text stringlengths 1 2.12k | source dict |
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which simplifies as:
$\log|y-2|-\log|y+2|=4(x+c),$ or
$$\left| \frac{y-2}{y+2} \right| = k e^{4x}$$ with $k > 0$ for the nonconstant solutions. The item inside absolute value signs is a linear fractional or Mobius transformation, once we decide about the $\pm$ signs the inverse is given by the matrix $$\left( \begin{... | {
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\begin{align} x &=\int\frac{\,\mathrm{d}y}{y^2-4}\\ &=\frac14\int\left(\frac1{y-2}-\frac1{y+2}\right)\,\mathrm{d}y\\ &=\frac14\log\left(\frac{y-2}{y+2}\right)+C \end{align} Therefore, $$\frac{y-2}{y+2}=ke^{4x}$$ or, solving for $y$, $$y=2\,\frac{1+ke^{4x}}{1-ke^{4x}}$$ We get your form of the answer by letting $k\lt0$.... | {
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# Binomial Distribution Probabilities - Need help with multiple requirements for k
Could someone please check if the procedure for part a) is correct and I need a bit of help in regards to part b). I don't have the answer to refer to. Thank you for your time in advance!
An airport reports that on a stormy day, with w... | {
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$$P_{complement} = P(0) + P(1) + P(9) + P(10) + P(11) + P(12)$$
Here you have to sum only $$6$$ probabilities and the probability of original event from question is:
$$P = 1 - P_{complement} = 1 - P(0) - P(1) - P(9) - P(10) - P(11) - P(12)$$
If this isn't an exam question you can also write a simple computer program... | {
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O(n+m) vs O(n+2m) time complexity
Are $$O(n+m)$$ and $$O(n+2m)$$ the same?
If $$m>n$$, then both complexities are $$O(m)$$. Likewise, if $$n>m$$, then they are $$O(n)$$. Is this correct?
• Since $n+m \le n+2m \le 2(n+m)$, assuming $n$ and $m$ are non-negative, $O(n+m) = O(n+2m)$ under whatever reasonable definition ... | {
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As case $$O(n+m) \subset O(n+2m)$$ is obvious, let me go to reverse subset direction: assume we have $$f\in O(n+2m)$$, which means $$f(n,m)\leqslant B_f (n+2m)$$ in appropriate conditions for some constant $$B_f$$. Now we need such $$C_f$$ for which holds $$B_f (n+2m) \leqslant C_f(n+m)$$. It's easy to see, that any $$... | {
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# Polar Plots and square roots
When I plot a polar plot of $r=\sin (3 \theta)$, and $r=\sqrt{\sin (3 \theta)}$ I get nearly identical graphs, both $3$ pedal rose type plots. In the case without the square root, it is easy to understand the plot. However, for the plot involving the square root of $\sin 3 \theta$, it is... | {
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• It's worth noting that the $r=\sqrt{\sin 3\theta}$ curve plots only non-negative $r$ values. This is because the square-root function itself never returns negative values. On the other hand, a related curve, $r^2 = \sin 3\theta$ would plot both a positive and a negative $r$ value for a given $\theta$, effectively cre... | {
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# Trouble with Lorentz transformations
1. Apr 8, 2010
### pc2-brazil
Good evening,
As an effort for trying to understand Lorentz transformations, I'm trying to use them to derive the "length contraction" result.
Consider two reference frames, O (non-primed) and O' (primed), moving with respect to each other with a ... | {
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Last edited by a moderator: Apr 25, 2017
3. Apr 9, 2010
### pc2-brazil
But why shouldn't t'2 equal t'1?
I thought this was implied, because, for the primed referential, the times of measurement of the ends of the rod were the same.
4. Apr 9, 2010
### starthaus
Because of relativity of simultaneity. You are measuri... | {
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Remember, gamma = 1/[SQRT(1 - v^2/c^2)]. Gamma is always >=1, so 1/gamma is always <=1.
This makes sense. You give us x'_2 and x'_1 so x_2 - x_1 is always <= x'_2 - x'_1
L' = (x'_2 - x'_1) and L = (x_2 - x_1)
Thus L' = gamma*L which is the way it should be...
The same as starthaus above...
Steve G
7. Apr 10, 2010
... | {
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### starthaus
When you write x'2 - x'1 it means that you are measuring in the primed frame.
One more time, L' = γL is wrong. If you did things correctly you should have gotten L' = L/γ
13. Apr 12, 2010
### stevmg
Then pc2-brazil had it right the first time...
L = $$\gamma$$L'
Starthaus, you are right... due to t... | {
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In the text Special Relativity by AP Frenchf M.I.T., on page 97, he goes over this exact problem presented by pc2-brazil. He assumes the measurement of x'_1 and x'_2 are done simultaneously in the S' frame of reference and he derives L' = x'_2 - x'_1 = L/gamma = (x_2 - x_1)/gamma. Now it would appear that by length con... | {
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As per convention let us assign x_1 = 0 and t_1 = 0 and x'_1 = 0. Now, we state that t_2 = 9. Thus, after 9 hours (t_2) in the S F.O.R. the coordinate of that rocket ship is x_2 = 9*0.8 = 7.2 lt-yr (which coincides with the position of that star.) Thus x_2 - x_1 = 7.2 - 0 = 7.2.
Because v (the velocity of S' is 0.6c, ... | {
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# Finding an Interval of Convergence
1. Sep 2, 2008
### Battlemage!
1. The problem statement, all variables and given/known data
Find the interval of convergence of the infinite series:
(x-1)n / 2n
n = 1
2. Relevant equations
Using the ration test. It converges if the absolute value of the limit of f(n+1)/f(n) a... | {
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Good, common question. As an example, suppose I have the series
$$\sum_{n=1}^\infty \frac{x^n}{n}$$
Work similar to what you did in your example shows the radius of convergence to be $$1$$, so I know the series converges for $$-1 < x < 1$$. What about the endpoints here?
First, consider $$x = 1$$. Simply plug this va... | {
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Solving an equation with fractional part function
Tags:
1. Apr 19, 2016
1. The problem statement, all variables and given/known data
If $\{ x \}$ denotes the fractional part of x, then solve:
$\{ x \} + \{ -x \} = x^2 + x -6$
It's provided that there are going to be 4 roots of this equation. And two of them will be ... | {
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5. Apr 19, 2016
BvU
Based on the first relevant equation I concluded $0.6$ ...
6. Apr 22, 2016
Sorry, I did a silly mistake there.
My teacher said, the fractional part function is defined as:
$\{ x \} = x - [x]$, where [x] is the greatest integer function.
@Ray Vickson , that makes $\{ -1.4 \} = +0.6$
So,
$\text{... | {
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# Write on my own my first mathematical induction proof
I am trying to understand how to write mathematical induction proofs. This is my first attempt.
Prove that the sum of cubic positive integers is equal to the formula $$\frac{n^2 (n+1)^2}{4}.$$ I think this means that the sum of cubic positive integers is equal t... | {
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Sorry for my soliloquy but it helps to understand and I would appreciate confirmation from you!
• "appartains at $N$" what on earth does that mean / have to do with anything? – MichaelChirico Oct 27 '15 at 12:04
• I am confused by your aside about odd numbers. 1 + 8 + 27 is not an odd number. – Eric Lippert Oct 27 '15... | {
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Which is the relation we set out to prove. So the method is to substitute $i=k+1$ into the formula you are trying to prove and then use the inductive assumption to recover the $\color{blue}{\mathrm{blue}}$ equation at the end.
• @Alwayslearning Good to see you making a positive start with inductive techniques, well do... | {
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It seems your issue is more conceptual than algebraic since you're stuck about which form of right hand side to use.
A proof by induction on a sum formula works by showing: (1) it holds in the base case, when the index is at its minimum; and (2) if it applies for the $n=k$ case, then it will also hold for the $n=k+1$ ... | {
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Which is precisely what you need.
• Thank you! But, perhaps, you could break your passages as the final ones go on the right column of the website where there are the related questions. Thus, I can't read all your passages. – Always learning Oct 27 '15 at 11:34
• Does this help? – Biouk Oct 27 '15 at 11:37
• yes, than... | {
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All distinct subgroups of $\mathbb{Z}_4 \times \mathbb{Z}_4$ isomorphic to $\mathbb{Z}_4$
This question is from a past exam.
Find all distinct subgroups of $\mathbb{Z}_4 \times \mathbb{Z}_4$ isomorphic to $\mathbb{Z}_4$
Attempt/Thoughts?
Since $\mathbb{Z}_4$ is cyclic we are looking for cyclic subgroups of the give... | {
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• This is exactly what I needed. Get my hands dirty and do these computations. Once I see the computations, I can actually understand what nrpeterson and julien are talking about. So Thank you. – minibuffer Jul 11 '13 at 18:56
• Nicholas: Just so you know I passed my exam :). Your answers/comments helped me a great dea... | {
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A cyclic group of order $4$ has two generators (each of order $4$ of course). And every order $4$ element is the generator a cyclic order $4$ subgroup.
Two cyclic subgroups of order $4$ are equal if and only if they share a generator.
So there is a natural partition of the set of order $4$ elements into pairs of gene... | {
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# Show that A is identical
• MHB
Gold Member
MHB
Hello! (Wave)
I want to prove that if a $m \times m$ matrix $A$ has rank $m$ and satisfies the condition $A^2=A$ then it will be identical.
$A^2=A \Rightarrow A^2-A=0 \Rightarrow A(A-I)=0$.
From this we get that either $A=0$ or $A=I$.
Since $A$ has rank $m$, it foll... | {
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Thus $A^2-A=0 \Rightarrow A(A-I)=0 \Rightarrow A^{-1}A(A-I)=0 \Rightarrow A-I=0 \Rightarrow A=I$.
Is it complete now? Could we improve something? (Thinking)
Homework Helper
MHB
So is it as follows?
Since the rank of the $m\times m$ matrix $A$ is $m$ we have that at the echelon form the matrix has no zero-rows. This ... | {
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# Prove $x<\sqrt{x^2+1}$.
Prove $$x<\sqrt{x^2+1}$$.
I am pretty sure this an easy question as the inequality seems obviously true, but I am not entirely convinced by my argument.
So I squared both sides (is this allowed?):
$$x^2, so $$0<1$$ so the inequality is obviously true.
However, I am unconvinced that this p... | {
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# How does $\left|\sqrt{2+3x_1} - \sqrt{2+3x_2}\right|$ become $\left| \frac{3(x_1-x_2)}{\sqrt{2+3x_1} + \sqrt{2+3x_2}} \right|$?
Could someone explain to me this transformation? It is used frequently in my uni course, and I do not understand what's happening:
$$\left|\sqrt{2+3x_1} - \sqrt{2+3x_2}\right| = \left| \fr... | {
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# a nice problem #2
#### sbhatnagar
##### Active member
Find the limit.
$$\lim_{n \to \infty} \left\{ \left( 1+\frac{1^2}{n^2}\right)\left( 1+\frac{2^2}{n^2}\right)\left( 1+\frac{3^2}{n^2}\right) \cdots \left( 1+\frac{n^2}{n^2}\right)\right\}^{\dfrac{1}{n}}$$
#### Fernando Revilla
##### Well-known member
MHB Math ... | {
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\displaystyle \begin{aligned} & \begin{aligned} \log (\ell) & = \lim_{n \to{+}\infty}\dfrac{1}{n}\;\displaystyle\sum_{k=1} ^n \log \left(1+(k/n)^2\right) \\& = \int_0^1 \log (1+x^2)\;dx \\& = \int_0^1\sum_{k \ge 0}\frac{(-1)^kx^{2k+2}}{k+1}\;{dx} \\& = \sum_{k \ge 0}\int_{0}^{1}\frac{(-1)^kx^{2k+2}}{k+1}\;{dx} \\& = \s... | {
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# Can a limit of an integral be moved inside the integral?
After coming across this question: How to verify this limit, I have the following question:
When taking the limit of an integral, is it valid to move the limit inside the integral, providing the limit does not affect the limits of integration?
For instance i... | {
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Taking the limit inside the integral is not always allowed. There are several theorems that allow you to do so. The major ones being Lebesgue dominated convergence theorem and Monotone convergence theorem.
The uniform convergence mentioned in the comments is a special case of Dominated convergence theorem.
• If you k... | {
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• Why the $g_n(x_0)$ is not equal to $e^{-x_0}\frac{x_0}{n}$ ? Sep 17, 2018 at 11:11
• Hi @Nameless! I'm obviously a newbie here. So pardon me to ask this. Why is it sufficient to show that $\| f_n-f\|_{\infty} \rightarrow 0$ to show that $f_n$ converges uniformly? :) Feb 28, 2019 at 2:56
• The expression $\|f_n - f\|_... | {
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# What is the maximum distance of k points in an n-dimensional hypercube?
For this question, I'm thinking only about the euclidean distance:
Let $p_1 = (x_1^{(1)}, \dots, x_n^{(1)})$ and $p_2 = (x_1^{(2)}, \dots, x_n^{(2)})$ be $n$-dimensional points. The euclidean distance of $p_1$ and $p_2$ is $$d(p_1, p_2) = \sqrt... | {
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## Arbitrary $n$
• $\alpha(n, k=2) = \sqrt{n}$
• $\alpha(n, 2^n) = 1$
What is $\alpha(n, k)$?
• if $k\leq n$ isn't it sufficent to consider the graph made up by the corners of the hypercube? – tired Oct 19 '16 at 20:33
• @tired: I'm not sure. This would mean $\alpha(2, 3) = 1$, but I'm relatively certain that you co... | {
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Let's start with the $$2$$D case.
Consider to have $$m$$ circles of radius $$r$$.
Packing them inside a square $$C_r$$ of size $$(1+2r) \times (1+2r)$$, will be equvalent to ask that their centers be inside the unit square and at mutual distance $$r \le d$$.
Then your question is equivalent to:
what is the maximum $... | {
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# Confusion on using LognormalDistribution in Mathematica
I would like to fit a LogNormalDistribution[μ,σ] based on some data that I have. Roughly speaking, would I need to take the Log of these data's y-values before calculating the mean and standard deviation in order to determine $\mu$ and $\sigma$ above, or can I ... | {
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As to which method (or some other method) is best, that would be a question for CrossValidated.
• Nice answer. I would suggest to use a defined SeedRandom when using RandomVariate so as to make your results replicable? – gwr Oct 11 '17 at 18:07
• @gwr Good suggestion! In fact, I thing I might open a question about tha... | {
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(dist6 = LogNormalDistribution @@ (#[Log[data]] & /@ {Mean,
StandardDeviation})) // InputForm
(* LogNormalDistribution[
2.904567990925176, 1.5064417937677634] *)
Using TransformedDistribution on the Log of the data
(dist7 = TransformedDistribution[E^x,
x \[Distributed] FindDistribution[Log[data]]]) // InputForm
(*... | {
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Find all solutions to the functional equation $f(x+y)-f(y)=\frac{x}{y(x+y)}$
Find all solutions to the functional equation $f(x+y)-f(y)=\cfrac{x}{y(x+y)}$
I've tried the substitution technique but I didn't really get something useful.
For $y=1$ I have
$F(x+1)-F(1)=\cfrac{x}{x+1}$
A pattern I've found in this examp... | {
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• Give me some time to fully comprehend the meaning of it. – Mr. Y Dec 27 '15 at 17:45
• @Mr.Y Sorry I mistyped it. Now the equation is the correct one and you should now be able to solve it. There are a few tricky parts left. – wythagoras Dec 27 '15 at 17:48
• If I rearrange your hint I have that $f(1)-f(y)=\cfrac{1}{... | {
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Another approach: I assume that $f$ is a function of a single real variable.Write the defining equation as $\frac{f(y+x) - f(y)}{(y+x)- y} = \frac{1}{y(x+y)}$ (for $x,y,(x+y) \neq 0).$ Take the limit as $x \to 0$. On the one hand this limit is $\frac{1}{y^{2}}.$ On the other hand, the limit is, by definition, the deriv... | {
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# How to Evaluate $\int \! \frac{dx}{1+2\cos x}$ ? [duplicate]
Possible Duplicate:
How do you integrate $\int \frac{1}{a + \cos x} dx$?
I have come across this integral and I tried various methods of solving. The thing that gets in the way is the constant $2$ on the $\cos(x)$ term. I tried the conjugate (works withou... | {
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$=\int \frac{\mathrm{d}x}{3\cos^2(x/2)-\sin^2(x/2)}$
Multiply the Nr and the Dr of the integrand by $\sec^2 (x/2)$.
You will get:
$\int \frac{\sec^2(x/2)\mathrm{d}x}{3-\tan^2(x/2)}$
Substitution:
$z=\tan(x/2)$
$\mathrm{d}z=1/2 \sec^2(x/2) \mathrm{d}x$
Therefore,
Integral=$\int \frac {2\mathrm{d}z}{3-z^2}$
$=\i... | {
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# Thread: Few more integral questions.
1. ## Few more integral questions.
First one, $\int x^4(x^5+2)^7dx$. Is that as straight forward as it seems? Expand that out $x^{39}+14x^{34}+84x^{29}+...$ and then integrate as normal? Or should I substitute, setting $u=x^5+2$ and $\frac{du}{5}=x^4dx$?
Second one, $\int \frac... | {
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5. Originally Posted by cinder
First one, $\int x^4(x^5+2)^7dx$. Is that as straight forward as it seems? Expand that out $x^{39}+14x^{34}+84x^{29}+...$ and then integrate as normal? Or should I substitute, setting $u=x^5+2$ and $\frac{du}{5}=x^4dx$?
Second one, $\int \frac{\cos(\sqrt{\theta})}{\sqrt{\theta}\sin^2(\sq... | {
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Is it possible to split the natural numbers into a finite number of sets so that no pair of numbers within a set adds up to a square?
My attempt:
$$\lbrace6,19,30\rbrace$$ is sufficient to show that two sets are impossible.
Using a computer program with a brute force method I found that separating the numbers $$1$$ ... | {
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In a previous version of my post I made a proof showing that the number of sets needed for Natural numbers $$1$$ to $$N$$ so that no pair of numbers in the same set sums to a square is no more than $$\lfloor\sqrt{2N-1}\rfloor$$. I realized that I can do significantly better than this. In order to explain the method tha... | {
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the new upper bound is $$\lfloor\sqrt{N+3}\rfloor-1$$ sets. This is the significant improvment from $$\lfloor\sqrt{2N-1}\rfloor$$ I mentioned at the beginning. | {
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End edits
For each natural number $$X$$ there are $$\lfloor\sqrt{2X-1}\rfloor-\lfloor\sqrt{X}\rfloor$$ numbers that are less than $$X$$ that when summed to $$X$$ results in a square. The expression: $$\lfloor\sqrt{2X-1}\rfloor-\lfloor\sqrt{X}\rfloor$$ increases as $$X$$ gets larger, because of this fact my guess is th... | {
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• I did some exploring with your method and I found that a set of numbers with with an odd cardinality where every pair of numbers in the set adds to a square is more "messy" to deal with than with a set of an even cardinality. There are three different ways to sort four numbers into pairs. This is why you needed a num... | {
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# Let $A\subseteq\mathbb{R}^k$, and let $A'=\{x\in\mathbb{R}^k \mid x\in\partial(A\setminus\{x\})\}$. How do I prove that $A'$ is closed?
I was asked this simple question a short while ago.
Let $$A\subseteq\mathbb{R}^k$$, and let $$A'=\{x\in\mathbb{R}^k \mid x\in\partial(A\setminus\{x\})\}$$. Prove that $$A'$$ is clo... | {
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I doubt my reasoning, but I am not sure what have I missed. Care to shed some light over the matter?
• It's not true that $A'=\partial A$ in general, and not every subset of a closed set is closed, so there's definitely still an argument to be made. It seems actually that your observations might more directly prove th... | {
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Let $$p \notin A'$$. This means that $$\exists r>0: B(p,r) \cap A \subseteq \{p\}$$. If in fact $$B(p,r) \cap A = \emptyset$$, for any $$x \in B(p,r)$$ we have a ball $$B(x,r') \subseteq B(p,r)$$ (open balls are open sets) and this witnesses also that $$B(x,r') \cap A= \emptyset$$ and so $$x \notin A'$$. So $$B(p,r) \s... | {
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Let's see: $$x\in\partial(A\setminus\{x\})$$ means that every neighborhood of $$x$$ intersects both $$A\setminus\{x\}$$ and its complement. However, $$x$$ surely belongs to the complement of $$A\setminus\{x\}$$, so the condition just reads $$\textit{every neighborhood of x intersects A\setminus\{x\}}$$ which is the sta... | {
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# Fibonacci Number Proof
How can I prove this statement? Would I use induction?
"Given $n \geq 11$, show that $a_n > (3/2)^{n}$. $a_n$ is the $n$th Fibonacci number."
-
Since $\sqrt[n]{a_n} \to \phi=(1+\sqrt5)/2 \approx 1.618$, there is not much improvement you can make to $3/2$. The proof given by Old John proves t... | {
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-
So should I select n = 11 and n+1 = 12? My friend mentioned something about choosing n = 11 and n-1 = 10, since the definition of Fibonacci number is F(n+1) = F(n) + F(n-1). Would this also work, or is the first option easier? – user41419 Sep 24 '12 at 23:03
For the induction step you do not need to specify the value... | {
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In your case $\rm\:a,b,c\, =\, 1,1,3/2\, >\, 0,\:$ and $\rm\:\color{#C00}{f(c)} = (3/2)^2\!-3/2-1 =\, \color{#C00}{-1/4} < 0,\:$ so it succeeds.
- | {
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# Thread: Triangles in 20 non-straight-line "dots"
1. ## Triangles in 20 non-straight-line "dots"
Twenty points/dots are given so that the three of them are never in a straight line. How many triangles can be formed with corners(as in vertex i think?) in the dots?
2. Calculate the number of ways to choose three poin... | {
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The number of ways to do this is $\binom{19}{2}=171$
Therefore, there are 171 triangles that can be drawn which include the leftmost point.
If you move on to the next point to the right and exclude the point previously chosen,
then you can draw another $\binom{18}{2}=153$ triangles.
These triangles do not include an... | {
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How to understand the choice of Krylov subspace orthonormal basis?
This semester, I study the Krylov subspace iterative methods (about Ax=b) using the book H. A. Van der Vorst. Iterative Krylov Methods for Large Linear Systems, volume 13. Cambridg... | {
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Methods to approximate obective function gradients from point cloud
Problem statement: Assume that I have an objective function $f(x)$ which takes as input a $D$-dimensional vector $x\in\mathbb{R}^D$, and that $f(x)$ is sufficiently smooth. Assume further that I ...
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Fast algorithm for computing low... | {
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Question regarding 1D implementation of the DG method
I'm pretty new to the DG method and have been writing a 1D code to help me understand the coding aspect. With respect a reference, I've been following these notes https://www3.nd.edu/~zxu2/...
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Why do not we choose the error solution norm as an iterative m... | {
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12X<=600 --> X<=50, But we rounded up, so the real number must be <50 --> 49 (C)
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# Math Help - Arithmetic Progression question
1. ## Arithmetic Progression question
Hey guys, I'm having problems with this question:
The sum of the first 100 terms in an arithmetic progression with first term a and common difference d is T. The sum of the first 50 even-numbered terms, i.e. 2nd, 4th, 6th, ..., 100th... | {
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3. Thanks ThePerfectHacker for your solution but I still dont quite get it.
EDIT: Got the answer, finally! Thanks once again!
RE-EDIT: No actually, what if I would to use formulas to solve this question? How would the solution be presented?
4. Originally Posted by margaritas
RE-EDIT: No actually, what if I would to ... | {
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# Conditional Probability (Baye's Rule) question with urns
There are two urns A and B. Urn A contains 1 red ball and 2 white balls, whereas urn B contains 2 red balls and 1 white ball. Calculate the conditional probability that a randomly chosen ball belonged to urn A given that it is white.
I know, that the answer i... | {
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# Finding the probability that red ball is among the $10$ balls
A box contains $20$ balls all of different colors including the red color. If we select $10$ balls randomly without replacement, what is the probability that the red ball will be among these $10$ balls?
What I think is that: If we let $X$ to be the numbe... | {
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Here's how the reasoning with $X$ could have gone: the chance the $n$-th ball came up red is the probability the other $n-1$ before it came up nonred and then you got the red ball out of the remaining ones (the latter colored red in the calculations below). The probability the first pick was nonred is $19/20$, the prob... | {
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Two quicker ways could have been the following:
1. Symmetry: For every way of taking 10 balls out of 20 and leaving the other 10 in the box, there is exactly one way of taking those other 10 out of the box and leaving the original 10 in, hence by symmetry the probability of choosing the red ball equals the probability... | {
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-
I don't think symmetry works in this case because I have a second part of the problem which generalize this to any box with n different objects. Thanks! – Kelly Sep 19 '11 at 2:30
@Kelly: Symmetry does in fact work "in this case" - is my description of it in this particular case not clear enough, should I elaborate ... | {
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# Tag Info
## Hot answers tagged convolution
42
Convolution is correlation with the filter rotated 180 degrees. This makes no difference, if the filter is symmetric, like a Gaussian, or a Laplacian. But it makes a whole lot of difference, when the filter is not symmetric, like a derivative. The reason we need convol... | {
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13
One reason you see people designing FIR filters, rather than taking a direct approach (like both 1 and 2) is that the direct approach usually fails to take into account the periodicity in the frequency domain, and the fact that convolution implemented using an FFT is circular convolution. What does this mean? Suppo... | {
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10
One problem is dealing with infinite length transforms that wrap-around when using a finite length FFT. The Fourier transform of a finite length frequency response is an infinite length impulse response or filter kernel. Most people would like their filter to finish before they die or run out of computer memory, so... | {
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Yes, you are correct. Multiplication in time domain means convolution in frequency domain and vice versa. Multiplying your signals $x[n]$ and $y[n]$ will give an output: \begin{align} z[n]&=\{2\cdot 5, 4\cdot 1, 1\cdot 8\}\\ &= \{10, 4, 8\}\end{align} Remember that this output is in time domain. When you convolve $x[n]... | {
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any order. This is not true if some of the involved sequences do not converge absolutely, which is the case for the given sequences$x_1[n]$and$x_2[n]$. Note that the convolution sum$x_1\star x_2$does not converge, i.e.,$x_3\star (... | {
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8
As a student I was involved in the same problem as you are. Let me explain to you in the simplest words without any math. Convolution: It is used to convolute two function. May sound redundant but I´ll put an example: You want to convolute (in a non math term to "combine") a unit cell (which can contain anything you... | {
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7
Linear convolution is the basic operation to calculate the output for any linear time invariant system given its input and its impulse response. Circular convolution is the same thing but considering that the support of the signal is periodic (as in a circle, hance the name). Most often it is considered because it i... | {
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7
Normally, the variables $t$ and $f$ are used for continuous time and frequency. But from your question I understand that you're talking about discrete time and frequency, and their relation via the Discrete Fourier Transform (DFT). If you have discrete-time sequences $a[n]$ and $b[n]$ and their DFTs $A[k]$ and $B[k]... | {
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# Convergence of fixed point iteration for polynomial equations
I'm looking for the solution $x$ of
$$x^n+nx-n=0.$$
Thoughts: From graphing it for several $n$ it seems there is always a solution in the interval $[\tfrac{1}{2},1)$. For $n=1$, the solution is the fraction $\tfrac{1}{2}$ and for higher $n$, the solutio... | {
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• – Mhenni Benghorbal Feb 6 '13 at 13:07
• @Nick Kidman: Not a solution, but a remark: the fact that there is always a solution on $(1/2,1)$ for $n>1$ follows directly from the fact that $f_n(1/2)=1/2^n+n/2-n=1/2^n-n/2<0$ and $f_n(1)=1+n-n=1>0$. So, by the intermediate value theorem, there exists $c\in(1/2,1)$ such tha... | {
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Isn't this simply the result of the Banach fixed-point theorem? In you case one must prove that there exists $q$ such that: $$d(F_n(x),F_n(y))=\frac{|y^n-x^n|}{n} \le q |x-y|$$ Which is true for all $n>1, x>0$, with say, $q=0.9$, So that it doesn't matter what $x_S$ you choose.
Regarding a closed form - convergence of... | {
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# If A = BCD show that $C^{-1}$ = $DA^{-1}B$
I came across this question in a past paper, If A = BCD show that $C^{-1}$ = $DA^{-1}B$. All these matrices are sqaure and have inverses.
I attempted a solution but I am not sure if-
1. The solution is correct and;
2. Is this the best solution
Here is my attempt: *
$A = ... | {
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# How to tell if a space is second-countable
A topological space is called second-countable iff it has a countable basis.
How to prove or at least make an assumption about whether a space does, or does NOT have countable basis? Which properties of a space can imply that it is, or isn´t second countable? Is second-cou... | {
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The minimal size of a base (weight) or dense set (density) etc. are just a few of the many ways to "measure" the "size" of a space. In analysis many spaces are indeed of countable base; but e.g. weak topologies on Banach spaces often are not.
• So is the $l^2$ second-countable, or not? You are saying only it is separa... | {
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» JavaScript » PHP Now this is pretty interesting, because how did we define these two? By using this website, you agree to our Cookie Policy. tcrossprod () function in R Language is used to return the cross-product of the transpose of the specified matrix. Properties of transpose » LinkedIn » C++ » Embedded Systems 'k... | {
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by changing the rows as columns and columns as rows. In this program, the user is asked to enter the number of rows r and columns c. Their values should be less than 10 in this program. In this tutorial, we are going to check and verify this property. The following properties hold: (AT)T=A, that is the transpose of the... | {
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Transposes of sums and inverses. Linear Algebra 11w: Introduction to the Transpose of a Matrix - Duration: 7:40. » Java » O.S. Transpose of the product of two matrices is equal to the product of their transposes taken in the reverse order, The transpose of the matrix products can be extended to several matrices The inv... | {
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is defined as “A Matrix which is formed by turning all the rows of a given matrix into columns and vice-versa.” If is an eigenvector of the transpose, it satisfies By transposing both sides of the equation, we get. returns the nonconjugate transpose of A, that is, interchanges the row and column index for each element.... | {
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flipping entries about the diagonal. » DOS Solved programs: The transpose of a matrix is an important phenomenon in the matrix theory. Languages: Example: If A= 1 2 3 4 5 6 , then AT = 2 4 1 4 2 5 3 6 3 5: Convention: From now on, vectors v 2Rn will be regarded as \columns" (i.e. » C#.Net & ans. » Ajax does not affect t... | {
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"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9715639694252315,
"lm_q1q2_score": 0.8475412764478711,
"lm_q2_score": 0.8723473730188542,
"openwebmath_perplexity": 7917.3825798895095,
"openwebmath_score": 0.21413716673851013,
"tag... |
# How to determine specific values of a function on each streamline
I was thinking if there is a way to obtain the values of a function from a streamline plot. This is what I mean:
Consider a typical streamline plot from the documentation: StreamPlot[{-1 - x^2 + y, 1 + x - y^2}, {x, -3, 3}, {y, -3, 3}, StreamPoints -... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971563967366828,
"lm_q1q2_score": 0.8475412698157195,
"lm_q2_score": 0.872347368040789,
"openwebmath_perplexity": 1430.8630016387176,
"openwebmath_score": 0.33564215898513794,
"tag... |
((0.0308851, 1.0987}, {0.00331156, 0.101788}}
The output can be dressed up a bit, if desired, by
Dynamic[StringForm["loc = , val = ",
loc = MousePosition["Graphics", {0, 0}],
val = {-1 - x^2 + y, 1 + x - y^2} /. {x -> First@loc, y -> Last@loc}]]
• Thanks, @bbgodfrey. This is what I actually meant.This concept could... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971563967366828,
"lm_q1q2_score": 0.8475412698157195,
"lm_q2_score": 0.872347368040789,
"openwebmath_perplexity": 1430.8630016387176,
"openwebmath_score": 0.33564215898513794,
"tag... |
# Lecture 015
Let $m\in \mathbb{N}$, define a relation on $\mathbb{Z}$ by $(\forall a, b \in \mathbb{Z})$: $$a \equiv b (\mod m) \iff m|a-b$$
read as "a is congruent to b modulo m" (a and b have the same meaning under modulo m)
Claim: this is an equivalence relation Proof: show reflexive, symmetric, transitive
• R:... | {
"domain": "kokecacao.me",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9970650478192279,
"lm_q1q2_score": 0.847476552796607,
"lm_q2_score": 0.8499711775577736,
"openwebmath_perplexity": 1264.1412800066726,
"openwebmath_score": 0.9393684267997742,
"tags": n... |
• $[(a, b)]_\sim \times [(c, d)]_\sim = [(ac+bd, bc+ad)]_\sim$
We check if this + and * is well defined
Table of Content | {
"domain": "kokecacao.me",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9970650478192279,
"lm_q1q2_score": 0.847476552796607,
"lm_q2_score": 0.8499711775577736,
"openwebmath_perplexity": 1264.1412800066726,
"openwebmath_score": 0.9393684267997742,
"tags": n... |
# Probability of a 4th die roll being higher than one of the first 3 rolls?
If I roll 3 dice with n sides, and then roll a 4th die of the same size, what are the odds of it being higher than at least one of the previous rolls?
I'm thinking it should be something like the odds of rolling a 1 in the first 3 dice times ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9879462233229674,
"lm_q1q2_score": 0.8474287634039871,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 187.7219510469037,
"openwebmath_score": 0.5211573839187622,
"tags... |
What is the probability that the 4th roll is not the smallest?
One of the 4 dice must be the smallest. It is just as likely to be the 4th die rolled as one of the previous 3.
But there is a possibility that there is a tie.
As n gets to be very large, I would expect our probability to approach $$\frac 14$$
There are... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9879462233229674,
"lm_q1q2_score": 0.8474287634039871,
"lm_q2_score": 0.857768108626046,
"openwebmath_perplexity": 187.7219510469037,
"openwebmath_score": 0.5211573839187622,
"tags... |
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