instruction stringlengths 12 30k |
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Here are the first $40$ integers for which $\dfrac{3^n+1}{n+1}$ is an integer. I've denoted their residue mod $72$ by color and a symbol.
$$\begin{array}{r}\dagger\,\color{violet}0,&\bullet\,\color{brown}1,&*\,\color{red}3,&27,&531,\\
1\,035,&4\,635,&6\,363,&11\,475,&19\,683,\\
4\,131,&80\,955,&*\,\color{red}{266\,4... |
It's been more than a year I'm in touch with iterations of functions. A lot of my posts are about it.
The problem being, it's not really a mathematical topic that gets a lot discussed, in books or in mathematical channels on YouTube (where all of my non - school/post high-school mathematical knowledge is from).
So... |
The result can be derived from the theorem below on surface integrals, by setting the integrand function $f$ to unity.
I have used the following conventions below :
1. The physics convention $(r, \theta, \varphi)$ for spherical coordinates, with $\theta$ the polar angle and $\varphi$ the azimuthal angle [SPH]. ... |
I have the set {$|z-2+5i|<7,\text{Im}(z)>0$}. I sketched it in Desmos, the overlapping region represents the set. I need to check the properties below of this set.
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/9U2o0.png
Interior points - Is this the set {$|z-2+5i<7$, Im(z) > 0}?
... |
I am doing section 3-4 exercise 12 and 13 of Differential Geometry of Curves and Surfaces by Do Carmo and was stuck on those problems. Problem 12 is:
> Prove that if w is a differentiable vector field on a compact surface S and $\alpha(t)$ is the maximal trajectory of w with $\alpha(0)=p\in S$, then $\alpha(t)$ is ... |
This is from Bachman's Harmonic Analysis book, exercise 8.3
>Let $\mu$ be a Haar measure in $G$ and let $E_1$ and $E_2$ be two compact subsets of $G$ such that $\mu(E_1)=\mu(E_2)=0$. Does this imply that $\mu(E_1E_2)=0$?
My attempt: $E_1E_2$ is compact. By outer regularity, consider an open neighborhood $U_2$ of $E... |
Let $a,b,c,d \in \mathbb R$ such that $0 < c < 2$ and $d\ge 1$. Let $i$ be the imaginary unit and $\Gamma$ the Euler Gamma.
Numerically it looks that
$$2^{(a+bi)}\Gamma\left(\frac{c-(a+bi)}{d} \right) \Gamma\left(\frac{(a+bi)}{2}\right) \asymp 2^{\frac{2(a+bi)}{d}} \Gamma\left(\frac{c-(a+bi)}{2}\right)\Gamma\le... |
I'm learning the maths behind error correction codes. For this purpose I made this question for myself:
Assume you have two random bits $x_0$, $x_1$, which are both i.i.d. and have a 50% chance of being 0 or 1 (the information bits) and an additional check bit $x_2 = x_0 \mathbin{\mathsf{XOR}} x_1$). You now transfe... |
Let $a,b,c,d \in \mathbb R$ such that $0 < c < 2$ and $d\ge 1$. Let $i$ be the imaginary unit and $\Gamma$ the Euler Gamma.
Numerically it looks that
$$2^{(a+bi)}\Gamma\left(\frac{c-(a+bi)}{d} \right) \Gamma\left(\frac{(a+bi)}{2}\right) \asymp 2^{\frac{2(a+bi)}{d}} \Gamma\left(\frac{c-(a+bi)}{2}\right)\Gamma\le... |
Let $a,b,c,d \in \mathbb R$ such that $c\ge 1$ and $0 < d < 2$. Let $i$ be the imaginary unit and $\Gamma$ the Euler Gamma.
Numerically it looks that
$$2^{(a+bi)}\Gamma\left(\frac{c-(a+bi)}{d} \right) \Gamma\left(\frac{(a+bi)}{2}\right) \asymp 2^{\frac{2(a+bi)}{d}} \Gamma\left(\frac{c-(a+bi)}{2}\right)\Gamma\le... |
I'm studying some scheme theory and commutative algebra, and the silly question in the title came to my mind [while reading this on the StacksProject](https://stacks.math.columbia.edu/tag/0AFW).
More specifically: let $A$ be a (commutative, with unity) ring and $M$ and $A$-module. We say that $M$ is a *locally free ... |
I'm studying some scheme theory and commutative algebra, and the silly question in the title came to my mind [while reading this on the StacksProject](https://stacks.math.columbia.edu/tag/0AFW).
More specifically: let $A$ be a (commutative, with unity) ring and $M$ and $A$-module. We say that $M$ is a *locally free ... |
I'm learning the maths behind error correction codes. For this purpose I made this question for myself:
Assume there are two random bits $x_0$, $x_1$, which are both i.i.d. and have a 50% chance of being 0 or 1 (the information bits) and an additional check bit $x_2 = x_0 \mathbin{\mathsf{XOR}} x_1$). You now transf... |
Firstly, we consider this problem
$I(a)=\int_{0}^{\pi/2}\frac{arctan(asin(x))}{sin(x)}dx$
In general, after implementing the above technique, we get:
$I(a) = \frac{\pi \ln(a+\sqrt{a^2+1})}{2}+C$
I've seen quite a few people use $I(0)=0$ and then use Newton to derive $I(a)$. But it's clear that that's only tru... |
Sorry I am newbie and probably not very knowledgible
Would it be correct to imply from Euler's function that for some integer x, we have
x (limit x -> infinity) = [math]\frac {(log x)^{π(x)}}{log 2 * log 3 * log 5 .....}[/math]
where π(x) is prime counting function?
(1/1 + 1/2 + 1/3 + …. upto x) = (1 + 1/2 + ... |
Sorry I am newbie and probably not very knowledgible
Would it be correct to imply from Euler's function that for some integer x, we have
x (limit x -> infinity) = (log x)^π(x)/(log 2 * log 3 * log 5 .....)
where π(x) is prime counting function?
(1/1 + 1/2 + 1/3 + …. upto x) = (1 + 1/2 + 1/4 + 1/8 + …) (1 + 1/... |
Equal pushforward measures imply existence of random variable copy with matching joint? |
I'm learning the maths behind error correction codes. For this purpose I made this question for myself:
Assume there are two random bits $x_0$, $x_1$, which are both i.i.d. and have a 50% chance of being 0 or 1 (the information bits) and an additional check bit $x_2 = x_0 \mathbin{\mathsf{XOR}} x_1$). You now transf... |
I am currently working on a project where I would like to define infintesimals that can be used in conjunction with the real numbers (similar to the hyperreals). Right now, I am working on an algebraic extension of $\mathbb{R}$ to allow for my infintesimals. I have a definition that works for all linear combinations of... |
Is it possible to extend the domain and range of a function that maps from R to R to other sets? |
I'm trying to understand the single-set category in Categories for the Working Mathematician , specifically the 2-category version, and I'm confused about the commutation of $s_0$ and $s_1$.
"Similarly a 2-category can be considered to be a single set $X$ considered
as the set of 2-cells (e.g., of natural transfor... |
It is well known how to parameterize rational solutions to $x^2+y^2=a$ for $a\in \mathbb N^+$: first we know which [$a$ admit integer solutions][1]; our characterization is strong enough to tell us that theses are [exactly the $a$][2] admitting rational solutions; then [if we have one rational solution][3] we can get a... |
I'm solving a coding problem, and I wanted to mathematically prove that the answer exists. Essentially, we are given a string of lowercase letters, and we want to know if it can be rearranged such that no character repeats. Example: `aab` can be written `aba`, but `aaab` can't be rearranged in such a way.
Let $N$ b... |
I was trying to solve the question: how many positive integers $n \leq 100$ can be written as the sum of 4 or more consecutive integers.
Here was my approach:
Let $m$ represent the number of consecutive integers considered, and let $k$ be the starting integer.
For $m=4$:
$ n = k + k + 1 + k + 2 + k + 3 = 4k... |
How many positive integers below 100 cannot be written as a sum of 4 or more consecutive integers? |
I want to write a mathematical paper that is tailored for applied mathematics. To be more specific, it is using extensively differential equations and integrals. I observed that there are two writing style. First one is more concise and abstract. It uses a lot of theorems, proofs, lemmas, definitions etc. This i... |
Given a set of points on a hexagonal lattice, is there an efficient way to compute a center point $c$ and a minimal integer radius $r$ such that all points are within Manhattan distance $r$ of $c$?
The problem for squares on a square lattice is easy—compute the bounding rectangle and lengthen the shorter sides... |
I've been reading recently about randomized and non-randomized statistical tests and their differences. If I understand the definition correctly, a nonrandomized test for all values of the test statistics rejects the null hypothesis with probability $1$ or $0$. For a randomized test, for certain values of the test stat... |
I came across the following change of variables to turn equation into an elliptic curve.
(Source: https://www.youtube.com/watch?v=6eZQu120A80&t=3022s&ab_channel=ImperialCollegeLondon, the part I'm talking about starts at 23:20)
We are given: $\frac{(t^2+1)^2}{c^2}=t-t^3$, where $t \in \Bbb R$ and $c$ is a positi... |
I'm looking for some bibliography about what I should/must have into account when I have a model and experimental data that can be expressed in a way such that I can use a linear regression method to fit and determine experimental parameters. I mean, what are the conditions I have to check between the variables or anyt... |
What considerations should I have into account when linearizing a non-linear model for linear regressions? |
I'm looking for some bibliography about what I should/must have into account when I have a model and experimental data that can be expressed in a way such that I can use a linear regression method to fit and determine experimental parameters. I mean, what are the conditions I have to check between the variables or anyt... |
Hi I have a specific problem, that is given a 1-d nonhomogeneous heat equation
$$\frac{\partial^2 T}{\partial x^2}+\frac{1}{k}g(x,t)=\frac{1}{\alpha}\frac{\partial T}{\partial t}$$
for any $t$, is there a way to get the location of maximum i.e. $x_{max}$ for ${T(x,t)_{max}}$ without solving the equation fully?
... |
The total derivative of a function $F:\mathbb{R}^n \to \mathbb{R}^m$ is defined in this way:
If a linear map $L:\mathbb{R}^n\to\mathbb{R}^m$ exists such that
\begin{equation}
\tag{*}
\lim_{\boldsymbol{h}\to\boldsymbol{0}} \frac{||F(\boldsymbol{x}+\boldsymbol{h})-F(\boldsymbol{x}) - L\boldsymbol{h}||}{||\boldsymbo... |
Are these definitions of the total derivative equivalent? |
Your answers to (a) and (b) are correct, and you correctly listed the subsets of $\{1,2,3\}$, but your count of true relations among them of the form $A\subseteq B$ is incorrect: **all** of the subsets, including the two-element ones, are subsets of $\{1,2,3\}$. Correct brute force counting will yield a total of $27$ t... |
Which element in Z2024 corresponds to the triple of numbers (1,3,22) in the product ℤ8 *ℤ11 * ℤ23? |
I was trying to solve the question: how many positive integers $n \leq 100$ can be written as the sum of 4 or more consecutive integers.
Here was my approach:
Let $m$ represent the number of consecutive integers considered, and let $k$ be the starting integer.
For $m=4$:
$$ n = (k) + (k + 1) + (k + 2) + (k ... |
When you know two sides and the angle between them, the so-called "side-angle-side" case, older textbooks would recommend using the [law of tangents](https://en.wikipedia.org/wiki/Law_of_tangents) to find the remaining angles, if you don't want to bother finding the remaining side. I think that's what you mean by "dir... |
The total derivative of a function $F:\mathbb{R}^n \to \mathbb{R}^m$ is defined in this way:
If a linear map $L:\mathbb{R}^n\to\mathbb{R}^m$ exists such that
\begin{equation}
\tag{*}
\lim_{\boldsymbol{h}\to\boldsymbol{0}} \frac{||F(\boldsymbol{x}+\boldsymbol{h})-F(\boldsymbol{x}) - L\boldsymbol{h}||}{||\boldsymbo... |
Recently I am learning about homology theory from the point of view of $R$-modules. However, I can not find a satisfactory proof for it. I have found a proof using the Barratt-Whitehead Lemma but could not find the proof of this lemma. Also tried to generalize the zig-zag Lemma for homology groups to $R$-modules doing ... |
Given a set of points on a hexagonal lattice, is there an efficient way to compute a center point $c$ and a minimal integer radius $r$ such that all points are within Manhattan distance $r$ of $c$?
The problem for squares on a square lattice is easy—compute the bounding rectangle and lengthen the shorter sides... |
I have a stochastic differential equation of the type:
$$ dX(t) = \mu(t) X(t)dt + \sigma(t) X(t) dW(t) \tag{1}
$$
However, my $\mu(t)$ is a complicated function of $t$ as well as $W(t)$, somewhat like:
$$ \mu(t) = t \cdot a(t) + b(t) + \sigma W(t) \tag{2}
$$
I feel like $W(t)dt$ that I get by substituting $(2... |
What are simple, easy-to-visualize examples of:
1. Two manifolds that are conformally equivalent but not projectively equivalent?
2. Two manifolds that are projectively equivalent but not conformally equivalent?
(I understand "conformal equivalence" to be, roughly, that there is a bijection that preserves angl... |
Simple examples distinguishing conformal equivalence from projective equivalence? |
I have a stochastic differential equation of the type:
$$ dX(t) = \mu(t) X(t)dt + \sigma(t) X(t) dW(t) \tag{1}
$$
However, my $\mu(t)$ is a complicated function of $t$ as well as $W(t)$, somewhat like:
$$ \mu(t) = t \cdot a(t) + b(t) + \sigma W(t) \tag{2}
$$
I feel like $W(t)dt$ that I get by substituting $(2... |
Sorry I am newbie and probably not very knowledgible
Would it be correct to imply from Euler's function that for some integer x, we have
x (limit x -> infinity) = $\frac {(log x)^{π(x)}}{(log 2 * log 3 * log 5 .....)}$
where π(x) is prime counting function?
(1/1 + 1/2 + 1/3 + …. upto x) = (1 + 1/2 + 1/4 + 1/8... |
It is easy to find a symmetric matrix with complex coefficients with order $n$ i.e. $A^n = I$, one can just consider the diagonal matrix with $n$-th roots of units.
But is there a symmetric matrix with integer coefficients with order $n$? |
Is there a symmetric matrix with integer coefficients of order n? |
Let $v_1,\dots,v_n$ be some $n$ orthogonal complex vectors, i.e. $v_k = a_k + ib_k$ for some real vectors $a_k,b_k$. Is then the set $\{a_1,b_1,\dots,a_n,b_n\}$ linearly independent, i.e. are the real and imaginary parts of the $v_n$s linearly independent of each other? We know that $$0 = \left<v_k,v_l\right> = \left<a... |
Are the real and imaginary parts of orthogonal complex vectors linearly independent of each other? |
Let $n$ be a positive integer. It is easy to find a symmetric matrix with complex coefficients with order $n$ i.e. $A^n = I$, one can just consider the diagonal matrix with $n$-th roots of units.
But is there a symmetric matrix with integer coefficients with order $n$? |
How Can I find the Differential of a Function of Matrices? |
Let $(X_i)$ be an iid sequence of random variables with $E[X_1] = \mu, Var(X_1) = \sigma^2.$ Let $\bar X_n = \frac{\sum_{k=1}^n X_n}{n}.$ I want to find an appropriate scaling of $\bar X_n - \mu$ such that the sequence of the scaled sequence converges in distribution to $N(\mu, \sigma^2).$ How do I find the scaling ? I... |
How can one find the appropriate scaling for a sequence to converge to $N(\mu.\sigma^2)$? |
I ran multiple simulations of the following function, and it seems to be fair shuffling, given that all permutations were roughly equal, but I don't understand why it works. It's just inserting at random positions within the current shuffled deck isn't it? I know about Fisher-Yates shuffling, for reference.
```
def... |
Problem statement: Assume $N=\sum_{i=-\infty}^{\infty} \delta_{\Gamma_i}$ is a homogeneous Poisson point process (hPPP) on the real line with positive rate $\lambda$. Let $N$ be independent of $(U_n)_{n\in\mathbb{Z}}$ which are i.i.d. uniform on the interval $\langle 0,1\rangle$. Now consider the process $N'=\sum_{\Gam... |
I'm supposed to show that for the Torus $T^d = (S^1)^d$ we have the formula:
$$ H_n(T^d) =\oplus_{i=0} ^d \oplus_{j=1}^{d \choose i} H_{n-i}(point) = \begin{cases} R^{d\choose n} &0\leq n \leq d \\ 0 & else \end{cases} $$
where $H$ is any homology theory. We have already proofed and its probably a consequenc... |
Imagine that you have a fraction, where in the denominator you can have numbers from 1-n the same goes for the numerator,let´s assume that the fraction is smaller than 1,. The question is how many uniqe values can you get with every single combination in relation to n and is there a function to describe it? So far I th... |
I am reading the paper of F. Beukers, A refined version of the Siegel-Shidlovskii theorem (here is the [link][1]). The author mentions the following result in Algebraic group theory without proof.
> Lemma 2.4. Let $G_1,...,G_n$ be linear algebraic groups and $H\subset G_1\times...\times G_n$ be an algebraic subgroup... |
Let $n$ be a positive integer. It is easy to find a symmetric matrix with complex coefficients with order $n$ i.e. $A^n = I$, one can just consider the diagonal matrix with $n$-th roots of units.
A matrix $A$ has order $n$ if it is the smallest positive integer such that $A^n = I$.
But is there a symmetric matrix... |
My question is a follow up to the famous problem 'Basketball Scores' from the green book. The question can be found [here][1].
I solved the problem using the factorial approach. I also understand how the recurrence is formed and the answer is verified using induction.
> My question is, **how do I solve for P(x, ... |
I am working on a problem which contains the following:
> Suppose that $u \in \mathcal{D}'(\mathbb{R}^n)$ satisfies $xu = 0$.
Where $\mathcal{D}'$ is the space of distributions. Now, I am not completely sure what this should mean. $x$ is a function, which is not even compactly supported, whereas $u$ is a distribu... |
How can one find the appropriate scaling for a sequence to converge to $N(\mu, \sigma^2)$? |
Let $(X_i)$ be an iid sequence of random variables with $E[X_1] = \mu, Var(X_1) = \sigma^2.$ Let $\bar X_n = \frac{\sum_{k=1}^n X_n}{n}.$ I want to find an appropriate scaling of $\bar X_n - \mu$ such that the sequence of the scaled random variables converges in distribution to $N(\mu, \sigma^2).$ How do I find the sca... |
The formula for continuos compound interest is
$$A = Pe^{(rt)}$$
Where $r$ is the interest rate per year.
Interestingly, even though the money is compounded continuosly , it still depends on the annual interest rate rather than instantaneous interest rate. Is there any intuition to understand this? |
> ... I don't understand why it works. It's just inserting at random
> positions within the current shuffled deck isn't it?
Imagine this. You have a deck of cards kept in front of you. You spread out one of my hands so you can create a deck on it. You reach out, pick the card at the top and place it on my hand.
... |
I'm not sure if I understand your question correctly. I think you are worried that the extension of the vector fields, which are initially a section into $D$ do not become vector fields in $\mathbb{R}^n$? In any case, I don't really see what is the problem here. These sections are still vector fields in $R^n$. We choos... |
Imagine that you have a fraction, where both the numerator and denominator take values in the set $\{1,2,\dots, n\}$. Let us assume that the fraction is smaller than 1. The question is how many unique values can you get with every single combination in relation to $n$ and is there a function to describe it?
So far,... |
> ... I don't understand why it works. It's just inserting at random
> positions within the current shuffled deck isn't it?
Imagine this. You have a deck of cards kept in front of you. You spread out one of your hands so you can create a deck on it. You reach out, pick the card at the top and place it on your hand.... |
Let $\sum x_n$ a series of positive terms such that
$$
\frac{x_{n+1}}{x_n} = \frac{\alpha n + \beta}{\alpha n + \gamma}
$$
I want to study the convergence of this series.
My attemp: because of the above relationship, it is clear that the quotient criterium fails, but we can apply Raabe's criterium and conclu... |
Algebraic points on circles/solutions to $x^2+y^2=a$ over $\overline{\mathbb Q}$: is there a point with odd degree field extension over $\mathbb Q$? |
The definition of the dot product of a second order tensor with a vector is:
$$ (\mathbf{a} \otimes \mathbf{b}) \cdotp \mathbf{x} = (\mathbf{b} \cdotp \mathbf{x})\mathbf{a} $$
The definition of the dot product of a third order tensor with a vector is:
$$ (\mathbf{a} \otimes \mathbf{b} \otimes \mathbf{c}) \cdotp \m... |
What is the definition for the dot product of a co-vector and a dyadic product of vectors? |
My textbook (Amann and Escher, *Analysis 1*) defines an operation $\circledast$ (i.e. a function $\circledast: X \times X \to X$) as *anticommutative* if it obeys
1. There is a right identity element $r:=r_X$, that is, $\exists r \in X: x\circledast r=x, x \in X$, and
2. $x \circledast y = r \iff (x \circledast y... |
Following this question:https://math.stackexchange.com/questions/1563669/show-that-every-lebesgue-integrable-function-can-be-approximated-in-norm-and-alm
Let $f\in L^1(\mathbb{R})$. Show that there exists a sequence of continuous functions $\{g_n\}$ so that $g_n\to f$ a.e. as $n\to \infty$.
In that answer, we f... |
Your question has two parts.
We fix a finite group $G$ of order $n$.
**(1) Does $G$ act transitively on a set $\Omega$ of size $n$?**
The answer is **yes**, and the reason is already found in the standard proof of Cayley's theorem, which considers the the action of $G$ on itself (so $\Omega = G$) via
$$g \bul... |
I'm supposed to show that for the Torus $T^d = (S^1)^d$ we have the formula:
$$ H_n(T^d) = \bigoplus_{i = 0}^d \bigoplus_{j = 1}^\binom{d}{i} H_{n-i}(\text{point}) = \begin{cases} R^\binom{d}{n} &0\leq n \leq d \\ 0 & \text{else} \end{cases} $$
where $H$ is any homology theory. We have already proofed and it... |
It is well known that the primes are distributed such that they occur with an approximate "likelihood" of $1/\log(n)$ around the integer $n$ - or more precisely, the number of primes up to $n$ is $\pi(n) \sim \int_1^n 1/\log(n)\;dn$.
**Question:** Are there other sequences that have a distribution such that the like... |
Examples of integer sequences that have a distribution approx $1/\log(n)$, like the primes do? |
Your question has two parts.
We fix a finite group $G$ of order $n$.
**(1) Does $G$ act transitively on a set $\Omega$ of size $n$?**
The answer is **yes**, and the reason is already found in the standard proof of Cayley's theorem, which considers the the action of $G$ on itself (so $\Omega = G$) via
$$g \bul... |
Consider $\sum \frac{n^2}{n!}$. How can I find the sum? My attemp: it is equal to $\sum \frac{n}{(n-1}!}$, and we know that $\sum \frac{1}{n!}=e$. |
As mentioned in Revuz-Yor pg.120, the main reason that we require $H_{s}$ to be predictable, is because we want the integral against an adapted increasing process $A_{s}$
$$H\cdot A:=\int H_{s}dA_{s}$$
to remain adapted and in turn preserve martingale too. So as mentioned in (2.2) Theorem part (a), if we don't h... |
I know that the spaces $W^{k,p}(\Omega)$ are separable and reflexive ($\Omega \subset \mathbb{R}^n$ open and bounded) for $p \in (1, \infty)$. I also learned that in a separable Banach space $X$ there exists a metric on the closed unit ball $ B \subset X^* $ which induces the weak* topology on $B$.
Now my thinking ... |
Is the weak topology in separable Sobolev spaces induced by a metric? |
I am reading the paper of F. Beukers, A refined version of the Siegel-Shidlovskii theorem (here is the [link][1]). The author mentions the following result in Algebraic group theory without proof.
> Lemma 2.4. Let $G_1,...,G_n$ be linear algebraic groups and $H\subset G_1\times...\times G_n$ be an algebraic subgroup... |
Consider $\sum \frac{n^2}{n!}$. How can I find the sum? My attemp: it is equal to $\sum \frac{n}{(n-1)!}$, and we know that $\sum \frac{1}{n!}=e$. |
For a convex function $f$, I am interested in bounds of the form $$\frac{f(y)-f(x)}{y-x}\geq (1-\lambda)f'(y)+\lambda f'(x),\quad x<y.$$
For $\lambda=1$, this becomes true of any convex function $f$. On the other hand, for $\lambda= 0$, this only holds for a line. So I'm wondering which functions satisfy the above f... |
I was reading through the appendix of Lee's *Introduction to Topological Manifolds* and came across the following exercise in the section on metric spaces:
> **Exercise B.11** Let $M$ be a metric space and $A \subseteq M$ be any subset. Prove that the following are equivalent:
> 1. A is bounded.
> 2. A is contai... |
I have a general assignment problem that assigns a set of payload tasks $T$ to a set of workers $A$, where $|T|$ >> $|A|$. Each task $T_i \in T$ consists of a tuple $(s_i, g_i)$, which represent the start and goal position of the task.
I would like a MILP formulation, similar to that which is used for the Hungarian... |
As @Sine of the Time suggests, wolfram is incorrect because the integral just oscillates, this can be observed due to the following : Take $M > \pi$, then $$ \int_{e^\pi}^{e^M} \frac{\sin{(\log{x})}}{x}dx \overset{\log{(x)} = t}{=} \int_{\pi}^{M} \sin{(t)} dt = -\cos{(t)}\big|_{\pi}^{M} = -2\cos^2{\left(M/2\right)}. $$... |
I was reading this [answer](https://math.stackexchange.com/a/125208/48606) to the question of the division by zero which proposes the idea of "warped" numbers.
So, am I correct, the following rules could be a definition of division by zero:
Let:
* $\infty_w$ be a type of infinity with the properties below.
* $Q... |
Your question has two parts.
We fix a finite group $G$ of order $n$.
**(1) Does $G$ act transitively on a set $\Omega$ of size $n$?**
The answer is **yes**, and the reason is already found in the standard proof of Cayley's theorem, which considers the the action of $G$ on itself (so $\Omega = G$) via
$$g \bul... |
I am currently studying the various characterizations of amenability, and I am in particular working with Følner sets: a group $G$ is amenable if and only if for every finite subset $S \subseteq G$ and every $\varepsilon >0$ there exists a non-empty finite subset $F \subseteq G$ such that $| sF \Delta F|\leq \varepsilo... |
I was reading this [answer](https://math.stackexchange.com/a/125208/48606) to the question of the division by zero which proposes the idea of "warped" numbers.
So, am I correct, the following rules could be a definition of division by zero:
Let:
* $\infty_w$ be a type of infinity with the properties below.
* $Q... |
The well-known identity for complex points forming an equilateral triangle reads
$$z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$$
I have a doubt concerning the uniqueness of this identity
> Is $z_1^2 + z_2^2 + z_3^2 - z_1 z_2 + z_2 z_3 + z_3 z_1$ the only quadratic form in 3 variables that when replacin... |
How special is $z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$? |
Your question has two parts.
We fix a finite group $G$ of order $n$.
**(1) Does $G$ act transitively on a set $\Omega$ of size $n$?**
The answer is **yes**, and the reason is already found in the standard proof of Cayley's theorem, which considers the the action of $G$ on itself (so $\Omega = G$) via
$$g \bul... |
I was reading this [answer](https://math.stackexchange.com/a/125208/48606) to the question of the division by zero which proposes the idea of "warped" numbers.
So, am I correct, the following rules could be a definition of division by zero:
Let:
* $\infty_w$ be a type of infinity with the properties below.
* $Q... |
Let $n \ge 1$ and R be a unital $\mathbb{R}$-subalgebra of $M_n(\mathbb{R})$. If $A \in R \cap GL_n(\mathbb{R})$, is there any criteria to guarantee that its inverse is also in R? Since R is a vector space over the reals, this can be simplified to finding out when the adjugate of A remains in R.
I initially expecte... |
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