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H: How to obtain the Klein bottle as a product of manifolds?
I know the Klein bottle $K$ is a fiber bundle over $S^1$, but my question is: is it possible to find a manifold $M$ such that $K = S^1 \times M$ without the need to take an equivalence relation afterwards?
My thoughts are, maybe something like taking $M$ as ... |
H: The function $f(x)=|x|^p,$ $x\in \mathbb{R}^{n}$ is strictly convex for $p>1$?
Let $p>1$. In the paper [1] below, it says that The function $f(x)=|x|^p,$ $x\in \mathbb{R}^{n}$ is strictly convex.
I would like to prove that. By definition, We need to show that the Hessian matrix
$H=\left(\frac{\partial^2 f}{\partial... |
H: Correct my intuition: every Galois group is $S_n$, and other obviously incorrect statements
(I hope that this question is acceptable and within the rules of math.stackexchange. If not, mods should edit at will and let me know if this question must be broken into several different questions. I ask these all together... |
H: What should I do if I can't see through abstractions in complex shapes
I am recently solving Problems & Solutions in Euclidean Geometry. I have learnt a lot of facts regarding to parallelogram, triangles, areas etc etc and solve quite a lot of questions. For the most part of questions, I can see through it either i... |
H: Prove $T \mathrm{ker}(A T) = \mathrm{ker}(A)$.
Let $T \in \mathbb{R}^{n\times n}$ (not necessarily invertible), $A \in \mathbb{R}^{m\times n}$. Prove the following equation
$$T \mathrm{ker}(A T) = \mathrm{ker}(A),$$
where $\mathrm{ker}(\cdot)$ gives the null space.
Here is my proof:
Let $\mathrm{ker}(A T) = \{x\col... |
H: Wronskian of two linearly independent differential functions. Show $c$ in [a,b] such that $g(c) = 0$ exists
Let $f,g: [a,b] → R$, two differential functions and suppose $f(a) = f(b) = 0$.
If $W(f,g): [a,b] → R$ and $W(f,g)(x) = f(x)g'(x) - g(x)f'(x)$ doesn't equal 0 for all $x$ in $[a,b]$, show that a $c$ in $[a,b]... |
H: Different approaches in evaluating the limit $\frac{(x^3+y^3)}{(x^2-y^2)}$ when $(x,y)\to(0,0)$.
Note that this question has been previously asked here. I understood the solutions available there but I have two different approaches to this problem, I'm not sure whether they are correct.
I need to know whether both ... |
H: Show that there exists a subsequence $\{E_{n_k}\}$ of $\{E_n\}$ such that $m(\cap_{k=1}^\infty E_{n_k})>\epsilon$ under these conditions....
Question: Let $\{E_n\}$ be a sequence of nonempty Lebesgue measurable subsets of $[0,1]$ such that $\lim_{n\rightarrow\infty}m(E_n)=1$. Show that for each $0<\epsilon<1$ ther... |
H: Does Riemann integrability implies integral mean value theorem?
We know that if $f$ is continuous on [a,b] and $f:[a,b] \to \mathbb{R}$, then there exists $c \in [a,b]$ with $f(c)(a-b) = \int_a^bf(x)dx$
If we change ''f is continuous on [a,b]'' to ''f is Riemann integrable'', does the mean value theorem for integra... |
H: Changing limit and derivative operator
I was trying to solve the following problem:
Let $f:\mathbb R\rightarrow \mathbb R$ be a differentiable function such that
$\lim_{x \to \infty} f(x)=1$ and
$\lim_{x \to \infty}f'(x)=a$.
Then find the value of $a$.
My brother who is not a pure math student suggested that take d... |
H: Possibly a variation of the increment theorem for functions of multiple variables
Suppose that $F:\mathbb R^n\to\mathbb R$ is $C^\infty$. I'd like to prove that $\forall a=(a_1,\ldots,a_n)\in\mathbb R^n$, there exist $C^\infty$ functions $G_i,i=1\ldots,n,$ such that $\forall x=(x_1,\ldots,x_n)\in\mathbb R^n$, we ha... |
H: the slower grape crusher
Two Grape Crushers take 4 days to crush certain amount of grapes.If one of them crushed half the grapes and the other crushed other half , then they complete the job in 9 days. How many days will it take slower crusher to do the job alone?
my solution :
the time taken when they both do 1/2 ... |
H: counting words with a condition
Suppose we are given a sequence $x_1x_2x_3x_4x_5x_6$ where the $x_i$ are digits 0 to 9, and we want to know how many of them do we have that satisfy $x_1<x_2<x_3<x_4<x_5<x_6$?
$discussion:$
Notice that $x_1$ can only be a number betwwen $0$ to $4$ so if $x_1=0$, then we reduce our pr... |
H: Is there a finite group which has two isomorphic maximal subgroups such that no automorphism can map one to the other?
Let $G$ be a finite group. Suppose $H$ and $K$ are two isomorphic maximal subgroups of $G$, then can we claim that there must be an automorphism $\alpha\in {\rm Aut}(G)$ such that $\alpha(H)=K$?
If... |
H: Interesting question regarding evolution of algorithm under certain conditions
On a random string of six digits containing numbers of the set $S=\{1,2,3,4,5,6\}$, repeat the following operation:
If $k$ is the first number of the string, then reverse the order of first $k$ numbers of the string.
For example: $342561... |
H: Bernoulli's Inequality for $-1 \leq x\leq 0$
My original goal was to prove that
$$\lim_{x\to 0}\frac{e^x-1}{x}=1$$
using the squeeze theorem as we haven't seen differentiability yet and thus I cannot use arguments such as Taylor series nor Bernoulli's theorem, nor can I use induction. For that I wanted to find a lo... |
H: How is $\mathbb{R}^n$ a compact subset of $\mathcal{H}(\mathbb{R}^n)$ (the set of all compact subsets of $\mathbb{R}^n$)?
I'm reading Essential Real Analysis by Michael Field. There's a definition of a metric space $(\mathcal{H}(\mathbb{R}^n),h)$, in which the set $\mathcal{H}(\mathbb{R}^n)$ is the set of all non-e... |
H: Proof for an adapted version of Gauss' Lemma
I am self learning abstract algebra. Today I watched a youtube video explaining the proof for an adapted version of Gauss' Lemma
In his proof, he claims that for any polynomial $f = gh \in \mathbb{Z}[x]$ and for any prime $p$, where $g,h \in \mathbb{Z}[x].$ If $p$ divide... |
H: A doubt about $\int_{0}^{1} f(x)~ \left(\int_0^x |f(t)| dt \right)~ dx=7$, if $\int_{0}^{1}f(x) dx=2, \int_{0}^{1} |f(x)| dx=4$
A question gives $f(x)$ as continuous and $f'(x)>0$ for all real values
of $x$ such that $\int_{0}^{1} f(x) dx=2, \int_{0}^{1} |f(x)| dx=4.$
So it is good to conclude that $f(x)=0$ will ha... |
H: Equivalent definition of Cohen-Macaulay Ring
We suppose all rings are commutative and unital. The most general defition for Cohen-Macaulayness goes as follows: A Noetherian local ring $R$ is $\textit{Cohen-Macaulay}$ if its depth is equal to its Krull dimenion. More generally a ring is called Cohen–Macaulay if it ... |
H: Product of sines when the angles form a sequence
I am wondering how to find the following value of $x$
$$x=\prod_{n=1}^{9}\sin\left(\frac{n\pi}{10}\right)$$
I notice that it has something to do with the de moivre's theorem as the angles are root angles of $1^\frac{1}{10}$
To my surprise, the value of the above prod... |
H: Product decomposition of a nonnegative matrix
Suppose $A\neq 0$ is a nonnegative matrix which can be decomposed as $BC$ with $B\neq 0$ nonnegative and $C$ orthogonal. Then, is $C$ also nonnegative?
I think yes. The case when $B$ is invertible is trivial. But, if $B$ is not invertible, it does not seem obvious. Any ... |
H: Minimum spanning tree formulation
I'm writing up a report for a solution of an energy grid problem for school, and after reading the CS-book multiple times (and a bit of googling), I can't seem to find the mathematical definition of the MST-problem. Is this correct?
Problem: Given an undirected weighted graph $G(V,... |
H: Show that the determinant of a matrix is nonzero
Suppose $u,v,w \in \mathbb{Q}$ with $u,v,w \neq 0$. Show that the
determinant of the following matrix is nonzero.
$$M = \begin{bmatrix} u & 2w & 2v \\ v & u & 2w \\ w & v & u \end{bmatrix}$$
Hint: Argue by contradiction, reduce to the case when $u,v,w$ are integers ... |
H: Proof that an equation is irrational
I hope you're keeping safe and well. I stumbled across this problem and wondered whether you could help.
Show that $\left(a+\sqrt b\right)\left(a-\sqrt b\right)^3$ is irrational if $a$ and $b$ are NOT square numbers.
Thank you so much for your help in advance.
Pac-Man
AI: Let'... |
H: Extracting the diagonal terms of a square matrix.
For a given square matrix $A\in\mathbb{R}^{m\times m}$ does there exist a matrix $B\in\mathbb{R}^{m\times m}$ such that for the product $C:=AB$ we have $C_{ii}=A_{ii},$ $1\leq i\leq m$, and $C_{ij}=0$ if $i\neq j$ ?
I think the question can be also restated in the f... |
H: An invertible linear map over C has a square root (Linear Algebra Done Right 8.33)
$T \in \mathcal{L}(V)$ is a complex finite-dim linear operator. 8.33 proves that if $T$ is invertible, it must have a square root.
It shows that $T |_{G(\lambda_i, T)} = \lambda_i (I + \frac{N_i}{\lambda_i})$ has a square root, where... |
H: Upper bound for the sinc function
Show that there exists a constant $0<c<1$ such that
$$
\frac{\sin x}{x} < c,\quad\textrm{for all }x\ge1.
$$
--Context--
In Probability Theory Lévy's Theorem is crucial to uncover probability measures from certain functions which are obtained as limits of characteristic functions... |
H: Topology question about a special subset in $\mathbb R^2$
Problem Statement:
Let $X = (\bigcup \limits_{n \in \mathbb N} \{\frac{1}{n}\} \times [0,1] ) \cup \{(0,0),(0,1)\}$ have a subspace topology as a subspace of $\mathbb R^2$. For any separation $U$ and $V$ of $X$, if $(0, 0) \in U$, then $(0, 1) \in U$ as wel... |
H: Question on poset of positive semi-definite matrices
Let $\Omega$ be a subset of the partially ordered set (poset) of $n\times n$ positive semi-definite matrices. I know that $\inf \Omega\in \bar{\Omega}$, where $\bar{\Omega}$ denotes the closure of $\Omega.$
If $\inf \Omega=X$, can I say that
\begin{array}{ll} \te... |
H: Exterior power on short exact sequence of modules with free middle term
Let $(R,\mathfrak m,k)$ be a Noetherian local ring. For a finitely generated $R$-module $M$, let $\wedge^j(M)$ denote its $j$-th exterior power. Recall that $\wedge^j(R^{\oplus j})\cong R,\forall j\ge 1$.
Now suppose we have an exact sequence o... |
H: What do you call an L1 regularization term involving a matrix vector product
Often times I have difficulties finding certain things when I do not use the correct terminology. Or there are many different terms for the same thing.
I would like to know what you would call the following optimization problem, so I have ... |
H: Why $\omega<2^{\omega}$ but $\omega^{\omega}\sim\omega$?
I understand the Cantor's proof for why $S<2^S$, but also we know that ordinals $\omega\sim\omega^2\sim\omega^3...$. This approaches to $\omega^{\omega}$, what should be at least not less than $2^{\omega}$. What do I miss with this logic?
AI: You are confusin... |
H: Sign of the line integral ($\int_{\vert z\vert=1} {1 \over z^2} \tan({\pi \over z}) dz$)
Find the value of the $$\int_{\vert z\vert=1} {1 \over z^2} \tan\left({\pi \over z}\right) dz$$
When we substitute $\omega = {1 \over z}$, then $d\omega = - {1 \over z^2}dz$, hence $\int_{\vert z\vert=1} {1 \over z^2} \tan({\p... |
H: Is $\mathbb Q \times \{0\}$ an integral domain?
Is $\mathbb Q \times \{0\}$ an integral domain?
I understand that $\mathbb Q \times \{0\}$ is a commutative ring with unity.
But there was no clear proof that it has no zero divisor.
How do I prove $\mathbb Q \times \{0\}$ is a zero divisor, if it's true that $\mathbb... |
H: Indefinite integral of $\frac{\sec^2x}{(\sec x+\tan x)^\frac{9}{2}}$
$$\frac{\sec^2x}{(\sec x+\tan x)^\frac{9}{2}}$$
My approach:
Since it is easy to evaluate $\int{\sec^2x}$ , integration by parts seems like a viable option.
Let $$I_n=\int{\frac{\sec^2x}{(\sec x+\tan x)^\frac{9}{2}}}$$
$$I_n=\frac{\tan x}{(\sec x+... |
H: Evaluate $\lim_{h\to 0}\frac{1}{h^2}\begin{vmatrix}\tan x&\tan(x+h)&\tan(x+2h)\\\tan(x+2h)&\tan x&\tan(x+h)\\\tan(x+h)&\tan(x+2h)&\tan x\end{vmatrix}$
Evaluate
$$
\lim_{h\to 0}\frac{\Delta}{h^2}=\lim_{h\to 0}\frac{1}{h^2}\begin{vmatrix}
\tan x&\tan(x+h)&\tan(x+2h)\\
\tan(x+2h)&\tan x&\tan(x+h)\\
\tan(x+h)&\tan(x+2h... |
H: 2-stable subsets of groups
In this post, multiplication of subsets of a group is defined by
$$ST= \{st| s\in S, t \in T\}$$
A subset of a group is called $n$-stable if there exists a natural number $n$ s.t. $$S = S\cdot \overbrace{S\cdot S\cdot S...}^{n \text{ times}}$$
The minimal $n$ for which it’s true determin... |
H: Uniqueness of the derivative of a differentiable function on a non-open set
If $E_i$ is a $\mathbb R$-Banach space and $\Omega_1\subseteq E_1$, then $f:\Omega_1\to E_2$ is called $C^1$-differentiable at $x_1\in\Omega_1$ if $$\left.f\right|_{O_1\:\cap\:\Omega_1}=\left.\tilde f\right|_{O_1\:\cap\:\Omega_1}$$ for some... |
H: Polynomial olympiad problem
Let $p(x)$ be a monic polynomial of degree four with distinct integer roots $a, b, c$ and $d$. If $p(r)=4$ for some integer $r$, prove that $r=\frac{1}{4}(a+b+c+d)$
My only idea was to let $p(x)=(x-a)(x-b)(x-c)(x-d)$, so that:
$4=(r-a)(r-b)(r-c)(r-d)$. But the casework here, looking for ... |
H: To prove that an operation is well-defined in modular arithmetic
I started to study the relation of congruence modulo n and a big important question came to me. In the book Poofs and Fundamentals, by Ethan D. Bloch, we have the definition:
Definition: Let $n \in \mathbb{N}$. Define operations $+$ and $\cdot$ on $\... |
H: interior and closure in metric spaces
let say we have $(\ell^{1}(\Bbb{N}),d_{1})$ as a metric space with $d_{1}((x_{n})_{n},(y_{n})_{n})=\sum_{n=0}^{\infty}|x_{n}-y_{n}|$. If $$D=\left\{x \in \ell^{1}(\Bbb{N}) \,\,\Big|\, \sum_{n=1}^\infty n|x_{n}|<\infty \right\}$$
I'm looking for the interior of $D$ and the closu... |
H: What does dimension in integer programing problem mean
In the problem P1 below described in a research paper I am reading, the authors say that the problem P1 below is a three dimensional integer programing programing problem. Can I ask what does 3-dimension means here? does it mean that it has three min/max functi... |
H: convex sets in convex optimization
How to prove that the following set is not convex?
$$M = \left\{ \mathbb{R}^{3}: x_{1}x_{2}x_{3}\le 1,x_{1}+x_{3}\ge 2,x_{1} \ge 0 \right\}$$
Thanks for any help.
I tried to write it down as intersection of two sets $\{x_1x_2x_3 \le 1\}$
and $\{x_1+x_3 \ge 2,x_1 \ge 0\}$. The sec... |
H: Groups of order $252 = 4 \cdot 7 \cdot 9$ are solvable
The goal is to prove that any group of order $252 = 36 \cdot 7$ is solvable, and because I managed to confuse myself, I'm asking here.
Let $G$ be a group of order $252$. By Sylow's Theorems, the number of $7$-Sylow subgroups of $G$ is either $1$ or $36$. If it ... |
H: If $A^2=\mathbb{I} (2\times 2$ identity) then $\mathbb{I} + A$ is invertible only if $A=\mathbb{I}$
I want to show that if $A^2=\mathbb{I}$ ($2\times2$ identity) then $\mathbb{I} + A$ is invertible only if $A=\mathbb{I}$.
I know that $A^2=\mathbb{I}$ means that $A=A^{-1}$
I've started by letting $A=\begin{bmatrix} ... |
H: If $|f'(c)|
We have a derivative function $f$ with for every $c$ element of $\mathbb{R}: |f'(c)|<M$. I tried to prove that
prove that $\displaystyle \left|\int_{0}^{1}f(x)\mathrm{d}x-\frac{1}n \sum_{k=0}^{n-1}f\left(\frac{x}n\right)\right|\leq\frac{M}{n}$.
I really have no idea how to start. I'm trying to use integ... |
H: Topological groups vs regular groups
I know group theory and I'm familiar with the concept and definition of Group.
Today I was reading an article about topology and discoverer the concept of "topological group". I read the definition and immediately the following question came to my mind:
What is the purpose of t... |
H: prove $\left(3, 1+\sqrt{-5}\right)$ is prime ideal of $\mathbb{Z}\left[\sqrt{-5}\right]$
How to prove that $(3, 1+\sqrt{-5})$ is prime ideal of $\mathbb{Z}[\sqrt{-5}]$?
attempt 1: use definition
Consider $a, b, c, d, k_1, k_2 \in \mathbb{Z}$ s.t. $$ac-5bd=3k_1+k_2,\, \, ad+bc=k_2.$$ To prove $\exists j_1, j_2 \in \... |
H: Maps Preserving Arc Connectedness?
We know that continuous functions do not preserve arc-connectedness (for an example, see this question I asked previously). So, the natural question that comes next is - which maps preserve arc-connectedness?
That is, if $X$ is arc connected, and $f:X\to Y$, then what are the weak... |
H: Find the number of other values of n for which $S_{n}$ = r is
Consider an arithmetic progression whose first term is $4$ and
the common difference is $-0.1.$ Let $S_{n}$ stand for the sum of the first
n terms. Suppose r is a number such that $S_{n}$ = r for some n. Then
the number of other values of n for which $S_... |
H: Validation of my proof for proving that the Sorgenfrey Line does not satisfies the second axiom of countability
In an exercice I am asked to prove the following:
Prove that the Sorgenfrey Line does not satisfies the second axiom of contability.
This is my second proof for this exercise because the first one was w... |
H: If $\left(\sqrt2\right)^x + \left(\sqrt3\right)^x = \left(\sqrt{13}\right)^{\frac{x}{2}}$ then the number of values of $x$ is?
If $\left(\sqrt2\right)^x + \left(\sqrt3\right)^x = \left(\sqrt{13}\right)^{\frac{x}{2}}$ then the number of values of $x$ is?
It is an exponent topic question tried squaring method but c... |
H: Notation for a k-partite graph
I am reading Graph Theory by Bondy and Murty. I always see the notation for a complete $k$-partite graph, but what is the notation for a $k$-partite graph? For example, my partite sets are of order $2$, $3$, and $4$, and it is not a complete $k$-partite graph. How can I write this gra... |
H: How do you find the Taylor series of an indefinite integral?
I am given the following problem :-
I know that I have to first find the Taylor series of the polynomial and then integrate each term, however I am having trouble finding the Taylor series before integrating because the derivative of $\frac{(e^t-e)}{(t-1... |
H: How to compute this integral over a small ball?
$$\int_{{\sqrt{x^2+y^2}}<1}x^2dxdy$$
I only know that $$z=(x, y)^T$$ is a small, directional vector.
AI: Not sure why $z$ would help here, but one way would be to do this directly, i.e.
$$
\int_{x=-1}^{x=1} \int_{y = -\sqrt{1-x^2}}^{y = \sqrt{1-x^2}} x^2 dy dx
= \int... |
H: If $x^2+y^2+xy=1$ then find minimum of $x^3y+xy^3+4$
If $x,y \in \mathbb{R}$ and $x^2+y^2+xy=1$ then find the minimum value of $x^3y+xy^3+4$
My Attempt:
$x^3y+xy^3+4$
$\Rightarrow xy(x^2+y^2)+4$
$\Rightarrow xy(1-xy)+4$ (from first equation)
$\Rightarrow xy-(xy)^2+4 =f(x)$
For minimum value, $\frac{df(x)}{dx}=0$... |
H: Counting the number of integers with given restrictions
Question: Consider the numbers $1$ through $99,999$ in their ordinary decimal representations. How many contain exactly one of each of the digits $2, 3, 4, 5$?
Answer: $720$.
Attempt at deriving the answer:
We have two cases: four digit numbers and five digit... |
H: Cannot solve linear system of equations
It would be nice if somebody could find my mistake for the following linear system of equations:
$$
\left\{\begin{array}{rcrcrcr}
-2x & - & 4y & - & z & = & -21
\\
-3x & + & y & + & 2z & = & -14
\\
x & - & 2y & - & z & = & a
\end{array}\right.
$$
It is solvable for all $a \in... |
H: Probability of bit strings
Suppose you pick a bit string of length $10$. Find the probability that the bit string has exactly two $1$'s, given that the string begins with a $1$.
Can someone please explain to me how to do it?
AI: Since you know that the first bit is $1$, you just want to find the probability that ex... |
H: How to efficiently sample edges from a graph in relation to its spanning tree
Consider a connected, unweighted, undirected graph $G$. Let $m$ be the number of edges and $n$ be the number of nodes.
Now consider the following random process. First sample a uniformly random spanning tree of $G$ and then pick an edge f... |
H: Mixed partials on the diagonal
Let $I\subseteq\mathbb{R}$ be an open interval and $\varphi\in C^2(I^2;\mathbb{R})$ be twice continuously-differentiable on $I^2:=I\times I$.
We call $\varphi$ separable if it can be written $\varphi\equiv\varphi(x,y) = \alpha(x) + \beta(y)$ for some $\alpha,\beta: I\rightarrow\mathbb... |
H: Improper integral of periodic function
I was given the following question:
Find all continuous periodic functions $f(x)$ for which the integral $ \int^\infty_0f(x)dx $ converges.
Now, I have a feeling this is only the function $ f(x) = 0$ , yet I have a problem formally proving it.
Any help will be appreciated!
A... |
H: Can I use induction with increments higher than 1?
Say I want to prove:
$a$ is odd $<=>$ $a^2$ is odd
However, instead of proving this in both directions, I want to show that this statement is true for all odd numbers.
Can I use induction with increments of 2 to get only the odd numbers? Does this require additio... |
H: An Application Kolmogorov's Three Series Theorem
I want to prove the following question, which is found in this practice exam:
My attempt so far is as follows - I just can't show that the $\sum E(Y_i)$ converges.
AI: Write $$0 = E X_i = E[ X_i 1_{|X_i| \leq 1}] + E[ X_i 1_{|X_i| > 1}] = E[Y_i] + E[ X_i 1_{|X_i| ... |
H: Determine $\sqrt{1+50\cdot51\cdot52\cdot53}$ without a calculator?
I've tried a lot of things but failed to do it, I've calculated the result inside the square root which is $7027801$ using substitution and factoring but $\sqrt{7027801}$ isn't possible to simplify.
AI: I used the following remarkable identity: $$(a... |
H: toom-cook algorithm matrix G
For this toom-cook algorithm at https://arxiv.org/pdf/1803.10986v1.pdf#page=6 , how do I get the value 4/2 in the matrix G ?
AI: We have
$$G = \begin{bmatrix} x_0^0 N^0 & x_0^1 N^0 & x_0^2 N^0 \\ x_1^0 N^1 & x_1^1 N^1 & x_1^2 N^1 \\ x_2^0 N^2 & x_2^1 N^2 & x_2^2 N^2 \\ x_3^0 N^3 & x_3^... |
H: How did the variance get calculated?
The Elm Tree golf course in Cortland, NY is a par 70 layout with 3 par
fives, 5 par threes, and 10 par fours. Find the mean and variance of par on this
course.
Mean was calculated as follows: Mean = 70/18 = 3.8888
Variance was found to be: second moment = (75 + 160 + 45)/18 = 28... |
H: Banach space of continuous and discontinuous functions on R
The set $C(\mathbb{R})$ of bounded continuous functions on $\mathbb{R}$ is a Banach space when equipped with the sup norm. In my understanding, it just follows from the fact that a Cauchy sequence of continuous functions converges uniformly to a continuous... |
H: Is $(\mathbb{Q},+,\cdot)$ a divisible semifield?
I know that $(\mathbb{Q},+,\cdot)$ is a semifield. But I would like to know that whether it is divisible with respect to the usual addition and multiplication. Please any idea or help? Thanks.
AI: The additive group $(\mathbb{Q},+)$ is divisible: $q \in \mathbb Q, n... |
H: Matrix multiplied by its pseudo-inverse doesn't give the identity matrix. Why?
Using Matlab, I randomly generate matrix $A \in \Bbb C^{2 \times 1}$ and compute its pseudo-inverse $A^{+}$. I notice that $AA^{+} \neq I$, and yet $\mbox{Tr}(AA^{+}) = 1$.
For other sizes it seems like the trace is equal to the smaller ... |
H: Inverse of cumulative distribution function
Let $F(x)$ is the cumulative distribution function and $P(x)$ is the (given) probability distribution function and $X$ is a random variable.
Can anybody please intuitively explain,
Why can the inverse of the CDF give us the random variable $X$?
Why can't we find the rand... |
H: Unclear limit in showing that $\ell^2$ spaces are complete
I'm working my way through the book Introduction to Hilbert Spaces with Applications and trying to follow the early example (1.4.6) showing that $\ell ^2$ spaces are complete (Cauchy sequences $(a_n) \in \ell ^2$ have their converge in $\ell ^2$). At this p... |
H: Prove that for all $x, y>0$, $\ln \left(\frac{x}{y}\right) \geqslant 4 \frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}}$
Prove that for all $x, y>0$,
$\ln \left(\frac{x}{y}\right) \geqslant 4 \frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}}$
Is there any role for
mean value theorem in the proof?,
Can we use the fact
$\ln (x)... |
H: How do we further simplify this expression involving a complex number?
I will state the problem here:
Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Compute
$\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 + \omega^3}.$
W ... |
H: How to account for increasing probability in odds of winning a competition when each round there is a random single decrease in the non-winner pool?
My Example:
Players 1 through n are playing a game and in each round there is one winner and after the winner is determined, one of the non-winners gets randomly taken... |
H: If a.i=4, then ,what is the value of (axj).(2j-3k) , where a is a vector
This is a question I saw in a question paper of a competitive exam but I was unable to solve it. Can anyone please assist me with any sort of hint to solve this problem and any type of explanation if needed in the hint. Any help will be highly... |
H: Column-sum and row-sum for fat matrices
For an $m \times n$ fat matrix ($m<n$)
$$A = \begin{pmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n}\\ a_{21} & a_{22} & \ddots & \cdots & a_{2n} \\ a_{31} & \cdots & \ddots& \ddots & \vdots \\ \vdots & \cdots & \cdots & \ddots & \vdots \\ a_{m1} & \cdots & \cdots & \c... |
H: probability question : what are the chances that at least 1 out of 150 passengers are infected
There are $10{,}000$ corona patients in a city of $7{,}500{,}000$ people.
$150$ people fly on a plane, what are the chances that at least $1$ person is infected?
I solved it using the below approach
select 1 infected pa... |
H: $(0)$ is the only minimal prime of $k[x,y]$
We define a prime ideal to be minimal if it is minimal with respect to inclusion. Given this, why $(0)$ is the only minimal prime of $k[x,y]$?, where $k$ is any field.
In particular, is there an easy way to see this? If I pick a nonzero prime ideal of $k[x,y]$, how do I k... |
H: How to find $a$, $b$, $c$ such that $P(x)=ax^3+bx^2+cx$ and $P\left(x\right)-P\left(x-1\right)=x^2$
I'm trying to find $a$, $b$ and $c$ such that $P(x)=ax^3+bx^2+cx$ and $P\left(x\right)-P\left(x-1\right)=x^2$.
After expanding the binomial in $P(x-1)$, I end up getting
$3ax^2-3ax+2bx+a-b=x^2$. What next? Using $3a ... |
H: If $X$ is homeomorphic to subset of set $Y$ and $Y$ is homeomorphic to subset of $X$ then are $X$ and $Y$ homeomorphic?
Let $(X, \tau_1)$ and $(Y, \tau_2)$ be topological spaces. Is it true that if $X$ is homeomoprhic to a subset of $Y$ and $Y$ is homeomorphic to a subset of $X$ then $X$ and $Y$ are homeomorphic sp... |
H: Proving a open and bounded subset of $\mathbb{R}$ is a union of disjoint open intervals
Below is the proof I have written up. I would love to get some feedback on it as I have not been able to readily spot any logical holes, but that might be because I'm missing something. What's more, I would love to know if there... |
H: How to correctly compate $f(n)$ and $g(n)$ when working through $O(n)$ notation?
Going through theory, missing the idea, need a bit of help. So, the initial state is:
$$f(n) = O(g(n))$$
Assume that $f$ and $g$ are both nondecreasing and always bigger than 1. And, from my understanding, $f$ must be less or equal to ... |
H: Prove/Disprove an inner product on a complex linear space restricted to its real structure is also an inner product
Let $V$ be an $n$-dimensional linear space and $(\cdot, \cdot)$ be an inner product on it. Define the conjugation map $\sigma: V \to V$ such that for any $\alpha, \beta \in V$ and $\lambda \in \mathbb... |
H: Negative exponential of an exponential random variable is a uniform random variable?
I know that the negative log of a uniform random variable is an exponential random variable. I am trying to prove the reverse, but I seem to have arrived at something that doesn't make sense.
Define $Y \sim \text{Exponential}(\lamb... |
H: Logarithm over complex numbers
The logarithm function is not certainly defined for every $\text{Re}(z)\leq0$, but the question is where is it defined?
I also know $\displaystyle \int_{C} \frac{1}{z} dz\neq0$ where $C$ is the unit circle defined by $ \gamma(t)=e^{it} $ for $0\leq t\leq 2\pi$, which implies that $\fr... |
H: The Nash equilibrium, an existence proof
I do not follow here in the -4th line
that the equality is achieved only if
$$p_i(s,\alpha)\leq 0$$
for every $s$.
AI: If $p_i(s,\alpha)\leq 0$ for every $s$, then $\alpha'_{i} = \alpha_{i}$, thus we get equality (proving "if" but not "only if").
Suppose there exists an $\ha... |
H: Calculate $\int_{0}^{1} \sin(x^2)$ with an error $\le 10^{-3}$
Calculate $\int_{0}^{1} \sin(x^2)$ with an error $\le 10^{-3}$
Let $f(x)= \sin(x^2) $ continuous in [0,1] so by the MVT for integrals we know $\int_{0}^{1} \sin(x^2) = \sin(c^2) \; \text{for} \; c \in [0,1]$. I don't really know if this is of any help. ... |
H: Determining group structure after computing homology: $\langle b,c \mid 2(b+c)=0, b+c=c+b \rangle.$
I am trying to determine what group this is a presentation of:
$$\langle b,c \mid 2(b+c)=0, b+c=c+b \rangle.$$
I am pretty sure it is $\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$, but am stuck on how to show it.
AI: Pu... |
H: Finding the function that satisfies to condition that the length of the curve is the same as the volume of rotation around the x-axis
I want to find the function that satisfies the following DE:
$$\pi y(x)^2=\sqrt{1+(y'(x))^2}$$
This comes from the fact that the left-hand side gives the volume of radiation around t... |
H: Equivalence of two 1-form defining the same hyperplane field on a manifold.
I have tried my best to explain my question.
Suppose M is n dimensional smooth manifold and $\xi$ is a smooth hyperplane field define on it. In addition, M has two 1-form define on it, $\alpha_1$ and $\alpha_2$ such that kernel($\alpha_1$)=... |
H: Convergence Range of $\sum\limits_{n=1}^{\infty} \frac{\sin(2n-1)x}{(2n-1)^2}$
$$\sum\limits_{n=1}^{\infty} \frac{\sin(2n-1)x}{(2n-1)^2}$$
My Attempt: I realize that $-1 \leq \sin(2n-1) \leq 1$. If I take the absolute value I can create the inequality:
$$\sum\limits_{n = 1}^{\infty} | \frac{\sin(2n-1)x}{(2n-1)^2}| ... |
H: Comparison of sequence of functions and function on $\mathbb{R}^2$
Let $\left(f_n\right)_{n\in\mathbb{N}}$ be a sequence of functions, where $f_n:\mathbb{R}\to\mathbb{R}$ and be $g:\mathbb{R}^2 \to \mathbb{R}$ some function. I know that if I plug in a sequence $\left(x_n\right)_{n\in\mathbb{N}}$ into $g$ I can cons... |
H: Heine Borel Theorem statement (a)
I have been following Prof Winston Ou's course on analysis on Youtube.
In the lecture on Heine Borel theorem, he mentioned that a set $E$ in $\mathbb R$ is closed and bounded implies that $E$ is a k-cell (hence $E$ is compact).
I don't understand how he came to this conclusion. For... |
H: Integral to Gamma function
For this function i need to convert it to either a Gamma or a Beta function
$$\int_0^1 \frac{3}{(1-x^3)^\frac{1}{3}} \,dx$$
I know I need to make the substitution $x^3=z$ but I am unsure where to go from here
AI: For simplicity, define
$$\mathcal I := \int_0^1 \frac{3}{(1-x^3)^\frac{1}{3}... |
H: Importance of Tannery's theorem
Tannery's theorem:
Let $S_n=\sum_{k=0}^\infty a_k(n)$ and $\lim_{n\to\infty}a_k(n)=b_k$. If $|a_k(n)|\le M_k$ and $\sum_{k=0}^\infty M_k\lt\infty$, then $\lim_{n\to\infty}S_n=\sum_{k=0}^\infty b_k$.
This can be used to prove that
$$\lim_{n\to\infty}\sum_{k=0}^n \frac{x^k}{k!}\prod_... |
H: Exponentials of the roots of polynomials $P(x)$ are always the roots of $P^*(x)?$
For any polynomial $P(x)$ with rational coefficients and no constant terms, are the exponentials of the roots of $P(x)=c_1x^n+c_2x^{n-1}+\cdot\cdot\cdot~+ c_kx$ always equal to the roots of $P^*(x)=e^{c_1\log^n(x)}+e^{c_2\log^{n-1}(x... |
H: Elementary combinatorics: how many lunch salads?
Here is the problem: a restaurant offers salads with the following options: choose five ingredients from a list of eight, plus two dressings from a list of four, but do not choose radishes (one of the ingredients) with peanut butter dressing (you can have radishes wi... |
H: Bounding sum by (improper) integral
I am trying to verify the following inequality that I came across while reviewing some analysis exercises online:
$$
\sum_{n=1}^{k} \left(1-\frac{n}{k}\right)n^{-1/7}\leq \int_{0}^{k}\left(1-\frac{x}{k}\right)x^{-1/7}\,dx, \hspace{3mm} k>1
$$
$\textbf{My question:}$ Why does the ... |
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