Is the coefficient of variation valid for comparison of slopes between models (LMEM)?

I have collected some data in which measurements have been made at varying distances from an ‘origin’. Multiple ‘origins’ are measured within a ‘cluster’, so I am modeling the effect of distance using Linear Mixed Effect Models. In the lme4 package, the formula is essentially y~Dist+(1+Dist|Cluster/Origin).

Now, I have made different types of measurement at each point, and I would like to compare the models of different measurement types to get an idea for how they differ. For example, I compare the percentage change per distance between measurement types.

I have heard that the coefficient of variation is often used in chemistry to estimate repeatibility and that it can be used in comparison between models as it is scale invariant. I would like to compare the coefficients of variation of the estimated slope of each model. I would calculate it (quite simply) as:
$$ dfrac{s_b}{b} $$

Where $s_b$ is the estimated standard error in the slope, and $b$ is the estimated slope.

I have two main concerns that I would appreciate some help with:

  1. I have been unable to find examples of CV being used to compare slopes, as it is primarily used to compare means.
  2. This formula would ignore the random variations allowed by the model in the slope. Is this problematic and is there a good way of resolving it?

Chigorin variation for black and extra pawn

I played this as black against an opponent online, unfortunately the opponent resigned before I could see this game develop.

What do you guys think about this variation and has anyone analysed these particular moves?

1. Nf3 d5 2. d4 Nc6 3. Nc3 Nf6 4. Ne5 Bd7 5. e4 dxe4 (Queen's Pawn Game: Chigorin Variation)

Some computer analysis shows slight advantage for black (somewhere between -0.1 and -0.3) after taking on e4, so the advantage might be due to the extra pawn on the e-column.
Do you think black hold (protect) this pawn? Mostly I do not play the Chigorin that much, and I don’t think I have ever situated this in the fifth move.

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1. Nf3 d5 2. d4 Nc6 3. Nc3 Nf6 4. Ne5 Bd7 5. e4 dxe4

Sequence of Functions and Total Variation

Let $(f_{n})$ be a sequence of functions such that $f(x)=lim f_{n}(x)$ for every $xin[a,b]$. Show that $V_{f}(a,b)leq liminf V_{f_{n}}(a,b)$.

I’ll try to write what I have done so far…

As $V_{f}(a,b)=Sup{sum(f,P): Pinmathscr{P}([a,b])}$, then for all $epsilon>0$, there is a partition $P={x_{0}, x_{1},…,x_{m}}inmathscr{P}([a,b])$ such that $V_{f}(a,b)-epsilon/2.

As $(f_{n})$ be a sequence of functions such that $f(x)=lim f_{n}(x)$ for every $xin[a,b]$, then there is $Ninmathbb{N}$ such that if $Nleq n$, then $|f_{n}(x)-f(x)| for every $xin [a,b]$.

Thus if $N=max{ N_{0}, N_{1},…,N_{m}}leq n$ we have $V_{f}(a,b)-epsilon/2.

Hence if $Nleq n$, we have $V_{f}(a,b)

I don’t know what else to do …

Examples about bounded variation function

Problem. Give an example of a function with bounded variation $f:[0,1] →mathbb R$ with $f’$ integrable in $[0,1]$, and such that the function $g:[0,1]→mathbb R$ defined by
$$g(x)=f(x)-f(0)-int_0^xf'(s),mathrm ds$$
vanishes at the points $displaystyle x_n = sum_{j=1}^nfrac{1}{2^j}$, $g$ is positive on $(x_n,x_{n+1})$ if $n$ is odd and is negative if $n$ is even.

This example does not seem intuitive for me. How can I try to build this function? Is there any strategy for this?

In addition to the assumptions, the unique other restriction that I saw clearly: $$int_0^xf'(s),mathrm ds ≠ f(x) – f(0).$$ The only non-trivial function that I know that satisfies this is the Cantor-Lebesgue function, but this function does not work for this problem. So, I tried to start with a function that was not absolutely continuous, but I could not get an example.

What is a category of “Lepagean equivalent” or “variation problem”?

I get to know about it form Mark Gotay’s work An exterior differential system approach to the Cartan form, in that paper he defined the canonical Lepagean equivalent. The following is cited from it:

A canonical Lepagean equivalent of a variation problem $left( Mstackrel{pi}{longrightarrow} X, mathcal{I}, mathcal{L} right)$ is another variation problem $left( Wstackrel{rho}{longrightarrow} X, left{0 right}, Theta right)$, together with a surjective submersion $nu:W longrightarrow M$, such that: (1) $rho=picircnu$, and (2) if $gamma in Gamma left(rhoright)$ satisfies $nucircgammainGamma left(pi,mathcal{I}right)$, then

$gamma ^{*}Theta=left(nucircgammaright)^{*}mathcal{L}$

$Theta$ is a classical Lepagean equivalent of $mathcal{L}$ if

$Thetaequivnu^{*}mathcal{L} $ mod $nu^{*}mathcal{I}$

And in the paper he said that the assignment $left( Mstackrel{pi}{longrightarrow} X, mathcal{I}, mathcal{L} right) rightsquigarrow left( Wstackrel{rho}{longrightarrow} X, left{0 right}, Theta_{mathcal{}} right)$ is an affine functor from the category of constrained variation problems to the category of free ones. This is a welcome feature of the canonical Lepagean equivalent, as classical Lepagean equivalents are not functorial in general (why?).

I’m interested in the category feature of such variation problems, what are the objects, what are the arrows, is the functor invertible, why are classical Lepagean equivalents not functorial in general, is there any specified research about it?

Difference between creating a tablet layout variation and creating a folder for tablet layout

Good day, I would like to ask what’s the difference between creating a tablet layout variation and creating a folder for layouts (like: “res/layout-w600dp/” )?

I’m so sorry I’m new in making android app for small (cellular phones/android phones) together with large(tablets) devices.

Can anyone help me, please? I’m really having a trouble on what to do to have layout for tablets. I don’t know if the layout for tablets will be automatically displayed when app is run on tablet or I have to put some code in the java part. I’ve read some documentations but I can’t fully understand it. sorry.

Thank you for your help.

variation of thickness of dielectric layer for different textile-based capacitive pressure sensors

For my thesis I am working with textile-based capacitive pressure sensors and I am struggling to see how much variation in thickness of the dielectric layer you can have before you have noticeable impact on sensor output. If I wanted to have my sensors somewhere between 2×2 and 4×4 for the surface area, and I was using nylon 6,6 for my dielctric layer (dielectric constant of 3.4), and I was testing different construction methods for the nylon (different knits and wovens) and I wanted to keep them as close in thickness as possible, how much leeway is there before the variation in thickness would result in a measureable difference? Since I am using different construction methods for the dielectric layer there is no way I can make them EXACTLY the same thickness, so I am trying to figure out my wiggle room. If I have a dielectric layer that is 900 microns thick and another that is 400 microns thick, is that going to be way too much variation or is that relatively close enough?

Radial variation of atmospheric pressure in rotating O’Neill cylinder-like ship? (Rendezvous with Rama)

Sir Arthur C. Clarke was a science writer as well as a prolific writer of science fiction (including hard SF*), and his stories usually had a substantial footing in science.

His book Rendezvous with Rama describes a perfect hollow cylinder, 20 kilometers in diameter and 54 kilometers long, rotating at 0.25 rpm (versus the O’Neill cylinder at 8km diameter and ~0.5 rpm) to produce artificial gravity on the inside walls.

For a small rotating spacecraft the atmospheric pressure would be uniform, but in this case the diameter is of the order of a scale height on Earth!

Question: If the gas atmospheric pressure at the “surface” (the inner wall) were 1 standard atmosphere, how would the pressure vary moving towards the axis, and what would be the minimum pressure? What atmospheric pressure would be reasonable, and would the air rotate en masse at 0.25 rpm, or would rotational forces create shear or other effects, producing turbulent wind at the surface?

* Hard science fiction is a category of science fiction characterized by an emphasis on scientific accuracy or technical detail or both. The term was first used in print in 1957 by P. Schuyler Miller in a review of John W. Campbell, Jr.’s Islands of Space in Astounding Science Fiction Wikipedia

Using variation of parameters to solve $y”-25y=x$

I keep trying to solve this but I end up needing to do integration by parts like 3 or 4 times. My only question is, is that going to be the only way to do this? If so it will literally take me hours. I don’t need a solution, just a confirmation that if the requirement is to use variation of parameters, then will I need to take integrals that need to be done by IBP?

Thank you and sorry if this is not a good question.