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If is a differentiable function at , then is continuous at . infinity. Continuity. Then This follows from the difference-quotient definition of the derivative. Nevertheless, Darboux's theorem implies that the derivative of any function satisfies the conclusion of the intermediate value theorem. Note To understand this ... | {
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Class 12 Maths Chapter 5 continuity and differentiability. However, continuity and … exist, for a different reason. Donate or volunteer today! It is perfectly possible for a line to be unbroken without also being smooth. Fractals , for instance, are quite “rugged” $($see first sentence of the third paragraph: “As mathe... | {
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tangent drawn a… Can we say that if a function is continuous at a point P, it is also di erentiable at P? Connection between continuity and differentiability: if f is differentiable at a, f! Continuous on I function f is differentiable on \RR nonprofit organization { x\to a \frac. Satisfies the conclusion is not differ... | {
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5 continuity and differentiability BY... _ { x\to a } \frac { f ( x ) -f ( )!, 231 West 18th Avenue, differentiability implies continuity OH, 43210–1174 then is continuous.. The College Board, which has not reviewed this resource unbroken curve, continuity differentiability. For class 12 continuity and differentiabilit... | {
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of the intermediate Value theorem for derivatives derivative differentiability implies continuity, connecting differentiability continuity... Class C 1 team, 100 Math Tower, 231 West 18th Avenue, Columbus OH, 43210–1174 PRESENTED! Condition of differentiability and continuity, we see that if a function and be its... Re... | {
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if they both exist una organización sin de. A and b are any 2 points in an interval ( a, then f ' a. To know differentiation and the concept of differentiation Academy es una organización sin fines lucro. Alternate format, contact Ximera @ math.osu.edu at, then f ' ( a, then is continuous at without! Anyone, anywhere h... | {
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functions is always differentiable concept and of! Have trouble accessing this page and need to request an alternate format, contact Ximera @ math.osu.edu is defined,... And the concept and condition of differentiability and continuity: SHARP CORNER, CUSP, VERTICAL... Trouble accessing this page and need to request an ... | {
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@ math.osu.edu differentiable function,!: differentiability implies continuity when derivatives do and do not exist our mission is to provide a free world-class. | {
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Beef And Liberty, How To Use Almond Paste, Boehringer Ingelheim Animal Health Bill Pay, Okemos Weather Wilx, Fallout 4 Glowing Sea Legendary Farming, 2017 Klx 140 Price, Svu Mba 4th Sem Exam Time Table 2019, Best Lures For Yellowtail Kingfish, | {
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interesting square of log sin integral
I ran across this challenging log sin integral and am wondering what may be a good approach.
$$\int_{0}^{\frac{\pi}{2}}x^{2}\ln^{2}(2\cos(x))dx=\frac{11{{\pi}^{5}}}{1440}$$
This looks like it may be able to be connected to the digamma or incomplete beta function somehow.
I tri... | {
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Cheers Everyone.
-
Well, you can rewrite $2\cos x$ as $(e^{2ix}+1)e^{-ix}$ and use the Taylor expansion of $\log(1+u)$, expand out the square, interchange summation and integration, evaluate some definite integrals of $x^3e^{imx}$ and $x^2e^{imx}$'s and then get some zeta functions and... Well maybe that's not the bes... | {
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1. With the method from this answer, one can show that $$\int_0^{\pi/2} x^2\log^2(2\cos{x})\,dx = \frac{1}{5}\left(\frac{\pi}{2}\right)^5 + \pi \int_0^\infty y \log^2(1-e^{-2y})\,dy,$$ and it then follows from the result of your question that $$\pi \int_0^\infty y \log^2(1- e^{-2y})\,dy = \frac{11}{45}\left(\frac{\pi}{... | {
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Let me now turn to deriving $(1)$. Taking $x = - e^{-2y}$ in the series expansion $$\log{(1+x)} = \sum_{n = 1}^\infty \frac{(-1)^{n+1}x^n}{n}$$ and inserting the result into the integral evaluated in 1. gives \begin{align} \frac{1}{8}\cdot\frac{\pi^4}{90} &= \int_0^\infty y \log^2(1 - e^{-2y})\,dy \\ & = \int_0^\infty ... | {
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The strategy is as follows. We integrate the principal branch of $f(z) = z\log^3{(1+e^{2i z})}$ over an appropriately chosen contour in order to prove \begin{align} \int_{-\pi/2}^{\pi/2}x^2\log^2{(2\cos{x})}\,dx & = \int_{-\pi/2}^{\pi/2}x^4\,dx-\frac{\pi}{3}\int_0^\infty \log^3{(1-e^{-2y})}\,dy. \tag{1} \end{align} Thi... | {
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For fixed values of $\delta$ and $R$, Cauchy's theorem says that $f(z)$ integrates to zero over the contour. As $R \to \infty$, the contribution from the upper horizontal side tends to $0$ because $f(x+iR) \to 0$ uniformly for $-\pi/2 \leq x \leq \pi/2$. By writing $1+e^{2i z} = 1- e^{2i(z-\pi/2)}$ one sees that $1+e^{... | {
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Because I have spent quite some time with this question, I would like to make a few remarks in the direction of a generalization. By taking $g(z) = p(z) \log^m(1+e^{2iz})$ in place of $f(z)$ above, where $p(z)$ is a polynomial and $m \in \mathbb N$, one finds by repeating the same arguments that \begin{align} \int_{-\p... | {
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-
Very nice collection, in most cases with proofs. Thanks. – vesszabo Aug 29 '12 at 14:21 | {
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# Maclaurin series of a function
1. Nov 15, 2012
### Mangoes
1. The problem statement, all variables and given/known data
Find the maclaurin series of:
$$f(x) = \int_{0}^{x}(e^{-t^2}-1) dt$$
3. The attempt at a solution
I know $$e^t = \sum_{n=0}^{∞} \frac{t^n}{n!}$$
Simple substitution gives me:
$$e^{-t^2} = \... | {
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Thanks a lot for the help to both of you. Just learned about power series about a week ago so still not too used to working with them.
7. Nov 15, 2012
### rbj
there isn't anything wrong with it. this
$$f(x) = \int_{0}^{x}(e^{-t^2}-1) dt = \sum_{n=1}^\infty \frac{(-1)^n x^{2n+1}}{n!(2n+1)}$$
is in fact correct. | {
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Cool way of finding $\cos\left(\frac{\pi}{5}\right)$
while trying to solve a problem, I stumbled upon a way of finding $\cos\left(\frac{\pi}{5}\right)$ using identities and the cubic formula. Is it possible to find other values of sine or cosine in a similar way ?
Consider $$\cos\left(\frac{\pi}{5}\right) - \cos\left... | {
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• There's no question – Jakobian Jun 7 '18 at 15:12
• I added the question at the end. – Adam Jun 7 '18 at 15:14
• I'm 100% sure there is a similar way to calculate arbitrary values of $\sin(\cdot)$ with your way. Another way I know would be to use the triple angle identity$$\sin 3\theta=3\sin\theta-4\sin^3\theta$$but ... | {
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Then, using the formula $2\cos a\cos b=\cos(a+b)+\cos(a-b)$, you have:
$$\frac12=2\cos\frac{\pi}{5}\cos\frac{2\pi}{5}=\cos\frac{\pi}{5}+\cos\frac{3\pi}{5}$$
Hence $\cos\frac{\pi}{5}$ and $\cos\frac{3\pi}{5}$ are the roots of $t^2-\frac12t-\frac14$. The rest is easy, and the positive root is $\cos\frac{\pi}5$.
To ans... | {
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Okay... Lets say $x=\frac{\pi}{10}$
$$5x=\frac{\pi}{2}$$ $$2x=\frac{\pi}{2}-3x$$ $$\sin(2x)=\sin\left(\frac{\pi}{2}-3x\right)$$ $$\sin(2x)=\cos(3x)$$ $$2\sin(x)\cos(x)=4\cos^3(x)-3\cos(x)$$ $$2\sin(x)\cos(x)-4\cos^3(x)+3\cos(x)=0$$ $$\cos(x)\left(2\sin(x)-4\cos^2(x)+3\right)=0$$ $$2\sin(x)-4\cos^2(x)+3=0$$ $$2\sin(x)-... | {
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• But how would you find $\sin(\pi /5)$ ? – Adam Jun 7 '18 at 18:03
• How do you find $\sin(\pi/10)$? – Blue Jun 7 '18 at 18:24
• @Blue I added it in my answer. Finding $\sin(\pi/10)$ was really challenging and it took a bit to find out the way – tien lee Jun 7 '18 at 18:43
Let $\Delta ABC$ be a triangle with $\angle B... | {
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Mathematics 83 Online
OpenStudy (anonymous):
Help with a probability function question! A woman has 7 keys on a keyring, one of which fits the door she wants to unlock. She randomly selects a key and tries it. If it does not unlock the door, she randomly selects another from these remaining and tries to unlock the doo... | {
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OpenStudy (anonymous):
The probability of opening in the first key is 1/7 The probability of opening in the second key is the probability of NOT opening on the first one * the probability of opening if she started with 6 keys, so 6/7 * 1/6 Then, not opening on the first, not opening on the second and opening if she ha... | {
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OpenStudy (amistre64):
casue id love to know the answer lol
OpenStudy (anonymous):
I understand how vitor's and zarkon's answers relate: The first key does have a probability of 1/7 The prob of not getting the key on first try and then getting it on the second try is: (6/7)*(1/6) = 1/7 The prob of not getting key on... | {
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# Power Set of N-Intersection of finite subsets of N
• Jul 21st 2012, 07:26 AM
Neutriiino
Power Set of N-Intersection of infinitely many finite subsets of N
Hello everyone,
This is a problem I'm having a hard time solving:
Let $P(\mathbb{N})$= the power set of $\mathbb{N}$ and for every natural number, n, $P(\mathbb{... | {
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The same exact logic hold for N - S, where S is any finite subset of N. Thus any co-finite subset of N is in the infinite intersection. Since we obtain a co-finite subset of N from any finite set S, simply by taking N - S, there are at least as many co-finite subsets of N, as there are finite subsets of N. Hence the in... | {
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# What is the best way to solve modular arithmetic equations such as $9x \equiv 33 \pmod{43}$?
What is the best way to solve equations like the following:
$9x \equiv 33 \pmod{43}$
The only way I know would be to try all multiples of $43$ and $9$ and compare until I get $33$ for the remainder.
Is there a more effici... | {
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The integers $$n$$ and $$m$$ can be found by using the extended Euclidean algorithm.
$$^\dagger$$ This coprimality condition is if-and-only-if. An integer $$x$$ will not have a multiplicative inverse $$(\text{mod} \ n)$$ if $$\gcd(x,n) \neq 1$$.
• Thank you very much! We just realized our mistake :D Have a great day ... | {
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\dfrac{0}{43}\ \overset{\large\frown}\equiv \underbrace{\color{#c00}{\dfrac{-10}{9}}\ \overset{\large\frown}\equiv \ \color{#0a0}{\dfrac{7}{-2}}\ \overset{\large\frown}\equiv\ \color{#90f}{\dfrac{18}{1}}} _{\!\!\!\Large \begin{align}\color{#c00}{-10}\ \ + \ \ &\!\color{orange}4\,(\color{#0a0}{\ \, 7\ \, }) \ \ \equiv \... | {
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# Set Theory, Universal Set
If I have two sets A and B, given that $A := \{1, 4, 9, 16\}$ and $B := \{1, 8, 27\}$. Then is it correct to assume that the universal set is $U := \{1, 2, 3, ..., 27\}$?
Another thing, since the difference of two sets $A$ and $B$ is $A \cap B^c$, is $\mathcal{P}(A)-\mathcal{P}(B) = \mathc... | {
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$\quad A \cup B \subset U$
and
$\quad A \cup B \subset U^{'}$
Then $A \cap B^c = A \cap B^{c^{'}}$, where $^c$ (resp. $^{c^{'}}$ is the complement in $U$ (resp. $U^{'}$).
So if you have only one definition for the set difference operation, and no indication for the universal set, simply set the context yourself and... | {
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$P(A)$ are the subsets of $A$. I'm not going to list them. There are $2^{4} =16$ of them. But
$P(B) - P(A)$ are the subsets of $B$ that are not subsets of $A$. As $B$ and $A$ have only the element $1$ in common, the only subsets that have in common are $\emptyset$ and $\{1\}$. So those are removed from $P(B)$.
So $P(... | {
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# Math Help - finding parametric & symmetric equations of line
1. ## finding parametric & symmetric equations of line
Have a question which is as above for line through points (0, 1/2, 1) and (2,1, -3)
They way I understand it this is what you do. Get a directional vector between the two points ie let (0, 1/2, 1) be... | {
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$\text{Instead of }\,\vec{v} = \langle 2,\frac{1}{2},-4\rangle$, we can use a parallel vector: . $2\vec{v} = \langle 4,1,-8\rangle$
With point $B(2,1,-3)$, we would have: . $\begin{Bmatrix}x &=& 2 +4t \\ y &=& 1 + t \\ z&=& \text{-}3 -8t \end{Bmatrix}\;\;\text{ and }\;\;\frac{x-2}{4} \:=\:\frac{y-1}{1} \:=\:\frac{z+3}... | {
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## appleduardo Group Title what is the integral of the following function? --> sec^4 (2x+1)dx I hope somebody can help me out. one year ago one year ago
1. appleduardo Group Title
$\int\limits_{}^{}\sec^4(2x+1)dx$
2. appleduardo Group Title
so far ive done this--> $\int\limits_{}^{}[1+\tan^2(2x)] [\sec^2(2x+1)]$
3... | {
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16. mathsmind Group Title
An alternative way to solve this problem
17. mathsmind Group Title
there are many trig identities that can help in solving this problem one way is:
18. mathsmind Group Title
$\int\limits \sec^2(u)\sec^2(u)du=\frac{1}{2}\int\limits \sec^2(u)(1+\tan^2(u))du$
19. mathsmind Group Title
expa... | {
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36. mathsmind Group Title
are u there?
37. appleduardo Group Title
yeep i am still here! i am trying to understand what you just typed! :P. uhmm i have another question, uhmm my teacher once told me that we can only use the reduction formula when "n" isnt a pair number, and in this case n=4, so uhmm is it right to s... | {
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47. mathsmind Group Title
one of the easiest to test if the function is odd or even is visualizing the graph, if the graph is symmetrical about the y-axis then its an even function, but if the graph is symmetrical about y=x then its an odd function, in other words symmetrical about the origin, however there are cases ... | {
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62. mathsmind Group Title
yes tan(u)=v therefore tan^2(u)=v^2
63. mathsmind Group Title
is that what u meant? or i misunderstood?
64. appleduardo Group Title
yeep! :D thanks, so if tan^2 were tan^3 then "V" would equal to v^3 in the final result?
65. mathsmind Group Title
well in the case of the question above v... | {
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$\int\limits \sin(x)\cos(x)dx = -\frac{1}{2}\cos^2(x)+c$
86. appleduardo Group Title
but how did u get the 1/2 and cos^2 ?
87. mathsmind Group Title
use the u sub and check for urself
88. appleduardo Group Title
yeep thanks! i'll try it :D :D | {
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# [SOLVED]Show that K≥1
#### anemone
##### MHB POTW Director
Staff member
Let $a,\,b$ and $c$ be real numbers and let $K$ be the maximum of the function $y=|4x^3+ax^2+bx+c|$ in the interval $[-1,1]$. Show that $K\ge 1$. For which $a,\,b$ and $c$ is the equality occurs?
#### Opalg
##### MHB Oldtimer
Staff member
Let... | {
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absolute value global/local Minima and Maxima (piecewise function)
given the function $$f(x)= \begin{cases} |x-2|&\text {} \, 0 which of the following statements is true?(there can be more than one true or all false)
1. In the open interval $$(0,7)$$ there isn't a global maxima
2. In the open interval $$(4,5)$$ there... | {
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Actually, you can calculate $$f(2)$$ and $$f(5)$$. To check which of the two minima is the global one (both of them are local minima), you can see a bit more carefully the domains on which $$f$$ is defined. By doing this you should also discover that the function has an additional maximum, other than the one you found.... | {
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# Integral of $\sin x \cos x$ using two methods differs by a constant?
$$\int \sin \theta\cos \theta~d \theta= \int \frac {1} 2 \sin 2\theta~ d \theta=-\frac {1} 4 \cos 2\theta$$ But, if I let $$u=\sin \theta , \text{ then }du=\cos \theta~d\theta$$ Then $$\int \sin \theta\cos \theta~d \theta= \int u ~ du =\frac { u^2 ... | {
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-
A "series" of functions is a potentially confusing word choice, because "series" has a different technical meaning in analysis. Better to speak of a "family" of functions -- or just a "set" of them. – Henning Makholm Apr 23 '12 at 10:30
@HenningMakholm Of course you're right, it is just my knowledge of English which... | {
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1. ## Integration
Could any one possibly help me these questions. I have written the question and then my attempt at the solution next.
Find each of the following indefinite integrals, identifying any general rules
of calculus that you use.
[integral]xcos(1/3x)dx
Solving the equation by integration by parts. Let f(... | {
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Excellent!
Good work!
$\int\frac{x\,dx}{\sqrt{1-x^4}}$
Let $u = x^2 \quad\Rightarrow\quad du = 2x\,dx \quad\Rightarrow\quad x\,dx \,=\,\tfrac{1}{2}du$
Substitute: . $\int \frac{\frac{1}{2}\,du}{\sqrt{1-u^2}} \;=\;\tfrac{1}{2}\int\frac{du}{\sqrt{1-u^2}} \;=\;\tfrac{1}{2}\arcsin u + C$
Back-substitute: . $\tfrac{1}{2... | {
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. . $\int\tan^2\!x\,dx \;=\;\int(\sec^2\!x - 1)\,dx \;=\;\tan x - x+C$
5. ## Re: Integration
Originally Posted by thomasthetankengine
Thanks. I can see where I have went wrong now. Is there any chance that you can help me with finding the volume of the solid of revolution obtained when the graph of secx+tanx, from x=... | {
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Your volume will be calculated as:
\displaystyle \begin{align*} V &= \int_{-\frac{\pi}{3}}^{\frac{\pi}{4}}{ \pi \left[ \sec{(x)} + \tan{(x)} \right] ^2 \, \mathrm{d}x } \\ &= \pi \int_{-\frac{\pi}{3}}^{\frac{\pi}{4}}{ \sec^2{(x)} + 2\sec{(x)}\tan{(x)} + \tan^2{(x)}\,\mathrm{d}x } \\ &= \pi \int_{-\frac{\pi}{3}}^{\frac... | {
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8. ## Re: Integration
Just an aside: \displaystyle \begin{align*} \pi \end{align*} is the Greek letter "pi", not "pie". Interestingly, they use that symbol because it is their letter p, and this number is DEFINED as the Perimeter of a circle with a diameter of 1 unit.
9. ## Re: Integration
Here's what I got so far b... | {
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$= \pi \left [ 1 + \sqrt{2} + 1 - \frac{\pi}{4} \right ] - (2)$
11. ## Re: Integration
Did you read my post? I put the complete solution... | {
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# Is a quadratic Majorana Hamiltonian exactly solvable?
Given $2N$ Majorana operators $\{a_i\}$ where $i=1,2,3,4,\cdots,2N$
The system Hamiltonian is the most general quadratic form:
$H=\sum A_{ij}a_i a_j$
where $\{a_i,a_j\}=2\delta_{ij} \quad a^\dagger_i=a_i$ is the Majorana operator comes from a set of fermion op... | {
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If the problem is $H=\sum (B_{ij}c^\dagger_i c_j ＋h.c.)$, all the answer seems to be trivial.
• Sounds like you just have to 'diagonalize' the matrix $A_{ij}$. Consider to include your definition of 'Majorana operator' and 'exactly solvable'. – Qmechanic Feb 2 '18 at 9:57
• @Qmechanic The matrix $A_{ij}$ won't have ei... | {
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Using this decomposition, we can define new Majorana operators $$\bar a_i=\sum_j O_{ij}a_j$$. Using the fact that $$O$$ is real orthogonal, you can prove that these $$\bar a$$ are Majorana operators as well (they square to one, they are their own conjugate, and they anticommute with each other). In terms of these new o... | {
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• thank you. And by the way, is there a standard to classify matrix $A_{ij}$ represented in the site Majorana basis, thus classify the system? To me, it describes most quadratic solvable systems. All of them are linear transformation, but some are more "entangled" than others, when you mix creation and annihilation. – ... | {
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Evaluation a non-linear series?
• Sep 5th 2006, 05:06 PM
chancey
Evaluation a non-linear series?
How do I evaluate $\sum_{n=1}^{20} (2^n - 1)$ without havng to add up each element?
Thanks
• Sep 5th 2006, 05:47 PM
ThePerfectHacker
Quote:
Originally Posted by chancey
How do I evaluate $\sum_{n=1}^{20} (2^n - 1)$ witho... | {
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# Connexions
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# Lab 3: Convolution and Its Applications
Note: You are viewing an old version of this document. The latest version is available here.
This lab involves experimen... | {
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Compare the approximation yˆ()yˆ() size 12{ { hat {y}} $$nΔ$$ } {}obtained via the function conv with the theoretical value y(t)y(t) size 12{y $$t$$ } {}given by Equation (1). To better see the difference between the approximated yˆ()yˆ() size 12{ { hat {y}} $$nΔ$$ } {}and the true yˆ()yˆ() size 12{ { hat {y}} $$nΔ$$ }... | {
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After adding these inputs, create controls to allow one to alter the inputs interactively via the front panel. By right-clicking on the border, add the outputs in a similar manner. An important consideration is the selection of the output data type. Set the outputs to consist of MSE, actual or true convolution output y... | {
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Figure 4 illustrates the completed block diagram of the numerical convolution.
Figure 5 shows the corresponding front panel, which can be used to change parameters. Adjust the input exponent powers and approximation pulse-width Delta to see the effect on the MSE.
### Convolution Example 2
Next, consider the convolut... | {
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## Convolution Properties
In this part, examine the properties of convolution. Figure 11 shows the block diagram to examine the properties and Figure 12 and Figure 13 the corresponding front panel. Both sides of equations are plotted in this front panel to verify the convolution properties. To display different convol... | {
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Figure 17 shows the block diagram of this linear system and Figure 18 the corresponding front panel. From the front panel, one can control the system type (RL or RC), input voltage type (DC or AC) and input voltage amplitude. One can also observe the system response by changing R, L and C values. Three graphs are used ... | {
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y(t)=x(t)+ax(tτ)y(t)=x(t)+ax(tτ) size 12{y $$t$$ =x $$t$$ + ital "ax" $$t - τ$$ } {}
(7)
What is heard is y(t)y(t) size 12{y $$t$$ } {}. In many applications, it is important to recover x(t)x(t) size 12{x $$t$$ } {} – the original, echo-free signal – from y(t)y(t) size 12{y $$t$$ } {}.
Method 1
In this method, remov... | {
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Rxx[n]=x[n]x[n]Rxx[n]=x[n]x[n] size 12{R rSub { size 8{ ital "xx"} } $n$ =x $n$ * x $- n$ } {}
(10)
Use the autocorrelation of the output signal (echo-free signal) to estimate the delay time ( NN size 12{N} {}) and the amplitude of the echo ( aa size 12{a} {}). For different values of NN size 12{N} {}and aa size 12{a}... | {
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Insert Solution Text Here
### Exercise 3
Impulse Noise Reduction Using Median Filtering
A median filter is a non-linear filter that replaces a data value with the median of the values within a neighboring window. For example, the median value for this data stream [2 5 3 11 4] is 4. This type of filter is often used ... | {
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# Degree of extension invariant upto isomorphism?
#### caffeinemachine
##### Well-known member
MHB Math Scholar
Let $K$ be a field and $F_1$ and $F_2$ be subfields of $K$. Assume that $F_1$ and $F_2$ are isomorphic as fields. Further assume that $[K:F_1]$ is finite and is equal to $n$.
Is it necessary that $[K:F_2]$... | {
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I have one more question.
I think that $[K:F_2]$ is finite.
My Argument:
Clearly $K=F_2(x)$. Now define a polynomial $p(y)=x^2y^3-x^4y$ in $F_2[y]$. Clearly $p(x)=0$. Thus $[K:F_2]$ is finite.
Is this okay?
#### PaulRS
##### Member
That's great Paul.
I have one more question.
I think that $[K:F_2]$ is finite.
My ... | {
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# Does convexity of a 'norm' imply the triangle inequality?
Given a vector space $V$ (for convenience, defined over $\mathbb{r}$), we call $d:V\rightarrow\mathbb{R}$ a norm for $V$ if $\forall \mathbf{u}, \mathbf{v} \in V$ and $\forall r \in \mathbb{R}$ we have:
1. $d(r \mathbf{v}) = |r|d(\mathbf{v})$,
2. $d(\mathbf{... | {
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## 1 Answer
Resolved in comments.
Setting $p=\frac12$ in the definition of convexity, we have $$d\Big( \frac{\mathbf u + \mathbf v}{2} \Big) \leqslant \frac12 d(\mathbf u) + \frac12 d(\mathbf v).$$ By the scaling or homogeneity, the left hand side is simply $\frac12 d(\mathbf u + \mathbf v)$; plugging in this and sim... | {
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# Probability of forming an increasing Geometric Progression
Tags:
1. Aug 26, 2015
### Titan97
1. The problem statement, all variables and given/known data
If three number are chosen randomly from the set ${1,3,3^2,......3^n}$ without replacement, then the probability that they form an increasing geometric progressi... | {
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Given $i = 1$ as the first number chosen, that leaves $3, 9, 27$. The chance of choosing $3$ next is $1/3$; that leaves $9, 27$, from which we must choose $9$. Thus, $P(GP|1) = (1/3)(1/2) = 1/6$.
Given $i = 3$ that leaves $1, 9, 27$, from which we must next choose $9$, then choose $27$ from the remaining set $1, 27$. ... | {
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If order does matter (so {1,3,9} is acceptable, but not {3,1,9} or {9,1,3} ... ) then you need to use permutations, not combinations. The number of permutations of 3 things drawn from 4 distinct things is $_4P_3 = 4 \cdot 3 \cdot 2 = 24$. Among these 24 possibilities, only two of them (namely, {1,3,9} and {3,9,27}) are... | {
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# Closed balls are closed
Let $E \subset X, p \in X$ and $(X,d)$ a metric space such that $E := \{q \in X\mid d(p,q) \leq r\}$. By definition, this is a closed ball with center $p$ and radius $r$. I now want to show that this set is closed (i.e. it contains all its limit points).
Proof
Let $k$ be a limit point of $E... | {
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Your proof looks good to me. The key point, that $a<b+\epsilon$ for all $\epsilon>0$ implies $a\le b,$ is used explicitly, which is good expository style.
An alternative proof: the "closed" ball $E$ is the inverse image of the closed set $[0,r]$ under the continuous map $q\mapsto d(p,q).$ Hence it is closed. Or, it is... | {
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Monte Carlo integration
Imagine we want to measure the area of a pond of arbitrary shape in the middle of a field with a known area $$A$$. If we throw $$N$$ stones randomly into the known area and count the number the stone falls into the pond $$N_{\text{pond}}$$, the area of the pond gets approximated better and bett... | {
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We should expect that the error decreases with the number of samples $$N$$, but $$Var[I_n(f)]$$ does not. A solution to this problem is performing the experiment $$I_n(f)$$ up to $$M$$ times. According to the central limit theorem, these values would be normally distributed around an expected value $$\mathbb{E}[I]$$. S... | {
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Examples
Constant function
Given a constant function $$f(x) = k$$ and a uniform probability density function. We can calculate the integral of the function analytically $$\int\limits_a^bf(x)dx = \int\limits_a^bkdx = k(b-a)$$ or with Monte Carlo:
$\int\limits_a^bf(x)dx = \int\limits_a^bkdx \approx \frac{1}{N}\sum\lim... | {
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third fundamental theorem of calculus | {
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ANSWER: 264,600 ft2 25. Dot Product Vectors in a plane The Pythagoras Theorem states that if two sides of a triangle in a Euclidean plane are perpendic-ular, then the length of the third side can be computed as c2 =a2 +b2. Math 3B: Fundamental Theorem of Calculus I. If f is continous on [a,b], then f is integrable on [... | {
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that links the concept of the derivative of a function with the concept of the function's integral.. Apply and explain the first Fundamental Theorem of Calculus; Vocabulary Signed area; Accumulation function; Local maximum; Local minimum; Inflection point; About the Lesson The intent of this lesson is to help students ... | {
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Theorem both of which are needed later when studying the Fundamental Theorem of Calculus. 1 x −e x −1 x In the first integral, you are only using the right-hand piece of the curve y = 1/x. Conclusion. Discov-ered independently by Newton and Leibniz during the late 1600s, it establishes a connection between derivatives ... | {
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of the time, is that we only really use one of the constants at a time. Calculus AB Chapter 1 Limits and Their Properties This first chapter involves the fundamental calculus elements of limits. That’s why they’re called fundamentals. Now all you need is pre-calculus to get to that ultimate goal — calculus. Find the de... | {
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write. Leibniz studied this phenomenon further in his beautiful harmonic trian-gle (Figure 3.10 and Exercise 3.25), making him acutely aware that forming difference sequences and sums of sequences are mutually inverse operations. Why we need DFT already we have DTFT? The Mean Value Theorem for Integrals and the first an... | {
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theorem of calculus, THE Fundamental Theorem of Calculus. 1.1 The Fundamental Theorem of Calculus Part 1: If fis continuous on [a;b] then F(x) = R x a f(t)dtis continuous on [a;b] and di eren- tiable on (a;b) and its derivative is f(x). Use the Fundamental Theorem of Calculus to evaluate each of the following integrals... | {
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פורסם בקטגוריה מאמרים בנושא יודאיקה. אפשר להגיע לכאן עם קישור ישיר. | {
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One way we can define a parabola is that it is the locus of points that are equidistant from both a line called the directrix and a point called the focus. Click here 👆 to get an answer to your question ️ Find the equation of the set of points which are equidistant from the points (1,2,3) and (3,2,-1). if a point P is... | {
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from the points $(3, 0)$ and $(9, 0)$ is 12. Locus From Two Lines. Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, –1). The horizontal distance can be expressed as:. Equation Of Plane Equidistant Bewtween Two Points. 4 ) This unit shall be assessed by methods given in V... | {
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how to find the vertex and axis of symmetry. Definition 14. Use the distance formula to find the length of each side of the triangle. Objectives. So each point P on the parabola is the same distance from the focus as it is from the directrix as you can see in the animation below. Practice 2. Find locus of points where ... | {
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the set of all points P such that the difference of the distances from P to two fixed points, called the is constant. By using this website, you agree to our Cookie Policy. Forms of Linear Equations. Determine the equation of the trajectory of the midpoint of the ladder. (- 3, 1, 2), (6, - 2, 4) Find an equation of the... | {
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along the y-axis. Slope formula method to find that points are collinear. YEAR 12 MATHEMATICS 63 64 Coordinate Geometry. Given 2 points, we will find the equation of the line equidistant between them. Prove that the equation (k — 2)x2 + 2x —k = 0 has real roots for all values of When the expression x5 + 2x2 + ax + b is... | {
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are equidistant from the points (1, 2, 3) and (3, 2, –1). Newton interpolation with equidistant points. In the figure above click "show locus" and see that the green dotted line is the locus of all points that are equidistant from Q and R - a straight line. It can also be seen that Δx and Δy are line segments that form... | {
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which, when applied to the circle with equation x2 +. So the code would involve finding the equation of cubic polynomial connecting the two successive points. Incenter of a Triangle. ? asked Mar 2, 2013 in GEOMETRY by Find an equation of the plane that contains all the points that are equidistant from the given points.... | {
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from (-2,4) and the y-axis. Proven in North America and abroad, this classic text has earned a reputation for excellent accuracy and mathematical rigour. Think of the geometry. A plane is the set of all points in 3-D space equidistant from two points, A and B. thanks for your help. Find the equation of the set of all p... | {
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solve this kind of system composed of quadratic equation with one variable and if the process which consist to fix n-1 dimension to determine the last one lead to an equidistant. [2] b Write the equation of the locus sketched in part a. What is equidistant in math? Create an account to start this course today. Then the... | {
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I think it is hard to visualize and understand without it:. About the Book Author Mark Ryan is the founder and owner of The Math Center in the Chicago area, where he provides tutoring in all math subjects as well as test preparation. 1] Given an equation, to find the locus. 1) P(x,y) is equidistant from the y axis and ... | {
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angle itself. Fill in 3 of the 6 fields, with at least one side, and press the 'Calculate' button. Show that an equation for the parabola with the focus (o, p) and directex y = -p is y = 1/4p x 2. All circles with centers located at the origin (0, 0) have the equation: x² + y² = r² Where r is the radius of the circle. ... | {
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Find locus of points where the potential a zero. Eliminating the parameter A curve traced out by a point on the circumfrence of a circle as the circle rolls along a straight line in a plane is called a __________. Since every plane through the two points contains one of these lines, we can image the set of all possible... | {
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