text stringlengths 1 2.12k | source dict |
|---|---|
Time Solution: As noted above, at the start, there was a $\frac{1}{5}$ chance of immediately matching with an average 3 minutes, and there was a $\frac{4}{5}$ chance of not matching while using an average 4 minutes. I just showed that from this latter stage, one would expect to need to use an additional mean 6.5 minu... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127437254887,
"lm_q1q2_score": 0.8439328612832796,
"lm_q2_score": 0.8539127585282744,
"openwebmath_perplexity": 1343.0153067454364,
"openwebmath_score": 0.7048300504684448,
"tags":... |
Happy New Year, everyone!
## Great Probability Problems
UPDATE: Unfortunately, there are a couple errors in my computations below that I found after this post went live. In my next post, Mistakes are Good, I fix those errors and reflect on the process of learning from them.
ORIGINAL POST:
A post last week to the ... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127437254887,
"lm_q1q2_score": 0.8439328612832796,
"lm_q2_score": 0.8539127585282744,
"openwebmath_perplexity": 1343.0153067454364,
"openwebmath_score": 0.7048300504684448,
"tags":... |
The second Puzzler investigated random geyser eruptions:
You arrive at the beautiful Three Geysers National Park. You read a placard explaining that the three eponymous geysers — creatively named A, B and C — erupt at intervals of precisely two hours, four hours and six hours, respectively. However, you just got there... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127437254887,
"lm_q1q2_score": 0.8439328612832796,
"lm_q2_score": 0.8539127585282744,
"openwebmath_perplexity": 1343.0153067454364,
"openwebmath_score": 0.7048300504684448,
"tags":... |
While the distributions are different, conveniently, there is still a time difference between 1 and 4 minutes when the total times aren’t equal. That means the second table shows the distribution for the 2nd and all future potential rounds until the siblings finally align. While this problem has the potential to exte... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127437254887,
"lm_q1q2_score": 0.8439328612832796,
"lm_q2_score": 0.8539127585282744,
"openwebmath_perplexity": 1343.0153067454364,
"openwebmath_score": 0.7048300504684448,
"tags":... |
$\frac{1}{5} *1 + \frac{4}{5}*5=4.2$ rounds
Time until Eating – In the first choice, there is a $\frac{1}{5}$ chance of ending in 3 minutes. If that doesn’t happen, there is a subsequent $\frac{1}{5}$ chance of ending with the second choice with no additional time. If neither of those events happen, there will be 1.... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127437254887,
"lm_q1q2_score": 0.8439328612832796,
"lm_q2_score": 0.8539127585282744,
"openwebmath_perplexity": 1343.0153067454364,
"openwebmath_score": 0.7048300504684448,
"tags":... |
# A proof that $\sqrt{2}$ is not a rational number.
Is this proof correct?
Suppose that $$\sqrt{2}=\frac{a}{b}$$, where $$a,b \in \mathbb{N}$$ and $$a$$ is as small as possible. Then $$\sqrt{2}b=a$$ which means $$2b=\sqrt{2} a$$. So we rewrite $$\sqrt{2}=\frac{a}{b}\cdot\frac{\sqrt{2}-1}{\sqrt{2}-1}=\frac{\sqrt{2}a-a... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127413158322,
"lm_q1q2_score": 0.8439328592256432,
"lm_q2_score": 0.8539127585282744,
"openwebmath_perplexity": 1612.1159535398947,
"openwebmath_score": 0.8867544531822205,
"ta... |
The numerator and denominator were each multiplied by $$(\sqrt k − q)\,$$ — which is positive but less than $$1$$ and then simplified independently. So the two resulting products, say $$a'$$ and $$b'$$, are themselves integers, which are less than $$a$$ and $$b$$ respectively. Therefore, no matter what natural numbers ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127413158322,
"lm_q1q2_score": 0.8439328592256432,
"lm_q2_score": 0.8539127585282744,
"openwebmath_perplexity": 1612.1159535398947,
"openwebmath_score": 0.8867544531822205,
"ta... |
This method generalizes to show the $$\,\Bbb Z\,$$ (or any PID) is integrally-closed, i.e. no proper fraction is a root of a polynomial that is monic (lead coef $$= 1),\,$$ i.e. the monic case of the Rational Root Test. You can find much further discussion of this and related ideas in my posts on denominator ideals. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127413158322,
"lm_q1q2_score": 0.8439328592256432,
"lm_q2_score": 0.8539127585282744,
"openwebmath_perplexity": 1612.1159535398947,
"openwebmath_score": 0.8867544531822205,
"ta... |
# Abelian group admitting a surjective homomorphism onto an infinite cyclic group
I am working on the following problem:
Let $G$ an Abelian group and $f: G \to \Bbb Z$ a surjective homomorphism. Prove that $G \cong \ker(f) \times \Bbb Z$
By means of the First Isomorphism Theorem, we can obtain that $G / \ker(f) \con... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127420043056,
"lm_q1q2_score": 0.8439328579764076,
"lm_q2_score": 0.8539127566694177,
"openwebmath_perplexity": 97.15963935734612,
"openwebmath_score": 0.967939019203186,
"tags... |
-
I am so very sorry for not understanding this immediately. Here is my point of confusion. I cannot see how $G = \ker(f) \times im(g) \implies G \cong \ker(f) \times \Bbb Z$, since $im(g) \subset G$ so you would get $G \cong \ker(f) \times im(g) \subset \ker(f) \times G$ – Orest Xherija Apr 16 '13 at 6:32
$g\colon \m... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127420043056,
"lm_q1q2_score": 0.8439328579764076,
"lm_q2_score": 0.8539127566694177,
"openwebmath_perplexity": 97.15963935734612,
"openwebmath_score": 0.967939019203186,
"tags... |
Now a short exact sequence is called split if one of the following equivalent properties hold:
a) $$B\cong A\oplus C$$ b) There exists a group homomorphism $i:C\to B$ such that $g\circ i=id$
c) There exists a group homomorphism $p:B\to A$ such that $p\circ f=id$.
So the direct way of showing that $G\cong \mathbb Z\o... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127420043056,
"lm_q1q2_score": 0.8439328579764076,
"lm_q2_score": 0.8539127566694177,
"openwebmath_perplexity": 97.15963935734612,
"openwebmath_score": 0.967939019203186,
"tags... |
# Will the convergence of $\frac{1}{N}\sum_{n=1}^{N}a_n$ imply the convergence of $\sum_{n=1}^{N}\frac{a_n}{n^2}$?
Assume that we have a positive sequence $\{a_n\}$ with its Cesàro mean converges: $$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}a_n<\infty\,.$$ I am wondering if the following summation converges: $$\lim_{N... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127440697254,
"lm_q1q2_score": 0.8439328579029641,
"lm_q2_score": 0.8539127548105611,
"openwebmath_perplexity": 504.1350469258439,
"openwebmath_score": 0.9079939126968384,
"tag... |
Given any real sequence $(a_n)_{n\ge 1}$, construct an auxillary sequence $(b_n)_{n\ge 0}$ by
$$b_n \stackrel{def}{=}\begin{cases}\frac1n\sum\limits_{k=1}^n a_k, & n > 0\\0, & n = 0\end{cases}$$ Whenever $b_n$ is bounded, i.e there is a $M > 0$ such that $|b_n| < M$ for all $n$, we can decompose the sum at hand into t... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127440697254,
"lm_q1q2_score": 0.8439328579029641,
"lm_q2_score": 0.8539127548105611,
"openwebmath_perplexity": 504.1350469258439,
"openwebmath_score": 0.9079939126968384,
"tag... |
Question
# Internal bisector of $$\angle \mathrm{A}$$ of triangle ABC meets side BC at D. A line drawn through $$\mathrm{D}$$ perpendicular to AD intersects the side AC at $$\mathrm{E}$$ and the side AB at F. If $$\mathrm{a},\ \mathrm{b},\ \mathrm{c}$$ represent sides of $$\Delta \mathrm{A}\mathrm{B}\mathrm{C}$$ then
... | {
"domain": "byjus.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127433812522,
"lm_q1q2_score": 0.8439328573150682,
"lm_q2_score": 0.8539127548105611,
"openwebmath_perplexity": 982.1148523721334,
"openwebmath_score": 0.7716891169548035,
"tags": null... |
# Math Help - Application of maxima and minima
1. ## Application of maxima and minima
Find the length of the longest rod which can be carried horizontally around a corner from a corridor 8m wide into one 4m wide. (Without involving angles if possible)
VIEW the attachment for the figure.
2. ## Re: Application of maxi... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.988312740283122,
"lm_q1q2_score": 0.843932856506667,
"lm_q2_score": 0.8539127566694178,
"openwebmath_perplexity": 2005.6053325893752,
"openwebmath_score": 0.8425948619842529,
"tags... |
I get (substituting for y into the square of the length) the following as the critical value:
$x=4\sqrt[3]{2}$
I began with:
$L^2=(8+x)^2+(4+y)^2$
$L^2=(8+x)^2+\left(4+\frac{32}{x} \right)^2$
$L^2=(8+x)^2+16\left(1+\frac{8}{x} \right)^2$
Implicitly differentiating with respect to $x$, we have:
$2L\cdot\frac{dL}{... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.988312740283122,
"lm_q1q2_score": 0.843932856506667,
"lm_q2_score": 0.8539127566694178,
"openwebmath_perplexity": 2005.6053325893752,
"openwebmath_score": 0.8425948619842529,
"tags... |
# 1) Derivation of the quadratic formula
This is note $1$ in a set of notes showing how to obtain formulas. There will be no words beyond these short paragraphs as the rest will either consist of images or algebra showing the steps needed to derive the formula mentioned in the title.
Suggestions for other formulas to... | {
"domain": "brilliant.org",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127447581985,
"lm_q1q2_score": 0.843932851142333,
"lm_q2_score": 0.8539127473751341,
"openwebmath_perplexity": 1398.7704980577782,
"openwebmath_score": 0.9564194679260254,
"tag... |
MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2
paragraph 1
paragraph 2
[example link](https://bril... | {
"domain": "brilliant.org",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127447581985,
"lm_q1q2_score": 0.843932851142333,
"lm_q2_score": 0.8539127473751341,
"openwebmath_perplexity": 1398.7704980577782,
"openwebmath_score": 0.9564194679260254,
"tag... |
Main
## Linearization of Differential Equations
Linearization is the process of taking the gradient of a nonlinear function with respect to all variables and creating a linear representation at that point. It is required for certain types of analysis such as stability analysis, solution with a Laplace transform, and ... | {
"domain": "apmonitor.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127399388854,
"lm_q1q2_score": 0.84393284702706,
"lm_q2_score": 0.853912747375134,
"openwebmath_perplexity": 2979.089223695814,
"openwebmath_score": 0.7025504112243652,
"tags": nul... |
#### Example
Part A: Linearize the following differential equation with an input value of u=16.
$$\frac{dx}{dt} = -x^2 + \sqrt{u}$$
Part B: Determine the steady state value of x from the input value and simplify the linearized differential equation.
Part C: Simulate a doublet test with the nonlinear and linear mode... | {
"domain": "apmonitor.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127399388854,
"lm_q1q2_score": 0.84393284702706,
"lm_q2_score": 0.853912747375134,
"openwebmath_perplexity": 2979.089223695814,
"openwebmath_score": 0.7025504112243652,
"tags": nul... |
# analytic solution with Python
import sympy as sp
sp.init_printing()
# define symbols
x,u = sp.symbols(['x','u'])
# define equation
dxdt = -x**2 + sp.sqrt(u)
print(sp.diff(dxdt,x))
print(sp.diff(dxdt,u))
# numeric solution with Python
import numpy as np
from scipy.misc import derivative
u = 16.0
x = 2.0
def pd_x(x):... | {
"domain": "apmonitor.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127399388854,
"lm_q1q2_score": 0.84393284702706,
"lm_q2_score": 0.853912747375134,
"openwebmath_perplexity": 2979.089223695814,
"openwebmath_score": 0.7025504112243652,
"tags": nul... |
plt.xlabel('x')
plt.ylabel('u')
plt.show()
Part C Solution: The final step is to simulate a doublet test with the nonlinear and linear models.
Small step changes (+/-1): Small step changes in u lead to nearly identical responses for the linear and nonlinear solutions. The linearized model is locally accurate.
Large... | {
"domain": "apmonitor.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9883127399388854,
"lm_q1q2_score": 0.84393284702706,
"lm_q2_score": 0.853912747375134,
"openwebmath_perplexity": 2979.089223695814,
"openwebmath_score": 0.7025504112243652,
"tags": nul... |
# Why is the range a larger set than the domain?
When we have a function $$f: \mathbb{R} \to \mathbb{R}$$, I can intuitively picture that and think that for every $$x \in \mathbb{R}$$, we can find a $$y \in \mathbb{R}$$ such that our function $$f$$ maps $$x$$ onto $$y$$.
I'm confused, however, when we have something ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9674102542943774,
"lm_q1q2_score": 0.8439178068074182,
"lm_q2_score": 0.8723473862936942,
"openwebmath_perplexity": 185.06046084182358,
"openwebmath_score": 0.9404276609420776,
"ta... |
• Thanks for explaining this. If I understood you correctly, that would mean that with such notation we tell more about the 'type' of the output (real, complex, etc.) than the actual range of the function. Mar 26 at 1:13
• @nocomment Yes that’s exactly the right way to think about the notation! Mar 26 at 1:22
• @DavidK... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9674102542943774,
"lm_q1q2_score": 0.8439178068074182,
"lm_q2_score": 0.8723473862936942,
"openwebmath_perplexity": 185.06046084182358,
"openwebmath_score": 0.9404276609420776,
"ta... |
The underlying idea of your question is the concept of cardinality. We say that two sets $$X$$ and $$Y$$ have the same cardinality iff the exists a bijection $$f:X\to Y$$ between them. Such relation is an equivalence relation, that is to say, it is reflexive, symmetric and transitive. Hence each cardinality is an equiv... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9674102542943774,
"lm_q1q2_score": 0.8439178068074182,
"lm_q2_score": 0.8723473862936942,
"openwebmath_perplexity": 185.06046084182358,
"openwebmath_score": 0.9404276609420776,
"ta... |
If $$A$$ is a finite set, then it is not possible for there to be a function $$f:A\to B$$ such that the range is a proper superset of the domain. However, this statement is not true for infinite sets, which is just one reason why they are so counter-intuitive. For instance, we can construct a surjective function $$f:\m... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9674102542943774,
"lm_q1q2_score": 0.8439178068074182,
"lm_q2_score": 0.8723473862936942,
"openwebmath_perplexity": 185.06046084182358,
"openwebmath_score": 0.9404276609420776,
"ta... |
# Trailing Zeros - How many trailing zeros are there in 100! (factorial of 100)?
Here's the question: How many trailing zeros are there in 100! (factorial of 100)?
Here's the solution: This is an easy problem. We know that each pair of 2 and 5 will give a trailing zero. If we perform prime number decomposition on all... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9674102542943774,
"lm_q1q2_score": 0.8439178052021413,
"lm_q2_score": 0.8723473846343394,
"openwebmath_perplexity": 329.67958230193784,
"openwebmath_score": 0.5877329111099243,
"ta... |
The 5s can appear only in numbers that are divisible by 5.
5, 10, 20, 25, 30, 35, 40, 45, 50 , 55, 60 , 65, 70, 75, 80, 85, 90, 95, 100
and the number of 5s in the numbers above are
1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2
Because for example
$15 = 3 \times 5$ so one single 5.
$50 = 2 \times 5 ^2$... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9674102542943774,
"lm_q1q2_score": 0.8439178052021413,
"lm_q2_score": 0.8723473846343394,
"openwebmath_perplexity": 329.67958230193784,
"openwebmath_score": 0.5877329111099243,
"ta... |
• Thank you very much for your comment. I definitely understand it better. There are two questions: 1. How can this concept be applied to a factorial? 2. Why does the statement 'the frequency of 2 will far outnumber the frequency of 5' matter? – Jun Jang Sep 4 '17 at 15:31 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9674102542943774,
"lm_q1q2_score": 0.8439178052021413,
"lm_q2_score": 0.8723473846343394,
"openwebmath_perplexity": 329.67958230193784,
"openwebmath_score": 0.5877329111099243,
"ta... |
Problem: Magic Numbers
Write a program that enters a single integer magic number and produces all possible 6-digit numbers for which the product of their digits is equal to the magical number.
Example: "Magic number" → 2
• 111112 → 1 * 1 * 1 * 1 * 1 * 2 = 2
• 111121 → 1 * 1 * 1 * 1 * 2 * 1 = 2
• 111211 → 1 * 1 * 1 *... | {
"domain": "softuni.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9674102571131691,
"lm_q1q2_score": 0.8439178012399987,
"lm_q2_score": 0.8723473779969194,
"openwebmath_perplexity": 286.15106899477524,
"openwebmath_score": 0.43166854977607727,
"tags": ... |
Writing a While Loop
Then we will start writing while loops.
• We will initialize first digit: d1 = 0.
• We will set a condition for each loop: the digit will be less than or equal to 9.
• In the beginning of each loop we set a value of the next digit, in this case: d2 = 0. In the nested for loops we initialize the v... | {
"domain": "softuni.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9674102571131691,
"lm_q1q2_score": 0.8439178012399987,
"lm_q2_score": 0.8723473779969194,
"openwebmath_perplexity": 286.15106899477524,
"openwebmath_score": 0.43166854977607727,
"tags": ... |
#### Loops and calculations
No. 77
##### Display all summands
Q:
In Figure 207, “Calculating values ” for a given value of limit only the first and last summand will show up: We just see 1 + ... + 5 = 15 rather than 1 + 2 + 3 + 4 + 5 = 15.
If LIMIT is equal to one the visible result is even worse:
1 + ... + 1 = 1... | {
"domain": "hdm-stuttgart.de",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9674102514755852,
"lm_q1q2_score": 0.8439178011378983,
"lm_q2_score": 0.8723473829749844,
"openwebmath_perplexity": 3636.7259905043843,
"openwebmath_score": 0.24367818236351013,
"ta... |
$( n k ) = n ! k ! ( n - k ) !$
Write an application which allows for determining the probabilistic success rates using this coefficient. For the German Glücksspirale a possible output reads:
Your chance to win when drawing 6 out of 49 is 1 : 13983816
Store parameters like 6 or 49 in variables to keep your softwar... | {
"domain": "hdm-stuttgart.de",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9674102514755852,
"lm_q1q2_score": 0.8439178011378983,
"lm_q2_score": 0.8723473829749844,
"openwebmath_perplexity": 3636.7259905043843,
"openwebmath_score": 0.24367818236351013,
"ta... |
We generalize this fraction cancellation example:
Equation 1. Calculating binomials by cancelling common factors.
$n ! k ! ( n - k ) ! = n ( n - 1 ) ... 1 k ( k - 1 ) ... 1 ( n - k ) ( n - k - 1 ) ... 1 = n ( n - 1 ) ... ( n - k + 1 ) k ( k - 1 ) ... 1$
And thus:
$( 49 6 ) = 49 × 48... | {
"domain": "hdm-stuttgart.de",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9674102514755852,
"lm_q1q2_score": 0.8439178011378983,
"lm_q2_score": 0.8723473829749844,
"openwebmath_perplexity": 3636.7259905043843,
"openwebmath_score": 0.24367818236351013,
"ta... |
# Why are inner products defined to be linear in the first argument only?
It seems to me that if the base field is the real numbers, then we have linearity in both arguments i.e. $\langle u + v, w + z\rangle = \langle u,w\rangle + \langle u,z\rangle + \langle v,w\rangle + \langle v,z\rangle$ because we know $\langle x... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471675459396,
"lm_q1q2_score": 0.8439127240829732,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 221.82920857106024,
"openwebmath_score": 0.965985119342804,
"tags... |
We might take an arbitrary bilinear from instead, but: We want to be able to define a norm (and ultimately a topology) from our inner product. First of all, this prevents us from talking about inner products in characteristic $\ne 0$. We also get difficculties if the groud field is larger than $\mathbb C$. Remains to c... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471675459396,
"lm_q1q2_score": 0.8439127240829732,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 221.82920857106024,
"openwebmath_score": 0.965985119342804,
"tags... |
• Thank you...this makes it clear. I figured that if we ONLY cared about real inner product spaces we could say that it's linear in both arguments but all of the definitions (I read) for an inner product space explicitly mentioned linearity only in the first argument and I wanted to make sure I wasn't missing anything ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471675459396,
"lm_q1q2_score": 0.8439127240829732,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 221.82920857106024,
"openwebmath_score": 0.965985119342804,
"tags... |
# Finding right inverse matrix
Given a $3\times 4$ matrix $A$ such as $$\begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ \end{pmatrix} ,$$ find a matrix $B_{4\times 3}$ such that
$$AB = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$$ Apart from simply multiplying $A$ with $B$ a... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471661250914,
"lm_q1q2_score": 0.8439127210756091,
"lm_q2_score": 0.8577681068080748,
"openwebmath_perplexity": 232.6729563492999,
"openwebmath_score": 0.6087852716445923,
"tag... |
$$B=\begin{bmatrix}0&0&0\\1&0&-1\\-1&1&1\\1&-1&0\end{bmatrix}$$
Of course there are infinite number of solutions | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471661250914,
"lm_q1q2_score": 0.8439127210756091,
"lm_q2_score": 0.8577681068080748,
"openwebmath_perplexity": 232.6729563492999,
"openwebmath_score": 0.6087852716445923,
"tag... |
# Determine if a system described by a differential equation is linear
A system ("A System is any physical set of components that takes a signal, and produces a signal") is described by this equation:
$\frac{dy(t)}{dt} + 3 \times y(t) = x(t)$
Where $x(t)$ is the input and $y(t)$ the output.
How to determine if this... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471616257376,
"lm_q1q2_score": 0.8439127190048129,
"lm_q2_score": 0.8577681086260461,
"openwebmath_perplexity": 181.94475578448282,
"openwebmath_score": 0.899189293384552,
"tag... |
-
Thank you, Teun. I thought I had to use something like S{ a * x_{1}(t) + b* x_{2}(t)} = a* S{x_{1}(t)} + b * S{x_{2}(t)} (In fact this is what you are using, but I wasn't sure which would be the system output).. – Chris Jan 27 '13 at 22:58
Glad to help! What you're saying is almost correct except for the fact that wh... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471616257376,
"lm_q1q2_score": 0.8439127190048129,
"lm_q2_score": 0.8577681086260461,
"openwebmath_perplexity": 181.94475578448282,
"openwebmath_score": 0.899189293384552,
"tag... |
The above equation can be written as $\dot{y} = f(y,x)$, where $f(y,x) = -3 y +x$. $f$ is globally Lipschitz in $y$, hence a globally unique solution exists passing through a given initial condition. Since the solution is unique, it is straightforward to verify (by differentiating and checking that it satisfies the ODE... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471616257376,
"lm_q1q2_score": 0.8439127190048129,
"lm_q2_score": 0.8577681086260461,
"openwebmath_perplexity": 181.94475578448282,
"openwebmath_score": 0.899189293384552,
"tag... |
# Classical Mechanics question
#### Klaas van Aarsen
##### MHB Seeker
Staff member
We are given the angular velocity $\omega = 7\cdot 10^{-5}\,rad/s$ and the mass $M=6\cdot 10^{24}\,kg$.
To achieve a free fall of $0\,m/s^2$ at radius $r$ we need that the centripetal acceleration is equal to the acceleration due to gr... | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.98384716257297,
"lm_q1q2_score": 0.8439127180287127,
"lm_q2_score": 0.8577681068080748,
"openwebmath_perplexity": 531.990208688609,
"openwebmath_score": 0.983230471611023,
"tags":... |
It means that answer 2 should be the correct answer.
Admittedly it's a bit strange that it is given as $4.4\cdot 10^7\,m$ instead of $4.3\cdot 10^7\,m$.
Since we're talking about earth, perhaps they used a mass and angular velocity with a higher precision than the ones given in the problem statement.
EDIT: Hmm... in th... | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.98384716257297,
"lm_q1q2_score": 0.8439127180287127,
"lm_q2_score": 0.8577681068080748,
"openwebmath_perplexity": 531.990208688609,
"openwebmath_score": 0.983230471611023,
"tags":... |
Can a countable group have uncountably many subgroups?
If $G$ is a countable group,can it have an uncountable number of distinct subgroups?
Let $V$ be a vector space of dimension $\aleph_0$ over a countable field $F$ (so $V$ is countable) and let $B$ be a basis for $V$ over $F$. Then every subset of $B$ spans a diffe... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471684931717,
"lm_q1q2_score": 0.8439127159524495,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 207.11658112225055,
"openwebmath_score": 0.9011984467506409,
"ta... |
Consider the direct sum of countably many $\mathbb{Z}/2\mathbb{Z}$ groups, which I'll denote by $$G = \displaystyle \bigoplus_{n = 1}^\infty \left( \mathbb{Z} / 2\mathbb{Z} \right)_n$$ and where the index is to keep track of each copy of $\mathbb{Z}/2\mathbb{Z}$. A set of subgroups of $G$ are formed by including or exc... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471684931717,
"lm_q1q2_score": 0.8439127159524495,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 207.11658112225055,
"openwebmath_score": 0.9011984467506409,
"ta... |
A finitely presented example: the free group $\mathbb{F}_2$ of rank $2$. Indeed, it contains the free group $\mathbb{F}_{\infty}$ of countable infinite rank. Let $\{x_1,x_2,\dots\}$ be a free basis for such a subgroup. For any sequence $\mathfrak{n} = (n_i)$ of positive integers, let $S( \mathfrak{n})$ denote the free ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471684931717,
"lm_q1q2_score": 0.8439127159524495,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 207.11658112225055,
"openwebmath_score": 0.9011984467506409,
"ta... |
• It is a good example, essentially $\mathbb{Z}^{(\mathbb{N})}$, that is, the multiplicative structure is not involved. – Orest Bucicovschi Oct 13 '14 at 18:25
• Indeed, the groups are familiar and I find monomials $x^n$ easier to think about than $\langle \frac{1}{p_n} \rangle$, where $p_n$ is the $n$th prime. – yatim... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471684931717,
"lm_q1q2_score": 0.8439127159524495,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 207.11658112225055,
"openwebmath_score": 0.9011984467506409,
"ta... |
• I do not understand how this answers the question. You have only shown that there are (finite or) countably many finite-index, normal subgroups of a finitely generated group (note: finitely generated, not countable, as mixedmath's answer demonstrates). Which is not awfully relevant... – user1729 Oct 14 '14 at 8:22
• ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471684931717,
"lm_q1q2_score": 0.8439127159524495,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 207.11658112225055,
"openwebmath_score": 0.9011984467506409,
"ta... |
# If $f(x+1) +f(x-1) =\sqrt{3}\,f(x)$ and $f(2) =2$, what is the value of $f(4)$?
My Attempt
$f(2)=2$. So, $f(1) + f(3)=2\sqrt{3}$ and $f(2) + f(4)=\sqrt{3}\,f(3)$. After solving these equations I got the value of $f(3)=2\sqrt{3}$ and $f(4)=4$. But are there any other methods than this? Any suggestions are welcome.
... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471642306267,
"lm_q1q2_score": 0.8439127140847803,
"lm_q2_score": 0.8577681013541613,
"openwebmath_perplexity": 159.769816088976,
"openwebmath_score": 0.9195312261581421,
"tags... |
If $c^2 > 4$, then $b$ has two possible values, one with $|b|>1$ and one with $|b|<1$.
If $c^2 < 4$, then $b$ has two possible complex values $b_1 =\dfrac{c+\sqrt{c^2-4}}{2} =\dfrac{c+i\sqrt{4-c^2}}{2}$ and $b_2 =\dfrac{c-i\sqrt{4-c^2}}{2}$. Note that $b_1b_2 =\dfrac{c+\sqrt{c^2-4}}{2}\dfrac{c-\sqrt{c^2-4}}{2} =\dfrac... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471642306267,
"lm_q1q2_score": 0.8439127140847803,
"lm_q2_score": 0.8577681013541613,
"openwebmath_perplexity": 159.769816088976,
"openwebmath_score": 0.9195312261581421,
"tags... |
How to properly choose a solution to a system of equations of trigonometric functions?
I worked my way to encounter a system of equations
$$\begin{cases} q=k_1\cos\phi_1+k_2\cos\phi_2& \\ 0=k_1\sin\phi_1+k_2\sin\phi_2& \end{cases}$$
I don't need to solve for specific angles $\phi_1,\phi_2$ because these equations wi... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471661250914,
"lm_q1q2_score": 0.8439127121325799,
"lm_q2_score": 0.8577680977182187,
"openwebmath_perplexity": 112.22244715368414,
"openwebmath_score": 0.9263508319854736,
"ta... |
$$\cos \phi_1 = \frac{q^2 + k_1^2 - k_2^2}{2qk_1}$$
which is indeed sign independent. But all of the terms with $\sin \phi_2$ are squared, so $\sin \phi_2$ is not sign independent. Solving the other way, that is, writing $q - k_2 \cos\phi_2 = \pm k_1 \sqrt{1-\sin^2 \phi_1}$ and $\mp k_2 \sqrt{1 - \cos^2 \phi_2} = k_1^... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471661250914,
"lm_q1q2_score": 0.8439127121325799,
"lm_q2_score": 0.8577680977182187,
"openwebmath_perplexity": 112.22244715368414,
"openwebmath_score": 0.9263508319854736,
"ta... |
This is a long-winded way of saying: You do get solutions that aren't unique when solving this. Specifically, you get 4 solutions, and 2 of them turn out to be fake solutions, leaving you with 2 actual ones. I can't tell where exactly in your question you went wrong, because you didn't explain your solution path, but h... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471661250914,
"lm_q1q2_score": 0.8439127121325799,
"lm_q2_score": 0.8577680977182187,
"openwebmath_perplexity": 112.22244715368414,
"openwebmath_score": 0.9263508319854736,
"ta... |
Polynomial Kernel. We show that for high-dimensional data, a particular framework for learning a linear transformation of the data based on the LogDet divergence can be efficiently kernelized to learn a metric (or equivalently, a kernel function) over an. be a linear transformation. Since n n matrices are linear transf... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
problem?. An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its range (or image) and the set of elements that are mapped to the zero vector, called the kernel of the map. Now, let $\phi: V\longrightarrow W$ be a linear mapping/tra... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
We have seen that any matrix transformation x Ax is a linear transformation. It is one-one if its kernel is just the zero vector, and it is. One thing to look out for are the tails of the distribution vs. Definition 6. Morphological transformations are some simple operations based on the image shape. Question: Why is a ... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
linear transformation T :P n → P n−1 is the derivative function that maps each polynomial p(x)to its derivative p′(x). To find the kernel of a matrix A is the same as to solve the system AX = 0, and one usually does this by putting A in rref. Proof: This theorem is a proved in a manner similar to how we solved the abov... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
vectors. Because is a composition of linear transformations, itself is linear (Theorem th:complinear of LTR-0030). And since we have a linear transformation that has the same properties of a subspace, the image and kernel of the linear transformation are subspaces of Rn. Define the transformation $\Omega: L(V,W) \to M_... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
From this, it follows that the image of L is isomorphic to the quotient of V by the kernel: ≅ / (). (Think of it as what vectors you can get from applying the linear transformation or multiplying the matrix by a vector. Answer to Find the kernel and image of each linear transformation in Problems a to c. We solve b... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
{x ∈ X: Ax = 0}. We build thousands of video walkthroughs for your college courses taught by student experts who got an A+. ) T: P 5 → R, T(a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 + a 5 x 5) = a 0. We de ne T Aby the rule T A(x)=Ax:If we express Ain terms of its columns as A=(a 1 a 2 a n), then T A(x)=Ax = Xn i=1 x i... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
with support from the following sponsors. Synonyms: kernel onto A linear transformation, T, is onto if its range is all of its codomain, not merely a subspace. The null space (or kernel) of consists of all vectors of the form , where are real numbers. (2) is injective if and only if. Sums and scalar multiples of linear... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
linear transformation T(x) = Ax is the span of the column vectors of A, that is the column space of matrix A. " • The fact that T is linear is essential to the kernel and range being subspaces. #20 Consider the subspace Wof R4 spanned by the vectors v1 = 1 1 1 1 and v2 = 1 9 −5 3. (c) Determine whether a given vector i... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
of addition and scalar multiplication. Remarks I The kernel of a linear transformation is a. Lesson: Image and Kernel of Linear Transformation Mathematics In this lesson, we will learn how to find the image and basis of the kernel of a linear transformation. Specifically, if T: n m is a linear transformation, then ther... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
point in Figure 5 (left) yields the linearly separable dataset Figure 5 (right). In this paper, we study metric learning as a problem of learning a linear transformation of the input data. Note that N(T) is a subspace of V, so its dimension can be de ned. You can even pass in a custom kernel. Intuitively, the kernel me... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
linear transformations, if and only if f* is symmetric with respect to G. Find the matrix and the eigenvectors of the transformation t. Thus, the kernel is the span of all these vectors. Up Main page Definition. To nd the image of a transformation, we need only to nd the linearly independent column vectors of the matri... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
transformation has the simplest possible representation. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Show that solutions take the form X + V where f(X)=Y and where V i... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
linear transformation. Finding matrices such that M N = N M is an important problem in mathematics. Problem: I can't find answer to a problem. KPCA with linear kernel is the same as standard PCA. Find a basis for the Ker(T). Find the kernel of the linear transformation. Linear combinations of normal random variables. T... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
which are annihilated by the transformation. We build thousands of video walkthroughs for your college courses taught by student experts who got an A+. The kernel of a linear transformation is a vector subspace. 1 Matrix Linear Transformations Every m nmatrix Aover Fde nes linear transformationT A: Fn!Fmvia matrix mult... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
of A. In both cases, the kernel is the set of solutions of the corresponding homogeneous linear equations, AX = 0 or BX = 0. Since the nullity is the dimension of the null space, we see that the nullity of T is 0 since the dimension of the zero vector space is 0. The kernel of T, ker (T), is the set of all vectors x in... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
of degree less. Conversely any linear fractional transformation is a composition of simple trans-formations. Of course we can. Choose a simple yet non-trivial linear transformation with a non-trivial kernel and verify the above claim for the transformation you choose. 1 2 -3 : 1/ 5 y 1 0 0 0 : - 7/. Because is a compos... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
in Rn of the form T(x) = Ax. Linear algebra - Practice problems for midterm 2 1. Using non-linear transformation, you can easily solve non-linear problem as a linear (straight-line) problem. Let L be de ned on P3 (the vector space of polynomials of degree less. 0:22 So, if I have one vector that goes to 0, that is the ... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
array([4,1,0, 1,4]) By combing the existing and new feature, we can certainly draw a line to separate the yellow purple dots (shown on the right). Remarks I The kernel of a linear transformation is a. (If all real numbers are solutions, enter REALS. (The dimension of the image space is sometimes called the rank of T, a... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
For example, the function defined on the real line is not linear, since whereas. Power Iterated Color Refinement. For example Let’s say we have a transformation x !˚ x. The challenge is to find a transformation -> , such that the transformed dataset is linearly separable in. THE PROPERTIES OF DETERMINANTS a. The next t... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
is linear Determining the representation matrix of a linear transformation Representation matrices Kernel of a linear transformation One-to-one linear transformations Onto linear transformations One-to-one and onto Other subjects: here you can put links to material on other subjects you found yourself. And if the trans... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
{/eq} is a polynomial. visualize what the particular transformation is doing. Construct a linear transformation f and vector Y so that the system takes the form f(X)=Y. This basis can be extended to. If m < n, then T cannot be one-to-one. As such, this theorem goes by the name of the Rank- nullity Theorem. If V is fini... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
of. Let T: V !W be a linear transformation. As such, this theorem goes by the name of the Rank- nullity Theorem. Finding a basis of the null space of a matrix. Find a basis for the kernel of T and the range of T. The dimension of the kernel of T is the same as the dimension of its null space and is called the nullity o... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
TRUE To show this we show it is a subspace Col A is the set of a vectors that can be written as Ax for some x. The linear transformation , from to , is both one-to-one and onto. The converse is also true. 0:22 So, if I have one vector that goes to 0, that is the kernel. This paper is organized as follows. Learn vocabul... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
classes of data ordered like a 5 on a dice. If w2 = 0, w3 = 1, then w1 = -1, and if w2 = 1 and w3 = 1, then w1 = 0. Introduction to Linear Algebra exam problems and solutions at the Ohio State University. The aim of our study of linear transformations is two-fold: • to understand linear transformations in R, R2 and R3.... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
class labels for each observation in the predictor data X based on the binary Gaussian kernel classification model Mdl. SVM algorithms use a set of mathematical functions that are defined as the kernel. We show that for high-dimensional data, a particular framework for learning a linear transformation of the data based... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
some vector, and we know that any linear transformation can be written as a matrix vector product, then the kernel of T is the same thing as the null space of A. The theorem relating the dimension of the kernel and image requires the vector spaces to be finite dimensional. In the linear map L : V → W, two elements of V ... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
new feature space:. Finding the kernel of a linear transformation involving an integral. Shed the societal and cultural narratives holding you back and let free step-by-step Linear Algebra: A Modern Introduction textbook solutions reorient your old paradigms. 4 Linear Transformations The operations \+" and \" provide a... | {
"domain": "tuning-style.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9838471647042428,
"lm_q1q2_score": 0.8439127109138213,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 420.58524934978,
"openwebmath_score": 0.7735112905502319,
"tag... |
# I draw a hand of 13 from a deck of 52 cards. What is the probability that I do not have a card from every suit?
I draw a hand of 13 from a deck of 52 standard playing cards. What is the probability that I do not have a card from every suit?
I count the number of ways I can draw 13 from 3 suits
$$\frac{{4\choose3}{... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 1,
"lm_q1q2_score": 0.8438951025545426,
"lm_q2_score": 0.8438951025545426,
"openwebmath_perplexity": 386.1807065294793,
"openwebmath_score": 0.5637345910072327,
"tags": null,
"url"... |
Is my reasoning sound? Have I made any mistakes? Is there a better solution?
• Good use of Inclusion/Exclusion. – André Nicolas May 5 '15 at 1:17
• @ whorl There are other ways of solving this, but they will be less elegant. Your way works perfectly fine. An example of a direct approach might have been: (For a simplif... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 1,
"lm_q1q2_score": 0.8438951025545426,
"lm_q2_score": 0.8438951025545426,
"openwebmath_perplexity": 386.1807065294793,
"openwebmath_score": 0.5637345910072327,
"tags": null,
"url"... |
Convergent sequence with odd terms decreasing and even terms increasing
Let $\left(a_n\right)$ is convergent sequence. $a_0=0, a_1=1,a_2,a_3,...$
Odd terms decrease and even terms increase and for all $n\ge1$ $$2\le \frac{a_n-a_{n-1}}{a_n-a_{n+1}}\le3.$$ Find the boundaries in which there can be a limit of this seque... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241965169938,
"lm_q1q2_score": 0.8438910025715247,
"lm_q2_score": 0.8705972734445508,
"openwebmath_perplexity": 175.61046105108053,
"openwebmath_score": 0.9852163195610046,
"ta... |
• This is not correct. In your solution $$b_n=-\frac13b_{n-1}=\frac13\cdot\frac12b_{n-2}$$ Then $$b_n=\frac16b_{n-2}$$ This is not a minimization. For example $$b_n=\frac13b_{n-1}=\frac13\cdot\frac13b_{n-2}$$ Then $$b_n=\frac19b_{n-2}$$ Jul 15 '17 at 14:33
• @Roman83 That does not show that my solution is incorrect. In... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241965169938,
"lm_q1q2_score": 0.8438910025715247,
"lm_q2_score": 0.8705972734445508,
"openwebmath_perplexity": 175.61046105108053,
"openwebmath_score": 0.9852163195610046,
"ta... |
The sum of the series $\hat{\beta}_{m,r}$ is $\sum_{0\leq k<m} b_k + r \sum_{k\geq m} b_k$. As shown above the last sum has the same sign as $b_m$ and can be made strictly smaller and larger by choosing suitable $r$ whenever $b_m/b_{m+1}\in (-\frac12,-\frac13)$ (an interior point). For example to minimize the sum, for ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241965169938,
"lm_q1q2_score": 0.8438910025715247,
"lm_q2_score": 0.8705972734445508,
"openwebmath_perplexity": 175.61046105108053,
"openwebmath_score": 0.9852163195610046,
"ta... |
This suggests that we consider the sequences $(m_n)_{n\ge0}$ and $(M_n)_{n\ge0}$ defined as follows: \begin{alignat*}{3} &m_0=0,m_1=1, \quad m_{2n+2}&&=(1-\alpha)m_{2n+1}+\alpha m_{2n},\quad m_{2n+3}&&=(1-\beta)m_{2n+2}+\beta m_{2n+1}.\\ &M_0=0,M_1=1, \quad M_{2n+2}&&=(1-\beta)M_{2n+1}+\beta M_{2n},\quad M_{2n+3}&&=(1-... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241965169938,
"lm_q1q2_score": 0.8438910025715247,
"lm_q2_score": 0.8705972734445508,
"openwebmath_perplexity": 175.61046105108053,
"openwebmath_score": 0.9852163195610046,
"ta... |
Now, if $X_n=\left[\begin{matrix} m_{2n}\\ m_{2n+1} \end{matrix}\right]$, and $Y_n=\left[\begin{matrix} M_{2n}\\ M_{2n+1} \end{matrix}\right]$ then \begin{equation*} X_{n+1}=\underbrace{\left[\begin{matrix} \alpha&1-\alpha\\ (1-\beta)\alpha&1-\alpha+\alpha\beta \end{matrix}\right]}_{A_{\alpha,\beta}}X_n,\qquad Y_{n+1}=... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241965169938,
"lm_q1q2_score": 0.8438910025715247,
"lm_q2_score": 0.8705972734445508,
"openwebmath_perplexity": 175.61046105108053,
"openwebmath_score": 0.9852163195610046,
"ta... |
\frac{1-\beta}{1-\alpha\beta} \end{equation*} Now, taking $a_n=m_n$ for all $n$, shows that the lower bound in the above inequality is the best possible, because it is attained, and taking $a_n=M_n$ for all $n$, shows that the upper bound in the above inequality is also the best possible, because it is attained. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241965169938,
"lm_q1q2_score": 0.8438910025715247,
"lm_q2_score": 0.8705972734445508,
"openwebmath_perplexity": 175.61046105108053,
"openwebmath_score": 0.9852163195610046,
"ta... |
Further, considering sequences $(a_n)_{n\ge0}$ of the form $a_n=tm_n+(1-t)M_n$ where $0<t<1$, shows that for any number $\ell$ in the interval $[\frac{1-\alpha}{1-\alpha\beta}, \frac{1-\beta}{1-\alpha\beta}]$ there exists a sequence $(a_n)_{n\ge0}$ satisfying the conditions of the problem and converging to $\ell$.
Rem... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9693241965169938,
"lm_q1q2_score": 0.8438910025715247,
"lm_q2_score": 0.8705972734445508,
"openwebmath_perplexity": 175.61046105108053,
"openwebmath_score": 0.9852163195610046,
"ta... |
# Antiderivative of an odd function
Is the antiderivative of an odd function even?
The answer given by the book is yes.
However, I found a counterexample defined in $$\mathbb{R}\setminus \{0\}$$: $$f(x)=\begin{cases}\ln |x|+1& x<0\\\ln |x|&x>0\end{cases}$$ Its derivative is $$\frac 1x$$, which is an odd function.
Qu... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9773708032626701,
"lm_q1q2_score": 0.8438537548058612,
"lm_q2_score": 0.8633916134888614,
"openwebmath_perplexity": 159.1561143404611,
"openwebmath_score": 0.9354775547981262,
"tag... |
Another interesting number theory tidbit
Hello,
I was browsing a set of number theory problems, and I came across this one:
"Prove that the equation a2+b2=c2+3 has infinitely many solutions in integers."
Now, I found out that c must be odd and a and b must be even. So, for some integer n, c=2n+1, so c2+3=4n2+4n+4=4... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9773708006261043,
"lm_q1q2_score": 0.843853754247241,
"lm_q2_score": 0.8633916152464016,
"openwebmath_perplexity": 891.5897578017862,
"openwebmath_score": 0.8383458256721497,
"tags... |
The case when k=0 has infinitely many solutions of which are all of the form $a=d(p^2-q^2)$, $b=2dpq$, $c=d(p^2+q^2)$ for integer p,q and an arbitrary constant d. The case k=3 makes the right hand side the square of 2n+2 when c=2n+1, and hence the case k=0 implies the case k=3. Applying the case when k=0 that I specifi... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9773708006261043,
"lm_q1q2_score": 0.843853754247241,
"lm_q2_score": 0.8633916152464016,
"openwebmath_perplexity": 891.5897578017862,
"openwebmath_score": 0.8383458256721497,
"tags... |
# Solve $(\sqrt{5+2\sqrt{6}})^{x}+(\sqrt{5-2\sqrt{6}})^{x}=10$.
Solve $(\sqrt{5+2\sqrt{6}})^{x}+(\sqrt{5-2\sqrt{6}})^{x}=10$
I square the both sides and get $(5+2\sqrt{6})^{x}+(5-2\sqrt{6})^{x}=98$. But I don't know how to carry on. Please help. Thank you.
-
Let $t=( \sqrt{5 + 2\sqrt{6}})^{x}\implies (\sqrt{5-2\sqr... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9773708006261042,
"lm_q1q2_score": 0.8438537542472409,
"lm_q2_score": 0.8633916152464016,
"openwebmath_perplexity": 1149.837368735485,
"openwebmath_score": 0.9436652064323425,
"tag... |
and so for $t=\left(\sqrt{5+2\sqrt{6}}\right)^x$......
-
Interesting. Firefox is rendering the first set of brackets as Floor brackets. Checked in IE and it's fine. Also, CTRL + and CTRL - (change font size) renders correctly. – Chris Cudmore Sep 25 '12 at 14:03
Hint $\$ Put $\rm\ b = 5 + 2\sqrt{6},\,\ a = b^{\,x/2}... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9773708006261042,
"lm_q1q2_score": 0.8438537542472409,
"lm_q2_score": 0.8633916152464016,
"openwebmath_perplexity": 1149.837368735485,
"openwebmath_score": 0.9436652064323425,
"tag... |
$(\sqrt{5+2\sqrt6})^x=5-2\sqrt 6$
$(\sqrt{5+2\sqrt6})^x=(5+2\sqrt 6)^{-2}$ $\Rightarrow$ $x=-2$
Definitly $x=2$, and $x=-2$ is solve.
-
Please could you tell me the reason why they are negative points – Madrit Zhaku Sep 25 '12 at 19:23 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9773708006261042,
"lm_q1q2_score": 0.8438537542472409,
"lm_q2_score": 0.8633916152464016,
"openwebmath_perplexity": 1149.837368735485,
"openwebmath_score": 0.9436652064323425,
"tag... |
# Train waiting time in probability
Let's say a train arrives at a stop every 15 or 45 minutes with equal probability (1/2). What is the expected waiting time of a passenger for the next train if this passenger arrives at the stop at any random time. This means that the passenger has no sense of time nor know when the... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9773708045809529,
"lm_q1q2_score": 0.8438537525085182,
"lm_q2_score": 0.8633916099737806,
"openwebmath_perplexity": 642.361616977668,
"openwebmath_score": 0.42805957794189453,
"tag... |
Picture in your mind's eye the whole train schedule is already generated; it looks like a line with marks on it, where the marks represent a train arriving. On average, two consecutive marks are fifteen minutes apart half the time, and 45 minutes apart half the time.
Now, imagine a person arrives; this means randomly ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9773708045809529,
"lm_q1q2_score": 0.8438537525085182,
"lm_q2_score": 0.8633916099737806,
"openwebmath_perplexity": 642.361616977668,
"openwebmath_score": 0.42805957794189453,
"tag... |
# Binary Search Introductions
Whenever a sorted array is given try to apply the binary search on that. This divides the array into two parts and only works on the other part. Recursion equation $$T(n) = T(\frac{N}{2}) + C$$
## Toy problem to start: Find Ceil
### Problem Statement
Find the ceil of a target number fo... | {
"domain": "theroyakash.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9773708012852457,
"lm_q1q2_score": 0.843853751380801,
"lm_q2_score": 0.8633916117313211,
"openwebmath_perplexity": 5941.5676078984625,
"openwebmath_score": 0.24568580090999603,
"tags... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.