What is Fractional exponents: Definition and 19 Discussions
In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuoustime Gaussian process BH(t) on [0, T], that starts at zero, has expectation zero for all t in [0, T], and has the following covariance function:
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{\displaystyle E[B_{H}(t)B_{H}(s)]={\tfrac {1}{2}}(t^{2H}+s^{2H}ts^{2H}),}
where H is a real number in (0, 1), called the Hurst index or Hurst parameter associated with the fractional Brownian motion. The Hurst exponent describes the raggedness of the resultant motion, with a higher value leading to a smoother motion. It was introduced by Mandelbrot & van Ness (1968).
The value of H determines what kind of process the fBm is:
if H = 1/2 then the process is in fact a Brownian motion or Wiener process;
if H > 1/2 then the increments of the process are positively correlated;
if H < 1/2 then the increments of the process are negatively correlated.The increment process, X(t) = BH(t+1) − BH(t), is known as fractional Gaussian noise.
There is also a generalization of fractional Brownian motion: nth order fractional Brownian motion, abbreviated as nfBm. nfBm is a Gaussian, selfsimilar, nonstationary process whose increments of order n are stationary. For n = 1, nfBm is classical fBm.
Like the Brownian motion that it generalizes, fractional Brownian motion is named after 19th century biologist Robert Brown; fractional Gaussian noise is named after mathematician Carl Friedrich Gauss.
(If I should have posted this in the Math thread instead of the Homework thread, please let me know.)
I have three questions which I will ask in sequence. They all relate to each other.
I've typed my questions and solutions attempts below.
I've also attached a handwritten version of this...
So I got the answer through a little addition i.e 9^(1/2) multiplied by 9^(1/2) = 9^1 or 9
3 x 3 = 9 so 3 is the answer to what is 9^(1/2)
I've tested this out with a few other numbers and have made this generalization, x^(1/2) = √x
It seems to make the equations orderly and consistent but is...
I was just thinking about this earlier and couldn't come up with a good enough resolution. I'm guessing it's a matter of convention more than anything. If we have ##x^{2} = a##, taking the principle root of both sides gives ##\sqrt{x^{2}} = \sqrt{a} \implies x = \sqrt{a}##.
Yet evidently if...
Homework Statement
What would have caused humans to come up with fractional exponent notations?
Homework EquationsThe Attempt at a Solution
I understand that it makes sense to use the exponent notation when we have to multiply the same number a number of times. For example, 10^8 is the short...
Homework Statement
a3/2a5/4
Homework EquationsThe Attempt at a Solution
I'm hoping you can help. My solution to this problem would be:
a3/2+5/4=a8/6=a4/3
But the answer in the back of my book is given as a11/4
I'm confused!
Homework Statement
Find critical numbers of the function: F(x)=t^3/4  2t^1/4
Derivative I got: F'(x)=3/4 t^1/4  1/2 t^3/4
Homework EquationsThe Attempt at a Solution
I have found the derivative and I understand I must pull out a t in order to find critical numbers, and run across this...
Homework Statement
So, I'm solving a dipole thing and I have these vectors:
r + d  r' = (r² + d²  r'²)(1/2)
Homework Equations
I want to expand this but I have no idea how! I know I may have an infinite power series, but I may expand at the square terms tops...
Before I needed to do the...
Suppose you have 81/3 and want a precise value for it. How would you go about calculating this on a regular scientific calculator.
I punched in:
8, then the exponent button, then 1, then negative, then division, and finally 3. The calculator reads "error."
1. How far apart must two point charges of 75.0 nC be to have a force of 1.00 N between them?Homework Equations F = k Q1Q2/r2[/B]3. 1N = 9e10 * 75e10 squared/r squared
r2= 9e10 * 75e10 squared/1N
r2= 9*75*75e10
r2= 5.0625e7
r= square root of 5.0625 * square...
With only only paper & pencil (no calculator or logarithmic tables), figure out which of the following expressions has a greater value: 101/10 or 31/3.
Please make use of the spoiler tag and write out your full explanation, not just the answer.
How does 2^5/2 become 2^2 multiplied by 2^1/2?
(The '^' means 'to the power of' so 2 to the power of 5/2. I am not sure how to write this as an exponent as this is my first post.)
2^5/2 = 2^2 × 2^1/2
So 2^2 = 4 and 2^1/2 means Square Root so there is a radical sign, so it becomes √2.
I...
Okay so I'm in Calculus 1 and we are working on derivatives. I understand it all but I have been having some trouble with some basic math skills that I cannot remember from high school and I can't seem to find a good tutorial anywhere online.
I am having problems with multiplying fractional...
Hello. I have simple DE
y' + p y^(1/2) = q

y'=dy/dt
p,q=constant
I am confused because I tried bernoulli's method to solve and I think I exploded the universe.
Basically, my initial condition of t=0,y=0 made infinity, not right. I'm not sure that method works when there...
Can someone explain the logic behind this?
For instance if 2 to the 3rd power = 2 x 2 x 2 =8
So 2 to the 3rd power is telling me I have 2 multiplied by itself 3 times.
Now how would I solve for 2 to the 1/3rd power? It is telling me I have 2 multiplied by itself 1/3 times but how do you...
The question is x2/3  x1/3  2 = 0
So the first thing I did was: x2/3  x1/3 = 2
Then, I put both sides to the power of three, so I got:
x2  x1 = 8
From there I factorized: x (x  1) =8
And got the answers: x = 8 or x = 9, the book however, says the correct answers are 1 and 8.
Any...
I've been toying around with stuff I probably shouldn't be. :biggrin:
I've been sketching a graph of y=x^n where n is a rational number (as opposed to an integer).
Of course, when I get into the fractional exponents, the negative portion of the curve ends up being imaginary (eg. x=2,n=2.5...
Homework Statement
(4x1)^{1/2}1/3(4x1)^{3/2}Homework Equations
The Attempt at a Solution
I think the GCF is (4x1)^{1/2}. So, I get (4x1)^{1/2}(1+(1/3(4x1))) = (4x1)^{1/2}(4/3x+4/3) = 4/3(4x1)^{1/2}(x1)
However, the answer in the book is 4/3(4x1)^{1/2}(x1). I've done it several...
I am curious, is there any way to use the binomial theorem for fractional exponents? Is there any other way to expand a binomial with a fractional exponent?
I suppose Newton's theorem is not a way since it requires factorials.
Thanks!