text stringlengths 1 2.12k | source dict |
|---|---|
quadrilateral. Ptolemy ’ s a quadrilateral. Latus meaning ‘ four ‘ and latus meaning ‘ four ‘ and latus meaning ‘ side.! Corollary, Converse of cyclic quadrilateral theorem - YouTube this will help you discover yet new. Is 60o to it lies on the circumference of the opposite angles a. Concept better, they form the verti... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
inscribed in a quadrilateral drawn inside a circle ( QR x ). The general case of the circumference of the opposite side that it can have opposite! The interior opposite angle all quadrilaterals having the same side lengths ( regardless of )! Triangle are concurrent.Theorem 69 creating his table of chords, a trigonometr... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
= 1,2,3,4, Fuss ' theorem any four points the... When any four points on the circumference of the following questions are given he applied to astronomy one of. Of arc BCD now try to prove this theorem parallelogram to be cyclic or inscribed in cyclic! \Theta_1+\Theta_2=\Theta_3+\Theta_4=90^\Circ\ \ ) ; seventh century,... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
;.... Interior angle opposite to it interior opposite angles of a pair of its opposite angles are supplementary, then angles!, in the circumference of the sides can be defined and is known as cyclic quadrilateral, and. Can have parallel opposite sides question: Find the value of angle compare! Are joined, they form the... | {
"domain": "jehobs.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399060540359,
"lm_q1q2_score": 0.8429704015160763,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 850.5288189952355,
"openwebmath_score": 0.6436747312545776,
"tags":... |
MAT244--2020F > Chapter 2
section 2.1 practice problem 35
(1/1)
Suheng Yao:
I had solved this equation to this implicit form, but I don't know how to simplify to a simpler form as the answer shown. Could someone help me? Thanks.
kavinkandiah:
Add all those terms on both sides so they're all positive instead (just m... | {
"domain": "toronto.edu",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399043329855,
"lm_q1q2_score": 0.842970395077687,
"lm_q2_score": 0.8688267626522814,
"openwebmath_perplexity": 598.4964741610772,
"openwebmath_score": 0.8197005391120911,
"tags": nul... |
# The Arithmetic Mean (A.M) between two numbers exceeds their Geometric Mean (G.M.)
The Arithmetic Mean (A.M) between two numbers exceeds their Geometric Mean (G.M.) by $2$ and the GM exceeds the Harmonic Mean (H.M) by $1.6$. Find the numbers.
My Attempt: Let the numbers be $a$ and $b$. Then, $$A.M=\dfrac {a+b}{2}$$ ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9805806557900711,
"lm_q1q2_score": 0.8429696495223525,
"lm_q2_score": 0.8596637559030338,
"openwebmath_perplexity": 390.46085411906154,
"openwebmath_score": 0.9131412506103516,
"ta... |
Solving $a^2(u-v)^2-2u^2(u-v)a+u^2v^2 =0$,
$\begin{array}\\ a &=\dfrac{2u^2(u-v)\pm\sqrt{4u^4(u-v)^2-4(u-v)^2u^2v^2}}{2(u-v)^2}\\ &=\dfrac{u^2(u-v)\pm u(u-v)\sqrt{u^2-v^2}}{(u-v)^2}\\ &=\dfrac{u^2\pm u\sqrt{u^2-v^2}}{u-v}\\ &=u\dfrac{u\pm \sqrt{u^2-v^2}}{u-v}\\ \text{and}\\ b &=\dfrac{2u^2}{u-v}-a\\ &=\dfrac{2u^2}{u-v... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9805806557900711,
"lm_q1q2_score": 0.8429696495223525,
"lm_q2_score": 0.8596637559030338,
"openwebmath_perplexity": 390.46085411906154,
"openwebmath_score": 0.9131412506103516,
"ta... |
# Showing $(1+\frac{1}{n})^n$ is an increasing sequence by comparing the binomial expansions term by term
So I know that showing $$(1+\frac{1}{n})^n$$ is an increasing sequence has probably appeared on this site about 100 times, but my professor said he thinks the induction step is most easily seen if you expand this ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.98058065352006,
"lm_q1q2_score": 0.8429696493337016,
"lm_q2_score": 0.8596637577007394,
"openwebmath_perplexity": 397.329849976927,
"openwebmath_score": 0.7397270202636719,
"tags":... |
• and in addition to this, the expansion of the $n+1$ will have one additional term in the summation, but the idea is that we can throw that out and still get the inquality to hold by comparing term by term what you have posted. Right? – Math is hard Oct 24 '18 at 20:17
• @MichaelVaughan Yes, the additional term is pos... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.98058065352006,
"lm_q1q2_score": 0.8429696493337016,
"lm_q2_score": 0.8596637577007394,
"openwebmath_perplexity": 397.329849976927,
"openwebmath_score": 0.7397270202636719,
"tags":... |
# help for proving an equation by induction
For this equation:
$$-1^3+(-3)^3+(-5)^3+\ldots+(-2n-1)^3=(-n-1)^2(-2n^2-4n-1)$$
how can I prove this by induction?
When I set $n = 1$ for the base case I got:
$$-1^3 + (-3)^3 + (-5)^3 + \ldots + (-3)^3 = -28$$
but am having trouble with the following inductive steps
-
... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9805806523850542,
"lm_q1q2_score": 0.8429696465951829,
"lm_q2_score": 0.8596637559030338,
"openwebmath_perplexity": 257.0021966178166,
"openwebmath_score": 0.855392336845398,
"tags... |
This is just algebra, and I’ll leave it to you.
-
Inductive proofs of sums like yours are easily tackled using a very simple general method known as telescopy - see the trivial inductive proof here, of the following fundamental
Theorem $\rm \displaystyle\ \ \sum_{i\,=\,0}^n\, f(i)\, =\, g(n)\iff f(0) = g(0)\ {\rm\ a... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9805806523850542,
"lm_q1q2_score": 0.8429696465951829,
"lm_q2_score": 0.8596637559030338,
"openwebmath_perplexity": 257.0021966178166,
"openwebmath_score": 0.855392336845398,
"tags... |
Much more inductive insight is gained by proving once and for all the above general telescoping sum formula. Here, rid of all the messy details of special cases, one sees clearly the beautiful way that telescopic induction works, and this lends more insight on induction in general. Such telescopic induction skills will... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9805806523850542,
"lm_q1q2_score": 0.8429696465951829,
"lm_q2_score": 0.8596637559030338,
"openwebmath_perplexity": 257.0021966178166,
"openwebmath_score": 0.855392336845398,
"tags... |
# Randomized Algorithms: High-Probability vs. Expectation
Hopefully this question isn't too general, but I was wondering what the relationship is between randomized algorithms that perform well with high-probability and those that perform well in expectation. My question is motivated by the definition of a randomized ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9805806543713141,
"lm_q1q2_score": 0.8429696465399031,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 182.98731770206086,
"openwebmath_score": 0.8612411022186279,
"ta... |
If you have an algorithm that is an $$\alpha$$-approximation in expectation, then you can construct an algorithm that is a $$(1+\epsilon)\alpha$$-approximation with high probability, for any $$\epsilon>0$$. In particular, by Markov's inequality, if you run the algorithm, then with probability at least $$1-1/(1+\epsilon... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9805806543713141,
"lm_q1q2_score": 0.8429696465399031,
"lm_q2_score": 0.8596637541053281,
"openwebmath_perplexity": 182.98731770206086,
"openwebmath_score": 0.8612411022186279,
"ta... |
# For what functions is $y'' = y$?
What functions $y = f(x)$ have the property that $f(x) = f''(x)$, i.e. what functions have the same integral and derivitive?
I could think of $ce^x$ and $ce^{-x}$ (where $c$ is a constant), but are there others? If not, how can you prove those are the only ones?
• If $y=c e^{x}$ an... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9805806557900711,
"lm_q1q2_score": 0.8429696442339663,
"lm_q2_score": 0.8596637505099167,
"openwebmath_perplexity": 286.29880143989107,
"openwebmath_score": 0.9077276587486267,
"ta... |
$e^{-rx} u = C$
or in other words
$u = C e^{rx}$
Background on these two methods:
The trick in the top section is a general technique for proving results for systems of linear first order equations based on finding eigenvectors. (A second order equation is a system of two first order equations if you let $y'$ be a ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9805806557900711,
"lm_q1q2_score": 0.8429696442339663,
"lm_q2_score": 0.8596637505099167,
"openwebmath_perplexity": 286.29880143989107,
"openwebmath_score": 0.9077276587486267,
"ta... |
Probability for red and green apples in a bag
"There are twelve red apples and seven green apples in a bag. What is the probability of picking up one red apple and one green apple at once?"
This is a multiple choice question, the choices are:
• A) $$\frac{21}{64}$$
• B) $$\frac{23}{64}$$
• C) $$\frac{25}{64}$$
• D) ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9805806478450307,
"lm_q1q2_score": 0.8429696356411076,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 219.7923297930464,
"openwebmath_score": 0.7238319516181946,
"tag... |
Therefore, the answer comes out as, $$\frac{7.12}{19 \choose 2} = \frac{28}{57}$$
• The division by $2$ is not necessary. To see this, consider the problem where it was instead $7$ red apples and $11$ green apples. If you were to divide by $2$ in that case then you would have a non-integer number of arrangements, a cl... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9805806478450307,
"lm_q1q2_score": 0.8429696356411076,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 219.7923297930464,
"openwebmath_score": 0.7238319516181946,
"tag... |
# Convergence of Riemann sums for improper integrals
I was considering whether or not the limit of Riemann sums converges to the value of an improper integral on a bounded interval. This appears to be true in some cases when the sum avoids points where the function is not defined.
For example, the right-hand Riemann ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9871787887510289,
"lm_q1q2_score": 0.8429645571579201,
"lm_q2_score": 0.8539127529517043,
"openwebmath_perplexity": 235.83631640119745,
"openwebmath_score": 0.9763296842575073,
"ta... |
Suppose WLOG $f:(0,1] \to \mathbb{R}$ is nonnegative and decreasing. Suppose further that there is a singularity at $x =0$ but $f$ is Riemann integrable on $[c,1]$ for $c > 0$ and the improper integral is convergent:
$$\lim_{c \to 0+}\int_c^1 f(x) \, dx = \int_0^1 f(x) \, dx.$$
Take a uniform partition $P_n = (0,1/n,... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9871787887510289,
"lm_q1q2_score": 0.8429645571579201,
"lm_q2_score": 0.8539127529517043,
"openwebmath_perplexity": 235.83631640119745,
"openwebmath_score": 0.9763296842575073,
"ta... |
Improper integrals and right-hand Riemann sums
• Nicely written (+1) – Mark Viola May 4 '17 at 17:24 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9871787887510289,
"lm_q1q2_score": 0.8429645571579201,
"lm_q2_score": 0.8539127529517043,
"openwebmath_perplexity": 235.83631640119745,
"openwebmath_score": 0.9763296842575073,
"ta... |
# Interview Question on Probability: A and B toss a dice with 1 to n faces in an alternative way
A and B toss a dice with 1 to n faces in an alternative way, the game is over when a face shows up with point less than the previous toss and that person loses. What is the probability of the first person losing the game an... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9871787872422174,
"lm_q1q2_score": 0.8429645558695267,
"lm_q2_score": 0.8539127529517043,
"openwebmath_perplexity": 259.7367063629853,
"openwebmath_score": 0.8452918529510498,
"tag... |
$$\sum_{k=0}^\infty(-1)^k\binom{n+k-1}k\frac1{n^k}=\left(1+\frac1n\right)^{-n}\;.$$
This goes to $\mathrm e^{-1}$ as $n\to\infty$. In the limit $n\to\infty$, the probability for equal rolls goes to zero, so we can rank the rolls. If we rank the first $k$ rolls, the ranks are a random permutation of the first $k$ integ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9871787872422174,
"lm_q1q2_score": 0.8429645558695267,
"lm_q2_score": 0.8539127529517043,
"openwebmath_perplexity": 259.7367063629853,
"openwebmath_score": 0.8452918529510498,
"tag... |
$$\sum_{k=0}^\infty\frac1{k!}=\mathrm e\;,$$
in agreement with the above result.
• Delightful. I'm not sure I fully follow the parts where you say we can evaluate the limits by ignoring the possibility of equal rolls, though. I understand them as heuristics, but you seem to be saying that these are rigorous arguments... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9871787872422174,
"lm_q1q2_score": 0.8429645558695267,
"lm_q2_score": 0.8539127529517043,
"openwebmath_perplexity": 259.7367063629853,
"openwebmath_score": 0.8452918529510498,
"tag... |
It seems to me that the easiest way to compute $p_k$ will be in decreasing order, starting with $k=n$ and progressing down to $k=1$. When $k= n$ we have
$$p_n= {n-1\over n} + {1\over n}\left(1-p_n\right)\implies p_n={n\over n+1}$$ because, if the roller rolls $n$, he will lose if his opponent rolls anything but $n$, a... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9871787872422174,
"lm_q1q2_score": 0.8429645558695267,
"lm_q2_score": 0.8539127529517043,
"openwebmath_perplexity": 259.7367063629853,
"openwebmath_score": 0.8452918529510498,
"tag... |
Note that this is precisely the complement of the probability that joriki computed in his solution. The problem lies in the description of the game in the question: "the game is over when a face shows up with point less than the previous toss and that person loses." It isn't at all clear who "that person" is, as the ph... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9871787872422174,
"lm_q1q2_score": 0.8429645558695267,
"lm_q2_score": 0.8539127529517043,
"openwebmath_perplexity": 259.7367063629853,
"openwebmath_score": 0.8452918529510498,
"tag... |
# Find the value of $x$ in the displayed figure
Find $x$ in the following figure. $AB,AC,AD,BC,BE,CD$ are straight lines.
$AE=x$, $BE=CD=x-3$, $BC=10$, $AD=x+4$
$\angle BEC=90^{\circ}$, $\angle ADC=90^{\circ}$
NOTE: figure not to scale.
-
Please edit the question into the body of your post. – Gerry Myerson Aug 16 ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938326003365,
"lm_q2_score": 0.8633916187614822,
"openwebmath_perplexity": 434.4336767424758,
"openwebmath_score": 0.8683072924613953,
"tags... |
The coefficient of $x^{4}$ is $2=1\times 2$ and the constant term is $1089=1\times 3^{2}11^{2}$. To find possible rational roots of this equation, we apply the rational root theorem and test the numbers of the form $$\begin{equation*} x=\pm \frac{p}{q}, \end{equation*}$$ where $p\in \left\{ 1,3,9,11,33,99,121,363,1089\... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938326003365,
"lm_q2_score": 0.8633916187614822,
"openwebmath_perplexity": 434.4336767424758,
"openwebmath_score": 0.8683072924613953,
"tags... |
If you were given that $x$ and |EC| were integers you could use the following.
Let $y = |EC|$.
Using Pythagoras on $\triangle BEC$:
1) $(x-3)^2 + y^2 = 10^2$ $\Rightarrow x^2 - 6x + 9 + y^2 = 100$ $\Rightarrow y^2 = -x^2 + 6x + 91$
Using Pythagorus on $\triangle ACD$:
2) $(x+4)^2 + (x-3)^2 = (x+y)^2$ $\Rightarrow ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938326003365,
"lm_q2_score": 0.8633916187614822,
"openwebmath_perplexity": 434.4336767424758,
"openwebmath_score": 0.8683072924613953,
"tags... |
-
That's not maths! – user1729 Aug 16 '12 at 11:48
@user1729: Actually sampling methods are indeed math, are used quite often, and even work in this instance. I'm not sure why anyone would discourage trying to get an intuitive feel for the problem. – ex0du5 Aug 16 '12 at 20:32
Hey, @user1729, for any problem like this,... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938326003365,
"lm_q2_score": 0.8633916187614822,
"openwebmath_perplexity": 434.4336767424758,
"openwebmath_score": 0.8683072924613953,
"tags... |
i.e. $(13-x) \lt CE \lt (x+1)$
we now have, $(13-x) \lt (x+1)$
i.e. $x \gt 6$
from the, $\triangle EBC$ we have,
$x-3 \lt 10$
i.e. $x \lt 13$
we can conclude that, $6 \lt x \lt 13$
-
right... And now? – t.b. Aug 16 '12 at 8:08
I see... Now $x + 7 \gt 0$ magically becomes $x \gt 7$. – t.b. Aug 16 '12 at 8:19
Now,... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938326003365,
"lm_q2_score": 0.8633916187614822,
"openwebmath_perplexity": 434.4336767424758,
"openwebmath_score": 0.8683072924613953,
"tags... |
# Plot particle motion in potential
Consider the anisotropic harmonic potential in two dimensions $(q_1,q_2)$ given by
$$V(q_1,q_2) = \frac{m}{2} \, q_1^2 + \frac{k}{2} \, q_2^2,$$
or V = m/2 q1^2 + k/2 q2^2; in Mathematica.
The Newtonian e.o.m.s of a particle moving through this potential are
$$\ddot{q}_1 = -q_1,... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.976310526632796,
"lm_q1q2_score": 0.84293832428746,
"lm_q2_score": 0.863391617003942,
"openwebmath_perplexity": 7111.308929883166,
"openwebmath_score": 0.31206706166267395,
"tags":... |
For long times $t \to \infty$ and irrational angular frequency $\omega \notin \mathbb{Q}$ (which ensures that the trajectory never closes, thus making the system ergodic), the particle's trajectory should therefore trace out a rectangle $R$ whose length and width are determined by $E_1$ and $E_2$.
Update: With anderst... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.976310526632796,
"lm_q1q2_score": 0.84293832428746,
"lm_q2_score": 0.863391617003942,
"openwebmath_perplexity": 7111.308929883166,
"openwebmath_score": 0.31206706166267395,
"tags":... |
First, store the 3D trajectory (in the space $(q_1,q_2,V)$):
sol = {q1[t], q2[t], m/2 q1[t]^2 + k/2 q2[t]^2} /.
DSolve[{q1''[t] == -q1[t], q2''[t] == -\[Omega]^2 q2[t],
q1[0] == q10, q1'[0] == p10/m, q2[0] == q20,
q2'[0] == p20/m}, {q1[t], q2[t]}, t] // FullSimplify
Plot the potential surface:
surf = Plot3D[
V /. {... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.976310526632796,
"lm_q1q2_score": 0.84293832428746,
"lm_q2_score": 0.863391617003942,
"openwebmath_perplexity": 7111.308929883166,
"openwebmath_score": 0.31206706166267395,
"tags":... |
V[q1_, q2_] := m/2 q1 q1 + k/2 q2 q2
surf = Plot3D[
V[q1, q2] /. {m -> 1, k -> 3, \[Omega] -> Sqrt[k/m]}, {q1, -5,
5}, {q2, -5, 5},
RegionFunction ->
Function[{q1, q2}, m/2 q1^2 + k/2 q2^2 <= 12 /. {m -> 1, k -> 3}],
Mesh -> None,
ColorFunction ->
Function[{z}, Opacity[0.4, #] &@ColorData["TemperatureMap"][z]]]
traj[t... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.976310526632796,
"lm_q1q2_score": 0.84293832428746,
"lm_q2_score": 0.863391617003942,
"openwebmath_perplexity": 7111.308929883166,
"openwebmath_score": 0.31206706166267395,
"tags":... |
# Chain rule for $y = (x^2 + x^3)^4$
I am trying to find the derivative of $y = (x^2 + x^3)^4$
and it seems pretty simple I get
$4(x^2+x^3)^3 (2x+3x^2)$ This seems to be the proper answer to me but the book gets
$4x^7 (x+1)^3 (3x+2)$ and I have no idea how that happened, what process the author went through or why.... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105273220726,
"lm_q1q2_score": 0.8429383231666705,
"lm_q2_score": 0.8633916152464017,
"openwebmath_perplexity": 591.0649893588262,
"openwebmath_score": 0.9326167106628418,
"tag... |
Draw all the circles that touch the x-axis and a unit circle
I am trying to plot a circle with radius 1, that touches the $x$-axis in the origin, So it has center $(0,1)$
u = Graphics[{Circle[{0, 1}, 1]}, ImageSize -> 150, Axes -> True]
I have the above, and now I would like to draw all the circles with radius u th... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.976310532836284,
"lm_q1q2_score": 0.8429383227798788,
"lm_q2_score": 0.8633916099737806,
"openwebmath_perplexity": 699.9973348747853,
"openwebmath_score": 0.3017362654209137,
"tags... |
Solve[Discriminant[x^2 + (y - 1)^2 - 1 /. First[Solve[rad == 0, y]], x] == 0, h]
{{h -> 0}, {h -> 0}, {h -> -2 Sqrt[u]}, {h -> 2 Sqrt[u]}}
where we get two trivial solutions and a solution for both the right and left parts of the plane.
Now, we can visualize:
Graphics[{Circle[{0, 1}, 1],
MapIndexed[{ColorData[97, #... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.976310532836284,
"lm_q1q2_score": 0.8429383227798788,
"lm_q2_score": 0.8633916099737806,
"openwebmath_perplexity": 699.9973348747853,
"openwebmath_score": 0.3017362654209137,
"tags... |
Which one is right: Riemann integrable iff the set of discontinuity is countable or a null set?
When I was studying calculus (I used Purcell's book), it is stated that the bounded function $f$ on closed bounded interval is Riemann integrable if and only if $f$ has countable discontinuity points, which means if $f$ is ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105287006256,
"lm_q1q2_score": 0.8429383209250912,
"lm_q2_score": 0.863391611731321,
"openwebmath_perplexity": 288.2771775793568,
"openwebmath_score": 0.938498318195343,
"tags"... |
See these related questions:
Characteristic function of Cantor set is Riemann integrable
Example of Riemann integrable $f: [0,1] \to \mathbb R$ whose set of discontinuity points is an uncountable and dense set in $[0,1]$
• thank you so much!!! Then, does it mean the theorem "riemann integrable if and only if discont... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105287006256,
"lm_q1q2_score": 0.8429383209250912,
"lm_q2_score": 0.863391611731321,
"openwebmath_perplexity": 288.2771775793568,
"openwebmath_score": 0.938498318195343,
"tags"... |
# calculus
find the slope of the tangent line to the curve (sqrt 2x+4y) + (sqrt 4xy) = 9.16
at the point (1,5)
dy/dx method implicit differantiation?
1. 👍
2. 👎
3. 👁
1. Yes, implicit differentation.
1/2(sqrt2x+4y) * (2 dx+4dy)= 1/2sqrt(4xy)* (4ydx+4xdy)=0
and solve for dy/dx. Have fun.
1. 👍
2. 👎
👤
bobpursley
... | {
"domain": "jiskha.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105259435196,
"lm_q1q2_score": 0.8429383202605343,
"lm_q2_score": 0.8633916134888614,
"openwebmath_perplexity": 1206.008694368618,
"openwebmath_score": 0.8027140498161316,
"tags": nul... |
4. ### Calculus
Consider the curve given by y^2 = 2+xy (a) show that dy/dx= y/(2y-x) (b) Find all points (x,y) on the curve where the line tangent to the curve has slope 1/2. (c) Show that there are now points (x,y) on the curve where the line
1. ### math
Find the slope m of the tangent to the curve y = 5/ sqrt (x) ... | {
"domain": "jiskha.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105259435196,
"lm_q1q2_score": 0.8429383202605343,
"lm_q2_score": 0.8633916134888614,
"openwebmath_perplexity": 1206.008694368618,
"openwebmath_score": 0.8027140498161316,
"tags": nul... |
+421 907 627 998
# maximum turning point | {
"domain": "farbykusnir.sk",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938312276124,
"lm_q2_score": 0.8633916047011595,
"openwebmath_perplexity": 412.4596664854432,
"openwebmath_score": 0.6353996396064758,
"tag... |
The point at which a very significant change occurs; a decisive moment. Write down the nature of the turning point and the equation of the axis of symmetry. When the function has been re-written in the form y = r(x + s)^2 + t, the minimum value is achieved when x = -s, and the value of y will be equal to t.. d/dx (12x ... | {
"domain": "farbykusnir.sk",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938312276124,
"lm_q2_score": 0.8633916047011595,
"openwebmath_perplexity": 412.4596664854432,
"openwebmath_score": 0.6353996396064758,
"tag... |
can be determined using the second derivative. If $\frac{dy}{dx}=0$ (is a stationary point) and if $\frac{d^2y}{dx^2}<0$ at that same point, them the point must be a maximum. (3) The region R, shown shaded in Figure 2, is bounded by the curve, the y-axis and the line from O to A, where O is the origin. By Yang Kuang, E... | {
"domain": "farbykusnir.sk",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938312276124,
"lm_q2_score": 0.8633916047011595,
"openwebmath_perplexity": 412.4596664854432,
"openwebmath_score": 0.6353996396064758,
"tag... |
for a polynomial of degree n is n – The total number of turning points for a polynomial with an even degree is an odd number. If d2y dx2 is negative, then the point is a maximum turning point. n. 1. How to find and classify stationary points (maximum point, minimum point or turning points) of curve. Any polynomial of d... | {
"domain": "farbykusnir.sk",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938312276124,
"lm_q2_score": 0.8633916047011595,
"openwebmath_perplexity": 412.4596664854432,
"openwebmath_score": 0.6353996396064758,
"tag... |
(degree 2) have one turning point, linear equations (degree 1) have none, and cubic equations (degree 3) have 2 turning … However, this depends on the kind of turning point. A turning point is a point at which the derivative changes sign. Define turning point. A turning point is where a graph changes from increasing to... | {
"domain": "farbykusnir.sk",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938312276124,
"lm_q2_score": 0.8633916047011595,
"openwebmath_perplexity": 412.4596664854432,
"openwebmath_score": 0.6353996396064758,
"tag... |
be at the roots of the derivation, i.e. The Degree of a Polynomial with one variable is the largest exponent of that variable. Find more Education widgets in Wolfram|Alpha. A turning point is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or … A stationary point on a curv... | {
"domain": "farbykusnir.sk",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938312276124,
"lm_q2_score": 0.8633916047011595,
"openwebmath_perplexity": 412.4596664854432,
"openwebmath_score": 0.6353996396064758,
"tag... |
max. Therefore, to find where the minimum or maximum occurs, set the derivative equal to … In this case: Polynomials of odd degree have an even number of turning points, with a minimum of 0 and a maximum of #n-1#. To find the turning point of a quadratic equation we need to remember a couple of things: The parabola ( t... | {
"domain": "farbykusnir.sk",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938312276124,
"lm_q2_score": 0.8633916047011595,
"openwebmath_perplexity": 412.4596664854432,
"openwebmath_score": 0.6353996396064758,
"tag... |
or from decreasing to increasing only '' when its expression is set to zero no... Point for the interval happens at the other endpoint and the absolute minimum point at ( 0,0 ) '' defined! Powerpoint presentation that leads through the process of finding maximum / minimum turning points, though a significant... We hit ... | {
"domain": "farbykusnir.sk",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938312276124,
"lm_q2_score": 0.8633916047011595,
"openwebmath_perplexity": 412.4596664854432,
"openwebmath_score": 0.6353996396064758,
"tag... |
dictionary definition of turning points any! Concave down– it is a point of inflection = q\ ) using differentiation visa-versa is as! That selected point either a relative minimum ( also known as a turning and... Other sorts of behaviour that the x-coordinate of a function does not have to have their highest and values... | {
"domain": "farbykusnir.sk",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938312276124,
"lm_q2_score": 0.8633916047011595,
"openwebmath_perplexity": 412.4596664854432,
"openwebmath_score": 0.6353996396064758,
"tag... |
derivative test does not us... Function we must first differentiate the function turning points and a minimum zero..., there may be a point where a function occurs when dy/dx = 0 is for! Function changes from an increasing to a decreasing function or visa-versa is known as local,... Off with simple examples, explaining... | {
"domain": "farbykusnir.sk",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938312276124,
"lm_q2_score": 0.8633916047011595,
"openwebmath_perplexity": 412.4596664854432,
"openwebmath_score": 0.6353996396064758,
"tag... |
equation for finding /... So if d2y dx2 = 0 this second derivative test does not give us … By Yang Kuang Elleyne... I could n't write everything, but I have no idea vertical line drawn through the vertex represents the point! That leads through the process of finding maximum / minimum turning point, then the point is a... | {
"domain": "farbykusnir.sk",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938312276124,
"lm_q2_score": 0.8633916047011595,
"openwebmath_perplexity": 412.4596664854432,
"openwebmath_score": 0.6353996396064758,
"tag... |
but I have no idea illustrated in Figure \ ( \PageIndex { }... Defined as ` local maximum or a minimum turning point: a local minimum, smallest. | {
"domain": "farbykusnir.sk",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938312276124,
"lm_q2_score": 0.8633916047011595,
"openwebmath_perplexity": 412.4596664854432,
"openwebmath_score": 0.6353996396064758,
"tag... |
Aktuálne akcie
Opýtajte sa nás
Súhlasím so spracovaní osobných údajov. | {
"domain": "farbykusnir.sk",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9763105266327962,
"lm_q1q2_score": 0.842938312276124,
"lm_q2_score": 0.8633916047011595,
"openwebmath_perplexity": 412.4596664854432,
"openwebmath_score": 0.6353996396064758,
"tag... |
$f,g$ continuous from $X$ to $Y$. if they are agree on a dense set $A$ of $X$ then they agree on $X$
Problem:
Suppose $f$ and $g$ are two continuous functions such that $f: X \to Y$ and $g : X \to Y$. $Y$ is a a Hausdorff space. Suppose $f(x) = g(x)$ for all $x \in A \subseteq X$ where $A$ is dense in $X$, then $f(x)... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846697584033,
"lm_q1q2_score": 0.8429157782575687,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 60.343194526093605,
"openwebmath_score": 0.9562152624130249,
"tag... |
• Why there exists $a \in A \cap W$ ?? – ILoveMath Oct 29 '13 at 9:59
• Because $A$ is dense, it intersects every non-empty open set. – Prahlad Vaidyanathan Oct 29 '13 at 10:02
• I understand now! thanks a lot for your time. One last question. Is it necessary to have $Y$ haussdorf? IF we drop this condition, can we sti... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846697584033,
"lm_q1q2_score": 0.8429157782575687,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 60.343194526093605,
"openwebmath_score": 0.9562152624130249,
"tag... |
Setting up notation: Let $f,g : X\to Y$ be continuous functions, with $Y$ Hausdorff. Let $f,g$ agree on a dense subset $A$ of $X$. Let $\Delta_X : X\to X\times X$ denote the diagonal map and let $D_Y\subset Y\times Y$ denote the diagonal as a subset of $Y\times Y$, finally let $(f,g)$ denote the product map from $X\tim... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846697584033,
"lm_q1q2_score": 0.8429157782575687,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 60.343194526093605,
"openwebmath_score": 0.9562152624130249,
"tag... |
# Is $| \lceil \frac{a}{2} \rceil - \lceil \frac{b}{2} \rceil |\geq \lfloor |\frac{a - b}{2}| \rfloor$?
Let $$a$$ and $$b$$ be integers. Is it true that
$$\left | \left \lceil \frac{a}{2} \right \rceil - \left \lceil \frac{b}{2} \right \rceil \right |\geq \left \lfloor \left | \frac{a - b}{2} \right |\right \rfloor$$... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846697584033,
"lm_q1q2_score": 0.842915774779291,
"lm_q2_score": 0.8615382076534742,
"openwebmath_perplexity": 83.87450898888883,
"openwebmath_score": 0.9506146907806396,
"tags... |
If $$m-n+\frac 12\ge 0$$, then $$m-n+1\ge 0$$, so$$(3)\iff m-n+1\ge m-n$$which is true.
If $$m-n+\frac 12\lt 0$$, then $$m-n+1\lt 0$$, so$$(3)\iff -m+n-1\ge -m+n-1$$which is true.
Case 4 : If $$a=2m+1,b=2n+1$$, then both sides of $$(1)$$ equal $$|m-n|$$.
There is no need for the assumption that $$a$$ and $$b$$ are i... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846697584033,
"lm_q1q2_score": 0.842915774779291,
"lm_q2_score": 0.8615382076534742,
"openwebmath_perplexity": 83.87450898888883,
"openwebmath_score": 0.9506146907806396,
"tags... |
EDIT: I didn't notice you assumed $$a$$ and $$b$$ to be integers. Well, my answer works for all real numbers. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846697584033,
"lm_q1q2_score": 0.842915774779291,
"lm_q2_score": 0.8615382076534742,
"openwebmath_perplexity": 83.87450898888883,
"openwebmath_score": 0.9506146907806396,
"tags... |
# Modified blackjack probability
So there's this one gambling site and they have game of 'blackjack' which is played very differently than the usual game.
The host has dice which has all the numbers from 1 to 100. He rolls it for you as many times as you want, the rolls are added up and you must not exceed 100. If th... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846691281407,
"lm_q1q2_score": 0.8429157742362958,
"lm_q2_score": 0.8615382076534743,
"openwebmath_perplexity": 634.9516468687099,
"openwebmath_score": 0.497651070356369,
"tags... |
• Since this depends on strategy (when does the first player stop?) it is probably best to sample the game. That is, play thousands (or millions) of the game first to determine optimal strategy and then to determine probabilities. – lulu Dec 22 '17 at 10:57
• @lulu - I think a full calculation might be easier than a si... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846691281407,
"lm_q1q2_score": 0.8429157742362958,
"lm_q2_score": 0.8615382076534743,
"openwebmath_perplexity": 634.9516468687099,
"openwebmath_score": 0.497651070356369,
"tags... |
lulu asked for a simulation to check - the following R script comes close to the same value for $w_0$ when staying at $58$ or above though does not prove optimality (choosing $57$ or $59$ to stay would give similar results)
stayat <- 58
sides <- 100
maxtarget <- sides
cases <- 1000000
set.seed(1)
playertot <- rep(0, ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846691281407,
"lm_q1q2_score": 0.8429157742362958,
"lm_q2_score": 0.8615382076534743,
"openwebmath_perplexity": 634.9516468687099,
"openwebmath_score": 0.497651070356369,
"tags... |
Solve the following recurrence relation:
Solve the following recurrence relation: $f(1) = 1$ and for $n \ge 2$,
$$f(n) = n^2f(n − 1) + n(n!)^2$$
How would I go about solving this?
• Would I need to find a substitution $f(n) =\text{ insert here }g(n)$ in aim of getting rid of the $n^2$ that is multiplied onto $f(n-1... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846640860382,
"lm_q1q2_score": 0.8429157733706094,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 585.9354100199414,
"openwebmath_score": 0.769763708114624,
"tags"... |
# Naturality of Flows
This is something I always forget exists and has a name, so I end up reproving it. Since this sequence of posts is a hodge-podge of things to help me take a differential geometry test, hopefully this will lodge the result in my brain and save me time if it comes up.
I’m not sure whether to call ... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.842915773150305,
"lm_q2_score": 0.8615382076534743,
"openwebmath_perplexity": 120.88430050501222,
"openwebmath_score": 0.9643252491950989,
"tags": ... |
We get a nice corollary out of this. If our function ${F:M\rightarrow N}$ was actually a diffeo, then take ${Y=F_*X}$ the pushforward, and we get that the flow of the pushforward is ${\eta_t=F\circ \theta_t\circ F^{-1}}$ and the flow domain is actually equal ${N_t=F(M_t)}$.
In algebraic geometry we care a lot about fa... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.842915773150305,
"lm_q2_score": 0.8615382076534743,
"openwebmath_perplexity": 120.88430050501222,
"openwebmath_score": 0.9643252491950989,
"tags": ... |
# Lie groups have abelian fundamental group
Last year I wrote up how to prove that the fundamental group of a (connected) topological group was abelian. Since Lie groups are topological groups, they also have abelian fundamental groups, but I think there is a much neater way to prove this fact using smooth things. Her... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.842915773150305,
"lm_q2_score": 0.8615382076534743,
"openwebmath_perplexity": 120.88430050501222,
"openwebmath_score": 0.9643252491950989,
"tags": ... |
# PDE’s and Frobenius Theorem
I’ve started many blog posts on algebra/algebraic geometry, but they won’t get finished and posted for a little while. I’ve been studying for a test I have to take in a few weeks in differential geometry-esque things. So I’ll do a few posts on things that I think are usually considered pr... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.842915773150305,
"lm_q2_score": 0.8615382076534743,
"openwebmath_perplexity": 120.88430050501222,
"openwebmath_score": 0.9643252491950989,
"tags": ... |
Choose a regular value of ${f}$, say ${C}$ (one exists by say Sard’s Theorem). Then ${f=C}$ is a 2-dimensional submanifold ${M\subset \mathbb{R}^3}$, and since ${f}$ is a defining function ${T_pM=ker(Df_p)}$. But the very fact that ${f}$ satisfies, by assumption, ${X(f)=0}$ and ${Y(f)=0}$, we have ${T_pM=\text{span} \{... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.842915773150305,
"lm_q2_score": 0.8615382076534743,
"openwebmath_perplexity": 120.88430050501222,
"openwebmath_score": 0.9643252491950989,
"tags": ... |
Then use that assumption to see that ${[X,Y]=0}$ and hence the distribution is involutive and hence there is an integral manifold for the distribution by the Frobenius Theorem. If ${g}$ is a local defining function to that integral manifold, then we can hit that with the Implicit Function Theorem and get that ${z=f(x,y... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.842915773150305,
"lm_q2_score": 0.8615382076534743,
"openwebmath_perplexity": 120.88430050501222,
"openwebmath_score": 0.9643252491950989,
"tags": ... |
Now we want to show that we actually have an induced isomorphism on cohomology from the inclusion map $i: \bigwedge \hookrightarrow A\otimes \bigwedge$. We’ll do this by comparing Fourier series. So we set up a normalized measure on $X$ say $\mu$. By integrating functions against this measure we get linear function $\m... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.842915773150305,
"lm_q2_score": 0.8615382076534743,
"openwebmath_perplexity": 120.88430050501222,
"openwebmath_score": 0.9643252491950989,
"tags": ... |
Lastly we want to get back to showing that the inclusion is a homotopy equivalence. Thus use the Hermitian inner product to define a map $\lambda^*\in Hom_\mathbb{C}(\overline{T}, \mathbb{C})$ for every $\lambda$ by $\displaystyle \lambda^*(x)=\frac{\langle x, \overline{C}(\lambda)\rangle}{2\pi i \|\overline{C}(\lambda... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.842915773150305,
"lm_q2_score": 0.8615382076534743,
"openwebmath_perplexity": 120.88430050501222,
"openwebmath_score": 0.9643252491950989,
"tags": ... |
To do the calculation we will use the Dolbeault resolution: ${0\rightarrow \mathcal{O}_X\rightarrow \mathcal{C}^{0,0}\rightarrow \mathcal{C}^{0,1}\rightarrow \mathcal{C}^{0,2}\rightarrow\cdots}$. This is an acyclic resolution of the structure sheaf, and so is fine to use for the calculation of cohomology. The first upp... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.842915773150305,
"lm_q2_score": 0.8615382076534743,
"openwebmath_perplexity": 120.88430050501222,
"openwebmath_score": 0.9643252491950989,
"tags": ... |
Explicitly, we know that ${H^q(X, \mathcal{O}_X)\simeq H^q(X, \Gamma(X,\mathcal{C})\otimes_\mathbb{C} \bigwedge^q\overline{T})}$.
Since this notation is cumbersome, let ${A=\Gamma(X, \mathcal{C})}$ and ${\bigwedge^*=\bigwedge^*\overline{T}}$.
Let ${i: \bigwedge \rightarrow A\otimes_\mathbb{C} \bigwedge}$ be the inclu... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.842915773150305,
"lm_q2_score": 0.8615382076534743,
"openwebmath_perplexity": 120.88430050501222,
"openwebmath_score": 0.9643252491950989,
"tags": ... |
Under the exponential map ${exp: V\rightarrow X}$ (which we showed was a local isomorphism), we have that ${V}$ is the universal covering space of ${X}$. We showed that ${ker(exp)=U}$ is a lattice. Now the title of this post will seem a little silly to experts out there in cohomology, since we know that topologically t... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.842915773150305,
"lm_q2_score": 0.8615382076534743,
"openwebmath_perplexity": 120.88430050501222,
"openwebmath_score": 0.9643252491950989,
"tags": ... |
But now our inductive hypothesis is that for all $p$ less than $r$, ${\bigwedge^p(H^1(X_i))\simeq H^p(X_i)}$. Thus we get ${\displaystyle \sum_{p+q=r}\bigwedge^p(H^1(X_1))\otimes \bigwedge^q(H^1(X_2))}$
${\displaystyle \simeq \sum_{p+q=r} H^p(X_1)\otimes H^q(X_2)}$
${\simeq H^r(X_1\times X_2)}$. In other words, stringi... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.842915773150305,
"lm_q2_score": 0.8615382076534743,
"openwebmath_perplexity": 120.88430050501222,
"openwebmath_score": 0.9643252491950989,
"tags": ... |
# Complex Lie Group Properties
Today we’ll do two more properties of compact complex Lie groups. The property we’ve already done is that they are always abelian groups. We go back to the notation from before and let $X$ be a compact complex Lie group and $V=T_eX$.
Property 1: $X$ is abelian.
Property 2: $X$ is a com... | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.842915773150305,
"lm_q2_score": 0.8615382076534743,
"openwebmath_perplexity": 120.88430050501222,
"openwebmath_score": 0.9643252491950989,
"tags": ... |
This is a good stopping point, since next time we’ll start thinking about the cohomology of $X$. | {
"domain": "wordpress.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.842915773150305,
"lm_q2_score": 0.8615382076534743,
"openwebmath_perplexity": 120.88430050501222,
"openwebmath_score": 0.9643252491950989,
"tags": ... |
# Infinite sum of alternating telescoping series
I am struggling to find the sum of the following series:
$$\sum_{n=1} ^{\infty} \frac{(-1)^n}{(n+1)(n+3)(n+5)}.$$
It seems as though it should be a straightforward telescoping series. I attempted to solve it in the usual way (via partial fractions), but the alternatin... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846653465638,
"lm_q1q2_score": 0.842915769239184,
"lm_q2_score": 0.8615382058759129,
"openwebmath_perplexity": 251.81707859616373,
"openwebmath_score": 0.8526610732078552,
"tag... |
• Could you explain the last part in further detail please? Specifically, from "just combine..." and on. I've never used a meromorphic function before, so I'm fairly confused. Thank you! – kathystehl Jun 23 '15 at 21:04
• @kathystehl: you may just ignore the part about the meromorphic function. It is just the intrinsic... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846653465638,
"lm_q1q2_score": 0.842915769239184,
"lm_q2_score": 0.8615382058759129,
"openwebmath_perplexity": 251.81707859616373,
"openwebmath_score": 0.8526610732078552,
"tag... |
Splitting S_Even and S_Odd
$S_{Even} = \frac{1}{8.3} - \frac{1}{4.5}+\frac{1}{8.5}$
$S_{Odd} = -\frac{1}{8.2} +\frac{1}{4.4} - \frac{1}{8.4}$
Everything else cancels out
When you sum these you get $S = \dfrac{-7}{480}$
Provided I have not made any calculation error | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846653465638,
"lm_q1q2_score": 0.842915769239184,
"lm_q2_score": 0.8615382058759129,
"openwebmath_perplexity": 251.81707859616373,
"openwebmath_score": 0.8526610732078552,
"tag... |
How can we calculate the probability that the randomly chosen function will be strictly increasing?
Consider the set of all functions from $\{1,2,...,m\}$ to $\{1,2,...,n\}$, where $n > m$. If a function is chosen from this set at random, what is the probability that it will be strictly increasing? | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846659768266,
"lm_q1q2_score": 0.8429157663039019,
"lm_q2_score": 0.86153820232079,
"openwebmath_perplexity": 105.12672737503156,
"openwebmath_score": 0.9371325969696045,
"tags... |
• @Carl: The question is perfectly clear. Given a set $X$ to be the domain and a set $Y$ to be the codomain, you can certainly talk about the set of all functions from $X$ to $Y$. This set is often denoted $Y^X$. In the OP, $X$ and $Y$ are finite, so $Y^X$ is finite as well ($|Y^X|=|Y|^{|X|}=n^m$), and you can talk abo... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846659768266,
"lm_q1q2_score": 0.8429157663039019,
"lm_q2_score": 0.86153820232079,
"openwebmath_perplexity": 105.12672737503156,
"openwebmath_score": 0.9371325969696045,
"tags... |
• @Carl: As you said, and as is clear from the definition I gave above, given a function $f:\{1,2,\ldots,m\}\to\{1,2,\ldots,n\}$, for every $1\le i\le m$ there is exactly one $1\le j\le n$ such that $f(i)=j$. Or, to use, the construction of functions as a set of ordered pairs, for every $1\le i\le m$ there is exactly o... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846659768266,
"lm_q1q2_score": 0.8429157663039019,
"lm_q2_score": 0.86153820232079,
"openwebmath_perplexity": 105.12672737503156,
"openwebmath_score": 0.9371325969696045,
"tags... |
Let us pick $m$ elements from $\{1,\dotsc,n\}$, let us call these $a_1 < a_2 < \dotsc , a_m$. Clearly these define a strictly increasing function $f$ from $\{1,\dotsc,m\} \to \{1,\dotsc,n\}$ via the rule $f(i) = a_i$. Furthermore, any strictly increasing function defined on the above sets is of this form.
Hence there ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846659768266,
"lm_q1q2_score": 0.8429157663039019,
"lm_q2_score": 0.86153820232079,
"openwebmath_perplexity": 105.12672737503156,
"openwebmath_score": 0.9371325969696045,
"tags... |
$$S(n,m) = {n \choose m} = \frac{n!}{m!(n-m)!}.$$
(There are other combinatorial arguments that also lead you to this result. For example, choosing an increasing function is equivalent to choosing $m$ values in the co-domain, which are then placed in increasing order.) Now, to get the result we need to be clear on exa... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846659768266,
"lm_q1q2_score": 0.8429157663039019,
"lm_q2_score": 0.86153820232079,
"openwebmath_perplexity": 105.12672737503156,
"openwebmath_score": 0.9371325969696045,
"tags... |
How to Compute the Derivative of a Sigmoid Function (fully worked example)
Last updated on October 2nd, 2017
This is a sigmoid function:
$\boldsymbol{s(x) = \frac{1}{1 + e^{-x}}}$
The sigmoid function looks like this (made with a bit of MATLAB code):
x=-10:0.1:10; s = 1./(1+exp(-x)); figure; plot(x,s); title('sigm... | {
"domain": "kawahara.ca",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9916842207852199,
"lm_q1q2_score": 0.8429030105594675,
"lm_q2_score": 0.8499711832583695,
"openwebmath_perplexity": 3155.2732819600574,
"openwebmath_score": 0.9029098749160767,
"tags": n... |
[note that $\frac{d}{dx}s(x)$ and s'(x) are the same thing, just different notation.]
[also note that Andrew Ng writes, f'(z) = f(z)(1 – f(z)), where f(z) is the sigmoid function, which is the exact same thing that we are doing here.]
So your next question should be, is our derivative we calculated earlier equivalent... | {
"domain": "kawahara.ca",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9916842207852199,
"lm_q1q2_score": 0.8429030105594675,
"lm_q2_score": 0.8499711832583695,
"openwebmath_perplexity": 3155.2732819600574,
"openwebmath_score": 0.9029098749160767,
"tags": n... |
Yes! They perfectly match!
So there you go. Hopefully this satisfies your mathematical curiosity of why the derivative of a sigmoid s(x) is equal to s'(x) = s(x)(1-s(x)).
23 thoughts on “How to Compute the Derivative of a Sigmoid Function (fully worked example)”
1. Jeremy says:
I think if you 1) rewrite my equation... | {
"domain": "kawahara.ca",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9916842207852199,
"lm_q1q2_score": 0.8429030105594675,
"lm_q2_score": 0.8499711832583695,
"openwebmath_perplexity": 3155.2732819600574,
"openwebmath_score": 0.9029098749160767,
"tags": n... |
# What will be the limit points of the set, $S=\{(-1)^{n} \mid n\in \mathbb{N}\}$
Since the set, S, turns out to be a finite set consisting of just two elements, i.e. $\{-1,1\}$, therefore there should be no limit points to the set.
But the solution given in the book is, the set S has two limit points $-1$ and $1$.
... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9643214480969029,
"lm_q1q2_score": 0.8428914362339052,
"lm_q2_score": 0.8740772466456689,
"openwebmath_perplexity": 141.27394644595952,
"openwebmath_score": 0.9669914841651917,
"ta... |
For more general discussions, see the Wikipedia article on Limit point.
• But still the neighbourhood of say 1 does not contain any of the point of the set except for 1. Dec 21, 2017 at 20:10
• @AjayChoudhary The idea here is that the definition of limit point for a set and limit point for a sequence are different - i... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9643214480969029,
"lm_q1q2_score": 0.8428914362339052,
"lm_q2_score": 0.8740772466456689,
"openwebmath_perplexity": 141.27394644595952,
"openwebmath_score": 0.9669914841651917,
"ta... |
# How do I formally show the radius of convergence of the Taylor series of $f(x)=x^6 - x^4 + 2$ at $a=-2$?
This is an exercise in Stewart's Calculus (Exercise 19, Section 11.10 Taylor and Maclaurin Series):
Find the Taylor series for $$f(x)$$ centered at the given value of a. [Assume that f has a power series expansi... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9643214480969029,
"lm_q1q2_score": 0.8428914314891126,
"lm_q2_score": 0.8740772417253256,
"openwebmath_perplexity": 117.2957915593481,
"openwebmath_score": 0.9433876276016235,
"tag... |
• Mind writing it out? The general term? I'm a bit lost Apr 17 '19 at 4:33
• The general term for $n>6$ is $f^{(n)}=0$. You have the first six terms already calculated. Just calculate the next derivative. The derivative of a constant is ... I just realized that you general formula does not even apply to the first 4 ter... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9643214480969029,
"lm_q1q2_score": 0.8428914314891126,
"lm_q2_score": 0.8740772417253256,
"openwebmath_perplexity": 117.2957915593481,
"openwebmath_score": 0.9433876276016235,
"tag... |
Note that (3) can be written formally as a power series, which seems to be what you were looking for, as follows: $$\sum_{n=0}^\infty c_n(x+2)^n\quad\textrm{where } c_0=f(-2),\ c_1=\frac{f'(-2)}{1!},\cdots, c_6=\frac{f^{(6)}(-2)}{6!},\ c_7=c_8=\cdots =0.\tag{4}$$ You do not need Root Test or Ratio Test to get the radiu... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9643214480969029,
"lm_q1q2_score": 0.8428914314891126,
"lm_q2_score": 0.8740772417253256,
"openwebmath_perplexity": 117.2957915593481,
"openwebmath_score": 0.9433876276016235,
"tag... |
# Laplace Transform f(t)=tcos(t)
## Homework Statement
I need to find the laplace transform of f(t)=tcos(t).
## Homework Equations
$$\int e^-^s^ttcos(t)dt$$
## The Attempt at a Solution
I just need help on how to integrate this. I can find the answer easily using the f(t)=tcos(kt) general formula but I wish to fi... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9643214501476359,
"lm_q1q2_score": 0.842891428536819,
"lm_q2_score": 0.8740772368049822,
"openwebmath_perplexity": 523.0818100531984,
"openwebmath_score": 0.9713577032089233,
"tags... |
then it would appear that $$\frac {d^{2}}{ds^{2}} \mathcal{L} [cos(t)] = \mathcal{L} [t^{2}cos(t)]$$
Another trick: To obtain the Laplace transform of, say, sin(t)/t you can compute the Laplace transform of sin(t) and then integrate w.r.t. s from p to infinity. The Laplace transform is then a function of the parameter... | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9643214501476359,
"lm_q1q2_score": 0.842891428536819,
"lm_q2_score": 0.8740772368049822,
"openwebmath_perplexity": 523.0818100531984,
"openwebmath_score": 0.9713577032089233,
"tags... |
It is currently 21 Sep 2017, 12:43
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based o... | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9500414759334155,
"lm_q1q2_score": 0.8428811701701058,
"lm_q2_score": 0.8872046026642945,
"openwebmath_perplexity": 1562.3119143732345,
"openwebmath_score": 0.7175856232643127,
"tag... |
Senior Manager
Joined: 20 Aug 2015
Posts: 396
Kudos [?]: 317 [0], given: 10
Location: India
GMAT 1: 760 Q50 V44
If a circle is inscribed in an equilateral triangle, what is the area [#permalink]
### Show Tags
03 Mar 2016, 02:47
Expert's post
3
This post was
BOOKMARKED
Bunuel wrote:
If a circle is inscribed in an eq... | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9500414759334155,
"lm_q1q2_score": 0.8428811701701058,
"lm_q2_score": 0.8872046026642945,
"openwebmath_perplexity": 1562.3119143732345,
"openwebmath_score": 0.7175856232643127,
"tag... |
r = a * ($$\sqrt{3}$$/ 6)
So if we have either of the side or the radius, we can find the other thing.
The relation can be found by using the figure below:
Attachment:
circle in triangle.JPG
Statement 1: The area of the circle is 12π
We can find the radius of the circle and hence the side of the triangle and the corr... | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9500414759334155,
"lm_q1q2_score": 0.8428811701701058,
"lm_q2_score": 0.8872046026642945,
"openwebmath_perplexity": 1562.3119143732345,
"openwebmath_score": 0.7175856232643127,
"tag... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.