text:
The Data Landscape

Amazon and University of Sheffield

<EMAIL>

There are three types of lies: lies, damned lies and statistics

??

There are three types of lies: lies, damned lies and statistics

<NAME>

There are three types of lies: lies, damned lies and statistics

<NAME> 1804-1881

Mathematical Statistics

• ‘Founded’ by <NAME> (1857-1936)

There are three types of lies: lies, damned lies and ‘big data’

<NAME> 1972-?

‘Mathematical Data Science’

• ‘Founded’ by ? (?-?)

Today

• Challenges of understanding and interpreting big data today are similar to those that Disraeli faced in with statistics in latter part of the 19th century.

• Data is elusive: it can promise much but deliver little.

<NAME>

• What’s the most common hat worn in the west?

The Three ’A’s

• Data Awareness

• Data Availability

• Data Analysis

Data Awareness

• What data you have and where its stored.

• May need to chance conception of what data is and how to obtain it.

• Production lines, smart phones.

• Locked away: manual log books, confidential data, personal data.

• An internal audit (you are ahead here!).

• The key to any successful campaign is a good map.

Data Availability

• How well are the data sources interconnected?

• How well curated are they?

• Curse of Disraeli was associated with unreliable data and unreliable statistics.

• Misrepresentation is worse than absence of data.

• Need an improved sense of data and its value.

Data Analysis

• accumulation of the necessary expertise to digest what the data tells us.

• data requires intepretation, and interpretation requires experience.

• Analysis is a bottleneck due to a skill shortage.

• Ideally, analysis should be carried out by individuals not only skilled in data science but also equipped with the domain knowledge.

“Embodiment Factors”

 compute ~10 gigaflops ~ 1000 teraflops? communicate ~1 gigbit/s ~ 100 bit/s embodiment(compute/communicate) 10 ~ 1013

Societal Effects

• Automated decision making in the computer based on data.

• Need to better understand our own subjective biases better.

• Particularly important where societal interventions are prescribed.

• But what is a societal intervention in the modern era? Much more subtle than before.

Societal Effects

• Shift in dynamic from the direct pathway between human and data to indirect pathway between human and data via the computer

• This change of dynamics gives us the modern and emerging domain of data science

Societal Challenges

1. Paradoxes of the Data Society

2. Quantifying the Value of Data

3. Privacy, Transparency, Fairness, Equality

• Able to quantify to a greater and greater degree the actions of individuals

• But less able to characterize society

• As we measure more, we understand less

What?

• Perhaps greater preponderance of data is making society itself more complex

• Therefore traditional approaches to measurement are failing

• Curate’s egg of a society: it is only ‘measured in parts’

Wood or Tree

• Can either see a wood or a tree.

Examples

• Election polls (UK 2015 elections, EU referendum, US 2016 elections)

• Clinical trials vs personalized medicine: Obtaining statistical power where interventions are subtle. e.g. social media

Also need

• More classical statistics!
• Like the ‘paperless office’
• A better characterization of human (see later)

• Larger studies (100,000 genome)
• Combined with complex models: algorithmic challenges

Challenge 2) Quantifying the Value of Data

There’s a sea of data, but most of it is undrinkable

We require data-desalination before it can be consumed!

Value

• How do we measure value in the data economy?
• How do we encourage data workers: curation and management
• Incentivization for sharing and production.
• Quantifying the value in the contribution of each actor.

Credit Allocation

• Direct work on data generates an enormous amount of ‘value’ in the data economy but this is unaccounted in the economy

• Hard because data is difficult to ‘embody’

• Value of shared data: Wellcome Trust 2010 Joint Statement (from the “Foggy Bottom” meeting)

Solutions

• Encourage greater interaction between application domains and data scientists

• Encourage visualization of data

• Implications for incentivization schemes

Privacy, Loss of Control and Marginalization

• Society is becoming harder to monitor

• Individual is becoming easier to monitor

• What does it mean if a computer can predict us better than we can ourselves?

Discrimination

• Potential for explicit and implicit discrimination on the basis of race, religion, sexuality, health status

• All prohibited under European law, but can pass unawares, or be implicit

Marginalization

• Credit scoring, insurance, medical treatment
• What if certain sectors of society are under-represented in our aanalysis?
• What if Silicon Valley develops everything for us?

Amelioration

• Work to ensure individual retains control of their own data
• We accept privacy in our real lives, need to accept it in our digital
• Control of persona and ability to project

• Need better technological solutions: trust and algorithms.

Societal Challenges

• Paradoxes of the Data Society

• Quantifying the value in the Data

• Privacy, Transparency, Fairness, Equality

Solutions

• Software: TensorFlow, scikit-learn (python), R, Spark.

• Education

• Automating the Data Science Pipeline:
• The Automatic Statistician
• AutoML

Conclusion

• Alone ‘big data’ promises much and delivers little.

• Data needs to be cared for: it needs to be curated and evaluated.

• Thee stages:
1. Awareness
2. Availability
3. Analysis


text:
Ask question

# A dead battery is charged by connecting it to the live battery ofanother car with jumper cables. Assume that V=9.5 V and R=0.90 Ohms. Determine the current in the starter. Your answer differs from the correct answerby orders of magnitude. A Determine the current in the dead battery. Your answer differs from the correct answerby orders of magnitude.

Question
Other
asked 2021-04-02
A dead battery is charged by connecting it to the live battery ofanother car with jumper cables. Assume that V=9.5 V and R=0.90 Ohms.
Determine the current in the starter.
Your answer differs from the correct answerby orders of magnitude. A
Determine the current in the dead battery.
Your answer differs from the correct answerby orders of magnitude.

## Answers (1)

2021-04-04
Assume current through starter be Is flowing downwards
Assume current through Dead battery be Id flowing downwards
So current through live battery is ($$\displaystyle{I}_{{s}}+{I}_{{d}}$$) flowing upwards
KVL for left loop -
$$\displaystyle+{9.5}{V}+{\left({0.9}\ {O}{h}{m}{s}\right)}{l}{d}-{\left({0.06}\ {O}{h}{m}{s}\right)}{l}{s}={0}$$
KVL for the big loop
$$\displaystyle+{12}{V}-{\left({0.01}\ {O}{h}{m}{s}\right)}{\left({l}{s}+{l}{d}\right)}-{\left({0.06}\ {O}{h}{m}{s}\right)}{l}{s}={0}$$
Solving these teo equation, we get
$$\displaystyle{l}_{{s}}={0.865}\ {A}$$
$$\displaystyle{l}_{{d}}={171.305}\ {A}$$

### Relevant Questions

asked 2021-05-09
The dominant form of drag experienced by vehicles (bikes, cars,planes, etc.) at operating speeds is called form drag. Itincreases quadratically with velocity (essentially because theamount of air you run into increase with v and so does the amount of force you must exert on each small volume of air). Thus
$$\displaystyle{F}_{{{d}{r}{u}{g}}}={C}_{{d}}{A}{v}^{{2}}$$
where A is the cross-sectional area of the vehicle and $$\displaystyle{C}_{{d}}$$ is called the coefficient of drag.
Part A:
Consider a vehicle moving with constant velocity $$\displaystyle\vec{{{v}}}$$. Find the power dissipated by form drag.
Express your answer in terms of $$\displaystyle{C}_{{d}},{A},$$ and speed v.
Part B:
A certain car has an engine that provides a maximum power $$\displaystyle{P}_{{0}}$$. Suppose that the maximum speed of thee car, $$\displaystyle{v}_{{0}}$$, is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power $$\displaystyle{P}_{{1}}$$ is 10 percent greater than the original power ($$\displaystyle{P}_{{1}}={110}\%{P}_{{0}}$$).
Assume the following:
The top speed is limited by air drag.
The magnitude of the force of air drag at these speeds is proportional to the square of the speed.
By what percentage, $$\displaystyle{\frac{{{v}_{{1}}-{v}_{{0}}}}{{{v}_{{0}}}}}$$, is the top speed of the car increased?
Express the percent increase in top speed numerically to two significant figures.
asked 2021-03-21
In the figure below, suppose that the switch has been closed for a length of time sufficiently long enough for the capacitor to become fully charged. (R = 13.0 k, R2 = 20.0 k, R3 = 5.00 k, ΔV = 9.30 V)

(a) Find the steady-state current in each resistor.
(b) Find the charge on the capacitor.
asked 2021-05-20
Assume that a ball of charged particles has a uniformly distributednegative charge density except for a narrow radial tunnel throughits center, from the surface on one side to the surface on the opposite side. Also assume that we can position a proton any where along the tunnel or outside the ball. Let $$\displaystyle{F}_{{R}}$$ be the magnitude of the electrostatic force on the proton when it islocated at the ball's surface, at radius R. As a multiple ofR, how far from the surface is there a point where the forcemagnitude is 0.44FR if we move the proton(a) away from the ball and (b) into the tunnel?
asked 2021-04-13
As depicted in the applet, Albertine finds herself in a very odd contraption. She sits in a reclining chair, in front of a large, compressed spring. The spring is compressed 5.00 m from its equilibrium position, and a glass sits 19.8m from her outstretched foot.
a)Assuming that Albertine's mass is 60.0kg , what is $$\displaystyle\mu_{{k}}$$, the coefficient of kinetic friction between the chair and the waxed floor? Use $$\displaystyle{g}={9.80}\frac{{m}}{{s}^{{2}}}$$ for the magnitude of the acceleration due to gravity. Assume that the value of k found in Part A has three significant figures. Note that if you did not assume that k has three significant figures, it would be impossible to get three significant figures for $$\displaystyle\mu_{{k}}$$, since the length scale along the bottom of the applet does not allow you to measure distances to that accuracy with different values of k.
asked 2021-02-14
The plane of a rectangular coil, 5.0-cm by 8.0-cm, is perpendicular to the direction of a magnetic field "B". If the coil has 75 turns and a total resistance of 8.0 Ohms, at what rate must the magnitude of "B" change to induce a current of 0.10 amps in the windings of the coil?
asked 2021-05-05
A person accidentally leaves a car with the lights on. If eachof the two headlights uses 43 W andeach of the two taillights 7.0 W,for a total of 1.0E2 W, how longwill a fresh 12-V battery last if it is rated at 8.0E1 A·h?
Assume the full 12 V appears across each bulb.
asked 2021-04-25
The unstable nucleus uranium-236 can be regarded as auniformly charged sphere of charge Q=+92e and radius $$\displaystyle{R}={7.4}\times{10}^{{-{15}}}$$ m. In nuclear fission, this can divide into twosmaller nuclei, each of 1/2 the charge and 1/2 the voume of theoriginal uranium-236 nucleus. This is one of the reactionsthat occurred n the nuclear weapon that exploded over Hiroshima, Japan in August 1945.
A. Find the radii of the two "daughter" nuclei of charge+46e.
B. In a simple model for the fission process, immediatelyafter the uranium-236 nucleus has undergone fission the "daughter"nuclei are at rest and just touching. Calculate the kineticenergy that each of the "daughter" nuclei will have when they arevery far apart.
C. In this model the sum of the kinetic energies of the two"daughter" nuclei is the energy released by the fission of oneuranium-236 nucleus. Calculate the energy released by thefission of 10.0 kg of uranium-236. The atomic mass ofuranium-236 is 236 u, where 1 u = 1 atomic mass unit $$\displaystyle={1.66}\times{10}^{{-{27}}}$$ kg. Express your answer both in joules and in kilotonsof TNT (1 kiloton of TNT releases 4.18 x 10^12 J when itexplodes).
asked 2021-03-28

The homogeneous plate ABCD is subjected to a biaxial loading asshown. It is known that $$\displaystyle\sigma_{{z}}=\sigma_{{o}}$$ and that the change in length of the plate in the x direction must be zero, that is $$\displaystyle\epsilon_{{x}}={0}$$. Denoting E the modulus of elasticity and by v Poisson's ratio, determine (a) the required magnitude of $$\displaystyle\sigma_{{x}}$$, and (b) the ratio $$\displaystyle{\frac{{\sigma_{{o}}}}{{\epsilon_{{z}}}}}$$
asked 2021-04-15
A car initially traveling eastward turns north by traveling in a circular path at uniform speed as in the figure below. The length of the arc ABC is 235 m, and the car completes the turn in 33.0 s. (Enter only the answers in the input boxes separately given.)
(a) What is the acceleration when the car is at B located at an angle of 35.0°? Express your answer in terms of the unit vectors $$\displaystyle\hat{{{i}}}$$ and $$\displaystyle\hat{{{j}}}$$.
1. (Enter in box 1) $$\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{i}}}+{\left({E}{n}{t}{e}{r}\in{b}\otimes{2}\right)}{P}{S}{K}\frac{{m}}{{s}^{{2}}}\hat{{{j}}}$$
(b) Determine the car's average speed.
3. ( Enter in box 3) m/s
(c) Determine its average acceleration during the 33.0-s interval.
4. ( Enter in box 4) $$\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{i}}}+$$
5. ( Enter in box 5) $$\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{j}}}$$
asked 2021-04-21
Car 1 has a mass of m1 = 65 ❝ 103 kg and moves at a velocity of v01 = +0.81 m/s. Car 2, with a mass of m2 = 92 ❝ 103 kg and a velocity of v02 = +1.2 m/s, overtakes car 1 and couples to it. Neglect the effects of friction in your answer.
(a) Determine the velocity of their center of mass before the collision m/s
(b) Determine the velocity of their center of mass after the collision m/s
(c) Should your answer in part (b) be less than, greater than, or equal to the common velocity vf of the two coupled cars after the collision? less than greater than equal to
...

format: plain


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# Monomial ideals: isomorphism problem for commutative algebras?

Let $I,J\unlhd K[x_1,\ldots,x_n]=K[x]$ be monomial ideals and $f\!: K[x]\to K[x]$ a graded isomorphism (given by a matrix $A=[\alpha_{i,j}]\in K^{n\times n}$, i.e. $x_i\mapsto\sum_j\alpha_{i,j}x_j$ is $f$), such that $f(I)=J$, i.e. $K[x]/I\cong K[x]/J$.

Question 1: How can I prove: There exists a bijection $\varphi\!: \{x_1,\ldots,x_n\}\to\{x_1,\ldots,x_n\}$ whose induced isomorphism $\overline{\varphi}\!: K[x]\to K[x]$ satisfies $\overline{\varphi}(I)=J$?

I suspect that $f$ suffices as $\varphi$, i.e. each column of $A$ contains only one nonzero entry, but I have problems proving this. Can a linear isomorphism send a monomial ideal to a monomial ideal without being a permutation of the variables? Let $I=\langle x^{a_1},\ldots,x^{a_k}\rangle$ and $J=\langle x^{b_1},\ldots,x^{b_l}\rangle$, so that $\langle f(x^{a_1}),\ldots,f(x^{a_k})\rangle =\langle x^{b_1},\ldots,x^{b_l}\rangle$. Here $f(x^a)=f(x_1^{a_1}\cdots x_n^{a_n})=$ $(\sum_j\alpha_{1,j}x_j)^{a_1}\cdots(\sum_j\alpha_{n,j}x_j)^{a_n}$, so every monomial of $f(x^a)$ has degree $|a|$. Since $\langle f(x^{a_1}),\ldots,f(x^{a_k})\rangle$ is a monomial ideal, every monomial of every $f(x^{a_i})$ is contained in $f(I)$, thus $f(I)$ is actually generated by all monomials of all $f(x^{a_1}),\ldots,f(x^{a_k})$. But I don't know how to prove that each $f(x^{a_i})$ has only one monomial, i.e. $f$ is a permutation of variables.

Basically I'm asking for an elementary proof of Theorem 5.27 in Polytopes, Rings, and K-Theory (<NAME> - 2009 - Springer SMM).

EDIT: It turns out (see mbrown's answer below) that my claim was wrong, so I change the question to look for a positive result.

Question 2: How can I prove the following claim Thm.5.27 using only elementary commutative algebra (no fancy shmancy Borel’s theorem):

Let $I\!\unlhd\!K[x]$ and $J\!\unlhd\!K[y]$ be monomial ideals with $\forall i,j\!: x_i\!\notin\!I,\, y_j\!\notin\!J$. Then t.f.a.e.:

1. $\exists$bijection $\varphi\!:\{x_1,\ldots,x_n\}\!\rightarrow\!\{y_1,\ldots,y_{n'}\}$ whose morphism $\overline{\varphi}\!: K[x]\!\rightarrow\! K[y]$ satisfies $\overline{\varphi}(I)\!=\!J$;
2. $K[x]/I \cong K[y]/J$ as $K$-algebras;
3. $K[x]/I \cong K[y]/J$ as $\mathbb{N}$-graded $K$-algebras.

(1)$\Rightarrow$(2) is clear.

(2)$\Rightarrow$(3) is shown here.

(3)$\Rightarrow$(1): Since the isomorphism $\overline{f}: K[x]/I\to K[y]/J$ is graded, it induces a $K$-linear isomorphism $f: K\{x_1,\ldots,x_n\}\to K\{y_1,\ldots,y_{n'}\}$ (so $n=n'$) that determines $\overline{f}$ and is given by an invertible matrix $A\in K^{n\times n}$. This induces an isomorphism $f:K[x]\to K[y]$ with $f(I)=J$. Now I don't know how to change $f$ to an isomorphism with $f(I)=J$, so that each column of $A$ has only one nonzero entry.

If there's an elementary proof in the literature, that would be a more than sufficient answer.

• In THM 5.27, it is required that $I,J$ do not contain any of the $x_i$. Do you want to include this as well?
– user55407
Oct 3 '13 at 3:03
• BTW: this implies Combinatorial Invariance of Stanley-Reisner rings (Bruns & Gubeladze): $\Delta\cong \Delta'$ iff $K[\Delta]\cong K[\Delta']$, for any simplicial complexes $\Delta,\Delta'$.
– Leo
Oct 3 '13 at 5:01
• Also on MO.
– Leo
Oct 6 '13 at 4:38

## 1 Answer

Say $n=2$. Define $f:k[x,y] \to k[x,y]$ by $x \mapsto x+y$, $y \mapsto y$. Say $I=J=(y^2)$ (I am squaring $y$ just in case you want to include the requirement I mentioned in the comments).

$f$ is a graded $k$-algebra automorphism, and it maps $I$ to $J$. But $f$ does not permute the variables.


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# Seyfert galaxy

The Circinus Galaxy, a Type II Seyfert galaxy

Seyfert galaxies are one of the two largest groups of active galaxies, along with quasars. They have quasar-like nuclei (very luminous, distant and bright sources of electromagnetic radiation) with very high surface brightnesses whose spectra reveal strong, high-ionisation emission lines,[1] but unlike quasars, their host galaxies are clearly detectable.[2]

Seyfert galaxies account for about 10% of all galaxies[3] and are some of the most intensely studied objects in astronomy, as they are thought to be powered by the same phenomena that occur in quasars, although they are closer and less luminous than quasars. These galaxies have supermassive black holes at their centres which are surrounded by accretion discs of in-falling material. The accretion discs are believed to be the source of the observed ultraviolet radiation. Ultraviolet emission and absorption lines provide the best diagnostics for the composition of the surrounding material.[4]

Seen in visible light, most Seyfert galaxies look like normal spiral galaxies, but when studied under other wavelengths, it becomes clear that the luminosity of their cores is of comparable intensity to the luminosity of whole galaxies the size of the Milky Way.[5]

Seyfert galaxies are named after <NAME>, who first described this class in 1943.[6]

## Discovery

NGC 1068 (Messier 77), one of the first Seyfert galaxies classified

Seyfert galaxies were first detected in 1908 by <NAME> and <NAME>, who were using the Lick Observatory to look at the spectra of astronomical objects that were thought to be "spiral nebulae". They noticed that NGC 1068 showed six bright emission lines, which was considered unusual as most objects observed showed an absorption spectrum corresponding to stars.[7]

In 1926, <NAME> looked at the emission lines of NGC 1068 and two other such "nebulae" and classified them as extragalactic objects.[8] In 1943, <NAME> discovered more galaxies similar to NGC 1068 and reported that these galaxies have very bright stellar-like nuclei that produce broad emission lines.[6] A year after, Cygnus A was detected at 160 MHz,[9] and detection was confirmed in 1948 when it was established that it was a discrete source.[10] Its double radio structure became apparent with the use of interferometry.[11] In the next few years, other radio sources such as supernova remnants were discovered. By the end of the 1950s, more important characteristics of Seyfert galaxies were discovered, including the fact that their nuclei are extremely compact (< 100 pc, i.e. "unresolved"), have high mass (≈109±1 solar masses), and the duration of peak nuclear emissions is relatively short (>108 years).[12]

In the 1960-1970s, research to further understand the properties of Seyfert galaxies was carried out. A few direct measurements of the actual sizes of Seyfert nuclei were taken, and it was established that the emission lines in NGC 1068 were produced in a region over a thousand light years in diameter.[13] Controversy existed over whether Seyfert redshifts were of cosmological origin.[14] Confirming estimates of the distance to Seyfert galaxies and their age were limited since their nuclei vary in brightness over a time scale of a few years; therefore arguments involving distance to such galaxies and the constant speed of light cannot always be used to determine their age.[14] In the same time period, research had been undertaken in order to survey, identify and catalogue galaxies, including Seyferts. Beginning in 1967, <NAME> published lists containing a few hundred galaxies distinguished by their very strong ultraviolet emission, with measurements on the position of some of them being improved in 1973 by other researchers.[15] At the time, it was believed that 1% of spiral galaxies are Seyferts.[16] By 1977, it was found that very few Seyfert galaxies are ellipticals, most of them being normal or barred spiral galaxies.[17] During the same time period, efforts have been made to gather spectrophotometric data for Seyfert galaxies. It became obvious that not all spectra from Seyfert galaxies look the same, so they have been subclassified according to the characteristics of their emission spectra. A simple division into types I and II has been devised, with the classes depending on the relative width of their emission lines.[18] It has been later noticed that some Seyfert nuclei show intermediate properties, resulting in their being further subclassified into types 1.2, 1.5, 1.8 and 1.9 (see Classification).[19][20] Early surveys for Seyfert galaxies were biased in counting only the brightest representatives of this group. More recent surveys that count galaxies with low-luminosity and obscured Seyfert nuclei suggest that the Seyfert phenomenon is actually quite common, occurring in 16% ± 5% of galaxies; indeed, several dozen galaxies exhibiting the Seyfert phenomenon exist in the close vicinity (≈27 Mpc) of our own galaxy.[3]

## Characteristics

Optical and ultraviolet images of the black hole in the centre of NGC 4151, a Seyfert Galaxy

An active galactic nucleus (AGN) is a compact region at the centre of a galaxy that has a higher than normal luminosity over portions of the electromagnetic spectrum. A galaxy having an active nucleus is called an active galaxy. Active galactic nuclei are the most luminous sources of electromagnetic radiation in the Universe, and their evolution puts constraints on cosmological models. Depending on the type, their luminosity varies over a timescale from a few hours to a few years. The two largest subclasses of active galaxies are quasars and Seyfert galaxies, the main difference between the two being the amount of radiation they emit. In a typical Seyfert galaxy, the nuclear source emits at visible wavelengths an amount of radiation comparable to that of the whole galaxy's constituent stars, while in a quasar, the nuclear source is brighter that the constituent stars by at least a factor of 100.[1][21] Seyfert galaxies have extremely bright nuclei, with luminosities ranging between 108 and 1011 solar luminosities. Only about 5% of them are radio bright; their emissions are moderate in gamma rays and bright in X-rays.[22] Their visible and infrared spectra shows very bright emission lines of hydrogen, helium, nitrogen, and oxygen. These emission lines exhibit strong Doppler broadening, which implies velocities from 500 to 4,000 km/s (310 to 2,490 mi/s), and are believed to originate near an accretion disc surrounding the central black hole.[23]

### Eddington luminosity

A lower limit to the mass of the central black hole can be calculated using the Eddington luminosity.[24] This limit arises because light exhibits radiation pressure. Assume that a black hole is surrounded by a disc of luminous gas.[25] Both the attractive gravitational force acting on electron-ion pairs in the disc and the repulsive force exerted by radiation pressure follow an inverse-square law. If the gravitational force exerted by the black hole is less than the repulsive force due to radiation pressure, the disc will be blown away by radiation pressure.[26]

The image shows a model of an active galactic nucleus. The central black hole is surrounded by an accretion disc, which is surrounded by a torus. The broad line region and narrow line emission region are shown, as well as jets coming out of the nucleus.

The luminosity of the black hole must be less than the Eddington luminosity.

$L < L_{eddington} = 1.26 \times 10^{38} \frac{M_{BH}}{M_{solar}} \, erg/sec$ $= 3.2 \times 10^{4} \frac{M_{BH}}{M_{solar}} L_{solar}$

In the above expression, L = luminosity, LEddington is the Eddington luminosity, MBH = mass of black hole, Msolar is the mass of the Sun, and Lsolar is the luminosity of the sun. Therefore, given the observed luminosity (which would be less than the Eddington luminosity), an approximate lower limit for the mass of the central black hole at the center of an active galaxy can be estimated.

The classical Eddington limit was originally derived for stars assuming spherical symmetry. This is a known approximation, but it is widely used. When the actual geometry of accretion discs is taken into account, it is found that the results can differ considerably from the classical value.[24]

### Emissions

The emission lines seen on the spectrum of a Seyfert galaxy may come from the surface of the accretion disc itself, or may come from clouds of gas illuminated by the central engine in an ionization cone. The exact geometry of the emitting region is difficult to determine due to poor resolution of the galactic center. However, each part of the accretion disc has a different velocity relative to our line of sight, and the faster the gas is rotating around the black hole, the broader the emission line will be. Similarly, an illuminated disc wind also has a position-dependent velocity. [27]

The narrow lines are believed to originate from the outer part of the AGN where velocities are lower, while the broad lines originate closer to the black hole. This is confirmed by the fact that the narrow lines do not vary detectably, which implies that the emitting region is large, contrary to the broad lines which can vary on relatively short timescales. Reverberation mapping is a technique which uses this variability to try to determine the location and morphology of the emitting region. This technique measures the structure and kinematics of the broad line emitting region by observing the changes in the emitted lines as a response to changes in the continuum. The use of reverberation mapping requires the assumption that the continuum originates in a single central source.[28] For 35 AGN, reverberation mapping has been used to calculate the mass of the central black holes and the size of the broad line regions.[29]

In the few radio-loud Seyfert galaxies that have been observed, the radio emission is believed to represent synchrotron emission from the jet. The infrared emission is due to radiation in other bands being reprocessed by dust near the nucleus. The highest energy photons are believed to be created by inverse Compton scattering by a high temperature corona near the black hole.[30]

## Classification

NGC 1097 is an example of a Seyfert galaxy. A supermassive black hole with a mass of 100 million solar masses lies at the centre of the galaxy. The area around the black hole emits large amounts of radiation from the matter falling into the black hole.[31]

Seyferts were first classified as Type I or II, depending on the emission lines shown by their spectra. The spectra of Type I Seyfert galaxies show broad lines that include both allowed lines, like H I, He I or He II and narrower forbidden lines, like O III. They show some narrower allowed lines as well, but even these narrow lines are much broader than the lines shown by normal galaxies. However, the spectra of Type II Seyfert galaxies show only both permitted and forbidden narrow lines. Forbidden lines are spectral lines that occur due to electron transitions not normally allowed by the selection rules of quantum mechanics, but that still have a small probability of spontaneously occurring. The term "forbidden" is slightly misleading, as the electron transitions causing them are not forbidden but highly improbable.[32]

In some cases, the spectra show both broad and narrow permitted lines, which is why they are classified as an intermediate type between Type I and Type II, such as Type 1.5 Seyfert. The spectra of some of these galaxies have changed from Type 1.5 to Type II in a matter of a few years. However, the characteristic broad emission line has rarely, if ever, disappeared.[33] The origin of the differences between Type I and Type II Seyfert galaxies is not known yet. There are a few cases where galaxies have been identified as Type II only because the broad components of the spectral lines have been very hard to detect. It is believed by some that all Type II Seyferts are in fact Type I, where the broad components of the lines are impossible to detect because of the angle we are at with respect to the galaxy. Specifically, in Type I Seyfert galaxies, we observe the central compact source more or less directly, therefore sampling the high velocity clouds in the broad line emission region moving around the supermassive black hole thought to be at the centre of the galaxy. By contrast, in Type II Seyfert galaxies, the active nuclei are obscured and only the colder outer regions located further away from the clouds' broad line emission region are seen. This theory is known as the "Unification scheme" of Seyfert galaxies.[34][35] However, it is not yet clear if this hypothesis can explain all the observed differences between the two types.[34]

### Type I Seyfert galaxies

Optical spectrum of the Type I Seyfert galaxy NGC 1275

Type I Seyferts are very bright sources of ultraviolet light and X-rays in addition to the visible light coming from their cores. They have two sets of emission lines on their spectra: narrow lines with widths (measured in velocity units) of several hundred km/s, and broad lines with widths up to 104 km/s.[36] The broad lines originate above the accretion disc of the supermassive black hole thought to power the galaxy, while the narrow lines occur beyond the broad line region of the accretion disc. Both emissions are caused by heavily ionised gas. The broad line emission arises in a region 0.1-1 parsec across. The broad line emission region, RBLR, can be estimated from the time delay corresponding to the time taken by light to travel from the continuum source to the line-emitting gas.[22]

### Type II Seyfert galaxies

Type II Seyfert galaxies have the characteristic bright core, as well as appearing bright when viewed at infrared wavelengths.[37] Their spectra contain narrow lines associated with forbidden transitions, and broad lines associated with allowed strong dipole or intercombination transitions.[34] In some Type II Seyfert galaxies, analysis with a technique called spectro-polarimetry (spectroscopy of polarised light component) revealed obscured type I regions. In the case of NCG 1068, nuclear light reflected of a dust cloud was measured, which led scientists to believe in the presence of an obscuring dust torus around a bright continuum and broad emission line nucleus. When the galaxy is viewed from the side, the nucleus is indirectly observed through reflection by gas and dust above and below the torus. This reflection causes the polarisation.[38]

### Type 1.2, 1.5, 1.8 and 1.9 Seyfert galaxies

In 1981, <NAME> introduced the notations Seyfert 1.5, 1.8 and 1.9, where the subclasses are based on the optical appearance of the spectrum, with the numerically larger subclasses having weaker broad-line components relative to the narrow lines. For example, Type 1.9 only shows a broad component in the line, and not in higher order Balmer lines. In Type 1.8, very weak broad lines can be detected in the lines as well as Hα, even if they are very weak compared to the Hα. In Type 1.5, the strength of the Hα and Hβ lines are comparable.[39]

### Other Seyfert-like galaxies

In addition to the Seyfert progression from Type I to Type II (including Type 1.2 to Type 1.9), there are other types of galaxies that are very similar to Seyferts or that can be considered as subclasses of them. Very similar to Seyferts are the low-ionisation narrow-line emission radio galaxies (LINER), discovered in 1980. These galaxies have strong emission lines from weakly ionised or neutral atoms, while the emission lines from strongly ionised atoms are relatively weak by comparison. LINERs share a large amount of traits with low luminosity Seyferts. In fact, when seen in visible light, the global characteristics of their host galaxies are indistinguishable. Also, they both show a broad line emission region, but the line emitting region in LINERs has a lower density than in Seyferts.[40] An example of such a galaxy is M104 in the Virgo constellation, also known as the Sombrero galaxy[41] A galaxy that is both a LINER and a Type I Seyfert is NGC 7213, a galaxy that is relatively close compared to other AGNs.[42] Another very interesting subclass are the narrow line Seyfert I galaxies (NLSy1), which have been subject to extensive research in recent years.[43] They have much narrower lines than the broad lines from classic Seyfert I galaxies, steep hard and soft X-ray spectra and strong Fe[II] emission.[44] Their properties suggest that NLSy1 galaxies are young AGNs with high accretion rates, suggesting a relatively small but growing central black hole mass.[45] There are theories suggesting that NLSy1s are galaxies in an early stage of evolution, and links between them and ultraluminous infrared galaxies or Seyfert II galaxies have been proposed.[46]

## Evolution

The majority of active galaxies we observe are very distant and show large Doppler shifts. This suggests that active galaxies occurred in the early Universe and, due to cosmic inflation, are receding away from us at very high speeds. Quasars are the furthest active galaxies, some of them being observed at distances 12 billion light years away. Seyfert galaxies are much closer than quasars.[47] Because light has a finite speed, looking across large distances in the Universe is equivalent to looking back in time. Therefore, the observation of active galactic nuclei at large distances and their scarcity in the nearby Universe suggests that they were much more common in the early Universe,[48] implying that active galactic nuclei could be early stages of galactic evolution. This leads to the question about what would be the local (modern-day) counterparts of AGNs found at large redshifts. It has been proposed that NLSy1s could be the small redshift counterparts of quasars found at large redshifts (z>4). The two have many similar properties, for example: high metallicities or similar pattern of emission lines (strong Fe [II], weak O [III]).[49] Some observations suggest that AGN emission from the nucleus is not spherically symmetric and that the nucleus often shows axial symmetry, with radiation escaping in a conical region. Based on this observations, models have been devised to explain the different classes of AGNs as due to their different orientations with respect to the observational line of sight. Such models are called unified models. Unified models explain the difference between Seyfert I and Seyfert II galaxies as being the result of Seyfert II galaxies being surrounded by obscuring toruses which prevent us from seeing the broad line region. Quasars and blazars can be fit quite easily in this model.[50] The main problem of such an unification scheme is trying to explain why some AGN are radio loud while others are radio quiet. It has been suggested that these differences may be due to differences in the spin of the central black hole.[36]

## Examples

Seyfert galaxy MRK 1513

The table below lists a few representative Seyfert galaxies from the Markarian catalog.[51]

Name Other names Longitude Latitude Right ascension Declination
Mark 205 MRK 0205 185.4338322 75.3106237 12h 21m 44.120s +75° 18′ 38.25″
Mark 231 MRK 0231 194.0593100 56.8736767 12h 56m 14.2344s +56° 52′ 25.236″
Mark 266 NGC 5256 204.573720 48.276093 13h 38m 17.69s +48° 16′ 33.9″
Mark 270 NGC 5283 205.2739946 67.6723111 13h 41m 05.759s +67° 40′ 20.32″
Mark 279 MRK 0279 208.2643618 69.3082128 13h 53m 03.447s +69° 18′ 29.57″
Mark 335 MRK 0335 1.5813306 20.2029144 00h 06m 19.519s +20° 12′ 10.49″
Mark 530 NGC 7603 349.7359060 0.2439521 23h 18m 56.617s +00° 14′ 38.23″
Mark 590 NGC 0863 33.6398442 -0.7666930 02h 14m 33.562s −00° 46′ 00.09″
Mark 686 NGC 5695 219.3421784 36.5678087 14h 37m 22.123s +36° 34′ 04.11″
Mark 744 NGC 3786 174.9272970 31.9092853 11h 39m 42.551s +31° 54′ 33.43″

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text:
This app was built to give people a sense of how the coronavirus epidemic is progressing in each country. The initial impetus was to move the conversation past daily updates on case numbers, and towards the near future. The hope was that doing so might help move us past the paralysis of daily shock, and towards important action.

As action is taken, the app gives us a sense of how well our actions are working. Difficult decisions have been made at all levels, from the individual to the nation, and it is important that we get feedback on those decisions as soon as possible.

Given how fast the epidemic was progressing, the app was built in a hurry at the University of Melbourne on the 12 March 2020. It has been continually refined since then, and work continues (see acknowledgements). The code is available here and contributions/suggestions are welcome. If you have data that you would like to see represented here, please get in contact.

The app relies on relatively simple heuristics rather than comprehensive analyses. The emphasis has been on speed, and continually updated data, rather than sophisticated analyses. Sophisticated analyses will likely come in time.

## Methods

### Raw case numbers

Our data is updated daily. Our global data come from the Johns Hopkins University dataset. Our Indian data comes from Ministry of Health and Family Welfare data, and are compiled by a team from Panjab university. Our German data comes from the Robert Koch Institut, and the csv is available here. Our Belgian, Colombian, Haitian, Italian, Japanese, Peruvian, and Swiss data come via the Covid-19 Data Hub. Belgian data is from Epistat, Belgian Infectious Diseases. Colombian data is from Ministerio de Salud y Protección Social de Colombia. Haitian data is from the Ministry of Public Health and Population of Haiti. Italian data is from Istituto Nazionale di Statistica. Japanese data is from the Japan COVID-19 Bulletin Board. Peruvian data is from COVID-19 Peru Data Repository. Swiss data is from Swiss Federal Statistical Office. Our Afghan and Brazilian data come via the R package covidregionaldata. Afghan data is from Afghanistan: Coronavirus(COVID-19) Subnational Cases. Brazilian data is from Número de casos confirmados de COVID-19 no Brasil.

These data report for each country the cumulative number of infections ($$I_t$$), number of deaths ($$D_t$$), and (for most countries) number of recoveries ($$R_t$$). The raw case numbers we report are simply a report of the JHU raw numbers for the selected country at the current date. We also provide aggregate tallies for each continent, and the world. Country to continent associations (for aggregating data to continent) were modified from this file. If a region is missing from the app it is either because the region is yet to reach 20 confirmed cases, or because there are currently serious issues with the data (we exclude regions when cumulative case numbers decrease by 25% between days).

### Current active cases

Active cases ($$A_t$$) are simply $$A_t = I_t-D_t-R_t$$ where $$t$$ denotes time. As of 23 March, however, JHU no longer report recoveries in the US and Canada at State/Province level, and recovery data at State level are not available for India. As a consequence, for these countries. we have had to assume that active cases take 22 days to resolve. Thus, $$A_t = I_t-D_t-(I_{t-22}-D_{t-22})$$.

### Growth models and forecast of active cases

We are interested in knowing how the number of active cases is going to change in the near term. We provide two alternative growth models. The growth dynamics of epidemics are complex, but we make the simplifying assumption that the epidemic is a long way from population saturation (i.e. that there is a ready supply of susceptible hosts) such that simple exponential growth provides a reasonable short-term approximation. The first model – the “constant growth” model – assumes a constant growth rate, in which case active cases follow an exponential growth model such that $$E(A_t) = A_0e^{rt}$$, where $$A_0$$ is the initial number of active cases, and $$r$$ is the (constant) intrinsic growth rate. The second model – “time-varying growth” – assumes that the growth rate changes linearly over time. Empirically, linear change in growth is what we have been observing in this epidemic as populations enact physical distancing and quarantine measures. Under this model, growth follows a Gaussian function such that $$E(A_t) = A_0e^{r_0t+\frac{b}{2}t^2}$$. Here, $$r_0$$ is the initial growth rate, and $$b$$ is the rate of change in growth rate over time.

To fit both models, we take the natural logarithm of both sides, yielding $$\ln A_t = rt + \ln A_0$$ (constant growth) and $$\ln A_t = \frac{b}{2}t^2+rt + \ln A_0$$ (time-varying growth). This shows us that we can fit a simple linear regression of $$\ln A_t$$ against $$t$$ in the constant growth scenario, and a quadratic function in the case of the time-varying model.

These fits give us an estimate of intrinsic growth rate, $$r$$, or $$r(t)$$.

We fit each model to the last $$n$$ days of $$A_t$$ data (where $$n$$ is determined by the user with the input slider) and extrapolate the fitted model to capture ten day into the future from now. Fitting and extrapolation is effected with the log-transformed model (lower plot on “10-day forecast” tab, with 95% confidence intervals) and the log of expected active case numbers is back-transformed to the original scale to produce the top plot on the “10-day forecast” tab. We provide a larger number of days for fit input for the time-varying growth model because this model estimates an additional parameter, so requires more data.

When public health interventions are rapidly changing the growth rate, this can be seen as deviations from the expected straight line on the log-plot. In these situations, when growth rate is declining rapidly (the curve is flattening), forecasts from the constant growth model (averaging growth over the last ten days) will be biased upwards. By altering the slider you can adjust the window over which growth rate is averaged, so you can get a sense of how recent shifts are affecting the forecast. The time-varying growth rate forecast should be less sensitive to changes in $$n$$, and is the better model when growth rates are changing in a steady linear manner.

### Doubling time

Under the constant growth model, the doubling time ($$t_d$$) is an intuitive measure of how fast a population is growing. It reports the number of days for the population to double in size and is calculated by setting $$A_t/A_0 = 2$$, yielding $$t_d = \frac{\ln 2}{r}$$. We calculate this directly from the previous estimate of $$r$$.

### Detection

One of the key sources of uncertainty in this pandemic has been undetected cases. Covid-19 has been quite a sneaky pathogen, and surveillance has often not been adequate, often resulting in a large number of infections in a country before it is noticed. We were interested to know how bad this problem might be for each country; deaths will be a function of the true number of cases, not the reported number.

Not all cases are detected and so the reported number of active cases is almost always lower than the true number of cases. There are two classes of non-detected cases:

1. Those that will be diagnosed soon (“Undiagnosed”). Individuals that are infected but yet to present symptoms, and;
2. Those that will never be diagnosed (“Undetected”). Individuals that present with mild symptoms, or are missed by surveillance.

The first class is caused by a delay between infection and diagnosis. However, if we know something about the length of this delay then we can use this information to inflate the observed case numbers to produce estimates of the number of undiagnosed cases. The most important determinant of the delay is the incubation period of the infection. This is known to be highly variable but can be modelled using a probability distribution, which is called the incubation distribution. Information about the incubation distribution exists from external sources. It is known that the average incubation period is 5-6 days and that 95% of infections have an incubation period of no more than 12-13 days. In our model, a log-normal distribution incorporating these features is used to model the incubation distribution. Given this model for the incubation distribution, the daily diagnosis counts reflect a larger and unknown number of Covid-19 infections in the population, aggregated with the incubation distribution. Using the observed diagnosis counts, we can therefore estimate the unknown number of Covid-19 infections by disaggregating the infection numbers from the incubation distribution. This process is known as statistical deconvolution, also called back-projection or back-calculation. From a technical perspective, the method is a non-negatively constrained linear Poisson regression analysis, where the $$y$$-variable is the observed diagnosis counts and the linear model is the linear convolution of the mean infection numbers and the incubation distribution. This is a non-standard analysis but reliable model fitting algorithms exist and are incorporated here. We use these to estimate undiagnosed cases, $$U_t$$.

The second class of non-detections, “Undetected” cases, are cases that are completely missed by surveillance. To estimate these, we use a heuristic that assumes: that deaths do not go undetected, that there is community transmission (and a closed population), that there is a case fatality ratio, $$f$$, for symptomatic cases (set by the user with a slider; lower numbers will cause our detection estimate to be lower), that detection is constant, and that there is a fixed time (here assumed to be 17 days) between onset and death. Observed new cases in the five days to time $$t$$ are calculated as $$N_t = A_t-A_{t-5}$$, and under the previous assumptions, the expected number of new cases $$\hat{N_t} = (D_{t+17}-D_{t+12})/f$$ (Our choice of a five day window for making this estimate reflects our interest in estimating detection early, when there are still few cases. A five-day window typically catches at least one death.) We then estimate detection probability, $$p(t)=N_t/\hat{N_t}$$. We average $$p(t)$$ over $$t$$ to estimate the total number of active symptomatic cases $$T_t = A_t/p$$, shown on the first tab of the site. Because detection changes over time also, as of 14 April we take this average over only the last five days of detection observations (typically 17-22 days in the past) to give the most up-to-date estimate for each region. We also show detection over time $$p(t)$$ for the selected region as a plot on the forecast tab.

The large number of assumptions behind this last method for estimating detection mean we should be very cautious with it, but it suffices to show that detection probability in some countries is very high (i.e. relatively few undetected cases), while in others it is very low indeed (many undetected cases). In countries with many undetected cases, there are many more deaths than there should be given reported case numbers. One obvious source of bias in both estimates is imported cases. When there is movement between countries and large differences in caseload, countries with small numbers of cases will have a large proportion of imported cases. These will cause us to estimate a larger number of non-detections than there are.

Detection is our weakest estimate in this app; please take it as indicative only. You can expect the estimates to have large uncertainty ranges, however, they do provide working estimates that give a sense of scale. At present we have not included uncertainty ranges such as confidence intervals but may consider this in future implementations if feasible. It would require computationally intensive methods such as bootstrapping. We note also that the estimates of undiagnosed infections can also be used as an alternative way to forecast new cases, since future diagnoses depend on past infections. This is another area of possible future development.

### Growth rate

Public health interventions are firmly aimed at reducing virus transmission and so reducing the growth in the number of active cases. The earliest indications of intervention success will manifest in lowered growth rates (around ten days after the intervention is made in the case of covid-19). The per-capita growth rate can be calculated each day as $$G_t = \frac{A_t-A_{t-1}}{A_{t-1}}$$. The top plot on the “Growth rates” tab shows this rate as a percentage, $$100\times G_t$$, expressing population growth in more broadly understood terms; as a compound interest rate. The plot uses a three-day moving average.

### Curve flattening

To capture the idea of “flattening the curve”, we made an index ($$C_t$$) that captures how growth rate changes over time. Conceptually, we were thinking of the second differential of $$\ln A_t$$ (scaled against the first differential), but the resultant index is a bit of a mouthful: $C_t = -\frac{(\ln A_t-\ln A_{t-1}) - (\ln A_{t-1}-\ln A_{t-2})}{\sqrt{(\ln A_{t-1}-\ln A_{t-2})^2}}$ Which could be simplified, but has been written so as to help you see that it is the change in growth rate divided by the magnitude of the growth rate, and then the whole multiplied by negative 1 so that positive values are “good”, negative values “bad”. It should rightly be thought of as an index of “suppression” rather than an index of “curve-flattening”.

The second plot on the “Growth rate and curve-flattening” tab places a smoother through this index across all time for which we have data. Times when countries are reducing their growth rate are indicated by the smoother sitting in positive territory.

The denominator in this index does make it very sensitive as growth rates come close to zero.

## The University

If you would like to support the University's work, please consider giving to our covid-19 emergency appeal.

### Media enquiries

University of Melbourne Newsroom, Phone +61 3 8344 4123; Email <EMAIL>

## Acknowledgements

First, to the people on the front line who are collecting these data. You are a legion of brave and selfless people. We hope this work has in some diffuse way made your job easier. Second, a big thanks to the team at Johns Hopkins University. Their early decision to make these data open access, and their ongoing commitment to curating a large dynamic dataset has provided a critical resource for helping the world understand and manage this pandemic.

The app was conceived by <NAME> at the University of Melbourne and has been greatly improved by Drs <NAME> and <NAME> (University of Melbourne). These three are the core development and maintenance team. State-level data for India are curated by Drs <NAME> and <NAME> (Panjab University). The app has been improved by contributions from <NAME> (Michigan State University); <NAME> (UoM); and several other contributors on github. <NAME> (University of Sydney) provided deconvolution code for estimating undiagnosed infection numbers. Valuable feedback has come from <NAME> (U Connecticut), <NAME> (Harvard), <NAME> (Memorial University of Newfoundland), <NAME> (Charles Darwin Uni), the Spatial Ecology and Evolution Lab (Melbourne), and a suite of people on the internet who are too numerous to name (but we thank them anyway).

Thanks to <NAME> from the University of Western Australia for the Turkish translation. Thanks also to the following, who are all from the University of Melbourne: Sanjana Ratan for the Hindi translation, <NAME>zmann for the German translation, <NAME> for the Spanish translation, and <NAME> for the Russian translation.


text:
# ddc lemma

Posted on May 4, 2016 by Dima

tags: Kahler manifolds

In this note I will prove the easy local case of $$dd^c$$ lemma.

Let $$(M,I)$$ be a complex manifold. Extend the action of the complex structure on exterior powers of the complexified tangent bundle forms by $\mathbf{I}: \bigwedge{}^* M \to \bigwedge{}^* M, \mathbf{I} := \sum i^{p-q} \cdot \Pi^{p,q}$ where $$\Pi^{p,q}: \bigwedge^{p+q} \to \bigwedge^{p,q}$$ is the natural projection. Define a dwisted differential $$d^c = I \circ d \circ I^{-1}$$. On the cotangent bundle define it to be $$\mathbf{I}(\alpha)(v_1, \ldots, v_n) := \alpha(\mathbf{I}v_1, \ldots, \mathbf{I}v_n)$$.

Proposition. $$\partial = \dfrac{d + i \cdot d^c}{2}, \bar \partial = \dfrac{d - i \cdot d^c}{2}$$, where $$\partial$$ and $$\bar\partial$$ are projections of $$d$$ on $$(\cdot +1, \cdot)$$ and $$(\cdot, \cdot+1)$$ components.

Proof. Indeed $\partial + \bar\partial + i \mathbf{I}^{-1} (\partial + \bar\partial) \mathbf{I} = \partial + \bar\partial + i (i^{q-p-1} \partial + i^{q-p+1}\bar\partial)i^{q-p}=\\ = \partial + \bar\partial + \partial + i^2\bar\partial=2\partial$ Similarly for the second equality. $$\square$$

In particular, $$\partial\bar\partial = -\dfrac{i}{2}dd^c$$ on a complex manifold.

Lemma (Poincaré lemma). If $$\alpha$$ is a closed form on a polydisc then it is exact.

Lemma (Poincaré-Dolbeault-Grothendieck lemma). If $$\alpha$$ is a $$\partial$$-closed, not holomorphic (i.e. $$\alpha \notin A^{n,0} M$$), form on a polydisc then it is $$\bar\partial$$-exact.

(this lemma means in particular that Dolbault resoltions of sheaves of holomorphic forms are acyclic)

Note that since $$\bar(\partial\alpha) = \bar\partial \bar\alpha$$, the PDG lemma also holds for $$\partial$$.

Lemma (local $$dd^c$$ lemma). Let $$\eta$$ be a $$(1,1)$$-form on a polydisc. The following are equvalent

• $$\eta$$ is exact, or
• $$\eta$$ is $$\partial$$-exact, $$\bar\partial$$-closed.
• there exists a function $$\chi$$ such that $$\eta = dd^c \chi$$.

Proof. $$3 \Rightarrow 1$$ is immedate.

$$1 \Rightarrow 2$$. Recall that $$d^2=\partial^2=\bar\partial^2=0$$, where $$d=\partial+\bar\partial$$, and therefore $$\bar\partial\partial=-\partial\bar\partial$$. If $$\eta = d\alpha$$ then there exist forms $$\beta \in A^{1,0}, \gamma\in A^{0,1}$$ such that $$\eta = \partial\beta + \bar\partial\gamma$$ and $$\bar\partial\beta=\partial\gamma=0$$. Then $$\partial(\eta) = \partial^2 \beta + \partial\bar\partial\gamma=-\bar\partial\partial\gamma=0$$. Similarly for $$\bar\partial\eta$$. Apply PDG and $$\partial$$-PDG to get $$\partial$$ and $$\bar\partial$$-exactness of $$\eta$$.

$$2 \Rightarrow 3$$. By Poincaré-Dolbeault-Grothendieck lemma, there exists a form $$\alpha \in A^{1,0}$$ such that $$\bar\partial\alpha=\eta$$.

Observe that $$\partial\alpha$$ is a closed form: $$(\partial + \bar\partial)(\partial\alpha) = \partial^2 \alpha - \partial\bar\partial\alpha=0 -\partial\eta=0$$. By Poincar'e lemma, there exists a form $$\alpha'$$ such that $$d\alpha' = \partial\alpha' + \bar\partial\alpha' = \partial\alpha$$. By grading considerations, $$\bar\partial\alpha' = 0$$.

Consider the form $$\beta = \alpha - \alpha' \in A^{1,0}$$. Then $$\partial\beta = \partial\alpha - \partial\alpha' = 0$$ and $$\bar\partial\beta = \bar\partial\alpha - \bar\partial\alpha' = \eta$$. By $$\partial$$-PDG, there exists a function $$\psi$$ such that $$\partial\psi = \beta$$, so $$\bar\partial\partial\psi=\eta$$. Put $$\chi=-i\psi$$. $$\square$$

Interestingly, this statement is true globally, i.e. for global $$(p,q)$$-forms on a manifold $$M$$, if the manifold is Kähler.

Lemma (global $$dd^c$$ lemma). Let $$\alpha$$ be a form on a Kähler manifold, and assume that $$\eta$$ is either $$d$$-, $$\partial$$- or $$\bar\partial$$-exact. Then $$\alpha$$ is $$dd^c$$-exact (or, which is the same, $$\partial\bar\partial$$-exact).

This result requires Hodge theory for K"ahler manifolds.

# The Picard scheme

Posted on April 24, 2016 by Dima

tags: moduli, Abelian varieties

## Functors $$Pic$$ and $$Div$$

We will denote base change with a subscript: $$X_T = X \times T$$.

If $$X$$ is a scheme, the Picard group of $$X$$ is defined to be the group of isomorphism classes of invertible sheaves on $$X$$. The relative Picard functor of an $$S$$-scheme $$X$$ is defined as ${\operatorname{Pic}}_{X/S}(T) := {\operatorname{Pic}}(X_T) / {\operatorname{Pic}}(T)$ where the embedding $${\operatorname{Pic}}(T) \hookrightarrow {\operatorname{Pic}}(X_T)$$ is given by the pullback along the structure maps $$X_T \to T$$.

Caveat: all representability results work with a sheafification of this functor in some topology, Zariski, étale, or fppf. Unless $$X \to S$$ is proper and has a section, they need not be isomorphic.

An effective divisor is a closed subscheme such that its ideal is invertible. If $$f: X \to S$$ is a morphism of schemes then a relative effective divisor on $$X$$ is an effective divisor $$D$$ such that $$D$$ is flat over $$S$$. For a morphism $$X \to S$$ define the functor of relative divisors ${\operatorname{Div}}_{X/S}(T) := {\{\ \textrm{ relative effective divisors on } X_T/T \ \}}$

We are interested in representability of this functor, so to this end we prove a little lemma.

Lemma. Let $$X \to S$$ be a flat morphism. Let $$D$$ be a closed subscheme of $$X$$ flat over $$S$$. Then $$D$$ is relative effective divisor in a neighbourhood of $$x \in X$$ if and only if $$D_s$$ is cut out in a neighbourhood of $$x$$ in $$X_s$$, where $$s$$ is the image of $$x$$, by a single non-zero element of $${\operatorname{\mathcal{O}}}_{X_s,x}$$ (which amounts to being an effective divisor in $$X_s$$ but we won’t prove it).

# Construction of Hilb and Quot

Posted on February 16, 2016 by Dima

tags: moduli

## Hilbert polynomial

If $$X$$ is a projective variety and $${\mathcal{F}}$$ is a sheaf on $$X$$ then the function $\chi(\mathcal F, n) = \sum (-1)^i H^i(X, \mathcal F(n))$ is called the Hilbert function of $$\mathcal F$$.

Lemma (Snapper lemma). It is a polynomial.

Proof. Induction on the dimension of support of $${\mathcal{F}}$$. Notice that $$\chi$$ is additive in extensions, i.e. $$\chi({\mathcal{F}}/{\mathcal{G}}) = \chi({\mathcal{F}}) - \chi({\mathcal{G}})$$, as follows from long exact sequence of cohomology.

Recall that for a divisor $$D$$ the sheaf $${\mathcal{O}}(D)$$ is defined as ${\mathcal{O}}(D)(U) := {\{\ f \in \mathcal{K}(U) \ \mid\ (f) + D \geq 0 \ \}}$ where $$\mathcal{K}$$ is the constant sheaf of rational functions on $$X$$. Let $$H$$ be a hyperplane in $${\mathbb{P}}^n$$, one has a natural short exact sequence $0 \to {\mathcal{O}}_{{\mathbb{P}}^n}(-H) \to {\mathcal{O}}_{{\mathbb{P}}^n} \to {\mathcal{O}}_H \to 0$ Assume that $$H$$ is chosen so that $$\dim {\operatorname{supp}}{\mathcal{F}}\cap H < \dim {\operatorname{supp}}{\mathcal{F}}$$. Tensoring with $${\mathcal{F}}$$, taking into account that $${\mathcal{O}}_{{\mathbb{P}}^n}(-H) \cong {\mathcal{O}}_{{\mathbb{P}}^n}(-1)$$, and passing to the long exact sequence we get $0 \to Tor_1({\mathcal{F}}, {\mathcal{O}}_H) \to {\mathcal{F}}(-1) \to {\mathcal{F}}\to {\mathcal{F}}\otimes {\mathcal{O}}_H \to 0$ Note that by looking at long exact cohomolgy sequences one establishes that Hilbert polynomial is an additive invariant. Read more (comments)

# Grauert's criterion of ampleness

Posted on October 1, 2015 by Dima

tags: line bundles, positivity

Here’s a writeup of a proof of Grauert’s criterion for ampleness (here is the link to the original paper). Since often proving that a variety is algebraic is not far from proving that it is projective, this criterion can be useful in judging algebraicity of a variety.

We start with some observations on cohomology of invertible sheaves which are powers of the invertible sheaf associated to hyperplane section.

Recall that if $$D$$ is a (Cartier) divisor on a variety $$X$$ then it gives rise to a sheaf ${\cal O}(D) := \{ f \in k(X) \mid (f) + D \geq 0 \}$ and a section $$s \in H^0(X, {\cal O}(D))$$ such that $$(s) = D$$. Let $$V = H^0(X, L)$$. Then a (generally speaking, partial) map $$\iota: X \dashrightarrow \mathbb{P}(V^\vee)$$ is defined: $$x \mapsto (f \mapsto f(x))$$. The value on the right, before projectivization, depends on trivialization, hence is only well-defined up to a constant, but since we projectivize, the map is well-defined. A line bundle (or a divisor $$D$$) is called very ample if $$\iota$$ is a closed embedding, ample if some multiple of it is ample.

If $$D$$ is ample, then $$H:=mD$$ is a hyperplane section for some $$m > 0$$. Then $$D_H^r$$ is the degree of $$X$$, hence positive.

Theorem. Let $$X=\mathbb{P}^d$$

• $$H^i(X, {\cal O}(n)) = 0$$, if $$0 < i < d$$ or $$i < 0$$
• $$H^i(X, {\cal O}(n)) \cong H^{n-i}(X, {\cal O}(-d-n-1))^\vee$$

A paranthesis on ampleness.

Lemma. Let $$X$$ be a scheme covered by finitely many affine schemes $$X_i$$ such that $$X_i$$ is the locus of points $$x$$ such that $$s_i$$ generates $${\operatorname{\mathcal{O}}}_{X,x}$$ for an $$s_i \in H^0(X, {\operatorname{\mathcal{L}}})$$. Let $$s_{ij} \in H^0(X_i, {\operatorname{\mathcal{L}}})$$ be sections such that $$s_{ij}/s_i$$ is a base in $$H^0(X_i, {\operatorname{\mathcal{O}}}_X)$$ for each $$i$$. (Note that $$s_i$$ generate $${\operatorname{\mathcal{L}}}$$. ) Then the morphism $$X \to {\operatorname{Proj}}[s_i, s_{ij}]$$ is an embedding.

Theorem. Let $${\operatorname{\mathcal{L}}}$$ be a sheaf such that for any finitely generated quasi-coherent $${\operatorname{\mathbb{F}}}$$ there exists an $$n_0$$ such that $${\operatorname{\mathbb{F}}}\otimes {\operatorname{\mathcal{L}}}^n$$ is generated by global sections for $$n \geq n_0$$. Then there exists an $$m$$ such that $${\operatorname{\mathcal{L}}}^m$$ is very ample (defines an embedding into $$P^n$$).

Proof ([Liu, 5.1.34, p.169]). Let $$U$$ be an affine neighbourhood of $$x$$ such that $${\operatorname{\mathcal{L}}}|_U$$ is free, and let $${\operatorname{\mathcal{I}}}$$ be the sheaf of ideals that cuts out the complement of $$U$$. There exists an $$n_0$$ such that $${\operatorname{\mathcal{I}}}{\operatorname{\mathcal{L}}}^n$$ is globally generated, so there is a section $$s \in H^0(X, {\operatorname{\mathcal{I}}}{\operatorname{\mathcal{L}}}^n) \subset H^0(X, {\operatorname{\mathcal{L}}}^n)$$ that doesn’t vanish at $$x$$. Since sectios of $${\operatorname{\mathcal{I}}}{\operatorname{\mathcal{L}}}^n$$ that vanish on the complement to $$U$$, $$X_s$$ is contained in $$U$$.

Now by compactness $$X$$ is covered by finitely many affines of the form $$X_{s_i}$$, $$s_i \in H^0(X, {\operatorname{\mathcal{L}}}^n)$$ with $$H^0(X_{s_i}, {\operatorname{\mathcal{O}}})$$ finitely generated. We can apply the previous lemma to conclude.

Theorem. Let $${\operatorname{\mathcal{L}}}$$ be a line bundle such that for any sheaf of ideals $${\operatorname{\mathcal{I}}}$$ there exists a number $$n$$ such that $$H^1(X, {\operatorname{\mathcal{I}}}\otimes {\operatorname{\mathcal{L}}}^n)$$ vanishes. Then $${\operatorname{\mathcal{L}}}$$ is ample, i.e. $${\operatorname{\mathbb{F}}}\otimes {\operatorname{\mathcal{L}}}^m$$ is globally generated for big enough $$m$$.


text: 2 added 28 characters in body edited Feb 12 '15 at 21:51 <NAME> 6,61111 gold badge1919 silver badges4747 bronze badges Definition 2. Let $$\gamma$$ be a geodesic. A pair of points $$p$$, $$q\in\gamma$$ are said to be conjugate if there exists a Jacobi field $$n$$ which is not identically zero but vanishes at both $$p$$ and $$q$$. Definition 2. A pair of points $$p$$, $$q\in\gamma$$ are said to be conjugate if there exists a Jacobi field $$n$$ which is not identically zero but vanishes at both $$p$$ and $$q$$. Definition 2. Let $$\gamma$$ be a geodesic. A pair of points $$p$$, $$q\in\gamma$$ are said to be conjugate if there exists a Jacobi field $$n$$ which is not identically zero but vanishes at both $$p$$ and $$q$$. 1 answered Feb 12 '15 at 21:40 <NAME> 6,61111 gold badge1919 silver badges4747 bronze badges The separation vector is a Jacobi field because it obeys the Jacobi equation. Here I will derive geodesic deviation from scratch because I find MTW's derivation hard to follow. (Much like everything else in that book.) Definition 1. Consider a family of timelike geodesics, having the property that in a sufficiently small open region of the Lorentz manifold $$(M,g)$$ precisely one geodesic passes through every point. Such a collection is called a congruence. The tangent field to this set of curves, parameterized by proper time $$s$$, is denoted by $$u$$ and is normalized as $$\langle u,u\rangle=-1$$. Let $$\gamma(t)$$ be some curve transversal to the congruence. This means that the tangent $$\dot\gamma$$ is never parallel to $$u$$ in the region under consideration. Imagine that every point on the curve $$\gamma(t)$$ moves a distance $$s$$ along the geodesics which passes through that point. Let the resulting point be $$H(s,t)$$. This defines a map $$H:O\longrightarrow M$$ for some open $$O\subset \mathbb{R}^2$$. For each $$t$$, the curve $$s\longmapsto H(s,t)$$ is a timelike geodesic with tangent vectors $$u(H(s,t))$$. We say that $$u$$ defines a vector field $$u\circ H$$ along which the map $$H$$ is tangential, i.e. of the form $$u\circ H=H_*\circ\frac{\partial}{\partial s}\quad\text{or}\quad u\big(H(s,t)\big)=T_{(s,t)}H\cdot\frac{\partial}{\partial s}$$ In other words, $$u$$ and $$\partial/\partial s$$ are $$H$$-related. We shall use for $$u\circ H$$ the same letter $$u$$. Let $$v$$ be the tangential vector field along $$H$$ belonging to $$\partial/\partial t$$ $$v=H_*\circ\frac{\partial }{\partial t}$$ The vectors $$v(s,t)$$ are tangent to the curves $$t\longmapsto H(s,t)$$ ($$s$$ fixed), and represent the separation of points which are moved with the same proper time along the neighboring geodesics of the congruence (beginning at arbitrary starting points). This is shown in the following diagram (taken from Straumann, General Relativity [2013]): Since $$\partial/\partial s$$ and $$\partial/\partial t$$ commute, the tangential fields $$u$$ and $$v$$ also commute. To see this, recall that if the vectors of two Lie brackets are $$H$$-related, then the Lie brackets themselves are $$H$$-related. To get the distance between curves, we create a projection operator $$id+u\otimes u$$ onto the subspace of the tangent space orthogonal to $$u$$. Thus the relevant infinitesimal separation vector $$n$$ is $$n=v+\langle v,u\rangle u$$ Note that $$\langle n,u\rangle=\langle v,u\rangle-\langle v,u\rangle=0$$, so $$n$$ is indeed perpendicular to $$u$$. We now show that $$n$$ is also Lie transported. We have \begin{align*} \mathcal{L}_un&=[u,n]=[u,v]+[u,\langle v,u\rangle u]=\big(u\langle v,u\rangle\big)u\\ u\langle v,u\rangle&=\frac{\partial}{\partial s}\langle v,u\rangle \end{align*} The normalization condition $$\langle u,u\rangle=-1$$ implies $$0=\frac{\partial}{\partial t}\langle u,u\rangle=2\langle \nabla_v u,u\rangle$$ Furthermore, since $$\nabla_vu=\nabla_uv$$, it follows that $$\frac{\partial}{\partial s}\langle v,u\rangle=\langle \nabla_uu,v\rangle+\langle u,\nabla_uv\rangle=\langle u,\nabla_vu\rangle=0$$ This shows that indeed $$\tag{1} \mathcal{L}_un=0$$ Next, we consider $$\nabla^2_uv=\nabla_u\nabla_uv=\nabla_u\nabla_vu=[\nabla_u,\nabla_v]u$$ However, the Riemann tensor for three vectors $$X,Y,Z$$ is $$R(X,Y)Z=[\nabla_X,\nabla_Y]Z-\nabla_{[X,Y]}Z$$. Due to $$[u,v]=0$$, we obtain $$\nabla^2_uv=R(u,v)u$$ This is called the Jacobi equation for the field $$v$$. We now show that $$n$$ also satisfies this equation. From (1) it follows that $$$$\nabla_un=\nabla_uv+\big(u\langle v,u\rangle\big)u+\langle v,u\rangle \nabla_uu=\nabla_uv$$$$ Furthermore, $$$$R(u,n)u=R(u,v)u+\langle v,u\rangle R(u,u)u=R(u,v)u$$$$ Putting this all together, we see that $$n$$ satisfies the Jacobi equation $$$$\nabla^2_un=R(u,n)u$$$$ The Jacobi field $$n$$ is everywhere perpendicular to $$u$$. In physics we call this the equation for geodesic deviation. For a given $$u$$ the right hand side defines at each point $$p\in M$$ a linear map $$n\longmapsto R(u,n)u$$ of the subspace of $$T_pM$$ perpendicular to $$u$$. In more familiar notation, we have $$\frac{D^2 n^a}{D\tau^2}=R^a_{\;bcd}u^b n^c u^d$$ So why even bother calling this equation a Jacobi equation? Perhaps there is some general result of Jacobi fields that we wish to apply to GR? This is indeed the case. Definition 2. A pair of points $$p$$, $$q\in\gamma$$ are said to be conjugate if there exists a Jacobi field $$n$$ which is not identically zero but vanishes at both $$p$$ and $$q$$. There exists a very powerful theorem regarding conjugate points (suitably generalized here to spacetime, but a similar version holds in positive-definite spaces): Theorem 1. Let $$\gamma$$ be a smooth timelike curve connecting two points $$p$$, $$q\in M$$. Then the necessary and sufficient condition that $$\gamma$$ locally maximizes proper between $$p$$ and $$q$$ over smooth one parameter variations is that $$\gamma$$ be a geodesic with no point conjugate to $$p$$ between $$p$$ and $$q$$. This theorem is the basis for the singularity theorems of GR. Since MTW doesn't discuss these theorems (AFAIK, haven't read the whole thing), they don't need to mention Jacobi fields or conjugate points. Indirectly, this property of Jacobi fields is used in the justification for Big Bang cosmology. For more in-depth discussions, I refer you to Wald, General Relativity (1984) Hawking & Ellis, The Large Scale Structure of Spacetime (1973) (in that order) and references therein.


text:
The below algorithms is given in the book Robust Adaptive Dynamic Programming.

## Problem statements¶

Given the linear system equation (1), design a linear quadratic regulator (LQR) in the form of that minimizes the following cost function

where .

The solution of this problem is which is also the solution of the Riccati equation, can be obtained by OpenControl.ADP_control.LTIController.LQR(). However, this approach requires the knowledge of the system dynamic (the matrix A and B). The model-free approach below will resolve this problem.

Lets begin with another control policy where the time-varying signal e denotes an artificial noise, known as the exploration noise. Taking time derivative of the cost function and integrate it within interval to obtain:

(1)

where .
We can see that this is the fix point equation, meaning that the K matrix can be solve by finite number of iteration. And because of no requirement of the system dynamic, it is a data-driven approach.

## On-policy learning¶

For computational simplicity, we rewrite (1) in the following matrix form:

(2)

where

Note

• To adopt the persistent excitation condition, one must choose a sufficiently large number of data to satisfy the following condition:

### Library Usage¶

Ctrl = LTIController(sys)
# set parameters for policy
Q = np.eye(3); R = np.array([[1]]); K0 = np.zeros((1,3))
explore_noise=lambda t: 2*np.sin(10*t)
data_eval = 0.1; num_data = 10

Ctrl.setPolicyParam(K0=K0, Q=Q, R=R, data_eval=data_eval, num_data=num_data, explore_noise=explore_noise)
# take simulation and get the results
K, P = Ctrl.onPolicy()

## Off-policy learning¶

Let define some new matrices

and defined by:

then for any given stabilizing gain matrix , (1) implies the same matrix form as (2)

### Library Usage¶

Setup a simulation section the same as the section then perform simulation by OpenControl.ADP_control.LTIController.offPolicy()

K, P = Ctrl.offPolicy()


text:
my teacher

🔥 | Latest

my teacher: awesomacious: I gave this picture to my teacher and he cried when i gave it to him and said “this is the nicest thing someone has done for me” and teared up

awesomacious: I gave this picture to my teacher and he cried when i gave it to him and said “this is the nicest thing someone has done f...

my teacher: I gave this picture to my teacher and he cried when i gave it to him and said “this is the nicest thing someone has done for me” and teared up

I gave this picture to my teacher and he cried when i gave it to him and said “this is the nicest thing someone has done for me” and tear...

my teacher: awesomacious: I have a tradition with my teacher to draw eachother memes on exams so here is another one

awesomacious: I have a tradition with my teacher to draw eachother memes on exams so here is another one

my teacher: awesomacious: I have a tradition with my teacher to draw eachother memes on exams so here is another one

awesomacious: I have a tradition with my teacher to draw eachother memes on exams so here is another one


text: Let $$S$$ be the sum of the first 2015 terms of the sequence $1,2,2,3,3,3,4,4,4,4,.......$ where $$n$$ occurs as $$n$$ terms . The sum of the digits of $$S$$ is


text:
# err.rate randomForest R

OOB error rate in the documentation is defined as

(classification only) vector error rates of the prediction on the input data, the i-th element being the (OOB) error rate for all trees up to the i-th. https://cran.r-project.org/web/packages/randomForest/randomForest.pdf

I need help in understanding this. I have 500 trees, and generally the error rate decreases as the number of trees increases. Why would this decrease as the number of trees increases?

When I print model$err.rate[,1], and we see the 500 oob error values, is this specifically for the last prediction, or is it some average? When you print(Forest) and it returns the OOB estimate of error rate, how does this single value relate to the 500 values printed previously, and how does the confusion matrix relate? ## 1 Answer Why would this decrease as the number of trees increases? From <NAME> & <NAME> "To tune or not to tune the number of trees in random forest?" we find an answer. The expected error rate can be a non-monotonous function of the number of trees. The expected error rate (equiv. $$\text{error rate} = 1 - \text{accuracy}$$) as a function of $$T$$ the number of trees is given by $$E(e_i(T)) = P\left(\sum_{t=1}^T e_{it} > 0.5\cdot T\right)$$ where $$e_{it}$$ is a binomial r.v. with expectation $$E(e_{it}) = \epsilon_i$$, the decision of a particular tree indexed by $$t$$. This function is increasing in $$T$$ for $$\epsilon_{i} > 0.5$$ and decreasing in $$T$$ for $$\epsilon_{i} < 0.5$$. The authors observe We see that the convergence rate of the error rate curve is only dependent on the distribution of the $$\epsilon_i$$ of the observations. Hence, the convergence rate of the error rate curve is not directly dependent on the number of observations $$n$$ or the number of features, but these characteristics could influence the empirical distribution of the $$\epsilon_i$$’s and hence possibly the convergence rate as outlined in Section 4.3.1 So, to directly answer your question, the decrease that you observe is related to the idiosyncratic component of the data. The decision of any tree is a certain kind of random variable, and the expected error rate for any number of trees $$T$$ is solely a feature of the $$\epsilon_i$$ which will be different for each data set. You may find that increasing the number of trees causes the error rate to oscillate a bit, perhaps because some datum $$j$$ has $$\epsilon_j$$ very near to $$\frac{1}{2}$$, so it's flipping between being classified one way and another way. Instead of tracking error rate or AUC as a function of $$T$$, I suggest tracking log-loss or Brier score as a function of $$T$$, because these measurements are monotonically decreasing in expectation as a function of $$T$$. See: Do we have to tune the number of trees in a random forest? When I print model$err.rate[,1], and we see the 500 oob error values, is this specifically for the last prediction, or is it some average?

This is the OOB rate for all trees with index less than or equal to the row index. That's what your quote from the documentation means when it says "the i-th element being the (OOB) error rate for all trees up to the i-th." Every time you add a tree, this could change the cumulative OOB error rate, so this is a convenient way to track how that change happens as you add trees.

When you print(Forest) and it returns the OOB estimate of error rate, how does this single value relate to the 500 values printed previously, and how does the confusion matrix relate?

The OOB rate printed by print(Forest) is the OOB rate for the entire ensemble. If you have T trees in Forest, it's Forest\$err.rate[T,1] expressed as a percent.