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H: Derivation of an integral function in $L^p$
I know that for any continuous function $f:[0,1]\to\mathbb{R}$
$$\frac{d}{dx} \int_0^x f(y)dy = f(x).$$
Let's say that $f\in L^p$ for $p>1$. Can I say that the equality still holds almost everywhere?
If not, what is the largest subset of $L^p$ such that the equality holds... |
H: Prove a relation $\mathcal R$ is reflexive if and only if its complement $\overline{\mathcal R}$ is irreflexive (strict).
Given a homogeneous binary relation $\mathcal R$ over a set $A$, $\mathcal{R}$ is reflexive if:
$$\forall a \in A:(a,a) \in \mathcal R$$
Prove a relation $\mathcal R$ is reflexive if and only if... |
H: 2 Cross Products?
Usually, if we want to find the cross product of 2 vectors $\vec{b}$ and $\vec{c}$, we want to find the vector which is perpendicular to both of them. Let's say the cross product of $\vec{b}$ and $\vec{c}$ is $\vec{d}$. Isn't $-\vec{d}$ then also perpendicular to $\vec{b}$ and $\vec{c}$? Does that... |
H: Grad Function Directional Derivative
Let $h(x, y)$ be some function from $\mathbb{R}^2$ to $\mathbb{R}$ which outputs height of a hill at a given point $(x, y)$. At a given moment, I am travelling up the hill with velocity $\mathbf{v}$ at angle $\theta$ to $\nabla h$. How do I prove that the rate at which my height... |
H: Condition of positive definiteness based upon diagonal elements of the original and inverse matrices
This is a sequel to this question in which I sought to expand on this question. Let me put it straight. Given a non-singular symmetric real matrix $A\in\mathbb{R}^{n\times n}$ such that $A_{ii}>0$.
Can we conclude t... |
H: Conditions for the existence of a lower bound of the operator norm $\|A^*A\|$ for a linear and continuous operator $A:X \to Y$
Setting: $X$, $Y$ Banach (or Hilbert) spaces, $A: X \to Y$ linear, continuous, injective, $A(X)$ is dense in $Y$ ($A$ is an imbedding operator in a Gelfand triple), $A^*$ is its adjoint
... |
H: Completeness of $\mathcal{l}^1$?
I am a bit confused with how $\mathcal{l}^1$ can be complete.
So we know that the sequence space $\mathcal{l}^1$, equipped with $||\cdot||_{\mathcal{l}^1}$ is complete. But the sequence of sequences
$$\left(x^{(n)}_k\right)_{k\in\mathbb{N}} = \left(\frac{1}{k^{1+1/n}}\right)_{k\in\m... |
H: number of pairs - combinatorics
if I want to know the number of pairs in a group of 2n elements
what is the difference between:
a. ${2n}\choose{2}$
b. $\frac{(2n)!}{2!^n\cdot n!}$
c.${2n}\choose{2}$ ${2n-2}\choose{2}$... ${2}\choose{2}$
I do not understand the differences between these options,
many thanks.
AI: For... |
H: Evaluate $\int_{(0,\infty)^n}\text{Sinc}(\sum_{k=1}^nx_k) \prod_{k=1}^n \text{Sinc}(x_k) dx_1\cdots dx_n$
In this post @metamorphy established this remarkable result (here Sinc$(x)$ denotes $\frac{\sin(x)}x$):
$$I(n)=\int_{(-\infty,\infty)^n}\text{Sinc}(\sum_{k=1}^nx_k) \prod_{k=1}^n \text{Sinc}(x_k) dx_1\cdots dx_... |
H: Splitting the set $A=\{1,2,...,n\}$ into at most $m$ non-empty disjoint subsets, whose union is $A$
How many different ways do there exist to split the set $A=\{1,2,...,n\}$ into at most $m$ non-empty disjoint subsets, whose union is $A$.
For example, if $m=3$ then we have the following:
$n=1: \quad$ there is only ... |
H: calculate $\oint_{|z|=1}z^{2018}e^{\frac{1}{z}}\sin\frac{1}{z}\text{dz}$
calculate $\oint_{|z|=1}z^{2018}e^{\frac{1}{z}}\sin\frac{1}{z}\text{dz}$
my try:
$
\begin{array}{c}
\oint_{|z|=1}z^{2018}e^{\frac{1}{z}}\sin\frac{1}{z}\text{dz}\\
\oint_{|z|=1}z^{2018}{\displaystyle \mathop{\sum_{0}^{\infty}}}\frac{\left(\frac... |
H: Is a square commutation matrix positive semidefinite?
Let $A \in \mathbb{R}^{n \times n}$ and denote the commutation matrix, made up of 0 and 1 such that each row and each column has exactly one 1, as $K_{n} \in \mathbb{R}^{n^2 \times n^2}$ , which is such that:
\begin{equation}
\operatorname{vec}(A^T) = K_{n} \ope... |
H: Smooth map between Riemannian manifolds of same dimension is local isometry iff. metric is preserved
I am just starting to read Lee's "Riemannian Manifolds" and one of the first exercises in the text (2.7) is the following: given a smooth map $\phi:(M,g)\to (\bar{M},\bar{g})$, prove that for $\dim M=\dim\bar{M}$ we... |
H: Minimization Problem: Deriving Dual Problem
Consider the following minimization problem $\min\{H(x,z) \equiv h_1(x) + h_2(z): Ax + Bz = c\}$, where $A \in \Bbb{R}^{m \times n}, B \in \Bbb{R}^{m \times p}$ and $c \in \Bbb{R}^{m}$ and $h_1, h_2$ are proper, closed and convex.
To find the dual problem of the optimizat... |
H: Show estimator is consistent
For a special case of the gamma distribution:
$$f(x)=\frac{x}{\theta^2}e^{-x/ \theta}$$
$$E(x) = 2\theta\\ V(x) = 2\theta^2$$
I find MLE of $\hat{\theta}$ to be $\sum_{i=1}^n \frac{x_i}{2n}$
To show consistency, I'd like to show that:
$\lim_{n\to \infty} E(\hat{\theta}) = \theta$
$\li... |
H: Let $p(x)$ be a polynomial with integer coefficients. Show that if $p(2)=3$ and $p(3)=5$ then $p(n)\ne0$ for all integers $n$.
Let $p(x)$ be a polynomial with integer coefficients. Show that if $p(2)=3$ and $p(3)=5$ then $p(n) \neq 0$ for all integers $n$.
I did manage to solve it using the fact that $a-b | p(a)-p(... |
H: given a set of 3d points and their covariance matrices finding the mean point
Say I have a set of observations of a 3d object in space. Knowing only the location of the observations my best guess for the location of the object would be the mean point.
But let's say I have the covariance matrix for each point based ... |
H: An Infinite non-nilpotent group whose every maximal subgroup is a normal subgroup.
It is widely known that for every finite group $G$, $G$ is nilpotent If and only if every maximal subgroup of $G$ is a normal subgroup.
But I don't know if there is an infinite non-nilpotent group whose every maximal subgroup is a no... |
H: Matrix A,B,C 2/3 not invertible
Let A,B,C be matrices such that the algebraic operations are defined.
Question: Disprove the following statement (by giving a counter example):
If AB=C and 2 of the 3 matrices are not invertible, then the third is not either
My struggle: I can't think of such an example, could I mayb... |
H: Confusion with Lagrange Multipliers
I am numerically solving an optimization problem of the form: Maximize $z$ subject to $f(\alpha,z)=c$.
Using the method of Lagrange Multipliers I first write down the Lagranian
$$
\mathscr L(\alpha,z,\lambda)=z-\lambda(f(\alpha,z)-c),
$$
for which upon setting the gradient equal ... |
H: Unitary transformation: order of $U^{\dagger}$ and $U$
If $U$ is a unitary matrix and $U^{\dagger} A U$ a unitary transformation, then also $U A U^{\dagger}$ is a unitary transformation. But are $U^{\dagger} A U$ and $U A U^{\dagger}$ necessarily equal in the general case?
AI: Counterexample:
$$
U=\pmatrix{0&-1&0\\... |
H: Prove: $\int_0^{\infty} \frac{\ln{(1+x)}\arctan{(\sqrt{x})}}{4+x^2} \, \mathrm{d}x = \frac{\pi}{2} \arctan{\left(\frac{1}{2}\right)} \ln{5}$
Prove: $$\int_0^{\infty} \frac{\ln{(1+x)}\arctan{(\sqrt{x})}}{4+x^2} \, \mathrm{d}x = \frac{\pi}{2} \arctan{\left(\frac{1}{2}\right)} \ln{5}$$
This might be a repeat question ... |
H: Open set implies the theorem, but does the condition imply the set is open?
Below is the theorem and my proof.
Theorem : $(X,d)$ is a metric space and $U,V \subseteq X$ are sets with $U$ open, such that $U \cap V=\varnothing$. Then $U \cap \overline V=\varnothing$ as well.
Proof : Suppose for the sake of contradict... |
H: Are singular foliations spanned by collinear vector fields equal?
Let $M$ be a compact $n$-manifold (let's say with boundary, but this isn't too important), $X$ a vector field on $M$ and $f:M \to \mathbb{R}$ a non-zero function on $M$.
My question: are the singular foliations spanned by $X$ and $fX$ equal? In other... |
H: How to properly substitute my variable so that these two integrals are equivalent?
I'm having troubles figuring out if what I'm doing is mathematically correct.
I replaced the $\theta$ term in my multidimensional numerical integral by giving it a functional dependence of the angle $\beta$ such as $\theta\left(\beta... |
H: Estimating Error due to replacing the sum $\sum\limits_{n=1}^{\infty} \frac{1}{n!} (\frac{1}{2})^n$ by the first $n$ terms
Question: Estimating Error due to replacing the sum $\sum\limits_{n=1}^{\infty} \frac{1}{n!} (\frac{1}{2})^n$ by the first $n$ terms
All I can really say at this point is that the remainder $R_... |
H: A Probability question on Bayes theorem
I am struggling to understand a problem from https://docplayer.net/6566428-Probability-exam-questions-with-solutions-by-henk-tijms-1.html
Problem :
On the island of liars each inhabitant lies with probability 2/3 . You overhear an inhabitant making a statement. Next you ask a... |
H: Find derivative of $\lfloor{x}\rfloor$ in distribution
Find derivative in distribution of $f(x)=\lfloor{x}\rfloor=E(x)$
$$E(x)≤x≤E(x+1)$$
Answer is :
$$\lfloor{x}\rfloor '=\displaystyle\sum_{k=-\infty}^{\infty}\delta_{k}$$
I don't have any idea about how to.
Can you assist?
I'm too thankful
AI: Let $f(x)=\lfloor ... |
H: Is there an irrational number that the digits never repeat anywhere and have all 10 digits appear everywhere?
Is there an irrational number that the digits never repeat anywhere and have all 10 digits appear everywhere?
let's look at one that doesn't work like $$\pi=3.141592653589793238462643383...$$ starting at th... |
H: how given a family of orthonormal functions on $[0..2\pi]$, modify this family to work on $[0..\ell]$
Suppose $\varphi_0(x), \varphi_1(x), ...$ are orthonormal functions on $[0..2\pi]$
How can I find an orthonormal family of functions working on $[0..\ell]$ and what is the intuition behind looking for such a modifi... |
H: show that $F= \underset{n\geqslant1}{\bigcap} \left\{ x \in X, d(x,F) < \frac{1}{n} \right\}.$
Let $F$ be a closed set from a metric space $(X,d)$ , show that $$F= \underset{n\geqslant1}{\bigcap} \left\{ x \in X, d(x,F) < \frac{1}{n} \right\}.$$
My attempt : $\Rightarrow)$
for all $x \in X$, $ x\in F\iff d(x,F)=0... |
H: Convergence of two complex valued sequence
[CMI PG2010, Part B] Let $\{a_n\}$ and $\{b_n\}$ be sequences of complex numbers such that each $\{a_n\}$ is non-zero, $\lim_{n\rightarrow\infty}a_n = \lim_{n\rightarrow\infty}b_n = 0$, and such that for every natural number $k$, $$\lim_{n\rightarrow\infty}\frac{b_n}{{a_n... |
H: Show that the $L^1$ and $L^2$ norms are not equivalent on the set of continuous functions from $[0,1]$ to $\mathbb{R}$
Let $E$ be the vector space of continuous functions on $[0,1]$.
Show that the $L^1$-norm is not equivalent to the $L^2$-norm.
My thought was that, given a sequence of functions $f_n\in E$ which con... |
H: How to mix 2 Fourier tables.
I am currently working on a homework problem in relation to the Fourier series. Here are the function's properties:
f(x)= {0, 0<x<1}
{1, 1<x<2}
{0, 2<x<3}
We are asked to find the Fourier equation from the Fourier tables. However, as you can see, there are 3 conditions, and all of the g... |
H: When is convolution not commutative?
Let $G$ be a locally compact Hausdorff group with a left Haar measure $\lambda$. Define the convolution of two functions $f,g \in L^1(G)$ by
$$(f \ast g)(x) = \int f(y) g(y^{-1}x) d\lambda (y), ~~~ \forall x \in G$$
If the group $G$ is abelian the convolution is commutative: $f ... |
H: A triangle is a compact set
Let's fix a triangle
$\Delta=\{t_1x_1+t_2x_2+t_3x_3 \in \mathbb{R}^2 \mid t_1,t_2,t_3 \ge 0 \quad \land \quad t_1+t_2+t_3=1\}$
of fixed vertices $x_1,x_2,x_3 \in \mathbb{R}^2$.
I want to show that $\Delta$ is compact in the plane. That's my attempt (the metric used here is the euclidean ... |
H: Is a function uniformly continuous on the union of two disconnected sets
Suppose $f: [0,1]\cup [2,3] \rightarrow \mathbb{R}$ is defined by
$$f(x) =\begin{cases} \sin(x), &0 \leq x \leq 1 \\x^2, &2 \leq x \leq 3.\end{cases}$$
Now $f(x)$ is uniformly continuous on $[0,1]$ and uniformly continuous on $[2,3]$. But is i... |
H: What is wrong with this definition of a truth predicate?
Tarski's theorem, interpreted in Peano Arithmetic, says there is no predicate $T$ such that $PA\vdash T(\phi)\leftrightarrow \phi$. However, we know that there are partial truth predicates for each $k< \omega$ such that, for all $\phi \in \Sigma_k$, $PA\vdash... |
H: A question on number of elements in a set
I am trying questions in permutations and combinations from an assignment and I was unable to solve this question.
Let D be a set of tuples $(w_{1} ,..., w_{10} )$ , where $w_{i} \in \{1,2,3\}$ , $1\leq i\leq 10$ and $w_{i}+w_{i+1} $ is an even number for each $i$ with $1... |
H: What is the importance of $R$ being a field in this question?
Here is the question I am trying to solve (Jeffery Strom, “Modern classical homotopy theory” on pg. 511):
Problem 22.39. Suppose $R$ is a field.
(a) Show that $h^n(?) = \operatorname{Hom}_R( H_n(? ; R), R)$ is a cohomology theory defined on (at least) t... |
H: Is the function $f = \sum_{n=0}^{\infty} 2^{-n}\chi_{[n,n+1)}$ Lebesgue integrable on $\mathbb{R}$?
Is the function $f = \sum_{n=0}^{\infty} 2^{-n}\chi_{[n,n+1)}$ Lebesgue integrable on $\mathbb{R}$? Justify your answer.
I came across this question on a past exam paper for a measure theory course I'm taking and I c... |
H: $3$ digit number being subtracted by its digits
This question is an AMC style question. The question is this:
If the integer $A$ is reduced by the sum of its digits, the result is $B$. If $B$ is increased by the sum of its ($B$'s) digits, the result is $A$. Compute the largest $3$-digit number $A$ with this proper... |
H: Is a ring homomorphism surjective if the restriction to the group of units is surjective?
Let $R_1,R_2$ be two rings and suppose that $f:R_1\to R_2$ is a ring homomorphism. Denote $R_1'$ and $R_2'$ for the groups of units of $R_1$ and $R_2$. Next, let $f':R_1'\to R_2'$ be the restriction of $f$ to $R_1'$. I've alre... |
H: Sigma-distributivity of the algebra of Baire sets modulo meagre sets
Let $A$ be the free $\sigma$-algebra with $\omega_1$ free $\sigma$-generators, $X$ its Stone space, and $Ba(X)/M$ the algebra of Baire subsets of $X$ modulo meagre sets. Then $A$ is $\sigma$-isomorphic to $Ba(X)/M$ by the Loomis-Sikorski theorem.
... |
H: Finite vs infinite Ramsey theorem - what's the difference?
The finite Ramsey theorem states that given a $k$ and an $r$, there exists an $N$ such that every $r$ coloring of the edges of $K_N$ contains a monochromatic clique of size $k$.
The infinite version says that every coloring $c:\binom{\mathbb{N}}{2}\mapsto [... |
H: calculate: $\int_{0}^{2\pi}e^{\cos\theta}(\cos(n\theta-\sin\theta))d\theta$
calculate: $\int_{0}^{2\pi}e^{\cos\theta}(\cos(n\theta-\sin\theta))d\theta$
my try:
$
\begin{array}{c}
\int_{0}^{2\pi}e^{\cos\theta}(\cos(n\theta-\sin\theta))d\theta\\
\int_{0}^{2\pi}e^{\cos\theta}(\frac{e^{-i(n\theta-\sin\theta)}}{2}+\frac... |
H: Counting irreducible polynomials of the form $x^2 - ax + 1$ over a finite field.
There are some variations of the question. Fixing the finite field ${\mathbb{F}}_q$, the number of all monic irreducible polynomials $x^2 - ax + b \in {\mathbb{F}}_q[x]$ is $(q^2 - q)/2$, which is easy to see just by dividing the numbe... |
H: A Probability question on picking right ball
I picked one example from https://docplayer.net/6566428-Probability-exam-questions-with-solutions-by-henk-tijms-1.html
The problem is -
Bill and Mark take turns picking a ball at random from a bag containing four red balls and seven white balls. The balls are drawn out o... |
H: Cauchy product of $1+2+4+8+16+32+\dots$ and $1-1+1-1+1-1+\dots$
Cauchy product of $1+2+4+8+16+32+\dots$ and $1-1+1-1+1-1+\dots$.
Then
$c_1=1\times 1=1$
$c_2=1\times(-1)+2\times 1=-1+2$
$c_3=1-2+4$
$c_4=-1+2-4+8$
$\dots$
so when $n$ is even $s_n=\sum^{n/2}_1 2^{2n-1}$, when $s$ is odd, $s_n=\sum^{(n+1)/2}_12^{2(n-... |
H: Integral of an Odd Function on $\Bbb{R}^{n}$
Given an odd integrable function $\Omega$ on $\Bbb R^n$, i.e. $\Omega \in L^1(\Bbb R^n)$ and $\Omega(-x) = -\Omega(x)$, how do I show that its integral over a symmetric set limited $C$ is zero, ie, if $C = - C$, then
$$\int_{C} \Omega(x)dx = 0.$$
It seems reasonable this... |
H: Is this function, constructed by taking the maximum values between continuous functions, still continuous?
For each natural number $n$, let $f_n : [0,1] \to [0,1]$ be a continuous function, and for each $n$ let $h_n$ be defined by $h_n(x) = \max\{f_1(x),\ldots,f_n(x)\}$. Show that for each $n$ the function $h_n$ i... |
H: $(0,1), [0,1), [0,1]$ are not homeomorphic
I need to show $(0,1)$, $[0,1)$ and $[0,1]$ are not homeomorphic using intermediate value theorem (without using connectedness).
I have already did proved that $(0,1)$, $[0,1]$ are not homeomorphic but I struggle with the 2 other couples.
My proof: assume there is an home... |
H: Is a locally compact hereditarily Lindelof Hausdorff space first countable?
Is a locally compact hereditarily Lindelof Hausdorff space first countable?
I was recently told that it is but I can't find any reference to what I would have thought would be a standard fact if it is correct.
AI: Let $X$ be a locally compa... |
H: Deriving Parallel and perpendicular vectors from triple vector product
How would one go about resolving the vector $\vec{p}$ into parallel and perpendicular vectors to the given vector $\vec{w}$
By considering - $\vec{w}\times(\vec{p}\times\vec{w})$
So far I have used the triple vector product however I seem ... |
H: For $1
For $1<p<2$ , Fourier transform $\mathscr{F}$ is not onto $L^p(\Bbb T) \to \ell^q(\Bbb Z)$ where $\frac{1}{p}+\frac{1}{q}=1$
For $1 \le p \le 2$, Hausdorff-Young inequality implies that $\mathscr{F}:L^p(\Bbb T) \to \ell^q(\Bbb Z)$ where $\frac{1}{p}+\frac{1}{q}=1$
Now for $p=1$ showing that $\mathscr{F}:L^... |
H: How do I show that $x$ is the supremum of set $S$? (decimal representation of reals)
Let $x$ be a fixed positive real number.
Let $l_0 = a_0$ be the largest integer less than x (that is, $a_0\in Z$ such that $a_0 \le x$), $a_1$ be the largest integer such that $l_1 = a_0+\frac{a_1}{10^1}\le x$, $a_2$ be the largest... |
H: Prove that the supremum of an affine function is concave
Suppose I have a function such that $F(\theta x+(1-\theta)y,z)=\theta F(x,z)+(1-\theta)F(y,z)$ for $\theta\in(0,1)$. I want to show that $F(x,z)$ is concave in the first argument when taking a supremum, that is $G(x)=\sup_z F(x,z)\text{ is concave.}$
Let $\th... |
H: prove or give counter example, for every holomorphic function on the unit disc there is $f(z)=z$
let f be a holomorphic function on $D=\{z\in \mathbb C:|z|<1\}$. and let $f$ be continuous on $cl(D)$ and $f[D]\subseteq D$.
Prove or give counter example,
$\exists z\in D\mathrm{.f(z)=z}$
AI: Counter example: Let $a$ w... |
H: Proving absolute continuity of the laplace transform
Suppose $f \in L^\infty(\mathbb{R})$ and define the laplace transform $F:(0,\infty)\rightarrow \mathbb{R}$ by $$F(s) = \int_0^\infty f(t)e^{-st}dt.$$
Prove that $F$ is absolutely continuous on $[a,b]$ for any $b>a>0$.
So, I'm thinking that I might be able to do t... |
H: Continuous Random Variable Conditional Proability
I need to answer two questions:
Find $P(Y|X)$;
$P(0<Y<1/2 | X=0.15)$.
For #1 I know I would have to use the double integral and find Pxy and I understand how to do #1.
However, I'm completely stuck on #2 and don't understand how to use the value of $X= 0.15$ beca... |
H: What is the number of connected components of a continuous image of some topological space?
We know that continuous image of a connected space is always connected i.e continuous image of a space with one component will always have one component.
Also a space with two components (2×2 invertible matrices) can have a ... |
H: Order 5 rational map?
$p(z) = 1-\frac{1}{z}$ has order 3: $p(p(p(z))) = z$
Is there an order 5 rational map with rational coefficients?
AI: Such a rational map would be a linear fractional transformation:
$$f(z)=\frac{az+b}{cz+d}.$$
Its $k$-th iterate is the identity iff $A^k=\lambda I$ for some $\lambda$
where $A=... |
H: Find $\lim inf A_n$ and $\lim sup A_n$
Let $A_n = (−1 + \frac{1}{n}, 2 − \frac{1}{n})$ if $n$ is odd and $[0, n]$ if $n$ is even. Find $\liminf A_n$ and $\limsup A_n$.
This is a question on a past paper for a measure theory module I'm taking and I'm not quite sure if my answer is correct. I have $\limsup A_n = [-1,... |
H: Proof that no number less than $b$ can be an upper bound.
The following are definitions used in the two proofs given below:
Definition: A subset $I$ of $\mathbb{R}$ is called an interval if, for any $a,b \in I$ and $x \in \mathbb{R}$ such that $a \le x \le b$, we have $x \in I$.
Definition: Let $a \le b$ be any two... |
H: Union of intersection of families
I'm studying Halmos' Naive Set Theory. In Section 9, Families, he (essentially) mentions a following exercise (on page 35).
Exercise. If $\{A_i\}$ and $\{B_j\}$ are both nonempty families, then
$(\bigcap_iA_i)\bigcup(\bigcap_jB_j)=\bigcap_{i,j}(A_i\bigcup B_j)$.
However, I think ... |
H: Real Signal Properties and $\cos(t)$ Fourier coefficients
I am struggling to conceptually understand why the Fourier coefficients for $\cos(t)$ are $a_1 = 1/2$ and $a_{-1} = 1/2$ in light of the fact that, for a real and even signal, the Fourier coefficients should be real and even (I am referring to a complex Four... |
H: Can the Chinese Remainder Theorem extend to an infinite number of moduli?
I've been trying to find info on this and have come up lacking. The CRT says that a system of congruences with coprime moduli always has a unique answer (modulo the product of the original moduli). And the generalizations I've seen defined sa... |
H: Contradicting Equations describing "Resultant" velocity
My definition of resultant velocity:
If a certain object, at some instant of time, moves with speed $v_x$ in the x-direction, and with speed $v_y$ in the y-direction, then it has a resultant velocity which is the hypotenuse of the triangle formed by the two ve... |
H: Need help with $\arccos$ equation
I have the equation
$$ \cos(2x + \frac{\pi}{9}) = 0.5$$
I know that in order to solve for $x\in \Bbb R$, I need to use
$$\arccos(0.5) = 2x + \frac{\pi}{9} $$
This yields
$$ 2x + \frac{\pi}{9} =
\begin{cases}
\frac{\pi}{3} + 2k\pi, & \text{Positive angle} \\
2 \pi - \frac{\pi}{3}+... |
H: Understanding the definition of a $G$-module
I took the following definition from Milne's Fields and Galois Theory (page 69):
The part I underlined is the one giving me trouble. In particular, I would like to know why a $G$-module by that definition is the same as giving a homomorphism $f: G \to \operatorname{Aut}... |
H: Determining linear independence by inspection
I was given a question asking me to "determine by inspection whether the following vectors are linear independent." I know how to determine if vectors are independent by putting in row-echelon form and looking for free columns, but I don't know if determining by inspect... |
H: Getting rid of not invertible matrices
I recently came across the follwing formulation:
$CYC'=M$
where all letters are matrices and the ' stands for transposed. I was wandering if there is a clever way to "isolate" $Y$ (in the sense of having $Y=\dots$) even if matrix $C$ is not invertible.
$Y$ is also diagonal and... |
H: Show that $X \sim Y$ implies $E[f(X)] = E[f(Y)]$ in any subset of $\Omega$
I know that given two identically distributed variables $X$, $Y$ and a measureable function $f$, the theorem holds when you integrate over the universe of events $\Omega$. However, I am not sure if it holds when you integrate over a smaller ... |
H: Let $A$ be a finite set and let $B$ be a subset of $A$ with $|A|=n$,$|B|=m$ and $0
Task is:
Let $A$ be a finite set and let $B$ be a subset of $A$ with $|A|=n$,$|B|=m$ and $0< m < n$.Find a formula for the number of subsets of $A$ that contain $B$ and prove your statement.
I assume this is $2^m$ + something. $2^m... |
H: Equality of Splitting Fields
I am well aware of the fact that any two splitting fields of a set of polynomials are isomorphic. However, I am wondering when two splitting fields are actually the same.
Fix an algebraic closure of $E$ of $F$. Then, if $\{ f_i \}$ are polynomials in $F[x]$ and $K_1$ and $K_2$ are split... |
H: Proof of Existence of Von Neumann Numerals in ZFC
Let us recall the recursive definition of the Von Neumann representation of the natural numbers:
$0=\emptyset, S(n)=n \cup \{n\}$
We know by the Axiom of Empty Set that $0$ exists, and we are now left with proving whether or not $1, 2, 3$ and so on exist.
$S(n)$ is ... |
H: Does this recursion problem have a typo?
Just wondering how to cast this sentence into the actual intention — #8
AI: The notation $(a_n,a_{n+1})=1$ can mean that the greatest common divisor of $a_n$ and $a_{n+1}$ is $1$;
i.e., $a_n$ and $a_{n+1}$ are relatively prime. |
H: Meaning of P-symmetric and O-symmetric
I'm working on some problem sets and I come across the phrases "P-symmetric" and "O-symmetric" which were referring to a region in the Cartesian plane. The only clue I have towards their meanings is one of the questions was asking if a region was P-symmetric "for some P in the... |
H: Showing Existence of Antiderivative for Complex-Valued Function
I am asked to show that for $z\in \mathbb{C} \setminus \{0,1\}$, there exists an analytic (single-valued) function, $F(z)$ on $\mathbb{C} \setminus \{0,1\}$, such that $F'=f$, where $$f(z) = \frac{(1-2z)\cos(2\pi z)}{z^2 (1-z)^2}$$
I know that if $$\in... |
H: $t$ derivative of Kirchhoff's solution
I know that the solution to the PDE
\begin{align*}
u_{tt} - \Delta u = 0, \quad \mathbb{R}^3\times[0, \infty)\\
u(x, 0) = 0, \quad x \in \mathbb{R}^3\\
u_t(x, 0) = g(x), \quad x \in \mathbb{R}^3
\end{align*}
is
$$u(x,t) = \mathrel{\int\!\!\!\!\!\!-}_{\partial B(0,1)}t g(x + tw... |
H: What is the difference between recursion and induction?
What is the difference between recursion and induction? I have heard those terms used interchangeably, but I was wondering if there is a difference between them, and if so, what the difference is.
AI: In my experience:
"Recursion" is a way of defining some ma... |
H: Confusion with Regards to General and Particular Solution Terminology in Differential Equations
I have been reading R. Kent Nagle's Fundamental's of Differential Equations textbook and I'm really confused as to the meaning of the terms of "Particular Solution" and "General Solution", specifically as they change fro... |
H: Reference for a combinatorial identity (likely redundant)
Fix two natural numbers $m$ and $n$ accordingly. Using Wolfram we have
$$\sum_{k=1}^m\binom{k+n}{n}=\dfrac{(m+1)\binom{m+n+1}{n}-n-1}{n+1}.$$
Question: Can someone give me either an appropriate reference in which this formula might have appeared in so I can ... |
H: sequence of functions converges pointwise at irrationals
Let $g:\mathbb{N}\to \mathbb Q$ be a bijection; let $x_n=g(n)$. Define the function $f:\mathbb{R}\to \mathbb{R}$ as $$x_n\mapsto 1/n \text{ for } x_n\in \mathbb Q$$
$$x\mapsto 0 \text{ for } x\notin \mathbb Q. $$
I proved that this function is continuous prec... |
H: Prime Factorization Proof - Find the unique integer k
Let $S = 1! 2! \dotsm 100!$ Prove that there exists a unique positive integer $k$ such that $S/k!$ is a perfect square.
I've seen this question asked before but the answers were quite confusing. Anyone have a simpler solution to this problem? I believe the idea ... |
H: Will the center of one be larger than the center of the other?(center of gravity)
Assume that you have $n$ positive values $C_1,C_2,\ldots,C_n$, and you have $n$ values $g_1,g_2,\ldots,g_n$ where each $g_t\in[0,0.1]$.
Do we then have that
$$\frac{\sum\limits_{t=1}^n\frac{C_tg_t}{(1+g_t)^t}}{\sum\limits_{t=1}^n\frac... |
H: Prove that $F$ is Lebesgue measurable and $\sum_{n=1}^\infty m(E_n)\geq Km(F)$ under these conditions...
Question: Suppose $E_n$, $n\in\mathbb{N}$, is a sequence of Lebesgue measurable subsets of $[0,1]$. Let $F$ be the set of all points $x\in[0,1]$ that belong to at least $K$ (some positive number) of the $E_n$'s... |
H: Does a right angle triangle ABC, right angled at A has A-symmedian?
A symmedian is defined to be the isogonal of a median in a triangle .
In EGMO , lemma 4.24 (Constructing the Symmedian),which states, "Let $X$ be the intersection of the tangents to $(ABC)$ at $B$ and $C$. Then line $AX$ is a symmedian."
My questio... |
H: Isomorphic Sylow p-subgroups of two finite abelian groups G and H
Let $G$ and $H$ be abelian groups of order $n$. I want to prove that $G$ is isomorphic to $H$ if and only if for every prime $p\mid n$, Sylow $p$-subgroup of $G$ is isomorphic to Sylow $p$-subgroup of $H$.
One direction is obvious.
How do I show that... |
H: If $|g| = k$ then $|g^m| = k / $lcm$(k, m)$
Here $g$ is in a group $G$.
The only proof I got uses the concept of cyclic groups, but this wasn't introduced yet. How can I prove it in a simpler way?
AI: Note: The result is
$$|g^m|=\frac{k}{\color{red}{\gcd(k,m)}}.$$
Let $|g^m|=t$, then $(g^m)^t=e$. This means $k | tm... |
H: How to solve a fraction with a numerator in exponential form and a denominator in numerical form without a calculator?
The question:
"Imagine unwinding (straightening out) all of the DNA from a single typical cell and laying it "end-to-end"; then the sum total length will be approximately $2$ meters. Assume the hum... |
H: Given $a^2 \equiv n\pmod q$ find $b$ such that $b^2 \equiv n\pmod {q^2}$
Given $a^2 \equiv n\pmod q$ find $b$ such that $b^2 \equiv n\pmod {q^2}$
$a,n,q$ are given. How to find $b$?
I know I am supposed to use Hensel's lemma and "lifting" $q$, I just don't know how to do it.
AI: You're looking for something of the ... |
H: Find the density of Z
I'm doing this problem from Carol Ash's The Probability Tutoring Book
Setup -
Let $Z = min(X,Y)$ where X and Y are independent random variables.
$$X \sim Exp(\lambda = 1) \\Y \sim Exp(\lambda = 1)$$
Find $F_Z(z)$
My attempt:
Since X and Y are independent: $f_{X,Y}(x,y) = f_X(x)f_Y(y) = e^{-x}e... |
H: Which of the following topological spaces are separable?
This is Exercise 6 from Section 2.2 on page 25 of Topology and Groupoids, by Brown.
Exercise:
A topological space is separable if it contains a countable, dense
subset. Which of the following topological spaces are separable?
$\mathbb{Q}$ with the order top... |
H: Dealing with Subspaces
Just having a little trouble understanding how subspaces work. I know that to be a subspace it has to hold for vector addition and scalar multiplication, which I assume is equivalent to
$u+v = v+u$ and
$ku = (ku$1, $ku$2)
But how does that work for showing something like $U = [x,y,z | 3x + y ... |
H: Which of the following functions is even?
Let $f(x)$ be a continuous function. Which of the following must be an even function?
$(1) \int_{0}^{x} f(t^2)\mathop{dt}$
(2) $\int_{0}^{x} f(t)^2\mathop{dt}$
(3) $\int_0^x t(f(t) - f(-t))\mathop{dt}$
(4) $\int_0^x t(f(t) + f(-t)) \mathop{dt}$.
I know an even function sat... |
H: Solving the complex equation $(1+z)^5=z^5$
I must to find $z\in\mathbb{C}$ such that:
$\boxed{(1+z)^5=z^5}$
Is the following equivalence correct?
$(1+z)^5=z^5\Leftrightarrow 1+z=z$
If this is not correct, how can solve this problem?
AI: Note that $z=0$ is not a solution, so you may divide both sides by $z^5$ and ge... |
H: How many $4$-digit numbers of the form $1a2b$ are divisible by $3$?
How many $4$-digit numbers of the form $\overline{1a2b}$ are divisible by $3?$
Hello I am new here so I don’t really know how this works. I know that for something to be divisible by 3, you add the digits and see if they are divisible by $3$. So th... |
H: pairwise relatively prime pairs
Let m be divisible by $1,2, ... , n$.
Show that the numbers $1+m(1+i)$ where $i = 0,1,2, ... , n$ are pairwise relatively prime.
My proof was as following let us have two different numbers $1+m(1+i)$ and $1+m(1+j)$, let d divides them. Thus $d\mid i-j$.
I feel this won't lead anywh... |
H: Proof by contradiction of a variant of PHP
Let $a_1, a_2,\ldots , a_n$ be positive integers.
Prove that if $(a_1+a_2+\ldots+a_n)-n+1$ pigeons are to be put in $n$ pigeonholes, then for some $i$, the statement "The $i^{th}$ pigeonhole must contain at least $a_i$ pigeons" must be true.
My approach:
Let us assume th... |
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