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Imagine a scenario where a forger attempts to produce fake currencies and the policeman has to try to distinguish those fake currencies from the real ones. At the beginning, both don’t have much experience, the forger will just come with a piece of paper with a dollar bill scribbled on it. Obviously, is that is a fake currency, but the unexperienced policeman still will struggle to figure out whether it is actually fake or not. After the first iteration, both two will learn how to improve in their own task thanks to the experience acquired in the previous iteration: if the policemen detected the fake currency, then the forger will learn how to make fake money more similar to the real ones. Otherwise, the policeman will develop new more sophisticated techniques to correctly classify fake and real currencies and so on.

This is equivalent to a zero-sum game where both parts have to improve their own strategy in order to succeed in their task and win over their opponent. Eventually, this competition will drive the forger to coin new currencies which are almost indistinguishable from the real ones and policemen will improve his detection methods until no fake currency can trick him.

What if we replace the forger with a generative model and the police with a classification model? This is analogous to how generative adversarial networks (GAN) work. The generative model is matched against a discriminative model whose task is to learn to determine whether a sample is from the model distribution (sample generated by the generative model) or comes from the data distribution (it’s a real sample). This is why this kind of models are also known as adversarial networks: two networks have to compete with each other in order to maximize their reward.

### Generative models

What I cannot create, I do not understand

Richard Feynman

Generative models, such as VAEs, are some kind of networks whose goal is to generate new samples from a distribution that is as similar as possible to the distribution of the original data using unsupervised learning techniques. The first step consists in collecting a certain amount of data and then, the model is trained to find certain parameters W defining a distribution resembling the distribution of the real data and eventually generating new samples extracted from it. Thus, generative model’s objective is to find some parameters W such that the distance between the estimated distribution and the distribution of the actual data is minimized.

One of their useful application consists in dealing with unbalanced datasets. Imagine a binary labeled dataset composed of 95% of positive labels and only the remaining 5% labels marked as negative. This is the typical case we can find in the anomaly detection field where, under normal conditions, anomalies are much less frequent than normal events such as player’s departure prediction in online video games or system anomalies detection. To relive this negative effect, we can use GAN to generate new samples, which have not been seen before, relative to the less frequent labels until negative and positive samples don’t have comparable size.

The goal of the generator is to learn the distribution pg over the target data x by mapping an input noise pz(z) to a new sample x‘ generated through the differentiable function G(z;θg) represented by the generative model G with parameters θg. On the other hand, the discriminator D(x;θd) is a model which outputs a single scalar value D(x) representing the probability that x came from the real data rather than from the distribution pg. Therefore, we simultaneously train D to maximize the probability of assigning the correct label to both training examples and samples from the generative model and train the generator to minimize the probability that D correctly detects a generated sample by the G. Overall, the loss function is the following minimax equation:

$min_G \, max_D V(D,G)=E_{x\sim p_{data(x)}}[logD(x)]+ E_{z\sim p_{z(z)}}[log(1-D(G(z)))]$

From the above function we can clearly observe that for correct predictions of the discriminator both terms tend toward log1 which is equal to 0. Conversely, when the generator successfully deceives the discriminator, the second term tends toward negative values by approaching close to log0, thus minimizing the value of the function.

Since in the early stages of the training phase G would be a very pool model with random parameters, D would detect fakes samples with a very high conﬁdence because the sample generated by G would be clearly different from the ones in the training data. Thus, the term log(1 − D(G(z))) would saturate since D(G(z)) will get every time very close to 0. To overcome this problem, rather than training G to minimize log(1−D(G(z))) we can train G to maximize logD(G(z)). This objective function has been proven to provide much stronger gradients early in learning.

### Training phase

During training, we alternate two phases: one in which we will train the discriminator and in the other one we will train the generator. Following, are listed the three steps involved in the first phase to update the weights of the discriminator:

• Sample a minibatch of m noise samples {z(1),…,z(m)} from the noise prior pg(z).
• Sample a minibatch of m examples {x(1),…,x(m)} from data generating distribution pdata(x).
• Update the discriminator by its stochastic gradient ascent:

$\nabla_{\theta_d} \frac{1}{m}\sum_{i=1}^{m}[logD(x^{(i)})+log(1-D(G(z^{(i)}))]$

Then, we alternate the process with the generative model training phase:

• Sample a minibatch of m noise samples {z(1),…,z(m)} from noise prior pg(z).
• Update the generator by descending its stochastic gradient:

 $\nabla_{\theta_g} \frac{1}{m}\sum_{i=1}^{m}log(1-D(G(z^{(i)}))$

As for the gradient update rules, we can use any standard gradient-based learning rule such as adam or SDG.

Ideally, we would like to train the GAN model until the discriminator D can no longer differentiate between real and fake samples because the estimated distribution is almost indistinguishable from the actual data distribution, that is, pg=pdata. At this point, D will output a 50% probability for every sample it encounters.

### Interesting projects using GAN

The next section will present some real applications of GAN models to both understand what kind of problems are GANs capable to deal and to give an idea of how powerful applications can be developed thanks to the contribution of artificial intelligence.

### U-GAT-IT (Face to Anime)

The adversarial network U-GAT-IT developed by some South-Korean developers in 2019 manages to convert real people’s faces into japanese Anime. More examples can be found in this gallery.

### FaceAPP (Face translation)

FaceApp is a face editing app that allows the user to edit its own face or someone else’s face to make it appear older or younger than their actual age. This App uses generative adversarial networks to train its program to create specific categories of realistic images. It then transfers the features to the photo uploaded by the user, giving it the desired category of the filter.

### AnimeGAN

AnimeGAN is a kind of deep convolutional GAN (DCGAN) trained to draw from scratch anime faces and it improves over time gradually generating faces resembling the ones drawn by japanese artists. A post about this project can be found on this link.


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This section contains some basic terms and concepts that are important in understanding radio emission measurement techniques.

5.1 Characteristics of the Signals

5.1.1 Electromagnetic Radiation: Rays, Waves, or Photons?

Wavelength and frequency ranges; what’s l , what’s n ?

The radio-frequency range is arbitrarily defined to be from frequencies (n ) beginning at the super-sonic, meaning just above the range of sound waves that humans can hear, 20 kHz or so, up to about 600 GHz, where the far infrared begins. The corresponding wavelength (l ) range is from many kilometers down to about 0.5 mm or 500 microns. Frequency and wavelength are related, as for any conventional wave phenomena, by:

n = c/l

where c is the speed of light, 299792.5 km/s.

Over this frequency range and at moderate equivalent temperatures, we can usually ignore photons; that is, we need not quantize the electromagnetic field. In most cases, this results in both simplification and insight that is lacking if one thinks photons instead of waves.

Many formulas used in radio astronomy (and electrical engineering) are based on the Rayleigh-Jeans approximation to the Planck black-body law. If, furthermore, the antenna structures are large compared to a wavelength, as is usually the case at centimeter and millimeter wavelengths, then we can often ignore waves also and use just ray-trace optics.

5.1.2 Law of large distances

Another helpful simplification involves the large distances and small angular sized of most astronomical objects. Angles in radians are, then, just the linear sizes divided by the distances.

5.1.3 Noise-like signals

A. Law of large numbers

Most astronomical sources are large in physical size, even though small in angular size, and radiation is emitted by a large number of statistically independent sources- atoms, molecules, or electrons. The resulting signals are noise-like, that is, the electric fields are Gaussian random variables with spectra that depend on the details of the emission mechanisms. Laboratory laser and maser oscillators are usually coherent because of the cavity in which they oscillate. Astronomical masers have never been found to have non-Gaussian statistics.

For similar reasons, many radio-astronomical sources are unpolarized; that is, signals in one polarization are statistically independent of signals in the orthogonal polarization. But some sources, especially those involving magnetic fields that extend over a significant part of the spatial extent of the source, can be polarized, and studying such polarization sometimes leads to significant insights.

B. Law of large sizes

As a rough rule with some exceptions, an incoherent source that is not resolved in angle can be seen to vary only on time scales long compared with the light travel time through the length of the emitting region. A black sphere with cyclic temperature variations, for example, will have these variations smoothed over as seen from a large distance if the cycle time is comparable to or less than the light travel time across a radius of the sphere. Some astronomical sources vary, and the time scales of the variations can sometimes be used to infer maximum sizes.

C. Significance for measurement techniques

Some of the techniques used in radio astronomy depend on these characteristics of the sources. One-bit autocorrelation spectroscopy, for example, depends on the signal voltage being a Gaussian random variable. And many-hour-long interferometry depends on the source being stable over that time.

5.2 Spectra

5.2.1 Continuum sources, black body or nn?

A black body at temperature T in the radio-frequency range has a specific intensity given by the Raleigh-Jeans approximation to the Planck black-body law:

I=2kT/l 2

where I is the specific intensity in, for example, watts/(m2Hz ster), k is Boltzman’s constant, 1.380 x 10-23 watts/Hz/K and l is wavelength.

The proportionality to temperature allows us to talk about intensities and powers in temperature units, K or Kelvins. If a source really were thermal emission from a black or gray body, then this radiation temperature would be independent of wavelength. More typically, however, sources are "colored", that is, radiation temperatures vary with wavelength and are not necessarily related to physical temperatures of the sources. A synchrotron-emission source, for example, has a spectral index n, usually defined by I µ  n-n, that is related to the energy distribution of the emitting electrons. Regardless of the emission mechanism or spectrum, we can use the Rayleigh-Jeans equation above to define a brightness temperature, Tb, proportional to intensity, but then Tb will be, in general, a function of frequency. Even in the high-frequency low-temperature range where Rayleigh-Jeans is no longer useful, we can use this equation to define a convenient fake brightness temperature. Figure 1 is a cartoon example of such spectra.

Figure 1

Electrical engineers implicitly use the Rayleigh-Jeans approximation also. The noise power per unit bandwidth from a warm resistor into a matched load is just kT provided that the frequency and temperature are in the range for which Rayleigh-Jeans is precise. To see this, imagine a resistor coupled to a feed looking at a black body. Even in the range where Rayleigh-Jeans is no longer useful; in thermal equilibrium, equal power must flow each way.

5.2.2 Atoms and molecules- emission and absorption lines

Emission and absorption lines in radio astronomy usually originate from atoms and small molecules or molecular ions in gaseous form, and molecular transitions at radio wavelengths are usually rotational. Emission lines result from warm gas overlying a cold background so that the intensity (or flux or radiation temperature) at the line frequency is sharply higher compared to nearby wavelengths. If such a gas cloud is optically thick (opaque), the specific intensity or "brightness" at the line frequency is given by the state temperature of the corresponding transition, which, at thermodynamic equilibrium, would be just the temperature. Thermodynamic equilibrium is, however, not very common in radio astronomy. Absorption lines result from cool gas overlying a hotter background source so that the intensity at the line frequency is sharply lower compared to nearby wavelengths. If such a gas cloud is optically thick, then the line center again gives the state temperature. Figure 2 is a cartoon of such spectra.

5.2.3 Doppler shifts and kinematics

Doppler shifts are very important in spectral-line radio astronomy. The non-relativistic form is usually written as

D n /n = -v/c

where D n is the change in frequency n due to the Doppler velocityv, defined as the rate of change of distance from source to observer (hence the minus sign), and c is the speed of light.

Even when speeds are relativistic, this non-relativistic formula is sometimes still used to define a convenient fake velocity.

Line widths and line shapes are influenced by several line-broadening mechanisms including a) natural line widths related to the lifetimes of the states involved in the transition, b) kinematic temperatures characterizing the small-scale random motions of the atoms or molecules, c) turbulence or larger-scale random motions, and d) kinematics, by which we mean large-scale ordered motions such as expansions, contractions, or rotations. A useful exercise is to estimate the spectrum that would be seen from some simple kinematic models such as a circumstellar shell or sphere that is expanding, contracting (infalling), or rotating and is unresolved in angle. In some cases the central star ionizes nearby gas, which makes a central continuum source. Such an object can then show both emission and absorption lines.

5.3.1 Parabolic-why?

Radio-astronomical sources are far away, so incoming signals often look like plane waves from a specific direction (point sources), and the first goal of a radio telescope is to catch as much energy as possible from such a wave and avoid as much as possible any other signals, especially local interference. The signal at the antenna in this case can be characterized by a flux density in Janskys (1 Jy = 10-26 w/m2/Hz), so the bigger the antenna, the more watts (well, pico-pico watts) we collect. A parabolic antenna (i.e., a parabola of revolution), which puts all this energy into a small spot where a feed can be placed, is usually an engineering optimum for centimeter and millimeter wavelengths.

5.3.2 Aperture efficiency and K/Jy

Consider a black sphere with diameter d and temperature T at a distance r from a circular receiving antenna with diameter D. Assume that r is much larger than either d or D. Figure 3 is a cartoon of this situation.

Figure 3

Then the power density (power per unit frequency interval) received by the antenna, P, can be calculated as its collecting area times the flux from the source at the antenna or as the specific intensity of the source times the solid angle of the antenna as seen from the source. Either way gives the same formula, namely,

P=I(p d2/4)(1/r2)(p D2/4)

The first three terms on the right are the source flux density, F, and A = p D2/4 is the antenna collecting area, which is usually smaller than its physical area due to various losses. If we characterize P in temperature units, P = 2kTR, as usual, then

TR /F=A/2k

is a figure of merit, sometimes called sensitivity, in Kelvins per Jansky for the antenna. That extra 2 is because the flux density refers to the total in both polarizations, but a single receiver can only receive one polarization. The ratio of the antenna collecting area from this formula to its physical area is called aperture efficiency usually expressed as a percentage and usually 60% or less.

5.3.3 Beam efficiency and beam dilution

Another figure of merit, appropriate for sources extended in angle, is the beam efficiency, crudely defined as the ratio of TR to the brightness temperature of the source. (There are more precise but less useful definitions.) Beam efficiency by this definition is, however, a function of the source angular size and shape, alas. A more useful parameter is the beam dilution, defined in the same way but for an assumed circular source with a specified diameter in beamwidths and as a function of this diameter. Planets with known brightness temperatures and angular sizes are candidate calibration sources for measuring aperture efficiency when they are small in angle compared with a beamwidth or points on the beam-dilution curve when they are larger in angle. Figure 4 is a cartoon of this situation.

5.3.4 Cassegrain-why?

A Cassegrain antenna comprises a parabolical main reflector and a concave hyperbolical subreflector near the prime focus of the main reflector to reflect incoming signals back to a spot near the center of the main reflector, the secondary focus, where the feed is placed. This feed is usually mounted on the front of a receiver box that fits through a hole in the center of the main reflector. This arrangement trades a little additional aperture blockage (the subreflector is larger than a prime-focus feed would be), for the ability to place additional equipment, such as a cryogenic refrigerator, near the secondary focus.

5.3.5 Beamwidth: l D

The beamwidth (full width to half power) of the radiation pattern of a circular antenna is approximately 1.2 l /D in radians, where l is the wavelength and D is the diameter of the antenna in, of course, the same units. The 1.2 factor depends somewhat on the feed illumination pattern, that is, on the pattern of the feed as seen from the main reflector. A circular antenna with circular illumination gives a circular beam.

When a finite antenna is used to map extended sources over a range of angles one can show the resulting maps are band limited in the sense that they contain no angular frequencies above a maximum (called the Nyquist limit) that depends only on the wavelength and the antenna diameter. Band-limited maps are, then, smooth continuous functions of two angles on the sky, but they can be specified or measured at a finite grid of evenly spaced points provided that these points are no farther apart than a Nyquist step, which is l /(2D) in radians. A Nyquist is typically a little less than half a beamwidth. Smooth maps can be obtained from finite grids of points by convolution.

5.3.6 Requirements for surface precision

The effective area of an antenna is less than its physical area because of various losses, one of which is due to the departure of the surface from an ideal parabolic shape. An antenna with a surface that is rough on the scale of a wavelength will be almost useless because of low aperture efficiency and also susceptibility to interference scattered into the feed. An ideal antenna would have at least so-called 1/20-wave optics, meaning that the surface is within a 1/20 of a wavelength of perfect. We must sometimes make do with antennas less than ideal; all antennas have some short-wavelength limit based on this criterion.

5.4 Interferometers

5.4.1 Why? Resolution: l /D

Two or more antennas looking at the same source at the same time can have their signals combined into an interferometer to give some of the information that would be obtained from a single antenna with a diameter equal the spacing between the interferometer antennas. The resolution of a two-antenna interferometer is approximately l/D (no 1.2), where D is now the spacing between antennas, but this resolution is only in the direction parallel to a line connecting the antennas and only for sources in a plane perpendicular to this line. A single measurement with a single pair of antennas gives a single datum to be placed in the so-called uvplane, which is the Fourier-transform plane of the angular distribution of the source on the sky. The uvplane is roughly comparable to the plane of the surface of an equivalent single antenna. Even with only two antennas, because of Earth's rotation, data can be obtained over time along two elliptical arcs in the uv plane. More antennas give more elliptical arcs, often crossing, but unless the antennas actually touch or overlap, there will always be gaps with no data in the uv plane. The corresponding synthesized antenna beam has side-lobes because of the missinguv data. There are helpful data-reduction techniques that amount to model fitting or to interpolating or extrapolating across these gaps.

5.4.2 Connected vs VLBI

Interferometers with antennas spaced up to, say, a few kilometers usually have cables or fibers connecting the antennas to a central site where the cross correlations are done in real time. With wider spacing, including transcontinental and intercontinental interferometers, the techniques of tape-recorder or very-long-baseline interferometry (VLBI) are used. At each antenna, data and microsecond timing information, usually derived from atomic clocks, are recorded on magnetic tapes to be shipped to a central site for cross correlation at a time that may be weeks or months later. This delay is a disadvantage, especially for troubleshooting, but there is almost no other way to do large data rates on megameter baselines with their milliarcsecond resolutions. Figure 5 is a sample uv coverage plot of 3C84 for a realistic VLBI experiment.

Figure 5.

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# Modern Physics: Chapter 8 – Normalizing a Wavefunction

I discussed in the previous chapter you the probability of a particle being detected in the continous range $(x_{0}, x_{1})$ can be calculated from its wavefunction using,

• $\int_{x_{0}}^{x_{1}} \psi^{*} \psi dx$

Now, assuming the particle actually exists, if we essentially look for it at every point in the entire universe we’re bound the detect it somewhere right? Hence the probability of finding the particle in the range $(-\infty, \infty)$ should be $1$. In other words,

• $\int_{-\infty}^{\infty} \psi^{*} \psi dx$ must equal $1$.

So, what if we get a wavefunction, say. $g(x)$ where $\int_{-\infty}^{\infty} g^{*} g dx$ doesn’t equal $1$ and equals some other number $k$ instead? Then we can make the function’s probability equal $1$ bu multiplying it with a constant $A$, such that the corrected wavefunction is:

• $\psi(x) = A \times g(x)$

This process is called normalizing the wavefunction. But how do we figure out what the value of $A$ is? Notice that now that $\psi(x) = A \times g(x)$,

• $\int_{-\infty}^{\infty} \psi^{*} \psi dx$
• $\Rightarrow \int_{-\infty}^{\infty} |A|^2 g^{*}(x) g(x) dx$
• $\Rightarrow |A|^2 \int_{-\infty}^{\infty} g^{*}(x) g(x) dx$

Now we already know that $\int_{-\infty}^{\infty} g^{*} g dx = k$, so the above expression simplifies to:

• $\Rightarrow |A|^2 \int_{-\infty}^{\infty} g^{*}(x) g(x) dx \Rightarrow |A|^2 \times k$

Now to ensure $\int_{-\infty}^{\infty} \psi^{*} \psi dx = 1$, we need a value for $A$ such that

• $|A|^2 \times k = 1$  so
• $|A|^2 = \frac{1}{k}$
• $\Rightarrow |A| = \frac{1}{\sqrt{k}}$

# Modern Physics: Chapter 7 – Introduction to Wavefunctions

XKCD/849

So I already mentioned that everything has a matter-wave associated with it. Developing the idea a bit further, we get the concept of a wavefunction which, by convention, is denoted by $\psi$.

Each physical system is described by a state function which determines all can be known about the system. … The wavefunction has to be finite and single valued throughout position space, and furthermore, it must also be a continuous and continuously differentiable function.

Furthermore, a wavefunction is generally a complex function i.e it consists of both a real part and an imaginary part.

• $\psi (x, t)$ – A wavefunction

To get the probably of a particle being at a position $x$, you have to get a real value out of the complex wavefunction. Hence the probably of finding a particle described by a wavefunction $\psi$ is the product of the function $\psi$ at $x$ and its complex conjugate $\psi^{*}$ at $x$.

• $\psi (x_{0}) \psi^{*}(x_{0})$ – probability of finding the particle at $x_{0}$.

To get the probability from a range $x \in (x_{0}, x_{1})$ where $x$ is a continous variable, you can take the integral of $\psi^{*}\psi$ from $x_{0}$ to $x_{1}$:

• $P(x) = \int_{x_{0}}^{x_{1}} \psi^{*}(x) \psi(x) dx$

# Modern Physics: Chapter 6 – The Heisenberg Uncertainty Principle

XKCD/824

One of the best explanations I’ve found of the Heisenberg uncertainty principle is in Volume III of the Feynman Lectures on Physics. Read the following sections (which I’ve linked) to understand the general concept behind it:

Essentially, there is an inverse relation between the width of a wave-packet, $\Delta x$ and the range of wavenumbers of the waves you’ll need to superpose to generate that wave-packet, $\Delta k$. In general, the more localized a wavepacket is, the more waves you need to add to create it. Here’s an animation to show what I mean:

We can describe this relation as:

• $\Delta x \Delta k \sim 1$

Now, since

• $p = \hbar k \Rightarrow \Delta p = \hbar \Delta k$

Multiplying the first relation with $\hbar$, we get

• $\Delta x \hbar \Delta k \sim \hbar$
• $\Rightarrow \Delta x \Delta p \sim \hbar$

or more precisely,

• $\Delta x \Delta p \ge \hbar$

Which is the Heisenberg Uncertainty Principle.

# Modern Physics: Chapter 5 – Fourier Transform I

XKCD – Fourier

To create a truly localized wavepacket we need to superpose not just two but an infinite amount of sinusoidal waves whose wavelengths and amplitudes vary in a continuous spectrum. To do that, we need to learn about what’s called the Fourier integral:

• $f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} a(k) e^{ikx} dk$

where

• $a(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-ikx} dx$

Why an exponential function instead of sine and cosine? This is because we just used Euler’s formula:

• $e^{ix} = cos(x) + isin(x)$

and it’s derivations:

• $cos(x) = \frac{e^{ix} + e^{-ix}}{2}$
• $sin(x) = \frac{e^{ix} - e^{-ix}}{2i}$

to replace the sines and cosines with the exponential function.

Essentially, this gives us a framework for expressing practically any function as a superposition of harmonic waves (just plug in the function $f(x)$ in the expression for $a(k)$ and then plug in $a(k)$ into the expression for the Fourier integral). This is known as the Fourier transform.


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# Non-surjectivity of $\exp$ map for non-compact Lie group.

I have often read that the exponential map from the Lie algebra of a non-compact Lie group is not surjective, however the product of exponentials involving the compact and non-compact generators of the algebra is surjective. At the moment such a proof is beyond my capability, might someone be able to give me a reference to this result (preferably the original paper, I think it may have been due to Cartan)?

For some non-compact Lie groups such as $\text{SL}_2(\Bbb R)$ it is the case that not every group element is a power of a Lie group element. See Proving that any element of $\text{SL}(2,\mathbb{R})$ can be expressed as $\pm\exp(z)$.
On the other hand, there are non-compact groups such as $\Bbb R^n$ (under addition) where every group element is a power of a Lie group element.
• Hi Lord Shark, that thread you linked was posted from my other account. Are you saying that non all elements of a non-compact lie group can be represented as a product of exponentials from the Lie algebra? The link below claims the contrary, that all elements of $\text{SL}(2,\mathbb{R})$ can be represented as a product of two exponentials. One involving the compact generator and one involving the non-compact generators of $\mathfrak{sl}(2,\mathbb{R})$. Although I haven't been able to find a reference for this claim... physics.drexel.edu/~bob/LieGroups/LG_07.pdf – SigmaAlpha Oct 19 '17 at 6:50
• I think $\text{SL}(2,\mathbb{R})$ is a special case... since we can express any $N\in\text{SL}(2,\mathbb{R})$ as $\pm\exp(z)$ for some $z\in\mathfrak{sl}(2,\mathbb{R})$ and since $-Id$ is in the image of the exponential, it follows that any element can be expressed as the product of two exponentials.. right ? But I am looking for something much more general than this, something about non-compact groups in entirety. – SigmaAlpha Oct 19 '17 at 6:55
• @SigmaAlpha That's not so special. In fact, every element of every real connected Lie group with Lie algebra $\mathfrak g$ can be written as $\exp(X).\exp(Y)$, for some $X,Y\in\mathfrak g$. This was proved in 2003 by Martin Moskowitz and Richard Sacksteder. – José Carlos Santos Oct 19 '17 at 12:48


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## Chemistry 9th Edition

Published by Cengage Learning

# Chapter 2 - Atoms, Molecules, and Ions - ChemWork Problems - Page 79: 108

#### Answer

(b), (d), (e) are true, while (a) and (c) are false.

#### Work Step by Step

a) Statement is false because Magnesium has symbol $Mg$, not $Mn$ ($Mn$ is a symbol of manganese) b) True. As all elements Phosphorus (P) , Arsenic (As) and Bismuth (Bi) are a part of group (family) 15. c) False. Gallium (Ga) $loses$ three electrons to form its $cation$, while Selenium (Se) and Bromine(Br) gain electrons to form anions. d) True. All elements that lie from group 3 -12 are called transition elements. Cobalt (Co), Nickel (Ni) and Mercury (Hg) are transition metals e) True . Correct name for $TiO_{2}$ $is$ Titanium dioxide

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


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# Duke University

The Duke Authenticity Project (DAP) - DAP is a collaboration between the first year residential campus and Duke's Center for Leadership Development and Social Action, which attempts to guide student exploration about how students' personal commitment to authenticity is the core to their ability to lead, and lead well.

## Format

• Audience: First-year students.
• When: Academic Year - both fall and spring semesters.
• Frequency: Fall Retreat (4 hours in afternoon), a series of three follow up, 1 hour-long small group discussions following the retreat that occur throughout the academic year (one in the fall semester, two in the spring semester).
• Approximate Number of Participants: 25-40 participants.
• Focus of Retreat: Student facilitated workshops that focus mostly on discussions about knowing one's authentic self, being authentic in relationships, and being an authentic leader.

## Funding & Operations

The Project is a collaboration between the first year residential campus and Duke's Center for Leadership Development and Social Action.

• External Support: None

### Assessment

This is the second year that Duke will assess learning from its leadership programs as they fall into one of three categories: Character, Collaboration, or Citizenship.  DAP falls into the "Character" framework, and so participants will receive the university-wide "Character" assessment at the end of the year.  After the fall retreat, students received an evaluation to measure program satisfaction.  Additionally, after the retreat and after each follow up conversation, students will answer one open-ended question intended to measure learning outcomes.  This will be the first year that we attempt to measure outcomes or impact with the "Character" assessment and with the open-ended questions.


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# How do you find the vertical, horizontal or slant asymptotes for f(x)=(2x^3+3x^2+x)/(6x^2+x-1)?

Oct 28, 2016

The vertical asymptote is $x = \frac{1}{3}$
The slant asymptote is $y = \frac{x}{3} + \frac{4}{9}$

#### Explanation:

Let's start by fatorising the numerator and the denominator
$2 {x}^{3} + 3 {x}^{2} + x = x \left(2 {x}^{2} + 3 x + 1\right) = x \left(2 x + 1\right) \left(x + 1\right)$
and $6 {x}^{2} + x - 1 = \left(3 x - 1\right) \left(2 x + 1\right)$

So $f \left(x\right) = \frac{2 {x}^{3} + 3 {x}^{2} + x}{6 {x}^{2} + x - 1} = \frac{x \cancel{2 x + 1} \left(x + 1\right)}{\left(3 x - 1\right) \cancel{2 x + 1}} = \frac{x \left(x + 1\right)}{\left(3 x - 1\right)}$

As we cannot divide by $0$, so the vertical asymptote is $x = \frac{1}{3}$

As the degree of the numerator is greater than the degree of the denominator, we would expect a slant asymptote. So we make a long division.
${x}^{2} + x$$\textcolor{w h i t e}{a a a a}$∣$3 x - 1$
${x}^{2} - \frac{x}{3}$$\textcolor{w h i t e}{a a a}$∣$\frac{x}{3} + \frac{4}{9}$
$0 + \frac{4 x}{3}$
$\textcolor{w h i t e}{a a a a}$$\frac{4 x}{3} - \frac{4}{9}$
$\textcolor{w h i t e}{a a a a a a}$$0 + \frac{4}{9}$

So $f \left(x\right) = \frac{x}{3} + \frac{4}{9} + \frac{\frac{4}{9}}{3 x - 1}$
The slant asymptote is $y = \frac{x}{3} + \frac{4}{9}$
There is no horizontal asymptote as limit of $f \left(x\right)$ as $x \to \pm \infty$ is $\pm \infty$
graph{(x^2+x)/(3x-1) [-7.02, 7.024, -3.51, 3.51]}


text:
## Introduction

The response of Atlantic meridional overturning circulation (AMOC) to global warming has profound impact on global climate1. Modeling studies have long suggested an AMOC weakening response to global warming induced by anthropogenic greenhouse gasses (GHGs)2,3,4. However, due to the presence of strong interdecadal variability and the shortness of direct measurements available5,6,7,8, whether this AMOC weakening has emerged in the real world has to be examined from a longer-term perspective, which can only be made indirectly with AMOC fingerprints. AMOC fingerprints used so far are located over the North Atlantic9,10,11,12,13,14,15,16,17,18, most of which are not specifically designed for detecting long-term trend9,10,11,12. Notably, however, the classical ‘warming hole’ fingerprint (TNA, Fig. 1e, Methods), which is characterized by a surface cooling over the subpolar North Atlantic (SPNA) relative to background global warming in response to reduced northward AMOC heat transport, has been used as an “all-climate” fingerprint to infer AMOC trend and variability13,14. In both CMIP5 and CMIP6 models, the forced responses of TNA do exhibit long-term trends largely consistent with multi-model ensemble mean (MMEM) AMOC (Fig. 1a, e), consistent with previous model analysis for TNA as an AMOC fingerprint13,14.

An application of TNA to the real world has been interpreted to show an AMOC weakening trend starting from the 1950s13,14 (Fig. 1e). However, this TNA weakening is reversed to strengthening from the 1990s to 2010s, largely overwhelmed by the 60-70-year Atlantic Multidecadal Oscillation (AMO, Fig. 1e, Methods) that has been identified in various fingerprints15,19. This dramatic difference between the observed TNA and model forced TNA response (Fig. 1e) likely suggests that the TNA in the real world contains strong “noise” of interdecadal variability, making it difficult to detect the global warming trend signal. Another challenge for TNA is the strong sensitivity of SPNA temperature to natural (volcanic) and anthropogenic aerosol radiative forcing, especially for the period of the 1950s–1990s20,21,22 (Fig. 1e, f). In addition, modeling studies suggest that the relationship between TNA and AMOC can be nonstationary and dependent on the forcing and processes that drive AMOC changes18,23,24.

The sparseness in AMOC measurements and the challenges on the TNA fingerprint lead to the following questions: (1) Has the anthropogenic AMOC weakening emerged in reality? (2) Is there an optimal fingerprint that can detect the AMOC trend more clearly than TNA?

## Results

### Accelerated AMOC weakening after the 1980s in models

While models suggest an AMOC weakening response to increased anthropogenic GHGs2,3,4, models also suggest an AMOC strengthening in response to anthropogenic aerosol cooling4,20,25,26,27. Before the 1980s, the modest GHGs forcing competes with the aerosol forcing, leaving a modest global warming3,4 and some uncertainty in the forced AMOC response, as seen in a weak decreasing trend in CMIP5 MMEM (−0.1 Sv/50years in 1920–1980) but a weak increasing trend in the CMIP6 MMEM (0.7 Sv/50years in 1920–1980) (Fig. 1a). The difference between the two MMEMs is likely caused by aerosol forcing, which remains highly uncertain28 and most likely overestimated in CMIP627,29 (Methods). In contrast, after the 1980s, the GHGs continued increasing while the aerosol forcing started decreasing4,30 (Fig. 1f), forcing an accelerated global sea surface warming at a rate of ~0.8 °C/50 years in model simulations (~0.1 °C/50 years in 1920–1980) and likely also in observations3,4. Along with the unprecedented warming rate, an accelerated AMOC weakening emerged since the 1980s in both CMIP5 MMEM (−1.2 Sv/50years in 1980–2005) and CMIP6 MMEM (−4.0 Sv/50 years in 1980-2014) (Fig. 1a), which can be also detected in ~80% of individual members (Fig. S1). Interestingly, this post-1980s AMOC weakening trend appears consistent with the AMOC observations and reanalyses after the 1990s31, including ECCOv4r4 estimate32 (Fig. 1a), GloSea5 reanalysis33 and a South Atlantic reconstruction34, as well as the direct RAPID measurements at 26.5°N since 20045,6,7,8 (Fig. 1a, Fig. S2b).

### Optimal fingerprint for forced AMOC change

The classic AMOC fingerprint TNA potentially contains strong “noise” of interdecadal variability, complicating the detection of long-term AMOC trend. Indeed, this “noise” effect can also be seen in the model historical runs in the trend correlation between TNA and AMOC for the long period of 1850-1985 by comparing cross-member and cross-model correlations27, with the latter favoring forced response because the ensemble mean in each model (~4 members for CMIP5 model and ~9 members for CMIP6 model, Table S1) suppresses internal variability. It is seen that the high cross-member trend correlation (~0.6) is largely reduced in the cross-model trend correlation for both CMIP5 (negative trend, to 0.02) and CMIP6 (positive trend, to 0.34) (Fig. 2e, f). Therefore, TNA cannot serve as a reliable fingerprint for long-term forced AMOC response.

Alternatively, we will show below that a salinity-based fingerprint firstly proposed in our previous study35 can be a better choice for detecting this long-term forced AMOC change. This potentially optimal AMOC fingerprint is characterized by a salinity pileup over the subtropical South Atlantic (STSA), which is associated with the anomalous salinity convergence caused by AMOC changes35 (Fig. 1b; Methods). This salinity fingerprint is defined as the sea surface salinity (SSS) difference between the STSA and the subtropical South Indo-Pacific (STSIP) $${S}_{S}=-({S}_{{STSA}}-{S}_{{STSIP}})$$, or approximately as $${{S}_{S} \sim S}_{{SA}}={-S}_{{STSA}}$$, with the minus sign to ensure the index change consistent with the sign of AMOC change. In contrast to all previous AMOC fingerprints that are located locally in the North Atlantic, our SSS fingerprint is unique in that it represents a remote AMOC response to the buoyancy flux forcing in the SPNA deep-convection region, which is known to force AMOC signal propagating southward coherently36,37,38. In response to a weakening AMOC, the northward salinity transport in the upper South Atlantic is reduced. Given the increase of mean climatological salinity from the subpolar to subtropics, the reduction in salinity transport is greater downstream (northern side of the STSA domain) than upstream (southern side of the STSA domain) in the upper branch of the AMOC, leading to a salinity pile-up in the South Atlantic35. The forced response of the SSS fingerprint as shown in MMEM (Fig. 1b, c) is overall consistent with that of AMOC (Fig. 1a) in both CMIP5 and CMIP6, consisting of a weak trend prior to ~1980s and an accelerated weakening afterwards. For the real world, the observed $${S}_{{S}}$$ or $${S}_{{SA}}$$ shows a slow decreasing trend in EN4 data39 prior to 1990s (albeit uncertain in the Ishii data40, which is too short), but a clearly accelerated weakening after the 1990s in both EN4 and Ishii data sets (Fig. 1b, c, Fig. S2a). We note that salinity observations in the earlier period in the South Atlantic tend to be sparse (with lower observation weight especially before the 1950s; Fig. 1d) and thus less reliable. Nevertheless, the lack of variability in the earlier period appears in both our salinity fingerprint and TNA, indicating that the stronger trend/variability in the later period relative to the earlier one is possibly still valid. Moreover, in contrast to the strong oscillation in TNA after the 1950s, the observed SSS indices (SS and SSA) decrease more smoothly and is less distorted by strong variability, notably, the AMO after the 1950s. As such, the observed SSS indices resemble the forced model AMOC trend signal (Fig. 1b) more than TNA and contains less variability noise. This visual impression can be quantified by defining the long-term linear trend as the “signal” and the remaining variability as “noise”. The signal/noise ratio thus calculated for the SSS fingerprint is about twice that of TNA (~0.8 vs ~0.4). Results remain robust if the annual TNA is replaced with the cold season TNA. This leads to our hypothesis that the SSS fingerprint is an optimal fingerprint that captures the real world AMOC trend better than TNA with a higher signal/noise ratio. The following section describes the multiple lines of evidence and the fundamental dynamics underpinning this hypothesis.

### Trend “Signal” and variability “Noise” in AMOC fingerprints

Our first modelling evidence for the optimal SSS fingerprint relative to TNA is their trend correlations with AMOC for the period of 1850-1985, which has been used for testing TNA14,27. In contrast to the reduced trend correlation from cross-member to cross-model between TNA and AMOC (Fig. 2e, f), the trend correlation between the SSS fingerprint and AMOC increases by nearly 50% in both CMIP5 and CMIP6 (Fig. 2b vs a or Fig. 2d vs c). Thus, suppressing internal variability by ensemble mean in each model increases the trend consistency between the SSS fingerprint and AMOC, but decreases the trend consistency between TNA and AMOC. This supports the SSS fingerprint containing more AMOC trend “signal” than TNA.

The second evidence of TNA containing more AMOC variability “noise” can be seen in the control simulations, which contains only internal variability under constant external forcing. Figure 3a shows the MMEM of lead-lag correlations between the annual northern AMOC transport (at ~30oN) and its fingerprints $${T}_{{NA}},$$ $${S}_{S}$$ and $${S}_{{SA}}$$ (for individual models see Fig. S3), with the maximum lagged correlations in each model shown in the scatter diagram Fig. 3d for TNA vs SS (magenta dot) and TNA VS SSA (orchid dot). Both TNA and SSS indices are correlated with AMOC at 95% significance level, confirming the validity of both fingerprints35. Relatively, however, TNA has a higher correlation (~0.6) and a short lag of 0-5 years, while the SSS indices have a lower correlation (~0.25) and decadal-scale lag (Fig. 3a, d, Fig. S3), supporting our hypothesis that TNA contains more AMOC internal variability than the SSS fingerprint. The longer lag of SSS indices can be understood as its delayed response to the AMOC change in the South Atlantic, which further lags the AMOC change in the North Atlantic by 5–8 years as seen in the lagged correlation (Fig. 3a, Fig. S3). Similar results can be found in the experiments forced by natural forcing of solar variability and volcanic eruption (Hist-nat), or forced by the stratospheric ozone (Hist-stratO3) (not shown) in the CMIP6 Detection and Attribution Model Intercomparison Project (CMIP6-DAMIP; Table S2; Methods). This is because the variable forcing in these two experiments are relatively weak such that model variability is still dominated by internal variability (green and grey lines in Fig. 4) as in the control simulations.

The third evidence for optimal SSS fingerprint can be seen in the simulations forced by long-term forcing of anthropogenic GHGs (Hist-GHG) and aerosols (Hist-aer) in CMIP6-DAMIP, in comparison with the control simulations. The maximum lagged correlation between AMOC and SSS-based AMOC indices nearly doubles in forced model runs compared with the pre-industrial control run (Fig. 3b, c vs a) and becomes comparable to that for TNA. This enhanced correlation is caused by the much-increased AMOC trend signal in the two forced experiments. Indeed, GHGs warming and aerosol cooling are the two leading long-term forcing, which nevertheless have the opposite climate impacts (brown and blue lines in Fig. 1f; Methods). The forced AMOC response exhibits a clear trend, decreasing to GHGs warming and increasing to aerosol cooling4,20,27 (Fig. 4a). Similarly, the forced SSS indices also show a clear trend response (orange and blue lines in Fig. 4b, c). The seemingly inconsistent SSS indices and AMOC after the 1990s under aerosol forcing is, we speculate, due to the lagged response of the South Atlantic salinity to the AMOC change, as discussed earlier. The SSS response, especially $${S}_{S}$$, to the GHGs or aerosols forcing is forced predominantly by the AMOC change, instead of the surface E-P forcing associated with the hydrological response35 (Fig. S4 and S5, Supplementary Text). In comparison, the forced TNA trend response, even using the MMEM, are still distorted substantially by internal variability (orange and blue lines in Fig. 4d). When all the forcing is combined, the competition between the impacts of GHGs and aerosols prior to the 1980s leads to the prevalence of natural AMOC variability41 (with only weak increasing trend), while the combination of continued increase of GHGs and the reduction of aerosols after the 1980s forces an accelerated AMOC weakening as in the CMIP6 historical runs27,41 (Fig. 4a, Fig. 1a). This final AMOC response is reasonably simulated by the SSS indices, especially, $${S}_{{SA}}$$ (Fig. 4b, c), but not by $${T}_{{NA}}$$. Finally, across simulation members in Hist-GHG and Hist-aer, the maximum lagged correlation with AMOC exhibits strong scatter with some even becoming negative for TNA, but has a much smaller scatter with all correlations positive for the SSS indices (Fig. 3e, f). The enhanced lead-lag correlation of SSS indices with AMOC and a small cross-member spread relative to TNA in response to long-term GHGs and aerosol forcing further support our hypothesis that the SSS fingerprint has a higher trend-signal/variability-noise ratio than TNA.

### Mechanisms for trend/variability ratio in AMOC fingerprints

The higher trend/variability ratio in the remote SSS fingerprint than TNA can be understood from the dynamic adjustment of AMOC response to buoyancy forcing over the SPNA. The AMOC response is determined by the basin-wide adjustment of westward propagating long Rossby waves42. Since Rossby wave speed increases towards the equator, the basin-wide adjustment is a decaying response43,44. In response to variable buoyancy forcing over the SPNA, higher-frequency AMOC variability decays faster southward, especially across the equator38,45,46, leaving more long-term trend response, or higher trend/variability ratio, remotely in the South Atlantic than the SPNA, and, in turn, in our SSS fingerprint than TNA.

This ocean dynamic mechanism is confirmed in our ocean general circulation model (OGCM) experiments forced by idealized heat flux forcing. Figure 5 shows one example, in which the North Atlantic is forced by variable heat flux forcing that consists of a centennial change (180-year period) and interdecadal variability (30 year variability) (Fig. 5a, Methods). The AMOC interdecadal variability is reduced significantly by ~70% from 30°N to 30°S (Fig. 5c), quantitatively consistent with theoretical studies45,46. In comparison, the AMOC centennial trend remains largely unattenuated (reduced by only ~30%) at 30°S (Fig. 5c). As a result, the AMOC trend/variability ratio, as defined as the ratio of standard deviation between centennial and interdecadal variability, increases monotonically southward by more than two times (Fig. 5h). The variation of TNA and Ss resembles that of the AMOC in the North and South Atlantic, respectively (Fig. 5b,c,e–g), as expected. Therefore, the trend/variability ratio of TNA and Ss also resembles that of AMOC in the North and South Atlantic, respectively (Fig. 5h). Finally, the lead-lag correlations of these fingerprints with the AMOC transport also show similar features to those in the forced CMIP6-DAMIP experiments (Fig. 3b, c), with the TNA and Ss of comparable correlations, and with the Ss lagging the TNA and the North Atlantic AMOC by nearly a decade (Fig. 5d).

In the real world, or even coupled general circulation models (CGCMs), the AMOC trend/variability could be complicated by other factors, such as the wind forcing and longer-term internal variability of centennial time scales. Nevertheless, given the stable climate in the STSA trade wind region relative to the storm track region of SPNA, wind forcing seems unlikely to distort the pattern of trend/variability ratio completely. Indeed, this increased trend/variability ratio in the STSA has been found in some CMIP6 experiments, for example, in one member of the historical GHGs forcing experiments in the GISS-E2-1-G model (hereafter GISS-GHGs) (Fig. 5i–l) (other members show similar results, Fig. S6). Similar to our OGCM experiment above, the AMOC in GISS-GHGs is dominated by a 30-yr internal variability along with a weakening trend in response to global warming (Fig. 5i). The trend/variability ratio in GISS-GHGs increases from 30°N to 30°S monotonically by nearly three times for AMOC, and, similarly, from TNA to SS for the fingerprints (Fig. 5i–l).

Besides the ocean dynamic mechanism above, atmospheric dynamics also favor the salinity fingerprint. TNA is very sensitive to volcanic forcing and stratospheric ozone as seen in the CMIP6-DAMIP experiments Hist-nat and Hist-straO3, where the responses are almost comparable to those to the anthropogenic GHGs and aerosol forcing (Fig. 4d). For example, the forced TNA response in Hist-nat shows two multi-decadal episodes (1870s–1930s, 1960s–1990s) of minimum (green, Fig. 4d), corresponding to the two observed cooling periods caused by volcanic eruptions22 (Fig. 1f). Similarly, the forced TNA response to stratospheric ozone variation shows strong multi-decadal variability between the 1980s and 2010s (gray, Fig. 4d), corresponding to the observed multi-decadal variability in spring Arctic stratospheric zone47. In comparison, there is little sensitivity to volcanic forcing and stratospheric ozone effect in AMOC (Fig. 4d vs a). This is consistent with studies showing that TNA can be driven by external radiative forcing, or more general, the atmospheric forcing without an explicit role of AMOC change21,22,48,49. In contrast to TNA, the SSS indices show little response to volcanic and stratospheric ozone forcing (Fig. 4b, c), due partly to the subtropical South Atlantic region being less susceptible to either volcanic eruption50,51 or Antarctic/Arctic stratospheric ozone variation47,52,53.

Finally, the stronger sensitivity of TNA than SS to fast variability forcing can also be seen in CMIP6 experiments of abrupt CO2 quadrupling of 150 years (abrupt-4xCO2, Table S1). In response to the abrupt CO2 increase, TNA exhibits a large opposite (negative) response in initial decades in about half of the models, while SS and AMOC are both dominated by a slow trend response across all models (Fig. S7). This provides a further support of our hypothesis.

Some studies have also suggested that changes in the South Atlantic could be related to interdecadal variability in the Pacific Ocean through atmospheric teleconnection54,55 or driven by the Southern Annular Mode (SAM) through its effect on Agulhas leakage56,57,58,59. Our results suggest, however, a strong control of northern Atlantic buoyancy forcing on AMOC on multi-decadal and longer timescales, with the long-term response emerged most clearly in the South Atlantic, as detected in both the control and forced simulations and supported by theoretical studies35,45,46.

## Discussion

Given the salinity pileup as the optimal AMOC fingerprint for detecting AMOC weakening in response to current global warming, we interpret the accelerated decline of the SSS index in the 1990s (Fig. 1b, c) as an evidence that an accelerated AMOC weakening forced by the combination of anthropogenic GHGs and aerosol forcing, as predicted by model experiments (Fig. 1a, Fig. 4a), may have occurred about one decade ago in the 1980s (while an overall weakening in AMOC probably started several decades earlier13,14). In the future, the combination of a continued increase of GHGs and decrease of anthropogenic aerosol will likely further accelerate the AMOC weakening. Our study, combined with recent analysis that AMOC may have evolved from relatively stable period to a point close to critical transition60, may provide early warning signals of accelerated anthropogenic AMOC weakening and its potential climate impact in the coming decades.

## Methods

### CMIP5 and CMIP6 model simulations

We use CMIP6 and CMIP5 experiments to evaluates the AMOC and its indices as well as their response to climate forcing (Table S1). CMIP6 employs the latest generation of climate models. We use the model output of 8 CMIP6 models with a total of 70 ensemble members and 8 CMIP5 models with a total of 31 ensemble members for all-forcing historical simulations. We also use 7 CMIP6 models (with one member for each) for PI control simulations. We found the spread among members within one model is comparable to that among different models. Therefore, the MMEM in the present study is calculated across all the available model members rather than across models’ ensemble means, although the results are similar.

We further investigate the attribution of AMOC and its fingerprints using simulations from the Detection and Attribution Model Intercomparison Project (DAMIP; Table S2). As a part of CMIP6, DAMIP isolates the individual effect of anthropogenic GHGs (Hist-GHG), anthropogenic aerosols (Hist-aer), natural forcing (e.g., solar activity and volcanic eruptions; Hist-nat) and stratospheric ozone (e.g., ozone depletion; Hist-stratO3). We define the anthropogenic aerosol forcing as the total absorbed short-wave radiation (ASR) in the Northern Hemisphere (ASR_NH) derived from the simulations forced by anthropogenic aerosols. Compared with CMIP5, more models with indirect aerosol forcing (e.g., aerosol-cloud microphysical effect) are included in CMIP6. The inclusion of indirect aerosol forcing reinforces the cooling effect of aerosols, leading to an increasing radiative cooling of ~2 W/m2 between the 1880s and the 1980s (Fig. 1f). After the 1980s, the cooling effect of anthropogenic aerosols decreases by ~0.5 W/m2. Our estimate of aerosol forcing agrees well with the gridded aerosol community datasets (CEDS)30,61. The larger magnitude (by ~0.3 W/m2) in our estimate compared with CEDS may be attributed partly to the overestimation of aerosol forcing in CMIP6 models and partly to the climate feedbacks in models, notably the sea ice-albedo feedback (i.e., more reflection of incoming short-wave radiation by increasing sea ice in response to aerosol’s cooling effect). The GHGs concentration, on the other hand, increases continually since the industrial revolution, notably a rapid increase in atmospheric CO2 from 290 to more than 400 ppm (Fig. 1f), corresponding to an increase in radiative warming effect of 3.5 W/m2 (ref. 61). We also use abrupt-4xCO2 experiments (Table S1) to study the response time scales of AMOC indices to abrupt CO2 quadrupling.

### Definition of AMOC strength, AMOC fingerprints and AMO index

AMOC intensity is defined as the maximum overturning streamfunction below 300 m over 30-50° N in the Atlantic. This is also referred to in the text as the northern AMOC. Similarly, the southern AMOC is defined as the maximum overturning circulation below 300 m over 10-34° S in the Atlantic. The “warming hole” SST based index TNA, is defined as annual mean, SPNA mean (15–40°W,46–60°N) SST minus annual mean, global mean SST. This definition is a simplified version of that of ref. 35 but with similar results27 (not shown). The “salinity pile-up” index SS, is defined as annual mean, STSA mean (averaged over 10–34°S) SSS minus annual mean, STSIP mean SSS at the same latitude band. The MMEMs for these two indices are calculated with each ensemble member equally weighted. The AMO index shown in Fig. 1e is calculated after ref. 14 as the weighted mean SST over the North Atlantic (0° N to 60° N) relative to the mean SST from the period 1900-1950 with the global mean SST (60° S to 60° N) removed.

### Ocean model and sensitivity experiments

The ocean model used in our study is POP262,63, which is the ocean-component of the coupled CESM model. Our version of POP2 has a uniform resolution of 3.6° in the zonal direction and a non-uniform resolution (0.6° near the equator, gradually increasing to the maximum of 3.4° at 35° N/S and then decreasing poleward) in the meridional direction. The model has 60 levels in the vertical, with a uniform resolution of 10 m in the upper 160 m, increasing to 250 m to the depth of 3500 m and then remaining the same towards depth. The control run is forced by the normal year forcing from the Co-ordinated Ocean–Ice Reference Experiments (CORE) dataset64, using the CORE experimental design as outlined in ref. 65. The CORE forcing and bulk formulas used here are the version 2 (COREv2) as defined in ref. 64. This model has been used for the demonstration of the mechanism of salinity pileup fingerprint35.

The CTRL run is integrated for 900 years when AMOC indices are found near quasi-equilibrium, from which the sensitivity experiment is launched. Sensitivity experiment is integrated 180 years. Anomalous surface heat flux is imposed to produce AMOC variability with different time scales. The magnitude of 30-year periodic forcing is 20 W/m2. Besides the periodic forcing, we also apply a linear heat flux anomaly over the North Atlantic (20°N–80°N), with its magnitude decreasing linearly from zero to -9 W/m2 in 90 years and then increasing linearly to 9 W/m2 from year 91 to180. The reversal of linear forcing causes a trend reversal in AMOC strength, resembling that observed in CMIP6. The wind stress remains unchanged in our sensitivity experiments so that there is no effect of wind induced circulation change. As such, the dynamic effect on salinity transport is caused by the buoyance-forced AMOC change only.


text:
# Multinacci Polynomial

1. Nov 3, 2006

### Diophantus

Can anyone please give me a hint on how I can prove that

$$g(x) = x^m - x^{m-1} - x^{m-2} - ... - x - 1$$

is irreudicble over the rationals for all natural numbers m?

Regards

Last edited: Nov 3, 2006
2. Nov 3, 2006

### Oxymoron

I would use an evaluation homomorphism, to change the variable, and then use the Eisenstein criterion for a prime p and show it is irreducible. I haven't actually done it (so this might be wrong), but this is what I would try first.

The evaluation homomorphism I would use would be of the form

$$\phi_{x+1}\,:\,\mathbb{Q}[x] \rightarrow \mathbb{Q}[x] \quad\quad \phi_{x+1}(g(x)) = g(x+1)$$

Hopefully you'll get a nice binomial expression to which you can check if p divides (by Eisenstein's Criteria).

3. Nov 8, 2006

### Diophantus

Alas, the Eisenstein criteria will not work here for every m.

Indeed $$\phi_{x+1}(x^4 - x^3 - x^2 - x - 1) = x^4 + 3x^3 + 2x^2 - 2x - 3$$

and then (2,3) = 1 implies that no suitable prime can be found.

Any other ideas?


text:
Tutorials
sql

# Aggregate Functions in SQL

Learn how to use aggregate functions for summarizing results and gaining useful insights about data in SQL.

Building reports from a given dataset is an essential skill if you are working with data. Because ultimately, you want to be able to answer critical business questions using the data at your disposal. Many times, these answers presented in the form of report charts. But sometimes, reports in the form of tables are also needed. In both cases, you might need to summarize the data using simple calculations. In SQL, you can summarize/aggregate the data using aggregate functions. With these functions, you will be able to answer questions like:

• What is the maximum value for the some_column_from_the_table? or
• What are the minimum values of some_column_from_the_table with respect to another_column_from_the_table?

and many more.

Let's get started to perform some data aggregation.

Note: To be able to follow along with this tutorial, you need to know how to write basic queries in PostgreSQL (which you are going to use as the RDBMS). This tutorial can serve as a good refresher.

## Setting up the database

Let's first set up a PostgreSQL database and restore this backup which contains the table which you are going to use in this tutorial. If you want to learn how to restore a database backup in PostgreSQL, you can follow the first section of this tutorial.

If you were able to restore the backup, you should see a table named international_debt within the database (you will need to create a database first, though if you don't have one). Let's quickly take a look at the first few rows of the table (a simple select query will help you do this):

The table contains information about the debt statistics of different countries across the globe for this current year in different categories (refer to the columns indicator_name and indicator_code). The debt column shows the amount of debt (in USD) a particular country has in a specific category. The data belongs to the domain of economics and is often used to analyze the economic conditions of different countries. The data was collected from the World Bank.

Now that you have set up the database successfully let's run some simple queries to know more about the data. Open up the pgAdmin tool and get started.

## Simple information matters

From the above figure, you can see that there are many duplicate entries for a single country but for different categories. A question that quickly gets raised from here is:

What are the different countries that the table contains records of?

If you simply run a select query with the country_name column, you will not get the right answer to this question because the result of the query will contain duplicates. Let's use the DISTINCT keyword to combat this.

select distinct country_name from international_debt;

And this should return something like this:

Now you have a fair answer to the above question. Now one last question before you proceed to aggregate functions:

How many different kinds of debt indicators are there in the table?

The query for answering should be similar to the above one. You just need to change the column name. Take it as an exercise, then? The result should be similar to the following:

## Aggregate functions

Let's start by running a query with an aggregate function and proceed accordingly. Along the way, you will learn more about the syntaxes and the kind of constructs you need to follow when applying aggregate functions in SQL.

select sum(debt) from international_debt;

And the result:

With the SUM() aggregate function, you can calculate the arithmetic sum across a column (that contains numeric values). With the above query, you got to know about the total debt that is pending by the countries listed in the table.

Note that SUM() does not take NULL values into consideration when calculating the sum. Now, let's find the answer to the question:

What is the maximum amount of debt?

Here comes the MAX() aggregate function to your rescue:

select max(debt) from international_debt;

Like the SUM() function, MAX() does not consider the NULL entries while doing its calculations. There's similar MIN() function as well. Let me know the minimum value of the debt column via the Comments section? Now, it will be a good idea to actually find out if there is an invalid entry in the debt column to ensure that the results are correct until now.

Note that you use these functions in lower case as shown above.

When you execute this query:select * from international_debt where debt is null;, you should get an empty result. Let's now find out the total number of distinct countries present in the table.

select count(distinct(country_name)) from international_debt;

And you see there is a total of 124 distinct countries present in the table. Pay close attention to the succession of functions you applied in the above query. Yes, it is allowed here, i.e. to connect more than one aggregate function in a logical way.

Now, suppose, you want to see the average value of the debt column. The function is AVG():

select avg(debt) from international_debt;

You see the value to be 1306633214.966397971 (USD). It is a good idea to present these results with suitable column names. From the above results, you can see that PostgreSQL changes the column name to aggregate function name (that is being included in the query) when returning the result. So, it is a good idea to give a proper alias to these columns. You can do this like so:

select avg(debt) as Average_Debt_By_A_Country from international_debt;

The result is much more interpretable:

Let's now take this to a bit more complex level. To be able to answer questions like What are the minimum values of some_column_from_the_table with respect to another_column_from_the_table?, you need to pair an aggregate function with GROUP BY clause. Let's see how.

## Aggregate functions + GROUP BY + more

Let's say you want to produce a report where you will be showing the country_name and the sum of their debts. Following is an example:

Reports like this are quite often used in the real world. So, what might be the query for getting a report like this? You will have to use the SUM() function on the debt. And you will also need to show the country_name with the sum of debts. Let's execute the following query:

select country_name, sum(debt) from international_debt;

Doesn't it produce the following error?

ERROR: column "international_debt.country_name" must appear in the GROUP BY clause or be used in an aggregate function

Let's now understand what it actually means. When you are using an aggregate function (like SUM() with an unaggregated column like country_name) you need to pass the unaggregated column to a GROUP BY clause. So, the correct query will be:

select country_name, sum(debt) as total_debt from international_debt group by country_name;

And the result is just right:

Note the use of aliasing in the query.

Now, suppose you need to sort this report in with respect to the total_debt in a descending manner. Remember ORDER BY clause? Yes, you can pair aggregate functions with ORDER BY clause as well:

select country_name, sum(debt) as total_debt from international_debt
group by country_name order by total_debt desc;

The result should be reversely sorted now:

Note the column that you used in the ORDER BY clause.

Now another important question:

What the highest amount of debt across different categories (in a reversely sorted manner)?

You will need to use the MAX() function here. The query to answer this question should not be a hard one to write now.

select indicator_code, max(debt) as maximum_debt from international_debt
group by indicator_code order by maximum_debt desc;

And you get a clean report:

You can also limit the number of rows in the reports like this. Say you just want to include the top five entries in the above report. You can do by using the LIMIT clause.

select indicator_code, max(debt) as maximum_debt from international_debt
group by indicator_code order by maximum_debt desc
limit 5;

Time for the final report for this tutorial. You need to include the names of the countries to the above report. How can you do it? The following query should let you do this:

select country_name, indicator_code, max(debt) as maximum_debt from international_debt
group by country_name, indicator_code order by maximum_debt desc;

Another good report:

In the above query, you added the country_name column after the SELECT clause and also added it after GROUP BY. You can extend this format up to any number.

The order of GROUP BY, ORDER BY and LIMIT is very important while generating reports like this. If you mistakenly change the order, you will have to face errors. See this for yourself:

select country_name, sum(debt) as total_debt from international_debt
order by total_debt desc group by country_name;

And you get:

ERROR:  syntax error at or near "group"
LINE 1: ... from international_debt order by total_debt desc group by c...

In the above query, you placed the ORDER BY clause before GROUP BY which is not permissible. In fact, it is also not applicable when you are not using aggregate functions. The correct order is - GROUP BY -> ORDER BY -> LIMIT. Always remember this.

## Taking things forward

Congrats! You made it till the end of this tutorial. In this tutorial, you got to know about the different aggregate functions in PostgreSQL and how to use them to generate useful reports. These are crucial skills for a data scientist indeed. To level up your SQL skills in a systematic way, you can take the following DataCamp courses:


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# desired velocity and the local space z axis

This question is related to my previous post here , but, even though I did get some insight, I didn't fully get it, and hence have another dilemma. Since the dead earth series is not publicly available, I'll try to explain as much as possible.

Initial around 21:35 timecode of the lesson 13 of the series, the author tries to find the direction the navMeshAgent should turn. For this, he applies the sign of the cross product between desiredVelocity vector and the local z-axis of the navMeshAgent. If the result was negative, he turns left, if positive he turns right. Note that here, he does not convert the desiredVelocity vector to the navMeshAgent's local space.

But, later, we come to a case where we require the angle between the desiredVelocity vector and the navMeshAgent's local z-axis. In this case, he does convert the desiredVelocity vector to the local space of the navMeshAgent using transform.InverseTransformVector(desiredVelocity). I don't really get the concept behind it. Why are we not converting the desiredVelocity vector to localSpace in the former case? Obviously the direction might change from conversion of one space to another, so the result should be affected. Please help me understand this. Thanks in advance.

This is all about apples to apples comparisons and what's the path of least resistance to the information we want.

When we just want to know whether one vector in the horizontal plane is to the left or right of another, checking the sign of the cross product is a nice quick way to do that, at just six multiplications and three subtractions — less than a matrix multiplication or quaternion rotation. (In fact, since we only care about the sign of the y component in that case, we can do it with two multiplications and a comparison — though the compiler might or might not figure out that optimization for us)

We could do this equally well in either world space or local space, since we'll get the same output vector either way, just expressed in a different basis. Since transform.forward already gives us the object's forward direction in world space, we can just use the world space velocity directly with that and not do any space conversions. So here, world space is the path of least resistance.

When we want to know the specific angle between two vectors, that can get a bit trickier. If we normalized the velocity first we could use the arcsine of the length of the cross product, but that's two square roots and divisions plus a transcendental, and it can give incorrect results if our vectors are not perfectly in-plane (say the velocity is tilted slightly uphill — we'd get an angle in a tilted plane instead of the pure horizontal angle we might expect)

But if we could express the velocity as a direction relative to the object's orientation, then we could use the convenient 2-argument arctangent function to get the exact planar angle we want. And here again, Unity offers a convenient path to get there with transform.InverseTransformVector, which provides exactly the needed transformation. Now we have the velocity in a coordinate scheme where z+ is the object's forward and x+ is its right, and finding the turning angle is as easy as expressing that vector in polar coordinates. So here, local space gives us the path of least resistance to what we want.

Note that if we need to do both in a single function, then we might as well use the latter trick with local space. That will get us both the angle and its sign at once. But if we're doing them separately, and either the velocity or the orientation may change in between, then using the shortest route to each data point makes sense.

The tutorial may also be deliberately trying to show you a variety of ways to find out about vector angles, to build up your toolbox.

• Nov 30 '21 at 16:09


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# Puzzles

## Archive

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## 22 December

22 is two times an odd number. Today's number is the mean of all the answers on days (including today) that are two times an odd number.
Clarification: You are taking the mean for answers on days that are two times an odd numbers; ie. the days are two times odd, not the answers.

## 16 December

Today's number is four thirds of the average (mean) of the answers for 13th, 14th, 15th and 16th December.
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# holani.net

Home > Error Function > Error Function Complex Number

# Error Function Complex Number

## Contents

Math. You can also select a location from the following list: Americas Canada (English) United States (English) Europe Belgium (English) Denmark (English) Deutschland (Deutsch) España (Español) Finland (English) France (Français) Ireland (English) calculus integration complex-analysis contour-integration share|cite|improve this question edited Mar 14 '14 at 22:49 Ron Gordon 109k12130221 asked Mar 14 '14 at 19:04 Sleepyhead 1385 add a comment| 3 Answers 3 active The system returned: (22) Invalid argument The remote host or network may be down. his comment is here

share|cite|improve this answer answered Mar 14 '14 at 19:28 GEdgar 46.9k153133 This might work, thanks –Sleepyhead Mar 14 '14 at 20:33 add a comment| up vote 1 down vote Negative integer values of Im(ƒ) are shown with thick red lines. real(erfz(1e-8 + 1i)) gives 3.1023...e-8, but the correct answer is 3.067...e-8 according to Mathematica. Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statement Mobile view Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and https://en.wikipedia.org/wiki/Error_function

## Hyperbolic Function Of Complex Number

J. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the A Course in Modern Analysis, 4th ed. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0521192255, MR2723248 External links MathWorld – Erf Authority control NDL: 00562553 Retrieved from

The error function is related to the cumulative distribution Φ {\displaystyle \Phi } , the integral of the standard normal distribution, by[2] Φ ( x ) = 1 2 + 1 Wall, H.S. Supancic, "On Bürmann's Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion," The Mathematica Journal, 2014. Complementary Error Function Another approximation is given by erf  ( x ) ≈ sgn  ( x ) 1 − exp  ( − x 2 4 π + a x 2 1

Join the conversation Find Code: All Words Any of the Words Exact Phrase Home:Code:Forums:Submit:Mailing List:About:Contact Code All VB.NET ASP.NET C# VB Classic ASP Classic Snippets Popular Resources SubmitCode Forums Articles Tips Generated Mon, 10 Oct 2016 02:05:33 GMT by s_ac4 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.10/ Connection Washington D.C., USA; New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. http://mathworld.wolfram.com/Erf.html The inverse error function is usually defined with domain (−1,1), and it is restricted to this domain in many computer algebra systems.

Johnson Steven G. Error Function Calculator ERFZ enhances ERF to evaluate the error function of complex numbers too. One should note that the number of terms needed to reach a tolerance depends on the value of $z$, and is fairly sensitive to $\Im{z}$. Mathematical Methods for Physicists, 3rd ed.

## Exponential Function In Complex Number

is the double factorial: the product of all odd numbers up to (2n–1). https://www.mathworks.com/matlabcentral/fileexchange/18312-error-function-of-complex-numbers The inverse complementary error function is defined as erfc − 1  ( 1 − z ) = erf − 1  ( z ) . {\displaystyle \operatorname ζ 8 ^{-1}(1-z)=\operatorname Hyperbolic Function Of Complex Number Properties Plots in the complex plane Integrand exp(−z2) erf(z) The property erf  ( − z ) = − erf  ( z ) {\displaystyle \operatorname − 6 (-z)=-\operatorname − 5 Analytic Function In Complex Numbers W.

Use sym to convert complex infinities to symbolic objects:[erfi(sym(i*inf)), erfi(sym(-i*inf))]ans = [ 1i, -1i]Handling Expressions That Contain Imaginary Error Function Many functions, such as diff and int, can handle expressions containing this content Compute the imaginary error function for elements of matrix M and vector V:M = sym([0 inf; 1/3 -inf]); V = sym([1; -i*inf]); erfi(M) erfi(V)ans = [ 0, Inf] [ erfi(1/3), -Inf] Soc. 3, 282-289, 1928. I was thinking using the stationary phase method, and for that i need to separate the erf into real and imaginary. Excel Complex Number Functions

2. Wolfram Problem Generator» Unlimited random practice problems and answers with built-in Step-by-step solutions.
3. A result found in Abramowitz & Stegun claims the following: $$\operatorname*{erf}(x+i y) = \operatorname*{erf}{x} + \frac{e^{-x^2}}{2 \pi x} [(1-\cos{2 x y})+i \sin{2 x y}]\\ + \frac{2}{\pi} e^{-x^2} \sum_{k=1}^{\infty} \frac{e^{-k^2/4}}{k^2+4 x^2}[f_k(x,y)+i g_k(x,y)] 4. Referenced on Wolfram|Alpha: Erf CITE THIS AS: Weisstein, Eric W. "Erf." From MathWorld--A Wolfram Web Resource. 5. Whittaker, E.T. 6. Handbook of Continued Fractions for Special Functions. Google search: Google's search also acts as a calculator and will evaluate "erf(...)" and "erfc(...)" for real arguments. Asking client for discount on tickets to amusement park Is masking before unsigned left shift in C/C++ too paranoid? New York: Gordon and Breach, 1990. weblink IDL: provides both erf and erfc for real and complex arguments. Positive integer values of Im(f) are shown with thick blue lines. Inverse Error Function Matlab provides both erf and erfc for real arguments, also via W. Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (2007), "Section 6.2. ## Consider the definition of the error function in the complex plane:$$\operatorname*{erf}{z} = \frac{2}{\sqrt{\pi}} \int_{\Gamma} d\zeta \, e^{-\zeta^2}$$where \Gamma is any path in the complex plane from \zeta = 0 New York: Random House, 1963. W. Seen this way, the error function of a complex number is equal to$$\operatorname*{erf}{(x+i y)} = \operatorname*{erf}{x} + i \frac{2}{\sqrt{\pi}} e^{-x^2} \int_0^y du \, e^{u^2} \cos{2 x u} \\ + \frac{2}{\sqrt{\pi}} Error Function Table Julia: Includes erf and erfc for real and complex arguments.

Retrieved 2011-10-03. ^ Chiani, M., Dardari, D., Simon, M.K. (2003). See AlsoMuPAD Functionserf | erfc | inverf | inverfc | stats::normalQuantile More AboutError Functions and Fresnel Functions Was this topic helpful? × Select Your Country Choose your country to get translated Taylor series The error function is an entire function; it has no singularities (except that at infinity) and its Taylor expansion always converges. check over here The imaginary error function has a very similar Maclaurin series, which is: erfi  ( z ) = 2 π ∑ n = 0 ∞ z 2 n + 1 n

Intermediate levels of Im(ƒ)=constant are shown with thin green lines. and Watson, G.N. Haskell: An erf package[18] exists that provides a typeclass for the error function and implementations for the native (real) floating point types. Wolfram Education Portal» Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more.

Sequences A000079/M1129, A001147/M3002, and A084253 in "The On-Line Encyclopedia of Integer Sequences." Referenced on Wolfram|Alpha: Erfi CITE THIS AS: Weisstein, Eric W. "Erfi." From MathWorld--A Wolfram Web Resource. Assoc. New York: Dover, pp.179-182, 1967. Weisstein. "Bürmann's Theorem" from Wolfram MathWorld—A Wolfram Web Resource./ E.

Speedwise, this implementation seems to be on par with Godfrey's (file ID: 3574) which I currently use for small to medium range of the complex argument. To do this, we take a detour through some Fourier theory. The inverse imaginary error function is defined as erfi − 1  ( x ) {\displaystyle \operatorname ∑ 8 ^{-1}(x)} .[10] For any real x, Newton's method can be used to Compute the imaginary error function for these numbers.

Why are so many metros underground? Step-by-step Solutions» Walk through homework problems step-by-step from beginning to end. I think these may be zeroes of the error function, but I have not yet investigated.) And now, here's a plot of some results; note that the plot of the effective Orlando, FL: Academic Press, pp.568-569, 1985.

The error function at +∞ is exactly 1 (see Gaussian integral). However, it can be extended to the disk |z| < 1 of the complex plane, using the Maclaurin series erf − 1  ( z ) = ∑ k = 0 Wolfram Demonstrations Project» Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. New York: Dover, pp.297-309, 1972.


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# Prove $\liminf \left(\frac{n!}{(n-k)! n^k}\right) = 1,\ k =$ constant.

I had this as an assumption in my textbook (in the section Poisson approximation to Binomial distribution) but couldn't prove it.

• $n! /(n-k)! \geq (n-k+1)^k$ – kingW3 Jun 7 at 16:39

You are trying to prove that $$\frac{n(n-1)...(n-k+1)}{n^k}\to 1$$ as $$n\to \infty$$. Note that the product above has a finite number (k) of terms, and so when finding the limit of the above product, you may look at the product of the limits: $$\lim_{n\to\infty}\frac{n(n-1)...(n-k+1)}{\underbrace{n\cdot n\cdot ...\cdot n}_{k}}\\=\bigg(\lim_{n\to\infty}\frac{n}{n}\bigg)\bigg(\lim_{n\to\infty}\frac{n-1}{n}\bigg)...\bigg(\lim_{n\to\infty}\frac{n-k+1}{n}\bigg)$$ Each of these limits equals $$1$$, so the limit is $$1$$.


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# 3.9: Settling of Spheres

## Introduction

This section deals with some basic ideas about settling of solid spheres under their own weight through still fluids. This is an important topic in meteorology (hailstones), sedimentology (sediment grains), and technology (cannon balls and spacecraft). In this section we will look at the terminal settling velocity of spheres as an applied problem. At the end I will make some comments about the complicated matter of the time and distance it takes for a sphere to attain its terminal settling velocity.

If placed in suspension in a viscous fluid, a solid body denser than the fluid settles downward and a solid body less dense than the fluid rises upward. A qualification is needed here, however: the body must not be so small that its submerged weight is even smaller than the random forces exerted on it by bombardment by the fluid molecules in thermal motion. Such small weights are generally associated only with the finest particles, in the colloidal size range of small fractions of a micron.

When a nonneutrally buoyant body is released from rest in a still fluid, it accelerates in response to the force of gravity. As the velocity of the body increases, the oppositely directed drag force exerted by the fluid grows until it eventually equals the submerged weight of the body, whereupon the body no longer accelerates but falls (or rises) at its terminal velocity, also called the fall velocity or settling velocity in the case of settling bodies (Figure $$\PageIndex{1}$$).

## Towing vs. Settling

I promised you in Chapter 2 that I would make some comments later about the differences between moving a sphere through still fluid and passing a moving fluid by a stationary sphere. This topic has relevance to the settling of spheres, so I will say some things about it at this point.

It should make sense to you that towing a sphere at velocity $$U$$ through a still fluid by exerting a constant force $$F_{D}$$ on it is equivalent to passing a steady and uniform stream of fluid at velocity $$U$$ around a sphere that is held fixed relative to the boundaries of the flow. This is largely true, but there are two complications. First, if the sphere is held fixed and the flow passes by it, the drag force can be influenced by even weak turbulence in the approaching flow, whereas if the sphere is towed through still fluid there can be no such effect. Second, you have seen that, in some ranges of relative velocity, eddies can form behind the sphere and break away irregularly; if the sphere is fixed and the fluid is flowing by, this causes the force to fluctuate about some average value but does not affect the relative velocity, whereas if the sphere is towed, either the velocity fluctuates along with the force or, if by definition we tow with a constant force, the velocity fluctuates but the force is steady.

Settling of a sphere through still fluid under its own weight is exactly like towing the sphere vertically downward by applying a constant towing force, namely the weight of the sphere, which is simply the Earth’s gravitational attractive force on the sphere. The weight of the sphere is constant and entirely independent of the state of motion, and the sphere responds by settling downward at some velocity through the fluid. (As noted above, this velocity may fluctuate slightly with time.) For spheres the differences between the fixed-sphere case and the settling-sphere case are usually assumed to be minor. Indeed, some of the data in Figure 2.3.1 for dimensionless drag force as a function of Reynolds number are from settling experiments and some are from wind-tunnel experiments with fixed spheres, and it can be seen that there is very little scatter of the combined experimental curve.

## Dimensional Analysis

To obtain an experimental curve for settling velocity we can simply transform the curve in Figure 2.3.1 for drag coefficient vs. Reynolds number for towed spheres into a curve based on settling velocity. In fact, much of this curve, especially for low Reynolds numbers, was obtained by settling experiments in the first place, with the experimental results recast into the form of drag coefficients.

When a sphere falls at terminal velocity the drag force $$F_{D}$$ is equal to the submerged weight of the particle, $$(1 / 6) \pi D^{3} \gamma^{\prime}$$, where $$\gamma^{\prime}$$ is the submerged weight per unit volume of the particle, equal to $$g\left(\rho_{s}-\rho\right)$$. Substituting this for $$F_{D}$$ in the definition of the drag coefficient $$C_{D}$$ in Equation 2.3.2, using settling velocity $$w$$ in place of $$U$$, and then solving for $$C_{D}$$,

$C_{D}=\frac{4}{3} \frac{\gamma^{\prime} D}{\rho w^{2}} \label{3.22}$

This expression for $$C_{D}$$, which can be viewed as the “settling drag coefficient”, can be used in the relationship for dimensionless drag force as a function of Reynolds number (Equation 2.3.2) for spheres moving through a viscous fluid:

$\frac{\gamma^{\prime} D}{\rho w^{2}}=f\left(\frac{\rho w D}{\mu}\right) \label{3.23}$

where the factor $$4/3$$ has been absorbed into the function, just for convenience. Figure $$\PageIndex{2}$$, which is the same as Figure 2.3.1 with axes relabeled and adjusted in scale to take account of the factor $$4/3$$, is the corresponding graph of this function. No data points are shown, because the curve is exactly the same as in Figure 2.3.2. Figure $$\PageIndex{2}$$ gives settling velocity was an implicit function of $$\rho$$, $$U$$, $$D$$, and $$\gamma^{\prime}$$.

The curve in Figure $$\PageIndex{2}$$ is still not very convenient for finding the settling velocity when the other variables are given. This is because both $$w$$ and $$D$$ appear in the dimensionless variables along both axes. Finding $$w$$ in an actual problem would necessitate laborious trial-and-error computation. To get around this problem the graph can be further recast into a more convenient form in which $$w$$ appears in only one of the two dimensionless variables. Also, because usually what is desired is $$w$$ as a function of $$D$$, or vice versa, it is convenient to arrange for $$D$$ to appear only in the other variable.

Recall from Chapter 2 that if you have a set of dimensionless variables for a problem you can multiply or divide any one of them by any others in the set to get a new variable to replace the old one. To get a dimensionless variable with $$w$$ but not $$D$$, invert the left-hand variable in Equation \ref{3.23} and multiply the result by the right-hand variable:

$\left(\frac{\rho w^{2}}{\gamma^{\prime} D}\right)\left(\frac{\rho w D}{\mu}\right)=\frac{\rho^{2} w^{3}}{\gamma^{\prime} \mu} \label{3.24}$

And to get a dimensionless variable with $$D$$ but not $$w$$, square the right-hand variable in Equation \ref{3.23} and multiply it by the left-hand variable:

$\left(\frac{\rho w D}{\mu}\right)^{2}\left(\frac{\gamma^{\prime} D}{\rho w^{2}}\right)=\frac{\rho \gamma^{\prime} D^{3}}{\mu^{2}} \label{3.25}$

It is convenient, but not necessary, to take these two variables to the one-third power, so that $$w$$ and $$D$$ appear to the first power; $$w\left(\rho^{2} / \gamma^{\prime} \mu\right)^{1 / 3}$$ can be viewed as a dimensionless settling velocity, and $$D\left(\rho \gamma^{\prime} / \mu^{2}\right)^{1 / 3}$$ as a dimensionless sphere diameter. Because these two new variables are equivalent to $$C_{D}$$ and $$\text{Re}$$, the functional relationship for $$C_{D}$$ vs. $$\text{Re}$$ can just as well be written

$\left(\frac{\rho^{2}}{\gamma^{\prime} \mu}\right)^{1 / 3} w=f\left(\frac{\rho \gamma^{\prime}}{\mu^{2}}\right)^{1 / 3} D \label{3.26}$

The usefulness of Equation \ref{3.26} is that the settling velocity appears only on the left side and the sphere diameter appears only on the right side.

It is now a simple matter to find $$w$$ for a given fluid, sphere size, and submerged specific weight by use of Figure $$\PageIndex{3}$$, which is a plot of dimensionless setting velocity vs. dimensionless sphere diameter. This curve is obtained directly from that in Figure $$\PageIndex{2}$$; you can imagine taking the original data points and forming the new dimensionless variables rather than the old ones to plot the curve in the new coordinate axes of Figure $$\PageIndex{3}$$. This emphasizes that these two curves are equivalent because they are based on the same set of experimental data.

If you are unsatisfied by the roundabout way of arriving at the functional relationship expressed in Equation \ref{3.24}, you might consider making a fresh start on dimensional analysis of the problem of settling of a sphere through a still fluid at terminal velocity (Figure $$\PageIndex{4}$$). Settling velocity $$w$$, the dependent variable, must depend on fluid density $$\rho$$, fluid viscosity $$\mu$$, sphere diameter $$D$$, and submerged weight per unit volume $$\gamma^{\prime}$$ of the sphere. Each of these must be included for the reasons given in Chapter 2. As before, acceleration of gravity and sphere density do not have to appear separately in the list of variables because they are important only by virtue of their combined effect on $$\gamma^{\prime}$$. The five variables $$w$$, $$\rho$$, $$\mu$$, $$D$$, and $$\gamma^{\prime}$$ should then combine into two dimensionless variables. You can conveniently arrange for one to contain $$w$$ but not $$D$$ and the other to contain $$D$$ but not $$w$$ by using the other three variables as the “repeating” variables. You might verify for yourself that this procedure leads to the two dimensionless variables in Equation \ref{3.26}.

## Settling at Low Reynolds Numbers

Remember from Chapter 3 that if the Reynolds number based on sphere diameter and relative flow velocity is less than about one, the drag force on the sphere is given exactly by Stokes’ Law, $$F_{D}=3 \pi \mu U D$$. This holds in particular for spheres settling under their own weight. Using Stokes’ Law it is easy to develop a useful formula for the settling velocity of spheres that is valid in the Stokes range ($$\text{Re} < 1$$). Write an equation that balances the submerged weight of the sphere, $$\left(\pi D^{3} / 6\right) \gamma^{\prime}$$, by the drag force, given by Stokes’ Law, 3$$\pi D \mu w$$, where I have used the settling velocity $$w$$ as the relative velocity of the fluid and the sphere. Solving for $$w$$,

$w=\frac{1}{18} \frac{\gamma^{\prime} D^{2}}{\mu} \label{3.27}$

This equation is widely cited and widely used in books and papers on settling of spheres and other bodies, like sediment particles, that have the approximate shape of a sphere, but keep firmly in mind that it applies only in the Stokes range of settling Reynolds numbers $$\rho w D / \mu$$.

## The Effect of Turbulence on Particle Settling

One clear effect of turbulence on particle setting is the possibility of dispersion. Think about arranging an experiment in which a batch or continuing supply of sediment particles is introduced at the free surface of a channel flow. If the flow is laminar, the particles (if they are identical in size, shape, and density) are identical also in their settling behavior as well, and they all land on the bottom boundary of the flow at the same point,. If the flow is turbulent, however, the settling particles land on the bottom over a wide streamwise range of points, the obvious reason being that each particles traverses (or, perhaps more accurately, finds itself within the domain of) a different set of eddies during its descent, and so it experiences a different set of local fluid velocities. Both the vertical and the horizontal components of fluid velocities within eddies act to spread out, or disperse, the particle trajectories around the overall average trajectory.

It might seem intuitively reasonable to you to assume that the average settling time of the particles in the turbulent flow is the same as the single, well- defined settling time in the laminar flow, because the particles in the turbulent flow are, when viewed on the scale of the particles themselves, just settling through surrounding fluid that is the same as in the laminar case, and the “ups” should balance out the “downs”, on average. That cannot be quite true, however, if only for the reason that the drag coefficient of a sphere moving through a fluid is non-negligibly affected by accelerations and decelerations of the fluid that is moving relative to the particle, and that is exactly what is happening when the particle is in a turbulent flow field.

But there is another effect, and one that cannot be ignored, that almost certainly lies outside of your intuition: under certain conditions, a settling particle that finds itself within a rotating eddy tends to become trapped within that eddy! This effect has been studied by Tooby et al. (1977) and Nielsen (1984). In an ideal vortex, rotating about a horizontal axis, a particle within the rising limb describes a circular orbit, which ideally would be closed and would not exhibit any net downward motion (Figure $$\PageIndex{5}$$). In Figure $$\PageIndex{5}$$, the tangential velocity of the fluid, u, is proportional to the distance from the center of the vortex.
Simple vector addition of $$u$$ and $$w$$, the settling velocity of the particle, products a circular trajectory of the sediment particle, with no net settling. In the experiment by Tooby et al. (1977), particles tended to spiral slowly outwards, so that they would ultimately diffuse out of the vortex, but sufficiently fine sediment would be trapped in vortices and would move with the vortex until it dissipated. The mechanism is not very sensitive to the size of the trapped particles, and it should tend to produce less vertical segregation by size than the classical diffusional theory predicts.

## References Cited

Lin, C.C., 1955, The Theory of Hydrodynamic Instability: Cambridge, U.K., Cambridge University Press, 155 p.

Nielsen, P., 1984, On the motion of suspended sand particles: Journal of Geophysical Research, v. 89, p. 616-626.

Reynolds, O., 1883, An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and the law of resistance in parallel channels: Royal Society [London], Philosophical Transactions, v. 174, p. 935-982.

Schubauer, G.B., and Skramstad, H.K., 1947, Laminar boundary-layer oscillations and stability of laminar flow: Journal of Aeronautical Science, 14 (2), p. 69-78.

Stokes, G.G., 1851, On the effect of the internal friction of fluids on the motion of pendulums: Cambridge Philosophical Society, Transactions, v. 9, no. 8, p. 287.

Tooby, P.F., Wick, G.L., and Isaacs, J.D., 1977, The motion of a small sphere in a rotating velocity field: a possible mechanism for suspending particles in turbulence: Journal of Geophysical Research, v. 82, p. 2096-2100.

Tritton, D.J., 1988, Physical Fluid Dynamics, 2nd Edition: Oxford, U.K., Oxford University Press, 519 p.