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H: All operators on subspace are scalar
I have an infinite dimensional unital $\mathrm{C}^*$-algebra $A\subset B(H)$ and a subspace $U\subset H$ such that for all $f\in A$, $f$ restricted to $U$ is scalar (there exists $\lambda\in\mathbb{C}$ such that:):
$$f(u)=\lambda u.$$
Can we conclude that $U$ is (zero or) one di... |
H: What's the point of obtuse angle trigonometry?
What is the point of $\sin$ or $\cos$ or $\tan$ of an obtuse angle? Don't we use these functions to find a missing side in a right angle triangle? So why would I use it on an obtuse angle?
I have also done a unit circle.
AI: If all you want to do is think about right t... |
H: What this language mean? I have to define a Turing machine for it.
I'm tasked with defining a Turing machine for the language below. However I simply do not understand what this language means due to its notation.
A Tm M = $( Q,\Sigma ,\Gamma ,\delta , q_0, q_a, q_r)$ is given. We want to define a Tm (not necessari... |
H: Questions in Proof of Lemma before theorem 7 of Section -6. 5 of Hoffman Kunze( Linear algebra)
While self studying Linear Algebra from Hoffman Kunze, I have some questions in proof of a lemma in section - Simultaneous Triangulation, Simultaneous Diagonalization.
As I have asked 4 questions here so I will give bou... |
H: quotient space with usual topology
let be $X$ be a closed with the usual topology, $C \subset X$ a closed interval. then the quotient space $X / C$ is homeomorhpic to the interval $X$ \ $ C$ with the usual topology?
AI: It suffices to define the continous surjective map $f\colon [0,5] \to [0,4]$ given by $$f(t) = ... |
H: What does $\Bbb{Z}_{18}$ mean in this question?
I am trying to solve this question
Let $G=\Bbb{Z}_{18}$ and $H$ a subgroup of G generated by $3$, Talk about normal subgroups and coset multiplication then find the cosets $(G/H,\bigotimes)$
Does $\Bbb{Z}_{18}$ mean $$\{0,1,2,3,...,16,17\}$$
or
$$\{1,a,a^2,...,a^{16... |
H: Prove that the following proposition is true.
Let a ∈ Z and let b ∈ Z. If n does not divide ab then n does not divide a and n does not divide b.
I am currently studying discrete math and I am unsure of how to format this proof in such a way to get my point across. If anyone could write it out for me that would be v... |
H: A question in a theorem of section 6.6 of Hoffman Kunze
I am reading Linear Algebra from Hoffman Kunze by myself and I have a question in a theorem of Section-" Invariant Direct Sums " :
I am confused by how authors prove uniqueness of expression of $\alpha$ in proving that direct sum representation exists as for... |
H: Convergence of double Integral
Let's assume that $f \in L^p(\mathbb{R})$ ($1 \leq p < \infty$). Does it then hold that
$$
\lim_{n \rightarrow \infty} \int_0^1 \int_0^1 \left \lvert f\left( \frac{\tilde{r}}{n} \right) -f\left(\frac{r}{n} \right) \right \rvert^p ~\mathrm{d}r \mathrm{d}\tilde{r} = 0 \quad ?
$$
And if ... |
H: How can we tell the relationship between two lim-sup sets?
Let $f_n(x)$ be sequence of functions, and $\epsilon>0$.
Denote two sets as follows
$$E(\epsilon) = \limsup_{n\rightarrow\infty}\{x:|f_n(x)| >\epsilon \}$$
$$F = \limsup_{n\rightarrow\infty}\{x:|f_n(x)| > 1/n \}.$$
Based on the definitions above, can we con... |
H: Prove that $P(\tau<\infty)=1$ and find the distribution of $X_\tau$
Let $\{X_n\}_{n\ge1}$ be a sequence of i.i.d. random variables with distribution $\mu$. For $A\in\mathcal{B}(\mathbb{R}):=\text{Borel $\sigma$-algebra of $\mathbb{R}$}$ with $\mu(A)\in(0,1)$, define
\begin{align*}
\tau=\inf{\{k\ge1, X_k\in A\}}
\e... |
H: Example of non-local homomorphism on local rings
Could someone give an example for a homomorphism between local rings which is not local?
I tried finding the example by defining a homomorphism from Z/9Z to Z/3Z which maps a+9Z to a+3Z. But it turns out that 0,3,6(i.e. non units of Z/9Z) are mapped to 0(non unit in ... |
H: Find the Taylor series for $f(z)=\frac{i}{(z-i)(z-2i)}$ about $z_0=0$.
Find the Taylor series for $f(z)=\frac{i}{(z-i)(z-2i)}$ about $z_0=0$ and the disk of convergence.
For the Taylor series I got
$-\frac{i}{2}+\frac{z(2+i)}{4}-\frac{z(2i+3)}{2!}...$, but I'm not super confident in it. Can someone confirm or deny ... |
H: combinatorics: 5 people picking 10 seats when there must be at least one space between them
I have this question: How many seating arrangements are there for $5$ people to sit in $10$ seats in a row, when $2$ people can't sit next to each other?
My idea:
If there must be at least one space between every 2 people, t... |
H: Bisection method for $f(x)=x^4-x-2$
Finding a root for the polynomial
$$f(x)=x^4-x-2$$
I just to double-check if I've done this correctly, as it's my first time doing so.
My work so far
In above function, $a=1$ and $b=2$ works, as $f(a)$ and $f(b)$ have opposite signs
$$f(1)=(1)^4-(1)-2=-2$$
$$f(2)=(2)^4-(2)-2=+12... |
H: Why is $\arctan(\tan(25\pi / 4)) = \pi/4$?
Why is $\arctan(\tan(25\pi /4)) = \pi/4$, and how can I get from the expression on the left to the one on the right?
AI: The ${\tan(x)}$ function is periodic, meaning it will not be injective, thus will not have an inverse. Unless you restrict the domain.
For example, in y... |
H: Calculating lim using polar coordinations?
How to calculate the following limit:
$$\lim_{(x,y)\to(0,0)}\frac{-2yx^3}{(x^2+y^2)^2}$$
I replaced $x$ with $r\cos(\theta)$ and $y$ with $r\sin(\theta)$ and got $-2\sin(\theta)\cos^3(\theta)$
and got stuck there... any help?
AI: There is no need to use polar coordinates h... |
H: Evaluating laplace transform of {x}
I came upon a question on Laplace transforms in my exams.
$${\scr L}[\{x\}]$$
where $\{x\}=x - [x]$ denotes the fractional part of $x$.
This is how I approached the problem.
$$f(x)=\{x\}=\sum_{n=-\infty}^\infty{(x-n)(u_n(x)-u_{n+1}(x))}$$
where $u_a(x)$ denotes the Heaviside func... |
H: Bijection between the power set of $\mathbb{Z}_+$ and the set of all infinite sequences $(x_n)_{n=1}^\infty$ with $x_n\in\{0,1\}$
Proposition. There is a bijective correspondence between $\mathscr P{(\mathbb{Z_+})}$, the power set of $\mathbb{Z_+}$, and $X^\omega$, the set of all infinite sequences of elements of ... |
H: Find all matrices that are invariant under base changes.
Today I was solving a problem about listing all pairs $(a,b)\in \mathbb{R}^2$ such tath there exist a unique symmetric $2\times 2$ matrix such that $det(A) = a$ and $trace(A)=b$, where $A$ is the matrix that fulfills such conditions.
The solution to this spec... |
H: Approximating minimum of $f(x)+g(x)$ by solving for $f(x)=g(x)$
I'm trying to make sense of the following argument from a paper I'm reading.
The author has the following upper bound:
$$R(T) \le N + O\left(T\sqrt{\frac{\log T}{N}}\right)$$
They then state the following:
"Recall that we can select any value for $N$ s... |
H: Show $\begin{bmatrix}d-\lambda\cr -c\end{bmatrix}$ is eigenvector of $\begin{bmatrix}a &b\cr c&d\end{bmatrix}$
Show $A=\begin{bmatrix}d-\lambda\cr -c\end{bmatrix}$ is eigenvector of $\begin{bmatrix}a &b\cr c&d\end{bmatrix}$
I first did $\operatorname{det}(A-\lambda I)=0$ and got $\lambda^2+(-a-d)\lambda+(ad-bc)=0$
... |
H: Is $f(x) = mx + c$ the only set of solutions to $f(x + 1) - f(x) = \text{constant}$ where $m$, $c$, and $x$ are integers?
Is $f(x) = mx + c$ the only set of solutions to $$f(x + 1) - f(x) = \text{constant}$$ where $m$, $c$, and $x$ are integers?
I was watching this video in Youtube (https://www.youtube.com/watch?... |
H: If $f$ is not bounded from above, then $\lim_{x \to b^{-}}f(x) = \infty$ - Feedback on attempted proofs
Let $f:[a,b) \to \mathbb{R}$ be a strictly monotone increasing continuous function on a half closed interval $[a,b)$, and let $d$ be a real number.
Claim: If $f$ is not bounded from above, then $\lim_{x \to b^{-}... |
H: If function is differentiable at a point, is it continuous in a neighborhood?
I was reading a proof for the multi-variable chain rule and in the proof the mean-value theorem was used. The use of the theorem requires that a function is continuous between two points.
Hence the motivation for the question, if a functi... |
H: Given i+j+k=n, find sum of all possible ijk
Just solved this question, correctly (made a computer program to, successfully, verify the formula that I got):
Let $S$ denote the set of triples $(i,j,k)$ such that $i+j+k=n$ and $i,j,k\in\mathbb{N}$.
Evaluate $$\sum_{(i,j,k)\in S}ijk$$
My solution was as follows:
I'll a... |
H: Proof by contradicion
Let $x_1, x_2, . . . , x_n$ be $n$ real numbers. Let the average of $$x = \dfrac{x_1 + x_2 + \cdots + x_n}{n}$$ be their average. Prove that at least one of $x_1, x_2, \cdots , x_n$ is greater than or equal to $x$.
I am pretty sure this proof can be proved with contrapositive and I think I ma... |
H: sum of digit of a large power
what is $P(P(P(333^{333})))$, where P is sum of digit of a number. for an example $P(35)=3+5=8$
a)18
b)9
c)33
d)333
f)5
I tried to find this but I couldn't. I started to find a pattern for an example the first few power of $333^{333}$ are:
$A=333*333=110889 \; \; \; \; \; \... |
H: Prove that (orthogonal) projections $P_{L_1}x=P_{L_1}(P_{L_2}x)$ for closed subspaces $L_1\subset L_2\subset H $
Let $L_1\subset L_2\subset H $ closed subspaces of $H$ and let $x_2=P_{L_2}x$ (*orthogonal projection of $x$ onto $L_2$). Prove $P_{L_1}x=P_{L_1}x_2$ ($P_{L_1}x=P_{L_1}(P_{L_2}x)$)
According to this func... |
H: Prove that $A\setminus B$ is non-countable where $A$ is a non-countable set and $B\subseteq A$ is a countable set.
Problem:
Let $A$ be a non-countable set and let $B\subseteq A$ be a countable set. Prove that $A\setminus B$ is non-countable.
Where, $(X \text{ is non-countable})\;\iff (|X|>|\mathbb{N}|)$.
Of course... |
H: Can you square both sides of an equation containing matrices?
For example A=λI ⇒ A²=λ²I where A is square, and λ∈ℝ.
Or more generally AB=CD ⇒ ABAB=CDCD.
Assuming all the necessary matrix products are possible, what other conditions would need to be fulfilled for this to hold?
AI: For my money, the answer you’re loo... |
H: D&D Probability of 2 Attempts
I wanted to get the full probability of 2 attempts made at 60% chance of success.
I was looking at a different chain of math and found my probability to hit an enemy is 60% per each attack but I was wondering how it would look at all the outcomes and the probability of it.
6/10 * 6/10... |
H: Let $G$ be a group with order $105 = 3 \cdot 5 \cdot 7$
(a) Prove that a Sylow $7$-subgroup of $G$ is normal
(b) Prove that $G$ is Solvable
Can anyone please tell me if I am correct?
(a) For the sake of contradiction suppose $G$ dose not have a normal Sylow $7$-subgroup.
We first show $G$ has a normal Sylow $5$-sub... |
H: $v_i$ is an eigenvector for $T$ with eigenvalue $\lambda _i$ then it's eigenvector for $T^*$ with eigenvalue $ \bar{\lambda}_i$ given normal $T$
Given an inner product space $V$ and a normal operator $T$, prove that $\ker T=\ker TT^*$
The solution I found mentions that using the fact that $T$ is normal we know it... |
H: show that there is a continuous function from [0, 1] onto a countable product of copies of [0, 1] with product topology
I only have difficulties in problem 11.
My efforts:
Follow the hint. We have a sequence of functions {$F_k(t)$}. Each $F_k(t)$ maps $t\in$ [0, 1] to a sequence $(t_1,t_2,...)$ in $\Pi_{n\geq 1}[0... |
H: A notation for distinctness (in logic)
I want to say formally that $C_0$, ..., $C_{n+1}$ are distinct. Two ways seem to be
(1) $\qquad\qquad\qquad\qquad (\forall x,y \in \Bbb N)(x \ne y \to C_x \ne C_y)$,
(2) $\qquad\qquad\qquad\qquad C_{0 \le i \le n} \space \ne \space C_{i+(1 \le j \le n+1)}$.
Are (1) and (2) int... |
H: Are we comparing the expectation of random variables in "convergence in probability"?
I was watching this and this to try to understand what convergence in probability means.
In the first video, I was confused at why, in the Excel demonstration, the random variables had their standard deviations depend on $n$. Isn'... |
H: Solve $\int\limits_0^{1/\sqrt{2}} \frac{au^2}{5(1-u^2)^2}du = 1$ for $a$
Problem:
The function $f_U(u) = \frac{au^2}{5(1-u^2)^2}$ is a probability density for the random variable $U$, which is non-zero on the interval $(0, \frac{1}{\sqrt{2}})$. I am supposed to find the value $a$.
I understand that that amounts to ... |
H: If $\lim_{x \to b^{-}}f(x) = \infty$ then the image of $f$ is the ray $[f(a),\infty)$ - Proof feedback
Let $f:[a,b) \to \mathbb{R}$ be a strictly monotone increasing continuous function on a half closed interval $[a,b)$, and let $d$ be a real number.
Claim: If $\lim_{x \to b^{-}}f(x) = \infty$ then the image of $f$... |
H: Describing the kernel of a group homomorphism
Let $\phi : \mathbb Z \to \mathbb Z_{75}$ be the function defined by $\phi(n) = 27n \mod 75$, for all $n \in \mathbb Z$.
I'm trying to describe the kernel of $\phi$ as simply as possible and so far I got...
$\phi (n) = 27n\mod {75}$ $\\$
$\phi (n) = 0$ $\\$
$27n \mod 75... |
H: If $X$ is a metric space and $E \subset X$, then $\overline{E}$ is closed.
I would like to understand how a certain statement below follows from the previous statements.
Theorem If $X$ is a metric space and $E \subset X$, then $\overline{E}$ is closed.
Proof. If $p \in X$ and $p \notin \overline{E}$ then $p$ is ... |
H: Proof that the limit is 0 using Fourier series
I have this problem and I guess I should solve it using Fourier because of the context:
let $f:\mathbb{R}\rightarrow\mathbb{R}$ a continuous function over $\mathbb{R}$ such that $\left|f\left(x\right)\right|\le x^4$ for all $x$.
Need to prove:
$\lim_{n\to\infty}\int_{... |
H: what is the probability that a prime number divides another prime plus 1?
what is the probability that a prime number divides another prime plus 1?
what I do know is that for 2 it's 100%
I can show this fact using a function
$f(x,y):=$ the number of primes between $1$ & $y$ that when you add 1 you can divide it by ... |
H: What would be the arithmetic/algebraic rules for solving the problem $500=\frac{66}{\sqrt{1 - \frac{V^2}{(3 \times10^8)^2}}}$
Every direction I take to solve this problem leaves me with a negative on one side of the equation and $V^2$ on the other. Arithmetic/algebraic rules were the cause of my last question on th... |
H: Evaluate $\lim_{x\to 0} \frac{f(x^3)}{x}$
Let $f:\mathbb{R}\to \mathbb{R}$ be a function such that $|f(x)|\leq 2|x|$ for every $x \in \mathbb{R}$. Evaluate $\lim_{x\to 0} \frac{f(x^3)}{x}$.
According to the answer key, it is $0$ (which matches mine). I am not so sure about my solution (below) though.
Rewritting $... |
H: what is the probability that you get a prime number when you pick two random numbers between 0 and 1 ,a and b, and divide b by a and round up?
what is the probability that you get a prime number when you pick two random numbers between 0 and 1, a and b, and divide b by a and round up?
let's say your random numbers ... |
H: Finding a particular continuous function
Let $f:[0,1]\rightarrow\mathbb R$ be a continuous function such that $\int_{0}^{1} f(x)(1-f(x)) dx =\frac {1} {4}$ . How many such functions exist?
I really have no idea where to start. How do we solve it?
AI: Hint
$$f(x)(1-f(x))-\frac{1}{4}=-\left(f(x)-\frac{1}{2}\right)^2.... |
H: Proving a group homomorphism
Let $\Bbb R [x]$ denote the group of all polynomials with real coefficients, under the operation of addition. Let $\psi : \Bbb R[x] \to \Bbb R$ be the function defined by $\psi (p(x)) = p(3)$, for all $p(x) \in \Bbb R [x]$.
I'm trying to prove this to be a group homomorphism and so far ... |
H: Prove that a function in $\mathbb R^n$ is surjective
I have a function $f$: $\mathbb R^n$$\,\to\,$$\mathbb R^n$ defined by $f(\hat{x}) = \hat{x} - 2(\hat{x} \cdot\hat{v})\hat{v}$, with $\mid \hat{v}\mid$ $= 1$. I'm looking to prove that this function is surjective.
I'm a bit rusty on vectors, however, so I'm strugg... |
H: If $B_n$ is bounded then $\bigcup_{n=0}^{\infty} \varepsilon_n E_n$ is bounded
Let $E$ be metrizable space, that is, there exists a basis $\mathcal{B}:=\{U_n \subset E \; ; \; n\in \mathbb{N}\}$ of neighborhoods of $0 \in E$. I want to prove that: if $\{B_n \subset E \; ; \; n \in \mathbb{N}\}$ is family of bounde... |
H: Existence of constant for a "Minkowski-like" inequality to hold on $L_p$ $p<1$.
I'm solving some problems to prepare for my phd qualifying exam on functional analysis and measure theory. I want to prove that given a measure space $(X,\mathcal{A},\mu)$ for every $0<p<\infty$, there exists a constant $C_p>0$ such th... |
H: continuous functions of finite dimension banach spaces in infinite dimension banach spaces
I have the following problem but I am not sure how to proceed.
I that could suppose that there is at least one continuous function, but afterwards I don't know how to continue.
Prove that there is no continuous function of a ... |
H: Determining if $\frac{(-1)^n2n+1}{n-2}$ converges
I know a series $S$ with general term $a_n$ diverges if $\lim_{n\to\infty} a_n \neq 0$. My series is $\left\{\frac{(-1)^n2n+1}{n-2}\right\}$.
$$\begin{align}
\lim_{n\to\infty} a_n &= \lim_{n\to\infty} \frac{(-1)^n2n+1}{n-2} \\
&= \lim_{n\to\infty} \left((-1)^n \frac... |
H: Continuous extension of a partially defined function on a finite subspace
Let $X$ and $Y$ be topological spaces, and let $x_1,...,x_m$ be distinct points in $X$. Let $f: \{x_1,...,x_m\} \to Y$ be a function.
Is there a continuous extension of $f$ from $\{x_1,...,x_m\}$ to $X$?
It's easy to see that this is true i... |
H: How was the vector magnitude derived?
The magnitude of a $n$-vector is defined as:
$$
\sqrt{a_1^2+a_2^2+...+a_n^2}
$$
or for those that prefer sigma notation:
$$
\sqrt{\sum_{i=1}^n a_i^2}
$$
How would this have been derived? Or was this one of those cases where mathematicians went trial-by-error to find a formula t... |
H: Why are isometries all and only $f:\mathbb{R}^n \rightarrow \mathbb{R}^n $ :$f(x)=Nx+p$,where $N$ ortogonal
A passage in my notes reads:
Isometries in metric spaces coincide with the usual ones when considering mappings $f:\mathbb{R}^n \rightarrow \mathbb{R}^m $ with the respective Euclidean metrics.
In particula... |
H: Equivalence relation in construction of Grothendieck group
Sorry if this has been asked before but I couldn't find the question I have.
Yesterday I read the wikipedia page for a Grothendieck group. It provided two explicit constructions given a commutative monoid $M$. One construction was taking the cartesion produ... |
H: The best way to reduce a set
I was always told that the best way to guess what number someone's thinking of is to split the set into half continuously until I reach one item (ex. guess number from 1-100; 1-50; 1-25; 1-12; 1-6; 1-3 ; 1) and that it will generally be guessed in $ \lceil \log_2 n \rceil $ where $n$ is... |
H: Let $f(x)=x^5+a_1x^4+a_2x^3+a_3x^2$ be a polynomial function. If $f(1)<0$ and $f(-1)>0$. Then
Let $f(x)=x^5+a_1x^4+a_2x^3+a_3x^2$ be a polynomial function. If
$f(1)<0$ and $f(-1)>0$. Then
$f$ has at least $3$ real zeroes
$f$ has at most $3$ real zeroes
$f$ has at most $1$ real zero
All zeroes are real
My attemp... |
H: A combination problem with repeated number
We know about the combination problems. However, if we say the numbers are $1,1,2,3,4,5$, then how many unique numbers with $6$ digits can be written with these numbers?
AI: first you put the 1's
6C2
then you multiply by 4! to permutate the others
another way to do it is:
... |
H: $\forall x \in \mathbb{R}^+ ( \exists M \in \mathbb{Z}^+ ( x > 1/M > 0))$: Cauchy Sequences
Synopsis
I'm sure this question can be found elsewhere on this website, but I haven't yet found a perspective though that uses the construction of the reals through Cauchy sequences to prove this statement. The reason why I'... |
H: Simple conditions on Radon-Nikodym derivative to obtain equivalent measures
Are there some simple conditions for the converse statement of the following statement:
If $\mu$ and $\nu$ are equivalent (i.e. $\mu \ll \nu$ and $\nu \ll \mu$) then
$$
\frac{d\mu}{d\nu}=\left(\frac{d\nu}{d\mu}\right)^{-1}
$$
Is strict po... |
H: Non existence of closure of an operator
A linear operator $\Omega$ on $C(X)$ is closed if its graph is a closed subset of $C(X)\times C(X)$. A linear operator $\bar{\Omega}$ is called the closure of $\Omega$ if $\bar{\Omega}$ is the smallest closed extension of $\Omega$. Not every linear operator has a closure. Th... |
H: General Gaussian integrals over the positive real axis.
Everyone has a special memory from their multivariable calc class deriving the famous Gaussian integral:
$$ \int_0^\infty e^{-x^2} \,dx = \frac{\sqrt\pi}{2}$$
A more general case is easy to find online and (not too hard to do yourself):
$$\int_{-\infty}^\infty... |
H: Let $G$ a graph of order $n$. Prove that $n\leq \mathcal{X}(G) \mathcal{X}(\overline{G})$ and $2\sqrt{n}\leq \mathcal{X}(G)+\mathcal{X}(\overline{G})$
Given a graph $G$, the $\textit{chromatic number}$ of $G$, denoted by $\mathcal{X}(G)$, is the smallest integer $k$ such that $G$ is $k-$colorable.
$\textbf{Problem.... |
H: How to find pair of reflected point of $f(x)$, $f^{-1}(x)$ and $g(x)$?
$f(x)=4+\sqrt{x-2}\\f^{-1}(x)=x^2-8x+18\\g(x)=x$
Obtain the graph of $f$ , $f^{-1}(x)$, and $g(x)=x$ in the same system of axes.
About what pair (a, a) are (11, 7) and (7, 11) reflected about?
After I graphed three equations, there is no pai... |
H: Whitney extension theorem for Hölder spaces
The usual Whitney extension theorem says that Whitney data with remainders like $R_\alpha=o(x-y)^{k-|\alpha|}$ extends to a $C^k$ function. If we also have $R_\alpha=o(x-y)^{k+\lambda-|\alpha|}$ for some $\lambda\in(0,1]$, do we get a $C^{k+\lambda}$ function? That is, a ... |
H: Prove the condition below mentioned.
Let $f(x)$, denote a polynomial in one variable with real coefficients, such that $f(a)=1$ for some real number
a. Does there exist a polynomial $g(x)$ with real coefficients, such that, if $p(x)=f(x) g(x),$ then $p(a)=1$ $p^{\prime}(a)=0$ and $p^{\prime \prime}(a)=0 ?$ Justify ... |
H: How to plot a plane with the given normal vector (numerically)?
I have some point and a normal vector. I want to plot a plane defined with this normal vector and a point. Analytically, we use formula (for simplicity, in 2d case):
$n_1 (x-x_0) + n_2(y-y_0) = 0$
So, we get an equation and that is it. But how can I do... |
H: Short, yet powerful proofs
One of my favorite proofs is the following:
Claim: There exists irrational numbers $\alpha$ and $\beta$ such that $\alpha^{\beta}$ is rational.
Proof: Let $\alpha = \sqrt{2}^{\sqrt{2}}$ and $\beta = \sqrt{2}$ so $\alpha, \beta \notin \mathbb{Q}$. Therefore, $$\alpha^{\beta} = (\sqrt{2}^{\... |
H: Showing $\operatorname{diag}(x)-xx'$ is positive definite on the tangent space of the unit simplex.
Let $x$ be in the unit simplex (i.e. $\sum_i x_i = 1, x_i \geq 0$ ).
I want to show that $\operatorname{diag}(x) - xx'$ is positive definite on the tangent space of the simplex. That is, $z'[\operatorname{diag}(x)-xx... |
H: If $\int f(x)dx =g(x)$ then $\int f^{-1}(x)dx $ is equal to
If $\int f(x)dx =g(x)$ then $\int f^{-1}(x)dx $ is equal to
(1) $g^{-1}(x)$
(2) $xf^{-1}(x)-g(f^{-1}(x))$
(3) $xf^{-1}(x)-g^{-1}(x)$
(4) $f^{-1}(x)$
My approach is as follows:
Let $f(x)=y$, therefore $f^{-1}(y)=x$, $\int f^{-1}(f(x))dx =g(f(x))$
On diffe... |
H: How to calculate $ \left| \sin x \right| $ derivative in a more elegant way?
I am trying to calculate the derivative of $\left| \sin x \right| $
Given the graphs, we notice that the derivative of $\left| \sin x \right|$ does not exist for $x= k\pi$.
Graph for $\left|\sin x\right|$:
We can rewrite the function as
$... |
H: Prove or disprove: a group with 3 different elements of order 6 can't be cyclic.
I have a question of prove or disprove as above: a group $G$ with $3$ different elements of order $6$ can't be cyclic.
I tried to find a group of order $6*n$ such that it has 3 different elements, but having trouble doing so. I know i... |
H: What are the odds of rolling a 1 and a 6 in four dice throws?
There are $\binom42 = 6$ different ways to arrange the two desired rolls in a sequence of $4$ dice, and $6^2$ possible results for the remaining two dice. There are $6^4$ possible total outcomes, so the odds of getting a 1 and a 6 should be $6^3/6^4 = 1/... |
H: What does $C([0,1))$ stand for in the picture below?
Could anyone tell me what $C([0,1))$ stands for in the picture below?
If I see "$f \in C[0,1)$" anywhere, I'd read it as "the function $f(x)$ is continuous for all $x\in[0,1)$". But this one should be different...
Any help is appreciated.
AI: $C(I)$ is the set (... |
H: volume of tilted ellipsoid
I'm supposed to calculate the volume of
$$(2 x+y+z)^2 + (x+2 y+z)^2 + (x+y+2 z)^2 \leq 1$$
simplifying it gives $$6 (x^2 + y^2 + z^2) + 10 (x y + y z + x z) \leq 1$$
After drawing it using GeoGebra, I saw that it's a tilted ellipsoid inside the unit sphere, but I'm unable to think of how... |
H: For any complex $z$, $|z-1|\leq |z-j|+|z-j^2|$
Let $z\in \mathbb C$. Prove that $|z-1|\leq |z-j|+|z-j^2|$
This inequality appears as an exercise in a book for highschoolers. It is marked as very difficult.
$j=\exp(2i\pi/3)$ denotes a third root of unity.
I tried squaring both sides, and making use of $j^2=-1-j$... |
H: Question on 'taking out' pointwise limit in the $L^p$ norm
In functional analysis, many properties of certain spaces are normally derived from taking a pointwise limit out of the norm, i.e.
$\lvert \lvert x \rvert \rvert=\lim\limits_{n\to \infty}\lvert \lvert x_{n} \rvert \rvert$ $(*)$.
The normal justification for... |
H: Does $\sum \frac{n!e^n}{n^n}$ converge or diverge
I have tried root and ratio test but they were inconclusive.
thank you very much
AI: Using Stirling's formula, you should find that
$$\frac{n!\,\mathrm e^n}{n^n}\sim_\infty\sqrt{2\pi n},$$
so it diverges trivially. |
H: Concavity/convexity of a function whose domain is a singleton set
Is a function which is defined on the singleton domain both convex and concave? This question got raised when I read the following from here: "If you look at the definition of concavity, you see that every function is concave on a domain consisting o... |
H: Calculate real matrix inverse of a complex matrix
Given a Hermitian positive semidefinite matrix $A \in \mathbb{C}^{n \times n}$.
If $B=A^{-1},\ D=\text{Re}(B),\ C=D^{-1}$.
where $D=\text{Re}(B)\Leftrightarrow{}d_{ij}=\text{Re}(b_{ij})$.
Can I calculate matrix $C$ from $A$ directly without calculating matrix invers... |
H: Ideas for calculating $K_0(l_{\infty})$ and $K_1(l_{\infty})$.
Thank you for answering my question. I'm a bit new to K-theory.
So I was wondering how can I calculate $K_0(l_{\infty})$ and $K_1(l_{\infty})$.
I think if we have one, then by using bott periodicity we can have the other one.
Can anyone help me?
AI: It ... |
H: How does $s'$ define from the theorem?
I encounter this theorem from my introductory algebra textbook. I'm quite not good at defining some of the definitions. Here is the theorem:
Theorem (3.2)
Let $m_{1}$, $m_{2}$ be positive integers. Let $d$ be a positive generator for the ideal generated by $m_{1}$ and $m_{2}$.... |
H: Continuity and derivability of a piecewise function $f(x,y)=\frac{x^2}{y}$
Study the continuity and the derivability of the function
$$f(x,y)=\begin{cases} \frac{x^2}{y}, \ \text{if} \ y \ne 0 \\ 0, \ \text{if} \ y=0 \end{cases}$$
For the continuity: $f$ is continuous where it is defined because it is a ratio of co... |
H: Proving that there exists an uncountable number of distinct basis of the euclidean topology on $\mathbb R$
In my general topology textbook there is the following exercise:
(i) - Let $\mathcal B$ be a basis for a topology $\tau$ on a non-empty set $X$. If $\mathcal B_1$ is a collection of subsets of $X$ such that ... |
H: If $\int_a^b f(x)dx=\left[F(x)\right]_a^b$, Is $\int_a^b \lvert f(x)\rvert dx= \left[\lvert F(x)\rvert \right]_a^b$ true?
I was brushing up on some basic inequalities and I attempted to derive an alternate form of the regular triangle inequality by using $(\left|b\right|-\left|a\right|)^2$. $$ \begin{align*}(\left|... |
H: $PSl_n(\mathbb{C})\cong PGl_n(\mathbb{C})$?
I was reading about projective linear groups because I was asked to show that $PSl_n(\mathbb{C})\cong PGl_n(\mathbb{C})$. Here $PSl_n(\mathbb{C})$ is the projective space of $Sl_n(\mathbb{C})$, i.e. $Sl_n(\mathbb{C})/(\text{scalar matrices in } Sl_n(\mathbb{C}))$ and $PGl... |
H: What does `Sitzber. Heidelberg Akad. Wiss., Math.-Naturw. Klasse. Abt. A' stand for?
I would like to cite an article from 1914 by Oskar Perron without any abbreviations. I am unable to figure out what `Sitzber. Heidelberg Akad. Wiss., Math.-Naturw. Klasse. Abt. A' is short for. Can anyone here perhaps help me with ... |
H: Joint distribution of $X$ and $Y-X$ (discrete case)
The joint distribution of the random variable $(X,Y)$ is $\mathbb{P}(X=x,Y=y)=\frac{e^{-2}}{x!(y-x)!},x=0,1,...,y=x=x+1,...$
Find the marginal distribution of $X$ and $Y$.
$\rightarrow \mathbb{P}(X=x)=\sum_{y=x}^{+\infty}\mathbb{P}(X=x,Y=y)=\frac{e^{-2}}{x!}\sum... |
H: is the limit as k approaches infinity of a Taylor Polynomial of order k, that approximates a function f, the same as the function itself?
Since Taylor polynomial approximation gets better as It's order gets bigger, so I was wondering, what happens when this order approaches infinity? Does the approximation equal th... |
H: Set of polynomials of degree less than $N$ that have value $0$ in $x=1$ as vector space?
How can I prove that all polynomials of degree less than $N$ that have value $0$ in $x=1$ are writable in this form?
\begin{equation}
p(x) = a_1 (x-1) + a_2 (x-1)^2 \dots + a_{N-1} (x-1)^{N-1}
\end{equation}
and so the set of p... |
H: Double integral of a shifted circle
Task: find a double integral $$\iint_D (x+y)dxdy,$$ where D is bound by $x^2 + y^2 = x + y$.
What I have done so far: turns out it's a circle $$(x-1)^2 + (y-1)^2 = 2$$
Calculating it as a common double integral is hard because I get something like this: $$\int_{1-\sqrt{2}}^{1+\sq... |
H: Any generalization of the complex conjugates in the theory of fields?
We know that for any complex number $z = x + \iota y$, where $x$ and $y$ are real numbers, there exists the complex number $\overline{z} = x - \iota y$, and the complex numbers $z$ and $\overline{z}$ are said to be the complex conjugates of each... |
H: Is there any real function that does not obey this rule on limit?
Consider the following rule
$\lim\limits_{h\to 0} f(x+h)= f(x)$
Do any real function exists that does not satisfy the above rule?
AI: Take the function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined by $f(0)=1$ and $f(x)=0$ for all $x\neq 0$. |
H: How to find the closest integer to a given number $y \in \mathbb{R}$ that has a perfect root $x \in \mathbb{Z}: \sqrt x = k \in \mathbb{Z}$?
Background Theory
An approximation of a number $x_0$ can be made using using derivatives, based on the formula $(1)$
$$f(x_0 + \Delta x) \approx f(x_0) + f'(x_0)*\Delta x\quad... |
H: Is the statement "$p$ implies $q$" logically equivalent to the statement "$p$ implies only $q$"?
I am confused if the statement "$p$ implies $q$" logically equivalent to the statement "$p$ implies only $q$"?
Assuming that the two said statement is logically equivalent, then the truth value of the statement ...
"If ... |
H: Clarification on proof of $\tau_1 \times \tau_2$ is $T_2$ iff $\tau_i $ is $T_2$ for $i=1,2$
Proposition: Let $(X, \tau)$ be a topological space
(1)If $\mathscr{B}$ is a base of $\tau$ and $p \in X$ , the set $\mathscr{B}_p=\{ B \in \mathscr{B}\mid p \in B\}$ is base of neighborhoods of $p$
(2) If for every $p \in ... |
H: Why is $\sqrt{x^2}$ is $|x|$?
I was trying a problem and was getting the wrong answer and when I saw the solution on the internet I found this statement written in square brackets $\sqrt{x^2}$[note square is on $x$] is $|x|$. Till now I have learned that by laws of exponents we can multiply the powers and obtain $\... |
H: Number of ways to color sides of square with rotation
The edges of a square are to be colored either red, blue, yellow, pink, or black. Each side of the square may only have one color, but a color may color many sides. How many different ways are there to color the square if two ways that can obtained from each ot... |
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