** subsystems and frameworks will be checked during the execution and testing stages. Last framework level tests will be performed to acknowledge the framework and exhibit the framework's status for creation administration. In any case, testing exercises won't end once the framework is in activity; testing will go on as the tasks and upkeep staff perform remedial, versatile, and other framework support exercises. What strategies are utilized to direct testing? There are Four fundamental confirmation techniques, as illustrated beneath.**

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The electron shell is comprised of heat energy. Energy is just heat expanding into area's of less heat. When energy enters the gravity field of the nucleus, it slows down because of the density of the gravity field. Energy reaches a point in the gravity field where its heat/density on the aether equals that of the gravity field and forms a shell of electricity, which is heat energy at a standstill. The energy still maintains the property of seeking out colder space even though its trapped in the shell. So when two atoms pull together because off gravity, the shell of electricity repels because the heat energy in either shell doesn't want to occupy the heat/density of the other atom's shell and they repel.

Check out the aether and the femto camera to hear about new evidence of the existence of the aether.

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Basically the paradox is this; In terms of Special Relativity, how does spinning disk work in special relativity, if the circumference of the rotating disk undergoes length-contraction (since it's parallel to motion) while its radius does not (since it's perpendicular to motion), and this would imply that [math]\frac{circumference}{diameter} \neq \pi [/math].

I checked bunch of similar questions of the same topic, and can't find a single person giving the correct answer on Quora. Instead I find all sorts of face-palm inducing nonsense like;

- Only the atoms length contract, but the space between them does not.
- The disk would tear into smaller pieces along the radius to give shorter total circumference.
- The disk would implode under pressure from the shrinking circumference.
- Centrifugal forces would counteract Lorentz contraction.
- There's no strong enough material to build such disk because of Born rigidity and elasticity, thus no paradox.
- You need to use General Relativity to solve the paradox.
- The geometry of the spinning wheel is non-euclidean. Just accept it.

And bunch of other answers going completely off on a tangent on topics like people inside a spinning train setting their clocks. Basically every single answer I can find tells me the author probably holds serious misconceptions about Special Relativity itself.

Okay, it's Quora so I shouldn't expect too much, but still I would have expected that at least someone would have given the solution to to something this simple, instead of seeing bunch of people with credentials compete with silly answers. Some of those people are citing their own book about the topic while giving a terrible answer... I mean I'm not that smart, but I solved this problem in my head, while driving. It's really that simple if you actually already understand Special Relativity properly.

What really surprised me was when I went on to check how does Wikipedia see this, and it also doesn't explain the proper solution. There is only one passing mention of the correct solution (kind of, possibly, can't really tell) in the "Brief History" section... with no actual explanation. I guess this is why no one in Quora also knows the answer, but still I'm quite dumbfounded to realize that the actual solution is apparently not very well known at all. I can't find any article actually explaining the correct solution.

Looking at all the bad answers, it seems to me that that there are few different reasons why most people get this so wrong.

One is that many people think about length contraction as something that happens to **objects**, when more accurately it's what happens to your coordinate system when you change your perspective from one inertial frame to another, and follow Einstein convention for isotropic C. If you think it happens to "objects" because they "move", you might be inclined to bring up stuff like "atoms shrink by space between them does not", and that is completely wrong perspective.

Second is that many authors start to analyze realistic materials and Born rigidity, which to some people perhaps seem like a way out of the paradox in some convoluted way. But that is also a complete red herring. The paradox is a thought experiment, and it has got nothing to do with realistic materials. It's about **geometry** in terms of special relativity, which ought to produce self-consistent results regardless of inertial frame. Solving Ehrenfest paradox by bringing up realistic materials and centrifugal forces is like solving twin paradox with "planet earth cannot produce enough energy to actually run that experiment".

Third reason is that a mathematical analysis in the framework of special relativity is easiest to do by making certain approximations, which are exactly the approximations leading into wrong answers. That misleading approximation is the idea of placing a number of **straight rods** along the circumference of the disk, and this approximation is exactly what will give you **wrong answers**. That's right, Einstein's own analysis is also flawed for the same reason, even though it led into insights that led into General Relativity.

Why that approximation produces a critically wrong answer, and what is the correct answer? I'll explain in a bit...

**The correct perspective**

First, just to convince the reader that this problem is in fact fully solvable in terms of Special Relativity without any hocus pocus about elastic materials, please be aware that the frame transformation from one inertial frame to another can be conceived as a sort of rotating / scaling of events in spacetime.

Like this;

https://en.wikipedia.org/wiki/File:Lorentz_transform_of_world_line.gif

The dots in that animation represent **events** as plotted on a spacetime diagram, and the "squishing" of the whole structure represents frame transformation from one inertial frame to another. Some events get pushed "towards the future" and some events get pushed "towards the past". Nothing actually happens to "objects" just because we choose to plot them in a different inertial frame; it's just about how we must plot events, if we are to assume isotropic C, and if we are to remain self-consistent in our mapping between frames.

It really is a good idea to view Special Relativity simply as self-consistent frame transformation rules, and you start seeing that the whole question of length contraction is **not** about how different observers "see things", or how they "measure things", or "what happens to objects", but rather about **how the universe must be plotted in spacetime diagrams when assuming different notions of simultaneity**.

In a nutshell, if we switch from one inertial frame representation to another - assuming unique simultaneity to each frame - we must plot the world state "ahead" of us as pushing towards the future (things that had not yet happened in old frame, have already happened in new frame), and conversely the world state "behind" us as pushing towards the past (something that had already happened in old frame, has not yet happened in new frame). Analyzing moving objects like this is what leads into the concept of "length contraction".

Since we are effectively molding the spacetime diagram around, but preserving the same exact light-like connections between events (the causality - the order of connected events - remains unchanged), it should be pretty easy for anyone to see that if it is possible to represent a spinning disk as a "set of events" in one frame, and it would have to also transform along with all the other events in self-consistent manner to any different frame without hiccups. From this perspective, the actual question behind the paradox is simple; **how would the spinning disk plot onto a spacetime diagram in terms of different notions of simultaneity?**

Even if you can't instantly figure out the exact solution, you should be able to already convince yourself that there is an exact solution out there which would just mold the (events making up the) spinning disk in consistent manner, along with everything else around the situation. What that exact solution is - let's get to it.**The common error**

Once the above is understood correctly, next it should be pretty easy to see how the "rigid rods along the circumference" analysis leads you down the wrong path, and at the same time get an glimpse of the correct solution.

- First, imagine a wheel-of-fortune, with pins sticking out from the outer circumference.
- Then we take a spoked wheel (a bicycle wheel), just proper size to snuggly fit inside the pins of the wheel-of-fortune.
- Last, let's enclose the whole two-disk setup inside a box with a snug fit.

*The purpose of this setup is to signal us if we are doing something inconsistent with our transformation - if the inner wheel fits inside the larger wheel, and if both wheels fit inside the box in one inertial frame, this must be so in all inertial frames. If it's not, we have performed an error in our analysis.*

Now let's take

Now let's set the

Since we have rod A spinning along, let's think about what happens if we shoot

At first glance it might seem like those two rods could be setup to become

The rod that is attached to the spinning wheel is - obviously - never moving in straight line; it is rotating. It's front end is always moving in different direction than its back end (each end is moving parallel to the part of the circumference it touches). So, the first question is,

If we plot the external box of the whole setup, in terms of the inertial frame of rod B, it's easy to propose relativistic speeds where the entire box gets plotted as length contracted to shorter length than rod B. The (non-rotating) wheel-of-fortune inside the box must also be mapped inside the box in every frame, and similarly squashed in the direction of motion - snuggly fitting inside the box. And the rotating bicycle wheel must fit also inside the pins of the wheel-of-fortune. It will get plotted also as snuggly fitting inside the wheel-of-fortune. Note though, the spokes will be plotted as curved because it is actively rotating and we are mapping it by a tilted simultaneity plane - this is just the flipside of the coin same coin that makes us map it as squashed.

Basically the internal configuration of our setup cannot change based on what inertial frame we map it from - rod A does not suddenly poke through the walls of the box just because we choose to plot the situation in different inertial frame. If we think it does, we are making an error in our analysis, or using invalid frame transformation. Basically it would imply an inconsistent change in the configuration of our system (some objects transforming in different ways than others - clearly invalid)

If we investigate a moment where the exact middle points of the rods meet in the same inertial frame, and we choose to plot this in terms of rod B's simultaneity, then the "front" end of rod A (in terms of direction of rotation) has already passed the "front" end of rod B (in terms of direction of motion of rod B in lab frame) some time ago. To be more accurate, since it's attached to a

And conversely, the world state behind us is plotted as pushing towards the past; the rear ends of the rods have not yet met. And since the rod is constantly rotating, the rear end of rod A is also plotted as curving "upwards", and moving towards rod B.

This is why, if you plot down the shape of the spokes of the wheel from the perspective of rod B, the end result looks like this;

https://en.wikipedia.org/wiki/File:Relativistic_wheels.gif

This is simply a result of plotting the events making up the "supposed world state" as transformed as per the Einstein convention of clock synchronization. A convention for plotting data. Nothing more, nothing less.

The error almost everyone makes is that they view length contraction as something actually occurring to

Also, since it was attached to the spokes of the rotating wheel, you'd have to plot the wheel also as having a larger length between two spokes than can be made to fit inside the box wheel-of-fortune, or inside the box. Obviously this result would mean your analysis is completely invalid, plain and simple.

And make no mistake about this - the same error happens no matter how short measurement rods you use. Shorter rods have smaller error, but you can always propose a speed where the error becomes obvious. And with smaller rods there's more of them so the end result of any full analysis is exactly as invalid. Basically you can't have an entire rod in a single inertial frame, while also being attached to the spinning wheel. These are mutually exclusive circumstances.

The same error exists in Einstein's co-rotating observer thought experiment, albeit in more subtle manner. But the point is, the co-rotating observer cannot have a measurement rod in any single inertial frame if that rod is to be also attached to the rotating wheel. The approximation necessary to imagine that situation will always make the analysis invalid for the same reasons as described above.

**The correct solution**

First clue to understanding how this situation really gets plotted correctly is this; Rod B only shares inertial frame with an **infinitesimally thin slice of the spinning wheel**. This is true by the very definition of "spinning". Also from the perspective of the lab frame (where the hub of the wheel is stationary), each infinitesimal slice of the spinning wheel is sitting in a different inertial frame, and does not have any "length" assignable to any single inertial frame. This is a simple mathematical fact arising from the very definitions behind special relativity and "spinning wheel".

Second key to understanding this is also associated with properly understanding length contraction as coordinate transformation. Remember when I said *"if we switch from one inertial frame representation to another, we must plot the world state "ahead" of us as pushing towards the future, and conversely the world state "behind" us as pushing towards the past."*. Note what happens in-between; **the world state in the infinitesimal slice exactly perpendicular to the motion does not transform at all!** This is btw also why the spokes at the bottom and at the top of the spinning wheel were plotted as straight in the relativistic wheel visualization above. (And I can show why the spokes are plotted as curved with another thought experiment too if anyone is interested)

This leads into the simple fact that, in the above experiment, at the moment when the middle part of rod A and rod B meet, **a non-rotating observer sitting at the hub of the wheel co-incides with this infinitesimal plane** that is cutting through the wheel, and that observer **will agree with simultaneity of all events that co-incide with that infinitesimal plane.**

We could repeat the same experiment in any direction, and get the same result, because the wheel is symmetrical. Thus we can see how

So getting back to the original Ehrenfest Paradox, the correct solution is simply to realize that the definitions of

To summarize;

The non-rotating observer (at rest with the hub of the spinning wheel) actually

If this still sounds like a strange claim to you, you are forgetting where length contraction comes from. It comes from dynamic notion of simultaneity, and only applies to how we plot

So the TL;DR solution is, the spinning wheel circumference

And as it turns out, all of the "commonly accepted" (or maybe there isn't one) solutions I can find are

Do note that this solution is all about how geometry gets plotted in terms of special relativity - it's not about how to set a wheel in rotation or about realistic materials. This solution simply arises from how objects get plotted into different inertial frames in self-consistent manner, following exactly the definitions of Special Relativity, and thus it is also exactly the correct solution to the original Ehrenfest Paradox.

Sorry about the length of this. I didn't want to just state what the correct solution is without explaining it in sufficient detail to give everyone a chance to convince themselves about this. Because it seems like the misconception here is so common that the actual end result probably just sound immediately wrong to most people until they think it through themselves.

And it says that "energy is quantized" at 11:40. And to illustrate this point the image on the screen splits into lots of uniform cubes.

This led me to conclude that everything is made of indivisible, uniform pieces. Like any image on this pixelated screen. No only because of they way they showed it but because they said that quantum mechanics governs the things that everything is made of. Also they said energy is quantized and some other sources have said that everything is made of energy. From this I concluded that everything is quantized.

Is this the case?

I found some sources that explain that the energy of light is quantized. Is that what Nova ment or do they mean all energy?

Also, I found some articles that say while light only comes in discrete chunks (quanta), the chunks can have any value depending on the circumstances. The light can't come in values of 1, then 3.5 then 2, only in one number. But depending on the light sources it can come in a steam of 1s, 3.5s 2s or any continuous value etc.

]]>Here is the description of the mentioned analogy from the book:

Each of the two remote observers - Alice and Bob - uses a device that has two buttons, labeled M and N, and a screen that can show either +1 or -1. During the experiment, Alice and Bob are unable to communicate with each other.

The source located roughly halfway between Alice and Bob sends them a couple of particles of some kind. Alice and Bob receive these particles and each insert them into their device. Then they select a random button on the device (M or N) and press it at the same time. Each device displays a value of +1 or -1, possibly related to the state of the generated particle. The entire operation described is called an event.

Both observers keep a record of the buttons they pressed and the numbers displayed. After receiving data on a large array of numbers, both parties meet and perform a correlation analysis of their records. Specifically, they estimate the value

Here, Ma, Mb, Na, Nb are the numbers that Alice and Bob receive after pressing the corresponding buttons. Each event only contributes to one of the values MaMb, MaNb, NaMb, NaNb. The book says that if | S | is greater than 2, then Bell's inequalities are violated.

This is how, if I understand it correctly, a typical experiment looks like:

I wrote a program that counts S for different algorithms of generating measurements of Alice and Bob. The following conclusions were obtained. Suppose Alice and Bob's buttons are completely random; Alice's measurement is also completely random, and Bob's measurement depends on Alice's measurement, but does not depend on the button that Alice or Bob pressed. Then S can take values from -2 to 2 (after averaging a large number of events).

Now suppose Alice's measurement is random, and Bob's measurement is defined as follows: if Alice pressed M and Bob pressed N, Bob's measurement is opposite to Alice's, otherwise Bob's measurement is the same as Alice's. Then S equals 4 - this is a violation of Bell's inequalities.

This leads to very interesting conclusions, but only under the condition that this whole analogy is correct. For the latter algorithm (for which S = 4) Bob's measurement indirectly depends on Alice's button, but does not correlate with it; therefore, Alice cannot convey information to Bob by pressing the button for a reason. This is consistent with what I have read in various sources about quantum entanglement - it does not allow information to be transmitted, but at the same time it cannot be called a complete absence of any interaction. Einstein called it “spooky action at distance”, and this characteristic is understandable, since particles located at different times can be entangled, so “spooky action through time” is an equivalent formulation.

I suppose, this analogue corresponds to this experiment:

https://en.wikipedia.org/wiki/CHSH_inequality

As far as I can see, when the polarizer a is set with 0 degrees angle, this corresponds to pressing M by Alice, and 45 degrees correspond to pressing N by Alice; when the polarizer b is set with 22.5 degrees angle, this corresponds to pressing M by Bod, and 67.5 degrees correspond to pressing N by Bob.

Please comment on whether my post is correct.

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My work,the semi classical Friedmann eq. And pre big bang phase

https://www.quora.com/q/vtikdwnzcqjjlems/The-Semi-Classical-Friedmann-Equation

The recently published" Independent" model

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I have heard that a rope is the model for these phenomena. You can start up a linear or circular wave through a rope. If you pass the rope through a slot, then the circular wave will turn into a linear one. If you put two perpendicular slots, then the wave will not pass; but if one more slot is placed between them at an angle of 45 degrees, then again part of the wave will pass. This illustrates that the slot, like the polarizer, “not only absorbs but also rotates”, I hope you understand what I mean. Is this correct?

I want to program a mathematical model that describes a polarizer for light or a slot for a rope, and so far I still don’t understand how to set a diagonal slot in this model. The point is obvious that if the vector [X=1;Y=0] is absorbed by the gap [1;1], then in the basis of this gap one of the components in the vector is set to zero, and in the original basis the component Y does not decrease, but increases, as the vector [1;0] is projected onto [1;1]. That is, as I understand it, one can say that "quantum magic is a consequence of mathematical magic" - in two-dimensional space, when a vector is absorbed, its individual components can increase. Is the written correct?

]]>Intuitively, it seems that this should not be so, and the model of the Conway's Game of Life looks more close to reality. In this game, the state of the system is described by discrete values, i.e. a finite amount of information is sufficient to describe the system. The question arises, are there any analogs of the Game of Life (cellular automata), in which the laws of conservation and the laws of thermodynamics work?

The Game of Life clearly reproduces reality very poorly, since it does not contain any of this. In addition, this game has a different arrow of time. In our reality, we experrience a psychological arrow of time: we remember the events of the past and predict the events of the future, and this knowledge about the past and the future is very asymmetric - information about the past is much more voluminous, more specific, detailed, more reliable than the information about the future. In the game Life, if there were intelligent beings, it would be the opposite: according to the state of the system at the moment of the present, it is possible to accurately predict the state of the system in the future, but it is impossible to recreate the state of the system in the past.

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An example of a Strong Nuclear Force confined composite particle being "Omega particles"

Using synthetic confinement of particles I think it would be possible to create higher order quarks than those in nature as the magnetic confinement would put additional pressure on the quarks keeping them from decaying and cohesive beyond what the Strong Nuclear Force would generally allow with it's strength, this additional pressure on the quarks keeping them stable. Possibly allowing for the creation of higher order composite particles such as other Pentaquarks.

The actual confinement in magnetic fields that stabilize the particles could happen in magnetic fields that are being used currently for fusion reactors allowing for a pressure to be placed upon the particles making a secondary bond between particles supporting their increased mass.

Note: If this does work then the particles can never leave confinement otherwise they will destabilize into energy.

It is like a oxygen tank, an oxygen tank can only hold so much oxygen before it will explode from the pressure but if you make the oxygen tank thicker, the tank can handle more pressure but in this instance, the oxygen is energy and the tank thickness is the additional confinement that holds the particle together by magnetic confinement increasing the "Thickness" of the Strong Nuclear Force or binding energy.

This can be explained by the simple equation P_{Spin} = P_{SNF} + P_{Magnetic}

Let us consider what field "blobs" can be, moving at the speed of light in a certain direction while maintaining their shape. That is, compact formations capable of traveling long distances compared to their size without significant changes in structure. Unlike dipole radiation, which propagates spherically in all directions. Perhaps such structure have emissions of atoms during the transitions of electron clouds to less energetic levels. Discussion of how justified use of the term "photon" in relation to such objects is beyond the scope of this article.

Let us take as basis the equations, existence of which in the real world is justified in the topic on dipole radiation:

The following symbols are used:

Scalar potential = a

Vector potential = **A**

Electrical field = **E**

Speed of light in vacuum = c

Time derivatives are denoted by singlequote '

a' = - c^{2} · div **A**

**A**' = - **E** - grad a

**E**' = c^{2} · rot rot **A**

The formulas are given in cylindrical coordinate system (ρ,φ,z),

associated with the point of space where the geometric center of field blob is located at the time of observation.

Let us put r^{2} = ρ^{2} + z^{2}

Motion occurs along z-axis at the speed of light and structure of field object remains unchanged,

that is, ∂/∂t = - c · ∂/∂z for all physical quantities.

Also, integral of internal energy throughout all the space must be finite, density of which is expressed by the law:

u = ε_{0}/2 · E^{2} + μ_{0}/2 · H^{2}

where E^{2} = E_{ρ}^{2} + E_{φ}^{2} + E_{z}^{2}, H^{2} = H_{ρ}^{2} + H_{φ}^{2} + H_{z}^{2}

**H** = 1/μ_{0} · rot **A**, **B** = rot **A**** **= μ_{0} · **H**

Let us put **J** = rot **B** = rot rot **A**

Let us start with the mathematically simplest descriptions possible from the point of view of field laws mentioned above. In cylindrically symmetric case, when ∂/∂φ = 0 for all physical quantities.

Basic equations are divided into two independent systems:

1. With circular electric field.

A_{φ}' = - c · ∂A_{φ}/∂z = - E_{φ}

→ E_{φ} = c · ∂A_{φ}/∂z

→ ∂E_{φ}/∂z = c · ∂^{2}A_{φ}/∂z^{2}

E_{φ}' = - c · ∂E_{φ}/∂z = c^{2} · J_{φ}

= c^{2} · (- ∂^{2}A_{φ}/∂z^{2} - ∂^{2}A_{φ}/∂ρ^{2} - ∂A_{φ}/∂ρ / ρ + A_{φ} / ρ^{2})

→ ∂E_{φ}/∂z = c · (∂^{2}A_{φ}/∂z^{2} + ∂^{2}A_{φ}/∂ρ^{2} + ∂A_{φ}/∂ρ / ρ - A_{φ} / ρ^{2})

Equating ∂E_{φ}/∂z from two equations, we get

∂^{2}A_{φ}/∂ρ^{2} + ∂A_{φ}/∂ρ / ρ - A_{φ} / ρ^{2} = 0

→ ∂/∂ρ (∂A_{φ}/∂ρ + A_{φ} / ρ) = 0

If A_{φ} is not zero in all the space,

so ∂A_{φ}/∂ρ + A_{φ} / ρ = 0, and A_{φ} is proportional to 1 / ρ, that gives infinite energy integral. Hence, such non-zero components of compact radiations can not exist. After artificial creation or computer modeling such structures will diverge in waves in all directions, instead of moving in one direction at the speed of light.

2. With circular magnetic field.

a' = - c · ∂a/∂z = - c^{2} · (∂A_{ρ}/∂ρ + A_{ρ} / ρ + ∂A_{z}/∂z)

→ ∂a/∂z = c · (∂A_{ρ}/∂ρ + A_{ρ} / ρ + ∂A_{z}/∂z)

A_{ρ}' = - c · ∂A_{ρ}/∂z = - E_{ρ} - ∂a/∂ρ

→ E_{ρ} = c · ∂A_{ρ}/∂z - ∂a/∂ρ

∂E_{ρ}/∂z = c · ∂^{2}A_{ρ}/∂z^{2} - ∂^{2}a/∂ρ/∂z

A_{z}' = - c · ∂A_{z}/∂z = - E_{z} - ∂a/∂z

→ E_{z} = c · ∂A_{z}/∂z - ∂a/∂z

∂E_{z}/∂z = c · ∂^{2}A_{z}/∂z^{2} - ∂^{2}a/∂z^{2}

E_{ρ}' = - c · ∂E_{ρ}/∂z = c^{2} · J_{ρ}

→ ∂E_{ρ}/∂z = c · (∂^{2}A_{ρ}/∂z^{2} - ∂^{2}A_{z}/∂ρ/∂z)

E_{z}' = - c · ∂E_{z}/∂z = c^{2} · J_{z}

→ ∂E_{z}/∂z = c · (∂^{2}A_{z}/∂ρ^{2} - ∂^{2}A_{ρ}/∂ρ/∂z - ∂A_{ρ}/∂z / ρ + ∂A_{z}/∂ρ / ρ)

Equating ∂E_{ρ}/∂z from the equations for A_{ρ}' и E_{ρ}', we get

c · ∂^{2}A_{ρ}/∂z^{2} - ∂^{2}a/∂ρ/∂z = c · (∂^{2}A_{ρ}/∂z^{2} - ∂^{2}A_{z}/∂ρ/∂z)

and conclude that a = c · A_{z} if we are talking about quantities decreasing to zero with distance from the center goes to infinity.

From the equation for a' then follows ∂A_{ρ}/∂ρ + A_{ρ} / ρ = 0,

which means A_{ρ} = 0 if A_{ρ} is not proportional to 1 / ρ with infinite energy integral.

From the equation for A_{z}' follows E_{z} = 0 at a = c · A_{z}

The following equations remain valid:

E_{ρ} = - ∂a/∂ρ = - c · ∂A_{z}/∂ρ

whereas from ∂E_{z}/∂z = c · (∂^{2}A_{z}/∂ρ^{2} + ∂A_{z}/∂ρ / ρ) = 0

it follows that with non-zero A_{z} must be A_{z} proportional to ln(ρ) and energy integral is infinite.

Thus, no valid expressions for field formations were found. The situation changes if we assume that div **E** ≠ 0 (non-zero charge density) and introduce additional terms into formulas for **E**' using the velocity field:

**E**′ = c^{2} · J - grad (**E** · **V**) - **V** · div **E**

where div **E** = ∂E_{ρ}/∂ρ + E_{ρ} / ρ + ∂E_{z}/∂z

in case of circular magnetic field, whereas case of circular electric field remains within previous calculations, since there div **E** = 0

Assuming that V_{z} = c is in the entire space around isolated field object, whereas V_{ρ} = 0 and V_{φ} = 0,

and since **E** · **V** = E_{z} · c, we get

E_{ρ}' = - c · ∂E_{ρ}/∂z = c^{2} · J_{ρ} - c · ∂E_{z}/∂ρ - 0 · div **E**

→ ∂E_{ρ}/∂z = ∂E_{z}/∂ρ - c · J_{ρ}

→ ∂E_{ρ}/∂z = ∂E_{z}/∂ρ - c · (∂^{2}A_{z}/∂ρ/∂z - ∂^{2}A_{ρ}/∂z^{2})

E_{z}' = - c · ∂E_{z}/∂z = c^{2} · J_{z} - c · ∂E_{z}/∂z - c · div **E**

→ ∂E_{z}/∂z = - c · J_{z} + ∂E_{z}/∂z + div **E**

→ c · J_{z} = div **E**

→ c · (∂^{2}A_{ρ}/∂ρ/∂z - ∂^{2}A_{z}/∂ρ^{2} + ∂A_{ρ}/∂z / ρ - ∂A_{z}/∂ρ / ρ) = div **E**

The following equations remain true

∂a/∂z = c · (∂A_{ρ}/∂ρ + A_{ρ} / ρ + ∂A_{z}/∂z)

E_{ρ} = c · ∂A_{ρ}/∂z - ∂a/∂ρ

E_{z} = c · ∂A_{z}/∂z - ∂a/∂z

From the expression for E_{z}' after substitutions it follows:

c · (∂^{2}A_{ρ}/∂ρ/∂z - ∂^{2}A_{z}/∂ρ^{2} + ∂A_{ρ}/∂z / ρ - ∂A_{z}/∂ρ / ρ)

= ∂E_{ρ}/∂ρ + E_{ρ} / ρ + ∂E_{z}/∂z = c · ∂^{2}A_{ρ}/∂ρ/∂z - ∂^{2}a/∂ρ^{2}

+ c · ∂A_{ρ}/∂z / ρ - ∂a/∂ρ / ρ + c · ∂^{2}A_{z}/∂z^{2} - ∂^{2}a/∂z^{2}

→ ∂^{2}a/∂ρ^{2 }+ ∂a/∂ρ / ρ + ∂^{2}a/∂z^{2} = c · (∂^{2}A_{z}/∂ρ^{2} + ∂A_{z}/∂ρ / ρ + ∂^{2}A_{z}/∂z^{2})

Which leads to the conclusion a = c · A_{z}

Then E_{z} = 0, also ∂A_{ρ}/∂ρ + A_{ρ} / ρ = 0, hence A_{ρ}_{ }= 0 to avoid infinity of energy integral.

As result we get:

a = c · A_{z}, A_{ρ} = 0, E_{z} = 0

E_{ρ} = - ∂a/∂ρ = - c · ∂A_{z}/∂ρ

Which corresponds to the equation derived earlier from E_{ρ}'

∂E_{ρ}/∂z = ∂E_{z}/∂ρ - c · (∂^{2}A_{z}/∂ρ/∂z - ∂^{2}A_{ρ}/∂z^{2})

Herewith B_{φ} = - ∂A_{z}/∂ρ = E_{ρ}/c

**Charge, spin and polarization**

If one looks in the direction of movement of field object, it is easy to notice that in the above version with annular magnetic field it is possible to orient this field clockwise or counterclockwise. Accordingly, radial electric field will be directed from z-axis outward or inward to this axis. To one type of field formations can be attributed conditional positive "spin", to the second negative.

Let us try to find out how intensity of fields can decrease at distance from the geometric center of object.

Let a = A_{0} / s, где A_{0} = amplitude constant,

and s^{2} = R^{2} + ρ^{2} + z^{2}, where R = object's scaling constant, possibly having an indirect relation to conditional "wavelength" in experiments. Note that ∂s/∂ρ = ρ / s, ∂s/∂z = z / s

Then A_{z} = A_{0} / c / s, A_{ρ} = 0, E_{ρ} = A_{0} · ρ / s^{3}, E_{z} = 0

div **E**** **= ∂E_{ρ}/∂ρ + E_{ρ} / ρ = A_{0} · (2 / s^{3} - 3 · ρ^{2} / s^{5})

The integral of charge density (divided by dielectric constant) over the entire space will be equal to

∫_{-∞}^{+∞}∫_{0}^{2·π}∫_{0}^{∞ }(2 / s^{3} - 3 · ρ^{2} / s^{5}) · ρ ∂ρ ∂φ ∂z = 0

That is, although charge density is not locally zero, the object as a whole is charged neutrally. This is natural, for example, for radiation arising from atoms and molecules, taking into account laws of conservation, since the particles located there will not give up part of their charge.

In general, when **E** = E_{ρ} = - ∂a/∂ρ, the subintegral expression

ρ · div **E** = ρ · (∂E_{ρ}/∂ρ + E_{ρ} / ρ) = ρ · (- ∂^{2}a/∂ρ^{2} - ∂a/∂ρ / ρ)

= - ρ · ∂^{2}a/∂ρ^{2} - ∂a/∂ρ = ∂/∂ρ (- ρ · ∂a/∂ρ)

Computing the integral ∫_{0}^{∞ }ρ · div **E** ∂ρ we get

for ρ = 0 the function - ρ · ∂a/∂ρ = 0,

for ρ = ∞ the function - ρ · ∂a/∂ρ = 0

if ∂a/∂ρ decreases by absolute value with a distance faster than 1 / s

Further computation of integrals by φ and z will not change zero result. The author of this article tested using MathCAD zero equality of the triple integral for a = A_{0} · ρ^{2} / s^{3} with E_{ρ} = A_{0} · (2 · ρ / s^{3 }- 3 · ρ^{3} / s^{5}), also for a = A_{0} · ρ^{4} / s^{5} with E_{ρ} = A_{0} · (4 · ρ^{3} / s^{5}^{ }- 5 · ρ^{5} / s^{7}), for a = A_{0} · ρ / s^{2}, a = A_{0} · z / s^{2}, a = A_{0} / s^{2}

Very wide range of such objects is neutrally charged in general, although it is likely that field formations are statistically inclined to take simplest geometric shapes, with minimum number of spatial extrema.

It should be noted that when a = A_{0} / s^{2} or s appears with even higher degrees, field formation receives significantly greater ability to penetrate matter than with a = A_{0} / s or a = A_{0} · ρ^{2} / s^{3}

Accordingly, the probability of registration of field object by measuring instruments is reduced. Which may be similar to the behavior of neutrinos in experiments.

Polarized field object can be described as follows:

s^{2} = R^{2} + X · x^{2} + Y · y^{2} + Z · z^{2}

where R, X, Y, Z are scaling constants

∂s/∂x = X · x / s, ∂s/∂y = Y · y / s, ∂s/∂z = Z · z / s

If a = A_{0} / s, where A_{0} is amplitude

A_{z} = A_{0} / c / s, A_{x} = 0, A_{y} = 0

E_{x} = A_{0} · X · x / s^{3}, E_{y} = A_{0} · Y · y / s^{3}, E_{z} = 0

B_{x} = - A_{0} / c · Y · y / s^{3}, B_{y} = A_{0} / c · X · x / s^{3}, B_{z} = 0

div **E** = ∂E_{x}/∂x + ∂E_{y}/∂y + ∂E_{z}/∂z

= A_{0} · (X / s^{3} - 3 · X · x^{2} / s^{5} + Y / s^{3} - 3 · Y · y^{2} / s^{5})

At the same time, all the above formulas for case of circular magnetic field remain true,

**E**′ = c^{2} · J - grad (**E** · **V**) - **V** · div **E**

E_{x}' = c^{2} · (∂B_{z}/∂y - ∂B_{y}/∂z) - 0 - 0 = 3 · A_{0} · c · X · Z · x · z / s^{5}

E_{y}' = c^{2} · (∂B_{x}/∂z - ∂B_{z}/∂x) - 0 - 0 = 3 · A_{0} · c · Y · Z · y · z / s^{5}

E_{z}' = c^{2} · (∂B_{y}/∂x - ∂B_{x}/∂y) - 0 - c · div **E **= 0

That is, there may be no cylindrical symmetry, with different X and Y, the field object will be stretched or extended along x- axis or y-axis. Compression or extension along z-axis is determined by multiplier Z. With significant differences between coordinate multipliers, structures arise with predominant orientation of fields in one direction (and the opposite also) in areas with high field energy density.

This topic can be seen as a preface:

This topic can be considered as a continuation:

]]>

**Fashion designing is a largely paid job and with the advancement of fashion designing sedulity, there is ample compass for amateur fashion introducers. Invented clothes, shoes, and accessories have come to be a symbol of status. A series of inventor stores have opened in the metropolitan cosmopolises of India where you would find a collection of extravagant inventor particulars and accessories. **

Essentially, I intend to build a sieve for figurate numbers, I'll code this in a programming language such as C or VB or something.

What I would like to be able to do is to enter any real number and have my algorithm, routine check the number to see what figurate number it comes closest too.

Note that the following link, has a table of some of the more common formulas to derive a figurate number series in forward expanding exponential series (gnomonic growth) ;

Figurate Number -- from Wolfram MathWorld

(Table copied and pasted below for your reference)

However, what I need to do, Im guessing, is to re-write these formula so that they process backwards, taking a large number and scailing it down.

For example ;

If using the triangular figurate formula n(n+1) /2

39 (39+1) / 2 = 780

However, I want to reverse this process so that I enter 780 and I get 39

I'd like to be able to reverse ALL of the formula shown below, each and every statement

the end goal is that I will have developed for myself a small software program that accepts any real number and then checks that numberagainst all these formula to see which ones it matches or comes very close to, I realzie this sounds odd, but I'd like to be able to even enter irrationals and remainders, non-whole numbers are 'ok', i.e. a number may be close to a triangular number, i.e. say I have 783, then the app returns that it is close to a triangular number 780 with remainder 3

What Im hoping to ask of you, is are you aware of any onlne source where these reversed formula have already been compiled and are available? If not, thoughts on whether this can be done, and if so, any tips?

If you feel inclined to take a crack at re-writing the tetrahedral, triangular, and pentagonal formula, I'd love to review your approach - and hope to be ebale to re-write all of these.

Much thanks and sincere regards to you!

]]>In this work are investigated details of an elementary electric and magnetic dipole radiation at distances much greater than size of the emitting element. Debatable conclusions are drawn.

Formulas are given in a cylindrical coordinate system (ρ,φ,z)

r^{2} = ρ^{2} + z^{2}

Given this, it is possible to write expressions differently for ρ and z

For example, 2 - 3 · ρ^{2} / r^{2} = 3 · z^{2} / r^{2} - 1

For all values ∂/∂φ = 0 (cylindrical symmetry)

Time derivatives are denoted by a quote '

**Electric elementary dipole**

Charge oscillates along z-axis near zero point with frequency ω, amplitude of dipole moment is P_{0}.

Dipole moment:

P_{z} = P_{0} · cos(ω·t)

Auxiliary functions:

COS = cos(ω·(t - r/c)), SIN = sin(ω·(t - r/c))

Scalar potential:

a = P_{0} / (4·π·ε_{0}) · z / r^{2} · (1 / r · COS - ω/c · SIN)

a' = - P_{0} / (4·π·ε_{0}) · ω · z / r^{2} · (ω/c · COS + 1 / r · SIN)

Vector potential:

A_{z} = - P_{0} · μ_{0}/(4·π) · ω / r · SIN

A_{z}' = - P_{0} · μ_{0}/(4·π) · ω^{2} / r · COS

div **A** = ∂A_{z}/∂z = P_{0} · μ_{0}/(4·π) · ω · z / r^{2} · (ω/c · COS + 1 / r · SIN)

a' = - c^{2} · div **A**

Scalar potential gradient:

∂a/∂ρ = P_{0} / (4·π·ε_{0}) · ρ · z / r^{3} · {(ω^{2}/c^{2} - 3 / r^{2}) · COS + ω/c · 3 / r · SIN}

∂a/∂z = P_{0} / (4·π·ε_{0}) / r^{2} · {1 / r · (ω^{2}/c^{2} · z^{2} + 1 - 3 · z^{2} / r^{2}) · COS + ω/c · (3 · z^{2} / r^{2} - 1) · SIN}

Magnetic induction:

B_{φ} = - ∂A_{z}/∂ρ = - P_{0} · μ_{0}/(4·π) · ω · ρ / r^{2} · (ω/c · COS + 1 / r · SIN)

B_{φ}' = - P_{0} · μ_{0}/(4·π) · ω^{2} · ρ / r^{2} · (1 / r · COS - ω/c · SIN)

Electric field:

E_{ρ} = - ∂a/∂ρ = - P_{0} / (4·π·ε_{0}) · ρ · z / r^{3} · {(ω^{2}/c^{2} - 3 / r^{2}) · COS + ω/c · 3 / r · SIN}

E_{z} = - A_{z}' - ∂a/∂z = P_{0} / (4·π·ε_{0}) / r^{2} · {(ω^{2}/c^{2} · ρ^{2} / r - 1 / r + 3 · z^{2} / r^{3}) · COS + ω/c · (1 - 3 · z^{2} / r^{2}) · SIN}

Electric field annular curl:

∂E_{ρ}/∂z - ∂E_{z}/∂ρ = P_{0} / (4·π·ε_{0}) · ω^{2}/c^{2} · ρ / r^{2} · (1 / r · COS - ω/c · SIN)

B_{φ}' = - (∂E_{ρ}/∂z - ∂E_{z}/∂ρ)

as it should be in equations of electromagnetic field.

div **E** = ∂E_{ρ}/∂ρ + E_{ρ} / ρ + ∂E_{z}/∂z = 0 (checked)

Magnetic field curl:

J_{ρ} = - 1/μ_{0} · ∂B_{φ}/∂z = - P_{0} / (4·π) · ω · ρ · z / r^{3} · {ω/c · 3 / r · COS - (ω^{2}/c^{2} - 3 / r^{2}) · SIN}

J_{z} = 1 / μ_{0} · (∂B_{φ}/∂ρ + B_{φ} / ρ) = P_{0} / (4·π) · ω / r^{2} · {ω/c · (1 - 3 · z^{2} / r^{2}) · COS - (ω^{2}/c^{2} · ρ^{2} / r - 1 / r + 3 · z^{2} / r^{3}) · SIN}

E_{ρ}' = - P_{0} / (4·π·ε_{0}) · ω · ρ · z / r^{3} · {ω/c · 3 / r · COS - (ω^{2}/c^{2} - 3 / r^{2}) · SIN} = J_{ρ}/ε_{0}

E_{z}' = P_{0} / (4·π·ε_{0}) · ω / r^{2} · {ω/c · (1 - 3 · z^{2} / r^{2}) · COS - (ω^{2}/c^{2} · ρ^{2} / r - 1 / r + 3 · z^{2} / r^{3}) · SIN} = J_{z}/ε_{0}

as it should be in equations of electromagnetic field.

**Magnetic**** ****dipole**

An annular current with small radius R changes direction according to periodic law.

Magnetic moment is directed along z-axis:

M_{z} = M_{0} · cos(ω·t), где M_{0} = π · R^{2} · I_{0}, I_{0} is current amplitude.

Auxiliary functions:

COS = cos(ω·(t - r/c)), SIN = sin(ω·(t - r/c))

Vector potential:

A_{φ} = M_{0} · μ_{0}/(4·π) · ρ / r^{2} · (1 / r · COS - ω/c · SIN)

Electric field:

E_{φ} = - A_{φ}' = M_{0} · μ_{0}/(4·π) · ω · ρ / r^{2} · (ω/c · COS + 1 / r · SIN)

E_{φ}' = M_{0} · μ_{0}/(4·π) · ω^{2} · ρ / r^{2} · (1 / r · COS - ω/c · SIN)

Magnetic induction:

B_{ρ} = - ∂A_{φ}/∂z = - M_{0} · μ_{0}/(4·π) · ρ · z / r^{3} · {(ω^{2}/c^{2} - 3 / r^{2}) · COS + ω/c · 3 / r · SIN}

B_{z} = ∂A_{φ}/∂ρ + A_{φ} / ρ = M_{0} · μ_{0}/(4·π) / r^{2} · {(ω^{2}/c^{2} · ρ^{2} / r + 2 / r - 3 · ρ^{2} / r^{3}) · COS - ω/c · (2 - 3 ·ρ^{2} / r^{2}) · SIN}

B_{ρ}' = - M_{0} · μ_{0}/(4·π) · ω · ρ · z / r^{3} · {ω/c · 3 / r · COS - (ω^{2}/c^{2} - 3 / r^{2}) · SIN} = - (- ∂E_{φ}/∂z)

B_{z}' = - M_{0} · μ_{0}/(4·π) · ω / r^{2} · {ω/c · (2 - 3 · ρ^{2} / r^{2}) · COS + (ω^{2}/c^{2} · ρ^{2} / r + 2 / r - 3 · ρ^{2} / r^{3}) · SIN} = - (∂E_{φ}/∂ρ + E_{φ} / ρ)

as it should be in equations of electromagnetic field.

Magnetic field curl:

J_{φ} = 1/μ_{0} · (∂B_{ρ}/∂z - ∂B_{z}/∂ρ) = M_{0} · μ_{0}/(4·π) · ω^{2}/c^{2} · ρ / r^{2} · {1 / r · COS - ω/c · SIN}

E_{φ}' = J_{φ}/ε_{0} (checked)

as it should be in equations of electromagnetic field.

**Conclusions**

Although divergence of electric field div(**E**) is zero everywhere (charge density is zero), scalar potential is urgently needed to describe radiation of electric dipole. To express time derivative a' is required vector potential **A**. At long distances, there is no question of lagging potentials of forcibly oscillating system, waves must propagate "by themselves" in wave zone. It begs the conclusion that potentials are objective physical reality, fundamental fields in vacuum, and are not mathematical abstractions. To describe dipole radiation, three fundamental fields are sufficient:

a' = - c^{2} · div **A**

**A**' = - **E** - grad a

**E**' = c^{2} · rot rot **A**

At the same time, Laplacian div grad (a) is fundamentally different from local charge density ε_{0} · div **E**, these are different quantities. Laplacian of scalar potential can be locally not zero in electric dipole radiation, unlike divergence of electric field. Formally, both of these quantities are "conserved", since it is possible to express derivatives in time as minus divergence of some known "flow" or current. But with respect to electric dipole, laplasian of scalar potential is preserved only globally, when positive density is emitted in one direction along z-axis, in opposite direction the same modulo negative goes. It cannot be said that scalar potential has significant value only in near zone of forced generation and lagging potentials. In far wave zone, its intensity, like time derivative, decreases proportionally to 1 / r along z-axis, the same applies to its gradient in some directions (ρ · z / r^{3}).

Electric and magnetic field decrease on average with distance as 1 / r, respectively, energy density decreases as 1 / r^{2}. That is, integral of energy density throughout space is infinite, and elementary dipoles cannot be used as basis for representing field objects with finite energy. The more time emitter works, more energy it loses with waves, without restrictions on final value.

This topic can be considered as a continuation:

]]>

I have been working on this over 10 years and just published today.

]]>

Does anyone know if other 'Generalized' Fibonacci numbers are present featurally in similar fractal figures? Or even more general patterns such as those presented by Metallic Means, Pisot-Vijayarhagavan numbers, etc.? Thanks.

]]>The experiment utilizes a Femto camera. A Femto camera takes a trillion frames per second and is capable of capturing light in slow motion as it leaves its source. The link below is a video of just that. By pausing the video where light has expanded into s sphere, one can then measure for space 'moving past' just as they did in the M&M experiment. If space is in fact the medium for light one would expect to be able to measure for the slight difference in speed along different directions in a paused image of the Femto camera.

When I measured with a ruler on the screen I did in fact find that light was travelling faster by a few mm per 25 cm in one direction over the other depending on how you want to look at it.

All waves are a denser part of a medium spreading out to a less dense part of that medium, so light is just that and its medium is space.

]]>

here's how i would handle the problem

so step 1, take the total population and divide by the number of seats. let's call this value D.

then assign each state 1 seat, and subtract from each, D

then again from top to bottom, assign 1 seat, subtracting D. if any are negative don't assign it a seat, obviously.

using the numbers in the video;

40075 population, 43 seats. that 932 representatives per seat.

new triangle 21878

circula 9713

squaryland 4167

octiana 3252

rhombus 1065

assign 1 seat for each state, and subtract 932.

new triangle 20946 reps 1

circula 8781 reps 1

squaryland 3,235 reps 1

octiana 2320 reps 1

rhombus 133 reps 1

---------------------

new triangle 20014 reps 2

circula 7849 reps 2

squaryland 2303 reps 2

octiana 1388 reps 2

rhombus 133 reps 1; no further reps.

---------------------

new triangle 19,082 reps 3

circula 6,917 reps 3

squaryland 1,371 reps 3

octiana 456 reps 3; no further reps.

rhombus 133 reps 1; no further reps.

---------------------

new triangle 18150 reps 4

circula 5985 reps 4

squaryland 439 reps 4 no further reps.

octiana 456 reps 3; no further reps.

rhombus 133 reps 1; no further reps.

---------------------

new triangle 442 reps 23

circula 393 reps 10

squaryland 439 reps 4

octiana 456 reps 3

rhombus 133 reps 1

with 2 reps left over. they go to the highest remaining. in this case, octiana and new triangle.

new triangle 442 reps 24

circula 393 reps 10

squaryland 439 reps 4

octiana 456 reps 4

rhombus 133 reps 1

now going from 43 to 44, we get the following

new triangle 14 reps 24

circula 603 reps 10

squaryland 523 reps 4

octiana 519 reps 3

rhombus 154 reps 1

again with 2 left over; they get assigned to curricular and squaryland.

]]>use a water wheel sort of thing that the sand falls on, and it tightens a spring.

after an hour, the spring releases energy to flip the hour glass over.

thoughts?

]]>I was sitting silently and took in the pitter patter of rain drops upon some metal flashing.. It was harmonic, music to the ears, and purposefully synchronized as well as perfect as a drummer keeping a timed beat.

I thought to myself, why and how can this be, other than the energy formed by each drop of rain creating its own splash of wisdom.

Once upon a time I was fortunate enough to see each snow flake fall individually as if to stop time in its tracks. I use this theory to observe each interaction with life as an individual one of a kind "no two blah blah are alike philosophy... And it will always hold true.

I sort of got off topic with my thesis , but every snow flake that falls is a one of a kind experience... Mathematics cannot duplicate it...therefore. We all hold an ocean in our eyes. Six degrees of separation can help explain this.

Im Canadian, I've shovelled a lot a snow, cheer go leafs go and love wu tang. Bet I can relate to some here , some there and some not at all. But if I was to guess, everyone drinks water...to me that's like saying we are "all in together".

In closing, "be water my friend", my water is James brown type funk n jazz n smiles. Others have crocodiles for smiles. Be one with nature. I think people really really lack this.

Anyway I look forward to feedback , criticism, conflict and harmony. To quote john Lennon " and the world can live as one/be as one...

Were into an intellectual future..let's all be on the same page, or at least in the same book...

peace Bobby

]]>If you have an ftl spacecraft you won’t be able to safely go anywhere unless you can actually see what’s ahead of you and jumping requires photography light doesn’t register quickly enough

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