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In the urn we have a white, b red and c black balls. We draw with replacement. Calculate the expected number of white balls, which are drawn until we drew black ball for the k-th time.
My solution:
The expected number of draws till be draw black ball for the k-th time is:
$$(\frac{a+b+c}{c})^k$$ That should be cor... |
Is the ring of smooth fcts over an open set $U$ the localization of the smooth functions on $M$ with respect to the nowhere vanishing fcts on $U$? |
Let $f:[1,\infty)\rightarrow \mathbb{R}$ be continuous and absolutely integrable. Show that $f^2$ is absolute integrable on $[1,\infty)$.
My solution:
Because $f$ is contiuous and absolutely integrable, $f$ will be eventually zero: $\lim_{x\rightarrow \infty}f(x)=0$. So $|f|^2<|f|$ is true for sufficient large val... |
I am reading the concept of [mapping cone of a resolution][1]. I need help with the following.
Let $I_1=\langle x_1^2-x_2 x_4, x_1 x_2-x_3 x_4, x_1 x_3-x_4^2,x_2^2-x_1 x_3, x_2 x_3-x_1 x_4,x_3^2-x_2 x_4 \rangle \subseteq R_1=\mathbb{K}[x_1,x_2,x_3,x_4]$
and
$I_2= \langle x_5^2-x_6 x_8, x_5 x_6 -x_7 x_8, x_5 x_... |
I'm faced with the problem
$\min x \in R^2: (1/2) x' M x$
subject to $a'x = b, c'x = 1, x \ge 0$
where $c = [1, 1, ..., 1]$ and $a = [a_1, a_2, ..., a_n]'$, where $a_j$ are in $R$, $b$ is in $R$ as well, and $M$ is a positive definite real symmetric matrix.
My first idea was setting up the Lagrangian
$L(... |
Suppose a particle starts at position 0 and at each time period particle $i$ moves one position to the right with probability $p_{i, j}$ or moves to the left with with probability $1-p_{i, j}$, where $j$ is the position of the particle before it moves. Let $X_{n, i}$ be the position of particle $i$ after $n$ moves. If ... |
If $m^2+n^2=1$, find the maxmum of $\dfrac{5-4m}{5-4n}$.
The original question is to find the maximum of $\dfrac{BD}{CD}$. By simplifying the formula through cosine theorem, I get the above formula. The value should be equal to $\sqrt{\dfrac{5-4m}{5-4n}}$.
How to find the value? Any elegant geometric solutions ar... |
Is the space $C_0^{k}(\Omega)$ a Montel space? |
In an art museum, there are $n$ paintings, $n \ge 33$, for which there are
used a total of $15$ different colors so that any two paintings have at least one common color and there are no two paintings that have exactly the same colors. Determine all possible values of $n \ge 33 $ so that anyway we color the painting... |
> Suppose that the $\mathbb{R}$-algebras $\mathcal{F}_1$ and $\mathcal{F}_2$, as vector spaces, are isomorphic to the plane $\mathbb{R}^2$. Let the multiplication in $\mathcal{F}_1$ and
> $\mathcal{F}_2$ be respectively given by the relations
>
> $(x_1, y_1) \cdot (x_2, y_2) = (x_1x_2, y_1y_2) $
>
> $(x_1, y_1) ... |
In the book, a first course in functional analysis by D.Somasundaram, it is mentioned that sigma(x) = {z:|z|≤||x||}
But the proof is given only for one side inclusion..that is.,
The spectrum of an element x in a Complex Banach Algebra is a subset of the closed disk with centre 0 and radius ||x||.
Is the reverse i... |
**Question**
Find all positive integers n for which$$N=\left({1^4+\frac{1}{4}}\right)\left({2^4+\frac{1}{4}}\right)...\left({n^4+\frac{1}{4}}\right)$$ is the square of a rational number.
**My attemp**
$$N=\prod_{k=1}^n\left({k^4+\frac{1}{4}}\right)$$
we have :
$$k^4+\frac{1}{4}=k^4+\frac{1}{4}+k^2-k^2=\left({k^... |
I have a polynomial over a specific the range. The first derivative has only two imaginary roots and no real roots. The first derivative is positive in the lower bound and upper bound. Does that mean that the function is monotone (increasing) over the range? |
almost sure convergence of positive random variables with bounded expectation implies uniform integrability? |
We have
$\frac{d}{dx}((1-x^2)^\frac{1}{2}\frac{dy}{dx}) + n^2(1-x^2)^\frac{1}{2}y=0$ for integer $n\geq0$,
and the question says to substitute $x=\cos z$, which leads to
$y''(z) + n^2y(z)=0$,
and then solving gives
$y(x) = A\cos(n\arccos(x)) + B\sin(n\arccos(x))$.
The question then says to sh... |
Consider the sequence $\{x_{n}\}_{n=1}^{\infty}$ consisting of positive numbers that satisfies the following conditions:
$(1).\lim_{n\to\infty}nx_n=\infty$;
$(2).\lim_{n\to\infty}x_n=0$.
Does there exist a $d\in(0,1)$ and two positive constants $K,L$ such that for all $n>N_0$, the inequality $0<K\leq \frac{x_n}{... |
**Question**
Find all positive integers $n$ for which$$N=\left({1^4+\frac14}\right)\left({2^4+\frac14}\right)...\left({n^4+\frac14}\right)$$ is the square of a rational number.
**My attempt**
$$N=\prod_{k=1}^n\left(k^4+\frac14\right)$$
we have :
$$k^4+\frac{1}{4}=k^4+\frac{1}{4}+k^2-k^2=\left({k^2+\frac{1}{2}}\... |
**Question**
Find all positive integers $n$ for which$$N=\left({1^4+\frac14}\right)\left({2^4+\frac14}\right)...\left({n^4+\frac14}\right)$$ is the square of a rational number.
**My attempt**
$$N=\prod_{k=1}^n\left(k^4+\frac14\right)$$
we have :
$$k^4+\frac{1}{4}=k^4+\frac{1}{4}+k^2-k^2=\left({k^2+\frac{1}{2}}\... |
>where is $Q$ in that case?
When you move $P$ to $T$, then $R$ moves to $R'$ where $\vec{RR'}=\vec{PT}$ implies $R'_x=R_x+T_x-P_x=-9.9$ and $R'_y=R_y+T_y-P_y=-233.5$.
The equation of the circle whose center is $R'$ with radius $250$ is given by
$$(x+9.9)^2+(y+233.5)^2=250^2$$
So, the slope of the tangent line... |
Let $L=\mathbb{Q}(x)$ be the field of rational functions over $\mathbb{Q}$ and let $\sigma,\tau\in\text{Aut}(L)$ be given by $\sigma(f(x))=f(1/x)$ and $\tau(f(x))=f(1-x)$ for $f(x)\in\mathbb{Q}(x)$.
Determine the field of invariants for both of these automorphisms. That is,
determine $L^{\langle\sigma\rangle}$ and ... |
Is a polynomial monotone when the first derivative has only imaginary roots? |
Two players play with a pile of 2024 chips placed on a table. Each in turn, a player removes a certain number of chips from the gaming table: at least one chip, but no more than half of the chips still on the table. The person who leaves only one chip on the table is declared the loser. Who will win the game, the playe... |
If $m^2+n^2=1$, find the maxmum of $\dfrac{5-4m}{5-4n}$.
The original question is to find the maximum of $\dfrac{BD}{CD}$. By simplifying the formula through cosine theorem, I get the above formula. The value should be equal to $\sqrt{\dfrac{5-4m}{5-4n}}$.
How to find the value? Any elegant geometric solutions ar... |
I have a question about this theorem:
" The closed unit ball of a normed space X is compact iff X is finite-dimensional."
if I take (R,+,.,Q)
R is a infinite dimensional vector space when we take rational numbers set as a field. but if we take Euclid norm. we know from Heine-Borel Theorem the closed unit ball i... |
### Question:
Prove using $\epsilon - \delta$ definition that if $g(x)$ continous in $a$ and that if $f(x)$ continuous in $g(a)$ then $f(g(x))$ is continuous in $a$
*Rem:* I know that there is a similar question [here][1] but this not the same question of mine as I am asking a proof by definition.
### M... |
### Question:
Prove using $\epsilon - \delta$ definition that if $g$ continous in $a$ and that if $f$ continuous in $g(a)$ then $f\circ g$ is continuous in $a$
*Rem:* I know that there is a similar question [here][1] but this not the same question of mine as I am asking a proof by definition.
### My Ans... |
Determine the general term of the sequence $(a_n)_{n\ge1}$, strictly decreasing, of strictly positive numbers, which satisfies the properties:
(a) $na_n \in \mathbb{N} \setminus \{0\}$ for every $n \in \mathbb{N} \setminus \{0\}$.
(b) $\sum_{k=1}^{n} (-1)^{k-1} a_k \binom{n}{k} = \sum_{k=1}^{n} a_k$.
I trie... |
### Question:
Prove using $\epsilon - \delta$ definition that if $g$ continous in $a$ and that if $f$ continuous in $g(a)$ then $f\circ g$ is continuous in $a$
*Rem:* I know that there is a similar question [here][1] but this not the same question of mine as I am asking a proof by definition.
### My Ans... |
For maps $f:X\to Z$ and $g:Y\to Z$ of topological spaces, we can define fibre product as $X\times_{Z}Y=\{(x,y)\in X\times Y \mid f(x)=g(y)\}$. I was wondering if there is a generalization of this concept. More precisely, if we have $f_i:X_i\to Z$ for $1\leq i\leq n$, then can we define generalized fibre product as a to... |
The [Lucas primality test][1] states that a number N is prime iff there exists a base $B, 1 < B < n$, such that:
[a] $\ B^{N-1} \equiv 1 \mod N$
...and...
[b] For all prime factors F of N-1 the congruence $B^{(N-1)/F} \equiv 1 \mod N$ does not hold.
Now we could select B at random but then the test wouldn'... |
> Let H be a Hilbert space and T a compact operator from H to H. Let ${e_n}$ be orthonormal series in H.
>Prove $Te_n \rightarrow 0$.
Hint: what do you know about $<Te_n,f>$ for $f\in H$
I'm taking a course in hilbert space and operators and stumbled on this problem that I can't see how to solve.
Can som... |
> Let $H$ be a Hilbert space and $T:H\to H$ a compact operator. Let $(e_n)$ be an orthonormal sequence in H.
>
>Prove $Te_n \rightarrow 0$.
Hint: what do you know about $\langle Te_n,f\rangle$ for $f\in H$?
I'm taking a course in Hilbert space and operators and stumbled on this problem that I can't see how t... |
I need to compute $z_{0.05}$. I check my statistics table and the value is $0.5199$ but in my example it says the value is $1.645$. Where am I going wrong? |
How to compute this Z-Value? |
The problem is as follows.
An insurance company sells 10,000 similar car insurance policies. They estimate that the amount paid out in claims on a typical policy has mean £240 and standard deviation £800. Estimate how much they need to put aside in reserves to be 99% sure that the reserve will exceed the total amou... |
As you can find in e.g. Section 3 of [O'Connor's SDG notes](https://arxiv.org/abs/0805.3307), the real line object $R$ of Smooth Infinitesimal Analysis satisfies the following: $$\forall x \in R.~x \neq 0 \rightarrow \exists y \in R. xy = 1$$
This means that if you can prove that some number $x$ is not equal to $0$,... |
The [Lucas primality test][1] states that a number N is prime iff there exists a base $B, 1 < B < n$, such that:
[a] $\ B^{N-1} \equiv 1 \pmod N$
...and...
[b] $B^{(N-1)/F} \not\equiv 1 \pmod N$ for all prime factors $F$ of $N-1$.
Now we could select B at random but then the test wouldn't be deterministic.... |
It is a problem found in Physics. From the edge of a (in distance r to the center) disk a (mathematical) pendulum is hanging. The disk spins with a velocity w, which also stimulates the hanging pendulum.(As can be seen in the picture below)
[![enter image description here][1]][1]
[![enter image description here][2]][... |
With the definition you quoted, the existence of a bilipschitz homeomorphism is utterly wrong. The simplest example is $X={\mathbb R}$ and $Y=(0,1)\subset \mathbb R$. One can give more interesting examples when both domains are bounded, for instance, $X$ is the unit disk in $\mathbb R^2$ and $Y$ is an annulus in $\math... |
My professor went over an example in class and I am kind of confused. We are giving a proof of the theorem that $\forall a \gt 0, \int_{0}^{\infty} e^{-(a+ib)^t} \, dt \, \colon= \lim_{R \to \infty} \int_{0}^{R} e^{-(a+ib)t} \, dt = \frac{1}{a+ib}$.
So, to prove, we integrated over the triangle with sides $\gamma_1 ... |
I'm in doubt about my arguments for this problem, in addition to having thought of an alternative argument after writing the solution. Please give me feedback on them, constructive criticism, and of course if they are valid.
Best regards.
**Suppose that f is a continuous function on [-1,1] such that $x^2+(f(x))^2... |
(Spivak - chap 7 - problem 6) my solution is wrong? I’m doubt about |
I was reading Introduction to Geometric Metric Theory by Leon Simon. Here's the full statement of the Riesz representation theorem for non-negative functionals.
> Suppose $X$ is a locally compact Hausdorff space, $\lambda:\mathcal{K}_{+}\to[0,\infty)$ with $\lambda(cf)=c\lambda(f)$, and $\lambda(f+g)=\lambda(f)+\... |
Let $B$ be a finite-dimensional, commutative Hopf algebra over a field $\mathbb{K}$ and let $M$ be a $B$-module. Is there a natural way to view $M^*$ as a $B$-comodule (or, equivalently, as a $B^*$-module)? Here, $*$ stands for the $\mathbb{K}$-linear dual.
|
$M$ a Hopf-module, is $M^*$ a Hopf comodule? |
I am studying QFT and renormalization of QED, and in a passage is exploited the following identity:
$$\frac{1}{ABC}=2∫_{0}^{1} dxdydzδ(x+y+z-1)\frac{1}{(Ax+By+Cz)^{3}}$$
for every $A,B,C\in\mathbb{C}$.Does someone know how to prove it? I know it's just to compute the integral but I'm struggling with the fact that I h... |
The cutting plane approach in convex optimization is a general recipe for minimizing a convex function. The argument relies on the fact that using the gradient vector we can cut the feasible set into two parts, and then by querying (some sort of) **center**, the volume of the new feasible set decreases geometrically. T... |
My professor went over an example in class and I am kind of confused. We are giving a proof of the theorem that $\forall a \gt 0, \int_{0}^{\infty} e^{-(a+ib)t} \, dt \, \colon= \lim_{R \to \infty} \int_{0}^{R} e^{-(a+ib)t} \, dt = \frac{1}{a+ib}$.
So, to prove, we integrated over the triangle with sides $\gamma_1 =... |
The cutting plane approach in convex optimization is a general recipe for minimizing a convex function. The argument relies on the fact that using the gradient vector we can cut the feasible set into two parts, and then by querying (some sort of) **center**, the volume of the new feasible set decreases geometrically. T... |
### Question:
Prove using $\epsilon$-$\delta$ definition that if $g$ continous in $a$ and that if $f$ continuous in $g(a)$ then $f\circ g$ is continuous in $a$
*Rem:* I know that there is a similar question [here][1] but this not the same question of mine as I am asking a proof by definition.
### My Ans... |
It wants us to prove this by integrating $f(z) = e^{-\pi z^2}$ over the rectangle with vertices $\pm n, \pm n + i \xi$ and taking the limit $n \to \infty$. I have tried to do this using Cauchy's Integral Formula, but it ends up with a nasty integral that results in $\text{erf}(x)$, which I know for a fact we cannot use... |
Suppose the following situation in Wordle. Please allow me to assume you know the rules.
You play the word SALET
You get SAL all green and ET both gray.
You deduce that there are five possible answers:
SALAD
SALSA
SALVO
SALON
SALLY
(And for the sake of this discussion, please assume that on... |
How to define the addition and multiplication of p-adic integers as infinite formal sums? |
I am trying to get acquainted with the [arithmetical hierarchy][1], and as I wrote down some examples, I got a bit confused.
Consider the language $L=\{+\cdot,<,=,0,1\}$ of $\mathsf{PA}$.
For example, the sentence $\phi:=\forall x\,\exists y\,(y<x\wedge x=y+1)$ is a $\Pi_{1}$-sentence because it is of the form $\... |
As a continuation of this question relating the [Minimum $k$ for which every positive integer of the interval $(kn, (k+1)n)$ is composite](https://math.stackexchange.com/questions/4877970/minimum-k-for-which-every-positive-integer-of-the-interval-kn-k1n-is-co) and this other one on the [divisibility of numbers in inter... |
I'm trying to solve the following definite integral:
$$
\int_{R-r}^{R}\sin^{-1}\left(\frac{x^{2} + R^{2} - r^{2}}{2xR}\right){\rm d}x
$$
I know that: $0<r<R$. But I'm stuck after that. Wolfram doesn't seem to have any ideas either. Any suggestions to proceed forward $?$. |
Atiyah-MacDonald exercises 20-23 in chapter 4 develop a theory of primary decomposition for modules, in analogy with the theory developed in the chapter for rings. Exercise 20 begins with this definition:
**Definition:** Given a (commutative, unital) ring $A$ and an $A$-module $M$, and a submodule $N\subset M$, the ... |
I'm faced with the problem
$\min x \in R^2: (1/2) x' M x$
subject to $a'x = b, c'x = 1, x \ge 0$
where $c = [1, 1, ..., 1]$ and $a = [a_1, a_2, ..., a_n]'$, where $a_j$ are in $R$, $b$ is in $R$ as well, and $M$ is a positive definite real symmetric matrix.
My first idea was setting up the Lagrangian
$L(... |
In the following figure, the line that touches the ellipse at only one point, called A, is the tangent line to the ellipse at that point. C is the center of the ellipse. Point $L'$ is the point where the perpendicular passing through C to the tangent line intersects the tangent line. Point L, instead, is the intersecti... |
When translation of a vector span the same space? |
Suppose $X_i$ is independently and identically distributed over $i$, and $E(|X_i|^d)$ with $d>0$ is not finite or undefined. I am wondering whether $\sum_{i=1}^N |X_i|^d$ is of any asymptotic order. Take [Cauchy distribution](https://en.wikipedia.org/wiki/Cauchy_distribution#Mean) as an example. If $d=1$, then $N^{-1}\... |
It wants us to prove that $\int_{-\infty}^{\infty}e^{-\pi x^2}e^{-2\pi ix \xi} dx = e^{-\pi \xi^2}$ by integrating $f(z) = e^{-\pi z^2}$ over the rectangle with vertices $\pm n, \pm n + i \xi$ and taking the limit $n \to \infty$. I have tried to do this using Cauchy's Integral Formula, but it ends up with a nasty integ... |
As a continuation of this question relating the [Minimum $k$ for which every positive integer of the interval $(kn, (k+1)n)$ is composite](https://math.stackexchange.com/questions/4877970/minimum-k-for-which-every-positive-integer-of-the-interval-kn-k1n-is-co) and this other one on the [divisibility of numbers in inter... |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...-83,-53,-23,7,37,67...\}.$$
I'm interested in particular with $$U:=(\mathbb Z/30\mathbb Z)^\times=\{\textbf{1},\textbf{7},\textbf{11},\textbf{13},\textbf{17},\textbf... |
Let $X$ be an reduced and irreducible complex analytic space of dimension $m$ and $Z$ an $n$-dimensional irreducible analytic subset of $X$ with reduced structure.
Then we say that ***$Z$ locally complete intersect at $x\in Z$*** if there exists an open neighborhood $U_x$ of $x$ in $X$ such that the ideal sheaf ... |
Is an irreducible analytic subset locally complete intersects at general point? |
let $X \sim Bin(10,p)$
We would like to test the following hypotheses:
$H_{0}: p=0.5 $ vs $H_{1} : p=0.3$
The Statistic $T(x)$ is defined as $T(x)=x$ and the critical (rejection) region is $R=\{x:T(x)\le4\}$
I want to calculate the conditional probability of the null hypothesis given that $T(x)\in ... |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...-83,-53,-23,7,37,67...\}.$$
I'm interested in particular with $$U:=(\mathbb Z/30\mathbb Z)^\times=\{\textbf{1},\textbf{7},\textbf{11},\textbf{13},\textbf{17},\textbf... |
let $X \sim Bin(10,p)$
We would like to test the following hypotheses:
$H_{0}: p=0.5 $ vs $H_{1} : p=0.3$
The Statistic $T(x)$ is defined as $T(x)=x$ and the critical (rejection) region is $R=\{x:T(x)\le4\}$
I want to calculate the conditional probability of the null hypothesis given that $T(x)\in ... |
Let $(\Omega, \mathcal{A},\mathbb{P})$ be a probability space and $Y:(\Omega, \mathcal{A})\rightarrow (\Omega', \mathcal{A}')$ be a random variable.
So usually conditional probability is defined via conditional expectation as
$\mathbb{P}(A|Y):=\mathbb{E}[\chi_A|Y]$, where $\chi_A$ is the indicator function of the s... |
Closure of quasi-projective variety: if $X=V(I)\setminus V(J)$, must $\overline{X}=V(I)$? |
Let $L=\mathbb{Q}(x)$ be the field of rational functions over $\mathbb{Q}$ and let $\sigma,\tau\in\text{Aut}(L)$ be given by $\sigma(f(x))=f(1/x)$ and $\tau(f(x))=f(1-x)$ for $f(x)\in\mathbb{Q}(x)$.
Determine the field of invariants for both of these automorphisms. That is,
determine $L^{\langle\sigma\rangle}$ and ... |
I am trying to evaluate $$\int_0^\infty\frac{x \sin(x)}{x^2+a^2} dx$$
I get $\pi\sin(ia)/4$ using residue theorem.
I integrated over the path that goes from -R to R along the real axis and then along a the semi-circle that goes back to -R in the upper half plane. By residue theorem, this is $$2i \pi \lim_{x\to ia... |
Not sure if this an appropriate venue for this question, please close question as opposed to migrating to Physics SE, because it is not appropriate there, thanks.
If we have a functional given by:
$$I=\int_1^2\bigg(L+\lambda f\bigg)\;dt,$$
the Euler-Lagrange equations are given by:
$${d\over dt}\bigg({\partial L\... |
I don't know if this is the right place to ask this Question, but I have previously asked a similar question where i asked how to write a simulation on this phenomenon. I got a great answer with a Mathematica code. I started to play around a little with the values and wrote a similar program myself in python. You can f... |
Suppose $X_i$ is independently and identically distributed over $i$, and $E(|X_i|^d)$ with $d>0$ is not finite or undefined:
- I am wondering whether $\sum_{i=1}^N |X_i|^d$ is of any asymptotic order. Take [Cauchy distribution](https://en.wikipedia.org/wiki/Cauchy_distribution#Mean) as an example. If $d=1$, then $N^{... |
How to ensure stability of 2nd order linear non homogeneous ODE? |
Does anyone know how to demonstrate that the stereographic projection is antiholomorphic? |
Let $(X_i, Y_i)_{i=1}^{\infty}$ be iid continuous random vectors, where $X_1$ have support $\mathcal{X}$. Let $B_n\subset \mathcal{X}$ be decreasing subsets such that $\cap B_n= x_0\in\mathcal{X}$.
I want to show that
$$
\sum_{i\leq n: X_i\in B_n}Y_i \overset{P}{\to} \mathbb{E}[Y_1\mid X_1=x_0].
$$
I assume ... |
I was doing some algebra today and I came across the term 'purely inseparable extension'.
However, I came across two different definitions of this term. The definitions go as follows:
> A field extension $E/K$ is purely inseparable if and only if for every $x \in E\setminus K$ the minimal polynomial of $x$ over $K$... |
Let $(M,g)$ be a Riemannian manifold. Let $p\in M$. Is it possible to find a chart $\varphi: p\in U \to \mathbb R^d$ such that the geodesic distance $d_M$ on $U$ is close to the euclidean distance $d_E$ on $\phi_i(U_i)$?
More precisely, if I take $\phi_i = \exp^{-1}_p: \{x\in M: d_M(x,p)\le \delta\} \to \{v\in \math... |
What conditions are necessary and sufficient for a smooth vector field $v: \mathbb R^2 \to \mathbb R^2$ to induce an ODE $y' = f(x,y)$?
By "induce" I mean: If we start at any point on the plane, and follow the trajectory of the vector field, we trace out a curve $y=f(x)$ with $y'$ a function of the vector field at t... |
Often in beginning set-theory courses, and in particular in Jech's book *Set Theory*, it is proved from scratch that given any two well-orderings, they are isomorphic or one is isomorphic to an initial segment of the other. This is the law of trichotomy for well-orderings, and while simple is not exactly trivial to pro... |
In this paper https://arxiv.org/pdf/1907.09605.pdf \\
let $\Omega \subset \mathbb{R}^n$ with $n \geq 1$ be a bounded Lipschitz domain with boundary $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ be an $L^2(\Omega)$ function (given datum), $K: L^2(\Omega) \rightarrow L^2(\Omega)$ be a bounded linear operator, an... |
By definition, the marginal pdf of $X_1$ is [given by](https://en.wikipedia.org/wiki/Joint_probability_distribution#Marginal_probability_distribution) :
$$ f_{X_1}(x) =\int_{x_2\in\mathbb R} f_{X_1,X_2}(x,x_2)\ d x_2 \quad\forall x\in\mathbb R\tag1$$
Where $f_{X_1,X_2}$ is the pdf of $(X_1,X_2)$. Of course, the margi... |
Suppose we want to prove that a property P is true for every integer in ℕodd = {1,3,5,7,9,...}. Consider the following induction mechanism:
1. Base case: Verify the property P(1)
2. Inductive step: Prove that for all k ≥ 1, P(k) ⇒ P(k + 1)
(a) Why might the above mechanism not constitute a valid proof?
(b)... |
The complex solutions always diverge, from what was tried, using
[Lagrange reversion](https://en.m.wikipedia.org/wiki/Lagrange_reversion_theorem), but the real solutions are:
$$\begin{align}\sin(x)^{\cos(x)}=2\mathop \iff^{x=\cos^{-1}(-\sqrt y)}y=1-4^{-\frac1{\sqrt y}}\implies x_k=\left(2k+\frac12\right)\pi+\sin^{-1... |
Suppose we want to prove that a property P is true for every integer in $ℕ_{odd}$ = {1,3,5,7,9,...}. Consider the following induction mechanism:
1. Base case: Verify the property P(1)
2. Inductive step: Prove that for all k ≥ 1, P(k) ⇒ P(k + 1)
(a) Why might the above mechanism not constitute a valid proof?
... |
In dimension $d = 2$ the function $g(x) = - (2 \pi)^{-1} \log |x|$ is the Green's kernel associated to Laplace's equation. For example, if $f \in C_c^\infty(\mathbb{R}^2)$ then the solution of $-\Delta u = f$ can be given by
$$ u(x) = \int g(x-y) f(y) dy $$
One also expects that the solution can be given by
$$ u... |
I have the parabola $x^2$, and a circle whose center is the point $(3,2)$. My goal was to figure out the minimum length of the circle's radius so that it would be tangent to the parabola. To do this, I decided to use the Pythagorean theorem to create a function that would show me the distances between $(3,2)$ and any p... |
Is it possible to write an equation that represents the root of this kind of function? |
Suppose we want to prove that a property P is true for every integer in $ℕ_{odd}$ = $\{1,3,5,7,9,...\}$. Consider the following induction mechanism:
1. Base case: Verify the property P(1)
2. Inductive step: Prove that for all k ≥ 1, P(k) ⇒ P(k + 1)
(a) Why might the above mechanism not constitute a valid proo... |
I have 2 equal sized groups of people and each group selected A, B, or C for 6 questions. The first group was given no guideline and the second group was given a guideline. I have a hypothesis that the guideline caused people to answer the questions a specific way, and thus the answers in the second group would be more... |
is there a statistical test for two categorical variables to compare how concentrated they are? |
Note that $P(1)$ is true, but $P(2)$ is not true, so you can't prove that $P(n)\implies P(n+1)$.
To prove $P(n)$ for all odd $n$, can you show that $P(n)\implies P(n+2)$
using $2^{n+2}+3^{n+2}=4(2^n)+9(3^n)=4(2^n+3^n)+5(3^{n})$? |
Suppose we want to prove that a property P is true for every integer in $ℕ_{odd}$ = $\{1,3,5,7,9,...\}$. Consider the following induction mechanism:
1. Base case: Verify the property P(1)
2. Inductive step: Prove that for all k ≥ 1, P(k) ⇒ P(k + 1)
(a) Why might the above mechanism not constitute a valid proo... |
Suppose we want to prove that a property $P$ is true for every integer in $ℕ_{odd}$ = $\{1,3,5,7,9,...\}$. Consider the following induction mechanism:
1. Base case: Verify the property $P(1)$
2. Inductive step: Prove that for all $k ≥ 1, P(k) ⇒ P(k + 1)$
(a) Why might the above mechanism not constitute a vali... |
Is $C_{p}^{\infty}(U)=C_{p}^{\infty}(M)$ as sets and $C_{p}^{\infty}(M)\cap C_{q}^{\infty}(M)=\emptyset$? |
I was doing some algebra today and I came across the term 'purely inseparable extension'.
However, I came across two different definitions of this term.
We consider algebraic extensions and write $p$ = char($K$). The definitions go as follows:
> A field extension $E/K$ is purely inseparable if and only if for eve... |
Let $(X_i, Y_i)_{i=1}^{\infty}$ be iid continuous random vectors, where $X_1$ have support $\mathcal{X}$. Let $B_n\subset \mathcal{X}$ be decreasing subsets such that $\cap B_n= x_0\in\mathcal{X}$.
I want to show that
$$
\sum_{i\leq n: X_i\in B_n}Y_i \overset{P}{\to} \mathbb{E}[Y_1\mid X_1=x_0].
$$
I assume ... |
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