text stringlengths 1 2.12k | source dict |
|---|---|
Given \begin{align} \theta_1\theta_2 & = \alpha = \frac{n_1} n \tag 2 \\[10pt] \text{and } \theta_1\theta_2^2 & = \beta = \frac{n_2} n \tag 3 \end{align} we can divide the left side of $(3)$ by the left side of $(2)$ to get $\theta_2,$ and doing the same with the right sides we get $\theta_2=n_2/n_1.$ We can divide the... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850837598123,
"lm_q1q2_score": 0.8413744910804595,
"lm_q2_score": 0.8558511414521923,
"openwebmath_perplexity": 234.58624607005808,
"openwebmath_score": 0.903330385684967,
"tag... |
# The independent random variables $X$ and $Y$ are uniformly distributed on the intervals [-1,1] for $X$ and $Y$…
The independent random variables $X$ and $Y$ are uniformly distributed on the intervals $[-1,1]$ for $X$ and $[0,2]$ for $Y$. Evaluate the probability that $X$ is greater than $Y$, $P(X>Y)$.
My solution: ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759654852756,
"lm_q1q2_score": 0.8413641234942559,
"lm_q2_score": 0.8577681104440172,
"openwebmath_perplexity": 180.60631159920337,
"openwebmath_score": 0.9274419546127319,
"ta... |
where $P(0<X<1)=\frac{1-0}{1-(-1)}=\frac12$, $P(0<Y<1)=\frac{1-0}{2-0}=\frac12$
Since $X$ and $Y$ are both uniform and identical distributed on $(0,1)$ we get $P(X>Y|0<X,Y<1)=\frac12$
Therefore $P(X>Y)=\frac12\cdot \frac12\cdot \frac12=\frac18$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759654852756,
"lm_q1q2_score": 0.8413641234942559,
"lm_q2_score": 0.8577681104440172,
"openwebmath_perplexity": 180.60631159920337,
"openwebmath_score": 0.9274419546127319,
"ta... |
# how to prove the relation between the floor function and the number of divisors
I am trying to get an intuitive meaning (a proof) or why the following is true. $$\sum_{k = 1}^n \sum_{d|k} 1 = \sum_{d = 1}^n \left[\frac{n}{d} \right]$$
I know that $f(x) = [x] = \sum_{n \leq x} 1$ but I can't see it in the case of a ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759649262344,
"lm_q1q2_score": 0.8413641230147282,
"lm_q2_score": 0.8577681104440172,
"openwebmath_perplexity": 314.1025840596626,
"openwebmath_score": 0.8311999440193176,
... |
This gives $$\sum_{d=1}^n \left[\frac{n}{d}\right]$$
-
Is 1 not a divisor? – Tyler Hilton Mar 23 '13 at 1:58
it is, it should be in the table too. I 've added it. – user58512 Mar 23 '13 at 2:05
For each positive integer $d\le n$ there are $\left\lfloor\dfrac{n}d\right\rfloor$ multiples of $d$ in the set $\{1,2,\dots... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759649262344,
"lm_q1q2_score": 0.8413641230147282,
"lm_q2_score": 0.8577681104440172,
"openwebmath_perplexity": 314.1025840596626,
"openwebmath_score": 0.8311999440193176,
... |
# Summation of logs
Are there any useful identities for quickly calculating the sum of consecutive logs? For example $\sum_{k=1}^{N} log(k)$ or something to this effect. I should add that I am writing code to do this (as opposed to doing this on a calculator) so N can be very large.
-
$\log a+\log b=\log (a\cdot b)$ ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759654852756,
"lm_q1q2_score": 0.8413641217110516,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 512.0193222854198,
"openwebmath_score": 0.913544774055481,
"tags"... |
For example, it is lngamma in GP, lgamma in C (math.h), LogGamma in Mathematica, lnGAMMA in Maple, LogGamma in Magma, gammaln in MatLab, lnGamma in Mathcad, log_gamma in Sage, math.lgamma in Python, and gammaln in Perl (Math::SpecFun::Gamma).
- | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759654852756,
"lm_q1q2_score": 0.8413641217110516,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 512.0193222854198,
"openwebmath_score": 0.913544774055481,
"tags"... |
# What is the abelianization of $\langle x,y,z\mid x^2=y^2z^2\rangle?$
Let $G=\langle x,y,z\mid x^2=y^2z^2\rangle$.
What is the abelianization of this group? (Also, is there a general method to calculate such abelianizations?)
Update: I know how to get a presentation of the abelianization by adding relations like $x... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759621310288,
"lm_q1q2_score": 0.8413641170506816,
"lm_q2_score": 0.8577681068080749,
"openwebmath_perplexity": 385.91434471406995,
"openwebmath_score": 0.8758471608161926,
"ta... |
You can rewrite your relator such that it has $0$ exponent sum in two of the generators, as the map $x\mapsto xyz, y\mapsto y, z\mapsto z$ is a Nielsen transformation: \begin{align*} \langle x, y, z\mid x^{2}=y^2z^2\rangle &\cong\langle x, y, z\mid x^{2}z^{-2}y^{-2}\rangle\\ &\cong\langle x, y, z\mid (xyz)^{2}z^{-2}y^{... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759621310288,
"lm_q1q2_score": 0.8413641170506816,
"lm_q2_score": 0.8577681068080749,
"openwebmath_perplexity": 385.91434471406995,
"openwebmath_score": 0.8758471608161926,
"ta... |
• Seems like overkill not to abelianize first and just work in the category of abelian groups. – C Monsour Aug 20 '18 at 13:35
• @CMonsour Probably, but this is what I did when I solved the problem :-) [Also, as I said in the post, the general idea of reducing exponent sums to $0$ has applications beyond abelianisation... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759621310288,
"lm_q1q2_score": 0.8413641170506816,
"lm_q2_score": 0.8577681068080749,
"openwebmath_perplexity": 385.91434471406995,
"openwebmath_score": 0.8758471608161926,
"ta... |
# Is this 3D curve a circle?
The following is a curve in $3$ dimensions:
$$\begin{eqnarray} x & = & \cos(\theta) \\ y & = & \cos(\theta - \pi/3) \\ z & = & \cos(\theta - 2\pi/3) \end{eqnarray}$$
Is the curve a circle?
If it is, what about this curve in $4$ dimensions?
$$\begin{eqnarray} x & = & \cos(\theta) \\ y &... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759604539051,
"lm_q1q2_score": 0.8413641156120983,
"lm_q2_score": 0.8577681068080749,
"openwebmath_perplexity": 291.2724173812747,
"openwebmath_score": 0.9572219252586365,
"tag... |
1. The curve lies on a sphere.
We have $x_i^2 = \cos^2(\theta-\pi i/n) = \frac12+\frac12\cos(2\theta-2\pi i/n)$. So $$\|\vec x\|^2 = \sum x_i^2 = \frac n2 + \frac12 \sum \cos(2\theta-2\pi i/n).$$ The latter term is zero because it is the sum of $n$ equally spaced sinusoids (it is equivalently the $x$-component of the ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759604539051,
"lm_q1q2_score": 0.8413641156120983,
"lm_q2_score": 0.8577681068080749,
"openwebmath_perplexity": 291.2724173812747,
"openwebmath_score": 0.9572219252586365,
"tag... |
Using trigonometric identities, we get $$\cos(\theta-\frac\pi3) = \cos\theta\cos\frac\pi3+\sin\theta\sin\frac\pi3=\frac12\cos\theta+\frac12\sqrt3\sin\theta,$$ $$\cos(\theta-\frac{2\pi}3) = \cos\theta\cos\frac{2\pi}3+\sin\theta\sin\frac{2\pi}3=-\frac12\cos\theta+\frac12\sqrt3\sin\theta.$$ Thus, if we write $\gamma=(x,y,... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759604539051,
"lm_q1q2_score": 0.8413641156120983,
"lm_q2_score": 0.8577681068080749,
"openwebmath_perplexity": 291.2724173812747,
"openwebmath_score": 0.9572219252586365,
"tag... |
Using formula 26 here for the curvature, we have
\begin{align*} \kappa&=\frac{\|\mathbf r^\prime\times \mathbf r^{\prime\prime}\|}{\|\mathbf r^\prime\|^3}\\ &=\frac{\left\|\langle-\sin\,\theta,\cos\left(\theta+\frac{\pi}{6}\right),\cos\left(\theta-\frac{\pi}{6}\right)\rangle\times\langle-\cos\,\theta,-\sin\left(\theta... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759604539051,
"lm_q1q2_score": 0.8413641156120983,
"lm_q2_score": 0.8577681068080749,
"openwebmath_perplexity": 291.2724173812747,
"openwebmath_score": 0.9572219252586365,
"tag... |
Also, satisfies the equation of the general circle in 3-D, $(x-a)^2+(y-b)^2+(z-c)^2=d^2$. $a=b=c=0, d^2=\frac{3}{2}$
In case of $x,y,z,w$,
$z=\cos(\theta-\frac{2\pi}{4})=\sin\theta$, $\sqrt2 y=\cos\theta+\sin\theta$, $\sqrt2 w=-\cos\theta+\sin\theta$
So,$x^2+z^2=1$ and $w^2+y^2=1$
$x^2+y^2+z^2+w^2=2$
Now $\sqrt2 (... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759604539051,
"lm_q1q2_score": 0.8413641156120983,
"lm_q2_score": 0.8577681068080749,
"openwebmath_perplexity": 291.2724173812747,
"openwebmath_score": 0.9572219252586365,
"tag... |
$$r^2 = \sum_{i=1}^N \left[ \cos^2\left( \theta + \frac{i-1}{N} \pi \right) \right] = \frac{N}{2}$$
-
Constant distance from origin implies a sphere not a circle. See this and this. – user2468 Aug 22 '12 at 18:11
@JenniferDylan: Yes this is correct, and if the curve was an offset circle then the distance would not be... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759604539051,
"lm_q1q2_score": 0.8413641156120983,
"lm_q2_score": 0.8577681068080749,
"openwebmath_perplexity": 291.2724173812747,
"openwebmath_score": 0.9572219252586365,
"tag... |
# Given that $a$ is an odd multiple of $1183$, find the greatest common divisor of $2a^2+29a+65$ and $a+13$.
Given that $a$ is an odd multiple of $1183$, find the greatest common divisor of $2a^2+29a+65$ and $a+13$.
I know there exists some slick technique to simplify this problem. Any hints are greatly appreciated.
... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759626900699,
"lm_q1q2_score": 0.8413641139638007,
"lm_q2_score": 0.8577681031721325,
"openwebmath_perplexity": 109.30670454801536,
"openwebmath_score": 0.8226304650306702,
"ta... |
### Factoring 2g^2+4g-15 Solution
The variable we want to find is g
We will solve for g using quadratic formula -b +/- sqrt(b^2-4ac)/(2a), graphical method and completion of squares.
${x}_{}=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$
Where a= 2, b=4, and c=-15
Applying values to the variables of quadratic equation -b, a an... | {
"domain": "b254.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759654852756,
"lm_q1q2_score": 0.8413641127950302,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 2795.8348856123985,
"openwebmath_score": 0.2706487774848938,
"tags": null... |
g2 + g + = 0
Solutions
how to factor polynomials?
Polynomials can be factored using this factoring calculator
how to factor trinomials
Trinomials can be solved using our quadratic solver
Can this be used for factoring receivables, business, accounting, invoice, Finance etc
No this cannot be used for that
If you... | {
"domain": "b254.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759654852756,
"lm_q1q2_score": 0.8413641127950302,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 2795.8348856123985,
"openwebmath_score": 0.2706487774848938,
"tags": null... |
# How does partial fraction decomposition avoid division by zero?
This may be an incredibly stupid question, but why does partial fraction decomposition avoid division by zero? Let me give an example:
$$\frac{3x+2}{x(x+1)}=\frac{A}{x}+\frac{B}{x+1}$$
Multiplying both sides by $x(x+1)$ we have:
$$3x+2=A(x+1)+Bx$$
w... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759618515083,
"lm_q1q2_score": 0.8413641096781006,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 149.29333552611186,
"openwebmath_score": 0.9398812055587769,
"ta... |
$$\rm\begin{eqnarray}\dfrac{f(x)}{h(x)} = \dfrac{g(x)}{h(x)} &\Rightarrow&\rm\ f(x) = g(x)\ \ for\ all\,\ x\in\mathbb R\, \ such\ that\ h(x)\ne 0\\ &\Rightarrow&\rm\ f(x) = g(x)\ \ for\ all\ \,x\in \mathbb R \end{eqnarray}$$
since $\rm\:p(x) = f(x)\!-\!g(x) = 0\:$ has infinitely many roots, viz. all $\rm\:x\in \mathbb... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759618515083,
"lm_q1q2_score": 0.8413641096781006,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 149.29333552611186,
"openwebmath_score": 0.9398812055587769,
"ta... |
$$\lim_{x \to 0} \frac{x^2}{x}$$
you observe that $x^2/x = x$ for all $x \neq 0$ so that
$$\lim_{x \to 0} \frac{x^2}{x} = \lim_{x \to 0} x$$
and then you apply the fact that $x$ is continuous at $0$ to obtain
$$\lim_{x \to 0} x = 0$$
-
Let $f(x)=p_1(x)/q_1(x)$, $g(x)=p_2(x)/q_2(x)$ be two rational functions. If $f... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759618515083,
"lm_q1q2_score": 0.8413641096781006,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 149.29333552611186,
"openwebmath_score": 0.9398812055587769,
"ta... |
# Open ball of radius, r = 0 is empty?
Is $B(a;0) = \{x : d(a, x) < 0\} = \varnothing$? And if so, is it always the case?
The reason I ask is because I want to know if the open interval $(a,a) = \varnothing$ when $a \in \mathbb{R}$.
Thank you.
Kind regards, Marius
-
Not every question involving sets has to do with... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9808759590563021,
"lm_q1q2_score": 0.8413641090636662,
"lm_q2_score": 0.8577681013541613,
"openwebmath_perplexity": 396.8606248092315,
"openwebmath_score": 0.9406955242156982,
"tag... |
# Lecture 7: KNN and Decision Trees¶
These demos illustrate various aspects of KNN and decision tree classification. They also show how to use built-in implementations of machine learning methods from scikit-learn. This is very useful!
## Imports¶
Run this cell.
In [52]:
import numpy as np
import matplotlib.pyplot ... | {
"domain": "umass.edu",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126500692716,
"lm_q1q2_score": 0.8413637944677987,
"lm_q2_score": 0.8596637577007393,
"openwebmath_perplexity": 6380.290869866497,
"openwebmath_score": 0.3988993763923645,
"tags": null... |
In [55]:
dist = DistanceMetric.get_metric('manhattan')
plot_distance_contours(dist)
### Chebyshev distance¶
We get another special case of the Minkowski distance called Chebyshev distance when $p \rightarrow \infty$. It looks like this.
In [56]:
dist = DistanceMetric.get_metric('chebyshev')
plot_distance_contours(d... | {
"domain": "umass.edu",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126500692716,
"lm_q1q2_score": 0.8413637944677987,
"lm_q2_score": 0.8596637577007393,
"openwebmath_perplexity": 6380.290869866497,
"openwebmath_score": 0.3988993763923645,
"tags": null... |
#Compute a prediction for every point in the grid
gyhat = clf.predict(gx)
gyhat = gyhat.reshape(gx1.shape)
#Plot the results
cmap_light = ListedColormap(['#FFAAAA', '#AAFFAA', '#AAAAFF'])
for i in [1,2,3]:
plt.plot(x[y==i,0],x[y==i,1],labels[i-1]);
plt.xlabel('area');
plt.ylabel('compactness');
plt.pcolormesh(gx1,gx2,... | {
"domain": "umass.edu",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126500692716,
"lm_q1q2_score": 0.8413637944677987,
"lm_q2_score": 0.8596637577007393,
"openwebmath_perplexity": 6380.290869866497,
"openwebmath_score": 0.3988993763923645,
"tags": null... |
### Observe that KNN can recover the ground truth¶
• Run the code below to fit KNN models and plot the resulting predictions for training sets of increasing size.
• Observe that as the training set gets bigger, the KNN predictions converge to the ground truth.
• But note: KNN predictions with smaller datasets are very... | {
"domain": "umass.edu",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126500692716,
"lm_q1q2_score": 0.8413637944677987,
"lm_q2_score": 0.8596637577007393,
"openwebmath_perplexity": 6380.290869866497,
"openwebmath_score": 0.3988993763923645,
"tags": null... |
In [51]:
#Load Data
x = data[:,[0,2]]
y = data[:,-1]
#Prepare grid for plotting decision surface
gx1, gx2 = np.meshgrid(np.arange(min(x[:,0]), max(x[:,0]),(max(x[:,0])-min(x[:,0]))/200.0 ),
np.arange(min(x[:,1]), max(x[:,1]),(max(x[:,1])-min(x[:,1]))/200.0))
gx1l = gx1.flatten()
gx2l = gx2.flatten()
gx = np.vstack((... | {
"domain": "umass.edu",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126500692716,
"lm_q1q2_score": 0.8413637944677987,
"lm_q2_score": 0.8596637577007393,
"openwebmath_perplexity": 6380.290869866497,
"openwebmath_score": 0.3988993763923645,
"tags": null... |
# Integrating $\int^2_0 xe^{x^2}dx$
Well what I was thinking was to integrate the indefinite integral first.
$$u=x^2$$, $$x=\sqrt u$$
$$du=2xdx = 2\sqrt {u} dx$$
$$dx= \frac{1}{2\sqrt{u}}du$$
$$\int xe^{x^2} dx = \int \sqrt{u}\frac{1}{2\sqrt{u}} du =\frac{1}{2}\int e^u du = \frac{1}{2}e^u =\frac{1}{2}e^{x^2} +C$$
... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126525529015,
"lm_q1q2_score": 0.841363793084011,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 365.8322709010761,
"openwebmath_score": 0.877921462059021,
"tags"... |
# Why aren't $\int_0^\pi\int_{-1}^1e^rdr\,d\theta$ and $\int_0^{2\pi}\int_0^1e^rdr\,d\theta$ equal? Doesn't this violate the Change of Variables thm?
Why aren't these two integrals equal?
$$\int_0^\pi \int_{-1}^{1} e^r \,dr\,d\theta \qquad\neq\qquad\int_0^{2\pi} \int_{0}^{1} e^r \,dr\,d\theta$$
Let me explain why I'... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126519319941,
"lm_q1q2_score": 0.8413637925502394,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 334.91910333217936,
"openwebmath_score": 0.9912928342819214,
"ta... |
\begin{aligned} \iint_D \frac{e^{\sqrt{x^2 + y^2}}}{\sqrt{x^2 + y^2}} dy dx &= \iint_{R_2} \frac{e^{|r|}}{|r|} |r| dr d\theta\\ &= \int_0^{\pi} \int_{-1}^1 e^{|r|} dr d\theta \end{aligned}
noting that the absolute value around the determinant of the Jacobian can also not be dropped in this case.
Now, noting that the ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126519319941,
"lm_q1q2_score": 0.8413637925502394,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 334.91910333217936,
"openwebmath_score": 0.9912928342819214,
"ta... |
The common integrand $$e^r$$ does not vary the same way over the two different integration domains (let's call them $$S_1$$ and $$S_2,$$ respectively), which merely have the same measure and geometric representation. Consequently, the two integrals are not guaranteed to be equal. Indeed, taking their difference: \begin... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126519319941,
"lm_q1q2_score": 0.8413637925502394,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 334.91910333217936,
"openwebmath_score": 0.9912928342819214,
"ta... |
why does this not violate the change of variables theorem?
This theorem isn't necessary here, but can be invoked via \begin{align}x&\color{red}=r\left|\cos\theta\right|,\\y&\color{red}=r\sin\theta,\\&f\Big(g(r,\theta),h(r,\theta)\Big)\det \left| \frac{\partial(x,y)}{\partial(r,\theta)} \right|\color{red}=e^{|r|}\ne e^... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126519319941,
"lm_q1q2_score": 0.8413637925502394,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 334.91910333217936,
"openwebmath_score": 0.9912928342819214,
"ta... |
# Capped binomial random variables
Consider a random variable $X = \sum_{i=1}^{m} X_i$, where each $X_i$ is an indicator random variable that is $1$ with probability $k/m$ and $0$ otherwise. We are interested in the quantity $S_X(m) = E[\min(X,k)]$. The motivation is that we have a bin of capacity $k$. At each step, a... | {
"domain": "mathoverflow.net",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126494483641,
"lm_q1q2_score": 0.84136379217459,
"lm_q2_score": 0.8596637559030338,
"openwebmath_perplexity": 179.7142072907256,
"openwebmath_score": 0.9838100671768188,
"tags":... |
A slight specialization of the corollary states:
Let $\newcommand{\E}{\mathbb E} Y = Y_1 + \cdots + Y_n$ be a sum of iid random variables taking values in $[0,1]$ with mean $\E Y_i = \mu$ and let $X \sim \mathrm{Bin}(n,\mu)$. For any convex function $g : [0,n] \to \mathbb R$, $$\E g(Y) \leq \E g(X) \ .$$
Your result ... | {
"domain": "mathoverflow.net",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126494483641,
"lm_q1q2_score": 0.84136379217459,
"lm_q2_score": 0.8596637559030338,
"openwebmath_perplexity": 179.7142072907256,
"openwebmath_score": 0.9838100671768188,
"tags":... |
-
$Y$ is smaller than $X$ in the convex order, by bounding from above any convex function on $[0,1]$ by the secant line passing through the points $(0,g(0))$ and $(1,g(1))$; this is Lemma 1 in cardinal's answer. And yes, convex order inequalities between between laws on the line (or a measurable vector space over the r... | {
"domain": "mathoverflow.net",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126494483641,
"lm_q1q2_score": 0.84136379217459,
"lm_q2_score": 0.8596637559030338,
"openwebmath_perplexity": 179.7142072907256,
"openwebmath_score": 0.9838100671768188,
"tags":... |
# Lorentz Contraction of Moving Line of Charge
1. Sep 13, 2013
### leonardthecow
1. The problem statement, all variables and given/known data
A point charge +q rests halfway between two steady streams of positive charge of equal charge per unit length λ, moving opposite directions and each at c/3 relative to the po... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.978712650690179,
"lm_q1q2_score": 0.8413637914826959,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 589.0182513351446,
"openwebmath_score": 0.8011196255683899,
"tags... |
Any help would be greatly appreciated, thanks!
2. Sep 13, 2013
### TSny
I think you have the right idea. In order to see where you might be making a mistake, we need to see more details of your calculation. What did you get for the speed of each line charged line in the new frame? EDIT: You are correct that you can ... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.978712650690179,
"lm_q1q2_score": 0.8413637914826959,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 589.0182513351446,
"openwebmath_score": 0.8011196255683899,
"tags... |
5. Sep 13, 2013
### leonardthecow
Ah I see it now!
So, because of length contraction,
λ=Q/L, L=L0
λ=Qγ/L0
λ/γ=Q/L=λ0.
For the moving line of charge,
γ=(1-(c/3)2/c2)-1/2
γ=(1-1/9)-1/2
γ=(8/9)-1/2
So, because λ10,
λ1=λ/γ
∴ λ=(8/9)-1/2λ1
λ=√(8)λ1/3.
I'm not sure what the sign error would be in the second line, my... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.978712650690179,
"lm_q1q2_score": 0.8413637914826959,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 589.0182513351446,
"openwebmath_score": 0.8011196255683899,
"tags... |
11. Sep 13, 2013
### TSny
Looks good.
Oops. Can you spot the error with the left hand side of this equation? Did you really want to use the charge density λ of the lines in the original frame where they are moving at c/3?
12. Sep 13, 2013
### yands
The rod moving with the frame will have the same value of its lin... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.978712650690179,
"lm_q1q2_score": 0.8413637914826959,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 589.0182513351446,
"openwebmath_score": 0.8011196255683899,
"tags... |
Show inclusion of $L^p$ spaces in a space of finite measure
Let $1 \leq p_1 \leq p_2 \leq +\infty$. Show that in a space of finite measure we have that $L^{p_2} \subset L^{p_1}$.
Could you give me some hints what I could do??
Let $F = |f|^{p_1}$ and $G = 1$. Apply the Holder inequality
$||FG||_1 \leq ||F||_p ||G||_... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126463438263,
"lm_q1q2_score": 0.8413637895057313,
"lm_q2_score": 0.8596637559030338,
"openwebmath_perplexity": 266.7642702726622,
"openwebmath_score": 0.90184086561203,
"tags"... |
$$\| f \|_{p_1} \leq C \| f \|_{p_2}$$
where $C = \left(\mu(X)^{1- p_2/p_1}\right)^{1/p_1}$.
• Could you explain me why we take $p=p_2/p_1$?? Also, how did you find $||G||_q$?? – Mary Star Nov 10 '14 at 10:57
• To show $L^{p_2} \subset L^{p_1}$ we want to show that $f \in L^{p_2} \rightarrow f \in L^{p_1}$. So if we ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126463438263,
"lm_q1q2_score": 0.8413637895057313,
"lm_q2_score": 0.8596637559030338,
"openwebmath_perplexity": 266.7642702726622,
"openwebmath_score": 0.90184086561203,
"tags"... |
• Could you explain why you take these conjugate exponents at Holders inequality?? Also you say for $f\in L^{p_2}$ and then you calculate the norm $p_1$, why?? – Mary Star Nov 10 '14 at 15:40 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126463438263,
"lm_q1q2_score": 0.8413637895057313,
"lm_q2_score": 0.8596637559030338,
"openwebmath_perplexity": 266.7642702726622,
"openwebmath_score": 0.90184086561203,
"tags"... |
How many different $3$-digit numbers can be formed if the digits $1$, $2$, $2$, $3$, $4$ are placed on separate cards?
Any suggestion on solving this problem to get rid of double/over counting?
The digits $1$, $2$, $2$, $3$, and $4$ are placed on separate cards. How many different $3$-digit numbers can be formed by a... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126457229186,
"lm_q1q2_score": 0.8413637889719595,
"lm_q2_score": 0.8596637559030338,
"openwebmath_perplexity": 213.00600709554945,
"openwebmath_score": 0.7702440619468689,
"ta... |
# Evaluating an indefinite integral with an inverse trigonometric function
I'm really stumped on a homework problem asking me to evaluate $\int \frac{ln\ 6x\ sin^{-1}(ln6x)}{x}dx$, and after a few hours of trying different approaches I'd definitely be appreciative for a bump in the right direction. As a caveat, I shou... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126500692716,
"lm_q1q2_score": 0.8413637874300498,
"lm_q2_score": 0.8596637505099168,
"openwebmath_perplexity": 189.96377343603348,
"openwebmath_score": 0.9940322041511536,
"ta... |
Hints; for the second integral try the substitution $$u=\sin(t)$$
When you get $$\sin^2(t)$$ in the integrand, use the identity $$\sin^2(t)=\frac12 -\frac12 \cos(2t)$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126500692716,
"lm_q1q2_score": 0.8413637874300498,
"lm_q2_score": 0.8596637505099168,
"openwebmath_perplexity": 189.96377343603348,
"openwebmath_score": 0.9940322041511536,
"ta... |
# Definition:Set Partition/Definition 2
## Definition
Let $S$ be a set.
A partition of $S$ is a set of non-empty subsets $\Bbb S$ of $S$ such that each element of $S$ lies in exactly one element of $\Bbb S$.
## Also defined as
Some sources do not impose the condition that all sets in $\Bbb S$ are non-empty.
This ... | {
"domain": "proofwiki.org",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126513110865,
"lm_q1q2_score": 0.8413637867381557,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 289.7059598196392,
"openwebmath_score": 0.9527775049209595,
"tag... |
# Systems of equations - further understanding
#### Yankel
##### Active member
Hello again,
I have a few more questions regarding systems of equations, I will collect them all here in one post since they are small.
1. The first is the following system:
x+2y-3z=a
3x-y+2z=b
x-5y+8z=c
I need to determine the relatio... | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126488274565,
"lm_q1q2_score": 0.841363784603069,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 703.2218095518988,
"openwebmath_score": 0.7576556205749512,
... |
2. A is a matrix over the R field with dimensions 3X4. The rank of A is 1. How many degrees of freedom (parameters, i.e. t,s,...) does the family of solutions of Ax=0 has ?
3. If Ax=b has infinite solution, then Ax=c has infinite solution or no solution. True or False ?
Thanks a lot !
Hello,
1. For it to be infinity... | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126488274565,
"lm_q1q2_score": 0.841363784603069,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 703.2218095518988,
"openwebmath_score": 0.7576556205749512,
... |
# Produce an explicit bijection between rationals and naturals?
I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but can anyone produce an explicit formula for such a bije... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126457229185,
"lm_q1q2_score": 0.8413637819342102,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 522.3228458510459,
"openwebmath_score": 0.9798880815505981,
"tag... |
Now, every positive integer can be written uniquely as $$p_1^{a_1}p_2^{a_2}\cdots$$, where the $$p_1=2,p_2=3,p_3=5,\dots$$ is the sequence of all primes, and the $$a_i$$ are non-negative integers, and are non-zero for only finitely many $$i$$s.
Similarly, every positive rational number can be written uniquely as $$p_1... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126457229185,
"lm_q1q2_score": 0.8413637819342102,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 522.3228458510459,
"openwebmath_score": 0.9798880815505981,
"tag... |
• Why hasn't this been upvoted more? Good answer! (+1) – fancynancy Feb 16 '15 at 17:46
• For your bijection between $\mathbb{N}$ and $\mathbb{Q}$, why have you defined it as $\rho_1$ instead of just $\eta$ or something else (I don't see anything to indicate significance of the index)? – fancynancy Feb 16 '15 at 17:52
... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126457229185,
"lm_q1q2_score": 0.8413637819342102,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 522.3228458510459,
"openwebmath_score": 0.9798880815505981,
"tag... |
This function enables us to say that the $6^{th}$ rational number is $2/3$. Moreover, this function is a bijection. For proof of this, see Theorem 5.1 here http://faculty.plattsburgh.edu/sam.northshield/08-0412.pdf.
Since $f$ is a bijection this implies that $f^{-1}$ exists. That means given a rational number we can f... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126457229185,
"lm_q1q2_score": 0.8413637819342102,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 522.3228458510459,
"openwebmath_score": 0.9798880815505981,
"tag... |
We already referenced a proof by Northshield showing that $g(z)=\dfrac{a_{z}}{a_{z+1}}$ if $z>0$ is a bijection from $\mathbb{N} \rightarrow \mathbb{Q}^{+}$. Equivalently, we may write this as $g$ is a bijection from $\mathbb{Z}^{+}$ to $\mathbb{Q}^{+}$ for $z>0$. Now, it follows by the symmetry of the problem that $g(... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126457229185,
"lm_q1q2_score": 0.8413637819342102,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 522.3228458510459,
"openwebmath_score": 0.9798880815505981,
"tag... |
We now define the function $h^{-1}: \mathbb{Z} \rightarrow \mathbb{N}$ as follows:
$$h^{-1}(z)= \begin{cases} 2z, & \text{if } z>0 \\ 1, & \text{if } z=0 \\ -2z-1, & \text{if } z<0 \\ \end{cases}$$
Then $h^{-1} \circ g^{-1}: \mathbb{Q} \rightarrow \mathbb{N}$ is the bijection we are looking for.
• Superb exposition!... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126457229185,
"lm_q1q2_score": 0.8413637819342102,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 522.3228458510459,
"openwebmath_score": 0.9798880815505981,
"tag... |
# Example
$$\Phi\Big(\frac{30}{43}\Big) = 2^03^15^27^3 = 25725$$ This is because $$\frac{30}{43} = \cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4}}}} := [1,2,3,4]$$ And vice-versa: $$\Phi^{-1}(225) = \frac{10}{13} = \cfrac{1}{1+\cfrac{1}{3+\cfrac{1}{3}}}$$ This is because $$225 = 2^03^25^2$$
Of course this works iff... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126457229185,
"lm_q1q2_score": 0.8413637819342102,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 522.3228458510459,
"openwebmath_score": 0.9798880815505981,
"tag... |
Therefore if we write $$S^n(x)$$ for $$n^{th}$$ iterate of $$S$$, then we obtain an explicit bijection $$F:\mathbb{N}\to \mathbb{Q}^{+}$$ by $$F(n)=S^n(0)$$. The proof is explained in the link I have mentioned above.
This is a bijection between the Stern-Brocot tree and the tree of Natural numbers. Every left node is ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126457229185,
"lm_q1q2_score": 0.8413637819342102,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 522.3228458510459,
"openwebmath_score": 0.9798880815505981,
"tag... |
while True:
#print('expand by phi(n) count on working range')
moreTicksList = []
curTick = moreTicks(curTick)
for i in range(negSide, posSide + 1):
for f in moreTicksList:
print(f + fractions.Fraction(i, 1), ', ', end='')
for f in moreTicksList:
phiList.append(f)
negSide = negSide - 1
print(negSide, ', ', end='')
for f... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126457229185,
"lm_q1q2_score": 0.8413637819342102,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 522.3228458510459,
"openwebmath_score": 0.9798880815505981,
"tag... |
0 , 1/2 , -1 , -1/2 , 1 , 3/2 , -2/3 , -1/3 , 1/3 , 2/3 , 4/3 , 5/3 , -2 , -3/2 , -5/3 , -4/3 , 2 , 5/2 , 7/3 , 8/3 , -7/4 , -5/4 , -3/4 , -1/4 , 1/4 , 3/4 , 5/4 , 7/4 , 9/4 , 11/4 , -3 , -5/2 , -8/3 , -7/3 , -11/4 , -9/4 , 3 , 7/2 , 10/3 , 11/3 , 13/4 , 15/4 , -14/5 , -13/5 , -12/5 , -11/5 , -9/5 , -8/5 , -7/5 , -6/5 ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126457229185,
"lm_q1q2_score": 0.8413637819342102,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 522.3228458510459,
"openwebmath_score": 0.9798880815505981,
"tag... |
, 25/8 , 27/8 , 29/8 , 31/8 , 33/8 , 35/8 , 37/8 , 39/8 , 41/8 , 43/8 , 45/8 , 47/8 , 49/8 , 51/8 , 53/8 , 55/8 , -7 , -13/2 , -20/3 , -19/3 , -27/4 , -25/4 , -34/5 , -33/5 , -32/5 , -31/5 , -41/6 , -37/6 , -48/7 , -47/7 , -46/7 , -45/7 , -44/7 , -43/7 , -55/8 , -53/8 , -51/8 , -49/8 , 7 , 15/2 , 22/3 , 23/3 , 29/4 , 3... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126457229185,
"lm_q1q2_score": 0.8413637819342102,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 522.3228458510459,
"openwebmath_score": 0.9798880815505981,
"tag... |
The sequence 'calibrates' the rational number 'tick marks' on our ideal 'measuring rod'. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126457229185,
"lm_q1q2_score": 0.8413637819342102,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 522.3228458510459,
"openwebmath_score": 0.9798880815505981,
"tag... |
1. ## finding numbers
i have a question to do with quadratic equations but what i'm asking isn't really about quadratics but i need to it to complete the question. basically i've been set the question factorise $54-15x-25x^2$. i know that you have to start by multiplying -25 and 54 which gives -1350. i then have to fi... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637808666667,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 347.6375996351792,
"openwebmath_score": 0.7521131634712219,
"tag... |
Mark
Make it positive so that it's easier to solve ..
$y^2+15y-1350=0$
$
(y+45)(y-30)=0
$
y=-45 , y=30
7. Originally Posted by mark
i have a question to do with quadratic equations but what i'm asking isn't really about quadratics but i need to it to complete the question. basically i've been set the question facto... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637808666667,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 347.6375996351792,
"openwebmath_score": 0.7521131634712219,
"tag... |
. . . $\begin{array}{cccccc}16\cdot90 &&&& 75 \\
18\cdot75 &&&& 57 \\
25\cdot54 &&&& 19 \\
30\cdot45 &&&& 15 & \Leftarrow\:\text{ There!}
\end{array}$
We want the middle term to be $+15x$, so we will use $-30x \text{ and }+45x$
We have: . $25x^2 -30x + 45x - 54$
Factor: . $5x(5x-6) + 9(5x-6)$
Factor: . $(5x-6)(5x+9... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637808666667,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 347.6375996351792,
"openwebmath_score": 0.7521131634712219,
"tag... |
# Shortcut for finding cube of the Numbers
Is there a shortcut for finding cube of a particular number like $68^3$ ? If anyone knows how to solve for two- and three digit numbers, can you please share the answer?
• there's a shortcut? – Gregory Grant Jul 26 '16 at 10:59
• I found some links by googling: youtube.com/w... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126482065489,
"lm_q1q2_score": 0.8413637805504227,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 234.91066406353437,
"openwebmath_score": 0.7544633150100708,
"ta... |
Arthur Benjamin wrote a paper called "Squaring, Cubing, and Cube Rooting" which contains a great shortcut for cubing. It's similar to the method posted by callculus, but the math is broken up in a different way.
Let's refer to the number to be cubed as $x$. The closest multiple of 10 to the given number will be called... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126482065489,
"lm_q1q2_score": 0.8413637805504227,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 234.91066406353437,
"openwebmath_score": 0.7544633150100708,
"ta... |
$$68^{3}=314,432$$
Practice this by cubing smaller numbers to get used to the pattern, and then work your way up to higher numbers as you get more comfortable with the process.
This works well for 2-digit numbers. It can be used for 3-digit numbers, as well, by defining $z$ as the nearest multiple of 100, but you obv... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126482065489,
"lm_q1q2_score": 0.8413637805504227,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 234.91066406353437,
"openwebmath_score": 0.7544633150100708,
"ta... |
MC: Ping-Pong Cubes
05-22-2018, 03:16 PM
Post: #1
Joe Horn Senior Member Posts: 1,460 Joined: Dec 2013
MC: Ping-Pong Cubes
You always thought that Ping-Pong™ only involved planes and spheres, right? Here's a programming mini-challenge that involves Ping-Pong Cubes!
Definition 1: A Ping-Pong Number is any multi-digit ... | {
"domain": "hpmuseum.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637791072295,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 1845.7878489250068,
"openwebmath_score": 0.45289504528045654,
"tags":... |
A much bigger challenge, which has eluded me thus far, is to find the 11th Ping-Pong Cube. It must be very large, if it even exists.
On the Prime the following program will solve the mini-challenge:
Code:
EXPORT PPCube() BEGIN PRINT(); FOR N FROM 10 TO 10000 DO IF ΠLIST(CONCAT(ΔLIST(ASC(STRING(N,1))),ΔLIST(ASC(STR... | {
"domain": "hpmuseum.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637791072295,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 1845.7878489250068,
"openwebmath_score": 0.45289504528045654,
"tags":... |
Looking for all such numbers less than 10,000 takes less than 5 seconds but finds no additional ones.
Quote:A much bigger challenge, which has eluded me thus far, is to find the 11th Ping-Pong Cube. It must be very large, if it even exists.
My program would find it in principle but it's limited to the 12-digit native... | {
"domain": "hpmuseum.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637791072295,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 1845.7878489250068,
"openwebmath_score": 0.45289504528045654,
"tags":... |
Code:
00 { 128-Byte Prgm } 01▸LBL "PPC" 02 STO "I" 03 1.009 04 XEQ 99 05 RTN 06▸LBL 97 07 RCL IND ST X 08▸LBL 99 09 STO IND ST Y 10 IP 11 1.00902 12 + 13 2 14 MOD 15 DSE ST Y 16 GTO 99 17▸LBL 01 18 RCL "I" 19 1ᴇ3 20 0 21▸LBL 14 22 % 23 RCL IND ST Z 24 IP 25 + 26 DSE ST Z 27 GTO 14 28 STO 00 29 ENTER 30 X↑2 31 × 32 XEQ ... | {
"domain": "hpmuseum.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637791072295,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 1845.7878489250068,
"openwebmath_score": 0.45289504528045654,
"tags":... |
The code within the list brackets above is a subroutine that takes a real number as input and returns a 1 if the input was valid, otherwise a 0. It simply checks the parity of each digit after the first to make sure it isn't the same as the previous one.
The basic approach was to execute a loop on sequential integers ... | {
"domain": "hpmuseum.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637791072295,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 1845.7878489250068,
"openwebmath_score": 0.45289504528045654,
"tags":... |
...and finally, to show how much time is spent in that laborious ping-pong validation, I replaced the validation subroutine from the previous SysRPL example with this Saturn code object to see how much improvement I could get. Using this brought the total time down to 1.26 seconds:
Code:
CODEM SAVE A=DAT... | {
"domain": "hpmuseum.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637791072295,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 1845.7878489250068,
"openwebmath_score": 0.45289504528045654,
"tags":... |
I've devised four or five different solutions using none, some or all of these techniques (the one above uses just the first kind) but they will be left an an exercise for the reader as I don't see much interest, if any, in HP-71B-coded solutions.
Have a nice weekend.
V.
.
05-26-2018, 06:47 AM (This post was last modi... | {
"domain": "hpmuseum.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637791072295,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 1845.7878489250068,
"openwebmath_score": 0.45289504528045654,
"tags":... |
In this particular case, the similarities with some other challenges provide some clues as to some possible optimizations. In my case, I didn't use any of them and instead simply wanted to use a "generic" solution as a basis, then refining the process with increasingly specialized translations of the code. Some here ha... | {
"domain": "hpmuseum.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637791072295,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 1845.7878489250068,
"openwebmath_score": 0.45289504528045654,
"tags":... |
The next program uses ISPP? To check for ping-pong numbers starting with 10, if a ping-pong number is found checks if the square of that number is a ping-pong number, if so stores it in a list and the process repeat ten times in a START-NEXT loop.
«
{ } 10 1 10
START
DO 1 +
UNTIL
IF DUP ISPP?
THEN
DUP SQ ISPP?
ELSE 0
E... | {
"domain": "hpmuseum.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637791072295,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 1845.7878489250068,
"openwebmath_score": 0.45289504528045654,
"tags":... |
Consider what I said above, think about it, and then decide who's the one being confrontational. This is no helpful or friendly way to address a fairly new member.
Have a nice weekend.
V.
.
05-27-2018, 05:06 AM
Post: #14
Joe Horn Senior Member Posts: 1,460 Joined: Dec 2013
RE: MC: Ping-Pong Cubes
(05-26-2018 05:01 PM... | {
"domain": "hpmuseum.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637791072295,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 1845.7878489250068,
"openwebmath_score": 0.45289504528045654,
"tags":... |
I see it differently.
Yes if the post count would equal the sum of all life experiences of a person, then I would be with you.
Instead I presume that here on the forum 99.9% of the userbase is 16 years old or older (I am much older than 16, I am near to EOL).
So independently from whom create the discussion (Joe or a... | {
"domain": "hpmuseum.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637791072295,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 1845.7878489250068,
"openwebmath_score": 0.45289504528045654,
"tags":... |
n1: I am the first that create a lot of math challenges that are trivial for many here.
n2: https://en.wikipedia.org/wiki/Four-sides_model
n3: of course they are not. I am amazed at the 71B capabilities that you expose nicely.
Wikis are great, Contribute :)
05-27-2018, 10:05 AM
Post: #16
brickviking Senior Member Pos... | {
"domain": "hpmuseum.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637791072295,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 1845.7878489250068,
"openwebmath_score": 0.45289504528045654,
"tags":... |
'nuff said from me.
(Post 232)
Regards, BrickViking
HP-50g |Casio fx-9750G+ |Casio fx-9750GII (SH4a)
05-27-2018, 12:30 PM (This post was last modified: 05-27-2018 12:47 PM by ijabbott.)
Post: #17
ijabbott Senior Member Posts: 596 Joined: Jul 2015
RE: MC: Ping-Pong Cubes
To clarify my previous point, I really meant i... | {
"domain": "hpmuseum.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637791072295,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 1845.7878489250068,
"openwebmath_score": 0.45289504528045654,
"tags":... |
Code:
#include <iostream> #include <gmpxx.h> using namespace std; static mpz_class nextpp(mpz_class n) { mpz_class a; mpz_class pow = 1; mpz_class d; n += 2; a = n; while ((d = (a % 10)) < 2) { pow *= 10; a /= 10; if (a >= 10) { n += pow; ... | {
"domain": "hpmuseum.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637791072295,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 1845.7878489250068,
"openwebmath_score": 0.45289504528045654,
"tags":... |
So if we simply look at answering that specific question, we very quickly find that a simple brute-force search will run for quite some time with no success. It is inevitable that optimizations need to be applied which will both speed up the test and limit the input to have any hope of being useful. Discovering the "ar... | {
"domain": "hpmuseum.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637791072295,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 1845.7878489250068,
"openwebmath_score": 0.45289504528045654,
"tags":... |
(05-26-2018 08:19 PM)Juan14 Wrote:
Code:
... THEN DUP SQ ISPP? ...
I did have to change the "SQ" above to "3. ^" in order to match the original problem, though. Just curious... did you find any interesting results from testing the squares?
05-27-2018, 04:48 PM
Post: #20
David Hayden Member Posts: 249 Joined: ... | {
"domain": "hpmuseum.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126444811033,
"lm_q1q2_score": 0.8413637791072295,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 1845.7878489250068,
"openwebmath_score": 0.45289504528045654,
"tags":... |
Find PowerPoint Presentations and Slides using the power of XPowerPoint. Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow. 4 The Magnetic Field of Filamentary Currents 141 3. doc), PDF File. Show through an example, how this law enables an easy evaluation of this m... | {
"domain": "animalhousesrl.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126438601955,
"lm_q1q2_score": 0.8413637768140202,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 527.8623371915256,
"openwebmath_score": 0.7762930393218994,
... |
I n the post dated 1 st February 2008 the equations to be noted in connection with magnetic fields were given. An ohm is equivalent to a volt per ampere. Using Ampere's circuital law for P B. Magnetic Boundary Conditions Magnetic boundary conditions are the conditions that a or (or ) field must satisfy at the boundary ... | {
"domain": "animalhousesrl.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126438601955,
"lm_q1q2_score": 0.8413637768140202,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 527.8623371915256,
"openwebmath_score": 0.7762930393218994,
... |
Pakistan. Do mathematically rigourous formulations of Ampère's law $(1)$ exist under more relaxed assumptions on $\boldsymbol{J}$, like the quoted case of $\boldsymbol{J}$ constant on a (bounded or unbounded) region and null outside of it, and, if they do, how can they be proved?. Both (a) and (b) d. This is the collog... | {
"domain": "animalhousesrl.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126438601955,
"lm_q1q2_score": 0.8413637768140202,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 527.8623371915256,
"openwebmath_score": 0.7762930393218994,
... |
states that. A solenoid is a long coil of wire closely wound in the form of helix as shown in Figure 3. Ampere’s Circuital Law; The Solenoid and the Toroid; Force between Two Parallel Currents, the Ampere; Torque on Current Loop, Magnetic Dipole; The Moving Coil Galvanometer; Class XII NCERT Physics Text Book Chapter 4... | {
"domain": "animalhousesrl.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126438601955,
"lm_q1q2_score": 0.8413637768140202,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 527.8623371915256,
"openwebmath_score": 0.7762930393218994,
... |
PPT presentation: "Magnetostatic Field: Ampere Circuital Law" is the property of its rightful owner. Basically, you select some loop (i. Solenoids and toroids are widely used in motors, generators, toys, fan-windings, transformers, electromagnets, etc. circuital 2015 December, “Hubo elecciones y las ganamos”, in El Nac... | {
"domain": "animalhousesrl.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126438601955,
"lm_q1q2_score": 0.8413637768140202,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 527.8623371915256,
"openwebmath_score": 0.7762930393218994,
... |
charges carrying their electric fields with them simplifies the derivation of the magnetic field of current in a straight infinitely long conductor. gov/grants/index. [2 marks] Still considering the electromagnet above, from Ampere's circuital law is there any region in the path of the magnetic flux where the magnitude... | {
"domain": "animalhousesrl.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126438601955,
"lm_q1q2_score": 0.8413637768140202,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 527.8623371915256,
"openwebmath_score": 0.7762930393218994,
... |
of the ampere, the SI unit of current, is as follows: For two thin, straight, stationary, parallel wires, the force per unit length one wire exerts upon the other in the vacuum of free space is ,. Electromagnetism. Ampere's law related the integrated magnetic field around a closed loop to the electric current passing t... | {
"domain": "animalhousesrl.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126438601955,
"lm_q1q2_score": 0.8413637768140202,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 527.8623371915256,
"openwebmath_score": 0.7762930393218994,
... |
Lecture 10 - Ampere's Law Overview. Physics Wallah - Alakh Pandey 254,964 views. (9) Chapter 3. The application of Ampere's circuital law involves finding the total current enclosed by a closed path. state ampere s circuital law - Physics - TopperLearning. Answer: Ampere's Circuital Law states the relationship between ... | {
"domain": "animalhousesrl.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126438601955,
"lm_q1q2_score": 0.8413637768140202,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 527.8623371915256,
"openwebmath_score": 0.7762930393218994,
... |
closed loops. 1 Faraday’s Law and Ampere’s Circuital Law 130. A solenoid is a long coil of wire closely wound in the form of helix as shown in Figure 3. txt) or read online for free. 2 Advanced texts often present it either without proof or as a special case of a complicated mathematical formalism. And yes, the Biot-Sa... | {
"domain": "animalhousesrl.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126438601955,
"lm_q1q2_score": 0.8413637768140202,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 527.8623371915256,
"openwebmath_score": 0.7762930393218994,
... |
concept of a line integral. Simulating Faraday's law in Matlab. Convection and Conduction currents, Dielectric constant, lsotropic and homogeneous Dielectrics. [ 9 ], §528). The magnetic field can be visualized in terms of flux lines, which form closed loops interlinking with the winding. English-German online dictiona... | {
"domain": "animalhousesrl.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126438601955,
"lm_q1q2_score": 0.8413637768140202,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 527.8623371915256,
"openwebmath_score": 0.7762930393218994,
... |
the integrated magnetic field around a closed loop to the electric current passing through the loop. Basically, you select some loop (i. doc), PDF File. The differential form of Ampere’s Circuital Law for magnetostatics (Equation \ref{m0118_eACL}) indicates that the volume current density at any point in space is propo... | {
"domain": "animalhousesrl.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126438601955,
"lm_q1q2_score": 0.8413637768140202,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 527.8623371915256,
"openwebmath_score": 0.7762930393218994,
... |
an electric field and the generating electric charges: The electric field points away from positive charges and towards negative charges. Ampere's Circuital Law Ampère's law relates magnetic fields to electric currents that produce them. 2 Advanced texts often present it either without proof or as a special case of a c... | {
"domain": "animalhousesrl.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126438601955,
"lm_q1q2_score": 0.8413637768140202,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 527.8623371915256,
"openwebmath_score": 0.7762930393218994,
... |
= µ0 I ∫B. Amperes law of force gives the magnetic force between two current carrying circuits in an otherwise empty universe. 대칭성이 있는 문제를 다룰 때 매우 유용하게 사용한다. Use this law to obtain the expression for the magnetic field inside an air cored toroid of average radius, having ‘n’ turns per unit length and carrying a steady ... | {
"domain": "animalhousesrl.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126438601955,
"lm_q1q2_score": 0.8413637768140202,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 527.8623371915256,
"openwebmath_score": 0.7762930393218994,
... |
free space. Simulating Faraday's law in Matlab. I want to acquire conductivity and I used Ampere's circuital law. He formulated the Ampere’s circuital law in 1826 , which relates the magnetic field associated with a closed loop to the electric current passing through it. dl for a closed curve is equal to µ0 times the n... | {
"domain": "animalhousesrl.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126438601955,
"lm_q1q2_score": 0.8413637768140202,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 527.8623371915256,
"openwebmath_score": 0.7762930393218994,
... |
equal to 0 times the net current i threading through the area enclosed by the curve i. Gauss's law: Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. In physics, Ampère's Circuital law, discovered by André-Marie Ampère, relates the ci... | {
"domain": "animalhousesrl.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126438601955,
"lm_q1q2_score": 0.8413637768140202,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 527.8623371915256,
"openwebmath_score": 0.7762930393218994,
... |
path is equal to the net current enclosed by that path. Ampère's circuital law is now known to be a correct law of physics in a magnetostatic situation: The system is static except possibly for continuous steady currents within closed loops. In was derived using hydrodynamics in 1861 by James Clerk Maxwell. The Biot–Sa... | {
"domain": "animalhousesrl.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126438601955,
"lm_q1q2_score": 0.8413637768140202,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 527.8623371915256,
"openwebmath_score": 0.7762930393218994,
... |
outline of the proof. Apply Ampere's circuital law to find magnetic field inside and outside of a toroidal solenoid. Links are added to Ampere's circuital law and Lorentz force and Biot-Savart law. Infinitely Long Line Current. The Biot-Savart law explains how currents produce magnetic fields, but it is difficult to ... | {
"domain": "animalhousesrl.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126438601955,
"lm_q1q2_score": 0.8413637768140202,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 527.8623371915256,
"openwebmath_score": 0.7762930393218994,
... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.