text stringlengths 1 2.12k | source dict |
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27. frx
Which form are you thinking about then?
28. sirm3d
29. frx
$-\frac{ a|\ln(t)|^{-a-1} }{ t }$ Can that be the correct derivation? If so it should give us the right answer $\lim_{t \rightarrow 0}\frac{ 1 }{ -\frac{ a|\ln(t)|^{-a-1} }{ t } } = \frac{ 1 }{- \infty }=0$ Can this be right?
30. sirm3d
we'll look... | {
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# Open and Closed mapping Examples
I am looking for three mappings f:X to Y any set of topology on X or Y. so very flexible. Can you help me find an example of a function that is (a) continuous but not an open or closed mapping (b) open but not closed or continuous (c) closed but not open of continuous
Thank you all
... | {
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• Thank you so much. As f(X)={0,1} in that example this maps an open set to an open set and there are no other open sets it must be an open mapping? and is not a closed mapping using the same logic as X is closed and it maps to an open set. sorry for the comment I am just trying to make sure my reasoning is okay. – Jm2... | {
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# set operations proofs | {
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Let x be an arbitrary element in the universe. \mathbf{R} = \mathbf{Q} \cup \overline{\mathbf{Q}}\,.\], Written $$A\cap B$$ and defined if and only if B and . (See section 2.2 example 10 for that. Back to Schedule I think either of those are valid proofs. This is a contradiction, so $$|A-B|\le|A|$$.∎. Thus, in particul... | {
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symbols. A \overline{A\cup B} and vice versa. We'll be careful for this one and manipulate the set builder notation. . , As an example, we can prove one of De Morgan's laws (the book proves the other). Let the sets $$S_1,S_2,\ldots ,S_n$$ be the students in each course. If , then and . B . Since we're doing the same ma... | {
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place many of the properties of set operations that we may use in later proofs. 3. Since , . Using set-builder notation, we can define a number of common sets and operations. 4. Here are some basic subset proofs about set operations. Since A x from the equivalences of propositional logic. The students taking, This is e... | {
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Sweet Potato Puffs Cinnamon Churro, Non Profit Training Courses, Prelude And Fugue In G Minor Bwv 861 Analysis, Lenovo Yoga 730 15 I7, Est Malayalam Meaning, Wilkinson Telecaster Saddles, Plastic Container With Lid, Mixed Vegetable Masala Dosa, Nonna Pia's Balsamic Glaze Strawberry Fig, Can Paint Kill You If You Eat It... | {
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# Calculus practice problem
My teacher assigned each student a practice problem yesterday for the new section we are starting, but I was absent, so I missed his explanation of how to do the problems. Can anyone explain to me how to solve it and provide the answer so I can practice it?
The problem is Suppose that $F(x... | {
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# What is the length of a sine wave from $0$ to $2\pi$?
What is the length of a sine wave from $0$ to $2\pi$? Physically I would plot $$y=\sin(x),\quad 0\le x\le {2\pi}$$ and measure line length.
I think part of the answer is to integrate this: $$\int_0^{2\pi} \sqrt{ 1 + (\sin(x))^2} \ \rm{dx}$$
Any ideas?
-
Does t... | {
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(where $m$ is a parameter):
$$4\sqrt{2}E\left(\frac12\right)$$
As Robert notes in a comment, different computing environments have different argument conventions for elliptic integrals; Maple for instance uses the modulus $k$ (thus, $E(k)$) instead of the parameter $m$ as input (as used by Mathematica and MATLAB), bu... | {
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return pi2 * s / v;
}
(make sure either your compiler does not (aggressively) optimize out the while (abs(v)+abs(w) != abs(v)) potion, or you'll have to use a termination criterion of the form abs(w) < tinynumber.)
Finally,
"I am also puzzled: a circle's circumference is $2\pi r$ and yet an ellipse's is an infinite... | {
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The arc length of the graph of a function $f$ between $x=a$ and $x=b$ is given by $\int_{a}^{b} \sqrt { 1 + [f'(x)]^2 }\, dx$. So, if you're considering $f(x)=\sin(x)$ then the correct integral is $\int_{0}^{2\pi} \sqrt { 1 + [\cos(x)]^2 }\, dx$. Unfortunately, this integral cannot be expressed in elementary terms. Thi... | {
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Responding to Henry, June 6, 2011, this equivalence emerges from a simple experiment given by Hugo Steinhaus in 'Mathematical Snapshots'. Take a roll of something (I use paper towelling) and saw through it obliquely, thus producing elliptic sections. Unroll it and you have a sine curve. (Tom Apostol and Mamikon Mnatsak... | {
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# Calculating Volume of Parallelpiped
I'm doing problems which have me calculate the volumes of parallelpipeds I'm slightly confused with this. I know the formula is:
$$V=\vec{a} \cdot (\vec{b}\times\vec{c})$$
where a and b form the base, with c being the "vertical" side. My issue is that when given three vectors wh... | {
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I guess that works. Thanks for the help.
HallsofIvy
It is worth remembering that with $\vec{u}= a\vec{i}+ b\vec{j}+ c\vec{k}$, $\vec{v}= d\vec{i}+ e\vec{j}+ f\vec{k}$, and $\vec{w}= x\vec{i}+ y\vec{j}+ z\vec{k}$
$$\vec{u}\cdot\left(\vec{v}\times\vec{w}\right)=$$$$\left(a\vec{i}+ b\vec{j}+ c\vec{k}\right)\left|\begin{a... | {
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# Evaluate the limit $\lim\limits_{n \to \infty} \frac{1}{1+n^2} +\frac{2}{2+n^2}+ \ldots +\frac{n}{n+n^2}$
Evaluate the limit
$$\lim_{n \to \infty} \dfrac{1}{1+n^2} +\dfrac{2}{2+n^2}+ \ldots+\dfrac{n}{n+n^2}$$
My approach :
If I divide numerator and denominator by $n^2$ I get :
$$\lim_{ n \to \infty} \dfrac{\frac... | {
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• brilliant! (all that follows is extra text inserted to satisfy the software censor's rule that "comments MUST BE at least 15 characters in length". I'm sure this excellent, and absolutely non-arbitrary rule must keep a lot of footling short comments from wasting certain peoples' very valuable time). I only wish there... | {
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• But, $$\lim_{n \to \infty} \frac1n\sum_{r=1}^n f\left(\frac rn\right)=\int_0^1f(x)dx$$ – lab bhattacharjee Dec 24 '13 at 16:24
• This is just Eric's answer. – Andrés E. Caicedo Dec 24 '13 at 18:23
• @lab (+1) thank you for that succinct statement. I have previously had to go through the angst of detail every time i h... | {
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$$\lim_{n \to \infty}\dfrac{1}{2}*(1+\dfrac{1}{n})$$
$$=\dfrac {1}{2}$$
• What happened from line 2 to 3? – Andrés E. Caicedo Dec 24 '13 at 16:43
• maybe he or she use limit first as $1/n^2\to 0$, big mistake! – mathlove Dec 24 '13 at 16:48
• Can I not use product of limits? – Satish Ramanathan Dec 24 '13 at 16:51
• ... | {
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# A problem regarding the in-equality of complex numbers .
$$\mathbf {The \ Problem \ is}:$$ Let, $$z_1,z_2 \cdots z_n$$ be such that the real and imaginary parts of each $$z_i$$ are non-negative . Show that $$\bigg|\sum_{i=1}^n z_i\bigg| \geq \frac{1}{\sqrt2} \sum_{i=1}^n |z_i|.$$
$$\mathbf {My \ approach} :$$ Actua... | {
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Let $$z_k = x_k + iy_k$$ for $$k=1, \ldots , n$$ where $$x_k,y_k\geq 0$$.
$$\left(\sum_{k=1}^n \lvert z_k \rvert\right)^2 =\left(\sum_{k=1}^n \sqrt{x_k^2 + y_k^2}\right)^2 \leq \left(\sum_{k=1}^n \left(x_k + y_k\right)\right)^2$$ $$= \left(\sum_{k=1}^n x_k + \sum_{k=1}^n y_k\right)^2$$ $$\leq 2\left(\left(\sum_{k=1}^n... | {
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I'll provide a different approach to the ones mentioned so far.
First let $$z_k=x_k+iy_k$$, then $$\vert z_1+\cdots+z_k\vert=\sqrt{(x_1+\cdots+x_k)^2+(y_1+\cdots+y_k)^2}$$ but $$h(t):=\sqrt{t},\, t\in\mathbb R\,$$ is a concave function, thus $$\sqrt{\frac{(x_1+\cdots+x_k)^2+(y_1+\cdots+y_k)^2}2}\stackrel{\color{red}{(... | {
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lottery algorithm formula pdf amazon basics wide ruled 85
# Laplacian in curvilinear coordinates
stunt cars 2
## shindai akuma boss drop
A curvilinear coordinate system is one where at least one of the coordinate surfaces is curved, e.g. in cylindrical coordinates the line between and is a circle. If the coordinate... | {
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Figure 2: Volume element in curvilinear coordinates. The sides of the small parallelepiped are given by the components of dr in equation (5). Vector v is decomposed into its u-, v- and w-components. 3 Divergence and laplacian in curvilinear coordinates Consider a volume element around a point P with curvilinear coordin... | {
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with provable stability on curvilinear multiblock grids. Hi if you have a simple ring, then you can define a cylindrical coordinate system at the ring centre (Model - Definition Coordinate systems - Cylindrical coordinates ) if this is then sys2, you have access to sys2.r (=sqrt((x-x0)^2+(y-yo)^2) if the axis is along ... | {
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## unibilt pricing
Orthogonal Curvilinear Coordinates Often it is convenient to use coordinate systems other than rectangular ones. You are familiar, for example, with polar coordinates (r, θ) in the plane. One can transform from polar to regular (Cartesian) coordinates by way of the transformation equations x = r cos... | {
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## joyride extended
nonorthogonal curvilinear coordinates. Outline: 1. Cartesian coordinates 2. Orthogonal curvilinear coordinate systems 3. Differential operators in orthogonal curvilinear coordinate systems 4. Derivatives of the unit vectors in orthogonal curvilinear coordinate systems 5. Incompressible N-S equation... | {
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## 2022 kawasaki z900 comfort seat
32,455 recent views. This course covers both the basic theory and applications of Vector Calculus. In the first week we learn about scalar and vector fields, in the second week about differentiating fields, in the third week about multidimensional integration and curvilinear coordina... | {
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## hugo blog themes
pq = 0 for p 6= q then the system of coordinates in orthogonal. We denote by G the matrix with elements g ik and g ik = (G 1) ik (the element (i;k) of the inverse of G ) and g = detjG j. The Laplacian in curvilinear coordinates is = 1 p g X ik=1;N @ @u i p gg ik @ @u k The quantization: a simple tw... | {
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## yellowtail irons
which is the ordinary Laplacian. In curvilinear coordinates, such as spherical or cylindrical coordinates, one obtains alternative expressions. ... is the D'Alembertian. Spherical Laplacian. The spherical Laplacian is the Laplace–Beltrami operator on the (n − 1)-sphere with its canonical metric of ... | {
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## 2m 70cm directional antenna
Feb 27, 2016 · I'm looking for a simple expression for the vector Laplacian $abla^2\mathbf{A}$ in orthogonal curvilinear coordinates. Actually, I don't require the whole thing, just the part of \$\mathbf{u}_i\.... Orthogonal Curvilinear Coordinates: Arc length, surface area and volume el... | {
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## edenpure air purifier thunderstorm
Vector identities; the Laplacian. Curvilinear coordinate system. Application: Maxwell equations and boundary conditions. Gauss' Law, Poisson and Laplace equations Electric field from point charge and charge distribution; Dirac delta function. Derivation of Gauss's Law from Coulomb... | {
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## eigen rotation matrix to roll pitch yaw
Laplacian — From the formulas ... In maple, a vector entry in curvilinear coordinates that has not been designated as a vector field consists of the values of the three coordinates of its head when its tail is placed at the origin, with the entries permanently identified as b... | {
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## panen138 slot
Laplacian — From the formulas ... In maple, a vector entry in curvilinear coordinates that has not been designated as a vector field consists of the values of the three coordinates of its head when its tail is placed at the origin, with the entries permanently identified as being in the coordinate sys... | {
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## rosebud movie
Curvilinear co-ordinates- Scale factors, Base vectors, Cylindrical-polar coordinates, Spherical-polar coordinates - Transformationsbetween Cartesian and curvilinear systems , Orthogonality. Elements of arc, area and volume in curvilinear system, Gradient, Divergence, Curl and Laplacian in orthogonal c... | {
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## epplus alternative
2. Electric Field in Curvilinear Coordinates Using the Generalized Functions Method The derivation presented in this section is similar to the one shown by Walsh and Donnelly (1987a) for the two-body scattering. Here, however, the system of equations is derived purely by using vector and dyadic c... | {
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## wills funeral service in northport obituary
The problem is considered in the curvilinear coordinate system (3.123).As it turns out, at b / a < ~ 2, the cross section of the surface φ = const by the planes z = const is almost exactly determined by the ellipse with the semi-axes given by curves 1 and 2 in Figure 38.(... | {
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## creepypasta x blind child reader
Equation of Poisson and cylindrical coordinates, Uncharged conducting Laplace, applications of Laplace's equation to and dielectric sphere in uniform electric field, problems (Conductors and dielectrics) having spherical cylindrical and Cartesian symmetry, Electrostatic Images, Poin... | {
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## the haunted house netflix
A number of coordinates are used for analyzing problems such as the colliding phenomena, magnetofluid dynamic equilibrium and stability of plasma or the analysis of magnetic field configuration, depending on each problem. Therefore, consideration on the vector anal. 32,455 recent views. Th... | {
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# Solving equation II
#### anemone
##### MHB POTW Director
Staff member
Solve the equation $\left\lfloor \dfrac{25x-2}{4} \right\rfloor=\dfrac{13x+4}{3}$.
##### Well-known member
Solve the equation $\left\lfloor \dfrac{25x-2}{4} \right\rfloor=\dfrac{13x+4}{3}$.
Subtract 1 from both sides
$\lfloor\frac{25x-6}{4}\rflo... | {
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# Solve Sudoku Puzzles Via Integer Programming: Problem-Based
This example shows how to solve a Sudoku puzzle using binary integer programming. For the solver-based approach, see Solve Sudoku Puzzles Via Integer Programming: Solver-Based.
You probably have seen Sudoku puzzles. A puzzle is to fill a 9-by-9 grid with i... | {
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This formulation is precisely suited for binary integer programming.
The objective function is not needed here, and might as well be a constant term 0. The problem is really just to find a feasible solution, meaning one that satisfies all the constraints. However, for tie breaking in the internals of the integer progr... | {
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Create an optimization variable x that is binary and of size 9-by-9-by-9.
x = optimvar('x',9,9,9,'Type','integer','LowerBound',0,'UpperBound',1);
Create an optimization problem with a rather arbitrary objective function. The objective function can help the solver by destroying the inherent symmetry of the problem.
s... | {
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sudsoln.x = round(sudsoln.x);
y = ones(size(sudsoln.x));
for k = 2:9
y(:,:,k) = k; % multiplier for each depth k
end
S = sudsoln.x.*y; % multiply each entry by its depth
S = sum(S,3); % S is 9-by-9 and holds the solved puzzle
drawSudoku(S)
You can easily check that the solution is correct.
### Function to Draw the S... | {
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# Math Help - Volume integral
1. ## Volume integral
Find the volume of the region bounded by the cone $x^2 + y^2 = z^2~ (z \ge 0)$ and the paraboloid $z =2x^2 + 2y^2$.
------
I'm having a hard time even visualising the integration region. The following is my attempt to plot the paraboloid (in blue) and the cone (in... | {
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How did you determine that the integrand should be the difference between the surface functions, I.e. $z_1 - z_2$?
to find the intersection: $z^2=x^2+y^2=\frac{z}{2}.$ thus $2z^2=z$ and hence $z=0, \frac{1}{2}.$ to answer your second question, recall that the volume of the region $D$ bounded by the two surfaces:
$z=f(... | {
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# An inequality on trace of product of two matrices
Suppose we have two $n \times n$ positive semidefinite matrices, $A$ and $B$, such that $\mbox{tr}(A), \mbox{tr}(B) \le 1$.
Can we say anything about $\mbox{tr}(AB)$? Is $\mbox{tr}(AB) \le 1$ too?
In the space of positive semi-definite matrices, trace is a proper i... | {
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Set $A=diag[1,0,0,\cdots,0]$ and $A=diag[0,1,0,\cdots,0]$, then $\mathrm{Tr}(AB)=0$.
Set $A=B=diag[1,0,0,\cdots,0]$, then $\mathrm{Tr}(AB)=1$.
Accoading to above, we can conclude that $\text{Range}(\mathrm{Tr}(AB))=[0,1]$.
• When does equality holds for $\text{Tr}(AB)\leq\frac{1}{2}[\text{Tr}(A^2)+\text{Tr}(B^2)]$?
... | {
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# lsqminnorm
Minimum norm least-squares solution to linear equation
## Description
example
X = lsqminnorm(A,B) returns an array X that solves the linear equation AX = B and minimizes the value of norm(A*X-B). If several solutions exist to this problem, then lsqminnorm returns the solution that minimizes norm(X).
e... | {
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Backslash 0 2.6667
lsqminnorm 8.8818e-16 2.2188
This figure illustrates the situation and shows which solutions each of the methods return. The blue line represents the infinite number of solutions to the equation ${\mathit{x}}_{2}=-\frac{2}{3}{\mathit{x}}_{1}+\frac{8}{3}$. The orange c... | {
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[Q,R,p] = qr(Anoise,0);
semilogy(abs(diag(R)),'o')
The solution to this issue is to increase the tolerance used by lsqminnorm so that a low-rank approximation of Anoise with error less than 1e-8 is used in the calculation. This makes the result much less susceptible to the noise. The solution using a tolerance is very... | {
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lsqminnorm computes the rank of A as the number of diagonal elements in the R matrix of the QR decomposition [Q,R,p] = qr(A,0) with absolute value larger than tol. If the rank of A is k, then the function forms a low-rank approximation of A by multiplying the first k columns of Q by the first k rows of R. Changing the ... | {
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MLE for Independent Exponentials
Here is the problem statement:
Let $(X_i,Y_i)$ be a random sample from a distribution with pdf $$f(x,y;\theta)= > \frac{1}{\theta^3}\exp\left(\frac{-x}{\theta}-\frac{y}{\theta^2}\right)\qquad > 0<x, 0<y.$$ (a) Find the MLE for $\theta$.
(b) Give the asymptotic distribution of $\sqrt{... | {
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• Some brackets inside your SUMS would make your notation clearer. Your derivation so far is correct. You need to break the sum of $(x_i+y_i)$ into separate terms for x and y. Then you can solve the quadratic in $\theta$. – wolfies Mar 2 '16 at 4:10
• The roots are not convenient to write and the second derivative is n... | {
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• Rose, C and Smith, M.D. (2000), Symbolic maximum likelihood estimation with Mathematica, Journal of the Royal Statistical Society, Series D: The Statistician, 49(2), 2000, 229-240. Download available: here
A more sophisticated version of same was then built into the mathStatica software package which we later develo... | {
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# Is there a weighted left-inverse of a matrix?
We can find a left inverse $A^{L} = (A^T A)^{-1}A^T$
In a situation $Ax=b$ using this left inverse I can obtain $x=A^{L}b$
This provides with "best-fit" solution for $x$, if I were to re-compute $b' = Ax$ I will get $b \ne b'$
This "best-fit" notion is based on a root... | {
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-
I think you mean $A^L = (A^T A)^{-1} A^T$ (assuming of course that $A^T A$ is invertible). – Robert Israel Sep 9 '11 at 19:20
that's what I meant! Fixed – Mikhail Sep 9 '11 at 19:21
So we're picturing $A$ having many more rows than columns. Then $M = (A^T A)^{-1}A^T$ is a left inverse of $A$ in the sense that $MA=\... | {
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-
It's not very intuitive (I'm not a math person) but I'll give it a try. Is there a way to perform this weighted derivation with SVD too? – Mikhail Sep 9 '11 at 20:33
Basically this is the same as the unweighted version if you replace $A$ by $W^{1/2} A$ and $b$ by $W^{1/2} b$. So if you know how to do unweighted leas... | {
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C. For the case of the ordinary inner product, the above optimality principle can be restated as $$A^T (b-Ax^*) = 0$$ which immediately gives you your least-squares solution; and for the case of the weighted inner product, it can be restated as $$A^T W (b-Ax^*)=0$$ which immediately gives you the weighted solution.
- | {
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1. ## Last two digit
Find the last digit two digit of the expression?
(201 * 202 * 203 *204 * 246 * 247 * 248 * 249)^2.
Is there any shortcut to find the last two digit of the above expression.
Any help would be appreciated.
Thanks,
Ashish
2. Do you know the chinese remainder theorem? If so let me know, and I will... | {
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Then there is 1 number mod ab that will satisfy this and this is how you figure out in general what it is.
Use the Euclidean Algorithm to find integers s,t, such that $\displaystyle as+bt=1$. Now take the number yas+xbt notice the sort of cross multiplying aspect of choosing this number.
now we consider $\displaystyl... | {
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Question
# There are two circles whose equations are $$x^{2}+y^{2}=9$$ and $$x^{2}+y^{2}-8x-6y+n^{2}=0,\ n\in Z$$. If the two circles have exactly two common tangents, then the number of possible values of $${n}$$ is
A
2
B
8
C
9
D
5
Solution
## The correct option is D $$9$$$$S_{1}:x^{2}+y^{2}=9$$$$C_{1}\equiv (0,0)... | {
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Eigenvalues and the Characteristic Equation
Given the following matrix,
$$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$$
assuming eigenvectors exist for $A$, they can be found by first solving for $\lambda$ (i.e. the roots of the equation) in the characteristic equation:
$$\text{det}(A-\lamb... | {
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• It's really confusing what you want to ask – Mathematician Sep 12 '13 at 12:53
You're nearly there.
A square matrix $B$ is non-invertible if and only if there exists a non-zero vector $v$ such that $Bv=0$. For example, necessity follows because $Bv=0$ implies that the function $f(v)=Bv$ is not injective (or "one-to... | {
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Finding an eigenpair $(\lambda, x)$ involves finding the eigenvalue $\lambda$ and its eigenvector $x$, such that \begin{gather} Ax = \lambda x. \end{gather}
This is equivalent to \begin{gather} Ax- \lambda x = 0. \end{gather}
This system can only have a nonzero solution $x$, if the matrix $(A-\lambda I)$ ($I$ being t... | {
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# How to solve this recurrence $K(n)=2K(n-1)-K(n-2)+C$?
The recurrence is $K(n)=2K(n-1)-K(n-2)+C$ where $C$ is a constant. What I have tried is substituting $2K(n-1)$ as we do in fibonnacical recurrences. It didn't gave me a fruitful expression! Can someone help in solving it? Not a homework problem.
$$\begin{bmatrix... | {
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• indeed Beautiful ! – Shubham Sharma Jul 9 '15 at 11:09
• Thanks! I'll admit that it wasn't obvious to me -- at first I was trying to be "creative" by turning this into a continuous equation, via converting differences into derivatives and seeing if I knew the solution to the (delay-?)differential equation. But as soo... | {
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$K_{part}$ we'll find in a form $K_{part}=\alpha n^2$ (polynom of degree $0$ and $1$ we used yet). Substitute it in $\eqref{1}$ and get $2\alpha=C$.
So, solution is $$K(n)=A+Bn+\frac{Cn^2}{2}.$$
If you prefer, we can take $K(0)=A$ and $K(1)=A+B+C/2$; hence, $$K(n)=K_0+(K_1-K_0)n+\frac{Cn(n-1)}{2}$$
• might be nice t... | {
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# Simplify : $( \sqrt 5 + \sqrt6 + \sqrt7)(− \sqrt5 + \sqrt6 + \sqrt7)(\sqrt5 − \sqrt6 + \sqrt7)(\sqrt5 + \sqrt6 − \sqrt7)$
The question is to simplify $( \sqrt 5 + \sqrt6 + \sqrt7)(− \sqrt5 + \sqrt6 + \sqrt7)(\sqrt5 − \sqrt6 + \sqrt7)(\sqrt5 + \sqrt6 − \sqrt7)$ without using a calculator .
My friend has given me this... | {
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$$\sqrt{s(s-a)(s-b)(s-c)}$$
where $s$ is the semi-perimeter $\frac12 (a + b + c)$.
Let $a, b, \text{and } c$ be $\sqrt{5}, \sqrt{6}, \text{and } \sqrt{7}$. Then the area is the square root of your expression divided by $4$. So, what is the area of this triangle? Use the law of cosines to find the cosine of the angle ... | {
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$$3=3\cdot1\cdot1\cdot1=P(1,1,1)=r(1+1+1)-(1+1+1)=3r-3$$
so $r=2$. The rest of the answer follows what the OP did:
$$P(\sqrt5,\sqrt6,\sqrt7)=2(5\cdot6+6\cdot7+7\cdot5)-(5^2+6^2+7^2)=104$$
Added 8/12/13: Eric Jablow's invocation of Heron's formula inspires one more approach:
Consider the triangle formed by the origi... | {
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$$A=a^2=5, B=b^2=6, C=c^2=7$$
$$2(AB+AC+BC) - (A^2+B^2+C^2) = 2(30+35+42) - (25+36+49) = 214-110 = 104$$ | {
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# rat
Rational fraction approximation
## Syntax
• ```[N,D] = rat(___)``` example
## Description
example
````R = rat(X)` returns the rational fraction approximation of `X` to within the default tolerance, `1e-6*norm(X(:),1)`. The approximation is a string containing the truncated continued fractional expansion.```... | {
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Express the elements of `X` as ratios of small integers using `rat`.
`[N,D] = rat(X)`
```N = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 D = 1 2 3 4 2 3 4 5 3 4 5 6 4 5 6 7```
The two matrices, `N` and `D`, approximate `X` with `N./D`.
View the elements of `X` as ratios using ```format rat```.
```format rat X```
```X = 1 1/2 1... | {
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The Ds are obtained by repeatedly picking off the integer part and then taking the reciprocal of the fractional part. The accuracy of the approximation increases exponentially with the number of terms and is worst when `X = sqrt(2)`. For `X = sqrt(2)` , the error with `k` terms is about `2.68*(.173)^k`, so each additio... | {
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# Prove the divergence of the sequence $\left\{ \sin(n) \right\}_{n=1}^{\infty}$.
I am looking for nice ways of proving the divergence of the sequence $\left\{x_n\right\}_{n=1}^{\infty}$ defined by $$x_n=\sin{(n)}.$$ One (not so nice) way is to construct two subsequences: one where the indexes are picked such that the... | {
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$$x_n^2 + y_n^2 = 1,$$
we must have
$$\alpha^2 + \beta^2 = 1,$$
a contradiction! Therefore $(y_n)$ cannot converge. ////
Of course, we can say much more on $(y_n)$. For example, we can show that the set of limit points of $(y_n)$ is exactly $[-1, 1]$, and the Cesaro mean of $(y_n)$ is 0 from Weyl's criterion.
(Thi... | {
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Imagine marching around the circumference of the unit circle in steps of arc length $1$. Then $\sin(n)$ is the $y$ coordinate at the $n$th step. Since $\pi\gt1$, the $y$ coordinate will be positive infinitely often and negative infinitely often. Consequently the limit of $\sin(n)$, if it exists, would have to be $0$. B... | {
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• You might wants to give a quick paraphrase/quote the comment text here. Even though we may not expect the MO like to rot, but comments are ephemeral. (Perhaps, equally important, this s a link-only answer). – The Long Night Nov 3 '18 at 12:24
Essentially, every point in the interval $[-1, 1]$ is a limit point for th... | {
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A known resultis that if {x_n} and {y_n} are two sequences such that y_n is divergent and x_n\y_n converges to 0 then x_n also diverges. Take x_n=sin (n) and y_n=n then the result follows.
• This is false. (What happens if you take $x_n = 1$ and $y_n = n$ with the same argument?) – mrf May 26 '14 at 7:59 | {
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# Find $q$ and $r$ with $0\leq r\leq |b|$, such that $a=qb+r$
Find $q$ and $r$, with $0\leq r\leq |b|$, such that $a=qb+r$ for
• $a=115,\ b=26$
• $a=400,\ b=-17$
• $a=-312,\ b=-64$
Sadly I missed the class where the prof went over this, so I have no idea what to do. Can somebody point me in the direction of the name... | {
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## eliassaab Group Title Let m and n be two positive integers. Show that (36 m+ n)(m+36 n) cannot be a power of 2. 2 years ago 2 years ago
1. sauravshakya
LET (36m+n)=2^x AND (m+36n)=2^y THEN, (36m+n)(m+36n)=2^(x+y) AlSO, x and y must me both integers
2. sauravshakya
Now, n=2^x - 36m THEN, m+36n=2^y m+36(2^x-36m)=2... | {
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The Pearson product-moment correlation coefficient (also referred to as Pearson’s r, or simply r) measures the strength of the linear association between two variables. That's fantastic !!! If one variable tends to increase as the other decreases, the correlation coefficient is negative. A correlation matrix is a table... | {
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elements of the correlation matrix, the main diagonal all comprises of 1. “Correlation” on the other hand measures both the strength and direction of the linear relationship between two variables. ⁄ Example 4.5.8 (Correlation-I) Let X have a uniform(0,1) distribution and Z have a uni-form(0,0.1) distribution. Referring... | {
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+1 and -1. However, the nonexistence of extreme correlations does not imply lack of collinearity. J. Ferré, in Comprehensive Chemometrics, 2009. 3.02.3.5.3(i) Correlation matrix. The Correlation Matrix Definition Correlation Matrix from Data Matrix We can calculate the correlation matrix such as R = 1 n X0 sXs where Xs ... | {
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i and j of X. A correlation matrix can be used as an input in other analyses. To estimate the market risk SCR, for example, six sub-risks (interest rate, equity, property, spread, currency and concentration risk) are aggregated using the market risk correlation matrix where the correlations between equity and the prope... | {
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matrix for a given dataset. Then select variables for analysis. Paste the formula below to N rows x N columns. First, let us calculate the matrix value for Sum of Squared Matrix. A correlation matrix is a table of correlation coefficients for a set of variables used to determine if a relationship exists between the var... | {
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Correlation matrix is a type of matrix, which provides the correlation between whole pairs of data sets in a matrix. Figure 1 – Reproduced Correlation Matrix. SSxy = ∑(xi - x̄) X (yi - ȳ) Coefficients have a range of -1 to 1; -1 is the perfect negative correlation while +1 is the perfect positive correlation. “Covaria... | {
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How To Pronounce Talisman, Highest Mileage Ford Transit, Amazon Flower Delivery, Very Puzzled Meaning In Urdu, Sleep Number Vs Tempurpedic Reddit, Rhino Rack Subaru Outback, Ge Clear Electrical Vinyl Tape, Taxi Snuff For Sale, How To Bold Numbering In Powerpoint, Uva Chemistry Phd, Dental Office Text Messaging, Best Pl... | {
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# Near-integer solutions to $y=x(1+2\sqrt{2})$ given by sequence - why?
EDIT: I've asked the same basic question in its more progressed state. If that one gets answered, I'll probably accept the answer given below (although I'm uncertain of whether or not this is the community standard; if you know, please let me know... | {
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Also, as seen in Fig. 1, there's a high degree of regularity here, with all the seemingly straight lines. If I remove the absolute value-part of the def. of $P_n$ the cross-pattern in Fig. 1 becomes (seemingly) straight, parallel declining lines. Why do these lines form? Could it be explained via some modular arithmeti... | {
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$$\left( 3 + \sqrt 8 \right)^n + \left( 3 - \sqrt 8 \right)^n$$ is an integer, while $$3 - \sqrt 8 = \frac{1}{3 + \sqrt 8}$$ has absolute value smaller than one.
My sequence would be $$x_{n+2} = 6 x_{n+1} - x_n$$ with $x_0 = 2,$ $x_1 = 6,$ $x_2 = 34$
I think I see what you did. Instead of taking powers $\left( 3 + \sqr... | {
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# What is meant by a polynomial that is “irreducible”? And a “prime” polynomial?
Let $P(\mathbb{F})$ be the set of all polynomials of one variable over a field $\mathbb{F}$. It is known that such a set, with usual polynomials operations, is a Euclidean domain, that is, a domain that has the Euclidean division algorith... | {
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All primes, $p$, are irreducible.
Proof: If $p=ab$, then $p \mid ab$, so $p\mid a$ or $p\mid b$, without loss of generality, we may assume $p\mid a$. Then $a=pv$ for some $v\in R$, and $p=pvb$. Since $R$ is a domain, we may cancel to get $1=vb$, so $b$ is a unit. Hence $p$ is irreducible.
On the other hand, it isn't ... | {
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• What you wrote applies only to domains, not general commutative rings. In non domains basic notions such as irreducible, associate etc, bifurcate into a few inequivalent notions, e.g. see here. – Bill Dubuque Aug 26 '17 at 22:48
• @BillDubuque Ah yes, that's a fair point, that was careless. Edited. Primes still ought... | {
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As for your next question, I think you are then asking why an irreducible polynomial cannot be constant since you are assuming that $p$ is irreducible, then asking why it cannot be constant. Irreducible polynomials are nonconstant by definition. The reason is that in $P(\mathbb{F})$, ever constant polynomial (except $0... | {
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I am not sure what exactly is going on with $f$ and $m$, but I hoped I answered your questions.
• I would argue for being prime being stronger just because if you exclude non-constant polynomials/ scalars like wikipedia does for irreducible you might find $3x^2+2x+1$ is irreducible on the integers, but only a subset o... | {
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• okay you got me, though quoting wikipedia it seems to also apply in rings: "In mathematics, an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the field or ring to which the coeff... | {
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# Is the graph $G_f=\{(x,f(x)) \in X \times Y\ : x \in X \}$ a closed subset of $X \times Y$?
I'm thinking about Hausdorff spaces, and how mappings to Hausdorff spaces behave. Suppose I have an arbitrary (continuous) function $f:X \longrightarrow Y$, where $Y$ is a Hausdorff space (I think it is irrelevant for my ques... | {
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