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H: Inequality string for comparison of series
I'd like to know if the following statement is a valid one. Specifically, I hope to use it in applying comparison tests for convergence/divergence of certain series.
For sufficiently large n,
$n, n^2, n^3, n^4, ... < 2^n, 3^n, 4^n, ... < n!$
The statement wasn't in my text... |
H: Converging Integral question
Is this integral convergent or divergent?
$$ \int_1^{\infty} \sqrt{x}\, \ln\!\left(1+ \frac{\sin(x)}{x}\right)\,\text {d}x$$
and
$$ \int_1^{\infty} \sqrt{x}\, \ln\!\left(1+ \frac{\cos(x)}{x}\right)\,\text {d}x$$
You cannot use the convergence tests because the $\ln$ changes sign ($-$ ... |
H: L'Hopital's rule conditions
I have seen easy geometrical argument why L'Hopital's rule ($\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$) works (local linearization). But, I still don't understand this:
why is rule defined just when limit is in form $\frac{0}{0}$ or $\pm \frac{\infty}{\infty... |
H: Why is it true that if the integral is finite then $\lim_{s\to\infty} \int_{s}^{\infty} f(x) dx=0$?
I saw it whilst reading my lecture notes that
Let $f(x)$ be integrable such that $\int_{-\infty}^{\infty} f(x) dx<\infty$ then it means that $\lim_{s\to\infty} \int_{s}^{\infty} f(x) dx=0.$
I can't seem to understa... |
H: Solvability by radicals
I study the book "Introduction To Field Theory" by Iain Adamson (https://archive.org/details/IntroductionToFieldTheory), and struggle with the Theorem 26.5. on page 166:
Let $F$ be a field of characteristic zero. If the polynomial $f$ in $F[x]$ is solvable by radicals, then the Galois group ... |
H: Fibonacci and tossing coins
Consider the following scheme starting with a sequence $\sigma_0 = \langle 1,1,\dots,1\rangle$ of length $k$, successively followed by sequences $\sigma_i$ of the same length but shifted by one to the right, where the first entry $\sigma_{i0}$ equals the sum of all values above, and $\si... |
H: Show that $\sum_{k=1}^\infty \frac{i}{k(k+1)}$ converges. Find its sum.
Show that $\sum_{k=1}^\infty \frac{i}{k(k+1)}$ converges. Find its sum.
The presence of the $i$ throws me off. Our professor taught us to find $a$ (the first term in the sequence) and $z$ (the multiplier to get the next term). Then find the mod... |
H: Show $l^p$ is not complete with the $q$ norm
I know the question has been asked here, but I do not understand the solution (Are $\ell_p$ spaces complete under the $q$-norm?)
I came up with my own solution and was wondering if it is correct.
Denote $l^p_q$ as $l^p$ with $q-$norm. Now if it is closed, then the identi... |
H: Prove that every set and subset with the cofinite topology is compact
Prove that every set with the cofinite topology is compact as well as every subset
Solution. Let $X$ be a nonempty set with the cofinite topology and let $ \mathscr{U}$ be an open cover of $ X $. Let $ U \in \mathscr{U}$. Then $X\setminus U$ is f... |
H: Verify $\tau=\{A \subseteq \mathbb{R}| |\mathbb{R}\setminus A| \leq |\mathbb{N}| \}$ is a topology
Consider the set $\tau=\{A \subseteq \mathbb{R}| |\mathbb{R}\setminus A| \leq |\mathbb{N}| \}$
Verify it is a topology
obviously $\mathbb{R} \in \tau$ since $ |\mathbb{R}\setminus \mathbb{R}|=0 \leq |\mathbb{N}| \... |
H: Calculate $\lim_{x \to 0} f(x)$
Let $f:\mathbb{R}\to \mathbb{R}$ be a function. Suppose $\lim_{x \to 0} \frac{f(x)}{x} = 0$. Calculate $\lim_{x \to 0} f(x)$.
According to the answer key, $\lim_{x \to 0} f(x) = 0$. I see $f(x)=x^2$ satisfies both limits, but is there a way "construct" an argument to prove this? I ... |
H: Contradicting the non-existence of a linear map $T: \Bbb R^5 \to \Bbb R^5$ and the Fundamental Theorem of Linear Algebra (from Axler Exercise 3.B(5))
I am asked to prove there does not exist a linear map $T:\Bbb R^5 \to \Bbb R^5$ such that $\operatorname{range}(T) = \operatorname{null}(T)$.
I think I understand th... |
H: What does it really mean for a model to be pointwise definable?
(Note: I'm only an amateur in logic, so I'm sorry for any weird
terminology or notation, or excessive tedious details. Most of what I
know is from Kunen's Foundations of Mathematics.)
I'm trying to learn a little about pointwise definable models. I'm... |
H: Show that there exists a metric $d$ on $\mathbb{R}$ such that $(\mathbb{R},d)$ is compact
I've come across this problem here and I've been trying to solve it. I've tried metrics like $d(x,y) = \ln(1+\frac{|x-y|}{1+|x-y|})$ but these end up not working (I believe this one does not give a totally bounded set). My thi... |
H: Simpler proof of van Kampen's theorem?
I've been trying to understand the proof of van Kampen's theorem in Hatcher's Algebraic Topology, and I'm a bit confused why it's so long and complicated.
Intuitively, the theorem seems obvious to me. Given a path $p$ in $A \cup B$, we can split it up into paths $p_1p_2...p_n$... |
H: Triangle problem with a simpler solution
Problem: In the triangle $\mathit{ABC}$ the angle $A$ is 60°. The “interior” circle has center $O$. If $|\mathit{OB}|=8$, $|\mathit{OC}|=7$, how long is $\mathit{OA}$?
“Solution”: Let the radius be $R$. Since $A=60°$; $|\mathit{OA}|=2R$ and (see image below) $|\mathit{AD}|... |
H: Lie Algebras of Covering of a Group is Isomorphic to the Lie Algebra of the Group.
If $\tilde{G} \to G$ is a covering of the lie group $G$, why are the associated lie algebras isomorphic? I.e., why $Lie(\tilde{G}) \cong Lie( G)$?
AI: There is natural identification of $T_eG\cong \mathfrak{g}$, and we know that a co... |
H: $\lim_{x \to +\infty} f(x) + g(x) = +\infty$. True or false?
True or false? If true, justify. If false, give counterexample. If $f,g : \mathbb{R} \to \mathbb{R}$ are functions such that $f$ is bounded and $\lim_{x \to +\infty} g(x) = +\infty$, then $\lim_{x \to +\infty} f(x) + g(x) = +\infty$.
I could not think o... |
H: brownian motion unbounded variation
I have been doing a little bit of reading regarding random processes and probability theory recently for some research I have been doing, and I have come across the claim in many places that Brownian motion cannot be treated with Riemannian integration due to the fact that it is ... |
H: Non-abelian group of order 165 containing $\mathbb{Z}_{55}$.
There is an example in this note http://www.math.mcgill.ca/goren/MATH370.2013/MATH370.notes.pdf (example 27.1.3 p. 56) that I cannot understand.
I attach the example.
Example 27.1.3 Is there a non-abelian group of order $165$ containing $\mathbb{Z}_{55}$?... |
H: Show the sequence $f_n(x)=\frac{1}{n}\chi_{[0,n]}$ has no weakly convergent subsequence in $L^1$.
Show the sequence $f_n(x)=\frac{1}{n}\chi_{[0,n]}$ has no weakly convergent subsequence in $L^1[R]$.
My observations:
Assume $f_{n_k} \to f$ weakly, then:
The sets where $f$ is positive or where it is negative have to ... |
H: Compute projection of vector onto nullspace of vector span
Say I have a matrix $\pmb{W}$ of $m$ vectors, each of length $n$: $\pmb{W} =\left[ \vec{W}_1, \dots, \vec{W}_m\right]$, where $\vec{W}_i \in \mathbb{R}^n$ for integers $1\leq i\leq m$. How would I go about computing the projection a new vector, $\vec{V} \in... |
H: Area of a Sector of a Circle Question
In the figure, $AB$ and $CD$ are two arcs subtended at center $O$. $r$ is the radius of the sector $AOB$. I was told to find the radius, $x$ (the angle), and the shaded area. I know $2\pi r\cdot\dfrac x{360} = 13$. And $\pi(r+4)^2 \cdot \frac{x}{360} - \pi r^2 \cdot \frac{x}{3... |
H: $\sum_{k=0} ^\infty (-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k+1}$ to differential equation
I want to use $S = \sum_{k=0} ^\infty (-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k+1}$ and get the relation $(a^2+1) S'=1−aS$. So far I am just getting $\frac{dS}{da} = \sum_{k=0} ^\infty (-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k} (2k+1)$, whi... |
H: If $f(x)= (x-a)(x-b)$ for then the minimum number of roots of equation $\pi(f'(x))^2 \cos(\pi(f(x))) + \sin(\pi(f(x)))f''(x) =0$
If $f(x)= (x-a)(x-b)$ for $a,b$ $\in \mathbb{R}$ then the minimum number of roots of equation
$$\pi(f'(x))^2 \cos(\pi(f(x))) + \sin(\pi(f(x)))f''(x) =0$$
in $(\alpha,\beta)$ where $f(\alp... |
H: Restriction of endomorphism on its image
Berkeley problems
Problem 7.4.7 Let $V$ be a finite-dimensional vector space and let $f:V\rightarrow V$ be a linear transformation. Let $W$ denote the image of $f$. Prove that the restriction of $f$ to $W$, considered as an endomorphism of $W$, has the same trace as $f:V\ri... |
H: Is there a simple rule defining the sequence $\frac 1 2, 1, -\frac 1 2, -1, \frac 1 4, \frac 1 2, \dots$?
I'm revisiting one of my old topology texts: "Introduction to Metric and Topological Spaces" by W.A. Sutherland, 1975 (the 1981 reprint with corrections), Oxford Science Publications.
One of the example illustr... |
H: Parametric solution of a Diophantine equation of three variables
I came across this Diophantine equation $$4x^2+y^4=z^2$$
Primitive solutions of this equation can be found by
\begin{align}
\begin{split}
x&=2ab(a^2+b^2)\\
y&=a^2-b^2\\
z&=a^4+6a^2b^2+b^4\\
\end{split}
\end{align}
where $a$, $b$ are relatively prime a... |
H: How to evaluate $\int _0^{\frac{\pi }{2}}x\ln \left(\sin \left(x\right)\right)\:dx$
How can i evaluate $$\int _0^{\frac{\pi }{2}}x\ln \left(\sin \left(x\right)\right)\:dx$$
I started like this
$$\int _0^{\frac{\pi }{2}}x\ln \left(\sin \left(x\right)\right)\:dx=\frac{x^2\ln \left(\sin \left(x\right)\right)}{2}|^{\fr... |
H: Showing $X_{(n)}$ is not complete for $\theta \in [1,\infty)$ when $X_i$'s are i.i.d $\text{Unif}(0,\theta)$
I am trying to show that the order statistic $X_{(n)}$ for a set of RV $\{X_i\}_{1}^{n}$ where $X_i\overset{iid}\sim \text{Unif}(0,\theta)$ is complete when $\theta \in (0,\infty)$ but not when $\theta \in ... |
H: Lottery probability -> Does winning affect others?
I came up with this question today since in italy somebody has won the national lottery:
(I know nothing about statistics)
there is a town with 10 spots where you can play lottery, and 1000 people play on each spot.
One of these guys wins. The probability that the ... |
H: Splitting Lemma: cokernel vs kernel being isomorphic
In Algebra: Chapter 0, the author proves the first part of the splitting lemma with the following: Proposition:
And the proof:
However, why do we have that coker $\phi$ $\cong$ ker $\psi$? I see that this would hold for when $\phi$ is surjective, as then the ke... |
H: Find a subgroup of index 3 of dihedral group $D_{12}$
Find a subgroup of index 3 in the dihedral group $D_{12}$. I know the number of elements in $D_{12}$ is 24 and also that is we have this subgroup of index 3, then we obtain that $|D_{12}:H|=8$, where $H$ is our wanted subgroup, but I don`t know how to go further... |
H: Yes/ No Is $(X,T)$ is connected?
Given $X= \{ a, b , c, d , e\}$ and $T= \{ X , \emptyset , \{a\} ,\{c,d\}, \{a , c, d\} , \{b ,c, d, e\} \}$. Is $(X,T)$ is connected ?
My attempt: I think yes take any two open set $\{a\}$ and $\{a,c,d\}$ , we have $\{a\} \cap \{a,c,d\} \neq \emptyset $ this implies that $(X,T)$... |
H: maximum eigen value of a square matrix whose rows are normalized (2-norm) to 1
Consider a positive definite square matrix $A$ of size $n\times n$, with rows $A_i$, such that $||A_i||_2=1$. For such a matrix, I have checked that the maximum eigen value is upperbounded by $\sqrt n$
How do i prove this?
Note: We can a... |
H: Inequality of a linear operator on Hilbert space.
Let $T \colon \mathcal H \to \mathcal H$ be a linear operator and let $x,y \in \mathcal H$. We assume that
$$ \forall \ z \in \mathcal H, \ \left \langle y-Tz,x-z \right \rangle \ge 0. $$
Show that $Tx=y$.
Any hints?
AI: Let $z=x-th$, where $t$ is real, then
$\langl... |
H: Evaluate $1-x+x^2-x^3+\cdots$
In some problem, I have to use the expression
$$\sum^\infty_{k=0}(-1)^kx^k=1-x+x^2-x^3+\cdots$$
I know about Taylor series, but I'm not sure how to find the equivalent to this. It's similar to the $log(1+x)$ series. Any help will be appreciated.
AI: For $|y| < 1$,
$$
1 + y + y^2 + \dot... |
H: Prove The following inequality $(ax+by)^2 \le ax^2+by^2$ for $a+b=1$
Prove The following inequality $(ax+by)^2 \le ax^2+by^2$ for $a+b=1, 0 \le a,b \le 1$
I tried expanding the equation and substituting $b=1-a$
\begin{equation}
(ax+by)^2=a^2x^2+2abxy+b^2y^2=a^2x^2+2axy-2a^2xy+b^2y^2
\end{equation}
The middle member... |
H: Let $X$ be a banach space, and let $U$ be a finite dimensional subspace, then there is a closed subspace $V$ s.t $X=U\bigoplus V$
Let $X$ be a banach space, and let $U$ be a finite dimensional subsapce, then there is a closed subspace $V$ s.t $X=U\bigoplus V$
MY attempt:
Let $U=Span\{v_1,...,v_n\}$ and consider the... |
H: An easy way to define the sequence $0$, $1$, $0$, $\frac12$, $1$, $0$, $\frac13$, $\frac23$, $1$, $0$, $\frac14$, $\frac24$, $\frac34$, $1$, $\ldots$?
Define $a_0=0$, $a_1=1$, $a_2=0$, $a_3=\frac 1 2$, $a_4=1$, $a_5=0$, $a_6=\frac 1 3$, $a_7=\frac 2 3$, $a_8=1$, $a_9=0$, $a_{10}=\frac 1 4 $, $a_{11}= \frac 2 4$, $a... |
H: Power series involving the Von Mangoldt-Function
I've been studying a proof of the Prime Number Theorem, given by D. V. Widder, where in one part he uses the identity $$\sum_{n=1}^{\infty}\frac{(\Lambda(n)-1)}{1-e^{-nx}}e^{-nx} = \sum_{n=1}^{\infty}(\log(n)-\tau(n))e^{-nx},$$ where $\Lambda(n)$ is the Von Mangoldt-... |
H: The tangent line is the best "linear" approximation to the graph of a differentiable function
I wanted to understand what it means that the tangent line is the best linear approximation to the graph of a differentiable function at the point of tangency.
I've looked in several books and I don't understand anything y... |
H: Why is $2$ considered a singular point for $f(x) = \frac{x-2}{x^2-x-2}$?
Let $$g(x) = \frac{1}{x^2-x-2} = \frac{1}{(x-2)(x+1)}$$
The domain of this function in apparently $D(g) = \{x \in \mathbb{R} : x \neq \{2,-1\}\}$
Now let $$f(x) = \frac{x-2}{x^2-x-2} = \frac{x-2}{(x-2)(x+1)}$$
The graph suggests that its domai... |
H: Question about proof of 'There are infinitely many primes $p$ with $p \equiv 2(\text{mod3})$'
I have read other proof, but I am stuck on the proof in my algebra class.
Hope someone could help me. Thanks a lot.
Prove by contradiction. Let $ \{ p_1,\dots p_n\} $ be our finite primes with $p_i \equiv 2 (\text{mod3})$ ... |
H: Is $x_n = (−1)^n$, $n ∈ \mathbb{N}$ convergent in $(\mathbb{R}, \cal{T} )$?
Let $\cal{T}$ = {$∅, \mathbb{R}$} ∪ {$(−a, a) : a ∈ (0, ∞)$} be a topology on $\mathbb{R}$.
Is the sequence $(x_n)_n∈\mathbb{N}$ defined by $x_n = (−1)^n$, $n ∈ \mathbb{N}$, convergent in $(\mathbb{R}, \cal{T} )$? In this case, what does it... |
H: multiplication of measurable functions in $L^p$ spaces
Let $(X, M, \mu)$ be a measure space, $q \in (0, +\infty]$ and $f,g : X \rightarrow \mathbb{C}$ in which $f \in L^{\infty} (\mu)$ and $g \in L^q (\mu)$. I want to show that $fg \in L^q (\mu)$.
For this, I showed that if $g$ is an $L^1$
function on $X$ and $f$ i... |
H: Why is there's a unique circle passing through a point?
I am trying to solve this problem:
We know that there's a circle with center$(m,h)$. And it passes through the points(1,0), (-1,0).
Show that there's a unique circle passing through the three points:$(1,0),(-1,0),(x_0,y_0)$.
I tried making substitution, and ge... |
H: How to prove there exists a positive integer $1\le i\le n$ so that $p^i(x)=x$ when $p:[n]\to[n]$ is permutation and $x\in[n]$
I am reading book A Walk Through Combinatorics and here is a Lemma and its proof.
Let $p:[n]\to[n]$ be a permutation, and let $x\in[n]$, then there exists a positive integer $1\le i\le n$ s... |
H: Are the solutions of $f(x+h)=f(x)f(h)$of the form $a^x$ even if we consider discontinuous functions
Let $$f(x):\mathbb{R}\to \mathbb{R} $$$$$$and$$f(x+h)=f(x)f(h)$$
If $f(x)$ is a continuous function then we can prove all solutions for ($f(x)$ not equal to zero at any point) are of the form $a^x$ .(Where $a^x$ i... |
H: How to compute $\int_0^1 \left\lfloor\frac2{x}\right\rfloor-2\left\lfloor\frac1{x}\right\rfloor dx$?
How to compute $$\int_0^1 \left\lfloor\frac2{x}\right\rfloor-2\left\lfloor\frac1{x}\right\rfloor dx\ ?$$
Now, what I did is break the integral so that $$\int_0^1 \left\lfloor\frac2{x}\right\rfloor dx-\int_0^12\l... |
H: Pull the limit inside the infinit serie in complex analysis?
Let $f: U \mapsto \Bbb C$ a holomorphic function and $U$ an open set of the complex plane. We have
$$f(z)=(z-z_0)^m\sum_{k=0}^{\infty}a_{k+m}(z-z_0)^k$$
with $m\geq 1$. In my course, it is written that the right hand side converges on some ball $B_r(z_0)$... |
H: Relationship Between Determinant and Matrix Rank
Let $n\in \mathbb{N}$, and $S\in \mathbb{R}^{n\times n}$ be a symmetric positive semi-definite (PSD) matrix with rank $r \triangleq \mathrm{rank(S)}\leq n$. Can $r$ be bounded in terms of the determinant of some function of $S$?
AI: At least a lower-bound is possible... |
H: Equal roots of a certain polynomial equation by changing the sign?
Is there a certain polynomial equations which when you change the sign of the equation the roots will still be the same? I wonder if there are, how can it be constructed using the algebraic properties?
AI: In general if
$$f(x)=f(-x)$$
the function $... |
H: Evaluate the limit $\lim\sqrt[n]{\frac{1}{n!}\sum(m^m)}$
In some problem, I need to evaluate this limit:
$$\lim_{n\rightarrow \infty}\sqrt[n]{\frac{1}{n!}\sum^n_{m=0}(m^m)}.$$
I know about Taylor series and that kind of stuff. I'm not sure where to start, maybe Stirling but after using it I still could not solve it... |
H: For an infinite sequence of functions $\Bbb{R}\to\Bbb{R}$, each function is a composition of a certain finite set of functions $\Bbb{R}\to\Bbb{R}$.
Given an infinite sequence of functions $\{g_1, g_2, \ldots, g_n, \ldots\}$ where $ g_n : \Bbb R \to \Bbb R$ prove there's a finite set of functions $ \{ f_1, f_2, \ld... |
H: Why $f(x)=x^2 \sin \frac{1}{x} $ Lipschitz but not continuously differentiable?
Let $f:[-1,1]\to \mathbb R$ such that $$f(x)=x^2 \sin \frac{1}{x} \quad (x\neq 0)$$ and $$f(0)=0.$$ It is clear to me that for $x\neq 0,$ $f$ is differentiable function (as being a product of two differentiable function). So $f'(x)= 2x... |
H: Which linear maps on a finite field are field multiplications?
I am mainly interested in the fields $\mathrm{GF}(2^n)$, but the question can be asked for any prime.
We can write out each element $x\in\mathrm{GF}(2^n)$ in base $2$ and note that its additive group combined with multiplication by elements of $\mathrm... |
H: The value of expression $x-y+2x^2y+2xy^2-x^4y+xy^4$
Let $x = \sqrt{3-\sqrt{5}}$ and $y = \sqrt{3+\sqrt{5}}$. If the value of expression $x-y+2x^2y+2xy^2-x^4y+xy^4$ can be expressed in the form $\sqrt{p}+\sqrt{q}$ where $p,q \in N$, then $(p+q)$ is equal to?
I have simplified the expression to $-11x+19y$ but don't k... |
H: Gaussian with zero mean dense in $L^2$
I have found in this article that linear combinations of Gaussian with fixed variance are dense in $L^2$. Can something similar be true for Gaussian of fixed mean and variable variance? Equivalently, can linear combination of this family of functions
$$
f(x,a) = e^{-(x/a)^2}... |
H: What is the cardinality of a vanishing set?
In Wiki's page on Chevalley–Warning theorem, under "Statement of the theorems", it's written that
Chevalley–Warning theorem states that [...] the cardinality of the
vanishing set of ${\displaystyle \{f_{j}\}_{j=1}^{r}}$ [...].
What does "the cardinality of the vanishing... |
H: Reverse order of a ring
When we think of an ordered structure with an order $\le$ we assume there is an opposite order $\le^{op}$ as well:
$a \le^{op} b \iff b \le a$.
I would suggest this is a fundamental principle for all ordered structures:
If a structure is ordered one way $(\le)$ it is also ordered the oppos... |
H: Properties of functions of mean zero
Let $f,g: \mathbb{R} \longrightarrow \mathbb{R}$ be differentiable functions and $a<b$ such that
$$\frac{1}{b-a}\int_{a}^{b}f(x)\;dx=0 \quad \text{and} \quad \frac{1}{b-a}\int_{a}^{b}g(x)\;dx=0 \tag{1}.$$
So, I think that I can conclude that
$$\int_{a}^{b}f'(x)\;dx=0 \tag{2}$$
M... |
H: How do we define power of irrational numbers?
power of rational numbers for me can be defined as multiplying $m$ times the $nth$ root of $x$. because we have:
$$
x^{\frac{m}{n}}
$$
when : $m,n \in \Bbb Z $.
Is this definition correct? if no what is the correct one and if yes, how can I extend this definition for ... |
H: fibonacci recurrence relation proof
I've been trying to prove the closed form solution of fibonacci recurrence sequence and achieve this
$a_n=\frac{1}{\sqrt{5}}[(\frac{1+\sqrt{5}}{2})^n−(\frac{1-\sqrt{5}}{2})^n]$
And so far I haven't achieve that, this is how I did it
$a_n=x(\frac{1+\sqrt{5}}{2})^n+y(\frac{1-\sq... |
H: Probability of black sock
this is an easy problem, but i feel i am stuck some whether conceptually.
A man has 3 pairs of black socks and 2 pairs of brown socks kept together in a box.If he dressed hurriedly in the dark, the probability that after he has put on a black sock, he will then put on another black socks i... |
H: Restricting a function in the disk algebra
Let $A$ be the disk algebra, i.e. continuous functions on the closed unit disk in $\Bbb{C}$ that are analytic on the interior of the disk. By the maximum-modulus theorem, we have an isometric morphism of algebras:
$$\varphi: A \to C(S^1): f \mapsto f\vert_{S^1}$$
The book... |
H: Bijection Cancellation rule for cartesian product
Suppose $A$, $B$ and $C$ are sets, and that there is a bijection between $C \times A$ and $C \times B$. Is there necessarily a bijection between $A$ and $B$?
I know this should work for finite sets - you can use a size argument to demonstrate $A$ and $B$ have the sa... |
H: Find the value of $\sum_{r=0}^{\infty} \tan^{-1}(\frac{1}{1+r+r^2})$
The given expression can be written as
$$\tan^{-1}(\frac{r+1+(-r)}{1-(-r)(r+1)})$$
$$=\tan^{-1}(r+1)-\tan^{-1}(r)$$
Therefore $$\sum =\tan^{-1}(1)-\tan^{-1}(0)+\tan^{-1}(2)....$$
Since it goes on to infinity, all the terms except $-\tan^{-1}(0)$ g... |
H: Proof that two charts on the tangent bundle $TM$ are $C^\infty(M)$-compatible
I'm struggling to understand a proof in the "Construction of the tangent bundle" section of the lecture notes downloadable here https://mathswithphysics.blogspot.com/2016/07/lectures-on-geometric-anatomy-of.html (Frederic Schuller's Lectu... |
H: Prove that every subsequence of a convergent real sequence converges to the same limit.
Here's the statement I want to prove:
Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of real numbers that converges to a real number $L$. Then, every subsequence $\{a_{n_k}\}_{k=1}^{\infty}$ converges to $L$.
Proof Attempt:
Let $\e... |
H: Adding differentials
Suppose I have a sum of two Indefinite integrals, $\int f(t)dx + \int f(t)dy$. Is it possible to write this in a singke form as $\int f(t) d(x+y)$, and vise versa?
It looks “okay” to me from a logical point of view, but I obviously have no rigorous reasoning of why this should be
Edit:
I now re... |
H: An equation for a graph which resembles a hump of a camel / pulse in a string?
Sorry if this question isn't valid. I just need to know an equation/function for a graph which resembles something close to
AI: $$\frac{a}{1+b(x-c)^2}+d$$Looks close to your graph.Try playing around with the constant to get the desired l... |
H: Find an angle between a triangle and a plane
The hypotenuse $AB$ of triangle $ABC$ lies in plane $Q$. Sides $AC$ and $BC $, respectively, create angles $\alpha$ and $\beta$ towards the plane Q (meaning they are tilted towards the plane $Q$ with such angles). Find the angle between plane $Q$ and the plane of the tr... |
H: Integer solutions of $2a+2b-ab\gt 0$
Let $a\in\mathbb{N}_{\ge 3}$ and $b\in\mathbb{N}_{\ge 3}$. What are the solutions of the Diophantine inequality
$$2a+2b-ab\gt 0?$$
By guessing, I found 5 solutions:
$$\text{1)}\, a=3,\, b=3$$
$$\text{2)}\, a=3,\, b=4$$
$$\text{3)}\, a=4,\, b=3$$
$$\text{4)}\, a=5,\, b=3$$
$$\tex... |
H: Invariant transformation's complement
Let V be an inner product space.
Let T : V-> V be linear and U a subspace of V . If T (U) ⊆ U, then T(U⊥)⊆ U⊥
I began with showing that (T(u), u') = 0, but didn't know how to show that (u, T(u')) = 0
AI: The statement is not true unless we are given further information about $T... |
H: Can someone explain how this integral of a third derivative works?
I'm reading some notes on the derivation of the Friedmann equation from Newton's formulas The paper reads:
The equation of motion for $R_s(t)$ can be obtained from the gravitational acceleration at the outer
edge of the sphere:
$$\frac{d^2R_s}{dt^2... |
H: Contour Integration to Evaluate Improper Integral
I am working on the problem above and have, for part a, that $D=\{z \in \mathbb{C} | 0< \Re(z) < 1\}$. What I'm working on now is part b.
My attempt so far: I have set up a rectangular contour, $\Gamma$, with its base sitting on the real axis, going from -R to R, a... |
H: Literature on bounds of Fubini's numbers
If anybody can suggest where I can find a literature for a known upper and lower bounds on Fubini numbers https://en.wikipedia.org/wiki/Ordered_Bell_number
AI: QING ZOU, "THE LOG-CONVEXITY OF THE FUBINI NUMBERS", http://toc.ui.ac.ir/article_21835_684378fec55e5c66c7fccd4321a8... |
H: Converse of $(A\rightarrow(B\rightarrow C))\rightarrow((A\rightarrow B)\rightarrow(A\rightarrow C))$
The following proposition in (1) is taken as an axiom in intuitionistic propositional logic.
$$(A\rightarrow(B\rightarrow C))\rightarrow((A\rightarrow B)\rightarrow(A\rightarrow C))\quad\quad(1)$$
What about its con... |
H: Does $f(x+yi):= \frac{xy}{x^2+y^2}$ have a continuous function in 0?
Does $f(x+yi):= \frac{xy}{x^2+y^2}$ have a continuous function in 0 ?
I would start by changing it to $f(z) = \frac{xy}{|z|^2}$ but i cant find anything for xy
AI: Hint: What is the limit as $x \to 0$ along the line $y=0$? What about the line $y=x... |
H: Proving that $\mathbb R^n$ satisfies the second axiom of countability
In a general topology exercise I am asked to prove the following:
A topological space $(X,\tau)$ is said to satisfy the second axiom of countability if there exists a basis $B$ for $\tau$, where $B$ consists of only a countable number of sets.
... |
H: Changing a double integral into a single integral - Volterra-type integral equations
I have a question regarding a calculation that i stumbled upon when proving that a Cauchy problem can be converted in a Volterra-type integral equation. Specifically, this equality:
\begin{equation*}
\int_0^t\int_0^sy(t) dt ds = \i... |
H: How can I approximate an arc of a circle with an ellipse?
If I know the center and radius of a massive circle C, how can I construct a smaller ellipse E to approximate the arc I'm interested in within a range of confidence R?
Approximate an Arc with an Ellipse
Basically, this is a Navigation problem relating to the... |
H: Find the matrix representation of the operator $A\in\mathcal L(G)$ in the basis $f$.
In the very beginning, I'm going to refer to my previous question where I applied the same method in a bit different vector space.
Let $G\leqslant M_2(\Bbb R)$ be the subspace of the upper-triangular matrices of the order $2$ and ... |
H: Bijection between $\mathbb{N}$ and $[0,\alpha]$
Suppose $\alpha<\omega_1$ is an ordinal. Can anyone give me an example of a bijection between $\mathbb{N}$ and $[0,\alpha]:=\{\gamma: \gamma\leq \alpha\}$. Is there an order preserving bijection between the two sets?
AI: There is a bijection iff $\omega\le\alpha<\omeg... |
H: What is the RN derivative of infinite product measure?
Suppose $\mu_k$ and $\nu_k$, $k=1,2,...$ are sigma-finite measures on spaces $(S_k,\mathcal F_k)$ such that $\nu_k<<\mu_k$ for each $k$. Let $f_k=\dfrac{d\nu_k}{d\mu_k}$ for each $k$. Then is it true that $\nu:=\prod_{k=1}^\infty \nu_k<<\prod_{k=1}^\infty \mu_... |
H: $ \lim_{n\to \infty} \int_0^1 e^{i\cdot n\cdot p(x)}~dx=0$ where $p(x)$ is a nonconstant polynomial with real coefficients
If $p(x)$ is a nonconstant polynomial with real coefficients, then how can we show that $$ \lim_{n\to \infty} \int_0^1 e^{i\cdot n \cdot p(x)}~dx=0 ?$$
The integrand $e^{i \cdot n \cdot p(x)}$... |
H: $ f(x)+ \sum \lambda_ig_i(x) \geq f(\bar x), \forall x \in \mathbb{R}^n.$
Suppose that $f,g_i : \mathbb{R}^n \to \mathbb{R}$ $(i=1,\ldots,m)$ are convex functions and $\exists x$ such that
$$g_i(x)<0 , \qquad i=1,\ldots,m.$$
Show $\bar x$ is optimized solution of
$$\min f(x)$$
$$\text{s.t. }g_i(x) \leq 0, \qquad i... |
H: Group action of the Baumslag-Solitar groups
The Baumslag-Solitar groups are defined by
$$G=BS(m,n)=\langle a,b: ba^{m}b^{-1}=a^{n}\rangle\,,$$
where $m,n$ are integers.
My question is: Is there a linear action of $G=BS(1,2)$ over $\mathbb{R}^{2}$ ?
AI: Yes, the matrices $\begin{bmatrix}2&0\\0&1\end{bmatrix}$ and $\... |
H: Evaluate a complex integral.
I looked around but I couldn't find if this question has been asked before.
Given two polynomials $P(z) = a_{n-1}z^{n-1}+\cdots+a_0$ and $Q(z)=z^n+b_{n-1}z^{n-1}_ + \cdots + b_0,$ prove for sufficently large $r>0$
$$ \int_{|z|=r} \frac{P(z)}{Q(z)} dz = 2 \pi a_{n-1} i$$
My attempt: I t... |
H: Continuity of $a^x+b$ with $a, b \in \mathbb R$
Let $a,b \in \mathbb{R}$ with $a > 0$. find $a$, $b$ so the function would be continuous
$$
f(x) = \begin{cases} a^x + b, & |x|<1 \\
x, & |x| \geq 1 \end{cases}
$$
I got $b = -a^x+x$ as my answer, but I'm unsure.
AI: Since $f(x) = a^x + b$ will be continuous on $|x| ... |
H: Find $L=\lim_{n\to \infty }\frac{1}{n}\sum_{k=1}^{n}\left\lfloor 2\sqrt{\frac{n}{k}} \right\rfloor -2\left\lfloor \sqrt{\frac{n}{k}} \right\rfloor$
Question:- Find Limit $$L=\lim_{n\to \infty }\frac{1}{n}\sum_{k=1}^{n}\left\lfloor 2\sqrt{\frac{n}{k}} \right\rfloor -2\left\lfloor\sqrt{\frac{n}{k}} \right\rfloor \tex... |
H: What is the fraction of customers lost in a finite queue with one server, M/M/1/k? k = four places and s = 1 server
What is the fraction of customers lost in a finite queue with one server, M/M/1/k? $k =$ four places and $s = 1$ server
$k=4, \lambda=\dfrac 1 {30}$, $\mu=\dfrac 1 {25}$
The steady-state probs are p... |
H: Dominated convergence theorem and Cauchy's integral formula
Let $U\subseteq \mathbb{C}$ be open and $\bar B(a,r) \subseteq U$. Let $\gamma(t) =a+ re^{it}$ with $t \in [0,1]$ be the boundary path of $B(a,r)$. By Cauchy's integral formula $f(w) = \frac{1}{2 \pi i}\int_{\gamma} \frac{f(z)}{(z-w)} dz$, where $w \in B(... |
H: Local diffeomorphism between a disk and a sphere
This may be a silly question, but I’ll make it anyway. Let $f: D^2 \to S^2$ be a local diffeomorphism between the closed unit disk and the unit sphere. Is it necessarily injective?
AI: No, I don't think so. For instance, think about stretching out $D^2$ into a very l... |
H: A question based on property of a function satisfying $f(1/n) =0$ for every $n \in\mathbb{N} $
I am trying quiz questions of senior year and was unable to solve this particular question.
It's image:
Unfortunately, I couldn't think which result in analysis I can use. I am totally confused and would really appreciate... |
H: How do smooth manifolds differ from manifolds embedded in $\mathbb{R}^n$?
Instead of defining a smooth manifold to be a manifold whose gluing functions are smooth, what would happen if we defined it as an $n$-manifold $M$ which has an embedding into $\mathbb{R}^{n +1}$?
A smooth map between manifolds $e_M : M \hook... |
H: Solving $f(x)$ in a functional equation
Find of general form for $f(x)$ given $f(x)+xf\left(\displaystyle\frac{3}{x}\right)=x.$
I think we need to substitute $x$ as something else, but I'm not sure. Will $x=\displaystyle\frac{3}{x}$ help me?
AI: Yes, it helps, as follows:
From
$$f(x)+xf(\frac{3}{x})=x\tag{*}$$
we... |
H: Let $S = \{1/n, n \in\mathbb N\}$ and we define a function $f : \{0\} \cup S \to \mathbb R$ as the formula below. is this function continuous at $0$?
the function is $f(x) = \begin{cases} \sin(\pi/x) & \text{ if } x\neq0 \\
0 & \text{ if } x=0 \end{cases}$
I know that t... |
H: Find a probability given specific cdf values
Given $X$ is uniform random variable, $P\{X>1\} = 0.6$ and $F(2) = 0.5$.
Find $P\{-1\leq X < 3\}$.
My solution is: $P\{X>1\} = F(\infty) - F(1) = 0.6$. So $F(1) = 0.4$
And now I assume that F grows linearly. I need to find $F(3) - F(-1)$. Using fact of linearity I can ... |
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