text stringlengths 83 79.5k |
|---|
H: Probability of the score of two rolls
I have a probability problem and this is the request:
What is the probability that the score of the first roll is higher than the score of a second roll. The rolls are equilibrated and independent. (I think one per time)
I have solved this problem with this method: $P(X>Y)=P(X-... |
H: Is it a convergent series?
If $f(x)$ is a continuous function, does
$$\lim\limits_{n\to\infty}\frac 1n\times \left[f\left(\frac{1}{n}\right) + f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n}{n}\right)\right]$$
converge?
AI: You should know that if $ f$ is integrable at $[a,b]$, then
$$\lim_{n\to+\infty}\frac{b-a... |
H: Find any values of $k$ for which $f$ is continuous
Sketch this function for $k = 1$. Is it continuous? Find any values of $k$ for which $f$ is continuous.
$$f(x)=
\begin{cases}
kx+3, & \text{$x≤1$} \\
(kx)^2-5, & \text{$x>1$}
\end{cases}$$
I would imagine that for the left side, I would get $4 (x+3$, which $1$ i... |
H: Prove you can weigh any number between 1 and $\frac{3^{n+1} -1}{2}$ using $n+1$ weights - Discrete
You have $n+1$ weights, with each weighing $1,3,9, \dots, 3^n$ (one of each)
Prove that you can weigh with a traditional scale ( The one with two bowls) each integer weight between $$1 ~~~~~\text{And }~~~~~\frac{3^{... |
H: Showing that $\|(x,y)\|_0=\sqrt{\|x\|^2+\|y\|^2}$ is norm if $\|\cdot\|$ is a norm.
The 3 properties are really easy to show but I cannot show that $\|(x,y)\|_0=\sqrt{\|x\|^2+\|y\|^2}$ satisfies the triangle inequality if $\|\cdot\|$ satisfies it. I tried to use Cauchy,S. inequality and etc.
My work:
Let $(x,y),(x'... |
H: Totally bounded set in a metric space $\implies$ bounded
I apologize if the question may be trivial, but it is a fact that my textbook does not even mention and I, studying as a self-taught, do not have so many certainties.
I believe the totally boundedness $ \implies $ boundedness implication is true in any metric... |
H: Matrix of a representation from character theory
we have learnt in class that a representation (of a finite group $G$) is completely determined by the characters on its conjugacy classes. First of all, I know that the characters can be used in a lots of ways:
(1) By the orthogonality relation for characters, we can... |
H: What are the steps to factor $x^2 - 1$ into $(x+1)(x-1)$?
Does $(x+1)(x-1) = x^2+1x-1x-1$? If so where are the $+1x$ and the $-1x$ when it is being factored from $x^2-1$ into $(x+1)(x-1)$?
What exactly are we dividing $x^2-1$ by to get $(x+1)(x-1)$ and how did you know what to divide it by?
AI: Method 0:
$+1x - 1x ... |
H: Random variables and geometric series
Q: Consider the following random variable $Y$. It takes only values of the form $\frac 1{2^k}$ for positive integers $k$ and $P\left(Y= \frac 1{2^k}\right) = \frac 1{2^k}$ for each $k$. Find the expected value of this random variable.
I did find out that $a = \frac12$ in the ge... |
H: Is there any difference in result between quadratic programming VS linear programming?
Assume that we want to solve this equation:
$$ Ax \leq b$$
So we can either use Quadratic Programming:
$$J_{max}: x^TQx + c^Tx$$
$$Ax \leq b$$
$$x \geq 0$$
Or Linear Programming:
$$J_{max}: c^Tx$$
$$Ax \leq b$$
$$x \geq 0$$
Where... |
H: On the existence of a holomorphic function
I have just encountered this exercise which has me stumped:
We are asked to prove that if $ f $ is holomorphic on the unit disk $ D $ and if $ f(z) \neq 0 $ on $ D $, that there is a holomorphic function $ g(z) $ on $ D $ such that $ f(z) = e^{g(z)} $. We are also asked t... |
H: A sufficient condition for a Lebesgue point
Let $f\in L^1(\Bbb R^n)$ and let $x\in \Bbb R^n$. $x$ is said to be a Lebesgue point of $f$ if $\lim_{r\to 0} \frac{1}{m(B(x,r))} \int_{B(x,r)} |f(y)-f(x)|~dm(y)=0$ where $m$ is Lebesgue measure on $\Bbb R^n$. Clearly the condition $\lim_{r\to 0} \int_{B(x,r)}f(y)~dm(y)=f... |
H: What is the dimension and base of the following vectors' sum and intersection?
I have 2 Vector Subspaces of $\mathbb{R}^3$, namely $U = \operatorname{Span}(\begin{pmatrix} 2 \\ 5 \\ 9\end{pmatrix}, \begin{pmatrix} 0 \\ -1 \\ -3\end{pmatrix})$ and $W = \operatorname{Span}(\begin{pmatrix} -3 \\ 1 \\ 6\end{pmatrix}, \... |
H: Number theory question involving primes
Prove that, if a, b are prime numbers $a > b$, each containing at least two digits,
then $a^4 - b^4$ is divisible by $240$. Also prove that, $240$ is the gcd of all the numbers
which arise in this way.
Looking at the prime factorisation $240=(2^4)*3*5$, i know i need to prove... |
H: Proof of $\sup|f(x)|=\left \| f \right \|_\infty $
Let $(\mathbb{R}^n,L_n,\lambda _n)$ be a measure space and let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be continuous and bounded, then $\sup\lvert f(x)\rvert = \lVert f \lVert_\infty $ .
Proof: we have $\lvert f(x)\rvert \leq \lVert f \rVert_\infty $ everywhere beca... |
H: Is $f(x, y) = x - y$ injective and surjective?
Consider $f(x,y) = x-y$ for $ℤ × ℤ^+ → ℤ$. Is this injective and surjective?
I think it's not injective because there are infinitely many values of $x$ and $y$ which can have the same difference, such as (50, 10), (49, 9) and so on.
But I think it's surjective because... |
H: Why is $\frac{\int_{64}^{65}1.04^xdx}{\int_{24.5}^{25.5}1.04^xdx}=1.04^{39.5}\;?$
I am not sure whether these two things are exactly equal or only approximately equal. Wolfram says the difference is zero. I also would like to know why they are equal (or approximately)
$$\frac{\int_{64}^{65}1.04^xdx}{\int_{24.5}^{25... |
H: Projective dimension of locally free sheaf on a Cartier divisor
Let $X$ be a projective scheme over $\mathbb{C}$, let $D\hookrightarrow X$ be an effective Cartier divisor. Is it true for any sheaf $\mathcal{F}$ which is locally free on $D$ (i.e. a pushforward of a locally free sheaf on $D$), that there exists a loc... |
H: Calculating number of elements in the union of three sets.
The three sets $A, B,$ and $C$ each have $2018$ elements. The intersection of any two of the sets has $201$ elements. The intersection of all three sets has $20$ elements. How many elements are there in the union of the three sets?
If it is saying there are... |
H: Fast Exponentiation question: I am not even sure how to read this question, let alone attempt to solve it.
The Question is
$$ N=12={ 2 }^{ 2 }+{ 2 }^{ 3 }\\ { M }^{ 2 }\equiv 51(mod\quad 59)\\ What\quad is\quad { M }^{ 12 }(mod\quad 59)? $$
The solution is in the book says its 7. I am not sure what I am supposed t... |
H: Notation question: dual space basis
I have an exercise that I am trying to decipher, but as I have never seen this notation before I do not know how to read it. The problem states:
The vectors $x_1=(1,1,1),x_2=(1,1,-1)$ and $x_3=(1,-1,-1)$ form a basis of $\mathbb{C}^3$. If ${y_1,y_2,y_3}$ is the dual basis, and i... |
H: how is the sum of 2 normally distributed random variables different to 2 times a normally distributed variable
Say we have X~N(10, 100). It seems to hold that X+X~N(20, 200), however, if we multiply X with constant we have to multiply the variance with the square of the constant. Take for example 2, then we have 2X... |
H: Is $\mathbb{Q}(\sqrt{pq})$ a subfield of $\mathbb{R}$?
Is $\mathbb{Q}(\sqrt{pq})$ a subfield of $\mathbb{R}$?
I have a feeling that the answer is no but I can't prove it. It clearly contains $0$ and $1$, and is closed under addition and multiplication. There is an inverse to addition and the inverse to multiplicati... |
H: Maximum Flow of a network $G$
We are supposed to find the maximum flow through this network $G$. We have that $val(f) \leq c(C)$ for every cut in the network, where $c(C)$ is the capacity of the cut $C$. So I understand I am supposed to find minimum cut. What confuses me is the definition that we have for cuts: a ... |
H: Finite numbers that cannot be represented with $4m+7k$ using complete induction
I need to prove that there exist finite amount of numbers that cannot be represented with $n=4m+7k~ |~ m,k \in \mathbb{N}$ .
Starting:
We say that $A = \{n \in \mathbb{N} | n=4m+7k \geq 18 ~~~ m,k \in \mathbb{N} \} \cup \{1,2, \do... |
H: Uniform Convergence of $\frac{n}{x+n}$
I was trying an exercise on uniform convergence of sequence of real-valued functions.
I got stuck in a problem in which I am supposed to prove that sequence defined by $f_n(x)=\frac{(n)}{(x+n)}$ is uniformly convergent on $[0,k]$.
I have found out its point-wise limit to be $1... |
H: How to expand $(\partial_\mu A^\mu)^2$
How would I expand the following:
$$(\partial_\mu A^\mu)^2 \tag{1}$$
My understanding of it makes me think it would be as simple as:
$$(\partial_\mu A^\mu)(\partial_\mu A^\mu)\tag{2}$$
but I recall in my lectures seeing something like:
$$\tag{3}(\partial_\mu A^\mu)^2 = (\parti... |
H: Proof regarding inverse images and unions of sets
I'm currently working through Analysis by Tao and I just did an exercise but I'm not sure if it's correct. The question is
Let $f: X \to Y$ be a function from one set X to another set $Y$,
and let $U, V$ be subsets of $Y.$ Show that $f^{−1}(U \cup V ) = f^{−... |
H: Show: Linear mapping or none
Could someone explain me if the following is a linear mapping or none:
$$\text{Of }f:\mathbb{R^3}\to\mathbb{R^3} \text{ is known: }$$
$$f(\begin{pmatrix}1\\2\\3 \end{pmatrix})= \begin{pmatrix} 5\\3\\1 \end{pmatrix}$$
$$f(\begin{pmatrix}3\\2\\1 \end{pmatrix})= \begin{pmatrix} 1\\3\\5 \en... |
H: Let F be a field of characteristic p and let $\alpha \in F$ be an element for which $f(\alpha)=0$. Prove that $f(\alpha^p)=0$
Let p be any prime and let $f(x) \in \Bbb{F}_p[x]$ be any polynomial with coefficients in $\Bbb{F}_p$. Let F be a field of characteristic p and let $\alpha \in F$ be an element for which $f... |
H: The strict topology is metrizable on bounded subsets
Let $A$ be a $\sigma$-unital C*-Algebra and $(x_{\lambda})_{\Lambda}$ be a norm bounded net in $\mathcal{M}(A)$ (the multiplier algebra of $A$). I proved that for a strictly positive element $h$ if $(x_{\lambda}h)_{\Lambda}$ and $(hx_{\lambda})_{\Lambda}$ are nor... |
H: Is this series convergent? $1 + 1/2 + 1/2 + 1/4 + 1/4 + 1/4 + 1/4 + ...$
Is this series convergent? How to prove?
$$1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{8} + ... + \frac{1}{8} \ (8 \times 1/8) + \frac{1}{16} + ...$$
It's equal to $\{1 + 1 + 1 + 1 + ...\}$... |
H: Square roots modulo $pq$ where $p$, $q$ are distinct primes - confusion with quote
I have a doubt about the following quote from a book:
The Chinese Remainder Theorem implies that, if $p$ and $q$ are
distinct primes, then $s$ is a square modulo $pq$ if and only if $s$
is a square modulo $p$ and $s$ is a square... |
H: How to solve equation with multiple trigonometric functions?
Solve for $x$:
$\arccos( \cos(x) y + z) = \arcsin( \sin(x) a+b)$.
AI: Let $\cos x = \alpha$, so $\sin x = \sqrt{1 - \alpha^2}$. Then take the $\cos (\cdot)$ of both sides and note that $\cos \left( \arcsin[q] \right) = \sqrt{1 - q^2}$. Then you get a qu... |
H: If $m
question: If $m<n$, show that there is a $1$-$1$ mapping $F:S_m\rightarrow S_n$ such that $F(fg)=F(f)F(g)$ for all $f,g\in S_m$. Where $S_n$ stands for symmetric group of degree $n$
my approach:
First I was thinking how to construct the $F$. Because if I did this properly then the only work is left to show th... |
H: The cyclic subgroups of $p^2$ order non-cyclic group are normal
I’m having a hard time on proving that every cyclic subgroup of $p^2$ order group is a normal subgroup, where $p$ is a prime number. I’m not going to use the truth that $p^2$ order group are abelian, since this is what I want to show through the proof.... |
H: Uniform convergence of $x^n$ using the definition
I have been trying to prove the uniform convergence of sequence of functions defined by $f_n(x)=x^n$ on $[0,k]$ where $k<1$ by the epsilon definition of uniform convergence.
I have found the point-wise limit of the same..i.e. $f(x)=0$
Now let $\epsilon>0$ be given.
... |
H: Exercise using Sard's theorem
Let $M$ be a compact $n$-dimensional differentiable manifold and $f:M\to\mathbb{R}^{n+1}$ differentiable with $0\notin f(M)$. Show that there is a straight line through the origin in $\mathbb{R}^{n+1}$ that intersects with $f(M)$ only finitely many times.
This was given as an exercise ... |
H: Parabola transformation
Find the real affine change of coordinates that maps the parabola in the $xy$-plane to the parabola in the $uv$-plane
$$4x^2 + 4xy + y^2 - y + 1 = 0$$
$$4u^2 + v = 0$$
My attempt:
Since there is an $xy$ term, we know that there is a rotation. Thus suppose there is a $x'y'$ coordinate system ... |
H: Showing that $\text{Hom}(M,\Gamma(X,\mathcal{F}))\simeq \text{Hom}(\widetilde{M},\mathcal{F})$ (exercise II.5.3 from Hartshorne)
This is exercise II.5.3 from Hartshore:
Let $X=\text{Spec}(A)$ be an affine scheme. Show that the functors $\widetilde{\,\,\,\,}$ and $\Gamma$ are adjoint, in the following sense: for an... |
H: The Monty Hall Three Door Puzzle
I was going through the Monty Hall Three Door Puzzle in "Discrete Mathematics and its Application" by Kenneth Rosen (5th Edition). While reading the excerpt from tbe book (given below) I could not quite convince myself about the solution as there was only verbal reasoning and not qu... |
H: Continuity of piecewise function using topology
I would like to check if what I'm doing is correct or if I'm missing something:
Given the two sets $$X = [0,1] \cup (2,3], \quad Y = [0,2],$$ both equipped with the standard topology, consider the function $f:X \to Y$ defined by $$f(x) = \begin{cases} x & \text{if $x ... |
H: Lp space Example
How are spaces connected $L_{\infty}(E)$ and $L_{p}(E)$, $|E| = \infty$?
$$f \in L_\infty, \text{ but } f\notin L_1 \quad f = \frac{1}{x} \quad E = [1, \infty)$$
What can we say about reverse insertion? I think there is a suitable example
AI: Neither containment holds for Lebesgue measure if $|E... |
H: X subset of space vector V, $Tv=Sv, v \in X$ then T=S
Given two linear transformations:$$S,T: V \to W$$
such that $X\subseteq V$, and it is true that $S(v)=T(v), \forall v\in X$, and the exercise requests to prove that $S=T$. I have started by writing the vectors in X $ \{v_{1},\ldots, v_m\} $ then relating them to... |
H: Proving that if $a$ is an element from a group and $|a|=n$, then $C(a)=C(a^k)$, when $k$ is relatively prime to n.
$C$ in question title denotes centralizer of an element $a$ in group (say, $G$) and is defined as follows:
$$C(b)=\{x\in G:bx=xb \;\;\forall x\in G\}$$
In order to prove the result in title, I proceede... |
H: Tu's An Introduction to Manifolds - Section 26.2 Cohomology of a circle, tabular form.
I'm trying to understand how to use the Mayer-Vietoris sequence to compute Cohomologies. There's a small chapter in Tu's Introduction to Manifolds explaining the basics, with some basic theory.
More specifically section 26.2 has ... |
H: Determine $p$ for which the hyperharmonic sequence $\sum_{k=1}^{\infty} \frac{1}{k^p}$ is convergent
I'm asked to determine $p \in \mathbb{R} $ for which the hyperharmonic series:
$$\sum_{k=1}^{\infty} \frac{1}{k^p}$$ is convergent.
I started using the Ratio Test, which gives me:
\begin{align}
L &= \lim_{k \to \inf... |
H: How does one find if $x^3 + 3x^2 - 8x + 12 = 0 \bmod 5k$ has any root for $k$ an integer?
Actually I want to show that there is no solution to the cubic equation for any value of $k$. I have tried some values of $k$, and I want to see if it can be shown to be true for all $k$.
AI: You could show $n^3+3n^2-8n+12\no... |
H: Show $(A\cup B) \setminus A = B \setminus (A\cap B) $
Show $(A\cup B) \setminus A = B \setminus (A\cap B) $:
My reasoning:
LHS: $$x\in((A\cup B)\setminus A)$$
$$\equiv x\in(A\cup B) \land x\notin A$$
$$\equiv(x\in A \land x\notin A) \lor(x\in B \land x\notin A)$$
$$\equiv x\in B \land x\notin A$$
$$\equiv x\in(B\se... |
H: prove non-continuity using open sets (topology)
In topology, continuity is defined as:
A function $f:X\rightarrow Y$ is continuous if the inverse image of an open set in $Y$ is an open set in $X$.
I have a problem to use it to check the non-continuous function. For example, in J.Munkres' book Topology (2nd Editio... |
H: How To Determine If $\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\left(\sum_{k=0}^{n-1}\binom{2k}{k}\binom{k}{n-k}\right)$ Converges or Diverges?
$$\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\left(\sum_{k=0}^{n-1}\binom{2k}{k}\binom{k}{n-k}\right)$$
Question : How do i determine if the above Series Converges to Diverges?
I have no ... |
H: Prove the equation combinatorially [full answer provided] - I need explanation for the answer
For every $N ∈ r, n$
$P(n,r) = \sum_{k=0}^{r}\binom{r}{k}P(n-m,k)P(m,r-k)$
Prove this combinatorially.
Answer:
The class has m boys and n - m girls. In what ways can r students be selected for different roles?
On the one... |
H: Finding a volume of a region defined by |x-y+z|+|y-z+x|+|z-x+y|=1
Find the volume of the region definded by |x-y+z|+|y-z+x|+|z-x+y|=1.
I'm having trouble approaching this problem. Could someone maybe give me a hint or a solution, it would be so helpful.
Thanks in advance and sorry for my bad english.
AI: Let's fin... |
H: When given $X = −\frac{\ln(1 − U)}{\lambda}$, why is $X \sim \mathrm{Exp}(\lambda)$ and not $\mathrm{Exp}(-\lambda)?$
When given $X = \frac{−\ln(1 − U)}{\lambda}$, why is the distribution of $X \sim \mathrm{Exp}(\lambda)$ and not $\mathrm{Exp}(-\lambda)?$ I solved for $X$ to get:
$-X=\cfrac{\ln(1-U)}{\lambda}$
$-\... |
H: Verifying the $N_k$ in Waring's Theorem (Probability)
I know that there are proofs on Waring's Theorem on StackExchange, but I plan on tackling it without looking at the proof. The statement follows from Grimmet and Stirzaker's Probability and Random Processes. This might be pretty silly, but I've actually had a pr... |
H: What sample size is needed to ensure a majority?
The results of a sample of voters showed that $55\%$ voted for a given candidate. It was determined that at a confidence level of $0.95$ that candidate would be the winner (i.e. would receive the majority of the votes). What sample size is needed to ensure the accur... |
H: Given an open cover of $X$, $f:I \rightarrow X$, $\exists 0\leq s_1 ..\leq s_n = 1$ such that $f([s_i, s_{i+1}]) \subset A_\alpha$, $A_\alpha$ unique
The problem:
You are given a continuous function $f: [0,1] \rightarrow X$.
We choose an open cover of $X$, $A_{\alpha}$.
Then we want to show that there exists $0= ... |
H: In a fraternal twin pregnancy what is the probability that both children inherit a particular chromosome?
Take a given chromosome that is only present in one of the parents of a child. Assume that the probability of one child inheriting this chromosome is ${1}\over{2}$. At the beginning of the pregnancy, before tes... |
H: $X_1,X_2, \ldots$ be i.i.d. Show that $\mathbb{E}|X_1| < \infty $ iff $ \frac{X_n}{n} \to 0$ a.s
Suppose $X_1,X_2, \ldots$ be i.i.d. Show that $\mathbb{E}|X_1| < \infty \Leftrightarrow \frac{X_n}{n} \to 0$ a.s
I tried using Markov but I don't know anything about the $ \mathbb{E}X $. I was also thinking of borel... |
H: Finding average from uniformly distributed values
Let's say, I am sending network packets at the rate of 450 packets per second.
The size of these packets is uniformly distributed between 100 to 500 bytes.
I want to know what's the average packet size / per second?
I tried to solve it:
Since its uniformly distribut... |
H: Proof on "No rectangles" on a grid
I was solving a problem of the UVA judge called "No rectangles". The problem is about picking points from an $n\times n$ grid such that $k$ points are chosen from each row and column but no $4$ of the points form a rectangle with sides parallel to the grid. They claim the followi... |
H: Why isn't $f(x)=0$ ever mentioned as a solution to $f'(x)=f(x)$?
I know that $f(x)=e^x$ is the accepted and useful solution to $f'(x)=f(x)$, but why isn't $f(x)=0$ ever mentioned as a solution as well? Is it simply because it's not useful?
AI: Since $f'(x) = f(x)$ is linear differential equation, if $f(x)$ is solut... |
H: How to find all forms of the fraction that would be in between two other fractions?
I've been going through lots of my math textbooks, and I'm able to solve a lot of them using some specific method or formula. But there's one problem I've come across quite a few times that I just couldn't figure out how to do. One ... |
H: Does $a=ea$ and $ae=eae$ for some $a\in R$, imply that $a=e$ for any idempotent $e$ in $R$?
Let $R$ be a ring with unity and $e^2=e\in R$. If $a=ea$ and $ae=eae$ for some $a\in R$, then prove that $a=e$.
Solution: Suppose that $a=ea$ and $ae=eae$, then $a^2=eaea=e(eae)a=eeaea=eaea=eaa=ea^2$. That is, $a^2=ea^2$... |
H: Let $x,y>1$ be coprime integers and $g>0$ a real number such that $g^x,g^y$ are both integers. Is it true that $g\in\mathbb N$?
Let:
$x, y\ $ be coprime integers greater than $1$
$g \in \mathbb{R}^+$
$g_,^x \ g^y \in \mathbb{N}$
Proposition: $g \in \mathbb{N}$
I have not managed to prove it. Via the fundamental t... |
H: $\operatorname{Cov}[\vec{X}\cdot({\bf{v}} \operatorname{Cov}[\vec{X},Y]), \vec{X}] = {\bf{v}\bf{v}}^{-1}\operatorname{Cov}[Y, \vec{X}]$?
In Shalizi's Advanced Data Analysis from an Elementary Point of View p.44, he writes that for a variable $Y$ with a $p$-dimensional vector of predictors $\vec{X}$, and $\bf{v}$ th... |
H: Proving non-differentiability of $f:\mathbb{R}^2 \to \mathbb{R}$
Question: Given $f:\mathbb{R}^2 \to \mathbb{R}$ defined by $f(x, y) =
\begin{cases}
x, & \text{if $y=x^2$} \\
0, & \text{otherwise}
\end{cases}$, show $f$ is not differentiable at $(0, 0)$.
Attempt: I know a few things about $f$: it is continuous at ... |
H: Uniqueness of the Frechet Derivative: the role of $x \in int_X(T)$
I'm currently trying to learn some functional analysis as a way to improve my ability to read economic theory papers. I've come across what I thought was a simple proof but on reflection I don't think I'm grasping it. I'm not a mathematician so I ap... |
H: Help proving limits don't exist.
I am not asking about any specific limits. For some reason rigorously proving a limit does not exist using an epsilon delta proof gives me a lot of trouble. Typically my book does this by negating the definition of a limit. Deciding on what to use as the $/epsilon$ is generally tric... |
H: Denseness of a sequence in 2-Torus
I want to show that if $\alpha$ and $\beta$ are rationally independent irrational numbers i.e. $\forall m,n \in\mathbb{Z}$ , $m\alpha + n\beta \not\in\mathbb{Z}$ , then the sequence
$\{ (n\alpha$ (mod 1) , $n\beta$ (mod 1) $\}_{n\in\mathbb{Z}}$ is dense in 2-Torus.
I managed to ... |
H: On the dimension of a vector space
Let $U, V, W$ be finite dimensional $\mathbb KG$-modules. Assume that the sequence of homomorphisms
$0 \to U \to V\to W\to 0$ is left exact but not right exact. Then, $\dim V \leq dim U + \dim W.$
How to prove this?
AI: This is a question about vector spaces, and the action of $G$... |
H: Why are hyperbolic functions defined by area?
I have successfully derived the hyperbolic functions in terms of exponentials from the graphical definition:
For area $u/2$ bound by the unit parabola ($x^2 - y^2 = 1$), a ray from the origin to a point $(a,b)$ on the hyperbola and the $x$-axis, $\cosh u = a$ and $\si... |
H: How to prove this identity on exponential generating function of harmonic numbers
I came across the following problem, let $N![z^N]A(z)$ denote the coefficient of an exponential generating function (EGF) $A(z)$. The EGF is similar to an ordinary generating function (OGF) $A'(z)$ except that instead of the series $A... |
H: Formula (not an algorithm, i.e. defining a function) for the nearest number of the form n(n+1), where n is an integer
What would be the formula which defines a function that returns the nearest number of the form n(n+1), where n is an integer?
AI: Given a positive real number $x$, you are really just solving
$$
x=... |
H: Deriving the time-dependent solution of the Schrödinger equation
I have the Schrödinger equation:
$$\dfrac{-\hbar^2}{2m} \nabla^2 \Psi + V \Psi = i \hbar \dfrac{\partial{\Psi}}{\partial{t}},$$
where $m$ is the particle's mass, $V$ is the potential energy operator, and $(-\hbar^2/2m) \nabla^2$ is the kinetic energy ... |
H: Simple function is measurable
I want to show that a simple function is measurable. I know that a simple function is a function whose range set is a finite set. Let $f$ be a simple function defined on a measurable set $E$ such that it's range set is $\{a_1,a_2,\ldots , a_n\}$. Then help me show that $f$ is measurabl... |
H: About $n! > (\frac{n}{e})^n$ in "Introduction to Algorithms 3rd Edition" by CLRS.
I am reading "Introduction to Algorithms 3rd Edition" by CLRS.
In Apendix C, the authors use the following inequality:
$n! > (\frac{n}{e})^n$.
My proof of this fact is the following:
base case:
$1! = 1 > \frac{1}{e} = (\frac{1... |
H: Looking to optimise my Runescape grind (probability)
I know there's gaming stackexchange for gaming questions, but I believe this is purely maths related.
I'll try to avoid using game jargon and keep it simple. I'm collecting keys in game, each key taking a fair bit of time to obtain. They open a chest, which gene... |
H: Defining the set of all algebraic numbers: Help with index for the union
I'm doing problem 2.2 from Rudin, which asks to show that the set of all algebraic numbers is countable. I was looking at the solution provided here (image also included), and understand the general idea of the proof, but I'm not sure why the ... |
H: Show the map $g:\mathbb{S}^1\to\mathbb{S}^1$ defined by $g(\cos(\theta), \sin(\theta)) = (\cos(a\theta), \sin(a\theta))$ is open where $a\in\Bbb N$.
For a function $g: \mathbb{S}^1 \to \mathbb{S}^1$ defined by $g(\cos(\theta), \sin(\theta)) = (\cos(a\theta), \sin(a\theta))$ where $a$ is an integer, how do I show $g... |
H: Average distance from a square's perimeter to its center
What is the average distance from any point on a unit square's perimeter to its center?
The distance from a square's corner to its center is $\dfrac{\sqrt{2}}{2}$ and from a point in the middle of a square's side length is $\dfrac{1}{2}$. A visual explanation... |
H: Let $V$ be a finite dimensional vector space and $W$ is proper subspace of $V$. Then show that Span of $(V/W)= V$.
Let $V$ be a finite dimensional vector space and $W$ is proper subspace of $V$.
Then show that Span of $(V/W)= V$.
I am trying to show that $V/W$ contain a basis of $V$ but How to proceed ? any hint
A... |
H: Define a linear transformation $T$, so that the null space is $z$-axis, and the range is the plane $x+y+z=0$
As stated in the title, it is requested to define a linear transformation $T:\Bbb R^3 \to \Bbb R^3$ such that the null space of $T$ is the $z$-axis, and the range of $T$ is the plane: $x+y+z=0$
I don't rea... |
H: In the laplace transform of 1, why is the s in the denominator?
Ive been using the Laplace transform for a little while now for some electrical engineering differential equations,
what I have never quite understood, is why is the $s$ in the denominator when you do the Laplace transform of 1. ie. this $\mathcal{L}\{... |
H: Evaluate $\int_{-1}^{1} [ \frac{2}{3} x^3 + \frac{2}{3}(2-x^2)^{3/2}] dx $
Evaluate $$\int_{-1}^{1} \left[ \frac{2}{3} x^3 + \frac{2}{3}(2-x^2)^{3/2}\right] dx $$
My attempt :$$ \frac{2}{3} \left[\frac{x^4}{4}\right]_{x=-1}^{x=1} + \frac{2}{3} \left[\frac{(2-x^2)^\frac{-1}{2}}{-1/2}\right]_{x=-1}^{x=1}=0$$
Is its ... |
H: Why does the infinite union exist in set theory?
I want to be clear that I am not asking about the axiom of union. I understand that for an infinite set $A$, $\bigcup A$ exists. My question is specifically about the more widely used (as far as I have seen) version of the notation:
$\bigcup_{b\in B} S_b$.
The techni... |
H: Show that $\frac{d}{dx}(\frac{\tan(x)}{1+\sec(x)})=\frac{1}{1+\cos(x)}$
Show that:
$\frac{d}{dx}(\frac{\tan(x)}{1+\sec(x)})=\frac{1}{1+\cos(x)}$
Applying the quotient rule I get
$\frac{\sec^2(x)(1+\sec(x))-\sec(x)\tan^2(x)}{(1+\sec^2(x))}$
From here I go into many directions but not towards the RHS. Guidance i... |
H: An integral involving trigonometric and exponential function
Prove that $$ \int_{0}^{\infty}x^{2019}\sin(\sqrt{3}x)e^{-3x}\mathrm{d}x=\dfrac{2019!\sqrt{3}}{2^{2021}\cdot 3^{1010}} $$
Hence generalize the integral for any other value than $2019$.I know it can be done by considering the integral $\displaystyle \int_{... |
H: Radius of convergence for binomial series (2)
I'm having trouble calculating the radius of convergence for for the following binomial series.
More in detail, I'm having trouble finding $c_k$ and $c_{k+1}$ for the following series:
$$ \sum_{k=0}^{\infty} \binom{3k}{k}x^{2k+1}$$
I'm not sure what to do with the $2k+1... |
H: Index of subgroups in a finite solvable group, with trivial Frattini subgroup (Exercise 3B.12 from Finite Group Theory, by M. Isaacs)
Let G be a finite solvable group, and assume that $\Phi(G) = 1$ where $\Phi(G)$ denotes the Frattini subgroup of G. Let M be a maximal subgroup of G, and suppose that $H \subseteq M$... |
H: Section to Skew-Symmetrization Map
Let $A$ be an $n\times n$ matrix skew-symmetric matrix. Define the map
$\mathbb{R}^{d^2}\to Skew_d$ by
$$
B\mapsto B^{\top} - B.
$$
Does this map have a continuous right inverse?
AI: The following addresses the question as first stated, which read, if I recall correctly:
"Let $A... |
H: Homework Help: Probability of 5 element subset having one prime and a single digit
Here's the question:
Determine the probability that a randomly chosen 5-element subset of numbers from 1 to 20 contains at least one single digit number and at least one prime number.
Hi. Currently stuck on this homework problem. I a... |
H: Can this function be defined in a way to make it continuous at $x=0$?
We have $$f=\frac{x}{\vert x-1 \vert - \vert x +1 \vert}$$
If we want to "define" this function to be continuous at $x=0$, it's limit at $0$ must equal $f(0)$. So we should find this limit and assign it to be equal to $f(0)$, then the function is... |
H: Reference for very basic books in Functional analysis
I'm confused about which books I have to read for Functional analysis for the beginner level.
I need references for very basic books in Functional analysis and that book must contain given Topics below
$1.$ Normed linear spaces,
$2.$ Banach spaces,
$3.$ Hilb... |
H: Proof verification: Fourier Inversion theorem
I want to prove Fourier Inversion theorem:
$$\int_{\mathbb{R}^n}\widehat{f}(\xi)e^{2\pi ix\cdot\xi}d\xi=f(x)$$
almost everywhere, where $f,\widehat{f}\in L^1(\mathbb{R}^n)$.
We can get a equation
$$\int_{\mathbb{R}^n}\widehat{f}(\xi)e^{2\pi ix\cdot\xi}e^{-\pi|\varepsilo... |
H: Showing that a relation is neither an equivalence relation nor a partial order
Say we have a relation $R$ on $\mathbb{Z} \times \mathbb{Z}$ such that $(a, b) R (c, d)$ if $a^2 + b^2 \leq c^2 + d^2$
So to prove that $R$ is not an equivalence relation we need to show that $R$
Is not one of reflexive, symmetric or t... |
H: Closed operator with non-closed range
I am trying to see that if I have $X$ a banach space and $S:X\rightarrow X$ a closed operator then it's image is closed in $X$?
We know since $X$ is closed that by the closed-graph theorem we will have that $S$ is bounded, but still I can't quite see why $S(X)$ would be closed... |
H: Let $f\in Hol(\mathbb{C}\backslash\{a_1\ldots a_N\})$ where ${a_1\ldots a_N\,\infty}$ are the poles of $f$ show that $f$ is rational function
Let $f\in Hol(\mathbb{C}\backslash\{a_1, \dots, a_N\})$ where ${a_1, \ldots ,a_N,\infty}$ are the poles of $f$. Show that $f$ is rational function.
I've tried to define $g(z)... |
H: If a function is Lipschitz, and differentiable, is its gradient also Lipschitz?
If $f(x)$ is Lipschitz, i.e.
$$||f(x) - f(y)|| \le L||x-y||$$ is it's gradient also Lipschitz?
$$||\nabla f(x) - \nabla f(y)|| \le K||x - y|| $$
And does $L = K$ ?
AI: By considering a function which depends only on the first coordina... |
H: Normal subgroup and singleton orbits
Here is the exercise 20.11 from Groups and Symmetry from Armstrong :
Let $H$ be a subgroup of $G$ and write $X$ for the set of left cosets of $H$ in $G$. We have the action : $$ g(xH) = gxH$$
Show that $H$ is a normal subgroup iff every orbit of the induced action of $H$ on $... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.