show that $U\otimes V\simeq V\otimes U$. Will someone explain what the notation $\simeq$ means in this case?
AI: It means that there exists a vector space isomorphism between $U... |
H: Find $\sigma$ for which $\sum _{i=1}^{n} \frac{1}{a_i\cdot a_{\sigma(i)}}$ is maximal
$$\text{Let: } 0 < a_1 \lt a_2 \lt \dots \lt a_n$$
$$\text{Find } \sigma \in S_n \text{ for which :}$$
$$\sum _{i=1}^{n} \frac{1}{a_i\cdot a_{\sigma(i)}} \text{ is maximal}$$
I think the maximum value of the sum will be reached if... |
H: To prove that $p$ is a prime number
I'm reading a book about proofs and fundamentals on my own and, currently, I'm having trouble proving this result.
Theorem: Let $p$ be a positive integer bigger than or equal to $2$ and such that, for any integers $a$ and $b$, if $p|ab$, then $p|a$ or $p|b$. Show that $p$ is a pr... |
H: EDIT: Can cdf have a set of continuity points which is a.s. different from $\emptyset$?
It can be easily proven that a cumulative distribution function (cdf)
$$F(x)=\mathbb{P}\left(X\le x\right)\hspace{0.3cm}\text{for }-\infty<x<+\infty$$
is right-continuous $\left(\text{that is }F(x)=F(x^+)\right)$, but NOT left-c... |
H: Direction Derviative, Different Results, Possible?
Given:
\begin{equation*}
f(x,y) = e^{xy}, \quad P=(1,1),\ \mathbf{v} = (-\frac{\sqrt{3}}{2}, \frac{1}{2})
\end{equation*}
I am trying to calculate the directional derivative of $f$ in direction of $\mathbf{v}$.
My problem is, that when I apply two different ways ... |
H: Why is that if $a\in\overline{A}$ then for every $n>0$ we have $B(a,1/n)\bigcap A \neq \emptyset$.
I don't quite understand one thing here:
Why is that if $a\in\overline{A}$ then for every $n>0$ we have $B(a,1/n)\bigcap A \neq \emptyset$.
I drew a picture below to visualize my understanding of this:.
Well, as y... |
H: Value of m for which the function will give integers as an output.
$F(m)=(2m^3+2m)/(m^2+1)$ and $g(m)=(m^4+1)/(m^2+1)$ What are the values of $m$ other than $1$ for which solution of both function will be integers. Please tell if there is any formula to find so or any technique?
AI: Whenever $m^2+1 \ne 0$, you find... |
H: Evaluate $\lim_{n\to \infty} \sum_{k=0}^n \frac{\sqrt {kn}}{n}$
I'm not sure which would be the best way to compute this limit. As you might have observed, if you expand the infinite sum and rearrange some terms you get:
$$\lim_{n\to \infty}\frac{\sqrt n}{n}\!\cdot\!\sum_{k=0}^n \sqrt k$$
it is easy to see that whe... |
H: What does it mean when a system is made dimensionless and what is the exact technique for that?
For school research I'm working on a system of ODEs to describe a chemical oscillator (the Oregonator). This system is described with the following system:
$$ \frac {dX}{dt}=k_1AY-k_2XY+k_3AX-2k_4X^2 $$
$$ \frac{dY}{dt}... |
H: What is the radius of the small circle inscribed in a square?
If we are given a square with sides of length 4cm. The smaller circle is tangent to the larger circle and the two sides of the square as shown in the photo below. How can i find the length of the radius of the smaller circle?
My approach :
The radius o... |
H: Poisson or exponential distribution
I have to complete the question for a homework task and I'm confused if it is a Poisson or exponential distribution. any insight would be appreciated.
cars arrive at a car wash in a town at an average rate of 50
per hour.
Q: If you arrive within a three minutes of another car , y... |
H: Covariant and contravariant components of vectors
I am struggling with the covariance and contravariance of vectors. In my physics classes, the professor explained that if covariant components transform with a certain matrix, then contravariant components transform with its inverse. However, I find the latter to be... |
H: If $A_t = cos(X_t)$ and $B_t = sin(X_t)$ find the infinitesimal increment for $Y_t = A_t^2 + B_t^2$
If $X_t$ is Brownian motion, I'm not sure how to apply Ito's lemma to get $d Y_t$ for $ Y_t = A_t^2 + B_t^2$ where $A_t = cos(X_t)$ and $B_t = sin(X_t)$ in particular, I get confused because $sin^2(x) + cos^2(x) = 1$... |
H: Second order ODE solution - help me spot a mistake
Solve following ODE:
$$
(1-x)x''+2(x')^2=0; x(0)=2, x'(0)=-1
$$
$$
x''=\frac{-2(x')^2}{1-x}
$$
substitute $x'=u(x)$ and assume $u \neq 0$
$$
uu'=\frac{-2(u)^2}{1-x} \\
\frac{dx}{1-x}=-\frac{du}{2u} \\
-\ln{|1-x|}=-\frac{1}{2}\ln{|u|}+c \\
|1-x|=\sqrt(|u|)e^c
$$
und... |
H: $\exp ^r z=\exp rz$ for all $z\in\mathbb{C}$ and $r\in\mathbb{R}$
Let $z\in\mathbb{C}$ and $r\in\mathbb{R}$. Assuming that $\exp ^r z$ can be a multi-valued function (and $\exp rz$ cannot), there always exists (for any given $z$ and $r$) some value of $\exp ^r z$ such that
$$\exp ^r z=\exp rz.$$
How I could I prove... |
H: Leftside limit of the Distribution function
Let $(\Omega,\mathrm{P})$ be a probabilty space and $X$ be a random variable.
Why is $\lim\limits_{s \uparrow t}\mathrm{P}(X < s)= \mathrm{P}(X \leq t)$?
I first thought that this follows from the continuity from below, but this doesn't works. If I consider $x_n \to t $ w... |
H: finding a power series for $f(z)=\frac{1}{1+z^2}$ centered at $0$
In an exercise I am asked to find a power series for $f(z)=\frac{1}{1+z^2}$ centered at $0$.
My approach was the following:
$f(z)=\frac{1}{1+z^2}=\frac{1-z^2}{(1+z^2)(1-z^2)}=(1-z^2)\frac{1}{1-z^4}=(1-z^2)\sum_{n\geq0}z^{4n}$
But this does not seem r... |
H: A proof question using Riemann Lemma
Let $l>0$, $f(x)$ is continuous on $[-l,l]$ and differentiable at $x=0$. Please use Riemann lemma to prove that
$$\begin{equation}
\lim_{n\rightarrow\infty}{\frac{1}{\pi}\int_{-l}^{l}{f(x)\frac{\sin{nx}}{x}}dx}=f(0)
\end{equation}$$
I am sorry for being stupid but I am stuck at ... |
H: Show that the function $f(x)=\frac{x-1}{2(x+1)}$ is continuous in $a=3$
What I've done is the following.
$$\biggl|\frac{x-1}{2(x+1)}-\frac{3-1}{2(3+1)} \biggr|<\epsilon$$
By some calculation, I got
$$\biggl|\frac{3x-5}{2(x+1)} \biggr|<\epsilon.$$
This is greater than zero when $x>5/3$ or $x<-1.$
So, then, we do the... |
H: Longest Rubik's cube algorithm - maximization problem
For a given Rubik's cube algorithm $A$ let $\mathfrak C(A)$ be the number of times, we have to repeat algorithm $A$ to get back to where we've started. For example if $A=RUR'U'$ then $\mathfrak C(A)=6$ The question is:
What is the greatest value of $\mathfrak C(... |
H: Not every matrix on V ⊗ W can be written as a tensor product of a matrix on V and another on W.
I am reading some notes about tensor product of vector spaces (those in here) where the following sentence can be found:
Note that not every matrix on V ⊗ W can be written as a tensor product
of a matrix on V and an... |
H: Uniform convergence of sequence of functions $\frac{2+nx^2}{2+nx}$ on [0,1]?
I have recently been trying some questions related to the uniform convergence of a sequence of functions. And meanwhile, I got stuck in one of the problems in which I have been supposed to discuss the point-wise and uniform convergence of ... |
H: Verification of proof: $f(x) = e^x$ is continuous at $a = 2$
$$|e^x-e^2|<\epsilon$$
So if $x<2$ then $2-x<\delta $
Calculations:
$$e^2-e^x<\epsilon$$
$$-e^x<\epsilon -e^2 $$
$$e^x>e^2-\epsilon$$
$$x>\ln(e^2-\epsilon)$$
$$-x<-\ln(e^2-\epsilon)$$
$$2-x<2-\ln(e^2-\epsilon)$$
Then we get:
$$2-x<2-\ln(e²-\epsilon) = \de... |
H: General solution of $tx''-x'+4t^3x=4t^3$
The task is to find general solution of:
$tx''-x'+4t^3x=4t^3$
The hint is to substitute $s=t^2$
My attempt: First I guessed that $x=1$ satisfies the equation and that is our particular solution. Now we have to find the solution to homogeneous eqaution:
$tx''-x'+4t^3x=0$
My... |
H: Integral $\int_{1/\sqrt{2}}^1 r^3 \sqrt{1/r^{2} -1}dr$
I have integral $\int_{1/\sqrt{2}}^1 r^3 \sqrt{\frac{1}{r^2}-1}dr$.
I calculated:
$$
\int_{1/\sqrt{2}}^1 r^3 \sqrt{\frac{1}{r^2}-1}dr= \int_{1/\sqrt{2}}^1 r^3 \sqrt{\frac{1-r^2}{r^2}}dr=\int_{1/\sqrt{2}}^1 r^2\sqrt{1-r^2} dr.
$$
But then I dont know how to proc... |
H: How to get every possible combination to choose k object from a set of n elements
How can I get the actual combinations to choose k objects of n elements.
The binomial coefficient only tells me how many possiblities exist, but I actually don't know the combinations. What methods are there?
AI: Brute force method: t... |
H: Solve this differential equation $x^2y''-5xy'+6y=0$
Solve this equation
$$
\begin{cases}
x^2y''-5xy'+6y=0 \\
y(-1)=3 \\
y'(-1)=2
\end{cases}
$$
I got
$$y=c_1x^{3+\sqrt3}+c_2x^{3-\sqrt3}$$
I have three little questions.
Could I solve the problem by substituting $(-1)^{\sqrt3}=\cos((\sqrt 3) \pi)+i\sin((\sqrt 3)\pi)... |
H: Guidance requested for vector dot product question.
Helping my child out with their year 11 exam preparation, specifically vectors and dot products, I think I may have figured out the answer but I'd like to get some confirmation or, more likely, a short sharp shock of education :-)
Keep in mind it's some thirty-plu... |
H: A question obout the sum of series in $[-\infty,+\infty]$
Suppose that the work set is $[-\infty+\infty]$, we suppose that $$\sum_{n=0}^{+\infty}a_n<+\infty.$$ Now can we says that $$\sum_{n=0}^{+\infty}(a_n-b_n)=\sum_{n=0}^{+\infty}a_n-\sum_{n=0}^{+\infty} b_n\quad$$
In the $[-\infty,+\infty]$ each series(well def... |
H: Find accumulation points of $a_n=\alpha(3-\frac{1}{n})^{n^{(-1)^{n}}}+\sqrt[n]{2^{n(-1)^n}+6^{n(-1)^{n+1}}}$ regarding $\alpha$
I solved it for $\alpha<0$ and $\alpha>0$. I can't find limit of $\alpha(3-\frac{1}{n})^n$ (the $n$ is even) when $\alpha = 0$. Solution says that this limit is equal to zero but i don't k... |
H: Find the value of function with given conditions
Let $f(x)$ be a fifth degree polynomial with leading coefficient unity.
If $f(1)=5, f(2)=4, f(3)=3, f(4)=2 , f(5)=1$ find $f(6)$
I know I can solve this by assuming a polynomial equation and then finding the coefficients of every term and finding the value of $f(6)$... |
H: "Sequentially compactness" of compact Riemann surface
I'm in trouble show that for any sequence $\{x_n\}_n$ in a compact Riemann surface, there exist a subsequence $\{ x_{n_k}\}_k$ of $\{x_n\}_n$, a chart $(U,\varphi)$ and an $x\in U$ such that the sequence $\{\varphi (x_{n_k})\}_k$ converges to $\varphi(x)$.
Any h... |
H: How to calculate $\int \frac{\sinh \left(x\right)}{\cosh \left(x\right)-\cos \left(y\right)}dy$?
How to calculate $$\int \frac{\sinh \left(x\right)}{\cosh \left(x\right)-\cos \left(y\right)}dy$$ I know that the answer is $2\arctan\left(\coth\frac{x}{2}\cdot\tan\frac{y}{2}\right)$ but don't have idea how to get it..... |
H: Choosing berries
You have 2 blue berries, 2 red berries and 2 black berries
In how many ways can you pick 2 berries?
My work:
you can choose two red berries or you can choose two blackberries or you can choose two blueberries or you can choose 1 blue and 1 red berry or you can choose 1 black and 1 blue berries or y... |
H: At what value does this integral reach its minimum?
The problem is to find the value of $a>1$ at which this integral $$\int_{a}^{a^2} \frac{1}{x}\ln\Big(\frac{x-1}{32}\Big)dx$$
reaches its minimum value. I don't want to know the number, I just want feedback on the ideas I'm trying. Considering that $$f(x)=\frac{1}{... |
H: Another proof (check) that the set of isolated points of a set in $\mathbb R^n$ is countable
Just thinking if this proof works, and i have some "not so explicit" ending so i would love if you suggest a way for refining this.
Theorem : $S \subset \mathbb R^n$ is a set. Then the set of isolated points of $S$ is coun... |
H: $ \sup_{x\in C}\|x\|=\sup_{x^*\in B^*}\sup_{x\in C}\langle x^*,x\rangle $
Let $X$ be a separable Banach space, the associated dual space is denoted by $X^*$ and the usual duality between $X$ and $X^*$ by $\langle , \rangle$. Let $B^*$ the closed unit ball of $X^*$.
Take $C$ nonempty weakly compact convex subsets o... |
H: Is there an easier way to solve the given problem?
If $x + 2 + \sqrt{3}i=0$ then find the value of $2x^4+3x^3-x^2-15x+36$
If you try to find the values of $x^2, x^3 $and $ x^4$, and then put the values in, we can find the solution to be 1. But is there a better way to solve this problem? Where you can probably brea... |
H: Is $x \approx x$?
If I write $x \approx y$, does this mean (a) $x$ is sufficiently close to $y$ for some practical purpose, or (b) $x$ is sufficiently close to $y$ for some practical purpose, but is not equal to $y$?
If (a) is true, then it appears $x \approx x$.
This question appears to have more importance when c... |
H: Average of a tensor with respect to a group action
Consider a smooth manifold $M$ (assume it boundary-free and orientable) and a tensor field $\mathcal{G}\in\Gamma(\otimes^hTM\otimes^kT^*M)$. Let $\Phi:\mathbb{T}^p\times M \rightarrow M$ be a torus action on the manifold. What is the definition of the average of th... |
H: Can you take the (Euclidean) norm of a scalar quantity (non-vector)?
Context: This question
My question is whether or not you can take the (Euclidean) norm of a scalar non-vector quantity. My intuition is no, but I just wanted to confirm it.
I understand that a norm in essence gives you a 'length' or 'magnitude' w... |
H: Showing $\mathbb{R}$ with the standard topology union irrational subsets is normal.
We are given that $\mathcal{T}$ is the standard topology on $\mathbb{R}$ and $\mathcal{O}=\{O\in\mathcal{P}(\mathbb{R}):O=U\cup A, U\in\mathcal{T}, A\subseteq\mathbb{R}\backslash\mathbb{Q}\}$. It can easily be shown that $(\mathbb{R... |
H: Affine function is a diffeomorphism?
Given an affine function $f:\mathbb{R}^n \to \mathbb{R}^n$ defined for all $x\in \mathbb{R}^n$ by $$f(x)=T(x)+a$$ such that $T$ is an invertible Linear map and $a\in \mathbb{R}^n$, is $f$ a diffeomorphism?
AI: It is a diffeomorphism iff $T$ is invertible.
It is easy to see that ... |
H: Why is a countable set closed?
If $T$ is an uncountable set, show that $$\mathcal{T}=\{\emptyset,T,\text{ all sets whose complements is at most countable}\}$$ forms a topology on $T$.
Answer:
We have $∅$ and $T$ in T . To check the other two axioms, it is easier to work with
complements, i.e. at most countable sets... |
H: Probability distribution and "inclusion"
Is there such thing as a probability distribution "included" in another?
For example, take two random variables X and Y, where Y takes it values in a set Sy included in Sx. How do you formalize this?
In practice, given observations on variables X and Y, can you test for such... |
H: How to show that $H=\langle a,b\rangle=\{a^m\cdot{b^n}\mid m,n\in{\mathbb{Z}}\}$ is a subgroup of $\langle G,\cdot\rangle$?
I have $a,b\in{G}$ and also $G$ is Abelian.
I thought that first I should show that $H\subset{G}$ and then show that $\forall{h_1,h_2}\in{H}$ we will have $ h_1\cdot{h_2}\in{H}$.
First to show... |
H: "generalised" gamma-like integral $\int_0^\infty x^ne^{-f(n)x}dx$
I have noticed that if we have an integral of the form:
$$I[f]=\int_0^\infty x^ne^{-f(n)x}dx=\frac{1}{f^{n+1}(n)}\int_0^\infty x^ne^{-x}dx=\frac{n!}{f^{n+1}(n)}$$
I was wondering what kind of restrictions would need to be applied to $f$ in order for ... |
H: Why is the dimension of a crystallographic group unique?
The algebraic definition of a crystallographic group goes as follows:
If a group $\Gamma$ fits into a short exact sequence
$$0 \to \mathbb{Z}^n \overset{i}{\to} \Gamma \overset{p}{\to} G \to 1$$
such that $i \left(\mathbb{Z}^n \right)$ is maximal abelian in $... |
H: Limit of p th norm of function as p tends to infinity
Let $f\colon[0,1]\to\mathbb R$ be continuous. Let $c_p=\left(\int_0^1|f(x)|^pdx\right)^{\frac{1}{p}}$. Then the limit $\lim_{p\to \infty}c_p$ is?
I know $\inf f(x) \leq c_p\leq \sup f(x)$ for $x\in [0,1]$. But no idea whether the sequence is increasing or decrea... |
H: general approach to comment on bijection of these functions whose graph can't be drawn
consider the following class of functions defined as:
If $f : \Bbb R \to \Bbb R$ be the function such that
$$ f(x)=x|x|-4 :x \in \Bbb Q$$
$$ f(x)=x|x|-\sqrt{3} :x \notin \Bbb Q$$
then $f(x)$ is there a general approach to comme... |
H: Tell me what it would look like to factor $x^2-1 = (x+1)(x-1)$ a different way
During the factoring of $x^2-1$ I saw a $+x$ and $-x$ were introduced but I wonder how the factoring would go if the $+x-x$ were added in reverse, like so $-x+x$.
I was shown $x^2-1$ can be factored to $(x+1)(x-1)$ thusly...
$x^2-1 =$
$x... |
H: Calculate area using double integral
I'm trying to calculate the area defined by the following curves: $y=x^2, 4x=y^2, y=6$ using double integrals.
I'm wondering whether my solution is correct:
Area = $\int^{6}_{0}\int^{\sqrt{4x}}_{0}1dydx - \int^{6}_{0}\int^{x^2}_{0}1dydx$
Is it correct?
Thanks!
AI: No, that is no... |
H: self-adjoint operator intuition
can someone please explain self-adjoint operator intuition to me.
And why when $T^* = T^{-1}$, $T$ preserves the inner product and therefore preserves the the orthonormal basis and the length and distance?
thank you
AI: Think about a symmetric matrix, it’s a simpler case. I think th... |
H: Simulate stars and bars problem
Suppose that I have $n$ balls to be divided among $k$ buckets so that each bucket has a non-negative number of balls. This is a classic stars and bars problem: the number of ways to do this division is $\binom{n+k-1}{k-1}$. (In my application, $k>>n$.)
I would like to run a simulatio... |
H: To find elements of truth set
STATEMENT : $x$ is a real number and $5$ $\in$ {$y$ is real number | $x^2+y^2 <50$ }.
Since $5$ is element of above set. so we have $x^2 + 25 < 50$. so we have $x^2 < 25$. So Truth set is {$-4,-3,-2,-1,0,1,2,3,4$}. Is my work correct? Thank you
AI: You assumed $x$ is an integer. But $... |
H: $\epsilon$-$\delta$ Proof of Limit in $\mathbb{R}^4$
Question: Determine $\lim_{(a,b,c,d) \to (0, 0, 0, 0)} \frac{a^2d^2 -2abcd + b^2c^2}{a^2 + b^2 + c^2 + d^2}$ and prove your result using the $\epsilon$-$\delta$ definition of a limit.
Attempt: I have completed the first part of the question using polar coordinate... |
H: Find the volume between the surface $x^2+y^2+z=1$ and $ z=x^2+(y-1)^2$
I'm trying to find the volume between the surface $x^2+y^2+z=1$ and $ z=x^2+(y-1)^2$ but nothing works for me.
I made the plot and it looks like this:
How could you start? Any recommendation?
AI: Try checking where the two surfaces intersect. S... |
H: What is the order of the subgroup of G (rubiks cube group) generated by ?
I got:
=1,FF,RR,FFRR,RRFF
But in my text book the answer is 12? Does any one else know the other elements?
I assumed since FFFFRRRR=1 there were no more?
AI: Let $f=FF$ and $g=RR$. Note that $f$ and $g$ have order $2$, and $fg$ and $gf$ have... |
H: Calculating distribution of items among cells with limited capacity
The questioning is:
Given 13 identical balls and 6 different cells, in how many ways can the balls be distributed between the cells so that there is not a single cell that has 3 balls?
My approach to the question was:
Calculate the amount of di... |
H: Use the Cauchy theorem to check whether a function is analytic or not in some region
Let $f $ be complex value defined in some bounded closed region $\gamma $ (rectangle) but we don't know whether it has poles or not in $\gamma$.
Let us consider the integral $I$ given by
$$I=\int_{\gamma} f(z)dz $$
Now we have ... |
H: How to find the PDF of a function of two random variables
Y is a uniform continuous random variable between [0,L] and X is a uniform continuous random variable given Y=y between [0,y]. What is the PDF of Z = X/Y.
AI: First of all you have to find the joint density $f_{XY}(x,y)$
You know:
Marginal $$f_Y(y)=\frac{1}... |
H: What mistake am I making while deriving the expansion of $\cos(\alpha + \beta)$
I was trying to derive the formula for expansion of $\cos (\alpha + \beta)$ by equating the ratio of lengths of two specific chords to the ratio of angles opposite to them but I'm not getting the correct results. Here's how I'm doing it... |
H: An application of Rouché's theorem
i need help to proof the next:
Let $f$ be analytic at $D$ minus a finite numbers of interior points where $f$ has poles. Show that if $0<|f(z)|<1$ over $\partial D$, then the number of poles of $f$ in $D$ is equal to the number of roots of the equation $f(z)=1$ in $D$.
I have trie... |
H: Evaluate the sum $\sum_{n=0}^{\infty} \frac{2n}{8^n}{{2n}\choose{n}}$
Evaluate the sum $\sum_{n=0}^{\infty} \frac{2n}{8^n}{{2n}\choose{n}}$
I am unable to find a way to solve this sum. I have never seen sums involving binomial coefficients multiplied by $n$. Help will be appreciated
AI: Note that
\begin{align}\bi... |
H: Birthday Problem: Probability that at least two also share the same weekday, if weekday uniformly distributed and independent of birth date?
Consider the following problem extension for the birthday problem: We now want know the probability that out of $n$ persons, at least two people were born on the same date and... |
H: Find a function satisfy certain condition
Find function $f:\mathbb{R}\to\mathbb{R}$ such that $f(2)=2$ and
$$\sum_{i=1}^nc_i\frac{\partial(x_1^2+\dots+x_n^2)^{\frac{f(y)}{2}}}{\partial x_i}=\frac{\partial(x_1^2+\dots+x^2_n)^{\frac{f(y)}{2}}}{\partial y}$$
This is what I got so far, firstly I tried to simplify the... |
H: On the codomain of a complex valued function
I want to denote a complex function that its outputs are real. So I must write:
$$f: \mathbb{C} \to \mathbb{R}$$
However, what happens to those $z$ that make $f(z)$ complex? Are they considered and then I cut off the imaginary output. For instance, let:
$$f(z) = z^2 + x_... |
H: matrix least square optimization with constraint
I want to Minimize the following equation (the error matrix) by
formulating a cost function and calculating the point where
its gradient equals zero.
\begin{equation}
\hat{X} = \arg \min_{X} \frac{1}{2} {\left\| AX - B \right\|}_{F}^{2}
\end{equation}
where ... |
H: Volume using double integrals
Calculate the volume of the solid bounded by the following surfaces:
$y=x^2, y=1, x+y+z=4, z=0$.
How does on set up the integral?
AI: $y=x^2$ is "parabola shifted infinitely" accordingly z-axis.
Your volume can be calculate by 2 and/or 3 dimensional integral:
$$\int_{-1}^{1}\int_{x... |
H: Verify the statements for Riemann-integrable function $f_n(x)$.
For each $n = 1, 2, \cdots$ a function $f_n(x)$ is defined so that it is Riemann-integrable on $[a, b]$ and the series $\sum_{n=1}^{\infty}f_n(x)$ converges $\forall \space x \in [a,b]$.
Which of the following statements are true?
$$\lim_{n\rightarrow... |
H: Suppose $[a], [b] \in \mathbb{Z}_n$ and $[a]\cdot[b] = [0]$. Is it necessarily true that either $[a] = [0]$ or $[b] = [0]$?
Let $n \in \mathbb{N}$. Let $R$ be the equivalence relation $\equiv \pmod{n}$. Suppose $[a], [b] \in \mathbb{Z}_n$ and $[a]\cdot[b] = [0]$. Is it necessarily true that either $[a] = [0]$ or $[... |
H: Find the distribution of numbers of arrivals of the Poisson process $N(t)$ in time interval $[t, t+\tau)$, $\tau \sim Exp(a)$.
Poisson process has rate $\lambda$ and $\tau \perp \!\!\! \perp N(t)$.
To find distribution i've started with $P(N(t+\tau)-N(t)=k) = P(N(\tau) = k)$. I know that $N(t) \sim Poiss(\lambda t... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.