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H: Proof verification: A certain process of redistribution stops after a finite number of steps.
QUESTION: There are $n\ge 3$ girls in a class sitting around a circular table, each having some apples with her. Every time the teacher notices a girl having more apples than both of her neighbours combined, the teacher t... |
H: What is the name of this property of a ring?
Let $R$ be a ring. For all $x,y\in R$, there exists a $z\in R$ such that $xy=yz$.
Does this property have a name? I do not assume that multiplication has an identity or that it is commutative, but I do assume that multiplication is associative
AI: Such rings (where $\for... |
H: Change of basis matrix from $\alpha$ to $\beta$ or from $\beta$ to $\alpha$?
In Peterson and Sochacki's Linear Algebra and Differential Equations they define (in section 5.3) the change of basis matrix from $\alpha$ to $\beta$, $[I]_{\beta}^{\alpha}$, as the matrix whose columns are the $\alpha$-coordinates of the ... |
H: Find all $z$ such that $|\cos z|^2+|\sin z|^2=4$
I need to solve for $z$ with $|\cos z|^2+|\sin z|^2=4$
I know $\cos z =\frac{1}{2}(e^{iz}+e^{-iz})$ and $\sin z = \frac{1}{2i}(e^{iz}-e^{-iz})$ but I'm not sure if this is helpful because I don't know how to split it into Re$(z)$ and Im$(z)$ to find $|\text{cos}\ z|$... |
H: Proving monotonocity and convergence for sequences $(s_n)$ and $(t_n)$
Let $X:=(x_n:n\in \mathbb N)$ be a bounded sequence, and for each $n\in \mathbb N$ let $s_n:=\sup\{x_k:k\geq n\}$ and $t_n:=\inf\{x_k:k\geq n\}$.
Prove that $(s_n)$ and $(t_n)$ are monotone and convergent.
My approach: Now as $X$ is bounded, it ... |
H: Is it true that $\frac{\ln(a)}2=\ln(\sqrt{a})$ for $a>0$? In particular, is $\frac{\ln(2)}{2}=\ln(\sqrt2)$?
I believe the following two identities are correct. For some reason, they look wrong to me. Are they?
$$ \frac{ \ln \left( 2 \right) } { 2 } = \ln( \sqrt{2} ) $$
$$ \frac{ \ln \left( a \right) } { 2 } = \ln( ... |
H: Approximation of $\frac {n-c \choose k} {n\choose k} $ using a radical
Is there a good approximation for
$\frac {n-c \choose k} {n\choose k} $, given large parameter n (k,c can be large as well, but k,c<n/2)?
I think I can express it as a simple exponent function f(n,k,c), but i am not sure.
I prove that this expr... |
H: Do I need Axiom of Choice for ZF construction of the Natural Numbers?
Using ZF axioms I have constructed the natural numbers like so:
0 = ∅
1 = {0} = {∅}
2 = {0,1} = {∅,{∅}}
3 = {0,1,2} = {∅,{∅},{∅,{∅}}}
4 = {0,1,2,3} = {∅,{∅},{∅,{∅}},{∅,{∅},{∅,{∅}}}}
etc.
I am trying to determine if I used the axiom of choice for ... |
H: Can you find a function that follows a rule while rational and it's differentiable everywhere? (relates to the Pythagorean theorem)
can you find a function that follows a rule while rational and it's differentiable?
let's call this function $\alpha(x)$
$x$'s simplest form is $\frac{a}{b}$ when $x$ is a fraction.
wh... |
H: Definition of subset
I am new to set theory, and even though I grasp the concept, I am having trouble with the formal definitions, specially with the subset one. The statement $A\subseteq B$ can be written as $\forall x(x\in A\rightarrow x\in B)$. Now, let $A = \left\{1, 2\right\}$ and $B = \left\{3, 4\right\}$ and... |
H: Why is my value for the length of daylight wrong?
I was watching a YouTube video where it showed how length of daylight changes depending on the time of year, and I was curious and wanted to try calculating the value of how long the daylight is in the Tropic of Cancer (23.5 degrees latitude) during the winter solst... |
H: Prove that $\int(x^2-1)^n \,dx = \frac{x(x^2-1)^n}{2n+1} - \frac{2n}{2n+1}\int(x^2-1)^{n-1} \,dx$
I have tried to solve the problem mainly with the LS of the equation.
I can not seem to get rid of the x variable within the resultant integrand.
ex. after the first integration by parts I am left with:
$x(x^2-1)^n - 2... |
H: Distribution of the mean of Brownian motions
Consider $Z_n = \sum_{k=1}^{n} \frac{B_k}{n}$ where each $B_k \sim N(0, k)$ is a Brownian motion. I'm trying to compute the distribution of $Z_n$. Obviously $EZ_n=0$. For $Var Z_n$, since $\mathbf{Cov}B_sB_t = \min\{s,t\},$ I derive
\begin{align}
Var Z_n &= \frac{1}{n^2}... |
H: Terence Tao Analysis I Proposition 4.4.5
In the book the proof for
Proposition 4.4.5: For every rational number $\epsilon > 0$, there exists a non-negative rational number $x$ such that $x^2 < 2 < (x + \epsilon)^2$
Proof:
Let $\epsilon > 0$ be rational. Suppose for the sake of contradiction that there is no non-neg... |
H: Convergence of $\int^{\pi/2}_0 x\sqrt{\sec x}dx$
At $x=\pi/2$, $\sec x$ goes to infinity, and $x$ is fixed, so $x\sqrt{\sec x}$ goes to infinity. It seems to diverge, but the solution says it converges. I don't know how to prove it. I cannot find the antiderivative of this function or suitable functions to apply co... |
H: Evaluating $\int _0^{\infty }W\left(\frac{1}{x^3}\right)\:\mathrm{d}x$
How can i evaluate $\displaystyle\int _0^{\infty }W\left(\frac{1}{x^3}\right)\:\mathrm{d}x$ in an easy manner i managed to end up with this
$$3\int _0^{\infty }\frac{W\left(\frac{1}{x^3}\right)}{W\left(\frac{1}{x^3}\right)+1}\:\mathrm{d}x$$
But ... |
H: Is it possible to subdivide a regular polygon of side-length $n$ into equilateral polygons of side-length $1$?
Suppose I have a regular polygon whose sides each measure $n$. I want to cut it up into smaller equilateral (but not necessarily regular) polygons whose sides each measure $1$.
Is this possible? If yes, wh... |
H: If$|f(x)-f(y)|\le (x-y)^2$, prove that $f$ is constant
(Baby Rudin Chapter 5 Exercise 1)
Let $f$ be defined for all real $x$, and suppose that
\begin{equation}\tag{1}
|f(x)-f(y)|\le (x-y)^2
\end{equation}
Prove that $f$ is constant.
My attempt:
Let $f$ be defined for all real-valued inputs. Let $x \in \mathb... |
H: Evaluating a binomial summation
I'm interested in evaluating the following summation, where the value of $n$ is known:
$$\sum_{i = 0}^{2n} \sum_{j = \max(0, i - n)}^{\min(i, n)} {i \choose j}.$$
In case you're wondering where the summation comes from, it is the answer to the following question: "How many binary str... |
H: Is there an "algebraic" way to construct the reals?
It's possible to construct $\mathbb{Q}$ from $\mathbb{Z}$ by constructing $\mathbb{Z}$'s field of fractions, and it's possible to construct $\mathbb{C}$ from $\mathbb{R}$ by adjoining $\sqrt{-1}$ to $\mathbb{R}$.
In both cases, the construction is done purely alge... |
H: How many different ways to fill a nonnegative integer matrix with fixed column and row sums
Given an $m$ by $n$ matrix, what is the general, closed-form formula to calculate how many different ways we can fill this matrix with nonnegative integers given the required sums of each row, $r_1, r_2, ..., r_m$ and of eac... |
H: Laplacian of a function has the same sign of the function itself
Here is the problem:
Let $U \subset \mathbb{R}^n$ be a connected open set with regular boundary, and $f:\mathbb{R}\to\mathbb{R}$ a function such that $tf(t)\geq0$ for all $t\in\mathbb{R}$. Show that every solution $u\in C^2(\overline{U})$ of the probl... |
H: How to solve $\int\frac{1}{\sqrt {2x} - \sqrt {x+4}} \, \mathrm{dx} $?
$$\int\frac{1}{\sqrt {2x} - \sqrt {x+4}} \, \mathrm{dx}$$
I have tried $u$-substitution and multiplying by the conjugate and then apply $u$-substitution. For the $u$-substitution, I have set $u$ equal to each square root term, set $u$ equal to... |
H: Using partial information to factor $x^6+3x^5+5x^4+10x^3+13x^2+4x+1.$
I wish to find exact expressions for all roots of $p(x)=x^6+3x^5+5x^4+10x^3+13x^2+4x+1.$ By observing that for the roots $x_0 \pm iy_0, x_0 \approx -0.15883609808599033632, y_0 \approx 0.27511219196092896700,$ we have that $x_0$ is the unique rea... |
H: Does $\mathrm{SO}_n \cong T^1\mathbb{S}^{n-1}$ for all $n \in \mathbb{Z}_+$?
Let $\mathbb{S}^{n-1}$ be the $(n-1)$-dimensional sphere and let $T^1\mathbb{S}^{n-1}$ be its unit tangent bundle. I have just learnt that $\mathrm{SO}_3 \cong T^1\mathbb{S}^2$. Here $\cong$ means 'homeomorphism'. Does it hold for all $n$ ... |
H: What does it mean when we say '$f$ is a function taking values on $\mathbb{R}$ $\cup$ {$-\infty, \infty$}'?
I was recently reading this post and noticed some terminology I am not familiar with. The title of the post is "Why is convex conjugate defined on functions taking values on extended real line?"
What does it ... |
H: Finding the equation of 4 circles given 3 tangents, one of which is oblique
The question is to find the equation of the four circles tangent to the x-axis, the y-axis and the line $x+y=2$. I have drawn out a diagram and have identified the 4 circles but I am stuck on how to find their equations.
AI: There are many ... |
H: I don't know why this is answer. $f\left( x\right) =\lim _{n\rightarrow +\infty }f_{n}\left( x\right) =0$
"n" is positive integer. An interval is [0,3].
$$f_{n}\left( x\right) =\begin{cases}n^{2}x\left( 0\leq x\leq \dfrac {1}{n}\right) \\ n\left( 2-nx\right) \left( \dfrac {1}{n} <x\leq \dfrac {2}{n}\right) \\ 0\le... |
H: If $p$ is not a limit point of $E$, then it has a neighborhood with at most 1 point of $E$ — why at most 1 point?
I have a doubt about the following proof from PMA Rudin:
Suppose $E \cap K$, and let $q \in K$ be a point which is NOT a limit point of $E$. Then $q$ has a neighborhood $N$ s.t. it has at most one poin... |
H: Given $y = x^3 − 2x$ for $x \geq 0$, find the equation of the tangent line to $y$ where the absolute value of the slope is minimized.
I tried finding the derivative of this, and promptly got $y=$ about $0.816$, but I have no idea how to put that into equation form or if I'm even correct.
AI: I assume you already go... |
H: Nonstandard models of PA
Reading The Incompleteness Phenomenon, by Goldstern and Judah, they show there are nonstandard models of PA by adding a constant greater than any natural number. They then show that any countable model consists of the standard naturals followed by a dense linear order of copies of the inte... |
H: Subgroup of a ring closed under multiplication?
Let $(R, +, \cdot)$ be a ring with identity $1$. Let $G \subset R$ be a group under addition. Then $G$ is a subset of $R$, so we can perform $R$'s multiplication on elements of $G$. Will multiplication in $G$ always be closed? What are some counterexamples?
If $R = \m... |
H: Show how to assume any matrix is upper triangular, and the concept of a basis in matrices?
How can you show that any given matrix can be assumed to be upper triangular? And what does the concept of a basis have to do with upper triangular matrices (or matrices in general)?
AI: A matrix represents a linear map with ... |
H: Using Correspondence Theorem for Rings
I was trying to solve a problem involving local rings and I did the following, which seems to lead to a contradiction but I cannot find where I have messed up:
For a field $\mathbb{K}$ we know that $\mathcal{M}=(x)/(x^3)$ is the unique maximal ideal in the quotient ring $\mat... |
H: Using ${\rm Lip}1$ to show that $C[0,1]$ is separable
I am studying for my PhD qualifying exams by going through the problems in Carothers, and I have come across this problem.
For each $n$, show that $$\{ f \in {\rm Lip}1 : \rVert f \lVert_{{\rm Lip}1} \leq n \}$$ is a compact subset of $C[0,1]$. Use this to give ... |
H: Composition Functions (Advanced Functions)
Question: a) Given the functions $f(x) = x + 2$ and $g(x) = 3^x$, determine an equation for (f ∘ g)(x) and (g ∘ f)(x).
b) Determine all values for $x$ for which $f(g(x)) = g(f(x))$.
*For part a), I got the equation:
$(f ∘ g)(x) = 3^x + 2$
and
$(g ∘ f)(x) = 3^{x+2}$
But I d... |
H: Show that $f(x)=\frac{1}{\sqrt{x}}$ is uniformly continuous on the domain $(1,\infty)$ but not on the domain $(0,1)$.
Show that $f(x)=\frac{1}{\sqrt{x}}$ is uniformly continuous on the domain $(1,\infty)$ but not on the domain $(0,1)$.
$\def\verts#1{\left\vert#1\right\vert}$
My Attempt
First we show that $\forall... |
H: Function of bounded variation whose reciprocal is not of bounded variation
Problem: Find an example of a positive function $f: [0,1] \to \mathbb{R}_{>0}$ that is of bounded variation, whose reciprocal $1/f$ is integrable but not of bounded variation.
One necessary condition for $f$ is that $\inf_{x \in [0,1]} f(x)=... |
H: Find the number of possible passwords that can be created
Suppose that you are assigned an e-mail account, you need to create your own password. The format of a password is three digits followed by five uppercase letters such that neither digit nor letter can be repeatedly used. Find the number of possible passwo... |
H: Solve for $(p,q)\in\mathbb{Z}$, $\frac{p}{\sqrt{3}-1}+\frac{1}{\sqrt{3}+1}=q+3\sqrt{3}$
The question says "find integers $p$ and $q$ such that $\frac{p}{\sqrt{3}-1}+\frac{1}{\sqrt{3}+1}=q+3\sqrt{3}$.
I tried solving it but couldn't quite get the grasp of it.
It's solved. Thank you.
AI: Hint: Assuming you mean $\fra... |
H: Find coordinates of a point Q on the graph $\sin (x) + \cos (y) = 0.5$ given that the gradient of its tangent is perpendicular to point P.
Note:
Point $P$ is on the $y$-axis and above the $x$-axis
$\frac{-\pi}{6}\le x \le\frac{7\pi}{6}$
$\frac{-2\pi}{3}\le y\le\frac{2\pi}{3}$
What I have done so far:
Solving for $P... |
H: Parametric representation of the intersection of spheres
Goal:
I am trying to find the curve of intersection of two spheres.
$\begin{align*}x^2+y^2+z^2 &= 9 \\ (x-3)^2+y^2+(z-1)^2 &= 4 \end{align*}$
What I have done:
One of the ways of achieving this is to do the following.
Eliminate one variable, in this case $y$... |
H: Confused about cyclic goups
The cyclic group $\mathbb Z_6$ has a subgroup of order $3$ so I can deduce that $\mathbb Z_3$ is a subgroup of $\mathbb Z_6$. On one hand, it seems false to me cause the group operations of the above groups are not the same. On the other hand, the subgroup of order $3$ is isomorphic to $... |
H: if the lcm is simply the product, then the integers are pairwise prime
I am trying to prove that
let $n_1,\ldots,n_k \in \Bbb Z\setminus\{0\}$. then $\gcd(n_i,n_j)=1 \forall i\neq j$ iff $\operatorname{lcm}(n_1,\ldots,n_k)=n_1\cdots n_k$
I can prove "$\Rightarrow$" this direction by the fact that $\gcd(n_1,n_1)\o... |
H: Problem related with semicircles on sides of triangle and common tangents through semicircles.
On the sides $ BC,CA,AB $ of a triangle $ABC$ semicircles $c_1,c_2,c_3$ are described externally.
If $t_1,t_2,t_3$ are the lengths of common tangents of $c_2,c_3;\;c_3,c_1$ and $c_1,c_2$ then $t_1t_2t_3$ in terms of semi... |
H: Showing the equation of a circle given diameter and Euclidean geometry
If $AB$ is the diameter of a circle and $P$ another point on the circumference, Euclidean geometry tells us that angle $APB = 90˚$. Use this fact to show that the equation of a circle whose diameter has endpoints $A(x_1,y_1)$ and $B(x_2,y_2)$ is... |
H: How to solve $x^{x^{x^x}} = 1/3^{\sqrt{48}}$
How to solve
$$x^{x^{x^x}} = \frac{1}{3^{\sqrt{48}}}$$
Attempt :
Let $x^{x^{\cdots}} = y$
$$\begin{align}
x^y &=y\\
y\ln(x) &= \ln(y)\\
-\ln(x) &= -\ln(y)e^{-\ln(y)}\\
-\ln(y) &= W(-\ln(x))\\
y &= e^{-W(-\ln(x))}
\end{align}$$
I'll stop this. Am i doing this right? I kno... |
H: Law of large numbers applicability
Can you define a series of $(X_n)$ so that for every such n, we get $X_n\sim\mathsf{Geo}\left(\frac{1}{7n}\right)$ and for a constant $c\in\mathbb{R}$, such that
$$\displaystyle{\mathbb{P}\left[ \lim_{n \to \infty}\frac{X_n}{n} = c \right] = 1}$$
if so, find $c$.
I believe ther... |
H: How many ways of a cup frozen yogurt can be dressed up?
Three of $16$ toppings can be selected for dressing up a cup of frozen yogurt. How many ways of a cup of frozen yogurt can be dressed up?
Select one:
a. $3360$
b. $560$
c. $48$
d. $45$
AI: Note: this is assuming that the order does matter
You have $16$ possi... |
H: Find the sum $\sum_{n=0}^{49} \sin((2n+1)x) $
This was an exercise in a chapter of a textbook on product to sum and sum to product trigonometric identities. The following question was asked with the given hint:
$$\sum_{n=0}^{49} \sin((2n+1)x) $$
Hint: multiply this sum by $2\sin(x)$
My attempt
$$\sum_{n=0}^{49} \si... |
H: Understanding the $\gamma$ rate of the SIR model
Relating the SIER model, I am trying to understand the intuition of the $\gamma$ paramater. This paramater is the recovery rate. $\gamma$ is fixed and biologically determined. Some authors conider for USA:
$$\gamma = \frac{1}{18}$$
and then asserts that $\frac{1}{\ga... |
H: Choosing pairs and singles out of $n$ students
Given $n$ different students, find the number of ways to divide them to $k$ pairs, and $n-2k$ "singles". No order in pairs/singles.
So my idea was to first choose $n-2k$ singles, then out of all possible pairs in the $2k$ students who are now determined, choose $k$ p... |
H: Show that the derived set of $A$ in a subspace $(Y, \mathcal{O_Y})$ is equal to $A^d \cap Y$.
I am reading "Set Theory and General Topology" by Fuichi Uchida.
There is the following problem in this book:
Let $(X, \mathcal{O})$ be a topological space.
Let $(Y, \mathcal{O_Y})$ be a subspace of $(X, \mathcal{O})$.
Le... |
H: Does the center of a perfect group not contain all elements of prime order?
Let $G$ be a finite perfect group (i.e. $G=G'$) and $Z(G)$ be its center.
I don't know whether this statement is correct:
There exists an element $x$ of prime order such that $x\notin Z(G)$.
A quick check on CFSG gives that this holds for... |
H: Prove that $\frac{1}{a_1 + 1} + \frac{1}{a_2 + 1} + \dots + \frac{1}{a_n + 1} < 2$ for all $n \ge 1.$
The sequence $a_n$ is defined by $a_1 = \frac{1}{2}$ and $a_n = a_{n - 1}^2 + a_{n - 1}$ for $n \ge 2.$
Prove that $\frac{1}{a_1 + 1} + \frac{1}{a_2 + 1} + \dots + \frac{1}{a_n + 1} < 2$ for all $n \ge 1.$
AI: You ... |
H: Expressing $\cos(\varphi)$ using $ z=e^{i \varphi} $
I need to express $ \cos(\varphi) $ with $z = e^{i \varphi}$ in order to use the Cauchy integral formula on the following integral:
$ \int^{2 \pi}_0 \frac{1}{3+2\cos(\varphi)} \,d \varphi $
I got:
$ \int_{|z|=1} \frac{e^{-i \varphi}}{3+2\cos(\varphi)} e^{i \varph... |
H: If $\sum_{r=0}^{n-1}\log _2\left(\frac{r+2}{r+1}\right)= \prod_{r = 10}^{99}\log _r(r+1)$, then find $n$.
If \begin{align}\sum_{r=0}^{n-1}\log _2\left(\frac{r+2}{r+1}\right) = \prod_{r = 10}^{99}\log _r(r+1).\end{align}
then find $n$.
I found this question in my 12th grade textbook and I just can't wrap my head ... |
H: Convex set contains line segment between that connects its interior and closure is contained in its interior
In this question, one of the answer (by Dimitris) uses the following lemma.
Lemma. Let $A\subset \mathbb{R}^N$ be a convex set. Suppose $\text{Int}A \ne \emptyset$. If $x \in \text{Int} A$ and $y \in \text{C... |
H: How to evaluate $\int x^5 (1+x^2)^{\frac{2}{3}}\ dx$?
I am trying to evaluate $\int x^5 (1+x^2)^{\frac{2}{3}} dx$
This is apparently a binomial integral of the form $\int x^m (a+bx^k)^ndx$. Therefore, we can use Euler's substitutions in order to evaluate it. Since $\dfrac{m+1}{k} = \dfrac{5+1}{2} = 3 \in \mathbb{Z... |
H: Find the values of α and β for which this series converges.
The given series is,
$$\sum\limits_{n\geq 1}(\sqrt{n+1}-\sqrt{n})^{\alpha}(\ln(1+1/n))^{\beta}$$
With $\alpha, \beta \in \mathbb{R}$.
I don't know how to begin, noreven which criterion use to find the values of $\alpha$ and $\beta$ for which this series co... |
H: Proving a graph doesn't have a perfect matching
Consider the following graph:
Find a perfect matching or prove one doesn't exist.
I don't think a perfect matching exists here, as the vertices $a_2, a_3$ and $a_4$ are problematic to us, but I'm having some trouble proving this. Using Hall's theorem, we can prove ... |
H: Given $dy/dx=-x/y$, how can I solve $d^2 y / dx^2$
The part that confuses me is the square being next to the $d$ vs being to the variable. My intuition tells me $d^2x$ is equivalent to $(dx)^2$ and $dx^2$ should be $d(x^2)$ (if that notation is valid)
AI: We have $$\frac{dy}{dx} = -\frac xy \\ \frac{d^2y}{dx^2} = \... |
H: How do I find the indefinite integral for $\int \frac{6}{2x-x^2}dx$?
How can I integrate this $\int\frac{6}{2x-x^2}dx$ ?
I know I have to integrate it using logarithms, I just don't know how to do this one.
Can someone help me out?
AI: HINT:
Apply partial fraction decomposition method.
$$\int\frac{6}{2x-x^2}dx=3\in... |
H: Is $\limsup\sqrt[n]{|a_{n+1}|}=\limsup\sqrt[n]{|a_n|}$?
Let $\{a_n\}_{n=1}^\infty$ be an arbitrary sequence of complex numbers. Does the equality
\begin{equation}
\limsup\sqrt[n]{|a_{n+1}|}=\limsup\sqrt[n]{|a_n|}
\end{equation}
hold?
I' m sure that this is the case, and it may seem a silly question, but it has been... |
H: why $\int_{0}^ {x} = \int_{\frac{-1}{2}}^{y}$ and $\int_{0}^{x} = \int_{y}^{1} ?$
I have some confusion in integration . My confusion marked in red and green circle as given below
Im not getting why $$\int_{0}^ {x} = \int_{\frac{-1}{2}}^{y}$$ and $$\int_{0}^{x} = \int_{y}^{1} $$ ?
Im not getting how its derive ?
... |
H: Open subgroup of ring of ring of integers
I am trying to understand following lemma from Milne's Class Field Theory: https://www.jmilne.org/math/CourseNotes/CFT.pdf#X.3.2.3 (the link will take you directly to the said lemma)
Let $L$ be a finite Galois extension of $K$ with Galois group $G$. Then there
exists an o... |
H: Relation between measure and outer measure
Let $(S, \Sigma, \mu)$ be a measure space and $\mu^*$ be an outer measure on $P(S)$ defined by $$\mu^*(A) = \inf \{\sum_{i = 0}^\infty \mu(E_i) \ | \ (E_i)_{i = 0}^\infty \textrm{ is a measurable cover of }A \}$$
I want to prove that for all $A \subseteq S$, there is a mea... |
H: If $T:(\mathbb{R}^2,\|\cdot\|_p) \to (\mathbb{R}^2,\|\cdot\|_q)$ is an onto linear isometry, then must it be $p=q$?
Question: Let $p,q\in [1,\infty)$ and suppose that that $T:(\mathbb{R}^2,\|\cdot\|_p) \to (\mathbb{R}^2,\|\cdot\|_q)$ is an onto linear isometry. Must it be $p=q$?
I think it is true as isometry prese... |
H: Integrating a function where the denominator is the square root of a second order polynomial
Below is a problem I did. The answer I got differed from the book by a factor of $2$. An online calculator get the book's answer. I would like to know where I went wrong.
Problem:
Evalaute the following integral:
$$ \int \f... |
H: How to solve $(1-x^2)y''-2xy'+2y=0$
I have not taken a course in differential equations but I decided to try and tackle this question I saw and solve for the general solution because why not. That said, I have some observations about the differential equation $(1-x^2)y'' - 2xy'+2y=0$.
$y=Ax$ is a solution to the eq... |
H: Prove that $\bigcap\mathcal H\subseteq(\bigcap\mathcal F)\cup(\bigcap\mathcal G)$.
Not a duplicate of
Prove that $∩\mathcal H ⊆ (∩\mathcal F) ∪ (∩\mathcal G)$.
This is exercise $3.5.17$ from the book How to Prove it by Velleman $($$2^{nd}$ edition$)$:
Suppose $\mathcal F$, $\mathcal G$, and $\mathcal H$ are nonempt... |
H: Does a function which is oscillating have to have not-continuous derivative?
If $f(x)$ is a differentiable function which is not $0$ everywhere and has the property that around any interval around $0$, $f$ is neither fully positive or negative. Then it can be proven that $f(0)=0$.
An example of such a function is... |
H: Why do I get Lie(matrix Lie group)=$\mathfrak{gl}_n(\mathbb{R})$?
There is something I don't understand about Matrix Lie groups. They are defined as closed subgroups of $GL_n(\mathbb{R})$ but we can show that a closed subgroup of a topological group is opened as well. So if $H$ is a Lie matrix group, $H$ is opened ... |
H: Triple negation implies double negation elimination?
I have a question about intuitionistic logic regarding the relationship between the triple negation elimination rule, i.e. $\neg\neg\neg A\leftrightarrow \neg A$, and the double negation elimination. We know from Brouwer (1925) that $\neg\neg\neg A\leftrightarrow... |
H: Finding area of $D =\{(x,y): x^2+y^2 \geq 1 , y \geq x-1 , y \leq 1, x \geq 0\}$
I want to calculate $$\iint_{D} dx \,dy$$ given that $$ D =\left\{(x,y): x^2+y^2 \geq 1 , y \geq x-1 , y \leq 1, x \geq 0\right\}$$
My attempt : I used polar cordinates $r,\theta$ such that $x =r \cos \theta , y= r \sin \theta$ and fr... |
H: How to simplify Kronecker delta with einstein summation?
I am trying to proof a vector identity. I have to prove the following;
I am bit confused how to simplify the following part..
$$\delta_{il} \delta_{jm} x_{j}y_{l}z_{m}$$
Any input is appreciated.
AI: Remember Kronecker $\delta$ "is" the identity matrix, so c... |
H: How do I prove a set is not simply connected?
The set in question is the unit open disc $U$ with the origin omitted. I know that this set is not simply connected as the path, say the circle centred at the origin with a radius of 1/2 is not homotopic to any point in $U$. However, how do I mathematically express this... |
H: Convex solution set
Let $y, z, a \in \mathbb{R}$. Is it true, that the solutions $x \in \mathbb{R}$ of the inequality
$$
\lvert x - y \rvert - \lvert x - z \rvert \leq a
$$
form a convex set (i.e. an interval)? I don't even know if this is true but I have not found a counterexample yet.
I tried to prove this by cho... |
H: How do I find all functions $F$ with $F(x_1) − F(x_2) \le (x_1 − x_2)^2$ for all $x_1, x_2$?
In calculus class we were given this so-called "coffin problem" originally from Moscow State University.
Find all real functions $F(x)$, having the property that for any $x_1$ and $x_2$ the
following inequality holds:
$$F(x... |
H: Product of two absolutely continuous measures
Suppose we have two probability measures on $\mathbb{R^{n}}$, $Q$ and $R$ absolutely continuous w.r.t $P$ and $\frac{dQ}{dP}$,$\frac{dR}{dP}$ is given. Is there a way to define multiplication for instance via Cauchy product such that we get a new absolutely continuous m... |
H: Does $\int_0^{\pi \over 2} \lfloor \tan(x) \rfloor\, dx$ converge?
My book follows the following method
Let
$$I=\int_0^{\pi \over 2} \lfloor \tan(x) \rfloor\, dx.$$
Then using King's rule $$\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx$$ we have
$$I=\int_0^{\pi \over 2} \lfloor \tan(\pi- x)\rfloor\, dx=\int_0^{\p... |
H: Prove, by contradiction, that if a and b are nonzero real numbers, and $a<\frac{1}{a}
Prove, by contradiction, that if a and b are nonzero real numbers, and $a<1/a<b<1/b$ then $a<−1$.
I understand that the first step is to assume that $a<1/a<b<1/b$ is true. Therefore, the hypotheses makes up:
$a ≠ 0$
$b ≠ 0$
$a<1/a... |
H: $\pi(v)=x+y$ implies $v=v_1+v_2,\; \pi(v_1)=x,\pi(v_2)=y$
Let $V$ be a vector space over some field $k$, let $U\subset V$ be a subspace of $V$ and consider the quotient space $V/U$, along with the projection $\pi:V\to V/U,\, x\mapsto x+U$.
Does it hold that $\pi(v)=x+y$ implies $v=v_1+v_2$ with $ \pi(v_1)=x,\pi(v_2... |
H: Does my claim work for a set?
I don't have enough reputation to comment on this neat answer so I ask here for some hints/answers. I take the set $A = \{0,1,2,3\}$ and so according to this answer, $A \sim \mathbb{Z}_4$. My question is: does that certain bijection allow to write the subset $\{0,2\}$ of $A$ is isomorp... |
H: Will optimal point of a convex function $f(x)$ under linear constraint lie on boundary?
I have a function $f(x)$ that is positive definite quadratic function. I have linear constraints , then will the optimal lie on boundary ?
My answer that I feel is "No" it will not lay at the boundary. But I am unable to give a ... |
H: Definite integral of $\int_{-2}^{2} \frac{5}{(x^2+4)^2}\,dx$ using the substitution of $x=2\tanθ$.
Can someone help with the integral $\int_{-2}^{2} \frac{5}{(x^2+4)^2}\,dx$?
I'm supposed to find the definite integral for this using the substitution $x=2\tanθ$.
This is what I've done so far:
$$\longrightarrow \frac... |
H: If $kx^2-4x+3k+1>0$ for at least one $x>0$ and if $k\in S$, find $S$
Options ; $A)~ (1,\infty)~~ B)~(0,\infty)~~C)~(-1,\infty)~~D)~(-\frac 14 , \infty)$
Obviously
$$16-4(k)(3k+1)<0$$
$$k\in (-\infty, -\frac 43)\cup (1,\infty)$$
And also $k>0$ so the answer should be A)
The answer is, however, A, B, D
I think it h... |
H: Prove that $\phi: G / F \rightarrow \operatorname{Sym}(X)$ is a monomorphism
I'm doing this exercise 11 in textbook Algebra by Saunders MacLane and Garrett Birkhoff.
If $G$ acts on $X$, and $F$ consists of those $g \in G$ fixing every $x \in X$, prove that $F \trianglelefteq G$. If $p: G \rightarrow G / F$ is the... |
H: How I can describe $v^{\perp}$ as $Rw$?
Let $v=(\alpha , \beta)\in \mathbb{R}^2$ be nonzero. Describe $v^{\perp}$ as R$w$ for a suitable $w$.
I am thinking to consider the orthogonal complement, and the plane must have an equation of the form $ax+by+cz=0$, This one should be perpendicular to the basis vectors.
Bu... |
H: Proving continuity at the end points of the extension of a continuous function
(Baby Rudin Chapter 4 Exercise 5)
I want to follow up on my previous question. (Previously, I asked if I needed to even prove that $g$ is continuous on the endpoints of $E$. Here, I want to ask about the actual methodology of proving tha... |
H: An Inequality in von Neumann algebras
In Section $9.9$ of the book 'Lectures on von Neumann algebras' by Strătilă and Zsidó, I am not getting how they get the following inequality:
Given a positive self-adjoint linear operator $A$ in the Hilbert space $\mathcal{H}$, we have $a=(1+A)^{-1}\in \mathcal{B}(\mathcal{H})... |
H: Find a basis $ {\{b_1,\cdots, b_n}\} $ of $ \mathbb{C}^n $ such that $ \langle b_j, b_k \rangle = 1 $ whenever $ j \neq k $
Find a basis $ {\{b_1,\cdots, b_n}\} $ of $ \mathbb{C}^n $ such that $ \langle b_j, b_k \rangle = 1 $ whenever $ j \neq k $ where $\left\langle (x_1,\cdots,x_n),(y_1,\cdots,y_n)\right\rangle:=... |
H: Eigenvectors matrix multiplied by its transpose $\boldsymbol{\chi} \boldsymbol{\chi}^T $
Let $V$ be the set of datapoint and assume that each point can be represented by a vertex. Then, given a similarity matrix $ \mathbf{M}$, we define a graph $G = (V, \mathbf{M})$ generated using $k$-nearest neighbors.
Let $\m... |
H: Martingale constructed from a random walk
I am trying to solve this problem for a while now but I am not coming to a solution. Could anyone help or give me a hint?
Let $S_n=\sum_{i=1}^{n}X_i$ be a random walk on $\mathbb{Z}$ with $S_0=0$, $\mathbb{P}(X_i=1)=\frac{2}{3}$, $\mathbb{P}(X_i=-1)=\frac{1}{3}$. For whic... |
H: Sequence in $\Bbb Q$ that converges to $0$ under $d_E$ and to $1$ under $d_2$
I am looking for a sequence in $\Bbb Q$ that converges to $0$ under the Euclidian metric and to $1$ under the 2-adic metric $d_2$. This metric is defined by the 2-adic absolute value as $d_2(x, y) = |x-y|_2 = 2^{-\text{ord}_2(x-y)}$ for ... |
H: why this operator $T$ is always diagonalizable?
Let $V = \mathbb{R}^3$ and $B=(v_1,v_2,v_3)$ ordered basis for $V$
Let $T:V \to V$ linear operator and given the representation matrix with respect to the basis $B$ $$[T]_B^B = {\left[\begin{array}{ccc} 3 & 0 & 8 \\ 0 & 0 & -1 \\ 8 & -1 & 5 \end{array}\right]}.$$
Why... |
H: Interpretation and use of the logarithmic scale for high school students
Often when we discuss on the logarithms in high school we also talk about a scale called logarithmic.
In the he logarithmic scale: the distance from $1$ to $2$ is the same as the distance from $2$ to $4$, or from $4$ to $8$ as the image below... |
H: About the hypotheses of Schauder Theorem
I know that Schauder Theorem says: $T: E \to F$ is an compact operator iff $T^{*}: F^{*} \to E^{*}$ is an compact operator.
My doubt is: what are the hypotheses about $E$ and $F$? Is it enough that they are just normed spaces or do they need to be Banach spaces? Or $E$ norme... |
H: Number of ternary strings of length n such that number of 0s is greater than or equal to number of occurrences of any other digit
I understand how to count this for a binary string of a fixed length using combinations, so I think the way to go with this problem is to use an exponential generating function for each ... |
H: What is the meaning of a probability distribution parameter?
Named probability distributions are often explicitly presented as having a specific number of parameters. For example, even though the Poisson distribution PMF equation $p_K (k) = \frac{\lambda^k}{k!e^\lambda}$ has two variables, $k$ and $\lambda$, only ... |
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