I'm currently digging in 'Finite and Instantaneous Screw Theory in Robotic Mechanism' by T. Sun, there are some mathematical equations which I couldn't understand. (Seems it's simple vector problem)
The problem is, when two dual quaternion D_a, D_b is exist,
which D_a = D_sa + D_va where D_sa means scalar of D_a, and D_va is vector of D_a
When multiply D_b and D_a (D_b D_a), does answer become
D_b D_a = D_sa D_sb + D_sb D_va + D_sa D_vb + D_vb X D_va - D_va (dot) D_vb ?
I can understand until D_sa D_sb + D_sb D_va + D_sa D_vb, but not D_vb X D_va - D_va (dot) D_vb
((dot) means dot product, and X means cross product)
Oh, I just got the answer.
it was because of nature of quaternion
(a + bi + cj + dk) * (w + xi + yj + zk) = aw -(bx + cy + dz) + ( ....
Related
So I have been given the following expression, but I cannot seem to solve it, can anyone do this and show the steps please?
Prove XY'Z + XYZ' + XYZ = XY + XZ
XY'Z + XYZ' + XYZ = XY + XZ
Notice X and Z are common factors between XY'Z and XYZ.
XZ(Y' + Y) + XYZ' =
Y' + Y is equal to 1 (if Y=0 then Y'=1 and so 0 + 1 = 1, that is 0 or 1 = 1. Similarly, if Y=1 then Y'=0 and so 1 + 0 = 1). Therefore, what you get is:
XZ·1 + XYZ' =
XZ·1 = XZ since A·1 = A (if A=0 then 0·1 is 0 and if A=1 then 1·1 = 1). Now the function is simplified to:
XZ + XYZ' =
Notice once again X is a common factor between XZ and XYZ'.
X(Z + YZ') =
Notice this time that Z + YZ' is a special case of the distributive law, which is A + A'B = A + B. This is because if we apply the general distributive law A + BC = (A + B)·(A + C) then we get A + A'B = (A + A')·(A + B) = 1·(A + B) = A + B. Following this reasoning we get to simplify the function even further:
X(Z + Y) =
All that's left is for us to use the distributive law and we finally arrive to the final result:
XY + XZ
Please note that nothing is written between variables, an AND operator (or "·" symbol) is assumed. It's just a way to save space.
Is this recurrence relation O(infinity)?
T(n) = 49*T(n/7) + n
There are no base conditions given.
I tried solving using master's theorem and the answer is Theta(n^2). But when solving with recurrence tree, the solution comes to be an infinite series, of n*(7 + 7^2 + 7^3 +...)
Can someone please help?
Let n = 7^m. The recurrence becomes
T(7^m) = 49 T(7^(m-1)) + 7^m,
or
S(m) = 49 S(m-1) + 7^m.
The homogeneous part gives
S(m) = C 49^m
and the general solution is
S(m) = C 49^m - 7^m / 6
i.e.
T(n) = C n² - n / 6 = (T(1) + 1 / 6) n² - n / 6.
If you try the recursion method:
T(n) = 7^2 T(n/7) + n = 7^2 [7^2 T(n/v^2) + n/7] + n = 7^4 T(n/7^2) + 7n + n
= ... = 7^(2i) * T(n/7^i) + n * [7^0 + 7^1 + 7^2 + ... + 7^(i-1)]
When the i grows n/7^i gets closer to 1 and as mentioned in the other answer, T(1) is a constant. So if we assume T(1) = 1, then:
T(n/7^i) = 1
n/7^i = 1 => i = log_7 (n)
So
T(n) = 7^(2*log_7 (n)) * T(1) + n * [7^0 + 7^1 + 7^2 + ... + 7^(log_7(n)-1)]
=> T(n) = n^2 + n * [1+7+7^2+...+(n-1)] = n^2 + c*n = theta(n^2)
Usually, when no base case is provided for a recurrence relation, the assumption is that the base case is something T(1) = 1 or something along those lines. That way, the recursion eventually terminates.
Something to think about - you can only get an infinite series from your recursion tree if the recursion tree is infinitely deep. Although no base case was specified in the problem, you can operate under the assumption that there is one and that the recursion stops when the input gets sufficiently small for some definition of "sufficiently small." Based on that, at what point does the recursion stop? From there, you should be able to convert your infinite series into a series of finite length, which then will give you your answer.
Hope this helps!
T(1) = T(2) = 1, and for n > 2, T(n) = T(n − 1) + T(n − 2) + 3.
What Ive done so far:
T(n-1) = T(n-2) + T(n-3) + 3 + 3
T(n-2) = T(n-3) + T(n-4) + 3 + 3 + 3
T(n) = T(n-2) + 2T(n-3) + T(n-4) + 3 + 3 + 3 + 3 + 3
T(n) = T(1) + 2T(2) + T(n-4) + 3(n + 2)
Im not sure if this is right, and if it is, how do I get rid of T(n-4).
These types of recurrences are tricky, and the repeated expansion method will unfortunately get you nowhere. Observing the recursion tree will only give you an upper bound, which is often not tight.
Two methods I can suggest:
1. Substitution + Standard Theorem
Make the following variable substitution:
This is in the correct form for the Akra-Bazzi method, with parameters:
2. Fibonacci formula
The Fibonacci series has an explicit formula which can be derived by guessing a solution of the form Fn = a^n. Using this as an analogy, substitute a similar expression for T(n):
Equating the constant and exponential terms:
Take the positive root because the negative root has absolute value less than 1, and will therefore decay to zero with increasing n:
Which is consistent with (1).
I'm interested in defining the following polynomial quotient ring in some CAS (Singular, GAP, Sage, etc.):
R = GF(256)[x] / (x^4 + 1)
Specifically, R is the set of all polynomials of degree at most 3, whose coefficients belong to GF(256). Two examples include:
p(x) = {03}x^3 + {01}x^2 + {01}x + {02}
q(x) = {0B}x^3 + {0D}x^2 + {09}x + {0E}
Addition and multiplication are defined as the per ring laws. Here, I mention them for emphasis:
Addition: The corresponding coefficients are XOR-ed (the addition law in GF(256)):
p(x) + q(x) = {08}x^3 + {0C}x^2 + {08}x + {0C}
Multiplication: The polynomials are multiplicated (coefficients are added and multiplicated in GF(256)). The result is computed modulo x^4 + 1:
p(x) * q(x) = ({03}*{0B}x^6 + ... + {02}*{0E}) mod (x^4 + 1)
= ({03}*{0B}x^6 + ... + {02}*{0E}) mod (x^4 + 1)
= ({1D}x^6 + {1C}x^5 + {1D}x^4 + {00}x^3 + {1D}x^2 + {1C}x + {1C}) mod (x^4 + 1)
= {01}
Please tell me how to define R = GF(256)[x] / (x^4 + 1) in a CAS of your choice, and show how to implement the above addition and multiplication between p(x) and q(x).
I wish to plot an ellipse by scanline finding the values for y for each value of x.
For a plain ellipse the formula is trivial to find: y = Sqrt[b^2 - (b^2 x^2)/a^2]
But when the axes of the ellipse are rotated I've never been able to figure out how to compute y (and possibly the extents of x)
In parametric form
x[t]= a Cos[t] Cos[psi] - b Sin[t] Sin[psi]
y[t]= b Cos[psi] Sin[t] + a Cos[t] Sin[psi]
Where psi is the rotation angle, and a and b the semi-axes.
The parameter t goes from 0 to 2 Pi.
Or if you prefer in Cartesian non-parametric form:
(a x^2+b y^2) Cos[psi]^2 + (b x^2 +a y^2) Sin[psi]^2 + (a-b) x y Sin[2 psi]==1
Which yields to the two possible solutions for y[x], equivalent to the two solutions for the square root in the non-rotated case:
y -> (-(Sqrt[2]*Sqrt[a + b - 2*a*b*x^2 + (-a + b)*Cos[2*psi]]) +
(-a + b)*x*Sin[2*psi]) / (2*(b*Cos[psi]^2 + a*Sin[psi]^2))
y -> (Sqrt[2]*Sqrt[a + b - 2*a*b*x^2 + (-a + b)*Cos[2*psi]] +
(-a + b)*x*Sin[2*psi])/ (2*(b*Cos[psi]^2 + a*Sin[psi]^2))
Well, you asked for it :)
Those functions give:
And the limits for x are:
LimitX= +/- Sqrt[a + b + (-a + b)*Cos[2*psi]]/(Sqrt[2]*Sqrt[a]*Sqrt[b])