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How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$? |
This question is taken from a contest in India.
- Prove that we can construct an infinite set of positive integers of the form $2^{n}-3$, where $n \in \mathbb{N}$, such that any two numbers from the set are relatively prime
I would like to have an answer for this question, and i would also like to know why... |
Using your differential calculus, you should be able to write the equation for the line tangent to the parabola at a given point on the parabola. So if $(x,y)$ satisfies $y=-x^2+4x+2$ what is the equation of the tangent line? Then you should be able to find the equation for the line perpendicular to the tangent at th... |
I didn't get this method.
$3a^2-9a+6 = 3(a^2-3a+2) = 3(a^2-1a*3+3^2-3^2+2)=$ $= 3[(a-3)^2-7]$, then? |
I have **differential calculus** problem where I must find the coordinates of a point C on a parabola. There is a point given which is D(8,4) and D is on a line that is intersecting with the parabola at C. Also, the line where there is the point D is perpendicular to the tangent of C.
----------
This is the e... |
How can you prove that $\lfloor{x + \dfrac{1}{2}}\rfloor$ is a $\theta(x)$ function?
I'm just practicing questions that could come up on an exam and this one was giving me a tough time trying to formally prove it.
$f(x)$ is $\theta(g(x)) \leftrightarrow f(x)$ is $O(g(x))$ and $f(x)$ is $\Omega(g(x))$ |
How can you prove that $\Bigl\lfloor{x + \dfrac{1}{2}}\Bigr\rfloor$ is a $\theta(x)$ function?
I'm just practicing questions that could come up on an exam and this one was giving me a tough time trying to formally prove it.
$f(x)$ is $\theta(g(x)) \leftrightarrow f(x)$ is $O(g(x))$ and $f(x)$ is $\Omega(g(x))$ |
This thing came up in a combinatorics course I am taking.
Choose a fixed set of primes $p_1,p_2,\dots,p_k$ and let $A_n$ be number of integers in $\\{1,2,\dots,n\\}$ which are not divisible by any of the $p_i$'s. $A_n$ is given by $ n - \sum_{1\leq i_1 \leq k} \lfloor \frac{n}{p_{i_1}} \rfloor + \sum_{1\leq i_1 < i_... |
You can write
$x - \frac{1}{2} \leq \lfloor x+\frac{1}{2} \rfloor \leq x+\frac{1}{2}$
Doesn't this give you the result?
|
You can write
$$x - \frac{1}{2} \leq \Bigl\lfloor x+\frac{1}{2}\Bigr\rfloor \leq x+\frac{1}{2}$$
Doesn't this give you the result?
|
If `q` and `r` are quaternions, and `p` is a point, applying `q` then `r` to `p` is:
`(q * r) * p * 1/(q * r)`
What if I want to go the other way? Instead of concatenating rotations, I want to remove them.
So, I have `q` and `m`, and I know that `q * x = m`, but I don't know what `x` is. How can I discover `x`... |
Suppose we have two sets of discrete events, $A$ and $B$. Then I think it is true that:
$$2\sum_{i \in A, j \in B}\Pr[i\ \textrm{AND}\ j] \leq \sum_{i \in A \cup B}\Pr[i] + \left(\sum_{i, j \in A}\Pr[i\ \textrm{AND}\ j] + \sum_{i, j \in B}\Pr[i\ \textrm{AND}\ j]\right)$$
My intuition for why this should be true i... |
Why does this inequality in probability theory hold? |
I am just beginning to read about the use of "Concave Programming" methods and use of the Karush-Kuhn-Tucker conditions to identify the maximum value of a non-linear objective function subject to inequality constraints.
The examples I have seen in the text I have at hand, all involve only linear constraints. Is this... |
if A = { Market research predicates strong demand } and B = { Market demand is strong }, can we reasonably assume that P(A or B) = P(A) * P(B)?
The problem is that I know
- P(B|A) = 0.8
- P(not B | not A) = 0.9,
- P(B) = 0.2
I need to calculate P(A) and P(not A).
For me, it seems that if P(A or B) !... |
Suppose we have two sets of discrete events, $A$ and $B$. Then I think it is true that:
$$2\sum_{i \in A, j \in B}\Pr[i\ \textrm{AND}\ j] \leq \sum_{i \in A}\Pr[i]+ \sum_{j \in B}\Pr[j] +\sum_{i, j \in A}\Pr[i\ \textrm{AND}\ j] + \sum_{i, j \in B}\Pr[i\ \textrm{AND}\ j]$$
My intuition for why this should be true ... |
if A = { Market research predicates strong demand } and B = { Market demand is strong }, can we reasonably assume that P(A or B) = P(A) * P(B)?
The problem is that I know
- P(B|A) = 0.8
- P(not B | not A) = 0.9,
- P(B) = 0.2
I need to calculate P(A) and P(not A).
For me, it seems that if P(A or B) !... |
If $q$ and $r$ are quaternions, and $p$ is a point, applying $q$ then $r$ to $p$ is:
$$
(qr)p\dfrac{1}{qr}
$$
What if I want to go the other way? Instead of concatenating rotations, I want to remove them.
So, I have $q$ and $m$, and I know that $qx =m$, but I don't know what $x$ is. How can I discover $x... |
I would like to find the apothem of the regular pentagon. It follows from
$$\cos \dfrac{2\pi }{5}=\dfrac{-1+\sqrt{5}}{4}.$$
But how can this be proved (geometrically or trigonometrically)? |
I was just wondering where the y'/(dy/dx) in implicit differentiation comes from.
x^2 + y^2 = 25
(d/dx) x^2 + (d/dy) y^2 **(dy/dx)** = 25 (d/dx)
2x + 2y (dy/dx) = 0
(dy/dx) = -x/y
Where does the bold part come from? Wikipedia says it's a byproduct of the chain rule, but it's just not clicking for me.
|
I've heard all these terms thrown about in proofs and in geometry, but what are the differences and relationships between them? Examples would be awesome! :) |
I know it is possible for a group $G$ to have normal subgroups $H, K$, such that $H\cong K$ but $G/H\not\cong G/K$, but I couldn't think of any examples with $G$ finite. What is an illustrative example? |
Finite group with isomorphic normal subgroups and non-isomorphic quotients? |
How about *combinatorially*? This follows from the following two facts.
- The eigenvalues of the <a href="http://en.wikipedia.org/wiki/Adjacency_matrix">adjacency matrix</a> of the <a href="http://en.wikipedia.org/wiki/Path_graph">path graph</a> on $n$ vertices are $2 \cos \frac{k \pi}{n+1}, k = 1, 2, ... n$.
... |
Further to Moron's answer, consider any sequence of the form x<sub>n</sub> = a<sup>n</sup> + b for fixed integers a,b. Is there an infinite set of the x<sub>n</sub> for which any two are coprime?
First, any common factor of a and b will divide every x<sub>n</sub>. Furthermore, for any n,
$$
x_n... |
Consider the probability space $([0,1]; B[0,1], L)$, where $B[0,1]$ contains the Borel sets
intersecting $[0,1]$ and $L$ is the Lebesgue measure. How do I find the sigma algebra generated by a random variable defined on this space, $X = 1_{[0,1/2]}$? Secondly, how do I determine whether random variables defined on thi... |
This is one of the descriptions I've seen online:
For any even n, say n = 2m, a row complete latin square of order n can be formed by writing down
0, 1, 2m - 1, 2, 2m - 2, 3,..., m + 1, m
as the first row and then developing subsequent rows by adding 1 modulo n.
I'm not quite clear on how that goes, such ... |
The sigma-algebra generated by $1_{[0,1/2]}$ is simply
$$
\bigl\\{\emptyset,[0,1],[0,1/2],(1/2,1]\bigr\\}.
$$
It consists of the preimages under the function $1_{[0,1/2]}$ of all Borel sets in the codomaion of the function $1_{[0,1/2]}$, namely, $(\mathbb R,B(\mathbb R))$.
The situation for $1_{[1/4,3/4]}$ is si... |
The sigma-algebra generated by $1_{[0,1/2]}$ is simply
$$
\bigl\\{\emptyset,[0,1],[0,1/2],(1/2,1]\bigr\\}.
$$
It consists of the preimages under the function $1_{[0,1/2]}$ of all Borel sets in the codomaion of the function $1_{[0,1/2]}$, namely, $(\mathbb R,B(\mathbb R))$. (Notice that the preimage $1_{[0,1/2]}^{-1... |
This thing came up in a combinatorics course I am taking.
Choose a fixed set of primes $p_1,p_2,\dots,p_k$ and let $A_n$ be number of integers in $\\{1,2,\dots,n\\}$ which are not divisible by any of the $p_i$'s. $A_n$ is given by $ n - \sum_{1\leq i_1 \leq k} \lfloor \frac{n}{p_{i_1}} \rfloor + \sum_{1\leq i_1 < i_... |
How do I use Lebesgue Dominating Convergence Theorem to evaluate
$\lim_{n \to \infty}\int_{[0,1]}\frac{n\sin(x)}{1+n^2\sqrt(x)}dx$
what dominating function to use here? |
I have trouble understanding Radon-Nikodym theorem. Here's an example.
Consider the measurable space $(\Omega,F)$, where $\Omega = R$, $F~$ is the $\sigma$-algebra of Borel sets. Let $P[dw] = \frac{1}{\sqrt(2\pi)}e^{-w^2/2}dw$ and $\tilde{P}[dw] = e^{-w}1_{[0,\infty)(w)}dw$ be two probability measures on $\omega$. H... |
How do I use Lebesgue Dominated Convergence Theorem to evaluate
$\lim_{n \to \infty}\int_{[0,1]}\frac{n\sin(x)}{1+n^2\sqrt(x)}dx$
what dominating function to use here? |
I want to refer to functions of the form $f(x) = \sum_{i=1}^n a_i x^{\alpha_i}$ where $\alpha_i < 1$.
This is not a polynomial, because $\alpha_i$ could be just real arbitrary numbers (though positive ones).
Is there a name for this class of functions that I can refer to? Wikipedia does claim these are not polynomi... |
polynomial of $x$? |
So, I don't like proofs.
To me building a proof feels like constructing a steel trap out of arguments to __make true__ what you're trying to assert.
Oftentimes the proof in the book is something that I get if I study, but hard to come up with on my own. In other words I can't make steel traps, but I feel fine bu... |
Suppose $B_t$ is a Brownian motion. As I understand, $B_2-B_1$ is independent of $B_3-B_2$ from properties of Brownian motion. Does it also mean that $B_1$ and $B_2$ are also independent? Can I use this independence to find the joint density of $B_1+B_2+B_3$ as each Brownian process is a normal process of mean 0 and va... |
As Arturo and Qiaochu have pointed out you get better at proving things by practicing a lot, solving exercises and by seeing how other people have proved things. There's a big part in learning how to prove statements by repeating certain strategies that may have worked for particular types of problems.
For instance... |
$B_1$ and $B_2$ are not independent.
Since $B_1$ and $B_2-B_1$ are independent,
$$0=\mathrm{Cov}(B_1,B_2-B_1)=E[B_1 (B_2-B_1)]=E[B_1 B_2]-E[B_1^2]=E[B_1 B_2] - t_1 \sigma^2$$
So,
$$\mathrm{Cov}(B_1,B_2)=E[B_1 B_2]=t_1 \sigma^2$$
You can use this result and the fact that linear combinations of normal variables... |
The usual action of $fg$ on $u⊗v$, where $f,g$ are elements in the Universal Enveloping Algebra $U(G)$ of a Lie algebra $G$ and $u,v$ are elements of a representation $V$ of $G$, is given by $fg(u⊗v)=fgu⊗v+fu⊗gv+gu⊗fv+u⊗fgv$, right? How to state this fact for $V^{\otimes n}$, i.e. $fg$ acting on $u⊗v$, where $u=⊗_... |
How to define the action of U(G) in this situation? |
To make the question more precise: there is a max static friction coefficient at points A and B. Calculating the perpendicular force at A and B is not hard, so the acceleration where it breaks loose is known. The acceleration increases exceeding this. So the stick starts to slide aft. Now you have a sliding frictio... |
To make the question more precise: there is a max static friction coefficient at points A and B. Calculating the perpendicular force at A and B is not hard, so the acceleration where it breaks loose is known. The acceleration increases exceeding this. So the stick starts to slide aft. Now you have a sliding frictio... |
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$ ? |
It seems you are asking two questions. The first is how to get better at doing proofs, and the previous answers are better than I can do. The second is why to do them, which has not been addressed. From your questions on this site, you seem more an engineer than a mathematician. This is not a negative-I am educated... |
How can I find the critical points of a square root function?
For example: What are the critical points of `f(x) = (7x^2 + 2x - 2)^(1/2)`? |
I'm setting this answer community wiki so other people can attach their own examples of curves constructed with the methods in this thread.
Now, Qiaochu's suggestion for building a curve $q(y)=p(x)$ with collinear singular points on the horizontal axis amounts to the construction of an appropriate [Hermite interpola... |
How can I find the critical points of a square root function?
For example: What are the critical points of $f(x) = \sqrt{7x^2 + 2x - 2}$? |
A friend of mine who is quite an aggressive Nominalist told me the other day:
"Mathematics and numbers are arbitrary; they can accurately predict physical systems in real life only because they are consistent, but any consistent framework will do."
I'm curious as to what other examples of "consistent frameworks" ... |
Exercise 1.14 of [the book][1] Rordam, Larsen and Laustsen "An introduction to K-theory for C*-algebras" asks to prove, that upper triangular matrix with elements from some C*-algebra $A$ is invertible in $M_n(A)$ iff all diagonal entries are invertible in $A$.
Trying to solve this I've found that if $a$ is invertib... |
Consider the scheme of random placing balls into N=1000 cells. We continue the procedure of placing balls as long as a last cell remains empty. The process terminates when a ball is placed into this cell. At this moment several cells (or a certain cell) contain(s) a maximum number of balls among all cells. What is the ... |
I've noticed a type of sum for the number of solutions of a type of quadratic form. For example, the number of solutions to $X^2-Y^2\equiv a\pmod{p}$ for $p$ an odd prime is given by the sum
$$\sum_{Y}(1+\left(\frac{Y^2+a}{p}\right))$$
where it seems that the trick is to isolate the $x^2$, add 1 to the RHS, and t... |
Formula for counting solutions to a quadratic form? |
I've noticed a type of sum for the number of solutions of a type of quadratic form. For example, the number of solutions to $X^2-Y^2\equiv a\pmod{p}$ for $p$ an odd prime is given by the sum
$$\sum_{Y=1}^p(1+\left(\frac{Y^2+a}{p}\right))$$
where $\left(\frac{Y^2+a}{p}\right)$ is the Legendre symbol. It seems that... |
Hi I have just found the proof that 90 equals 95 and was wondering if I have made some mistake. If so, which step in my proof is not true?
**Definitions:**
1. $\angle ABC=90^{\circ}$
2. $\angle BCD=95^{\circ}$
3. $|AB|=|CD|$
4. $M:=$ the center of $BC$
5. $N:=$ the center of $AD$
6. $l:=$ a ... |
i have an object and a 3d direction vector and position for it . I would like to know how do i determine if a certain point X is in the space below the plan determined by my direction ?
Here is an image that i have drawn to make it more clear . In this image i've made the vector 2d
http://yfrog.com/53imgukp
P... |
i have an object and a 3d direction vector and position for it . I would like to know how do i determine if a certain point X is in the space below the plan determined by my direction ?
Here is an image that i have drawn to make it more clear . In this image i've made the vector 2d
![diagram][1]
Please Help!
... |
Much effort has been expended on a famous unsolved problem about the [Riemann Zeta function $ \zeta(s) $][1]. Not surprisingly, it's called the Riemann hypothesis, which asserts:
$$ \zeta(s) = 0 \Rightarrow \text{Re } s = 1/2 \text{ or Im } s = 0. $$
Now there are numerical methods for approximating $ \zeta(s) $... |
Exercise 1.14 of [the book][1] Rordam, Larsen and Laustsen "An introduction to K-theory for C*-algebras" asks to prove, that upper triangular matrix with elements from some C*-algebra $A$ is invertible in $M_n(A)$ iff all diagonal entries are invertible in $A$.
Trying to solve this I've found that if $a$ is invertib... |
As far as i know LU decomposition allow you to calculate matrix determinant following easy and cheap formula: Det[A] = Det[L] Det[U] = Det[U]
Trying this out in Mathematica 7 gives me correct result by absolute value, i.e. it ignores negative determinants and transforms them to positive ones.
Sample code:
matr... |
As far as i know LU decomposition allow you to calculate matrix determinant following easy and cheap formula: Det[A] = Det[L] Det[U] = Det[U]
Trying this out in Mathematica 7 gives me correct result by absolute value, i.e. it ignores negative determinants and transforms them to positive ones.
Sample code:
... |
One of the most famous 2D-curves are [logarithmic spirals][1] (or Spira mirabilis). They can be constructed by using a machinery that ensures a constant angle between the tangent and the radial lines all the time while plotting it.<br><br>
![alt text][2]
**My question**<br>
I can see it in the picture that the sp... |
Why does an equiangular spiral become logarithmic (intuitively)? |
57^46 divide by 17
What is best and quickest way to solve such kind of problem manually? Approximate answer will do |
what is 57^46 divide by 17 ?? |
I would like to find the apothem of a regular pentagon. It follows from
$$\cos \dfrac{2\pi }{5}=\dfrac{-1+\sqrt{5}}{4}.$$
But how can this be proved (geometrically or trigonometrically)? |
Why not simply do the computation? The most time-consuming part is to compute the $46$th power of $57$, but that can be done quite fast using [repeated squaring](http://en.wikipedia.org/wiki/Exponentiation_by_squaring) in 6 multiplications. That results in a number with 81 digits. Dividing by 17 using long division is ... |
The coordinate transformation (due to Beukers, Calabi and Kolk)
$$x=\frac{\sin u}{\cos v}$$
$$y=\frac{\sin v}{\cos u}$$
transforms the square domain $0\lt x\lt 1$ and $0\lt y\lt 1$ into the triangle domain $u,v>0,u+v<\pi /2$ (in *Proofs from the Book* by M. Aigner and G. Ziegler).
Since the invert transform... |
Why not simply do the computation? The most time-consuming part is to compute the $46$th power of $57$, but that can be done quite fast using [repeated squaring](http://en.wikipedia.org/wiki/Exponentiation_by_squaring) in 6 multiplications. That results in a number with 81 digits. Dividing by 17 using long division is ... |
Let $G$ be a group and let $G'$ be the derived subgroup, defined as the subgroup generated by the commutators of $G$.
Is there an example of a finite group $G$ where not every element of $G'$ is a commutator? $G'$ is only generated by commutators, but with all of the properties of commutators (ie: what happens unde... |
Let $Z$ be a countable set. Let $f_1,....,f_n$ be a collection of real functions over $Z$. Let $z_1,...,z_m,...$ be an enumeration of elements of $Z$.
Define $V_1 = { (f_1(z),...,f_n(z)) | z \in Z }$ a set of $n$ dimensional vectors.
Define $V_2 = { (f_i(z_1),...,f_i(z_m),..., )) | i \in { 1,...,n} }$ a set of infi... |
do two vector spaces have the same linear dimension? |
Let $Z$ be a countable set. Let $f_1,....,f_n$ be a collection of real functions over $Z$. Let $z_1,...,z_m,...$ be an enumeration of elements of $Z$.
Define $V_1 = \\{ (f_1(z),...,f_n(z)) | z \in Z\\}$ a set of $n$ dimensional vectors.
Define $V_2 =\\{ (f_i(z_1),...,f_i(z_m),..., )) | i \in\\{ 1,...,n\\} \\}$ a se... |
Question 1: Setting $u=v=\pi/4$ in your first set of formulas gives $(x,y)=(1,1)$.
Question 2: I have no idea. (Luck? Inspiration? Trial and error?) |
Question 1: Setting $u=\pi/2-v$ in your first set of formulas gives $(x,y)=(1,1)$, so that point corresponds to the whole hypotenuse $u+v=\pi/2$.
Question 2: I have no idea. (Luck? Inspiration? Trial and error?) |
**HINT** $\:$ Both $\:\rm e^{ix}\:$ and $\:\rm cos(x) + i \; sin(x) \:$ are solutions of $\;\rm y' = i \; y,\;\; y(0) = 1, \;$ so they are equal by the **uniqueness theorem**. Alternatively, **bisect** into even and odd parts the power series for $\:\rm e^{ix} \:,\;$ i.e.
$$\begin{align}
\rm f(x) \ \ &=\ \ \rm\fr... |
So this is some I partially understand, I'm not sure what I dont and do understand because most of my understanding is based on assumtions... sorry if I sound a little stupid!
The equation $6 \cos x - \sin x = 5 $ needs to be turned into the form $ \Re \cos (x + \alpha) $ then solved for **in the interval** $ -1/2\... |
So this is some I partially understand, I'm not sure what I dont and do understand because most of my understanding is based on assumtions... sorry if I sound a little stupid!
The equation $6 \cos x - \sin x = 5 $ needs to be turned into the form $ \Re \cos (x + \alpha) $ then solved for **in the interval** $ -1/2\... |
The coordinate transformation (due to Beukers, Calabi and Kolk)
$$x=\frac{\sin u}{\cos v}$$
$$y=\frac{\sin v}{\cos u}$$
transforms the square domain $0\lt x\lt 1$ and $0\lt y\lt 1$ into the triangle domain $u,v>0,u+v<\pi /2$ (in *Proofs from the Book* by M. Aigner and G. Ziegler).
Since the invert transform... |
As the cosine is periodic, there are many values of $\theta$ which have the same $cos(\theta)$. So they are just asking for all the values between $-\pi/2$ and $\pi/2$ that solve the equation. Your solution got one of them, $cos(0.605)$ does equal $5/\sqrt(37)$. But so does $cos(-.605)$ You were supposed to find th... |
ok so I did all the revision problems and noted the ones I couldn't do today and Im posting them together, hope thats not a problem with the power that be?
I have exhibit A:
$e^{-x} -x + 2 $
So I differentiate to find where the derivative hits 0:
$-e^{-x} -1 = 0 $
Now HOW do I figure when this _hits_ zer... |
So this is some I partially understand, I'm not sure what I dont and do understand because most of my understanding is based on assumtions... sorry if I sound a little stupid!
The equation $6 \cos x - \sin x = 5 $ needs to be turned into the form $ \Re \cos (x + \alpha) $ then solved for **in the interval** $ -1/2\... |
As the cosine is periodic, there are many values of $\theta$ which have the same $\cos(\theta)$. So they are just asking for all the values between $-\pi/2$ and $\pi/2$ that solve the equation. Your solution got one of them, $\cos(0.605)$ does equal $5/\sqrt(37)$. But so does $\cos(-.605)$ You were supposed to find... |
So this is some I partially understand, I'm not sure what I dont and do understand because most of my understanding is based on assumtions... sorry if I sound a little stupid!
The equation $6 \cos x - \sin x = 5 $ needs to be turned into the form $ \Re \cos (x + \alpha) $ then solved for **in the interval** $ -1/2\... |
I need calculate ray (line) intersection with torus for my ray-tracing program (I know, its to graphics, but i need math behind it).
I can solve equation of order x^4, but thats too way slow (Cardano's method). So is there better way, how to calculate this ?
My torus has center [0,0,0] and R, r diameters
Line (R... |
This thing came up in a combinatorics course I am taking.
Choose a fixed set of primes $p_1,p_2,\dots,p_k$ and let $A_n$ be number of integers in $\\{1,2,\dots,n\\}$ which are not divisible by any of the $p_i$'s. $A_n$ is given by $ n - \sum_{1\leq i_1 \leq k} \lfloor \frac{n}{p_{i_1}} \rfloor + \sum_{1\leq i_1 < i_... |
What kind of speed are you looking for? In general, smart interval-based solvers work well for this sort of thing: the core idea is that by using interval arithmetic you can bound the values of the 'potential function' (the quartic polynomial whose roots you're trying to find) within any given interval, and then fast-... |
Let $r$, $s$ be positive real numbers and $\theta$, $\phi$ real numbers with $|\theta -\phi|<\pi$. Then an argument of $re^{i\theta}+se^{i\phi}$ lies between $\theta$ and $\phi$.
Can someone give a short, clean proof of the statement above that doesn't rely on geometric intuition? Of course the proof may use the not... |
If I have a doubly stochastic matrix, how can I find the set of all BFS?
http://en.wikipedia.org/wiki/Doubly_stochastic_matrix |
Suppose we observe 84 alchoholics with cirrhosis of the liver, of whom 29 have hepatomas, that is, a liver-cell carcinoma. Suppose we know, based on a large sample, that the risk of hepatoma among alcoholics without cirrhosis of the liver is 24%.
What is the probability that we observe exactly 29 alcoholics with cir... |
Parabola attributes - custom curve? |
Don Knuth's Volume 4, Fascicle 2, of *The Art of Computer Progamming* has a long section on generating all permutations. I found a draft [here][1] online.
Then, going from a permutation to a permutation matrix is fairly straightforward. For example, suppose you have the permutation 1342 of the numbers 1, 2, 3, and... |
Don Knuth's Volume 4, Fascicle 2, of *The Art of Computer Progamming* has a long section on generating all permutations, including algorithms for doing so. I found a draft [here][1] online.
Then, going from a permutation to a permutation matrix is fairly straightforward. For example, suppose you have the permutati... |
I need to come up with an equation for a curve but I have forgotten most of my calculus by this point. What I need very close to a parabola that opens up. I'm currently using y = (0.03*x)^2 but this does not have a fast enough slope on the ends.
I need the slope in the middle to be very gradual, and then near the en... |
I'm searching for an algorithm that forms a balanced (or quasi-complete) latin square, in which every element is a horizontal neighbor to every other element exactly twice, and a vertical neighbor to every other element exactly twice.
I've found one example (for **n = 5**), but am not clear about a couple of steps:
... |
I'm searching for an algorithm that forms a balanced (or quasi-complete) latin square, in which every element is a horizontal neighbor to every other element exactly twice, and a vertical neighbor to every other element exactly twice.
I've found one example (for **n = 5**), but am not clear about a couple of steps:
... |
When pricing a european option by monte carlo over 30 days for instance, what's the difference between one big 30 day jump vs 30 one day steps? |
The coordinate transformation (due to Beukers, Calabi and Kolk)
$$x=\frac{\sin u}{\cos v}$$
$$y=\frac{\sin v}{\cos u}$$
transforms the square domain $0\lt x\lt 1$ and $0\lt y\lt 1$ into the triangle domain $u,v>0,u+v<\pi /2$ (in *Proofs from the Book* by M. Aigner and G. Ziegler).
Since the invert transform... |
Would $y=\cot(3x+0.07)$ for $0\le x\le 1$ work?
![graph][1]
Or $y=-\frac{x-0.5}{(x+0.1)(x-1.1)}$?
![graph][2]
[1]: https://i.stack.imgur.com/hXtwi.png
[2]: https://i.stack.imgur.com/KspqB.png |
How does norm(A) <= p(A) + e imply norm(A) < 1 if p(A) < 1 ? |
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