instruction stringlengths 12 30k |
|---|
Are there solutions of $a^n+n+b^n=c^n$ for $n>2$? |
Show that for every $x \in \mathbb{R}^{+}$
$$x-\frac{x^{2}}{2} < \log (x+1) < x,$$
and as an application, compute the limit
$$\lim_{n\to \infty}\prod_{k=1}^{n}\left(1+\frac{k}{n^{2}}\right).$$
My attempt:
Consider the map $f:\mathbb{R}_{0}^{+} \to \mathbb{R}$ defined by $f(x) = \log (x+1) -x+\frac{x^{2... |
Let’s consider five pairwise-distinct points $A$, $B$, $C$, $D$, and $E$ in a metric space over $\mathbb{R}^4$.
Can we somehow semi-automatically determine if the metric has a following feature—that regardless of the choice of those five points, any given distances $d(A, x)$, $d(B, x)$, …, $d(E, x)$ will either be i... |
What measures allow to distinguish points by their known distances? |
How to compute degree of $f: S^1\to S^1$ defined as $e^{ix}\mapsto e^{iax}$?
It is known that $f$ induces a vector space isomorphism $H^1(f): \mathbb R\to \mathbb R$ which is multiplication by a number that we call degree $(f)$.
The following theorem is also known to me:
If $M,N$ are oriented, compact, connect... |
How to compute degree of $f: S^1\to S^1$ defined as $e^{ix}\mapsto e^{iax}$? |
Show that for every $x \in \mathbb{R}^{+}$
$$x-\frac{x^{2}}{2} < \log (x+1) < x,$$
and as an application, compute the limit
$$\lim_{n\to \infty}\prod_{k=1}^{n}\left(1+\frac{k}{n^{2}}\right).$$
My attempt:
Consider the map $f:\mathbb{R}_{0}^{+} \to \mathbb{R}$ defined by $f(x) = \log (x+1) -x+\frac{x^{2... |
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 $A$ be an affine $k$-algebra and $\Omega_{A|k}$ the module of Kähler differentials of $A$? Under what conditions on $A$ is $\Omega_{A|k}$ a reflexive $A$-module? Clearly, $A$ being smooth is sufficient since this is equivalent to $\Omega_{A|k}$ being projective. What else can be said? |
When is the module of Kähler differentials reflexive? |
There's Wenshen's [Youtube Video Lectures][1], then there's Infinity Learn's Intro videos [one][2], [two][3], and [three][4]. Then there's Ezikian Acadamy and [his videos][5] (which aren't under a playlist, unfortunately), and then there's Dr. Bazett's [video][6]
[1]: https://youtube.com/playlist?list=PLbxFfU5GKZ... |
I have a question with my reasoning (or lack thereof) -- I understand what the correct answer is, but I do not see the gap in my logic.
**Premise:**
We have n balls and we randomly allocate them to n buckets with equal probability. What is the probability that exactly one bucket remains empty?
**My Attempt:**
L... |
How to compute degree of $f: S^1\to S^1$ defined as $e^{ix}\mapsto e^{iax}$?
It is known that $f$ induces a vector space isomorphism $H^1(f): \mathbb R\to \mathbb R$ which is multiplication by a number that we call degree $(f)$.
The following theorem is also known to me:
If $M,N$ are oriented, compact, connect... |
I think what is confusing you is that you are "starting out" with a planar graph of $n+1$ nodes.
But note that you are not yet saying anything about its colorability. All you're trying to show here is that $P(n) \to P(n+1)$, or:
"Any planar graph with $n$ nodes is $6$-colorable" $\to$ "Any planar graph with $n+1$... |
I have a question with my reasoning (or lack thereof) -- I understand what the correct answer is, but I do not see the gap in my logic.
**Premise:**
We have n balls and we randomly allocate them to n buckets with equal probability. What is the probability that exactly one bucket remains empty?
**My Attempt:**
L... |
I am reading an interesting paper [One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational](https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=427&option_lang=eng) by V.V. Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive integer parameters satis... |
I'm having a hard time solving an analysis problem. I tried really hard to find an answer online, I have downloaded numerous solutions manual for the book that proposed this question but I could not find it solved anywhere.
Let $y_n$ be a sequence of non zero positive numbers and let the limit of its series be $+\inft... |
I think what is confusing you is that you are "starting out" with a planar graph of $n+1$ nodes.
But note that you are not yet saying anything about its colorability. All you're trying to show here is that $P(n) \to P(n+1)$, or:
"Any planar graph with $n$ nodes is $6$-colorable" $\to$ "Any planar graph with $n+1$... |
Roger has 101 dalmatian dogs. Each dog has a unique number of black spots from the set {1, 2, 3, . . . , 101}. We choose any 52 of the 101 dogs. We want to prove that any set of 52 dogs satisfies the following addition property:
Addition property: The numbers of spots on two of the dogs add up to exactly the number ... |
I think what is confusing you is that you are "starting out" with a planar graph of $n+1$ nodes.
But note that you are not yet saying anything about its colorability. All you're trying to show here is that $P(n) \to P(n+1)$, or:
"Any planar graph with $n$ nodes is $6$-colorable" $\to$ "Any planar graph with $n+1$... |
This question is from QuantGuide(Busted 6 II):\
Suppose you play a game where you continually roll a die until you obtain either a 5 or a
6. If you receive a
5, then you cash out the sum of all of your previous rolls (excluding the
5). If you receive a
6, then you receive no payout. You have the decision to ca... |
I realized that the function $p(y;b)$ cannot be a log-concave for all fixed $y$. Let me show this for $y=0$. In this case,
$$p(0;b)=\int_{0}^{\infty}\dfrac{1}{\sqrt{b+\frac 5u}} u^{1/2}\exp\left(-\dfrac{2u}{3}\right)du.$$
is **both convex and log-convex** over $b>0$.
Considering that the function $$\frac{1}... |
I realized that the function $p(y;b)$ cannot be a log-concave for all fixed $y$. Let me show this for $y=0$. In this case,
$$p(0;b)=\int_{0}^{\infty}\dfrac{1}{\sqrt{b+\frac 5u}} u^{1/2}\exp\left(-\dfrac{2u}{3}\right)du.$$
is **both convex and log-convex** over $b>0$.
Considering that the function $$\frac{1}... |
I think what is confusing you is that you are "starting out" with a planar graph of $n+1$ nodes.
But note that you are not yet saying anything about its colorability. All you're trying to show here is that $P(n) \to P(n+1)$, or:
"Any planar graph with $n$ nodes is $6$-colorable" $\to$ "Any planar graph with $n+1$... |
I want to prove that for a graph with $n$ vertices, the size of the largest set of cycles that don't share edges with each other is at least $\frac{(|E(G)| - |V(G)|)}{2\lfloor \log_2(n+1)\rfloor}$.
It is trivial to prove this for all forests and trees. I can also observe that if we can prove this inequality for all ... |
Show that there is no homeomorphism $f:\mathbb{R}^2\setminus\mathbb{S}^1\rightarrow \mathbb{R}^2\setminus\mathbb{S}^1$ such that $f(0, 0) = (2, 0)$. I proceed as follows (everything has its usual topology):
Suppose there is such a homeomorphism $f$. Then the restriction of $f$ to $\{(x,y)\in\mathbb{R}^2:x^2+y^2<1\}$... |
The naive way I think about rotations is by imagining a finger sliding the sphere and in doing so it traces a spherical curve. Let's start now with a spherical curve which I assume to be rectifiable. Can I recover the rotation I performed to produce the curve?
I think we should impose the condition that when the curve... |
Among 101 dalmatian dogs, each dog has a unique number of black spots from the set {1, 2, 3, . . . , 101}. We choose any 52 of the 101 dogs. We want to prove that any set of 52 dogs satisfies the following addition property:
Addition property: The numbers of spots on two of the dogs add up to exactly the number of s... |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\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 $f(z)=\lambda(-\frac{1}{1+z})=\frac{1}{1-\lambda(z)}$, where $\lambda(z)$ is the modular lambda function, $\lambda(z)=16q-128q^2+704q^3+\ldots$, where $q=e^{\pi i z}$ (https://en.wikipedia.org/wiki/Modular_lambda_function).
What is the growth rate of the values of the higher order derivatives of $f(z)$ for fixed... |
I'm currently working on a problem related to particle motion in the plane \( \mathbf{R}^2 \) and could use some guidance. The particle moves along a curve, and at any given time \( t \) seconds after it starts, its position is described by the vector \( \mathbf{r}(t) = (2 \cos \pi t, \sin \pi t) \), with the units in ... |
Q) How to prove that the sequence:$ 2,4,6,8,...,1000$ is an A.P.$($$Arithmetic$ $Progression$)?
First of all, the $1^{st}$ term of this sequence is $2$ and the common difference of this sequence is also $2$. Therefore this sequence is in A.P.
**Doubt:**
I can't understand that how we simply tell that the given... |
I'm currently working on a problem related to particle motion in the plane $( \mathbf{R}^2 )$ and could use some guidance. The particle moves along a curve, and at any given time $( t )$ seconds after it starts, its position is described by the vector $( \mathbf{r}(t) = (2 \cos \pi t, \sin \pi t) )$, with the units in ... |
The naive way I think about rotations is by imagining a finger sliding the sphere and in doing so it traces a spherical curve. This processes is then possibly repeated starting from a different point and so on. Let's start now with a spherical curve which I assume to be rectifiable. Can I recover the rotation I perform... |
I was given that a random sample size n was taken from the following populations, N($\mu_{1}$,2) and N($\mu_{2}$,5). I am to find the MLE of $\tilde{\theta}$ for $\theta= \mu_{1}-\mu_{2}$. I have shown previously that for each individual distribution, the MLE would be $\hat{\mu_{1}}=\bar{x}$ and $\hat{\mu_{2}}=\bar{y}... |
The original question is
If $A, B$ are real $n \times n$ matrices such that $A$ is symmetric and positive definite and $B$ is such that $B^{T} = -B$, then is $A + B$ invertible ?
#### MY APPROACH
Let $ C = A + B$, then we have $C^{T} = A^{T} + B^{T} = A - B$. So $C$ is not a symmetric matrix. Considering
$$... |
For a real matrix A, if $x^{T}Ax > 0$ for all x then is A invertible? |
Define a function g from the set of real numbers to S by the following formula:
For all real numbers x,
[![function][1]][1]
Prove that g is a one-to-one correspondence. (It is possible to prove this statement either with calculus or without it.) What conclusion can you draw from this fact?
My question is th... |
I am trying to read G.E. Sacks's book on higher order recursion theory, and he has this result (where $O$ is Kleene's $O$ and $H_x$ is a hyperarithmetic set, i.e. where we have sets of the form $H_{2^a} = (H_e)'$ for some $e$ and $'$ is the jump).
Chapter 2, Theorem 1.3: Each of the following predicates is $\Pi_1^1$... |
Is there any specific reason for which integrals start at zero? Logically when we think of an integral as a summation of area, it must start from negative infinity right? Why do indefinite integrals start at zero? Like when I plug in a value in the indefinite integral, we get the area from 0 to the plugged in value. Th... |
Let $I,J$ be ideals of ring $R$. The product of two ideals is defined by $\sum_{\text{finite}}ab : a\in I,b\in J$.
My question is why do we take finite sums when defining the product of ideals?
For example: Consider $R=\mathbb Z$ ,$I= 5\mathbb Z$ and $J=7\mathbb Z$ then
$IJ=\sum a_i b_j:a_i\in I,b_j\in J=\sum (5... |
Why do we take finite sums when defining the product of ideals? |
The original question is
If $A, B$ are real $n \times n$ matrices such that $A$ is symmetric and positive definite and $B$ is such that $B^{T} = -B$, then is $A + B$ invertible ?
#### MY APPROACH
Let $ C = A + B$, then we have $C^{T} = A^{T} + B^{T} = A - B$. So $C$ is not a symmetric matrix. Considering
$$... |
Let $\mathcal{C}$ be an error-correcting code of length $n$ and $q$ the number of distinct symbols of the alphabet. We define the information rate of $\mathcal{C}$ by
$$\frac{\log_q\left(|\mathcal{C}|\right)}{n}.$$
I know it generalizes the concept which is connected to the time needed for the transmission of informa... |
Why is the information rate of an error-correcting code defined by $\frac{\log_q\left(|\mathcal{C}|\right)}{n}$? |
I'm trying to determine the complete set of smooth group actions of $\mathrm{SO}(3)$ on $S^2$. That is, I'm trying to determine all smooth $\sigma: \mathrm{SO}(3) \rightarrow \mathrm{Diff}(S^2)$. The two most obvious actions are the trivial action $\sigma_t(R) = I$ where all elements of $\mathrm{SO}(3)$ act by the iden... |
I'm currently studying the Global Torelli Theorem for K3 surfaces and I encountered the following section in Huybrechts book.
> If $\mathcal{X}\rightarrow S$ is a smooth proper morphism over a connected base $S$ with $X=\mathcal{X}_t$ for some fixed $t \in S$, then the *monodromy* of this family is the image of the ... |
What is a monodromy representation? |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\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... |
Define a function $g$ from the set of real numbers to $S$ by the following formula:
For all real numbers $x$,
$$
g(x) = \frac12\biggl( \frac x{1+|x|} \biggr) + \frac12.
$$
Prove that $g$ is a one-to-one correspondence. (It is possible to prove this statement either with calculus or without it.) What conclusi... |
Let’s consider five non-colinear points $A$, $B$, $C$, $D$, and $E$ in a metric space over $\mathbb{R}^4$.
Can we somehow semi-automatically determine if the metric has a following feature—that regardless of the choice of those five points, any given distances $d(A, x)$, $d(B, x)$, …, $d(E, x)$ will either be imposs... |
Show that $f(x, y)=(-1)^y x$ is an isomorphism from $\mathbb{R}^{+} \times \mathbb{Z}_2$ to $\mathbb{R}^*$.
(REMARK: To combine elements of $\mathbb{R}^{+}\times \mathbb{Z}_2$, one multiplies first components, adds second components.) Conclude that $\mathbb{R}^* \cong \mathbb{R}^+ \times \mathbb {Z}_2$.
I am stuck ... |
I have seen in two places, different requirements for a Markov chain to converge to its stationary/invariant distribution:
1. Irreducibility and aperiodicity. As mentioned [here](https://math.stackexchange.com/q/588251/762889)
2. Irreducibility and recurrence.
***
Is it true that under irreducibility and recu... |
Show that $f(x, y)=(-1)^y x$ is an isomorphism from $\mathbb{R}^{+} \times \mathbb{Z}_2$ to $\mathbb{R}^*$.
(REMARK: To combine elements of $\mathbb{R}^{+}\times \mathbb{Z}_2$, one multiplies first components, adds second components.) Conclude that $\mathbb{R}^* \cong \mathbb{R}^+ \times \mathbb {Z}_2$.
I am stuck ... |
Consider two functions $f_1(n)$ and $f_2(n)$ that grow to infinity at the same speed, say $$\lim_{n\to \infty} \frac{f_1(n)}{f_2(n)}=c $$ for some $c>0$. I am studying the squared difference of $f_1$ and $f_2$. Can the squared difference of the floor functions of $f_1$ and $f_2$ be estimated by that of the original fun... |
> Any suggestions on where I made an error in my logic?
The problem with your logic is that you are introducing an extra restriction. You are allowing only the last ball to be paired with another ball. In other words you are forcing the last ball to necessarily be in the bin with $2$ balls. This is a restriction tha... |
This question is from QuantGuide(Busted 6 II):\
Suppose you play a game where you continually roll a die until you obtain either a 5 or a
6. If you receive a
5, then you cash out the sum of all of your previous rolls (excluding the
5). If you receive a
6, then you receive no payout. You do not have the decisio... |
I've just been (as of two days ago) introduced to the Bochner integral, and I've [read][1] that Fubini's theorem holds for it, but I haven't been able to find its version for the said integral. So here's my question:
Let $(X,\Sigma,\mu),(Y,\Omega,\upsilon)$ be $\sigma$-finite measure spaces, $E$ a Banach space and $... |
I have seen elsewhere that every positive integer greater than 1 that is not a power of 2 can be written as the sum of some number of consecutive positive integers; however, given a specific x and y, where x and y are positive integers, can one find a formula to determine if x can be written as the sum of y consecutive... |
Can a positive integer x be written as the sum of y consecutive positive integers? |
How to evaluate the series which looks like a telescopic series but isn't one? |
I originally asked this in the Physics SE, but the question was closed because it was not about the underlying physics of it, so I'll try asking here.
Let $Q(q,p),P(q,p)$ be a canonical transformation with valence $\lambda$. The following is intended to be a proof of the following relation:
$$\lambda = \frac{\partial... |
There is not necessarily any "middle" of a four-torus, nor is there necessarily any "hole" in any place you might call the "middle".
When we try to visualize space in cases where the space is not Euclidean, there's a strong inclination to try to do this by embedding the space in a Euclidean space of higher dimension... |
Evaluate the series which looks like a telescopic series but isn't one? |
I am working on:
Prove that the Lie bracket of two symplectic vector fields is a Hamiltonian vector field. More
precisely, show that for all $X, Y ∈ \scr X(M, \omega)$ we have
$i_{[X,Y ]}\omega = −dH$ where $H = \omega(X, Y )$
My attempt:
**My definition of Hamiltonian and symplectic vector fields are:**... |
I recently came across [this][1] paper by Friedgut that shows low sensitivity boolean functions are close to juntas. This was an unintuitive result to me, as I thought I could easily imagine a function on the boolean hypercube that is a linear combination of many low-degree monomials, and therefore has low sensitivity,... |
Let $a$, $b$, $c$ and $d$ be arbitrary positive real numbers. Can all the products
$4a(1-b)$, $4b(1-c)$, $4c(1-d)$ and $4d(1-a)$ be greater than $1$?
Any idea how to solve this, because I don't know anything else except that $a,b,c,d>0$ and $4a(1-b)$, $4b(1-c)$, $4c(1-d)$, $4d(1-a)>1$, if and only if such numbers e... |
Cann all $4a(1-b)$, $4b(1-c)$, $4c(1-d)$ and $4d(1-a)$ be greater than $1$? |
Why is the Inner Product Induced by a Gaussian Matrix a Gaussian Process? |
I have a question with my reasoning (or lack thereof) -- I understand what the correct answer is, but I do not see the gap in my logic.
**Premise:**
We have n balls and we randomly allocate them to n buckets with equal probability. What is the probability that exactly one bucket remains empty?
**My Attempt:**
L... |
Define a function $g$ from the set of real numbers to $S$ by the following formula:
$$
g(x) = \frac12\biggl( \frac x{1+|x|} \biggr) + \frac12,\quad x\in\mathbb{R}.
$$
Prove that $g$ is a one-to-one correspondence. (It is possible to prove this statement either with calculus or without it.) What conclusion can y... |
I'm trying to calculate about $dg \wedge \star df$.
By the definition of Hodge-star, I write
$$\star df=
\left(\partial_\mu f\right) g^{\mu \nu} \sqrt{|g|} \epsilon_{\nu \rho_1 \ldots \rho_{n-1}} d x^{\rho_1} \wedge \ldots \wedge d x^{\rho_{n-1}}
$$
and
$$dg=\partial_i g dx^i.$$
But I had problem calculating
... |
I have a question with my reasoning (or lack thereof) -- I understand what the correct answer is, but I do not see the gap in my logic.
**Premise:**
We have $n$ balls and we randomly allocate them to $n$ buckets with equal probability. What is the probability that exactly one bucket remains empty?
**My Attempt:*... |
We have $n$ balls and we randomly allocate them to $n$ buckets with equal probability. What is the probability that exactly one bucket remains empty? |
Let $X$ and $Y$ be centered normal (Gaussian) random variables. Let $\rho:= \frac{E[XY]}{\sigma_X \sigma_Y}$ where $\sigma_X^2 = E[X^2]$ and $\sigma_Y^2 = E[Y^2]$. It is known that if $(X,Y)$ is a Gaussian vector then
$$ E[X|Y] = \frac{\rho \sigma_X}{\sigma_Y} Y. $$
However, I came across an exercise saying that th... |
Is it true that $E[X|Y] = \rho \frac{\sigma_X}{\sigma_Y} Y$ if $X$ and $Y$ are not jointly gaussian? |
**Background**
Assume that we have a solid ellipse with uniform density, and that it rolls along a curve.
In the following <a href="https://mathoverflow.net/questions/418486/along-what-curve-does-an-ellipse-roll-down-the-fastest">MO question</a>, I asked along what curve an ellipse rolls down fastest. It was poi... |
I am working on:
Prove that the Lie bracket of two symplectic vector fields is a Hamiltonian vector field. More
precisely, show that for all $X, Y ∈ \scr X(M, \omega)$ we have
$i_{[X,Y ]}\omega = −dH$ where $H = \omega(X, Y )$
My attempt:
**My definition of Hamiltonian and symplectic vector fields are:**... |
This can be shown by noting
\begin{equation*}
[a,x][b,x] = [a,x]^b[b,x]= [ab,x]
\end{equation*}
The middle equality follows because $[a,x]\in A$, which is Abelian. This together with
\begin{equation*}
[a,x^{i+j}] = [a,x^j][a,x^i]^{x^j} = [a,x^j][a^{x^j},x^i]
\end{equation*}
is enough (do you see why?) |
suppose we have, in the ring $\mathbb{Z} [x,y,z,w]$, the following polynomials:
$f=xw-yz$,
$g=x^2z-y^3$,
$h=yw^2-z^3$,
$k=xz^2-y^2w$.
The question is to prove that $I=(f,g,h,k)$ is the radical ideal of $J=(f,g,h)$.
I tried several things to no avail: by imposing $x>y>z>w$ I tried to divide $k^2$ by $J... |
I recently came across [this][1] paper by Friedgut that shows low sensitivity boolean functions are close to juntas. This was an unintuitive result to me, as I thought I could easily imagine a function on the boolean hypercube that is a linear combination of many low-degree monomials, and therefore has low sensitivity,... |
I am having trouble understanding a definition regarding tempered distributions from Bahouri's "Fourier Analysis and Nonlinear PDEs". In definiton 1.26, the author defines the subspace $\mathcal{S}^{'}_h$ of tempered distributions through a condition related to an $L^\infty$ norm of a convolution of a Schwartz function... |
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width">
<title></title>
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
<script id="MathJax-script" async
src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml... |
Consider some unital commutative algebra $A$ such that for all its elements we have
$$(ab)c + a(bc) = 2 b (ac) $$
Does this imply the algebra is associative ?
or in symbols :
$$(ab)c + a(bc) = 2 b (ac) \implies^? x(yz) = (xy)z$$ |
$(ab)c + a(bc) = 2 b (ac) \implies^? x(yz) = (xy)z$? |
Let $L$ be a number field, $p$ a prime number and $\alpha$ an algebraic integer in $L$ such that $L=\mathbb{Q}(\alpha)$. If we call $\mathcal{O}_L$ the ring of integers of $L$, I want to show that $p\mathcal{O}_L$ is coprime with the conductor of $\mathbb{Z}[\alpha]$ if and only if $p$ does not divide the Index $[\mat... |
Let $X_1, \ldots, X_N \in \mathbb{R}^p$ and $Y_1, \ldots, Y_N \in \mathbb{R}$. Define
$$
X=\left[\begin{array}{c}
X_1^{\top} \\
\vdots \\
X_N^{\top}
\end{array}\right] \in \mathbb{R}^{N \times p}, \quad Y=\left[\begin{array}{c}
Y_1 \\
\vdots \\
Y_N
\end{array}\right] \in \mathbb{R}^N
$$
Let
$$
\ell_i(\t... |
Let’s consider five non-colinear points $A$, $B$, $C$, $D$, and $E$ in a normed vector space $\mathbb{R}^4$.
Can we somehow semi-automatically determine if the metric has a following feature—that regardless of the choice of those five points, any given distances $d(A, x)$, $d(B, x)$, …, $d(E, x)$ will either be impo... |
Can all $4a(1-b)$, $4b(1-c)$, $4c(1-d)$ and $4d(1-a)$ be greater than $1$? |
I understand the notion of Linearity typically applied to define Linear Programs (here I will capitalize "Linear" when and only when I use it in this sense). I want to instead inquire about the notion of linearity as applied in math more broadly, with an eye toward scaling and adding multiple Mathematical Programs.
... |
[![enter image description here][1]][1]
It is known that the if the product of two line segments OA,OB drawn from any point O to a curve is a constant, then the curve is a circle (black). The product is the square of the geometric mean.
However what is another curve (green) if harmonic mean of OA,OB is a consta... |
Let $X$ and $Y$ be centered normal (Gaussian) random variables. Let $\rho:= \frac{E[XY]}{\sigma_X \sigma_Y}$ where $\sigma_X^2 = E[X^2]$ and $\sigma_Y^2 = E[Y^2]$. It is known that if $(X,Y)$ is a Gaussian vector then
$$ E[X|Y] = \frac{\rho \sigma_X}{\sigma_Y} Y. $$
However, I came across an exercise saying that th... |
I'm currently delving into a problem that involves analyzing the motion of a particle moving along an elliptical path in the plane, and I could really use some help figuring out a couple of aspects. The motion of the particle is defined by the position vector $$\mathbf{r}(t)=(3 \cos 2 \pi t, 2 \sin 2 \pi t),$$ where \(... |
Let $L$ be a number field, $p$ a prime number and $\alpha$ an algebraic integer in $L$ such that $L=\mathbb{Q}(\alpha)$. If we call $O_L$ the ring of integers of $L$, I want to show that $pO_L$ is coprime with the conductor of $\mathbb{Z}[\alpha]$ if and only if $p$ does not divide the Index $[O_L:\mathbb{Z}[\alpha]]$... |
I recently saw a poster claiming free WiFi if you could answer the integral. This is the integral: $$\int_{-2}^2{\left(x^3\cos\frac x 2+\frac 1 2\right)\sqrt{4-x^2dx}}$$the only problem is that the dx is under the square root. You could say this is a mistake and integrate and you would get $\pi$. But that answer is not... |
![Picture][1]
[1]: https://i.stack.imgur.com/N88ay.jpg
The picture shows an example of solving the integer problem with a decomposition method. However, what I am trying to ask is about the dual function part instead of the integer part. As you can see, the integer variables are relaxed. However, I cannot fin... |
Let's first consider solving it directly. Then discuss some point you miss.
Since $\arg\left(\dfrac{z-1}{z}\right)=\dfrac{\pi}{2}$, this means $\dfrac{z-1}{z}\in\mathbb{R}^+i$, this means there is $k\in\mathbb{R}^+$ such that $$\dfrac{z-1}{z}=ki\iff z=\dfrac{1}{1-ki}=\dfrac{1+ki}{1+k^2}$$ So what is the locus of thi... |
I need to find how distant the center of an orthogonal rectangle is from the center of a circle, given a specific angle.
The dimensions of the rectangle are proportional to the circle radius, so they use a 0-1 range, where 1 is the radius of the circle. The objective is to get the position of the center of that rect... |
I'm trying to calculate the departing angle of a geodesic using several different methods in two different coordinate spaces.
Currently I'm testing the following geodesic terminal points:
- Start (80°W, 40°N)
- End (40°E, 21°N)
The two different coordinate spaces are:
- Earth-ellipsoidal (WGS-84, $a = 6,37... |
Is it possible (using ZFC or any other axiom system which is known to be as consistent as ZF) to prove the existence of an ultrafilter $\mathcal F$ on the set of prime numbers s.t. there is an embedding $\mathbb R\hookrightarrow \tilde{\mathbb F}=\prod_p \mathbb F_p/\mathcal F$, but without this field $\tilde{\mathbb F... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.