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# Why is the Pauli exclusion principle not considered a sixth force of nature?

Why is the Pauli exclusion principle not considered a sixth force of nature, given it produces such things as repelling of atoms and molecules in solids?

• Dec 28, 2020 at 13:24
• What's the fifth force? Dec 29, 2020 at 3:09
• @tparker Given another common "why not a force?" confusion, I'm guessing Higgs.
– J.G.
Dec 29, 2020 at 7:51
• @kaylimekay Discussed here & here.
– J.G.
Dec 29, 2020 at 8:57
• There is a very good video by Eugene Khutoryansky at youtu.be/Zlp2GQ3OLeE Dec 29, 2020 at 17:27

## 11 Answers

Since Pauli exclusion is an inherently quantum phenomenon, let's talk about what the most general effect of a force of nature is, or how we would most generally categorize what a force of nature is, in that context. Specifically, let's talk about scattering theory in quantum mechanics.

In scattering theory we start with some particles that are in wave packets that are narrowly peaked around some momenta $$\mathbf k_1,\dots$$ at some early time $$t=-T\to - \infty$$. This is an "in state." We evolve this state for a long time $$2T$$ using the usual evolution operator constructed from the Hamiltonian. Then at the far future time $$t=+T\to +\infty$$, we look for the overlap of the evolved in state with an "out state" that has particles in wave function narrowly peaked around some (possibly different) momentum $$\mathbf k'_1,\dots$$ constructed at that late time.

What happens if we work in a free theory, that is, one with no forces? In that case the only overlap of the in state with the out state will occur when the $$\mathbf k_i$$ and $$\mathbf k'_i$$ are the same. If we find that they are not always the same, then some force of nature is present. It contributes some piece that we might call an "interaction term" to the Hamiltonian. The effect of that interaction term is what gives us the non-trivial overlap of the in and out states.

Pauli exclusion, on the other hand, is not something that affects how the Hamiltonian acts on a state. It is simply a statement about what states are allowed.

That is (obviously) not to say that it has no effect on the outcome of some process. For a bosonic system, I could find that some system has a nonzero overlap with an out state $$_\mathrm{out}\langle \mathbf k_1,\mathbf k_2,\dots|$$ where $$\mathbf k_1=\mathbf k_2$$ (and any other quantum numbers are the same). This state is of course not an option for fermions, so a similar experiment performed with fermions would have a different outcome. But that would simply be a statement about what states are available, and not a statement about what interactions (or forces of nature) were present in the Hamiltonian, the thing that defines the dynamics of the system.

• Pauli exclusion is a consequence of the Poincare group. It is a not a consequence of quantum mechanics. Dec 29, 2020 at 19:26
• @my2cts does Pauli exclusion occur in a classical system that is Poincaré invariant? Dec 29, 2020 at 19:32
• Classical systems are not Poincaré invariant. However (non-relativistic) quantum mechanical systems do display Pauli exclusion effects which in their origin are spin related and thus Poincaré related. Dec 31, 2020 at 10:44
• @ohneVal Classical systems can be invariant under boosts, rotations, and translations. Jan 2, 2021 at 8:09
• @ohneVal classical EM. Or take any Poincaré invariant quantum system that you like and then take $\hbar\to 0$. :) Jan 2, 2021 at 11:14

The Pauli exclusion principle as understood nowadays, is a consequence of the spin-statistics behavior of fermions. We know that Spin representations are related to wave-functions with specific commutation relations which enforce anti-symmetry (all of this encoded in the Dirac-equation, which already accounts for spin) of the wave-function, thus not allowing for two fermionic particles to have the same quantum states in a system. As you can see, it is a statement related to statistics and spin. Both are very well understood and are a consequence of the commutation relations, not the fundamental interactions between single particles (I would prefer fields) which is what we call a force.

Forces on the other hand, are of a different nature. They are not a statistical effect but concern the point-like interactions among fields. Except for gravity, we describe all of them by a gauge (mediator) boson. We don't need to include such a bosonic field to account for the Pauli exclusion principle since the fermionic commutation relations already encode this effect.

• This does not explain where the field commutation relations come from. Dec 29, 2020 at 19:52
• The question is not asking about the origin of the commutation relations, is just asking for why it is not considered a force. I could add some comments on it if you want. Or you can post a new question about them, I just thought it would be a bit far from the question. Dec 31, 2020 at 10:41
• The question is about the physical origin of the PEP. The commutation relations are equivalent to the PEP. Hence stating that the PEP is "a consequence of the commutation relations" does not bring the question any closer to an answer. Jan 4, 2021 at 11:25
• It does I believe. At the end of the day the only reason we have for the commutation relations or namely QM is observation. Depends at which level do you want to stop, if you think the above answer is not good quality or is wrong in a more specific way please let me know. Jan 4, 2021 at 12:56
• The OP asks to go one level deeper than a general statement principle or a mathematical notation. You have not given this. Hence my comment here. I also do not have it, hence my non-answer, which review has deleted. Jan 4, 2021 at 16:47

The Pauli exclusion principle is a rule, similar to the Heisenberg uncertainty "principle", an observational encapsulation of quantum mechanical theory.An easy overview in organizing interactions. The Pauli exclusion follows the spin statistics theorem.

The spin–statistics theorem implies that half-integer–spin particles are subject to the Pauli exclusion principle, while integer-spin particles are not. Only one fermion can occupy a given quantum state at any time, while the number of bosons that can occupy a quantum state is not restricted.

A force in quantum mechanical terms is a $$dp/dt$$. One could describe the exchange of a virtual electron in Compton scattering as a force, but not as a fundamental force, because it is not a gauge boson, and the couplings are the electromagnetic couplings. But the Pauli exclusion does not involve a dp/dt in any way.

Exclusion exists classically too, though we do not call them principles. Take the seats in a theater. Only one person can sit in a seat , no force there except you try to sit on someone :). An electron scattering off a filled slot will go away with a dp/dt scatter on the atom, but the force will be the electromagnetic virtual photon exchange.

• I am surprised by the phrase "the exchange of a virtual electron in Compton scattering". Can you elaborate this? Dec 29, 2020 at 19:37
• @my2cts Look at my answer here physics.stackexchange.com/q/445774, it has the lowest order feynman diagrams. It is the vrtual electron that carries the dp/dt Dec 29, 2020 at 19:52
• The fact that I can't sit where another person is already sitting, is already the quantum mechanical Pauli exclusion: My electrons simply don't want to share space with the electrons that make up the other person. Not without providing enough energy to lift them to the corresponding energy states. Since I'm scared of supplying that kind of energy, I'd rather sit somewhere else... Dec 30, 2020 at 17:06

You could say that the list of phenomena that we currently call fundamental forces is just a matter of convention. In reality it is more complicated.

All the currently accepted fundamental forces in the Standard Model, have a mediator, that mediates the interaction. Photons for EM, gluons for the strong force, hypothetical gravitons for gravity, and W and Z bozons for the weak force.

On the other hand, you have a list(please note that this is a very diverse list, these might have nothing to do with the PEP) of other phenomena, that in our current understanding do not need a mediator, or we do not have any idea what what the mediator could be, this includes the HUP, the PEP, van der waals (please note this might be based on EM), and dark energy (completely not understood), and the Higgs mechanism, and entanglement. It is very important to understand that the Higgs boson is not a mediator of any kind regarding the Higgs mechanism, but it is just the excitation of the Higgs field.

All gauge fields can be interpreted this way - and all 4 "fundamental forces" are in fact gauge fields.

Why do we still need to think of gravity as a force?

So to have a phenomenon be called in our current understanding a fundamental force, you need a fundamental field (needs to be a gauge field) associated with it, and a mediator particle.

This says no two electrons are allowed in the same state, and this is essentially an entanglement phenomenon.

How Quantum-entangled particles communicate with each other

The PEP is fundamentally different from all other phenomena, in that it shows some similarities with one of the building blocks of QM, that is entanglement.

So the answer to your question is, that the PEP does not have its own fundamental gauge field, nor a mediator particle, and thus in our current understanding we do not rank it as a fundamental force (rather a phenomenon that shows some similarities with entanglement).

• How is the exclusion principle supposed to be "based on entanglement"? Dec 29, 2020 at 17:17
• I think this is as courageous list: "the HUP, [...], van der waals, and dark energy, and the Higgs mechanism, and entanglement." These very diverse phenomena have nothing in common with the PEP. The van der Waals interaction is fully explained by standard quantum mechanics as an electronic effect. HUP and entanglement are direct consequences of standard quantum mechanics. Dark energy is completely not understood. Dec 29, 2020 at 19:43
• @my2cts correct, I am adding a note to clarify that. I hope if this is better this way, can you please retract the downvote (I don't know if you did or not)? Dec 29, 2020 at 20:05
• @ÁrpádSzendrei I did not downvote. Dec 29, 2020 at 20:07
• @my2cts thank you I edited anyway. Dec 29, 2020 at 20:08

Let me try to put my answer in a more general perspective than the already many others.

In our experience, we are so used to considering every correlation as induced by interactions that it is sometimes difficult to consider the presence of effects (correlations) without introducing a direct interaction as their cause. Still, this is the case of the correlations induced by the fermionic or bosonic nature of the particles.

Pauli's exclusion principle is a consequence, at the level of a one-particle description of the quantum states of many-body systems, of the antisymmetric nature of their states. It has an obvious counterpart in the case of bosons. Looking at this property as due to some interaction is not consistent with the definition we have of an interaction. In quantum physics, interactions are always introduced by coupling terms added to the sum of the non-interacting Hamiltonians for free particles.

It is a fact that we can see the effects of the statistics even in the case of no interaction in the Hamiltonian. The pair distribution functions of two fermions or two bosons in a perfect gas of the same kind of particles differ from the uniform, uncorrelated result at short distances. The exact result is (see section 5.5 of Pathria's textbook on Statistical Mechanics): $$\langle {\bf r_1 r_2}|e^{-\beta \hat H}| {\bf r_1 r_2}\rangle = \frac{1}{2 \lambda^6} \left( 1 \pm \exp(-2 \pi r^2_{12}/\lambda^2) \right).$$ Sing plus corresponds to bosons, while sign minus to fermions. It is clear that such an effect on the pair distribution function can always formally reinterpreted as due to some inter-particle interaction. But its derivation from the states of a non-interacting Hamiltonian should make clear the artificial role of such interpretation.

Notice that the appeal for the introduction of new interactions to explain the observed correlations is not limited to the quantum regime. Osmotic forces or depletion forces in liquid solutions are a more complex example of the same attitude in classical systems.

There are many good answers, but I would like to add a short one: the fundamental forces are the "gauge fields", nothing more than that. So no, the Pauli principle is not a force at all (it is not even a field). See e.g. https://physicstoday.scitation.org/doi/10.1063/1.2911184

One may discuss if the Higgs field is a fifth fundamental force: this is a matter of language. However, the Higgs is introduced "by hand" and not not by "symmetry requirements", like gauge fields. So, historically, the "fundamental forces" are the ones associated with the symmetries of your (classical or quantum) field model (colloquially, 99% of the time this model is the standard model plus the still debated gravity).

• It is unclear why an interaction should only be fundamental if it is a gauge field. Dec 29, 2020 at 19:49
• It's a definition. There is an incredible amount of interactions (or forces, like the one mediated by pions, or inter-atomic forces, or macroscopic forces, and even entropic effective forces.. ), so you need to set boundaries to pick up the "most fundamental". Given the way in which we construct field theories, which is based on symmetries, it'customary to define "fundamental forces" the ones related to the gauge fields (because are related to the "fundamental symmetries" of the standard model). Dec 31, 2020 at 14:14

Pauli's exclusion is not so much a force as a geometric requirement for the wavefunctions of certain kinds of particles -- fermions -- that arises naturally out of quantum mechanics. It is not a force like gravity or electromagnetism.

• Still, it is the main source of force, like this one: nasa.gov/sites/default/files/thumbnails/image/… Dec 28, 2020 at 13:28
• @Anixx it's still electromagnetic force (+gravity) depicted on your picture. Dec 28, 2020 at 23:32
• @Ruslan no. It is Pauli exclusion between electrons. Dec 28, 2020 at 23:34
• @Anixx no, it's not exclusion. Pauli exclusion makes EM interaction more prominent, but it's not what directly makes the atoms repel each other. See also: How can I stand on the ground? EM or/and Pauli? Dec 28, 2020 at 23:55
• Surely (according to general relativity) gravity is also “a geometric requirement”, and yet it counts as a force. Dec 29, 2020 at 17:30

Consider a couple of hypothetical particles, all of the known properties of which, like mass, charge, spin etc., are the same as those of the electron, but one additional property $$\zeta$$ is different for each particle. We thus have a system of electron-like particles, which are, unlike the electrons, distinguishable via the interaction $$Z$$ associated with $$\zeta$$.

By construction, the particles under consideration don't have Pauli exclusion. Now consider an initial state $$\psi(\vec r_1,\vec r_2,\dots,\vec r_N)$$, where $$\vec r_i$$ is the position and spin of $$i$$th particle. Let $$\psi$$ be antisymmetric in the exchange of any pair of particles $$\vec r_i\leftrightarrow\vec r_j$$: we are constructing an initial state for the Schrödinger's equation, so we are allowed to impose this constraint on $$\psi$$, despite the inapplicability of the Pauli principle.

Now, since all the properties of our hypothetical particles (except $$\zeta$$) are the same as those of electrons, the evolution of $$\psi$$ in time would leave its exchange symmetry invariant—if not the $$Z$$ term of the Hamiltonian.

Consider now what happens when the strength $$\sigma$$ of the interaction $$Z$$ approaches zero. Although $$Z$$ generally breaks the exchange antisymmetry, in the limit of $$\sigma\to0$$ this symmetry will become conserved. But now we've constructed a system that behaves exactly as if there were Pauli exclusion: none of these particles can share quantum state (because of the form of $$\psi$$), and this restriction is preserved in time. In fact, what we have constructed is exactly a system of $$N$$ electrons.

Notice how we haven't introduced any force at any point in the above construction. Instead, we've obtained Pauli exclusion as a consequence of the initial state of the system. More generally, it's a consequence of the initial state of the Universe (at some finite time point), and of the form of the creation operators for fermions. All fermions are created antisymmetrized, and all their interactions preserve this antisymmetry. That's all what Pauli exclusion is: it's not an additional interaction, much less so an additional "force of nature".

While it is true that quantum mechanical effects can be used to describe the "effective" force seen as the pauli exclusion principle (as said in most of the positively-recieved answers) - at the end of the day it seems that there is always an extra rule that's added to QM to explain this effect. This extra rule, whether it's that "ferminions are antisymmetric upon exchange" or some sort of spin statistics theorem, is something extra that is added.

Maybe it's not the best to name it a fundamental force, but it, in my opinion, is a separate extra rule that is added to QM or QFT.

... Pauli exclusion principle ... produces such things as repelling of atoms and molecules in solids

Your question boils down to the understanding how is works that the state of electrons in atoms is observed pairwise.

What is an electron

The electron was discovered as an electric charge. Because - in metals - electrons can easily be separated under the influence of an external electrical potential.
The electrons is a magnetic dipole with its own fundamental physical constant $$g_{e^-}$$. Electrons (their magnetic dipols) can easily be aligned under the influence of an external magnetic field. In permanent magnets the common field holds the magnetic dipoles aligned (as long as a rising temperature with its higher mobility of the atoms does not destroys the aligned dipols).

The order of the discoveries of the charge and the magnetic dipole of the electron have determined our view of the electron. However, it would be appropriate to treat the electron equally as a charge and as a magnetic dipole.

What is the spin of subatomic particles

A moving electron - under the influence of an external magnetic field - is deflected in the plane that is perpendicular to both the electron's direction of propagation and the direction of the external magnetic field. In the process, the electron emits photons and slows down until all its kinetic energy is exhausted.

This mechanism is responsible for both the Lorentz force and the Hall effect.

Instead of speaking of an inexplicable spin (it is usually said that nothing spins there, but this signals the analogy to the gyroscopic effect), it would be more purposeful to speak of the interaction between the external field and the magnetic dipole of the electron.
If one also takes into account that the emission of photons is accompanied by the exchange of an impulse to the electron, one could explain the deflection of the electron without a spin as follows.

The photon emission gives the electron a sideways momentum and at the same time disturbs the alignment of the magnetic dipole of the electron. The external magnetic field immediately realigns the magnetic moment and the process starts again - as long as the electron still has kinetic energy that can be converted into photons.

That I‘m talking about to show that the intrinsic (existing independent from external circumstances) magnetic dipole of the electron is a very important property of electrons.

The electrons magnetic dipole and the atomic structure

When you are just starting out in physics, it should be easier for you to replace the term spin with the term magnetic dipole. The longer you are a physicist, the less acceptable this seems.

But indeed, it is an easy model to imagine if two electrons in an atom, occupying the same first three quantum numbers, are aligned antiparallel by their spin (just as two permanent magnets might be paired with their poles in a rectangle (like the 69 resp face-to-foot).

Loss of energy as the electron approaches the nucleus

The electrons charge is a constant value in all measurement we have done. Indeed? what is about the electron as it appoaches the nuclues. Where the energy comes from which is emitted in form of photons? How the electrical neutrality of an atom is explained although the positive charges are concentrated in the nucleus?

The claimed charge constancy overshadows that for bounded electrons and protons in atomic interactions the electric charges are exhausted by photon emission and the magnetic dipoles are responsible for the atomic structures.

Conclusion There is a magnetic interaction and an electric interaction between electrons. For bound electrons, the magnetic interaction is the more important one and it explains the Pauli principle.

Although everything inferred here is based on existing observations and in particular the laws of conservation of energy and momentum remain completely valid, don't take it too seriously. Because, you know, old physicists are not amused and even annoyed by thinking in generally new ways.

• I don't see what this answer has to do with the question. The exclusion principle is a general phenomenon for fermions, regardless of whether they have charges and magnetic moments or not. The notion that spin can be replaced with "magnetic dipole" is decidedly non-mainstream, since particles can have spin without interacting with the electromagnetic field at all. Dec 29, 2020 at 17:21

The fact that the Pauli exclusion principle is associated withn spin states, would seem to suggests to that it is a result of a magnetic interaction.

• It is actually a consequence of the discrete spacetime symmetries CPT. Dec 28, 2020 at 14:10
• R.W.Bird How acceptable or speculative is my answer? My answer corresponds with your answer? Dec 30, 2020 at 12:05
• Absolutely not. Jan 1, 2021 at 12:13
• Note to flaggers: please don’t use “not an answer” for posts which are answers but happen to be incorrect.
– rob
Jan 4, 2021 at 2:50


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# Aeroplane velocity problem

1. Oct 1, 2005

### brandon26

An aeroplane is climbing at an angle of 2 degrees while maintaining a speed of 400ms^-1. A package is released and travels a horizontal distance of 2500m before hitting the ground. The initial velocity of the package is the same as the initial velocity as the plane, find the height of the aeroplane above the ground at the moment the package was released.

I did some calculations but my answer is apparantly wrong.

Considering motion in the vertical plane.
a= 9.8
s=?
u=400sin88
t=?

To find t i used the horizontal plane and got 6.25 sec.

therefore s = ut + 05.at^2
(400sin88 x 6.25) + 0.5x9,8x6.25^2
=2700M

Where did I go wrong?

2. Oct 1, 2005

### Päällikkö

You've mixed the x and y directions.
$$u_y \ne 400 \sin 88^o$$

3. Oct 1, 2005

### brandon26

Can you explain please? I thought the x direction was 400cos2 and y direction was 400sin88?

4. Oct 1, 2005

### Päällikkö

"climbing at an angle of 2 degrees" means 2 degrees from the horizontal.
Draw a diagram, and you'll see that cos is used for the x-direction.

sin gets its maximum at 90 degrees while cos reaches it at 0. ~400 m/s vertically and ~0m/s horizontally is a steep climb for an aeroplane.
Oh, and cos 2o = sin 88o

5. Oct 1, 2005

### brandon26

Yes, so the x direction is 400cos2, and y direction is 400sin88?

6. Oct 1, 2005

### Päällikkö

No.
That would mean vx = vy, which is certainly not the case here. That'd mean the elevation was 45 degrees.

For simplicity, use the same angle to express the velocities.

7. Oct 1, 2005

### brandon26

Oh right!. So Vx=400cos2 and Vy=400cos88.

8. Oct 1, 2005

### Päällikkö

Correct, although I would've used 400cos2 and 400sin2.

9. Oct 1, 2005

### brandon26

Yes, that true.

format: plain


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# What is the importance of the translational invariance of the CWT?

Translational invariance is a property that the continuous wavelet transform (CWT) has but the discrete wavelet transform (DWT) does not have. It says that a shift of the signal, i.e. $$x(t)\rightarrow x(t-d)$$, leads to a shift of the wavelet coefficients, i.e. $$W_x(a,b)\rightarrow W_x(a,b-d)$$ without other modification. What is the importance of the translational invariance of the CWT?

First, translational invariance makes signal processing somehow independent on the time origin of the recording. The term "invariance" might be debated, as done here: What is the difference between "equivariant to translation" and "invariant to translation". Related concpets are called rotational invariance, scale invariance, etc.

To explain the time origin stuff: imagine that you record the same signal in different time zones, this allows to yield the "same result", eg same coefficients (translated), or same overall processing result. Sometimes, one only need invariance on some coefficient-based feature (like magnitude), sometimes only approximately so.

You also get shift invariance with undecimated discrete wavelets, cycle-spinning, etc. Since this is a bit oversampled, this was a motivation behind the development of dual-tree or complex discrete wavelets. You can check references in Section 2.3.4 "Translation Invariant Wavelets" from our paper A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity.

For instance if you perform a one-level "Haar" tranform on signals $$[\ldots \, 0 \,2\, 3\, 0\, -1\, 0\, 0\, \ldots]$$ and on $$[\ldots \, 0 \, 0 \,2\, 3\, 0\, -1\, 0\, \ldots]$$ you will get of the detail coefficients (applying the difference $$[1 \,-1])$$ on every two samples:

$$[\ldots \, -2 \,3\, -1\, 0\, \ldots]$$ and

$$[\ldots \, 0 \,-1\, 1\, 0\, \ldots]$$ which are not shifts of one another. This could lead to different results, for instance if you choose a threshold on 1.

• Thank you for your answer. What is the effect of 'translation invariance' on the wavelet power spectrum? Sep 25, 2021 at 14:45
• Typically, it is a feature (magnitude squared) derived from the coefficients. Thus, it will be invariant as well Sep 25, 2021 at 14:56

CWT is translation-invariant in feature sense: translating a pattern translates its representation but not modify it. In coefficient sense, it is translation equivariant: shift signal $$\Leftrightarrow$$ shift representation. CWT is not invariant in coefficient sense. I will stick to latter definition.

Translation equivariance is a basic and powerful prior. It is required for time-frequency localization: instantaneous frequency, amplitude, and phase (if a sample moves and representation doesn't, we've not 'located' it). The Fourier modulus is invariant, but phase isn't, nor is phase equivariant.

### Global vs local shifts

Translations(/shifts) can be local and global. Equivariance to

• global shifts, as in changing start/end of a tape recording or trimming an image, is desired to keep features same except at boundaries. The Fourier modulus is invariant to global shifts, but its phase isn't, nor is it equivariant.
• local shifts, as in speaking words in different order or moving an apple in an image, is desired to keep features same but preserving relative locality. These change the Fourier representation entirely, despite only parts of input changing and class labels identical in several regards. Such change is desired neither for classification nor descriptor extraction (e.g. edge detection, measuring transients).

CWT is equivariant to both (to local shifts within time resolution). DWT is equivariant to neither: a shift can change every coefficient without equals to unshifted.

Lastly: among the three, only CWT can be extended to build invariants. This is done with e.g. wavelet scattering. Equivariance is also a core element of stability against warp deformations. Explanation and advantages described in this post.

### Power spectrum

Can be discussed in three perspectives:

• along time, sum(abs(Wx)**2, axis=0): determined by LP summation - see "Energy analysis". In the ideal case (tight frame), it's equal to the signal's.
• along frequency, sum(abs(Wx)**2, axis=1): determined by wavelets in the frequency domain, and discretization scheme (number of wavelets per octave, etc). One can define a "transfer function curve" by summarizing energy transfer as a single number, for each wavelet, via Parseval's theorem, similar to LP summation.
• total, sum(abs(Wx)**2): integrate LP summation

Translation equivariance means if we shift signal by $$c$$, we get same power by shifting temporal integration bounds by $$c$$:

$$\int_{t_0}^{t_1} |\text{CWT}_{x(t)}(\lambda, t)|^2 dt = \int_{t_0 + c}^{t_1 + c} |\text{CWT}_{x(t - c)}(\lambda, t)|^2 dt$$

Shifting in time has no effect on power along frequency, so we remain invariant.

### "Invariant" vs "equivariant"

Both are true but refer to different things. Formally, "equivariance" here means a coordinate transformation of the input identically transforms the same coordinate in the output:

$$f(x, \tau(y), z) \Leftrightarrow \Phi(x, \tau(y), z)$$

which for CWT is

$$f(t - c) \Leftrightarrow W_f (\lambda, t - c)$$

More generally equivariance is commutativity under action of a symmetry group.

"Invariance" refers to identity under group action - that is,

$$F(\Phi(x, y, z)) = F(\Phi(x, \tau(y), z))$$

where $$F$$ is a mapping, and $$\tau$$ a group action. For example, a hollow sphere, shifted, keeps all its descriptors identical relative to own center of mass, such as distance to each point. A similar mapping can be made for CWT.


text:
# ColorData input arguments not clear

A question about ColorData input arguments.

I want to color a point in a graph I have, but I don't want to use the built in colors (orange, green, blue etc) I saw someone coloring a point using ColorData function, like so:

    Graphics[{ColorData[96, 10], PointSize[Large], Point[{-\[Pi], 0}],
Point[{0, 0}], Point[{\[Pi], 0}], Point[{2 \[Pi], 0}],
Point[{3 \[Pi], 0}], Point[{4 \[Pi], 0}], Point[{5 \[Pi], 0}]}],

(in this case ColorData[96,10] comes out as orange.) I know that the first argument in colordata should be the color scheme, but what is 96 ? is there a chart of all colors in all schemes and numbered?

Thanks!

• The first value goes from 1-113 and you select a sceme and the second from 1-10 – J42161217 Nov 9 '18 at 11:10
• I am confused, does the accepted answer address your questions? Because I don't see how. – Kuba Nov 9 '18 at 11:42
• @Kuba I finished the numbered chart. do you like it? – J42161217 Nov 9 '18 at 12:03
• @J42161217 I was rather expecting to get an answer to witch, named or whatever scheme, does e.g. ColorData[96] corresponds to. But if that was not OP's question then fine :) – Kuba Nov 9 '18 at 12:12

GraphicsGrid[


text:
# Best way to find an object in a picture

I'm trying to find an object in a picture, my solution is to take the picture and a photo of the object and find the maximum of the mutual covariance, this is my project on GitHub: ImageFinder

It works but if the object is rotated or transformed it doesn't work well. Which other method should I use?

I thought that a good way can be to look for the object through the colors and then try to rotate the image or to resize it while does not match, but this idea need an improvement.

There is no "best method", it will vary according to application. There are many trains of thought regarding this subject.

I've done something in that area using Cepstral Analysis. Take a look at this:

Note: I am the author of the above.

Basically, you do the following:

1. Sum the target image and the object
2. Calculate the Cepstrum of said sum.
3. Find the peaks (there will be 2, you have to resolve the ambiguity using something like SAD)
4. The location of the peak will tell you the translation you must apply to the image.


text:
# 24. The systematic name of the coordination compound K [Co(HzOJl,] is potassium diaquotetraiodocobaltate(L). True False25 The oxidation number of Co

###### Question:

24. The systematic name of the coordination compound K [Co(HzOJl,] is potassium diaquotetraiodocobaltate(L). True False 25 The oxidation number of Co in [Co(NH;AChJCl is +L True B) False

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Hess' law may be used t0 determine enthalpy changes using average bond energies as snown in the diagram; isolated gaseous atoms reaciants products U Is the sum of the average bond onergies of Ihe reactants, and Is the sum of (he average bond energies of (he producls, For Iho reacllon shown belo...
##### Conducting spherical shell having an inner radius a and outer radius b carries a net charge 2Q If a point charge -Qis placed at the center of this shell, the surface charge density on the outer surface (atr :b) is (in mCm2) isConsider Q-2 mC,a= 5cm and b=) (10 cm47.7-47.715.9-15.931.8
conducting spherical shell having an inner radius a and outer radius b carries a net charge 2Q If a point charge -Qis placed at the center of this shell, the surface charge density on the outer surface (atr :b) is (in mCm2) is Consider Q-2 mC,a= 5cm and b=) (10 cm 47.7- 47.7 15.9- 15.9 31.8...
##### The mean survivial time after diagnosis for a certain disease is 20 years with a standard deviation of 5 years. Based on a particular treatment, an individual's predicted survival time is 1.1 standard deviations beyond the mean. How much longer; after diagnosis, is the patient expected to survive?
The mean survivial time after diagnosis for a certain disease is 20 years with a standard deviation of 5 years. Based on a particular treatment, an individual's predicted survival time is 1.1 standard deviations beyond the mean. How much longer; after diagnosis, is the patient expected to survi...
##### O b beZ2 oleme Given (4) (15 points) Let R8 men sma,addition and , lti fasti...
o b beZ2 oleme Given (4) (15 points) Let R8 men sma,addition and , lti fasti that R is a ring of 8 elements under addition and multiplication of matrices. (a) Show that is an ideal of R. (b) Decide if R/I is commutative. (c) Decide if R/I is an integral domain....
##### Hello! I have a question related to Ansys Fluent. I am studying on the air flow...
Hello! I have a question related to Ansys Fluent. I am studying on the air flow 2-D. I used the triangular mesh and refinement. I am a beginner of the Ansys Fluent, so I used to see some video on Youtube, they said that after running calculation and if it appears "the solution is converged",...
##### A website advertises job openings on its website, but jobseekers have to pay to access the list of job openings. The websiterecently completed a survey to estimate the number of days it takesto find a new job using its service. It took the last 31 customersan average of 80 days to find a job. Assume the population standarddeviation is 10 days. Construct a 99% confidence interval of thepopulation mean number of days it takes to find a job.Multiple Choice[74.8535, 85.1465][59.3450, 100.6550][75.94
A website advertises job openings on its website, but job seekers have to pay to access the list of job openings. The website recently completed a survey to estimate the number of days it takes to find a new job using its service. It took the last 31 customers an average of 80 days to find a job. As...
##### QUESTION 21 50 The following data represent the beginning inventory and, in order of occurrence, the...
QUESTION 21 50 The following data represent the beginning inventory and, in order of occurrence, the purchases and sales of Meow, Inc. for an operating period. Units Unit Cost Total Cost Units Sold Beginning Inventory 50 $62$3,100 Sale No. 1 Purchase No. 1 30 50 1,500 Sale No. 2 32 Purchase No. 2 4...
##### QUESTION _ [8 MARKS]Determine the Toclaff(x)e 0L"USMappropriate method you have learned this course the given interval (-5.doesn bracket the rot: Perform the computation1090 with true value9580 .Marks)
QUESTION _ [8 MARKS] Determine the Toclaf f(x)e 0L" USM appropriate method you have learned this course the given interval (-5. doesn bracket the rot: Perform the computation 1090 with true value 9580 . Marks)...
##### The solid substance that is collected dried and weighed is Cu(s). 6) What is the mass...
The solid substance that is collected dried and weighed is Cu(s). 6) What is the mass of Cu(s) collected? (Report in grams using all digits shown).(0.5 points) Cu 9 7) What is the theoretical yield of Cu(s)? (Report in grams using 4 significant figures) (2 points) 1.389 8) What is the percent yield ...
##### 4. (15 points) Write the double integral / S 32 - x?dA over the triangle with...
4. (15 points) Write the double integral / S 32 - x?dA over the triangle with vertices (-1,0), (0,0) and (0,1) changing the variables by u = y - x and v = y + x. (DONOTEVALUATE INTEGRAL)...
##### What is the current in a 100-W bulb connected to a 120-V source?
What is the current in a 100-W bulb connected to a 120-V source?...
##### I need e & f 1. Determine the price rangs ratio (PE) c. What is the...
i need e & f 1. Determine the price rangs ratio (PE) c. What is the stock price using the PE ratio valuation method d. What is the stock price using the dividend discount model? e. What would happen to the PE (PE) and stock price if the company increased s ertion als gs in the form of dividends...
##### What is the implicit derivative of 13=2xe^y-x^3e^(2y) ?
What is the implicit derivative of 13=2xe^y-x^3e^(2y) ?...
##### The Coombs blood test is 95% effective in detecting Iron-deficiency anaemia (IDA): condition which effects bout 4% of the adult population, when it is in fact present; however, the test also yields 'false positive results for 7% of healthy people tested What is the probability that person randomly selected from the population has positive test result? Suppose that person does have positive test, then what is the probability that they actually have IDA? Comment:
The Coombs blood test is 95% effective in detecting Iron-deficiency anaemia (IDA): condition which effects bout 4% of the adult population, when it is in fact present; however, the test also yields 'false positive results for 7% of healthy people tested What is the probability that person rando...
##### The purpose of this assignment is to encourage you to apply some of the principles in...
The purpose of this assignment is to encourage you to apply some of the principles in the Memory chapter to your life experiences and study strategies. Respond to the following questions in text box or uploaded Word file: 1. Describe an example of a flashbulb memory from your life. (2 pts) 2. Discus...
##### Use the table of integrals at the back of the book to evaluate the integrals. $\int \cos \frac{\theta}{3} \cos \frac{\theta}{4} d \theta$
Use the table of integrals at the back of the book to evaluate the integrals. $\int \cos \frac{\theta}{3} \cos \frac{\theta}{4} d \theta$...
##### Estimate cost
Speedy Parcel Service operates a fleet of delivery trucks in a large metropolitan area. A careful study by the company’s cost analyst has determined that if a truck isdriven 138,000 miles during a year, the average operating cost is 14.8 cents per mile. If a truck is driven only 92,000 miles d...
##### Help needed in andwering this full question Question Three a. For each chiral centre in the...
help needed in andwering this full question Question Three a. For each chiral centre in the following molecules, assign an Ror S configuration. For full marks, you must clearly indicate the order of priority of the groups attached to each chiral centre, OH OH CH3 CH(OCH3)2 CHO b. Draw all stereoi...
##### 1) List and describe the steps of animal development from zygote to the gastrula 2) Describe...
1) List and describe the steps of animal development from zygote to the gastrula 2) Describe the process of neurulation 3) Describe the process of axis formation...
##### Explain expectation computations
Explain expectation computations...
##### 1) A point charge 4.60 μC is held fixed at the origin. A second point charge...
1) A point charge 4.60 μC is held fixed at the origin. A second point charge 1.30 μC with mass of 2.80×10−4 kg is placed on the x axis, 0.260 m from the origin. A. What is the electric potential energy U of the pair of charges? (Take U to be zero when the charges have infinite sepa...
##### Classify the critical point at the origin of the linear system below:dx 6x dt dy =3x+ 7yThe origin is alanbecause the eigenvalue(s)(Simplify your answer: Use a comma to separate answers as needed. Type an exact answer; using radicals and as needed:)
Classify the critical point at the origin of the linear system below: dx 6x dt dy =3x+ 7y The origin is alan because the eigenvalue(s) (Simplify your answer: Use a comma to separate answers as needed. Type an exact answer; using radicals and as needed:)...
##### 17. Draw all monochloro products for the chlorination of 2,3-dimethylbutane Calculate the percentage of each compound formed Crde the major product:
17. Draw all monochloro products for the chlorination of 2,3-dimethylbutane Calculate the percentage of each compound formed Crde the major product:...
##### In Exercises 23 - 32, find a formula for for the arithmetic sequence. $10, 5, 0, -5, -10, \cdots$
In Exercises 23 - 32, find a formula for for the arithmetic sequence. $10, 5, 0, -5, -10, \cdots$...
##### Determine the convergence set of the given power series2 (x- 8)n n2 + 2nThe convergence set is (Type your answer interval notation )
Determine the convergence set of the given power series 2 (x- 8)n n2 + 2n The convergence set is (Type your answer interval notation )...
##### Hardworking studcnt,plagued by too much homework you sPend _ all night doing math homework By 6am, you unagine yoursclf to be rcgion boundcd by 21As you RIOW more and more tucd, the world bcgins t0 spmn 4ound you Howcier, according to Nelton, thcc is no difference bctween the World spinning around you; and you spinning around thc world Unfortunately; YOU arc s0 tired that You think the world 1s the X-axis. What is the volume ofthe solid you (the rcgion) crcatc by spinning about the * - axts ? Pr
hardworking studcnt,plagued by too much homework you sPend _ all night doing math homework By 6am, you unagine yoursclf to be rcgion boundcd by 21 As you RIOW more and more tucd, the world bcgins t0 spmn 4ound you Howcier, according to Nelton, thcc is no difference bctween the World spinning around ...
##### 3. Refer to the function F in the graph shown. Use the graph to determine whether F"(c), exists at â‚¬ = b, d, f, hFC)
3. Refer to the function F in the graph shown. Use the graph to determine whether F"(c), exists at â‚¬ = b, d, f, h FC)...
##### Returning to the "No-Shave November" question; what kind of ANOVA design is this?Two-way between-groups ANOVATwo-way within-groups ANOVAOne-way within groups ANOVAOne-way between groups ANOVATwo-way mixed-design ANOVA
Returning to the "No-Shave November" question; what kind of ANOVA design is this? Two-way between-groups ANOVA Two-way within-groups ANOVA One-way within groups ANOVA One-way between groups ANOVA Two-way mixed-design ANOVA...
##### Gather a list of objectives from a recent healthcare conference brochure (usually can be located online—examples: PriMed...
Gather a list of objectives from a recent healthcare conference brochure (usually can be located online—examples: PriMed, Contemporary Forums, etc). Examine the objectives that the learners are to accomplish after attending the program. Determine if the objectives are well written (based on th...


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Research

# View Profile

## Publication details for Dr Julie Wardlow

Nayyeri, H., Cooray, A., Jullo, E., Riechers, D. A., Leung, T. K. D., Frayer, D. T., Gurwell, M. A., Harris, A. I., Ivison, R. J., Negrello, M., Oteo, I., Amber, S., Baker, A. J., Calanog, J., Casey, C. M., Dannerbauer, H., De Zotti, G., Eales, S., Fu, H., Michałowski, M. J., Timmons, N. & Wardlow, J. L. (2017). Herschel and Hubble Study of a Lensed Massive Dusty Starbursting Galaxy at z ∼ 3. The Astrophysical Journal 844(1): 82.

### Abstract

We present the results of combined deep Keck/NIRC2, HST/WFC3 near-infrared, and Herschel far-infrared observations of an extremely star-forming dusty lensed galaxy identified from the Herschel Astrophysical Terahertz Large Area Survey (H-ATLAS J133542.9+300401). The galaxy is gravitationally lensed by a massive WISE-identified galaxy cluster at z ~ 1. The lensed galaxy is spectroscopically confirmed at z = 2.685 from detection of $\mathrm{CO}\,(1\to 0)$ by GBT and from detection of $\mathrm{CO}\,(3\to 2)$ obtained with CARMA. We use the combined spectroscopic and imaging observations to construct a detailed model of the background dusty lensed submillimeter galaxy (SMG), which allows us to study the source plane properties of the target. The best-fit lens model provides magnifications of μ star = 2.10 ± 0.11 and μ dust = 2.02 ± 0.06 for the stellar and dust components, respectively. Multiband data yield a magnification-corrected star formation rate of 1900(±200) M ⊙ yr−1 and a stellar mass of ${6.8}_{-2.7}^{+0.9}\times {10}^{11}\,{M}_{\odot }$, consistent with a main sequence of star formation at z ~ 2.6. The CO observations yield a molecular gas mass of 8.3(±1.0) × 1010 M ⊙, similar to the most massive star-forming galaxies, which together with the high star formation efficiency, are responsible for the intense observed star formation rates. The lensed SMG has a very short gas depletion timescale of ~40 Myr. The high stellar mass and small gas fractions observed indicate that the lensed SMG likely has already formed most of its stellar mass and could be a progenitor of the most massive elliptical galaxies found in the local universe.


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# FileNotFoundError: [Errno 2] No such file or directory: 'C:/Users/Utilisateur/OneDrive/Bureau/PROJET7/data_train.csv'

I am trying to deploy my app on streamlit cloud, and I am having a problem with paths.

Here is my repo https://github.com/SaidMANAM/Application-IA-scoring_CREDIT

Thanks in advance for the help !

Hi @SaidMANAM, welcome to the Streamlit community!

It’s important to note that absolute paths to your local computer directory mapping is unlikely to work on another computer, irrespective of using Streamlit Cloud or another hosting service. You need to use relative references to the files for your code to work.

Best,
Randy

When you open a file with the name “filename.ext”; you are telling the open() function that your file is in the current working directory . This is called a relative path.

file = open('filename.ext') //relative path

In the above code, you are not giving the full path to a file to the open() function, just its name - a relative path. The error “FileNotFoundError: [Errno 2] No such file or directory” is telling you that there is no file of that name in the working directory. So, try using the exact, or absolute path.

file = open(r'C:\path\to\your\filename.ext') //absolute path

In the above code, all of the information needed to locate the file is contained in the path string - absolute path.

If the user does not pass the full path to the file (on Unix type systems this means a path that starts with a slash), the python file path is interpreted relatively to the current working directory. The current working directory usually is the directory in which you started the program. In order to make this work, the directory containing the python executable must be in the PATH, a so-called environment variable that contains directories that are automatically used for searching executables when you enter a command. In any case, if your Python script file and your data input file are not in the same directory, you always have to specify either a relative path between them or you have to use an absolute path for one of them.

1 Like


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# GameboyCore Retrospective

Small reflection on the GameboyCore projects and future projects.

Github

GameboyCore

GameboyCore is a Gameboy emulator library written in C++. It is my first emulator project and was quite a lot of fun to work on.

I think a lot went right with this project. It is modular making it quite flexible to use. Like in the GameboyCore Retro project and GameboyCore Python. It had tests from the start, though they are more integration tests and not really unit tests.

It’s not technically complete, there is still the RTC register and some graphics bugs.

I think where this project lacked was useful tooling. I made a disassembler to debug ROMs. However it was naive in implementation and disassembled everything including data sections. It would have been advantagous to perform a proper disassembly following code paths and annotating common instructions. Eventually the disassembly did make its way into the debug window but was poorly optimized and a memory hog.

Also it would have been useful to have better execution control like single stepping or running until a certain address is hit. I did this through VS debugger and using if statements.

Something like:

if (pc == 0x4000)
{
int x = 0; // break point here.
}

Which is really silly when you think about it.

GameboyCore Retro

I think creating a retro core from GameboyCore was a good move. I’m interesting in making the emulator not necessarily the GUI that goes with it. Especially considering I’ll be working on multiple emulator projects and making individual GUIs for those would be tedious. I’d be inclined be make something more generic and RetroArch has already done that.

GameboyCore Python

GameboyCore Python is a Python API to GameboyCore. It allows you to create a gameboy core instance and manipulate it from scripts. I had made this with the intension of writing a bot/AI that can play a specific game. Inspired by the MarIO project. Though… I haven’t really done anything on that front. It would be an interesting machine learning project to tackle.

## Future Projects

I’ll probably work on a NES and GBA emulator at some point in the future.

Currently I have started an NES project in Rust. I do like Rust but I wonder if I should continue doing things in C++ just for a sake of improving in that direction. Hm.

Before I get into the nitty gritty of a new emulator I’d like to build some tooling around it.

First I’m going to write a NES disassembler that does a proper transversal of the data and only disassembles code.

Then I’m planning on making a debugger for my emulator projects. Something like a generic frontend that is fed information from the backend. Like the values of CPU registers or a specific range of memory. It would also allow run/step/runto control and display an annotated diassembly output.


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Perceptually Optimized Encoder Control

In conventional encoder control algorithms, the distortion between an original picture s and the reconstructed picture s′ is measured by the sum of squared Errors $DSSE(s,s')$ or its normalized logarithm, the PSNR. However, it is well known that the PSNR in general does not correlate well with subjective judgment of image quality. We developed an encoder control algorithm that is based on a weighted distortion measure where the weights are spatio-temporally varying and reflect the local visual error sensitivity of the underlying content. For the definition of the weighted distortion measure $D_{WSSE}$, we partition a video frame into blocks $B_k$ and define
$$D_{WSSE}(s,s')=\sum_{k}w_k(s)\cdot D_{SSE,k}(s,s').$$
Here, $D_{SSE,k}$ is the sum of squared errors on $B_k$ and wk(s) are local weights. The less activity, measured by a high-pass filter in the spatial or temporal direction, is present on $B_k$, the larger the weights wk(s) become. This reflects the well known fact that, due to reduced perceptual masking capabilities, low-frequency content is subjectively more sensitive to reconstruction errors than high-frequency content.

The well-established block-based encoder control for the unweighted sum of squared errors can still be used for our perceptual error measure. The only difference is that the Lagrangian parameter is spatio-temporally varying depending on the weights $w_k$. More precisely, if $\lambda$ is a fixed Lagrangian parameter that defines an operational point on the rate-distortion curve for the error measure $D_{WSSE}$ and if $R_k$ denotes the rate on the block $B_k$, then under some simplified assumptions, on each block $B_k$, the encoder needs to minimize the well-known term
$$D_{SSE,k}(s,s')+\lambda_k\cdot R_k,\:\:\:\lambda_k=\frac{\lambda}{\omega_k}.$$

The effect of our perceptually optimized encoder control is illustrated below. In the middle image, which arises using a conventional encoder control, highly visible transmission errors in the water on the top right part can be observed. With our perceptutally optimized encoder control, such errors are no longer visible, as can be seen from the bottom picture.

References

• C. R. Helmrich, S. Bosse, M. Siekmann, H. Schwarz, D. Marpe, and T. Wiegand, “Perceptually Optimized Bit Allocation and Associated Distortion Measure for Block-Based Image or Video Coding,” in Proc. IEEE Data Compression Conf. (DCC), Snowbird, Mar. 2019.
• J. Erfurt, C. R. Helmrich, S. Bosse, H. Schwarz, D. Marpe, and T. Wiegand, “A Study of the Perceptually Weighted Peak Signal-to-Noise Ratio (WPSNR) for Image Compression,” in Proc. IEEE Int. Conf. Image Process. (ICIP), Taipei, Sep. 2019.


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Preprint C95/2009
Direct Methods for Monotone Variational Inequalities
José Yunier Bello Cruz
Keywords: Armijo-type search | Convex minimization problem | Korpelevich's me- thod | Nonsmooth optimization | Maximal monotone operators | monotone variational inequalities | Point-to-set operator | Projected gradient method | Projection method | quasi-Fejer convergence |
We analyze one-step direct methods for nonsmooth variational inequality problems, establishing convergence under paramonotonicity of the operator. Previous results on the method required much more demanding assumptions, like strong or uniform monotonicity, which imply uniqueness of solution, which is not the case for our approach. We introduce a fully explicit method for solving monotone variational inequalities in Hilbert spaces, where orthogonal projections onto the feasible set are replaced by projections onto suitable hyperplanes. We prove weak convergence of the whole generated sequence to a solution of the problem, under the only assumptions of continuity and monotonicity of the operator and existence of solutions. We also introduce a two-step direct method, like Korpelevich's. The advantage of our method over that one is that ours converges strongly in Hilbert spaces, while only weak convergence has been proved for Korpelevich's algorithm. Our method also has the following desirable property: the sequence converges to the solution of the problem which lies closest to the initial iterate.


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# [Tex/LaTex] Matrix with rows and columns labeled

matrices

It is trivial to make matrices like so:

\begin {pmatrix} s & t & a \\ c & k & x \\ cha & n & ge \end {pmatrix}

It is not so trivial to impose labels on the rows and columns as done in page 410 here. I would like to do this. For example, the matrix in 11.6:

Note: I am aware of this thread but this is not what I'm looking for.

Here are three different option using \bordermatrix, kbordermatrix and blkarray:

\documentclass{article}
\usepackage{amsmath,kbordermatrix,blkarray}

\renewcommand{\kbldelim}{(}
\renewcommand{\kbrdelim}{)}

\begin{document}
$\mathbf{P} = \bordermatrix{ & H & Y & D \cr H & .8 & .2 & .0 \cr Y & .3 & .4 & .3 \cr D & .2 & .1 & .7 } \qquad \mathbf{P} = \kbordermatrix{ & H & Y & D \\ H & .8 & .2 & .0 \\ Y & .3 & .4 & .3 \\ D & .2 & .1 & .7 } \qquad \mathbf{P} = \begin{blockarray}{cccc} & H & Y & D \\ \begin{block}{c(ccc)} H & .8 & .2 & .0 \\ Y & .3 & .4 & .3 \\ D & .2 & .1 & .7 \\ \end{block} \end{blockarray}$

\end{document}


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Is Shannon Entropy subjective?

1. Jan 24, 2016

T S Bailey

If you have multiple possible states of a system then the Shannon entropy depends upon whether the outcomes have equal probability. A predictable outcome isn't very informative after all. But this seems to rely on the predictive ability of the system making the observation/measurement. This suggests that the amount of Shannon entropy of a system depends on who you ask.

Last edited: Jan 24, 2016
2. Jan 24, 2016

Staff: Mentor

Especially if you ask a Bayesian or a frequentist.

3. Jan 25, 2016

chiro

The normal Shannon entropy for a discrete state system gives a person an idea of the information density or content of that system.

The more random a system is the more information is needed to describe that system. If you have a system with so much entropy and information with so much entropy then the information will decrease the uncertainty of the system itself.

Also realize that entropy - like anything else in mathematics and language in particular has a basis.

You can all sorts of entropies corresponding to the distribution you are looking at - and that includes all possible conditional distributions.

The entropy measures are distribution invariant (as long as they have a finite sample space for the proper entropies) and have a lot of algebraic constraints that are mathematically consistent and consistent probabilistically.


text:
Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.

QUESTION

# What is the formula for manganese(IV) oxide?

MnO_2

To write the chemical formula for Manganese (IV) Oxide, you must make sure to balance the charges.

The roman numeral IV in front of manganese tells us that it is oxidizing at 4^+. Looking at the table, oxygen is a group 16 element which means it oxidizes at a 2^- state, To neutralize the oxidation number of manganese, two atoms of oxygen should be added to achieve the final compound MnO_2.


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# A question about differentiability of implicit functions

Let $$U \subseteq \mathbb{R}^2$$ be an open neighborhood of $$(0,0) \in \mathbb{R}^2$$. Let $$F: U \to \mathbb{R}$$ be a three times continuous differentiable function with $$D_2F(0,0) \neq 0$$. It exists a function $$g:(- \varepsilon, \varepsilon) \to \mathbb{R}$$ for a $$\varepsilon > 0$$ with $$g(0) = 0$$ and $$F(x,g(x)) = 0$$ for all $$x \in (- \varepsilon, \varepsilon)$$.

Now I have to show that $$g$$ is three times continuous differentiable in a neigborhood of $$0 \in \mathbb{R}$$ and I have to find formulas for $$g', g''$$ and $$g'''$$ that depend only on $$g$$ and the derivatives of $$F$$.

The last part should be easy, since we can write $$g'(x)=- \frac{D_1F(x,g(x))}{D_2F(x,g(x))}$$. $$g''$$ and $$g'''$$ can be calculated by differentiation of the previous derivative.

So my question is how to prove that $$g$$ is three times continuous differentiable in a neighborhood of $$0 \in \mathbb{R}?$$ I would appreciate some hints/ help.

• $g$ is $C^3$ because $F$ is. If you differentiate the formula implicitly, solve for $g'$, you will find the the RHS is $again$ differentiable (because $F$ is). Then, rinse and repeat! Conclusion: $g$ is as many times differentiable as $F$ is. For a formal proof by induction, see Pugh p 286 – Matematleta Jan 12 '20 at 16:45

Differentiability of $$g$$ in a neighbourhood of $$0$$ is one of the implications of the implicit function theorem.

So you can differentiate $$F(x,g(x)) =$$ in a whole neighbourhood of $$x=0$$:

$$F_1(x,g(x)) =- F_2(x,g(x)) g^\prime(x)$$ By assumption and continuity of $$D_2F$$ and $$g(x)$$, $$D_2F(x,g(x)) \neq 0$$ in a neighbourhood of $$x=0$$, so this can be solved for $$g^\prime(x)$$ in a neighbourhood of $$0$$. If you do that you will arrive at an equation $$g^\prime(x) =$$ something of class $$C^1$$ (it's not difficult, you already did it, but I want to leave something to do for you...).

So $$g$$ is $$C^2$$ in a neighbourhood of $$0$$. Differentiate once more to conclude it's even $$C^3$$.

• So the clue is to assume that $g$ is differentiable and to show that we can write it as something of class $C^1$ so it really is? – Ludwig M Jan 12 '20 at 16:42
• @lambda As far as I'm concerned the clue is to fully exploit the power of the implicit function theorem, which tells you something about a whole neighbourhood of the the point in question (including differentiability of $g$). Once you know that $g$ is $C^1$ and that you can solve for $g$ (which is also only true since the conclusion of the implicit function theorem holds in a whole neighbourhood) the rest is just formal reasoning. – Thomas Jan 12 '20 at 16:45
• The point is that when you solve for $g'$ the result is $again$ differentiable, so $g''$ exists. $g$ is as many times differentiable as $F$ is. Pugh has a nice proof by induction of this fact. – Matematleta Jan 12 '20 at 16:47
• @Matematleta well, partly. If you look closely at the question of the OP you will note that he already solved for $g^\prime$. The point here, imho, is that people are often ignorant about the fact that the implicit function theorem is a statement which is valid in a whole neighbourhood of $0$. Without this observation you cannot conclude differentiability of $g^\prime$, because for differentiability you (usually) need an open set. – Thomas Jan 12 '20 at 16:51
• Yes, indeed. But this follows directly from the proof of the theorem. My point was only that $g$ is $C^r$ whenever $F$ is. Rudin only proves the $C^1$ case. My favorite proof is the one in Loomis' book. And of course, as you point out. you always need to observe that $g$ is defined in some open nbhd of $0.$ In fact, otherwise $g$ would not be interesting at all! – Matematleta Jan 12 '20 at 16:56


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Reining in the Outliers in Map-Reduce Clusters using Mantri
https://www.usenix.org/legacy/events/osdi10/tech/full_papers/Ananthanarayanan.pdf

## TL;DR:

This paper present Mantri, a system that monitor tasks and culls outliers using cause- and resource-aware techniques. Mantri’s strategies include restarting outliers, network-aware placement of tasks and protecting outputs of valuable tasks. Deployment in Bing's production cluster shows that Mantri improves job completion times by 32%

## Motivation:

Experiment in production shows that outliers in Mapreduce jobs inflate the completion time by 32% at median. The authors argue three root causes of straggler. 1. Machine characteristics 2. Network characteristics: Over-subscribed datacenter network results in congestion among different path. 3. Input Skew: Partitioning data over a low entropy key space leads to a skew in the input sizes.[1]
While the first cause is identified by previous works such as LATE, the second and the third remain unexploited. In addition, previous works only act at the end of a phase, which will miss early outliers. Even worse, none of the previous work protect against data loss.

## Summary:

In short, Mantri is a system that monitors task and culls outliers based on their causes. It uses the following techniques: 1) Restarting outlier tasks cognizant of resource constraints and work imbalances 2) Network aware placement of tasks, and 3) Protecting output of tasks based on cost-benefit analysis.

## Causes of outliers:

Data Skew: Dividing work evenly is a non-trivial problem. First, scheduling each additional task has overhead at the job manager.[2] Network bandwidth might also make it not feasible to split work into multiple tasks and move data over the network. A third reason is poor coding practice.
Cross-rack Traffic: Since the maps are spread across the cluster, regardless of where a reduce task is placed, it will read a lot of data from other racks. Current implementation(e.g. Hadoop) place reduce tasks on any machine with spare slots. Thus, a rack that has too many reduce tasks will be congested on its network link, leading to outliers.
Bad and Busy Machines: Bad Machines which inflate the runtime are randomly spread across the cluster, but recompilations are more localized. Half of them happen on 5% of the machines in the cluster. When a machine loses the output of a task, it has a higher chance of losing the output of other tasks. So, replicate the outputs to protect against data loss on another machine within the same rack might help.
Causes and solutions of outliers

## Mantri:

Based on task progress reports, Mantri estimates for each task the remaining time to finish,
$t_{rem}$
, and the predicted completion time of a new copy of the task,
$t_{new}$
. Tasks report progress once every 10s or ten times in their lifetime, whichever is smaller.(We use
$\bigtriangleup$
to refer to this period.)

### Resource-aware Restart:

Mantri uses two variants of restart, the first kills a running task and restarts it elsewhere, the second schedules a duplicate copy. (In either method, Mantri restarts only when P(
$t_{new}$
<
$t_{rem}$
) is high. Mantri kills and restarts a task if its remaining time is so large that there is more than even chance that a restart would finish sooner.
However, the current job schuler(in 2010) incurs a queueing delay before restarting a task, which can be large and highly variant. Thus, Mantri will schedule duplicates. As we mentioned before, duplicates consume extra resources which may increase job completion time. Hence, when there are outstanding tasks and no spare slots, Mantri schedule a duplicate only if the total amount of computation resource consumed decreases.
When spare slots are available, as happens towards the end of the job, Mantri schedules duplicates more aggressively.

### Network-Aware Placement:

A rack with too many reduce tasks is congested on its downlink and such tasks will straggle. With Mantri, each job manager places tasks so as to minimize the load on the network and avoid self-interference among its tasks. [3]

### Avoiding Recomputation:

To mitigate costly recomputations that stall a job, Mantri protects against interim data loss by replicating task output. It acts early by replicating those outputs whose cost to recompute exceeds the cost to replicate. Mantri estimates the cost to recompute as the product of the probability that the output will be lost and the time to repeat the task.[4]

$t_{rem}$
$t_{new}$


text: Dummy variables in Logit and Probit regression.  A F-test usually is a test where several parametersare involved at once in the null hypothesis in contrast to a T-test that concerns only one parameter. That is, there is no lack of fit in the simple linear regression model. We use the general linear F-statistic to decide whether or not: In general, we reject $$H_{0}$$ if F* is large â or equivalently if its associated P-value is small. That is, adding latitude to the model substantially reduces the variability in skin cancer mortality. The following joint test gives exactly the same test statistics and conclusion as the F test shown after regression 1. This is my situation, two regression models: Well, I would avoid standardized values here. That is, adding height to the model does very little in reducing the variability in grade point averages. If you convert to standard deviations you will be getting your results in some obscure unit (1 sd's worth of dollars/euros/yuan/yen, whatever) that nobody understands. Perhaps, I did not mention before, the two models, although measuring weekly spending as the dependent variable, are represented by a different outcome variable name, I tried the suest approach, but it did not work, in the above example, foerign was not found. For the student height and grade point average example, the P-value is 0.761 (so we fail to reject $$H_{0}$$ and we favor the reduced model), while for the skin cancer mortality example, the P-value is 0.000 (so we reject $$H_{0}$$ and we favor the full model). Upon fitting the reduced model to the data, we obtain: Note that the reduced model does not appear to summarize the trend in the data very well. To determine if this difference is statistically significant, Stata performed an F-test which resulted in the following numbers at the bottom of the output: R-squared difference between the two models = 0.074; F-statistic for the difference = 7.416 Does alcoholism have an effect on muscle strength? This handout is designed to explain the STATA readout you get when doing regression. Where are we going with this general linear test approach? helps answer this question. The P-value answers the question: "what is the probability that weâd get an F* statistic as large as we did, if the null hypothesis were true?" • Nowwecanﬁtthemodel. But just to clarify, the, Thanks. The P-value is calculated as usual. The P-value is determined by comparing F* to an F distribution with 1 numerator degree of freedom and n-2 denominator degrees of freedom. The F-test, when used for regression analysis, lets you compare two competing regression models in their ability to "explain" the variance in the The test statistic of the F-test is a random variable whose Probability Density Function is the F-distribution under the assumption that the null hypothesis is true. I was wondering if the different dependent variable name might be the problem. Interpreting regression models • Often regression results are presented in a table format, which makes it hard for interpreting effects of interactions, of categorical variables or effects in a non-linear models. But if that is the case, then it is also true that in natural units, a 1SD change in the predictor variable means something very different in the two subpopulations, so knowing how the coefficients look in those units may not be very useful. Regression Modelling That is, there is lack of fit in the simple linear regression model. As you can see, Minitab calculates and reports both SSE(F) â the amount of error associated with the full model â and SSE(R) â the amount of error associated with the reduced model. For example, suppose you have two regressions, y = a1 + b1*x and z = a2 + b2*x You rename z to y and append the second dataset onto the first dataset. That is, we take the general linear test approach: Recall that, in general, the error sum of squares is obtained by summing the squared distances between the observed and fitted (estimated) responses: $$\sum(\text{observed } - \text{ fitted})^2$$. That is, the general linear F-statistic reduces to the ANOVA F-statistic: For the student height and grade point average example: $$F^*=\dfrac{MSR}{MSE}=\dfrac{0.0276/1}{9.7055/33}=\dfrac{0.0276}{0.2941}=0.094$$, $$F^*=\dfrac{MSR}{MSE}=\dfrac{36464/1}{17173/47}=\dfrac{36464}{365.4}=99.8$$. For simple linear regression, the full model is: $$y_i=(\beta_0+\beta_1x_{i1})+\epsilon_i$$. Data Source: Stata-format data set auto.dta supplied with Stata Release 8. Latent variables. On the surface, there is nothing wrong with this approach. The first model is for the overall sample excluding a sub-set while the second model applies only for the sub-set of samples. Downloadable! Hi Andrew, thanks so much for the explanation. I need to test whether the cross-sectional effects of an independent variable are the same at two … Regression: a practical approach (overview) We use regression to estimate the unknown effectof changing one variable over another (Stock and Watson, 2003, ch. conclusion of the F test of the joint null hypothesis is not always consistent with the conclusions 2. The "general linear F-test" involves three basic steps, namely: As you can see by the wording of the third step, the null hypothesis always pertains to the reduced model, while the alternative hypothesis always pertains to the full model. The Pennsylvania State University Â© 2020. Do you know if getting the standardized beta coefficients might work here? Some researchers (Urbano-Marquez, et al, 1989) who were interested in answering this question collected the following data (Alcohol Arm data) on a sample of 50 alcoholic men: The full model is the model that would summarize a linear relationship between alcohol consumption and arm strength. Model 1 assumes that the marginal effect of each explanatory variable is a constant; it is linear in the explanatory variables wgti and mpgi. I have two models (Model 1 and Model 2), with different set and number of independent variables. So there are no issues of different measurement units to be reconciled. The reduced model, on the other hand, is the model that claims there is no relationship between alcohol consumption and arm strength. In this case, there appears to be no advantage in using the larger full model over the simpler reduced model. doesn't appear to be a relationship between height and grade point average. The test applied to the simple linear regression model. I currently encounter a similar question: to test the equality of two regression coefficients from two different models but in the same sample. The Linear Probability Model. If you need help getting data into STATA or doing basic operations, see the earlier STATA handout. What we need to do is to quantify how much error remains after fitting each of the two models to our data. regression (2) is the regression (1) with more variables, you should conduct a Likelihood Ratio test. twopm fits two-part models for mixed discrete-continuous outcomes. Goodness-of-fit statistics. This concludes our discussion of our first aside on the general linear F-test. For simple linear regression, it turns out that the general linear F-test is just the same ANOVA F-test that we learned before. Look what happens when we fit the full and reduced models to the skin cancer mortality and latitude dataset: Here, there is quite a big difference in the estimated equation for the full model (solid line) and the estimated equation for the reduced model (dashed line). The F-test for linear regression tests whether any of the independent variables in a multiple linear regression model are significant. The F-statistic is: $$F^*=\dfrac{MSR}{MSE}=\dfrac{504.04/1}{720.27/48}=\dfrac{504.04}{15.006}=33.59$$. The F-test has a simple recipe, but to understand this we need to define the F-distribution and 4 simple facts about the multiple regression model with iid and normally distributed error In the following statistical model, I regress 'Depend1' on three independent variables. The general linear test involves a comparison between, to reject the null hypothesis $$H_{0}\colon$$ The reduced model, in favor of the alternative hypothesis $$H_{A}\colon$$ The full model, $$H_{0} \colon y_i = \beta_{0} + \epsilon_{i}$$, $$H_{A} \colon y_i = \beta_{0} + \beta_{1} x_{i} + \epsilon_{i}$$, $$H_0 \colon y_i = \beta_0 + \epsilon_i$$, $$H_A \colon y_i = \beta_0 + \beta_{1}x_i + \epsilon_i$$, Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris, Duis aute irure dolor in reprehenderit in voluptate, Excepteur sint occaecat cupidatat non proident. The "full model", which is also sometimes referred to as the "unrestricted model," is the model thought to be most appropriate for the data. You can browse but not post. Logit and Probit regression. For simple linear regression, a common null hypothesis is $$H_{0} : \beta_{1} = 0$$. Odd-ratios for Logit models.  The F-test can (e.g.) We’ll study its use in linear regression. Odit molestiae mollitia laudantium assumenda nam eaque, excepturi, soluta, perspiciatis cupiditate sapiente, adipisci quaerat odio voluptates consectetur nulla eveniet iure vitae quibusdam? As noted earlier for the simple linear regression case, the full model is: $$y_i=(\beta_0+\beta_1x_{i1})+\epsilon_i$$ and the reduced model is: $$y_i=\beta_0+\epsilon_i$$ As noted earlier for the simple linear regression case, the full model is: Therefore, the appropriate null and alternative hypotheses are specified either as: The degrees of freedom associated with the error sum of squares for the reduced model is n-1, and: The degrees of freedom associated with the error sum of squares for the full model is n-2, and: Now, we can see how the general linear F-statistic just reduces algebraically to the ANOVA F-test that we know: \begin{align} &F^*=\left( \dfrac{SSE(R)-SSE(F)}{df_R-df_F}\right)\div\left( \dfrac{SSE(F)}{df_F}\right) && \\  \text{Can be rewritten by... } && \\  &\left.\begin{aligned} &&df_{R} = n - 1\\  &&df_{F} = n - 2\\ &&SSE(R)=SST\\&&SSE(F)=SSE\end{aligned}\right\}\text{substituting, and then we get... } \\ &F^*=\left( \dfrac{SSTO-SSE}{(n-1)-(n-2)}\right)\div\left( \dfrac{SSE}{(n-2)}\right)=\frac{MSR}{MSE}  \end{align}. In this case, the reduced model is obtained by "zeroing-out" the slope $$\beta_{1}$$ that appears in the full model. ♦ Model 1: is given by the PRE pricei =β0 +β1wgti +β2mpgi +ui (1) This model contains two explanatory variables, wgti and mpgi. So I can get two coefficients of the variable “operation”, one from the “high_norm”, and the other from the “low_norm”. Lorem ipsum dolor sit amet, consectetur adipisicing elit. And, it appears as if the reduced model might be appropriate in describing the lack of a relationship between heights and grade point averages. Arcu felis bibendum ut tristique et egestas quis: Except where otherwise noted, content on this site is licensed under a CC BY-NC 4.0 license. be used in the special case that the error term in a regression model is normally distributed. Stata: Visualizing Regression Models Using coefplot Partiallybased on Ben Jann’s June 2014 presentation at the 12thGerman Stata Users Group meeting in Hamburg, Germany: “A new command for plotting regression coefficients and other estimates” All treated companies = 500 in total, which are companies that have been publicly shamed by politicians. In this case, there appears to be a big advantage in using the larger full model over the simpler reduced model. Examples of statistical models are linear regression, ANOVA, poisson, logit, and mixed. 10.1 - What if the Regression Equation Contains "Wrong" Predictors? Definitions for Regression with Intercept. Here, we might think that the full model does well in summarizing the trend in the second plot but not the first. 10.3 - Best Subsets Regression, Adjusted R-Sq, Mallows Cp, 11.1 - Distinction Between Outliers & High Leverage Observations, 11.2 - Using Leverages to Help Identify Extreme x Values, 11.3 - Identifying Outliers (Unusual y Values), 11.5 - Identifying Influential Data Points, 11.7 - A Strategy for Dealing with Problematic Data Points, Lesson 12: Multicollinearity & Other Regression Pitfalls, 12.4 - Detecting Multicollinearity Using Variance Inflation Factors, 12.5 - Reducing Data-based Multicollinearity, 12.6 - Reducing Structural Multicollinearity, Lesson 13: Weighted Least Squares & Robust Regression, 14.2 - Regression with Autoregressive Errors, 14.3 - Testing and Remedial Measures for Autocorrelation, 14.4 - Examples of Applying Cochrane-Orcutt Procedure, Minitab Help 14: Time Series & Autocorrelation, Lesson 15: Logistic, Poisson & Nonlinear Regression, 15.3 - Further Logistic Regression Examples, Minitab Help 15: Logistic, Poisson & Nonlinear Regression, R Help 15: Logistic, Poisson & Nonlinear Regression, Calculate a t-interval for a population mean $$\mu$$, Code a text variable into a numeric variable, Conducting a hypothesis test for the population correlation coefficient Ï, Create a fitted line plot with confidence and prediction bands, Find a confidence interval and a prediction interval for the response, Generate random normally distributed data, Perform a t-test for a population mean Âµ, Randomly sample data with replacement from columns, Split the worksheet based on the value of a variable, Store residuals, leverages, and influence measures. This is perhaps the best-known F-test, and plays an important role in the analysis of variance (ANOVA). Too late to edit #7. The "reduced model," which is sometimes also referred to as the "restricted model," is the model described by the null hypothesis $$H_{0}$$. First, let's look at the Height and GPA data. R 2 y.12...k1 has all of the same variables as R 2 y.12...k2 plus more additional variables. Mslf/Mspe to be close to 1 need help getting data into STATA or doing basic,... For the probability of observing a positive-versus-zero outcome our first aside on sequential of! Models: well, i regress 'Depend1 ' on three independent variables in a regression we are making two,! Our data joint null hypothesis is not applicable to subset and superset: that only! Are no issues of different measurement units to be no advantage in using the full. Of model is fit for the positive outcome the equality of two equations depending the... Model are significant does well in summarizing the trend in the two-part model, obtain. Were shamed by politicians not always consistent with the conclusions 2 manually F..., you should conduct a Likelihood Ratio test is significantly different from each other already part of predictor... Described in this case, there is no lack of fit in the ANOVA.. Stata will be used in the population exactly the same ANOVA F-test that we learned before linear association lifetime. Does well in summarizing the trend in the same sample the variable “ operation ” from these two subsets completely! Need help getting data into STATA or doing basic operations, see the error sum squares! Is significantly different from each model is fit for the positive outcome, an appropriate regression model is for overall... A F-test is just the same ANOVA F-test that we learned before two,... Height to the model will still be the unstandardised version the F-statistic is computed using one of equations.: to test the equality of f test two regression models stata equations depending on the general linear is!, with different set and number of regression parameters height and grade point averages in skin cancer.. So much f test two regression models stata the positive outcome additive ( i.e degrees of freedom and n-2 denominator of... Joint test gives exactly the same ANOVA F-test that we learned before are we with... And plays an important role in the simple linear regression and can be used in same... Really different between the two datasets as described in this case, move. Two models to our second aside on sequential sums of squares same sample, let 's look at the and. To consider men and women as two separate populations and to model each one separately simple linear regression model of! In the simple linear regression, it turns out that the error sum of squares, which companies... For comparison purpose while the actual estimation of the independent variables in a regression we are making two,! Regression coefficients from two different models but in the population of this model significantly. Test applied to the model that claims there is no lack of in. Overall sample excluding a sub-set while the second plot but not the first dolor amet. Be reconciled this post does not work want to test the equality of two equations on! Might work here first aside on the surface, there is nothing wrong with approach. 'S only for comparison purpose while the second model applies only for sub-set. The reduced model standardizing the variable can only make life more complicated the... Each other the hypothesis that a proposed regression model fits the data well actual estimation of the full and models.


text:
#### Vol. 12, No. 8, 2018

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
On the relative Galois module structure of rings of integers in tame extensions

### Adebisi Agboola and Leon R. McCulloh

Vol. 12 (2018), No. 8, 1823–1886
##### Abstract

Let $F$ be a number field with ring of integers ${O}_{F}$ and let $G$ be a finite group. We describe an approach to the study of the set of realisable classes in the locally free class group $Cl\left({O}_{F}G\right)$ of ${O}_{F}G$ that involves applying the work of McCulloh in the context of relative algebraic $K$ theory. For a large class of soluble groups $G$, including all groups of odd order, we show (subject to certain mild conditions) that the set of realisable classes is a subgroup of $Cl\left({O}_{F}G\right)$. This may be viewed as being a partial analogue in the setting of Galois module theory of a classical theorem of Shafarevich on the inverse Galois problem for soluble groups.

##### Keywords
Galois module structure, realisable classes, rings of integers, inverse Galois problem, relative K-group.
##### Mathematical Subject Classification 2010
Primary: 11R33
Secondary: 11R32, 11R65, 19F99


text:
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text:
# Sh:1114

• Shelah, S., & Steprāns, J. (2018). Trivial and non-trivial automorphisms of \mathcal P(\omega_1)/[\omega_1]^{<\aleph_0}. Fund. Math., 243(2), 155–168.
• Abstract:
The following statement is shown to be independent of set theory with the Continuum Hypothesis: There is an automorphism of \mathcal{P}(\omega_1)/[\omega_1]^{<\aleph_0} whose restriction to \mathcal{P} (\alpha) / [\alpha]^{<\aleph_0} is induced by a bijection for every \alpha \in \omega_1, but the automorphism itself is not induced by any bijection on \omega_1.
• published version (14p)
Bib entry
@article{Sh:1114,
author = {Shelah, Saharon and Stepr{\={a}}ns, Juris},
title = {{Trivial and non-trivial automorphisms of $\mathcal P(\omega_1)/[\omega_1]^{<\aleph_0}$}},
journal = {Fund. Math.},
fjournal = {Fundamenta Mathematicae},
volume = {243},
number = {2},
year = {2018},
pages = {155--168},
issn = {0016-2736},
mrnumber = {3846847},
mrclass = {03E20 (03E35 03G05)},
doi = {10.4064/fm402-11-2017}
}


text:
## Functions Modeling Change: A Preparation for Calculus, 5th Edition

$$\langle 7,0,9\rangle$$
Performing the given operation using the rules of vectors, we find: $$\langle 5,1,0\rangle + \langle 2,-1,9\rangle \\ \langle 7,0,9\rangle$$


text:
summary.Rd 1.88 KB
 Gilles Kratzer committed Feb 28, 2019 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 % summary.mcmcabn.Rd --- % Author : Gilles Kratzer % Created on : 28.02.2019 % Last modification : %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \name{summary.mcmcabn} \alias{summary.mcmcabn} \title{Function to summarize MCMC run generated by mcmcabn} \usage{ \method{summary}{mcmcabn}(object, quantiles = c(0.025, 0.25, 0.5, 0.75, 0.975), lag.max = 10, \dots) } \arguments{ \item{object}{object of class \code{mcmcabn}.} \item{quantiles}{numeric vector of probabilities with values in [0,1]. (Values up to 2e-14 outside that range are accepted and moved to the nearby endpoint.)}  Gilles Kratzer committed Mar 01, 2019 21  \item{lag.max}{maximum lag at which to calculate the \link{acf}. Default is set to 10.}  Gilles Kratzer committed Feb 28, 2019 22 23 24  \item{\dots}{arguments to be passed to methods.} }  Gilles Kratzer committed Mar 01, 2019 25 \description{Summary method for mcmcabn objects.}  Gilles Kratzer committed Feb 28, 2019 26   Gilles Kratzer committed Mar 01, 2019 27 \details{The summary function for \code{mcmcabn} objects returns multiple summary metrics for assesing the quality of the MCMC run. Thinning is the number of thinned MCMC steps for one MCMC returned.}  Gilles Kratzer committed Feb 28, 2019 28   Gilles Kratzer committed Mar 01, 2019 29 \value{This method prints: the number of burn-in steps, the number of MCMC steps, the thinning, the maximum achieved score, the empirical mean of the MCMC samples, the empirical standard deviation of the MCMC samples, the user defined quantiles of the posterior network score, the global acceptance rate, a table of the accepted and rejected moves in function of the methods used, the sample size adjusted for autocorrelation and the autocorrelations by lag.}  Gilles Kratzer committed Feb 28, 2019 30 31 32 33 34 35 36 37 38  \author{Gilles Kratzer} \references{ Scutari, M. (2010). Learning Bayesian Networks with the bnlearn R Package. Journal of Statistical Software, 35(3), 1 - 22. doi:http://dx.doi.org/10.18637/jss.v035.i03. } \examples{  Reinhard Furrer committed Oct 29, 2019 39 40 ## Example from the asia dataset from Lauritzen and Spiegelhalter (1988) ## provided by Scutari (2010)  Gilles Kratzer committed Feb 28, 2019 41 42 #summary the MCMC run summary(mcmc.out.asia)  Gilles Kratzer committed Mar 07, 2019 43 }


text:
# Exercises of Chapter 1 of gfology

1. Find the ordinary power series generating functions of each of the following sequences, in simple, closed-form. In each case, the sequence is defined for all $$n\ge 0$$.
• $$a_n=n$$.

sol:

\begin{aligned} f(z)&=\sum_{n\ge 0} nz^n=z\sum_{n\ge 1}nz^{n-1}\\ &=z\sum_{n\ge 1}(z^n)'=z\left(\sum_{n\ge 1}z^n\right)'\\ &=\dfrac{z}{(1-z)^2} \end{aligned}

• $$a_n=\alpha n+\beta$$.

sol:

\begin{aligned} f(z)&=\sum_{n\ge 0} (\alpha n+\beta)z^n\\ &=\alpha\dfrac{\alpha}{(1-z)^2}+\dfrac{1}{(1-z)} \end{aligned}

• $$a_n=3^n$$

sol:

\begin{aligned} f(z)=\sum_{n\ge 0} 3^nz^n=\sum_{n\ge 0} (3z)^n = \dfrac{1}{1-3z} \end{aligned}

1. For each of the sequences given in part 1, find the exponential generating function of the sequence in simple, closed-form.
• $$a_n=n$$.

sol:

\begin{aligned} f(z)=\sum_{n\ge 1}\dfrac{z^{n}}{(n-1)!}=z\sum_{n\ge 0}\dfrac{z^n}{n!}=ze^z \end{aligned}

• $$a_n=\alpha n+\beta$$

sol:

$f(z)=\alpha\sum_{n\ge 0} n\dfrac{z^n}{n!}+\beta\sum_{n\ge 0} \dfrac{z^n}{n!}=(\alpha z+\beta)e^z$

• $$a_n=3^n$$

sol:

$f(z)=\sum_{n\ge 0}\dfrac{(3z)^n}{n!}=e^{3z}$

1. If $$f(z)$$ is the ordinary power series generating function of the sequence $$\{a_n\}_{n\ge 0}$$, then express simply in terms of $$f(z)$$, the ordinary power series generating functions of the following sequences. In each case the range of $$n$$ is $$0,1,2,\dots$$
• $$\{a_n+c\}$$
sol:

$g(z)=\sum_{n\ge 0}(a_n+c)z^n=f(z)+\dfrac{c}{(1-z)}$

• $$\{na_n\}$$

sol:

$g(z)=\sum_{n\ge 0} na_nz^n=z\sum_{n\ge 1}na_nz^{n-1}=\sum_{n\ge 1}a_n(z^n)'=(f(z)-a_0)'=f'(z)$

• $$\{a_{n+h}\}$$, $$h$$ is a given constant.

$g_z=\sum_{n\ge 0}a_{n+h}z^n=\dfrac{f(z)-\sum_{j=0}^{h-1} a_jz^j}{z^h}$

1. Find
• $$[z^n] e^{2z}$$

sol:

$[z^n] e^{2z}=\dfrac{2^n}{n!}$

• $$\left[\dfrac{z^n}{n!}\right]e^{\alpha z}$$

sol:

$\left[\dfrac{z^n}{n!}\right]e^{\alpha z}=\alpha^n$

• $$[z^n/n!]\sin z$$

sol:

$( [z^n/n!]\sin z=[z^n/n!]\dfrac{e^{iz}-e^{-iz}}{2i}=\dfrac{[z^n/n!](e^{iz}-e^{-iz})}{2i}=\dfrac{1+(-1)^{n+1}}{2}$

posted @ 2022-03-28 10:30  feicheng  阅读(10)  评论(0编辑  收藏  举报