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It's worth pointing out that you need to go up to the quaternions - a four-dimensional space - because a factorization of the sort you're describing doesn't 'work' in just three dimensions. Suppose you had $z=a+bi+cj$ (and so $\bar{z}=a-bi-cj$) and $w=d+ei+fj$ (with $\bar{w}=d-ei-fj$). Then $x=zw$ would be of the form ... | {
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# Area of surface
1. Feb 24, 2015
### Incand
1. The problem statement, all variables and given/known data
Calculate the area of the surface $x^2+y^2+z^2 = R^2 , z \ge h , 0 \le h \le R$
2. Relevant equations
$A(S_D) = \iint_D |\mathbf r'_s \times \mathbf r'_t|dsdt$
where $S_D$ is the surface over $D$.
3. The attem... | {
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3. Feb 25, 2015
### HallsofIvy
Staff Emeritus
The only two variables you have are "$\phi$" and "$\theta$" and you have already integrated with respect to $\phi$ so the only possible remaining variable is $\theta$. $\theta$ is the angle a line from (0, 0, R) to (x, y, z) makes with the z-axis. When (x, y, z)= (0, 0, R... | {
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# What does $A^{-1}=A^T$ have to do with “orthogonality”?
Whenever I read some use of the term “orthogonal”, I have been able to find some way in which it is at least metaphorically similar to the idea of two orthogonal lines in euclidean space.
E.g. orthogonal random variables, etc.
But I cannot see how $A^{-1}=A^T... | {
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• This is an exceptionally clear answer +1 – Karl May 26 '18 at 20:32
Beacuase a matrix has that property if and only if all columns are orthonormal. This is also equivalent to the assertion that all rows are orthonormal.
• Also, a matrix of change of basis of two orthonormal basis has this property. This can also be... | {
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To expand on Akiva Weinberger's comment - an orthogonal matrix describes a transformation that preserves the geometry. Distances and angles in inner-product spaces are defined by the inner product, and so to say a matrix $A$ defines a geometry-preserving transformation (a "rigid" or "orthogonal" transformation) is to r... | {
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# Condition for common roots of two Quadratic equations: $px^2+qx+r=0$ and $qx^2+rx+p=0$
The question is:
Show that the equation $px^2+qx+r=0$ and $qx^2+rx+p=0$ will have a common root if $p+q+r=0$ or $p=q=r$.
How should I approach the problem? Should I assume three roots $\alpha$, $\beta$ and $\gamma$ (where $\alph... | {
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Thus, since $s = 1$, substituting back into either polynomial gives $p+q+r=0$.
Easy-peasy!
But wait ... The equations $k^3 = 1$ and $s^3 = 1$ have fully three solutions: namely, $\omega^{0}$, $\omega^{+1}$, $\omega^{-1}$, where $\omega = \exp(2i\pi/3) = (-1+i\sqrt{3})/2$. Nobody said coefficients $p$, $q$, $r$, or co... | {
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${\bf Assumptions}$: This answer assumes $p,q\not=0$ and real polynomials; $p,q,r\in\mathbb{R}$. The first condition is assumed since the polynomials is said to be quadratic and otherwise the only-if statement is not true since $p=0\implies x = -\frac{r}{q}$ is a common root for all $q,r$ and if $q=0$ we need $p=r=0$ t... | {
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# Thread: Positive Integers x & y
1. ## Positive Integers x & y
If positive integers x and y are NOT both odd, which of the following must be even?
A. xy
B. x + y
C. x - y
D. x + y - 1
E. 2(x + y) - 1
My Effort:
I decided to experiment by letting x be 3 and y be 2.
Doing this quickly revealed the fact that choice ... | {
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5. ## Re: Positive Integers x & y
Originally Posted by harpazo
If positive integers x and y are NOT both odd, which of the following must be even?
A. xy______B. x + y______C. x - y______D. x + y - 1______E. 2(x + y) - 1
Why is choice D not the answer?
To harpazo, I cannot understand how this can be so mysterious.
Lear... | {
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# Using the Intermediate Value Theorem and derivatives to check for intersections.
I have the following question:
Prove that the line $y_1=9x+17$ is tangent to the graph of the function $y_2=x^3-3x+1$. Find the point of tangency.
So, what I did was:
Let's construct a function $h$, such that $$h(x)=x^3-3x+1-(9x+17)=... | {
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I have a comment, and I know this has been looked at:
The statement involving Bolzano's Theorem, where you find that there is $c\in (0,5)$ where $h(c)=0$, implies that the graphs of the two functions intersect. It doesn't imply tangency, and it's actually a distraction, in that it gives you a point in $(0,5)$, which i... | {
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Of course, the less complicated alternative is to check that $h(x)$ has a double-root at $x=-2$, which it does, since $h(x) = (x+2)^2(x-4)$, which guarantees a point of tangency there. (This method is exactly that of the other answer, where you find roots of $h(x)$, and then check to see which of those is a root of $h'... | {
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If the curves are tangent at a point of intersection, then not only $f(c) = g(c)$ is true but $f'(c) = g'(c)$ as well. That is, $f$ and $g$ share the same tangent line (point and slope) at $c$. The equation $f'(c) = g'(c)$ is equivalent to $h'(c) = 0$. So we need to solve the equations $h(x) = 0$ and $h'(x) = 0$.
One ... | {
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# Find a formula for $\sum_{i=1}^n (2i-1)^2 = 1^2+3^2+....+(2n-1)^2$
Consider the sum $$\sum_{i=1}^n (2i-1)^2 = 1^2+3^2+...+(2n-1)^2.$$
I want to find a closed formula for this sum, however I'm not sure how to do this. I don't mind if you don't give me the answer but it would be much appreciated. I would rather have ... | {
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hint
We have
$$1^2+3^2+5^2+... (2n-1)^2=$$
$=\sum$ odd$^2$=$\sum$ all$^2$-$\sum$even$^2=$
$$\sum_{k=1}^{2n} k^2-(2^2+4^2+...4n^2)=$$
$$\sum_{k=1}^{2n}k^2-4\sum_{k=1}^n k^2=$$
$$\boxed {\color {green}{\frac {n(4n^2-1)}{3}}}$$
for $n=2$, we have $10$ , for $n=3$, we find $35$ and for $n=4$, it is $84$.
• i don't ... | {
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# How to find the corner point of a non-function equation?
Consider the equation
$$-242.0404+0.26639x-0.043941y+(5.9313\times10^{-5})\times xy-(3.9303\times{10^{-6}})\times y^2-7000=0$$
with $$x,y>0$$. If you plot it, it'll look like below:
Now, I want to find the corner point/inflection point in this equation/grap... | {
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The wikipedia page on the general hyperbola will be our guide, here.
Step 1. According to the wiki page, we must write the hyperbola in the form $$A_{xx}x^2+2A_{xy}xy+A_{yy}y^2+2B_xx+2B_yy+C=0.$$ We have $$0x^2+\left(5.9313\times 10^{-5}\right)xy-\left(3.9303\times 10^{-6}\right)y^2 + 0.26639x-0.043941y-7242.0404=0,$$... | {
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Step 3. The formula for $$a^2$$ is $$a^2=-\frac{\Delta}{\lambda_1 D},$$ where \begin{align*} \Delta&=\left|\begin{matrix}A_{xx}&A_{xy}&B_x\\A_{xy}&A_{yy}&B_y\\B_x&B_y&C\end{matrix}\right|=6.26559\times 10^{-6} \\ 0&=\lambda^2-(A_{xx}+A_{yy})\lambda+D. \end{align*} Unfortunately, the wiki page fails to distinguish betwe... | {
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• Thank you so much @AdrianKeister. This is definitely a very detailed and great answer. I've actually implemented your approach using Matlab and it gives a "VERY CLOSE" coordinates. I've updated my answer and uploaded a new image with the point that your approach finds. Do you have any idea why it has a little bit err... | {
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We first simplify the expression, and then solve for $$x:$$ \begin{align*} -242.0404+0.26639x-0.043941y+\left(5.9313\times10^{-5}\right)xy-\left(3.9303\times{10^{-6}}\right) y^2-7000&=0 \\ 0.26639x-0.043941y+\left(5.9313\times10^{-5}\right)xy-\left(3.9303\times{10^{-6}}\right) y^2-7242.0404&=0 \end{align*} \begin{align... | {
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Suppose I have data $$(a_i,b_i)_{i=1}^n$$ on two random variables $$A$$ and $$B$$. I store my data as vectors a and b, and compute their correlation using the cor function in R:
cor(a, b)
## [1] 0.4326075
Now suppose I append a mirrored version of my data by defining the vectors
alpha = c(a, b)
beta = c(b, a)
so... | {
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cor(scale(a), scale(b))
## [1] 0.4326075
The scale function de-means its argument and scales it to have unit variance. These operations don’t change the correlation of a and b. But they do change the correlation of alpha and beta:
alpha = c(scale(a), scale(b))
beta = c(scale(b), scale(a))
cor(alpha, beta)
## [1] ... | {
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c(mean(d1), mean(d2))
## [1] 7.487179 8.051282
c(var(d1), var(d2))
## [1] 25.94139 32.23110
These differences come from el listing each edge only once: it includes a row c(i, j) for the edge between nodes $$i$$ and $$j\not=i$$, but not a row c(j, i). Whereas assortativity_degree accounts for edges being undirected... | {
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# Why is $\mathbb{Z} [\sqrt{24}] \ne \mathbb{Z} [\sqrt{6}]$?
Why is $\mathbb{Z} [\sqrt{24}] \ne \mathbb{Z} [\sqrt{6}]$, while $\mathbb{Q} (\sqrt{24}) = \mathbb{Q} (\sqrt{6})$ ?
(Just guessing, is there some implicit division operation taking $2 = \sqrt{4}$ out from under the $\sqrt{}$ which you can't do in the ring?)... | {
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# Finding variance of the sample mean of a random sample of size n without replacement from finite population of size N.
I encountered this problem in the book "Introduction to the Theory of Statistics" (by Mood, Graybill and Boes) and I have not been able to solve part (c): | {
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"A bowl contains five chips numbered from 1 to 5. A sample of two drawn without replacement from this finite population is said to be random if all possible pairs of the five chips have an equal chance to be drawn.
(a) What is the expected value of the sample mean? What is the variance of the sample mean?
(b) Suppose t... | {
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It is time to upgrade the methods you rely on, since listing every possibility yields the result for small values of $n$ and $N$ but this approach (1) leads to a dead end for general values (as you realized), and (2) provides no insight.
So... let us attack directly (c), considering $N$ chips numbered from $1$ to $N$, ... | {
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Sanity check: If $n=N$, then $E(\bar X^2)=E(\bar X)^2$ (do you see why?).
• Hi thank you very much!! I followed every step and seen that it works. The "sanity check" was very useful for the conclusion (when n=N, $\bar{X}$ will always be the same, hence its variance is equal to zero and the second moment equals the squa... | {
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# Find all functions $f$ such that $f(x)+f(\frac{1}{1-x})=x$
I would like to find all functions $$f:\mathbb{R}\backslash\{0,1\}\rightarrow\mathbb{R}$$ such that
$$f(x)+f\left( \frac{1}{1-x}\right)=x.$$
I do not know how to solve the problem. Can someone explain how to solve it?
In one of my attempts I did the follo... | {
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$$2f(x)=x-\frac{1}{1-x}+1-\frac{1}{x}\to f(x)=\frac12\left(x-\frac{1}{1-x}+1-\frac{1}{x}\right)$$
• [+1] I am amazed at the way you have found your way with a machete toward the hidden unique solution. My answer, which in fact parallels yours, is guided by a group attached to the functional equation. – Jean Marie May ... | {
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It suffices now to make the following combination of equations (1)+(2)-(3) (the parts in red) to obtain:
$$f(x)=\frac12\left(x+1-\frac{1}{x}-\frac{1}{1-x}\right)$$
Remark: the group of functions $\phi_k$ has been recognized by Kummer in the mid-nineteenth century in connection with hypergeometric differential equatio... | {
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What can I say with mean, variance and standard deviation?
I wrote this code in R:
getinfoNumeric <- function(attr) {
cat(min(attr), " ")
cat(max(attr), " ")
cat(mean(attr), " ")
cat(var(attr), " ")
cat(sd(attr), " ")
}
When I apply it to an attribute, it gives me the following result:
• 50
• 100
• ... | {
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Your histogram looks roughly normal, which makes for an easy interpretation of standard deviation.
In a normal distribution, 68% of observations are $$\pm$$ one standard deviation of the mean, 95% of observations are within $$\pm$$ two standard deviations of the mean, and 99.7% of observations are within $$\pm$$ three... | {
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# Transforming a function by a sequence geometric operations on its graph.
I am solving the following problem:
Let $f(x) =\sqrt{x}$. Find a formula for a function $g$ whose graph is obtained from $f$ from the given sequence of transformations:
• shift right $3$ units
• horizontal shrink by a factor of $2$
• shift up... | {
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But then, so would the other.
• Does the order of the sequence of transformations give any insight into where the origin of the shrink would make the most sense? – Jonny Mar 31 '15 at 23:31
• @Jonny I don't think there is any ambiguity, especially when you consider the transformations of a single ordered pair as I do ... | {
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# §3.5(i) Trapezoidal Rules
The elementary trapezoidal rule is given by
3.5.1 $\int_{a}^{b}f(x)dx=\tfrac{1}{2}h(f(a)+f(b))-\tfrac{1}{12}h^{3}f^{\prime\prime}% (\xi),$ Symbols: $dx$: differential of $x$ and $\int$: integral A&S Ref: 25.4.1 (second relation only) Permalink: http://dlmf.nist.gov/3.5.E1 Encodings: TeX, ... | {
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with a function $f$ that is analytic in a strip containing $\Real$. For further information and examples, see Goodwin (1949b). In Stenger (1993, Chapter 3) the rule (3.5.5) is considered in the framework of Sinc approximations (§3.3(vi)). See also Poisson’s summation formula (§1.8(iv)).
If $k$ in (3.5.4) is not arbitr... | {
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3.5.9 $E_{n}(f)=c_{1}h^{2}+c_{2}h^{4}+\dots+c_{m}h^{2m}+\mathop{O\/}\nolimits\!\left(% h^{2m+2}\right),$ Symbols: $\mathop{O\/}\nolimits\!\left(x\right)$: order not exceeding, $E_{n}(f)$: error term and $c_{m}$: coefficients Referenced by: §3.5(iii) Permalink: http://dlmf.nist.gov/3.5.E9 Encodings: TeX, pMML, png
wher... | {
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for some $\xi\in(a,b)$. For the Bernoulli numbers $\mathop{B_{m}\/}\nolimits$ see §24.2(i).
When $f\in\mathop{C^{\infty}\/}\nolimits$, the Romberg method affords a means of obtaining high accuracy in many cases with a relatively simple adaptive algorithm. However, as illustrated by the next example, other methods may ... | {
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Rules of closed type include the Newton–Cotes formulas such as the trapezoidal rules and Simpson’s rule. Examples of open rules are the Gauss formulas (§3.5(v)), the midpoint rule, and Fejér’s quadrature rule. For the latter $a=-1$, $b=1$, and the nodes $x_{k}$ are the extrema of the Chebyshev polynomial $\mathop{T_{n}... | {
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3.5.19 $E_{n}(f)=\gamma_{n}f^{(2n)}(\xi)/(2n)!,$
where
3.5.20 $\gamma_{n}=\int_{a}^{b}p_{n}^{2}(x)w(x)dx,$ Symbols: $dx$: differential of $x$, $\int$: integral, $\gamma_{n}$: coefficients, $w$: weight and $p_{n}$: set of monic polynomials A&S Ref: 25.4.29 (for general weight function) Referenced by: §3.5(vi) Permali... | {
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The nodes $x_{k}$ and weights $w_{k}$ for $n=5$, $10$ are shown in Tables 3.5.1 and 3.5.2. The $p_{n}(x)$ are the monic Legendre polynomials, that is, the polynomials $\mathop{P_{n}\/}\nolimits\!\left(x\right)$18.3) scaled so that the coefficient of the highest power of $x$ in their explicit forms is unity.
# Gauss–Ch... | {
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and
3.5.25 $\displaystyle x_{k}$ $\displaystyle=\pm\mathop{\cos\/}\nolimits\!\left(\frac{2k}{2n+1}\pi\right),$ $\displaystyle w_{k}$ $\displaystyle=\frac{4\pi}{2n+1}{\mathop{\sin\/}\nolimits^{2}}\!\left(\frac{k}{% 2n+1}\pi\right),$ $\displaystyle\gamma_{n}$ $\displaystyle=\frac{\pi}{2^{2n}},$ $\alpha=-\beta=\pm\tfrac... | {
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# Gauss–Hermite Formula
3.5.28 $\displaystyle(a,b)$ $\displaystyle=(-\infty,\infty),$ $\displaystyle w(x)$ $\displaystyle=e^{-x^{2}},$ $\displaystyle\gamma_{n}$ $\displaystyle=\sqrt{\pi}\,\frac{n!}{2^{n}}.$
The nodes $x_{k}$ and weights $w_{k}$ for $n=5,10$ are shown in Tables 3.5.10 and 3.5.11. The $p_{n}(x)$ are t... | {
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Let $\mathbf{v}_{k}$ denote the normalized eigenvector of $\mathbf{J}_{n}$ corresponding to the eigenvalue $x_{k}$. Then the weights are given by
3.5.32 $w_{k}=\beta_{0}v_{k,1}^{2},$ $k=1,2,\dots,n$, Symbols: $\mathbf{v}_{k}$: normalized eigenvector and $w_{k}$: weights Permalink: http://dlmf.nist.gov/3.5.E32 Encodin... | {
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For the Bromwich integral
3.5.35 $I(f)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}e^{\zeta}\zeta^{-s}f(\zeta)d\zeta,$ $s>0$, $c>c_{0}>0$, Symbols: $dx$: differential of $x$, $e$: base of exponential function, $\int$: integral and $I(f)$: Bromwich integral Referenced by: §3.5(viii), §3.5(ix), §3.5(viii) Permalink: ht... | {
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with appropriate conditions. The pair
3.5.40 $\displaystyle g(t)$ $\displaystyle=\mathop{J_{0}\/}\nolimits\!\left(t\right),$ $\displaystyle G(p)$ $\displaystyle=\frac{1}{\sqrt{p^{2}+1}},$
where $\mathop{J_{0}\/}\nolimits\!\left(t\right)$ is the Bessel function (§10.2(ii)), satisfy these conditions, provided that $\s... | {
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3.5.42 $\mathop{\mathrm{erfc}\/}\nolimits\lambda=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i% \infty}e^{\zeta-2\lambda\sqrt{\zeta}}\frac{d\zeta}{\zeta},$ $c>0$, Symbols: $\mathop{\mathrm{erfc}\/}\nolimits z$: complementary error function, $dx$: differential of $x$, $e$: base of exponential function, $\int$: integral and $\la... | {
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A second example is provided in Gil et al. (2001), where the method of contour integration is used to evaluate Scorer functions of complex argument (§9.12). See also Gil et al. (2003b).
If $f$ is meromorphic, with poles near the saddle point, then the foregoing method can be modified. A special case is the rule for Hi... | {
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# Convergence of methods
#### evinda
##### Well-known member
MHB Site Helper
Hello
Could you tell me,why both of the Gauss-Seidel and Jacobi method,when we apply them at the tridiagonal matrix with the number 4 at the main diagonal and the number 1 at the first diagonal above the main and also the number 1 at the fir... | {
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Your 4-1 band matrix is easily strictly diagonally dominant, so it converges.
Your 2-(-1) band matrix is not strictly diagonally dominant, so it is an edge case.
Combined with large enough matrices and rounding errors, it will probably diverge.
I checked that with a 4x4 matrix A and some choices for b and the initial ... | {
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Your Hilbert matrix is symmetric positive-definite, so it will converge if the matrix is small enough.
Thanks to Opalg we already know that large Hilbert matrices have a determinant of near-zero, making it ill-conditioned, meaning that the rounding errors will become too large if the matrix is large enough.
Another che... | {
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#### evinda
##### Well-known member
MHB Site Helper
Also,is the Hilbert matrix strictly dominant,so that the Jacobi methodi would converge,but it doesn't because of the fact that the matrix is ill-conditioned??Or isn't it strictly dominant?
#### Klaas van Aarsen
##### MHB Seeker
Staff member
So,the Jacobi method won... | {
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"... |
#### Klaas van Aarsen
##### MHB Seeker
Staff member
Also,is the Hilbert matrix strictly dominant,so that the Jacobi methodi would converge,but it doesn't because of the fact that the matrix is ill-conditioned??Or isn't it strictly dominant?
The Hilbert matrix is not strictly diagonally dominant.
You may want to look u... | {
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Perhaps you want to try if for dimensions up to 10?
You might want to do the same thing for the Hilbert matrix.
I think I am not able to do this,because I haven't get taught how to do this in matlab So,if I am asked to tell why the methods do not converge,is the right answer that they would converge because the spectra... | {
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"... |
So,the Hilbert matrix diverges because the spectral radius of the iteration matrix is larger than 1,and the the 2-(-1) band matrix bacause of the rounding errors??
- - - Updated - - -
Here's my result for n=5 for the Hilbert matrix.
View attachment 1748
I found the max eigenvalue 3.4441 using the Jacobi method,and 1... | {
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#### Klaas van Aarsen
##### MHB Seeker
Staff member
Btw, it still only means that the methods might diverge.
It still depends on the actual b vector and the initial guess what will happen.
The best we can say is that they are not guaranteed to converge.
Only the 4-1 band matrix is guaranteed to converge.
For the 4-1 ... | {
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# Can we combine two reactions and then calculate the equilibrium concentrations?
At a certain temperature, $$\ce{N2O5}$$ dissociates as: \begin{align} \ce{N2O5 (g) &<=> N2O3 (g) + O2 (g)} & K_1 &= 4.5 \end{align} At the same time, $$\ce{N2O3}$$ also dissociates as: \begin{align} \ce{N2O3 (g) &<=> N2O (g) + O2 (g)} & ... | {
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Is it allowed to first combine the reactions and then use the combined reaction?
• I think there is sufficient information about where the OP's attempt to solve this went awry. It might help to see the "combined reaction", though. And maybe which species are in the flask after equilibrium is established. – Karsten The... | {
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I have two unknowns, $$x$$ and $$y$$. Unless one of them is much bigger than the other (in which case I can first neglect the smaller one and come back to it later), I have to solve a system of two equations for $$x$$ and $$y$$ simultaneously. The first equation is already buried in the ICE table (subscript $$eq$$ is f... | {
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Is it allowed to first combine the reactions and then use the combined reaction?
Yes, but we have to consider all species simultaneously because none of them are minor species. When calculating, for example, hydronium and hydroxide concentrations in a solution of a weak acid, we can set aside the autodissociation of w... | {
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Well, you can apply conservation of matter in terms of oxygen and nitrogen.
For nitrogen:
$$2\ce{[N2O5]} + 2\ce{[N2O3]} + 2\ce{[N2O]} = 2\times(4\ \mathrm{M})\tag{1}$$
For oxygen:
$$5\ce{[N2O5]} + 3\ce{[N2O3]} + \ce{[N2O]} + 2 \ce{[O2]} = 5\times(4\ \mathrm{M})\tag{2}$$
Then, there are the equilibrium expressions:... | {
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Probability Theory, work check
Homework Statement
Hi all, could someone give my working a quick skim to see if it checks out? Many thanks in advance.
Suppose that 5 cards are dealt from a 52-card deck. What is the probability of drawing at least two kings given that there is at least one king?
The Attempt at a Solu... | {
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Hi all, could someone give my working a quick skim to see if it checks out? Many thanks in advance.
Suppose that 5 cards are dealt from a 52-card deck. What is the probability of drawing at least two kings given that there is at least one king?
The Attempt at a Solution
Let ##B## denote the event that at least 2 kin... | {
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Your method is OK. The method of Perok in #3 is faster. However, I have one quibble: you ought to keep more significant figures when doing calculations that involve subtractions, so as to avoid "subtractive error magnification". In your case you do not do too badly, getting 0.1237 instead of 2257/18472 ≈ 0.12218, but t... | {
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Q&A
# Is Pythagorean theorem really valid in higher dimensional space?
+3
−0
I saw that someone was writing Pythagorean theorem in 3 dimensional space. The equation was :
$$c=\sqrt{x^2+y^2+z^2}$$ If it's really correct than Pythagorean should work in higher dimensional space either. So I can write that
$$c=\sqrt{\... | {
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This generalizes to higher dimensions: an $n$-dimensional simplex (a shape made of $n + 1$ vertices in $n$-dimensional space, all the edges that connect them, all the triangular faces that those edges create, all the tetrahedra that those triangles create, and so on up to $(n - 1)$-dimensional hyperfaces) with a vertex... | {
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Why does this post require moderator attention?
+2
−0
Consider a vector, $\mathbf v=(a,b,c)$, in $3$-space. We can project this onto the $xy$-plane, say, producing the vector $(a, b, 0)$ which we can identify with the $2$-vector, $(a, b)$. This two vector corresponds to the hypotenuse of a right triangle whose side l... | {
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That said, there are many other (probably better) ways of thinking about this. For example, any two vectors (in any dimension) gives rise to a (potentially degenerate) triangle in the plane spanned by those vectors (which will be ambiguous if they are linearly dependent). Namely, given vectors $\mathbf u$ and $\mathbf ... | {
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# Existence of a prime ideal in an integral domain of finite type over a field without Axiom of Choice
Let $A$ be an integral domain which is finitely generated over a field $k$. Let $f \neq 0$ be a non-invertible element of $A$. Can one prove that there exists a prime ideal of $A$ containing $f$ without Axiom of Choi... | {
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This answer builds on Qiaochu's and uses the same definition as Qiaochu, to wit: A ring $R$ is noetherian if, for any nonempty collection of ideals $\mathcal{I}$, there is some $I \in \mathcal{I}$ which is not properly contained in any $J \in \mathcal{I}$.
Theorem: If $R$ is noetherian, then $R[x]$ is noetherian.
Thi... | {
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Note that no element of $\mathcal{I} \setminus \mathcal{J}$ contains an element of $\mathcal{J}$, by the maximality of $\bar{J}$, so it is enough to show that $\mathcal{J}$ has a maximal element.
Choose an ideal $K \in \mathcal{J}$. (Making one choice does not use AC.) Let $m$ be an index such that $s_m(K) = \bar{J}$.... | {
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Let $f \in I_{\leq d}$ and let the leading term of $f$ be $r x^d$. Then $r \in s_d(I) = s_d(J)$ so there is some $g \in J_{\leq d}$ with leading term $r$. Since $I \supseteq J$, we have $g \in I$ and hence $f-g \in I$. Since $\deg(f-g) < d$, by the induction hypothesis, we have $f-g \in J$. So $f = (f-g)+g \in J$. QED
... | {
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Fields are obviously Noetherian. Note that Noetherian rings contain maximal ideals by definition.
Proposition: Let $R$ be a Noetherian ring and $f : R \to S$ a surjective ring homomorphism. Then $S$ is Noetherian.
Proof. Let $I_i$ be a non-empty collection of ideals in $S$. Then $f^{-1}(I_i)$ is a non-empty collectio... | {
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However, as far as I can tell, we cannot prove the converse without dependent choice.
-
Beat me to it! I think the proof of Hilbert basis theorem which I know, arguing about ideals of leading coefficients, does not use choice (explicitly...). – Andrew Jul 11 '12 at 18:03
@Andrew: see my edit. – Qiaochu Yuan Jul 11 '... | {
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By Noether normalization lemma(this can be proved without AC), there exist algebraically independent elements $x_1, ..., x_n$ in $A$ such that $A$ is a finitely generated module over the polynomial ring $A' = k[x_1,..., x_n]$. Let $K$ and $K'$ be the fields of fractions of $A$ and $A'$ respectively. Let $L$ be the smal... | {
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# Where do summation formulas come from?
It's a classic problem in an introductory proof course to prove that $\sum_{ i \mathop =1}^ni = \frac{n(n+1)}{2}$ by induction. The problem with induction is that you can't prove what the sum is unless you already have an idea of what it should be. I would like to know what the... | {
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• Generally, I imagine, they come from people figuring out a pattern for some small number of iterations and then proving it continues with induction. In particular, you might want to look up the story of Gauss as a little kid supposedly figuring out exactly the sum you wrote here in the case of $n=100$. – user137731 M... | {
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We start out with this equation:
$$n^k = \sum_{i=1}^n i^k-(i-1)^k.$$
This holds for $k \geq 1$. This follows by telescoping; the $n^k$ term survives, while the $0^k$ term is zero and all the other terms cancel. Using the binomial theorem:
\begin{align*} n^k & = \sum_{i=1}^n i^k-\sum_{j=0}^k {k \choose j} (-1)^j i^{k... | {
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Like Gauß did:
Write first n numbers in ordered direction in first row. Write same numbers in opposite direction in second row. Then add column-numbers. You get $n$ times the number $n+1$. This product $n \cdot (n + 1)$ is two times the sum of first $n$ numbers. Divide by two, and you get the sum of the first $n$ numb... | {
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# What does $\sum\limits_{n=1}^{N-1} \frac{1}{n} - \sum_{n=3}^{N+1} \frac{1}{n}$ simplify to?
A solution to one of the exercises in my text states:
$$\sum\limits_{n=1}^{N-1} \frac{1}{n} - \sum_{n=3}^{N+1} \frac{1}{n} = \frac{1}{1} + \frac{1}{2} - \frac{1}{N} - \frac{1}{N+1}$$
I have no idea how to get the right hand... | {
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$=\frac{1}{1}+\frac{1}{2}-\frac{1}{N}-\frac{1}{N+1}$
As for the harmonic series, this isn't a harmonic series per se, rather it is a difference of finite harmonic series with just different boundaries of summation, which should clear up why such a closed form is possible.
First thing to notice:
$$\sum_{n=1}^{N-1}\fr... | {
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# Is there a formula for solving the congruence equation $ax^2 + bx + c=0$?
Using the quadratic formula, we have either 0, 1, or 2 solutions. I wonder if we could use it this formula for congruence? Or is there a formula to solve quadratic equation for congruence?
Edit Assume that $ax^2 + bx + c \equiv 0 \pmod{p}$, w... | {
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After the above step the problem reduces to the prime power case $n = p^k$. In this case the question of what solutions look like is completely answered by Hensel's lemma. Again the case $p = 2$ is special.
-
@Quiaochu Yuan: Thank you. How about if $(a, p) = 1$, is it a special case? – Chan Apr 4 '11 at 2:35
@Chan: if... | {
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Here (link) is a thorough discussion of the steps in reducing general moduli quadratic equation problems to those of prime moduli, including the case $p=2$.
- | {
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Arrangements of people in a circle
Suppose there are $20$ kids. $10$ boys and $10$ girls.
What is the probability that the kids will arrange in this way: the circle will include exactly $5$ separate couples of girls and between each $2$ girls couple there will be at least one boy.
What I tried to do is to separate t... | {
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As for the favorable cases, there are two ways to choose on which side of Anne another girl will sit. Once that choice is made, the positions of the girls are determined by how many boys sit between each pair of girls. Let $x_k$ be the number of boys in the $k$th group to Anne's left as we proceed clockwise around the ... | {
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To see why this is the case, let's label our groups as $G_1, G_2, B_1$, followed by $G_3, G_4, B_2$, and so on... up to $G_9, G_{10}, B_5$ and having the remaining groups be $B_6$ and $B_7$ up to $B_{10}$.
One of the $9!$ permutations you list has the following sequence of people in it: $B_6, B_7, G_1, G_2, B_1$ and t... | {
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"openwebmath_score": 0.7187069654464722,
"ta... |
## Discrete Mathematics with Applications 4th Edition
$g(x)=\frac{2x^{3}+2x}{x^{2}+1}=\frac{(2x)(x^{2}+1)}{x^{2}+1}=2x=f(x)$ Yes, the functions $f$ and $g$ are equivalent.
Note that $x^{2}+1\ne0$; therefore, $g$ has the same domain as $f$. | {
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"openwebmath_score": 0.8153178691864014,
"tags":... |
# Is the limit of sequence enough of a proof for convergence?
I have a sequence $a_{n}=\frac{(n+1)(n^{2})}{(2n+1)(3n^{2}+1)}$ ,
and limit of it when $n$ goes to infinity is $\frac{1}{6}$.
Because limit is some number, is that enough of a proof that this sequence is convergent, or should i do something more?
• It is... | {
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"lm_q2_score": 0.8596637451167997,
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"openwebmath_score": 0.9490106105804443,
"ta... |
Theorems on limits lift up the burden of doing $\varepsilon$-$\delta$ proofs. If you know that $\lim_{n\to\infty}a_n=l$ and $\lim_{n\to\infty}b_n=m$ (real $l$ and $m$), then you can also say \begin{gather} \lim_{n\to\infty}(a_n+b_n)=l+m \tag{Theorem 1}\\[6px] \lim_{n\to\infty}(a_nb_n)=lm \tag{Theorem 2}\\[6px] \lim_{n\... | {
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"id": null,
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"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9793540698633748,
"lm_q1q2_score": 0.8419151874941286,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 303.41175652877035,
"openwebmath_score": 0.9490106105804443,
"ta... |
The definition of $\lim_{n\to \infty} a_n = a$ is something like this $$\forall \epsilon>0 \;\exists n_0\in \mathbb N,\; \forall n \geq n_0: |a_n - a| < \epsilon$$
Now, if you had just that and nothing else, you would indeed need to prove that your sequences converges to $1/6$ using the definition, i.e. "that $\epsilo... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9793540698633748,
"lm_q1q2_score": 0.8419151874941286,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 303.41175652877035,
"openwebmath_score": 0.9490106105804443,
"ta... |
# Polynomial form of $\det(A+xB)$
Let $A$ and $B$ be two $2 \times 2$ matrices with integer entries. Prove that $\det(A+xB)$ is an integer polynomial of the form $$P(x) = \det(A+xB) = \det(B)x^2+mx+\det(A).$$
I tried expanding the determinant of $\det(A+xB)$ for two arbitrary matrices, but it got computational. Is th... | {
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"tags... |
First step: Assume that $A=I$ and let $\lambda_1, \lambda_2$ denote the eigenvalues of $B$. (They might not be distinct, this does not affect the argument). Then $$\tag{1}\det(I+xB)=(1+x\lambda_1)(1+x\lambda_2)=1 + x\,\mathrm{trace} (B) + x^2 \det B.$$ Here we use the fact that the eigenvalues of $I+xB$ are $1+x\lambda... | {
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Therefore, using the same argument as before, we obtain in the case of invertible $A$ the formula $$\det(A+xB)=\det(A) + \mathrm{trace}(\det A A^{-1}B) x+ \det A p_2( A^{-1}B)x^2+\ldots+ \det A p_{n-1}( A^{-1}B)x^{n-1} + \det (B) x^n.$$
To remove the invertibility assumption on $A$, the formula one should use is the f... | {
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"openwebmath_score": 0.9301890134811401,
"tags... |
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