## What does non-degeneracy of of the diffusion coefficient in the context of a SDE mean?

In the introduction of a paper I was reading the author writes without elaborating that

” In the case of a non-degenerate diffusion coefficient, Stroock and Varadhan , proved the existence of a unique weak solution for a SDE with bounded measurable drift , and bounded continuous diffusion coefficients. “

So if the SDE is $$dX_t=\mu(X_t)dt+\sigma(X_t)dB_t$$

So what does non-degeneracy of the diffusion coefficient mean here? Does it mean that $$\sigma(X_t)\sigma(X_t)^T$$ is invertible for all $$t\ge 0$$ almost surely? Or does it say something directly about the function $$\sigma$$?

## engineering derivations for lateral earth pressure coefficient on a retaining wall

my text book on soils engineering advises that the lateral earth pressure coefficient is determined by: (1-sin(x))/(1+sin(x)), however it also advises that by simple trigonometry this can also be written as tan^2(pi/4-(x)/2). how do you get from the first to equal the second using trigonometry?

## Derivation of correlation coefficient using bell curve

You can use python to add up all those distances between the data point and the line of best fit, given by the fact that -x/a is perpindicular to ax.

I plugged the sum of all the distances in to the gaussian function so that the f(infinity) –> 0 and f(0) –> 1. Thus the closer the data are to the line of best fit, the higher the correlation coefficient.

Although, I don’t know if this is the correct way to calculate coefficient

## How to compute the correlation coefficient?

The question is:

One package of potatoes contains 10 potatoes and weighs exactly 500 grams. Denote by $$X_{1}, \dots, X_{10}$$ the weights of each potato.

Are the random variables $$X_{1}, \dots, X_{10}$$ independent?

Compute the correlation coefficient of $$\rho(X, Y)$$ where $$X=X_{1}$$ and $$Y = \sum_{i=2}^{10} X_{i}$$

I know this formula $$\rho=\frac{cov(X,Y)}{\sigma_{X} \sigma_{Y}}$$ and that $$cov(X,Y)=E[XY] – E[X]E[Y]$$

So it seems that it is just to plug in the right values and compute. But Im not sure how to calculate $$E[X]$$ and $$E[Y]$$..

I think it is something along with: I know that $$E[X]=xf(x)$$ and here $$x=X_{1}$$ and $$f(x) = 1$$ soo this equals $$X_{1}$$? This is true (since this set only contains this potato so therefore we must always get it when we choose). But the answer should be a number, not a random variable…

The same goes for $$E[Y]$$.

I know from the solutions that the answer is: $$\rho(X,Y)=-1$$ and thus they are in correlation.

## How to Integrate Parametric Curves With Algebraic Coefficient?

I have researched all over the place to get answers to this question; albeit to no avail. Most information that I could find focused on simple sin / cosine values – as such I make this request for help.

I have this information:

x = acos3t , y = asint, 0 ≤ t ≤ pi/6 

I need to integrate this between limits of a and 0. I used the general formula for integrating parametrics albeit the answer seems to go on forever. I can’t seem to manipulate the answer to look like the original and then set that part to I (if that makes sense).

A photo of the question is attached if that helps. Check here.

Many thanks to anyone who helps.

## Fixed point iteration with the same power coefficient

I have a function where $$f(x) = x^3cos(x)-x^3/10$$, with that said, how do i find the fixed point iteration formula for it. I have tried adding an unknown to it and get the $$x$$ but it does not converge to a point.

## $q$-factorial coefficient asymptotics

Consider the $$[n]!_q = \prod\limits_{k = 1}^{n} \frac{q^k – 1}{q – 1} = \sum\limits_{k = 0}^{\binom n 2} c_k q^k$$ and let $$\{f_n\}_{n \in \mathbb{N}}$$ be the sequence of the functions on $$[0; 1]$$ defined by the following $$f_n(x) = \frac{c_{\lfloor \binom n 2 x \rfloor}}{n!}$$ Is there a formula for $$\lim\limits_{n \rightarrow \infty} f_n(x)$$? Roughly speaking, what is the limit distribution of its coefficients?

## Calculus: Finding an unknown coefficient in continuous equations

I am watching a video trying to understand continuity in calculus. The author first starts with the $$ax – 4, x < 1$$, and we are trying to find the value of “a”.

The author subsequently derives $$ax – 4$$ as $$7x – 4$$, where $$a = 7$$.

Why did he come to a conclusion where $$a$$ is $$7$$, where $$x < 1$$?

## HELP CALCULATING CORRELATION COEFFICIENT FOR MY DATA

I need help knowing if I am calculating the correlation coefficient correctly for my research. I have 3 sets of high order factors scores (one score/factor for each participant) which needs to be correlated with the pressure they felt to confess (rates on a scale from 1-10). How can this be calculated in excel or R? Supplied the data via picture. enter image description here

Instead of downvoting me help me.

## Vectors , finding coefficient

if (x,y,z) unequal to (0,0,0) and (i + j + 3k)x + ( 3i – 3j +k)y + (-4i + 5j)z = a(xi + yj + zk), then the value of a is

Attempt: I wrote $$(x+3y-4z) i + (x-3y+5z) j + (3x+y) k = a (x i + y j + z k)$$ Which gave me $$x+3y-4z=ax\ x-3y+5z=ay\ 3x+y=az$$ Now these are 4 unknown and only 3 equations.

Also, i dont know determinant , so please suggest me another way to solve it without using determinants.