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# The Hypnotic World of Degenerate Spirals

Recently, I was looking for a way to add a spiral to my website1. My search led me to this CodePen, which I found fascinating. For one, the code for simulating the spiral was very simple. But what really caught my attention was what happened when I started playing with the provided "angle" slider. Incredibly complex, intricate shapes were being drawn in front of my eyes from a few simple lines of code, and they didn't even look like spirals after some point.

I fell down an enjoyable rabbit hole trying to understand what was actually going on, and then I ended up finding and playing around with different types of spirals to see what other effects I could observe. I'd love to share all my findings with you, but if you don't care for the details then feel free to jump directly to the visualisations!

## What is a Spiral?

In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.

I think it's useful to start by looking at a simple circle. Even though we usually think of circles as static, we could also think of them as curves revolving around a point, always staying at the exact same distance to that point.

As the radius $$r$$ is kept constant while the angle $$\theta$$ keeps increasing, this motion traces a perfect circle after $$360^{\circ}$$ and goes on to loop over itself forever.

When we look at a circle from this perspective, the only difference of a spiral is that the radius is not kept constant, but changes according to some function of the angle: $r = f(\theta)$ Different functions lead to different spirals, some of which we will explore later on. For now, let's start with the simplest one, the Archimedean Spiral: $r = a\theta$

In this example, I chose $$a = 0.1$$ so the radius is always equal to $$\theta / 10$$ (with $$\theta$$ given in degrees). Changing this parameter does not change the fundamental shape of the spiral, it only changes the distance between consecutive loops.

To change the shape of the spiral, we need to change the relation between the angle and the radius. For example, Fermat's Spiral has loops that get closer to each other as the curve gets farther from the center: $r = a \sqrt{\theta}$

With a small change, we can even get a spiral converging towards a point instead of diverging away from one, such as a lituus: $r = \frac{a}{\sqrt{\theta}}$

## How to Draw a Spiral

There are multiple ways of drawing a spiral using code, and you can see different implementations in many languages at Rosetta Code. We are interested in the approach using line segments, since that is the one that will lead to those interesting patterns at the start of this post.

Let's say we want to draw a spiral that has completed 5 full rotations, so $$\theta$$ has gone from $$0^\circ$$ to $$1800^\circ$$. We are going to approximate this curve by picking lots of points that would be on this spiral and connecting them with straight line segments.

This is basically how I've been drawing all the spirals so far, but without showing you the actual points. To create an animation of an expanding spiral, all you need to do is increase $$\theta$$ a little every frame and draw the corresponding spiral2.

In the above example, I chose to pick a point every $$5^\circ$$ from $$0^\circ$$ to $$1800^\circ$$, which I think creates a pretty reasonable approximation. Of course, if we decrease this gap we will pick more points and get a better approximation, and similarly if we increase this gap we will get a worse approximation.

If you played around a bit with the slider above, you might already see where we're going with this.

## Spiraling out of Control

What if, instead of committing to picking points at regular intervals (like every $$5^\circ$$), we limit ourselves to approximate the spiral with a fixed number of points (say, 100) no matter how big the spiral gets. In this case, as the spiral expands, the few points have to spread apart further and further to cover the entire shape, making our approximation progressively worse.

And there we are! Now we just have to choose some arbitrary number of points, let $$\theta$$ grow and watch our spiral degenerate slowly into these cool, intricate patterns as our approximation fails spectacularly.

In the next section, I will remove the explanatory elements like the coordinate axes and the line/point showing the end of the spiral, so you can enjoy the hypnotic visualisations of what I'm now calling degenerate spirals.

## Hypnotic Visualisations

Archimedean Spiral

Fermat's Spiral

Lituus

## Bonus: The Second Branch (and Beyond)

I've only been drawing a single branch of these spirals, in fact there are more! We obtained the spirals so far by starting at $$\theta = 0$$ and slowly increasing it; we can get another one by moving in the other direction. Here are both branches of the Archimedean Spiral and their romantic dance, starting with a small heart:

This actually adds a whole new layer of beautiful complexity to our degenerate spirals, leading to lots of flowery patterns!

There is another way of adding a secondary branch to a spiral, and that is by drawing its reflection with respect to the origin. Here is an example using Fermat's spiral:

Of course, you don't have to choose one or the other, you can have four branches at the same time. So here, to end this post, are all four branches of a lituus gracefully degenerating into an end mark.

## Final Remarks

If you'd like to dive deeper into the world of spirals, the Wikipedia article is a good place to start with lots of general information and links to different types of spirals. All visualisations on this page were made with the amazing p5.js library. If you'd like to play around with code, I've made a template degenerate spiral sketch on the p5.js Web Editor that you can edit and run! If you would instead prefer to look at the code of the sketches here, check out the source for this post.

The format of this post is inspired by the Explorable Explanations movement, which I enjoy so much. If you liked this at all, you'll probably love some of the posts there. This is the first time I'm exploring and explaining a topic in this way, so if you have any feedback (or just want to say hi), feel free to write to me!

#### Footnotes

1. As one does. ↩

2. If you are just trying to draw a proper spiral that keeps expanding, it is obviously inefficient to redraw the entire spiral every frame; you should just add a few new line segments to the end at every frame. However, we are not trying to draw proper spirals and redrawing every frame is essential for the effect we are pursuing here. ↩


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Astrometric Reductions in CCD Scan Observing
Session 73 -- Astrometric Instruments and Results
Display presentation, Friday, January 14, 9:30-6:45, Salons I/II Room (Crystal Gateway)

## [73.05] Astrometric Reductions in CCD Scan Observing

R.C. Stone (U.S. Naval Obs., Flagstaff Station)

CCD scan" observing is a popular method for observing large regions of the sky in a rapid manner, and the ad hoc assumption is often made that the transit of a star across the chip can be well represented by uniform and rectilinear motion. Although this is a good assumption near the celestial equator, it fails at higher declinations and in a manner dependent on the scale of the telescope, the physical size of the CCD, and by the declination being observed. Standard astrometric reductional techniques are not appropriate in diurnal scan observing, since the projection of the sky onto the CCD is not spherical. Methods for reducing CCD scan data have been developed at the U.S. Naval Observatory at Flagstaff for high precision astrometry using numerical techniques to model the distortions inherent in scan observing and tested on the observatory's 20-cm transit telescope. Scan observations have been made of regions at declinations $\delta$ $\sim$ 0$^o$, 44$^o$, and 70$^o$ with accurately known star positions. With the corrections applied, there is very good agreement ($\sigma \sim \pm$0.1 arcsec) between these positions and those determined with the transit telescope. Without the corrections applied, the agreement is very poor for the higher declinations. This paper describes these techniques and predicts the sizes of these distortions, based on model results, for a wide range of telescope scales, CCD dimensions, and declinations being observed. \vfill\eject\bye

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Episode #125 of the Stack Overflow podcast is here. We talk Tilde Club and mechanical keyboards. Listen now

# Tag Info

3

Quite simple approach is the scattering model, which considers reflected and transmitted wave as a scattering pattern by the dipoles induced in the second medium. This model is originally due to Sagnac, but it has been included in a number of textbooks. A good resource for this model is a paper Doyle, W. T. (1985). Scattering approach to Fresnel’s equations ...

1

Not sure I understand what exactly you mean by timing the measurement. If we reduce the frequency enough so that we are dealing with low frequency radio waves, then one can measure the direction of the field based on the force it would exert on a charged particle. However, for light, the frequency is too high to do that. The integration time of a photo ...

1

To properly describe the polarization state of a photon, it is useful to introduce some concepts of quantum optics. The polarization is in fact one of the possible degrees of freedom of a photon. As a matter of convenience, we can assume as a base for the polarization state of the photons the one formed by two orthogonal axis (e.g. an horizontal axis and a ...

1

The wording of the question is problematical. A vertically polarized photon can pass only once through a polarizer, so it makes no sense to ask about "half of the time." Here's what quantum mechanics says: Consider an experiment in which a single photon is prepared in a state of vertical polarization. The photon interacts with a diagonal (45 degrees ...

1

The Dirac spinor can be interpreted in terms of left/right chiral spinors because the Dirac representation is a reducible representation of the Lorentz group, $$(\tfrac12,0)\oplus (0,\tfrac12)\tag1$$ On the other hand, the vector representation of the Lorentz group is irreducible, $$(\tfrac12,\tfrac12)\tag2$$ and, as such, no analogous interpretation is ...

Only top voted, non community-wiki answers of a minimum length are eligible


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# Revs per min from metres per second

by Taryn
Tags: metres, revs
 P: 66 okay so here is the problem and here is wat I did! I am just tryin to study up for exams now... and this is one of the problems! A sample of blood is placed in a centrifuge of radius 17.5 cm. The mass of a red corpuscle is 3.00×10-16 kg, and the magnitude of the force required to make it settle out of the plasma is 4.08×10-11 N. At how many revolutions per second should the centrifuge be operated? Basically wat I did is found the velocity in metres per second first which I find is 154m/s which I found by usin F=(mv^2)/r But here is the simple problem that I am confused about... how do I change this to revs/sec. Is it somethin to do with C=2PIr, thats all I can think of! Thanks for your help!
Emeritus
PF Gold
P: 9,772
 Quote by Taryn Basically wat I did is found the velocity in metres per second first which I find is 154m/s which I found by usin F=(mv^2)/r But here is the simple problem that I am confused about... how do I change this to revs/sec. Is it somethin to do with C=2PIr, thats all I can think of! Thanks for your help!
You could do it that way, however it is easier to remember that if we resolve newton's second law radially we achieve;

$$F = m\alpha$$

Where alpha is centripetal acceleration and $\alpha = r\omega^2$ where omega is angular velocity (rads/s). Thus;

$$\fbox{F = mr\omega^2}$$

You method is completely valid, but is a bit long winded . You can convert radians per second into revolutions per second by dividing by $2\pi$.
 P: 66 okay I did that and now I am so far off... I did 123686.127rev/sec. so wat I did was w^2=f/mr but then I got that in rads per second =7.77E5 which is bigger then I expected. So I then divided by 2*PI... then it was wrong... the answer is meant to be 140.3
Emeritus
PF Gold
P: 9,772
Revs per min from metres per second

 Quote by Taryn okay I did that and now I am so far off... I did 123686.127rev/sec. so wat I did was w^2=f/mr but then I got that in rads per second =7.77E5 which is bigger then I expected. So I then divided by 2*PI... then it was wrong... the answer is meant to be 140.3
You forgot to square root the 7.77x105. If you square root that value then divide by $2\pi$ you sould obtain the correct answer.
 P: 66 thanks a lot I appreciate it, I thought I did sqaure root the answer but now I got the right answer.! ;P
Emeritus
PF Gold
P: 9,772
 Quote by Taryn thanks a lot I appreciate it, I thought I did sqaure root the answer but now I got the right answer.! ;P
No problem
HW Helper
P: 6,679
 Quote by Hootenanny You could do it that way, however it is easier to remember that if we resolve newton's second law radially we achieve; $$F = m\alpha$$ Where alpha is angular acceleration and $\alpha = r\omega^2$ where omega is angular velocity (rads/s). Thus; $$\fbox{F = mr\omega^2}$$
I am sure you meant to say that the quantity $r\omega^2 = v^2/r$ is the centripetal acceleration not angular acceleration.

AM
Emeritus
 Quote by Andrew Mason I am sure you meant to say that the quantity $r\omega^2 = v^2/r$ is the centripetal acceleration not angular acceleration. AM


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# cfdm.CoordinateReference.equals¶

CoordinateReference.equals(other, rtol=None, atol=None, verbose=None, ignore_type=False)[source]

Whether two coordinate reference constructs are the same.

Equality is strict by default. This means that:

• the datum and coordinate conversion components must have the same string and numerical parameters.

The dimension coordinate, auxiliary coordinate and domain ancillary constructs of the coordinate reference constructs are not considered, because they may only be correctly interpreted by the field constructs that contain the coordinate reference constructs in question. They are, however, taken into account when two fields constructs are tested for equality.

Two real numbers x and y are considered equal if |x-y|<=atol+rtol|y|, where atol (the tolerance on absolute differences) and rtol (the tolerance on relative differences) are positive, typically very small numbers. See the atol and rtol parameters.

Any type of object may be tested but, in general, equality is only possible with another coordinate reference construct, or a subclass of one. See the ignore_type parameter.

NetCDF elements, such as netCDF variable and dimension names, do not constitute part of the CF data model and so are not checked.

New in version (cfdm): 1.7.0

Parameters
other:

The object to compare for equality.

atol: number, optional

The tolerance on absolute differences between real numbers. The default value is set by the cfdm.atol function.

rtol: number, optional

The tolerance on relative differences between real numbers. The default value is set by the cfdm.rtol function.

verbose: int or str or None, optional

If an integer from -1 to 3, or an equivalent string equal ignoring case to one of:

• 'DISABLE' (0)

• 'WARNING' (1)

• 'INFO' (2)

• 'DETAIL' (3)

• 'DEBUG' (-1)

set for the duration of the method call only as the minimum cut-off for the verboseness level of displayed output (log) messages, regardless of the globally-configured cfdm.log_level. Note that increasing numerical value corresponds to increasing verbosity, with the exception of -1 as a special case of maximal and extreme verbosity.

Otherwise, if None (the default value), output messages will be shown according to the value of the cfdm.log_level setting.

Overall, the higher a non-negative integer or equivalent string that is set (up to a maximum of 3/'DETAIL') for increasing verbosity, the more description that is printed to convey information about the operation.

ignore_type: bool, optional

Any type of object may be tested but, in general, equality is only possible with another CoordinateReference instance, or a subclass of one. If ignore_type is True then cfdm.CoordinateReference(source=other) is tested, rather than the other defined by the other parameter.

Returns
bool

Whether the two coordinate reference constructs are equal.

Examples:

>>> c = cfdm.CoordinateReference(
...     coordinates=['dimensioncoordinate0']
... )
>>> c.equals(c)
True
>>> c.equals(c.copy())
True
>>> c.equals('not a coordinate reference')
False


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# Problem of the Week #79 - September 30th, 2013

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#### Chris L T521

##### Well-known member
Staff member
Thanks again to those who participated in last week's POTW! Here's this week's problem!

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Problem: If the ellipse $x^2/a^2 + y^2/b^2 = 1$ is to enclose the circle $x^2+y^2=2y$, what values of $a$ and $b$ minimize the area of the ellipse?

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Remember to read the POTW submission guidelines to find out how to submit your answers!

#### Chris L T521

##### Well-known member
Staff member
This week's question was correctly answered by anemone (you just made it in at the last minute ) and MarkFL. I'm going to post both of their fine solutions below.

anemone's solution:
Since the ellipse is to enclose the circle, we know the ellipse will intersect the circle for only one value of $x$(this means also for two different $y$ values since the ellipse is to enclose the circle), that is, if we substitute the equation of the ellipse into the equation of the circle, the discriminant of the quadratic equation that we got must be zero.

So, if we substitue $$\displaystyle x^2=\left( 1-\frac{y^2}{b^2} \right) \left(a^2 \right)$$ into the equation $$\displaystyle x^2+y^2=2y$$, we get

$$\displaystyle x^2+y^2=2y$$

$$\displaystyle \left( 1-\frac{y^2}{b^2} \right) \left(a^2 \right)+y^2=2y$$

$$\displaystyle (b^2-y^2)a^2 +b^2y^2=2b^2y$$

$$\displaystyle (b^2-a^2)y^2 -2b^2y+a^2b^2$$

Set the discriminant equals to zero we get

$$\displaystyle (-2b^2)^2-4(b^2-a^2)(a^2b^2)=0$$

$$\displaystyle 4b^2(b^2-a^2(b^2-a^2)=0$$

$$\displaystyle 4b^2(b^2-a^2b^2+a^4)=0$$

Since $$\displaystyle b^2 \ne 0$$, it must be $$\displaystyle b^2-a^2b^2+a^4=0$$ or $$\displaystyle b^2=\frac{a^4}{a^2-1}$$

Now, bear in mind that we're asked to find the values of [FONT=MathJax_Math]a[/FONT] and [FONT=MathJax_Math]b[/FONT] so that the area of the ellipse is at its minimum, this simply means we need to find an equation the defines the area of the ellipse(A) in terms of $a$ and the area of the ellipse could be found using calculus technique. Since the ellipse has 4 symmetric quarter, we can find the area of any one quarter and multiply it by 4 in order to get the total area.

$$\displaystyle A=4\int_0^b \frac{a}{b}\sqrt{b^2-y^2} dy$$ [SIZE=+0]
[/SIZE]

We now make the substitution $$\displaystyle y=b\sin \theta$$ so that $$\displaystyle dy=b\cos \theta d \theta$$and the area is now given by

$$\displaystyle A=4\frac{a}{b}\int_0^{\frac{\pi}{2}}(\sqrt{b^2-b^2\sin^2 \theta}) b\cos \theta d \theta$$ [SIZE=+0]
[/SIZE]

$$\displaystyle A=4\frac{a}{b}(b^2)\int_0^{\frac{\pi}{2}} \cos^2 \theta d \theta$$ [SIZE=+0]
[/SIZE]

$$\displaystyle A=4ab\int_0^{\frac{\pi}{2}} \left( \frac{\cos 2\theta}{2}+\frac{1}{2} \right) d \theta$$ [SIZE=+0]
[/SIZE]

$$\displaystyle A=4ab \left( \frac{\sin 2\theta}{4}+\frac{\theta}{2} \right)_0^{\frac{\pi}{2}}$$ [SIZE=+0]
[/SIZE]

$$\displaystyle A=4ab \left( 0+\frac{\pi}{4}-0 \right)$$ [SIZE=+0]
[/SIZE]

$$\displaystyle A=ab\pi$$[SIZE=+0]
[/SIZE]

or $$\displaystyle A^2=a^2b^2\pi^2=a^2(\frac{a^4}{1-a^2})\pi^2 =\left(\frac{a^6}{a^2-1} \right)\pi^2$$

To minimize A, we differentiate the equation of $A$ with respect to $a$ and obtain

$$\displaystyle 2A\frac{dA}{da}=\left(\frac{(1-a^2)(6a^5)-(a^6)(-2a)}{(1-a^2)^2} \right)\pi^2$$

$$\displaystyle 2A\frac{dA}{da}=\frac{a^5\pi^2(4a^2-6)}{(1-a^2)^2}$$

Since $A \ne 0$, $$\displaystyle \frac{dA}{da}=0$$ is possible iff $$\displaystyle a^2=\frac{6}{4}=\frac{3}{2}$$ and since $a>0$, we must have $$\displaystyle a=\sqrt{\frac{3}{2}}$$ and when $$\displaystyle a=\sqrt{\frac{3}{2}}$$, $$\displaystyle b^2=\frac{(\frac{3}{2})^2}{\frac{3}{2}-1}=\frac{9}{2}$$ which gives $$\displaystyle b=\frac{3}{\sqrt{2}}$$.

Now, to determine the nature of the inflexion point where $(a, b)=\left(\sqrt{\frac{3}{2}}, \frac{3}{\sqrt{2}} \right)$ gives the minimum/maximum area of the ellipse, we consider the gradient of the graph of $A$ versus $a$ from left to right and we see that

 a 1.1 $\sqrt{\dfrac{3}{2}}\approx 1.2247$ 2 A 9.12 0 33.5 $\dfrac{dA}{da}$ -17.2 0 5.23 sign of $\dfrac{dA}{da}$ - 0 + change of slope of tangent \ _ /

Therefore, we can conclude that the minimal area of the ellipse is attained at the point $(a, b)=\left(\sqrt{\frac{3}{2}}, \frac{3}{\sqrt{2}} \right)$.

MarkFL's solution:
First, we want to define our objective function $A(a,b)$, the area of the ellipse, in terms of the parameters $a$ and $b$:

We know the ellipse is centered at the origin, and will pass through the points $(0,b),\,(a,0)$ and so by symmetry, we may state:

$$\displaystyle A(a,b)=4b\int_0^a \sqrt{1-\left(\frac{x}{a} \right)^2}\,dx$$

using the substitution:

$$\displaystyle \frac{x}{a}=\sin(\theta)\,\therefore\,dx=a\cos( \theta)\,d \theta$$

we now have:

$$\displaystyle A(a,b)=4ab\int_0^{\frac{\pi}{2}}\sqrt{1-\sin^2(\theta)}\cos(\theta)\,dx$$

$$\displaystyle A(a,b)=4ab\int_0^{\frac{\pi}{2}}\cos^2(\theta)\,dx$$

Using a double-angle identity for the cosine function, we obtain:

$$\displaystyle A(a,b)=2ab\int_0^{\frac{\pi}{2}}1+\cos(2\theta)\,dx$$

Applying the FTOC, there results:

$$\displaystyle A(a,b)=2ab\left[\theta+\frac{1}{2}\sin(2\theta) \right]_0^{\frac{\pi}{2}}=\pi ab$$

Thus, our objective function is:

$$\displaystyle A(a,b)=\pi ab$$

Now, for the constraint, lets begin with the equations of the circle and the ellipse:

$$\displaystyle x^2+y^2=2y$$

$$\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$

Obviously there will have to be at least one point where the two curves are tangent. At these tangent points, the slope of the two curves will have to be the same as well.

If we multiply the second equation by $-a^2$ and then add the two equations, we eliminate $x$ to obtain:

$$\displaystyle y^2-\frac{a^2}{b^2}y^2=2y-a^2$$

Writing this equation in standard quadratic form in $y$, we have:

$$\displaystyle \left(b^2-a^2 \right)y^2-2b^2y+a^2b^2=0$$

Next, implicitly differentiating the equation of the circle, we find:

$$\displaystyle 2x+2y\frac{dy}{dx}=2\frac{dy}{dx}$$

$$\displaystyle \frac{dy}{dx}=\frac{x}{1-y}$$

And doing the same for the equation of the ellipse, we obtain:

$$\displaystyle \frac{2x}{a^2}+\frac{2y}{b^2}\frac{dy}{dx}=0$$

$$\displaystyle \frac{dy}{dx}=-\frac{b^2x}{a^2y}$$

Equating the two slopes, we then get:

$$\displaystyle \frac{x}{1-y}=-\frac{b^2x}{a^2y}$$

$$\displaystyle y=\frac{b^2}{b^2-a^2}$$

Substituting this into the quadratic we found in $y$, we now have:

$$\displaystyle \left(b^2-a^2 \right)\left(\frac{b^2}{b^2-a^2} \right)^2-2b^2\left(\frac{b^2}{b^2-a^2} \right)+a^2b^2=0$$

And this gives us our constraint function:

$$\displaystyle g(a,b)=b^2\left(a^2-1 \right)-a^4=0$$

Using Lagrange multipliers, we obtain the system:

$$\displaystyle \pi b=\lambda\left(2ab^2-4a^3 \right)$$

$$\displaystyle \pi a=\lambda\left(2a^2b-2b \right)$$

Solving both equations for $\lambda$ and equating, we obtain:

$$\displaystyle \lambda=\frac{\pi b}{2a\left(b^2-2a^2 \right)}=\frac{\pi a}{2b\left(a^2-1 \right)}$$

$$\displaystyle \frac{b}{a\left(b^2-2a^2 \right)}=\frac{a}{b\left(a^2-1 \right)}$$

$$\displaystyle b^2\left(a^2-1 \right)=a^2\left(b^2-2a^2 \right)$$

$$\displaystyle a^2b^2-b^2=a^2b^2-2a^4$$

$$\displaystyle b^2=2a^4$$

Substituting for $b^2$ into the constraint function, we find:

$$\displaystyle 2a^4\left(a^2-1 \right)-a^4=0$$

Taking $0<a$ and dividing through by $a^4$, we obtain:

$$\displaystyle 2\left(a^2-1 \right)-1=0$$

$$\displaystyle a^2=\frac{3}{2}$$

$$\displaystyle a=\sqrt{\frac{3}{2}}$$

And thus:

$$\displaystyle b=\sqrt{2}a^2=\frac{3}{\sqrt{2}}$$

And so, we may conclude that the area of the ellipse is minimized for:

$$\displaystyle (a,b)=\left(\sqrt{\frac{3}{2}},\frac{3}{\sqrt{2}} \right)$$

Here is a plot of the circle and the ellipse:

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# The resultant of an arbitrary polynomial and a cyclotomic polynomial

This is a natural generalization of this question.

Let $f$ be a monic irreducible polynomial over $\mathbb Z$. Let $S_f$ be the set of natural numbers $n$ such that one of the three equivalent conditions hold:

a. There exists some prime $p$ such that a root of $f$ in $\bar{\mathbb F}_p$ is also a primitive $n$th root of unity.

b. The ideal in $\mathbb Z[x]$ generated by $f$ and the $n$th elementary cyclotomic polynomial is not the unit ideal.

c. The resultant of $f$ and the $n$th elementary cyclotomic polynomial is not $\pm 1$.

Is $S_f$ cofinite for all $f$ with Mahler measure larger than $1$?

What I know:

It is easy to check that $S_f$ is infinite, since for each prime a root of $f$ is the $n$th root of unity for some $n$, but each prime can only show up finitely many times for a fixed $n$, since the resultant is nonzero and thus has finitely many prime factors.

If $f$ has no complex roots of norm $1$, then $S_f$ is cofinite. This is an extension of the argument in the linked mathoverflow question. The resultant of $f$ and the $n$th elementary cyclotomic polynomial can be computed by inclusion-exclusion from the resultants of $f$ and $x^{n/d}-1$, for $d$ squarefree dividing $n$. Since the resultant of $f$ and $x^n-1$ is $a^{n+o(1)}$ for $a>1$ equal to the Mahler measure of the $f$, the result of $f$ and the elementary cyclotomic polynomial is $a^{\Phi(n)+O(2^k)}$, where $k$ is the number of primes dividing $n$. Since $2^k/\Phi(n)$ is small for large $n$, the exponential is larger than $1$ for large $n$, therefore the resultant equals $1$ only finitely many times.

But this argument fails if a root is norm 1. In fact, purely analytic methods of estimating the resultant should fail in this case. There are bad points on the unit circle that get too close to roots of unity too often. To prove $S_f$ cofinite, you would need to use some algebraic method to prove that those points are not algebraic integers.

-
I think that it's not cofinite for $f(x)=x$ or $f(x)=x-1$ or $f(x)=x+1$ or $f(x)=x^2+1$ or... . What did you mean to ask? Or did I misunderstand? –  Kevin Buzzard May 27 '12 at 22:12
If $f$ is not cyclotomic then its Mahler measure is $> 1$ and your analytic argument goes through. –  Felipe Voloch May 27 '12 at 22:23
@Kevin: Clarified. Ignoring $f(x)=x$ and every cyclotomic polynomial. @Felipe: Let $\alpha$ be a root of norm $1$. Couldn't $\alpha^n$ be extremely close to $1$? This would make the error in the estimate of the resultant very large. I don't see why the error should be small enough to ensure that the resultant is bigger than $1$. –  Will Sawin May 27 '12 at 22:34
@Will: I haven't tried to write out the estimates, so I can't guarantee anything, it's just my feeling. If you want to have an estimate for $|\alpha^n -1|$ you can use linear forms in logarithms à la Baker (Gelfond-Schneider really since it's only one log). –  Felipe Voloch May 27 '12 at 23:54
If you take the resultant of $f$ with $x^n-1$ rather than with the cyclotomic polynomial, I think this follows from Skolem-Mahler-Lech. Let $\alpha_1$, ..., $\alpha_r$ be the roots of $f$, so the resultant above is $s_n:=\prod (\alpha_i^n-1)$. Then $s_n$ obeys a linear recursion with integer coefficients so the only way for it to be $1$ infinitely often is if it is $1$ on an entire arithmetic progression. I feel like there should be an easy contradiction here, although I haven't actually found one. The actual cyclotomic question strikes me as harder, although Felipe's suggestion sounds good. –  David Speyer May 28 '12 at 0:20

For an algebraic number $\gamma$ which is not a root of unity, Baker's theorem gives a bound (uniform in $n$) of the form $|\gamma^n - 1| > n^{-C}$ for some constant $C$ (only depending on $\gamma$).

In particular, if $s_n:=\prod |\alpha^n_i - 1|$, then Baker's method gives the following estimate uniform in $n$ (for $\alpha_i$ not a root of unity): $$n \log \mathcal{M}(\alpha) + A \ge \log|s_n| \ge n \log \mathcal{M}(\alpha) - A - B \log|n|,$$ where $\mathcal{M}(\alpha)$ is the Mahler measure of $\alpha$. Proof: for $|\alpha_i| > 1$, one has $\log|\alpha^n_i - 1| \sim n \log|\alpha_i|$ up to $O(1)$, which gives rise to the term $n \log \mathcal{M}(\alpha)$; the term $\log|\alpha^n_i - 1|$ for $|\alpha_i| \le 1$ is trivial to bound from above and can be bound from below by Baker's Theorem. The logarithm of the Mahler measure is the sum of $\log|\alpha_i|$ for the roots $|\alpha_i|> 1$. By a theorem of Kronecker this sum is positive if $\alpha$ is not a root of unity.

OTOH, if $\Phi_n(x)$ is the $n$th cyclotomic polynomial and $t_n:=\prod |\Phi_n(\alpha_i)|$, then we deduce that $\sum_{d|n} \log(t_n) = \log(s_n)$, and hence $$\log(t_n) = \sum_{d|n} \log(s_{n/d}) \mu(d) \ge \varphi(n) \log \mathcal{M}(\alpha) - d(n)(A + B \log(n)),$$ where $d(n)$ is the number of divisors of $n$. The bounds $\phi(n) \gg n^{1-\epsilon}$ and $d(n) = n^{\epsilon}$ easily give the asymptotic relation $$\log(t_n) \sim \varphi(n) \log \mathcal{M}(\alpha) \gg 1$$ as $n$ goes to infinity.

FWIW, the bounds of Baker (and the other bounds used above) are effective, so for any particular $\alpha$ one could in principle find all $n$ with $t_n = 1$. (n.b. Gelfond's estimate would give $\epsilon \cdot n$ instead of $B \log|n|$ as an error term, which is enough to show that $s_n \rightarrow \infty$ but not $t_n$.)

-


text:
# What is particle under potential in quantum field theory?

Let's say we solve the Schrodinger equation with infinite well. We can quantize the field by the resonance state and can get the annihilation and creation operator. So we will get the sort of particle corresponding to the resonance state. The resonance state would have mass corresponding to the energy and the mass will exist as the multiple of the minimum energy level.

Is the resonance state a new type of particle? If it's not, what does the creation operator create?

Let's consider we put a photon inside the infinite well. If the photon is not in the resonance state, the photon will be represented by the superposition of the resonance states. So when it goes inside the well, photon seems not a fundamental particle but a complex particle constructed from more basic states.

And if the resonance is the particle, what happens to the photon then? If we put a photon detection device inside the infinite well, can we detect the photon? Or is the photon's identity vanished completely and it's meanless to talk of the photon anymore?

• The problem is that the Schrödinger equation in QFT is only applicable to systems of states of fields. Photons are quantization of the electromagnetic field, there is nothing such as potential for the photons, because the equation describing the photons is not the Schrödinger equation, but instead, $\square A_\mu=J_\mu$ (where $A_\mu$ is the electromagnetic quadri-potential field and $J_\mu$ a source of this field). – Jeanbaptiste Roux Jul 15 at 8:56
• @JeanbaptisteRoux Thanks! It's no wonder I was talking something nonsense. Could I replace the photons with the electrons to make it a bit more sense? – kevin012 Jul 15 at 9:33
• I am not the expert of the experts, but I don't think it will have a sense even with a Dirac bispinor field: there is no meaning to quantization when we are in presence of interactions (because of the potential). We can do a series expansion of the S matrix and then we can talk about virtual particles, but there is no "quantization of an interacting field" only interactions via quantization of fields, due to a small structure constant (compared to 1). – Jeanbaptiste Roux Jul 15 at 9:47


text:
# Equation for the distance traveled by a falling object

The velocity of a falling object (or the derivative with respect to time of the distance traveled by the falling object) is given by

$$\dot{s} = \frac{rm}{k}\left(\frac{e^{rt}-e^{-rt}}{e^{rt}+e^{-rt}}\right),$$

where $r = \sqrt{gk/m}$.

Here, $g$ is acceleration due to gravity, $m$ is the mass of the object, and $k$ is a constant that captures air resistance.

I need to integrate this to obtain an exact formula for $s$. Can anyone give me some guidance for how to do this? I'm not sure what strategy is appropriate.

The answer should come out to be

$$s(t) = \frac{V^2}{g}\ln\left(\cosh\frac{gt}{V}\right),$$

where $V=\sqrt{\frac{mg}{k}}$ is the terminal velocity.

• You would obtain $s$ by integration, not differentiation. – Math1000 Mar 3 '17 at 22:36
• Consider $$\tanh(rt)=\frac{e^{rt}-e^{-rt}}{e^{rt}+e^{-rt}}$$ – Mengchun Zhang Mar 3 '17 at 22:37
• @dbliss Bingo :) you got exactly the right answer – Mengchun Zhang Mar 3 '17 at 23:18

The limiting velocity is the value of $\dot{s}$ as $t\rightarrow\infty$, so $V^2=\frac{rm}{k}$
Integrating the expression you have gives (assuming $v=0$ when $t=0$) $$s=\frac Vr\ln\cosh rt$$
But $\frac{V^2}{g}=\frac mk=\frac Vr$
Hence $$s=\frac{V^2}{g}\ln\cosh\frac{gt}{V}$$


text:
• ### Synchrotron emission from nanowire-array targets irradiated by ultraintense laser pulses(1802.07520)

Feb. 21, 2018 physics.plasm-ph
We present a numerical study, based on two-dimensional particle-in-cell simulations, of the synchrotron emission induced during the interaction of femtosecond laser pulses of intensities $I=10^{21}-10^{23}\,\mathrm{Wcm}^{-2}$ with nanowire arrays. Through an extensive parametric scan on the target parameters, we identify and characterize several dominant radiation mechanisms, mainly depending on the transparency or opacity of the plasma produced by the wire expansion. At $I=10^{22}\,\mathrm{Wcm}^{-2}$, the emission of high-energy ($>10\,\mathrm{keV}$) photons attains a maximum conversion efficiency of $\sim 10\%$ for $36-50\,\mathrm{nm}$ wire widths and $1\,\mu\mathrm{m}$ interspacing. This maximum radiation yield is similar to that achieved in uniform plasma of same average (sub-solid) density, but nanowire arrays provide efficient radiation sources over a broader parameter range. We examine the variations of the photon spectra with the laser intensity and the wire material. Finally, we demonstrate that the radiation efficiency can be further enhanced by adding a plasma mirror at the backside of the nanowire array.
• ### Baby MIND: A magnetized segmented neutrino detector for the WAGASCI experiment(1705.10406)

July 16, 2017 physics.ins-det
T2K (Tokai-to-Kamioka) is a long-baseline neutrino experiment in Japan designed to study various parameters of neutrino oscillations. A near detector complex (ND280) is located 280~m downstream of the production target and measures neutrino beam parameters before any oscillations occur. ND280's measurements are used to predict the number and spectra of neutrinos in the Super-Kamiokande detector at the distance of 295~km. The difference in the target material between the far (water) and near (scintillator, hydrocarbon) detectors leads to the main non-cancelling systematic uncertainty for the oscillation analysis. In order to reduce this uncertainty a new WAter-Grid-And-SCintillator detector (WAGASCI) has been developed. A magnetized iron neutrino detector (Baby MIND) will be used to measure momentum and charge identification of the outgoing muons from charged current interactions. The Baby MIND modules are composed of magnetized iron plates and long plastic scintillator bars read out at the both ends with wavelength shifting fibers and silicon photomultipliers. The front-end electronics board has been developed to perform the readout and digitization of the signals from the scintillator bars. Detector elements were tested with cosmic rays and in the PS beam at CERN. The obtained results are presented in this paper.
• ### Baby MIND Experiment Construction Status(1704.08917)

April 28, 2017 physics.ins-det
Baby MIND is a magnetized iron neutrino detector, with novel design features, and is planned to serve as a downstream magnetized muon spectrometer for the WAGASCI experiment on the T2K neutrino beam line in Japan. One of the main goals of this experiment is to reduce systematic uncertainties relevant to CP-violation searches, by measuring the neutrino contamination in the anti-neutrino beam mode of T2K. Baby MIND is currently being constructed at CERN, and is planned to be operational in Japan in October 2017.
• ### Baby MIND: A magnetised spectrometer for the WAGASCI experiment(1704.08079)

April 26, 2017 hep-ex, physics.ins-det
The WAGASCI experiment being built at the J-PARC neutrino beam line will measure the difference in cross sections from neutrinos interacting with a water and scintillator targets, in order to constrain neutrino cross sections, essential for the T2K neutrino oscillation measurements. A prototype Magnetised Iron Neutrino Detector (MIND), called Baby MIND, is being constructed at CERN to act as a magnetic spectrometer behind the main WAGASCI target to be able to measure the charge and momentum of the outgoing muon from neutrino charged current interactions.
• ### Isothermal anisotropic magnetoresistance in antiferromagnetic metallic IrMn(1607.07999)

July 27, 2016 cond-mat.mtrl-sci
Antiferromagnetic spintronics is an emerging field; antiferromagnets can improve the functionalities of ferromagnets with higher response times, and having the information shielded against external magnetic field. Moreover, a large list of aniferromagnetic semiconductors and metals with N\'eel temperatures above room temperature exists. In the present manuscript, we persevere in the quest for the limits of how large can anisotropic magnetoresistance be in antiferromagnetic materials with very large spin-orbit coupling. We selected IrMn as a prime example of first-class moment (Mn) and spin-orbit (Ir) combination. Isothermal magnetotransport measurements in an antiferromagnetic-metal(IrMn)/ferromagnetic-insulator thin film bilayer have been performed. The metal/insulator structure with magnetic coupling between both layers allows the measurement of the modulation of the transport properties exclusively in the antiferromagnetic layer. Anisotropic magnetoresistance as large as 0.15 % has been found, which is much larger than that for a bare IrMn layer. Interestingly, it has been observed that anisotropic magnetoresistance is strongly influenced by the field cooling conditions, signaling the dependence of the found response on the formation of domains at the magnetic ordering temperature.
• ### Engineering the microstructure and magnetism of La$_2$CoMnO$_6$ thin films by tailoring oxygen stoichiometry(1407.2072)

July 8, 2014 cond-mat.mtrl-sci
We report on the magnetic and structural properties of ferromagnetic-insulating La$_2$CoMnO$_6$ thin films grown on top of (001) STO substrates by means of RF sputtering technique. Careful structural analysis, by using synchrotron X-ray diffraction, allows identifying two different crystallographic orientations that are closely related to oxygen stoichiometry and to the features (coercive fields and remanence) of the hysteresis loops. Both Curie temperature and magnetic hysteresis turn out to be dependent on the oxygen stoichiometry. In situ annealing conditions allow tailoring the oxygen content of the films, therefore controlling their microstructure and magnetic properties.
• ### Intrinsic robust antiferromagnetism at manganite surfaces and interfaces(1302.5293)

Feb. 21, 2013 cond-mat.mtrl-sci
Ferromagnetic/metallic manganese perovskites, such as La2/3Sr1/3MnO3 (LSMO)are promising materials for the design and implementation of novel spintronic devices working at room temperature. However, their implementation in practical applications has been severely hampered due to the breakdown of their magnetotransport properties at temperatures well below their magnetic transition temperature. This breakdown has been usually associated to surface and interface related problems but its physical origin has not been clearly established yet. In this work we investigate the interface between La2/3Sr1/3MnO3 (LSMO) thin films and different capping layers by means of x-ray linear dichroism and transport measurements. Our data reveal that, irrespective to the capping material, LSMO/capping layer bilayers exhibit an antiferromegnetic/insulating phase at the interface, likely to originate from a preferential occupancy of Mn 3d 3z2-r2 eg orbitals. This phase, which extends ca. 2 unit cells, is also observed in an uncapped LSMO reference sample thus, pointing to an intrinsic interfacial phase separation phenomenon, likely to be promoted by the structural disruption and symmetry breaking at the LSMO free surface/interface. These experimental observations strongly suggest that the structural disruption at the LSMO interfaces play a major role on the observed depressed magnetotransport properties in manganite-based magnetic tunneling junctions and it is at the origin of the so-called dead layer.
• ### Mn valence instability in La2/3Ca1/3MnO3 thin films(0909.3037)

Sept. 16, 2009 cond-mat.mtrl-sci
A Mn valence instability on La2/3Ca1/3MnO3 thin films, grown on LaAlO3 (001)substrates is observed by x-ray absorption spectroscopy at the Mn L-edge and O K-edge. As-grown samples, in situ annealed at 800 C in oxygen, exhibit a Curie temperature well below that of the bulk material. Upon air exposure a reduction of the saturation magnetization, MS, of the films is detected. Simultaneously a Mn2+ spectral signature develops, in addition to the expected Mn3+ and Mn4+ contributions, which increases with time. The similarity of the spectral results obtained by total electron yield and fluorescence yield spectroscopy indicates that the location of the Mn valence anomalies is not confined to a narrow surface region of the film, but can extend throughout the whole thickness of the sample. High temperature annealing at 1000 C in air, immediately after growth, improves the magnetic and transport properties of such films towards the bulk values and the Mn2+ signature in the spectra does not appear. The Mn valence is then stable even to prolonged air exposure. We propose a mechanism for the Mn2+ ions formation and discuss the importance of these observations with respect to previous findings and production of thin films devices.
• ### Exchange bias in laterally oxidized Au/Co/Au nanopillars(0902.1687)

Feb. 10, 2009 cond-mat.mtrl-sci
Au/Co/Au nanopillars fabricated by colloidal lithography of continuous trilayers exhibit and enhanced coercive field and the appearance of an exchange bias field with respect to the continuous layers. This is attributed to the lateral oxidation of the Co interlayer that appears upon disc fabrication. The dependence of the exchange bias field on the Co nanodots size and on the oxidation degree is analyzed and its microscopic origin clarified by means of Monte Carlo simulations based on a model of a cylindrical dot with lateral core/shell structure.
• ### Magnetic dead layers in La2/3Ca1/3MnO3 thin films probed by X-ray magnetic circular dichroism in reflection(0706.0821)

Surface magnetic properties of perovskite manganites have been a recurrent topic during last years since they play a major role in the implementation of magnetoelectronic devices. Magneto-optical techniques, such as X-ray magnetic circular dichroism, turn out to be a very efficient tool to study surface magnetism due to their sensitivity to magnetic and chemical variations across the sample depth. Nevertheless, the application of the sum rules for the determination of the spin magnetic moment might lead to uncertainties as large as 40% in case of Mn ions. To overcome this problem we present an alternative approach consisting of using X-ray magnetic circular dichroism in reflection geometry. Fit of the data by using a computer code based in a 4X4 matrix formalism leads to realistic results. In particular, we show that surface and interface roughness are of major relevance for a proper description of the experimental data and a correct interpretation of the results. By using such an approach we demonstrate the presence of a narrow surface region with strongly depressed magnetic properties in La2/3Ca1/3MnO3 thin films.
• ### Formation of Mn2+ in La2/3Ca1/3MnO3 Thin Films due to Air Exposure(cond-mat/0509513)

Sept. 20, 2005 cond-mat.mtrl-sci
We report on the chemical stability of La2/3Ca1/3MnO3 thin films. X-ray absorption spectroscopy at the Mn L-edge and O K-edge makes evident deviations from the nominally expected (2/3-1/3) Mn3+/Mn4+ ratio after the growth of thin films on LaAlO3 substrates. As-grown thin films, exhibiting Curie temperature, TC, well below that of the LCMO bulk material, develop an unexpected Mn2+ contribution after a few days of air exposure which increases with time. Moreover, a reduction of the saturation magnetization, MS, is also detected. The similarity of the results obtained by electron yield and fluorescence yield demonstrates that the location of the Mn valence anomalies are not confined to a narrow surface region of the film but can extend throughout the film thickness in case of granular films. High temperature annealing not only improves the magnetic and transport properties of such as-grown films but also recovers the expected 2/3-1/3 Mn3+/Mn4+ ratio, which thereafter is stable to air exposure. Similar results on La2/3Ca1/3MnO3 films grown on SrTiO3 and NdGaO3 substrates demonstrate that there is no direct relation between the observed Mn valence instability and the strain state of the films due to their lattice mismatch with the substrate. A mechanism for the formation of Mn2+ ions formation is discussed.
• ### Evidence of strong antiferromagnetic coupling between localized and itinerant electrons in ferromagnetic Sr2FeMoO6(cond-mat/0205187)

Magnetic dc susceptibility ($\chi$) and electron spin resonance (ESR) measurements in the paramagnetic regime, are presented. We found a Curie-Weiss (CW) behavior for $\chi$(T) with a ferromagnetic $\Theta = 446(5)$ K and $\mu_{eff} = 4.72(9) \mu_{B}/f.u.$, this being lower than that expected for either $Fe^{3+}(5.9\mu_{B})$ or $Fe^{2+}(4.9\mu_{B})$ ions. The ESR g-factor $g = 2.01(2)$, is associated with $Fe^{3+}$. We obtained an excellent description of the experiments in terms of two interacting sublattices: the localized $Fe^{3+}$ ($3d^{5}$) cores and the delocalized electrons. The coupled equations were solved in a mean-field approximation, assuming for the itinerant electrons a bare susceptibility independent on $T$. We obtained $\chi_{e}^{0} = 3.7$ $10^{-4}$ emu/mol. We show that the reduction of $\mu_{eff}$ for $Fe^{3+}$ arises from the strong antiferromagnetic (AFM) interaction between the two sublattices. At variance with classical ferrimagnets, we found that $\Theta$ is ferromagnetic. Within the same model, we show that the ESR spectrum can be described by Bloch-Hasegawa type equations. Bottleneck is evidenced by the absence of a $g$-shift. Surprisingly, as observed in CMR manganites, no narrowing effects of the ESR linewidth is detected in spite of the presence of the strong magnetic coupling. These results provide evidence that the magnetic order in $Sr_{2}FeMoO_{6}$ does not originates in superexchange interactions, but from a novel mechanism recently proposed for double perovskites.
• ### Cationic ordering control of magnetization in Sr2FeMoO6 double perovskite(cond-mat/0007248)

July 14, 2000 cond-mat
The role of the synthesis conditions on the cationic Fe/Mo ordering in Sr2FeMoO6 double perovskite is addressed. It is shown that this ordering can be controlled and varied systematically. The Fe/Mo ordering has a profound impact on the saturation magnetization of the material. Using the appropriate synthesis protocol a record value of 3.7muB/f.u. has been obtained. Mossbauer analysis reveals the existence of two distinguishable Fe sites in agreement with the P4/mmm symmetry and a charge density at the Fe(m+) ions significantly larger than (+3) suggesting a Fe contribution to the spin-down conduction band. The implications of these findings for the synthesis of Sr2FeMoO6 having optimal magnetoresistance response are discussed.


text:
Over the long haul populations grow and decay. To describe the growth and decay of civilization's dependence on nuclear and fossil fuels, M. King Hubbert <ref>1956, M.King Hubbert, Nuclear energy and fossil fuels, Publication 95, Shell Oil Company, http://www.hubbertpeak.com/hubbert/1956/1956.pdf</ref>, <ref>1962, National Academy of Sciences Publication 1000-D</ref> chose a simple equation. <ref>Kenneth S. Deffeyes, 2006, Beyond Oil: The view from Hubbert's Peak: Hill and Wang ISBN 0-8090-2957-X.</ref>

Introduce bacteria to food and their population will grow exponentially until there no longer is food. As we remove all the fish from the ocean our regular catch will be proportional to the number of remaining fish. Hubbert's equation models both exponential growth and decay with a single equation of three parameters to be chosen from the data. He claimed 52 years ago that worldwide oil production would be peaking about now (2008). It is.

Hubbert's math has four different expressions which we examine before showing they are mathematically equivalent.

Basic definitions

We define:

t  is time in years

Q(t)   is cumulative production in billion barrels at year t.

<latex>Q_\infty</latex> is the ultimate recoverable resource.

P(t)=dQ/dt  is production in billion barrels/year at year t.

?  is the year at which production peaks.

?  is an inverse decay time (imaginary frequency).

The four forms of Hubbert's equation

The Hubbert's equation can be expressed in four forms. First, the differential form

: <latex>\frac{dQ}{dt}=P=\omega\ Q\left(1 - \frac{Q}{Q_\infty}\right) \qquad \mbox{(1)} \!</latex>

This equation is non-linear in Q but it reduces to familiar linear equations near the beginning and <latex>Q \approx 0</latex> and near the end at <latex>Q \approx Q_{\infty}</latex>. As production begins and <latex>Q/Q_\infty</latex> is small, equation (1) reduces to dQ/dt=?Q which displays exponential growth at a rate ?. As production ends near <latex> Q \approx Q_\infty</latex> the non-linear equation reduces to exponential decay. To prove this fact change variables from Q to q where <latex> Q=Q_\infty - q</latex> and evaluate the result at small q.

Dividing equation (1) by Q we get the second form of the Hubbert equation sometimes called the Hubbert Linearization.

: <latex>\frac{P}{Q}= \omega\ \left(1 - \frac{Q}{Q_\infty}\right) \qquad \mbox{(2)} \!</latex>

The important thing about this equation is that it is linear in the two variables Q and P/Q. If you have historical measurements of Pi and Qi, you can plot these points in the (Q,P/Q)-plane and hope for them to reasonably fit a straight line. Fitting the best line to the scattered points we can read the axis intercepts. At Q=0 with equation (2) we can read off the value of the growth/decay parameter <latex>\omega = (P/Q)_{\rm intercept}</latex>. For world oil, according to Deffeyes it is 5.3%/year. At the other intercept, P/Q=0 we must have <latex>Q=Q_{\infty}</latex>. Again, according to Deffeyes, <latex>Q_{\infty}</latex> is two trillion barrels.

The third form of Hubbert's equation is the one best known. It looks like a Gaussian, but it isn't. (A Gaussian decays much faster.) The current production P=dQ/dt is

:<latex>

              P(t) =
Q_\infty \ \omega \
{
1
\over
(e^{-(\omega/2) (\tau-t)} + e^{(\omega/2) (\tau-t)})^2
}
\qquad \mbox{(3)} \!</latex>
			 This is the equation of a blob, also known as ''Hubbert's pimple'', symmetric
it decreases (or increases) exponentially towards its maximum
value at the center at t=?.
The function resembles a gaussian but exponential decay
is much weaker than gaussian decay.
Exponential growth is common in ecological systems
which may also decay exponentially as resources are depleted or predator
numbers grow exponentially.
			 All that remains is to figure out ?.
The Hubbert curve is symmetrical and reaches its maximum
when half the oil is gone.
That happens when <latex>Q=Q_\infty/2</latex>.
In the case of USA production which has passed its peak
we can find the year that ''Q'' reached that value (about 1973).
There is some debate about what year world production peaks, but general agreement is that it is about now (2008).
Under Hubbert assumptions the decline curve is a mirror of the rise curve. That means we start down gently over the next decade, but about 25 years from now we hit the inflection point and see a 5%/year decline every year thereafter.
			 In real life there is no reason for the decay rate to match the growth rate.   The decay could be faster because of [[horizontal drilling]].
The decay could be slower because we tax to conserve or successfully invest in technologies.
As liquid oil depleats, society is switching to mining [[tar sands]].
			 The Hubbert equation, in all its forms, follows as a consequence of the definition of
the ''logistic'' function ''Q(t)''.  It ranges from ''0'' in the past to <latex>Q_\infty</latex> in the future.
			 :<latex>Q(t) = {Q_\infty \over 1 + e^{\omega (\tau-t)}}  \qquad \mbox{(4)} \!</latex>
			 ==== Verification the four forms are equivalent ====
If you buy the idea that your data scatter in (Q,P/Q)-space is a straight line, then you have bought equation (2).  If you buy any one of equations (1),(2),(3), or (4), then you have bought them all because they are mathematically equivalent.
Starting from the definition (4) using the rule from calculus that ''d(1/v)/dt= -(dv/dt)/v 2''  yields equation (3).
			 :<latex>
{dQ\over dt} = P(t) =
Q_\infty \ \omega\
{
e^{\omega (\tau-t)}
\over
(1 + e^{\omega (\tau-t)})^2
}
\qquad \mbox{(5)} \!</latex>
					  :<latex>
P(t) =
Q_\infty \ \omega \
{
1
\over
(e^{-(\omega/2) (\tau-t)} + e^{(\omega/2) (\tau-t)})^2
}
which is equation (3).
									   Equation (4) allows us to eliminate the denominator in
equation (5) getting equation (2)
									   :<latex>P/Q = (Q/Q_\infty) \ \omega \ e^{\omega (\tau-t)} </latex>
									   :<latex>P/Q = (Q/Q_\infty) \ \omega \ ((1+ e^{\omega (\tau-t)}) -1)  </latex>
									   :<latex>P/Q = (Q/Q_\infty) \ \omega \ (Q_\infty/Q-1)  </latex>
									   :<latex> P/Q = \omega \ (1-Q/Q_\infty) \!</latex>
									   which is equation (2).
Multiplying both sides by Q gives equation (1).
									   ==See also==
{{EnergyPortal}}
{{Portal|Sustainable development|Sustainable development.svg}}
*'''[[:Category:Peak oil]]'''
*[[Hubbert curve]]
									   ==References==
{{Reflist}}


text:
# Properties

 Label 1936.2.a.l Level $1936$ Weight $2$ Character orbit 1936.a Self dual yes Analytic conductor $15.459$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1936 = 2^{4} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1936.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.4590378313$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 88) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10})$$ q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 6 * q^9 $$q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 6 q^{9} - 9 q^{15} + 6 q^{17} + 4 q^{19} - 6 q^{21} - q^{23} + 4 q^{25} + 9 q^{27} + 8 q^{29} + 7 q^{31} + 6 q^{35} - q^{37} - 4 q^{41} + 6 q^{43} - 18 q^{45} + 8 q^{47} - 3 q^{49} + 18 q^{51} + 2 q^{53} + 12 q^{57} + q^{59} - 4 q^{61} - 12 q^{63} + 5 q^{67} - 3 q^{69} - 3 q^{71} - 16 q^{73} + 12 q^{75} + 2 q^{79} + 9 q^{81} - 2 q^{83} - 18 q^{85} + 24 q^{87} + 15 q^{89} + 21 q^{93} - 12 q^{95} - 7 q^{97}+O(q^{100})$$ q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 6 * q^9 - 9 * q^15 + 6 * q^17 + 4 * q^19 - 6 * q^21 - q^23 + 4 * q^25 + 9 * q^27 + 8 * q^29 + 7 * q^31 + 6 * q^35 - q^37 - 4 * q^41 + 6 * q^43 - 18 * q^45 + 8 * q^47 - 3 * q^49 + 18 * q^51 + 2 * q^53 + 12 * q^57 + q^59 - 4 * q^61 - 12 * q^63 + 5 * q^67 - 3 * q^69 - 3 * q^71 - 16 * q^73 + 12 * q^75 + 2 * q^79 + 9 * q^81 - 2 * q^83 - 18 * q^85 + 24 * q^87 + 15 * q^89 + 21 * q^93 - 12 * q^95 - 7 * q^97

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 3.00000 0 −3.00000 0 −2.00000 0 6.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1936.2.a.l 1
4.b odd 2 1 968.2.a.a 1
8.b even 2 1 7744.2.a.b 1
8.d odd 2 1 7744.2.a.bk 1
11.b odd 2 1 176.2.a.c 1
12.b even 2 1 8712.2.a.x 1
33.d even 2 1 1584.2.a.q 1
44.c even 2 1 88.2.a.a 1
44.g even 10 4 968.2.i.j 4
44.h odd 10 4 968.2.i.i 4
55.d odd 2 1 4400.2.a.a 1
55.e even 4 2 4400.2.b.b 2
77.b even 2 1 8624.2.a.c 1
88.b odd 2 1 704.2.a.b 1
88.g even 2 1 704.2.a.l 1
132.d odd 2 1 792.2.a.g 1
176.i even 4 2 2816.2.c.i 2
176.l odd 4 2 2816.2.c.d 2
220.g even 2 1 2200.2.a.k 1
220.i odd 4 2 2200.2.b.a 2
264.m even 2 1 6336.2.a.k 1
264.p odd 2 1 6336.2.a.h 1
308.g odd 2 1 4312.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.a.a 1 44.c even 2 1
176.2.a.c 1 11.b odd 2 1
704.2.a.b 1 88.b odd 2 1
704.2.a.l 1 88.g even 2 1
792.2.a.g 1 132.d odd 2 1
968.2.a.a 1 4.b odd 2 1
968.2.i.i 4 44.h odd 10 4
968.2.i.j 4 44.g even 10 4
1584.2.a.q 1 33.d even 2 1
1936.2.a.l 1 1.a even 1 1 trivial
2200.2.a.k 1 220.g even 2 1
2200.2.b.a 2 220.i odd 4 2
2816.2.c.d 2 176.l odd 4 2
2816.2.c.i 2 176.i even 4 2
4312.2.a.l 1 308.g odd 2 1
4400.2.a.a 1 55.d odd 2 1
4400.2.b.b 2 55.e even 4 2
6336.2.a.h 1 264.p odd 2 1
6336.2.a.k 1 264.m even 2 1
7744.2.a.b 1 8.b even 2 1
7744.2.a.bk 1 8.d odd 2 1
8624.2.a.c 1 77.b even 2 1
8712.2.a.x 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1936))$$:

 $$T_{3} - 3$$ T3 - 3 $$T_{5} + 3$$ T5 + 3 $$T_{7} + 2$$ T7 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T + 3$$
$7$ $$T + 2$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T - 6$$
$19$ $$T - 4$$
$23$ $$T + 1$$
$29$ $$T - 8$$
$31$ $$T - 7$$
$37$ $$T + 1$$
$41$ $$T + 4$$
$43$ $$T - 6$$
$47$ $$T - 8$$
$53$ $$T - 2$$
$59$ $$T - 1$$
$61$ $$T + 4$$
$67$ $$T - 5$$
$71$ $$T + 3$$
$73$ $$T + 16$$
$79$ $$T - 2$$
$83$ $$T + 2$$
$89$ $$T - 15$$
$97$ $$T + 7$$


text:
# Fenwick, the internal auditor for Discount Department Stores, hated traveling but since he covere...

Fenwick, the internal auditor for Discount Department Stores, hated traveling but since he covered all 14 discount stores in his district, he was on the road constantly. This time he was in Grapevine, Texas. Located near the end of Dallas/ Fort Worth International Airport runways, the Grapevine’s store windows shook every few minutes as each jet lumbered into the air. It was evening and the store was empty except for the piles of financial records surrounding him. The Grapevine operation had lost money for three straight years, had closed the previous month, and its remaining inventory had been divided among other locations.

Fenwick's instructions were clear: Find out what went wrong? He was about to reach for his last steaming cup of coffee—filled to the brim— when a jolt from yet another jet, the loudest of the day, spilled his coffee. This, in turn soaked the bank slips on his desk. Fenwick got up and spread the dozen or so deposit slips on the carpet to dry. He then cleared the rest of the mess and returned his attention to the stained papers on the floor. He stacked the slips one by one, glancing at them casually.

After looking at just a few, he spotted a distinct abnormality. Fenwick had been around the other stores enough to know that currency and other checks represented about half the money going into their bank accounts; credit card deposits made up the rest. However, the documents in his hands were a different mix—three quarters credit card, one quarter cash. Also, Fenwick spotted a refund of exactly $300 on one day’s tape and a refund of$400 on the next day’s tape (Wells, 2002).

Questions:

Discuss some commonalities of the two types of asset misappropriation schemes.

Why would a perpetrator choose one scheme over another?

Are there any common red flags? What are they?

Are there any common detection and prevention strategies? If so, what are they? If not, why not?

## Plagiarism Checker

Submit your documents and get free Plagiarism report

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text:
## Introduction

Modern metabolomics experiments yield high-dimensional datasets with hundreds to thousands of measured metabolites in large human studies with thousands of participants1. Such datasets are routinely generated to profile the molecular phenotype of disease and identify the underlying pathological mechanisms of action2,3,4,5. Extracting systemic effects from high-dimensional datasets requires dimensionality reduction approaches to untangle the high number of metabolites into the processes in which they participate. To this end, linear dimensionality reduction methods, such as principal component analysis (PCA) and independent component analysis (ICA), have been extensively applied to high-dimensional biological data6,7,8,9,10. However, metabolic systems, like most complex biological processes, contain non-linear effects which arise due to high-order enzyme kinetics and upstream gene regulatory processes11,12. For example, metabolite ratios are an intuitive and widely used approach to detect non-linear effects in metabolomics data, approximating the steady state between reactants and products of metabolic reactions13,14. Extending this concept, systematic methods that take nonlinearities into account are required to correctly recover the functional interplay between metabolites in an unbiased fashion.

Autoencoders (AEs) are a type of neural network architecture developed as an unsupervised dimensionality reduction method that can capture non-linear effects15. AEs reduce high-dimensional data into latent variables through an encoding/decoding process which recreates the input data after passing through a lower dimensional space. Once the model is fitted, the latent variables represent a compact, often easier-to-interpret version of the original data. While AEs have been successfully used for prediction tasks on biological datasets16, they tend to learn latent spaces specifically fitted to the input dataset and are therefore not generalizable to biological phenomena absent from the training data17. To address this, Variational Autoencoders (VAEs) were introduced as a probabilistic extension of the AE architecture that constrains the latent variables to follow a predefined distribution18. With this extension, the VAE not only reconstructs the input data, but infers the generative process behind the data, leading to high generalizability across datasets. The VAE architecture has, for example, proven effective for predicting cell-level response to infection from transcriptomic data in cell types not used during training, and predicting drug response from gene expression data where drug response information was sparse17,19,20. In addition, latent dimensions of VAE models trained on gene expression data have been shown to strongly associate with clinical parameters21 and have outperformed other common low-dimensional representation methods (PCA and k-means clustering) in capturing biologically meaningful pathways22.

The application of deep learning architectures to metabolomics datasets has considerably lagged behind most other omics23 due to the unavailability of large metabolomics cohorts. One previous study used (non-variational) AEs on a small set of metabolomics samples (n = 271), demonstrating superior predictability of ER status in breast cancer patients using the latent representations of the model over common machine learning methods24. By applying VAE architectures to large-scale metabolomics data, we have the potential to learn more accurate latent dimension representations that systematically account for non-linear effects. In addition, the probabilistic structure of VAEs will learn latent dimensions that are generalizable across multiple datasets.

In this paper, we trained a VAE model on 217 metabolite measurements in 4644 blood samples from the TwinsUK study25 and evaluated our model performance in comparison to linear and non-linear PCA models (Fig. 1a). To investigate the biological relevance of the learned VAE and PCA latent dimensions, we employed the Shapley Additive Global Importance (SAGE) method26, which determines the contribution of each input to each latent dimension. We calculated SAGE values at different granularities, i.e., metabolites, sub-pathways, and super-pathways (Fig. 1b). We then applied the models on three additional blood metabolomics datasets to test their ability to recover disease phenotypes in unseen datasets: Type 2 Diabetes diagnosis in The Qatar Metabolomics Study on Diabetes (QMDiab, n = 358), therapy response in an acute myeloid leukemia dataset (AML, n = 85), and schizophrenia diagnosis in a third validation dataset (n = 201) (Fig. 1c).

## Results

### VAE model construction and fitting

Our VAE architecture consisted of an input/output layer, an intermediate layer and a latent layer. We split the TwinsUK cohort into an 85% training and a 15% test set, and the training set was used to optimize the hyperparameters in the VAE model. Keras Tuner27 identified the following optimal hyperparameters: Intermediate layer dimensionality = 200, learning rate = 0.001, and Kullback-Leibler (KL) divergence weight = 0.01 (see Methods). With these parameters fixed, we optimized the dimensionality d of the latent layer z by calculating the reconstruction MSE of the correlation matrix (CM-MSE) of metabolites (Fig. 2a, b). We observed that the CM-MSE curve plateaus after d = 18, indicating that increasing the latent dimensionality beyond this value only marginally improved the models. The final architecture of the model consisted of a 217-dimensional input/output layer (the number of metabolites in our datasets), a 200-dimensional intermediate layer, and an 18-dimensional latent layer (Fig. 2c).

We used principal component analysis (PCA) as a baseline model to compare the VAE to a linear latent variable embedding method. To this end, we fitted a PCA on the TwinsUK train data and extracted the first d = 18 dimensions, i.e., principal components. To further compare our VAE results to a non-linear latent variable method, we used the same fit and extraction procedures using kernel (K)PCA models with cosine similarity, sigmoid, RBF, and polynomial kernel transformations. On training data, neither PCA nor VAE outperformed KPCA data matrix reconstruction. On testing data, however, both PCA and VAE outperformed the KPCA models, with PCA showing better reconstruction accuracy of the data matrix than the VAE (Supplementary Figs 1 and 2). All non-linear models, KPCA and VAE, outperformed PCA in correlation matrix reconstruction via CM-MSE in both the TwinsUK train and test set (Fig. 2d, Supplementary Fig. 3).

This difference between the MSE on the correlation matrix and the more commonly used sample-wise MSE18 outlines ambiguities in the methods to assess reconstruction performances. Notably, other authors have shown previously that sample reconstruction performance does not necessarily imply better model performance19,20, as evidenced by the large discrepancy of KPCA sample reconstruction between training and test sets. However, our results suggest that while the non-linear methods do not reconstruct the original data matrix precisely, they are superior to PCA at preserving metabolite correlations.

### Interpretation of VAE latent space dimensions in the context of metabolites and pathways

We evaluated the composition of all latent dimensions in the context of metabolic pathways. For each metabolite in our dataset, a “sub-pathway” and “super-pathway” annotation was available (see Methods). Sub-pathways refer to biochemical processes such as “TCA Cycle” and “Sphingolipid Metabolism”, while super-pathways are broad groups such as “Lipid” and “Amino acid”. To provide insights into the processes represented by different VAE dimensions, we computed SAGE scores26, a measure of model feature relevance, at the level of metabolites, sub-pathways and super-pathways (Fig. 3 and Supplementary Fig. 4). PCA and KPCA results can be found in Supplementary Figs 59.

The VAE sub-pathway heatmap (Fig. 3a) shows that nearly all dimensions have major contributions by lipid and amino acid super-pathways. The prevalence of the two super-pathways can be attributed to the fact that those groups contain the largest number of metabolites in the dataset. Note that we deliberately omitted the “Unknown” molecule group, which refers to unidentified metabolites that could originate from any pathway.

Inspecting the SAGE values in the other direction, almost all sub-pathways are predominantly represented by a single VAE dimension that captures the respective pathway the most (Fig. 3b, red square marks). For instance, “glycolysis, gluconeogenesis and pyruvate metabolism” is functionally related to the sub-pathway pathway “Alanine and Aspartate Metabolism” through the glucose-alanine cycle28, and both were represented by VAE dimension 12. Another interesting example is VAE dimension 14, which captured essential catabolic processes, such as ketone bodies, fatty acid metabolism, and the TCA cycle. Taken together, these results show that VAE latent dimensions capture a complex mix of functionally related sub-pathways, thus capturing major metabolic processes in the dataset.

In contrast, PCA and KPCA dimensions 1 to 5, which by construction represent the highest variations in the data, nonspecifically captured various sub-pathways. Most other PCA and KPCA dimensions also primarily contained unrelated sub-pathways (Supplementary Figs 59).

### VAE latent space captures signals in unseen diabetes, schizophrenia, and cancer metabolomics datasets

We investigated whether VAE latent dimensions learned on the TwinsUK data contained information that is generalizable to other datasets. To this end, we encoded metabolomics data from three clinical datasets, type 2 diabetes, acute myeloid leukemia (AML), and schizophrenia using the VAE, PCA, and KPCA encoders trained on the TwinsUK dataset. For each VAE, PCA, and KPCA latent dimension, we performed a two-sided t-test between diabetic vs. non-diabetic individuals, full vs. no response in an AML clinical trial, and schizophrenic vs. non-schizophrenic individuals, respectively. Across all datasets, the best performing VAE dimensions associated substantially stronger with the patient groups than any of the PCA or KPCA dimensions (Fig. 4a–f). The strength of associations between VAE dimensions and disease parameters were comparable to single metabolite associations (Supplementary Data 1 and 2). However, unlike the VAE dimensions, these univariate associations do not represent system-level mechanisms related to the diseases. Notably, results did not differ substantially for different dimensionalities than d = 18 (Supplementary Fig. 15).

To obtain a better understanding of the driving factors of the VAE associations, we ranked pathways and metabolites by their calculated SAGE values (Fig. 4g–i):

#### Type 2 diabetes

VAE latent dimension 12 showed the highest association with type 2 diabetes, with a considerably stronger signal than the second highest association by PCA dimension 16 (p = 1.6 × 10−22 vs. p = 2.1 × 10−20, respectively; Fig. 4d). The top sub-pathways were “glycolysis, gluconeogenesis, and pyruvate metabolism”, “Lysolipid”, and “Glycine, Serine, and Threonine metabolism”. The top-ranking metabolite in dimension 12 was glucose, which is a main factor directly affected by the disease, and thus served as a positive control. Other high-ranking metabolites included serine, mannose, and glycine (Fig. 4g). Reduced levels of the glucogenic amino acids serine and glycine have repeatedly been reported in Type 2 Diabetes cases as impaired glucose uptake in insulin-resistant cells triggers hepatic gluconeogenesis, consuming these amino acids29,30,31. Furthermore, plasma levels of mannose have also been found to associate with type 2 diabetes diagnosis32. We then correlated dimension 12 with clinical lab measurements from the QMDiab study and found a strong association between this dimension and HbA1c, a widely used diabetes biomarker33,34 (p = 6.2 × 10−45 compared to PCA p = 1.1 × 10−30, and the strongest associated KPCA dimension p = 6.0 × 10−38, Supplementary Figs 1012). This finding demonstrates how a quantitative disease biomarker can carry more information than a crude disease yes/no classification, and further highlights the higher information content in the VAE latent dimensions compared to PCA.

#### Acute myeloid leukemia (AML)

AML response groups associated an order of magnitude stronger with VAE dimension 12 than with RBF KPCA dimension 10 (p = 0.027 vs. p = 0.12, respectively; Fig. 4f). Note that the p-value would not withstand multiple testing correction (Supplementary Data 1); the detected signal is thus merely suggestive and requires replication in future studies. In addition to the glycolytic signals described above, dimension 12 was also driven largely by the “Lipid” super-pathway, with the “Lysolipid” sub-pathway ranking highly for this dimension. The lysolipid sub-pathway has prominent contributions from the metabolite 1-palmitoleyolglycerophosphocholine, belonging to the glycerophosphocholine (GPC) family, which are products of cellular membrane breakdown35. In the clinical trial from which the AML data was taken, the clinical arms were given varying doses of the drug daunorubicin, an anthracycline antibiotic whose mechanism of action includes targeting the cell membrane36,37,38,39. In fact, the localization of daunorubicin in cellular membranes depends on the daunorubicin/lipid ratio, and therefore differential lipid concentrations between patients could affect drug response40,41. In addition, the sub-pathway “Polyunsaturated Fatty Acid (n3 and n6)” was another lipid-related driver of dimension 12. The upregulation of polyunsaturated fatty acids (PUFA) is a trait of anthracyclin treatment42, indicating that differential abundance of PUFA may reflect treatment efficacy and thus response in the clinical trial arms. We furthermore investigated correlations of the latent dimensions with 21 major AML-related mutations; the analysis revealed only one significant result, with the NPM1 mutation associating significantly with polynomial KPCA model dimension 11 (t-test, p < 0.05, Supplementary Figs 13 and 14). However, this genetic association did not translate to the clinical association with drug response.

#### Schizophrenia

VAE dimension 11 had a stronger association with schizophrenia than the second-ranked RBF KPCA dimension 2 (p = 3.0 × 10−6 vs. p = 3.6 × 10−6, respectively; Fig. 4e). The top scoring super-pathway for this dimension (Fig. 4h) was amino acids, with the Phenylalanine and Tyrosine metabolism sub-pathway and its associated metabolites 3-(4-hydroxyphenyl)lactate, phenyllactate (PLA), and tyrosine being highly ranked in this dimension. Phenylalanine and tyrosine are the amino acid precursors for the synthesis of the main neurotransmitter implicated in the neurobiology of schizophrenia, dopamine43. Many studies have shown altered transport of tyrosine across the blood-brain barrier as a prominent mechanism in schizophrenic patients44,45,46. In addition, inflammation, increasingly implicated in schizophrenic patients, reduces the activity of the enzyme responsible for converting phenylalanine to tyrosine, such that phenylalanine is instead converted to phenylpyruvic acid, a precursor metabolite to PLA43,47. Finally, dimension 11 was strongly driven by the amino acid creatine. Studies have shown that schizophrenia patients have significantly reduced brain levels of creatine compared with healthy controls48, and creatine-producing enzymes in serum have long been implicated in schizophrenic psychosis49,50.

Taken together, these results suggest that our VAE has learned representations of metabolic processes that are essential in unseen clinical outcomes.

## Discussion

In this study, we trained a VAE on metabolomics data from the TwinsUK population cohort and applied the learned latent representations on unseen data. Notably, non-linear dimensionality reduction methods, including kernel PCA and our VAE model, outperformed the linear PCA model in metabolite correlation matrix reconstruction, underlining the importance of non-linear relationships in metabolomics data. Interpretation of VAE latent dimensions at the metabolite, sub-pathway, and super-pathway level revealed that these dimensions represent functionally related and distinct cellular processes. Moreover, VAE latent dimensions showed substantially stronger disease associations than PCA and even the non-linear KPCA models in unseen Type 2 Diabetes, AML, and schizophrenia datasets. This demonstrates that the VAE learned a latent representation of metabolomics data that is biologically informative and transferable across different cohorts.

The generalizability of the VAE across different datasets is especially remarkable given the vastly different underlying populations of the datasets analyzed here. The VAE was trained on the TwinsUK population cohort, a European-ancestry population cohort consisting predominantly of British women (~92%), while the validation datasets are mixed-gender and multi-ethnic cohorts from the US and Qatar. Despite the existence of these variations in our datasets, our VAE learned a generalized representation of metabolomics data which was able to identify disease-related differences.

The main limitation of our study is the size of the TwinsUK training dataset with n = 4644. This is a general issue with human subject metabolomics studies, where even the largest cohorts reach only about n = 15,00051. Deep learning models are currently more popular in larger datasets of n = 60,000 samples or more, such as single cell transcriptomic52,53,54,55, image56,57,58, and text sources59. Learning the variation in such large datasets allows these models to significantly outperform their linear counterparts. Large metabolomics datasets, such as that of the UK BioBank with a sample size of up to n = 500,00060, will be available in the near future, and will enable the creation of more expressive and deeper VAE models.

To the best of our knowledge, this is the first study to construct a universal latent representation of large-scale metabolomics data using VAEs. Our results show that VAEs are well-suited for metabolomics data analysis and can potentially replace dimensionality reduction approaches, such as PCA, in creating a universal, systems-level understanding of metabolism.

## Methods

### Datasets

The TwinsUK registry is a population-based study of around 12,000 volunteer twins from all over the United Kingdom. The participants have been recruited since 1992 and are predominantly female, ranging in age from 18 to 103 years old25. For our study, we included data from 4644 twins (4256 females, 388 males), the subset of TwinsUK for which plasma metabolomics measurements were available. Ethical approval was granted by the St Thomas’ Hospital ethics committee and all participants provided informed written consent.

The QMDiab study was conducted between February and June of 2012 at the Dermatology Department of Hamad Medical Corporation (HMC) in Doha, Qatar. The study population was between the ages of 23 and 71, predominantly of Arab, South Asian, and Filipino descent61. For this study, we included plasma data of 358 subjects (176 females, 182 males; 188 diabetic, 177 non-diabetic). The study was approved by the Institutional Review Boards of HMC and Weill Cornell Medicine-Qatar (WCM-Q). Written informed consent was obtained from all participants.

The cohort of patients with acute myeloid leukemia (AML) comes from the ECOG-ACRIN Cancer Research Group phase 3 trial NCT00049517. This study was conducted between December 2002 and November 2008, recruiting 657 patients with AML between the ages of 17 and 60. A subset of these patients had follow-up profiling to determine their response to therapy. For this study, we included the serum metabolomics measurements of 85 subjects of which 43 responded to therapy and 42 did not (34 females, 51 males). The study was approved by the institutional review board at the National Cancer Institute and each of the study centers, and written informed consent was provided by all patients.

For the schizophrenia analysis, metabolomics samples were taken from an antipsychotics study conducted in Qatar between December 2012 and June 201462. A total of 226 participants between the ages of 18 and 65 years of age were recruited, predominantly of Arab descent. For our study, we included plasma metabolomics measurements from 201 subjects (76 females, 125 males; 97 schizophrenic, 104 non-schizophrenic). Approval for the study was obtained from the HMC and WCM-Q Institutional Review Boards, and all participants provided written informed consent.

### Metabolomics measurements and metabolite annotations

Metabolic profiling for all four cohorts was performed using non-targeted ultrahigh-performance liquid chromatography and gas chromatography separation, coupled with mass spectrometry on the Metabolon Inc. platform as previously described63. Notably, the AML dataset was based on serum samples, while TwinsUK, QMDiab, and schizophrenia metabolomics were run on plasma samples. However, previous studies have shown that these two sample types are comparable, as shown by high correlations and good reproducibility between plasma and serum measurements in the same blood sample64.

For each metabolite measured on the Metabolon platform, a super-pathway and sub-pathway annotation was provided. For super-pathways, we have nine annotations referring to broad biochemical classes, namely “Amino acid”, “Carbohydrate”, “Cofactors and vitamins”, “Energy”, “Lipid”, “Nucleotide”, “Peptide”, “Xenobiotics”, and “Unknown”. Note that “Unknown” is assigned to unidentified metabolites. Furthermore, we have 54 sub-pathway which represent more functional metabolic processes, such as “Carnitine metabolism”, “TCA Cycle”, and “Phenylalanine and Tyrosine Metabolism”.

### Data processing and normalization across datasets

For each dataset, metabolite levels were scaled by their cohort medians, quotient normalized58 and then log-transformed. Samples with more than 30% missing metabolites and metabolites with more than 10% missing samples were removed. Missing values were imputed using a k-nearest neighbors imputation method65. Datasets with BMI measurements (Schizophrenia, QMDiab, and Twins) were corrected for that confounder and then mean-scaled. 217 metabolites were overlapping between the 4 datasets and were kept for further analysis.

Semi-quantitative, non-targeted metabolomics measurements are inherently challenging to compare across datasets due to heterogeneity between studies. This prevents any machine learning model from being directly transferable from one study to the other. Thus, to ensure comparability, datasets were normalized using a uniform group of participants as a reference set. This group was selected as follows: Male, within a 20-year age range (30–50 for TwinsUK, QMDiab, and schizophrenia, 40–60 for AML due to low sample size of younger participants), BMI between 25 and 30 (not available for AML data, thus not filtered for that dataset), and in the respective control group. Each metabolite in each dataset was then scaled by the mean and standard deviation of their respective uniform sample groups. The assumption of this approach is that the uniform group of reference participants has the same distributions of metabolite concentrations.

### Variational autoencoder fitting

To train our VAE model, we first split the TwinsUK data into 85% training and 15% test sets. We then fixed our VAE architecture to be composed of an input/output layer, an intermediate layer which contains non-linear activation functions, and a d-dimensional latent layer. The latent layer consists of a mean vector μ and a standard deviation vector σ, both of length d, which parametrize each latent dimension as a Gaussian probability distribution. This latent space, denoted by z, is constructed by the simultaneous learning of the μ and σ encoder through the use of a reparameterization trick that enables back propagation during training18. The d x d covariance matrix ∑ of the underlying multivariate Gaussian is assumed to be diagonal (i.e., no correlation across latent dimensions), allowing the covariance matrix to be represented by a single vector σ of length d.

For the parameter fitting procedures, all weights were initialized using Keras’ default model weight initialization, i.e., Glorot uniform66. Leaky rectified linear units (ReLUs)67 were used for non-linear activation functions. The VAE models were trained for 1000 epochs using MSE loss for sample reconstruction and a batch size of 32.

To select the latent dimensionality d of our VAE model, we initially fixed this value to d = 50. We then optimized the model hyperparameters using Keras Tuner27 and the TwinsUK training set with MSE as the objective metric to minimize. We identified the following optimized values: Intermediate layer dimensionality = 200, learning rate = 0.001, and Kullback-Leibler (KL) divergence weight = 0.01. Note that despite our hyperparameter choices, other optimal hyperparameters exist and could be chosen through Keras Tuner. Using these hyperparameters, we then optimized d by calculating the reconstruction MSE of the correlation matrix (CM-MSE) of metabolites for d = 5, 10, 15, 18, 20, 30, 40, 60, 80, 100, 120, 160, and 200 on the TwinsUK test set. Our final model consisted of a 217-dimensional input/output layer (the number of metabolites in our datasets), a 200-dimensional intermediate layer, and an 18-dimensional latent layer. For all sample encodings in the study, we used their respective μ values.

We further performed sensitivity analysis to determine the effect of the choice of d on clinical parameter association. We trained 100 models using the architecture above, varying the latent layer dimensionality for d = 10, 13, 15, 16, 17, 18, and 20. For each model, we extracted the p-value of the highest associated latent dimension, creating a p-value distribution for each d, which was then compared to the PCA and KPCA associations (Supplementary Fig. 15).

All models were computed on a deep learning-specific virtual machine running on Google Compute Engine with two NVIDIA Tesla K80 GPU dies and 10 virtual CPUs.

### PCA and kernel PCA embedding and reconstructions

We used PCA with d = 18 latent dimensions as a baseline model. On the mean-centered TwinsUK training set data matrix with n = 3947 samples (rows) and k = 217 metabolites (columns), we calculated the rotation matrix Q, a k x k matrix of eigenvectors ordered by decreasing magnitudes of eigenvalues. To embed a new m x k dataset X with m samples into the m x d PCA latent space A, we first calculated XQ = A and subsetted to the first d columns, denoted by A, d. To simulate the process of encoding and decoding in PCA for dataset X, we calculated the reconstructed dataset as

$${\hat{\boldsymbol X}}={\boldsymbol A}_{\ast},{}_{d}{{\boldsymbol Q}^{-1}}_{d,\ast}$$
(1)

We used kernel PCA with d = 18 latent dimensions as comparative non-linear models. Using the Scikit-learn python package68, we applied the four available non-linear kernels to the mean-centered TwinsUK training set data matrix: cosine similarity, sigmoid, radial basis function (RBF), an d polynomial transformations. KPCA hyperparameters were chosen for each individual kernel using the grid search method based on their ability to minimize model MSE. Once transformed, the decomposition, embedding, and encoding/decoding processes were performed identically to those of linear PCA.

### Model assessments

We assessed our KPCA, PCA and VAE models using sample reconstruction mean squared error (MSE) and metabolite-wise correlation matrix MSE (CM-MSE). We calculated CM-MSE by first computing the metabolite-wise correlation matrix of an input dataset and reconstructed input dataset. Afterwards, we calculated the MSE between the upper triangular matrix of the two symmetric correlation matrices.

To calculate a confidence interval for both MSE and CM-MSE between our input and reconstructed data, we randomly sampled the same samples with replacement from the two datasets and then calculated MSE and CM-MSE. We performed this for 1000 iterations.

### Model interpretation

In order to interpret each latent dimension for our VAE, PCA, and KPCA models, we calculated Shapley Additive Global Importance (SAGE) values26 for metabolites, sub-pathways, and super-pathways. As contributions of metabolites, sub-pathways, and super-pathways to latent dimensions can vary between multiple trained VAE models, this analysis was performed on a VAE model with average performance from the clinical association analysis. Briefly, SAGE is a model-agnostic method that quantifies the predictive power of each feature in a model while accounting for interactions between features. This is achieved by quantifying the decrease in model performance when combinations of model variables are removed. Since there are exponentially many combinations of variables, a common approach is to sample the feature combination space sufficiently. For each of the tested combinations, a loss function, such as MSE, is used to quantify the decrease in performance compared to the model output (here each VAE, PCA, and KPCA latent dimension) computed using the full model. Then, the mean of all MSEs is calculated, which represents the contribution of the model variables to a latent dimension. To calculate pathway-level SAGE values, metabolites were grouped into pathways and each pathway was treated as a single variable. For each of our VAE, PCA, and KPCA models, we ran SAGE using our TwinsUK test set with default parameters, e.g., marginal sampling size of 512, as suggested by Covert, et al. 26. We used the SAGE code from https://github.com/iancovert/sage.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.


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# Two independent samples

Description of activity and Excel instructions:Two independent samples

Dataset: sleep.xls

## Always look at the data

Even though the sample sizes are large and OLI does not instruct you to create descriptive statistics, it's a good idea to do so anyway:

• At a minimum, calculate means and standard deviations for each sample.
• Create a histogram for each sample.

## Calculate t-test value and p-value

Excel has a built in data analysis package that runs basic statistical analyses, which Calc does not have. However, it's only a few extra steps in Calc to do the calculations for this test.

It is important to review the summary statistics for the two samples of data.

• Calculate the mean and standard deviation for each group.

Calc does not report the t-test value. You can program the formula in Calc, but I don't recommend taking the time to do so unless you are interested in this kind of thing. The formula for this test is:

$t = {\overline{y}_1 - \overline{y}_2 \over s_{\overline{y}_1 - \overline{y}_2}}$

where

$s_{\overline{y}_1 - \overline{y}_2} = \sqrt{{s_1^2 \over n_1} + {s_2^2 \over n_2}}.$

The calculated t-test statistic for this data is -1.23042.

Calc provides a formula for calculating the p-value which can be used with a few types of t-tests. Type=3 is for use in this situation: unequal samples sizes and unequal variances. Note that Mode=1 indicates a one-sided test:

• To calculate the p-value, enter the formula "=TTEST([range y1],[range y2],1,3)".

Return to the OLI Two Independent Samples page to complete the interpretive exercises.


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# Understanding ergodic capacity: how to prove : two channels c1, c2 - capacities, but we do not know what channel has c1, what c2 nevertheless capacity is c1+c2

As far as I understand if we have channel which has random capacity for each discrete time moment (for example Rayleigh fading ), then

the capacity is just average capacity.

I am trying to understand it.

It seems the following simpliefied situation should be understand first: consider the two channels assume we know that capacity of one of them is c1 , capacity of another is c2, but we do not know what channel has c1, what channel has c2. Nevetheless it seems that capacity is c1+c2, the same as as if we know what channel has c1, what channel has c2...

For example if we consider AWGN channels - can one suggest some simple proof of this ?

(Similar to proof of capacity for AWGN with balls filling )

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# Thread: Showing Transformation is True

1. ## Showing Transformation is True

I'm confused by this question:
A transformation T in the plane consists of a reflection in the line x + y = 0 followed by a translation in which the point (x,y) is transformed to the point (x+h,y+k).

(a). Show that the matrix representing T is

$\displaystyle \begin{bmatrix} 0 & -1 & h \\ -1 & 0 & k \\ 0 & 0 & 1 \end{bmatrix}$
The question only mentions x and y so I assume this is supposed to be an $\displaystyle R^2$ transformation but the matrix is a 3x3.. don't see how the multiplication would work.

I tried transforming the unit square but didn't do much to help me understand.

Thanks

2. this is in $\displaystyle R^3$

T transforms (x,y,z) to (-y+zh,-x+zk,z) so that the z coordinate is unchanged

the first two rows reflect in the line y=-x and then add zh to the x and zk to the y coordinate respectively, the third row leaves the z coordinate unaltered.

if (x,y) is transformed to (x+h,y+k) then we are in the plane z=1.

So i guess
$\displaystyle \begin{bmatrix} 0 & -1 & -h \\ -1 & 0 & -k \\ 0 & 0 & 1 \end{bmatrix}$

is the same transformation but in the plane z=-1

3. This is an example of "projective coordinates". Essentially, a point (x, y) is represented by (x, y, 1) with the provision that the triple (x, y, a) represents the same point as (x/a, y/a, 1).

Working in two dimensions, we can represent a rotation about the origin, through angle $\displaystyle \theta$ as a matrix multiplication:
$\displaystyle \begin{bmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos( theta)\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}x cos(\theta)- y sin(\theta) \\ x sin(\theta)+ y cos(\theta)\end{bmatrix}$
but translations have to be represented as a vector addition:
$\displaystyle \begin{bmatrix}x \\ y\end{bmatrix}+ \begin{bmatrix}h \\ k\end{bmatrix}$$\displaystyle = \begin{bmatrix} x+ h \\ y+ k\end{bmatrix}$

Using projective coordinates we can combine both into a single matrix:
$\displaystyle \begin{bmatrix} cos(\theta) & -sin(\theta) & h \\ sin(\theta) & cos(\theta) & k \\ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}x \\ y \\ 1\end{bmatrix}= \begin{bmatrix}x cos(\theta)- y sin(\theta)+ h \\ x sin(\theta)+ y cos(\theta)+ k \\ 1\end{bmatrix}$

"Projective Coordinates" are derived from "Projective Geometry". If, in three dimensional Euclidean geometry, you define two points to be "equivalent" if and only if they lie on the same straight line through the origin and define "points" in your new geometry to be the equivalence classes, then you have the "Projective Plane".

4. Thanks for the reply, looking over it now. Think I need to read over this a few thousand more times.