text stringlengths 83 79.5k |
|---|
H: Intuition behind Doob's Optional Stopping theorem
I would like to ask about intuitively interpreting the results of Doob's Optiona Stopping Theorem applied to standard Brownian motion.
The theorem provides three conditions, under which a stopped process is a martingale. One of these conditions is that the stopping ... |
H: The minimal number with a given number of divisors
My question comes here https://doi.org/10.1016/j.jnt.2005.04.004. Let $A(n)$ which assigns to each number $n$ the smallest number with exactly $n$ divisors. Is it true that $A(n)<A(n+1)$?
AI: No. The smallest number with exactly $5$ divisors is $2^4=16$. The smal... |
H: Inequality in a Hilbert Space: $\sup_{||f||_{2}\leq 1}||fg||_{2}\leq C ||g||_{2}$
Let $f,g \in L^2(0,1)$ My question is the following: is there a constant $C>0$ such that
$$\sup_{||f||_{2}\leq 1}||fg||_{2}\leq C ||g||_{2},$$
all I know is that we have
$$\sup_{||f||_{2}\leq 1}<f.g>\leq C ||g||_{2},$$
using cauchy sc... |
H: Closure of a topological space $Y$
Let $X$ be a topological space, and let $Y \subseteq X$. Prove that $\overline{Y} = \displaystyle\bigcap_{F \textrm{ is closed and }Y\subseteq F} F$.
$x \in \overline{Y}$ iff for all $U$ open set in $X$ such that $x \in U$ satisfy $Y \cap U \neq \emptyset$.
AI: Let $x \in \overlin... |
H: $T^2$ is measure preserving but $T$ is not measure preserving
I have had trouble with this exercise. Can anyone help?
Give an example of a probability space $(\Omega, \mathcal{F}, P)$ and a measurable mapping
$T : \Omega \rightarrow \Omega$ such that $T^2$ is measure preserving but $T$ is not measure preserving.
Th... |
H: Can I square both sides of inequality for these functions?
I have two functions $f(x)$ and $g(x)$ for $x>0$. Both functions are monotonically increasing and $f(x)>5$ and $g(x)>0$ .
I know that $f(x)>\sqrt{g(x)}$.
Then, can I conclude that $f(x)^2>g(x)$ for $x>0$?
AI: Yes, because both $f$ and $\sqrt g$ are positive... |
H: If $T(p(t)) = p(t+1)$ then find its minimal polynomial where $T$ is a linear operator from $\Bbb{P_n} \rightarrow \Bbb{P_n}$
I tried substituting $p(t)$ with $p(t-1)$ and then taking the transformation to get some kind of annhilating polynomial but that just gave me trivial solutions. Also after spending an hour I ... |
H: Elements of the ring of multipliers are integral
For $K$ a number field, $\{\alpha_1, \dots, \alpha_n\}$ a basis of $K/\mathbb{Q}$ and $M = \mathbb{Z}\alpha_1 + \dots + \mathbb{Z}\alpha_n$, the corresponding ring of multipliers is defined as
$$ \mathcal{O} = \{ \alpha \in K : \alpha M \subseteq M \}. $$
I want to p... |
H: Prove that a Tower of Height $H$ can be built if $H*(H+1)/2 = R + G$
Let us define a Red-Green Tower:
Each level of the red-green tower should contain blocks of the same
color.
At every Increase in the level, number of blocks in that level is one less then previous level.
Prove that a Red-Green Tower of Height $H... |
H: Questions about counting the number of triples arranged in geometric progression
Problem: Three tickets are chosen from a set of $100$ tickets numbered $1,2,3,\ldots,100$. Find the number of choices such that the numbers on the three tickets are in geometric progression.
Solution: Let $k, n \in \mathbb Z_+$ s.t... |
H: Remainder of $15^{81}$ divided by $13$ without using Fermat's Little theorem.
I was requested to find the congruence of $15^{81}\mod{13}$ without using Fermat's theorem (since that is covered in the chapter that follows this exercise). Of course I know that by property $15^{81} \equiv 2^{81} \pmod{13}$, but how cou... |
H: Large N limit of a particular sum
I'm working through a statistical physics book and one of the problems makes the claim that the quantity:
$$H =\frac{1}{N}\sum_{n=1}^N\frac{1}{\frac{a}{N} + \frac{b}{N^{5/3}}(n^{5/3}-(n-1)^{5/3})}$$
can be expressed in the large $N$ limit as the integral:
$$H = N\int_0^1 \frac{\te... |
H: Partial Fraction Decomposition of $\frac{1}{x^2(x^2+25)}$
I have been reviewing some integration techniques and have been searching for tough integrals with solutions online. When I was going through the solution, however, I found a discrepancy between my solution and theirs and think what I did was correct instead... |
H: $ker(T)^{\bot} = \overline{im(T^*)}$ if $T$ is a linear operator between Hilbert spaces
Let $T$ be a linear operator.
For any underlying normed spaces it holds that
$$ker(T)^{\bot} \subset \overline{im(T^*)},$$
but if they are both Hilbert spaces we get
$$ker(T)^{\bot} = \overline{im(T^*)}.$$
Now my question is:
Ho... |
H: Convergence in distribution - Gamma distribution/degenerate distribution
I got a gamma distribution which is defined as followed
$G_{n,\lambda}=\frac{\lambda e^{-\lambda x}(\lambda x)^{n-1}}{(n-1)!}$.
The paper I am currently reading says, that die $G_{n, n/t}$ converges in distribution to the degenerate distributi... |
H: Understanding the double integral $\int_0^\infty\int_0^t f(x)g(t-x)dxdt$
I am dealing with the integral $\int_0^\infty\int_0^t f(x)g(t-x)dxdt$ and $g(x)=0$ if $x<0$.
I need to arrive to $\int_0^\infty f(x)dx\int_0^\infty g(t)dt$.
Using Fubini, I have:
$$\int_0^\infty\int_0^t f(x)g(t-x)dxdt=$$
$$\int_0^t f(x)\int_0^... |
H: Integration with absolute value and a constant range in it
The question is
$\int_1^4 |a^2-x^2|dx=\cdots$ for $1<a<4$
Now that I am confuse since I don't have any idea on how to separate into several definite integrals. I mean, when $a=2$, then we can separate it into $\{1,2\}$ and $\{2,4\}$. However, when $a=3$, ... |
H: Proof of the inclusion sets
I would know why if I have $A \subseteq B ,\:$ I obtain $P(B)\geq P(A)$
Thanks to everyone
AI: Hint:
Write $B= A\cup (B\smallsetminus A)$ and use the axioms of a probability measure. |
H: is there a function $\gamma(x)$ where when $a$ & $b$ and $a+1$ & $b+1$ are co-prime, $\gamma(\frac{a}{b})>\gamma(\frac{a+1}{b+1})$
is there a function $\gamma(x)$ where when $a$ & $b$ and $a+1$ & $b+1$ are co-prime, $\gamma(\frac{a}{b})>\gamma(\frac{a+1}{b+1})$
when you start with $\gamma(\frac{1}{2})$ you get an i... |
H: every root of unity $x$ determines a degree one representation of $G$
I was told that if $G=\langle g \rangle$ is a cyclic group then every root of unity $x$ determines a degree one representation of $G$. But what is this representation? $\rho(x)=?$
AI: It is only $n$th roots of unity, where $g$ has order $n$. Then... |
H: Calculate residues at all isolated singularities of $f(z)=\frac{z^2+4}{(z+2)(z^2+1)^2}$.
Calculate residues at all isolated singularities of $f(z)=\frac{z^2+4}{(z+2)(z^2+1)^2}$.
So I found the isolated singularities to be $z=-2$ and $z=i$.
Then I found the residue at $z=-2$.
$\operatorname{res}(f,i)=\lim_{z\to -2... |
H: Is every commutative ring isomorphic to a product of directly irreducible rings?
In the following, all rings are assumed to be commutative, with multiplicative identity.
A ring $R$ is said to be directly irreducible if it is not isomorphic to the direct product of two non trivial rings. An equivalent condition is t... |
H: Solve for x, such that: $\sum_{n=0} \frac{x}{4^n} = \sum_{n=0} \frac{1}{x^n}$
The question is stated as the following to solve for x: $$x+\frac{x}{4}+\frac{x}{16}+\frac{x}{64}+...=1+\frac{1}{x}+\frac{1}{x^2}+...$$
I have so far determined their infinite series as shown in the title and have been treating both these... |
H: Is the map $T \mapsto T^*T$ lower-semicontinuous in the weak operator topology?
Let $H$ be a separable Hilbert space and $\mathcal{L}(H)$ the set of bounded operators on $H$.
If $D \in \mathcal{L}(H)$ is a positive operator, $D \geq 0$, is true that the set
$$\{T \in \mathcal{L}(H)\mid T^*T \leq D\}$$ is closed in ... |
H: Derivative of $h(x,t)=g\left(\frac{x}{t^2}\right)$
Can someone please explain me the method to get the partial derivative of a function like:
$h(x,t)=g\left(\frac{x}{t^2}\right)$ where $g$ is a differentiable function from $\mathbb{R}\to\mathbb{R}$.
I know that if we have a function $f(x(t,s),y(t,s))$ then
$\frac... |
H: Orthonormal columns of block matrices expanded with Kronecker products
Let $W_{i,1},W_{i,2},W_{i,3} \in \mathbb{R}^{n \times n}$, $i \in {1,2}$ be such that
$$
\eqalign{
\Big[\matrix{W_{i,1}^T & W_{i,2}^T & W_{i,3}^T}\Big]
\left[\matrix{W_{i,1}\\W_{i,2}\\W_{i,3}}\right]
= I
}
$$
where $I$ is the ident... |
H: What is the order and structure of Aut(SL$_2(p)$)?
Here $p$ is a prime. We know that for $Z(SL_2(p)) = \{ \pm I \}$ and $\lvert SL_2(p)\rvert= p^3-p$ so there are $\frac{p^3-p}{2}$ inner automorphisms. What is the outer automorphism group?
AI: It is $\mathrm{PGL}_2(p)$ for $p\geq 5$, which I assume you are thinking... |
H: How can I prove $A − A(A + B)^{−1}A = B − B(A + B)^{−1}B$ for matrices $A$ and $B$?
The matrix cookbook (page 16) offers this amazing result:
$$A − A(A + B)^{−1}A = B − B(A + B)^{−1}B$$
This seems to be too unbelievable to be true and I can't seem to prove it. Can anyone verify this equation/offer proof?
AI: \begin... |
H: Why is the solution to a non-homogenous linear ODE written in terms of a general fundamental solution and not a matrix exponential?
Why is the solution to a non-homogenous linear ODE written in terms of a general fundamental solution and not a matrix exponential? Generally, I see the solution to a non-homogenous l... |
H: Ideal of whole numbers generates whole ring, can we find a linear combination of 1 with coefficients always whole numbers?
Let $R$ be a unitary ring. For $a,b \in \mathbb{Z}$ we can embed $a,b$ in $R$. Now consider the ideal $I:=(a,b)$ in $R$. If $(a,b )= R$ we find $c,d\in R$ such that $ac+bd = 1$. Can we always c... |
H: What does it mean for F(Lim X) to be 'naturally isomorphic' to Lim(F(X))
Let F be a left adjoint functor. Given any diagram X, what does it mean for F(Lim X) to be naturally isomorphic to Lim(F(X)). What are the functors here which are naturally isomorphic?
AI: The functors are $$\mathcal{C}^I\overset{F^I}{\longrig... |
H: Distance between two closed subsets of $\mathbb{R}$
Suppose $A$ and $B$ are two closed subsets of $\mathbb{R}$, and assume that : $$\text{dist}(A,B)=\inf_{(a,b)\in A\times B}|a-b|=0.$$
Is it true that $A\cap B\neq\varnothing$ ?
I found a counter-example using arithmetics : $A=\mathbb{N}^\star$ and $B=\pi\mathbb{N}^... |
H: Understanding matrix norm and quadratic equations
Suppose I have two quadratic expressions
$$(a - b)^2$$ and $$(a-c)^2$$
where $a, b, c$ are real numbers, then I think the following holds:
$$(a - b)^2 < (a - c)^2$$
iff $|a - b| < |a - c|$.
Now suppose I have two expressions in matrix form:
$$((A - B)v)^T((A-B)v)$$
... |
H: Calculating second derivatives with curves
I always thought that the following is true:
Let $M$ be a smooth manifold, $f \in C^2(M)$, and $X \in \Gamma(TM)$. Then
$$ X^2f(x) = \frac{d^2}{dt^2} (f \circ \gamma)(0) $$
for any curve $\gamma$ with $\gamma(0) = x, \gamma'(0) = X_x$.
Here, the LHS should be interpreted... |
H: I don't understand Gödel's incompleteness theorem anymore
Here's the picture I have in my head of Model Theory:
a theory is an axiomatic system, so it allows proving some statements that apply to all models consistent with the theory
a model is a particular -- consistent! -- function that assigns every statement t... |
H: Question concerning prime ideals of $\mathbb{C}[x,y]$
I know that $(0)$, $(x-a,y-b)$ for $(a,b)\in\mathbb{C}^2$ and $(f(x,y))$ for $f(x,y)$ irreducible in $\mathbb{C}[x,y]$ are all prime ideals in $\mathbb{C}[x,y]$.
What I'd like to understand is why they are the only prime ideals. In particular, I'd like to know w... |
H: Let $A\cup B$ be open, disconnected in $\Bbb{R}^2$ where $A,B$ are non-empty, disjoint. Are both $A,B$ open in $\Bbb{R}^2$?
I have tried it in the following manner-
Assume $A$ is not open. Then $\exists x\in A$ such that $x\notin A^\circ$ i.e. $\forall \epsilon>0, B(x,\epsilon)\not\subset A$ .
Now $x\in A\cup B$, o... |
H: How do we apply the dominated convergence theorem to conclude the proposed claim?
Definition
Let $\{f_{\lambda}:\lambda\in\Lambda\}$ be a collection of functions in $L^{1}(\Omega,\mathcal{F},\mu)$. Then for each $\lambda\in\Lambda$, by the dominated convergence theorem and the integrability of $f_{\lambda}$,
\begin... |
H: Solve the initial value problem: $\frac{dy}{dx} = e^{x+y}$, given $y(0)=0$.
I have attempted the question several times so far, and I have always reached the same answer that differs from the solution, any advice would help greatly!
My attempt
$$\frac{dy}{e^y} = e^x dx$$
Taking the integral, I got $$-e^{-y} = e^x +... |
H: If evey linear program can be transformed to an unconstrained problem, then the optimum is unbounded because the objective is linear?
Since optimization problems with linear equality constraints can be converted into an unconstrained problem this should apply for linear programs in standard form, right?
But doesn't... |
H: Prove $\sum_{n=1}^\infty \frac{1}{a_n}$ is divergent if $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ are both convergent
In my practice midterm there is a multiple choice question that I thought was relatively straight forward but the solutions gave an answer that was unexpected to me.
Question: If $\sum_{n=... |
H: If $f(2n)=\frac1{f(n)+1}$ and $f(2n+1)=f(n)+1$ for all $n\in\Bbb N$, then find $n$ such that $f(n)=14/5$.
The set $\mathbb{N}$ is the set of nonnegative integers. Let $ f : \Bbb{N} \rightarrow \Bbb{Q}$ be defined such that
1.) $f(2n) = \dfrac{1}{f(n)+1}$ for all integers $n>0$, and
2.) $f(2n + 1 ) = f(n)+1$ for a... |
H: How does the process of simplifying imaginary numbers actually work?
Sorry in advance if this is a really stupid question
In class I've been told that $$\sqrt{-25} = 5j $$
Converting $\sqrt{-25} $ into $5j$ is straightforward for me, but I don't understand how it works
Doesn't the property of $\sqrt{xy} = \sqrt{x}... |
H: If $\text{ord}_m(c)=n$, find $\text{ord}_m(c^2)$
I'm trying to solve the following problem:
if $\text{ord}_m(c)=n$, find $\text{ord}_m(c^2)$
Here's what I have so far (admittedly, not much):
We know that $c^n \equiv 1 \pmod{m}$, equivalently, $c^n = 1 + mp$, $p\in\mathbb{Z}$
We know $c^2$ = $c\cdot c$.
And, of c... |
H: Kernels of commuting linear operators on infinite dimensional vector space
If $S$ and $T$ are commuting operators on an infinite dimensional vector space $V$, it is in general true that
$$\ker S + \ker T \subseteq \ker(ST),$$
but in general equality does not hold. A simple example is given by $S = T = \frac{d}{dx}$... |
H: Linear programming with min of max function
I have to write the linear program which minimizes this function :
$$y = \max_j \sum_{i=1}^{n}c_{ij}x_{ij}$$
My book says that this is not a linear function but it can be trasformed into one using the minimizing program $\min y$ with the conditions :
$$ \sum_{i=1}^{n}c_{... |
H: L'hopital rule fails with limits to infinity?
$$ \lim_{n \to \infty} \frac{1 +cn^2}{(2n+3 + 2 \sin n)^2} = ? $$
if I factor the $n^2$ out of denominator,
$$ \lim_{n \to \infty} \frac{ 1 + cn^2}{ n^2 ( 2 + 3n^{-1} + 2 \frac{ \sin n}{n} )^2}$$
And take limit directly, I get the answer as
$$ \frac{c}{4}$$
However, I... |
H: Finding the height of a Pyramid where the sides are given by an equation
Problem:
The vertex of a pyramid lies at the origin, and the base is perpendicular to the x-axis at $x = 4$. The cross sections of the pyramid perpendicular to the x-axis are squares whose diagonals run from the curve $y = -5x^2$ to the curve ... |
H: Sequence of Lebesgue integrable functions bounded in norm converges pointwise
I've come across a problem which states:
Given a sequence of integrable functions $\{f_k\}$ ($k≥1$) on $[0,1]$ with the property that $||f_k||_1 ≤ \frac{1}{2^k} $, then $f_k \rightarrow 0$ pointwise almost everywhere on $[0,1]$.
I'm not ... |
H: How should one understand the "indefinite integral" notation $\int f(x)\;dx$ in calculus?
In calculus, it is said that
$$
\int f(x)\; dx=F(x)\quad\text{means}\quad F'(x)=f(x)\tag{1}
$$
where $F$ is a differentiable function on some open integral $I$. But the mean value theorem implies that any differentiable funct... |
H: I need to help with a formula to find the length of a line.
I have a rectangle which I know the width and height of it.
I need to draw a line inside the rectangle and the information that I have include knowing the starting point of the line as well as the angle of the line.
My question is how can I calculate the l... |
H: How to evaluate the following limit: $\lim_{x\to 0}\frac{12^x-4^x}{9^x-3^x}$?
How can I compute this limit
$$\lim_{x\to 0}\dfrac{12^x-4^x}{9^x-3^x}\text{?}$$
My solution is here:
$$\lim_{x\to 0}\dfrac{12^x-4^x}{9^x-3^x}=\dfrac{1-1}{1-1} = \dfrac{0}{0}$$
I used L'H$\hat{\mathrm{o}}$pital's rule:
\begin{align*}
\lim_... |
H: Show that $\mathfrak{m}_p$ is an ideal in $\mathcal{O}_V.$
I'm working through Algebraic Geometry: A Problem Solving Approach by Garrity et al, and I have found myself stuck on Exercise 4.13.1, which is the section Points and Local Rings.
Let $V = V(x^2 + y^2 - 1) \subset \mathbb{A}^2(k).$ Let $p = (1, 0)
\in V.$... |
H: What is the difference between a semiconnected graph and a weakly connected graph?
These are the definitions I have found:
Semiconnected: if, and only if, for any pair of nodes, either one is reachable from the other, or they are mutually reachable.
Weakly connected: if, and only if, the graph is connected when the... |
H: A good description of the set of inner points of a polyhedron.
How can I get a good description of the inner points of a polyhedron? I am trying to calculate the volume of a polyhedron by change of variables, but I can't describe the set of points of the polyhedron properly (given its vertices).
I look for a descri... |
H: Inequality concerning functions quadratic decay of distribution $m(|f|>\lambda)\leq C\lambda^{-2}$
This was a problem in a prelim that I was not able to solve back in the day.
Let $m$ denote the Lebesgue measure on $\mathbb{R}^n$. Suppose $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue measurable function such ... |
H: Character table of quaternions $Q_8$
I wanted to compute the character table of the quaternion group Q_8, and to this end I was reading this answer. For the degree 1 representation I used this question but for the degree 2 representation I don't understand how the hint from the first link works.
AI: The hint is to ... |
H: Directly Calculating Birthday Paradox Probabilites
I am trying to calculate the probability of at least 2 people sharing a birthday in a group of 4 people. I understand that calculating it as 1-P(no shared birthdays) is simpler, but I would like to understand the counting method by doing it directly.
My attempt for... |
H: What is the ordinary differential equation for double exponential summation?
Given the following ordinary differential equation (ODE)
$$\frac{dy}{dx} = a y$$
its general solution is $y = c e^{a x}$, where $c$ is a constant. If we know
$$y = c_{1} e^{a_{1} x} + c_{2} e^{a_{2} x}$$
what is the corresponding ordinary ... |
H: What does the "$\bigwedge$" symbol mean in "$\bigwedge_{j=1,\ldots,M,j\neq i}\Delta_i(x)>\Delta_j(x)"$?
What does the "$\bigwedge$" symbol mean in the following?
I assume it means "and". Am I right?
AI: It is the conjunction of all predicates evaluated where the index is in the indicated domain.
EG: $\lower{1ex}{\... |
H: Proving a linear map is surjective
Suppose $V_1, \dots, V_m$ are vector spaces. Prove that
$\mathcal{L}(V_1 \times \dots \times V_m, W)$ is isomorphic to
$\mathcal{L}(V_1, W) \times \dots \times \mathcal{L}(V_m, W).$ (Note that $V_{i}$'s can be infinite-dimensional.)
I am having trouble showing that $\varphi$ def... |
H: How to find the dominant eigenvalue of a Next generation matrix
I have been working with a compartmental model and I am now trying to compute the basic reproduction number. To do this I must find the spectral radius (dominant eigenvalue) of the following matrix.
$$K=\begin{bmatrix}\frac{\beta_{HH}( \pi_H/ \mu_H)}{(... |
H: Prove that $5^{3^n} + 1$ is divisible by $3^{n + 1}$ for all nonnegative integers $n.$
Prove that $5^{3^n} + 1$ is divisible by $3^{n + 1}$ for all nonnegative integers $n.$
I tried to use Proof by Induction, but I'm stuck on the case for when $n=k+1.$
AI: Hint:
$5^{3^{k+1}}+1=(5^{3^k})^3+1=(5^{3^k}+1)(5^{2\cdot3^k... |
H: why $2\pi= c$ and $c=\pi ?$
Let $T:V\rightarrow V$ be the linear transformation defined as follows: If $f\in V, g=T(f)$means that $$g(x)=\int_{-\pi}^{\pi}\{1+\cos(x-t)\}f(t)~dt$$
Find all real $c \neq 0$ and all nonzero $f$ in $V$ such that $T(f)=cf$.
My attempt : I got the answer but i didn't understand the answ... |
H: Lebesgue measure of boundary of an open set.
Let $g$ be a continuous function on $\mathbb{R}^n$ and $$O=\{x\in \mathbb{R}^n:g(x)\neq 0\}.$$
Is it true that Lebesgue measure of Boundary of $O$ always zero?
AI: In $\Bbb R$ let $C$ be a fat Cantor set. This is constructed
in a similar way to the usual Cantor set, but ... |
H: Bijection between $\mathrm{Hom}(D,-)$ functor exact sequnce and $F \leftrightarrow (g,f) $
The following is a paragraph from Dummit and Foote CH-10 (pp. 388).
Let $ 0 \rightarrow L \xrightarrow{\psi} M \xrightarrow{\varphi} N \rightarrow 0 $ be an exact sequence. Then,
\begin{equation}\label{key}
0 \rightarrow \... |
H: What is the equation of the quadratic function whose vertex of the graph is on the $x$-axis and passes through the two points $(1,4)$ and $(2,8)$?
What is the equation of the quadratic function whose vertex of the graph is on the $x$-axis and passes through the two points $(1,4)$ and $(2,8)$?
Here is my attempt:
Us... |
H: Must a local homomorphism from a Noetherian local ring to an artinian local ring factor through a power of its maximal ideal
Let $f : B\rightarrow A$ be a local homomorphism of Noetherian local rings where $A$ is moreover Artinian. Must $f$ factor through $B/m_B^n$ for some $n$?
AI: Yes. Note that $m_A^n=0$ for som... |
H: Show that $m(\{x\in[0,1]:\text{$x$ lies in infinitely many $E_j$}\})\geq\frac{1}{2}$ when $m(E_j)\geq\frac{1}{2}$
Question: Suppose $E_1, E_2,\ldots$ is a sequence of measurable subsets of $[0,1]$ with $m(E_j)\geq\dfrac{1}{2}$. Show that $m(\{x\in[0,1]:\text{$x$ lies in infinitely many $E_j$}\})\geq\dfrac{1}{2}$,... |
H: Unspecified Constraint
In the following question-
Let a, b, c be positive real numbers. Prove that $$\sum_{cyc} {a^3\over a^3+b^3+abc} \ge 1.$$
In here, there is no constraint given.
But in the solution, the author assumes cyclic substitutions of $x={b\over a}$.
But that means xyz = 1
How can this happen if no co... |
H: Suppose $z$ and $\omega$ are two complex numbers such that $|z|≤1$ and $|\omega|≤1$ and $|z+i\omega|=|z-i\omega|=2$. Find $|z|$ and $|\omega|$.
Suppose $z$ and $\omega$ are two complex numbers such that $|z|≤1$ and $|\omega|≤1$ and $|z+i\omega|=|z-i\omega|=2$. Find $|z|$ and $|\omega|$.
My attempt:
I squared the tw... |
H: Where is the copy of $\mathbb{N}$ in the constructible hierarchy relative to a real closed field?
Let $X$ be a real closed field. Let us define a constructible hierarchy relative to $X$ is defined as follows. (This is slightly nonstandard terminology.). Let $L_0(X)=X$. For any ordinal $\beta$, let $L_{\beta+1}(X... |
H: Find the coefficient of $x^{24}$ in the binomial equation
Find the coefficient of $x^{24}$ in the equation ${\left( {1 - x} \right)^{ - 1}}.{\left( {1 - {x^2}} \right)^{ - 1}}.{\left( {1 - {x^3}} \right)^{ - 1}}$
My approach is as follow
The equation used is ${\left( {1 - x} \right)^{ - n}} = \sum\limits_{r = 0}^\i... |
H: Are these true for a martingale? $E\left[ \frac{X_{n+1}}{X_n} \right] = 1, E\left[ \frac{X_{n+2}}{X_n} \right] = 1 $
Let $(X_n)_{n \in \mathbb{Z}_+}$
be a martingale, $X_n(\omega) \neq 0$, and $X_{n+1}/X_n, X_{n+2}/X_n \in L^1 (n \in \mathbb{Z}_+)$
Do the following hold for $n \in \mathbb{Z}_+$?
$E\left[ \frac{X_{n... |
H: What does distance of a point from line being negative signify?
When we take distance from the line, we take
$$ d = \frac{ Ax_o + By_o + C}{ \sqrt{A^2 +B^2}}$$
usually with a modulus on top, now my question is if I evaluate this distance as negative what does it mean? Can I decide on which half-plane a point using ... |
H: One-step probability transition matrix
Start by rolling one die
If the outcome is even, roll two dice on the next turn
If odd, roll one die on the next turn
If two dice are rolled and sum is odd, roll one die next turn
If two dice are rolled and sum is even, roll two die next turn
Game ends, when a sum of 7 or 12 ... |
H: Is $X$ Hausdorff if its quotient space is Hausdorff?
Let $X$ be a topological space and $Y$ be a set.
Let $f:X\longrightarrow Y$ be a surjection and endow $Y$ with the quotient topology.
If $Y$ is Hausdorff, can I say that $X$ is also Hausdorff?
AI: If $X$ is any non-Hausdorff space, $Y=\{0\}$ and $f$ is the obviou... |
H: Isomorphism between a power set $\{0, 1\}^n$ (regarded as a ring) and n digit binary number?
We can treat a power set of a set as ring, $A+B = A\cup B-A\cap B, A\times B = A\cap B$.
Can we use 'modified' modular binary additions and multiplications to represent addition and multiplication of a power set of finite s... |
H: Calculate new tile coordinates for 90 degree rotated square object
So I have a square (all sides even) with even AxA tiles (for example 3x3). Every tile has its x,y coordinates according its placement. After I rotate the object by 90 degrees around the center, the these world coordinates are no more valid (see imag... |
H: Capped Geometric Distribution
The Geometric Distribution is defined as the number of trials it takes for the first success to appear in a sequence of Bernoulli trials.
My question is what happens when the number of trials is capped at a number $N$? That is what would the distribution of $Y$ be if we define it as
$$... |
H: Example of Separable Product Space with cardinality greater than continuum?
In Willard, it's given that, for Hausdorff non-singleton spaces -
$\prod_{\alpha\in A}X_\alpha$ is separable iff $X_\alpha$ is separable $\forall\alpha\in A$ and $|A|\le\mathfrak{c}$
From reading the proof, I found that we could prove $\p... |
H: find $f(n)$ that isn't little $o(n^2)$ and isn't $w(n)$
im having a problem with a question that tells me to find $f(n)$ so that
$f(n) \neq o(n^2)$ meaning also that $\lim_{\rightarrow\infty} \frac{f(n)}{n^2} \neq 0$
and $f(n) \neq w(n)$ meaning also that $\lim_{n\rightarrow\infty} \frac{f(n)}{n} \neq \infty$
I hav... |
H: Relation between number of edges and the sum of number of vertices of degree $k$
Let $P$ be a polyhedron and let $G$ be its associated graph. Suppose $P$ has $V$
vertices, $E$ edges, and $F$ faces. For each $k$, let $V_k$ be the number of vertices of
degree $k$, and let $F_k$ be the number of faces of $P$ (or regio... |
H: Point of intersection of pair of straight lines
We have our general equation is second degree in two variables:
$ax^2+2hxy+by^2+2gx+2fy+c=0$
Let's say this represents a pair of straight lines
Our professor told that we could find the point of intersection by partially differentiating it twice, once with x and once ... |
H: If $x$ is a fixed point of a continuous function $f$, there is an open neighborhood $N$ of $x$ with $f(N)\subseteq N$
Let $\Omega$ be an open subset of a topological space $E$ and $x\in\Omega$ be a fixed point of a continuous function $f:\Omega\to E$.
How can we show that there is an open neighborhood $N$ of $x$ w... |
H: How to verify this given homeomorphism in Munkres on stereographic projection
In munkres topology for theorem 59.3, he provided a homeomorphism between $S^n- p$ and $\mathbb{R}^n$,
where
$$f(x) = \frac{1}{1 - x_{n+1}} (x_1, \dots , x_n)$$
and
$$f^{-1}(y) = (t(y)y_1, \dots , t(y)y_n, 1- t(y))$$
given $t(y) = \frac{2... |
H: An area preserving diffeomorphism between a disk and an ellipse
This is a self-answered question. I post it here since (embarrassingly) it took me some time to realize that the solution is obvious.
Let $D \subseteq \mathbb R^2$ be the closed unit disk and let $E$ be an ellipse with the same area, i.e. with minor a... |
H: Find the area between $f(x) = x^2+3x+7 $ and $g(x) = xe^{x^3+4}$ for $x \in [3,5]$.
Calculate the area between the two functions, $f(x)$, $g(x)$, for $x \in [3,5]$.
$$f(x)=x^2+3x+7$$
$$g(x)=xe^{x^3+4}$$
To determine the area between the functions I used the formula $A= \int_a^b|f(x)-g(x)|dx$. Therefore, I have:
\... |
H: Representations of the formulas with the elements of the linear representations of $GF(2^2)$
The problem statement is as below.
Represent the following formulas with the elements which are of the linear representations of $GF(2^2)$ where the root $\alpha$ satisfies $x^2+x+1=0$ .
$(1)$ $\alpha^4$
$(2)$ $\alpha^2-(\... |
H: Finding $\frac{\cot\gamma}{\cot \alpha+\cot\beta}$, given $a^2+b^2=2019c^2$
This is a question that appeared in the $2018$ Southeast Asian Mathematical Olympiad:
In a triangle with sides $a,b,c$ opposite angles $\alpha,\beta,\gamma$, it is known that $$a^2+b^2=2019c^2$$ Find $$\frac{\cot\gamma}{\cot\alpha+\cot\bet... |
H: Properties of Lebesgue measure.
Let $A,B\subset \mathbb{R}$ such that $A$ is a set of positive Lebesgue measure and $B$ is a set of zero Lebesgue measure (hence $B^c$ is dense in $\mathbb{R}$). Is it true that
$$\overline{A\setminus B}=\overline{A}?$$
($\overline{A} $ denotes the closure of $A$ in $\mathbb{R}$)
AI:... |
H: Compactness Interpretation?
We define a set $X$ as compact if: "for every open cover of $X$, there exists a finite subcover."
An Open Cover $C$ of $X$ is defined as the union of a collection of Open Sets:
$$C = \bigcup_{i \in I} A_i$$
and a finite subcover as the union over a finite subcollection:
$$F=\bigcup_{i \i... |
H: Do infinitely many points on earth have the same temperature as their antipodal?
Let $X=S^2$ be the unit sphere in $\mathbb{R}^3$ and $T:X\rightarrow \mathbb{R}$ be a continuous function.
My topology textbook claims that the set $A=\{x \in X\ |\ T(x)=T(-x)\}$ has an infinite number of elements.
The fact that $A$ is... |
H: Problem with distribution of random variable which is a sum of function's values
I need to find distribution of the random variable $Y=\sum\limits_{i=1}^{n}f(U_i)$, where
$$
f(x_1,x_2) = \left\{ \begin{array}{ll}
1 & \textrm{when $x_1>x_2$}\\
0 & \textrm{otherwise}
\end{array} \right.
$$
and $U_1, U_2, \ldots, U_n$... |
H: Help in finding $\int \frac{x+x\sin x+e^x \cos x}{e^x+x\cos x-e^{x} \sin x} dx$
I want to find
$$\int \frac{x+x\sin x+e^x \cos x}{e^x+x\cos x-e^{x} \sin x} dx.$$
But since algebraic, exponential and trigonometric functions are involved I am not able to solve it. Please help in finding it by hand.
AI: Note the law o... |
H: Show $h(x) \in F[x]$
$Q)$ There are fields $\mathbb{Q} \subset F \subset K$ (Here the $\mathbb{Q}\subset $F and $\mathbb{Q}\subset $ K are Galois extension)
Say the $f(x) \in F[x]$ and $g(x) \in \mathbb{Q}[x]$ with $f(\alpha_1) = g(\alpha_1)$ for some $\alpha_1 \in K$. Plus $f$ has roots $\alpha_1, \alpha_2$ and $... |
H: What is the maximum value of the $4 \times 4$ determinant composed of 1-16?
If 1-9 is filled in the $3 \times 3$ determinant, and each number appears once,then the maximum value of the determinant is $412$.
For example, the following determinant can take the maximum value of $412$:
$$\left|
\begin{array}{ccc}
1 &... |
H: Characterization of Integral Domains
We can show that in an Integral Domain I if $x^2=1$ then $x=\pm 1$.
Is the converse true?
i.e if for any x in I with $x^2=1 \implies x=\pm1$ then I is an Integral Domain.
AI: No. Take $\mathbb{Z}/4$, for example, where it is well-known all squares are 0 or 1 (depending on how t... |
H: If $a_{1}=1$ and for n>1, $a_{n}=a_{n-1}+\frac{1}{a_{n-1}}$ , then $a_{246}$ lies between two integers, what are they?
I tried to use the characteristic equation technique which basically led me to (n-2) roots.
what I did was put $a_n$ as $x^n$ and same for n-1. Taking $x^{(n-2)}$ led me to a higher degree polynomi... |
H: $L_2$ convergence maintains the sign
Let $X$ a finite dimensional space such that $X\subset H^1.$ Let a sequence of non-negative functions $f_n\in X,\,n\geq1$ and a function $f\in H^1$ such that
\begin{equation}
\|f_n - f\|_{L_2} \to0,\;\;\;n\to \infty.
\end{equation}
It is sufficient to conclude to the fact that $... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.