text stringlengths 1 2.12k | source dict |
|---|---|
the Math Dude podcast on Quick and Dirty Tips. I used to get ideas from here. n!/(n-r)!r! One of the best known features of Pascal's Triangle is derived from the combinatorics identity . This website is so useful!!! Thanks. After that it has been studied by many scholars throughout the world. 204 and 242).Here's how it... | {
"domain": "beerlak.hu",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688145,
"lm_q1q2_score": 0.8421477261723621,
"lm_q2_score": 0.8723473813156294,
"openwebmath_perplexity": 541.3227765729619,
"openwebmath_score": 0.5620970129966736,
"tags": nul... |
a triangular array of the binomial coefficients. Struggling Ravens player: 'My family is off limits' McConaughey responds to Hudson's kissing insult All values outside the triangle are considered zero (0). Pascal Triangle in Java at the Center of the Screen. Following are the first 6 rows of Pascal’s Triangle. Discover... | {
"domain": "beerlak.hu",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688145,
"lm_q1q2_score": 0.8421477261723621,
"lm_q2_score": 0.8723473813156294,
"openwebmath_perplexity": 541.3227765729619,
"openwebmath_score": 0.5620970129966736,
"tags": nul... |
Pascal’s triangle, each number is the sum of the two numbers directly above it. Pascal's triangle. The Pascal's Triangle was first suggested by the French mathematician Blaise Pascal, in the 17 th century. It is named after the 1 7 th 17^\text{th} 1 7 th century French mathematician, Blaise Pascal (1623 - 1662). 3 hour... | {
"domain": "beerlak.hu",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688145,
"lm_q1q2_score": 0.8421477261723621,
"lm_q2_score": 0.8723473813156294,
"openwebmath_perplexity": 541.3227765729619,
"openwebmath_score": 0.5620970129966736,
"tags": nul... |
easily calculate probabilities when tossing coins, it’s time to dig a bit deeper and investigate the properties of the triangle itself. Pascal's Triangle, named after French mathematician Blaise Pascal, is used in various algebraic processes, such as finding tetrahedral and triangular numbers, powers of two, exponents ... | {
"domain": "beerlak.hu",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688145,
"lm_q1q2_score": 0.8421477261723621,
"lm_q2_score": 0.8723473813156294,
"openwebmath_perplexity": 541.3227765729619,
"openwebmath_score": 0.5620970129966736,
"tags": nul... |
of 2 (i.e., 1, 2, 4, 8, 16, etc.) He had used Pascal's Triangle in the study of probability theory. One color each for Alice, Bob, and Carol: A cas… Scientific American and Quick & Dirty Tips are both Macmillan companies. World finally discovers one thing 'the Rock' can't do. Step 1: Draw a short, vertical line and wri... | {
"domain": "beerlak.hu",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688145,
"lm_q1q2_score": 0.8421477261723621,
"lm_q2_score": 0.8723473813156294,
"openwebmath_perplexity": 541.3227765729619,
"openwebmath_score": 0.5620970129966736,
"tags": nul... |
specifically binomials, it is used for expanding binomials. Golden Ratio, Phi and Fibonacci Commemorative Postage Stamps, The Golden Ratio in Character Design, Cartoons and Caricatures, Golden ratios in Great Pyramid of Giza site topography, Michelangelo and the Art of the Golden Ratio in Design and Composition, Google... | {
"domain": "beerlak.hu",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688145,
"lm_q1q2_score": 0.8421477261723621,
"lm_q2_score": 0.8723473813156294,
"openwebmath_perplexity": 541.3227765729619,
"openwebmath_score": 0.5620970129966736,
"tags": nul... |
infinite. Pascal Triangle is named after French mathematician Blaise Pascal. Pascal’s Triangle Last updated; Save as PDF Page ID 14971; Contributors and Attributions; The Pascal’s triangle is a graphical device used to predict the ratio of heights of lines in a split NMR peak. Hi, just wondering what the general expres... | {
"domain": "beerlak.hu",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688145,
"lm_q1q2_score": 0.8421477261723621,
"lm_q2_score": 0.8723473813156294,
"openwebmath_perplexity": 541.3227765729619,
"openwebmath_score": 0.5620970129966736,
"tags": nul... |
Do not count the 1’s. Donald Duck visits the Parthenon in “Mathmagic Land”, “The Golden Ratio” book – Author interview with Gary B. Meisner on New Books in Architecture. Which meant that soon after publishing his 1653 book on the subject, “Pascal’s triangle” was born! 260. 30 seconds . Pascal's Triangle or Khayyam Tria... | {
"domain": "beerlak.hu",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688145,
"lm_q1q2_score": 0.8421477261723621,
"lm_q2_score": 0.8723473813156294,
"openwebmath_perplexity": 541.3227765729619,
"openwebmath_score": 0.5620970129966736,
"tags": nul... |
Is $\exp(x)$ the same as $e^x$?
For homework I have to find the derivative of $\text {exp}(6x^5+4x^3)$ but I am not sure if this is equivalent to $e^{6x^5+4x^3}$ If there is a difference, what do I do to calculate the derivative of it?
• There is no difference; they are alternative notations for the same thing. Apr 8... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688145,
"lm_q1q2_score": 0.8421477245704521,
"lm_q2_score": 0.8723473796562744,
"openwebmath_perplexity": 414.6225045550749,
"openwebmath_score": 0.8039519786834717,
"tag... |
• Agreed, they are two different ways of looking at the same thing... one way $e^x$ kind of the "mathy" way to view it, and $\text{exp}(x)$ is more the "programatical" way to view it. Apr 8, 2015 at 21:39
• @TravisJ It has nothing to do with programming and everything to do with the clarity of written mathematics, as t... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688145,
"lm_q1q2_score": 0.8421477245704521,
"lm_q2_score": 0.8723473796562744,
"openwebmath_perplexity": 414.6225045550749,
"openwebmath_score": 0.8039519786834717,
"tag... |
$$\prod_i e^{x_i} = \exp \sum_i x_i$$
• Allow me to add to your last line that most physicists do not come across situations where $e$ might be ambiguous and therefore they use $e$ for exponentials (except when the argument gets too long). Apr 9, 2015 at 11:25
• Yes, I'm with @Hrodelbert: the statement that "exp" is p... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688145,
"lm_q1q2_score": 0.8421477245704521,
"lm_q2_score": 0.8723473796562744,
"openwebmath_perplexity": 414.6225045550749,
"openwebmath_score": 0.8039519786834717,
"tag... |
In manifold theory (most particularly Lie Group theory or Riemannian geometry), the exponential map $\exp$ is a map from a tangent space to the manifold itself. For Lie groups, it expresses the local group structure and allows to lift many problems from the group to the tangent space (the Lie algebra). It also defines ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688145,
"lm_q1q2_score": 0.8421477245704521,
"lm_q2_score": 0.8723473796562744,
"openwebmath_perplexity": 414.6225045550749,
"openwebmath_score": 0.8039519786834717,
"tag... |
• I am a bit surprised be the final sentence. Can you give an example of situation where $e^t$ is defined but different from $\exp(t)$? Apr 9, 2015 at 13:45
• @MarcvanLeeuwen No I cannot. I guess what I remembered is that for disconnected Lie groups, $\exp$ does not cover the entire group, implying that not all group e... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688145,
"lm_q1q2_score": 0.8421477245704521,
"lm_q2_score": 0.8723473796562744,
"openwebmath_perplexity": 414.6225045550749,
"openwebmath_score": 0.8039519786834717,
"tag... |
Exponentiation can be replaced by using $\exp$ and $\ln$ together via $a^b=\exp(b\ln a)$, and the ambiguities arise from the $\ln$ part of the replacement. So it can be expedient to work with just $\exp$ when the task does not require anything else. Informally, $e^x$ is used equivalently to $\exp(x)$ anyway but the lat... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688145,
"lm_q1q2_score": 0.8421477245704521,
"lm_q2_score": 0.8723473796562744,
"openwebmath_perplexity": 414.6225045550749,
"openwebmath_score": 0.8039519786834717,
"tag... |
• Why would $a^z$ be multivalued? If $a>0$, we define $a^z = e^{z \cdot \log a}$. Apr 9, 2015 at 20:28
• What if $a$ isn't real? @goblin To write $x^y$ in general, you require multivalued functions or some horrible branch cut. Apr 9, 2015 at 20:29
• Being real isn't strong enough; we want $a \in \mathbb{R}_{>0}$. Other... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9653811591688145,
"lm_q1q2_score": 0.8421477245704521,
"lm_q2_score": 0.8723473796562744,
"openwebmath_perplexity": 414.6225045550749,
"openwebmath_score": 0.8039519786834717,
"tag... |
# Calculating overlapped area between random circles
Suppose I am plotting random circles like in the following example: m=125;
g10=Graphics[{Table[Circle[pt[i],r],{i,m}]},Axes-
>True,PlotRange->{{-5000,5000},{-5000,5000}}]
Some of the circles may overlap. How can I calculate the whole overlapped area (Integrate it)... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9579122708828602,
"lm_q1q2_score": 0.8421454674048952,
"lm_q2_score": 0.8791467580102418,
"openwebmath_perplexity": 3119.0007980635746,
"openwebmath_score": 0.5219480395317078,
"ta... |
• great answer +1. It was interesting to see how this question could be interpreted in different ways, depending on the r value. – Rashid May 21 '16 at 12:46
• @Rashid, thank you for the upvote. I chose the lazier/easier way out in interpreting the question :) – kglr May 21 '16 at 12:51
@kglr beat me to posting and ha... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9579122708828602,
"lm_q1q2_score": 0.8421454674048952,
"lm_q2_score": 0.8791467580102418,
"openwebmath_perplexity": 3119.0007980635746,
"openwebmath_score": 0.5219480395317078,
"ta... |
EDIT Here is a Module form that runs in 6-10 seconds on my laptop for m=125 and r=100:
overlapAreaModule[shapes_] :=
Module[{totalArea, shapePairs, overlappingArea, areaRatio},
totalArea = Total[Map[Area, shapes]];
shapePairs =
Map[RegionIntersection, Subsets[shapes, {2}]] /.
EmptyRegion[2] -> Nothing;
overlappingArea... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9579122708828602,
"lm_q1q2_score": 0.8421454674048952,
"lm_q2_score": 0.8791467580102418,
"openwebmath_perplexity": 3119.0007980635746,
"openwebmath_score": 0.5219480395317078,
"ta... |
2,238 views
Three dice are rolled independently. What is the probability that the highest and the lowest value differ by $4$?
1. $\left(\dfrac{1}{3}\right)$
2. $\left(\dfrac{1}{6}\right)$
3. $\left(\dfrac{1}{9}\right)$
4. $\left(\dfrac{5}{18}\right)$
5. $\left(\dfrac{2}{9}\right)$
2/9 is correct.
Its a beautiful qu... | {
"domain": "gateoverflow.in",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357259231532,
"lm_q1q2_score": 0.8421015967957214,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 1248.0759004312215,
"openwebmath_score": 0.5599600672721863,
"tags"... |
suppose three dices are there A,B,C, three are rolled independently, see we get 4 difference between max and min when we get either (1,5) or (2,6) in any of the two dices. okay?
1st case: we get (1,5), 1 and 5 can be the result of any of the two dice in 3C2=3 ways, either (A-B) or (B-C) and (C-A) and also they can be ... | {
"domain": "gateoverflow.in",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357259231532,
"lm_q1q2_score": 0.8421015967957214,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 1248.0759004312215,
"openwebmath_score": 0.5599600672721863,
"tags"... |
Case 2: Minimum value on the dice is 2 and the maximum value is 6
Now when 2 and 6 have appeared the third number can be anyone from $\{2,3,4,5,6\}$ so that the difference between the minimum and the maximum number remains 4.
Now, this case is symmetrical to case 1 where there are two possibilities when the number on... | {
"domain": "gateoverflow.in",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357259231532,
"lm_q1q2_score": 0.8421015967957214,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 1248.0759004312215,
"openwebmath_score": 0.5599600672721863,
"tags"... |
Fourier theorems under various conventions
There are several slightly different ways to define a Fourier transform. This means that when you look up a theorem about the Fourier transform you have to ask yourself which convention the source is using. All the common conventions can be summarized in the following definit... | {
"domain": "johndcook.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357221825193,
"lm_q1q2_score": 0.8421015953718922,
"lm_q2_score": 0.8577681104440172,
"openwebmath_perplexity": 541.1312549454769,
"openwebmath_score": 0.8439435362815857,
"tags": ... |
Converting between definitions
For a function f(x), let F(f)(ω) be its Fourier transform. You can convert between the eight possible definitions by applying three equations. Here a * stands for any particular choice of a parameter, as long as the same choice is applied on both sides of the equation:
Another way to co... | {
"domain": "johndcook.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357221825193,
"lm_q1q2_score": 0.8421015953718922,
"lm_q2_score": 0.8577681104440172,
"openwebmath_perplexity": 541.1312549454769,
"openwebmath_score": 0.8439435362815857,
"tags": ... |
Plancherel
Plancherel’s formula is Parseval’s formula with g = f. This says a function and its Fourier transform have the same L2 form for definitions F+τ1, F-τ1, F+1τ, and F-1τ. For definitions F+11 and F-11 the norm of the Fourier transforms is larger by a factor of √2π. For definitions F+ττ and F-ττ the inner produ... | {
"domain": "johndcook.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357221825193,
"lm_q1q2_score": 0.8421015953718922,
"lm_q2_score": 0.8577681104440172,
"openwebmath_perplexity": 541.1312549454769,
"openwebmath_score": 0.8439435362815857,
"tags": ... |
# Why can linear independence of $e^{ax +by^2}$ be proven without considering $y$?
Linear independence of $$e^{at}$$ has been answered multiple times. My favorite one is by Marc van Leeuwen in this one: Proof of linear independence of $e^{at}$. The answer uses the property that $$e^{at}$$ are eigenfunctions of the dif... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357259231532,
"lm_q1q2_score": 0.842101595010954,
"lm_q2_score": 0.8577681068080748,
"openwebmath_perplexity": 305.62483752007046,
"openwebmath_score": 0.9506980776786804,
"tag... |
edit: assume all $$a_k$$ and $$b_k$$ are distinct.
• You have to take in consideration that the linear independence you present, is towards variable $x$, judging from the operator you are using. A 2 variable function $f(x,y)$ would need a 2-dimension operator. So what your assumption is that $e^{ax+bx^2}$ is linear to... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357259231532,
"lm_q1q2_score": 0.842101595010954,
"lm_q2_score": 0.8577681068080748,
"openwebmath_perplexity": 305.62483752007046,
"openwebmath_score": 0.9506980776786804,
"tag... |
The proof is straight forward: If $$c_1, \ldots, c_n \in \Bbb R$$ with $$\sum_{j=1}^n c_j g_j(x) h_j(y) = 0 \text{ for all } (x, y) \in A \times B$$ then in particular $$\sum_{j=1}^n \bigl( c_j h_j(y_0) \bigr) g_j(x) = 0 \text{ for all } x \in A$$ Since the $$g_j$$ are linearly independent it follows that $$c_j h_j(y_0... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357259231532,
"lm_q1q2_score": 0.842101595010954,
"lm_q2_score": 0.8577681068080748,
"openwebmath_perplexity": 305.62483752007046,
"openwebmath_score": 0.9506980776786804,
"tag... |
# Reducing fractions?
I want to reduce the two following fractions:
$$\frac{2x + 2y}{x + y}$$
$$\frac{3ab^2}{12ab}$$
I fully understand the concept of reduce fractions of this type:
$$\frac{15}{20}$$
but i do not know what steps to take for reducing fractions like the two above. Anyone that can explain the steps ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357205793903,
"lm_q1q2_score": 0.8421015939967793,
"lm_q2_score": 0.8577681104440172,
"openwebmath_perplexity": 834.4442369894616,
"openwebmath_score": 0.9999967813491821,
"tag... |
# Finding equation of parabola when focus and equations of two perpendicular tangents from any two points on the parabola are given
If the focus of a parabola and the equations of two perpendicular tangents at any two points $P$ and $Q$ on the parabola are given, can we find the equation of the given parabola?
If not... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.981735721648143,
"lm_q1q2_score": 0.842101593128754,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 206.86105324240975,
"openwebmath_score": 0.708640456199646,
"tags":... |
• The reason could be that the vertex is the midpoint of focus and directrix and since you are taking a reflection, slop of the tangent really doesn't matter. So if you consider a particular case of a tangent with slope equal to - 1/(slope of axis), that is tangent at vertex, you obviously get a point on the directrix.... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.981735721648143,
"lm_q1q2_score": 0.842101593128754,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 206.86105324240975,
"openwebmath_score": 0.708640456199646,
"tags":... |
# Evaluate $\int 7\tan^5x\sec^2 x\,dx$
How do you evaluate this trigonometric integral: $$\int 7\tan^5x\sec^2 x\,dx$$? Please help. Thank you in advance for your help.
• Put $u = \tan x$ – Sandeep Thilakan Feb 14 '14 at 5:39
Given $\int 7\tan ^5 x \sec ^2 x dx$, let $\tan x=t$. Then $\sec ^2 x dx=dt$. Hence we must ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357248544007,
"lm_q1q2_score": 0.8421015905246779,
"lm_q2_score": 0.8577681031721325,
"openwebmath_perplexity": 379.0040046264015,
"openwebmath_score": 0.9302329421043396,
"tag... |
[At this point, we have a perfectly acceptable "polynomial in secants"; but we should show that this in fact is equivalent to a simpler expression.]
$$= \ 7 \ (\tan^2 x + 1) \ [ \ \frac{1}{6} (\tan^2 x + 1)^2 \ - \ \frac{1}{2} (\tan^2 x + 1) \ + \ \frac{1}{2} \ ] \ + \ C$$
$$\ = \ 7 \ (\tan^2 x + 1) \ [ \ \frac{1}{6}... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357248544007,
"lm_q1q2_score": 0.8421015905246779,
"lm_q2_score": 0.8577681031721325,
"openwebmath_perplexity": 379.0040046264015,
"openwebmath_score": 0.9302329421043396,
"tag... |
# Interesting pattern within $m^n+1\equiv0\pmod n$
In recent days, I have been studying the properties of $$m^n+h\equiv0\pmod n$$ where $$m,n\in\mathbb{N}$$ and $$h\in\mathbb{Z}$$, and I have noticed that for the equation $$m^n+1\equiv0\pmod n$$, some even numbers n have solutions and some don't.(If $$n$$ is odd then ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357248544007,
"lm_q1q2_score": 0.8421015869551431,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 75.53636669791322,
"openwebmath_score": 0.9069632291793823,
"tag... |
Nice observation! Something else you might notice, which turns out to imply your observation, is that all of the odd prime factors of your numbers are $$1 \bmod 4$$: $$\{ 5, 13, 17, 29, \dots \}$$. And a final thing you might notice is that all of your numbers are themselves congruent to $$2 \bmod 4$$, or equivalently ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357248544007,
"lm_q1q2_score": 0.8421015869551431,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 75.53636669791322,
"openwebmath_score": 0.9069632291793823,
"tag... |
$$(x^2 + y^2)(z^2 + w^2) = (xz - yw)^2 + (yz + xw)^2$$
(which again admits several proofs) shows that a product of numbers of the form $$x^2 + y^2$$ is again of the form $$x^2 + y^2$$. To show that we can always arrange for $$\gcd(x, y) = 1$$ is slightly more annoying but still doable. If the $$\gcd$$ isn't equal to $... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357248544007,
"lm_q1q2_score": 0.8421015869551431,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 75.53636669791322,
"openwebmath_score": 0.9069632291793823,
"tag... |
$$m^{2 p^k q} \equiv -1 \bmod p^k.$$
To do this recall that as above, since $$p \equiv 1 \bmod 4$$ we know that there exists a solution to $$x^2 \equiv -1 \bmod p$$. By Hensel's lemma this solution lifts to a solution to $$x^2 \equiv -1 \bmod p^k$$. Call it $$i$$ (since it's a primitive $$4^{th}$$ root of unity). Then... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357248544007,
"lm_q1q2_score": 0.8421015869551431,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 75.53636669791322,
"openwebmath_score": 0.9069632291793823,
"tag... |
# How to understand the concept of norm equivalence?
I'm mainly dealing with $\mathbb{R}^n$.
$\nu(\cdot)$ and $\mu(\cdot)$ are equivalent iff
there exist constants $c_1,c_2>0$ such that for every $x \in \mathbb{R}^n$, $c_1\nu(x)\leq \mu(x)\leq c_2\nu(x)$.
I understand the definition. What I don't understand is why ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357179075082,
"lm_q1q2_score": 0.8421015863506224,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 134.6929996733618,
"openwebmath_score": 0.9408715963363647,
"tag... |
• Note that two norms are equivalent if there exist constants c1, c2 > 0 such that for every x... not the other way around. Oct 24 '16 at 10:36
• Also, what do you mean by "property"? Can you make an example of norm-dependent properties that fit your question? Oct 24 '16 at 10:38
• Notice that when a vector space is eq... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357179075082,
"lm_q1q2_score": 0.8421015863506224,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 134.6929996733618,
"openwebmath_score": 0.9408715963363647,
"tag... |
• ... and since the topology in the present case determines which sequences are convergent one gets: a sequence is convergent with respect to $\nu$ if and only if it is convergent with respect to $\mu$. The same statement holds for the convergence of series or for continuity of functions. Oct 24 '16 at 11:07
• @HagenKn... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357179075082,
"lm_q1q2_score": 0.8421015863506224,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 134.6929996733618,
"openwebmath_score": 0.9408715963363647,
"tag... |
You want to prove a similar statement for $\|\cdot\|_2$.
Thus, given $\epsilon >0$, by the equivalence of norms, you can find $r>0$ s.t. $rB^1_{\epsilon}=\{r\cdot x\mid x\in B^1_{\epsilon}\}\subseteq B^2_{\epsilon}$ (we're just rescaling the ball). If $n$ is great enough, you have that $x_n\in rB_{\epsilon}^1$, thus a... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357179075082,
"lm_q1q2_score": 0.8421015863506224,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 134.6929996733618,
"openwebmath_score": 0.9408715963363647,
"tag... |
(Oliver Heaviside, quoted by Kline) In this chapter, we apply our results for sequences to series, or in nite sums. ANALYSIS I 13 Power Series 13. Series may diverge by marching off to infinity or by oscillating. Then the series converges when (c) Give an example of a. 01 Single Variable Calculus, Fall 2005 Prof. These... | {
"domain": "konoozargan.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357227168957,
"lm_q1q2_score": 0.8421015851216596,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 710.5052515824704,
"openwebmath_score": 0.8989253640174866,
"ta... |
without bound. The barrier between convergence and divergence is in the middle of the -series::" " " " " " " " "8 8x $# 8 8 8 8 ¥ ¥ â ¥ ¥ ¥ â ¥ ¥ ¥ â ¥ ¥ ¥ â ¥ 8 8 8 # "Þ" È8 ln convergent divergent ». Example Use the integral test to determine whether the series is. -1 and 1 are called cluster points of the sequence, ... | {
"domain": "konoozargan.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357227168957,
"lm_q1q2_score": 0.8421015851216596,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 710.5052515824704,
"openwebmath_score": 0.8989253640174866,
"ta... |
sum is called convergent series. Example The sequence a_n= (-1)^n is not convergent. This is a geometric series with ratio, r = 4/5, which is less than 1. Jason Starr. If the sequence of partial sums is a convergent sequence (i. transform D. ( 7) Alternating series test ( A. Determine whether the series is convergent o... | {
"domain": "konoozargan.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357227168957,
"lm_q1q2_score": 0.8421015851216596,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 710.5052515824704,
"openwebmath_score": 0.8989253640174866,
"ta... |
diverges. Relevant theorems, such as the Bolzano-Weierstrass theorem, will. 4 Sequence and Series of Real Numbers M. Absolute Convergence and the Ratio and Root Tests Example 2 shows that the alternating harmonic series is conditionally convergent. By inspection, it can be difficult to see whether a series will converg... | {
"domain": "konoozargan.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357227168957,
"lm_q1q2_score": 0.8421015851216596,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 710.5052515824704,
"openwebmath_score": 0.8989253640174866,
"ta... |
termscancel out previousterms in the sum? May have to use partial fractions, properties of logarithms, etc. Definition, using the sequence of partial sums and the sequence of partial absolute sums. Also the series X1 n=1 1 n1=2 diverges. Key Concepts The in nite series X1 k=0 a k converges if the sequence of partial su... | {
"domain": "konoozargan.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357227168957,
"lm_q1q2_score": 0.8421015851216596,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 710.5052515824704,
"openwebmath_score": 0.8989253640174866,
"ta... |
sequence (i. When the terms of the series live in ℝ n , an equivalent condition for absolute convergence of the series is that all possible series obtained by rearrangements of the terms are also convergent. Geometric Series: THIS is our model series A geometric series converges for. It requires us to learn things in o... | {
"domain": "konoozargan.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357227168957,
"lm_q1q2_score": 0.8421015851216596,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 710.5052515824704,
"openwebmath_score": 0.8989253640174866,
"ta... |
OF SEQUENCES BECKY LYTLE Abstract. Alternating sequences change the signs of its terms. The barrier between convergence and divergence is in the middle of the -series::" " " " " " " " "8 8x$ # 8 8 8 8 ¥ ¥ â ¥ ¥ ¥ â ¥ ¥ ¥ â ¥ ¥ ¥ â ¥ 8 8 8 # "Þ" È8 ln convergent divergent ». If and then Theorem 2. But if these huge mass... | {
"domain": "konoozargan.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357227168957,
"lm_q1q2_score": 0.8421015851216596,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 710.5052515824704,
"openwebmath_score": 0.8989253640174866,
"ta... |
translation, English dictionary definition of convergent. a sequence that is convergent only has one cluster point, in. If, for increasing values of n, the sum approaches indefinitely a certain limit s, the series will be called convergent, and the limit in question will be called the sum of the series. Definition Impro... | {
"domain": "konoozargan.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357227168957,
"lm_q1q2_score": 0.8421015851216596,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 710.5052515824704,
"openwebmath_score": 0.8989253640174866,
"ta... |
Leibniz formula for pi):. For example, the sequence 1, 2, 3, 4, 5, 6, 7, diverges since its limit is infinity (∞). Convergent thinking refers to the ability to put a number of different pieces from different perspectives of a topic together in some organized, logical manner to find a single answer. (a) ∑a Divergent n: ... | {
"domain": "konoozargan.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357227168957,
"lm_q1q2_score": 0.8421015851216596,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 710.5052515824704,
"openwebmath_score": 0.8989253640174866,
"ta... |
time after this, divergent series were mostly excluded from mathematics. How To Reconstruct Linked, Convergent and Serial Arguments with Argunet by Gregor Betz , Wednesday, June 12th, 2013 Linked, convergent and serial argumentation are basic notions of argument structure in Critical Thinking and Informal Logic. 258 Ch... | {
"domain": "konoozargan.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357227168957,
"lm_q1q2_score": 0.8421015851216596,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 710.5052515824704,
"openwebmath_score": 0.8989253640174866,
"ta... |
sn as a telescoping sum (as in Example 7). We can use convergence to describe things that are in the process of coming together,. 2 Tests for Convergence Let us determine the convergence or the divergence of a series by comparing it to one whose behavior is already known. Such a finite value is called a regularized sum... | {
"domain": "konoozargan.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357227168957,
"lm_q1q2_score": 0.8421015851216596,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 710.5052515824704,
"openwebmath_score": 0.8989253640174866,
"ta... |
) Suppose 0 • an • bn for n ‚ k for some k: Then. 2 Geometric Series. Relevant theorems, such as the Bolzano-Weierstrass theorem, will. A divergent question is asked without an attempt to reach a direct or specific conclusion. X1 n=0 (2n)! (n!)2 Thinking about the problem: Which test should I use to determine whether t... | {
"domain": "konoozargan.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357227168957,
"lm_q1q2_score": 0.8421015851216596,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 710.5052515824704,
"openwebmath_score": 0.8989253640174866,
"ta... |
was introduced by Cauchy in his "Cours d'Analyse" in order to avoid frequent mistakes in working with series. Divergent thinking has been hot recently. The series diverges if there is a divergent series of non -negative terms with 2. Divergent series - Wikipedia. This is because it is difficult to show that a series no... | {
"domain": "konoozargan.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357227168957,
"lm_q1q2_score": 0.8421015851216596,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 710.5052515824704,
"openwebmath_score": 0.8989253640174866,
"ta... |
how to use it. One example of convergent thinking is school. Divergent Series. The sum of a convergent geometric series can be calculated with the formula a ⁄ 1 – r, where “a” is the first term in the series and “r” is the number getting raised to a power. Absolute Convergence, Conditional Convergence and Divergence; G... | {
"domain": "konoozargan.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357227168957,
"lm_q1q2_score": 0.8421015851216596,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 710.5052515824704,
"openwebmath_score": 0.8989253640174866,
"ta... |
x_0 x 0 exists if no matter how x 0 x_0 x 0 is approached, the values returned by the function will always approach. (a) If a n > b n for all n, what can you say about P a n? Why? (b) If a n < b n for all n, what can you say about P a. If then we write If the sequence s n is not convergent then we say that the series i... | {
"domain": "konoozargan.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357227168957,
"lm_q1q2_score": 0.8421015851216596,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 710.5052515824704,
"openwebmath_score": 0.8989253640174866,
"ta... |
and the Vindhya and Satpura horsts in India. Every series uniquely defines the sequence of its partial sums. You appear to be on a device with a "narrow" screen width (i. b) If there is a divergent series ∑ b n and an ≥ b n for all n, then ∑ a n diverges. Similarities Between Convergent and Divergent Thinking. This doe... | {
"domain": "konoozargan.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357227168957,
"lm_q1q2_score": 0.8421015851216596,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 710.5052515824704,
"openwebmath_score": 0.8989253640174866,
"ta... |
t series examples: Infinite series: An infinite series is the sum of infinite sequence of terms which we denote : That is, given an infinite sequence of real numbers, a 1, a 2, a 3,. There exist numerous classes of divergent series that converge in some generalized sense, since to each such divergent series some "gener... | {
"domain": "konoozargan.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357227168957,
"lm_q1q2_score": 0.8421015851216596,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 710.5052515824704,
"openwebmath_score": 0.8989253640174866,
"ta... |
Home Misc Find the first N prime numbers. (Method 4) [Sieve of Eratosthenes]
# Find the first N prime numbers. (Method 4) [Sieve of Eratosthenes]
0 comment
Question: Given an integer N, find the prime numbers in that range from 1 to N.
Input: N = 25
Output: 2, 3, 5, 7, 11, 13, 17, 19, 23
We have several ways of fi... | {
"domain": "studyalgorithms.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357232512719,
"lm_q1q2_score": 0.842101583795263,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 3963.036829112114,
"openwebmath_score": 0.2162545919418335,
"ta... |
.hljs {
box-sizing: border-box;
}
.hljs.shcb-code-table {
display: table;
width: 100%;
}
.hljs.shcb-code-table > .shcb-loc {
color: inherit;
display: table-row;
width: 100%;
}
.hljs.shcb-code-table .shcb-loc > span {
display: table-cell;
}
.wp-block-code code.hljs:not(.shcb-wrap-lines) {
white-space: pre;
}
.wp-bl... | {
"domain": "studyalgorithms.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357232512719,
"lm_q1q2_score": 0.842101583795263,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 3963.036829112114,
"openwebmath_score": 0.2162545919418335,
"ta... |
#### You may also like
This site uses Akismet to reduce spam. Learn how your comment data is processed.
This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept Read More | {
"domain": "studyalgorithms.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357232512719,
"lm_q1q2_score": 0.842101583795263,
"lm_q2_score": 0.8577680977182186,
"openwebmath_perplexity": 3963.036829112114,
"openwebmath_score": 0.2162545919418335,
"ta... |
# How many bit strings of length $8$ start with $00$ or end with $1?$
How many bit strings of length $8$ start with $00$ or end with $1$?
I know about product rule and sum rule but I'm unsure how to incorporate it into this.
Would it be like this$? (x$ being either $1$ or $0):$
For starting with $00:$ $0 0 x x x x ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357173731318,
"lm_q1q2_score": 0.842101578753182,
"lm_q2_score": 0.8577680977182187,
"openwebmath_perplexity": 161.97624879476672,
"openwebmath_score": 0.6210427284240723,
"tag... |
$$2^6 + 2^7 - 2^5$$
• But the question says "How many bit strings of length 8 start with 00 OR end with 1?" Emphasize the "or". In that case, is finding the number of the combinations of 00xxxxx1 still the right answer? – user1766555 Apr 22 '13 at 16:34
• Yes...we are using the inclusive sense of "or", that means one ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357173731318,
"lm_q1q2_score": 0.842101578753182,
"lm_q2_score": 0.8577680977182187,
"openwebmath_perplexity": 161.97624879476672,
"openwebmath_score": 0.6210427284240723,
"tag... |
Edit: In other words, $A\cup B$ means $A$ or $B$ (or both) and $A\cap B$ means $A$ and $B$ ;)
• Thank you for illustration :) – user1766555 Apr 22 '13 at 16:50
• You're welcome ! Hope it made it clearer how you could find the formulas yourself if you need them again ;) – Dolma Apr 22 '13 at 16:53 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9817357173731318,
"lm_q1q2_score": 0.842101578753182,
"lm_q2_score": 0.8577680977182187,
"openwebmath_perplexity": 161.97624879476672,
"openwebmath_score": 0.6210427284240723,
"tag... |
# Distribution of sum of exponentials
Let $$X_1$$ and $$X_2$$ be independent and identically distributed exponential random variables with rate $$\lambda$$. Let $$S_2 = X_1 + X_2$$.
Q: Show that $$S_2$$ has PDF $$f_{S_2}(x) = \lambda^2 x \text{e}^{-\lambda x},\, x\ge 0$$.
Note that if events occurred according to a ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9907319868927341,
"lm_q1q2_score": 0.8420936354259576,
"lm_q2_score": 0.8499711794579723,
"openwebmath_perplexity": 2290.753148109162,
"openwebmath_score": 1.0000100135803223,
"tag... |
MGF Approach
This approach uses the moment generating function (MGF).
\begin{align} M_{S_2}(t) &= \text{E}\left[\text{e}^{t S_2}\right] \\ &= \text{E}\left[\text{e}^{t(X_1 + X_2)}\right] \\ &= \text{E}\left[\text{e}^{t X_1 + t X_2}\right] \\ &= \text{E}\left[\text{e}^{t X_1} \text{e}^{t X_2}\right] \\ &= \text{E}\left... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9907319868927341,
"lm_q1q2_score": 0.8420936354259576,
"lm_q2_score": 0.8499711794579723,
"openwebmath_perplexity": 2290.753148109162,
"openwebmath_score": 1.0000100135803223,
"tag... |
SectionSLTSurjective Linear Transformations¶ permalink
The companion to an injection is a surjection. Surjective linear transformations are closely related to spanning sets and ranges. So as you read this section reflect back on Section ILT and note the parallels and the contrasts. In the next section, Section IVLT, w... | {
"domain": "ups.edu",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9907319866190857,
"lm_q1q2_score": 0.8420936314281889,
"lm_q2_score": 0.8499711756575749,
"openwebmath_perplexity": 264.7288847323717,
"openwebmath_score": 0.9572227001190186,
"tags": null,
... |
To show that a linear transformation is not surjective, it is enough to find a single element of the codomain that is never created by any input, as in Example NSAQ. However, to show that a linear transformation is surjective we must establish that every element of the codomain occurs as an output of the linear transfo... | {
"domain": "ups.edu",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9907319866190857,
"lm_q1q2_score": 0.8420936314281889,
"lm_q2_score": 0.8499711756575749,
"openwebmath_perplexity": 264.7288847323717,
"openwebmath_score": 0.9572227001190186,
"tags": null,
... |
Proof
Let us compute another range, now that we know in advance that it will be a subspace.
In contrast to injective linear transformations having small (trivial) kernels (Theorem KILT), surjective linear transformations have large ranges, as indicated in the next theorem.
SubsectionSSSLTSpanning Sets and Surjective... | {
"domain": "ups.edu",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9907319866190857,
"lm_q1q2_score": 0.8420936314281889,
"lm_q2_score": 0.8499711756575749,
"openwebmath_perplexity": 264.7288847323717,
"openwebmath_score": 0.9572227001190186,
"tags": null,
... |
SubsectionSLTDSurjective Linear Transformations and Dimension¶ permalink
Proof
Notice that the previous example made no use of the actual formula defining the function. Merely a comparison of the dimensions of the domain and codomain are enough to conclude that the linear transformation is not surjective. Archetype O... | {
"domain": "ups.edu",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9907319866190857,
"lm_q1q2_score": 0.8420936314281889,
"lm_q2_score": 0.8499711756575749,
"openwebmath_perplexity": 264.7288847323717,
"openwebmath_score": 0.9572227001190186,
"tags": null,
... |
C20
Example SAR concludes with an expression for a vector $\vect{u}\in\complex{5}$ that we believe will create the vector $\vect{v}\in\complex{5}$ when used to evaluate $T\text{.}$ That is, $\lteval{T}{\vect{u}}=\vect{v}\text{.}$ Verify this assertion by actually evaluating $T$ with $\vect{u}\text{.}$ If you do not ha... | {
"domain": "ups.edu",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9907319866190857,
"lm_q1q2_score": 0.8420936314281889,
"lm_q2_score": 0.8499711756575749,
"openwebmath_perplexity": 264.7288847323717,
"openwebmath_score": 0.9572227001190186,
"tags": null,
... |
Solution
C27
Let $\ltdefn{T}{\complex{3}}{\complex{4}}$ be given by $\lteval{T}{\colvector{a\\b\\c}} = \colvector{a + b -c\\ a - b + c\\ -a + b + c\\a + b + c}\text{.}$ Find a basis of $\rng{T}\text{.}$ Is $T$ surjective?
Solution
C28
Let $\ltdefn{T}{\complex{4}}{ M_{22}}$ be given by $\lteval{T}{\colvector{a\\b\\c\... | {
"domain": "ups.edu",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9907319866190857,
"lm_q1q2_score": 0.8420936314281889,
"lm_q2_score": 0.8499711756575749,
"openwebmath_perplexity": 264.7288847323717,
"openwebmath_score": 0.9572227001190186,
"tags": null,
... |
Solution
T20
Suppose that $A$ is an $m\times n$ matrix. Define the linear transformation $T$ by \begin{equation*} \ltdefn{T}{\complex{n}}{\complex{m}},\quad \lteval{T}{\vect{x}}=A\vect{x}\text{.} \end{equation*} Prove that the range of $T$ equals the column space of $A\text{,}$ $\rng{T}=\csp{A}\text{.}$
Solution | {
"domain": "ups.edu",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9907319866190857,
"lm_q1q2_score": 0.8420936314281889,
"lm_q2_score": 0.8499711756575749,
"openwebmath_perplexity": 264.7288847323717,
"openwebmath_score": 0.9572227001190186,
"tags": null,
... |
# Discrete Probability Distribution Problem
• Jun 29th 2009, 01:28 AM
ose90
Discrete Probability Distribution Problem
Hi everyone, I hope you can help me with the following problem, especially the part 2 of the problem.
Three dice are thrown in a game. If 1 or 6 turns up, you will be paid 1p. If neither 1 nor 6 turn ... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9907319877136793,
"lm_q1q2_score": 0.8420936304759744,
"lm_q2_score": 0.8499711737573762,
"openwebmath_perplexity": 1336.5276195260062,
"openwebmath_score": 0.7499650716781616,
"ta... |
$EV \;=\;\left(\frac{19}{27}\right)(+1) + \left(\frac{8}{27}\right)(\text{-}5) \;=\;-\frac{7}{9}$
In 9 games, your expectation is: . $(9)\left(\text{-}\frac{7}{9}\right) \;=\;-7$
You can expect to lose \$7.
Quote:
Say you are given opportunity to change the rule of the game if 1 or 6 appears.
To make the game worth... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9907319877136793,
"lm_q1q2_score": 0.8420936304759744,
"lm_q2_score": 0.8499711737573762,
"openwebmath_perplexity": 1336.5276195260062,
"openwebmath_score": 0.7499650716781616,
"ta... |
# Inverse of a certain unit upper triangular matrix
I don't know if there is a certain name for this matrix. but I want to show
$$\begin{pmatrix}1&\gamma&\gamma^2& \ldots & \gamma^n\\ &1&\gamma&\ddots&\vdots\\ &&\ddots&\ddots&\gamma^2\\ &&&1&\gamma\\ &&&&1\end{pmatrix}^{-1}= \begin{pmatrix}1&-\gamma&& & \\ &1&-\gamma... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986151393040476,
"lm_q1q2_score": 0.8420872563576934,
"lm_q2_score": 0.8539127585282744,
"openwebmath_perplexity": 118.47505196117896,
"openwebmath_score": 0.9342688322067261,
"tag... |
We can find a nice formula for $$M^{-1}$$, where $$M = \begin{pmatrix}1&-\gamma&& & \\ &1&-\gamma&&\\ &&\ddots&\ddots&\\ &&&1&-\gamma\\ &&&&1\end{pmatrix}$$ In particular, it is useful to note that $$M = I - N$$, where $$N = \begin{pmatrix}0&\gamma&& & \\ &0&\gamma&&\\ &&\ddots&\ddots&\\ &&&0&\gamma\\ &&&&0\end{pmatrix... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986151393040476,
"lm_q1q2_score": 0.8420872563576934,
"lm_q2_score": 0.8539127585282744,
"openwebmath_perplexity": 118.47505196117896,
"openwebmath_score": 0.9342688322067261,
"tag... |
• @AndreuPayne I see now that you're particularly concerned with having a detailed, rigorous proof for the formula. With that in mind, I've added a proof of the formula for $N^k$. I hope this suffices. – Omnomnomnom Oct 3 '18 at 15:44 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986151393040476,
"lm_q1q2_score": 0.8420872563576934,
"lm_q2_score": 0.8539127585282744,
"openwebmath_perplexity": 118.47505196117896,
"openwebmath_score": 0.9342688322067261,
"tag... |
# In a $n \times n$ grid of points, choosing $2n-1$ points, there will always be a right triangle
$$\textbf{Question:}$$ Consider a $$n×n$$ grid of points. Prove that no matter how we choose $$2n-1$$ points from these, there will always be a right triangle with vertices among these $$2n-1$$ points.
This question inde... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513889704251,
"lm_q1q2_score": 0.842087251049111,
"lm_q2_score": 0.8539127566694178,
"openwebmath_perplexity": 148.9946776252967,
"openwebmath_score": 0.8826546669006348,
"tags... |
Base case: Prove it for $$n = 2$$ and all $$m\geq 2$$.
This is left to the reader (Consider the sum of degrees $$d(m_1) + d(m_2) = n + 1$$.)
Suppose for $$n, m \geq 3$$, that there is such a graph with no path of length 3 for $$n, m \geq 2$$.
There is a vertex (WLOG $$c_1$$) of degree $$d \geq 2$$.
If $$d = m$$, clear... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513889704251,
"lm_q1q2_score": 0.842087251049111,
"lm_q2_score": 0.8539127566694178,
"openwebmath_perplexity": 148.9946776252967,
"openwebmath_score": 0.8826546669006348,
"tags... |
P.S. A translation of this proof into graph theory would go like this. A bipartite graph has bipartition $$(V_1,V_2)$$, $$|V_1|=m\ge2$$, $$|V_2|=n\ge2$$, and it has $$m+n-1$$ edges. If there is no path of length $$3$$, then each edge has an endpoint of degree $$1$$. Therefore there are at least $$m+n-1$$ vertices of de... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513889704251,
"lm_q1q2_score": 0.842087251049111,
"lm_q2_score": 0.8539127566694178,
"openwebmath_perplexity": 148.9946776252967,
"openwebmath_score": 0.8826546669006348,
"tags... |
# Finding where the tail starts for a probability distribution, from its generating function
Suppose we generate "random strings" over an $m$-letter alphabet, and look for the first occurrence of $k$ consecutive identical digits. I was with some effort able to find that the random variable $X$, denoting the time until... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513901914406,
"lm_q1q2_score": 0.8420872502586375,
"lm_q2_score": 0.8539127548105611,
"openwebmath_perplexity": 236.30588985281662,
"openwebmath_score": 0.9307999014854431,
"ta... |
• A formal solution is provided by Fourier Transform, i.e. if $z=e^{i\theta}$ the $Pr(x=m)=\int d\theta P(e^{i\theta})e^{-im\theta}$ Feb 10, 2016 at 16:49
• @Marcel: Thanks, can that be used to compute the bounds quickly? A priori it doesn't seem like doing the integration for every $m$ will be any faster than just com... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513901914406,
"lm_q1q2_score": 0.8420872502586375,
"lm_q2_score": 0.8539127548105611,
"openwebmath_perplexity": 236.30588985281662,
"openwebmath_score": 0.9307999014854431,
"ta... |
Upon finding $C$ and $\alpha$, we have approximations for $N_1(m,k)$ and $N_2(m,k)$ by solving $1 - C(\frac{1}{\alpha})^n = p$. The general solution is $$N_p(m,k) \approx \frac{\ln C -\ln(1-p)}{\ln(\alpha)}.$$For convenience, I used $C = 1$ to check against your values. This gave approximations of $N_1 = 7, 76, 768, 76... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513901914406,
"lm_q1q2_score": 0.8420872502586375,
"lm_q2_score": 0.8539127548105611,
"openwebmath_perplexity": 236.30588985281662,
"openwebmath_score": 0.9307999014854431,
"ta... |
• You can check how far $\alpha$ is away from the true root by computing $\alpha-\alpha_* \approx q(\alpha)/q'(\alpha)$ ($q$—the denominator), like in a Newton step, it is in fact $\sim m^{-3k}$, so extremely close. Also, $C$ is equal to $$-\frac{(m-\alpha)\alpha^{k-1}}{(\alpha-1)q'(\alpha)}$$ which you can derive by e... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513901914406,
"lm_q1q2_score": 0.8420872502586375,
"lm_q2_score": 0.8539127548105611,
"openwebmath_perplexity": 236.30588985281662,
"openwebmath_score": 0.9307999014854431,
"ta... |
First, the generating function of the cumulative distribution $\sum {\rm Pr}(X\le n)z^n$ is formed from the generating function of the exact distribution $\sum {\rm Pr}(X=n)z^n$ by multiplying by a factor if $1/(1-z)$. So, define $$Q(z)=\frac{P(z)}{1-z}=\frac1{1-z}\cdot\frac{(m-z)z^k}{m^k(1-z)+(m-1)z^k}.$$ For an arbit... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513901914406,
"lm_q1q2_score": 0.8420872502586375,
"lm_q2_score": 0.8539127548105611,
"openwebmath_perplexity": 236.30588985281662,
"openwebmath_score": 0.9307999014854431,
"ta... |
As a side note, my preference for obtaining the original generating function $P(z)$ would be the Goulden-Jackson cluster method. This method is more general than the one you referenced, it's easier to learn, and it has a wide range of applications. For sure, it's one of my favorites.
• Thank you for this excellent and... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513901914406,
"lm_q1q2_score": 0.8420872502586375,
"lm_q2_score": 0.8539127548105611,
"openwebmath_perplexity": 236.30588985281662,
"openwebmath_score": 0.9307999014854431,
"ta... |
# Recurrence relations and Generating functions - how to find the initial conditions?
Best to ask by example. Given the recurrence relation $a_{n}=a_{n-1}+a_{n-2}$, and some given initial conditions, we can find a similar relation for the generating function for the sequence, $f(x)=\sum_{n=0}^\infty a_nx^n$: $$f(x)=xf... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986151392226466,
"lm_q1q2_score": 0.8420872483301434,
"lm_q2_score": 0.8539127510928476,
"openwebmath_perplexity": 324.09421014773903,
"openwebmath_score": 0.9247905015945435,
"tag... |
Now multiply $(2)$ through by $x^n$ and sum:
\begin{align*} \sum_na_nx^n&=\sum_na_{n-1}x^n+\sum_na_{n-2}x^n+(a_1-a_0)\sum_n[n=1]x^n+a_0\sum_n[n=0]x^n\\ &=x\sum_na_nx^n+x^2\sum_na_nx^n+(a_1-a_0)x+a_0\;. \end{align*}
Thus, if your generating function is $A(x)=\displaystyle\sum_na_nx^n$, you have $$A(x)=xA(x)+x^2A(x)+(a... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986151392226466,
"lm_q1q2_score": 0.8420872483301434,
"lm_q2_score": 0.8539127510928476,
"openwebmath_perplexity": 324.09421014773903,
"openwebmath_score": 0.9247905015945435,
"tag... |
We get $$f'(x)=pxf'(x)+pf(x)+qx^2f'(x)+2qxf(x)+c'(x),$$ so $$c'(0)=f'(0)-pf(0)=a_1-pa_0.$$ Since everything is determined once $a_0$ and $a_1$ are known, $c(x)$ is a polynomial of degree $\le 1$, and therefore $c(x)=c'(0)x+c(0)=(a_1-pa_0)x+a_0$.
Repeated differentiation also has to work for your more general question,... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.986151392226466,
"lm_q1q2_score": 0.8420872483301434,
"lm_q2_score": 0.8539127510928476,
"openwebmath_perplexity": 324.09421014773903,
"openwebmath_score": 0.9247905015945435,
"tag... |
# Remainder when large numbers divided
• June 27th 2011, 01:13 PM
Zalren
Remainder when large numbers divided
What is the remainder when $2001^{2001}$ is divided by $26$?
I think I have to do something with $26$, like break it up into $13 * 2$. Try to use Fermat's Theorem somehow. I know that $(2001, 13) = 1$ and $(2... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513893774303,
"lm_q1q2_score": 0.8420872458973154,
"lm_q2_score": 0.8539127510928476,
"openwebmath_perplexity": 1172.1188057173451,
"openwebmath_score": 0.6643934845924377,
"ta... |
$2001^{2001} = 2001^{1024} 2001^{512} 2001^{256} 2001^{128} 2001^{64} 2001^{16} 2001 \equiv 1 \cdot 1 \cdot 1 \cdot 1 \cdot 1 \cdot 1 \cdot 25 \equiv 25 \pmod {26}$
I think you made it harder than it has to be (both of you). Isn't it true that $\displaystyle 2001^{2001}\equiv 25^{2001}\equiv (-1)^{2001}\equiv -1\equiv... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513893774303,
"lm_q1q2_score": 0.8420872458973154,
"lm_q2_score": 0.8539127510928476,
"openwebmath_perplexity": 1172.1188057173451,
"openwebmath_score": 0.6643934845924377,
"ta... |
# Double Inequalities
• March 5th 2010, 12:46 AM
anonymous_maths
Double Inequalities
Need help to solve this:
0 < √1-x^2 < 1
< x < 1
How does that work?
Thanks.
• March 5th 2010, 01:27 AM
Prove It
Quote:
Originally Posted by anonymous_maths
Need help to solve this:
0 < √1-x^2 < 1
< x < 1
How does that work?
... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.98615138856342,
"lm_q1q2_score": 0.8420872452022217,
"lm_q2_score": 0.8539127510928476,
"openwebmath_perplexity": 1217.9644568594867,
"openwebmath_score": 0.9278259873390198,
"tags... |
If [/tex]0= \sqrt{1- x^2}[/tex] then $0= 1- x^2$, $x= \pm 1$.
If $\sqrt{1- x^2}= 1$ then $1- x^2= 1$, x= 0.
The whole point of that is that the three points, -1, 0, and 1 separate intervals where $\sqrt{1- x^2}$ is less than 0, greater than 0, or does not exist. Try one value of x in each of the intervals x< -1, -1< ... | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.98615138856342,
"lm_q1q2_score": 0.8420872452022217,
"lm_q2_score": 0.8539127510928476,
"openwebmath_perplexity": 1217.9644568594867,
"openwebmath_score": 0.9278259873390198,
"tags... |
# Factoring $x^2 + x +1 > 0$ from Spivak Calculus exercise
Hi!! I found me in trouble when I saw the solution of a simple inequality, that can be found at the end of the first chapter, that is the exercise 4 - (viii): $x^2+x+1 > 0$. Very easy to solve I know, but I also saw the related solution on the Spivak Calculus ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513897844355,
"lm_q1q2_score": 0.8420872444117482,
"lm_q2_score": 0.8539127492339909,
"openwebmath_perplexity": 180.31110784050162,
"openwebmath_score": 0.839210033416748,
"tag... |
I hope someone could help me
• Look up: completing the square. – Zain Patel Jul 20 '15 at 19:06
• Among the various methods for motivating how one might discover how to complete the square, here's one you might not come across. Note that $x^2+x+1=x(x+1)+1,$ where the two variable factors are "not balanced" (the zeros ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513897844355,
"lm_q1q2_score": 0.8420872444117482,
"lm_q2_score": 0.8539127492339909,
"openwebmath_perplexity": 180.31110784050162,
"openwebmath_score": 0.839210033416748,
"tag... |
As suggested, this was achieved through completing the square:
$$x^2 +x+1 = x^2 + x +\left(\frac{1}{2}\right)^2 + 1 -\frac{1}{4}$$
$$= \left(x+\frac{1}{2}\right)^2 + \frac{3}{4}$$
You need to go and read up completing the square to understand how I got the above.
$$ax^2+bx+c=a\left(x^2+\frac{b}{a}x+\frac{c}{a}\righ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513897844355,
"lm_q1q2_score": 0.8420872444117482,
"lm_q2_score": 0.8539127492339909,
"openwebmath_perplexity": 180.31110784050162,
"openwebmath_score": 0.839210033416748,
"tag... |
# $f, g$ entire functions with $f^2 + g^2 \equiv 1 \implies \exists h$ entire with $f(z) = \cos(h(z))$ and $g(z) = \sin(h(z))$
I am studying for a qualifier exam in complex analysis and right now I'm solving questions from old exams. I am trying to prove the following:
Prove that if $f$ and $g$ are entire functions s... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513918194609,
"lm_q1q2_score": 0.8420872443163682,
"lm_q2_score": 0.8539127473751341,
"openwebmath_perplexity": 160.22667943963717,
"openwebmath_score": 0.9407651424407959,
"ta... |
Thanks for any help.
-
Since $f-ig$ is the reciprocal of $f+ig$, you know that $e^{-ih(z)}=f(z)-ig(z)$. Average this with $e^{ih(z)}$ to finish. (I am not familiar with the holomorphic logarithm theorem you mention though.) – anon Jul 6 '12 at 4:21
@anon Maybe that name is not the usual one. I have only seen it by th... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513918194609,
"lm_q1q2_score": 0.8420872443163682,
"lm_q2_score": 0.8539127473751341,
"openwebmath_perplexity": 160.22667943963717,
"openwebmath_score": 0.9407651424407959,
"ta... |
Tags
#m249 #mathematics #open-university #statistics #time-series
Question
The 1-step ahead forecast error at time t, which is denoted et, is the difference between the observed value and the 1-step ahead forecast of Xt:
et = xt - $$\hat{x}_t$$
The sum of squared errors, or SSE, is given by
SSE = $$\large \sum_{t=1}^ne... | {
"domain": "buboflash.eu",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513914124559,
"lm_q1q2_score": 0.8420872439688214,
"lm_q2_score": 0.853912747375134,
"openwebmath_perplexity": 3995.280183672095,
"openwebmath_score": 0.8339410424232483,
"tags": nu... |
#### Parent (intermediate) annotation
Open it
is given by SSE = $$\large \sum_{t=t}^ne_t^2 = \sum_{t=t}^n(x_t-\hat{x}_t)^2$$ Given observed values x 1 ,x 2 ,...,x n ,the optimal value of the smoothing parameter α for simple exponential smoothing is the value that <span>minimizes the sum of squared errors.<span><body><... | {
"domain": "buboflash.eu",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9861513914124559,
"lm_q1q2_score": 0.8420872439688214,
"lm_q2_score": 0.853912747375134,
"openwebmath_perplexity": 3995.280183672095,
"openwebmath_score": 0.8339410424232483,
"tags": nu... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.