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

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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.

$q$-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$ ?

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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.