text:
# I How does Bell make this step in his proof?

1. Oct 4, 2016

### N88

From drchinese website http://www.drchinese.com/David/Bell_Compact.pdf on page 406 there are 2 equations at the top. How do you get from the top one to the second one? There is a hint about using (1) but I think it cannot be done. You might be able to do it with other assumptions but I think they will be wrong too. Any suggestions?

2. Oct 4, 2016

### Staff: Mentor

I have taken the liberty of changing the thread title to reflect the question that's actually being asked.

3. Oct 5, 2016

### zonde

$A(\vec{a},\lambda)$ is either 1 or -1 so $[1-A(\vec{b},\lambda)A(\vec{c},\lambda)]$ is either 0 or 2. If for particular $\lambda$ it's 0 then expression in the first integral for the same $\lambda$ is 0 as well. If it's 2 then in first integral it's 2 or -2. So there is no way how module of the first integral can get bigger than the second integral.

4. Oct 5, 2016

### N88

Are you discussing how to get from the second equation to the third equation? My question is how to get from the first to the second.

5. Oct 5, 2016

### zonde

You mean how to see that $A(\vec{a},\lambda)A(\vec{b},\lambda)-A(\vec{a},\lambda)A(\vec{c},\lambda)=-A(\vec{a},\lambda)A(\vec{b},\lambda)[A(\vec{b},\lambda)A(\vec{c},\lambda)-1]$ ?

6. Oct 5, 2016

Thanks. Yes.

7. Oct 5, 2016

### zonde

It's easy to see in opposite direction. Just notice that from (1) you get that $A(\vec{b},\lambda)^2=1$

8. Oct 5, 2016

### N88

But in the second equation, the first $A(\vec{b},\lambda)$ was tested by Bob when Alice's detector was set to $\vec{a}$. The second $A(\vec{b},\lambda)$ in the second equation can be tested by Bob when Alice's detector is at setting $\vec{c}$. How does Bob come to have the same particles to test to get $A(\vec{b},\lambda)^2=1$? Because doesn't that relation $A(\vec{b},\lambda)^2=1$ only hold for tests on the same λ?

Looks like there's another assumption somewhere?

9. Oct 5, 2016

### zonde

$\lambda$ is independent of measurement settings. If follows from assumption that is stated at the top of page 404 (Alice's result does not depend on Bob's measurement setting and vice versa).

10. Oct 5, 2016

### DrChinese

If (and this sequence works in either direction):
$-[A(\vec{a},\lambda)A(\vec{b},\lambda)-A(\vec{a},\lambda)A(\vec{c},\lambda)]=A(\vec{a},\lambda)A(\vec{b},\lambda)[A(\vec{b},\lambda)A(\vec{c},\lambda)-1]$

Then:
$-[A(\vec{a},\lambda)A(\vec{b},\lambda)-A(\vec{a},\lambda)A(\vec{c},\lambda)]=A(\vec{a},\lambda)A(\vec{b},\lambda)[-1 + A(\vec{b},\lambda)A(\vec{c},\lambda)]$

Then:
$A(\vec{a},\lambda)A(\vec{c},\lambda)=A(\vec{a},\lambda)A(\vec{b},\lambda)[A(\vec{b},\lambda)A(\vec{c},\lambda)]$

Then we are finally saying:
$1[A(\vec{a},\lambda)A(\vec{c},\lambda)]=A(\vec{b},\lambda)A(\vec{b},\lambda)[A(\vec{a},\lambda)A(\vec{c},\lambda)]=A(\vec{a},\lambda)A(\vec{b},\lambda)[A(\vec{b},\lambda)A(\vec{c},\lambda)]$

Which I think Bell took for granted we would follow, I guess with some reference to (1) - not sure. As zonde already mentioned, $A(\vec{b},\lambda) A(\vec{b},\lambda)$ is 1 classically. The above is a relationship that should be true if $\vec{a}, \vec{b}, \vec{c}$ are all simultaneously well defined. That assumption is essentially the assumption of realism - that $\vec{a}, \vec{b}, \vec{c}$ are well-defined at all times. If they are not, it is like dividing by zero and you get results that themselves are suspect. Ergo a Bell Inequality, which does not always hold.

11. Oct 5, 2016

### N88

But here you have assumed on the RHS the equality that I want to derive from the LHS. My earlier post remains valid:

On RHS, the first $A(\vec{b},\lambda)$ was tested by Bob when Alice's detector was set to $\vec{a}$. The second $A(\vec{b},\lambda)$ in the second equation can be tested by Bob when Alice's detector is at setting $\vec{c}$. How does Bob come to have the same particles to test to get $A(\vec{b},\lambda)^2=1$? Because doesn't that relation $A(\vec{b},\lambda)^2=1$ only hold for tests on the same λ?

Looks like there's another assumption somewhere?[/QUOTE]

12. Oct 5, 2016

### N88

I agree. But to make sense of their test results, they need to be comparing paired results from tests on the same particle pair. And Bell and QM do not allow that you can test the same pair twice.

13. Oct 6, 2016

### zonde

We can do that in Bell's model. In his model outcome of measurement is fully determined by measurement angle and $\lambda$ and of course we can copy value (or set of values) of lambda.
Of course we can't rewind back reality, change something and then look what is different. But our models are not copies of reality, they are just models. And we can do such things with models.

Last edited: Oct 6, 2016
14. Oct 6, 2016

### rubi

We can do it only in classical models. In quantum models, non-commuting observables don't simultaneously have well-defined outcomes, so that it's not possible to consistently assign values to unobserved quantities. This phenomenon is called contextuality. It's very apparent in the GHZ state, where any assignment of values to unbserved quantities is inconsistent.

15. Oct 6, 2016

### stevendaryl

Staff Emeritus
I'm not sure I understand your objection. Bell is assuming (for the purpose of showing that the assumption leads to a contradiction with the predictions of quantum mechanics) that when a twin-pair of particles is created, a value of $\lambda$ is associated with each particle, and that Alice's measurement result is a deterministic function of $\lambda$. She deterministically gets the result $A(\vec{a}, \lambda)$, where $\lambda$ is the hidden variable associated with the twin pair, and $\vec{a}$ is the axis she uses for the measurement. Similarly, Bob's measurement result is a different function of $\lambda$: He gets the result $B(\vec{b}, \lambda)$, where $\vec{b}$ is the axis he uses.

So $A(\vec{a}, \lambda)$ and $B(\vec{b}, \lambda)$ are assumed to be ordinary functions of two parameters. His proof then just amounts to showing that there are no such functions reproducing the predictions of quantum mechanics.

In reasoning about $A$ and $B$, you can forget that they correspond to measurements, and just think of them as mathematical functions.

16. Oct 6, 2016

### zonde

Quantum contextuality basically means that certain measurement results of individual particles can't be predetermined by hidden variables. This is the same conclusion that Bell reaches with his theorem.
However it does not mean that we can't calculate statistical predictions for alternative scenarios.

17. Oct 6, 2016

### rubi

It means that we cannot calculate statistical predictions for scenarios in which non-commuting observables have simultaneously well-defined outcomes. Hence the answer to post #12 is that this is where Bell's assumptions contradict QM, which eventually allows for the inequality violation.

18. Oct 6, 2016

### zonde

OP has doubts whether Bell reasoning does not make additional assumption. So what is your answer? Is there additional assumption that does not follow from assumptions he has stated?

19. Oct 6, 2016

### rubi

Yes, Bell's paper is not very explicit about the assumptions he makes. This is the reason for the big confusion about his assumptions. Bell never states explicitely that the existence of values for unobserved variables is an assumption. On the contrary, he believes that this follows from the EPR argument. That's why I was pointing at the GHZ state, which shows that the EPR argument itself is inconsistent.

20. Oct 6, 2016

### zonde

Right at the start Bell says: "It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty."
Then later on he provides his reasoning how he arrives at "more complete specification of the state" described as lambda.
Then again he gives his assumption at the top of page 404 (that B's result does not depend on A's measurement setting and vice versa).

But as I understand you maintain position that Bell's reasoning is faulty and there can be independent measurements on two distant particles that with certainty will give (UP,DOWN) or (DOWN,UP) but not (UP,UP) nor (DOWN,DOWN) and yet it does not follow that we can meaningfully talk about UP or DOWN measurement for each particle separately. So it is reasonable option that combination of two independent parts taken together is meaningful while each independent part taken separately is not.

21. Oct 6, 2016

### DrChinese

$\lambda$ is the same set of parameters or functions for settings a, b c. The "other" assumption is one I spelled out: realism. That a, b and c yield simultaneously well defined outcome functions. That assumption is the heart of the Bell proof, as can be seen.

You also say: "But to make sense of their test results, they need to be comparing paired results from tests on the same particle pair. And Bell and QM do not allow that you can test the same pair twice."

That statement is incorrect. Bell assumes you could have made a counterfactual measurement. That assumption leads to a contradiction with experiment (and with the predictions of QM). Yours is a red herring argument that is frequently made when discussing Bell. Bell assumes the classical position - in which case your objection wouldn't matter. The QM position is irrelevant to that view other than the fact that it makes a different prediction. But QM could be wrong and Bell would still set limits (the inequality), and that would still be contradicted by experiment. (Of course Bell didn't know that for sure back in the day, but we do.)

22. Oct 6, 2016

### rubi

Yes, this is where he misses the possibility of contextuality. It was later shown by Stapp and Eberhard that this is an essential assumption of every Bell-type theorem. Theorems like GHZ or Kochen-Specker show that it is violated by QM. From Bell's theorem, we can only conclude that contextuality or locality (or both) is violated by QM.

No, it's not faulty. He just doesn't spell out the assumptions very explicitely. Original papers are rarely a good source for learning something.

Bell assumes that there is a value up/down for each particle and each value of the angle $\vec a$, even for those that aren't measured. That's the assumption that isn't made very explicit in the paper and which is violated by QM, independently of Bell's result.

23. Oct 6, 2016

### DrChinese

I agree with rubi on this. Although the 2 equations at the top page of page 406 is where the realism assumption is introduced, Bell hardly makes note of it. He casually says "let c be another unit vector" when Bell might have said:

"Let's assume there are an infinite number of other unit vectors, c, d, e, etc. that are also defined by $\lambda$. Those would correspond to elements of reality as defined in the EPR paper. Using vector c, we would have:"

Had he done that, it would have clarified things for many a subsequent reader. However, at the time he wrote there were only a relatively small group of persons he was making his argument to, and he expected they would see his reasoning as presented. Apparently that was correct.

24. Oct 6, 2016

### DrChinese

As mentioned, I agree with rubi's take on Bell. Forget Bell's statement "It is the requirement of locality ... that creates the essential difficulty." The requirement of locality is not the essential difficulty, if anything it is the requirement of realism that is essential. That can be better seen in the GHZ proof, which does not depend on locality. The requirement of locality is of course needed for the Bell conclusion, as the EPR formulation required 2 separated particles and you must assume there is no interaction between them that allows $\lambda$ to include a remote measurement setting.

25. Oct 6, 2016

### zonde

I will just quote Bell (emphasis mine): "Since we can predict in advance the result of measuring any chosen component of $\vec\sigma_2$, by previously measuring the same component of $\vec\sigma_1$, it follows that the result of any such measurement must actually be predetermined."


text:
How to include math symbols?

Most of the time you will want an "inline" math equation: the x/2 button above the edit box. You can then put a LaTex command for the symbols you want.
The symbols we will most need are:

 LaTeX command What you get \geq, \leq $\geq, \leq$ \times $\times$ \cup, \cap $\cup, \cap$ \mathbb{R}, \mathbb{Z}, etc. $\mathbb{R}, \mathbb{Z}$, etc. \frac{a}{b} $\frac{a}{b}$ \alpha, \pi, \theta, etc. $\alpha, \pi, \theta$, etc.

Here are some other resources: The WikiDot help page on equations (scroll down); the tables of symbols under Producing Mathematical Formulae in LaTeX (David Wilkins). I also created a discussion forum on this page for LaTeX questions and insights.

page revision: 2, last edited: 10 Mar 2010 03:12

format: plain


text:
# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 27th 2012

I added an Idea-section to element in an abelian category and added a reference by George Bergman.

This links back to the new Idea section at abelian categories - embedding theorems. Check if you agree with the wording.

• CommentRowNumber2.
• CommentAuthorZhen Lin
• CommentTimeAug 28th 2012

The same definition/technique also found in CWM. It’s a nice substitute for having enough projectives.

• CommentRowNumber3.
• CommentAuthorIngoBlechschmidt
• CommentTimeAug 28th 2012
• (edited Aug 28th 2012)

I added the CWM reference to the article.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeAug 28th 2012
• (edited Aug 28th 2012)

Hm, “the same definition” refers to that of Gelfand-Manin, which I guess is in fact adopted from MacLane. But Bergman’s definition which I had announced above is explicitly different. He comments on that at the very end.

This probably deserved to be expanded on, but for the moment I just added this paragraph to the entry:

However, beware that the passage to equivalence classes does not respect the abelian group structure and hence generalized elements in this sense cannot be added or subtracted. A more natural approach is discussed in (Bergman) where the actual generalized elements are remembered but a refinement of their domain is allowed, much as familiar from topos theory.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeAug 28th 2012

Hi Ingo,

since you are making substantial contributions, you should have a page in “category: people”. I have created one: Ingo Blechschmidt. Hopefully I identified your webpage correctly. Please correct and/or add material as need be.

1. Urs: Thanks! The webpage is indeed mine (but quite dated and in need of quite a few updates).

I created a stub at diagram chase (nLab, Forum) to summarize the different approaches.


text:
### ALMBS State Primary Election Results

********** (NATIONALIST) **********

GOVERNOR
DALLAS KOKOSKA. . . . . . . . . 44 23.66

SECRETARY OF STATE
(Vote For Not More Than ) 1
JACOB "Mr Gravy" SMITH . . . . . . 107 58.47

AUDITOR
(Vote For Not More Than ) 1
ALEX "A-Mac" MACDONALD . . . . . . 50 27.17

TREASURER
(Vote For Not More Than ) 1
MICHAEL FOUTS . . . . . . . . . 95 51.91

COMMISSIONER OF AGRICULTURE
(Vote For Not More Than ) 1
WILLIAM G SLONAKER . . . . . . . 88 47.83

ATTORNEY GENERAL
(Vote For Not More Than ) 1
DONALD "The Waterboy" BOUCHER. . . . 77 41.40

JUSTICE OF THE SUPREME COURT
(Vote For Not More Than ) 5
PHILIP WILSON
TOY REID
ZAC WILCOXEN
DAVID-MICHAEL BUCKMAN
WILL SHAFFER

********** (FEDERALIST) **********

GOVERNOR
(Vote For Not More Than ) 1
DAMIAN COOKE . . . . . . . . . 45 26.32

SECRETARY OF STATE
(Vote For Not More Than ) 1
JACK MCGEE . . . . . . . . . . 168 100.00

AUDITOR
(Vote For Not More Than ) 1
JOSH "J Watts" WATSON . . . . . . 99 58.93

TREASURER
(Vote For Not More Than ) 1
CHARLIE "Not the Hunchback" HAGEBO . . 51 29.82

COMMISSIONER OF AGRICULTURE
(Vote For Not More Than ) 1
AARON "Tater Salad" WHITTAKER. . . . 71 41.28

ATTORNEY GENERAL
(Vote For Not More Than ) 1
JOEL M. WHETZEL . . . . . . . . 48 28.24

JUSTICE OF THE SUPREME COURT
(Vote For Not More Than ) 5
AUSTIN PATTERSON
ZACH PARRUCCI
TOMMY CAIN
J.R HAMILTON
SETH BLAKE


text:
# Survival analysis with ridge regression in R give same results with different random seeds

I am doing survival analysis using ridge regression. I'm using this R command:

coxph(Surv(time, status) ~ ridge(x1, x2, x3), data=DATA)

As far as I know, lambda (the regulation parameter) is estimated using cross validation, but then this R code should result in different results with different random seeds. But I got always the same coefficients; how can that happen?

Is coxph(Surv()~.) not a commonly used approach? Should I use glmnet or any other functions?

• Perhaps it would help if you told us what package "ridge" is in. – alex Jul 1 '13 at 19:44
• Not knowing the above, one hypothesis is that the lambda sequence is a deterministic function of the entire data set. So no matter how data is randomly allocated into the cross-validation sets, the same optimal lambda is chosen. Then the model is refit using that lambda on the entire data set. – alex Jul 1 '13 at 19:46
• The R package I am using is "survival": stat.ethz.ch/R-manual/R-devel/library/survival/html/ridge.html – Jenny Jul 1 '13 at 19:51

## 1 Answer

The documentation you linked combined with the method signature indicates that one chooses either theta, or it is chosen for you as a function of df. If the latter is not specified, then it defaults to half the number of variables.

As far as I can tell, no cross-validation is occurring. Why do you think that it is?


text:
# Plane equation

• June 25th 2012, 08:46 PM
M.R
Plane equation
Hi,

I am trying to understand the following problem:

Find the parametric vector equation of the line through $(2, -1, 2)$ and perpendicular to the plane $x-2z=5$

The equation of the line is just the point + a vector in that direction. The answer shows:

$(2, -1, 2) + t(1, 0, 2)$

But how do you know it's perpendicular?
• June 25th 2012, 09:34 PM
M.R
Re: Plane equation
I figured it out!

From the equation of the plane, the coefficients yield the components of the normal vector
• June 26th 2012, 07:10 AM
biffboy
Re: Plane equation
So I think it should be t(1,0,-2) not t(1,0,2)
• June 26th 2012, 01:34 PM
M.R
Re: Plane equation
Yes, Thank you.


text:
# Tag Info

12

Another way of solving such problems is to go to another reference frame, where you obviously don't have enough energy. For example you've got a $5 MeV$ photon, so you think that there is plenty of energy to make $e^-e^+$ pair. Now you make a boost along the direction of the photon momentum with $v=0.99\,c$ and you get a $0.35 MeV$ photon. That is not ...

8

In the beginning, let's investigate the pair production. We know from relativity that mass can be equivalent to energy and if we set the most important physical constants to one $\hbar = c = \varepsilon = 1$, then we have the following relation (assuming that we produce electron-positron pair): $$E = \omega = \gamma_1 m_{e^-} + \gamma_2 m_{e^+}$$ Where ...

5

The answer by @gns-ank covers the kinematics of why. Below I tackle the Why does a photon "split" into an electron and positron, and not just bounce off the nucleus in your comment to his answer. In general physics can answer "why" in a nested way, like russian dolls. In the end, the kernel answer is "because it does". In this case though we are in ...

5

Pair production of electron/positron happens in the electric field of the atoms to satisfy conservation laws and the same will be true for off mass shell Z0 going into a neutrino antineutrino pair, interacting with the weak field of the atoms. The equivalent to the photon weak interaction mediator is the Z0 and neutrino antineutrino pairs can be formed ...

5

Basics You need to be able to generate a pair-creation event and be able to image it well enough to know what it was. Getting a pair conversion event Pair creation calls for the highest energy gamma you can get and as much mass in the chamber as you can arrange. The odds of getting a pair-conversion event are graphed in figure 31.17 of the 2013 ...

5

Other way to see why this is impossible is to look at inverse process: why annihilating positron and electron can't give up only one photon? Imagine these two particles at rest near each other (or look at center-of-mass system). They will annihilate giving 1MeV of energy, but single photon can't pick this energy up by itself because it would also have E/c of ...

4

There's a simple argument that massive bodies that are not black holes cannot emit Hawking radiation. Consider a single proton in its ground state. It does not emit radiation because if it were, it would have to decay to some lower-energy state. This eventually may happen when the proton decays, but the radiation is not black-body radiation, so it can't be ...

4

look at energy-momentum conservation: $$p_\gamma = p_1+p_2$$ the photon has invariant mass 0 wheras the electron and positron have mass $m_e$ $$p_\gamma^2 = (p_1+p_2)^2 = p_1^2+p_2^2+2p_1\cdot p_2$$ $$0 = 2m_e^2 + 2p_1\cdot p_2$$ $$-m_e^2= p_1\cdot p_2 = E_1E_2-|\vec{p_1}||\vec{p_2}|cos\theta > E_1E_2-|\vec{p_1}||\vec{p_2}| = E_1E_2(1-\beta_1\beta_2) ... 4 Check if momentum can be conserved. That ought to do the trick. 4 The general diffeomorphism symmetry in the target space is not a symmetry of the world line theory or, analogously, the world sheet theory! A general spacetime diffeomorphism changes the metric tensor g_{\mu\nu}(X^\alpha) which plays the role of the "coupling constants" (coefficients defining the action, e.g. your exponent) in the world line or world sheet ... 4 This process is the result of the cooperation of two theories of nature: (i) Special relativity: This is a huge topic to study but we shall only need a small part of it, and perhaps the most famous one, which tells us this E=mc^2. This equation shows us that matter and energy are equivalent and interchangeable. For example, if an amount of energy E = ... 3 The pair production is only possible due to relativistic quantum physics and one needs to describe all these processes by the so-called "quantum field theory" or its generalization (well, string theory is the only example). In quantum field theory, electromagnetic fields are, just like all other fields, quantized. All of the configurations of the ... 3 Question 3: One time and two space dimensions for simplicity (t, x, y). Photon travelling in +x direction. Photon four momentum is (\frac{E}{c}, p_x, 0). It's null so$$ \frac{E^2}{c^2}-p_x^2=0$$So$$p_x = \frac{E}{c} $$E=h\nu, so photon four momentum is (\frac{h\nu}{c}, \frac{h\nu}{c}, 0)  To keep it simple, assume the electron/positron are ... 3 The reply by Emilio Pisanty to your other question also pertains here. But to prove the impossibility of pair production of a photon in vacuum it is not necessary to go into the mathematics of the Lorenz transformations further than you have already done. When one uses valid algebra and from two paths reaches a different answer one has already proven that ... 3 Pair production is not the same as decay of a particle. A particle can decay into two components according to its decay probability without needing an extra interaction. A lamda in its rest frame will decay into a proton and a pion, for example, within a predictable decay time . There is no rest frame for the photon since its mass is 0 and it is always ... 3 My previous answer is beside the point now that the question has been edited. There is a simpler example of a question of this type which has been analyzed in great detail in the literature, and that involves an electric charge which is stationary in a gravitational field. Since the power radiated is nonzero if a charge is accelerated one might, by the ... 3 There is no tree level vertex for a neutrino scattering off of a photon, so you have two choices: Weak Drell-Yan if the progenitors are leptons. That is$$l + \bar{l} \to Z^0 \to \nu + \bar{\nu} \,. A multi-step process involving a weak radiative correction. Say running a $W^\pm$ across the out-going lepton lines in a normal Drell-Yan diagram. Unlike the ...

3

You can't simultaneously conserve energy and linear momentum. Let the photon have energy $E_{\gamma} = p_{\gamma} c$ and the electron have energy $E_{-}^{2} = p_{e}^{2}c^2 + m_{e}^{2}c^4$ and an analogous expression for the positron. Suppose the electron and positron depart from the interaction site with an angle $2\theta$ between them. Conservation of ...

2

Let us say that the photon has created pair of massive particles. There must exist a reference frame ("center of mass") in which the total momentum of the two particles is zero. So by momentum conservation, in the same frame the photon had to have momentum zero. But photon cannot have momentum zero, because then it would have zero energy. So momentum is not ...

2

Gamma decay is the emission of photons. You are thinking of $\beta$ decay. When the particle is decaying, if it emits a $W^{-}$boson, it will subsequently decay and create an electron ($e^-$), and an electron antineutrino ($\overline{v}_e$), the antimatter particle to an electron neutrino. It will also flip a neutron into a proton. This is known as ...

2

If you look at a photon in a vacuum, there is no preferred inertial frame of reference. And the energy of the photon is dependent upon which frame is chosen. So some frames have more than the needed energy, others less. It would be really awkward to have pair production in reference frames that lack the needed energy.

2

Electron and positron can annihilate in free space, but a single photon cannot be turned into electron-positron pair because conservation of energy-momentum cannot be satisfied. The nucleus in pair production absorbs some of the momentum.

2

Annihilation can happen when all the quantum numbers of two colliding particles add up to zero. It might be electron on positron, proton on antiproton, neutron on antineutron , quark on antiquark etc.The force responsible depends on the possible interactions of the annihilating particles. In the case of electron positron annihilation it is primarily the ...

2

The answer is no, because there is no horizon. A time-independent gravitational field emits no particles, because energy is conserved, just like a stationary charge distribution does not radiate. This is consistent with the equivalence principle, as explained below. In the Unruh effect, the particle emmission is from the "black wall" acceleration horizon ...

2

The Heisenberg uncertainty principle applies only to operators satisfying canonical commutation rules. This is the case for corresponding components of position and momentum operators, but not for the energy operator (Hamiltonian), which has no associated conjugate partner. (Conjugate pairs of selfadjoint operators necessarily have unbounded spectrum, while ...

2

Conservation of energy and conservation of momentum cannot really be separated, since energy and momentum are just different components of a relativistic 4-vector; different inertial observers will "split" this 4-momentum into energy and momentum in different ways, much like they will "split" spacetime into space and time in different ways. The real reason ...

2

It depends what you mean by "electrostatic energy". When we are talking of pair production we are talking of physics at the quantum mechanics framework. FEYNMAN DIAGRAMS for pair production by a gamma ray (left) or an electron (right). These represent the processes in the preceding sketch. Lets take the simplest diagram on the left: a photon ...

2

These electron-positron pairs are created by gamma rays. I don't know anything about how to make a cloud chamber, but detecting cosmic gamma rays at the surface of the Earth is very very rare. The atmosphere is very opaque to gamma rays (Source). Cosmic gamma rays burst are commonly detected on satellites orbiting the Earth, but very few make it to the ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
The Theory of Matrices, Volume 1

Publisher:
American Mathematical Society/Chelsea
Number of Pages:
384
Price:
45.00
ISBN:
0-8218-1376-5
Wednesday, August 23, 2006
Reviewable:
Yes
Include In BLL Rating:
Yes
F. R. Gantmacher
Publication Date:
2000
Format:
Hardcover
Category:
Monograph
BLL Committee
11/4/2010
BLL Rating:

• I. Matrices and operations on matrices: 1. Matrices. Basic notation; 2. Addition and multiplication of rectangular matrices; 3. Square matrices; 4. Compound matrices. Minors of the inverse matrix
• II. The algorithm of Gauss and some of its applications: 1. Gauss's elimination method; 2. Mechanical interpretation of Gauss's algorithm; 3. Sylvester's determinant identity; 4. The decomposition of a square matrix into triangular factors; 5. The partition of a matrix into blocks. The technique of operating with partitioned matrices. The generalized algorithm of Gauss
• III. Linear operators in an $n$-dimensional vector space: 1. Vector spaces; 2. A linear operator mapping an $n$-dimensional space into an $m$-dimensional space; 3. Addition and multiplication of linear operators; 4. Transformation of coordinates; 5. Equivalent matrices. The rank of an operator. Sylvester's inequality; 6. Linear operators mapping an $n$-dimensional space into itself; 7. Characteristic values and characteristic vectors of a linear operator; 8. Linear operators of simple structure
• IV. The characteristic polynomial and the minimal polynomial of a matrix: 1. Addition and multiplication of matrix polynomials; 2. Right and left division of matrix polynomials; 3. The generalized Bézout theorem; 4. The characteristic polynomial of a matrix. The adjoint matrix; 5. The method of Faddeev for the simultaneous computation of the coefficients of the characteristic polynomial and of the adjoint matrix; 6. The minimal polynomial of a matrix
• V. Functions of matrices: 1. Definition of a function of a matrix; 2. The Lagrange-Sylvester interpolation polynomial; 3. Other forms of the definition of $f(A)$. The components of the matrix $A$; 4. Representation of functions of matrices by means of series; 5. Application of a function of a matrix to the integration of a system of linear differential equations with constant coefficients; 6. Stability of motion in the case of a linear system
• VI. Equivalent transformations of polynomial matrices. Analytic theory of elementary divisors: 1. Elementary transformations of a polynomial matrix; 2. Canonical form of a $\lambda$-matrix; 3. Invariant polynomials and elementary divisors of a polynomial matrix; 4. Equivalence of linear binomials; 5. A criterion for similarity of matrices; 6. The normal forms of a matrix; 7. The elementary divisors of the matrix $f(A)$; 8. A general method of constructing the transforming matrix; 9. Another method of constructing a transforming matrix
• VII. The structure of a linear operator in an $n$-dimensional space: 1. The minimal polynomial of a vector and a space (with respect to a given linear operator); 2. Decomposition into invariant subspaces with co-prime minimal polynomials; 3. Congruence. Factor space; 4. Decomposition of a space into cyclic invariant subspaces; 5. The normal form of a matrix; 6. Invariant polynomials. Elementary divisors; 7. The Jordan normal form of a matrix; 8. Krylov's method of transforming the secular equation
• VIII. Matrix equations: 1. The equation $AX=XB$; 2. The special case $A=B$. Commuting matrices; 3. The equation $AX-XB=C$; 4. The scalar equation $f(X)=O$; 5. Matrix polynomial equations; 6. The extraction of $m$-th roots of a non-singular matrix; 7. The extraction of $m$-th roots of a singular matrix; 8. The logarithm of a matrix
• IX. Linear operators in a unitary space: 1. General considerations; 2. Metrization of a space; 3. Gram's criterion for linear dependence of vectors; 4. Orthogonal projection; 5. The geometrical meaning of the Gramian and some inequalities; 6. Orthogonalization of a sequence of vectors; 7. Orthonormal bases; 8. The adjoint operator; 9. Normal operators in a unitary space; 10. The spectra of normal, hermitian, and unitary operators; 11. Positive-semidefinite and positive-definite hermitian operators; 12. Polar decomposition of a linear operator in a unitary space. Cayley's formulas; 13. Linear operators in a euclidean space; 14. Polar decomposition of an operator and the Cayley formulas in a euclidean space; 15. Commuting normal operators
• X. Quadratic and hermitian forms: 1. Transformation of the variables in a quadratic form; 2. Reduction of a quadratic form to a sum of squares. The law of inertia; 3. The methods of Lagrange and Jacobi of reducing a quadratic form to a sum of squares; 4. Positive quadratic forms; 5. Reduction of a quadratic form to principal axes; 6. Pencils of quadratic forms; 7. Extremal properties of the characteristic values of a regular pencil of forms; 8. Small oscillations of a system with $n$ degrees of freedom; 9. Hermitian forms; 10. Hankel forms
• Bibliography
• Index
Publish Book:
Modify Date:
Thursday, November 4, 2010


text:
Q. The maximum value of θ until which the approximation sinθ ≈ θ holds to within 10%
error is
(A) 10 degree
(B) 18 degree
(C) 50 degree
(D) 90 degree

according to me answer should be B i.e. 18 degree but my teacher gave answer of this C i.e. 50 degree but this is unacceptable for me as in this approximation error would be more than 13%.


text:
# Homework Help: Double integration

1. Jan 23, 2008

### rolylane

Hi
I've been working on a problem and I'm nearly there but I'm struggling with the integration part at the end and was hoping you might be able to help if you have the time. The original question was

$$\int \int (y^2 z^2 + z^2 x^2 + x^2 y^2) \: dS$$
Evaluated on the region of $$z^2 = x^2 + y^2$$ between z=1 and z=2.
.
.
.
.
Then by substituting in x=r*cos(θ) and y=r*sin(θ), and multiplying by 'r' (the Jacobian determinant), I got $$\sqrt{2} \int_{0}^{2\pi} \int_{1}^{2}(r^5 + r^3 \cos(\theta)^2 \sin(\theta)^2) \: dr \: d\theta$$

and then I'm stuck. Any help or advice would really be appreciated
Thanks

2. Jan 23, 2008

### NateTG

This can be cracked with trig identities, or done by parts.

3. Jan 23, 2008

### unplebeian

I haven't done the steps but are you sure the integrand contains cos (theta^2) and sin (theta^2) or is it cos^2 (theta) (cos squared theta) ?

Last edited: Jan 23, 2008
4. Jan 23, 2008

### blochwave

Same thing, he didn't write cos(theta^2), they're outside the parentheses

5. Jan 23, 2008

### CompuChip

It says $\cos(\theta)^2$ which I read as $\cos^2(\theta)$ as opposed to $\cos(\theta^2)$.

You might replace $\cos^2\theta = 1 - \sin^2\theta$, then two of the integrals become very easy, and one of them becomes $\sin^4\theta$ for which there are tricks.

6. Jan 23, 2008

### unplebeian

Well, then just go brute force and then apply by parts. I got int( sin^2(2 x theta))

After simplyfying and using by parts I get int( 2*theta*sin(4theta))

By parts again.

7. Jan 23, 2008

### NateTG

Actually, I was thinking...
$$\cos \theta \sin \theta = \frac{1}{2} \sin \left( 2 \theta \right)$$


text:
# FAQ - frequently asked questions

Refer to Vespa Support for more support options.

### Ranking

#### Does Vespa support a flexible ranking score?

Ranking is maybe the primary Vespa feature - we like to think of it as scalable, online computation. A rank profile is where the application’s logic is implemented, supporting simple types like double and complex types like tensor. Supply ranking data in queries in query features (e.g. different weights per customer), or look up in a Searcher. Typically, a document (e.g. product) “feature vector”/”weights” will be compared to a user-specific vector (tensor).

#### Where would customer specific weightings be stored?

Vespa doesn’t have specific support for storing customer data as such. You can store this data as a separate document type in Vespa and look it up before passing the query, or store this customer meta-data as part of the other meta-data for the customer (i.e. login information) and pass it along the query when you send it to the backend. Find an example on how to look up data in album-recommendation-docproc.

#### How to create a tensor on the fly in the ranking expression?

Create a tensor in the ranking function from arrays or weighted sets using tensorFrom... functions - see document features.

#### How to set a dynamic (query time) ranking drop threshold?

Pass a ranking feature like query(threshold) and use an if statement in the ranking expression - see retrieval and ranking. Example:

rank-profile drop-low-score {
function my_score() {
expression: ..... #custom first phase score
}
rank-score-drop-limit:0.0
first-phase {
if(my_score() < query(threshold), -1, my_score())
}
}

#### Does Vespa support early termination of matching and ranking?

Yes, this can be accomplished by configuring match-phase in the rank profile, or by adding a range query item using hitLimit to the query tree, see capped numeric range search.
Both methods require an attribute field with fast-search. The capped range query is faster but beware that if there are other restrictive filters in the query one might end up with 0 hits. The additional filters are applied as a post filtering step over the hits from the capped range query. match-phase on the other hand is safe to use with filters or other query terms, and also supports diversification which the capped range query term does not support.

### Documents

#### What limits apply to json document size?

There is no hard limit. Vespa requires a document to be able to load into memory in serialized form. Vespa is not optimized for huge documents.

#### Can a document have lists (key value pairs)?

E.g. a product is offered in a list of stores with a quantity per store. Use multivalue fields (array of struct) or parent child. Which one to chose depends on use case, see discussion in the latter link.

#### Does a whole document need to be updated and re-indexed?

E.g. price and quantity available per store may often change vs the actual product attributes. Vespa supports partial updates of documents. Also, the parent/child feature is implemented to support use-cases where child elements are updated frequently, while a more limited set of parent elements are updated less frequently.

#### What ACID guarantees if any does Vespa provide for single writes / updates / deletes vs batch operations etc?

See the Vespa Consistency Model. Vespa is not transactional in the traditional sense, it doesn’t have strict ACID guarantees. Vespa is designed for high performance use-cases with eventual consistency as an acceptable (and to some extent configurable) trade-off.

#### Does vespa support wildcard fields?

Wildcard fields are not supported in vespa. Workaround would be to use maps to store the wildcard fields. Map needs to be defined with indexing: attribute and hence will be stored in memory. Refer to map.

#### Is there any size limitation in multivalued fields?

No limit, except memory.

#### Can we set a limit for the number of elements that can be stored in an array?

Implement a document processor for this.

#### How to auto-expire documents / set up garbage collection?

Set a selection criterion on the document element in services.xml. The criterion selects documents to keep. I.e. to purge documents “older than two weeks”, the expression should be “newer than two weeks”. Read more about document expiry.

#### How to increase redundancy and track data migration progress?

Changing redundancy is a live and safe change (assuming there is headroom on disk / memory - e.g. from 2 to 3 is 50% more). The time to migrate will be quite similar to what it took to feed initially - a bit hard to say generally, and depends on IO and index settings, like if building an index for ANN. To monitor progress, take a look at the multinode sample application for the clustercontroller status page - this shows buckets pending, live. Finally, use the .idealstate.merge_bucket.pending metric to track progress - when 0, there are no more data syncing operations - see monitor distance to ideal state. Nodes will work as normal during data sync, and query coverage will be the same. Nodes will be busier, though.

#### How does namespace relate to schema?

It does not, namespace is a mechanism to split the document space into parts that can be used for document selection - see documentation.

### Query

#### Is hierarchical facets supported?

Facets is called grouping in Vespa. Groups can be multi-level.

#### Is filters supported?

Add filters to the query using YQL using boolean, numeric and text matching.

#### How to query for similar items?

One way is to describe items using tensors and query for the nearest neighbor - using full precision or approximate (ANN) - the latter is used when the set is too large for an exact calculation. Apply filters to the query to limit the neighbor candidate set. Using dot products or weak and are alternatives.

#### Stop-word support?

Vespa does not have a stop-word concept inherently. See the sample app for how to use filter terms.

#### How to extract more than 400 hits / query and get ALL documents?

Trying to request more than 400 hits in a query, getting this error: {'code': 3, 'summary': 'Illegal query', 'message': '401 hits requested, configured limit: 400.'}.

• To increase max result set size, configure maxHits in a query profile, e.g. <field name="maxHits">500</field> in search/query-profiles/default.xml (create as needed). Query timeout can be increased, but it will still be costly and likely impact other queries - large limit more so than a large offset. It can be made cheaper by using a smaller document summary, and avoiding fields on disk if possible.
• Using visit in the document/v1/ API is usually a better option for dumping all the data.

#### How to make a sub-query to get data to enrich the query, like get a user profile?

See the UserProfileSearcher for how to create a new query to fetch data - this creates a new Query, sets a new root and parameters - then fills the Hits.

#### How to create a cache that refreshes itself regularly

public class ConfigCacheRefresher extends AbstractComponent {
...
private final ScheduledExecutorService configFetchService = Executors.newSingleThreadScheduledExecutor();
private Chain<Searcher> searcherChain;
...
void initialize() {
Runnable task = () -> refreshCache();
searcherChain = executionFactory.searchChainRegistry().getChain(new ComponentId("configDefaultProvider"));
}
...
public void refreshCache() {
Execution execution = executionFactory.newExecution(searcherChain);
Query query = createQuery(execution);
...
public void deconstruct() {
super.deconstruct();
try {
configFetchService.shutdown();
configFetchService.awaitTermination(1, TimeUnit.MINUTES);
...
}

#### Is it possible to query Vespa using a list of document ids?

The best article on the subject is multi-lookup set filtering. Refer to the weightedset-example - also see weightedset for writing a YQL query to select multiple IDs. The ID must be a field in the document type.

#### How to count hits / all documents without returning results?

Count all documents using a query like select * from doc where true - this counts all documents from the “doc” source. Using select * from doc where true limit 0 will return the count and no hits, alternatively add hits=0. Pass ranking.profile=unranked to make the query less expensive to run. If an estimate is good enough, use hitcountestimate=true.

#### Must all fields in a fieldset have the same type?

Yes - a deployment warning with This may lead to recall and ranking issues can be emitted if not. This warning comes whenever fields with different kinds of tokenization are put in the same fieldset. This is because a given piece of text searching one fieldset is tokenized just once, so there’s no right choice of tokenization in this case. More details on stack overflow.

#### Searchers and timeout questions

During multi-phase searching, is the query timeout set for each individual searcher or is the query timeout set for the entire search chain? Also, if we asynchronously execute several search chains, can we set different query timeouts for each of these chains plus a separate overall timeout for the searcher that performs the asynchronous executions?

The timeout is for the entire query (and most Searchers don’t check timeout - use getTimeLeft). E.g. if a search chain has 3 searchers, it is OK for 1 searcher to take 497 ms and 2 searchers to each take 1 ms for a query timeout of 500 ms. You can set a different timeout in each cloned query you send to any of those chains, and you can specify the timeout when waiting for responses from them.

### Feeding

#### How to debug document processing chain configuration?

This configuration is a combination of content and container cluster configuration, see indexing and feed troubleshooting.

#### I feed documents with no error, but they are not in the index

This is often a problem if using document expiry, as documents already expired will not be persisted, they are silently dropped. Feeding stale test data with old timestamps can cause this.

#### How to feed many files, avoiding 429 error?

Using too many clients can generate a 429 response code. The Vespa sample apps use the vespa-feed-client which uses HTTP/2 for high throughput - it is better to stream the feed files through this client.

#### Does Vespa support addition of flexible NLP processing for documents and search queries?

E.g. integrating NER, word sense disambiguation, specific intent detection. Vespa supports these things well:

#### Does Vespa support customization of the inverted index?

E.g. instead of using terms or n-grams as the unit, we might use terms with specific word senses - e.g. bark (dog bark) vs. bark (tree bark), or BCG (company) vs. BCG (vaccine name). Creating a new index format means changing the core. However, for the examples above, one just need control over the tokens which are indexed (and queried). That is easily done in some Java code. The simplest way to do this is to plug in a custom tokenizer. That gets called from the query parser and bundled linguistics processing Searchers as well as the Document Processor creating the annotations that are consumed by the indexing operation. Since all that is Searchers and Docprocs which you can replace and/or add custom components before and after, you can also take full control over these things without modifying the platform itself.

#### Does vespa provide any support for named entity extraction?

It provides the building blocks but not an out-of-the-box solution. We can write a Searcher to detect query-side entities and rewrite the query, and a DocProc if we want to handle them in some special way on the indexing side.

#### Does vespa provide support for text extraction?

You can write a document processor for text extraction, Vespa does not provide it out of the box.

#### How to do Text Search in an imported field?

Imported fields from parent documents are defined as attributes, and have limited text match modes (i.e. indexing: index cannot be used). Details.

#### Why is closeness 1 for all my vectors?

If you have added vectors to your documents and queries, and see that the rank feature closeness(field, yourEmbeddingField) produces 1.0 for all documents, you are likely using distance-metric: innerproduct, but your vectors are not normalized, and the solution is normally to switch to distance-metric: angular.

With non-normalized vectors, you often get negative distances, and those are capped to 0, leading to closeness 1.0. Some models, such as models from sbert.net, claim to be normalized but are not.

### Programming Vespa

#### Is Python plugins supported / is there a scripting language?

Plugins have to run in the JVM - jython might be an alternative, however Vespa Team has no experience with it. Vespa does not have a language like painless - it is more flexible to write application logic in a JVM-supported language, using Searchers and Document Processors.

#### How can I batch-get documents by ids in a Searcher

A Searcher intercepts a query and/or result. To get a number of documents by id in a Searcher or other component like a Document processor, you can have an instance of com.yahoo.documentapi.DocumentAccess injected and use that to get documents by id instead of the HTTP API.

### Performance

Vespa has a near real-time indexing core with typically sub-second latencies from document ingestion to being indexed. This depends on the use-case, available resources and how the system is tuned. Some more examples and thoughts can be found in the scaling guide.

#### Is there a batch ingestion mode, what limits apply?

Vespa does not have a concept of “batch ingestion” as it contradicts many of the core features that are the strengths of Vespa, including serving elasticity and sub-second indexing latency. That said, we have numerous use-cases in production that do high throughput updates to large parts of the (sometimes entire) document set. In cases where feed throughput is more important than indexing latency, you can tune this to meet your requirements. Some of this is detailed in the feed sizing guide.

#### Can the index support up to 512 GB index size in memory?

Yes. The content node is implemented in C++ and not memory constrained other than what the operating system does.

#### Get request for a document when document is not in sync in all the replica nodes?

If the replicas are in sync the request is only sent to the primary content node. Otherwise, it’s sent to several nodes, depending on replica metadata. Example: if a bucket has 3 replicas A, B, C and A & B both have metadata state X and C has metadata state Y, a request will be sent to A and C (but not B since it has the same state as A and would therefore not return a potentially different document).

#### How to keep indexes in memory?

Attribute (with or without fast-search) is always in memory, but does not support tokenized matching. It is for structured data. Index (where there’s no such thing as fast-search since it is always fast) is in memory to the extent there is available memory and supports tokenized matching. It is for unstructured text.

It is possible to guarantee that fields that are defined with index have both the dictionary and the postings in memory by changing from mmap to populate, see index > io > search. Make sure that the content nodes run on nodes with plenty of memory available, during index switch the memory footprint will 2x. Familiarity with Linux tools like pmap can help diagnose what is mapped and if it’s resident or not.

Fields that are defined with attribute are in-memory, fields that have both index and attribute have separate data structures, queries will use the default mapped on disk data structures that supports text matching, while grouping, summary and ranking can access the field from the attribute store.

A Vespa query is executed in two phases as described in sizing search, and summary requests can touch disk (and also uses mmap by default). Due to their potential size there is no populate option here, but one can define dedicated document summary containing only fields that are defined with attribute.

The practical performance guide can be a good starting point as well to understand Vespa query execution, difference between index and attribute and summary fetching performance.

#### Can one do a partial deploy to the config server / update the schema without deploying all the node configs?

Yes, deployment is using this web service API, which allows you to create an edit session from the currently deployed package, make modifications, and deploy (prepare+activate) it: deploy-rest-api-v2.html. However, this is only useful in cases where you want to avoid transferring data to the config server unnecessarily. When you resend everything, the config server will notice that you did not actually change e.g. the node configs and avoid unnecessary noop changes.

#### Deployment fails / nothing is listening on 19071

Make sure all Config servers are started, and are able to establish ZooKeeper quorum (if more than one) - see the multinode sample application. Validate that the container has enough memory.

#### Startup problems in multinode Kubernetes cluster - readinessProbe using 19071 fails

The Config Server cluster with 3 nodes fails to start. The ZooKeeper cluster the Config Servers use waits for hosts on the network, the hosts wait for ZooKeeper in a catch 22 - see sampleapp troubleshooting.

#### Starting Vespa using Docker on M1 fails

Using an M1 MacBook Pro / AArch64 makes the Docker run fail:

WARNING: The requested image’s platform (linux/amd64) does not match the detected host platform (linux/arm64/v8)
and no specific platform was requested

Make sure you are running a recent version of the Docker image, do docker pull vespaengine/vespa.

#### How fast can nodes be added and removed from a running cluster?

Elasticity is a core Vespa strength - easily add and remove nodes with minimal (if any) serving impact. The exact time needed depends on how much data will need to be migrated in the background for the system to converge to ideal data distribution.

#### Should Vespa API search calls be load balanced or does Vespa do this automatically?

You will need to load balance incoming requests between the nodes running the stateless Java container cluster(s). This can typically be done using a simple network load balancer available in most cloud services. This is included when using Vespa Cloud, with an HTTPS endpoint that is already load balanced - both locally within the region and globally across regions.

#### Supporting index partitions

Search sizing is the intro to this. Topology matters, and this is much used in the high-volume Vespa applications to optimise latency vs. cost.

#### Can a running cluster be upgraded with zero downtime?

With Vespa Cloud, we do automated background upgrades daily without noticeable serving impact. If you host Vespa yourself, you can do this, but need to implement the orchestration logic necessary to handle this. The high level procedure is found in live-upgrade.

#### Can Vespa be deployed multi-region?

Vespa Cloud has integrated support - query a global endpoint. Writes will have to go to each zone. There is no auto-sync between zones.

#### Can Vespa serve an Offline index?

Building indexes offline requires the partition layout to be known in the offline system, which is in conflict with elasticity and auto-recovery (where nodes can come and go without service impact). It is also at odds with realtime writes. For these reasons, it is not recommended, and not supported.

#### Does vespa give us any tool to browse the index and attribute data?

No. Use visiting to dump all or a subset of documents. See dumping-data for a sample script.

#### What is the response when data is written only on some nodes and not on all replica nodes (Based on the redundancy count of the content cluster)?

Failure response will be given in case the document is not written on some replica nodes.

#### When the doc is not written to some nodes, will the document become available due to replica reconciliation?

Yes, it will be available, eventually. Also try Multinode testing and observability.

#### Does vespa provide soft delete functionality?

Yes just add a deleted attribute, add fast-search on it and create a searcher which adds an andnot deleted item to queries.

#### Can we configure a grace period for bucket distribution so that buckets are not redistributed as soon as a node goes down?

You can set a transition-time in services.xml to configure the cluster controller how long a node is to be kept in maintenance mode before being automatically marked down.

Grouped distribution is used to reduce search latency. Content is distributed to a configured set of groups, such that the entire document collection is contained in each group. Setting the redundancy and searchable-copies equal to the number of groups ensures that data can be queried from all groups.

#### How to set up for disaster recovery / backup?

Refer to #17898 for a discussion of options.

#### How to check Vespa version for a running instance?

Use /state/v1/version to find Vespa version.


text:
Categories

# Number Theory of Primes | AIME I, 2015

Try this beautiful problem from American Invitational Mathematics Examination, AIME, 2009 based on geometric sequence. Use hints to solve the problem.

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2015 based on Number Theory of Primes.

## Number Theory of Primes – AIME 2015

There is a prime number p such that 16p+1 is the cube of a positive integer. Find p.

• is 307
• is 250
• is 840
• cannot be determined from the given information

### Key Concepts

Series

Theory of Equations

Number Theory

AIME, 2015

Elementary Number Theory by Sierpinsky

## Try with Hints

Notice that 16p+1must be in the form $(a+1)^{3}=a^{3}+3a^{2}+3a$, or $16p=a(a^{2}+3a+3)$. Since p must be prime, we either have p=a or a=16

p not equal to a then we have a=16,

p\(=16^{2}+3(16)+3=307


text:
# Signed total Italian k-domination in graphs

Document Type : Original paper

Author

RWTH Aachen University

Abstract

Let k ≥ 1 be an integer, and let G be a finite and simple graph with vertex set V (G)
A signed total Italian k-dominating function (STIkDF) on a graph G is a function
f : V (G) → {−1, 1, 2} satisfying the conditions that $sum_{xin N(v)}f(x)ge k$ for each vertex
v ∈ V (G), where N(v) is the neighborhood of $v$, and each vertex u with f(u)=-1 is adjacent to
a vertex v with f(v)=2 or to two vertices w and z with f(w)=f(z)=1. The weight of an STIkDF f is
$omega(f)=sum_{vin V(G)}f(v)$. The signed total Italian k-domination number $gamma_{stI}^k(G)$ of G is the
minimum weight of an STIkDF on G. In this paper we initiate the study of the signed total Italian k-domination
number of graphs, and we present different bounds on $gamma_{stI}^k(G)$. In addition, we determine the
signed total Italian k-domination number of some classes of graphs. Some of our results are extensions of
well-known properties of the signed total Roman $k$-domination number $gamma_{stR}^k(G)$,
introduced and investigated by Volkmann [9,12].

Keywords

Main Subjects


text:
# Presentation Archive

### Fifth Force Searches in Galaxies

#### Harry Desmond (Oxford)

July 23, 2018

Abstract: Extra (“fifth”) forces generically follow from new dynamical fields, and are therefore ubiqituous in extensions to the standard models of particle physics and cosmology. Universally-coupled fifth forces are sufficiently strongly constrained by lab and Solar System tests as to be virtually irrelevant for cosmology. However, broad classes of Lagrangians exhibit “screening mechanisms” which hide the fifth force in high-density environments such as the Milky Way while allowing it to remain operative on larger scales. I will describe the search for screened fifth forces on the scale of galaxies and their environments. This is achieved in two stages. First, we model the gravitational environments of the local (\$z \lesssim 0.05\$) universe to determine the screening properties of real galaxies and the strength of the fifth-force field over space. Second, we use this information to forward-model two signals — displacement between stellar and gas mass centroids and warping of stellar disks — and hence constrain the fifth-force strength and range with a Bayesian likelihood formalism. Taking \$\sim\$ 10,000 HI detections from the ALFALFA survey and \$\sim\$ 4,000 images from the Nasa Sloan Atlas, we demonstrate sensitivity to new forces of strength \$~\% that of gravity for ranges O(1-10 Mpc). Both signals give evidence for a chameleon- or symmetron-screened fifth force of range ~1.8 Mpc and strength ~0.02 \$G_N\$, but we caution that unmodelled systematics such as baryonic physics may impact the inference at this level.


text:
Deepak Scored 45->99%ile with Bounce Back Crack Course. You can do it too!

# Solve this following

Question:

Let the volume of a parallelopiped whose coterminous edges are given by $\overrightarrow{\mathrm{u}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}, \overrightarrow{\mathrm{v}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{w}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ be $1 \mathrm{cu}$. unit. If $\theta$ be the angle between the edges $\overrightarrow{\mathrm{u}}$ and $\overrightarrow{\mathrm{w}}$, then $\cos \theta$ can be

1. $\frac{7}{6 \sqrt{3}}$

2. $\frac{5}{7}$

3. $\frac{7}{6 \sqrt{6}}$

4. $\frac{5}{3 \sqrt{3}}$

Correct Option: 1

Solution:


text:
## Time ordering operator

As I understand it, the time ordering operator works as follows (for $$t<0$$):
$$T c^\dagger(t) c(0) = -c(0) c^\dagger(t)$$ for fermions and
$$T c^\dagger(t) c(0) = c(0) c^\dagger(t)$$ for bosons.

Now suppose instead of these creation/annihilation operators, I had a more general commutation relation, ie $$[d,d^\dagger] = S$$, how does the time ordering operator behave?

Edit: After rereading that, I should be more specific. I'm trying to formulate DMFT equations in a non-orthogonal basis. So the creation/annihilation operators anti-commute to give [d_a,d_b^\dagger]_+ = S_{a,b}. The Green's function is usually defined as $$G(\tau) = \langle T c(\tau) c^\dagger(0) \rangle$$ and I need to understand exactly what the time ordering operator does, but I'm not even totally sure how its really defined, because as I understand it above, it works differently for fermions and bosons.
 PhysOrg.com physics news on PhysOrg.com >> Is there an invisible tug-of-war behind bad hearts and power outages?>> Penetrating the quantum nature of magnetism>> Rethinking the universe: Groundbreaking theory proposed in 1997 suggests a 'multiverse'

 Similar Threads for: Time ordering operator Thread Forum Replies Quantum Physics 17 Quantum Physics 5 Quantum Physics 9 High Energy, Nuclear, Particle Physics 2 General Physics 1


text:
3 years ago

# Measurements of $\pi^{\pm}$, $K^{\pm}$ and proton yields from the surface of the T2K replica target for incoming 31 GeV/c protons with the NA61/SHINE spectrometer at the CERN SPS.

V. I. Kolesnikov, B. Baatar, N. Abgrall, H. Cherif, V. A. Kireyeu, K. Kadija, M. Baszczyk, N. Kargin, A. Aduszkiewicz, A. Brandin, K. Grebieszkow, M. Gaździcki, V. Klochkov, M. Ćirković, R. Engel, A. Garibov, A. Blondel, V. N. Kovalenko, W. Bryliński, W. Dominik, M. Bogomilov, Z. Fodor, T. Antićić, J. Brzychczyk, A. Ereditato, T. Czopowicz, O. Busygina, S. R. Johnson, A. Korzenev, T. Kobayashi, F. Guber, J. Dumarchez, S. Kowalski, E. V. Andronov, A. Bzdak, S. A. Bunyatov, M. Deveaux, S. N. Igolkin, S. Bhosale, E. Kashirin, M. Golubeva, T. Hasegawa, A. Krasnoperov, N. Davis, A. Damyanova, K. Kowalik, A. Haesler, A. Ivashkin, O. Golosov, A. Bravar, G. A. Feofilov, M. Koziel, E. Kaptur, S. Ilieva, P. Dorosz, D. Kolev, L. Fields, M. Kiełbowicz, A. E. Hervé

Measurements of the $\pi^{\pm}$, $K^{\pm}$, and proton double differential yields emitted from the surface of the 90-cm-long carbon target (T2K replica) were performed for the incoming 31 GeV/c protons with the NA61/SHINE spectrometer at the CERN SPS using data collected during 2010 run. The double differential $\pi^{\pm}$ yields were measured with increased precision compared to the previously published NA61/SHINE results, while the $K^{\pm}$ and proton yields were obtained for the first time. A strategy for dealing with the dependence of the results on the incoming proton beam profile is proposed. The purpose of these measurements is to reduce significantly the (anti)neutrino flux uncertainty in the T2K long-baseline neutrino experiment by constraining the production of (anti)neutrino ancestors coming from the T2K target.

Publisher URL: http://arxiv.org/abs/1808.04927

DOI: arXiv:1808.04927v1

You might also like
Discover & Discuss Important Research

Keeping up-to-date with research can feel impossible, with papers being published faster than you'll ever be able to read them. That's where Researcher comes in: we're simplifying discovery and making important discussions happen. With over 19,000 sources, including peer-reviewed journals, preprints, blogs, universities, podcasts and Live events across 10 research areas, you'll never miss what's important to you. It's like social media, but better. Oh, and we should mention - it's free.

Researcher displays publicly available abstracts and doesn’t host any full article content. If the content is open access, we will direct clicks from the abstracts to the publisher website and display the PDF copy on our platform. Clicks to view the full text will be directed to the publisher website, where only users with subscriptions or access through their institution are able to view the full article.


text:
# Determining mole fraction of carbon dioxide knowing Kp only

I've been assigned the following problem at school:

Given the following reaction:

$$\ce{C (s) + CO2 (g) <=> 2 CO (g)}$$

Find the mole fraction of $$\ce{CO2}$$ provided that, in equilibrium, $$K_\mathrm{p} = 14.1$$.

Our teacher's solution is $$x(\ce{CO2}) = 0.324$$ (mole fraction).

I suspect that the problem as is provides insufficient information to arrive at the solution above. Assuming that the total pressure of the system is $$\pu{10 atm}$$, the resulting mole fraction is indeed $$0.324$$.

Is the problem truly incomplete or can it be solved without knowing the total pressure?

• You seem to be comfortable enough working out the answer when you're given a pressure. Why not try working it out without explicitly choosing a pressure? Use $p$ as a symbol instead of putting in something like 10 atm, and if you find that all the $p$'s cancel out in your answer, then it follows that you don't actually need to know the pressure. – orthocresol Jan 11 at 20:42
• I have attempted this. The final expression is: 14.1x = (1-x)^2*PT. The total pressure does not seem to cancel out. Here, x is the mole fraction. X seems to be a function of PT. The value of X changes with other values of PT. – Grego Jan 11 at 20:44
• Took me a while to understand what that equation meant, but yeah, you are correct! So, the question is indeed incomplete. [More common notation would be something like $p_\text{tot}$; PT suggests some pressure multiplied by temperature...] – orthocresol Jan 11 at 20:48
• You can format mathematical and chemical expressions on Chemistry.SE using MathJax; this post contains further details. – orthocresol Jan 11 at 20:50
• Ok! Thanks for the quick reply! My bad. I'll keep this in mind for the next question. – Grego Jan 11 at 20:51


text:
# 2016 Q4 Macroeconomics (Kobe Univ.)

## Supplementary Materials

• Computation of Optimal Growth based on Euler Equation
• Computation of Optimal Growth based on Dynamic Programming

## RStudio

To run Rmd files above and to compile (knit) them to pdf or other document format, install RStudio > 1.0.

## R Resources

If you are new to R and RStudio, or more generally to programing languages, the following short online material by Jenny Bryan is a good starting point. It will help you streamline your daily workflow. It is important in particular to understand the concept of “working directory” from the earliest stage of your learning.

Need a more comprehensive guidebook? Here is a website I’d recommend:

Because of my intensive using of functions and operators defined in dplyr and tidyr, my codes might look obscure for those who learned R years back and haven’t updated their coding style, and for those who don’t have background in (relational) database systems. Chapter 5 of Grolemund and Wickham will help.


text:
# FURTHER MATHEMATICS OF THE RECIPROCAL SYSTEM

This paper will present in the most concrete, explicit manner the mathematics of space–time, radiation, and matter of the Reciprocal System.  Readers without special knowledge of the Reciprocal System are first urged to study Larson’s books, especially Nothing But Motion¹ before undertaking the study of this paper.

I.  Mathematics of Space–Time

A.   Rectangular Coordinates

Starting from any reference point x0, y0, z0, t0 in the 0–system, the space–time progression is a spherical expansion.  In rectangular coordinates the equation is

 (x–x0)² + (y–y0)² + (z–z0)² = c² t² (1)

where c is the speed of light.  If we choose the reference point to be x0 = 0, y0 = 0, z0 = 0, t0 = 0, then the equation is simply

 x² + y² + z² = c² t²

Now consider a second system, the 0´–system, moving translationally with respect to the 0´–system in the x–direction.  What is the equation for the progression in the 0´–system?  From the inverse Lortentz transformations,

 [x´+vt´]² 1 x²= ————— = ———— (x´²+v²t´²+2v x´t´) (2) [n1-(v²/c²)]² 1-(v²/c²)

 [t´+(v/c²)x´]² 1 t²= ————— = ———— (v²/c4)x´²+t´²+(2v/c²)x´t´) (3) [n1-(v²/c²)]² 1-(v²/c²)

 y² = y´² (4)

 z² = z´² (5)

Upon substitution, we obtain

 x² + y² + z² – c²t² = x´² + y´² + z´² – c²t´² (6)

But since the left side of the equation equals zero, so must the right side:

 x´² + y´² + z´² = c²t´² (7)

Thus the progression as determined by 0' is also spherical.  And so the equation for the progression is invariant under a Lorentz transformation.

B. Polar Coordinates

In polar coordinates the equation is simply

 r – r0 = c(t – t0) (8)

 Or, letting r0 = 0, t0 = 0, (9)

 r/t = c

In the Reciprocal System the speed of light is the natural unit of velocity and so r and t must take equal natural values.  The space–time progression is thus ¹/1, ²/2, 3/3, etc.  Thus one unit of space is equivalent to one unit of time.  If there are an infinite number of space units, there must be an infinite number of time units; if there are a finite number of space units, there must be a finite number of time units.

In the Reciprocal System radiation is the combined motion of a simple harmonic oscillation in one dimension and a uniform translation in a perpendicular direction.

A.   Simple Harmonic Oscillation

The equation for a simple harmonic oscillation in one dimension (say the y direction) is

 y   =   A*SIN(–2pfost) (10)

where A is the amplitude and fos is the frequency.  Since the oscillation takes place over one natural space unit, the amplitude must be one–half a natural space unit and thus is

 A   =    .5*Snat = .5 x 4.558816 x 10–8m = 2.279408 x 10–8m (11)

for all photons.  In observation from the time–space region this value is reduced by the interregional ratio142.222 to 1.6027 x 10–10m = 1.6027 Å

The other variable to be determined in eq.(10) is the frequency, fos.  In one cycle the oscillation travels one space unit up and one space unit down, for a total of two units.  The average velocity of the oscillation is th

 vos  =    (2*snat/cycle) * fos (cycles/sec) (12)

The natural unit of frequency must occur when the average velocity is c.

 vos  =    c = 2*snat * fos nat (13) But   c    =    snat/tnat´ so  snat/tnat = 2*snat * fos nat Solving for fos nat we have

 fos nat      = 1/(2*tnat) = 1/(2*1.520655 * 10–16sec) = 3.2880575 * 1015 cycles/sec (14)

the Rydberg frequency.  (Actually, Larson derived the natural unit of time from the Rydberg frequency, but I think it was instructive to do the reverse, and this method will be used to calculate rotational and rotational vibration frequencies as well.  Of course, this method assumes that the natural unit of time can be found by some other means.)  Because of the discrete nature of the Reciprocal System, it is only possible to have integer multiples or reciprocal integer multiples of the Rydberg frequency.

Putting the values of A and fos in eq.(10) we have

 y    =    1.6027*SIN(–2pn * 3.2880575 * 1015 * t)Å (15)

where

 n    =    1, 2, 3, ...or (16) n    =    ½, ¹/3, ¼, ....

B.  Perpendicular Translation

Perpendicular to the oscillation is a translation at unit velocity (the speed of light).  Let x be perpendicular to y.  Then

 x    =    c * t (17)

C.  Combined Motion

From eq.(17) t can be found in terms of x and c and put in eq.(15).  The result is

 y    =    1.6027 * SIN(–2pn * 3.2880575 * 1015 * x/2.997930 * 108) (18) or   y    =    1.6027 * SIN (–6.8912465 * 107 * nx)Å

if x is given in meters.  This is the equation for a monochromatic wave of radiation in the Reciprocal System.

III.  Mathematics of Matter

Particles of matter consist of rotating photons.  Subatoms have one rotating photon; atoms have two rotating photons (both photons rotate about the same central point). The rotational motion has a translational effect, which will be discussed after the mathematics of the rotation has been worked out.

A.   Rotation

1single systems–particles

A photon can rotate around either of two horizontal axes passing through its midpoint, and also around itself.  In the Reciprocal System the true physical zero is motion at unit speed.  Anything physical must have a motion either greater or less than unit speed.  This deviation is called a speed displacement by Larson.  The first particle has 1 speed displacement around one horizontal axis of the photon and is called the rotational base.  Actually there are two rotational bases:  one with one speed displacement above unity, the other with one speed displacement below unity.  As will be discussed later, the one displacement unit neutralizes the translational motion of the photon in the original dimension, but the progression now continues in the remaining dimension, so the effective displacement is zero.  In the ground state condition, the photon that is being rotated is one vibrational displacement away from unity (either 2R or (1/2)R).  Here is a table giving the photon frequency, the rotational displacement, the effective rotational displacement, and rotational speed of the cosmic rotational base and the material rotational base:

 Photon  frequency Rotational  displacement Effective  rotational  displacement Rotational  speed ———— ————— ———————— —————— Cosmic rotational  base ½R (1)-0-0 0-0-0 2-1-1 Material rotational  base 2R 1-0-0 0-0-0 ½-1-1

In the above table the speeds are calculated from  the  displacements  as  follows.   For  displace

ments of np, ns, and nE, the speeds are (np+1), (ns + 1), and (nE + 1) for a cosmic particle, and 1/(np + 1), 1/(ns + 1), and 1/(nE + 1) for a material particle. (Of course material particles could have high speed electric displacement, and cosmic particles could have low speed electric displacement).
These speeds can be converted to conventional units, such as revolutions per second, as follows.  In one rotation of a photon about a horizontal axis the tip of the photon coverse a distance of p* snat´ a circumference.  The speed of the rotation is then

 Vrot =    [(p * snat) /rev] * frot (rev/sec) (19)

The natural frequency of rotation must occur when the speed is c.

 c   =  ¹ * snat * frot nat But  c   =  snat/tnat´ so solving for frot nat, we get frot nat        =  1/(¹tnat) But  tnat      =  1/(2R) so   frot nat  =  2R/¹ (20)

where R is the Rydberg frequency, as before.  In these terms, then, the cosmic rotational base is a photon that has a vibrational oscillation of 1.6440288 x 1015 cycles/sec and is rotating at 4.1864848 x 1015 revolutions/sec around one axis, and 2.0932424 x 1015 revolutions/sec around the other two axes.  Likewise, the material rotational base is a photon that has a vibrational oscillation of 6.576115 x 1015 cycles/sec and is rotating at 1.0466212 x 10 15 revolutions/sec around one axis, and 2.093242 x 1015 revolutions/sec around the other two axes.
All the other particles have photon vibrational frequencies, rotational displacements, effective rotational displacements, rotational speeds, and rotational frequencies.  Here is a complete tabulation:

 Photon  frequency Rotational  displacement Effective rot. displacement Rotational  speed Rotational frequency ———— ————— ————— ———————— —————— M–positron 2R 1–0–1 0–0–1 ½-1–½ R/p-2R/p-R/p C–positron ½R (1)–0–(1) 0–0–(1) 2–1–2 4R/p-2R/p-4R/p M–electron 2R 1–0–(1) 0–0–(1) ½-1–2 R/p-2R/p-4R/p C–electron ½R (1)–0–1 0–0–1 2–1–½ 4R/p-2R/p-R/p

 Photon  frequency Rotational  displacement Effective rot. displacement Rotational  speed Rotational frequency ———— ————— ————— ———————— ———————— M-massless neutron 2R 1–1–0 ½ –½–0 ½–½-1 R/p-R/p-²R/p C-massless neutron ½R (1)–(1)–0 (½)–(½)–0 2–2–1 4R/p-4R/p-²R/p M-neutrino 2R 1–1–(1) ½–½–(1) ½-½–2 R/p-R/p-4R/p C–neutrino ½R (1)–(1)–1 (½)–(½)–1 2–2–½ 4R/p-4R/p-R/p

Many more permutations appear to be possible, but the probability principles keep eccentricity to a minimum.  Since none of the above particles has an effective displacement of 1 or more, they are all massless (aside from the mass contribution of an electric charge).  The diameter of all the particles is one natural space unit, reduced by the (one–photon) interregional ratio, or 3.2054 Å.  However, because these particles do not exert any force in the uncharged state, a particle–measuring probe would not be able to detect any size of these particles at all.

2.  intermediate systems

Intermediate particles have two rotating photons, but one of the two sets has no effective displacement and thus contributes no primary mass.  The two intermediate particles are the neutron and the mass one hydrogen isotope (and their cosmic analogs).  There are only two kinds of rotations that can combine to form this kind of particle, the proton type and the neutrino type  We identify the combination of the material proton rotation and the material neutrino rotation as the mass one hydrogen atom; the combination of the material proton rotation and the cosmic neutrino rotation as the neutron; the combination of the cosmic proton rotation and the cosmic neutrino rotation as the mass one atom of cosmic hydrogen; and the combination of the cosmic proton rotation and the material neutrino rotation as the cosmic neutron. The proton is a single system with displacements 2–1–(1), effective displacements 1–1–(1), speeds ½-½-2, and  rotational frequencies 2R/3p–R/p–4R/p.  Then we would have the following table for the neutron and mass one hydrogen.

 Photon  frequency Rotational  displacement Effective rot. displacement Rotational  speed Rotational frequency ———— ————— ————— ———————— ———————— Neutron { 2R 2–1–(1) 1–1–(1) ¹/² –½-2 2R/³p-R/p-4R/p ½R (1)–(1)–1 (½)–(½)–1 2–2–½ 4R/p-4R/p-R/p H¹ { 2R 2–1 > (1) 1–1 > (1) ¹/³-½ > 2 2R/³p-R/p >4R/p 2R 1–1 ½–½ ½ –½ R/p-R/p

The new notation makes clear the two photons involved and the five rotations (to be discussed next).

3.  Atomic cycles

Atoms have two rotating photons, but here both systems have effective displacements and both systems ordinarily have the same velocities.  Let the first photon be called A and the second be called B.  A and B are mutually perpendicular.  We have the following five rotations:

(i)    the rotation of A about B produces disk a;

(ii)   the rotation of B about A produces disk b;

(iii)  then disk a can be rotated about A;

(iv)  then disk b can be rotated about B;

(v)   finally the whole structure can be rotated in the electric dimension.

This last rotation is in the scalar direction opposite to that of the previous rotations.  Cosmic atoms have speeds above unity for the first four types of rotations, whereas material atoms have speeds below unity for the first four types.  The electric rotation may be above or below unity for both cosmic and material atoms.

The first particle with two effective rotating systems is deuterium, the second is helium, etc.  A table similar to that for the intermediate particles can be made.

 Photon  frequency Rotational  displacement Rotational  speed Rotational frequency ———— ————— ——————— ———————— Deuterium { 2R 2–1 > (1) ¹/³ –½ > 2 2R/³p-R/p >4R/p 2R 2–1 2³–½ 2R/³p-R/p Helium { 2R 2–1 > 0 ¹/³-½ > 1 2R/³p-R/p >2R/p 2R 2–1 ½³ –½ R/³p-R/p

All other atoms can be given appropriate values in the same manner.  In the solid state, however, the values that govern the physical properties are not the actual rotations, but the relative rotations, and the different values there are not due to inherent differences in the rotational speeds, but to differences in the orientations of the interacting atoms, and this will be discussed further later.

4.  Electric charges and magnetic charges

According to the Reciprocal System an electric charge is a rotational vibration about the electric axis, and the magnetic charge is a rotational vibration about one of the magnetic axes.  Both charges have the same natural frequency, calculated as follows.  In one cycle the motion covers a distance of ¹ * snat one way and p * snat back, for a total of 2p * snat.  So we have

 vch =  (2p * snat/cycle) * fch (cycles/sec) (21)

At the unit level, vch = c = snat/tnat, so

 snat/tnat = (2p * snat/cycle) * fch nat (22)

Solving for fch nat and recalling that tnat = 1/(2R),

 fch nat = R/p (23)

This frequency is one–half that of a full rotation and can thus be considered to be effective in one direction only half the time.  One negative electric charge is a rotational vibration of R/2p = 5.233106 x 1014 cycles/sec.  One positive electric charge is a rotational vibration of 2R/p = 2.093242 x 1015 cycles/sec.  Similarly one unit of magnetic charge is a rotational vibration of 2R/p = 5.233106 x 1014 cycles/sec, whereas one unit of isotopic charge is a rotational vibration of R/2p = 5.233106 * 1014 cycles/sec.  The isotopes of atoms result from the addition of isotopic charges.

B.  Translation

The rotational motion of particles has a translational effect.  The maximum inward translation is two full units, giving one net inward unit.  In terms of rotation we can have 2³ = 8 one–dimensional rotational electric displacements or 4 two–dimensional rotational magnetic displacements.   Note that since 1³ = 1, the first magnetic rotational displacement, which is ½ unit rotational speed, produces one unit of inward translation and thus neutralizes the original translational motion of the photon, but the progression still continues in the third dimension.  Thus the rotational base and all the single system massless particles previously discussed move at the speed of light.  Additional magnetic and electric displacements produce a net inward motion, and the inward motion of a group of atoms is termed gravitation.

For atoms with magnetic displacements of less than 4 and electric displacements of less than 8, the frequency of the rotating photons is normally one displacement above unity, or 2R (the frequency of photons in cosmic atoms is (½)R).  When the magnetic displacement reaches 4 or the electric displacement reaches 8, the rotation must be extended to a second vibrational displacement unit–which means that the frequencies of the photons are now 3R (or (¹/3)R for cosmic atoms).  As Larson points out, though, it is possible to have these higher frequency photons even when the rotational displacements are less than 4 or 8, in which case we can say that the atom is “excited”.
After the change to vibration two, two units of vibrational displacement exist to be rotated, and so each added unit of rotational displacement corresponds to only one–half unit of added specific speed.  Thus the speeds corresponding to magnetic displacements can be listed as follows:

 MagneticDisplacement MagneticSpeeds 1 ½ ½ ½ 2 ¹/3 ²/5 ¹/3 3 ¹/4 ¹/3 ¹/4 4 ¹/9 ²/7 ¹/5 5 ²/5 ¹/4 in one displacement axis only

And the speeds corresponding to electric displacements can be listed as follows:

 Electric Displacement Electric Speeds Electric Displacement Electric Speeds 1 ½ ½ 9 ¹/9 ¹/6 2 ¹/3 ²/5 10 ²/19 ²/13 3 ¹/4 ¹/3 11 ¹/10 ¹/7 4 ¹/5 ²/7 12 ²/21 ²/15 5 ¹/6 ¹/4 13 ¹/11 ¹/8 6 ¹/7 ²/9 14 ²/23 ²/17 7 ¹/8 ¹/5 15 ¹/12 ¹/9 8 ²/17 ²/11 16 ²/25 ²/19

In the solid state, the values for electric rotation can be further altered.  Larson states that a combination of one atom of electric displacement x with another atom of electric displacement 8–x results in a neutral bond.  This bond gives rise to an electric speed of ¹/10 for vibration one, and ¹/5 for vibration two.  Also there can be a combination of two 8–x atoms, which Larson calls a secondary positive bond.  In this case the rotational speed comes to ¹/(18–2x).
One final set of complications involves the lower group elements.  Here there is just one subordinate magnetic displacement unit and thus these elements have less rotational force and thus are closer together in the solid state.  The force is proportional to ln t, where t is the inverse of the magnetic speed, and since ln 2 is less than 1, atoms that have magnetic speed greater than ¹/3 in any dimension have no effective force in that dimension.  The number of “active” dimensions is given in the following tabulation.

Lower Group Atomic Table in Solid Stat

 Atom.  No Elem. Oscillation Frequency Rotational Displ. Bond Rotational Speed Rotational Frequency Active Dim. { 2R 2-1 > (1) ¹/³–½ > ¹/10 2R/³p-R/p > R/5p 1 H Neutral 1 2R 2-1 ¹/³–½ 2R/³p-R/p
 { 2R 2-1 > 0 ¹/³–½ > 1 2R/³p-R/p > 2R/p 2 He Zero 1 2R 2-1 ¹/³–½ 2R/³p-R/p
 { 3R 2-1 > 1 ²/5–²/5 > ½ 4R/5p-4R/5p > R/p 3 Li Positive 2 3R 2-1 ²/5–²/5 4R/5p-4R/5p
 { 3R 2-1 > 2 ¹/³–½ > ²/5 2R/³p-R/p > 4R/5p 4 Be Positive 2 3R 2-1 ¹/³–½ 2R/³p-R/p
 { 3R 2-1 > 3 ¹/³–½ > ¹/5 2R/³p-R/p > 2R/5p 5 B Neutral 1 3R 2-1 ¹/³–½ 2R/³p-R/p
 { 2R 2-2 > (5) ¹/³–¹/³ > ¹/10 2R/³p-2R/³p > R/5p or B Neutral 1 2R 2-2 ¹/³–¹/³ 2R/³p-2R/³p
 { 2R 2-1 > 4 ¹/³–½ > ¹/10 2R/³p-R/p > R/5p 6 C Neutral 2 2R 2-1 ¹/³–½ 2R/³p-R/p
 { 2R 2-2 > ¹/³–¹/³ > 1 2R/³p-2R/³p > 2R/p or C Zero 3 2R 2-2 ¹/³–¹/³ 2R/³p-2R/³p
 { 2R 2-2 > (3) ¹/³–¹/³ > ¹/10 2R/³p-2R/³p > R/5p 7 N Neutral 1½ 2R 2-2 ¹/³–¹/³ 2R/³p-2R/³p
 { 2R 2-2 > (3) ¹/³–¹/³ > 1 2R/³p-2R/³p > 2R/p or N Zero 3 2R 2-2 ¹/³–¹/³ 2R/³p-2R/³p
 { 2R 2-2 > (2) ¹/³–¹/³ > ¹/10 2R/³p-2R/³p > R/5p 8 O Neutral 1½ 2R 2-2 ¹/³–¹/³ 2R/³p-2R/³p
 { 2R 2-2 > (2) ¹/³–¹/³ > 1 2R/³p-2R/³p > 2R/5p or O Zero 3 2R 2-2 ¹/³–¹/³ 2R/³p-2R/³p
 { 2R 2-2 > (1) ¹/³–¹/³ > ¹/10 2R/³p-2R/³p > R/5p 9 F Neutral 2 2R 2-2 ¹/³–¹/³ 2R/³p-2R/³p

The reader can continue the table all the way to element 118.  Again, one must first determine the kind of bond involved before the electric rotational speed can be determined.

Since different atoms have different rotational speeds and thus different rotational forces, a particle probe of equal energy shot at atoms of different elements would “penetrate” to different depths.  Thus experimenters have concluded that “nuclear” size is proportional to atomic weight.  Actually what they are measuring is atomic size, and according to the Reciprocal System this is constant (2.914 Å diameter)–but the force is proportional to the atomic weight of the atom.  Also, where interatomic distances are less than 2.914 Å, the atoms are partially merged; where distances are greater than 2.914Å, the atoms are separate.

Reference

1. Dewey B. Larson, Nothing But Motion (North Pacific Publishers: Portland, Oregon, 1979)

International Society of  Unified Science
Reciprocal System Research Society

Salt Lake City, UT 84106
USA

Theme by Danetsoft and Danang Probo Sayekti inspired by Maksimer


text:
# ON NILPOTENCE INDICES OF SIGN PATTERNS

• Erickson, Craig (Department of Mathematics and Statistics, Minnesota State University) ;
• Kim, In-Jae (Department of Mathematics and Statistics, Minnesota State University)
• Published : 2010.01.31
• 72 20

#### Abstract

The work in this paper was motivated by [3], where Eschenbach and Li listed four 4 by 4 sign patterns, conjectured to be nilpotent sign patterns of nilpotence index at least 3. These sign patterns with no zero entries, called full sign patterns, are shown to be potentially nilpotent of nilpotence index 3. We also generalize these sign patterns of order 4 so that we provide classes of n by n sign patterns of nilpotence indices at least 3, if they are potentially nilpotent. Furthermore it is shown that if a full sign pattern A of order n has nilpotence index k with $2{\leq}k{\leq}n-1$, then sign pattern A has nilpotent realizations of nilpotence indices k, k + 1, $\ldots$, n. Hence, the four 4 by 4 sign patterns in [3, page 91] also allow nilpotent realizations of nilpotence index 4.

#### Keywords

Jordan block;nilpotence index;potentially nilpotent sign pattern

#### References

1. R. A. Brualdi and B. L. Shader, Matrices of Sign-solvable Linear Systems, Cambridge University Press, New York, 1995.
2. M. S. Cavers and K. V. Vander Meulen, Spectrally and inertially arbitrary sign patterns, Linear Algebra and its Applications 394 (2005), 53–72. https://doi.org/10.1016/j.laa.2004.06.003
3. C. A. Eschenbach and Z. Li, Potentially nilpotent sign pattern matrices, Linear Algebra and its Applications 299 (1999), 81–99. https://doi.org/10.1016/S0024-3795(99)00163-9
4. Y. Gao, Z. Li, and Y. Shao, Sign patterns allowing nilpotence of index 3, Linear Algebra and its Applications 424 (2007), 55–70. https://doi.org/10.1016/j.laa.2006.03.038
5. I.-J. Kim, D. D. Olesky, B. L. Shader, P. van den Driessche, H. Van Der Holst, and K. N. Vander Meulen, Generating potentially nilpotent full sign patterns, Electronic Journal of Linear Algebra 18 (2009), 162–175.

#### Cited by

1. Tree sign patterns that allow nilpotence of index 4 vol.63, pp.5, 2015, https://doi.org/10.1080/03081087.2014.914930
2. Potentially nilpotent patterns and the Nilpotent-Jacobian method vol.436, pp.12, 2012, https://doi.org/10.1016/j.laa.2011.05.017