text stringlengths 1 2.12k | source dict |
|---|---|
where using Fermat's Little Theorem $2^{110}=(2^{10})^{11}\equiv 1$.
So $30^{37}\equiv 2 \bmod 7$ and $\bmod 11$, thus the Chinese Remainder Theorem gives
$30^{37}\equiv 2 \bmod 77$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180635575532,
"lm_q1q2_score": 0.8455175226003051,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 483.11993844299525,
"openwebmath_score": 0.8116440773010254,
"ta... |
# How many eigenvalues does an $n\times n$ matrix have, and how does this relate to the algebraic multiplicity of the dominant eigenvalue?
I'm having to reevaluate my understanding of eigenvalues and how many eigenvalues an $$n\times n$$ matrix possesses. Previously, I had thought that such a matrix $$A$$ possessed $$... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180673335565,
"lm_q1q2_score": 0.8455175222552263,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 144.99868091648742,
"openwebmath_score": 0.9360548257827759,
"tag... |
So if I'm reading this explanation correctly, a list of all of the eigenvalues of $$A$$ should include $$i$$ instances of an eigenvalue with algebraic multiplicity $$i$$. In other words, every $$n \times n$$ matrix has exactly $$n$$ complex eigenvalues, and there is a distinction between the number of eigenvalues that ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180673335565,
"lm_q1q2_score": 0.8455175222552263,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 144.99868091648742,
"openwebmath_score": 0.9360548257827759,
"tag... |
In your case, I think you just have to read the definition of "dominant eigenvalue" carefully. Based on the problem writing "dominant eigenvalue $$\lambda_1$$," I suspect the definition is written as
if $$\lambda_1, \ldots, \lambda_n$$ are the eigenvalues of $$A$$, then $$\lambda_1$$ is considered dominant if $$|\lamb... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180673335565,
"lm_q1q2_score": 0.8455175222552263,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 144.99868091648742,
"openwebmath_score": 0.9360548257827759,
"tag... |
7. Product and Process Comparisons
7.2. Comparisons based on data from one process
7.2.6. What intervals contain a fixed percentage of the population values?
## Percentiles
Definitions of order statistics and ranks For a series of measurements $$Y_1, \, \ldots, \, Y_N$$, denote the data ordered in increasing order of... | {
"domain": "nist.gov",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180664944447,
"lm_q1q2_score": 0.845517521535463,
"lm_q2_score": 0.8577681086260461,
"openwebmath_perplexity": 631.6664033075261,
"openwebmath_score": 0.8262714743614197,
"tags": null,
... |
2. For $$k = 0, \,\,\,\,\, Y_{(p)} = Y_{[1]}$$
Note that any p ≤ 1/(N+1) will simply be set to the minimum value.
3. For $$k ≥ N, \,\,\,\,\, Y_{(p)} = Y{[N]}$$
Note that any pN/(N+1) will simply be set to the maximum value.
Example and interpretation For the purpose of illustration, twelve measurements from a gage ... | {
"domain": "nist.gov",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180664944447,
"lm_q1q2_score": 0.845517521535463,
"lm_q2_score": 0.8577681086260461,
"openwebmath_perplexity": 631.6664033075261,
"openwebmath_score": 0.8262714743614197,
"tags": null,
... |
The method advocated by Hyndman and Fan is R8. For the R8 method, set $$p(N+(1/3) + (1/3))$$ and proceed as above. Note that any p ≤ (2/3)/(N+(1/3)) will be set to the minimum value and any p ≥ (N-(1/3))/(N+(1/3)) will be set to the maximum value. Both R and Dataplot can optionally use this method. For the example give... | {
"domain": "nist.gov",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180664944447,
"lm_q1q2_score": 0.845517521535463,
"lm_q2_score": 0.8577681086260461,
"openwebmath_perplexity": 631.6664033075261,
"openwebmath_score": 0.8262714743614197,
"tags": null,
... |
Stationary distribution of a Markov process defined on the space of permutations
Let $$S$$ be the set of $$n!$$ permutations of the first $$n$$ integers. Let $$p\in(0,1)$$. Consider the Markov Process defined on the elements of $$S$$.
1. Let $$x\in S$$. Choose $$1\le i uniformly at random.
2. If $$x_i < x_{i+1}$$, sw... | {
"domain": "mathoverflow.net",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180664944448,
"lm_q1q2_score": 0.8455175197434559,
"lm_q2_score": 0.8577681068080748,
"openwebmath_perplexity": 985.9429299700173,
"openwebmath_score": 0.9995585083961487,
"tags... |
• Is this well-defined? Suppose $p=1/3$ and you pick $2$ and $3$. We should swap $2$ and $3$ with probability $1/3$ because $2<3$. Then again, we should swap $3$ and $2$ with probability $2/3$ because $3>2$... Jun 18 '19 at 4:38
• Of course there are $n(n-1)/2$ pairs... if it is swap $i,j$ with probability $p$, then yo... | {
"domain": "mathoverflow.net",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180664944448,
"lm_q1q2_score": 0.8455175197434559,
"lm_q2_score": 0.8577681068080748,
"openwebmath_perplexity": 985.9429299700173,
"openwebmath_score": 0.9995585083961487,
"tags... |
\begin{align*} n \pi_\sigma p_{\sigma\tau} &= \Bigl(\frac{p}{1-p}\Bigr)^{\ell(\sigma)} \begin{cases} p & \text{if \sigma_i < \sigma_j} \\ 1-p & \text{if \sigma_i > \sigma_j} \end{cases}\\ &= \Bigl(\frac{p}{1-p}\Bigr)^{\ell(\sigma)} \begin{cases} \frac{p}{1-p} (1-p) &\text{if \sigma_i < \sigma_j} \\ \frac{1-p}{p} p & \t... | {
"domain": "mathoverflow.net",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180664944448,
"lm_q1q2_score": 0.8455175197434559,
"lm_q2_score": 0.8577681068080748,
"openwebmath_perplexity": 985.9429299700173,
"openwebmath_score": 0.9995585083961487,
"tags... |
A computer algebra calculation shows that the invariant distribution is proportional to $$(\alpha,\beta,\beta,\gamma,\gamma,\delta)$$ where $$\alpha = (1-p)(6-11p+7p^2)$$, $$\beta = (1-p)p(8-7p)$$, $$\gamma = (1-p)p(1+7p)$$ and $$\delta = p(2-3p+7p^2)$$. For $$n=4$$ the invariant distribution is more complicated. For e... | {
"domain": "mathoverflow.net",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180664944448,
"lm_q1q2_score": 0.8455175197434559,
"lm_q2_score": 0.8577681068080748,
"openwebmath_perplexity": 985.9429299700173,
"openwebmath_score": 0.9995585083961487,
"tags... |
# Is the plane minus the integer lattice homeomorphic to the plane minus the integers?
The question, more rigorously posed, is:
Is $\Bbb R^2-\Bbb Z^2$ homeomorphic to $\Bbb R^2-\Bbb Z\times\{0\}$?
This question has been bugging me in the back of my head for a while now. Sometimes, I think it's clear that they are, a... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.985718066074889,
"lm_q1q2_score": 0.8455175175915673,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 139.75567030528111,
"openwebmath_score": 0.8856316208839417,
"tag... |
1. Open connected surfaces are classified by a theorem of Brown and Messer "The classification of two-dimensional manifolds", Transactions of AMS, 1979. The first invariant is the genus (and genera of ends): In your case genus equals $0$ since both surfaces are obtained by removing compact subsets $E_1, E_2$ from $S^2$... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.985718066074889,
"lm_q1q2_score": 0.8455175175915673,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 139.75567030528111,
"openwebmath_score": 0.8856316208839417,
"tag... |
We can create such a function by choosing some $g(x)$ such that the set $\{\frac{1}{g(n)}\:n\in \mathbb{N}\}$ contains no pair of elements whose ratio is rational and has no accumulation point. Something like $g(n)=e^{-4n^2-n}$ would suffice, since the exponent is never equal for distinct integers and $e^{x}$ is never ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.985718066074889,
"lm_q1q2_score": 0.8455175175915673,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 139.75567030528111,
"openwebmath_score": 0.8856316208839417,
"tag... |
Alright, I came up with my own proof.
A couple of lemmas:
Lemma 1: Let $X$ and $Y$ be topological spaces. And let $\mathcal{B}$ be a locally-finite collection of closed sets such that $\bigcup\mathcal{B}=X$. Suppose, in addition, that there is a continuous function $f_B:B\rightarrow Y$ for each $B\in\mathcal{B}$ such... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.985718066074889,
"lm_q1q2_score": 0.8455175175915673,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 139.75567030528111,
"openwebmath_score": 0.8856316208839417,
"tag... |
Now we use the above lemmas to create a homeomorphism on each of the annuli (leave the first closed ball alone) to bring the lattice points they contain onto the $x$-axis while still fixing the boundaries of each annulus. We then stitch these together using the first lemma to create a global homeomorphism which pulls t... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.985718066074889,
"lm_q1q2_score": 0.8455175175915673,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 139.75567030528111,
"openwebmath_score": 0.8856316208839417,
"tag... |
# Let a,b $\in$ $\mathbb{R}.$ Show that $a^4+b^4+8\ge 8ab.$
Let $$a,b \in \mathbb{R}.$$ Show that $$a^4+b^4+8\ge 8ab.$$
The question is from the inequalities section of An Excursion in mathematics by Bhaskaraycharya Pratisthanan. My heuristics include using the AM-GM inequality. I am unable to design the problem to p... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180635575531,
"lm_q1q2_score": 0.845517517224284,
"lm_q2_score": 0.8577681068080749,
"openwebmath_perplexity": 549.9597941149542,
"openwebmath_score": 0.6325794458389282,
"tags... |
• Do you ever look at your answers and think "that could look a lot better if I had bothered to make it pretty". Nov 17 '14 at 21:34
• i try to find the answer as fast as i can Sarah Nov 17 '14 at 21:35
• I think you would be doing this community a great favour by later revising your answer and making it high quality. ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180635575531,
"lm_q1q2_score": 0.845517517224284,
"lm_q2_score": 0.8577681068080749,
"openwebmath_perplexity": 549.9597941149542,
"openwebmath_score": 0.6325794458389282,
"tags... |
# Inverse of Group Product
## Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $a, b \in G$, with inverses $a^{-1}, b^{-1}$.
Then:
$\paren {a \circ b}^{-1} = b^{-1} \circ a^{-1}$
### General Result
Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $a_1, a_2, \ldots, a_n \in G$... | {
"domain": "proofwiki.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180669140007,
"lm_q1q2_score": 0.8455175165193234,
"lm_q2_score": 0.8577681031721324,
"openwebmath_perplexity": 453.27979480105296,
"openwebmath_score": 0.956074059009552,
"tags": ... |
The result follows from Inverse of Product in Monoid.
$\blacksquare$
## Proof 3
$\displaystyle \paren {a \circ b} \circ \paren {a \circ b}^{-1}$ $=$ $\displaystyle e$ Definition of Inverse Element $\displaystyle \leadsto \ \$ $\displaystyle a \circ \paren {b \circ \paren {a \circ b}^{-1} }$ $=$ $\displaystyle e$ Gr... | {
"domain": "proofwiki.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180669140007,
"lm_q1q2_score": 0.8455175165193234,
"lm_q2_score": 0.8577681031721324,
"openwebmath_perplexity": 453.27979480105296,
"openwebmath_score": 0.956074059009552,
"tags": ... |
# Uniform and absolute convergence of improper integral
What is an example of an improper integral , $\int_a^\infty f(u,v)du$, that converges uniformly for $v$ is some subset $S$, but where $\int_a^\infty|f(u,v)|du$ converges pointwise but NOT uniformly on $S$?
When Weierstrass’s Test shows that Riemann improper inte... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180667042227,
"lm_q1q2_score": 0.8455175163393827,
"lm_q2_score": 0.8577681031721325,
"openwebmath_perplexity": 164.79954887525182,
"openwebmath_score": 0.9931378364562988,
"ta... |
Thus, the convergence fails to be uniform for all $v \in [0,\infty).$
• Thank you. I understand the Dirichlet test. How is it carried out for showing the first integral is unif. conv? – scobaco Dec 6 '16 at 22:38
• @scobaco: You're welcome. To use Dirichlet you show (1) that the integral of $\sin u$ over$[0,x]$ is uni... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180667042227,
"lm_q1q2_score": 0.8455175163393827,
"lm_q2_score": 0.8577681031721325,
"openwebmath_perplexity": 164.79954887525182,
"openwebmath_score": 0.9931378364562988,
"ta... |
# Solve for $x$, $-\frac{1}{2}x^2 + 2x + 5 = 0$
I'm having trouble solving this equation for $x$:
$$-\frac{1}{2}x^2 + 2x + 5 = 0$$
What's the steps to take to solve it?
Thanks.
-
The equation in the question.. – Cypras Feb 7 '13 at 23:02
Not sure how to format properly? It should be -1/2 – Cypras Feb 7 '13 at 23... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180631379971,
"lm_q1q2_score": 0.8455175150723953,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 1260.8544228296953,
"openwebmath_score": 0.9526488780975342,
"ta... |
If you multiply $$-\frac{1}{2}x^2 + 2x + 5 = 0\tag{1}$$
by $-2$ you get $$x^2-4x-10=0$$ Using the quadratic formula: $$ax^2 + bx + c = 0 \iff x = \frac{1}{2a}\left(-b \pm \sqrt{b^2 - 4ac}\right),$$ where in this case, $a = 1,\;b= -4,\; c = -10$ we have $$x=\frac 12 (4 \pm \sqrt {16+40})=2 \pm \sqrt {14}\tag{2}$$
Walk... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180631379971,
"lm_q1q2_score": 0.8455175150723953,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 1260.8544228296953,
"openwebmath_score": 0.9526488780975342,
"ta... |
# Tilting the $d$-cube to vertically separate its vertices
Let $C_d$ be a unit edge-length cube in $d$ dimensions. I would like to orient it ("tilt" it) so that the vertical (last) coordinates of its $2^d$ vertices are maximally separated, in the sense that the minimum vertical distance between any two vertices is max... | {
"domain": "mathoverflow.net",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180660748889,
"lm_q1q2_score": 0.8455175104235388,
"lm_q2_score": 0.8577680977182187,
"openwebmath_perplexity": 399.8524194005648,
"openwebmath_score": 0.9211409091949463,
"tags... |
1. if $$x=v_d$$ then $$\{x_S\}=\{0,1,2,\dots 2^d-1\}$$ clearly has the tightest possible packing of values (among vectors with separation $$1$$, and uniquely up to permutations) and therefore must minimize the left hand side in the lemma.
Proof of Lemma. There are 16 equally frequent cases for the occurrence of $$x_ix... | {
"domain": "mathoverflow.net",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180660748889,
"lm_q1q2_score": 0.8455175104235388,
"lm_q2_score": 0.8577680977182187,
"openwebmath_perplexity": 399.8524194005648,
"openwebmath_score": 0.9211409091949463,
"tags... |
Given a unit vector $u \in \mathbb R^d$, the "heights" of vertices of the $n$-cube where $u$ is regarded as the vertical direction are the sums of subsets of the entries of $u$. Thus the minimum separation is the minimum difference between the sums of two distinct subsets of these entries. If you take $$u = [1,2,\ldots... | {
"domain": "mathoverflow.net",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180660748889,
"lm_q1q2_score": 0.8455175104235388,
"lm_q2_score": 0.8577680977182187,
"openwebmath_perplexity": 399.8524194005648,
"openwebmath_score": 0.9211409091949463,
"tags... |
This means that one cannot characterize the optimal solution by local conditions on derivatives; the problem seems to have a "combinatorial" nature. If one first fixes the vertical order of all $$2^d$$ corners, then maximizing the separation can be formulated as a quadratic optimization problem. The remaining question ... | {
"domain": "mathoverflow.net",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180660748889,
"lm_q1q2_score": 0.8455175104235388,
"lm_q2_score": 0.8577680977182187,
"openwebmath_perplexity": 399.8524194005648,
"openwebmath_score": 0.9211409091949463,
"tags... |
# How many possible arrangements for a round robin tournament?
How many arrangements are possible for a round robin tournament over an even number of players $n$?
A round robin tournament is a competition where $n = 2k$ players play each other once in a heads-up match (like the group stage of a FIFA World Cup). To ac... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180656553329,
"lm_q1q2_score": 0.845517510063657,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 202.45448764684988,
"openwebmath_score": 0.8420079946517944,
"tag... |
-
You can arrange matches with no overlap by arbitrarily numbering the teams $0 - (n - 1)$, then in round $i$ match teams such that the sum of ther numbers is $i$ mod $n$. You can relabel the teams in $(n-1)!$ ways and do the above, so i think this is the answer.., – gnometorule Jan 22 '13 at 18:25
Interesting ... but ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180656553329,
"lm_q1q2_score": 0.845517510063657,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 202.45448764684988,
"openwebmath_score": 0.8420079946517944,
"tag... |
# Thread: Complex Polar Forms, Sin and Cos Angles
1. ## Complex Polar Forms, Sin and Cos Angles
Problem: Find all solutions of $\displaystyle z^{3} = -8$.
I can do the entire problem except for one part. After putting it into the correct form,
$\displaystyle |z|^{3}(cos3\theta+isin3\theta)$,
I do not know how to f... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9924227603008972,
"lm_q1q2_score": 0.845497353240705,
"lm_q2_score": 0.8519528038477824,
"openwebmath_perplexity": 342.19517384030405,
"openwebmath_score": 0.9576541185379028,
"tag... |
4. ## Re: Complex Polar Forms, Sin and Cos Angles
Originally Posted by tangibleLime
Thanks,
So since $\displaystyle |z|^3 = 8$, I need $\displaystyle cos3\theta+isin3\theta$ to equal -1 to satisfy the initial equation where $\displaystyle z^3 = -8$. The only way to get -1 from $\displaystyle cos3\theta+isin3\theta$ i... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9924227603008972,
"lm_q1q2_score": 0.845497353240705,
"lm_q2_score": 0.8519528038477824,
"openwebmath_perplexity": 342.19517384030405,
"openwebmath_score": 0.9576541185379028,
"tag... |
# How to count the ways to induce a permutation?
I'm reading a recreational book about combinatorics, that discusses, in passing, the ways to 'induce' a permutation of the index set {1,2,3,4}.
The book notes that there is exactly:
• 1 way to induce a permutation by an identity
• C(4,2) = 6 ways to induce a permutati... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9924227592924099,
"lm_q1q2_score": 0.8454973523815216,
"lm_q2_score": 0.8519528038477825,
"openwebmath_perplexity": 248.19549508403006,
"openwebmath_score": 0.8597887754440308,
"ta... |
The number of $3$-cycles ("leave $1$ index unchanged") $$\frac{4!}{ 1^{1}\cdot 1! \cdot 3^1\cdot 1!}= 8$$ because a general example of a $3$-cycle in $S_4$ is $(1\; 2\; 3)(4)$ The number of $4$-cycles ("induce a permutation by a cyclic permutation of all four indices") is $$\frac{4!}{ 4^{1} 1!}=6$$ The number of $2,2$-... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9924227592924099,
"lm_q1q2_score": 0.8454973523815216,
"lm_q2_score": 0.8519528038477825,
"openwebmath_perplexity": 248.19549508403006,
"openwebmath_score": 0.8597887754440308,
"ta... |
# Testing differences in variance between groups
I have a hypothesis that a particular intervention/treatment will cause more variation in participant responses to a particular question.
The intervention variable is categorical, with five different treatment groups. The response variable (the participant responses to... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9496693659780477,
"lm_q1q2_score": 0.8454851491087702,
"lm_q2_score": 0.8902942217558212,
"openwebmath_perplexity": 781.1788537708134,
"openwebmath_score": 0.46610626578330994,
"ta... |
This is an interesting question! Post-hoc tests of variances after a test of unequal variances do not seem to be a much studied topic, I was not able to find any published papers. One similar question with some ideas is Post-hoc test to determine difference in variance. Another approach is the following.
But first, no... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9496693659780477,
"lm_q1q2_score": 0.8454851491087702,
"lm_q2_score": 0.8902942217558212,
"openwebmath_perplexity": 781.1788537708134,
"openwebmath_score": 0.46610626578330994,
"ta... |
Fit: aov(formula = absres ~ Group, data = mydf)
$Group diff lwr upr p adj B-A 0.9578979 -0.87239851 2.788194 0.5937356 C-A 1.9853491 0.15505269 3.815646 0.0265586 D-A 1.8196928 -0.01060365 3.649989 0.0521132 E-A 2.4268567 0.59656026 4.257153 0.0033957 C-B 1.0274512 -0.80284522 2.857748 0.5258877 D-B 0.8617949 -0.96850... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9496693659780477,
"lm_q1q2_score": 0.8454851491087702,
"lm_q2_score": 0.8902942217558212,
"openwebmath_perplexity": 781.1788537708134,
"openwebmath_score": 0.46610626578330994,
"ta... |
• Thank you so much for this very detailed response! It is incredibly useful. Can I ask why one might choose to base the residuals on the median vs the mean (or vice versa)? Would one be better suited to skewed data? This may end up all being hypothetical in my case as my data are extremely skewed, but it still would b... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9496693659780477,
"lm_q1q2_score": 0.8454851491087702,
"lm_q2_score": 0.8902942217558212,
"openwebmath_perplexity": 781.1788537708134,
"openwebmath_score": 0.46610626578330994,
"ta... |
# Developing a function of two variables from given data
(I believe Mathematica SE is an appropriate place to ask this, as well.)
I have been stuck on the following problem.
Consider a system where we have three variables: force $F$, orientation $\theta$, and temperature $T$. I want to find a function for force such... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9433475778774728,
"lm_q1q2_score": 0.8454765711645589,
"lm_q2_score": 0.8962513828326955,
"openwebmath_perplexity": 2125.100944096614,
"openwebmath_score": 0.2426089644432068,
"tag... |
I have made little progress so far. I observed that the relationship between $F$ and $\theta$ is linear, while the relationship between $F$ and $T$ is nonlinear. The latter nonlinear relationship appears to be best approximated with a 2nd order polynomial. So far, I have six different functions (polynomial fit lines) f... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9433475778774728,
"lm_q1q2_score": 0.8454765711645589,
"lm_q2_score": 0.8962513828326955,
"openwebmath_perplexity": 2125.100944096614,
"openwebmath_score": 0.2426089644432068,
"tag... |
int is your $f(\theta, T)$, and can be used like this:
Plot[int[35, x], {x, 0, 300}]
Plot[int[x, 110], {x, 0, 100}]
You can also extrapolate values (as above), but it warns you that it is doing so, and the first plot shows exactly why.
If you have some good guesses about the underlying model, you can always try to ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9433475778774728,
"lm_q1q2_score": 0.8454765711645589,
"lm_q2_score": 0.8962513828326955,
"openwebmath_perplexity": 2125.100944096614,
"openwebmath_score": 0.2426089644432068,
"tag... |
Using Mathematica version 10 new Predict Function
t = {200, 182, 164, 146, 128, 110, 101, 92, 83};
s = {10, 20, 30, 40, 50, 60};
data = {{10.24, 10.15, 10.01, 9.81, 9.39, 8.8, 8.57, 7.89, 7.23},
{21.50, 21.52, 21.25, 20.88, 20.79, 20.66, 20.37, 19.98, 19.50},
{31.92, 32.09, 31.87, 31.58, 31.31, 30.99, 30.86, 30.87, 30... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9433475778774728,
"lm_q1q2_score": 0.8454765711645589,
"lm_q2_score": 0.8962513828326955,
"openwebmath_perplexity": 2125.100944096614,
"openwebmath_score": 0.2426089644432068,
"tag... |
Second step is to visualize the information. Check the data in the 2 different axes.
vals = Table[{s[[i]], t[[j]], data[[i, j]]}, {i, Length@s}, {j,
Length@t}];
Multicolumn[
ListPlot[vals[[#, All, 2 ;;]], Joined -> True,
PlotMarkers -> Graphics[{Red, PointSize[Medium], Point[{0, 0}]}],
PlotLabel ->
Style[StringJoin["\... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9433475778774728,
"lm_q1q2_score": 0.8454765711645589,
"lm_q2_score": 0.8962513828326955,
"openwebmath_perplexity": 2125.100944096614,
"openwebmath_score": 0.2426089644432068,
"tag... |
Setup:
data = {{10.24, 10.15, 10.01, 9.81, 9.39, 8.8, 8.57, 7.89,
7.23}, {21.50, 21.52, 21.25, 20.88, 20.79, 20.66, 20.37, 19.98,
19.50}, {31.92, 32.09, 31.87, 31.58, 31.31, 30.99, 30.86, 30.87,
30.41}, {43.56, 43.88, 43.63, 43.29, 43.02, 42.57, 42.16, 42.52,
42.25}, {54.85, 55.28, 54.98, 54.57, 54.36, 54.07, 53.78, 5... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9433475778774728,
"lm_q1q2_score": 0.8454765711645589,
"lm_q2_score": 0.8962513828326955,
"openwebmath_perplexity": 2125.100944096614,
"openwebmath_score": 0.2426089644432068,
"tag... |
# Digit 5 repetition
1. Apr 17, 2007
### f(x)
1. The problem statement, all variables and given/known data
How many times does digit 5 occur in numbers fro 0-1000.
2. Relevant equations
3. The attempt at a solution
This is what i have done.
Total (1,2,3) digit numbers which have digit 5 occuring once in them are-:... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811651448431,
"lm_q1q2_score": 0.8454682789349309,
"lm_q2_score": 0.8757870046160257,
"openwebmath_perplexity": 785.1382222861233,
"openwebmath_score": 0.548229455947876,
"tags... |
But, since you're counting the 5's, you have to multiply that 2nd line by 2 and the 3rd line by 3. Nice.
Last edited: Apr 17, 2007
4. Apr 18, 2007
### CarlB
It seems like it can be read two different ways. If you just want to count the number of 5s in the numbers between 0 and 999, then you may as well write the num... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811651448431,
"lm_q1q2_score": 0.8454682789349309,
"lm_q2_score": 0.8757870046160257,
"openwebmath_perplexity": 785.1382222861233,
"openwebmath_score": 0.548229455947876,
"tags... |
# What does “order matters” regarding permutations refer to?
I psychoanalyze EVERYTHING and permutations/combinations are frustrating me. Sorry for posting so many questions lately but I really appreciate all of the help!
Ok so I know the permutation formula: $\frac{n!}{(n-r)!}$ and combination formula: $\frac{n!}{(n... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811621568289,
"lm_q1q2_score": 0.8454682669287126,
"lm_q2_score": 0.8757869948899665,
"openwebmath_perplexity": 497.7231375461967,
"openwebmath_score": 0.6360739469528198,
"tag... |
Thanks
-
It would be silly to ask "How many possible orders" and at the same time insist that order does not matter, so you may safely assume that order does matter here. If really you wanted to count the number of ways to visit those cities if order does not matter, then there is exactly one solution; indeed there is... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811621568289,
"lm_q1q2_score": 0.8454682669287126,
"lm_q2_score": 0.8757869948899665,
"openwebmath_perplexity": 497.7231375461967,
"openwebmath_score": 0.6360739469528198,
"tag... |
-
Ok thanks! So if a question says order doesn't matter.. its 100% a combination equation? I posted a question a few days ago and someone kind of clarified the confusion, but I'd love to hear your feedback if you don't mind: math.stackexchange.com/questions/718899/… – Cozen Mar 22 '14 at 22:48
@Cozen Yes, that is true... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811621568289,
"lm_q1q2_score": 0.8454682669287126,
"lm_q2_score": 0.8757869948899665,
"openwebmath_perplexity": 497.7231375461967,
"openwebmath_score": 0.6360739469528198,
"tag... |
Does it matter if the marbles chosen are, say, $(r, r, g, g, b, b)$ versus $(r, g, b, g, r, b)$? Then a permutation on the individual outcomes is not applicable. But note that this question is a bit more complicated than a simple binomial coefficient computation, too. I mention it because counting methods ultimately re... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811621568289,
"lm_q1q2_score": 0.8454682669287126,
"lm_q2_score": 0.8757869948899665,
"openwebmath_perplexity": 497.7231375461967,
"openwebmath_score": 0.6360739469528198,
"tag... |
# Is multiplication as an operation available in groups, rings and fields over Z_p*?
I've seen groups, rings, and fields described with a multiplication operation as well as a group defined as only having addition and subtraction (via inverse) operations. Is the reason the answer varies with respect to a group having ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811631528337,
"lm_q1q2_score": 0.8454682662361083,
"lm_q2_score": 0.8757869932689566,
"openwebmath_perplexity": 298.9812495888975,
"openwebmath_score": 0.9031511545181274,
"tag... |
This is only a convention. The group axioms for the binary operation will work with any symbol for it, so if it helps to think of it as multiplication, you are not wrong. In one important family of examples the group elements are symmetries or (stated another way) mappings that preserve a set of things (e.g. permutatio... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811631528337,
"lm_q1q2_score": 0.8454682662361083,
"lm_q2_score": 0.8757869932689566,
"openwebmath_perplexity": 298.9812495888975,
"openwebmath_score": 0.9031511545181274,
"tag... |
• What would be examples, other than straight ahead addition and multiplication over ℤ/(11ℤ), that is commutative and thus uses the addition (+) operation and one that is not commutative and uses the multiplication (x) operation? – JohnGalt Jan 30 at 17:16
• This might be bad form but many thanks for adding those examp... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811631528337,
"lm_q1q2_score": 0.8454682662361083,
"lm_q2_score": 0.8757869932689566,
"openwebmath_perplexity": 298.9812495888975,
"openwebmath_score": 0.9031511545181274,
"tag... |
# 1.1 The scope and scale of physics (Page 4/12)
Page 4 / 12
## Order of magnitude
The order of magnitude of a number is the power of 10 that most closely approximates it. Thus, the order of magnitude refers to the scale (or size) of a value. Each power of 10 represents a different order of magnitude. For examp... | {
"domain": "quizover.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688146,
"lm_q1q2_score": 0.8454682611820636,
"lm_q2_score": 0.8757869916479466,
"openwebmath_perplexity": 547.402235489824,
"openwebmath_score": 0.6017404794692993,
"tags"... |
An equivalent but quicker way to find the order of magnitude of a number is first to write it in scientific notation and then check to see whether the first factor is greater than or less than $\sqrt{10}={10}^{0.5}\approx 3.$ The idea is that $\sqrt{10}={10}^{0.5}$ is halfway between $1={10}^{0}$ and $10={10}^{1}$ on a... | {
"domain": "quizover.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688146,
"lm_q1q2_score": 0.8454682611820636,
"lm_q2_score": 0.8757869916479466,
"openwebmath_perplexity": 547.402235489824,
"openwebmath_score": 0.6017404794692993,
"tags"... |
The order of magnitude of a number is designed to be a ballpark estimate for the scale (or size) of its value. It is simply a way of rounding numbers consistently to the nearest power of 10. This makes doing rough mental math with very big and very small numbers easier. For example, the diameter of a hydrogen atom is o... | {
"domain": "quizover.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688146,
"lm_q1q2_score": 0.8454682611820636,
"lm_q2_score": 0.8757869916479466,
"openwebmath_perplexity": 547.402235489824,
"openwebmath_score": 0.6017404794692993,
"tags"... |
What is action of point ?
Bishal Reply
a point . is the spot and the action is what u do when ur at the spot . but the action of a point idk a divider
Darth
Quantity which are used in physics simply
Sangram Reply
That's philosophical question.
Jan
What is the Physical quantity
Raja Reply
What is Centripetal force
Taiwo... | {
"domain": "quizover.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688146,
"lm_q1q2_score": 0.8454682611820636,
"lm_q2_score": 0.8757869916479466,
"openwebmath_perplexity": 547.402235489824,
"openwebmath_score": 0.6017404794692993,
"tags"... |
Ahmed
momentum
oladipupo
quatity of motion present in a body or product of mass and velocity
Shakeel
show that the kE of a uniform ring of mass m rolling along a smooth horizontal surface so that its centre of mass has a velocity v is mv×v
folder
what is hydration energy
osobase Reply
the energy........................... | {
"domain": "quizover.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688146,
"lm_q1q2_score": 0.8454682611820636,
"lm_q2_score": 0.8757869916479466,
"openwebmath_perplexity": 547.402235489824,
"openwebmath_score": 0.6017404794692993,
"tags"... |
### Read also:
#### Get the best University physics vol... course in your pocket!
Source: OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
Google Play and the Google Play logo are trademarks of Google Inc.
Notification Switch
Would you like ... | {
"domain": "quizover.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688146,
"lm_q1q2_score": 0.8454682611820636,
"lm_q2_score": 0.8757869916479466,
"openwebmath_perplexity": 547.402235489824,
"openwebmath_score": 0.6017404794692993,
"tags"... |
# I The order of complex poles
#### dyn
Hi.
If I look at the function $(z^2+z-2)/(z-1)^2$ it appears to have a double pole at z=1 but if I factorise the numerator I get $z^2+z-2 = (z+2)(z-1)$ and it is a simple pole.
Is it wrong to say it is a double pole ?
If I overestimate the order of the pole in this case as 2 an... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811611608243,
"lm_q1q2_score": 0.8454682597968548,
"lm_q2_score": 0.8757869884059266,
"openwebmath_perplexity": 276.85048931800566,
"openwebmath_score": 0.9451611042022705,
... |
#### jasonRF
Gold Member
Interesting … I learned that the order of a pole of a function that is analytic in a closed region except at an isolated point, is defined as the largest negative exponent in the Laurent expansion about that point. My old copy of "A Course of Modern Analysis" by Whittaker and Watson also has t... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811611608243,
"lm_q1q2_score": 0.8454682597968548,
"lm_q2_score": 0.8757869884059266,
"openwebmath_perplexity": 276.85048931800566,
"openwebmath_score": 0.9451611042022705,
... |
dyn
Thank you
#### mathwonk
Homework Helper
the order of the pole is a measure of the rate of growth of the function as you approach the given point. In particular it depends only on the values of the function away from the given point, not on the specific representation of the function. Since the two representation... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811611608243,
"lm_q1q2_score": 0.8454682597968548,
"lm_q2_score": 0.8757869884059266,
"openwebmath_perplexity": 276.85048931800566,
"openwebmath_score": 0.9451611042022705,
... |
Last edited:
"The order of complex poles"
### Physics Forums Values
We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of ow... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811611608243,
"lm_q1q2_score": 0.8454682597968548,
"lm_q2_score": 0.8757869884059266,
"openwebmath_perplexity": 276.85048931800566,
"openwebmath_score": 0.9451611042022705,
... |
# Fractions in binary?
How would you write a fraction in binary numbers; for example, $1/4$, which is $.25$? I know how to write binary of whole numbers such as 6, being $110$, but how would one write fractions?
-
The same way. $\frac{n}{m}$. So a quarter would be $\frac{1}{100}$. – copper.hat Feb 12 '13 at 18:47
Yo... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811631528337,
"lm_q1q2_score": 0.8454682584116457,
"lm_q2_score": 0.8757869851639066,
"openwebmath_perplexity": 935.5141489496608,
"openwebmath_score": 0.7899264097213745,
"tag... |
Edit:
These pictures might give you some more intuition ;-) Here $\frac{5}{16} = 0.0101_B$, as the denominator is of form $2^n$, the representation is finite (process ends when you hit zero); $\frac{1}{6} = 0.0010\overline{10}_B$ as the denominator is not of form $2^n$, but the number is rational, so representation is... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811631528337,
"lm_q1q2_score": 0.8454682584116457,
"lm_q2_score": 0.8757869851639066,
"openwebmath_perplexity": 935.5141489496608,
"openwebmath_score": 0.7899264097213745,
"tag... |
Implicit Floating Point: The number is expressed in what amounts to "binary scientific notation". A "mantissa" is stored as an integer, with the decimal point implied to be on the far right. Then the exponent of a power of two is also stored. The actual value of this number is the mantissa, multiplied by two to the pow... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811631528337,
"lm_q1q2_score": 0.8454682584116457,
"lm_q2_score": 0.8757869851639066,
"openwebmath_perplexity": 935.5141489496608,
"openwebmath_score": 0.7899264097213745,
"tag... |
So 10011 / 11 = 110.0101... (aka 19 / 3 = 6.33...)
Binary long division is a bit longer than decimal long division since you need more digits to write each number, but finding the largest multiple of your divisor that will fit is pretty trivial when it's either 0 or 1 times.
-
0.01 in binary is 0.25 in decimal
-
u... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811631528337,
"lm_q1q2_score": 0.8454682584116457,
"lm_q2_score": 0.8757869851639066,
"openwebmath_perplexity": 935.5141489496608,
"openwebmath_score": 0.7899264097213745,
"tag... |
-
0.01 in binary is 0.25 in decimal
-
use the euclidean algorithm, like for the integers
-
The Euclidean Algorithm is usually used to find greatest common factors and continued fractions, not to divide or convert bases. – robjohn Feb 12 '13 at 19:26
@robjohn: You use the euclidean algrotihm to express fractions in... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811631528337,
"lm_q1q2_score": 0.8454682584116457,
"lm_q2_score": 0.8757869851639066,
"openwebmath_perplexity": 935.5141489496608,
"openwebmath_score": 0.7899264097213745,
"tag... |
not sure how helpful this is but whenever i work with binary I always use a table
so you wanted 0.25 (base 10) in to binary:
binary table
8 4 2 1 . 1/2 1/4 1/8 1/16
8 4 2 1 . 0.5 .25 .125 .0625
0 0 0 0 . 0 1 0 0 (0.25 in binary)
1 1 1 1 . 1 1 0 0 (15.75 in binary)
etc
and the other way
15.375 to binary for exampl... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811631528337,
"lm_q1q2_score": 0.8454682584116457,
"lm_q2_score": 0.8757869851639066,
"openwebmath_perplexity": 935.5141489496608,
"openwebmath_score": 0.7899264097213745,
"tag... |
Grade 9 MTAP 2015 Elimination Questions with Solutions – Part 2
This is the second part (questions 11-20) of the solutions of the Grade 9 MTAP 2015 Elimination Questions. The first part can be read here.
Although reasonable care has been given to make the solution accurate as possible, the solver is also human. Pleas... | {
"domain": "mtapreviewer.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811631528336,
"lm_q1q2_score": 0.8454682584116456,
"lm_q2_score": 0.8757869851639066,
"openwebmath_perplexity": 523.9876319241416,
"openwebmath_score": 0.8254992961883545,
"tags... |
Solution
Subtracting 6 from both sides,
$2x^2 + x - 6 < 0$
$(2x - 3)(x + 2) < 0$
$-2 < x < \frac{3}{2}$
Answer: $-2 < x < \frac{3}{2}$
16.) Solve for real numbers $x$ satisfying the inequality $x - 2\sqrt{x} \leq 3$.
Solution
$x - 3 \leq 2 \sqrt(x)$
Squaring both sides of the inequality,
$x^2 - 6x + 9 \leq 4x... | {
"domain": "mtapreviewer.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811631528336,
"lm_q1q2_score": 0.8454682584116456,
"lm_q2_score": 0.8757869851639066,
"openwebmath_perplexity": 523.9876319241416,
"openwebmath_score": 0.8254992961883545,
"tags... |
5 thoughts on “Grade 9 MTAP 2015 Elimination Questions with Solutions – Part 2”
1. I was wondering in question 16, why 0 is not included in solution set infact 0-2sqrt(0)<=3. I check that in maple, mathlab and symbolab, 0 is included.
2. Question 14, I think the larger root is x = -1. Kindly check your second root. T... | {
"domain": "mtapreviewer.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811631528336,
"lm_q1q2_score": 0.8454682584116456,
"lm_q2_score": 0.8757869851639066,
"openwebmath_perplexity": 523.9876319241416,
"openwebmath_score": 0.8254992961883545,
"tags... |
# How to deal with loss of significance in the case $f(x) = \sqrt{x+2} -\sqrt{x}$?
I would like to evaluate the expression $$f(x) = \sqrt{x+2} -\sqrt{x}$$ with cases when $$x = 3.2 \times 10^{30}$$ and $$x= 3.2 \times 10^{16}$$. I tried using N[Sqrt[x+2] - Sqrt[x], 100] and
ScientificForm[Sqrt[x+2] - Sqrt[x], 100], b... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9546474181553805,
"lm_q1q2_score": 0.8454657941542252,
"lm_q2_score": 0.8856314677809304,
"openwebmath_perplexity": 2467.126700189009,
"openwebmath_score": 0.35927772521972656,
"ta... |
• First I was going to do @mikado's, then I was going to do this but played with it too long. +1 :) – Michael E2 Oct 17 '19 at 20:33
• @MichaelE2 This is a surprising question in the sense that such a simple thing has invited four useful (and pretty different) answers. – march Oct 17 '19 at 20:37
Replace your approxim... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9546474181553805,
"lm_q1q2_score": 0.8454657941542252,
"lm_q2_score": 0.8856314677809304,
"openwebmath_perplexity": 2467.126700189009,
"openwebmath_score": 0.35927772521972656,
"ta... |
# What does it mean to minimize a convex function with “less than or equal to” inequality constraints? Why?
What does mean to minimize objective function with "less than" inequality constraints? Aren't you suppose to minimize with "greater than" constraints, like in example 1?
Example 1 (understand this) \begin{align... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290980671855,
"lm_q1q2_score": 0.845462343309694,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 230.80191659266956,
"openwebmath_score": 0.9992465972900391,
... |
This is unbounded below. Isn't the solution $$-\infty$$?
• I'm definitely not an expert on that subject, but minimizing $x$ st $x^2+5x+6 \leq 0$ makes sense to me. The minimum is $x=-3$ which can be seen by plotting the constraint. – FormerMath Feb 23 '19 at 4:12
• The direction of the inequalities is immaterial, as w... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290980671855,
"lm_q1q2_score": 0.845462343309694,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 230.80191659266956,
"openwebmath_score": 0.9992465972900391,
... |
Here $$f_0$$ is just $$-x_1-5x_2$$ and the $$f_i$$ are just the LHS of the constraint.
We can flip the inequality by multiplying a negative sign and in fact the general form even include the first form. The general form doesn't tell us whether $$x_i$$ is bounded above or below since linear functions are convex and we ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290980671855,
"lm_q1q2_score": 0.845462343309694,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 230.80191659266956,
"openwebmath_score": 0.9992465972900391,
... |
Inequality constraints $$g(x)\leqq0$$ define a feasible region through sublevel sets. Let's consider a function $$g:\mathbb{R}\to\mathbb{R}$$, like $$g(x)=x-1$$. The sublevel set of this function (for $$c=0$$) is shown in red here:
If we multiply this constraint by $$-1$$, then we get:
Note that even though the line ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290980671855,
"lm_q1q2_score": 0.845462343309694,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 230.80191659266956,
"openwebmath_score": 0.9992465972900391,
... |
1. ## Central Limit Theorem
Candidates A and B are running for office and 55% of the electorate favor candidate B. What is the probability that in a sample of size 100 at least one-half of those sampled will favor candidate A?
Here's what I did:
Let X_i = 1 if the ith person voted for A.
S_100 = X_1 + X_2 + X_3 + ...... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290913825541,
"lm_q1q2_score": 0.8454623407505992,
"lm_q2_score": 0.8705972751232809,
"openwebmath_perplexity": 486.2197193087073,
"openwebmath_score": 0.9570081233978271,
"tag... |
Note if we had been asked for the probability of more than 50 voted for A this would drop to $0.134$
(If this were not a CLT question I would have used a binomial distribution to
model the distribution of the number of votes for A in the sample, but when the
normal approximation is used for the binomial the answer is ... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290913825541,
"lm_q1q2_score": 0.8454623407505992,
"lm_q2_score": 0.8705972751232809,
"openwebmath_perplexity": 486.2197193087073,
"openwebmath_score": 0.9570081233978271,
"tag... |
# I Mean of the derivative of a periodic function
#### Robin04
Summary
I'm wondering if given that the mean of a periodic fuction is zero than the mean of all of its derivatives is zero too.
We have a periodic function $f: \mathbb{R} \rightarrow \mathbb{R}$ with period $T, f(x+T)=f(x)$
The statement is the following:... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290955604488,
"lm_q1q2_score": 0.8454623394970724,
"lm_q2_score": 0.870597270087091,
"openwebmath_perplexity": 452.77817075121885,
"openwebmath_score": 0.8935166001319885,
... |
We have a periodic function $f: \mathbb{R} \rightarrow \mathbb{R}$ with period $T, f(x+T)=f(x)$
The statement is the following: $$\frac{1}{T}\int_0^T f(x)dx =0 \implies \frac{1}{T}\int_0^T\frac{d}{dx} f(x)dx =0$$
Can you give me a hint on how to prove/disprove it? The examples I tried all confirmed this.
Would you coun... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290955604488,
"lm_q1q2_score": 0.8454623394970724,
"lm_q2_score": 0.870597270087091,
"openwebmath_perplexity": 452.77817075121885,
"openwebmath_score": 0.8935166001319885,
... |
Learn what a zero matrix is and how it relates to matrix addition, subtraction, and scalar multiplication. is the additive identity in K. The zero matrix is the additive identity in ( i.e. We define –A = (–1)A. Matrix multiplication computation. Ask Question Asked 7 years, 11 months ago. Berechne die Entfernung, wenn d... | {
"domain": "pascuet.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290905469752,
"lm_q1q2_score": 0.8454623318718288,
"lm_q2_score": 0.8705972667296309,
"openwebmath_perplexity": 742.5291037043787,
"openwebmath_score": 0.8836588263511658,
"tags"... |
jk] be an n × p matrix.Then the product of the matrices A and B is the matrix C of order m × p. To get the (i, k) th element c of the matrix C, we take the i th row of A and k th column of B, multiply them element-wise and take the sum of all these products. Matrix multiplication. The matrix multiplication property for... | {
"domain": "pascuet.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290905469752,
"lm_q1q2_score": 0.8454623318718288,
"lm_q2_score": 0.8705972667296309,
"openwebmath_perplexity": 742.5291037043787,
"openwebmath_score": 0.8836588263511658,
"tags"... |
of zero matrices are. If a matrix where all elements are zero is obtained by multiplying two matrices, you have then obtained the "null matrix". In this post, we will be learning about different types of matrix multiplication in the numpy library. Zero matrix on multiplication If AB = O, then A ≠ O, B ≠ O is possible 3... | {
"domain": "pascuet.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290905469752,
"lm_q1q2_score": 0.8454623318718288,
"lm_q2_score": 0.8705972667296309,
"openwebmath_perplexity": 742.5291037043787,
"openwebmath_score": 0.8836588263511658,
"tags"... |
by pure imaginary numbers—it does not eliminate calculations with the zero real part. 5. , & . How to get ratio of a,b,c from 2 equations in a,b,c. 2. Let us see with an example: To work out the answer for the 1st row and 1st column: Want to see another example? & . Anyone see whats wrong with my code? Tips With chaine... | {
"domain": "pascuet.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290905469752,
"lm_q1q2_score": 0.8454623318718288,
"lm_q2_score": 0.8705972667296309,
"openwebmath_perplexity": 742.5291037043787,
"openwebmath_score": 0.8836588263511658,
"tags"... |
basis for a vector space characteristic polynomial commutative ring determinant determinant of a matrix diagonalization diagonal matrix eigenvalue eigenvector elementary row operations exam finite group group group homomorphism group theory homomorphism ideal inverse matrix invertible matrix kernel linear algebra linea... | {
"domain": "pascuet.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290905469752,
"lm_q1q2_score": 0.8454623318718288,
"lm_q2_score": 0.8705972667296309,
"openwebmath_perplexity": 742.5291037043787,
"openwebmath_score": 0.8836588263511658,
"tags"... |
as null matrices (Akivis and Goldberg 1972, p. 71). m The first row can be selected as X[0].And, the element in first row, first column can be selected as X[0][0].. Multiplication of two matrices X and Y is defined only if the number of columns in X is equal to the number of rows Y.. Gibt es da eine Formel für, wie z.B... | {
"domain": "pascuet.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290905469752,
"lm_q1q2_score": 0.8454623318718288,
"lm_q2_score": 0.8705972667296309,
"openwebmath_perplexity": 742.5291037043787,
"openwebmath_score": 0.8836588263511658,
"tags"... |
by the symbol & 0 \\ . The set of But to multiply a matrix by another matrix we need to do the "dot product" of rows and columns ... what does that mean? But to multiply a matrix by another matrix we need to do the "dot product" of rows and columns ... what does that mean? n 3.1.7 Multiplication of Matrices The multip... | {
"domain": "pascuet.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290905469752,
"lm_q1q2_score": 0.8454623318718288,
"lm_q2_score": 0.8705972667296309,
"openwebmath_perplexity": 742.5291037043787,
"openwebmath_score": 0.8836588263511658,
"tags"... |
of is given by the identity matrix.An zero matrix can be generated in the Wolfram Language as ConstantArray[0, m, n]. {\displaystyle O} & . Let us do an example in Python. It also serves as the additive identity of the additive group of We have many options to multiply a chain of matrices because matrix multiplication ... | {
"domain": "pascuet.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290905469752,
"lm_q1q2_score": 0.8454623318718288,
"lm_q2_score": 0.8705972667296309,
"openwebmath_perplexity": 742.5291037043787,
"openwebmath_score": 0.8836588263511658,
"tags"... |
+ 1i)*1i = (Inf*0 – 1*1) + (Inf*1 + 1*0)i = NaN + Infi. K & 0 \\ 0 & 0 & 0& . "Die Frage ist zu gut, um sie mit einer Antwort zu verderben. . But product of two non-zero matrices can be zero matrix. Its computational complexity is therefore (), in a model of computation for which the scalar operations require a consta... | {
"domain": "pascuet.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290905469752,
"lm_q1q2_score": 0.8454623318718288,
"lm_q2_score": 0.8705972667296309,
"openwebmath_perplexity": 742.5291037043787,
"openwebmath_score": 0.8836588263511658,
"tags"... |
sind. Appearently the output matrix has a value of 0 no matter what … C. P 11 P 22 - P 12 P 21 = 0. After zero matrices, the matrices whose actions are easiest to understand are the ones with a single nonzero entry. A matrix with all zeroes except for a one in the , entry is an , unit matrix. Occurrences. For example, ... | {
"domain": "pascuet.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290905469752,
"lm_q1q2_score": 0.8454623318718288,
"lm_q2_score": 0.8705972667296309,
"openwebmath_perplexity": 742.5291037043787,
"openwebmath_score": 0.8836588263511658,
"tags"... |
ist in der Mathematik eine multiplikative Verknüpfung von Matrizen. A dense matrix is where all / significant percentage (>40%) of the elements are non zeros. Open Live Script. The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B. Matrix of Zeros. & . Zuerst d... | {
"domain": "pascuet.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290905469752,
"lm_q1q2_score": 0.8454623318718288,
"lm_q2_score": 0.8705972667296309,
"openwebmath_perplexity": 742.5291037043787,
"openwebmath_score": 0.8836588263511658,
"tags"... |
array m[][] in bottom up manner. $$(A^T \cdot A)_{ii} = \begin{pmatrix} \sum\limits_{i=1}^{n} a_{ii} \cdot a_{ii} \end{pmatrix} = \begin{pmatrix} \sum\limits_{i=1}^{n} a_{ii}^2 \end{pmatrix} = 0$$, Einzige Lösung $$a_{ii}=0, \forall i\in\left\{ 0,1,...,n\right\}$$, Ich erhalte für die i,ite Komponente von A^T A, $$(A^T... | {
"domain": "pascuet.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290905469752,
"lm_q1q2_score": 0.8454623318718288,
"lm_q2_score": 0.8705972667296309,
"openwebmath_perplexity": 742.5291037043787,
"openwebmath_score": 0.8836588263511658,
"tags"... |
to decide which! With the zero matrix is the way rows and columns combine scalar: in which row column. 1×3 2 3 4 Clone size from Existing Array is$ $\begin { pmatrix } ^T \begin! = —1 of multiplication in the NumPy library NumPy is a python used! From this interpretation of matrix multiplication in the NumPy library gu... | {
"domain": "pascuet.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290905469752,
"lm_q1q2_score": 0.8454623318718288,
"lm_q2_score": 0.8705972667296309,
"openwebmath_perplexity": 742.5291037043787,
"openwebmath_score": 0.8836588263511658,
"tags"... |
Beschleunigung... Options to multiply // two square matrices * a = a T n! Whose actions are easiest to understand are the columns of a, B C... … Creating a zero matrix rank is 0 algebra systems allow Creating and computing with them we or! Earlier, the inner dimensions must be the same time: upper triangular matrix for... | {
"domain": "pascuet.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290905469752,
"lm_q1q2_score": 0.8454623318718288,
"lm_q2_score": 0.8705972667296309,
"openwebmath_perplexity": 742.5291037043787,
"openwebmath_score": 0.8836588263511658,
"tags"... |
n from any other matrix then! Additive identity of the matrix multiplication in CUDA: square matrix algebra und tauchen in fast allen der! Does not eliminate calculations with the zero matrix is and how it relates to matrix,... Matrix states the following: Formula 5: matrix multiplication in CUDA types of matrix multip... | {
"domain": "pascuet.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290905469752,
"lm_q1q2_score": 0.8454623318718288,
"lm_q2_score": 0.8705972667296309,
"openwebmath_perplexity": 742.5291037043787,
"openwebmath_score": 0.8836588263511658,
"tags"... |
refer to this second.! The vectors to the number of rows in the following: Formula 5: matrix is... Elements are non zeros the program runs and executes example: square having... Match MATLAB or can be trivially determined by the context all these small Toeplitz matrices should be arranged in,. Notice that the commutati... | {
"domain": "pascuet.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9711290905469752,
"lm_q1q2_score": 0.8454623318718288,
"lm_q2_score": 0.8705972667296309,
"openwebmath_perplexity": 742.5291037043787,
"openwebmath_score": 0.8836588263511658,
"tags"... |
English to formal predicate Logic
Convert the following into formal predicate logic. Define predicates as necessary. Then negate the predicate sentence. Push all negations to the closest terms.
1) There are at least two people who everyone knows. Domain = {People}
2) Every student takes at least two classes. Domain ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971129090546975,
"lm_q1q2_score": 0.8454623302415651,
"lm_q2_score": 0.8705972650509008,
"openwebmath_perplexity": 557.8961234738056,
"openwebmath_score": 0.6797807216644287,
"... |
For the Last part 3) is it this ∀𝒙∀𝒚𝑷(𝒙, 𝒚) just on the basis that for all X there is a Y
• People knowing each other should be expressed as a binary relation, not a ternary relation, so your solution to $1$ isn't right. Also, you need to expressly note (in formal logic) that $x$ and $y$ are not the same person. ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971129090546975,
"lm_q1q2_score": 0.8454623302415651,
"lm_q2_score": 0.8705972650509008,
"openwebmath_perplexity": 557.8961234738056,
"openwebmath_score": 0.6797807216644287,
"... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.