text stringlengths 1 2.12k | source dict |
|---|---|
# Plot results
fig, ax = plt.subplots()
ax.plot(theta_hat_grid, errors, '-o', label='MSE', c='C1')
ax.axvline(theta, color='g', ls='--', label=r"$\theta_{True}$")
ax.axvline(theta_hat, color='r', ls='-', label=r"$\hat{{\theta}}_{MSE}$")
ax.set(
title=fr"Best fit: $\hat{{\theta}}$ = {theta_hat:.2f}, MSE = {best_error:.2... | {
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"openwebmath_score": 0.6878114938735962,
"tags": n... |
We will then use your function to compute $$\hat{\theta}$$ and plot the resulting prediction on top of the data.
def solve_normal_eqn(x, y):
"""Solve the normal equations to produce the value of theta_hat that minimizes
MSE.
Args:
x (ndarray): An array of shape (samples,) that contains the input values.
y (ndarray): ... | {
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Notation¶
(47)\begin{align} x_{i} &\quad \text{input, independent variable}\\ y_{i} &\quad \text{measurement, dependent variable}\\ \mathbf{x} &\quad \text{vector of input values}\\ \mathbf{y} &\quad \text{vector of measurements}\\ \hat{y}_{i} &\quad \text{estimate of dependent variable}\\ \epsilon_{i} &\quad \text{me... | {
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"lm_q1q2_score": 0.8431536976940098,
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"openwebmath_score": 0.6878114938735962,
"tags": n... |
# Convergence of infinite series $\sum_{k=2}^{\infty}\frac{(-1)^k}{k\ln(k)}$
In a recent assignment, as an intermediate step, I need to show that $$\sum_{k=2}^{\infty}\frac{(-1)^k}{k\ln(k)}$$ converges. It is not hard to see that $$\sum_{k=2}^{\infty}\frac{1}{k\ln(k)}$$ is divergent, therefore I think one has to deal ... | {
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"lm_q2_score": 0.8539127566694178,
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"openwebmath_score": 0.88047856092453,
"tags... |
# Finding eigenvalues of $A^{10} + A^7 + 5A$.
Problem: Let $A = \begin{pmatrix} 1 & 2 & -1 \\ 0 & 5 & -2 \\ 0 & 6 & -2 \end{pmatrix}$.
1) Compute the eigenvalues of $A^{10} + A^7 + 5A$.
2) Compute $A^{10} X$ for the vector $X = \begin{pmatrix} 2 \\ 4 \\ 7 \end{pmatrix}$.
Attempt at solution: I first computed the ei... | {
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• The theorem can be further generalized to polynomial case (your guess is right and not hard to verify). – Zhanxiong Jul 13 '15 at 21:16
• If $Ax = \lambda x$ then $(A^{10} + A^7 + 5A)(x) = A^{10}x + A^7x + 5Ax = \lambda^{10}x + \lambda^7 x + 5 \cdot \lambda x = (\lambda^{10} + \lambda^7 + 5 \cdot \lambda) \cdot x$. –... | {
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• I don't understand how $v_3$ can be a different eigenvector from $v_2$, when the eigenvalue is $1$ in both cases? How did you compute $v_3$? – Kamil Jul 13 '15 at 22:35
• If an eigenvalue $\lambda$ has multiplicity $> 1$, such as $\lambda = 1$ in our case, it can happen that there are two linearly independent eigenve... | {
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## Questions:
There is an island which is represented by square matrix nxn.
A person on the island is standing at any given co-ordinates (x,y). He can move in any direction one step right, left, up, down on the island. If he steps outside the island, he dies.
Let the island be represented as (0,0) to (n-1,n-1) (i.e ... | {
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"openwebmath_perplexity": 801.352328315279,
"openwebmath_score": 0.5746873021125793,
"tags": null... |
Python code:
from itertools import product as prod
def prob_death(island_size, steps):
if island_size < 1 or steps < 0: raise ValueError
new_prob = [[0. for i in range(island_size)] for j in range(island_size)]
if steps == 0:
return new_prob
old_prob = prob_death(island_size, steps - 1)
directions = [(0, -1), (1, 0),... | {
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"openwebmath_score": 0.5746873021125793,
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>>> mpr(prob_death(5, 5))
0.806641 0.666016 0.622070 0.666016 0.806641
0.666016 0.437500 0.349609 0.437500 0.666016
0.622070 0.349609 0.261719 0.349609 0.622070
0.666016 0.437500 0.349609 0.437500 0.666016
0.806641 0.666016 0.622070 0.666016 0.806641
The probability of dying in 5 steps. I can’t verify the exact value... | {
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"openwebmath_perplexity": 801.352328315279,
"openwebmath_score": 0.5746873021125793,
"tags": null... |
Question
# Let $$f(x)$$ and $$g(x)$$ be differentiable for $$0\leq x \leq1$$, such that $$f(0)=0$$, $$g(0)=0$$, $$f(1)=6$$. Let there exists a real number $$c$$ in $$(0,1)$$ such that $$f'(c)=2g'(c)$$. Then the value of $$g(1)$$ must be
A
1
B
3
C
2
D
1
Solution
## The correct option is C $$3$$From lagrange's theore... | {
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"tags": nu... |
# Points of Intersection for Two Logarithmic Spirals
So I have two logarithmic spirals in parametric form $$x(t) = ae^{bt}\cos t \\ y(t)=ae^{bt}\sin t$$ and $$x'(t) = \alpha e^{\beta t}\cos t \\ y'(t)=\alpha e^{\beta t}\sin t$$ With $\beta$ and $b$ having opposite signs so the spirals grow in opposite directions. Sett... | {
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"openwebmath_score": 0.9046685099601746,
"ta... |
In this reply I will give a solution for the intersection of two logarithmic spirals.
Consider two spirals in the complex plane with different angular regions, e.g.,
$$z=e^{(b+i)u}\\ w=e^{(\beta+i)v}$$
and we seek all the points where $z=w$. We can expand these and separate the real and imaginary parts to obtain
$$... | {
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"ta... |
• Thank you so much! This is precisely what I was trying to do. After your comment last night I played around a bit and realized I should switch to a complex valued function, but then today I was still thoroughly stumped. I don't think I could have figured most of that out on my own, maybe back while taking complex var... | {
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"openwebmath_perplexity": 156.84178299424192,
"openwebmath_score": 0.9046685099601746,
"ta... |
# how to solve for a continuous function?
I am having a midterm review in school and there's one concept that I forgot how to solve, and that is solving for continuous functions?
More precisely, what does a variable have to be for the following to be continuous. For example, the problem I am dealt with solving is $$F... | {
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... |
As $x$ approaches $2$ from the right, $f(x)$ approaches $a^2(2^2)-7(2)$. We want this "limit from the right" to be $f(2)$, that is, $10$. This will be the case precisely if $$4a^2 -14=10.$$ Solve for $a$. We get $a=\pm\sqrt{6}$.
-
ok, so using what you told me, wouldn't a (in my problem) be 1? – Ronnie.j Jan 17 '12 a... | {
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... |
# Confused about how to solve basic combinatorial problem
The following exercise comes from a book. I do not know how to approach it. I get the sense that it is of the "stars and bars" flavor, but is really a "stars and squares and bars" problem - which is to say one must divide multiple groups of indistinguishable "i... | {
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$$\binom{5+6-1}{6-1}=\binom{10}5=252\;.$$
Similarly, there are
$$\binom{3+6-1}{6-1}=\binom85=56$$
distinguishable outcomes for the red balls. Since any outcome of the blue balls can be combined with any outcome of the red balls, there are altogether $252\cdot56=14,112$ distinguishable outcomes.
Note that although y... | {
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# For $s_n \leq t_n$ above specific n, prove $\lim_{n\to\infty} s_n \leq \lim_{n\to\infty} t_n$
The question: Suppose there exists $$N_0$$ such that $$s_n \leq t_n$$ for all $$n > N_0$$. Prove that if $$\lim_{n\to\infty} s_n$$ and $$\lim_{n\to\infty} t_n$$ exists, then $$\lim_{n\to\infty} s_n \leq \lim_{n\to\infty} t_... | {
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I'm certain I got lost in the details somewhere. Could someone point out errors that I've made?
Your idea of the proof is correct, though you made it a bit too complicated with the technical part. Here is a clear way to write it. Assume $$s>t$$. Then there exists $$\epsilon>0$$ such that $$s-\epsilon>t+\epsilon$$. Now... | {
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# Solution Check-Finding The Radon Nikodym Derivative
I was hoping to get my solution to part $$\textbf{i}$$ of this qual question regarding the Radon-Nikodym derivative checked for rigor and correctness. Then I was hoping to get advice on proceeding with part $$\textbf{ii}$$. Here is the question:
Let $$m$$ be the L... | {
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Let $$\mu_1(X)=m\{x \in [0,\frac {\pi} 2]: \sin y \in X\}$$. Verify that the derivative of $$\sin^{-1} x$$, namely $$\frac 1 {\sqrt {1-x^{2}}}$$ is the RND of $$\mu_1$$ w.r.t. $$m$$. [ For this show that $$\int_a^{b} \frac 1 {\sqrt {1-x^{2}}} dx=\mu_1 (a,b)$$ for $$a ].
Now consider $$\mu_2(X)=m\{x \in [\frac {\pi} 2, ... | {
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# replacing Inequalities
I encountered a problem today:
Prove that:
$$\frac{a^3+b^3+c^3}{a^2+b^2+c^2} \ge \frac{a+b+c}{3}$$
for all $a,b,c>0$
I used the RMS-AM inequality to replace the LHS with
$$\frac{\sqrt{a^2+b^2+c^2}}{\sqrt{3}}$$
and replaced the RHS using AM-GM inequality $$\frac{3abc}{a^2+b^2+c^2}$$
I ca... | {
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It is equivalent to $$(a^2-b^2)(a-b)+(a^2-c^2)(a-c)+(b-c)(b^2-c^2)\geq 0$$ and this is true.
Note that \begin{align*} a^3+b^3+c^3 &= \frac{a^4}{a}+\frac{b^4}{b}+\frac{c^4}{c}\\ &\ge \frac{(a^2+b^2+c^2)^2}{a+b+c}. \end{align*} Then \begin{align*} \frac{a^3+b^3+c^3}{a^2+b^2+c^2} &\ge \frac{a^2+b^2+c^2}{a+b+c}\\ &\ge \fr... | {
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# Homework Help: Algebra II Equations Containing Radicals
1. Jun 16, 2012
### velox_xox
...As for the problem, I tend to get problems with this same form (but with slight differences such as two separate binomials underneath individual square roots) wrong. I'm not quite sure what I'm doing wrong, so I was hoping for... | {
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This equation is easy to solve for y.
4. Jun 16, 2012
### Staff: Mentor
Here's another method to consider. Multiply both sides of the equation by
$\sqrt{y+5}$ - $\sqrt{}y$
You get
$\sqrt{y+5}$ - $\sqrt{}y$ = 1
If you subtract this equation from the original equation, you get
2$\sqrt{}y$ = 4
5. Jun 18, 2012
##... | {
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Splitting a circle into 3 equal parts using 2 lines
The picture probably explains my question best.
I need to find a way to divide a circle into 3 parts of equal area with only 2 lines that intersect each other on the outline of the circle.
Also I need to check, if whatever diameter is between those lines, also splits... | {
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Numerical Details
The iteration from this answer, applied to $$\theta+\sin(\theta)=\frac\pi3$$, is $$\theta_{n+1}=\frac{\pi/3-\sin(\theta_n)+\theta_n\cos(\theta_n)}{1+\cos(\theta_n)}\tag7$$ Here is the result of this iteration starting at $$0$$; $$\begin{array}{l|l} n&\theta_n\\\hline 0&0\\ 1&0.5\color{#AAA}{235987755... | {
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"ta... |
If you want to solve $$\theta+\sin(\theta)=\frac\pi3$$ what you can do is to expand the lhs as a Taylor series and use series reversion. Using ths simple $$y=\theta+\sin(\theta)=2 \theta -\frac{\theta ^3}{6}+\frac{\theta ^5}{120}-\frac{\theta^7}{5040}+O\left(\theta ^9\right)$$ this would give $$\theta=\frac{y}{2}+\frac... | {
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"openwebmath_score": 0.7216383814811707,
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3. Observe the essential difference between the cases with even and odd $$n$$. Whereas in former case the diameter drawn from the common intersection point is one of the "red" lines, in the latter case it is the bisector of the central angle. Let $$x_k$$ be the angle of a red line with the diameter. Then the following ... | {
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Probability over time
So given say, a deck of cards, I draw cards until I pick out a specific one, in this case, the ace of spades. Once the ace of spades has been drawn, you restart. Now, this continues till infinity. On average, how many tries does it take to get the card? Is there a solution to a deck of size $n$?
... | {
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Hopefully it is more clear now that the result is the same as before.
-
I think I explained the problem wrong, cards are selected without replacement. – dzk87 Nov 26 '12 at 20:13
Ok, but you say "this continues till infinity". I interpret that to mean "pick a card, stop when you get the ace of spades, write down the a... | {
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# What is the angle of $\angle BPC$ in $\triangle BPC$
In $\triangle ABC$, the internal bisector of $\angle ABC$ and the external bisector of $\angle ACB$ meet at $P$. If $\angle BAC = 40^\circ$ what is the measure of $\angle BPC$?
My try: i) Sum of angles of a triangle is $180^\circ$.
ii) Vertical opposite angles a... | {
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Extend $BC$ to $D$, so that $\angle ACD$ is an exterior angle of the triangle. Thus, $$\angle PCD = \angle ACD/2= 90-C/2$$
Using the exterior angle sum property in $\Delta PBC$, $$\angle BPC+\angle PBC = \angle PCD$$ $$\angle BPC+B/2=90-C/2$$ $$\angle BPC=90-(B+C)/2$$ $$\angle BPC=90-(180-A)/2=A/2=20^{\circ}$$ | {
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• @amWhy but then, $B4 and$C$could be confused with the angles of$\Delta ABC$. I've edited the post though – Prathyush Poduval Mar 29 '18 at 18:12 • I've deleted my comment, given your edit. I was showing how you need to be consistent. None of the angles in this case should be represented by one letter; So using$\angle... | {
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# Math Help - Relative Rates
1. ## Relative Rates
I've added an attachment to this post to explain the problem.
Two carts, A and B, are connected by a rope 39 feet long that passes over a pulley P. The point Q is on the floor h = 12 ft directly beneath P and between the carts. Cart A is being pulled away at a speed ... | {
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Thanks.
3. Hello, joatmon!
Two carts, A and B, are connected by a rope 39 feet long that passes over a pulley P.
The point Q is on the floor h = 12 ft directly beneath P and between the carts.
Cart A is being pulled away at a speed of 2.5 ft/s.
How fast is cart B moving toward Q at the instant when cart A is 5 ft fro... | {
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$\frac{dAP}{dt} = \frac{AQ}{AP} \frac{dAQ}{dt} = \frac{5}{13} (2.5) = \frac{25}{26} ft/sec$
Again, if $\frac{dAP}{dt} = \frac {25}{26}$, then $\frac{dPB}{dt} = \frac{-25}{26}$ since the length of this segment is constant. THIS IS A KEY ASSUMPTION! IS IT CORRECT???? (sorry for the caps)
Yes, that is correct. AP+ PB= 39... | {
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# Example of Partial Order that's not a Total Order and why?
I'm looking for a simple example of a partial order which is not a total order so that I can grasp the concept and the difference between the two.
An explanation of why the example is a partial order but not a total order would also be greatly appreciated.
... | {
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Check the above is a partial not total order.
• My favourite set is $\varnothing$. It represents the emptiness of existence! :-) – Asaf Karagila Apr 20 '13 at 19:39
• @AsafKaragila Do you mean existence of emptiness? – Git Gud Apr 20 '13 at 19:40
• @GitGud: No.${}$ – Asaf Karagila Apr 20 '13 at 19:41
• @Peter: While y... | {
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• if in the same example, I take my set to be {2,3,7,11} then is it partially ordered? I see, no element is related to any other under this relation. – Aditi Narware Jan 8 '17 at 11:00
A total order is a partial order, but a partial order isn't necessarily a total order.
A totally ordered set requires that every elem... | {
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Also $\mid$ is an order relation.
1. reflexivity is obvious: $a=a\cdot 1$, so $a\mid a$;
2. also transitivity is easy: if $a\mid b$ and $b\mid c$, then $b=ax$ and $c=by$ for some $x$ and $y$; therefore $c=a(xy)$ and $a\mid c$;
3. antisymmetry is a bit more difficult, but not so much:
Assume $a\mid b$ and $b\mid a$;... | {
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A simple example is a set with four elements $S = \{a, b, c, d\}$. We'll define a partial order so that $a$ is the smallest element, $d$ is the largest element, and $b$ and $c$ are intermediate elements that are incomparable with each other. The relation $R \subset S \times S$, where $(x,y) \in R \Leftrightarrow x \leq... | {
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# Finding the uniqueness of longest increasing subsequence
I have a problem related to a common dynamic programming problem LIS. I got LIS function that takes arr as an input and returns the length of the longest increasing subsequence.
def LIS(arr):
n = len(arr)
# initialize LIS values for all indexes
lis = [1]*n
... | {
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# Length is 1 for all subsequences of single element.
lis = [1] * n
# unique_lis[i] will tell whether the LIS that ends at index i is unique.
unique_lis = [True] * n
# Compute LIS values in bottom up manner
for i in range(1, n):
for j in range(0, i):
if arr[i] > arr[j]:
if lis[i] < lis[j] + 1:
lis[i] = lis[j] + 1
uniq... | {
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# Condition probabilities with three events
The events A, B, C have the probabilities: P(A|B)=0.25, P(C|B)=0.5, P(A∩C|B)=0.10. Given that B has happened, find the following probabilities:
a) That only C has happened 
b) That only C or only A has happened, but not both of them
c) That C or A has happened
I have a... | {
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algebra
A wire to be attached to support a telephone pole. Because of surrounding buildings, sidewalks and roadways, the wire must be anchored exactly 21 feet from the base of the pole. Telephone company workers have only 28 feet of cable, and 2 feet of that must be attached to the cable to the pole and to the stake o... | {
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1. maths
two poles 6m and 15m tall are 20 m apart a wire is attached to the top of each pole and is also staked to the ground some where b/n two poles where should be the wire be staked so that the minimum amount of wire be used?
2. trig
A 52-foot wire running from the top of a tent pole to the ground makes an angle... | {
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## Wednesday, November 25, 2009
### SICP Exercise 1.12: Pascal's Triangle
From SICP section 1.2.2 Tree Recursion
Exercise 1.12 asks us to write a procedure that computes the elements of Pascal's triangle by means of a recursive process.
The numbers along the left and right edges of the triangle are all 1, and each ... | {
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To finish up the procedure, we need to recursively calculate values from the interior of the triangle. An interior value is defined as the sum of the two values directly above in the triangle. So, for example, to calculate Pascal(4, 2) we would sum Pascal(3, 1) and Pascal(3, 2). Each of these values (since there are bo... | {
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(define (pascal n)
(if (= n 0) n
(+ (* (pascal (- n 1)) 2) 1)))
Bill the Lizard said...
sethreno,
"a procedure that computes elements of Pascal's triangle" is a little ambiguous. If I were teaching a course using SICP, I'd accept that interpretation of the question.
However, your solution is off by one.
> (pascal 0... | {
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Anonymous said...
The elements of the Pascal's triangle are the Binomial coefficients, i.e., they are the coefficients of the terms in the expansion of (x + y)^n, where n is a natural number.
Since the coefficients of the terms not in the expansion are zero, the values outside the Pascal's triangle should be 0 not #f.... | {
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;Value 43: (1 4 6 4 1)
1 ]=> (pascal-triangle 5)
;Value 44: (1 5 10 10 5 1)
1 ]=> (pascal-triangle 6)
;Value 45: (1 6 15 20 15 6 1)
1 ]=>
Dylan Beckwith said...
Seeing as "Pascal's Triangle" is a listed representation of the binomial coefficients.
i defined and ran a (choose n k) procedure thru a coefficient pro... | {
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# Find a (nontrivial) linear system of equations satisfied by any vector minimizing the energy
Here is an exercise 1.5 from the book Numerical Algorithms: Methods for Computer Vision, Machine Learning, and Graphics (by J. Solomon):
Suppose $A,B \in R^{n \times n}$ and $\vec{a},\vec{b} \in R^{n}$. Find a (nontrivial) ... | {
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• Use the identity $\|v\|_2^2 = v^Tv$, expand the result, take the gradient. – user856 Jan 16 '17 at 16:59
• I've updated original question. Could you tell me please, is it correct? – devforfu Jan 24 '17 at 7:19
• expand the norms by using $|X|_2^2 = X^TX$ for any expression $X$ that occurs, then just sum up and differ... | {
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• So do you think that the system of equations that was asked to be found in this exercise i.e. $C\vec{x}=\vec{c}$ is actually a formulation of least squares problem when there are two minimization objectives, right? – devforfu Jan 24 '17 at 7:16
• I made an edit, hope it helps. – WalterJ Jan 24 '17 at 10:40
• Yes, it ... | {
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# Magnetic Field of Rotating Circular Ring
1. Jul 7, 2015
### zero1342
1. The problem statement, all variables and given/known data
Find the magnetic field at position z (z=0 in the plane of the ring) along the rotation axis for a circular ring of radius r, carrying a uniform linear charge density λ, and rotating ab... | {
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Does dl = 2πrdr?
4. Jul 8, 2015
### ShayanJ
To find the magnetic field, you need to use Biot-Savart's law $\vec B=\frac{\mu_0}{4\pi} \int \frac{I d \vec l \times \vec r}{r^3}$.
Where $d\vec l$ is the differential length(so its dimension is length not length2 so rdr can't be right) along the wire in the direction of ... | {
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# Explaining the product of two ideals
My textbook says that the product of two ideals $I$ and $J$ is the set of all finite sums of elements of the form $ab$ with $a \in I$ and $b \in J$. What does this mean exactly? Can you give examples?
• The particular case of $I$ and $J$ being principal is easy to understand and... | {
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• This is a good example, 1+! Almost the same, of course, works with $R[X,Y]$ and the two ideals $(X,Y)$ and $(1-X,Y)$. Geometrically, $Y$ vanishes on $\{(0,0),(1,0)\} \subseteq \mathbb{A}^2_R$, but cannot be written as a product of two polynomials which vanish on $(0,0)$ resp. $(1,0)$. Are there $1$-dimensional exampl... | {
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For example, in $\mathbb{Q}[x,y]$, one computes $(x,y) \cdot (x^2,y^2)=(x^3,x y^2,x^2 y,y^3)$.
In general, one observes that $I \cdot J \subseteq I \cap J$. This is not an equality in general; in the above example the intersection is just $(x,y)$. However, one has (in the commutative case) $\sqrt{I \cdot J} = \sqrt{I ... | {
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Make a new GeoGebra worksheet. By writing $A$1, you make an absolute reference to A1 when dragging both along a column and along a row. GeoGebra (www.geogebra.org) is free dynamic mathematics software for all levels of education that brings together geometry, algebra, spreadsheets, graphing, statistics and calculus in ... | {
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The Advanced tab in Object Properties : 9. Join us! Join us! A spreadsheet is a program that can be used to organize data in tables and perform mathematical computations. After entering the function and the parameter values in the appropriate cells, select the rectangular area of the desired Operation Table with the mo... | {
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and $$E$$ to $$A2$$. That is, geogebra spreadsheet does not seem to recognize these as geogebra commands/formulae that should then be displayed appropriately in graphics view. In that way they will be shown in the spreadsheet. You need JavaScript enabled to view it. The basic feature of any spreadsheet is that you can ... | {
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within the spreadsheet. A rabbit is running along a river. GeoGebra provides several Math Apps for learning and teaching at all levels. Make a slider $$b$$ of integer values between 0 and 100 representing Bertil's annual percentage raise of his monthly salary. It can be shown that this perpendicular bisector is a tange... | {
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in GeoGebra has most of the regular spreadsheet-features. Thereby, the name of the object matches the name of the Spreadsheet Cell used to initially create it (e.g. By dragging the filled in rectangle, you make so called relative copies. Put a trace on the perpendicular bisector and drag $$D$$. Most parts of GeoGebra a... | {
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and students. This view is part of GeoGebra user interface. If possible, GeoGebra immediately displays the graphical representation of the object you entered in a Spreadsheet Cell in the Graphics View as well. The object-oriented way of doing things in GeoGebra however, makes the spreadsheet a very useful tool. Constru... | {
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# Show that the order of a cycle is the length of the cycle
## Homework Statement
Prove that if ##\sigma## is the m-cycle ##(a_1 ~a_2~ \dots ~ a_m)##, then for all ##i \in \{1,2, \dots , m \}##, ##\sigma^i (a_k) = a_{k+i}##. Deduce that ##\sigma^m (a_k) = a_k##
## The Attempt at a Solution
I will try to do this by ... | {
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Mr Davis 97
It is correct and you've proven it for all ##n \in \mathbb{N}##. However, the question is poorly worded. It should have been ##\sigma^i (a_k) = a_{k+i~\operatorname{mod}m}## since ##\sigma(a_m)=a_1## and not ##a_{m+1}##. Hence the induction basis is already wrong without the modulo on the index.
I see what ... | {
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Prove that if ##\sigma## is the m-cycle ##(a_1 ~a_2~ \dots ~ a_m)##, then for all ##i \in \mathbb{N}##, ##\sigma^i (a_{k \mod m}) = a_{k+i \mod m}##. Deduce that ##\sigma^m (a_{k \mod m}) = a_{k \mod m}## and so ##| \sigma | = m##.
fresh_42
Mentor
2021 Award
So would a less sloppy problem statement be
Prove that if #... | {
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2) Prove that if ##\sigma## is the m-cycle ##(a_1 ~a_2~ \dots ~ a_m)##, then for all ##i \in \mathbb{N}##, ##\sigma^i (a_k) = a_{k+i}##, where ##k+i## is replaced by its least positive residue mod m.
fresh_42
Mentor
2021 Award
Formally: the second implies the first; within the context: none. As ##\sigma^m=1##, with th... | {
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Mr Davis 97 | {
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# Evaluate derivative of Lagrange polynomials at construction points
Assume, that we have points $$x_i$$ with $$i=1,...,N+1$$. We construct the Lagrange basis polynomials as \begin{align} L_j(x) = \prod_{k\not = j} \frac{x-x_k}{x_j-x_k} \end{align}
Now according to my computation and the results by Yves Daoust here, ... | {
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which gives the matrix
D =
10.5000 0 0 0 0 0 0
0 -0.0000 0 0 0 0 0
0 0 0.0000 0 0 0 0
0 0 0 -0.0000 0 0 0
0 0 0 0 0.... | {
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function y = alternative_dl(j,x,z)
y = 0;
n = length(x);
for l=1:n
if not(l==j)
k = 1/(x(j)-x(l));
for m=1:n
if not(m==j) && not(m==l)
k = k*(z-x(m))/(x(j)-x(m));
end
end
y = y + k;
end
end
end
Which agrees with the finite difference computation.
So it seems to me, that simplifying the above formula includes some ... | {
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And this is the same as your new derived formula.
• But your formula contains $L_i(x)$ on the right hand side. Now if I evaluate $L'_i(x_j)$ I get $L_i'(x_j) = L_i(x_j)\cdot \sum...=0\cdot \sum$ since $L_i(x_j)=0$ if $i\not = j$. – k1next Jan 15 '15 at 11:39
• Okay I see, that I copied the formula wrong, but the probl... | {
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# High school contest math question (number theory) - prove:
Reposting with Mathjax - sorry, first time!
Let $$S = \{4,8,9,16,...\}$$ be the set of integers of the form $$m^k$$ for integers $$m, k \ge 2$$. For a positive integer $$n$$, let $$f(n)$$ denote the number of ways to write $$n$$ as the sum of (one or more) ... | {
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For part b, note that all multiples of $$4$$ can be represented because you have all the powers of $$2$$ except $$1,2$$. Express any multiple of $$4$$ in binary and read off the numbers to add to get it. All numbers equivalent to $$1 \bmod 4$$ that are at least $$9$$ can be expressed because the number minus $$9$$ is a... | {
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• So this answer assumes that you can pick non-distinct elements? I guess this might be the question's intention, as someone noted that 34 can't be represented by picking distinct elements – Benjamin Wang Dec 14 '20 at 10:08
• No, it does not assume you can duplicate elements. You can get $34$ as $25+9$. I had thought ... | {
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# What's the difference between “$\exists F_\epsilon :m^*(E\setminus F_\epsilon)<\epsilon$ for each $\epsilon$,” and “$\exists F:m^*(E\setminus F)=0$”?
Let $E\subseteq \mathbb{R}$ be given, and let $m^*$ denote the outer measure. For each $\epsilon$, there exists a closed set $F_\epsilon\subseteq E$ such that $m^*(E\s... | {
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The 2nd statement becomes: $\;\; \left( \exists F \subseteq {\mathbb R} \right )\left(\forall \, \epsilon > 0 \right):$ $\;\;\;F$ is closed and $m^{*}\left(E-F\right) < \epsilon$
Of course, the fact that the 2nd statement is a logically stronger $\exists \; \forall$ uniform statement doesn't mean that, in this specifi... | {
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Can someone please tell me how to do these?
Suppose that f is given for x in the interval [0,12] by
x= 0 2 4 6 8 10 12
f(x)= -14 -10 -6 -5 -6 -9 -11
A. Estimate f(2) using the values of f in the table.
f'(2)=_______
B. For what values of x does f'(x) appear to be positive?
_________
(Give your answer as an interva... | {
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2. Physics
Suppose that a simple pendulum consists of a small 66.0 g bob at the end of a cord of negligible mass. Suppose that the angle between the cord and the vertical is given by θ = (0.0800 rad) cos[(4.30 rad/s)t + ϕ] (a) What is the
3. Calculus
Suppose that p(x) is the density function for heights of American ... | {
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# How to find the solution for $\frac{2x-3}{x+1} \leq 1$?
I have the following inequality:
$$\frac{2x-3}{x+1}\leq1$$
so, considering $x \neq -1$, I started multiplying $x+1$ both sides:
$$2x-3\leq x+1$$
then I subtracted $x$ both sides:
$$x-3\leq1$$
and then sum $3$ both sides:
$$x\leq4$$
Therefore, my solutio... | {
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Case 2: $x+1<0$. Then we get $$2x-3\geq x+1$$ (notice that since $x+1<0$ the sign had to switch directions) We then solve to find $x\geq 4$. Since for this case we also had $x+1<0$, which is the same as $x<-1$ we conclude no such $x$ exists. (A number cannot be less than -1 and greater than 4)
Hope that helps,
-
You ... | {
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# Square matrix with rational coefficients having $k$-th root
Let $A\in M_{n}(\mathbb{Q})$, meaning that $A$ is $n\times n$ matrix with entries in the rational numbers $\mathbb{Q}$. Suppose that $A$ satisfies two conditions:
• $\det(A)\neq 0$
• For every integer $k$, there exists a $B\in M_{n}(\mathbb{Q})$ such that ... | {
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Remark 2. It is tempting to modify the above argument for the case of $\mathbb{Q}$, but it doesn't seem to work immediately. Again using the fact that $\det(A)\neq 0$, there are only finitely many primes appearing in the numerator and denominator of $\det(A)$, so we can pick a prime $p$ which doesn't appear there. Redu... | {
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• Try $A=\pmatrix{1&1\\0&1}$. – Lord Shark the Unknown Jul 7 '18 at 17:23
• @LordSharktheUnknown Very nice! Indeed, if we let $B = \pmatrix{1 & \frac{1}{k} \\ 0 & 1}$, then $B^k = A$. You should post this as an answer when you get a chance. Also, I wonder if these are the only examples. In other words, do you think tha... | {
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# Overview
Let $T_n(b)$ denote an $n$ by $n$ matrix with all entries zero except for top right entry $b$. Then for each $n\geq 2$ there is an infinite family of exceptional matrices in $M_n(\mathbb Q)$, of the form $$A = S(I_n+T_n(b))S^{-1}$$ where $S$ can be any invertible matrix in $M_n(\mathbb Q)$. The solutions $B... | {
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For the former case, if $a=1$ then we have the construction given at the beginning of this answer. For the latter case, if $a=2$ then we have the other construction given at the start.
For all the rest of the cases, we show that we cannot take squareroots infinitely, hence contradicting that $A$ is exceptional. (For o... | {
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Proof. By the Cayley-Hamilton theorem in dimension $n=2$, for any $R_i$ we have $$(R_i)^2 - tr(R_i)R_i + \det(R_i)I = 0$$ Therefore a necessary condition for $R_j = (R_{j+1})^2$, together with $\det(R_{j+1})=1$, is \begin{align*} R_j - tr(R_{j+1})R_{j+1} + I &= 0\\ tr(R_{j+1})R_{j+1} &= R_j + I \end{align*} Let $R_j = ... | {
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Hence by induction we can see that the denominator is squareroot-ed every iteration. If $R_j\not \in \mathbb Z$ for some $j$, then this iterative squareroot process must result in an irrational denominator eventually, i.e. $tr(R_m) \not\in \mathbb Q$ for some large enough $m>j$. This contradicts $R_m\in M_2(\mathbb Q)$... | {
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# C2. Case $n=2$ and $c\neq 0$
Choosing the conjugation matrices $$S= \begin{pmatrix} 0 & 1 \\ c & -a \end{pmatrix}, S^{-1} = \begin{pmatrix} a/c & 1/c \\ 1 & 0 \end{pmatrix},$$ we get a new exceptional matrix $A'$ $$A'= SAS^{-1} = \begin{pmatrix} a+d & 1 \\ bc-ad & 0 \end{pmatrix} = \begin{pmatrix} a+d & 1 \\ -1 & 0 ... | {
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Since $r_0\in\mathbb Z$, this leaves the final case of $r_0 = -1$. For this case we use a numeric solver to show directly that there are no solutions to $$B^3 = \begin{pmatrix} -1 & 1\\ -1 & 0 \end{pmatrix}$$ Edit 1: Algebraic proof
Let $B$ be a solution to $A=B^3$ and $t=tr(B)$. Then $$\det(B)^3 = \det(A) = 1$$ Since ... | {
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# find all real and imaginary roots
• Sep 29th 2009, 06:29 AM
absvalue
find all real and imaginary roots
I need to find all real and imaginary roots of this equation:
$2x^7 - 2x^6 + 7x^5 - 7x^4 - 4x^3 + 4x^2 = 0$
Using synthetic division, I found that 1 is a root.
$(x - 1)(2x^6 + 7x^4 - 4x^2) = 0$
I'm not really s... | {
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"lm_q1_score": 0.9850429129677614,
"lm_q1q2_score": 0.8430501123080064,
"lm_q2_score": 0.8558511524823263,
"openwebmath_perplexity": 538.6870097376623,
"openwebmath_score": 0.9220497608184814,
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Example of CT convolution
This is an example of convolution done two ways on a fairly simple general signal.
$x(t) = u(t)\$
$h(t) = {e}^{-\alpha t}u(t), \alpha > 0\$
Now, to convolute them...
1. $y(t) = x(t)*h(t) = \int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau$
2. $y(t) = \int_{-\infty}^{\infty}u(\tau){e}^{-\alpha (... | {
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"openwebmath_perplexity": 1306.297873807096,
"openwebmath_score": 0.7508476972579956,
"tags"... |
Name --dennis.m.snell.1, Mon, 01 Oct 2007 16:51:27
The authorship issue was not an issue. It was mentioned in class, but by a student asking about it. There is a log of every action and every edit on this kiwi that can be reviewed each week. You are safe in leaving out your name. Sometime soon the editing will be rewo... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429160413819,
"lm_q1q2_score": 0.8430501113168498,
"lm_q2_score": 0.8558511488056151,
"openwebmath_perplexity": 1306.297873807096,
"openwebmath_score": 0.7508476972579956,
"tags"... |
The roots of the equation are the values of x at which ax² + bx + c = 0. Solve an equation of the form a x 2 + b x + c = 0 by using the quadratic formula: x = − b ± √ b 2 − 4 a c: 2 a: Step-By-Step Guide. When only one root exists both formulas will give the same answer. The root of a quadratic equation Ax 2 + Bx + C =... | {
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"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429116504952,
"lm_q1q2_score": 0.8430501093697634,
"lm_q2_score": 0.8558511506439707,
"openwebmath_perplexity": 413.96140376611504,
"openwebmath_score": 0.8033725023269653,
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real root. Let's check these values: (-3)^2 +8*-3 +15 = 9 - 24 + 15 = 0 and (-5)^2 + 8*-5 +15 = 25 - 40 + 15 = 0. When a is negative, this parabola will be upside down. So when you want to find the roots of a function you have to set the function equal to zero. The quadratic formula gives two solutions, one when ± … Th... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429116504952,
"lm_q1q2_score": 0.8430501093697634,
"lm_q2_score": 0.8558511506439707,
"openwebmath_perplexity": 413.96140376611504,
"openwebmath_score": 0.8033725023269653,
"tag... |
is a value calculated from a quadratic equation. So we have a single irrational root in this case. There are however some field where they come in very handy. Value of determinant B2 – 4AC, defines the nature of roots of a Quadratic Equation Ax2 + Bx + C = 0. It is also called an "Equation of Degree 2" (because of the ... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9850429116504952,
"lm_q1q2_score": 0.8430501093697634,
"lm_q2_score": 0.8558511506439707,
"openwebmath_perplexity": 413.96140376611504,
"openwebmath_score": 0.8033725023269653,
"tag... |
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