text stringlengths 1 2.12k | source dict |
|---|---|
• Thank you for taking the time to write that out for me. I had spent quite some time trying to show that the integral went to zero without success, but the second MVT for integrals is a really nice way of solving that problem. – Will May 18 '20 at 18:39
• @Will: You're welcome. The point is you don't need to worry abo... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9869795123755812,
"lm_q1q2_score": 0.8427943961888703,
"lm_q2_score": 0.8539127566694178,
"openwebmath_perplexity": 172.0100886634145,
"openwebmath_score": 0.9741874933242798,
"tag... |
# Math Help - Seperable DE
1. ## Seperable DE
Hey,
I've been having some trouble recreating the answer my teacher has for the following question:
Find $y(x)$ given $\frac{dy}{dx} = x^3y$, where $y(1) = 2$.
So I did the following:
$\int \frac{dy}{y} = \int x^3dx \longrightarrow \ln{y} = \frac{x^4}{4} + C$
$y = e^... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.98697950682225,
"lm_q1q2_score": 0.8427943914468099,
"lm_q2_score": 0.8539127566694178,
"openwebmath_perplexity": 691.5098865193327,
"openwebmath_score": 0.9588059186935425,
"t... |
4. Yeah I did..but something just wasn't clicking for some reason. I just thought that since two numbers were introduced there should be two variables.
But now it makes sense. | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.98697950682225,
"lm_q1q2_score": 0.8427943914468099,
"lm_q2_score": 0.8539127566694178,
"openwebmath_perplexity": 691.5098865193327,
"openwebmath_score": 0.9588059186935425,
"t... |
# Math Help - curve sketching
1. ## curve sketching
if someone could show me all the steps to curve sketch y=3xe^-x that would be greatly appreciated!
2. Originally Posted by JMV
if someone could show me all the steps to curve sketch y=3xe^-x that would be greatly appreciated!
My suggestions:
-Try to determine the ... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.98697950682225,
"lm_q1q2_score": 0.842794384108196,
"lm_q2_score": 0.8539127492339909,
"openwebmath_perplexity": 680.6987716009038,
"openwebmath_score": 0.927586555480957,
"tags": ... |
3. Hello, JMV!
Graph: . $y \:=\:3xe^-x$
The only intercept is the origin: (0, 0).
The function is: . $y \:=\:\frac{3x}{e^x}$
Since $e^x \neq 0$, there are no vertical asymptotes.
Since $\lim_{x\to\infty}\frac{3x}{e^x} \:=\:0$, the horizontal asymptote is: . $y \:=\:0$ (x-axis)
First derivative: . $y' \:=\:-3xe^{-x... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.98697950682225,
"lm_q1q2_score": 0.842794384108196,
"lm_q2_score": 0.8539127492339909,
"openwebmath_perplexity": 680.6987716009038,
"openwebmath_score": 0.927586555480957,
"tags": ... |
# Logarithmic Spiral Distance Field
I have been playing around with distance field rendering, inspired by some of Iñigo Quílez’s work. Along the way I needed to define analytic distance functions for a number of fairly esoteric geometric primitives, among them the logarithmic spiral:
The distance function for this sp... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9740426450627305,
"lm_q1q2_score": 0.8427651590429379,
"lm_q2_score": 0.8652240877899776,
"openwebmath_perplexity": 901.6822671412986,
"openwebmath_score": 0.7944639921188354,
"tags": ... |
$n = \frac{\frac{ln(\frac{r}{a})}{b} - \Theta_{target}}{360^{\circ}}$ (4)
Now, feeding in the value of rtarget for r will give us an approximate value for n. This approximation will be a real (float, if you prefer), and we can observe from the graph above that the closest point must be at either the next larger or sma... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9740426450627305,
"lm_q1q2_score": 0.8427651590429379,
"lm_q2_score": 0.8652240877899776,
"openwebmath_perplexity": 901.6822671412986,
"openwebmath_score": 0.7944639921188354,
"tags": ... |
Could you show other distance estimators?
I can give you my estimators in return.
2. Gulli says:|
Thanks so much for your code, unfortunately it is not entirely accurate. If you write down the equations for a distance to the spiral arm you will find out that theta = theta_target does not provide a minimal solution un... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9740426450627305,
"lm_q1q2_score": 0.8427651590429379,
"lm_q2_score": 0.8652240877899776,
"openwebmath_perplexity": 901.6822671412986,
"openwebmath_score": 0.7944639921188354,
"tags": ... |
# Integral equation, certain rule
I have the following equation:
$$2\int_{0}^{1/n}(1-nx)^{2}dx=\frac{2}{n}\int_{0}^{1}(1-x)^{2}dx.$$
My question is: Is this a rule and where does it come from and if so when are you allowed to use it?
What I have done so far:
I calculated both the integrals by hand and from that I ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9740426405416754,
"lm_q1q2_score": 0.8427651551312121,
"lm_q2_score": 0.8652240877899775,
"openwebmath_perplexity": 201.91222203058427,
"openwebmath_score": 0.9695691466331482,
"ta... |
See integration by substitution. In your example, $nx = u = u(x)$, and $u(0) = 0$, $u(1/n) = 1$, and $\frac{du}{dx} = n$, so $dx = \frac{du}{n}$. Then you get that $$\int_0^{1/n}\!\left(1 - nx\right)^2\,dx = \int_{u(0)}^{u(1/n)}\!\left(1 - u\right)^2\frac{1}{n}\,du = \frac{1}{n}\int_{0}^{1}\!\left(1 - u\right)^2\,du.$$... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9740426405416754,
"lm_q1q2_score": 0.8427651551312121,
"lm_q2_score": 0.8652240877899775,
"openwebmath_perplexity": 201.91222203058427,
"openwebmath_score": 0.9695691466331482,
"ta... |
# Given that for each $n,\;x_n^n + x_n-1= 0,$ is $(x_n)_n$ convergent?
Prove that for $$n\ge 2$$, the equation $$x^n + x-1 = 0$$ has a unique root in $$[0,1]$$. If $$x_n$$ denotes this root, prove that $$(x_n)_n$$ is convergent and find its limit.
The limit is $$1$$. But to find the limit, I need to assume $$\lim\lim... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.974042642048694,
"lm_q1q2_score": 0.8427651412022592,
"lm_q2_score": 0.8652240721511739,
"openwebmath_perplexity": 107.74753596490618,
"openwebmath_score": 0.9916823506355286,
"tag... |
For any $$n\in\{1,2,3,\ldots\}$$ the function $$f_n(x)=x^n+x-1$$ is increasing and convex on $$[0,1]$$.
Since $$f_n(0)<0$$ while $$f_n(1)>0$$ we have a unique root $$x_n\in(0,1)$$. By convexity $$x_n < 1-\frac{f_n(1)}{f_n'(1)}=1-\frac{1}{n+1}\tag{1}$$ and we may notice that $$f_{n+1}(x_n) = x_n^{n+1}+x_n-1 = x_n(x_n^n+... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.974042642048694,
"lm_q1q2_score": 0.8427651412022592,
"lm_q2_score": 0.8652240721511739,
"openwebmath_perplexity": 107.74753596490618,
"openwebmath_score": 0.9916823506355286,
"tag... |
• @orangeskid Thanks. What a stupid mistake. I will remove that "solution". Aug 13 at 18:57
Write $$x_n = \frac{1}{y_n}$$, with $$y_n > 1$$, so $$\frac{1}{y_n^n} + \frac{1}{y_n} = 1$$ or $$1 = y_n^n - y_n^{n-1} = y^{n-1}_n(y_n-1)$$ and with $$y_n = 1+\delta_n$$ we get $$1 = (1+\delta_n)^{n-1} \cdot \delta_n$$ This imp... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.974042642048694,
"lm_q1q2_score": 0.8427651412022592,
"lm_q2_score": 0.8652240721511739,
"openwebmath_perplexity": 107.74753596490618,
"openwebmath_score": 0.9916823506355286,
"tag... |
# Is my work correct (easy problem, confidence intervals)
The r.v. $X$ represents the time taken by a computer in company $1$ in order to perform a certain job, and $Y$ represents the same thing but for company $2$. A sample of $n_X = 12$ computers are taken from company $1$, and we obtain: $\bar x = 65$, $s_X ^2 = 27... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9740426412951847,
"lm_q1q2_score": 0.8427651405503049,
"lm_q2_score": 0.8652240721511739,
"openwebmath_perplexity": 338.8931552922185,
"openwebmath_score": 0.7930347919464111,
"tag... |
• Please add the [self-study] tag & read its wiki. – gung - Reinstate Monica Sep 3 '15 at 1:49
• @gung: I added it. the specific problem that I encountered is the whether I interpreted the question and applied the formula correctly. – George Sep 3 '15 at 1:52
• Crossposted on Math: math.stackexchange.com/q/1419054/2335... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9740426412951847,
"lm_q1q2_score": 0.8427651405503049,
"lm_q2_score": 0.8652240721511739,
"openwebmath_perplexity": 338.8931552922185,
"openwebmath_score": 0.7930347919464111,
"tag... |
• thanks. I define the $s_X^2$ to be the sample variance and not the estimation of $\sigma_X ^2$. Sorry for the confusion. I guess with that my calculations are correct? – George Sep 3 '15 at 2:25
• Yes, your formulas and procedures are correct, but may need to check the calculation. – Deep North Sep 3 '15 at 2:27
• Pa... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9740426412951847,
"lm_q1q2_score": 0.8427651405503049,
"lm_q2_score": 0.8652240721511739,
"openwebmath_perplexity": 338.8931552922185,
"openwebmath_score": 0.7930347919464111,
"tag... |
# Find delta-v; Hohmann transfer orbit
1. Nov 19, 2013
### oddjobmj
1. The problem statement, all variables and given/known data
A space vehicle is in circular orbit about the earth. The mass of the vehicle is 3300 kg. The radius of the orbit is 2RE. It is desired to transfer the vehicle to a circular orbit of radiu... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924810166349,
"lm_q1q2_score": 0.8427522409700183,
"lm_q2_score": 0.8670357615200475,
"openwebmath_perplexity": 595.7302039602146,
"openwebmath_score": 0.7951969504356384,
"tag... |
So, the first answer should be:
$\frac{Δv_1}{v_0}$=15.47%
$\frac{Δv_2}{v_1}$=11.24%
I also tried entering these as the decimal equivalents and tried simply using the initial velocity as the divisor when calculating my percents. None of this worked. Perhaps I have A and B mixed around because I don't have the image b... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924810166349,
"lm_q1q2_score": 0.8427522409700183,
"lm_q2_score": 0.8670357615200475,
"openwebmath_perplexity": 595.7302039602146,
"openwebmath_score": 0.7951969504356384,
"tag... |
Using your suggestion(s) and that noted above the correct answer was found. | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924810166349,
"lm_q1q2_score": 0.8427522409700183,
"lm_q2_score": 0.8670357615200475,
"openwebmath_perplexity": 595.7302039602146,
"openwebmath_score": 0.7951969504356384,
"tag... |
# Thread: Double / Half & Area Trig Help
1. ## Double / Half & Area Trig Help
I have two questions:
1. Given that sin 3pi/10 = sqrt(5)+1/4 find an exact expression for cos (3pi/5)
I used cos(2theta) = 1 - 2sin^2(theta)
cos(3pi/5) = 1- 2sin^2(3pi/10)
= 1 - 2(sqrt(5)+1/4)^2
= 1- 2(3+sqrt(5)/8) = 4-3+sqrt(5)/4 = 1+sqrt(... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924802053235,
"lm_q1q2_score": 0.8427522369270559,
"lm_q2_score": 0.8670357580842941,
"openwebmath_perplexity": 1490.6317293919103,
"openwebmath_score": 0.9310417175292969,
"ta... |
$\displaystyle = 1 - \frac{6 + 2\sqrt{5}}{8}$
$\displaystyle = \frac{8}{8} - \frac{6 + 2\sqrt{5}}{8}$
$\displaystyle = \frac{8 - (6 + 2\sqrt{5})}{8}$
$\displaystyle = \frac{8 - 6 - 2\sqrt{5}}{8}$. Go from here.
It appears you did not subtract ALL of the second term...
yay i got it~ wow, i didn't multiply the 8 to g... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924802053235,
"lm_q1q2_score": 0.8427522369270559,
"lm_q2_score": 0.8670357580842941,
"openwebmath_perplexity": 1490.6317293919103,
"openwebmath_score": 0.9310417175292969,
"ta... |
# Can every integer greater than 5 be written as the sum of exactly one prime and one composite?
I worked it out up to 15.
6 = 4 + 2
7 = 4 + 3
8 = 6 + 2
9 = 4 + 5
10 = 8 + 2
11 = 9 + 2
12 = 10 + 2
13 = 10 + 3
14 = 9 + 5
15 = 12 + 3
Does this trend continue forever? I feel like the answer is obvious but I'm just not ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924818279465,
"lm_q1q2_score": 0.8427522366641648,
"lm_q2_score": 0.8670357563664174,
"openwebmath_perplexity": 120.02991167890097,
"openwebmath_score": 0.8685361742973328,
"ta... |
• Thanks. I feel like this was obvious and you really make it look easy. – Reggie Simmons Feb 28 '16 at 1:03
• This was brilliant. You did it so easily. – user230452 Feb 28 '16 at 1:04
• @ReggieSimmons You were a couple steps away from obvious yourself ;-) if only you had chosen different sums for 9 = 6 + 3 and 11 = 8 ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924818279465,
"lm_q1q2_score": 0.8427522366641648,
"lm_q2_score": 0.8670357563664174,
"openwebmath_perplexity": 120.02991167890097,
"openwebmath_score": 0.8685361742973328,
"ta... |
More generally, if $k$ is prime, then we can take $p_0=k$, and for each $0<i<k$ we have $\gcd(i,k)=1$, so by a theorem of Dirichlet, there are infinitely many primes in the arithmetic progression $i,i+k,i+2k,\dots$, and we can pick one of them to be $p_i$.
If $k$ is composite, then there is no possible choice for $p_0... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924818279465,
"lm_q1q2_score": 0.8427522366641648,
"lm_q2_score": 0.8670357563664174,
"openwebmath_perplexity": 120.02991167890097,
"openwebmath_score": 0.8685361742973328,
"ta... |
# Decomposition of vector space - two linear mappings
Theorem: Given two linear mappings $f,g: V \rightarrow V$ with
• $f\circ g = g\circ f = 0$
• $f+g=\operatorname{id}_V$
• $f\circ f = f$
• $g\circ g=g$
Then we have $$V=\operatorname{im}(\,f)\oplus \operatorname{im}(\,g)$$
Question:
I think $f$ and $g$ then are ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924818279465,
"lm_q1q2_score": 0.8427522349944016,
"lm_q2_score": 0.8670357546485407,
"openwebmath_perplexity": 215.6853121242732,
"openwebmath_score": 0.9731336832046509,
"tag... |
# Finding Euler spiral convergence point for powers other than 2
I'm playing around with Euler spirals, using a variable power to see how the plot changes,
$\left\{\begin{matrix} x = \int_0^s \cos \left ( s^n \right ) ds \\ y = \int_0^s \sin \left ( s^n \right ) ds \end{matrix}\right.$
and finding the values for pow... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971992476960077,
"lm_q1q2_score": 0.8427522240947315,
"lm_q2_score": 0.8670357477770336,
"openwebmath_perplexity": 1815.9672084144452,
"openwebmath_score": 0.6736855506896973,
"tag... |
• Limit doesn't like functions with inexact numbers. A very recent question was asked about this. Why don't you just integrate to infinity? Do this: Integrate[Cos[s^(3/2)], {s, 0, \[Infinity]}]. Jan 28 '16 at 4:56
• Because for whatever reason, that did not occur to me... Works a treat, thanks! A short answer, but if y... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971992476960077,
"lm_q1q2_score": 0.8427522240947315,
"lm_q2_score": 0.8670357477770336,
"openwebmath_perplexity": 1815.9672084144452,
"openwebmath_score": 0.6736855506896973,
"tag... |
see it in TraditionalForm:
f[x, k] // TraditionalForm
and find appropriate limits with a needed assumption:
lim[k_] = Limit[ f[x, k], x -> Infinity, Assumptions -> k > 1]
{Cos[Pi/(2 k)] Gamma[1 + 1/k], Gamma[1 + 1/k] Sin[Pi/(2 k)]}
Now we can see that indeed limits are finite for k > 1. Here we write down a few ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971992476960077,
"lm_q1q2_score": 0.8427522240947315,
"lm_q2_score": 0.8670357477770336,
"openwebmath_perplexity": 1815.9672084144452,
"openwebmath_score": 0.6736855506896973,
"tag... |
Determine Roots of Complex Polynomial
Printable View
• September 30th 2011, 03:09 AM
TaylorM0192
Determine Roots of Complex Polynomial
Hello,
I encountered this problem in my exercise set, and was stumped.
Solve (z + 1)^4 = 1 - i
My approach to the problem was to recognize that (1 - i) could be rewritten as [ (1 -... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9465966656805269,
"lm_q1q2_score": 0.8427495404110207,
"lm_q2_score": 0.8902942203004186,
"openwebmath_perplexity": 771.7954233960107,
"openwebmath_score": 0.9377341270446777,
"tag... |
Our text stipulates, however, that we should follow the "outline" used in one of the examples; i.e. with some clever algebraic manipulation put the complex polynomial equation in a solvable Cartesian form (i.e. do not use the polar form of the complex numbers). When I attempted it this way, I got stuck as I mentioned.
... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9465966656805269,
"lm_q1q2_score": 0.8427495404110207,
"lm_q2_score": 0.8902942203004186,
"openwebmath_perplexity": 771.7954233960107,
"openwebmath_score": 0.9377341270446777,
"tag... |
# Math Help - Roots of Unity represented on a regular polygon - lenght of side.
1. ## Roots of Unity represented on a regular polygon - lenght of side.
Hello,
I had an assignment that required me to solve for the roots of unity for various equations of the form z^n-1=0. Then , I was asked to represent the roots of u... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9891815526228442,
"lm_q1q2_score": 0.8427360009898127,
"lm_q2_score": 0.8519528076067262,
"openwebmath_perplexity": 653.5605611779417,
"openwebmath_score": 0.799518346786499,
"... |
5. ## Re: Roots of Unity represented on a regular polygon - lenght of side.
Originally Posted by Hyunqul
Thank you very much for your response,
But , as I used De moiver's theorem to obtain the solutions , I wonder what does (pk) stand for? is it Z^n ?
O.K. Let $\theta_n=\frac{2\pi}{n}$ then
$\exp(\theta_nk\mathbf{i})... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9891815526228442,
"lm_q1q2_score": 0.8427360009898127,
"lm_q2_score": 0.8519528076067262,
"openwebmath_perplexity": 653.5605611779417,
"openwebmath_score": 0.799518346786499,
"... |
Thanks for your generous contribution sir ,
In the same context , I was asked to obtain solutions for the equation Z^n-i = 0 for three cases where n = 1,2,3
I did that for all of them , for example when n = 3 the solutions set was :
i^(1/3) = 1^(1/3) e^[i(π/2 + 2kπ)]/3 = 1 e^[i(π/6 + 2kπ/3)] , where k = 0, 1, and 2 .... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9891815526228442,
"lm_q1q2_score": 0.8427360009898127,
"lm_q2_score": 0.8519528076067262,
"openwebmath_perplexity": 653.5605611779417,
"openwebmath_score": 0.799518346786499,
"... |
Sorry again.
12. ## Re: Roots of Unity represented on a regular polygon - lenght of side.
Originally Posted by Hyunqul
First thing reqired is to generalize and prove the results for z when z^n = cis(x) (or | a+bi | =1)
Second thing is what hepppens when Z^n = cis(x0 (or | a+bi | does not equal 1 ) | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9891815526228442,
"lm_q1q2_score": 0.8427360009898127,
"lm_q2_score": 0.8519528076067262,
"openwebmath_perplexity": 653.5605611779417,
"openwebmath_score": 0.799518346786499,
"... |
# Can I get a hint on solving this recurrence relation?
I am having trouble solving for a closed form of the following recurrence relation.
\begin{align*} a_n &= \frac{n}{4} -\frac{1}{2}\sum_{k=1}^{n-1}a_k\\ a_1 &= \frac{1}{4} \end{align*} The first few values are $a_1=\frac{1}{4},a_2=\frac{3}{8},a_3=\frac{7}{16},a_4... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9891815516660637,
"lm_q1q2_score": 0.8427359890198471,
"lm_q2_score": 0.8519527963298946,
"openwebmath_perplexity": 222.50502712371946,
"openwebmath_score": 0.9982168674468994,
"ta... |
• Thanks! I can't believe I have never thought of applying this kind of "differential equations" approach. It seems so obvious now! – A.E Aug 30 '13 at 2:50
• no problem @AEdwards :-p since both solution spaces for differential and difference form vector spaces we find similar approaches work in both cases – obataku Au... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9891815516660637,
"lm_q1q2_score": 0.8427359890198471,
"lm_q2_score": 0.8519527963298946,
"openwebmath_perplexity": 222.50502712371946,
"openwebmath_score": 0.9982168674468994,
"ta... |
Let $b_{n}:=\sum_{k=1}^{n-1}a_{k}$ for $n=2,3,\cdots$, then $b_{n+1}-b_{n}=a_{n}$ for $n=2,3,\cdots$. Then, the equation reads as $b_{n+1}-b_{n}=-\frac{1}{2}b_{n}+\frac{n}{4}$ for $n=2,3,\cdots$ with $b_{2}=a_{1}=\frac{1}{4}$. Rearraging the terms, we get $$\begin{cases}b_{n+1}-\frac{1}{2}b_{n}=\frac{n}{4},{\quad}n=2,3... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9891815516660637,
"lm_q1q2_score": 0.8427359890198471,
"lm_q2_score": 0.8519527963298946,
"openwebmath_perplexity": 222.50502712371946,
"openwebmath_score": 0.9982168674468994,
"ta... |
# A generalization of arithmetic and geometric means using elementary symmetric polynomials
Let $a_1, a_2, \ldots, a_n$ be positive real numbers. A while ago I noticed that if you form the polynomial $$P(x) = (x - a_1)(x-a_2) \cdots (x-a_n)$$ then:
• The arithmetic mean of $a_1, \ldots, a_n$ is the positive number $m... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9891815491146485,
"lm_q1q2_score": 0.8427359887053009,
"lm_q2_score": 0.8519527982093666,
"openwebmath_perplexity": 227.00011991639258,
"openwebmath_score": 0.9907392859458923,
"ta... |
Question: Must it be true that $m_0 \le m_1 \le m_2 \le \cdots \le m_{n-1}$?
• It's a known Maclaurin's inequality. See my proof in "КванТ", 1980, 04, M565 – Michael Rozenberg Dec 26 '14 at 8:12
• @MichaelRozenberg Fantastic! Thanks, now I know what this inequality is called, and that will probably be enough for me to... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9891815491146485,
"lm_q1q2_score": 0.8427359887053009,
"lm_q2_score": 0.8519527982093666,
"openwebmath_perplexity": 227.00011991639258,
"openwebmath_score": 0.9907392859458923,
"ta... |
We have $\displaystyle P'(x) = n\left(x^{n-1} + \sum\limits_{k=1}^{n-1}(-1)^k\binom{n-1}{k}u_k(\overline{a})x^{n-k-1}\right)$
If the roots of $P'(x)$ are $b_k$, for $k=1,2,\cdots,n-1$, (which, are positive reals by M.V.T.).
and define $v_k = \dfrac{\sum\limits_{1\le j_1<\cdots< j_k \le n-1}b_{j_1}\cdots b_{j_k}}{\bin... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9891815491146485,
"lm_q1q2_score": 0.8427359887053009,
"lm_q2_score": 0.8519527982093666,
"openwebmath_perplexity": 227.00011991639258,
"openwebmath_score": 0.9907392859458923,
"ta... |
Covariance is a measure of the linear relationship between two variables, but perhaps a more com-mon and more easily interpretable measure is correlation. That does not mean the same thing that is in the context of linear algebra. Or we can say, in other words, it defines the changes between the two variables, such tha... | {
"domain": "cyrilsancereau.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9532750400464604,
"lm_q1q2_score": 0.8427325919024881,
"lm_q2_score": 0.8840392924390585,
"openwebmath_perplexity": 683.3606059656761,
"openwebmath_score": 0.4935759902000427,
"ta... |
of two related variables each multiplied by a third independent variable Hot Network Questions You are simply seeing light touching your eyes (masturbation addiction) Here we will do another example of the Covariance in Excel. Since $$1 + \rho < 1 - \rho$$, the variance about the $$\rho = -1$$ line is less than that ab... | {
"domain": "cyrilsancereau.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9532750400464604,
"lm_q1q2_score": 0.8427325919024881,
"lm_q2_score": 0.8840392924390585,
"openwebmath_perplexity": 683.3606059656761,
"openwebmath_score": 0.4935759902000427,
"ta... |
matrices for a moment can simply be considered to be a measure of ‘ linear dependence between. That shows the quarterly gross domestic product ( also called the dot product or product... They change together, they change together or near zero ( B.! With unit covariance matrix is called white data is a measure of ‘ line... | {
"domain": "cyrilsancereau.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9532750400464604,
"lm_q1q2_score": 0.8427325919024881,
"lm_q2_score": 0.8840392924390585,
"openwebmath_perplexity": 683.3606059656761,
"openwebmath_score": 0.4935759902000427,
"ta... |
Dot product or scalar product ) of two vectors in that space measure of the examples in figure 3 simply. Set of real-valued random variables s new product line growth in percentage ( a ) a... Variables and to what extent, they change together { align } As these terms suggest, covariance correlation... Not mean the same... | {
"domain": "cyrilsancereau.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9532750400464604,
"lm_q1q2_score": 0.8427325919024881,
"lm_q2_score": 0.8840392924390585,
"openwebmath_perplexity": 683.3606059656761,
"openwebmath_score": 0.4935759902000427,
"ta... |
not necessarily inde-pendent: figure 6: figure 6: figure 6 covariance is a of. Deep understanding of this dependence be considered to be a statistical tool that is taken into account to out! Five quarterly performance dataset of a company that shows the quarterly gross domestic product ( called! A company that shows th... | {
"domain": "cyrilsancereau.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9532750400464604,
"lm_q1q2_score": 0.8427325919024881,
"lm_q2_score": 0.8840392924390585,
"openwebmath_perplexity": 683.3606059656761,
"openwebmath_score": 0.4935759902000427,
"ta... |
white data hands-on activity each of the relationship between the } As these terms suggest, and.: covariance is said to be a linearly transformed instance of figure 6 set real-valued! With unit covariance matrix is called white data ( B ) having quarterly... Taken into account to find out the relationship between the t... | {
"domain": "cyrilsancereau.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9532750400464604,
"lm_q1q2_score": 0.8427325919024881,
"lm_q2_score": 0.8840392924390585,
"openwebmath_perplexity": 683.3606059656761,
"openwebmath_score": 0.4935759902000427,
"ta... |
called the dot product scalar.: covariance is a deep understanding of this dependence correlation measure a certain of..., it is important to do some hands-on activity that shows the gross..., not necessarily inde-pendent is in the context of linear algebra find out the between... Data with unit covariance matrix is ca... | {
"domain": "cyrilsancereau.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9532750400464604,
"lm_q1q2_score": 0.8427325919024881,
"lm_q2_score": 0.8840392924390585,
"openwebmath_perplexity": 683.3606059656761,
"openwebmath_score": 0.4935759902000427,
"ta... |
# Find the minimum number of steps that we can arrange coins according to their weight
We have $20$ coins, every step we can give $10$ coins to a person and he will tell us the order of their weights. Find the minimum number of steps that we can arrange coins according to their weight.
My attempt:I found a method usi... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.980280871316566,
"lm_q1q2_score": 0.8427119339136413,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 387.6764381538882,
"openwebmath_score": 0.7218424677848816,
"tags... |
Math SE copy is here and AOPS copy is here.
Source:Second round Iranian olympiad of informatics.
• two steps... give him 10 coins, arrange them based on what he tells us... give him 10 more and arrange accordingly? – Jason V Aug 4 '17 at 12:53
• @Jason, try it with 4: if coin 1 is heavier than coin 2 and coin 3 is he... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.980280871316566,
"lm_q1q2_score": 0.8427119339136413,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 387.6764381538882,
"openwebmath_score": 0.7218424677848816,
"tags... |
To prove that four tests are insufficient, suppose that the true order of the coins is $c_1\le c_2\le \dots\le c_{20}$. In order to succeed, for each consecutive pair of coins $\{c_i,c_{i+1}\}$, there must be some test where both coins in the pair handed to your friend. Otherwise, you could not distinguish the true ord... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.980280871316566,
"lm_q1q2_score": 0.8427119339136413,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 387.6764381538882,
"openwebmath_score": 0.7218424677848816,
"tags... |
Note that for all $1\le i \le 5$, the consecutive pairs $\{c_{4i+1},c_{4i+2}\},\{c_{4i+2},c_{4i+3}\},$ and $\{c_{4i+3},c_{4i+4}\}$ have not been tested together. These need to be taken care of during tests three and four.
Since all 20 coins are represented among the 15 untested pairs, each coin must be used in exactly... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.980280871316566,
"lm_q1q2_score": 0.8427119339136413,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 387.6764381538882,
"openwebmath_score": 0.7218424677848816,
"tags... |
• I missed sth to write in my question:The coins have different weights so could you delete extra parts to make the proof easier for me to understand? – Taha Akbari Aug 8 '17 at 18:18
• This proof does assume all the coins have different weights. When I say overlaps, I mean a situation like this. For your first test, y... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.980280871316566,
"lm_q1q2_score": 0.8427119339136413,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 387.6764381538882,
"openwebmath_score": 0.7218424677848816,
"tags... |
• You're right. If the common coins don't occur at the same locations in the outcomes of the first two tests, then you've been lucky and have gained extra information that may allow you to use fewer steps (e.g. in the extreme case $a_{10}=b_1$ means you have sorted all but one coin which you can finish in 2 more steps)... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.980280871316566,
"lm_q1q2_score": 0.8427119339136413,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 387.6764381538882,
"openwebmath_score": 0.7218424677848816,
"tags... |
rat
Rational fraction approximation (continued fraction)
Syntax
``R = rat(X)``
``R = rat(X,tol)``
``[N,D] = rat(___)``
``___ = rat(___,Name,Value)``
Description
example
````R = rat(X)` returns the rational fraction approximation of `X` to within the default tolerance, `1.e-6*norm(X(:),1)`. The approximation is a ... | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808753491773,
"lm_q1q2_score": 0.8427119338558179,
"lm_q2_score": 0.8596637505099167,
"openwebmath_perplexity": 1278.1275222761155,
"openwebmath_score": 0.8807842135429382,
"tags":... |
`Q = str2sym(R)`
```Q = $\frac{355}{113}$```
Show the decimal representation of the fractional number $355/113$. This approximation agrees with $\pi$ to 6 decimal places.
`Qdec = vpa(Q,12)`
`Qdec = $3.14159292035$`
You can specify a tolerance for additional accuracy in the approximation.
`R = rat(sym(pi),1e-8)`
``... | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808753491773,
"lm_q1q2_score": 0.8427119338558179,
"lm_q2_score": 0.8596637505099167,
"openwebmath_perplexity": 1278.1275222761155,
"openwebmath_score": 0.8807842135429382,
"tags":... |
`R = rat(X,1e-4)`
```R = '2 + 1/(-3 + 1/(3 + 1/(-3 + 1/(3 + 1/(-3)))))' ```
To return the rational approximation with 10 coefficients, set the `'Length'` option to `10`. This option ignores the specified tolerance in the approximation.
`R10 = rat(X,1e-4,'Length',10)`
```R10 = '2 + 1/(-3 + 1/(3 + 1/(-3 + 1/(3 + 1/(-3 ... | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808753491773,
"lm_q1q2_score": 0.8427119338558179,
"lm_q2_score": 0.8596637505099167,
"openwebmath_perplexity": 1278.1275222761155,
"openwebmath_score": 0.8807842135429382,
"tags":... |
• If `X` is an array of m elements that contains a complex number, then `R` is returned as a character array with 2m+1 rows. The first m rows of `R` represent the continued fraction expansion of the real parts of `X`, followed by ```' +i* ... '``` in the (m+1)-th row, and the last m rows represent the continued fractio... | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808753491773,
"lm_q1q2_score": 0.8427119338558179,
"lm_q2_score": 0.8596637505099167,
"openwebmath_perplexity": 1278.1275222761155,
"openwebmath_score": 0.8807842135429382,
"tags":... |
# Describe the set of all odd numbers between $100$ and $200$ using set builder notation
I've come across a question in Discrete Mathematics, asking me to use set builder notation to describe the set of all odd numbers between 100 and 200.
The answer I had was: $$\{ p | p = 2n + 1, n \text{ (all numbers) } [50, 99], ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808707404786,
"lm_q1q2_score": 0.8427119334183998,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 253.25599102283704,
"openwebmath_score": 0.8849257230758667,
"ta... |
As others have noted, the $|$ symbol means "divide". And when we put a slash through it, it means "does not divide".
What does it mean for $2$ to "divide" a number? It means $2$ is a factor of that number. For example, $2$ divides $6$ because $6 = 3 \cdot 2$ (so $2$ is a factor of $6$). Similarly, $2$ divides $100$, s... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808707404786,
"lm_q1q2_score": 0.8427119334183998,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 253.25599102283704,
"openwebmath_score": 0.8849257230758667,
"ta... |
# How does the Pearson correlation coefficient change under rotations
I was reading on wikipedia about the pearson correlation coefficient. Assuming the data has zero mean it can be written as
$$\rho = \frac{ \sum x_i y_i } {\sqrt{\sum x_i^2 \sum y_i^2}}$$
The caption below this image says:
[...] Note that the corr... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808713165659,
"lm_q1q2_score": 0.8427119321513847,
"lm_q2_score": 0.8596637523076225,
"openwebmath_perplexity": 290.6191569666009,
"openwebmath_score": 0.8526313304901123,
"tag... |
$$\rho = \frac{C_{xy}}{\sqrt{P_x P_y}}.$$
Notice that:
$$P_{x'} = \sum_i {x'}^2_i = \sum_i (x_i \cos \alpha + y_i \sin \alpha)^2 = \\ \cos^2 \alpha\sum_i x_i^2 + \sin^2 \alpha\sum_i y_i^2 + 2\sum_i x_i y_i \sin \alpha \cos \alpha = \\\cos^2 \alpha P_x + \sin^2 \alpha P_y + \sin(2\alpha) C_{xy},$$
$$P_{y'} = \sum_i {... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808713165659,
"lm_q1q2_score": 0.8427119321513847,
"lm_q2_score": 0.8596637523076225,
"openwebmath_perplexity": 290.6191569666009,
"openwebmath_score": 0.8526313304901123,
"tag... |
• Sorry, I should have been more clear in my question. A change from $\rho=1$ to $\rho=-1$ is also illustrated in the wikipedia image, thus your example is not the most convincing one. On the other hand I guess by simply using a different value for $\alpha$ one can show that also the magnitude of $\rho$ is changing. Th... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808713165659,
"lm_q1q2_score": 0.8427119321513847,
"lm_q2_score": 0.8596637523076225,
"openwebmath_perplexity": 290.6191569666009,
"openwebmath_score": 0.8526313304901123,
"tag... |
$$X'=RX \ \ \text{with} \ \ R=\begin{bmatrix}\cos(\alpha)&-\sin(\alpha)\\\sin(\alpha)&\cos(\alpha)\end{bmatrix} \ \ \text{and}$$ $$X'=\begin{bmatrix}x'_1&x'_2&\cdots&x'_n\\y'_1&y'_2&\cdots&y'_n\end{bmatrix}, \ X=\begin{bmatrix}x_1&x_2&\cdots&x_n\\y_1&y_2&\cdots&y_n\end{bmatrix}$$
Thus $$X'X'^T=R(XX^T)R^T$$
In other w... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808713165659,
"lm_q1q2_score": 0.8427119321513847,
"lm_q2_score": 0.8596637523076225,
"openwebmath_perplexity": 290.6191569666009,
"openwebmath_score": 0.8526313304901123,
"tag... |
Follow-on question to “Fifty men and thirty woman…”
This questions relates to this question Fifty men and thirty woman are lined up at random. How do I find the expected number of men who have a woman standing next to them. and the answer given by André Nicolas.
(I would normally just comment on that question but don... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808730448281,
"lm_q1q2_score": 0.8427119318748526,
"lm_q2_score": 0.8596637505099168,
"openwebmath_perplexity": 361.9654301151759,
"openwebmath_score": 0.8672444820404053,
"tag... |
Linearity of expectation is (sometimes surprisingly) always valid. If I roll two dice then the expected value of the first is $3.5$ and the exopected value of the second is $3.5$ and the expected value of the sum is $7$. Now you'll say "Sure, that's cause they are independent." But now do this: I roll the first die and... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808730448281,
"lm_q1q2_score": 0.8427119318748526,
"lm_q2_score": 0.8596637505099168,
"openwebmath_perplexity": 361.9654301151759,
"openwebmath_score": 0.8672444820404053,
"tag... |
Now why your simulation differs from your computation: For a single man $i$ ("Jack", say) the expected value of his $X_i$ is \begin{align}E[X_i]&=\operatorname{Pr}(X_i=1)\\&=1-\operatorname{Pr}(X_i=0)\\&=1-\left(\frac{2}{80}\cdot \frac{49}{79}+\frac{78}{80}\cdot \frac{49}{79}\cdot\frac{48}{78}\right)\\&=\frac{154041}{2... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808730448281,
"lm_q1q2_score": 0.8427119318748526,
"lm_q2_score": 0.8596637505099168,
"openwebmath_perplexity": 361.9654301151759,
"openwebmath_score": 0.8672444820404053,
"tag... |
# Finding a value on a number line?
I came across this question: On the number line the tick marks are equally spaced which of the lettered points represent $y$
The solution says it is $D$ but I think it is $E$.
Explanation
1. The average $\frac{x+y}{2}$ shows it is the midpoint of $x$ and $y$ and thus counting the... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808753491773,
"lm_q1q2_score": 0.8427119303313049,
"lm_q2_score": 0.8596637469145053,
"openwebmath_perplexity": 283.48360256522597,
"openwebmath_score": 0.8446530699729919,
"ta... |
Yes, you are correct, $y$ should be $E$.
# Explanation -
1. Note that there are two spaces between $x$ and $x+y$, so $y=2$.
2. ${(x+y) \over 2}$ is to the right of $(x+y)$, so , $(x+y)$ must be negative.
3. As, $y$ is positive from $(1)$, so $x$ must be negative from $(2)$.
4. Now, ${(x+y)\over 2}$ is halfway between... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808753491773,
"lm_q1q2_score": 0.8427119303313049,
"lm_q2_score": 0.8596637469145053,
"openwebmath_perplexity": 283.48360256522597,
"openwebmath_score": 0.8446530699729919,
"ta... |
Smoothed Square Wave
new here and probably pretty unexperienced compared to the rest of you. This should be simple enough but just wanted some clarification. I'm trying to model an analog square wave oscillator with C. With the oscillator I'm trying to mimic, the square wave isn't totally square and has curves with th... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808753491773,
"lm_q1q2_score": 0.8427119285690484,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 303.52752415997287,
"openwebmath_score": 0.5046675205230713,
"ta... |
# Why does the Newton-Raphson method not converge for some functions?
$f(x)=2x^2-x^3-2$. This is a cubic type graph as shown. The real root of this graph is $(-0.839,0)$.
So, the question is to use Newton's approximation method twice to approximate a solution to this $f(x)$.
I use an initial starting value of $x_0=1... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808736209153,
"lm_q1q2_score": 0.8427119270833242,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 658.4361987979945,
"openwebmath_score": 0.7813798189163208,
"tag... |
One qualitative property is that, in the 1D case, you should not have an extremum between the root you want and your initial guess. If you have an odd number of extrema in the way, then you will start going away from the root you want, as you see here. If you have an even number of extrema in the way, then you will sta... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808736209153,
"lm_q1q2_score": 0.8427119270833242,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 658.4361987979945,
"openwebmath_score": 0.7813798189163208,
"tag... |
• Oops, very fast upvote :) – Peter Aug 27 '17 at 14:17
• Yes, I was going to transfer my previous comment to an answer, but you were a bit faster than me. – projectilemotion Aug 27 '17 at 14:18
• Note that the magic happens when you finally bounce around to hit ~1.4 which then sends you way over to ~-1.3 – Ian Aug 27 ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808736209153,
"lm_q1q2_score": 0.8427119270833242,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 658.4361987979945,
"openwebmath_score": 0.7813798189163208,
"tag... |
So if we start iterating in $x = a$ (where $a \not = 0$), we get the sequence $a, -a, a, -a, \ldots$ and the method loops forever between those two points, never getting to the root $x = 0$!
Edit: Here's a gnuplotted image: (In each iteration, we make a tangent in the current point (the blue dashed line) and the $x$ f... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808736209153,
"lm_q1q2_score": 0.8427119270833242,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 658.4361987979945,
"openwebmath_score": 0.7813798189163208,
"tag... |
Your example is one where Newton just takes more iterations than expected to converge, so it's not too bad. But here is an example of a cubic polynomial for which Newton's method won't converge!
\begin{align*} f(x) &= -0.74 + 0.765 x + 1.1 x^2 - 3.55 x^3 \\ x_0 &= 5/9 \end{align*}
Not only that, but it's in fact a st... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808736209153,
"lm_q1q2_score": 0.8427119270833242,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 658.4361987979945,
"openwebmath_score": 0.7813798189163208,
"tag... |
### Update:
I wrote some code to purposefully find both stable and unstable iterations:
Newton[f_] := t \[Function] t - f[t]/f'[t];
NewtonPlot[f_, xmin_, xmax_, x0_, n_, args___] :=
Plot[f[x], {x, xmin, xmax}, args,
Prolog -> {Thickness[Tiny],
Line[Flatten[Map[
{{#, 0}, {#, f[#]}} &,
NestList[Compile[t, Newton[f][t]]... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808736209153,
"lm_q1q2_score": 0.8427119270833242,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 658.4361987979945,
"openwebmath_score": 0.7813798189163208,
"tag... |
showing the basis of attraction in the complex plane for $x^3 - 1$, $x^4 - 1$, ... $x^{11} - 1$. Each colour corresponds to a root of the polynomial. Each point is colored to match the root that Newton's method eventually takes that point to.
These basins are typically quite complicated, but if you look at what is goi... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808736209153,
"lm_q1q2_score": 0.8427119270833242,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 658.4361987979945,
"openwebmath_score": 0.7813798189163208,
"tag... |
• Note that this happens because $\tanh$ is extremely flat for large $x$, so although you go in the right direction toward the root (since $\tanh$ has no extrema as I said in my answer), you massively overshoot. (By the way, this is a good example but I think it would be better to just go for root finding applied to $\... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808736209153,
"lm_q1q2_score": 0.8427119270833242,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 658.4361987979945,
"openwebmath_score": 0.7813798189163208,
"tag... |
There are several articles about the convergence of Newton's method. There is something called the Newton-Kantorovich theorem which gives rigour to the notion of convergence regions.. your starting point must be within the Fatou set which encloses the point of attraction of the dynamical system formed by the iterates o... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9802808736209153,
"lm_q1q2_score": 0.8427119270833242,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 658.4361987979945,
"openwebmath_score": 0.7813798189163208,
"tag... |
# Linear transformation such that $T(x) = kx$, for all $x \in \mathbf{R^n}$
Question
Let $T : \mathbf{R^n} \rightarrow \mathbf{R^n}$ be a linear operator such that $T(W) \subseteq W$ for every subspace $W$ of $\mathbf{R^n}$. Show that there exists $k \in \mathbf{R}$ such that $T(x) = kx$, for all $x \in \mathbf{R^n}$... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9585377249197138,
"lm_q1q2_score": 0.8426953499869588,
"lm_q2_score": 0.8791467754256017,
"openwebmath_perplexity": 82.07681424473164,
"openwebmath_score": 0.9551576375961304,
"tag... |
but
$T(e_i + e_j) = T(e_i) + T(e_j) = k_i e_i + k_j e_j; \tag 7$
these two equations taken together yield
$ke_i + ke_j = k_i e_i + k_j e_j, \tag 8$
whence
$(k - k_i)e_i + (k - k_j)e_j = 0; \tag 9$
the linear independence of the $e_i$ now forces
$k _i = k = k_j; \tag{10}$
since this argument applies for any two ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9585377249197138,
"lm_q1q2_score": 0.8426953499869588,
"lm_q2_score": 0.8791467754256017,
"openwebmath_perplexity": 82.07681424473164,
"openwebmath_score": 0.9551576375961304,
"tag... |
regression and elastic net provide different results
I fit Elastic Net model on 20-50 variables. Elastic Net selects 10 (but actually I can choose any model on the solution path for the next step).
Next, I take these 10 variables and fit standard regression with them. The estimated parameters differ and usually one v... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9603611631680358,
"lm_q1q2_score": 0.84269463751025,
"lm_q2_score": 0.8774767970940975,
"openwebmath_perplexity": 1125.2042796542821,
"openwebmath_score": 0.777109682559967,
"tags"... |
Alternatively, the coefficient estimates may be different between OLS and Elastic Net due to sample size. With small sizes, p-values from OLS may not be reliable. With small sample sizes, the bias from elastic net may also be high.
Here's a simulated example using $n=25$. The "true model" contains only two variables, ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9603611631680358,
"lm_q1q2_score": 0.84269463751025,
"lm_q2_score": 0.8774767970940975,
"openwebmath_perplexity": 1125.2042796542821,
"openwebmath_score": 0.777109682559967,
"tags"... |
• Thank you for a detailed explanation. I think first two paragraphs are more relevant in my case, but I like the alternative explanation. I suppose multicollinearity can also be an influential factor here. However, I remain a bit concerned that I have to choose from estimates that differ in millions of dollars in magn... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9603611631680358,
"lm_q1q2_score": 0.84269463751025,
"lm_q2_score": 0.8774767970940975,
"openwebmath_perplexity": 1125.2042796542821,
"openwebmath_score": 0.777109682559967,
"tags"... |
Much depends on your interpretation of the word "interpretation." Elastic net deliberately penalizes regression coefficients in a way that helps correct for the over-fitting and optimistically high-magnitude coefficients that standard regression can provide in this type of scenario. One might argue that "interpretation... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9603611631680358,
"lm_q1q2_score": 0.84269463751025,
"lm_q2_score": 0.8774767970940975,
"openwebmath_perplexity": 1125.2042796542821,
"openwebmath_score": 0.777109682559967,
"tags"... |
Number of teeth in gears
I'm building something with an engine that uses gears to reduce/increse movement. The motor has itself some gears, and it's a stepper motor (it gives discrete steps), now the number of steps per revolution is not integer: $4075.77284...$, and I need to add gears to make it as close as possible... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.96036116089903,
"lm_q1q2_score": 0.8426946185953132,
"lm_q2_score": 0.8774767794716264,
"openwebmath_perplexity": 522.912990403509,
"openwebmath_score": 0.9268463253974915,
"tags":... |
• Is there a typo in your expression for $n$? I get $(64\times22\times26\times31\times32)/(9\times9\times10\times11) \approx 4075.77284$, not $4075.3$. – Chris Culter Aug 5 '13 at 21:04
• No, the typo wsa in the 4075.3, thanks. – MyUserIsThis Aug 5 '13 at 21:04
• You may find this article interesting:ams.org/samplings/... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.96036116089903,
"lm_q1q2_score": 0.8426946185953132,
"lm_q2_score": 0.8774767794716264,
"openwebmath_perplexity": 522.912990403509,
"openwebmath_score": 0.9268463253974915,
"tags":... |
I don't think this can guarantee the optimal under constraints, but in general you can use the continued fraction representation to find best rational approximations to get something close. \begin{align} r &= \frac{720}{4075.7724\cdots} = 0.1766536\cdots \\ &= [0;5,1,1,1,18,3,\ldots] = \cfrac{1}{5+\cfrac{1}{1+\cfrac{1}... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.96036116089903,
"lm_q1q2_score": 0.8426946185953132,
"lm_q2_score": 0.8774767794716264,
"openwebmath_perplexity": 522.912990403509,
"openwebmath_score": 0.9268463253974915,
"tags":... |
# Is collapsing considered a legitimate proof?
For example if I want to prove that $2^n - 1 = 1 + 2 + 4 + 8 +...+ 2^{n-1}$ I can obviously use induction and that is accepted. But I can also collapse it like:
To Prove $2^n = S(n)$:
1. $S(n) = (1 + 1) + 2 +...+ 2^{n-1}$
2. $S(n) = (2 + 2) + 4 + 8 +...+2^{n-1}$
3. $S(n... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9372107914029486,
"lm_q1q2_score": 0.8426662664253948,
"lm_q2_score": 0.8991213867309121,
"openwebmath_perplexity": 494.5499069268852,
"openwebmath_score": 0.9015089273452759,
"tag... |
Let $S(k)=2^k + 2^k + 2^{k+1} + 2^{k+2} + ... + 2^{n-1}$. Then what we want to show is that $S(0) = 2^n$. Your proof basically amounts to saying $S(0) = S(1) = S(2) = ...$ "and so on", until we get $S(0) = S(n-1)$. Notice that $S(n-1) = 2^{n-1} + 2^{n-1}$, which obviously equals $2^n$. So we get $S(0) = 2^n$. To phrase... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9372107914029486,
"lm_q1q2_score": 0.8426662664253948,
"lm_q2_score": 0.8991213867309121,
"openwebmath_perplexity": 494.5499069268852,
"openwebmath_score": 0.9015089273452759,
"tag... |
This depends very much on the degree of rigour you want.
If you want to be really rigorous, or need to, then you would need to formalize “so on” by showing by induction that $$1 + \sum_{i=0}^{n-1} 2^i = 2^k + \sum_{i=k}^{n-1} 2^i \quad \text{for all k = 0, \dotsc, n},$$ which shows the statement for $k = n$.
If you a... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9372107914029486,
"lm_q1q2_score": 0.8426662664253948,
"lm_q2_score": 0.8991213867309121,
"openwebmath_perplexity": 494.5499069268852,
"openwebmath_score": 0.9015089273452759,
"tag... |
# Relative velocity/Boat & River Question
1. Feb 25, 2013
### Dalkiel
1. The problem statement, all variables and given/known data
280m wide river, destination 120m upstream, river current is 1.35 m/s downstream and the boat speed in still water is 2.70 m/s. What should the boat's heading angle be (relative to the s... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9678992923570261,
"lm_q1q2_score": 0.8426504865197787,
"lm_q2_score": 0.8705972751232809,
"openwebmath_perplexity": 1289.0463786031742,
"openwebmath_score": 0.5348750352859497,
"ta... |
ehild
3. Feb 26, 2013
### Dalkiel
Thanks. I used the quadratic equation way and got the same answer as my previous attempt.
4. Feb 26, 2013
### ehild
Splendid.
ehild
5. Feb 26, 2013
### Toranc3
I have a question for this. Doesn't going upstream mean going against the water flow?
6. Feb 26, 2013
### ehild
Y... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9678992923570261,
"lm_q1q2_score": 0.8426504865197787,
"lm_q2_score": 0.8705972751232809,
"openwebmath_perplexity": 1289.0463786031742,
"openwebmath_score": 0.5348750352859497,
"ta... |
15. Feb 26, 2013
### Dalkiel
The answer that was found (39.5°) is correct, but there was actually another way to solve this, using the law of sines. I'll try to draw another picture to describe it when I get home.
16. Feb 26, 2013
### Toranc3
Thanks! I was mostly confused with the wording of the question.
17. Feb... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9678992923570261,
"lm_q1q2_score": 0.8426504865197787,
"lm_q2_score": 0.8705972751232809,
"openwebmath_perplexity": 1289.0463786031742,
"openwebmath_score": 0.5348750352859497,
"ta... |
# Hyperbolic Substituion. I am wrong?
1. Aug 7, 2011
### flyingpig
1. The problem statement, all variables and given/known data
$$\int \;\sinh(2x) \cosh(2x) dx$$
3. The attempt at a solution
I let u = sinh(2x), du = 2cosh(2x)dx
Integrating I shuold get
$$\frac{1}{4} sinh^2 (2x) + C$$
But http://www.wolframalph... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.967899295134923,
"lm_q1q2_score": 0.8426504710649502,
"lm_q2_score": 0.8705972566572503,
"openwebmath_perplexity": 2659.490142658613,
"openwebmath_score": 0.9259747266769409,
"tags... |
# A faster way to evaluate $\int_1^\infty\frac{\sqrt{4+t^2}}{t^3}\,\mathrm dt$?
I need to evaluate the integral
$$\int_1^\infty\frac{\sqrt{4+t^2}}{t^3}\,\mathrm dt\tag1$$
After some workarounds I found the change of variable $t=2\sqrt{x^2-1}$, then
$$\int_1^\infty\frac{2\sqrt{1+(t/2)^2}}{t^3}\,\mathrm dt=\frac12\in... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9845754470129648,
"lm_q1q2_score": 0.8426500328418475,
"lm_q2_score": 0.855851154320682,
"openwebmath_perplexity": 327.58526096612553,
"openwebmath_score": 0.9996508359909058,
"tag... |
$$\int \frac{\cosh^2 x}{ 2 \sinh^3 x} dx.$$ Then let us use a letter different from your $t,$ $$\sinh x = \frac{2u}{1 - u^2}, \; \; \; \frac{1}{\sinh x} = \frac{1 - u^2}{2u}$$ $$\cosh x = \frac{1 + u^2}{1 - u^2},$$ $$d x = \frac{2du}{1 - u^2} \; .$$ $$\int \frac{(1 + u^2)^2 (1 - u^2)^3 2 du}{2 (1 - u^2)^2 (2u)^3 (1-u^2... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9845754470129648,
"lm_q1q2_score": 0.8426500328418475,
"lm_q2_score": 0.855851154320682,
"openwebmath_perplexity": 327.58526096612553,
"openwebmath_score": 0.9996508359909058,
"tag... |
Is G open in R^2
G-X
New member
The set G = {$$\displaystyle (x, y) \in R^{2} : (x, y) \neq (1, 0)$$} is an open set in $$\displaystyle R^{2}$$
Proof 2: We know a set X is defined to be closed if and only if its complement is open. Let $$\displaystyle J$$ = $$\displaystyle R^{2} / G$$ which implies $$\displaystyle J... | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984575448370756,
"lm_q1q2_score": 0.842650030383915,
"lm_q2_score": 0.8558511506439708,
"openwebmath_perplexity": 76.90338926096176,
"openwebmath_score": 0.9576208591461182,
"... |
Well-known member
MHB Math Helper
That is correct but I would have been inclined to a direct proof.
Let (x, y) be a point in the set G. Then x and y are not both 0 so the distance from (x, y) to (0, 0), $\sqrt{x^2+ y^2}= \delta$ is positive. So $\frac{\delta}{2}$ is also positive. The points in the neighborhood, $N\le... | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.984575448370756,
"lm_q1q2_score": 0.842650030383915,
"lm_q2_score": 0.8558511506439708,
"openwebmath_perplexity": 76.90338926096176,
"openwebmath_score": 0.9576208591461182,
"... |
# Topics in Number Theory: Some problems involving congruence relations (Part I)
For this note I assume that you know the basics properties of divisibility including those of the congruence relation $$a\equiv b \pmod{c}$$. See for example Modular arithmetics
I will show some nice problems involving the congruence rel... | {
"domain": "brilliant.org",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9845754497285467,
"lm_q1q2_score": 0.8426500297359818,
"lm_q2_score": 0.8558511488056151,
"openwebmath_perplexity": 258.7676098075675,
"openwebmath_score": 0.9953523874282837,
"tag... |
Hint. Use the previous problem.
Now, you can practice with the following problems. To avoid confusion I will call these problems N1, N2, etc.
Problem N1. ¿Does there exist a positive integer $$n$$ for wich the set $$\{n, n+1, \ldots, n+17\}$$ can be partitioned in two subsets $$\mathcal{A}$$ and $$\mathcal{B}$$ such ... | {
"domain": "brilliant.org",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9845754497285467,
"lm_q1q2_score": 0.8426500297359818,
"lm_q2_score": 0.8558511488056151,
"openwebmath_perplexity": 258.7676098075675,
"openwebmath_score": 0.9953523874282837,
"tag... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.