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 continuous-time 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:

























{\displaystyle E[B_{H}(t)B_{H}(s)]={\tfrac {1}{2}}(|t|^{2H}+|s|^{2H}-|t-s|^{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: n-th order fractional Brownian motion, abbreviated as n-fBm. n-fBm is a Gaussian, self-similar, non-stationary process whose increments of order n are stationary. For n = 1, n-fBm 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.

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  1. A

    Powers with fractional exponents that have an even denominator

    (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 hand-written version of this...
  2. hackedagainanda

    Why do fractional exponents result in the square root operator?

    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...
  3. E

    B Resolving index laws for fractional exponents

    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...
  4. M

    Intuitive explanation of fractional exponents?

    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...
  5. S

    Multiplying Algebraic Fractional Exponents

    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!
  6. Cjosh

    Pulling fractional exponents out of an expression

    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...
  7. P

    Binomial expansion for fractional power

    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...
  8. B

    Doing Fractional Exponents on Basic Scientific Calculator

    Suppose you have 8-1/3 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."
  9. snowjoe

    How far apart are two point charges....

    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 * 75e-10 squared/r squared r2= 9e10 * 75e-10 squared/1N r2= 9*75*75e-10 r2= 5.0625e-7 r= square root of 5.0625 * square...
  10. Greg Bernhardt

    Challenge 23: Fractional exponents

    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.
  11. B

    Fractional Exponents (How is it done?)

    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...
  12. D

    Add, sub, multiply, and dividing w/ fractional exponents & radicals

    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...
  13. H

    Solving nonlinear first order DE w/ fractional exponents

    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...
  14. G

    Solve Fractional Exponents: Logic Explained

    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...
  15. P

    Solving Equations with Fractional Exponents

    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...
  16. DaveC426913

    Fractional exponents of negative

    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...
  17. D

    Factoring Algebraic Expressions with Fractional Exponents

    Homework Statement (4x-1)^{1/2}-1/3(4x-1)^{3/2}Homework Equations The Attempt at a Solution I think the GCF is (4x-1)^{1/2}. So, I get (4x-1)^{1/2}(1+(-1/3(4x-1))) = (4x-1)^{1/2}(-4/3x+4/3) = -4/3(4x-1)^{1/2}(x-1) However, the answer in the book is 4/3(4x-1)^{1/2}(x-1). I've done it several...
  18. J

    Binomial theorem for fractional exponents?

    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!
  19. DaveC426913

    Fractional exponents of negative numbers?

    I was just playing around in my head. I wanted to plot this graph: y=x^2.5; x=-2 This is valid right? My calc says it's invalid input.