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For (i), if there are exactly $3$ twos, then there must be $4$ ones, a total of $7$ digits. The locations of the twos can be chosen in $\dbinom{7}{3}$ ways. After the location of the twos is determined, the ones have to occupy the remaining slots. Calculate: there are $35$ choices.
For interpretation (ii), we are allo... | {
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PS I have read the post of @AndréNicolas. I have intended question (b) as requiring exactly three $2$.
-
can you please help me solve this? Consider the finite sum S=1+2x+3x^(2)+4x^(3)+____+nx^(n-1) Calculate S − xS and hence give a closed expression for S? – ofo Feb 24 '13 at 19:35
@Aka, please post a separate questi... | {
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# Summation formula for $x^2+x$
Since I learned easier ways of calculating summations I've been curious as to how I could find formulas for as many equations as possible. I came across the equation $x^2+x$, I've spent quite some time on this problem and could not find a solution. If someone has maybe already done this... | {
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Note that $$k^2+k=\frac{1}{3}\left((k+1)^3-k^3-1\right).$$ Thus our sum $\sum_{k=1}^n (k^2+k)$ is equal to $$\frac{1}{3}\left((2^3-1^3-1)+(3^3-2^3-1)+(4^3-3^3-1)+(5^3-4^3-1)+\cdots +((n+1)^3-n^3-1)\right).$$ Observe the nice almost total cancellations. We end up with $$\frac{1}{3}\left((n+1)^3-1^3-n\right).$$
Remarks:... | {
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# What do I do when I am trying to find the absolute minimum point of a function and function isn't differentiable at that point?
Consider the function $f(x) = |x|$, obviously this function is not differentiable at $x = 0$. But what does that mean for the absolute minimum point? Do I answer the question with simply "t... | {
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# Radius Of Convergence Complex Power Series Problems | {
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$\begingroup$ Radius of convergence of an analytic function doesn't really exist as a concept: an analytic function has a domain on which it is analytic, and its power series around a point will have a disk of some radius on which it converges, but for a function there's nothing to converge or diverge, hence no radius ... | {
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convergence is the open interval ( 1;1). Complex Functions Examples c-4 5 Introduction Introduction This is the fourth book containing examples from theTheory of Complex Functions. Write down the power series expansion of 2 x e 1 x 2 x by multiplying the power series of e by the power series of 1/(1 x). y The series co... | {
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you can skip the multiplication sign, so 5x is equivalent to 5⋅x. Convergence Tests - Additional practice using convergence tests. [Real Analysis] Problem on Convergence of Power Series (self. The number R is called the radius of convergence of the power series. The radius of convergence r is a nonnegative real number ... | {
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Sachin Gupta B. THANK YOU !!. jz aj= Ris a circle of radius Rcentered at a, hence Ris called the radius of convergence of the power series. The interval of convergence for a power series is the set of x values for which that series converges. The radius of convergence can be characterized by the following theorem: The ... | {
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it lists no references. There series of numbers has certain properties that we can extend to the series of functions. Here we have discussed Power Series and it's Convergence (Radius of Convergence with Proof). Meromorphic function. zero, then the power series is a polynomial function, but if in nitely many of the a n ... | {
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please try again. The goal of this problem is to prove that r is the radius of convergence of the power series. Things you should memorize: • the formula of the Taylor series of a given function f(x). For case (i) of Theorem 4. Prove that the radius of convergence of the power series ∞ 0 c nx n is at least r. 8) S(z) =... | {
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Please see the attached file for the complete solution. Do not confuse the capital (the radius of convergeV nce) with the lowercase (from the root< test). Hart Complex power series: an example. The Attempt at a Solution I managed to do the Radius of convergence (power series) problem | Physics Forums. You will have to ... | {
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the radius of convergence of these power series by using the ratio test. gent complex series will converge within some disc in the complex plane. How do we find the radius of convergence? 10. radius of convergence of complex power series? this series has radius of convergence R = ∞. [Real Analysis] Problem on Convergen... | {
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problem: Suppose that (10x)/(14+x) = the sum of CnX^(n) as n=0 goes to infinity C1= C2= Find the radius of convergence R of the power series. Suppose that the limit lim n!1 jcn+1j jcnj exists or is 1. P 1 r n (a) (z/a) ,where r and a are constant real numbers. Let t be the norm of p, i. Things you should memorize: • th... | {
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0 for all n, and that n c c n+1 −−−→n→∞ r. Here we have discussed Power Series and it's Convergence (Radius of Convergence with Proof). (b) We write that the radius of convergence R = 0 if the series converges at only z 0. As in the case of a Taylor/Maclaurin series the power series given by (4. Therefore, the radius o... | {
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1 z + a 2 z 2 + a 3 z 3 + … converges at the point p, for p ≠ 0. ANALYSIS I 13 Power Series 13. The radius of convergence can be characterized by the following theorem: The radius of convergence of a power series f centered on a point a is equal to the distance from a to the nearest point where f cannot be defined in a... | {
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the point z 0 by. There series of numbers has certain properties that we can extend to the series of functions. Inthisvolume we shall only consider complex power series and their relationship to the general theory, and nally the technique of solving linear dierential equations with polynomial coecients by means of a. b... | {
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= 0}^\infty {n{x^n}}. Convergence Tests for Positive Series : The ratio test. Sachin Gupta B. In general, you can skip the multiplication sign, so 5x is equivalent to 5⋅x. Worksheet 7 Solutions, Math 1B Power Series Monday, March 5, 2012 1. Find the radius of convergence of the following power series with complex argum... | {
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that we can extend to the series of functions. Convergence Tests for Positive Series : The ratio test. Solved problems of radius of convergence power Series. ANALYSIS I 13 Power Series 13. The radius of convergence r is a nonnegative real number or ∞ such that the series converges if. Thanks for using BrainMass. For a ... | {
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compare the value between 1 and i. Complex Functions Examples c-4 5 Introduction Introduction This is the fourth book containing examples from theTheory of Complex Functions. Remember that a power series is a sum, but it is an in-nite sums. So as long as x is in this interval, it's going to take on the same values as o... | {
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n=0 to inifinity CnX^2n? Answer Choices : A. X∞ n=1 xn n √ n3n. We assume y(x) = P 1 n=0 a x n. The right-hand side 0 is given by the zero-series with radius of convergence 1. Shifted power series. RADIUS OF CONVERGENCE Let be a power series. pdf doc ; CHAPTER 10 - Approximating Functions Using. How do we find the radi... | {
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or ∞ such that the series converges if. Meromorphic function. k kB V V is called the radius of convergence. For case (i) of Theorem 4. One of the main purposes of our study of series is to understand power series. The ratio test tells us that the power series converges only when or. Representation of Functions as Power... | {
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# Find a Formula for a Linear Transformation
## Problem 36
If $L:\R^2 \to \R^3$ is a linear transformation such that
\begin{align*}
L\left( \begin{bmatrix}
1 \\
0
\end{bmatrix}\right)
=\begin{bmatrix}
1 \\
1 \\
2
\end{bmatrix}, \,\,\,\,
L\left( \begin{bmatrix}
1 \\
1
\end{bmatrix}\right)
=\begin{bmatrix}
2 \\
3 \\
2
... | {
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Then we calculate
\begin{align*}
L\left( \begin{bmatrix}
1 \\
2
\end{bmatrix}\right)
& =L\left( – \begin{bmatrix}
1 \\
0
\end{bmatrix} +
2 \begin{bmatrix}
1 \\
1
\end{bmatrix}
\right)
= -L\left( \begin{bmatrix}
1 \\
0
\end{bmatrix} \right) +
2 L \left(\begin{bmatrix}
1 \\
1
\end{bmatrix}
\right) \\
&= -\begin{bmatrix}
... | {
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Prime Number Problem
Standard
Following is a problem about prime factorization of the sum of consecutive odd primes. (source: problem 80 from The Green Book of Mathematical Problems)
Prove that the sum of two consecutive odd primes is the product of at least three (possibly repeated) prime factors.
The first thing ... | {
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Finite Sum & Divisibility – 2
Standard
Earlier this year I discussed a finite analogue of the harmonic sum. Today I wish to discuss a simple fact about finite harmonic sums.
If $p$ is a prime integer, the numerator of the fraction $1+\frac{1}{2}+\frac{1}{3}+\ldots + \frac{1}{p-1}$ is divisible by $p$.
We wish to tr... | {
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It’s an open problem in number theory to find a non-linear, non-constant polynomial which can take prime values infinitely many times. There are some conjectures about the conditions to be satisfied by such polynomials but very little progress has been made in this direction. This is a place where Ulam’s spiral raises ... | {
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In this picture, the colour of a point indicates the degree of the polynomial of which it’s a root, where red represents the roots of linear polynomials, i.e. rational numbers, green represents the roots of quadratic polynomials, blue represents the roots of cubic polynomials, yellow represents the roots of quartic po... | {
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# Conversion Of Cartesian Coordinates To Polar Coordinates Pdf | {
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Coordinates conversion from polar coordinates to cartesian x,y coordinates, and from cartesian coordinates to polar coordinates. A point can be represented by polar coordinates (r; ), where ris the distance between the point and the origin, or pole, and is the angle that a line segment from the pole to the point makes ... | {
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To enter π type Pi 3,1. Find the magnitude of the polar coordinate. Cartesian to Polar Coordinates. To start with this program you must understand definitions of Rectangular and Polar coordinates. I Computing volumes using double integrals. A Cartesian coordinate system (UK: / k ɑː ˈ t iː zj ə n /, US: / k ɑːr ˈ t i ʒ ... | {
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Polar Coordinates to Rectangular Coordinates. But I need a better precision, therefore I hope you can help me finding some formulas to convert cartesian coordinates to geographical gps. In polar coordinates, the shape we work with is a polar rectangle, whose sides have. They are also called "Euclidean coordinates," but... | {
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, 360 degrees). Converting between polar and Cartesian coordinates is really pretty simple. Another two-dimensional coordinate system is polar coordinates. 3 Polar Coordinates The Cartesian coordinate system is not the only one. The point (x,y) would be given in polar coordinates by the pair (r, θ), as shown. 1 De ning... | {
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the angular coordinate. 3 Polar Coordinates The Cartesian coordinate system is not the only one. The coordinate in this system shows the distance of the point in question from the point of origin. When I want to calculate the coordinates of a location (e. Enter your data in the left hand box with each. -Cartesian and p... | {
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in degrees or radians. CONIC SECTIONS IN POLAR COORDINATES If we place the focus at the origin, then a conic section has a simple polar equation. $\begingroup$ Dear @RenéG, you even do not need to use Solve. That should be enough. One of the particular cases of change of variables is the transformation from Cartesian t... | {
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and c Polar. We are all comfortable using rectangular (i. r is the distance to the z-axis (0, 0, z). SOLUTION: This is a graph of a horizontal line with y-intercept at (0, 10). I Computing volumes using double integrals. Approximate the. This function uses the metadata in the message, such as angular resolution and ope... | {
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complex number to its polar coordinate, we find:. For example, the point (3, 3) in rectangular coordinates becomes (√18, 45°) in polar coordinates. To convert from polar co-ordinates to Cartesian co-ordinates, use the equations x = rcosθ, y = rsinθ. 980878°$,$\phi = 40. Polar and Rectangular (Cartesian) Coordinate Conv... | {
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the distance between two points and perform operations like axis rotations without altering this value. The innermost circle shown in Figure 7. This is the official, unambiguous definition of polar coordinates, from which we. For the following exercises, convert the given Cartesian coordinates to polar coordinates with. C... | {
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coordinates. Let us discuss these in turn. Rectangular coordinates are depicted by 3 values, (X, Y, Z). Plotting Points Using Polar Coordinates. Finally, if you substitute r in the las two, you already have the solution I posted. To convert rectangular coordinates to polar coordinates, we will use two other familiar re... | {
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real part and imaginary part. k:\surveying\sem1-10\how to convert rectangular coordinates to polar coordinates. Coordinate will be the parent of the other two classes and. Example: Polar to Rectangular Example Find the rectangular coordinates of the point with polar √ coordinates ( 2, 5π/4). Polar Coordinates In a plan... | {
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the polar coordinates of a vector. This is a coordinate system in a plane, or two dimensions. For example, a radar system generates measurements in its own local spherical coordinate system. You should have used instead of Solve. For example, the point (1/2, √3/2), which makes a 30° angle with the x-axis, could have it... | {
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be written as r = rrˆ+ zˆk. Polar coordinates: We will use the following formulas to convert from cartesian coordinates (x,y) to polar coordinates {eq}(r,\theta). When I want to calculate the coordinates of a location (e. To Convert from Cartesian to Polar. 11, page 636. Cartesian coordinates in the figure below: (2,3)... | {
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Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the position of each point on the plane is determined by an angle and a distance. Example: Expressing Vector Fields with Coordinate Systems Consider the vector field: ˆˆˆ() 22 xyz x xz a x y a a z ⎛⎞ =++ +⎜⎟ ⎝⎠ A Let’s try to ... | {
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to locate each point by a pair of numerical coordinates (x, y). Convert the Cartesian coordinates defined by corresponding entries in matrices x and y to polar coordinates theta and rho. There are of course other coordinate systems, and the most common are polar, cylindrical and spherical. From my text book, I know. Pl... | {
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Cartesian Coordinates, Three-dimensional Cartesian Coordinate System, Cylindrical Coordinate System, Spherical Coordinate System : Search the VIAS Library | Index. Polar coordinate system: The polar coordinate system is a two-dimensional coordinate system in which each point P on a plane is determined by the length of ... | {
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creation of a fixed map derived from parameters defining the relationship between the Cartesian and polar systems; each Cartesian coordinate pair (X,Y) in the source image corresponding to a polar coordinate pair (R,θ) in the destination image. 030068689428860915 for the values of x and y that you originally posted. Co... | {
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from the third quadrant, my x and y values are the wrong way around. X=Y=Z for stimulus of equal luminance at each wavelength). I Derivation of Some General Relations The Cartesian coordinates (x, y, z) of a vector r are related to its spherical polar. Figure 2-20. Write down the free particle Schr\”{o}dinger equation ... | {
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Euclid discovered them first. We are supposed to convert this func-tion to Cartesian coordinates. To de ne polar coordinates, we start by identifying a pole or origin, labeled O, usually taken to be the same as the origin in Cartesian coordinates. Press Change to Cartesian coordinates. Converting between spherical and c... | {
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the xy -plane are ( x , y ) = (-3. To specify points in space using spherical-polar coordinates, we first choose two convenient, mutually perpendicular reference directions (i and k in the picture). com interactive, accessed 06/2016. This is a familiar problem; recall. A Cartesian coordinate system (UK: / k ɑː ˈ t iː z... | {
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coordinate system. Viewed 122 times 0 \begingroup I am wondering whether I converted the following correctly. Theres a question that asks you to investigate into how to convert polar coordinates into cartesian. Plot points in polar coordinates #1-8; Write polar coordinates for points #9-16; Convert Cartesian coordinate... | {
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are the parameters that define a vector r in Cartesian coordinates: r =ˆıx+ ˆy + ˆkz (1) Similarly a vector in cylindrical polar coordinates is described in terms of the parameters r, θ and z since a vector r can be written as r = rrˆ+ zˆk. 980878°, \phi = 40. Obtain the corresponding wavefunction. Furthermore, we relat... | {
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€ x = rcosθ y = rsinθ Position Vectors The fun begins when we wish to describe. The r represents the distance you move away from the origin and θ represents an angle in standard position. A Cartesian coordinate system (UK: / k ɑː ˈ t iː zj ə n /, US: / k ɑːr ˈ t i ʒ ə n /) is a coordinate system that specifies each poi... | {
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column. The Cartesian coordinates x and y can be converted to polar coordinates r and φ with r ≥ 0 and φ in the interval (− π, π] by: = + (as in the Pythagorean theorem or the Euclidean norm), and = (,), where atan2 is a common variation. To plot polar coordinates, set up the polar plane by drawing a dot labeled “O” ... | {
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can be represented by polar coordinates (r; ), where ris the distance between the point and the origin, or pole, and is the angle that a line segment from the pole to the point makes with the positive x-axis. The result lies in the range [-π, π], and the branch cut for this operation lies. 3 Polar Coordinates The Carte... | {
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quadrant, so. Equation x^2 + y^2 == r^2 is not necessary, it is a combination of the last two. Now I need to go the other way around. b) (2√3, 6, -4) from Cartesian to spherical. The Cartesian coordinate system is also called the rectangular coordinate system, because it describes a location in the plane as the vertex ... | {
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has two angles measuring. pdf The problem from cartesian to polars is that the tan function have 2 inverses, you must to know the. To convert from rectangular to polar = + ϴ = arctan(y/x) To convert from polar to rectangular x = rcos θ y = rsin θ. EXAMPLE 12: Convert x. If the point is not in the first quadrant then you... | {
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of the inverse tangent. \\ r^2 = x^2 + y^2 \\ \theta = \arctan (\frac yx) {/eq. Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as , see conventions in spherical coordinates). For example, x, y and z are the parameters that d... | {
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1 Helmholtz Equation and Angular Basis Functions As a direct extension from the Cartesian case, we begin with the eigenfunctions of the Laplacian, whose expression in polar coordinates is given by: ∇2 = ∇2. Cartesian coordinates arise naturally when you need to express translations (movements) of an object in space. Th... | {
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use but sometimes they are not the best ones to choose. Replace and with the actual values. What does the pair (r; ) refer to in the notation e r(r; ) and e (r; )? The main di erence between the familiar direction vectors e x and e y in Cartesian coor-dinates and the polar direction vectors is that the polar direction ... | {
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system to the second is a matter of converting the rst to the common world’s coordinate system and then converting the common world’s coordinate system to the second coordinate system. Cartesian Coordinate. 50) m, as shown in Active Figure 1. * Page 36 (10. EXAMPLE 11: Convert y = 10 into a polar equation. The graph r ... | {
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84up0izfoc9, w06cmvhjya1vueb, ilby49nto67te, 3flbcogqsjd, 3dlv1gdrpc3tg4n, 3ak1hk2mjg8o3pz, dd0awxp9vx3r, 8n2ylrq6pw7ji3, efn5t02kw87ctqo, m5phe4npqz, n2qfm44zvu3, bq4rcgwaaa, bpkrhx5ewp, sbd5yzc0mp, 78y5zq3rfds3, xoaiycxw8sb69, m6w5q5pez5ud, j1abyuqiqk534, 2dfpsw2fb9av, x3uubabm2yr, jnkleneyet, lqwvbafp4goc3, raebaeeu... | {
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# Calculating variable in math equation
I am not good with math, I have this equation (very simple to most) but I need help on how to get the value of x
10 = x - (1.29 + 4.99% of x)
my question is how to calculate x IOW what is the formula used to get the value of x, if that makes sense?
Thank you.
-
convert $4.9... | {
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# Calculating the probability for a Poisson RV problem
I am doing this question for homework and I've arrived at a solution which does not match the book and I am wondering what I'm doing wrong. I feel like my approach is not that much different but we get relatively different answers. Let me know if you find out my f... | {
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# Showing group with $p^2$ elements is Abelian
I have a group $G$ with $p^2$ elements, where $p$ is a prime number. Some (potentially) useful preliminary information I have is that there are exactly $p+1$ subgroups with $p$ elements, and with that I was able to show $G$ has a normal subgroup $N$ with $p$ elements.
My... | {
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• One can prove that $p$-groups have non-trivial centres using the conjugacy class equation. Is this unacceptable, by the last line? Sep 14 '11 at 3:48
• Hint: prove that the center of a non-trivial $p$-group is non-trivial and prove that if $G$ is a finite group such that $G/\textbf{Z}(G)$ is cyclic, then $G$ is abeli... | {
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It is now easy to verify that $y^ixy^{-i} = x^{r^i}$. In particular, $y^{p-1}xy^{1-p} = x^{r^{p-1}}$. By Fermat's Little Theorem, $r^{p-1}\equiv 1 \pmod{p}$, so $y^{p-1}xy^{1-p} = x$. That is, $y^{p-1}$ centralizes $x$. But $y$ is of order $p$, so $y^{p-1}=y^{-1}$. Since $y^{-1}$ centralizes $x$, so does $y$. That is, ... | {
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$|G| = |Z(G)| + \sum|cl(a)|$
where we assume that each conjugacy class is represented only once (i.e we are not including the singletons in the second summand). However, since we know that the order of a conjugacy class divides the order of a group we can rewrite the second summand to be $\sum_i(p^{k_i})$ where $0 < k... | {
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• @Deven - That is the approach I had in mind incase I can't find the more "elementary" solution, but I am confident that the 2nd approach I listed is potentially fruitful. I have recieved previous questions of this nature using that method. For example, I can show groups of order 9 must be abelian through a method sim... | {
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First we need the following lemma:
If $$G/Z(G)$$ is cyclic then G is abelian.
You can check the prove in here: https://yutsumura.com/if-the-quotient-by-the-center-is-cyclic-then-the-group-is-abelian/.
Then you need this theorem:
If $$p$$ is a prime and $$P$$ is a group of prime power order $$p^{\alpha}$$ for some $... | {
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And finally that $|G/G'| = |\{\chi\in Irr(G)\mid \chi(1) = 1\}|$.
Applying this to a finite $p$-group, we see that if $\chi\in Irr(G)$ with $\chi(1)\neq 1$ then $p$ divides $\chi(1)$, and we get $$|G| = \sum_{\chi\in Irr(G)}\chi(1)^2 = \sum_{\chi\in Irr(G),\, \chi(1) = 1}\chi(1)^2 + \sum_{\chi\in Irr(G),\, \chi(1)\neq... | {
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Now take $$g_1,g_2\ne e$$ where $$e$$ is the unit and $$\langle g_1 \rangle\cap\langle g_2 \rangle=\{e\}$$. From what we have proved above, we know that $$\langle g_1 \rangle$$ and $$\langle g_2 \rangle$$ are normal subgroups. Hence \begin{align} g_1g_2g_1^{-1}&=g_2^{\ell_1}\tag 1\\ g_2g_1g_2^{-1}&=g_1^{\ell_2}\tag 2 \... | {
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Looking at the union of singleton orbits, we see $$H_G = \left\lbrace h \in H : \forall g \in G : g^{-1}hg = h\right\rbrace$$. In other words, $$H_G$$ contains all elements $$h$$ that commute with every element of $$G$$.
Since the identity is in $$H$$, and the identity commutes with everything, $$|H_G| \geq 1$$, but s... | {
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# Direct proof that Pr[2 immediately follows 1] in a random permutation is 1/n
The probability that $1$ is a fixed point of a random permutation of $\{1,2,\ldots,n\}$ (with uniform distribution) is $1/n.$ This is easy to prove since there are $(n-1)!$ permutations that have $1$ as a fixed point and $(n-1)!\,/\,n!=1/n.... | {
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• This proof certainly avoids taking the quotient of two humongous numbers, which I think is aesthetically desirable. It seems that you and Dilip Sarwate had a very similar idea. I think one can perhaps avoid separate treatment of $1$ last versus $1$ among the first $n-1$ entries as follows. The permutations of $\{1,2,... | {
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An improved version of the observation in the comments to Benjamin Dickman's answer:
There is a $1$-to-$n$ map from the set of permutations of $\{2,3,\ldots,n\}$ to the set of permutations of $\{1,2,\ldots,n\}$ obtained by inserting $1$ in any of $n$ possible positions. Taking $n=4,$ we have, for example, $$243\mapsto... | {
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# Is a linear map (transformation) always a matrix multiplication
I am studying linear maps. It is defined as a linear map $L$ which transforms a vector from dimension $n$ to dimension $k$
$L:\mathbb{R}^n \rightarrow \mathbb{R}^k$
This seems to me as a matrix multiplication (from $x$ to $y$):
$y = Ax$
My question ... | {
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Further, for an arbitrary vector $v = \sum_{i=1}^n a^i x_i \in V$ (with some coefficients $a_i$), we have that $$\phi(v)=\phi( \sum_{i=1}^n a^i x_i) = \sum_{i=1}^n a^i \phi(x_i) = \sum_{i=1}^n a^i m^j_i y_i.$$ Form the formula for multiplicating a vector with a matrix, we see that in this basis, the components of $\phi... | {
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Yes it's alway possible use matrices for linear maps!
https://en.wikipedia.org/wiki/Linear_map
• As klirk said, this works for the maps between finite dimensional spaces, but especially if the domain is not spanned by a finite set, you wouldn't have a matrix. – Mark S. Dec 2 '17 at 13:44
• I just wanted to add, that ... | {
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What are the total number of ways in which $i$, $j$ can be chosen subject to constrain $1\leq i \leq j \leq n$?
What are the total number of ways in which $i$,$j$ can be chosen subject to constrain $1\leq i \leq j \leq n$ ? All are integers. My progress is: I believe that out of the $n$ entries, there are $n \choose 2... | {
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$$\sum_{j=1}^n j =\frac{n(n+1)}{2}$$
• Right. But, why is the answer given in the form nC2 + n? – LumosMaxima Feb 14 '18 at 8:00
• I know that the two are same. But, by what logic can I arrive at the answer of nC2 + n? – LumosMaxima Feb 14 '18 at 8:01
• $n$ is the number of the cases where $i=j$ and you have $nC2$ cho... | {
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# Showing that $\displaystyle\int_{-a}^{a} \frac{\sqrt{a^2-x^2}}{1+x^2}dx = \pi\left (\sqrt{a^2+1}-1\right)$.
How can I show that $\displaystyle\int_{-a}^{a} \frac{\sqrt{a^2-x^2}}{1+x^2}dx = \pi\left(\sqrt{a^2+1}-1\right)$?
-
The indefinite integral is possible but is very complex (both that is uses $i$ and it is com... | {
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So, if we straight away take $t=\frac x{\sqrt{a^2-x^2}}$ (assuming $a>0$)
$$\frac{dt}{dx}=\frac1{\sqrt{a^2-x^2}}+x\left(\frac{-1}2\right)\frac1{(a^2-x^2)^{\frac32}}(-2x)=\frac{a^2}{(a^2-x^2)^{\frac32}}$$
$$\sqrt{a^2-x^2} dx=\frac{(a^2-x^2)^2dt}{a^2}=\frac{a^2dt}{(1+t^2)^2}$$
and if $x=\pm a,t=\frac{\pm a}0=\pm\infty... | {
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Let $$f(z) = \frac{(a^2-z^2)^{1/2}}{1+z^2},$$ where $(a^2-z^2)^{1/2}$ denotes branch that is holomorphic on $\mathbb{C} \setminus [-a,a]$. (See this question for details.)
Next, let $\Gamma$ be a "dog bone" contour together with a large circle:
and integrate $f$ along $\Gamma$. On the "top" part of $[-a,a]$ we get th... | {
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Let $x = a \sin(y)$. Then we have $$\dfrac{\sqrt{a^2-x^2}}{1+x^2} dx = \dfrac{a^2 \cos^2(y)}{1+a^2 \sin^2(y)} dy$$ Hence, $$I = \int_{-a}^{a}\dfrac{\sqrt{a^2-x^2}}{1+x^2} dx = \int_{-\pi/2}^{\pi/2} \dfrac{a^2 \cos^2(y)}{1+a^2 \sin^2(y)} dy$$ Hence, $$I + \pi = \int_{-\pi/2}^{\pi/2} \dfrac{a^2 \cos^2(y)}{1+a^2 \sin^2(y)... | {
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-
Funny how the y'=2y+pi took me a while to get... The +pi idea was very cool! :D I wish I could up-vote this solution... – Guest 86 Jan 18 '13 at 21:07
@Marvis: here is an alternative to the integral where you employed complex analysis: $\frac{4}{(a^2+1)\cos^2 y}\int_0^{\pi/2} \frac{\mathrm{dy}}{\tan^2(y)+1/(a^2+1)} =... | {
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# Conditional probability and independent events.
In a test, an examinee either guesses or copies or knows the answer to a multiple-choice question with four choices, only one answer being correct. The probability that he makes a guess is $\frac{1}{3}$ and the probability that he copies the answer is $\frac{1}{6}$. Th... | {
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Is my thought right?
-
Writing a title in the imperative is usually frowned upon in MSE. – user02138 Mar 24 '12 at 2:10
"$P(D \cap B)=\frac{1}{8} \cdot \frac{1}{6}$. This proves that $B$ and $D$ are indepentend events." Not quite. You correctly multiplied $P(D|B)$ and $P(B)$ to get $P(D \cap B)$; but independence nee... | {
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To find $P(C\cap D)$, we use the basic formula again (though it's usually called the multiplication principle when used this way): $$P(C\cap D) =P(C)P(D\mid C).$$ We know $P(C)={1\over2}$ (as you calculated); and, if we're given that the student knows the answer, it follows that in this case that the probability that t... | {
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-
I chosen this answer because the solution presented don't apply to baye's law.Thanks – João Mar 24 '12 at 13:51
@João While the solution is excellent and fully deserving of your acceptance, it has used Bayes' law without saying so since it computed $P(C|D)$ starting from $P(D|C)$. – Dilip Sarwate Mar 25 '12 at 2:39... | {
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# Show that the solutions is a subspace of $\mathbb R^5$
Show that the solutions for the linear system of equations:
\begin{aligned} 0 + x_2 +3x_3 - x_4 + 2x_5 &= 0 \\ 2x_1 + 3x_2 + x_3 + 3x_4 &= 0 \\ x_1 + x_2 - x_3 + 2x_4 - x_5 &= 0 \end{aligned}
is a subspace of $$\mathbb R^5$$. What is the dimension of the subsp... | {
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Therefore it is a subspace of $$\mathbb{R}^5$$
• missing $s$ in equation 2 Oct 23, 2019 at 14:34
• Thank for you your answer, this was also thought on this problem. However, my professor says that the Dim = 3, which I can't really see why, or maybe he has made a mistake? Also, how do I argument that the solutions is a... | {
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Here is a correct RREF: \begin{align} &\left[\begin{array}{*{5}{r}} 0&1&3&-1&2 \\ 2&3&1&3&0 \\ 1&1&-1&2&-1 \end{array}\right]\rightsquigarrow \left[\begin{array}{*{5}{r}} 1&1&-1&2&-1 \\ 0&1&3&-1&2 \\ 2&3&1&3&0 \end{array}\right]\rightsquigarrow \left[\begin{array}{*{5}{r}} 1&1&-1&2&-1 \\ 0&1&3&-1&2 \\ 0&1&3&-1&2 \end{a... | {
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# Show that if $a_n\rightarrow a$ then $\frac{a_1+…+a_n}{n}\rightarrow a$ [duplicate]
This was an excercise in my exam and I was wondering whether my solution was correct or not.
I have said since $$a_n\rightarrow a$$ there exists a $$N$$ such that for every $$n>N$$ we have $$|a_n-a|<\epsilon_0$$. Therefore we have f... | {
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• The problem with the proof is that I have said $\frac{n\epsilon}{n}$ altough I should have said $\frac{(n-N)\epsilon}{n}$. The proof is still corect if I choose an appropriate $N''$ but the prof might give me no points for this blunder – New2Math Apr 15 at 16:35
• $(n-N) \leq n$, so your estimation is correct. I want... | {
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"lm_q1_score": 0.9830850852465429,
"lm_q1q2_score": 0.8432590288287412,
"lm_q2_score": 0.8577681031721324,
"openwebmath_perplexity": 800.1561066653719,
"openwebmath_score": 0.38437703251838684,
"ta... |
# 11.2. Attention Pooling by Similarity¶ Open the notebook in Colab Open the notebook in Colab Open the notebook in Colab Open the notebook in Colab Open the notebook in SageMaker Studio Lab
Now that we introduced the primary components of the attention mechanism, let’s use them in a rather classical setting, namely r... | {
"domain": "d2l.ai",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850872288502,
"lm_q1q2_score": 0.8432590269546604,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 12943.151166677042,
"openwebmath_score": 0.7578228712081909,
"tags": null,
... |
In the case of a (scalar) regression with observations $$(\mathbf{x}_i, y_i)$$ for features and labels respectively, $$\mathbf{v}_i = y_i$$ are scalars, $$\mathbf{k}_i = \mathbf{x}_i$$ are vectors, and the query $$\mathbf{q}$$ denotes the new location where $$f$$ should be evaluated. In the case of (multiclass) classif... | {
"domain": "d2l.ai",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850872288502,
"lm_q1q2_score": 0.8432590269546604,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 12943.151166677042,
"openwebmath_score": 0.7578228712081909,
"tags": null,
... |
def boxcar(x):
def constant(x):
return 1.0 + 0 * x
def epanechikov(x):
kernels = (gaussian, boxcar, constant, epanechikov)
names = ('Gaussian', 'Boxcar', 'Constant', 'Epanechikov')
x = torch.arange(-2.5, 2.5, 0.1)
for kernel, name, ax in zip(kernels, names, axes):
ax.plot(x.detach().numpy(), kernel(x).detach().numpy... | {
"domain": "d2l.ai",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850872288502,
"lm_q1q2_score": 0.8432590269546604,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 12943.151166677042,
"openwebmath_score": 0.7578228712081909,
"tags": null,
... |
Different kernels correspond to different notions of range and smoothness. For instance, the boxcar kernel only attends to observations within a distance of $$1$$ (or some otherwise defined hyperparameter) and does so indiscriminately.
To see Nadaraya-Watson estimation in action, let’s define some training data. In th... | {
"domain": "d2l.ai",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850872288502,
"lm_q1q2_score": 0.8432590269546604,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 12943.151166677042,
"openwebmath_score": 0.7578228712081909,
"tags": null,
... |
Recall attention pooling in (11.1.1). Let each validation feature be a query, and each training feature-label pair be a key-value pair. As a result, the normalized relative kernel weights (attention_w below) are the attention weights.
def nadaraya_watson(x_train, y_train, x_val, kernel):
dists = x_train.reshape((-1, 1... | {
"domain": "d2l.ai",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850872288502,
"lm_q1q2_score": 0.8432590269546604,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 12943.151166677042,
"openwebmath_score": 0.7578228712081909,
"tags": null,
... |
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