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# Curvature of Space: What is it?

1. Jan 15, 2014

### rushikesh

Curvature of Space-time: What is it?

General relativity talks about curvature of space-time due to mass. What does it actually mean by 'curve'. Is the space made of something that we can say is curving? If space is purely 'empty', then what is getting curve? Or is it that curving is just an analogy used?

Last edited: Jan 15, 2014
2. Jan 15, 2014

### pervect

Staff Emeritus
Both space and space-time are curved in general relativity. The easiest way to describe how to measure curvature involves being able to measure angles, but you can also infer and compute curvature by only being able to measure distances.

For simplicity I'll talk about how to measure the curvature of space.

The "easy way" involves selecting three nearby points, and drawing a triangle made out of geodesics (geodesics are the shortest lines connecting those points*). You then measure and sum the interior angles of the triangle.

If the sum is not exactly 180 degrees, you have curvature. The amount of the curvature is the ratio of the angular excess divided by the area enclosed by the triangle.

(For a proof on the sphere, see http://nrich.maths.org/1434)

*This is actually true in the normal convex neighborhood of the points, if the points are too far apart, the line of shortest distance between two points will still be a geodesic, but the presence of multiple geodesics connecting the two points means that some of the geodesics might not minimize the distance.

Because you can measure distances (and angles) even in empty space, you don't need a "thing" to curve. The ability to measure distances between points is necessary and sufficient to define the geometry, and its curvature.

The technique above is one way of measuring what is called the "sectional curvature" of a surface, see for instance the wiki article http://en.wikipedia.org/wiki/Sectional_curvature.

The descriptions of curvature used in GR are the Riemann curvature tensor and various contractions of it (such as the Ricci and Einstein tensors). It's possible in principle to determine the full Riemann curvature tensor from a knowledge of all the sectional curvatures for the various plane "slices" of the geometry, though I don't know the details offhand.

3. Jan 15, 2014

### Staff: Mentor

To add to what pervect posted, in GR, curvature of *spacetime* is the same as tidal gravity. We measure tidal gravity by looking at the motion of freely falling objects that are close together, to see if there is relative acceleration between them.

For example, suppose we have two objects that start out momentarily at rest at slightly different altitudes above the Earth. As the objects free-fall towards the Earth, their radial separation will increase (because the one that is slightly closer to the Earth will accelerate downward, relative to an observer at rest, slightly more). This shows that tidal gravity is present.

The reason tidal gravity, as just described, can be interpreted as spacetime curvature is that free-falling objects move on geodesics of spacetime--the spacetime analogues of curves like great circles on a sphere. So the two free-falling objects that I described above are moving on nearby geodesics which start out parallel--meaning parallel in spacetime, because they are both at rest at some instant of time, so if we drew a spacetime diagram with time, according to an observer at rest relative to the Earth, on one axis, and radial distance from Earth on the other axis, and drew the worldlines of both objects on the diagram, they would both cross the "t = 0" line at right angles. But these geodesics don't *stay* parallel--the objects don't stay at rest relative to each other, as I showed above. That means spacetime must be curved.

Note that this is a different definition of "curvature" than pervect gave, but the two definitions are actually equivalent. For example, on a sphere, triangles drawn on the sphere's surface have angles summing to more than 180 degrees; but also, if you take two great circles (which are the geodesics on a sphere) that start out parallel--for example, two meridians of longitude both crossing the equator at right angles--they don't stay parallel; the meridians get closer together, eventually meeting at the poles. It can be shown mathematically that the two manifestations of curvature always occur together: any space or spacetime that shows one will also show the other. So you can use either one to tell whether curvature is present.

4. Jan 15, 2014

### pervect

Staff Emeritus
Ill add a bit too. "Parallel transport" of a vector along a closed curve is actually the way that most textbooks in GR discuss curvature, but I felt that the angle-summing method was more intuitive.

Geodesic deviation is the technical name for the tendency of the distance between nearby parallel geodesics to accelerate relative to one another. This value of the accleration can be expressed by the Riemann curvature tensor, see for instance the wiki article http://en.wikipedia.org/wiki/Geodesic_deviation

Last edited: Jan 15, 2014
5. Jan 15, 2014

### A.T.

Some pictures to explain intrinsic curvature:

Positve curvature (circumference / diameter < PI) :

Negative curvature (circumference / diameter > PI) :

For triangles (three points connected geodesically) you have:

Positve curvature : inner angle sum > PI
Negative curvature : inner angle sum < PI
Flat : inner angle sum = PI

http://www.ams.org/samplings/feature-column/fcarc-sphericon2
http://abyss.uoregon.edu/~js/lectures/cosmo_101.html

However, as others said, this intrinsic curvature is related to tidal effects, or the non-uniformity of gravity. In a small area, where tidal effects are negligible, the Equivalence Principle applies, and gravity can be modeled without intrinsic curvature, just like in a rocket accelerating in flat space time. This is shown in the animation below. Note that the cone surface has no intrinsic curvature, because it can be rolled out flat. Unlike a sphere or a saddle surface.

Over a larger area you need intrinsic curvature to make those local cone-like patches fit together. This looks something like this:

It is "just an analogy" in the sense that it is an abstract mathematical model, like anything in physics. The mathematical concept of curvature doesn't require space to be filled with something, in order to be curved. Curvature is a certain distortion of distances between coordinates.

Last edited: Jan 15, 2014
6. Jan 15, 2014

### drumjonny

Y
Hay

7. Jan 16, 2014

### Jonathan Scott

The curvature mentioned so far here is intrinsic curvature, which is like the curvature of part of a ball and causes local geometry to diverge from flat space. This is effectively present wherever mass density is present, but not outside the mass in empty space.

When free fall motion through empty space under gravity is described as following the curvature of space-time, this is a bit different. This is mathematically more like the curvature of a cone of paper, which makes lines appear curved from some points of view (when watching from a distance) but to someone moving around the cone, the local geometry is like ordinary flat paper. This is related to the way in which a free-falling object appears to be accelerating and/or following a curved path to a distant observer, but an observer with the object would not feel any acceleration.

Note that free fall motion under gravity is normally dominated by curvature of space with respect to time. That is, as a particle travels around an orbit or otherwise follows a free fall path, a line showing its motion in space plotted against time curves towards the gravitational source. Gravity also curves space by the same curvature (in units where c=1) but this only has a significant effect on motion at relativistic speed, including light, which is deflected equally by the curvature with respect to time and space, so it ends up being deflected by twice the amount that Newtonian gravity theory would predict (as that only includes the time part).

Anyway, the important point to note is that there's more than one sort of curvature involved, and when empty space is described as curved, that's a different sort of curvature from the curvature at a location with a non-zero mass density.

8. Jan 16, 2014

### Staff: Mentor

This is not correct. Spacetime curvature in GR is tidal gravity, and tidal gravity is present even in empty space. There is a sense in which the curvature in the presence of mass density (or more generally stress-energy) is different from the curvature present in empty space, but it's not a matter of "intrinsic" curvature vs. some other kind. See further comments below.

I don't think this is a very good analogy. The cone of paper does have extrinsic curvature, but the only reason we know this is that we can perceive and measure the higher-dimensional space that the cone is embedded in. We can't perceive or measure any higher-dimensional manifold that spacetime is embedded in, so the only curvature of spacetime we can measure is intrinsic curvature. The intrinsic geometry of the cone is flat globally (except at the apex of the cone), not just locally; but the intrinsic geometry of spacetime is only flat locally, not globally.

That's because free-falling objects follow geodesics. But spacetime curvature--meaning intrinsic curvature, which, as above, is the only kind we can measure in spacetime--is not a matter of whether or not objects are following geodesics. It's a matter of whether or not tidal gravity is present. As I described in my earlier post, you can directly measure tidal gravity using only free-falling objects, following geodesics; each individual geodesic is "straight", but the relationship between them shows the presence of spacetime curvature.

As I noted above, this isn't true; spacetime curvature in general is just tidal gravity, and is present everywhere.

However, relativists often divide up spacetime curvature into two types: Weyl curvature and Ricci curvature. The kind of curvature I described in my earlier post, which is what we usually think of as "tidal gravity", is Weyl curvature; it can be present in empty space. Ricci curvature, however, can only be present when nonzero stress-energy is present; that's because Ricci curvature is the kind that appears in the Einstein Field Equation, which relates curvature to stress-energy. In the idealized case of a perfectly spherically symmetric gravitating body, Ricci curvature is only present inside the body, and Weyl curvature is only present in the empty space surrounding the body (at least, I think the latter is true, but I'd have to check to be sure). But both of them are "intrinsic" curvature.

9. Jan 16, 2014

### WannabeNewton

Indeed as Peter noted there is obviously tidal gravity in vacuum space-times (excluding trivial solutions) and this is due to the non-vanishing Riemann curvature; the Ricci curvature does not determine the Riemann curvature in 4 dimensions so just because the Ricci curvature vanishes in vacuum space-times doesn't mean the Riemann curvature will also vanish. Indeed the Weyl curvature (which comes from the Weyl tensor i.e. the conformal tensor) will still be non-vanishing.

Given a space-time $(M,g_{\mu\nu})$ and a geodesic congruence $\xi^{\mu}$, the expansion scalar $\theta := \nabla_{\mu}\xi^{\mu}$ will measure the convergence (resp. divergence) of neighboring geodesic integral curves of $\xi^{\mu}$ towards (resp. from) a reference geodesic from the same congruence. For example the congruence of geodesics in Scwarzschild space-time defined by test particles freely falling from rest at infinity has $\theta = -\frac{3}{2}(\frac{2M}{r^3})^{1/2}$ for ingoing geodesics as we would expect.

Tidal gravity is second order in $\nabla_{\mu}$ since it relates to relative accelerations between integral curves of $\xi^{\mu}$. Tidal gravity governs the proper time evolution of $\theta$ along the reference geodesic as well as the proper time evolution of the vorticity and shear of the congruence. Imagine for example a vector $\eta^{\mu}$ that extends from the reference geodesic to a neighboring geodesic in the congruence such that $\eta_{\mu}\xi^{\mu} = 0$ everywhere along the reference geodesic and such that $\eta^{\mu}$ remains locked to the neighboring geodesic (we can arrange for both of these by Lie transporting $\eta^{\mu}$ along $\xi^{\mu}$: $\mathcal{L}_{\xi}\eta^{\mu} = 0$). Then the acceleration of the neighboring geodesic relative to the reference geodesic is simply given by the equation of geodesic deviation: $a^{\mu} = \xi^{\gamma}\nabla_{\gamma}(\xi^{\nu}\nabla_{\nu}\eta^{\mu}) = -R_{\gamma \delta \nu}{}{}^{\mu}\eta^{\delta}\xi^{\gamma}\xi^{\nu}$.

Clearly this tidal gravity (relative acceleration) will manifest itself in non-trivial vacuum space-times such as Schwarzschild space-time and it will fully determine the Riemann curvature, which is a purely intrinsic measure of curvature. GR does not deal with extrinsic curvature for obvious reasons.

Free fall motion is characterized by the geometric statement that $\xi^{\nu}\nabla_{\nu}\xi^{\mu} = 0$; this is an invariant statement about the vanishing path curvature of geodesics. You're confusing this with the shape of the resulting integral curves of $\xi^{\mu}$ as represented in different coordinate systems. This has nothing to do with extrinsic curvature it is simply a result of using coordinate systems corresponding to different physical observers: the integral curves of $\xi^{\mu}$ are straight lines in locally inertial frames but, for example, will appear curved in the proper reference frame of a static observer in Schwarzschild space-time.

EDIT: I would recommend reading section 13.6 of MTW and working through exercise 13.14.

Last edited: Jan 16, 2014
10. Jan 16, 2014

### Naty1

A complemenatry description...In another discussion I had posted a view from Roger Penrose,

PeterDonis posted there:
This also appeared in that discussion but I did not record the source:

11. Jan 17, 2014

### Jonathan Scott

I have no major disagreement with the technical details in this thread, but the original post seemed to be at a very basic level and I thought it could be useful to illustrate the ideas informally at a lower level.

I hoped it would be clear that I was of course using the "ball" curvature analogy (intrinsic curvature in 3D) to relate to Ricci, and I was using the "cone" analogy for linear curvature in the sense of the shape of a ruler or light beam or the path of a falling object mapped relative to time using a typical coordinate system, as this is the common usage outside GR.

Of course in a real gravitational situation, there is tidal curvature everywhere, but I consider that a less important and higher order effect.

12. Jan 17, 2014

### Staff: Mentor

I'm not sure I understand this. "Tidal curvature" is the only kind of spacetime curvature there is. It's only "higher order" in the sense that it depends on second derivatives of the metric; but the first derivatives of the metric (the connection coefficients) aren't properly described by the term "curvature", and they don't form a tensor anyway.

13. Jan 17, 2014

### Naty1

Back to some real basics for rushikesh if he is still reading:

from his original post

yes. Gravitational spacetime curvature is also caused by other forms of energy. You can see a visualization of different sources of gravity here:

http://en.wikipedia.org/wiki/Stress-energy_tensor

In GR spacetime does curve, but there are multiple measures, multiple components, of such curvature. Tidal gravity is a physical way to think about 'curvature' in GR.

Another way to think about it was explained to me some years ago in these forums: SR is modeled on flat graph paper; You might have 'curves' on that graph paper illustrating space and time but those are simply coordinate type curves: expressing a relationship in curved coordinates [like spherical coordinates] just changes the picture, not the physics. When the graph paper itself is curved [in such a way it cannot be flattened without tearing] THAT is instrinsic spacetime gravitational curvature.

Dynamic illustration of curvature:

It's spacetime that is curving not just space. But nobody really agrees on what spacetime 'is' anymore than they can explain what an electron 'is'. The best we do is to predict actions and confirm those predictions in experiments [make observations].

Here are a few attempts to describe aspects of space and time which I like:

John Archibald Wheeler,

rushikesh:
Even a 'vacuum' is not necessarily 'empty', depending on what you mean. Spacetime has vacuum energy, a cornerstone of quantum mechanics. And in GR there is an ever present sort of 'negative' pressure', very tiny, but very important, also called the cosmological constant and called by some 'dark energy'.

Curving of space time is the geometrical interpretation originated by Einstein. Other physical physical forces are modeled via field theory. Einstein's view of gravity is unique and distinct. It works everywhere we can so far imagine except at the big bang and the center of black holes.
If there is a better 'analogy' it may come from our next theory....quantum gravity.

14. Jan 17, 2014

Please note that although general relativity talks about curvature of space-time , but actually there is another representation of geometry of space-time called Weitzenbock geometry , which the curvature of space-time is exactly zero, but information of gravity is hidden in a non-symmetric connection. This interpretation of gravity is also called Teleparallel gravity , the space time doesn't have curvature but instead the torsion(anti symmetric part of connection) is non-zero. So we encounter a convention, which do you like? the gravity as curvature of space-time or the torsion of it?

15. Jan 28, 2014

### nalA

The youtube video posted above was great for visualizing why the apple falls from the tree's point of view - thanks! In fact all those links to visualizations were really helpful.

Got some questions ...

When spacetime is 'bent' by mass is it being bent through a higher dimension than the 3+1 of spacetime itself? Or are the 3+1 dimensions all you need in this interpretation of gravity?

On the Einstein vs Newton video it looks like the time axis was being bent into the space axis from the external point of view (and vice versa): is that true? Is that what's happening?

Can I think of time like a spacial dimension at right angles (in a direction I can't perceive or point in) to the 3 of space? Or is that wrong?

If responding to me please use simple words and pictures as much as is possible ;-) More links to brilliant visual descriptions like those one above would be perfect!

16. Jan 28, 2014

### Mordred

17. Jan 28, 2014

### the_pulp

The intuitive examples of curvature made here are related to curvature of space, but I didnt notice anyone related to the curvature of spacetime. I mean, the idea of curvature of space is mathematically given by connections $\Gamma$ijk that mean something like when you go around the jk plane how an "i" arrow changes (Leonard Susskind explains it perfectly, I have to check again his course). My doubt is, what does the connection intuitively mean when i, j or k is the time direction?

18. Jan 28, 2014

### WannabeNewton

19. Jan 28, 2014

### FactChecker

Remember that Post #2 explained that if the angles of a triangle do not add up to 180, then you are in curved space. There is no need, or reason to believe, that there are more dimensions that the space is "curved in". The diagrams of triangles in Post #5 are shown on a surface a higher dimensional space, but that can be misleading. If you make a triangle with "straight" lines and the angles do not add up to 180, then you know that "straight" lines in your world are really curved. Imagine a bug on the surfaces on Post #5. He thinks those grid lines are all straight lines and he knows nothing about higher dimensions. But a smart bug would still know he is on a curved surface.

20. Jan 29, 2014

### nalA

I feel like I'm still missing something. I think you're right - perhaps it's the diagrams that are confusing me because they all seem to be showing spacetime (represented by a 2 dimensional surface) being curved through a higher dimension to produce the geometrical effects mentioned above ... or are those diagrams representing 2 spacial dimensions and the 'higher' dimension that they're curving through is time? Is that it?

If not, what other ways are there to explain why, when there are large masses about, the angles of triangles don't add up to 180 degrees and parallel lines meet other than with a higher dimension that we can't perceive to curve spacetime through?


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# Overlaying a point cloud on a two-dimensional probability distribution such that the local point density corresponds to the local probability density

Say I sample a set of points $(p_1,p_2,...) \in P$ from a probability distribution $f(x,y)$, e.g. a bivariate normal distribution, such that my sampling process chooses a point in the distribution with a probability proportional to the value $f(x,y)$ at that point. In particular, let's say we use the Metropolitan-Hastings algorithm to sample the set of points $P$ via a biased random walk.

Now, I take $P$, and I apply some arbitrary geometric rotational and translational geometric transform that acts uniformly on all points $p_i \in P$. I provide this transformed set of points $P^*$ to you, and your job is to reverse the geometric transform and map the coordinates back to their original positions to the best of your ability.

What is the optimal way to proceed?

A thought is that we can sample a new set of points $P_2$ from the distribution $f(x,y)$, and then use something like the RANSAC algorithm (http://en.wikipedia.org/wiki/RANSAC) to find a geometric transform to align the $P^*$ and $P_2$. This, however, seems to be a bit wasteful?

Motivation - Say we know that the maxima of the distribution $f(x,y)$ occurs at some $(x_k,y_k)$. What is $(x_k,y_k)^*$, i.e. where is this point after applying the geometric transform that mapped $P$ to $P^*$? We could also ask about other "distinct" points of interest in $f(x,y)$ so that it isn't simply a matter of computing something like a geometric mean (or perhaps geometric median) of the point set $P^*$ to calculate $(x_k,y_k)^*$. Can we figure this out without the kind of silly solution I propose the employs the RANSAC algorithm?

(In response to R. B.'s comments): I mean "up to symmetry" here - apologies for the confusion.

• Suppose $f(x,y)$ is indeed the bivariate normal distribution. Now I take $P$ and rotate it. The result still is a bivariate normal distribution. How do you expect to find the original points? I don't see anything that could help you figure the rotation, even if $f$ is known to you. – R B Mar 13 '14 at 8:34
• @RB I don't need to recover the exact original transformation, just one that properly overlays the set of points $P^*$ on the original distribution in a manner that best approximates sampling $||P||$ points from this distribution. Also, the bivariate normal distribution has a lot of symmetry, but I'd like this to work for less symmetric probability distributions as well. – O1155 Mar 13 '14 at 11:32
• @RB I'm wondering if this can be done in a smarter manner than simply resampling from $f(x,y)$ and then aligning these resampled points with $P^*$ using something like RANSAC? – O1155 Mar 13 '14 at 11:35
• I think @RB's point is that the two distributions (the original and the rotated one) are identical. There's no way to tell them apart. You have to add some proviso to your question of the form "upto symmetries". So the bivariate normal is a particularly bad case. And in fact any radially symmetric distribution would be impossible to recover except "upto symmetries" – Suresh Venkat Mar 13 '14 at 17:15

This seems to depend heavily on the form of $f$. If $f$ has a small number of distinct features (say, a handful of well-separated sharp peaks), then you can attempt to identify those features in the point cloud and then use SVD to find the Euclidean transformation that best takes the observed features back to their original locations. If $f$ is spread out and has no obvious landmarks (e.g., $f(x,y)\propto e^{-\beta(x^2+y^2)}\left(1+\varepsilon\sin^2(x+y)\right)$, where $\beta$ and $\varepsilon$ are small), then there will be no way to find the maximum-likelihood transformation without trying a large number of possibilities. In general it becomes an optimization problem that could have many, many local optima.

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# Series Transistor Regulator

Discussion in 'General Electronics Chat' started by Roam, May 5, 2011.

1. ### Roam Thread Starter New Member

May 5, 2011
3
0
I'm new to these forums so just saying hi before I make my post. The following is the diagram of a series transistor regulator. Could anyone explain, what factors here would limit the maximum current that could be supplied by this transistor-regulated supply? Any explanation is greatly appreciated.

$R_L$ is the resistive load and $V_o$ is the output.

2. ### R!f@@ AAC Fanatic!

Apr 2, 2009
8,943
803
Welcome dear.

MAX current is always limited by the source.
In other words, the tx supplies the current demand and the series element regulates the current.The smoothing cap comes into play when the rectified AC dips..which is known as ripple.

Last edited: May 5, 2011
Roam likes this.

Dec 26, 2010
2,147
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The transistor will also have limitations due to its own current rating, and the amount of heat it can safely dissipate. If either of these results in a lower limit than other factors, it may define the limit overall.

There is also a more basic limit in this simple circuit. Rz, 100Ω, supplies the Zener current and thr transistor base current. If we know the rectifier output voltage, the Zener current and voltage, and the transistor current gain we can estimate a maximum output current.

Example: Rectifier output =10V, Zener voltage = 6.8V Zener current =10mA, transistor current gain = 50 times.

Current in Rz = (10V - 6.8V)/100Ω = 32mA. If the Zener needs 10mA, the current available for the transistor base is 32mA - 10mA = 22mA.

Thus the maximum output current in this made-up example would be 22mA * 50 = 1100mA or 1.1A

Roam likes this.
4. ### THE_RB AAC Fanatic!

Feb 11, 2008
5,435
1,306
About the only useful information in your schematic is that the main filter cap is 1000uF so that would *suggest* maximum amps at about 1 amp, possibly 0.5 amp.

Roam likes this.
5. ### BillO Distinguished Member

Nov 24, 2008
985
136
There is no current limiting designed into this circuit. Shorting the output will likely take the transformer, the pass transistor, the bridge or any combination thereof. If you were to build this design around a specific current capability, Id suggest a fuses on both input and output.

Edit: That being said...the transformer, the bridge and the pass transistor all have to meet the current requirements with a little head room to keep everything cool and happy. For instance, it you required a 12v output @ 3A I would specify a 12.7v 4A transformer, a 12.7V 1W zener, Rz at 1/2W, a 6A bridge, a 6,800uf 25v Cs, and a TIP35 pass transistor with a decent heat sink enough to dissipate 10W or more. I'd also put a 4A fast blow fuse on the output and a 1/2A sow blow on the transformer primary.

Yes, I over design a bit for my own purposes. I hate stuff failing. If this is for a commercial product, you might want to tighten up on the capacitor, rectifier and transistor, but only if it saves considerable money.

Last edited: May 5, 2011
Roam likes this.


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# EventBus in Android lessons learned

[ ]

Event bus can be very helpful when you need to pass events cross multiple objects back and forth. In such cases, the listeners may get messy.

For example, you have object A and B, which need to communicate with each other, then you need to have listeners from A to B and also listeners from B to A. However, image you have more than two objects, you will need o(n^2) listenrs. If you don’t remove all listeners, you will likely to run to memory leak.

With event bus, the life is much easier, where each object only need to register to the event bus to recevice events and post events to event bus:

Pros

• You can effectively decouple your Fragments and Activities and get rid of really complex listener-based architectures.
• The ability to change the receiver thread could be quite useful, specially on Android.
• Is easy to pass small objects in the events for quick UI updates.

Cons

• Is really easy to abuse it and make your code unreadable and untestable.
• You need to fill your code with as many onEvent methods as receivers you want, making it less easy to understand for new people.
• You loss control over who’s notifying who unless you use external plugins.

Go to EventBus in Android: lessons learned to read more.

Written on August 18, 2017


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LinearSolver::Schwarz::Schwarz< FieldsTag, OptionsGroup, SubdomainOperator, SourceTag > Struct Template Reference

An additive Schwarz subdomain solver for linear systems of equations $$Ax=b$$. More...

#include <Schwarz.hpp>

## Public Types

using operand_tag = FieldsTag

using fields_tag = FieldsTag

using source_tag = SourceTag

using options_group = OptionsGroup

using component_list = tmpl::list<>

using observed_reduction_data_tags = observers::make_reduction_data_tags< tmpl::list< async_solvers::reduction_data, detail::reduction_data > >

using initialize_element = tmpl::list< async_solvers::InitializeElement< FieldsTag, OptionsGroup, SourceTag >, detail::InitializeElement< FieldsTag, OptionsGroup, SubdomainOperator > >

using register_element = tmpl::list< async_solvers::RegisterElement< FieldsTag, OptionsGroup, SourceTag >, detail::RegisterElement< FieldsTag, OptionsGroup, SourceTag > >

template<typename ApplyOperatorActions , typename Label = OptionsGroup>
using solve = tmpl::list< async_solvers::PrepareSolve< FieldsTag, OptionsGroup, SourceTag, Label >, detail::SendOverlapData< FieldsTag, OptionsGroup, SubdomainOperator >, detail::ReceiveOverlapData< FieldsTag, OptionsGroup, SubdomainOperator >, detail::SolveSubdomain< FieldsTag, OptionsGroup, SubdomainOperator >, detail::ReceiveOverlapSolution< FieldsTag, OptionsGroup, SubdomainOperator >, ApplyOperatorActions, async_solvers::CompleteStep< FieldsTag, OptionsGroup, SourceTag, Label > >

## Detailed Description

### template<typename FieldsTag, typename OptionsGroup, typename SubdomainOperator, typename SourceTag = db::add_tag_prefix<::Tags::FixedSource, FieldsTag>> struct LinearSolver::Schwarz::Schwarz< FieldsTag, OptionsGroup, SubdomainOperator, SourceTag >

An additive Schwarz subdomain solver for linear systems of equations $$Ax=b$$.

This Schwarz-type linear solver works by solving many sub-problems in parallel and combining their solutions as a weighted sum to converge towards the global solution. Each sub-problem is the restriction of the global problem to an element-centered subdomain, which consists of a central element and an overlap region with its neighbors. The decomposition into independent sub-problems makes this linear solver very parallelizable, avoiding global synchronization points altogether. It is commonly used as a preconditioner to Krylov-type solvers such as LinearSolver::gmres::Gmres or LinearSolver::cg::ConjugateGradient, or as the "smoother" for a Multigrid solver (which may in turn precondition a Krylov-type solver).

This linear solver relies on an implementation of the global linear operator $$A(x)$$ and its restriction to a subdomain $$A_{ss}(x)$$. Each step of the algorithm expects that $$A(x)$$ is computed and stored in the DataBox as db::add_tag_prefix<LinearSolver::Tags::OperatorAppliedTo, operand_tag>. To perform a solve, add the solve action list to an array parallel component. Pass the actions that compute $$A(x)$$, as well as any further actions you wish to run in each step of the algorithm, as the first template parameter to solve. If you add the solve action list multiple times, use the second template parameter to label each solve with a different type. Pass the subdomain operator as the SubdomainOperator template parameter to this class (not to the solve action list template). See the paragraph below for information on implementing a subdomain operator.

Subdomain geometry:
The image below gives an overview of the structure of an element-centered subdomain:
Fig. 1: An element-centered subdomain

Fig. 1 shows part of a 2-dimensional computational domain. The domain is composed of elements (light gray) that each carry a Legendre-Gauss-Lobatto mesh of grid points (black dots). The diagonally shaded region to the left illustrates an external domain boundary. The lines between neighboring element faces illustrate mortar meshes, which are relevant when implementing a subdomain operator in a DG context but play no role in the Schwarz algorithm. The dashed line gives an example of an element-centered subdomain with a maximum of 2 grid points of overlap into neighboring elements. For details on how the number of overlap points is chosen see LinearSolver::Schwarz::overlap_extent. Note that the subdomain does not extend into corner- or edge-neighbors, which is different to both [63] and [69]. The reason we don't include such diagonal couplings is that in a DG context information only propagates across faces, as noted in [63]. By eliminating the corner- and edge-neighbors we significantly reduce the complexity of the subdomain geometry and potentially also the communication costs. However, the advantages and disadvantages of this choice have yet to be carefully evaluated.

Subdomain operator:
The Schwarz subdomain solver relies on a restriction of the global linear problem $$Ax=b$$ to the subdomains. The subdomain operator $$A_{ss}=R_s A R_s^T$$, where $$R_s$$ is the restriction operator, essentially assumes that its operand is zero everywhere but within the subdomain (see Section 3.1 in [63]). Therefore it can be evaluated entirely on an element-centered subdomain with no need to communicate with neighbors within each subdomain-operator application, as opposed to the global linear operator that typically requires nearest-neighbor communications. See LinearSolver::Schwarz::protocols::SubdomainOperator for details on how to implement a subdomain operator for your problem.
Algorithm overview:
In each iteration, the Schwarz algorithm computes the residual $$r=b-Ax$$, restricts it to all subdomains as $$r_s=R_s r$$ and communicates data on overlap regions with neighboring elements. Once an element has received all data on its subdomain it solves the sub-problem $$A_{ss}\delta x_s=r_s$$ for the correction $$\delta x_s$$, where $$\delta x_s$$ and $$r_s$$ have data only on the points of the subdomain and $$A_{ss}$$ is the subdomain operator. Since all elements perform such a subdomain-solve, we end up with a subdomain solution $$\delta x_s$$ on every subdomain, and the solutions overlap. Therefore, the algorithm communicates subdomain solutions on overlap regions with neighboring elements and adds them to the solution field $$x$$ as a weighted sum.
Applying the global linear operator:
In order to compute the residual $$r=b-Ax$$ that will be restricted to the subdomains to serve as source for the subdomain solves, we must apply the global linear operator $$A$$ to the solution field $$x$$ once per Schwarz iteration. The global linear operator typically introduces nearest-neighbor couplings between elements but no global synchronization point (as opposed to Krylov-type solvers such as LinearSolver::gmres::Gmres that require global inner products in each iteration). The algorithm assumes that the action list passed to solve applies the global linear operator, as described above.
Subdomain solves:
Each subdomain solve is local to an element, since the data on the subdomain has been made available through communication with the neighboring elements. We can now use any means to solve the subdomain problem that's appropriate for the subdomain operator $$A_{ss}$$. For example, if the subdomain operator was available as an explicit matrix we could invert it directly. Since it is typically a matrix-free operator a common approach to solve the subdomain problem is by means of an iterative Krylov-type method, such as GMRES or Conjugate Gradient, ideally with an appropriate preconditioner (yes, this would be preconditioned Krylov-methods solving the subdomains of the Schwarz solver, which might in turn precondition a global Krylov-solver - it's preconditioners all the way down). Currently, we use LinearSolver::Serial::Gmres to solve subdomains. It supports preconditioning, and adding useful subdomain preconditioners will be the subject of future work. Note that the choice of subdomain solver (and, by extension, the choice of subdomain preconditioner) affects only the performance of the Schwarz solver, not its convergence or parallelization properties (assuming the subdomain solutions it produces are sufficiently precise).
Weighting:
Once the subdomain solutions $$\delta x_s$$ have been found they must be combined where they have multiple values, i.e. on overlap regions of the subdomains. We use an additive approach where we combine the subdomain solutions as a weighted sum, which has the advantage over "multiplicative" Schwarz methods that the subdomains decouple and can be solved in parallel. See LinearSolver::Schwarz::element_weight and Section 3.1 of [63] for details. Note that we must account for the missing corner- and edge-neighbors when constructing the weights. See LinearSolver::Schwarz::intruding_weight for a discussion.
Possible improvements:
• Specify the number of overlap points as a fraction of the element width instead of a fixed number. This was shown in [62] to achieve scale-independent convergence at the cost of increasing subdomain sizes.
• Subdomain preconditioning (see paragraph on subdomain solves above)

The documentation for this struct was generated from the following file:
• src/ParallelAlgorithms/LinearSolver/Schwarz/Schwarz.hpp


text:
# Thread: Symmetric domain means the function is a sum of even and odd?

1. ## Symmetric domain means the function is a sum of even and odd?

Hello! I have an exercise that's bugging the out of me:

It's from Robert Adam's Calculus, sixth edition. Chapter P5, exercise 35. It goes like this:

"Let f be a function whose domain is symmetric about the origin, that is, -x belongs to the domain whenever x does. Show that f is the sum of an even function and an odd function

f(x) = E(x) + O(x)

where E is an even function and O is an odd function"

Then, in the second part of the exercise, we have "Show that there is only one way to write f as the sum of an even and an odd function".

Now, I don't have any issues with the calculation of the proof. Let E(x) = (f(x) + f(-x))/2 and go from there. It's the conclusion as such that I just can't get to fit in my head! I mean, take for example f(x) = x^2. By convention it's domain is the entire real line, so in other word it has a domain that's symmetrical about the origin, right? But just how is this supposed to work?

f(x) = E(x) + O(x)

I could take

f(x) = x^2 + 0x^3,

or I could go with

f(x) = x^2 + 0x^5.

Nothing unique about it. Really, what is E(x) and O(x) supposed to be in any given specific example of f?

So the conclusions of this exercise just confuses me. Could anyone maybe give me a hand in understanding how a symmetrical domain automatically gives that the function has to be a sum of an even and an odd dito? And thus help me untangle where I go wrong in my thinking with the example of f(x) = x^2?

2. ## Re: Symmetric domain means the function is a sum of even and odd?

Your $O(x)$ is identically zero in both cases - as you can tell from your proof where $O(x) = \frac{f(x)-f(-x)}2$.

3. ## Re: Symmetric domain means the function is a sum of even and odd?

Fair enough, but to be honest I'm still not sure I buy that. I mean, is the zero function, g(x) = 0, actually an odd function? My instinct would be to consider it even. I mean, 0 is an even number, and any constant function could be written as h(x) = Cx^0, right?

4. ## Re: Symmetric domain means the function is a sum of even and odd?

the function $f(x)=0$ with the given domain $D$ is both even and odd

it can be shown that it is the only such function

and that's what you need to show uniqueness:

If $f(x)=g(x)+h(x)=k(x)+m(x)$ where $g,k$ are even and $h,m$ are odd then

$g=k$ and $h=m$


text:
THE RADIAN MEASURE OF AN ANGLE The following discussion includes some paper and pencil exercises which will help you to understand the radian measure of an angle. You will need a compass, a protractor, and a flexible ruler (one that can bend to allow

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Illustrated definition of Radian: The angle made by taking the radius and wrapping it round the circle. One Radian is (180pi) degrees, which The angle made by taking the radius and wrapping it round the circle. One Radian is (180/ π) degrees, which is

Heat radiation (as opposed to particle radiation) is the transfer of internal energy in the form of electromagnetic waves. For most bodies on the Earth, this radiation lies in the infrared region of the electromagnetic spectrum.

20/9/2017 · So a radian is about 360 /(2 * pi) or 57.3 degrees. Now don’t be like me, memorizing this thinking “Great, another unit. 57.3 degrees is so weird.” Because it is weird when you’re still thinking about you! Moving 1 radian (unit) is a perfectly normal distance to travel.

In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Spectral radiance is the radiance of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. These are directional

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Online calculator to convert degrees to radians (deg to rad) with formulas, examples, and tables. Our conversions provide a quick and easy way to convert between Angle units. The following is a list of definitions relating to conversions between degrees and radians.

14/2/2015 · How do you convert radians to meters?? Is it even possible? That would be converting an angular displacement to a linear displacement. You’d need more data to find a length given an angle in radians.

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Well, the most obvious one, if you kind of view a radian as another way of saying radiuses, or I guess radii. Well, you say, look, this is subtended by an arc of one radius. So why don’t we call this right over here one radian, which is exactly how a radian is

$\begingroup$ You would have to put a conversion coefficient with a value of one radian in front of $\vec{r}\times\vec{F}$. It would be possible to reformulate all the equations of physics in this way to explicitly include radians, but it would make things messier than

A radian is a unit of measuring angles. It is shown by the symbol “rad” or, less often, c (for circular measure). The radian was once an SI supplementary unit, but was done away with in 1995. A “milliradian” is 1/1000 of a radian.

And if you wanna really think about, well how many degrees are there per radian you can divide both sides of this by (pi). So if you divide both sides of this by (pi), you would get one radian it would have to go from plural to singular one radian is equal to 180/pi

25/11/2012 · A jet plane flies overhead at Mach 1.75 and at a constant altitude of 953 m. A. What is the angle in radian of the shock-wave cone? B. How much time after the plane passes directly overhead do you hear the sonic boom? Neglect the variation of the speed of sound

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›› Definition: Radian In mathematics and physics, the radian is a unit of angle measure. It is the SI derived unit of angle. It is defined as the angle subtended at the center of a circle by an arc of circumference equal in length to the radius of the circle. Angle

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18/2/2017 · [徵求] Radian Mastering NSS Physics 3 and 4 (Exercise + Solution) yuentung_lau 提示: 作者被禁止或刪除 內容自動屏蔽 發表於 2013-6-11 01:39 PM 1 # yuentung_lau 禁止訪問 帖子 260 積分 340 金幣 0 註冊時間 2010-6-20 1 # 大 中 小 發表於 2013-6-11 01

1 degree = 60 minute 1 minute of an arc = 1/60 degree 180 degree = pi radian 1 degree = pi/180 radian therefore 1/60 of pi/180 = 1minute 0r 1 minute = pi /( 60 .180) radian

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27/12/2017 · All in all, I didn’t find anything pertaining to my particular comment. Not in the article, nor in the comments on it. The article does say that angular frequency is different from frequency, although I do consider that ambiguous. For starters, angular velocity seems to be a better term than angular frequency, but that’s not something we can change.

Revolution/Second : The revolution per second is a metric unit of angular velocity (rotational speed). It also is a unit of angular frequency. Its symbols are r•s⁻¹ and r/s. It is equal to 6.28318530718 radian/second. Radian/Second : The radian per second is the SI unit of angular velocity (rotational speed), commonly denoted by the greek letter ω (omega).

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Radian vs Degree Degrees and radians are units of angular measurement. Both are commonly used in practice, in fields such as mathematics, physics, engineering, and many other applied sciences. Degree has a history running back to ancient Babylonian

The radian is useful for physics widely when the requirement of the angular measurement arises. For example, angular velocity is typically measurable in radians per second. (Rad/s). One revolution per second gets equals to 2 pie radians per second. Ques.

Sal converts the radian measures π and -π/3 to degrees. We’re asked to convert pi radians and negative pi/3 radians to degrees. And the first question I’ll ask you: If you do one revolution, You have an angle that went all the way around once.

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Radian to Degree Measure The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. In radians, one complete counterclockwise revolution is 2 π and in degrees, one complete counterclockwise revolution

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Physics Dept) o o 1 0.017 radian 180 180 1 radian 57.3 sr 180 radianso NAME: .. CIRCULAR MOTION AQA [Inc. additional What is if s = arc length and r = radius (same UNIT as r) then (radians) is DEFINED as = s r i.e. so given an

One radian is the angle traced by an arc of length equal to radius of circle. Length of circular track of radius r is 2p r.Therefore numbers of radians in a circle of radius r will be =2p r / r = 2p Angle at the centre of circle in one complete rotation = 360 o = 2p rad

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Metric (SI) multipliers Prefix Abbreviation Value peta P 10 15 tera T 10 12 giga G 10 9 mega M 10 6 kilo k 10 3 hecto h 10 2 deca da 10 1 deci d 10 –1 centi c 10 –2 milli m 10 –3 micro P 10 –6 nano n 10 –9 pico p 10 –12 femto f 10 –15 Unit conversions 1 radian

Radian – DSE Physics 分享了 1 條連結。2016年4月27日 · 在這段視頻中，我們會探討水的沸騰。這段視頻的內容並非完全屬於DSE的課程要求之內，校內試亦不會考核。然而，對於具有好奇心的同學來說，這段視頻是值得一看的。特別要提的是，這段視頻會

7/4/2005 · Introductory Physics Homework Help Converting rev to rad/s Thread starter hotmail590 Start date Apr 7, 2005 Apr 7, 2005 #1 hotmail590 52 0 Hello I have a question about converting rev to rad/s. If i had 1.13 revolutions for every 2.08s, how do I convert that into

Radian measure is just different unit of measure. Just as we can measure a football field in yards or feet — we can measure a circle in degrees (like the good old days) or in radians (welcome to the big leagues!) Think about what the word radian sounds like

You can convert RPM (revolutions per minute) to rad/s (radians per second) or rad/s to RPM using a two-stage process. First, you convert between seconds and minutes, by either dividing or multiplying by 60. Second, you convert the number of revolutions to radians

Degrees are warm, friendly, familiar. Why ditch them in favor of this bizarre radian? Here’s the best answer I can give you: Degrees are fine for everyday measurements. But Trigonometry marks a turning point in math, when the student lifts his gaze from the

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7/6/2010 · Can anyone explain when i should use which mode? I’m taking a physics class and i keep switching back and forth to get the right answer but I really don’t understand why and when? When you’re graphing a trigonometric function, often you want to use radian

This calculation library is so good! I will recommend this site to my friends. I suggest that whenever I give an angle in such form, for instance radians, you must also give the answer in radian form, or in terms of pi. I hope that this site will grow so that more people

Radian definition, the measure of a central angle subtending an arc equal in length to the radius: equal to 57.2958 . Abbreviation: rad See more. Commas mark divisions in sentences. Periods end declarative sentences. Apostrophes show possession. Easy, right?

MINUTE TO RADIAN (‘ TO rad) FORMULA To convert between Minute and Radian you have to do the following: First divide (Math.PI/(180*60)) / 1 = 0.00029089 Then multiply the amount of Minute you want to convert to Radian, use the chart below to guide you.

Cos 5π/2 radian mode because there is no degree symbol Sec 3π radian mode because there is no degree symbol I should mention that for those first 4 problems, I think the point is actually not to use a calculator. The second part says ¨Find the function values.

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steradian: The steradian (symbolized sr) is the Standard International ( SI ) unit of solid angular measure. There are 4 pi , or approximately 12.5664, steradians in a complete sphere.


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## anonymous 3 years ago How many solutions does the system have? 3x + 12y = 20 y = –1/4 x + 5/3

1. anonymous

Do you know how to solve systems such as this?

2. anonymous

Hint: multiply$y=\frac{-x}{4}+\frac{5}{3}$by 12.

3. TheViper

If $\Large{\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}}$ Then it has INFINITELY MANY solutions. $\Large{\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2}}$ Then it has NO solutions. $\Large{\dfrac{a_1}{a_2}\neq\dfrac{b_1}{b_2}}$ Then it has only ONE solution.

4. TheViper

Here, $\Large{\color{red}{a_1=3, a_2=\dfrac{-1}{4}, b_1=12, b_2=1, c_1=20, c_2=\dfrac{5}{3}.}}$

5. anonymous

oh alright i think i see


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Cosmological redshift

# Introduction

SKIRT allows explicitly configuring and simulating astrophysical objects at non-zero redshift. In addition to actually shifting the wavelengths in the recorded synthetic observations, this enables SKIRT to perform the appropriate relativistic flux and surface brightness calibrations and to simulate the effect of the cosmic microwave background on the dust temperature.

This concept note summarizes the related configuration options and the corresponding physical mechanisms and numerical recipes. The discussion is organized in sections as follows:

# Redshift and cosmology

SKIRT assumes that the complete simulated model (or, more precisely, the model coordinate frame) is at a single, given redshift $$z$$ within the range $$0\le z \le 15$$, i.e. including the epoch of reionization. The MonteCarloSimulation class offers the cosmology property for specifying the cosmology related parameters of the input model in addition to the redshift. The cosmology property simply points to an instance of a Cosmology subclass. The current implementation provides the following two Cosmology subclasses:

Class name Description Properties
LocalUniverseCosmology The model is in the Local Universe at redshift zero None
FlatUniverseCosmology The model is at some nonzero redshift in a flat universe $$z,h,\Omega_\mathrm{m}$$

LocalUniverseCosmology is the default "cosmology", reproducing the "Local Universe" behavior of SKIRT as if redshift were not implemented. On the other hand, FlatUniverseCosmology specifies a standard spatially-flat $$\Lambda\mathrm{CDM}$$ cosmology model. In addition to the redshift, this cosmology has two parameters, namely the reduced Hubble constant $$h=H_0/(100 \,\mathrm{km} \,\mathrm{s}^{-1} \,\mathrm{Mpc}^{-1})$$ and the matter density fraction $$\Omega_\mathrm{m}$$. These are summarized in the following table. The default values are compatible with the latest observations (Planck 2018) and close to the values used by many recent cosmological simulations. If $$z=0$$, the behavior is the same as that of a LocalUniverseCosmology.

Property Description Default
redshift Redshift $$0\le z \le 15$$ of the model coordinate frame $$z=1$$
reducedHubbleConstant Reduced Hubble constant $$h$$ $$h=0.675$$
matterDensityFraction Cosmological matter density fraction $$\Omega_\mathrm{m}$$ $$\Omega_\mathrm{m}=0.310$$

# Instruments and calibration

## Distant instruments

Instruments using parallel projection are implemented by DistantInstrument subclasses and include instances of SEDInstrument, FrameInstrument, and FullInstrument. If the model redshift is non-zero, a distant instrument can be placed in either the model rest-frame (at a given distance) or the observer frame (honoring the redshift). To allow this selection, the DistantInstrument class allows its distance property to have a zero value. Specifically,

• If distance is non-zero, the instrument is placed in the rest frame of the model at the given (non-relativistic) distance. Wavelengths are not shifted and fluxes are calibrated as usual.
• If distance is zero, the instrument is placed in an observer frame corresponding to the redshift specified for the model. The wavelength of each detected photon packet is shifted accordingly before the packet's contribution is recorded in the instrument, and flux/surface brightness calibration takes into account the relevant relativistic effects including the luminosity distance.
• If distance is zero and the model redshift is zero as well, a fatal error is issued during setup.

This approach allows the on-the-fly convolution for a broadband-based instrument to occur in either the rest frame or the observer frame depending on the instrument's settings. A configuration might even include both type of instruments at the same time.

## Local instruments

The other SKIRT instruments include instances of AllSkyInstrument and PerspectiveInstrument. These instruments are "local" in the sense that they are commonly positioned inside or very near the input model. As a consequence, these instruments are always considered to be in the rest frame of the input model.

## Calibration

We consider the calibration for a distant instrument assuming a standard spatially-flat $$\Lambda\mathrm{CDM}$$ cosmology. This means that the line-of-sight and transverse comoving distances are identical. We further ignore the radiation density, which is justified for the allowed redshift range $$z\le15$$. With these assumptions, the comoving distance $$d_\mathrm{M}(z)$$ corresponding to redshift $$z$$ can be obtained from the matter density fraction $$\Omega_\mathrm{m}$$ and the Hubble constant $$H_0=h \times 100 \,\mathrm{km} \,\mathrm{s}^{-1} \,\mathrm{Mpc}^{-1}$$ using

$d_\mathrm{M}(z) = \frac{c}{H_0} \int_0^z \frac{\mathrm{d}z'}{\sqrt{\Omega_\mathrm{m}(1+z')^3 + (1-\Omega_\mathrm{m})}}$

where $$c$$ is the speed of light. The integral can easily be evaluated numerically. Because this evaluation happens just once during setup, speed is not important and there is no need to use more complicated methods.

The angular-diameter distance $$d_\mathrm{A}(z)$$ and the luminosity distance $$d_\mathrm{L}(z)$$ are then obtained from

$d_\mathrm{A}(z) = (1+z)^{-1} \, d_\mathrm{M}(z)$

and

$d_\mathrm{L}(z) = (1+z) \, d_\mathrm{M}(z).$

The angular-diameter distance converts a proper transverse separation $$\mathrm{d}l$$ to the corresponding observed angular separation $$\mathrm{d}\psi$$,

$\mathrm{d}\psi = \frac{\mathrm{d}l}{d_\mathrm{A}(z)}.$

This is used for surface brightness calibration and to properly calculate the angular pixel size, which is written to the FITS file header representing the instrument data cube.

The luminosity distance converts a total luminosity $$L_\mathrm{tot}$$ to the corresponding observed total flux $$F_\mathrm{tot}$$,

$F_\mathrm{tot} = \frac{L_\mathrm{tot}}{4\pi\,d_\mathrm{L}^2(z)}$

or a neutral-style monochromatic luminosity $$\lambda L_\lambda$$ to the corresponding observed neutral-style flux,

$(1+z)\lambda \, F_\lambda[(1+z)\lambda] = \frac{\lambda L_\lambda[\lambda]}{4\pi\,d_\mathrm{L}^2(z)}$

where $$\lambda$$ is the emitted wavelength and $$(1+z)\lambda$$ is the observed wavelength. This is used for both flux density and surface brightness calibration.

# Dust heating by the CMB

The temperature of the cosmic microwave background (CMB) increases with redshift. Therefore, at higher redshifts, the CMB radiation may contribute significantly to the heating of cold dust grains. To allow including this effect in SKIRT dust heating and emission calculations, the DustEmissionOptions class offers a Boolean property called includeHeatingByCMB. This property is available even for models at redshift zero, although in that case CMB heating will be insignificant except for very contrived simulation models.

If CMB dust heating is turned on in the configuration, rather than performing actual radiative transfer for the CMB radiation, the dust heating/emission calculations in the EquilibriumDustEmissionCalculator and StochasticDustEmissionCalculator classes include an additional source term corresponding to the CMB spectrum. The extra radiation field for a model at redshift $$z$$ is given by

$B_\lambda(\lambda,[1+z]\,T_\mathrm{CMB}^{z=0}) \quad \mathrm{with} \quad T_\mathrm{CMB}^{z=0} = 2.725\,\mathrm{K}$

where $$B_\lambda(\lambda,T)$$ is the Planck function. The dust emission calculators pre-calculate and add this spectrum to the radiation field $$J_\lambda(\lambda)$$ provided for each calculation.

Implementing CMB dust heating as a "forced" source term in the dust heating calculations has some implications worth considering. An important benefit is that the mechanism consumes a trivial amount of memory and processing time. On the other hand, it assumes that the opacity of the medium at CMB wavelengths is sufficiently low for the CMB to be homogeneous across the spatial domain. Taking into account the actual opacity of the medium would require performing a full radiative transfer simulation (i.e., shooting photon packets through the medium). And lastly, while the effects of the CMB radiation on the dust emission are included, the CMB radiation itself never reaches the instruments. In other words, the "observed" fluxes do not include the CMB background itself, just its effects on the dust emission spectrum.

The figure below illustrates the effect of enabling dust heating by the CMB for a toy model including a central point source in spherical dust cloud placed at redshift 5. The model specifies the Themis dust mix and employs stochastic dust heating. The average dust temperature is about 8 K when CMB heating is disabled, and about 16 K when it is enabled. The unrealistically low dust temperature of course magnifies the effect of the CMB heating.


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# Bernoulli distribution

#### Dear Wikiwand AI, let's keep it short by simply answering these key questions:

Can you list the top facts and stats about Bernoulli distribution?

In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,[1] is the discrete probability distribution of a random variable which takes the value 1 with probability ${\displaystyle p}$ and the value 0 with probability ${\displaystyle q=1-p}$. Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are boolean-valued: a single bit whose value is success/yes/true/one with probability p and failure/no/false/zero with probability q. It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails", respectively, and p would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and p would be the probability of tails). In particular, unfair coins would have ${\displaystyle p\neq 1/2.}$
Parameters Probability mass function Three examples of Bernoulli distribution: .mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}  ${\displaystyle P(x=0)=0{.}2}$ and ${\displaystyle P(x=1)=0{.}8}$   ${\displaystyle P(x=0)=0{.}8}$ and ${\displaystyle P(x=1)=0{.}2}$   ${\displaystyle P(x=0)=0{.}5}$ and ${\displaystyle P(x=1)=0{.}5}$ ${\displaystyle 0\leq p\leq 1}$ ${\displaystyle q=1-p}$ ${\displaystyle k\in \{0,1\}}$ ${\displaystyle {\begin{cases}q=1-p&{\text{if }}k=0\\p&{\text{if }}k=1\end{cases}}}$ ${\displaystyle {\begin{cases}0&{\text{if }}k<0\\1-p&{\text{if }}0\leq k<1\\1&{\text{if }}k\geq 1\end{cases}}}$ ${\displaystyle p}$ ${\displaystyle {\begin{cases}0&{\text{if }}p<1/2\\\left[0,1\right]&{\text{if }}p=1/2\\1&{\text{if }}p>1/2\end{cases}}}$ ${\displaystyle {\begin{cases}0&{\text{if }}p<1/2\\0,1&{\text{if }}p=1/2\\1&{\text{if }}p>1/2\end{cases}}}$ ${\displaystyle p(1-p)=pq}$ ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle {\frac {q-p}{\sqrt {pq}}}}$ ${\displaystyle {\frac {1-6pq}{pq}}}$ ${\displaystyle -q\ln q-p\ln p}$ ${\displaystyle q+pe^{t}}$ ${\displaystyle q+pe^{it}}$ ${\displaystyle q+pz}$ ${\displaystyle {\frac {1}{pq}}}$


text:
Article Text

One principle and a fourth fallacy of disability studies
Free
1. John Harris
1. Correspondence to:
 Professor John Harris, Institute of Medicine, Law and Bioethics, School of Law, University of Manchester, Manchester M13 9PL, UK;
 John.M.Harris{at}man.ac.uk

## Abstract

This brief paper shows that the idea of benefits to the subject compensating for the harms of disability is at best self defeating and at worst sinister. Equally benefits to third parties while real are dubious as compensating factors. This shows that disabilities are just that, a net loss and not a net gain.

• Disability
• impairment
• handicap
• equality
• social conception of disability
• harmed condition conception of disability

## Statistics from Altmetric.com

I feel privileged to have taken part in this symposium and to have had the benefit of this exchange with Tom Koch. However, his clarification raises more problems than it solves.1 Koch suggests that Steven Hawking's condition may have contributed positively to his work; that may well be true but what follows? The lessons Koch believes we may learn are clearer from two comments Koch makes. First he suggests, “harm' (resulting from a distinguishing condition) is balanced, and in some cases outweighed, by the resulting benefit'.” What does “balanced” mean here—does it mean that overall the disability is worth having for the good that it does? If it does not mean this what does it mean? If Koch's remarks are taken at face value it might be right to inflict such a condition—for example on children for the sake of the balancing good, rather in the way that I balance the pain and inconvenience of surgery against the resulting good both for myself and my children. This looks as though it means that the disability is somehow acceptable or justified by the balancing good. It seems close to saying that suffering is balanced by the character-improving qualities that it has. But if the benefit of disability can balance the harm then it can over-balance it, and it could be true that one legitimate way to achieve those goods might be to impose the suffering or the disability on children for the sake of the benefit. Or to make a weaker claim, to say that the disability or suffering doesn't matter because of the good that it brings. Of course someone with Motor Neurone Disease could, in the event of a cure being found, freely choose to remain as he is for the sake of the benefits to his work, but it is quite another thing to choose to impose disease on a child in the expectation both of compensating Hawking-like benefits and in the expectation that this new individual will make the same cost benefit analysis. If Koch is serious and if we care that our children have all the benefits we can provide then this might surely be not only right but mandatory.

If “harm” resulting from a distinguishing condition may be balanced by a good or benefit, that seems to imply something else, namely that the good or benefit could be adequate compensation for the condition. Suppose there is someone who, through fault of another, has been disabled. A court might decide, following Koch, that she had no claim for compensation because the condition had already been compensated for by the balancing benefit. If “balance” here doesn't mean that this is both possible and possibly legitimate, what does it mean?

Secondly, Koch takes a different tack, he talks about the benefits to family members (and presumably friends) of persons with these “life-enhancing conditions”. When Koch says: “in these cases the experience (of harm) is formative and in some cases positively transforming” I must confess I feel a distinct chill. How are we to think of these formative and positively transforming benefits derived from other people's illness? I have no objection to people making the best of a bad job, but we mustn't lose sight of the fact that it really is a very bad job that we are talking about. Remember the context of the symposium, which gave rise to this exchange. It was primarily about the desirability of preventing and curing disease and disability, about whether what people like me term “disabilities” are really bad at all and as Koch terms it, about “the protection of persons of difference”. In that context, if disabilities are harms to the individuals who suffer (sorry!) from them then the “secondary gain” of others from the illness is surely a highly problematic compensating factor. It implies that somehow the harm of the disease is mitigated by factors such as these and that it might conceivably be right to decline to prevent or cure the disease or disability for fear of losing these “positively transforming” benefits for the friends and relatives. That is not, of course, to say that no benefits to third parties would justify refusal to protect from disability. But such benefits would have to be massive, urgent, and unobtainable by other more moral means. Here, surely, the beneficial “feel good factors” could not conceivably be of sufficient weight or urgency and are in any event obtainable by other means.

Of course if “disabilities” are really only “physically or cognitively distinguishing features” as Koch claims then it might well look as though such secondary gain is well worthwhile. And if they are, as Koch quotes a Canadian jurist as claiming, a “life enhancing condition”, then how could it possibly be wrong to confer such a benefit on children even without the various forms of secondary gain?

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# Is This A Derivative?

I am in a little over my head. This all began with my reading how each level of pascals triangle adds to $2^n$, where n=row# starting with n=0. I then though, "wouldn't it be clever if the rows added to something else--like say $3^n$ instead?" Or even better generalize it for any constant, $a^n$.

All that was needed was to Multiply any given number in pascal's triangle by $a^n/2^n$

$$\begin{array}{rcccccccccc} & & & & & & 1\\\ & & & & & \frac{a}{2} & & \frac{a}{2}\\\ & & & & \frac{a^2}{4} & & \frac{a^2}{2} & & \frac{a^2}{4}\\\ & & & \frac{a^3}{8} & & \frac{3*a^3}{8} & & \frac{3*a^3}{8} & & \frac{a^3}{8}\\\ & & \frac{a^4}{16} & & \frac{a^4}{4} & & \frac{3*a^4}{8} & & \frac{a^4}{4} & & \frac{a^4}{16}\\\ & \frac{a^5}{32} & & \frac{5*a^5}{32} & & \frac{5*a^5}{16} & & \frac{5*a^5}{16}& & \frac{5*a^5}{32} & & \frac{a^5}{32}\\\ & \frac{a^6}{64}& &\frac{3*a^6}{32}& &\frac{15*a^6}{64}& &\frac{5*a^6}{16}& &\frac{15*a^6}{64}& &\frac{3*a^6}{32}& &\frac{a^6}{64}\\\ & & ... & & & &... & & & & ... & \end{array}$$

Adding any row should give $a^n$. Placing said coefficients in front of a binomial expansion and solving for the binomial expression yields $$(a^n/2^n)*(x+y)^n$$

Letting a=2 makes pascals triangle, but every other value of "a" distorts every value and relation (except n=0 row for obvious reasons).

This triangle creates some interesting relations that are shared with Pascal's triangle and are immediately obvious: every term in the middle column may be divided by "a" to yield the term above and to the right or left of it--just like Pascals triangle (a=2).

Next is a really fascinating fluke: $$\frac{\partial \frac {a^4}{4}}{\partial a}=a^3$$, which is the sum of the line above it. $$\frac{\partial \frac {a^2}{4}}{\partial a}=\frac {a}{2}$$, which is found in the line above it.

I realize that "a" must have a definite value as a coefficient in order to have meaning, and it is not itself a function, but it seems curious that derivative relationship would show up in the relations between the coefficients of this modified triangle. This serendipitous relation fascinates me to the point of asking "whats up?" here. Is this a derivative? Have derivative relationships popped up organically elsewhere in function theory?

-
I don't understand the question. –  Ethan Jun 3 '13 at 3:00
Binomial origin of a derivative: en.wikipedia.org/wiki/Binomial_theorem#Geometric_explanation –  JoeHobbit Jun 7 '13 at 17:20
Legendre Polynomials are closely related as demonstrated by $\frac{1}{2^n}$ in the solutions. –  JoeHobbit Jun 12 '13 at 9:07
–  JoeHobbit Jul 4 '13 at 1:29

Pascal's triangle is fascinating and encodes many profound mathematical relationships. However, I think all you've observed is that if $f(a)=a^n$ then $f'(a)=na^{n-1}$. This explains why if you differentiate a term $\frac{a^n}{2^n}{n\choose k}$ in one row with respect to $a$ you'll get something resembling something in the row right above.

In particular you get $a^{n-1} \frac{n!\cdot n}{2^n(n-k)!k!}$. For small n it might seem that there are patterns, particularly in the rows that are powers of 2 since the $2^n$ might go away.

In general, however, this is not a meaningful manipulation of Pascal's triangle, especially since you've stolen away one of its most beautiful features: the fact that ${{n-1}\choose k-1}+{{n-1}\choose k}={n \choose k}$, equivalently that each entry is the sum of the two entries directly above.

As to your derivative question, I'm not sure what you're expecting... I think what you've observed unfortunately happens to be purely incidental. But it's great that you're noticing patterns - that's how most great ideas in math come about! Derivatives have wide-ranging and incredibly useful applications. If by "derivative relationships" you mean differential equations, then heck yeah, they come up all the time in analysis and applied mathematics.

-

You may rewrite your $$\frac{a^n}{2^n}(1+1)^n$$ as $$\left(\frac{a}{2} + \frac{a}{2}\right)^n$$

Unfortunately neither of the two examples you mention appear to be specific examples of general patterns.

-
What does this answer exactly? –  Ethan Jun 3 '13 at 3:04
I didn't say it wasn't an answer, I asked what it answered. –  Ethan Jun 3 '13 at 3:12
I believe the OP's questions were " Is this a derivative? Have derivative relationships popped up organically elsewhere in function theory?" and I don't see this as an answer to either of those questions... –  DanZimm Jun 3 '13 at 3:22
@Dann The first, and in my opinion, most salient question the OP had was "what's up?" He found what may be a pattern and wanted to know why. –  vadim123 Jun 3 '13 at 3:24
@vadim123 I suppose I still don't see how this answers what's up - I may be ignorant (as usual) –  DanZimm Jun 3 '13 at 3:31


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33rd M. Smoluchowski Symposium on Statistical Physics

3-4 December 2020
Europe/Warsaw timezone

On zero-delay synchronization in a network of timed automata modeling cardiac tissue

4 Dec 2020, 16:30
30m
30 minute lecture

Speaker

Danuta Makowiec (Institute of Theoretical Physics, University of Gdańsk)

Description

A network of timed automata (NTA), inspired by Greenberg-Hastings cellular automata, can efficiently and faithfully reflects the work of the real pacemaker [1]. In this system, each automaton cyclically switches between three states of a certain length of time: from $F$ firing state of length $f$, to $R$ refractory state of length $r$ and then to $A$ activity state of length $a$, what closely follows the phenomenologically known oscillations of myocytes. As a results of interactions between neighboring cells, the self-organization of the oscillating units to a common oscillation emerges which then leads to forming a signal of initiating a contraction of the whole heart. Thus NTA establishes an appropriate platform to model a natural human pacemaker, in which we gain insight into conditions and stages of the synchronization process [2].

The visible effect of synchronization in NTA is the excitation wave, i.e. the emergence of steady move of a cluster formed by automata staying in $F$ state.
The synchronization can be of frequency locking type, when some automata force the neighboring automata to faster switch the state $A$ to $F$ than it is expected by intrinsic automaton's cycle, or of phase-locking type, when neighboring automata exhibit the common difference in oscillatory phase. Thus, the frequency-locking synchronizations result in the fast rhythms. The visible effect of such synchronization is a spiral origin of the excitation wave. The phase-locking synchronization occurs when neighboring automata are one step late in the advancement of the cyclic transitions of intrinsic time periods. A visible effect of such synchronization is propagation of the excitation wave with rhythm comparable to the period of oscillating automata.

The strength of synchronization is due to the reduction of uncorrelated degrees of freedom to a collective mode of operation, which then enables the long-term ordering and coordination of biological processes

The emergence of zero-delay phase-locking synchronization, the marching squad synchronization, has been observed in NTA when in response to the $F$ state of neighbors, the length of the refractory phase $r$ was increased [1].
On the other hand, in NTA system in which the vagal regulation was modeled as increase in the length $a$ of the activity state in response to the vagal activity, such synchronization is also observed but only when the activity of vagal system often changes [2].
The mechanisms behind synchronizing without any delay in spatially distributed systems are not clear and debated. The presentation will be devoted to analyze possible mechanisms leading to the marching squad synchronization in the NTA system modeling the pacemaker.

[1] D. Makowiec "Pacemeaker rhythm through networks of pacemaker automata - a review" Acta Physica Polonica B Proceedings Supplement, 7(2014) 347
[2] D. Makowiec, W. Miklaszewski, J. Wdowczyk, A. Lawniczak "From cellular automata model of vagal control of human right atrium to heart beats patterns" Physica D, to appear, (2020)

Primary author

Danuta Makowiec (Institute of Theoretical Physics, University of Gdańsk)

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text:
# Beta refraction

June 13, 2012, 6:38 pm
 Topics:

Beta refraction is an effect that results from the latitudinal variation of Rossby wave phase speed which is, in turn, due to the beta effect. If a line of Rossby waves were started along a straight eastern ocean boundary, then those at low latitudes would arrive at the western boundary before those at high latitudes.

 This article is written at a definitional level only. Authors wishing to expand this entry are inivited to expand the present treatment, which additions will be peer reviewed prior to publication of any expansion.

Glossary


text:
# Delta Hat

An experience-led company specialising in the technical aspects of evidence-based medicine and health technology assessment

## Who are $$\hat{\delta}$$?

At Delta Hat ($$\hat{\delta}$$) we are health economists and statisticians who provide support to the pharmaceutical industry. We focus on the technical aspects of evidence-based medicine and health technology assessment – from meta-analysis to cost-effectiveness modelling, and everything in between.

## Why are we called $$\hat{\delta}$$?

The name of the company represents what we offer - in statistics Greek letters are commonly used to denote unknown parameters. The letter $$\delta$$ is commonly used to represent the difference. Placing a circumflex, ^, or ‘hat’ over a letter denotes the estimator of this value. Delta hat, $$\hat{\delta}$$, therefore means ‘estimated difference’.

Often in comparative effectiveness and health technology assessment this is the parameter we are trying to estimate – the added value of the new intervention. As the true value can never be observed (at best we conduct a trial to estimate this) further work is often needed to make robust estimates of the benefit of treatment. At $$\hat{\delta}$$, we have the skills to perform such analyses (even our website is built in R), with the experience to help understand the implications of the findings.

Please get in touch, as we are more than happy to talk through what solutions we might recommend and what services we may be able to offer.

## How can you contact $$\hat{\delta}$$?

Details are available here or just click here to send us an email.

Modern slavery statement


text:
# Student-Faculty Research in Computer Science at Grinnell (Fall 2016 Edition)

I’ve had some prospective students ask about student-faculty research opportunities at Grinnell. Our student-faculty-research landscape is changing with a new research for all initiative, so I expect that I will need to write a new version of this essay in a year. In addition, the particulars of each faculty member’s research agenda and approaches also change from year to year. So I’ll probably write a new version every year.

Grinnell has excellent support for student-faculty research, most significantly through the Mentored Advanced Project (MAP) program and a few variants thereof [1]. The MAP program allows students to stay and work with faculty in the summer and receive both credit and compensation (roughly $3400 for 10 weeks of work [2]). Computer Science takes good advantage of these resources. This past summer, I think we had twenty five students working with faculty in the department for part or whole of the summer. I had eight full-time Grinnell students (two students between their first and second years, four students between their second and third years, and two students between their third and fourth years), two full-time students from other institutions, and one half-summer student. Peter-Michael Osera had six students (I think), mostly between their second and third years. Charlie Curtsinger also had six students. John Stone had two students as assistant system administrators, which, while not research in the traditional sense, gave students a valuable experience. My students have submitted two papers [3] which are currently under review, Charlie’s students had papers accepted and will be presenting this fall (in Amsterdam, I think), and Peter-Michael’s plan to submit papers. While we don’t have twenty-five students every summer (in fact, I think we only had four last summer), as I look toward the future I think it likely that we will support at least a dozen each summer, provided Grinnell continues to make appropriate resources available. The computer science faculty do a variety of kinds of research. Charlie is primarily interested in systems issues. He had one team of students looking at how to analyze the power consumption of individual parts of programs so that one could optimize a program to use less energy. His other team of students was exploring how the layout of data in memory affects the running time of a program. Peter-Michael is interested in issues of program generation and verification. In the past, he had built a system that automatically generates a function given a few sample inputs and outputs from that function. One group of students looked at both improving the system and designing a user interface that allows a user to provide feedback as the system builds the function. A second group of students was exploring an interactive interface to help people prove programs correct. I’m exploring issues in CS education, particularly in how we diversify the group of people doing computing. My students worked together to design and offer a code camp for middle schoolers. Along the way, they did a fairly large scale study on models of code camps across the country and a comparison between academic and commercial models of code camps. Mr. Stone had one student writing a secure backup system (one that is even secure from system administrators) and another exploring how to upgrade our Web site from Drupal 6 to Drupal 8 (more complicated than you might expect, not least because we have custom modules). Faculty beyond the computer science department also involve CS students in their research. The College has a fairly large digital liberal arts initiative, and our students have helped with some of those projects. I appreciate that those projects really do help people think differently about the value of computing and about approaches to understanding texts. A few of our Math/Stats faculty do work that readily involves computer science students. And one or more of our Physics faculty have students program as part of their research and so are happy to hire CS majors who also know some physics. In the paragraphs above, I focused primarily on summer research. Some CS faculty also take research students during the year, and some faculty from other departments want CS students as research assistants (or as research programmers) throughout the year. Our academic-year projects can also have tangible outcomes. For examples, I know that Mr. Walker is taking students to a conference in a week or two to present research they did with him last spring. Needs and approaches vary from semester to semester. Some students work for credit. Some work for compensation. This fall is pretty busy, so I don’t think any of the CS faculty have research students [4]. I’ve already talked to one about a project for spring. I will note that in most cases, CS faculty supervise student-faculty research that falls within their broader research agenda. That is, most of us propose a framework and let students design within that framework. Few of us will supervise a completely student-designed MAP because it requires significantly more effort on our part, often for significantly less benefit. However, most students find more than enough freedom working within our broader research agendas. And, as we heard at a recent internship discussion, most of our students leverage their summer Grinnell CS experiences into internships and more. [1] We also have Mentored Introductory Projects (MIPs/299s), targeted at students finishing their first year, and summer [2] I don’t think$3,200 is enough. Students doing REUs get money and housing (and, sometimes, support for transportation). But Grinnell students do get credit and, if their papers are accepted, support for travel to conferences. So, maybe it is.

[3] I originally wrote my students have submitted two students, which made Michelle laugh much too hard.

[4] Toby Baratta tells me that she’s doing a MAP with Charlie Curtsinger this semester, so I’m clearly wrong.

Version 1.0.2 of 2016-09-18.


text:
# Confidence intervals question

I'm just unclear what $\sigma$ represents here. Am looking for a confidence interval where $\mu=\sigma$? or looking for the confidence interval where $\sigma=\sqrt{\sigma^2/n}$?? any hints on where to start this?

-

$\sigma$ is the standard deviation of your initial sample $X_1,\ldots,X_n$, and $S$ is the usual estimator of the standard deviation of a sample where also the mean is unknwon. What you're told is that $$S\approx \mathcal{N}\left(\sigma,\frac{\sigma^2}{2n}\right)$$ where $\approx$ means approximately distributed as. So yes, the mean of $S$ is $\sigma$ where $\sigma$ is the standard deviation of $X_i$. Note that the above implies that $$\frac{S-\sigma}{\sigma}=\frac{S}{\sigma}-1\approx\mathcal{N}\left(0,\frac{1}{2n}\right)$$ which enables you to arrive at that confidence interval.


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This site is supported by donations to The OEIS Foundation.

# Decomposition of numbers into weight × level + jump

The main goal of this page is to provide a new way to see certain numbers by their decomposition into weight × level + jump. This decomposition can be seen as a generalization of the sieve of Eratosthenes (which is the particular case of the sequence of natural numbers). Applied to prime numbers, this decomposition is used to obtain a new classification of primes.

arXiv:0711.0865 [math.NT]: Decomposition into weight * level + jump and application to a new classification of prime numbers, 2007 - 2010.

## Principles

Principle of decomposition: we choose the smallest weight such that in the Euclidean division of a number by its weight, the remainder is the jump (first difference, gap). The quotient will be the level.

Principles of classification: if the number is not decomposable, it is not classified. If the weight is greater than the level then the number is classified by level, if not then it is classified by weight.

## Definitions

Let ${\displaystyle \scriptstyle {\{a(n)\}}_{n=i_{\min }}^{\infty }\,}$ be a strictly increasing integer sequence.

### Jump

The jump (first difference, gap) of ${\displaystyle \scriptstyle a(n)\,}$ is defined as

${\displaystyle d(n):=a(n+1)-a(n).\,}$

### l(n)

l(n) is defined as

${\displaystyle l(n):={\begin{cases}a(n)-d(n)&{\text{if }}a(n)-d(n)>d(n),\\0{\rm {~otherwise}}.\end{cases}}\,}$

#### Alternative definition with mod function

${\displaystyle l(n):={\rm {largest~}}l{\rm {~such~that~}}a(n+1)=a(n)+a(n){\rm {~}}mod{\rm {~}}l,{\rm {~or~}}0{\rm {~if~no~such~}}l{\rm {~exists.~}}}$

### Weight

The weight of ${\displaystyle \scriptstyle a(n)\,}$ is defined as

${\displaystyle k(n):={\begin{cases}{\rm {smallest~}}k>d(n){~s.t.~}k|l(n),\\0{\rm {~if~}}l(n)=0.\end{cases}}\,}$

#### Alternative definition with mod function

${\displaystyle k(n):={\rm {smallest~}}k{\rm {~such~that~}}a(n+1)=a(n)+a(n){\rm {~}}mod{\rm {~}}k,{\rm {~or~}}0{\rm {~if~no~such~}}k{\rm {~exists.~}}}$

### Level

The level of ${\displaystyle \scriptstyle a(n)\,}$ is defined as

${\displaystyle L(n):={\begin{cases}{\frac {l(n)}{k(n)}}&{\text{if }}k(n)>0,\\0&{\text{if }}k(n)=0.\end{cases}}\,}$

### Decomposition criterion

A strictly increasing integer sequence, ${\displaystyle \scriptstyle {\{a(n)\}}_{n=i_{\min }}^{\infty }\,}$ can be decomposed into weight × level + jump when ${\displaystyle \scriptstyle l(n)\,}$ is different from 0 which can be rewritten when:

${\displaystyle a(n)-d(n)>d(n),\,}$

or

${\displaystyle d(n)<{\frac {1}{2}}\times a(n),\,}$

or

${\displaystyle a(n+1)<{\frac {3}{2}}\times a(n)\,}$

### Unique decomposition

The weight ${\displaystyle \scriptstyle k(n)\,}$ is the smallest such that in the Euclidean division of ${\displaystyle \scriptstyle a(n)\,}$ by its weight ${\displaystyle \scriptstyle k(n)\,}$, the quotient is the level ${\displaystyle \scriptstyle L(n)\,}$, and the remainder is the jump ${\displaystyle \scriptstyle d(n)\,}$. We have the unique decomposition

${\displaystyle a(n)=k(n)\times L(n)+d(n)={\rm {weight}}\times {\rm {level}}+{\rm {jump}}\,}$

### Principles of classification

• If for ${\displaystyle \scriptstyle a(n)\,}$,
${\displaystyle l(n)=k(n)=L(n)=0\,}$

then ${\displaystyle \scriptstyle a(n)\,}$ is not classified.

• If for ${\displaystyle \scriptstyle a(n)\,}$,
${\displaystyle k(n)>L(n)\,}$

then ${\displaystyle \scriptstyle a(n)\,}$ is classified by level, if not then it is classified by weight.

## Algorithms

Naive algorithm (PARI/GP):

 decompnaive(n,n1)={ /*strictly increasing*/ if(n>=n1,print("n1 must be greater than n");return); /*jump*/ d=n1-n; /*l=n-d if n>2*d else the number is not decomposable*/ if(n>2*d,l=n-d,print(d, ", 0, 0");return); /*we look for the weight to jump+1 until l*/ for(k=d+1,l,if(n%k==d,print(n," = ",k," * ",l/k," + ",d);return)); }

Algorithm "newSieve":

 decompsieve(n,n1)={ /*strictly increasing*/ if(n>=n1,print("n1 must be greater than n");return); /*jump*/ d=n1-n; /*l=n-d if n>2*d else the number is not decomposable*/ if(n>2*d,l=n-d,print(d, ", 0, 0");return); /*we look for the weight to jump+1 until sqrt(l)*/ for(k=d+1,sqrt(l),if(n%k==d,print(n," = ",k," * ",l/k," + ",d);return)); /*we look for the level to jump until 1 (--)*/ forstep(le=d,1,-1,if(n%floor(l/le)==d,print(n," = ",l/le," * ",le," + ",d);return)); }

Algorithm "newSieve" is the fastest for numbers classified by level.

## General remarks

 ${\displaystyle \scriptstyle n}$ Numbers of decompositions into weight × level + jump 1 0 2 0 3 1 4 1 5 2 6 2 7 3 8 3 9 4 10 4 11 5 ... ...

The numbers of decompositions possible of ${\displaystyle \scriptstyle n}$ into weight × level + jump is A004526 (with an offset at 1).

The sequence with the the greatest growth that can be decomposed is A003312.

Throughout this page, the jump is the first difference but we can take the second, third... difference. See A133346 and A133347 for primes.

## Decomposition of natural numbers, the sieve of Eratosthenes

If the decomposition is possible (i.e. if ${\displaystyle n>2}$), we have:

The weight is the smallest prime factor of ${\displaystyle \scriptstyle n-1\,}$ and the level is the largest proper divisor of ${\displaystyle \scriptstyle n-1\,}$. The natural numbers classified by weight are the ${\displaystyle composites+1}$ and the natural numbers classified by level are the ${\displaystyle primes+1}$. The decomposition into weight × level + jump of natural numbers is thus a reformulation of the sieve of Eratosthenes.

Plot of log(A020639) vs log(A032742) for n ≤ 10^4, the sieve of Eratosthenes (OEIS graph):

## Decomposition of prime numbers

${\displaystyle \scriptstyle p(1)\,=\,2}$, ${\displaystyle \scriptstyle p(2)\,=\,3}$ and ${\displaystyle \scriptstyle p(4)\,=\,7}$ are the only primes not decomposable[1]. Except for ${\displaystyle \scriptstyle p(1)\,=\,2}$, ${\displaystyle \scriptstyle p(2)\,=\,3}$ and ${\displaystyle \scriptstyle p(4)\,=\,7}$, the decomposition into weight × level + jump of prime numbers is:

Plot of log(A117078) vs log(A117563) for n ≤ 10^4 (OEIS graph):

### Primes of level (1;i)

Principle of classification for primes of level 1:

If for p(n), l(n) is a prime says prime(n-i) then p(n) is of level (1;i).

### Direct relations

For ${\displaystyle \scriptstyle p(n)\,}$ different from 2, 3 and 7, we have:

• ${\displaystyle l(n)=p(n)-g(n)=2\times p(n)-p(n+1)=k(n)\times L(n),\,}$
• ${\displaystyle p(n)=l(n)+g(n)=k(n)\times L(n)+g(n),\,}$
• ${\displaystyle \gcd(g(n),2)=2,\,}$
• ${\displaystyle \gcd(p(n),g(n))=\gcd(p(n)-g(n),g(n))=\gcd(l(n),g(n))=\gcd(L(n),g(n))=\gcd(k(n),g(n))=1,\,}$
• ${\displaystyle 3\leq k(n)\leq l(n),\,}$
• ${\displaystyle 1\leq L(n)\leq l(n)/3,\,}$
• ${\displaystyle 2\leq g(n)\leq k(n)-1,\,}$
• ${\displaystyle 2\times g(n)+1\leq p(n).\,}$

#### Primes classified by weight

For primes classified by weight (Cf. A162175) (primes for which ${\displaystyle \scriptstyle k(n)\,\leq \,L(n)\,}$), we have:

• ${\displaystyle g(n)+1\leq k(n)\leq {\sqrt {l(n)}}\leq L(n)\leq {\frac {l(n)}{3}}.\,}$

82,89 % of primes ${\displaystyle \scriptstyle p(n)\,}$ are classified by weight for ${\displaystyle \scriptstyle n\leq 5\cdot 10^{7}\,}$.

We can see that by definition, the primes classified by weight follow Legendre's conjecture and Andrica's conjecture.

#### Primes classified by level

For primes classified by level (Cf. A162174) (primes for which ${\displaystyle \scriptstyle k(n)\,>\,L(n)\,}$), we have:

• ${\displaystyle L(n)<{\sqrt {l(n)}}
• ${\displaystyle L(n)+2\leq g(n)+1\leq k(n)\leq l(n).\,}$

17,11 % of primes ${\displaystyle \scriptstyle p(n)\,}$ are classified by level for ${\displaystyle \scriptstyle n\leq 5\cdot 10^{7}\,}$.

Knowing that the primes are rarefying among the natural numbers and according to the numerical data, we make the following conjecture:

### Lesser of twin primes

If ${\displaystyle \scriptstyle p(n)}$ is a lesser of twin prime greater than ${\displaystyle 3}$ then ${\displaystyle \scriptstyle p(n)}$ has a weight of ${\displaystyle 3}$. If ${\displaystyle \scriptstyle p(n)}$ has a weight of ${\displaystyle 3}$ then ${\displaystyle \scriptstyle p(n)}$ is a lesser of twin prime greater than ${\displaystyle 3}$[1].

### Conjectures

The well-known conjecture on the existence of an infinity of twin primes can be rewritten as:

• Conjecture 1: The number of primes with a weight equal to 3 is infinite.

To extend this conjecture we make these two conjectures:

• Conjecture 2: The number of primes with a weight equal to k is infinite for any ${\displaystyle \scriptstyle k\,\geq \,3\,}$ which is not a multiple of 2;
• Conjecture 3: The number of primes of level ${\displaystyle \scriptstyle L\,}$ is infinite for any ${\displaystyle \scriptstyle L\,\geq \,1\,}$ which is not a multiple of 2.

• Conjecture 4: Except for p(6) = 13, p(11) = 31, p(30) = 113, p(32) = 131 et p(154) = 887, primes which are classified by level have a weight which is itself a prime.

The conjecture on the existence of an infinity of balanced primes can be rewritten as:

That we can easily generalize by:

• Conjecture 7: If the jump g(n) is not a multiple of 6 then l(n) is a multiple of 3. (trivial)
• Conjecture 8: If l(n) is not a multiple of 3 then jump the g(n) is a multiple of 6. (trivial)

Knowing that the primes are rarefying among the natural numbers and according to the numerical data, we make the following conjecture:

## Decomposition of odd numbers

If the decomposition is possible, we have:

Plot of log(A090368) vs log(A184726) for n ≤ 10^4 (OEIS graph):

## Decomposition of even numbers

If the decomposition is possible, we have:

Plot of log(A090369) vs log(A184727) for n ≤ 10^4 (OEIS graph):

## Decomposition of composite numbers

If the decomposition is possible, we have:

Plot of log(A130882) vs log(A179621) for n ≤ 10^4 (OEIS graph):

## Decomposition of semiprimes

If the decomposition is possible, we have:

Plot of log(A130533) vs log(A184729) for n ≤ 10^4 (OEIS graph):

## Decomposition of 3-almost primes

If the decomposition is possible, we have:

Plot of log(A130650) vs log(A184753) for n ≤ 10^4 (OEIS graph):

## Decomposition of lucky numbers

If the decomposition is possible, we have:

Plot of log(A130889) vs log(A184828) for n ≤ 10^4 (OEIS graph):

## Decomposition of prime powers

If the decomposition is possible, we have:

Plot of log(A184829) vs log(A184831) for n ≤ 10^4 (OEIS graph):

## Decomposition of squarefree numbers

If the decomposition is possible, we have:

Plot of log(A184832) vs log(A184834) for n ≤ 10^4 (OEIS graph):

## Decomposition of triangular numbers

If the decomposition is possible, we have:

Plot of log(A130703) vs log(A184219) for n ≤ 10^4 (OEIS graph):

## Decomposition of squares

If the decomposition is possible, we have:

Plot of log(A133150) vs log(A184221) for n ≤ 10^4 (OEIS graph):

## Decomposition of pentagonal numbers

If the decomposition is possible, we have:

Plot of log(A133151) vs log(A184751) for n ≤ 10^3 (OEIS graph):

## Sequences

Sequences related to the decomposition.