text stringlengths 83 79.5k |
|---|
H: Find rank of matrix
If $X$ is a $N \times D$ matrix with $(D\gg N)$ with $\operatorname{rank}(X) = N$, what is $\operatorname{rank}(X^T \cdot X)$ where $X^T$ is the transpose matrix of $X$?
I am little new to linear algebra and I am not having any approach, I faced this problem in other context(linear regression).
... |
H: Image of intersections is equal to intersections of images if and only if the inverse of the image is the set
I'm working through some exercises dealing with mappings and the properties of its image and pre-image.
Let $f: S \to T$ be function. I want to prove that the statements
$f$ is injective
$f(A\cap B) = f(A)... |
H: Composite function with integral
Suppose $f(z)=\int_{0}^{z} g(y)dy, h(x)=x^2$, then the composite $f\circ h=\int_{0}^{x^2} g(y)dy$. Is this correct?
AI: Yes that's right. $f \circ h = f(h(x))$ so you just plug in $h(x)$ for $z$ to get $\int_{0}^{x^2} g(y) dy$. |
H: How can a vector field act on a Lie Algebra element?
We have the definition of a vector field as a smooth section of the tangent bundle $$X:P\longrightarrow TP,$$ where $(TP,\pi',P)$ is the tangent bundle over the total space of the principal G-bundle $(P,\pi,M)$. I.e, a vector field is an assignment of a vector ($... |
H: Computing the number of $\sigma$-algebra.
Let $X$ be a set. How many $\sigma$-algebras of subsets of $X$ contain exactly $m$ elements?
Any hints for how to begin a solution to this problem are greatly appreciated.
My initial approach is as follows:
Let $|X| = n$, then $|P(X)| = 2^n$
Thus our count is given by
$\bin... |
H: Simplifying trigonometry function by substitution
I want to simplify the following expression: $\frac{{\left( {1 + \sqrt 3 \tan {1^{\circ}}} \right)\left( {1 + \sqrt 3 \tan {2^{\circ}}} \right)\left( {\tan {1^{\circ}} + \tan {{59}^{\circ}}} \right)\left( {\tan {2^{\circ}} + \tan {{58}^{\circ}}} \right)}}{{\left( {1... |
H: A set S is a subspace of an inner product space $V$ iff $(S^{\perp})^{\perp}=S$?
$S$ is any set of a finite-dimensional inner product space $V$. Prove or disprove that $S$ is a subspace of $V$ iff $(S^{\perp})^{\perp}=S$.
My approach was to consider a minimum set $B$ of linearly independent vectors which spans $S... |
H: On the double series $\sum_{(m,n)\in\mathbb{Z}^2\setminus\{(0,0)\}}\frac{m^2+4mn+n^2}{(m^2+mn+n^2)^s}$
I'm having a difficult time calculating the series
$$\sum_{(m,n)\in\mathbb{Z}^2\setminus\{(0,0)\}}\frac{m^2+4mn+n^2}{(m^2+mn+n^2)^s} \quad , \quad s>2$$
I don't even know where to start. Truth be told I don't have... |
H: Definition and intuition of a tubular neighborhood of a submanifold
Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$ and $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary$^1$.
Now let $$T_xM:=\left\{v\in\mathbb R^d\mid\exists\varepsilon>0,\gamma\in C^1((-\varepsilon,\varepsilon),M):\gamma(... |
H: Is the product of two Cesaro convergent series Cesaro convergent?
Let $\{a_n \}_{n \geq 1}$ and $\{b_n \}_{n \geq 1}$ be two sequences of real numbers such that the infinite series $\sum\limits_{n=1}^{\infty} a_n$ and $\sum\limits_{n=1}^{\infty} b_n$ are both convergent in the Cesaro sense i.e. \begin{align*} \lim... |
H: Set of possible reflections of a vector.
How many vectors can one construct by by reflecting a vector $b\in\mathbb{R}^d$ for $b\neq 0$?
Reflections can be described by Householder matrices $H=I-2vv^T/||v||_2^2$.
In other words, I'm interested in the subset $S(b):=\{v\mid \exists \text{ Householder Matrix } H: v=Hb\... |
H: Common root of a cubic and a biquadratic equation
The value of $a$ given that the cubic equation
$$x^3+2ax+2=0$$
and the biquadratic equation
$$x^4+2ax^2+1=0$$
have a common root.
I know how to use common root condition for two quadratic equations, But I don't know how to solve this...
AI: Assume $r$ is the common ... |
H: Question about Chinese Remainder Theorem
This is a homework problem that I'm confused about. I understand the solution until it says "With the Chinese Remainder Theorem and some computation this shows that $n \equiv 301 \pmod{420}.$ I'm not sure how to use Chinese Remainder Theorem and how to get this, any explanat... |
H: Surjectivity of stalks of holomorphic functions
Let $X=\mathbb{C}$ with the classical topology, let $\mathcal{O}_X$ be the sheaf of holomorphic functions, and let $\mathcal{O}^*_X$ be the sheaf of invertible (nowhere $0$) holomorphic functions. I'd like to understand why $\rm exp: \mathcal{O}_X\rightarrow \mathcal{... |
H: Find volume under given contraints on the Cartesian plane.
The constrains are given as
$$x^2+y^2+z^2 \leqslant 64,\,x^2+y^2\leqslant 16,\,x^2+y^2\leqslant z^2,\,z\geqslant 0\,.$$
With the goal of finding the Volume.
Personally, I have trouble interpreting the constrains in terms of integrals to find the Volume. How... |
H: If the angle between the line $x=y=cz$ and the plane $z=0$ is $45^\circ$, find all values of $c$.
If the angle between the line $x=y=cz$ and the plane $z=0$ is $45^\circ$, find all values of $c$.
Before actually doing the calculations I thought I would get $c=1$ since the angle between $x=y=z$ and the $xy$-plane ... |
H: What does this function do? (why do we xor?)
I stumbled across some source code on GitHub, but the author did not provide any documentation so I'm trying to figure it out by myself.
def s(k,x,mul=gf832_mul):
if k==0:
return x
tmp = s(k-1,x,mul)
result = mul(tmp,tmp)
return result ^ tmp
It's my ... |
H: Let $G$ be a finite non-solvable group, each of whose proper subgroups is solvable.
Show that $G/\Phi(G)$ is a non-abelian simple group, where $\Phi(G)$ denotes the Frattini subgroup of $G$
So $G/\Phi(G)$ can't be abelian since if it were then is would be solvable and since $\Phi(G)$ is a solvable normal subgroup o... |
H: A compact normal operator is diagonalisable.
Consider the following proof in Murphy's book: "$C^*$-algebras and operator theory":
I try to understand why $u(K^\perp) \subseteq K^\perp$. Equivalently, one can prove $u^*(K) \subseteq K$.
I tried to calculate $\langle u(x), y \rangle$ with $x \in K^\perp, y \in K$ an... |
H: Calculate Hessian of a "weird" function
let be $L $ an invertible matrix, $b \in \mathbb{R}^{n}$ and $\mu \in \mathbb{R}$
and $J: \mathbb{R}^{n} \rightarrow \mathbb{R}$ defined as $J(x)=\|L x\|_{2}^{2}+\mu x^{t} x-b^{t} x$
How do I calculate its hessian matrix?
I cant operate as a usual function and calculate its... |
H: Projective algebraic sets
I want to find a projective algebraic set which is connected but not irreducible? I try to find but stuck at some points if it is not irreducible then it is also not connected
AI: It is probably worth looking at the notions of connectedness and irreducibility more attentively. A space is n... |
H: How to prove this matrix is positive semi-definite?
Define $K:\mathbb{R}^{2}\times\mathbb{R}^{2} \to \mathbb{R}$ by
$$K(a,b)=K\left((a_{1},a_{2}),(b_{1},b_{2})\right)=a^{T}b+\cos\left(\frac{(a_{2}-b_{2})\pi}{3}\right)=a_{1}b_{1}+a_{2}b_{2}+\cos\left(\frac{(a_{2}-b_{2})\pi}{3}\right).$$
Let $a_{1},a_{2},\cdots a_{n}... |
H: Rotating a quarter circle -- how long has a point traveled.
Question: see below quarter circle $AOB$. $P$ is the midpoint of $AO$. $OM$ is considered as the "ground surface". We keep rotating $AOB$ to the right, until $OB$ sits on the ground surface again. How long has $P$ travaled during this time?
This puzzle rem... |
H: A donut shop sells 12 types of donuts. A manager wants to buy six donuts, one for himself and 5 for his employees.
I'm reading Advanced Combinatorics by Mitchell T. Keller and William T. Trotter and it's missing an answer book (or, at least, I couldn't find one.) I've been doing the exercises but want to make sure ... |
H: How to prove that $-|z| \le \Re (z) \le |z|$ and $-|z| \le \Im (z) \le |z|$?
I am reading Ahlfors' "Complex Analysis". Early in the book, he uses the fact that for $z \in \mathbb{C}$ we have
$$
-\lVert z\rVert \le \Re (z) \le \lVert z\rVert\qquad \text{and} \qquad -\lVert z\rVert \le \Im (z) \le \lVert z\rVert
$$
... |
H: Does $g(v_n) \longrightarrow g(0)$ for all $v_n \text{s.t.} ||v_{n+1}|| \leq ||v_n||$ imply $g$ continuos at $0$?
The question is pretty much summed up in the title:
Let $V,W$ be normed vector spaces and $$ g: V \to W$$.
Suppose g fulfills $$g(v_n) \longrightarrow g(0) $$ for all sequences $v_n$ such that $$v_n \lo... |
H: Use linearisation of a certain function to approximate $\sqrt[3]{30}$
Background
Find the linearisation of the function
$$f(x)=\sqrt[3]{{{x^2}}}$$
at
$$a = 27.$$
Then, use the linearisation to find
$$\sqrt[3]{30}$$
My work so far
Applying the formula
$${f\left( x \right) \approx L\left( x \right) }={ f\left( a \rig... |
H: Are final functors stable under pullback?
Recall the notion of a final functor, which is a sort of colimit-preservation property.
Is such class of functors stable under pullbacks in Cat? Namely, is the pullback of a final functor along any other functor still final? If not, what is a counterexample?
A reference wou... |
H: Different ways for stating the recursion theorem
I've been trying to understand the recursion theorem for quite a while now and I still don't think I understand it 100%, I've checked multiple books and pdfs and I've noticed that this theorem is often stated in two different ways
Let $A$ be a set, $a ∈ A$, and $r :... |
H: How to prove that there is closure in NP for the reverse operation on strings?
I have a language A which is known to be in NP. I want to know that if I have the language $A^R$ which takes in $w^R$ which is a word of A but read in reverse will it still be in NP?
I tried to prove it by writing that: because A is in N... |
H: What should $n$ be equal to, so that $5^{2n+1}2^{n+2} + 3^{n+2}2^{2n+1}$ is completely divisible by $19$?
What should $n$ be equal to, so that the number:
$$5^{2n+1}2^{n+2} + 3^{n+2}2^{2n+1}$$
is completely divisible by 19? I broke it into this:
$$20\cdot 2^{n}\cdot 25^{n}+18\cdot 3^{n}\cdot 4^{n}$$ But what should... |
H: If $ \lim_{x \to +\infty}f(x) = A $ and $ \lim_{x \to +\infty}f'(x) = B $, prove that $B = 0$
Problem: Let $ f: \mathbb{R} \to \mathbb{R} $ be a function of class $ C^1 $ such that $ \lim_{x \to +\infty}f(x) = A $ and $ \lim_{x \to +\infty}f'(x) = B $ for $ A, B \in \mathbb{R} $. Prove that $B = 0$.
I need help in ... |
H: Show that $\sum_n (1-e^{-1/n})$ diverges
I'm stuck in showing that $\sum_n (1-e^{-1/n})$ diverges. The ratio and the root test are inconclusive. Possibly the comparison test is the way to go, but I can't find a proper bound. I got
$$
1-\frac{1}{e^{1/n}}>1-\frac{1}{2^{1/n}}
$$
but still I can't see here if the last ... |
H: How many "types" of hypothesis tests are there?
I don't know why but I find textbooks to be so inconsistent on describing the types of hypothesis tests there are (at an intro level); I want to organize my thoughts a little bit.
When you repeatedly take samples of size $n$ from a certain population distribution with... |
H: Let $G$ be a finite group such that if $A, B\le G$ then $AB\le G$. Prove $G$ is a solvable.
Let $G$ be a finite group satisfying the following property: (*) If $A, B$ are subgroups of $G$ then $AB$ is a subgroup of $G$. Prove $G$ is solvable.
So I felt like a good place to start is to let $|G| = p_1^{a_1}p_2^{a^2... |
H: Geometric proof for the half angle tangent
Using the fact that the angle bisector of the below triangle splits the opposite side in the same proportion as the adjacents sides, I need to give a geometric proof of the half-angle tangent $$ \tan \frac{\beta}{2} \ \ = \ \ \frac{\sin \beta}{1 \ + \ \cos \beta} \ \ . ... |
H: Constants in matrix integration
Suppose you have an integral of a matrix-valued function of the form:$$\int_a^b B A(t) C dt$$
In this case, the notation $A(t)$ is to denote a matrix that depends on the integrated variable $t$ (for example $A(t) = e^{tA}$), where the matrices $B$ and $C$ are independent of $t$. Is t... |
H: Showing the support of a sheaf may not be closed (Liu 2.5)
This is question 2.5 of Qing Liu.
I am new in algebraic geometry and really stuck on it and can't do anything to solve it.
The question:
Let $F$ be a sheaf on $X$. Let $\operatorname{Supp} F=\{x\in X:F_x\neq 0\}$. We want to show that in general, $\operator... |
H: Let $0\leq a \leq b \leq 1$. Then we have for all natural numbers $m\geq 2$ the inequality $b^{\frac m2}-a^{\frac m2} \leq\frac m2(b-a)$
Let $0\leq a \leq b \leq 1$. Then we have for all natural numbers $m\geq 2$ the inequality $b^{\frac{m}{2}}-a^{\frac{m}{2}} \leq\frac{m}{2}\left(b-a\right)$.
My first idea was to ... |
H: Let $x=\begin{bmatrix}3\cr4\end{bmatrix}$ and $A=\begin{bmatrix}0&x^T\cr x&0\end{bmatrix}$ is A diagonizable?
I had a problem: let $x=\begin{bmatrix}3\cr4\end{bmatrix}$ and $A=\begin{bmatrix}0&x^T\cr x&0\end{bmatrix}$ is A diagonizable?
But when I plug in the matrix x and its transpose into A the dimensions don't w... |
H: In △ABC, ∠A= 60$^∘ $, BC=12, BD⊥AC, CE⊥AB and∠DBC = 3∠ECB. Find EC in the form $a(\sqrt{b}+\sqrt{c})$
Given that $m \angle A= 60^\circ$, $BC=12$ units, $\overline{BD}\perp\overline{AC}$, $\overline{CE} \perp \overline{AB}$, and $m \angle DBC = 3m \angle ECB$, the length of segment $EC$ can be expressed in the form ... |
H: Is $(I \circ A - I \circ B)$ positive semi-definite if $A$, $B$ and $A - B$ are positive semi-definite?
Let $A$ and $B$ are positive definite and positive semi-definite matrices, respectively. $A - B$ is positive semi-definite.
Is it true that $(I \circ A - I \circ B)$ is positive-semidefinite?
I believe this sta... |
H: $f^{*}$ is surjective if and only if $f$ is injective
I’m having a hard time understanding this proof and I hope someone could help me.
Theorem: Let $f: A \rightarrow B$ a map. Think of this map as inducing the map $f^{*}: \mathcal{P}(B) \rightarrow \mathcal{P}(A)$. Then, $f^{*}$ is surjective if and only if $f$ i... |
H: Is there any visual representation on why (certain) trigonometric functions have infinite derivatives.
As far as I understand, the first derivative of a function gives you the slope at a particular point.
The second derivative would give the concavity.
The third derivative would give the rate of change of the conca... |
H: Find the area of a triangle $\triangle FGH$
In triangle $\triangle FGH$, $GM$ is a median that lies on $4x-y=27$; height $HA$ lies on $x-y+3=0$; $F$ is $(4,5)$. Find the area of the triangle $\triangle FGH$.
My attempt:
F is not on any of the lines. Intersection of the lines (10,13). Then, i struggle.
AI: With GF$\... |
H: How to use general recursion to generate a set of words?
I am just getting to recursion in one of my classes, and I'm a bit confused on how to go about generating a set of words and the notation.
Given the following question, how would I go about generating this set?
(assuming ∑={a,b}), the set of strings with twic... |
H: Prime ideals of $\Bbb C[x, y]$
In the exercise 3.2.E of Vakil's "Foundations of Algebraic Geometry", it is asked to prove that all the prime ideals of $\Bbb C[x, y]$ are of the form $(0)$, $(x-a, y-b)$ or $(f(x, y))$, where $f$ is an irreducible polynomial. In order to do so, it is suggested to consider a non-princ... |
H: Determine the lie algebra of the subgroup of SO(4)
Let $G\subset SO(4)$be the subgroup given below:
$$G=\left\{
\begin{pmatrix}
a & -b & -c &-d\\
b & a & -d & c\\
c & d & a & -b \\
d &-c & b &a
\end{pmatrix} : a,b,c,d\in \mathbb{R}, a^2+b^2+c^2+d^2=1\right\}$$
Find the lie algebra $\mathfrak{g}$.
I know that if $X\... |
H: Show that a martingale $\{X_n\}$ is bounded in $L^2$ if and only if $EX_n^2<\infty$ for each $n$ and $\sum_{n\ge1}E(X_{n+1}-X_n)^2<\infty$
A martingale $\{X_n\}$ is bounded in $L^2$ by definition if $\sup\limits_nEX_n^2<\infty$. Show that a martingale $\{X_n\}$ is bounded in $L^2$ if and only if $EX_n^2<\infty$ fo... |
H: arctan of ratio of two normal variables is uniform
Say $X, Y$ are independent standard normals, and $\theta = \arctan(Y/X)$. Prove that $\theta$ is uniformly distributed over it's range.
It is pretty intuitive that the distribution of $\theta$ would be uniform given a scatter plot of $X,Y$, but how can I mathematic... |
H: Translation of perfect set still perfect set
Let $P\subset\mathbb R$ be a perfect set. For each nonzero $r\in\mathbb R$, define $$ D_r=r\cdot P$$ Without checking the details $D_r$ still perfect set. I think, it is true. Does that correct ?
AI: $x \to rx$ is a homeomorphism so it certainly maps perfect sets to per... |
H: Simple Linear Regression Machine Learning Course
Your friend in the U.S. gives you a simple regression fit for predicting house prices from square feet. The estimated intercept is -44850 and the estimated slope is 280.76. You believe that your housing market behaves very similarly, but houses are measured in square... |
H: Can $y=10^{-x}$ be converted into an equivalent $y=\mathrm{e}^{-kx}$?
I was dealing with the values:
| Digits | Expression | Value |
|--------|------------|-----------------------|
| 1 | 10⁻¹ | 0.1 |
| 2 | 10⁻² | 0.01 |
| 3 | 10⁻³ |... |
H: In a quiz with 13 people, what are the probabilities that exactly 12, exactly 11 and exactly 10 people answer correctly?
A group of 13 people, p1, p2, p3, ... , p13 answers the same question. The probability that a person answers correctly are known but differs between the people. For example the probability that p... |
H: Proof of The Third Isomorphism Theorem
Here's what I'm trying to prove right now:
Let $V$ be a vector space over $\mathbb{F}$. Let $M$ be a linear subspace of $V$ and $N$ be a linear subspace of $M$. Prove that the mapping $x+N \mapsto x+M$ between the quotient spaces $V/N \to V/M$ is linear with kernel $M/N$. Then... |
H: Is this set of linear maps valid?
Notation: The set of all linear maps from a vector space $V$ to a vector space $W$ (over a field $\mathbb{F}$) is denoted $\mathcal{L}(V, W)$.
The question states:
Show that $\{ T \in \mathcal{L}(\mathbb{R}^5, \mathbb{R}^4) : dim(null(T)) > 2\}$ is not a subspace of $\mathcal{L}... |
H: Calculate $\sum_{r=0}^n \cosh(\alpha+2r\beta)$
I try to calculate $\sum_{r=0}^n\cos(\alpha+2r\beta)$ first to get some insights.
First we prove that $\sum_{r=0}^\infty\cos(\alpha+2r\beta)=0$. (The statement and the proof is flawed, I will edit it later.)
$Proof$: For $\sum_{r=0}^\infty\cos(\alpha+2r\beta)$ is the r... |
H: prove that $f(x)=\sum_{|\alpha|\leq k}\frac{1}{\alpha !}D^{\alpha}f(0)x^{\alpha} + O(|x|^{k+1})$
If $\alpha$ is multiindex and $f$ is smooth,prove that $f(x)=\sum_{|\alpha|\leq k}\frac{1}{\alpha !}D^{\alpha}f(0)x^{\alpha} + O(|x|^{k+1})$. The hint is to use taylors form for $g(t)=f(tx)$. If i do this i will found t... |
H: Necessary condition for x>0 being an integer
I was trying to solve a number theory problem and then I realized that I was needing to verify (prove or disprove) the following ''fact'' about numbers. I would appreciate any help.
Q: Suppose $x >0$ is such that $x^n \in \mathbb{Z}$ for all $n \geq 2$, then $x$ must be ... |
H: Does the Null spaces of matrix $n\times n$ matrix $A$ and matrix $BA$ equal to each other if the matrix $B$ is invertible?
As the title says, both $A$ and $B$ are $n\times n$ matrices, I want to prove the Null spaces $Null (A)$ = $Null(BA)$.
I do not know if the statement is correct,
and I am not sure whether my de... |
H: Hausdorff and non-discrete topology on $\mathbb{Z}$
Construct a topology $\mathfrak{T}$ on $\mathbb{Z}$ such that $\mathbb{Z}$ is Hausdorff and non-discrete with respect to $\mathfrak{T}$.
$\textbf{My idea}$ : We know that $\mathbb{Q}$ is Hausdorff and non-discrete with respect to the topology inherited from $\math... |
H: Rearrangement diverges then original series also diverges?
Question: if $\sum y_n$ is "any" rearrangement of series $\sum x_n$ , where $\sum x_n$ is series of positive terms. Then, if $\sum y_n$ diverges then original series $\sum x_n$ also diverges?
I think yes. Because series $\sum x_n$ is series of positive t... |
H: Given a basis $\mathcal{B}$, can I assume that $\mathcal{B}$ is orthonormal?
Let $E$ be a vector space over $\mathbb{C}$ such that $\text{dim}(E)=n \in \mathbb{N}$. Let $\mathcal{B}:=\{e_1,e_2,\cdots, e_n\}$ be a basis of $E$. I know that if $ E $ is vector space with inner product $\langle \cdot, \cdot \rangle : E... |
H: Convergence in L2 (up to a constant) implies convergence in probability?
Suppose we have a sequence of random variables $\{X_n\}$ such that $\mathbb{E} [|X_n|^2] = a_n + c$ where $a_n \to 0$ is a decreasing sequence of positive real numbers and $c > 0$ is a constant. Then can we say anything about the convergence i... |
H: Absolutely continuous function with bounded derivative on an open interval is Lipschitz
I've come across a question which states that one can prove $f:\mathbb{R} \rightarrow \mathbb{R}$ is Lipschitz iff $f$ is absolutely continuous and there exists $M \in \mathbb{R}$ such that $|f'(x)|≤ M$ almost everywhere.
I've o... |
H: Proving only certain methods of Integration work on certain problems
I have not studied Pure Maths (Stuff which includes rigorous proving) yet however I did take classes on Integration.
There were certain problems which we could do only by certain ways to get the results for instance the integral of $ln(x)$ can be ... |
H: How to solve this ODE: $y'(x) e^x = y^2(x)$?
I am trying to solve the differential equation
$$y'(x) e^x = y^2(x) \quad (DE) $$
This is a Bernoulli form DE i.e $y'(x) + a(x)y(x) = b(x)y^r(x)$, where $r = 2, a(x) = 0, b(x) = \frac1e $
Let $u(x) = y^{1-r} = y^{-1} \iff u'(x) = -y^{-2}(x) y'(x)$
Then for $y \neq 0$: ... |
H: Is "An undirected graph $G(V,E)$ has at least $|V|-|E|$ connected components" a true statement?
I'm taking this Coursera's course on Graph Theory, which is part of a specialization in discrete math for CS, offered by University of California, San Diego: https://www.coursera.org/specializations/discrete-mathematics
... |
H: Evaluating $\lim_{n\rightarrow\infty}\int_0^n\frac{(1-\frac{x}{n})^n}{ne^{-x}}dx$
Question: Find $\lim_{n\rightarrow\infty}\int_0^n\frac{(1-\frac{x}{n})^n}{ne^{-x}}dx$.
My thoughts: First, I'd like to bring the limit inside the integral, because $\lim_{n\rightarrow\infty}\frac{(1-\frac{x}{n})^n}{ne^{-x}}=\frac{e^{-... |
H: Find pdf of transformation of two random variables using CDF
Let $X,Y \sim$ Uniform$(0,1)$ be independent. Find the PDF for $X/Y$.
Let $Z=X/Y$. We want to find $F_z(z)=P(Z \leq z)=P(X/Y \leq z)$.
We can make $Y$ super small with fixed $X$, and conversely we can make $X$ really small with fixed $Y$. Thus it appear... |
H: Number theory question: Prove $27\mid a+b+c$ if $(a-b)(b-c)(c-a)=a+b+c$.
Integers $a,b$ and $c$ satisfy $(a-b)(b-c)(c-a)=a+b+c$. Prove $27\mid a+b+c$.
AI: First, assume that $3 \nmid (a+b+c)$. Then, $3 \nmid (a-b)(b-c)(c-a)$, which means that $a,b,c$ must be distinct modulo $3$. However, we still get $a+b+c \equiv ... |
H: Examples of closed manifolds?
In Spivak's Diff Geom (vol.1), p.19, he says a closed manifold is non-bounded and compact (A point in boundary has a neighborhood homeomorphic to half-space). I don't know a non-trivial example of that.
For example, compact subset of $\mathbb{R}^2$ is usually closed set and has boundar... |
H: Integration by parts on manifold without boundary
Suppose $M$ is a compact Riemannian manifold without boundary. Does the integration by part hold? For example, do we have
$$
\int_M\nabla_g u\nabla_g v dx=\int_M-\Delta_g uvdx?
$$
Here, $\nabla_g$ and $\Delta_g$ are defined w.r.t the Riemannian metric $g$ on the man... |
H: How to solve this ODE: $x^3dx+(y+2)^2dy=0$?
I am trying to solve $$ x^3dx+(y+2)^2dy=0 \quad( 1)$$
Dividing by $dx$, we can reduce the ODE to seperate variable form, i.e
$$ (1) \to (y+2)^2y'=-x^3 $$
Hence,
$$ \int (y(x)+2)^2y'(x) dy = \int -x^3dx = - \frac{x^4}{4} + c_1$$
This LHE seems to be easy to solve using i... |
H: Non-negative convergent series $a_n$ where $\lim\sup na_n >0$
From Carothers, Chapter 1, Exercise 34:
Suppose that $a_n \geq 0$ and $\sum_{n=1}^\infty a_n< \infty$. Give an example showing that $\lim\sup_{n\to \infty} n a_n > 0$ is possible.
Looking at the sequence $n a_n$, there is a subsequence $n_k a_{n_k}$ th... |
H: Is there a third "version" of difference-of-means hypothesis testing?
I'm following Schaum's Outlines for statistics as well as taking a course and I'm getting mixed up with the way hypothesis testing is done for differences of means.
First the class described a "two-sample unpaired t-test":
$$
t = \frac{(\bar{x}_1... |
H: A double check of my answer. Involves geometry and algebra
The $2$ slices cut into by Lee combine to make one big triangle. I used the Pythagorean theorem to find the length of the missing side (hypotenuse) to be $\sqrt{128}$, and I used the Pythagorean theorem again to find the length of the side John's slice bor... |
H: Definition of vertex in graph theory
What is the definition of vertex in graph theory?
Is it just an endpoint of an edge.If we consider it like that then won't there be an uncountably many number of vertex in every graph because every point can be considered as a vertex right also won't there be an uncountably many... |
H: Is there a general formula for infinite series of a rational function
Is there some sort of formula to calculate $$\sum_x \frac{P_1(x)}{P_2(x)}$$
In particular, what is $$\sum_x \frac{1}{ax^2+bx+c}$$
And what is $$\sum_x \frac{1}{x^3+1}$$
AI: $$S_p=\sum_{x=0}^p \frac{1}{ax^2+bx+c}=\frac 1a\sum_{x=0}^p \frac{1}{(x-r... |
H: A prime ideal is either maximal right ideal or small right ideal.
Definition:- A right ideal $I$ of a ring $R$ is called small right ideal if $I+J=R\implies J=R$ for any right ideal $I$ of $R$.
My Question:- A prime ideal is either a maximal right ideal or a small right ideal.
I have tried to find counterexamples b... |
H: How to evaluate $\int_{0}^{\infty} x^{\nu} \frac{e^{-\sqrt{x^2+a^2}}}{\sqrt{x^2+a^2}} \, dx$?
$$\int_{0}^{\infty} x^{\nu} \frac{e^{-\sqrt{x^2+a^2}}}{\sqrt{x^2+a^2}} \, dx$$
Is it possible to calculate this for $a>0$ and $\nu=0, 2$ ?
I think the result seems to include exponential integral function, but I failed to ... |
H: Calculation of $\left(\frac{1}{\cos^2x}\right)^{\frac{1}{2}}$
Shouldn't $\left(\frac{1}{\cos^2x}\right)^{\frac{1}{2}} = |\sec(x)|$?
Why does Symbolab as well as my professor (page one, also below) claim that $\left(\frac{1}{\cos^2x}\right)^{\frac{1}{2}} = \sec(x)$, which can be negative? Also, the length of a vecto... |
H: MIN-FORMULA $\in$ NP
I am thinking about the following problem: Show that MIN-FORMULA $\in$ NP.
MIN-FORMULA is the set of minimal boolean formulas, i.e. formulas such that there is no shorter formula that computes the same boolean function.
I would like to have an advise whether or not the following approach/idea i... |
H: Action of the $n$-th roots of unity on $\mathbb{A}^2$
The following appears as Example 1.9.5 (c) in Fulton's Intersection theory.
Let $X$ be the quotient of $\mathbb{A}^2$ obtained by identifying $(s,0)$ with $(\mu s,0)$ for all $n$-th roots of unity $\mu$; equivalently, $$X = \operatorname{Spec} K[s^n, st, s^2t, ... |
H: How can we prove mgf of sample proportion of binomial distribution converges to exp(pt)?
$S_{n}$ follows Binomial(n,p).
$X_{n}$ is the sample proportion which is $X_{n} = S_{n}/n$.
How can we prove $\lim_{n \to +\infty} M_n{(t)} = e^{pt}$ ?
What I found is
\begin{align}
\lim_{n \to +\infty} M_n{(t)} &=\lim_{n \to +... |
H: Interchange of diagonal elements with unitary transformation
I have a matrix:
$$\left(\begin{array}{lll}
a & 0 & 0 \\
0 & b & 0 \\
0 & 0 & c
\end{array}\right)$$
Which I want to change to:
$$\left(\begin{array}{lll}
a & 0 & 0 \\
0 & c & 0 \\
0 & 0 & b
\end{array}\right)$$
How can I do that with a unitary transforma... |
H: Is $\mathbb{Q}\;\cong\; (\prod_{n\in\omega}\mathbb{Z}/p_n\mathbb{Z})/\simeq_{\cal U}$?
Let ${\cal U}$ be a non-principal ultrafilter on $\omega$, and for each $n\in\omega$, let $p_n$ denote the $n$th prime, that is $p_0 = 2, p_1=3, \ldots$.
Next we introduce the following standard equivalence relation on $\big(\pro... |
H: What's the difference between a Proof System and a Theory?
I found this question Is the axiom of induction required for proving the first Gödel's incompleteness theorem?
I am apparently under the (wrong) impression that a theory and a proof system are synonyms. I'm pointing this out because from the accepted answer... |
H: Why do we consider the zeroes of the expression when solving rational inequalities?
When finding the solution set of rational inequalities (of the form $ {x-3\over x+2} \geq 0 $ etc.) , why do we consider the "zeroes" of the expression.
So I've been taught to solve this as follows:
First,
$x-3 = 0 \implies x = 3 \... |
H: Hankel function expansion for large arguement
The leading order behaviour of the Hankel function for large arguments is known to be
$$
H_{n}^{(1)}(z)\sim\sqrt{\frac{2}{\pi z}}e^{i\left(z-\frac{n\pi}{2}-\frac{\pi}{4}\right)}
$$
as $z\to\infty$. I would like to know what the analytical form of the full expansion is i... |
H: Finding $P[X+Y > 1, X > 1]$
I'm trying to solve the following problem: i have two independent exponentially distributed r.v. $X$ and $Y$ both with $\lambda = 1$. I want to know the probability $P[X+Y > 1, X > 1]$. Since they are independent, i wrote the joint p.d.f. as $f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y) = e^{-x}\c... |
H: Bessel functions in terms of the confluent hypergeometric function
I am familiar with the representation of the Bessel functions as
$$J_m(x) = \left(\frac{x}{2}\right)^m \sum_{k = 0}^{\infty} \frac{(-1)^k}{k! (k + m)!} \left(\frac{x}{2}\right)^{2k},$$
for some integer $m$.
Now, I have come across the representation... |
H: Uncountability of $\mathbb{R}$
I was trying to follow this proof of the uncountabilty of R and at first it seemed clear, but when I tried to explain it to myself I realized I didn't really understand one of the steps.
The proof is by contradiction, using the nested intervals theorem:
Assume $\mathbb{R}$ is countabl... |
H: A right triangle has a certain angle twice of another angle in the triangle. Find the maximum number of integer side lengths it has.
A right triangle has a certain angle twice of another angle in the triangle. Find the maximum number of integer side lengths it has.
How I tried working on the problem:
There are $2$... |
H: Is $(a/b)-1$ approximately equal to $\log_e (a/b)$
I was reading an article where in one of the steps we were trying to calculate the daily return. It said
Return = (a / b) – 1
It then said, this equation can be approximated to:
Return = Log e (a/b)
Could someone explain a proof around how these are equal? Why $\... |
H: Prove that $\text{tr} (\phi \otimes \psi) = \text {tr} \phi \text {tr} \psi $.
Let $E,F$ be two vector spaces of dimension $n$ and $m$ respectively and let $\phi : E \rightarrow E$,
$\psi : F \rightarrow F $ be two linear transformations. Prove that $\text{tr} (\phi \otimes \psi) = \text {tr} \phi \text {tr} \psi $... |
H: All norm in a finite dimensional topological space are equivalent
Definition
If $V$ is a finite dimensional vector space then we say that the norm $||\cdot||_1$ and $||\cdot||_2$ are equivalent if and only if there exist two positive constant $m$ and $M$ such that
$$
m||v||_1\le||v||_2\le M||v||_2
$$
for any $v\in... |
H: How to prove that these functions do not intersect?
I want to prove that these two functions $f(x)$ and $g(x)$ do not intersect for $x>1$:
$$f(x)=\cosh \left(\frac{2 \sqrt{2} \pi x \left(x^2-1\right) \cosh (\pi x)}{\sqrt{x^4+6 x^2+\left(x^2-1\right)^2 \cosh (2 \pi x)+1}}\right)$$
$$g(x)=\frac{4 x^2+\left(x^2-1\r... |
H: Periodic functions for the definite integral
Considering this question where there is this integral:
$$\int_{0}^{4\pi} \ln|13\sin x+3\sqrt3
\cos x|\mathrm dx \tag 1$$
Easily all the periodic function $$a'\sin(x)+b\cos(x)+c=0 \tag 2$$ can be written as:
$$A\sin(x+\phi)+c=0, \ A=\sqrt{a'^2+b^2}\quad \text{ or }\quad... |
H: How to prove that there is an infinite number of primitive pythagorean triples such as $b=a+1$ and $2 | a$?
I need to prove that there is an infinite primitive pythagorean triples such as $b=a+1$ and $2 | a$ but I don't know how.
I know that $(2st, s^2-t^2, s^2+t^2)$ is a primitive pythagorean triple, then I tried ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.