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# what is the meaning of 'tail' of kurtosis?

There are two kurtosis types : positive(leptokurtic) and negative(platykurtic). leptokurtic is heavy tailed, and platykurtic is thin tailed. But leptokurtic is more thinner and pointy than platykurtic so I think leptokurtic is thin tailed... But it is heavy-tailed. So I found that I misunderstood the meaning of 'tail'

What does the tail mean?

• Can you show an example of what you mean by leptokurtic being thin-tailed (compared to some platykurtic case)? – Glen_b -Reinstate Monica Sep 15 '15 at 0:29

## 3 Answers

There are two parts to address here -- 1. what does it mean for something to be heavy-tailed? and 2. does higher kurtosis mean a heavier tail and vice-versa?

1. What's heavy-tailed mean?

a. What heavier tail means in a "handwavy" sense -- most people picture it this way:

but because the tail is quite small, it's better to look at the log-density (which preserves the ordering of height of density)

(that handwavy sense fails, however, in a variety of cases, including the rather basic situations of when you try to encompass distributions which are not continuous and unimodal)

b. More strictly, a good definition (about which, see whuber's answer here) for heavy-tailedness would be that if $Y$ is heavier-tailed than $X$, as $t$ becomes sufficiently large, then $S_Y(t)>S_X(t)$ for all $t>t_0$ fr some $t_0$, where $S$ is the survivor function, $1-F$. [This is of course, for heavier-tailed on the right; there'd be a similar definition for left-tail heavy tailedness in terms of $F$ rather than $S$. When both tails are under consideration, as they would need to be for comparing kurtosis, "heavier tailed" would apply if it was heavier in both tails.]

(Again, however, if you're trying to look at it, a log-scale comparison would often be more useful than a direct comparison of $S(t)$)

2. How does heavy-tailedness relate to kurtosis?

Now we have a definition of what a heavy tail is, the premise (which is not your own, since it's in countless textbooks) in your question is false -- there is no absolutely general connection between heavy-tailedness and higher 4th standardized moment. It tends to be the case that higher kurtosis goes with heavier tail and vice-versa -- we see that when comparing a t(5) with a normal, for example -- but it's not always the case; one can readily find a lighter-tailed distribution by that definition in 2.b. which nevertheless has higher kurtosis. See this example -- t with 10d.f. vs Laplace:

(in this case, because $f$ is always higher past a certain point, the survivor function must also be higher at least in that region so we can just stick with density for the comparison. Note that these two densities cross six times. the Laplace has a higher tail between about 2.5145 and 12.8, as well as a higher peak; that gives it higher kurtosis even though its extreme tail is lower)

• There is not just one measure of tails. E{Z^4*I(|Z| >1)} is one, E{Z^10 *I(|Z| > 5} another. For a sequence of distributions where kurtosis tends to infinity, E{Z^4 *I(|Z| > b)}/kurtosis -> 1, for any b; hence, large kurtosis is mostly determined by the tail. Further, the contribution of |Z| <1 to kurtosis is small, as noted below, for all distributions. Hence, kurtosis itself is a measure of tailweight, although not identical to other tailweight measures. Similarly, other tailweight measures do not measure the same thing as kurtosis. There are, after all, infinitely many tailweight measures. – Peter Westfall Nov 9 '17 at 13:19
• Also, the given "definition" of heavier-tailed is actually quite silly. By that definition, the N(0,1) distribution is heavier-tailed than the .9999*U(-1,1) + .0001*U(-1000,1000) distribution. The latter distribution is extremely outlier-prone and has very high kurtosis, the former is not outlier-prone at all and has low kurtosis. The argument is a complete red herring, because it is not even a logical measure of "tailedness". – Peter Westfall Nov 20 '17 at 0:27

Comments from probably deleted duplicate question that are useful:

In fact, higher kurtosis is associated with both increased peakedness and heavier extreme tails, but there's no necessary relationship in either case (you can find counterexamples to higher peak or heavier extreme tails, even though both are typical with higher kurtosis). A variety of shapes is possible in either case. – Glen_b♦ Sep 17 '13 at 11:10

Kendall and Stuart (I think in Vol I) present a collection of counterexamples to the various combinations of peak and tails. See also the discussion here – Glen_b♦ Sep 17 '13 at 11:16

Also see the paper by Kevin P. Balanda and H.L. MacGillivray. "Kurtosis: A Critical Review". The American Statistician 42:2 [May 1988], pp 111–119. (And yes, generally speaking, fat-tailed and heavy-tailed are synonyms, except for contexts where one or both has been given an explicit definition.) – Glen_b♦ Sep 17 '13 at 11:23

You might want to read the following paper: DeCarlo, L. T. (1997). On the meaning and use of kurtosis. Psychological Methods, 2(3), 292. – Alecos Papadopoulos Sep 17 '13 at 23:31

A caveat about the DeCarlo paper - the very first sentence of the abstract is wrong! Even for symmetric, unimodal distributions, high kurtosis does not mean "peakedness," and low kurtosis does not mean flatness. There are symmetric, unimodal flat-topped distributions with extremely high (even infinite) kurtosis; there are also infinitely peaked distributions with negative excess kurtosis. See specific examples given here: en.wikipedia.org/wiki/… – Peter Westfall Oct 21 '17 at 12:20

Glen_b, your "you can find counterexamples to heavier extreme tails, though both are typical with higher kurtosis" is only correct if you define "heavier extreme tails" in a particular way out of infinite ways. There is no counterexample to the mathematical facts that (i) as kurtosis tends to infinity, E(Z^4*I(|Z| >b))/kurtosis ->1, for all b; and (ii) kurtosis is within +- .5 of E(Z^4*I(|Z| > 1)) +.5; both of these statements explain how kurtosis is related to tails of the distribution, and not the peak. If you define "heavier tails" as larger "E(Z^4*I(|Z| > 1))" then there you have kurtosis. – Peter Westfall Nov 15 '17 at 23:32

Glen_b, your comment "In fact, higher kurtosis is associated with both increased peakedness ..." seems patently false in the face of numerous counterexamples. Can you provide a theorem to justify it? – Peter Westfall Nov 20 '17 at 1:43

The above, and the question itself are rooted in what kurtosis is. It may be useful to review a physical problem for which kurtosis is defined. For that, the fourth moment about the mean is analogous to a thin sheet of material rotated about its mean mass, or vertical balance point, as a rate of change of a rate of change of torque. This says very little about the thin sheet's height at any particular distance from the axis of rotation, but rather is an integral behaviour over the whole sheet.

• Very helpful collection of notes (+1). Often overlooked is Kaplansky, I. 1945. A common error concerning kurtosis. Journal, American Statistical Association 40: 259 only. This gave concrete examples showing that higher and lower kurtosis (in the specific sense of dimensionless ratio based on fourth and second moments around the mean) can contradict any word summary based on peaks or tails. If the name seems familiar somehow, Irving Kaplansky is better known as an outstanding algebraist (groups, rings, and what not). – Nick Cox Oct 20 '18 at 9:54
• Nick, this is not a correct characterization of Kaplansky's paper as regards tails. He mentions peak only, and nothing about tails. Specifically, he showed that, among distributions having mean=0 and variance=1, larger kurtosis does not correspond to a higher peak. As you know, I gave three theorems in my 2014 TAS paper relating kurtosis to a distribution's tails, and I mention a fourth such theorem in my post above. So indeed, kurtosis measures tails. – Peter Westfall Oct 25 '18 at 18:46
• @PeterWestfall The greater the moment, the greater the tail bias from the integral definition of the $n^{\text{th}}$ moment; $\int (x-\mu)^n f(x) dx$, provided that the range of $|x-\mu|>1$. When the range of $|x-\mu|<1$, the opposite should be true. – Carl Oct 26 '18 at 22:46
• Hi Carl. State as a theorem (or conjecture) please? In particular, "tail bias" does not seem to be defined. You might check my various theorems and counterexamples first though. – Peter Westfall Oct 27 '18 at 13:07
• @PeterWestfall Please provide a link to you paper. I used the word bias improperly for stats, I get it from more from EE, apologies. In the region for which $|x-\mu|>1$ then $|(x-\mu)^{n+1})|>|(x-\mu)^n|$, i.e., the weighting of the tail is relatively heavier. This outlines a tendency for tail influence. To see if this influence dominates $f(x)$ behaviour for $n$ depends on all of $f(x)$, not just $|x-\mu|>1$. – Carl Oct 27 '18 at 19:59

There is most definitely a direct relationship connection between kurtosis and the tails of distribution. One simple connection is as follows: Let $$X$$ be any random variable with finite fourth moment, let $$Z = (X - \mu)/\sigma$$, and let $$U = Z^4$$. Then the kurtosis of $$X$$ is $$E(U)$$.

A common way of teaching and understanding expectation is using the "point of balance" concept, where the expected value of a random variable is equal to the point on the horizontal axis where the distribution plot balances.

Consider the plot of the probability distribution function (pdf) of $$U$$, and locate the value 3.0 on the horizontal axis. This is the kurtosis of the normal distribution. If the pdf is too heavy to the right of 3.0, so that it is pulled down to the right when attempted to balance at 3.0, then the kurtosis is greater than 3.0, and the distribution of $$X$$ can be said to be "heavier tailed than the normal distribution." Conversely, if the pdf of $$U$$ is too light to the right of 3.0, so that the curve is pulled down to the left when attempted to balance at 3.0, then the kurtosis is less than 3.0, and the distribution of $$X$$ can be said to be "lighter tailed than the normal distribution."

In light of the above discussion, "tail leverage" is perhaps a better characterization of kurtosis than "tail mass." It is possible that more mass in the tails corresponds to less kurtosis, and that less mass in the tails corresponds to greater kurtosis. It all depends on the remoteness of the mass; i.e., it depends on the leverage.

Precise mathematical logic showing that kurtosis is mostly determined by the tails is given as follows:

For all possible distributions of random variables $$X$$ having finite fourth moment (discrete, continuous, skewed, empirical), kurtosis is necessarily between

$$\int Z^4 I(|Z| > 1) dF(Z)$$

and

$$\int Z^4 I(|Z| > 1) dF(Z) + 1$$,

where $$Z = (X-\mu)/\sigma$$.

For continuous distributions in this class with density of $$Z^2$$ decreasing on the interval $$[0,1]$$, the "$$+1$$" is reduced to "$$+.5$$" or less.

Finally, for any sequence of distributions where the kurtosis $$k$$ tends to infinity,

$$\int Z^4 I(|Z| > b) dF(Z) / k \rightarrow 1$$,

for all real $$b$$.

These are all mathematical theorems so there can be no counterexamples.

Thus, it is primarily the values in the tails of the $$Z$$ distribution that determine the value of kurtosis.

Yes, there are other tailweight measures that are not kurtosis, in fact there are infinitely many. So it is a red herring to state that kurtosis does not measure tailweight if it does not correspond to a particular tailweight measure. It is an unsupported generalization to say that, just because kurtosis does not measure one of infinitely many tailweight measures, that kurtosis itself does not measure tailweight.

I have seen no theorems generally linking kurtosis to the peak, or to the probability concentration within the $$\mu \pm \sigma$$ range, but I have seen plenty of counterexamples to such notions. Can anyone point to such a theorem?

The three theorems I mentioned above are proven in Westfall, P.H. (2014). Kurtosis as Peakedness, 1905 – 2014. R.I.P. The American Statistician, 68, 191–195. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4321753/

Edit, 7/18/2018: Yet another theorem to connect kurtosis to the tails (and not the peak). Take any distribution (or data set). Remove the data or mass within one standard deviation of the mean. Replace the data/mass in any way you like, but keep the mean and standard deviation the same as with the original data/distribution. Then the difference between the maximum and minimum kurtosis for all such replacements is $$\le$$ 0.25.

If you play the same game with the tails, the difference is unbounded.

• Another direct connection of kurtosis to tail is this: Consider the (discrete or continuous) distribution of $V = Z^4$, say $p(v)$. (Here, $Z = (X-\mu)/\sigma$, where $X$ is the original data or random variable.) The function $p(v)$ balances precisely at the kurtosis of the distribution of $X$. Place a fulcrum at 3.0 under the graph of $p(v)$; 3.0 is the kurtosis of the normal distribution. If the graph of $p(v)$ falls to the right of the fulcrum at 3.0, then the pdf of $X$ is heavier-tailed than the normal distribution. If it falls to the left, then it is lighter-tailed. – Peter Westfall Aug 8 '19 at 13:54


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## § Weird canonical example of monic and epic: left/right shift

• Consider the function right over an infinite sequence a_n which is defined as right(a[:])[i] = 0 if i == 0 else a[i-1]. That is, it shifts a sequence to the right.
• See that this is injective, and not surjective: for example, there is no pre-image to any sequence that starts with a non-zero value, such as 1, 0, 0, ....
• Its dual, left(a[:])[i] = a[i+1] is surjective, but not injective.
• This makes it ideal as an "extreme case" to test the monic/epic conditions as left/right cancellable.

#### § Monic

• We know that right is monic. Is it cancellable if we run it before or after?
• We should run it after --- that way, if right(f(a[:])) equals right(g(a[:])) for all a[:], we know that the sequences are (0, f1, f2, ...) which equals (0, g1, g2, ...) which implies (f1, f2, ...) equals (g1, g2, ...). So we can conclude that f = g from right . f = right . g.
• On the other hand, if we consider f(right(a[:])) and g(right([a:]), we will only test the equality of f and gat sequences of the form (0, a1, a2, ...) which is insufficient, this we cannot conclude f = g. So we cannot conclude f = gfrom f . right = g . right.

#### § Epic

• We know that left is epic. Is it cancellable if we run it before or after?
• Suppose f . left = g . left. Since left is epic, this tests f and g on every possible input. Thus f = g.
• On the other hand, suppose left . f = left . g. This is insufficient, since we will only test the equality of (f2, f3, ...)with (g2, g3, ...) leaving f1 =? g1 untested. Thus, we cannot conclude f = g from left . f = left . g.

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# 23.4 Electric generators

 Page 1 / 6
• Calculate the emf induced in a generator.
• Calculate the peak emf which can be induced in a particular generator system.

Electric generators induce an emf by rotating a coil in a magnetic field, as briefly discussed in Induced Emf and Magnetic Flux . We will now explore generators in more detail. Consider the following example.

## Calculating the emf induced in a generator coil

The generator coil shown in [link] is rotated through one-fourth of a revolution (from $\theta =0º$ to $\theta =\text{90º}$ ) in 15.0 ms. The 200-turn circular coil has a 5.00 cm radius and is in a uniform 1.25 T magnetic field. What is the average emf induced?

Strategy

We use Faraday’s law of induction to find the average emf induced over a time $\Delta t$ :

$\text{emf}=-N\frac{\Delta \Phi }{\Delta t}\text{.}$

We know that $N=\text{200}$ and $\Delta t=\text{15}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{ms}$ , and so we must determine the change in flux $\Delta \Phi$ to find emf.

Solution

Since the area of the loop and the magnetic field strength are constant, we see that

$\Delta \Phi =\Delta \left(\text{BA}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\theta \right)=\text{AB}\Delta \left(\text{cos}\phantom{\rule{0.25em}{0ex}}\theta \right)\text{.}$

Now, $\Delta \left(\text{cos}\phantom{\rule{0.25em}{0ex}}\theta \right)=-1\text{.}0$ , since it was given that $\theta$ goes from $\text{0º}$ to $\text{90º}$ . Thus $\Delta \Phi =-\text{AB}$ , and

$\text{emf}=N\frac{\text{AB}}{\Delta t}.$

The area of the loop is $A={\mathrm{\pi r}}^{2}=\left(3\text{.}\text{14}\text{.}\text{.}\text{.}\right)\left(0\text{.}\text{0500}\phantom{\rule{0.25em}{0ex}}\text{m}{\right)}^{2}=7\text{.}\text{85}×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}$ . Entering this value gives

$\text{emf}=\text{200}\frac{\left(7\text{.}\text{85}×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}\right)\left(1\text{.}\text{25}\phantom{\rule{0.25em}{0ex}}\text{T}\right)}{\text{15}\text{.}0×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}\text{s}}=\text{131}\phantom{\rule{0.25em}{0ex}}\text{V.}$

Discussion

This is a practical average value, similar to the 120 V used in household power.

The emf calculated in [link] is the average over one-fourth of a revolution. What is the emf at any given instant? It varies with the angle between the magnetic field and a perpendicular to the coil. We can get an expression for emf as a function of time by considering the motional emf on a rotating rectangular coil of width $w$ and height $\ell$ in a uniform magnetic field, as illustrated in [link] .

Charges in the wires of the loop experience the magnetic force, because they are moving in a magnetic field. Charges in the vertical wires experience forces parallel to the wire, causing currents. But those in the top and bottom segments feel a force perpendicular to the wire, which does not cause a current. We can thus find the induced emf by considering only the side wires. Motional emf is given to be $\text{emf}=\mathrm{B\ell v}$ , where the velocity v is perpendicular to the magnetic field $B$ . Here the velocity is at an angle $\theta$ with $B$ , so that its component perpendicular to $B$ is $v\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta$ (see [link] ). Thus in this case the emf induced on each side is $\text{emf}=\mathrm{B\ell v}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta$ , and they are in the same direction. The total emf around the loop is then

$\text{emf}=2\mathrm{B\ell v}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta \text{.}$

This expression is valid, but it does not give emf as a function of time. To find the time dependence of emf, we assume the coil rotates at a constant angular velocity $\omega$ . The angle $\theta$ is related to angular velocity by $\theta =\mathrm{\omega t}$ , so that

$\text{emf}=\text{2}\mathrm{B\ell v}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\mathrm{\omega t}\text{.}$

Now, linear velocity $v$ is related to angular velocity $\omega$ by $v=\mathrm{r\omega }$ . Here $r=w/2$ , so that $v=\left(w/2\right)\omega$ , and

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text:
# Understanding relative compactness of a sequence of functions

I'm trying to understand the following definition of pre-compactness (from Wikipedia):

A relatively compact subspace (or relatively compact subset, or precompact) $Y$ of a topological space $X$ is a subset whose closure is compact. Since closed subsets of a compact space are compact, every subset of a compact space is relatively compact. In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence in $Y$ has a subsequence convergent in $X$.

I'm thinking of $X$ as a functional space, e.g., the space of all right continuous functions with left limits and of $Y$ as a sequence of functions in $X$ indexed by $n$, say $\{f^{n}(t)\}$.

In this case, according to the definition, the criterion for relative compactness is that ${\it any}$ sequence in $Y$ has a convergent subsequence. But what does that mean in the above case where $X$ itself is a sequence of functions? Is it simply that the sequence $\{f^{n}(t)\}$ must have a convergent subsequence? Could you give an example where the ${\it any}$ part makes sense?

As the space of all right continuous functions with left limits, I believe $X$ is not necessarily a metric space, since it is not generally bounded. Therefore, the equivalence between sequential compactness and compactness does not necessarily hold for $Y$.
If you add the condition that the functions in $X$ be bounded, then then space can be expressed as a metric space, equipped with the $sup$ norm
By construction, subset $Y$ itself is a sequence of functions where $Y$ is defined as $\{f^{n}(t)\}$ and so you are correct in that $\{f^{n}(t)\}$ must have a convergent subsequence.
Note $\{f^{n}(t)\}$ itself need not converge as it is the closure of $Y$ that is compact.
• Thanks a lot for your answer. Yes, I'm concerned with a specific set of functions that are bounded on compact sets. I'm still not sure however what the ${\it any}$ subsequence means in the definition? Would that make sense when $Y$ is a certain family of functions and not necessarily a sequence of functions? In that case what would be those subsequences? Could you give an example? – Submartingale Sep 15 '16 at 3:02


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# Flux problem

1. Apr 27, 2008

### kasse

1. The problem statement, all variables and given/known data

Evaluate the surface integral of F*n (the flux) where n is the upward-pointing unit normal vector to the plane z = 3x+2 that lies within the cylinder x^2 + y^2 = 4. F = <0, 2y, 3z>

2. The attempt at a solution

The normal vector is <3,0,1> so that F*n = 3z. dS equals sqrt(10). I transform to polar coordinates and integrate 3*sqrt(10)*(3r^2*cos(theta) + 2r) over the cylinder. The answer I get is 24sqrt(10)*pi, while the correct answer is 24*pi. Does this mean that dS=1? Why?

2. Apr 27, 2008

### Dick

You said n is a unit vector. Then you said n=<3,0,1>. That's not a unit vector. <3,0,1>/sqrt(10) is.

3. Apr 28, 2008

### HallsofIvy

Staff Emeritus
Time for me to rush to battle again! I fight constantly against the very notation "$\vec{f}\cdot\vec{n}d\sigma$" because if you follow that literally, computing $\vec{n}$ and $d\sigma$ separately you wind up calculating the length of a vector twice and then watch them cancel out! (Unless you forget one of them!)

In this problem, we can write the position vector of a point on the plane as $\vec{r}(x,y)= x\vec{i}+ y\vec{j}+ (3x+ 2)\vec{k}$.
Differentiating: $\vec{r}_x= \vec{i}+ 3\vec{k}$ and $\vec{r}_y= \vec{j}$. The cross product of those two vectors, $-3\vec{i}+ \vec{k}$, is the "fundamental vector product" for the plane and the vector differential of area, $d\vec{\sigma}$ is $(-3\vec{i}+ \vec{k})dxdy$.

Now $\int\int \vec{F}\cdot\vec{d\sigma}$ $= \int\int (2y\vec{j}+ 3z\vec{k})\cdot(-3\vec{i}+ \vec{k})dxdy$ $= \int\int 3z dxdy= 3\int\int (3x+ 2)dxdy$.

Integrate that over the circle of radius 2.

Obviously the way to do that is to switch to polar coordinates. We could do that right at the beginning:
write $\vec{r}(r,\theta)= rcos(\theta)\vec{i}+ rsin(\theta)\vec{j}+ (3rcos(\theta)+ 2)\vec{k}$, differentiate with respect to r and $\theta$ and the "r" you need for rdrd$\theta$ pops out automatically!

[Here endeth the sermon]

4. Apr 28, 2008

### Dick

Amen. It is clearer to keep the volume and the direction information in the same package.


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Subscribe
Issue No.12 - December (1999 vol.10)
pp: 1252-1261
ABSTRACT

Abstract—The main results of this paper are algorithms for time-optimal gossip of large packets in noncombining full-duplex all-port 2-D tori and meshes of any size <tmath>$m\times n$</tmath>. The gossip algorithms define the structure of broadcast trees and lock-step scheduling schemes for packets that make the broadcast trees time-arc-disjoint. The gossip algorithm for tori is also buffer-optimal—it requires routers with auxiliary buffers for at most three packets. The gossip algorithm for meshes requires routers with auxiliary buffers for <tmath>$O(m+n)$</tmath> packets.

INDEX TERMS
All-to-all broadcast, gossip, 2D torus, 2D mesh, time-arc-disjoint broadcast trees, buffer requirements.
CITATION
Michal Soch, Pavel Tvrdík, "Time-Optimal Gossip of Large Packets in Noncombining 2D Tori and Meshes", IEEE Transactions on Parallel & Distributed Systems, vol.10, no. 12, pp. 1252-1261, December 1999, doi:10.1109/71.819947


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# Debugging Auto-Device Plugin¶

## Using Debug Log¶

In case of execution problems, just like all other plugins, Auto-Device provides the user with information on exceptions and error values. If the returned data is not enough for debugging purposes, more information may be acquired by means of ov::log::Level.

There are six levels of logs, which can be called explicitly or set via the OPENVINO_LOG_LEVEL environment variable (can be overwritten by compile_model() or set_property()):

ov::Core core;

// compile a model on AUTO and set log level to debug
ov::CompiledModel compiled_model = core.compile_model(model, "AUTO", ov::log::level(ov::log::Level::DEBUG));

// or set log level with set_property and compile model
core.set_property("AUTO", ov::log::level(ov::log::Level::DEBUG));
ov::CompiledModel compiled_model2 = core.compile_model(model, "AUTO");

    core = Core()
# compile a model on AUTO and set log level to debug
compiled_model = core.compile_model(model=model, device_name="AUTO", config={"LOG_LEVEL":"LOG_DEBUG"});
# set log level with set_property and compile model
core.set_property(device_name="AUTO", properties={"LOG_LEVEL":"LOG_DEBUG"});
compiled_model = core.compile_model(model=model, device_name="AUTO");

When defining it via the variable,
a number needs to be used instead of a log level name, e.g.:

Linux
export OPENVINO_LOG_LEVEL=0

Windows
set OPENVINO_LOG_LEVEL=0

The property returns information in the following format:

[time]LOG_LEVEL[file] [PLUGIN]: message

in which the LOG_LEVEL is represented by the first letter of its name (ERROR being an exception and using its full name). For example:

[17:09:36.6188]D[plugin.cpp:167] deviceName:MYRIAD, defaultDeviceID:, uniqueName:MYRIAD_

## Instrumentation and Tracing Technology¶

All major performance calls of both OpenVINO™ Runtime and the AUTO plugin are instrumented with Instrumentation and Tracing Technology (ITT) APIs. To enable ITT in OpenVINO™ Runtime, compile it with the following option:

-DENABLE_PROFILING_ITT=ON

### Analyze Code Performance on Linux¶

You can analyze code performance using Intel® VTune™ Profiler. For more information and installation instructions refer to the installation guide (PDF) With Intel® VTune™ Profiler installed you can configure your analysis with the following steps:

1. Open Intel® VTune™ Profiler GUI on the host machine with the following command:

cd /vtune install dir/intel/oneapi/vtune/2021.6.0/env
source vars.sh
vtune-gui

2. select Configure Analysis

3. In the where pane, select Local Host

4. In the what pane, specify your target application/script on the local system.

5. In the how pane, choose and configure the analysis type you want to perform, for example, Hotspots Analysis : identify the most time-consuming functions and drill down to see time spent on each line of source code. Focus optimization efforts on hot code for the greatest performance impact.

6. Start the analysis by clicking the start button. When it is done, you will get a summary of the run, including top hotspots and top tasks in your application:

7. To analyze ITT info related to the Auto plugin, click on the Bottom-up tab, choose the Task Domain/Task Type/Function/Call Stack from the dropdown list - Auto plugin-related ITT info is under the MULTIPlugin task domain:


text:
How can I compute an exponent with a rational number, using no calculator?

I would like to note before hand I am trying to improve my math by making a calculator application.

Anyways, let's say I would like to compute $2^{2.534}$.

Beforehand, I know that:

1. $$a^x=e^{xln(a)}$$

I already have a working algorithm on finding $ln(a)$, but ultimately in the end this still gives me a rational number and without actually knowing how to compute it, I arrive to my original problem again.

At this point, I figure I can store a table of already computed values for $e^{x}$, and then just interpolate values? However, I feel that my accuracy would greatly depend on how many values I store and it makes me come to hte question of how did anyone create these tables in the first place?

• you can break $2^{2.534}$ as $2^2 * 2^\frac{5}{10}* 2^\frac{3}{100}..$ – Raghukul Raman Oct 7 '17 at 17:43
• @RaghukulRaman Let's assume in place of $2.534$, I had an irrational number such as $√2$. What would I do then? – user382540 Oct 7 '17 at 17:46
• you can only find an approximate answer, due to limitations of memory and computations,btw $\sqrt 2$ can also be computed – Raghukul Raman Oct 7 '17 at 17:49

The logarithm and trigonometric tables were tools I still used in the early 70s when I attended high school and calculators were too expensive ans/or not allowed in classroom. The mathematicians of precomputer era calculated tables by hand. They had polynomial expansion like Taylor or MacLaurin formulae.

Logarithm standard formula

$$\log(1+x)=x-x^2/2+x^3/3-x^4/4+x^5/5-x^6/6+x^7/7-x^8/8+\ldots$$

is too slow, and soon were invented accelerated methods for alternating series like this.

Essentially the calculation were done by hand

Example. For small $x$

$\sin x\approx x-\frac{x^3}{6}+\frac{x^5}{120}$

Try $x=0.15$

$\sin 0.15 \approx 0.15 -\frac{0.15^3}{6}+\frac{0.15}{120}=\frac{3}{20}-\frac{27}{8000}+\frac{243}{3200000}= \color{red}{0.149438132}8125$

My calculator gives $\color{red}{0.149438132}4736$

Impressive, isn't it? Only three terms, such a precision... but caution if we look for values "far" from zero, like $x=2$ we get a bad result

$\sin 2\approx 2-8/3+32/120\approx 0.93$

while $\sin 1\approx 0.909297$

Hope this can be useful

• Could you explain how I could compute $e^x$ where $0<x<1$? – user382540 Oct 7 '17 at 22:11
• Use $$e^x\approx 1+x+x^2/2+x^3/6+x^4/24$$ If you want more precision just add $x^5/5!+x^6/6!\ldots$. Anyway with $5$ terms you get $3$ exact decimals – Raffaele Oct 8 '17 at 9:23


text:
8/19/2012

## Some Geometry Questions

Some friends have requested for some help for EOY questions in the HCI paper. Here are the solutions.
Q15)

a)
angle EAB = 65 (base of isoc triangle)
angle AEF = angle EAB = 65 (alt. angles)
cos 65 = FE/EA
6 cos 65 = FE = roughly 2.54 (3sf)

therefore FE = roughly 2.54 (3sf)
b)
AF = roughly 5.44 (3sf)
sin angle D = sin 55 = 5.44/AD

therefore AD = roughly 6.64 (3sf)

Q16)
The question:
ABC is a right-angled triangle with angle ABC - 90 degrees. Given that  AD = 10cm, DB = 6cm, BE = 8cm and EC = 10cm, find the shaded area.

See the solution below. We'll make use of menelaus theorem

Click to set custom HTML


text:
# File group provider#

Trino can map user names onto groups for easier access control and resource group management. Group file resolves group membership using a file on the coordinator.

## Group file configuration#

Enable group file by creating an etc/group-provider.properties file on the coordinator:

group-provider.name=file
file.group-file=/path/to/group.txt

The following configuration properties are available:

Property

Description

file.group-file

Path of the group file.

file.refresh-period

How often to reload the group file. Defaults to 5s.

## Group files#

### File format#

The group file contains a list of groups and members, one per line, separated by a colon. Users are separated by a comma.

group_name:user_1,user_2,user_3


text:
# 1.7: Surface Tension

Figure 1.17. Surface tension control volume analysis describing principal radii.

The surface tension manifested itself by a rise or depression of the liquid at the free surface edge. Surface tension is also responsible for the creation of the drops and bubbles. It also responsible for the breakage of a liquid jet into other medium/phase to many drops (atomization). The surface tension is force per length and is measured by [N/m] and is acting to stretch the surface.

Surface tension results from a sharp change in the density between two adjoined phases or materials. There is a common misconception for the source of the surface tension. In many (physics, surface tension, and fluid mechanics) books explained that the surface tension is a result from unbalanced molecular cohesive forces. This explanation is wrong since it is in conflict with Newton's second law (see Example 1.14). This erroneous explanation can be traced to Adam's book but earlier source may be found.

Example 1.14

Fig. 1.18 Surface tension erroneous explanation.

"The cohesive forces between molecules down into a liquid are shared with all neighboring atoms. Those on the surface have no neighboring atoms above, and exhibit stronger attractive forces upon their nearest neighbors on the surface. This enhancement of the intermolecular attractive forces at the surface is called.'' Explain the fundamental error of this explanation (see Figure 1.18).

Solution 1.14

It amazing that this erroneous explanation is prevalent in physics and chemistry (check the standard books for general chemistry in any college). In fact, even in Wikipedia this erroneous explanation appears. The explanation based on the unbalance of the top layer of molecules. Consider the control volume shown in Figure 1.18. The control volume is made from a molecule thickness and larger width. If this explanation was to be believed it must obey Newton's Laws. However, as it will be shown, this explanation violates Newton's Laws and hence it is not valid. The entire liquid domain is in a static equilibrium and hence every element is static equilibrium including the control volume. The pulling on the left of control volume is balanced with forcing that pulling to the right. However, the control volume is pulled by the molecules below while there counter force to balance it. There are no molecules about to balance it. If this explanation was correct the top layer (control volume) was supposed to be balanced. According to Newton second Law this layer should move down and the liquid cannot be at rest ever. Obviously, the liquid is at rest and this explanation violates Newton second law. In the Dimensional Analysis Chapter, provide another reason why this explanation violate all what is known experimentally about the surface tension.

The relationship between the surface tension and the pressure on the two sides of the surface is based on geometry. Consider a small element of surface. The pressure on one side is $$P_{i}$$ and the pressure on the other side is $$P_{0}$$. When the surface tension is constant, the horizontal forces cancel each other because symmetry. In the vertical direction, the surface tension forces are puling the surface upward. Thus, the pressure difference has to balance the surface tension. The forces in the vertical direction reads $\left(P_{i}-P_{0}\right)dll_{1}dll_{2} = \Delta P dll_{1}dll_{2} = 2\sigma dll_{1} \sin(\beta_{1}) + 2\sigma dll_{2} \sin(\beta_{2})\tag{47}$ For a very small area, the angles are very small and thus $$\left(\sin(\beta) \sim \beta \right)$$ Furthermore, it can be noticed that $$dll_{i} \sim 2R_{i}d\beta_{i}$$. Thus, the equation (47) can be simplified as $\Delta P = \sigma \left(\frac{1}{R_{1}} + \frac{1}{R_{2}}\right)\tag{48}$ Equation (48) predicts that pressure difference increase with inverse of the radius. There are two extreme cases: one) radius of infinite and radius of finite size. The second with two equal radii. The first case is for an infinite long cylinder for which the equation (48) is reduced to $\Delta P = \sigma\left(\frac{1}{R}\right)\tag{49}$ Other extreme is for a sphere for which the main radii are the same and equation (48) is reduced to $\Delta P = \frac{2\sigma}{R}\tag{50}$ Where $$R$$ is the radius of the sphere. A soap bubble is made of two layers, inner and outer, thus the pressure inside the bubble is $\Delta P = \frac{4\sigma}{R}\tag{51}$

Example 1.15

A glass tube is inserted into bath of mercury. It was observed that contact angle between the glass and mercury is $$55^{\circ}$$.

Fig. 1.19 Glass tube inserted into mercury.

Estimate the force due to the surface tension (tube is depicted in Figure 1.19). It can be assume that the contact angle is the same for the inside and outside part of the tube. Estimate the depression size. Assume that the surface tension for this combination of material is 0.5 [N/m]}

Solution 1.15

The mercury as free body that several forces act on it. $\label{tubeMercury:inside} F = \sigma 2\,\pi \cos55^\circ\,\left( {D_i} + {D_o} \right)$ This force is upward and the horizontal force almost canceled. However, if the inside and the outside diameters are considerable different the results is $\label{tubeMercury:horizontal} F = \sigma 2\,\pi \sin55\circ\,\left( {D_o} - {D_o} \right)$ The balance of the forces on the meniscus show under the magnified glass are $\label{tubeMercury:balance} P \, \overbrace{\pi\,r^2}^{A} = \sigma\,2\,\pi\,r + \cancelto{\sim0}{W}$ or $\label{tubeMercury:balanceF} g\,\rho\,h \, \pi\,r^2 = \sigma\,2\,\pi\,r + \cancelto{\sim0}{W}$ Or after simplification $\label{tubeMercury:balancea} h = \dfrac{2\,\sigma}{ g\,\rho\,r}$

Example 1.16

A Tank filled with liquid, which contains $$n$$ bubbles with equal radii, $$r$$. Calculate the minimum work required to increase the pressure in tank by $$\Delta P$$. Assume that the liquid bulk modulus is infinity.

Solution 1.16

The work is due to the change of the bubbles volume. The work is
$w = \int_{r_0}^{r_f} \Delta P(v) dv \label{intro:eq:bubbleWork} \tag{52}$
The minimum work will be for a reversible process. The reversible process requires very slow compression. It is worth noting that for very slow process, the temperature must remain constant due to heat transfer. The relationship between pressure difference and the radius is described by equation (50) for reversible process. Hence the work is
$w = \int_{r_0}^{r_f} \overbrace{\dfrac{2\,\sigma}{r}}^{\Delta P} \overbrace{4\,\pi\,r^2\,dr}^{dv} = 8\, \pi\,\sigma \int_{r_0}^{r_f} r dr = 4\,\pi\,\sigma \left( {r_f}^2 - {r_0}^2 \right) \label{intro:eq:bubbleWork1} \tag{53}$
Where, $$r_0$$ is the radius at the initial stage and $$r_f$$ is the radius at the final stage. The work for n bubbles is then $$4\,\pi\,\sigma\,n\,\left( {r_f}^2 - {r_0}^2 \right)$$. It can be noticed that the work is negative, that is the work is done on the system.

Example 1.17

Fig. 1.20 Capillary rise between two plates.

1.20. Notice that previously a rise for circular tube was developed which different from simple one dimensional case. The distance between the two plates is $$ll$$ and the and surface tension is $$\sigma$$. Assume that the contact angle is $$0^{\circ}$$ (the maximum possible force). Compute the value for surface tension of $$0.05[N/m]$$, the density $$1000[kg/m^3]$$ and distance between the plates of $$0.001[m]$$.}

Solution 1.17

In Figure 1.20 exhibits the liquid under the current study. The vertical forces acting on the body are the gravity, the pressure above and below and surface tension. It can be noted that the pressure and above are the same with the exception of the curvature on the upper part. Thus, the control volume is taken just above the liquid and the air part is neglected. The question when the curvature should be answered in the Dimensional analysis and for simplification this effect is neglected. The net forces in the vertical direction (positive upwards) per unit length are
$\label{2DTubeRise:eq:gov} 2 \sigma \cos\,0^{\circ} = g\,h\,ll\,\rho \Longrightarrow h = \dfrac {2\,\sigma}{ll\,\rho\,g} \tag{54}$
Inserting the values into equation (54) results in
$\label{2DTubeRise:eq:resutlts} h = \dfrac {2\times 0.05}{0.001 \times 9.8\times \times 1000 } = \tag{55}$

Example 1.18

Develop expression for rise of the liquid due to surface tension in concentric cylinders.

Solution 1.18

The difference lie in the fact that "missing''cylinder add additional force and reduce the amount of liquid that has to raise. The balance between gravity and surface tension is $\label{hConcentricTube:gov} \sigma \, 2\,\pi\,\left(r_i\,\cos\theta_i + r_o\cos\theta_o\right) = \rho\,g\,h\, \left( \pi (r_o)^2 - \pi (r_i)^2 \right)$ Which can be simplified as $\label{hConcentricTube:results} h = \dfrac{2\,\sigma \left(r_i\,\cos\theta_i + r_o\cos\theta_o\right) } { \rho\,g\, \left((r_o)^2 - (r_i)^2 \right) }$ The maximum is obtained when $$\cos\theta_i = \cos\theta_o = 1$$. Thus, the previous equation can be simplified $\label{hConcentricTube:resultsF} h = \dfrac{2\,\sigma } { \rho\,g\, \left(r_o - r_i \right) }$

### Contributors

• Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.


text:
## Main

The advent of free-electron laser (FEL) facilities, providing femtosecond light pulses in the gigawatt regime at extreme-ultraviolet (XUV) or X-ray wavelengths, has opened up many prospects for experiments in isolated atoms and molecules in the gas phase14,15. Over the past decade, pioneering results concerning multi-photon ionization of atoms16 and small molecules17 have been obtained using pulses from self-amplified spontaneous emission (SASE) FEL sources8. However, these pulses are prone to a low degree of coherence and poor shot-to-shot reproducibility owing to the instability inherent to the SASE process. As a result, despite theoretical predictions to observe Rabi dynamics at short wavelengths18,19,20,21, its effects on the measured spectra were only indirect22,23. Instead, XUV pulses from a SASE FEL have been used as a pump that allowed subsequent ultrafast Rabi dynamics to be driven by laser pulses at near-infrared wavelengths24. In this regard, XUV pulses from a seeded FEL, such as FERMI (Free Electron laser Radiation for Multidisciplinary Investigations)10, with its high temporal and spatial coherence, and large peak intensity can allow the study of coherent light–matter interactions25 and phase-dependent interference effects of the wavefunction2.

According to the Rabi model1, if a two-level atom initially in its ground state $$\left|a\right\rangle$$, is subjected to an interaction with a field of frequency ω that couples it to the excited state $$\left|b\right\rangle$$, the probability for excitation varies sinusoidally in time as $${P}_{b}(t)=| \frac{\varOmega }{W}{| }^{2}{\sin }^{2}\left(\frac{Wt}{2}\right)$$. The oscillating population leads to a symmetric structure in the frequency domain, known as an Autler–Townes (AT) doublet. The splitting is given by the generalized Rabi frequency $$W=\sqrt{{\varOmega }^{2}+\Delta {\omega }^{2}}$$, where Δω = ω − ωba, is the detuning of the field with respect to the atomic transition frequency, ωba. The Rabi frequency for light–matter interaction within the dipole approximation is Ω = eE0zba/ħ, with E0 being the electric field amplitude, zba the transition matrix element, ħ the reduced Planck constant and e the elementary charge. In addition to the periodic population transfer Pb(t), the coherent dynamics is further associated with sign changes of the oscillating amplitudes for the two states. For fermions, such sign changes of the wavefunction can be connected to rotations in real space1 that have been measured for neutron beams in magnetic fields26. Analogous sign changes in quantum optics were studied using Rydberg atoms to determine the number of photons in a cavity27. Recently, the sign changes in Rabi amplitudes have been predicted to strongly alter AT doublet structures in photo-excited atoms, when probed by attosecond XUV pulses28.

Here we investigate the Rabi dynamics at XUV wavelengths in helium atoms induced by an intense pulse from the FERMI seeded FEL that couples the two levels $$\left|a\right\rangle =1{s}^{2}$$ (1S0) and $$\left|b\right\rangle =1s4p$$ (1P1), with ħωba = ϵb − ϵa = 23.742 eV (ref. 29). The term ϵa (ϵb) denotes the energy of the state $$\left|a\right\rangle$$ ($$\left|b\right\rangle$$). The dynamics is probed in situ by recording photoelectrons ejected from the state $$\left|b\right\rangle$$ or $$\left|a\right\rangle$$ during the ultrashort interaction, with one or two XUV-FEL photons, as illustrated in Fig. 1a. To interpret this nonlinear dynamics, we have developed an analytical model by partitioning the Hilbert space into the two-level system and its complement. We expand the time evolution in the form of a Dyson series, where in the zeroth order, the two-level system undergoes Rabi oscillations. The photoionization dynamics from the excited (ground) state is treated by first (second)-order time-dependent perturbation theory describing the FEL interaction with the complement of the two-level system (see Supplementary Information for details). The resulting AT doublet structure depends on whether the photoelectron is originating from the ground state, $$\left|a\right\rangle$$, or the excited state, $$\left|b\right\rangle$$, as shown in Fig. 1b. The narrow spectral bandwidth of the XUV-FEL pulse (20–65 meV; Methods) enables efficient coupling of $$\left|a\right\rangle$$ and $$\left|b\right\rangle$$ with the dipole element zba = 0.1318a0, with a0 being the Bohr radius30. As shown in Fig. 1c, several physical effects can impede the duration of Rabi cycling: spontaneous emission sets a fundamental limit at about 4 ns (ref. 31), whereas one-photon ionization from the excited state (I) and two-photon non-resonant ionization from the ground state (II) restrict Rabi cycling at progressively higher intensities. The present experiment (diamond) can be driven coherently, as it takes place over an ultrafast time duration, with the estimated full-width at half-maximum (FWHM) of the driving FEL pulse duration being 56 ± 13 fs (ref. 32). This is three orders of magnitude shorter than the lifetime for photoionization of the Rabi cycling atom τa+b ≈ 100 ps (see Supplementary Information for details). Thus, the Hamiltonian for a two-level system $$H=\frac{1}{2}\hslash {\omega }_{ba}{\hat{\sigma }}_{z}+\hslash \varOmega \cos (\omega t){\hat{\sigma }}_{x}$$, where $${\hat{\sigma }}_{z}$$ and $${\hat{\sigma }}_{x}$$ are Pauli operators, can be satisfied by a time-dependent wavefunction of the form $$| \varPsi (t)\rangle \,=$$ $$a(t){{\rm{e}}}^{-{\rm{i}}{{\epsilon }}_{a}t/\hbar }| a\rangle +b(t){{\rm{e}}}^{-{\rm{i}}{{\epsilon }}_{b}t/\hbar }| b\rangle$$. Within the rotating-wave approximation, the amplitudes of the ground and excited states are expressed as:

$$\left\{\begin{array}{rcl}a(t) & = & \left[\cos \frac{Wt}{2}-{\rm{i}}\frac{\Delta \omega }{W}\sin \frac{Wt}{2}\right]\,\exp ({\rm{i}}\Delta \omega t/2)\\ b(t) & = & -{\rm{i}}\frac{\varOmega }{W}\sin \frac{Wt}{2}\,\exp (-{\rm{i}}\Delta \omega t/2),\end{array}\right.$$
(1)

provided that the electric field can be approximated as a flat-top shape in time. The sign changes associated with these Rabi amplitudes are essential to understand the ultrafast build-up of AT doublets from $$\left|a\right\rangle$$ or $$\left|b\right\rangle$$, by absorption of two or one resonant XUV-FEL photons. The AT doublet emerges owing to a destructive interference effect between photoelectrons ejected before and after the first sign change, which is found to occur at 1/2 and 1 Rabi period for the amplitudes a(t) and b(t) with Δω = 0, respectively. This is in agreement with the results from the analytical model presented in Fig. 1b. The contribution from the excited state, $$\left|b\right\rangle$$, can be understood as the Fourier transform of the time-dependent amplitude b(t). Similarly, the contribution from the ground state, $$\left|a\right\rangle$$, can be related to the Fourier transform of a(t) through a non-resonant wave packet composed of complement states, $$\left|c\right\rangle$$, as illustrated in Fig. 1a. Thus, the observation of a doublet from either state $$\left|a\right\rangle$$ or state $$\left|b\right\rangle$$ is a direct measurement of Rabi dynamics in the energy domain.

Measured photoelectron spectra, shown in Fig. 2a, exhibit an AT splitting of ħΩ = 80 ± 2 meV (see Methods for details about the blind deconvolution procedure used here; the reported uncertainty is obtained from a fit of the symmetric AT doublet with two Voigt profiles with the same width). The corresponding Rabi period 2π/Ω ≈ 52 fs, given its proximity to the FWHM of the XUV-FEL pulse, suggests that the experiment was performed in a regime of ultrafast AT doublet formation close to a single Rabi cycle. A slight blue detuning of the XUV light, by about 11 meV relative to the atomic transition29, is required to record a symmetric AT doublet (black squares in Fig. 2a). This blueshift is one of the major experimental results, and identifying its physical origin is among the main objectives of the theoretical efforts presented in this work. A strong asymmetry is observed when the FEL frequency is detuned to the red (red circles in Fig. 2a) or blue (blue diamonds in Fig. 2a) side of the symmetric doublet. The asymmetry of the AT doublet is qualitatively well reproduced by ab initio numerical simulations for helium within the time-dependent configuration-interaction singles (TDCIS) approximation33, as shown in Fig. 2b. Gaussian pulses were used with parameters chosen to match the experimental conditions with an effective intensity of 2 × 1013 W cm−2 (as obtained from Ω) and a pulse duration (FWHM) of 56 fs (see Methods for details). It is worth noting that the Rabi dynamics is sensitive to the exact shape of the driving pulse. For instance, a Gaussian pulse can induce more Rabi oscillations than a flat-top pulse with same FWHM by a factor of $$\sqrt{{\rm{\pi }}/(2ln2)}\approx 1.5$$ as follows from the area theorem34. Thus, the calculated photoelectron spectra from the analytical model using flat-top pulses in Fig. 1b for 3/2 Rabi periods agree well with those from the TDCIS calculations using Gaussian pulses with a FWHM close to a single Rabi period in Fig. 2b. Clearly, the AT doublet manifests itself between 1 and 3/2 Rabi periods. The difference in kinetic energy (about 0.4 eV) of the symmetric AT doublet between experiment and theory (Fig. 2a and Fig. 2b, respectively) is attributed to electron correlation effects not included in the TDCIS calculations that increase the binding energy beyond the Hartree–Fock level. The observed asymmetry in the AT doublet cannot be explained by a breakdown of the rotating-wave approximation because the experiment is performed at a resonant weak-coupling condition6: ω ≈ ωba and Ω/ωba = 0.34%. Instead, we express the Rabi amplitudes from equation (1) in terms of their frequency components and find that a(t) has two asymmetric components that are proportional to (1 ± Δω/W), whereas b(t) has two symmetric components that are proportional to ±Ω/W. Using the analytical model with 3/2 Rabi periods, we confirm that the AT doublet from $$\left|b\right\rangle$$ is symmetric, whereas that from $$\left|a\right\rangle$$ is asymmetric, as shown in Fig. 2c and Fig. 2d, respectively. Quite remarkably, the observed asymmetry in the experiment suggests that the photoelectron signal contains significant contributions from the two-photon ionization process from $$\left|a\right\rangle$$. We question how this is possible given that the electric field amplitude E0 = 0.02388 atomic units implies an ionization-probability ratio of 104:1 in favour of the one-photon process from $$\left|b\right\rangle$$.

We propose that the two-photon signal from $$\left|a\right\rangle$$ can compete with the one-photon signal from $$\left|b\right\rangle$$ owing to superposition of intermediate (complement) states, which are illustrated as grey bound and continuum states in Fig. 3a. This leads to a giant localized wave, $$\left|{\rho }_{\ne b}\right\rangle$$, compared with the normalized wavefunction for $$\left|b\right\rangle$$, as shown in Fig. 3b. The largest contributions to the giant wave come from the dipole-allowed complement states that are close to the one-photon excitation energy. The scaling factor owing to atomic effects is calculated to be about 1:104 in favour of the two-photon process (see Extended Data Table 1 for the matrix elements). Thus, we can explain why the XUV-FEL pulse is intense enough for the non-resonant two-photon process from the ground state to be comparable to the one-photon process from the resonant excited state. In general, addition of two pathways leads to quantum interference that depends on their relative phase. From Fig. 3b, we notice that the giant wave oscillates out of phase with the excited state close to the atomic core, which affects the signs of the matrix elements (Extended Data Table 1). The ultrafast build-up of the AT doublet can be used to study the resulting interference phenomenon in time. To understand this phenomenon, we have used the analytical model to perform calculations where the one- and two-photon contributions are added coherently to simulate the angle-integrated measurements. In Fig. 3c, we show how the resonant case (Δω = 0) leads to a strongly asymmetric AT doublet after one Rabi period. Figure 3d indicates that a blue detuning (Δω = 62 meV) leads to the symmetric AT doublet at an earlier time between 0.5 and 1 Rabi periods. The advancement of the AT doublet in time follows from the faster Rabi cycling at the rate of generalized Rabi frequency. Thus, we have found that the blueshift of the symmetric AT doublet is due to quantum interference between the one-photon (I) and two-photon (II) processes. A general loss of contrast in the AT doublet structures is found by considering the effect of an extended gas target in our model. However, the two-photon doublet was found to be less sensitive to the volume averaging effect when compared with the one-photon doublet (see Supplementary Information for details), allowing us to clearly observe the AT doublet in the measured photoelectron signal.

To provide further evidence in support of the coherent interaction between the helium atoms and the XUV-FEL pulses, we show that the ultrafast emergence of the AT doublet can be interpreted in terms of the dressed-atom picture with coupled atom–field energies ϵ± = (ϵa + ϵb + ħω ± ħW)/2. One photon energy above these coupled energies implies final photoelectron kinetic energies $${{\epsilon }}_{\,\pm }^{{\rm{k}}{\rm{i}}{\rm{n}}}={{\epsilon }}_{\pm }+\hslash \omega$$, where Ip = −ϵa = 24.5873 eV is the ionization potential of helium. In Fig. 4a, kinetic energies are labelled with the uncoupled atom–field states35, $$\left|a,1\right\rangle$$ and $$\left|b,0\right\rangle$$. The experimental results in Fig. 4b can be understood as one photon above $$\left|a,1\right\rangle$$ at large detuning of the XUV-FEL pulse. This is because the interaction is weak far from the resonance with the atom remaining mostly in its ground state $$\left|a\right\rangle$$, such that two photons are required for photoionization. The region closer to the resonance is influenced by quantum interference between the one-photon (I) and two-photon (II) processes that leads to suppression of $$\left|b,0\right\rangle$$ and enhancement of $$\left|a,1\right\rangle$$, with both coupled energies appearing briefly to form an avoided crossing in kinetic energy. It is noted that the avoided crossing appears at a blue detuning from the resonant transition, Δω = 0 (denoted by the dashed vertical line), revealing the quantum interference between the two pathways from the ground state $$\left|a\right\rangle$$ and the excited state $$\left|b\right\rangle$$. Similar results were obtained from the TDCIS simulations (Fig. 4c) and the analytical model with contributions from both $$\left|a\right\rangle$$ and $$\left|b\right\rangle$$ for 3/2 Rabi periods (Fig. 4d). The observed blue detuning for the experimental avoided crossing (about 11 meV) is well reproduced by TDCIS calculations (about 14 meV). The enhanced shift of the AT doublet to blue detuning in the analytical model is an effect of the pulse envelope that can be reproduced with TDCIS using smoothed flat-top pulses. This indicates that the amount of blueshift of the AT doublet can be coherently controlled by the exact profile of the FEL pulse.

Our results show that it is now possible to simultaneously drive and interrogate ultrafast coherent processes using XUV-FEL pulses. Previous attempts to understand Rabi dynamics at short wavelengths have relied on the strong-field approximation, where the influence of the atomic potential is neglected, leading to an inconsistent AT doublet when compared with numerical simulations19. In contrast, our analytical model includes the full effect of the atomic potential and Rabi dynamics in the two-level subspace, whereas the remaining transitions to and within the complement of the Hilbert space are treated by time-dependent perturbation theory. Consequently, we could establish a unique mechanism in the form of a giant Coulomb-induced wave from the ground state to explain why the non-resonant two-photon process can compete with the resonant one-photon process and generate quantum interference effects at the high intensities provided by the XUV-FEL beam. With this model, we now understand how ultrafast Rabi dynamics at short wavelengths are imprinted on photoelectrons from weakly ionized atoms. Our experimental approach of using photoionization as an in situ probe of Rabi dynamics does not rely on any additional laser probe field, and, hence, is easily applicable to other quantum systems. We think that such photoionization processes are of interest to different domains: the one-photon domain connected to the symmetric excited-state dynamics (I) and the two-photon domain connected to the asymmetric ground-state dynamics (II), as shown in Fig. 1c. At the boundary between these domains, quantum interference is manifested in the photoelectron signal. The experimental blueshift of the symmetric AT doublet, reported in our work, is an observation of this type of effect. Owing to substantially different angular distributions from domains I and II (Extended Data Table 1), we predict that the boundary regime will exhibit intricate angular dependencies of the photoelectrons that could be the subject of upcoming experiments. Studying more complicated Rabi dynamics, affected by rapid photoionization36 or autoionization decay37, are natural extensions of our work. Coherent population inversion with core-excited states and ultrafast core ionization with simultaneous ionic-state excitation21 are examples of configurations achievable at short wavelengths. Given the ongoing developments of seeded FEL facilities around the world38,39 capable of providing light pulses down to few-ångström wavelength, our findings can inspire future studies involving coherent control in multi-electronic targets, such as molecules, and nano-objects with site specificity, opening up pathways for steering the outcomes of photo-induced processes across ultrafast timescales.

## Methods

### Experiment

The experiment was carried out at the low-density-matter beamline of FERMI40. A pulsed Even–Lavie valve, synchronized with the arrival of the FEL pulse served as the target source. The target gas jet was estimated to be a cone with a diameter of 2 mm at the interaction region. We measured the photoelectron spectra at and around the 1s2 → 1s4p transition in helium, using a 2-m-long magnetic bottle electron spectrometer (MBES). The gas jet, FEL beam and magnetic bottle axes are mutually perpendicular, with the first two being on the horizontal plane of the laboratory, and the last one in the vertical direction. Before entering into the flight tube of the MBES, the photoelectrons were strongly retarded to below 1-eV kinetic energy to achieve high spectral resolution (EE ≈ 50). To suppress any short-term fluctuation arising from the instability of the FEL, we performed a ‘round trip’ scan across the wavelength range, 52.50 nm ↔ 51.80 nm. Empirically, the FWHM of the XUV-FEL pulse duration (τxuv) can be approximated32 to be in between $$\left({\tau }_{{\rm{seed}}}/{n}^{1/2}\right)$$ and $$\left(7{\tau }_{{\rm{seed}}}/6{n}^{1/3}\right)$$. Here τseed ≈ 100 fs is the duration (FWHM) of the seed pulse (wavelength 261.08 nm) and n = 5 is the harmonic order for the undulator. It leads to τxuv = 56 ± 13 fs, which matches well the FWHM of about 66 fs, obtained from the simulation of the FEL dynamics using PERSEO41. The spectral bandwidth (FWHM) of the pulse was estimated using PERSEO to be around 0.13 nm at the central wavelength of λ = 52.216 nm. Extended Data Fig. 1a and Extended Data Fig. 1b show the simulated spectral and temporal profiles of the FEL pulse, respectively. At best focus, the spot size (FWHM) was estimated to be 12 μm. We measured the energy per pulse at the output of the FEL undulator to be around 87 μJ, which refers to the full beam including all photons contained in the transverse Gaussian distribution. To consider those, we used 4σ as the FEL beam diameter at best focus, where σ = 12/2.355 ≈ 5.1 μm. Hence, the beam waist (w0) is given by w0 = 2σ = 10.2 μm, along with a Rayleigh length of $${\rm{\pi }}{w}_{0}^{2}\,/\lambda \approx 6.3\,{\rm{mm}}$$. The peak intensity was estimated to be about 1.4 × 1014 W cm−2. This clearly shows that the FEL pulses were intense enough to drive the coherent Rabi dynamics. However, this peak intensity alone does not directly correlate to the ultrafast Rabi dynamics, as the Rabi frequency is proportional to the E-field strength, and Rabi cycling requires sufficient area of the pulse for the AT doublet to emerge. Furthermore, given the shot-to-shot fluctuations of the FEL pulse parameters, we have used the observed AT splitting to extract the average interaction strength in a reliable manner without any need for simulated values of the pulse duration or experimental mean pulse energy.

### Data analysis

To filter the measured photoelectron spectra on a shot-to-shot basis, we used the photon spectrum recorded by the Photon Analysis Delivery and REduction System (PADRES) at FERMI to determine the bandwidth (FWHM) of the XUV pulse. Any shot without the photon spectrum was rejected: out of 355,000 shots, 354,328 shots were retained. All the shots with more than 65-meV FWHM width were discarded (Extended Data Fig. 2a). It is noted that the simulated value of the photon bandwidth (59 meV) lies within the filtering window of 20–65 meV. In addition, we chose only the shots with integrated spectral intensities ranging from 0.8 × 105 to 1.6 × 105 in arbitrary units (Extended Data Fig. 2b). The filtered shots were sorted into 30 photon-energy bins, uniformly separated from each other by about 13 meV and covering the entire photon energy window of the wavelength scan (Extended Data Fig. 2c). Overall, only 304,192 shots (filtering ratio of 0.857) out of the raw data were retained. The measured photoelectron spectra, following shot-to-shot filtering, are shown in Extended Data Fig. 3. The avoided crossing is only faintly visible here. To obtain the clear avoided crossing from Fig. 4b, we deconvoluted the photoelectron spectra for three photon energies near the 1s2 → 1s4p transition using the Richardson–Lucy blind iterative algorithm42. To reduce the noise introduced during the deconvolution, we incorporated the Tikhonov–Miller regularization procedure into the algorithm43. The outcomes are shown in Extended Data Fig. 4. Following deconvolution, the values of FWHM for the Gaussian instrument-response functions were found to be 70.9 ± 1.2 meV, 69.6 ± 2.4 meV and 69.4 ± 1.4 meV, for the three photon energies. These values match well the combined resolution of about 65 meV, obtained from the photon bandwidth and the kinetic energy resolution of the MBES. No filter, either metallic or gaseous, was used along the path of the FEL beam. Hence, a minor contribution (<5%) from the second-order light can be noticed as an asymmetric tail close to 22.8-eV kinetic energy (Extended Data Fig. 4a,b). To rule out any artefact from the fluctuations of the FEL pulse properties, we used another filtering criterion for the photon bandwidth (1–45 meV) and the integrated spectral intensity (1 × 105 to 3 × 105 a.u.). The corresponding deconvoluted photoelectron spectra at 23.753 eV is shown in Extended Data Fig. 5, along with that from Fig. 2a. No significant change in the AT doublet structure due to change in filtering criteria could be seen. Finally, for a transform-limited Gaussian pulse, τxuv can vary between 30 fs and 90 fs from shot to shot that encompasses its empirical value of 56 ± 13 fs. As $${\tau }_{{\rm{xuv}}}^{2}$$ is significantly higher than the absolute value of the simulated group-delay dispersion of the FEL pulse of −690 fs2, no effect due to the linear chirp was considered in the theoretical calculations.

### Numerical simulations using TDCIS

The ab initio numerical simulations are performed using the time-dependent (TD) configuration-interaction singles (CIS) method33,44,45,46 in the velocity gauge. The CIS basis for helium is constructed using Hartree–Fock orbitals that are computed using B-splines. Exterior complex scaling is used to dampen spurious reflections during time propagation of TDCIS47. The vector potential of the XUV-FEL pulse is defined as

$$A(t)={A}_{0}\sin (\omega t)\exp \,\left[-2{\rm{l}}{\rm{n}}(2)\frac{{t}^{2}}{{\tau }^{2}}\right].$$
(2)

The central frequency, ω, is set close to the CIS atomic transition frequency, ωba = 0.887246 atomic units = 24.1432 eV, between the Hartree–Fock ground state $$\left|a\right\rangle =1{s}^{2}$$ (1S0) and the singly excited state $$\left|b\right\rangle =1s4p$$ (1P1). The duration of the pulse is set to τ = 56 fs and the peak intensity is set to $$I={(\omega {A}_{0})}^{2}\,[{\rm{a.u.}}]\times 3.51\times 1{0}^{16}\,[{\rm{W}}\,{{\rm{cm}}}^{-2}]=2\times 1{0}^{13}\,{\rm{W}}\,{{\rm{cm}}}^{-2}$$. The CIS dipole matrix element between the ground state and the excited state is given by zba = 0.124a0 and the ionization potential is related to the 1s orbital energy in Hartree–Fock Ip = −ϵa = 24.9788 eV, in accordance with Koopmans’ theorem. Photoelectron distributions are captured using the time-dependent surface flux (t-SURFF)48 and infinite-time surface flux (iSURF)49 methods. The high kinetic energy of the photoelectrons ensures a proper description of the physics by the surface methods.


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# Hall set

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A Hall set over generators $A=\{a_1,a_2,\dots\}$ is a totally ordered subset $H$ of the free magma $M(A)$, i.e. the free non-associative structure over $A$ (cf. also Associative rings and algebras). The elements of $M(A)$ correspond to completely bracketed words over $A$ (or rooted planar binary trees with leaves labelled by generators $a_1,a_2,\dots$; cf. also Binary tree). These are defined recursively as brackets $t=[t',t'']$, where $t',t''$ are bracketed words of lower weight; the bracketed words of weight one correspond to the generators $a_1,a_2,\dots$.

The total order $<$ on $H$ is required to satisfy the condition:

$H_0$) for any $t=[t',t'']\in H$, $t<t''$.

Let $H\subset M(A)$ be a totally ordered subset such that condition $H_0$) is fulfilled. Then the set $H$ is a Hall set if and only if the following conditions are satisfied:

$H_1$) $A\subset H$;

$H_2$) $t=[t',t'']\in H$ if and only if

$H_{2.1}$) $t',t''\in H$ with $t'<t''$; and

$H_{2.2}$) either $t'\in A$ or $t'=[s',s'']$, and then $s''\geq t''$.

This definition is from [a1]. M. Hall's original definition replaces condition $H_0$) by simply imposing that brackets of lower weight be smaller with respect to $<$. Many authors proposed generalizations of Hall's construction by relaxing condition $H_0$) in many different ways. Viennot's definition acted as a unifying framework for the various proposed generalizations; see [a2] for details. Observe that the definition of a Hall set admits symmetries, with respect to the order $<$; for instance, one could invert the order $<$ and write $>$ instead. It must be noted, however, that conditions $H_1$) and $H_2$) must be coherent; that is, the sequence of elements $t'',s'',t'$ must either be increasing (as is the case here) or decreasing. Hence, there are four possible equivalent definitions for Hall sets.

An example of a Hall set is the set of basic commutators (cf. Basic commutator), the basic commutator Hall set. It has the total order reversed with respect to the definition above. Another symmetry involves writing bracketed words from right to left instead of from left to right. For example, the "Hall set" described in [a4], Sect. 2.10, arises from the basic commutator Hall set in this way. Another Hall set can be defined using Lyndon words (cf. Lyndon word).

Elements of a Hall set are used to obtain a basis of the free Lie algebra over $A$ (cf. Lie algebra, free; Hall polynomial). X. Viennot [a1] has shown that his definition not only gives a unifying framework but is also, in a certain sense, optimal: Any subset $H\subset M(A)$ satisfying $H_1$) and $H_2$), leading to a basis of the free Lie algebra, must satisfy condition $H_0$).

Hall sets may be shown to coincide with Lazard sets (cf. Lazard set). The combinatorics of Hall sets is also studied in [a2] and [a3].

See also Hall word.

#### References

 [a1] X. Viennot, "Algèbres de Lie libres et monoïdes libres" , Lecture Notes in Mathematics , 691 , Springer (1978) [a2] G. Melançon, "Combinatorics of Hall trees and Hall words" J. Combin. Th. , 59A : 2 (1992) pp. 285–308 [a3] C. Reutenauer, "Free Lie algebras" , London Math. Soc. Monographs New Ser. , 7 , Oxford Univ. Press (1993) [a4] N. Bourbaki, "Groupes et algèbres de Lie" , II. Algèbres de Lie libres , Hermann (1972)
How to Cite This Entry:
Hall set. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hall_set&oldid=35333
This article was adapted from an original article by G. MelanÃ§on (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article


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# Unity 2D animated tiles library: IsSelected variable not working or not present within RuleTileEditor.cs

It told me to download this library in order to work the animated tiles: https://github.com/Unity-Technologies/2d-extras

After this the tutorial told me to unzip the library and place it into my assets folder. It loaded into my assets folder and then got the following error:

Assets\2d-extras-master\2d-extras-master\Editor\Tiles\RuleTile\RuleTileEditor.cs(277,26): error
CS1061: 'ReorderableList' does not contain a definition for 'IsSelected' and no accessible extension
method 'IsSelected' accepting a first argument of type 'ReorderableList' could be found (are you
missing a using directive or an assembly reference?)

Does anyone perhaps know why this error is occuring.

1. Open Unity
2. On top of Unity window, you will see your version number. Will look something like, Unity 2020.3.8f1
3. Go to: https://github.com/Unity-Technologies/2d-extras
4. On Unity 2d Extras Github, check to see if your Unity version above matches the current branch. If you're unsure, do Control + F and search the word Branches to find the branches link on Github, click this. This will take you to all the branches of 2d extras.
5. Select the branch that matches your Unity Version. Ex: If my version is 2020.3, I would select the 2020.3 branch. Click the blue link to your branch.
6. Click the green code button on top right of the branch page, download zip.
8. Open your Unity project, go to Window -> Package Manager.
9. Click the + in package manager, select Add package from disk.
10. Locate your unzipped folder for 2d extras and select the package.json inside it, click open.
You are most likely on an Unity version where their ReorderableList no longer provides the IsSelected method. Looking at the GitHub repository it seems that there are different branches for different versions of Unity. For an example the branch "2020.3" no longer seems to contain a reference to IsSelected, so you need to change to an appropriate branch of the repository that works with your version of Unity.


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## Precalculus (6th Edition) Blitzer

The parabola is opening to the left. Thus the equation should have a form of $y^2=4px$, or answer (a).