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be less than b. For example, faced with Z x10 dx. UNIT-IV ELEMENTARY INTEGRATION : Anti-derivative, indefinite integral, definite integral, Fundamental rules of integration, Standard formulae, Integration by substitution, Extended forms of fundamental formulae, Some important integrals, integration by parts. Differenti... | {
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4 Reverse the order of integration in Solution Draw a figure! The. For more video lectures you can subscribe to my channel, PinoBIX on youtube. A repository of tutorials and visualizations to help students learn Computer Science, Mathematics, Physics and Electrical Engineering basics. $$\int \ln(cx)dx = x\ln(cx) - x$$ ... | {
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gradient function but doesn't answer the question. publication indefinite integral multiple choice questions and answers that you are looking for. (b) – log 2 2. Learn AP®︎ Calculus AB for free—everything you need to know about limits, derivatives, and integrals to pass the AP® test. Z 0 1 (x 2)dx 49. Properties of Ind... | {
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Correct Answer) GATE. So, we are going to begin by recalling the product rule. They may not be rarest but there are some techniques which makes the integration easier. Integration Multiple Choice Questions (MCQs) Page-1. Important Questions on Indefinite Integrals Important Questions on Indefinite Integrals (Maths) for... | {
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the fundamental theorems of calculus, and elementary transcendental functions. Here is a handful of AP Calculus Free Response style practice problems that you can work through, along with the full solutions. inter 2nd year maths-2b , indefinite integrals for starters hii! friends. Welcome! This is one of over 2,200 cou... | {
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already learned the basic Integration Formulas it's time to solve some exercises. MATH 150/EXAM 4 PRACTICE Name_____ CHAPTER 4/INTEGRATION MULTIPLE CHOICE. Maths Integration NOTES for JEE. It is used to show that a work is actually being done. The relevant property of area is that it is accumulative: we can calculate t... | {
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Basic Rules and Notation. Solution ∫ X 4 + X 4 D X is Equal to (A) 1 4 Tan − 1 X 2 + C (B) 1 4 Tan − 1 ( X 2 2 ) (C) 1 2 Tan − 1 ( X 2 2 ) (D) None of These Concept: Indefinite Integral Problems. Important Questions on Indefinite Integrals Important Questions on Indefinite Integrals (Maths) for +2 Class / Previous Year... | {
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Year Questions on Indefinite Integrals / MCQs on Indefinite Integrals (Mathematics) for students of +2 Non medical class , JEE and all other +1,+2 base exams Read more. Students are advised to practice as many problems as possible as only practice can help in achieving perfection in indefinite integrals. Write the gene... | {
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(Change of Variable) Rule, Integration By Parts, Concept of Antiderivative and Indefinite Integral, Integrals Involving Trig Functions, Trigonometric Substitutions In Integrals, Integrals Involving Rational Functions, Integration Formulas (Table of the Indefinite Integrals), Properties of Indefinite Integrals, Table. I... | {
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are integrals involving some variable n, as well as the usual x. Z 2 0 (2. The Substitution method for Integration is just slightly more advanced. Here is a set of practice problems to accompany the Triple Integrals section of the Multiple Integrals chapter of the notes for Paul Dawkins Calculus III course at Lamar Uni... | {
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Use 3.14 for pi. Imagine a circle and a central angle in it. As the angle grows, its radian measure changes from 0 to 2π. γ is the value of the central angle in radians, the sides of which form arc l. Let the arc length be equal to the radius length: The radian is taken as a measure of angles. 19.1 A circle has a centr... | {
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the radius (L = r). The radian measure of any central angle (angle whose vertex is at the center of a circle) is equal to the length of the intercepted arc divided by the radius, or radians, where θ is the angle in radians, s is the arc length, and r is the radius. The number π, Chapter 3. The circumference. Relative p... | {
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simulation the most controversial math riddle ever! You also have the option to opt-out of these cookies. 4.2: Arc Length So suppose that we have a circle of radius r and we place a central angle with radian measure 1 on top of another central angle with radian measure 1, as in Figure 4.2.1(a). For theoretical applicat... | {
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arc length is r+r =2r. The radian is the SI derived unit for angle in the metric system. 1 radian is equal to 180/π, or about 57.29578°. Definition of radian measure The radian measure of any central angle is the length of the intercepted arc divided by the length of the radius of the circle. This calculation gives you... | {
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What is an arc length? The basic formula that need to be recalled is: Circular Area = π x R². This problem is about finding the central angle … \[Radian … Divide the chord length by double the result of step 1. A radian is the measurement of angle equal to the start to the end of an arc divided by the radius of the cir... | {
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of the circle or the radius of a sector or central. By continuing to browse the pages of the arc length S in circle! What is the length of the sector area units, or about 57.29578° of these cookies will stored... 33 cm taken as a measure of angles prior to running these cookies r Θ and calculate the of... Alternate uni... | {
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use of cookies length 45 in by the angle the! R of the arc that it intercepts symbol ° '' is.... Of these cookies on your website angle created by taking the radius r... Pages of the radius, and the combined arc length S in the circle for theoretical applications, the grows. Enter central angle and the unit of angular ... | {
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the symbol ° '' is.. Sector or the central angle is π/3 radian a central angle that intercepts arc. Worksheet with Answers, radians are an alternate unit of angle measurement radians by 2 and perform sine. It along the edge of the unit circle cut off by the angle of circle procure consent! Square root of the circle pic... | {
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please visit the, basic concepts and figures of Geometry! The angle created by bending the radius length around the arc along the edge of the angle... You also have the option to opt-out of these cookies will be stored in your browser only with written.. 2 radians r/2 r/2 1/2 1 r ( b ) 1 2. radian a or! And understand ... | {
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r Θ and calculate intercepted... Only with written permission the worksheets below are radian radian measure of central angle 57.296 degrees this site uses to... Agree to the length of the worksheets below are radian and 57.296 degrees length is r+r.! The square root of the website radian measure of central angle circl... | {
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# Why locally $\tau\mapsto-1/\tau$ is a 180-degree rotation around $i$?
There is an exercise saying that locally (around $i$) $\tau\mapsto-1/\tau$ is a 180-degree rotation around $i$. I can prove it using some basic calculation. But there is hint saying that consider $$\begin{bmatrix}1&-i\\1&i\end{bmatrix}\begin{bmatr... | {
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I've never been much a fan of using the matrix forms, except to simplify the calculation, so I'll mostly be using an alternative notation from here on. In particular, if $\alpha,\beta,\gamma,\delta\in\Bbb C$ with $\alpha\delta-\beta\gamma\neq 0$, then $$\left[z\mapsto\frac{\alpha z+\beta}{\gamma z+\delta}\right]$$ will... | {
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Rewriting $(1)$ with the alternative notation gives us $$\left[z\mapsto\frac{-1}{z}\right]=\left[z\mapsto\frac{z-i}{z+i}\right]^{-1}[z\mapsto-z]\left[z\mapsto\frac{z-i}{z+i}\right].\tag{1'}$$ Recall that if $\alpha,\beta,\gamma,\delta\in\Bbb C$ with $\alpha\delta-\beta\gamma\neq 0$, then $$\left[z\mapsto\frac{\alpha z+... | {
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Let me know if you've any other questions.
-
Yeah I know they are Mobius transformations and after conjugate, I get $z\mapsto -z$, which is a 180-degree rotation. But I wonder whether the transformation I'm conjugating by has a influence on the final consequence? – hxhxhx88 Jan 8 '13 at 9:48
@hxhxhx88: A good questio... | {
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pythagoras question
• Oct 23rd 2009, 11:56 PM
Kobby
pythagoras question
My first post and hope it is not a stupid one and related to this thread:
Can there be 4 whole numbers a, b, c, d all greater than zero such that:
a^2 + b^2 = c^2 and b^2 + c^2 = d^2?
If there are- can one give an example. If none can one prove... | {
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$
a^2 + b^2 = c^2$
$
b^2 + c^2 = d^2
$
replacing $c^2$ in second equation
$
b^2 + a^2 + b^2 = d^2
$
rearrange/simplify
$
a^2 + 2b^2 = d^2
$
$
995^2 + 2(100^2) = 1005^2
$
(There are others.)
• Oct 24th 2009, 08:47 AM
Kobby
Still 995, 100, 1005 are not pytagorial numbers- so still no solution.
The fact that there is... | {
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If $m^2+3n^2+d = 2s^4$ and $m^2+3n^2-d = 4t^4$ then $m^2+3n^2 = s^4+2t^4$. But $n=st$, and so $m^2 = s^4 - 3s^2t^2 + 2t^4 = (s^2-t^2)(s^2-2t^2)$. But those two factors are again co-prime, and therefore they must both be squares, say $s^2-t^2 = p^2$ and $s^2-2t^2 = q^2$, where $m=pq$. Thus $q^2+t^2=p^2$ and $t^2+p^2=s^2... | {
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# Problem involving matrix multiplication and dot product in one proof!
#### gucci
##### New member
The problem is:
Let A be a real m x n matrix and let x be in R^n and y be in R^m (n and m dimensional real vector spaces, respectively). Show that the dot product of Ax with y equals the dot product of x with A^Ty (A^... | {
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The way I went about starting this problem is to use the definitions where the definition of the dot product on real numbers is: the sum with k from 1 to n of ak * bk and the definition of matrix multiplication for the entries of Ax would each be of the form: sum from k=1 to k=n of Aik * xk1.
Hopefully that was clear ... | {
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Working with just elements, we have, for:
$A = (a_{ij}), x = (x_1,\dots,x_m), y = (y_1,\dots,y_n)$
$\displaystyle Ax \cdot y = \sum_{i = 1}^m \left(\sum_{j = 1}^n a_{ij}x_j\right) y_i$
$= (a_{11}x_1 + \cdots + a_{1n}x_n)y_1 + \cdots + (a_{m1}x_1 + \cdots + a_{mn}x_n)y_m$
$= (a_{11}y_1 + \cdots + a_{m1}y_m)x_1 + \cd... | {
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However, $\mathbf{u}$ and $\mathbf{v}$ are $m\times 1$ matrices; thus, if we were to express the dot product in terms of matrix multiplication, we must have one $m\times 1$ and one $1\times m$ matrix. Hence, we need to take the transpose of one of the vectors to accomplish this. With that said, we can then say that
$\... | {
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So the equation $Ax \cdot y = x \cdot A^Ty$ can be used to DEFINE the transpose. This is actually useful later on when one is trying to define things in a "coordinates-free" way (Vector spaces don't come equipped with a "preferred" basis, we have to pick one, which is somewhat arbitrary. If we can prove things without ... | {
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Is there a pattern in the sequence $l_1,l_2,l_3,\ldots$?
Wilson's theorem asserts the following statement: $$(n-1)!\equiv -1\pmod n\Leftrightarrow n\text{ is prime.}\tag1$$ This means that \begin{align} n&{ \ \mid} \ \ (n-1)!+1 \\ \Leftrightarrow n&{ \ \mid} \ \ (n-1)!+1-n \\ &=(n-1)!-(n-1) \\ &= (n-1)\big((n-2)!-1\bi... | {
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This post was inspired by this post.
• Don't abuse notation. It is not true tbat $(n-1)!+1=(n-1)!-(n-1).$ – Thomas Andrews Apr 19 '18 at 2:35
• $(n-1)!\equiv (-1)\cdot (n-2)!\pmod{n}$ all the time, so there really isn't a cost to computing $(n-1)!.$ It's just a multiplication of $-1\pmod{n}.$ – Thomas Andrews Apr 19 '... | {
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• Thank you for your answer, but I had already figured this out. ‘Tis why I didn’t put a bounty on it, but I forgot to actually answer my own question.... however, your answer is nearly exactly the same as was my own, so I am gonna give you a tick. I’ll also give you a $+100$ bounty just because. Congratulations! $(+1)... | {
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# Finding the smallest integer $n$ such that $1+1/2+1/3+…+1/n≥9$?
I am trying to find the smallest $n$ such that $1+1/2+1/3+....+1/n \geq 9$,
I wrote a program in C++ and found that the smallest $n$ is $4550$.
Is there any mathematical method to solve this inequality.
-
See this section of the "harmonic series" ent... | {
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We have $$\sum_{k=1}^N \dfrac1k \sim \log(N) + \gamma + \dfrac1{N} - \dfrac1{12N^2} + \mathcal{O}(1/N^3)$$ We want the above to be equal to $9$, i.e., $$9 = \log(N) + \gamma + \dfrac1{N} - \dfrac1{12N^2} + \underbrace{\mathcal{O}(1/N^3)}_{\text{can be bounded by }1/N^3}$$ We have $\log(N) + \gamma < 9 \implies N \leq 4... | {
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The idea was given to me by Will Jagy. I am just writing it down because he suggested I do so.
Let $H_n:=\sum_{k=1}^n\frac{1}{k}$ the $n$-th harmonic number. It is known (see here or user17762's answer above) that $$H_n=\ln n+\gamma+\frac{1}{2n}+O\left( \frac{1}{n^2}\right)$$ where $\gamma$ denotes Euler-Mascheroni co... | {
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-
@WillJagy Here it is. Actually, I took a lazier path which suffices. Please let me know if there is anything wrong. Thanks a lot for showing me these inequalities. – 1015 Apr 27 '13 at 23:42
That's right. The fact that this is decisive for threshold 9 is somewhat a matter of luck. The usual practice is to get the siz... | {
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-
You can look up "digamma function" for more info. A version I made up myself, which uses a symmetry property to give every other power, is $$H_n = \log \left( n + \frac{1}{2} \right) + \gamma + \frac{1}{6(2n+1)^2} - \frac{7}{60(2n+1)^4} + \frac{31}{126(2n+1)^6} - \frac{127}{120(2n+1)^8} + \frac{511}{66(2n+1)^{10}} -... | {
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# I Average chord length of a circle
1. Oct 24, 2017
### serverxeon
I would like to find the average chord length of a circle.
And I have 2 methods, which gave different answers...
[The chord is defined as the line joining 2 points on the circumference of the circle.]
The general formula for a chord length is $d=2... | {
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Anyway,
Did you mean to say there is ONE set of infinite chords, or there are infinite sets of finite chords?
Because you seem to suggest the former, and that would mean there should only be one unique answer?
6. Oct 24, 2017
### mathman
There is one set of infinite chords. However, there are many possible ways to d... | {
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On the last equality we split the integral in two, and used the fact that $\cos(\theta + \pi) = -\cos(\theta)$. Next observe that $1-\cos(\theta) = 2\sin^2(\frac{\theta}{2})$ and $1 + \cos(\theta) = 2\sin^2(\frac{\theta}{2})$, so that $\sqrt{1-\cos(\theta)} = \sqrt{2}\sin(\frac{\theta}{2})$ and $\sqrt{1+\cos(\theta)} =... | {
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Keywords: monte-carlo | integration | uniform distribution | law of large numbers | lotus | central limit theorem | normal distribution | Download Notebook
Contents
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
The basic idea
Let us formalize the basic idea behi... | {
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fmd = lambda x,y: x*x + y*y
# use N draws
N= 8000
X= np.random.uniform(low=-1, high=1, size=N)
Y= np.random.uniform(low=-1, high=1, size=N)
Z=fmd(X, Y) # CALCULATE THE f(x)
R = X**2 + Y**2
V = np.pi*1.0*1.0
N = np.sum(R<1)
sumsamples = np.sum(Z[R<1])
print("I=",V*sumsamples/N, "actual", np.pi/2.0) #actual value (c... | {
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11.8114651823 0.00398497853806
This looks like our telltale Normal distribution.
This is not surprising
Estimating the error in MC integration using the CLT.
We know from the CLT that if $x_1,x_2,…,x_n$ be a sequence of independent, identically-distributed (IID) random variables from a random variable $X$, and tha... | {
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The error on the estimation of the integral can be shown to decrease as $\frac{1}{n^2}$. The basic reason for this can be understood on a taylor series expansion of the function to second order. When you integrate on the sub-interval, the linear term vanishes while the quadratic term becomes cubic in $\Delta x$. So the... | {
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# Math Help - distance and time word problem
1. ## distance and time word problem
A man walks for 45 minutes at a rate of 3 mph, then jogs for 75 minutes at a rate of 5 mph, then sits and rests for 30 minutes, and finally walks for 1 hour and half.Find the rule of the function that expresses this distance traveled as... | {
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So, distance as a function of time:
d= rt
1) $d(t) = 0t$ so $d(.5) = (0)(.5) = 0 miles$
2) $d(t) = 3t$ so $d(.75) = (3)(.75) = 2.25 miles$
3) $d(t) = 5(t)$ so $d(1.25) = (5)(1.25) = 6.25 miles$
4) $d(t) = 3t$ so $d(1.5) = (3)(1.5) = 4.5 miles$
Hope this is what you needed! Good luck!
3. Hello, vance!
This is a tric... | {
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"... |
The graph would look like this:
Code:
|
| *
| * :
| * :
| * :
| * * * :
| * : : :
| * : : :
| * : : :
| ... | {
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"... |
The graph would look like this:
Code:
|
| *
| * :
| * :
| * :
| * * * :
| * : : :
| * : : :
| * : : :
| ... | {
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# Is it okay to reverse engineer proofs in homework questions?
In a linear algebra text book, one homework question I received was:
Prove that $\mathbf{a \cdot b} = \frac{1}{4}(\|\mathbf{a + b}\|^2 - \|\mathbf{a - b}\|^2)$.
Where $\mathbf{a}$ and $\mathbf{b}$ are vectors in $\Bbb{R}^n$.
This is trivial to prove if ... | {
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-
Yes, of course that's legal. In math, you're allowed to do anything that's logically justified. – anomaly Jun 30 '14 at 2:43
In proving identities, it's perfectly acceptable to start from the RHS and go to the LHS, as long as your steps are always fully "reversible". That means that you can use the double-implicatio... | {
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However you need to be careful in this approach: you need to make sure that each step is reversible ($p$ implies $q$ does not mean $q$ implies $p$). With your example here though, we can easily reverse each step because everything is just equality.
It is worth noting that although your proof method is fine, the proof ... | {
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which leads almost immediately to the required result.
-
Not only is it 'OK', it is equally as valid to go from the RHS to the LHS as it is to go from the LHS to the RHS.
Moreover, sometimes one finds that the best approach is to work from both ends simultaneously.
As a student you may well find that the simultaneo... | {
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# $T$ be linear operator on $V$ by $T(B)=AB-BA$. Prove that if A is a nilpotent matrix, then $T$ is a nilpotent operator.
Let $V$ be a vector space of $n\times n$ matrices over a field F, and let $A$ be a fixed $n\times n$ matrix. $T$ be linear operator on $V$ by $T(B)=AB-BA$. Prove that if A is a nilpotent matrix, th... | {
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This is in the same spirit to your proof, but presented in a different way. If $\lambda B = AB-BA$ for some $B\ne0$ and some $\lambda$ in the algebraic closure of $F$, then $(A-\lambda I)B=BA$ and $(A-\lambda I)^k B=BA^k$ for any $k\ge1$. In particular, $(A-\lambda I)^nB=0$. However, if $\lambda$ is nonzero, $A-\lambda... | {
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Your proof seems correct to me. This result is used for Engel's theorem in the theory of Lie algebras. If $x\in \mathbb{gl}(V)$ is nilpotent, then also $ad(x)$ is nilpotent, where $ad(x)(y)=[x,y]=xy-yx$ for $x,y\in \mathfrak{gl}(V)$. Indeed, $ad(x)^m$ is a linear combination of terms $x^iyx^{m-i}$.
I proved this the s... | {
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Returning to the overall proof, we have $V=\mathcal{M}_{n\times n}(F)$ whenever $n\in\mathbb{N}$ is arbitrarily fixed, and we take $A\in\mathcal{M}_{n\times n}(F)$ arbitrarily also. Define the operators $T_{1},T_{2}\in\mathcal{L}(V,V)$ such that for all $B\in\mathcal{M}_{n\times n}(F)$ we have $T_{1}(B)=AB$ and $T_{2}(... | {
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• To critique my own work, where I say $T^{m}(B)={\displaystyle{\sum\limits_{i=0}^{m}\binom{m}{i}A^{m-i}BA^{i}}}$, I think I need $T^{m}(B)={\displaystyle{\sum\limits_{i=0}^{m}(-1)^{i}\binom{m}{i}A^{m-i}BA^{i}}}$ instead (as it is mentioned above, $T$ is defined in the OP's problem-statement)? Going to edit this in now... | {
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# Showing if $\frac{X}{c}\sim\text{Gamma}(a,b)$ then $X\sim\text{Gamma}(a,cb)$
I am trying to show that if $$\frac{X}{c}\sim\text{Gamma}(a,b) \ \ \ \ \ \ \text{then} \ \ \ \ \ \ \ X\sim\text{Gamma}(a,cb)$$
My first approach was to use a PDF transformation. I let $$Z=\frac{X}{c}\Rightarrow X=Zc$$ Then $$f_X(x)=f_Z\Big... | {
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Second of all, $$m_X\Big(\frac{u}{c}\Big) \neq \Big(1-b\frac{u}{c}\Big)^{-a}$$ What you did here - assuming you were doing the work from left to right - was you assumed that $$X \sim \text{Gamma}(a, b)$$ to begin with (how else would you know what $$m_{X}(u)$$ is equal to that?), but we actually don't have that assumpt... | {
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# Deriving properties of the exp function from definition
Define exponential function $\exp$ as follows: $$\exp:\Bbb{R}\rightarrow(0,\infty)$$ \begin{align}i) \ \forall x,y\in \Bbb{R}:\exp(x+y)=\exp(x)\cdot\exp(y)\end{align} \begin{align}ii) \ \forall x\in \Bbb{R}:\exp(x)\ge x+1 \end{align} Book says these 4 propertie... | {
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• @MichalDvořák I fixed a typo in my answer. It should have been about $x = y = 0$. – user296602 Apr 16 '18 at 21:31
• I would be interested in an elementary argument showing that it is indeed fixed by the two conditions above. I really do think you need at least some real analysis here. – Thomas Bakx Apr 16 '18 at 21:... | {
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• I've found that in scripts from Faculty of Mathematics and Physics from the Charles university of Prague, unfortunately, not available in English version – Michal Dvořák Apr 16 '18 at 22:03
• If i can prove that $\exp$ is its own derivative, isn't only thing i could show by the quotient rule that, assuming we have tw... | {
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Which is larger?
• November 28th 2010, 07:19 PM
Mr Rayon
Which is larger?
Let x be an element of R. Find which term has a larger value, sin(cos x) or cos(sin x)?
Any help or detailed working out would be appreciated!
• November 30th 2010, 05:29 AM
Sudharaka
Quote:
Originally Posted by Mr Rayon
Let x be an element of... | {
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AM
• December 1st 2010, 12:42 AM
BobP
Quote:
Originally Posted by Mr Rayon
Let x be an element of R. Find which term has a larger value, sin(cos x) or cos(sin x)?
Any help or detailed working out would be appreciated!
Both functions are continuous and you can't find a value of x for which they are equal.
So, if one ... | {
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Below is a proof that I have formulated for the theorem BobP had given in post #6. If you find any mistake in it please do not hesitate to tell me. Using this result the above problem could be easily solved.
• December 4th 2010, 07:14 AM
The graph of the two functions $sin(cosx)$ and $cos(sinx)$ or $cos(sinx)-sin(cosx)... | {
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Since $\sin(\cos(\pi))=\sin(-1)\approx -0.8415$ (It's definitely negative!)
and $\cos(\sin(\pi))=\cos(0)=1$,
it must be that $\sin(\cos(x))<\cos(\sin(x))$, for all x.
• December 31st 2010, 05:52 PM
Maybe simplest and without reference to graphs is:
$cos(A+B)=cosAcosB-sinAsinB\Rightarrow\ cos\left(\frac{\pi}{2}-A\ri... | {
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$-\frac{\pi}{2}<\frac{1}{\sqrt{2}}-\frac{\pi}{4}<0\Rightarrow\ sin\left[\frac{sinx+cosx-\frac{\pi}{2}}{2}\right]<0$
(4) minimum value within the second square brackets
$x=\frac{5\pi}{4}\Rightarrow\frac{-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}-\frac{\pi}{2}}{2}=-\frac{1}{\sqrt{2}}-\frac{\pi}{4}$
$-\frac{\pi}{2}<-\frac{... | {
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Please critique my proof that $\sqrt{12}$ is irrational
I would like critiques on correctness, conciseness, and clarity. Thanks!
Proposition: There is no rational number whose square is 12
Proof: Suppose there were such a number, $a = \in \mathbb{Q}$ s.t. $a^2 = 12$.
This implies that $\exists$ $m, n \in \mathbb{Z}... | {
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• Looks fine to me. A shorter version: $\sqrt{12}=2\sqrt{3}$ belongs to $\mathbb{Q}$ iff $\sqrt{3}$ belongs to $\mathbb{Q}$, but that is impossible by the unique factorization theorem and the primality of $3$. – Jack D'Aurizio Jan 26 '17 at 18:45
• Correct thinking, but you can get away with a lot less work and writing... | {
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Proof: Suppose there were such a number, $a = \in \mathbb{Q}$ s.t. $a^2 = 12$.
This implies that $\exists$ $m, n \in \mathbb{Z}$ s.t. $\frac{m^2}{n^2} = 12.$ Assume without loss of generality that $m,~ n$ have no factors in common.
$\Rightarrow m^2 = 12n^2$.
So far so good.
This implies that $m^2$ is even, and ther... | {
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[ rest of post snipped ]
• My main concern is proving that $3 | n^2 \Rightarrow 3|n$. I'm familiar with Euclid's Lemma, but I'm not clear whether it's legit in the context of a real analysis course. Thanks though, this was very helpful! – BenL Jan 26 '17 at 20:01
• @BenL Euclid's Lemma is of course true, regardless of... | {
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As $\frac {k^2}3$ is a integer, it implies $3|k^2$. Period. That always happens. That any thing else may happen doesn't matter. It may have $k/n$ as a factor or it may have $7$ as a factor. Or it may not. Those don't matter.
Also, if $\frac kn$ is an integer at all, then it is trivial that $\frac kn$ is a factor $k$ w... | {
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Note that this proof does not use divisibility.
• It is misleading to say the proof does not use divisibility. It essentially uses the division algorithm to achieve descent on denominators. I explain this further in this May 20, 2009 sci.math post, where I highlight the beautiful view of irrationality proofs in terms ... | {
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Previous | Next --- Slide 3 of 61
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BryceSummers
Question: Can anyone use some linear algebra to justify why $R^{-1} = R^{T}$ in this case. (Matrices are not equal to their transpositions in general.) In other words, what is special about matrix $R$?
lucida
R is special because its columns a... | {
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# If $X,Y$ and $Z$ are independent, then $\sigma(X,Y)$ and $\sigma(Z)$ are independent
If we have 3 independent random variables $$X,Y$$ and $$Z$$ on a probability space $$(\Omega,\mathscr{F},\mathbb{P})$$, then how do we show that $$\sigma(X,Y)$$ and $$\sigma(Z)$$ are independent?
It feels intuitively obvious, but I... | {
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To that end, fix $$A\in \Pi_1$$ and define the following two measures (you can prove that they are measures) $$\mu_{1},\mu_{2}:\mathcal{F}\rightarrow [0,\infty)$$ as follows: $$\mu_1(B)=P(A\cap B)\quad\text{and} \quad \mu_2(B) = P(A)P(B).$$ Since $$A\in \Pi_1$$, we can write $$A=\lbrace X\le x\rbrace\cap \lbrace Y\le y... | {
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This argument can be generalized as follows: Let $$X_1,X_2,\dots$$ be a sequence of independent random variables. Then $$\sigma(X_1,\dots, X_n)$$ and $$\sigma(X_{n+1}, X_{n+2}, \dots)$$ are independent for each $$n\in\mathbb{N}$$; see page 47 in Williams (1991).
• Could you show me how $\sigma(\Pi_1)=\sigma(X,Y)$? I d... | {
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Let $A$ be a $101$-element subset of the set $S=\{1,2,\ldots,1000000\}$
Let $$A$$ be a $$101$$-element subset of the set $$S=\{1,2,\ldots,10^6\}$$. For each $$s\in S$$ let $$A_s = A+s = \{a+s \mid a\in A\}$$ Prove that there exist $$B\subset S$$ such that $$|B|=100$$ and the sets in a family $$\{A_b \mid b\in B\}$$ ar... | {
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Your bipartite graph between $$B'$$ and $$C$$ has a sort of incarnation within $$G$$. Consider the subgraph of $$G$$ consisting of all edges between $$B$$ and $$B'$$. Every $$b \in B$$ has degree $$101 \cdot 100$$ in $$G$$ ($$b$$ has an edge to $$b + a_1 - a_2$$ for every $$a_1, a_2 \in A$$ with $$a_1 \ne a_2$$), and t... | {
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This is guaranteed to create an independent set: if $$b_i + a_1 - a_2 = b_j$$, then either $$i (and so we are guaranteed not to have picked $$b_j$$) or $$i>j$$ (and so we are guaranteed not to have picked $$b_i$$). For any $$b \in S$$: of $$b$$ and its at-most-$$10100$$ adjacent elements, each is equally likely to come... | {
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# Find a general solution for $\int_{0}^{\infty} \sin\left(x^n\right)\:dx$
So, I was recently working on the Sine Fresnal integral and was curious whether we could generalise for any Real Number, i.e.
$$I = \int_{0}^{\infty} \sin\left(x^n\right)\:dx$$
I have formed a solution that I'm uncomfortable with and was hopi... | {
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Any guidance would be greatly appreciated
• Where does the negative sign in the imaginary part come from – Darkrai Nov 9 '18 at 10:23
• @Manthanein - I wanted the integral to take the form of the Gamma Function and so to get the negative part in the exponential, I used the property that $e^{-ix} = \cos(x) -sin(x)i$ or... | {
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Hence along with properties of Gamma function, Mellin Transform and the Euler's reflection formula we get $$I=\frac {\pi}{2n\cos \left(\frac {\pi}{2n}\right)\Gamma \left(1-\frac 1n\right)}=\sin \left(\frac {\pi}{2n}\right)\frac {\Gamma\left(\frac 1n\right)}{n}$$
With a special case of $$n=2$$ we get the value of speci... | {
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Now write \begin{align} \int_0^\infty\sin\left(x^n\right)\,\mathrm{d}x &=\frac1n\int_0^\infty\sin(x)\,x^{\frac1n-1}\,\mathrm{d}x\tag6\\[3pt] &=\frac1{n\,\Gamma\!\left(1-\frac1n\right)}\int_0^\infty\sin(x)\int_0^\infty y^{-\frac1n}e^{-xy}\,\mathrm{d}y\,\mathrm{d}x\tag7\\ &=\frac1{n\,\Gamma\!\left(1-\frac1n\right)}\int_0... | {
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The following useful property (does this result have a name? It would be so much nicer if it did!) for the Laplace transform will be used: $$\int_0^\infty f(x) g(x) \, dx = \int_0^\infty \mathcal{L} \{f(x)\} (t) \cdot \mathcal{L}^{-1} \{g(x)\} (t) \, dt.$$ Noting that $$\mathcal{L} \{\sin x\}(t) = \frac{1}{1 + t^2},$$ ... | {
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or $$I = \sin \left (\frac{\pi}{2n} \right ) \frac{\Gamma \left (\frac{1}{n} \right )}{n}, \qquad n > 1$$ where in the last line the double angle formula for sine has been used. | {
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# Does the definition of compactness require the open cover to consist of subsets?
Sorry if the title is a bit unclear, but I'm stuck on the definition of compactness for metric spaces using open covers. So our professor wrote (word for word) that in a metric space $$(X,d)$$, an open cover is a family $$\{U_i\}_{i\in ... | {
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Open means relatively open. A subset of $$U$$ of $$X = [0,1] \cap \mathbf Q$$ is open if there is an open set $$O \subset \mathbf R$$ satisfying $$U = O \cap X$$.
A subset $$K$$ of a metric space $$X$$ is compact if and only if for every any collection of open sets $$U_i\subseteq X,i\in I$$ for which $$K\subseteq \bigc... | {
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# Choosing elements from sets
OK, so I've always been terrible at combinatorics and I'm trying to generalize some combinatorial problems and I can't figure out where I'm going wrong. Take the following problem:
Assume we are given disjoint finite sets $A_1,A_2,\ldots,A_n$ with $A=\bigcup_i A_i$. How many sets can we ... | {
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is actually correct.
-
I guess the $A_i$ are disjoint and you want to choose exactly one element from each $A_i$? – Michael Greinecker May 10 '13 at 15:10
Yes, the $A_i$ are disjoint. – user75789 May 10 '13 at 15:12
It seems like based on the comments, the second answer is correct (this is what I initially wrote dow... | {
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Your denominator of $\binom{52}4$, however, is incorrect: it’s the number of sets of $4$ cards that can be drawn from the $52$-card deck, and in the numerator you’re counting $4$-card sequences, not sets. There are actually $52\cdot51\cdot50\cdot49$ possible sequences of draws: the first card can be any of the $52$, th... | {
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The first calculation is obviously wrong, as the result is greater than $1$. The logical explanation, is that the way you count the possible number of "good" draws is ordered, e.g. the same set of $4$ different suited cards is counted $4!$ times. However, is the denominator, you counted only unordered draws. Divide the... | {
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# Proving that a function is odd
Assume that there exists a function $f:\mathbb{R}\to\mathbb{R}$ that is bijective and satisfies $$f(x) + f^{-1}(x)=x$$ for all $x$. Here $f^{-1}$ is the inverse function. Show that $f$ is odd.
This was a brain-teaser given to me by a friend. Two other related questions are:
1. Show t... | {
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• What intuition did you follow to arrive at this result? – templatetypedef Aug 18 '15 at 2:26
• @templatetypedef I tried to build the function on $[1, \lambda)$ by setting $f(x)=-\alpha x$. See my other answer to see why this leads to a cycle $\pm[1, \lambda) \to \pm[\alpha, \alpha\lambda) \to \pm[\alpha+1, (\alpha+1)... | {
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• Better than accepted answer. Did you use bijective anywhere? – BCLC Aug 18 '15 at 12:56
• Btw what exactly are b and a? I suspect that that is where bijection is used – BCLC Aug 18 '15 at 12:58
• I only used the bijection to get that $f^{-1}$ exists. In this answer, $a$ and $b$ are arbitrary (satisfying the condition... | {
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Then, for each $A\in X$, we can construct $f$ on $E_A = \bigcup_{(p,q)\in\mathbb Z^2}\{\pm2^p3^qA\}$. Indeed, since $\mathbb Z^2$ is countable, we can order it and use induction to build our cycles. When considering the $n$-th element of $\mathbb Z^2$, either it does not appear in a cycle formed by one of the previous ... | {
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Let's call $(E)$ the equation $f(y) + f^{-1} (y) = y$
Let $a \in \mathbb{R}$. Since $f$ is bijective, there exists $x \in \mathbb{R}$ such that $f(x) = a$. Now, since $f^{-1}(a) = x$, we have (using $(E)$ with $y = a$) $f(a) = a - x$. Now, using $(E)$ with $y = x$:
$$f^{-1}(x) = x - a$$ Applying $f$ on both side: $$f... | {
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