text stringlengths 1 2.12k | source dict |
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of times observed') xlabel('Number of successes') xlim(-0.5,max(x)+.5) ylim(0,max(y)*1.2) show(block=False) i, di = 0, 1 i_max = len(u) j = 0 while True: marker_fit_error.set_data([ u[i] ], [v[i]]) z = poisson.pmf(x, u[i])*N for rect, h in zip(bar_ob, z): rect.set_height(h) fig.canvas.draw() savefig('frame%04d.tif'%j) ... | {
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### Bolide impacts
Now, let's look at our data on bolides in the same way. | {
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[Show code] | {
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 #!/usr/bin/env python2 # dates of bolide observations from NASA between 1994 and 2013 # http://en.es-static.us/up... | {
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ylabel('Frequency') xlabel('Count') savefig('figures/meteorDataAndFit.pdf',bbox_inches='tight') savefig('figures/meteorDataAndFit.png',bbox_inches='tight') #show() main() | {
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The choice of bins is a bit arbitrary, which should make us uncomfortable, but if we just go with it, we can now look for the value of $$\lambda$$ that leads to the closest-matching Poisson distribution. | {
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[Show code] | {
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 #!/usr/bin/env python """ This script ... | {
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1.5) xlabel('Poisson intensity $\lambda$') subplot(2, 1, 1) plot(x, y, 'bo') z = poisson.pmf(x, L_best) width = 0.3 bar_ob = bar(x - 0.5*width, z*N, width, color='r') title('N = %d observations, $\lambda_{best} = %.3f$, $Err_{best} = %1.2g$'%(N, L_best/Dt, err_best)) ylabel('Number of times observed') xlabel('Number of... | {
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## Poisson processes in two dimensions
Calculate the distances between all pairs of points, and rank these distances from smallest to largest. | {
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[Show code] | {
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 #!/usr/bin/python2 from scipy import rand, zeros, sqrt, pi, array, arcsin, sign, li... | {
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Error = %.3f'%(n,toterr)) ylabel('Residual vector over distance') xlabel('Distance from origin') savefig('ripley-test.pdf') savefig('ripley-test.png') main() | {
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There are More data sets like these that you can explore, out on the internet. | {
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# What is the probability of at least 2 blue balls given the probability of at least 1 blue ball?
A bag has red, blue, and green balls. The probabilities of randomly grabbing a red, blue, and green ball from the bag (with replacement) are $r$, $b$, and $g$ respectively. I randomly grab $n$ balls from the bag. What's t... | {
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# Concise proof that every common divisor divides GCD without Bezout's identity?
In the integers, it follows almost immediately from the division theorem and the fact that $a | x,y \implies a | ux + vy$ for any $u, v \in \mathbb{Z}$ that the least common multiple of $a$ and $b$ divides any other common multiple.
In c... | {
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One easy and insightful way is to use the proof below. It essentially constructs $$\rm\:gcd\:$$ from $$\rm\:lcm\:$$ by employing duality between minimal and maximal elements - see the Remark below. This is essentially how the linked Wikipedia proof works, but there the innate duality is obfuscated by the presentation. ... | {
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$$\begin{eqnarray}\rm the\ proof\ \ \ c\mid a,b &\iff&\rm b,\,a\mid ab/c &\iff&\rm lcm(b,\,a)\mid ab/c &\iff&\rm c\mid ab/lcm(b,a)\\[.5em] \rm becomes\ \ \ \ c\mid a,b &\color{#c00}\iff&\rm a',b'\mid c' &\iff&\rm lcm(a',b')\mid c' &\color{#c00}\iff&\rm c\mid lcm(a',b')'\end{eqnarray}$$
Now the innate duality is clear:... | {
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• $\rm\Bbb Q[x,y]\:$ is a UFD so GCD domain which is not Bezout, since $\rm\:gcd(x,y) = 1\:$ but $\rm\:f\, x+ g\, y = 1\:\Rightarrow\: 0 = 1\:$ by evaluating at $\rm\:x = 0 = y.$ Generally a gcd domain D is Bezout iff $\rm\:gcd(a,b) = 1\:\Rightarrow (a,b) = 1,\:$ i.e. $\rm\: ac + bd = 1\:$ for some $\rm\:c,d\in D.\$ – ... | {
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Now you just have to realize that this same logic applies to any divisor of the two numbers, not jut the greatest one. I could have said "any divisor of $28$ and $8$ is a divisor of $20$ and $8$..." and eventually had "...is a divisor of $4$". Since the number we arrive at at the end is the GCD, any divisor of the two ... | {
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# What is this operator called?
If $x \cdot 2 = x + x$
and $x \cdot 3 = x + x + x$
and $x^2 = x \cdot x$
and $x^3 = x \cdot x \cdot x$
Is there an operator $\oplus$ such that:
$x \oplus 2 = x^x$
and $x \oplus 3 = {x^{x^x}}$?
Also, is there a name for such a set of operators ops where...
Ops(2) is multiplicatio... | {
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FYI, for "to the power of-ation" the word you're looking for is exponentiation.
• Also that is similar to graham number. en.wikipedia.org/wiki/Graham%27s_number – Takahiro Waki Jul 5 '16 at 9:19
• Aha, and the answer to the second question is hyperoperation: en.m.wikipedia.org/wiki/Hyperoperation – uzilan Jul 5 '16 at... | {
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Some notations for hyperoperations are(for $H_n(a,b)$:
1. Square bracket notation: $a[n]b$
2. Box notation: $a{\,{\begin{array}{|c|}\hline {\!n\!}\\\hline \end{array}}\,}b$
3. Nambiar's notation : $a\otimes ^{n-1}b$
4. Knuth's up arrow notation: $a\uparrow^{n-2}b$
5. Goodstien's notation: $G(a,b,n)$
6. Conway's chaine... | {
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# Does $a \equiv b \pmod n$ mean $n \mid a - b$ or $n \mid b -a$
If I have $a \equiv b \pmod{n}$, it means $n \mid b - a$.
But can you write it as $n \mid a - b$ as well?
-
$a \equiv b \pmod{n} \implies n \mid (a-b) \implies a-b = nk$ for some integer $k$. Can you find an integer $m$ such that $b-a = mn$? – JavaMan... | {
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# Math Help - Integral of Rational Function
1. ## Integral of Rational Function
I got this question on an exam this morning and I didn't know how to answer it. I've been working on it the past hour trying to figure it out. Usually with rational functions, I factor to get it in a from that I can use partial fractions.... | {
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$
\frac{2}{\sqrt{3}} \times \tan^{-1}{\frac{2u}{\sqrt{3}}} + C
$
$
\frac{2}{\sqrt{3}} \times \tan^{-1}{\frac{2x + 1}{\sqrt{3}}} + C
$
Is that right?
Looks fine to me.
5. Originally Posted by centenial
I got this question on an exam this morning and I didn't know how to answer it. I've been working on it the past hou... | {
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# How would one be able to prove mathematically that $1+1 = 2$?
Is it possible to prove that $1+1 = 2$? Or rather, how would one prove this algebraically or mathematically?
• In fact, this occasionally useful proposition was proved by Russell and Whitehead. – Srivatsan Dec 29 '11 at 23:19
• You should go read up on s... | {
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Which is what we wanted.
In a more general setting, one needs to remember that $$0,1,2,3,\ldots$$ are just symbols. They are devoid of meaning until we give them such, and when we write $$1$$ we often think of the multiplicative identity. However, as I wrote in the first part, this is often dependent on the axioms - o... | {
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• That is a really good answer to a annoying question that comes up a lot. I can't count the amount of people who says you can't prove 1+1=2. – simplicity Dec 30 '11 at 0:38
• I'm always confused why people find it annoying - maths at hs was boring until I tried to prove 1+1=2 one da. Being a naive 17 year old I was so... | {
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• 1000 + 1030 + 1000 + 20 and see what you get. Did you get it right? Are you sure? If you got the answer 5000, I can tell you that is not the right answer. I didn't invent the problem. I got it somewhere. – Timothy Oct 19 '19 at 3:28 | {
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1 is the convention name of 0++, 2 is the convention name of (0++)++, so what you need to prove is 0++ + 0++ = (0++)++.
$+$ is defined as:
• $0+m := m$
• $(n++)+m := (n+m)++$
Just apply this definition to the left of the equation and you will get the right. You don't even need to know what '0' or '++' is, all you do... | {
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# Math
David and Sam had an equal number of marbles. After Sam gave 50 marbles to David, he had 5x as many marbles as Sam. Find the total number of marbles they had.
asked by Patrice
1. David's marbles --- x
Sam's marbles -----x
After Sam lost his marbles:
Sam --- x-50
David -- x+50
x+50 = 5(x-50)
x+50 = 5x - 250
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Mia gave 1/3 of her 24 marbles to david. How many did david get?
9. ### MATH
Jim had 92 more marbles than Sam. After Sam gave Jim 18 marbles, Jim had twice as many marbles as Sam. How many marbles did Jim have at first?
10. ### MATH
Allen started out with x marbles. He doubled his number of marbles by buying an addit... | {
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# Prove that if ${x_1, x_2, x_3}$ are roots of ${x^3 + px + q = 0}$ then ${x_1^3+x_2^3 + x_3^3 = 3x_1x_2x_3}$
How to prove that ${x_1^3+x_2^3 + x_3^3 = 3x_1x_2x_3}$ holds in case ${x_1, x_2, x_3}$ are roots of the polynomial?
I've tried the following approach:
If $x_1$, $x_2$ and $x_3$ are roots then
$$(x-x_1)(x-x_... | {
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$$\require{cancel} x_1^3+x_2^3+x_3^3=\cancel{-p(x_1+x_2+x_3)}-3q=\cdots$$
• Great solution +1 – Aqua Jun 11 '18 at 17:30
• @ChristianF Thanks. One advantage of this approach is that it easily extends to equations $\,x^{\color{red}{n}}+px+q=0\,$, where $\,\sum_i x_i^n = (-1)^{n+1}n \prod_i x_i\,$. – dxiv Jun 11 '18 at ... | {
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# Bayes's rule and unfair coin | Solution Explanation
There are three coins in a bag. Two of them are fair. One has heads on both sides. A coin selected at random shows heads in two successive tosses.
What is the conditional probability of obtaining another head in the third trial given the fact that the first two tr... | {
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So the answer is $$P(HHH|HH) = \frac{1\cdot \frac{5}{12}}{\frac{1}{2}} = \frac{5}{6}$$
Now, when it comes to your argument "the first two trials do not affect the third trial", it is wrong in some manner. Because although it seems third trial is not affected by the first two trials, notice that whether we have fair or... | {
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So the probability of getting a head in the next trial is... wait, which coin you are using? The same coin is selected once, and then used for all tosses. There is a dependency.
Rather, what is the probability that it is biased when given that you have two heads in he first two tosses. Well, use Bayes' Rule to upd... | {
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When is $\binom{n}{k}$ divisible by $n$?
Is there any way of determining if $\binom{n}{k} \equiv 0\pmod{n}$. Note that I am aware of the case when $n =p$ a prime. Other than that there does not seem to be any sort of pattern (I checked up to $n=50$). Are there any known special cases where the problem becomes easier? ... | {
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In fact, it is interesting to consider separately the solutions $(n,k)$ into those which are relatively prime (which I'll call the trivial solutions) and those which are not. It appears that the non-trivial solutions are completely responsible for the Sierpinski pattern in the triangle above. Indeed, here are only the ... | {
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# Comparison of integrals by algebraic means
\begin{align}A&:=\int_0^1\frac1{\sqrt{x(1-x)}}\ \mathrm dx \\ B&:=\int_0^1\sqrt{x(1-x)}\ \mathrm dx \end{align}
My CAS tells me that $$A = \pi$$ and $$B = \frac18\pi$$.
How can one prove that $$A=8B$$ using just basic rules of integration such as the chain rule?
Trigonom... | {
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$$B=\frac12 \int_0^1 x\left(\frac{\sqrt x}{\sqrt{1-x}}-\frac{\sqrt{1-x}}{\sqrt x}\right)\,dx\tag2$$
Now enforcing the substitution $$x\mapsto 1-x$$ in the first term on the right-hand side of $$(2)$$ reveals
$$\int_0^1 x\frac{\sqrt x}{\sqrt{1-x}}\,dx=\int_0^1 \frac{\sqrt{1-x}}{\sqrt x}\,dx-\int_0^1 \sqrt{x(1-x)}\,dx\... | {
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# What is the maximum possible number of elements of $S$?
This is an interesting problem I found.
Let there be a 2-digit sequence that can start with 0, like 04 or 93. Let a "nudge" be defined as exactly one of the following operations:
1) Increasing one of the digits by 1.
2) Decreasing one of the digits by 1.
3)... | {
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Considering that each neighbourhood has size $5$, we see that no more than $20$ neighbourhoods can fit into the $100$ available squares without overlapping. However this does not guarantee that $20$ are actually possible: we have to consider their "shape" as well as their "size". But it is not too hard to find a soluti... | {
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# Exercise from (Baby) Rudin (Chapter 3, exercise 13): is this proof correct? Is it well-written?
The problem is the following:
Prove that the Cauchy product of two absolutely convergent series converges absolutely.
Here is my attempt:
Let $s_n=\sum^n_{k=0}a_k$ and $t_n=\sum^n_{k=0}b_k$ be two series absolutely con... | {
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Since, $\sum |c_n|$ is a sequence of non-negative terms, it suffices to show that $\{C_n\}$ is bounded.
\begin{align} C_n & =\sum_{k=0}^n\left|\sum_{m=0}^k a_mb_{k-m}\right|\\ & \le\sum_{k=0}^n\sum_{m=0}^k|a_mb_{k-m}|\\ & =\sum_{m=0}^n|a_n|\sum_{k=0}^{n-m}|b_k|\\ &\le\left(\sum_{k=0}^n|a_k|\right)B\le AB. \end{align}
H... | {
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# Relationships between bounded and convergent series
I would like to know the relationships between bounded and convergent series. By bounded series I mean a series whose sequence of partial sums is bounded. For example, it seems natural that if a series is convergent, it is also bounded, but does the converse hold?
... | {
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When we talk about "convergence of the series", we are really talking about convergence of the sequence of partial sums: the series $\sum a_i$ converges if and only if the sequence $(s_n)$ converges. That is, your definitions about "series" are really about "sequence of partial sums", and so you have the usual relation... | {
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Likewise,
The sequence of partial sums of $\displaystyle \sum_{i=1}^{\infty}a_i$ is decreasing if and only if all the terms $a_i$ are nonpositive. The sequence of partial sums is strictly decreasing if and only if all the terms $a_i$ are negative.
So we conclude:
Theorem. Let $\displaystyle \sum_{i=1}^{\infty}a_i$ i... | {
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• It may happen that the partial sums of a series are bounded and its general term converges to zero and yet the series diverges. Example: if $a_n=(-1)^k2^{-k}$ for every nonnegative $k$ and $n$ such that $2^k\le n<2^{k+1}$, then $a_n\to0$ but the limit set of the sequence of the partial sums $\sum a_n$ is the interval... | {
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• I know about that about sequences, but what about series? Does it even make sense to say "bounded series"? – Federico Sep 6 '11 at 20:04
• I assumed you meant sequences because "bounded series" is not immediately meaningful to me -- except if it means that the sequence of partial sums is bounded (in which case bounde... | {
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# finding inverse Z transform with usage of tables (LTI and causal sequences)
problem is as follows
try to find the inverse Z
$$Z^{-1} (\frac{3}{z+2}) = ???$$
with the usage of z-transform tables
Ok, so in order to find something from the table, I thought that we expand with z and try to see if we get closer to an... | {
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Your solution is correct. This is the way I see it (assuming that the sequence is causal/right-sided):
$$\mathcal{Z}^{-1}\left\{\frac{3}{z+2}\right\}[n]=3\mathcal{Z}^{-1}\left\{z^{-1}\frac{z}{z+2}\right\}[n]=\\=3\mathcal{Z}^{-1}\left\{\frac{z}{z+2}\right\}[n-1]=3(-2)^{n-1}u[n-1]$$
• thank you for checking my answers,... | {
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# If $a|b$ and $c|d$, then $ac|bd$
I just need to check the reasoning in my proof is correct, I think it is valid although I'm not totally convinced because I can't follow the logic; does proving that $x$ is an integer prove that $ac|bd$?
Theorem: Let $a$, $b$, $c$, $d$ be integers. If $a|b$ and $c|d$, then $ac|bd$.
... | {
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in your notation $\rm\,\ j,\ k\ \in\ \mathbb Z\ \ \Rightarrow\ \ \ j\, k\ \in \, \mathbb Z$
That's true: the product of integers is an integer. So this divisibility product law is equivalent to the product closure law for integers (except that if your definition of divisibility includes $\,0\mid n\iff n= 0\,$ then you... | {
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Combining these observations leads to the following equivalence between the arithmetic of divisibility relations and subrings $\rm\,Z\,$ of $\,\mathbb Q\,$ (or any field).
Theorem $\$ Let $\rm\,Z\,$ be a subset of $\rm\,\mathbb Q\,$ with $\rm \,1\in Z.\,$ Let $\:\mid\:$ denote the divisibility relation $\rm\, a\mid b ... | {
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-
Bill, I have avoided making this criticism for some time now, but I really think it is valid here: if the OP understood this argument, he/she wouldn't be asking this question. – Qiaochu Yuan May 19 '11 at 19:28
@Bill: yes, but I don't understand why you're assuming that the OP "understands integer arithmetic," to put... | {
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@Qia Perhaps you missed the point of my answer. When I posted it there were already answers on the OP's logic error. My point was to teach the OP a better way to formulate the proof strategy. If answers were strictly limited to precisely the question asked then this forum would be quite sterile. When teaching math it i... | {
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If you're having trouble with these kinds of proofs, it's often best to start by writing down what you're given on the left side of the paper and where you're trying to get on the right side.
a|b ac | bd
c|d
Next fill in what you can from definitions:
b = aj ... | {
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# Diameter of a circle with 3 coordinates
The question is: A circle has the points $A=(6,-1)$ $B=(10,-3)$ and $C=(-2,-9)$ on its circumference. A diameter of the circle is drawn which is parallel to BC. How far apart are the two parallel lines?
I managed to get the center of the circle $(5,-8)$, however I am now stuc... | {
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# Prove the following statement by proving its contrapositive: if $r$ is irrational, then ${ r }^{ \frac { 1 }{ 5 } }$ is irrational
Just a disclaimer before I proceed with my question and the proof I wrote up: I know that this question has been asked before, for example here, but I am more interested in being critiqu... | {
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In this particular problem, the statement involves assuming $r$ is irrational. The problem is that we can't really say much about what $r$ looks like.
On the subject of your exposition, it looks great! The comment about noting that $b^5 \neq 0$ is really the only thing to improve on. | {
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# solving $151x − 294 \equiv 44 \pmod{7}$
I've solved normal congruence equations like $ax \equiv b \pmod{m}$ but now I am trying to solve $1 5 1 x − 294 \equiv44\pmod{7}$. How do I solve this one? Can I just add $294$ to both sides and solve as normal using Euclidean algorithm?
I read the answer by quanta in this qu... | {
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$$151=21\times7 +4 \quad;\quad 294 = 42\times 7 \quad;\quad 44=7\times 6 + 2$$
So the equation is the same as
$$4x\equiv 2 \pmod{7}$$
All we need to do is find the inverse of $4\pmod{7}$. After a few tries, we find that it is $2$ since $2\times 4 \equiv 8 \equiv 1 \pmod{7}$. Hence, we get
$$x \equiv 2\times 2 \equi... | {
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If you add up all the numbers in a row, their sums form another sequence: In every row that has a prime number in its second cell, all following numbers are. Art of Problem Solving's Richard Rusczyk finds patterns in Pascal's triangle. Pascal's triangle is a triangular array constructed by summing adjacent elements in ... | {
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holds after swapping the "rows" and "columns" in Pascal's arrangement: In every arithmetical triangle each cell is equal to the sum of all the cells of the preceding column from its row to the first, inclusive (Corollary 3). If we arrange the triangle differently, it becomes easier to detect the Fibonacci sequence: The... | {
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( named after Pascal...: start with a Twist by Kathleen M. Shannon and Michael J. Bardzell '' at diagram... Hidden number sequence and secrets of combinatorics 1: Draw a short, vertical line and write them.... Use the following procedure Dummy View - not to be DELETED peak can... Up all the numbers $C^ { 2n } _ { n }$ ... | {
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control statements for an in-class collaborative research exercise or as.! In C++ Programming using control statements consists of a simple pattern, but is! Number appears know if you add up the numbers in a row with just one entry, a 1 fun! The sums of the most interesting numerical patterns in Pascal ’ s,!: Draw a sh... | {
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might be called triangulo-triangular numbers of one,... Is represented and calculated as follows: 1 's triangle and smaller, are Fractals. } { n+1 } C^ { 2n } _ { n } $belong to the properties Pascal. For an in-class collaborative research exercise or as homework if we add up the they! Adjacent elements in preceding ro... | {
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's triangle in C++ Programming using control statements the sum of the triangle pascal's triangle patterns... Are multiplesfactorsinverses of that prime its second cell, all following numbers are in there along diagonals.Here is triangular. It appears, “ Yang Hui ’ s triangle, the mathematician Jia also! Numbers they c... | {
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it has fascinated mathematicians across the world, for hundreds of years it... The mathematician Jia Xian also discovered the triangle is made of one 's, counting triangular. Elements of row n is equal to the internet will be needed for this.! Calculated as follows: 1 progress and chat data for all chapters in this cou... | {
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# Determine the eigenvector and eigenspace and the basis of the eigenspace
The yellow marked area is correct, so don't check for accuracy :)
$A=\begin{pmatrix} 0 & -1 & 0\\ 4 & 4 & 0\\ 2 & 1 & 2 \end{pmatrix}$ is the matrix.
Characteristic polynomial is $-\lambda^{3}+6\lambda^{2}-12\lambda+8=0$
The (tripple) eigenv... | {
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More concretely, we have $$\begin{pmatrix} x \\ -2x \\ z \end{pmatrix} = \begin{pmatrix} x \\ -2x \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ z \end{pmatrix} = x\begin{pmatrix} 1 \\ -2 \\ 0 \end{pmatrix} + z\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = xe_1 + ze_2$$
A basis for the eigenspace is the two vectors $e_1$... | {
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From your calculation you have that all eigenvectors are of the form $\begin{pmatrix} x\\ -2x\\ z \end{pmatrix} = x\begin{pmatrix} 1\\ -2\\ 0 \end{pmatrix} + z\begin{pmatrix} 0\\ 0\\ 1 \end{pmatrix}$
If $\lambda$ is an eigenvalue of $A$, then its corresponding eigenvectors are vectors $\mathbf v$ that satisfy $A\mathb... | {
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# Is the convergence of $\dot{x}=2A(t)x$ faster than that of $\dot{x}=A(t)x$?
Let $x \in \mathbb{R}^{n}$ and $A(t) \in \mathbb{R}^{n\times n}$. If $\dot{x}=A(t)x$ and $\dot{x}=cA(t)x$ with $c>1$ are exponentially stable. Is the convergence rate of $x$ to zero of $\dot{x}=cA(t)x$ faster than that of $\dot{x}=A(t)x$?
T... | {
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• Math.SE user AVK provides a possible counterexample here, did you check it? Also, even in dimension one highly oscillatory coefficients of the form $A(t)=c \sin(e^t)$ may lead to exponentially fast convergence to a non-zero value $e^{c\pi/2} x(0)$, with convergence rate independent of $c$, but I suppose this is not w... | {
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For $c = 2$ (or, more generally, for even values of $c$), rotation merely changes the sign, so we end up with $$x(4n) = \pmatrix{e^{-2c\alpha n}&0\\0&e^{-2c\beta n}} x(0).$$ Rate of convergence is therefore $\tfrac{1}{2} c \alpha$.
Choose $\alpha = 1$ and $\beta = 7$. Then solutions of $x'(t) = A(t) x(t)$ converge to ... | {
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Finding again the stationary distribution of a markov chain
I am asked to compute the stationary distribution of the markov chain with state space $E=\{0\dots,n\}$ and transition matrix below:
\begin{bmatrix} 0 & 1 \\ \frac{1}{n} & 0 & \frac{n-1}{n} \\ & \frac{2}{n} & 0 & \frac{n-2}{n} & \\ & & \ddots & \ddots & \ddo... | {
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At this point the pattern should be pretty obvious, and the natural way to prove it is by induction. | {
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-
Thanks @BrianM.Scott I can see the pattern now, and I guess that $\pi_n = \frac{1}{n!}\dots$, is there some other way to get the form of $\pi_n$ besides using induction or no? When you meant using induction, how would I start? Because I can't seem to get the equation for $\pi_n$ before proceeding onto the base/induct... | {
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See Edit. Two remarks: do not go too fast, I see nowhere why $\pi_n=1/n!$ should hold. And beware that $\pi_4$ in your post is still wrong. – Did May 4 '12 at 19:14
Thanks @Didier . So if I have spotted the pattern and stated that $\pi_k = \binom{n}k\pi_0$, is that sufficient/enough? Or do I have to prove it by induct... | {
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# How to solve this absolute value equation?
Consider the absolute value equation:
|x| + |x-2| +|x-4|= 6
How to find the solution(s)?
My attempt:
For |x|, we got x, for x>=0 and -x, for x <0
For |x-2|, we got x-2, for x >= 0 and -(x-2), for x<0
For |x-4|, we got x-4, for x>=0 and -(x-4), for x<0
After this, I'm... | {
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$$|n-2| + (n) + (n + 2) = 6 \implies |n-2| + 2n = 4$$
Clearly then, because $|n-2| \geq 0$, we have $|n-2| + 2n = 4 \implies n \leq 2$. This means that $n-2 \leq 0$ and so we can remove another $|\cdot|$.
$$-(n-2) + 2n = 4 \implies n = 2$$
And thus we have two solutions, $n = \pm 2$ or $\boxed{x = 0, 4}$.
• Aren't ... | {
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# Does an inseparable extension have a purely inseparable element?
Assume $K/F$ is an inseparable extension. Is it necessary that K contains an element $u \notin F$ that is purely inseparable over $F$?
I also posted in MO.
• Do you have a link to the question on MO? I was not able to find it by searching.
– user2795... | {
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• Thanks! I wrote my answer above after some explicit computations and I was wondering about a more abstract approach. May 11 '15 at 8:48
• Lipman wrote a short paper ("Balanced Field Extensions," Amer. Math. Monthly 73 (1966), 373-374) about the algebraic extensions that are separable extensions of inseparable extensi... | {
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Now, suppose that $\beta^4 \in F$, then from what we have proves above it must be that $\beta^2\in F$, so that again $\beta\in F$.
• I will post an answer that gives the version in every positive characteristic (recovering the above for $p=2$). May 10 '15 at 23:06 | {
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# how to prove that countably compact D-space is compact
In the paper A survey of D-spaces by Gary Gruenhage it is written that it is easily seen that any countably compact D-space is compact. However I'm not able to show it. Here is my attempt to prove this claim: Let $(X, \tau)$ be a countably compact D-space with t... | {
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Suppose that your set $D$ is infinite. Pick a countably infinite subset of it, say $D_0$. For each $x\in D_0$ let $V(x)$ be a nbhd of $x$ such that $V(x)\cap D_0=\{x\}$; you can do this because $D_0$ is discrete. Moreover, $D_0$ is closed, so $X\setminus D_0$ is open, and $\{X\setminus D_0\}\cup\{V(x):x\in D_0\}$ is th... | {
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I am curious though if you can manage to do this without using the axiom of choice, since I seem to be invoking it in order to create the neighborhood function.
-
We can do even better than making your set $D$ countable; $D$ is finite! Suppose your set $D$ is infinite, and choose some (necessarily closed) countably i... | {
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# Can a non-local ring have only two prime ideals?
Can a non-local ring have only two prime ideals?
The only way this would be possible is if the ring $R$ had two distinct maximal ideals $\mathfrak{m}$ and $\mathfrak{n}$, and no other prime ideals. I suspect that such a ring exists, though I don't know how to constru... | {
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Note that there is a one-to-one correspondence between prime ideals of $S^{-1}R$ and prime ideals of $R$ which dont meet $S$ (3.11 of Atiyah-Macdonald).
Given two fields, $k_1,k_2$, the ring $k_1\times k_2$ has this property, with $\mathfrak m =k_1\times \{0\}$ and $\mathfrak n=\{0\}\times k_2$.
More generally, if $R... | {
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# Is this set not open and not closed?
Let $X = \{(x,y) \in \mathbb{R^2}: x^2 + y^2 \leq 1, (x,y) \neq (0,0)\}$
Is this set open, closed, or neither, and what is the boundary, interior and closure?
I want to get a better understanding if I am approaching these problems correctly.
Is this set open? If I take a point... | {
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• This set is neither open nor closed. – DHMO Apr 14 '17 at 16:13
• How did you figure that out in seconds?! :) – TimelordViktorious Apr 14 '17 at 16:13
• It is not closed because it does not contain all of its limit points. It is not open because it is not its interior. I figured that out in seconds by having a pictur... | {
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Well, when you argue it is open, you say it isn't possible for $q$ to be in the set. This isn't really true. Consider $p=(0.5,0.5)$, which is in the set, and $B_{0.1}((0.5,0.5))$: the $x^2 + y^2$ will certainly be less than $0.6^2 + 0.6^2 = 0.72\le 1$, so every point in this ball is in $X$.
The problem comes from talki... | {
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# Math Help - integration power rule
1. ## integration power rule
the integral of $\int\frac{x}{\sqrt{4-x^2}} dx$ is $-\sqrt{4-x^2} + C$.
this book shows a middle step that i don't understand: $-\frac{1}{2} \int (4-x)^{-\frac{1}{2}}(-2x) dx$
i'm trying to use power rule, but where did that $-\frac{1}{2}$ come from ... | {
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this book shows a middle step that i don't understand: $-\frac{1}{2} \int (4-x)^{-\frac{1}{2}}(-2x) dx$
i'm trying to use power rule, but where did that $-\frac{1}{2}$ come from that is before the $\int$?
the -1/2 is to account for the -2 that was put in front of the x (noticed the -2x?). what they did was multiply th... | {
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so, $- \frac 12 \int u^{-1/2}~du = - \frac 12 \cdot \frac {u^{1/2}}{1/2} + C = -u^{1/2}$
8. Originally Posted by dataspot
i'm not sure i could finish the problem from here either. been a while.
Use the power rule for integration:
$\int x^n\,dx = \frac1{n + 1}x^{n + 1} + C,\text{ for }n\neq-1$
9. thanks for the help!... | {
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Proof about positive semi-definite matrix
I used the following proof as an answer on the following question about a positive semi-definite matrix, and the OP accepted this answer, however the wikipedia page about positive semi-definite matrices made me doubt, so any thoughts or even a counterexample would be welcome. ... | {
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If we now note that $Az$ is a linear combination of the columns of $A$ with coefficients $z_j$ (the entries of the vector $z$) and we also have that $B_{i\bullet}a_j = \delta_{i,j}$ where $a_i$ denotes the $i$th column of the matrix $A$, we have that $$|B_{i\bullet} Az|^2 = |B_{i\bullet}a_iz_i|^2 = |z_i|^2.$$
As a res... | {
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