logical deduction - Four is Cosmic!

This is a little puzzle I heard a while back from one of my mathematically inclined friends- I get the sense that it's bounced around a little, so forgive me if you've heard it.

There is a sort of function that, given any positive whole number, will return another positive whole number. Interestingly, there is one number that will return itself- the number 4. There are no other loops of any size, so you can start at any other number and, by repeatedly plugging in the output of the function, you will eventually reach 4.

Given this diagram showing a few examples of those succession chains, determine the function.

(If anyone feels like they really need more information, you can post a number in the comments and I'll give you its chain. However, the givens are sufficient to find a solution.)

Answer

Four is cosmic because:

It has four letters.
Each number is followed by the number of letters in its spelling, but four leads back to itself indefinitely!
Four is the only cosmic number because it is the only number with its own number of letters in its English spelling, and there are no cycles of two or more words that have each other's letter counts.

And it should probably go: 18 --> 8 --> 5 --> 4 (from Doge's comment)

astronomy - How can a full moon be seen south of an observer's location?

I know this seems like a simple question, but I'm trying to debate with a flat earth theorist. I asked him to explain why can the ISS visibly be seen orbiting the Earth with the naked eye, and he put this question to me instead.

He asked:

"How can a full moon be observed south of an observer's location, despite the fact that if the moon is illuminated by the sun, an observer has to be almost directly between the sun and the moon to observe a full moon?"

quantum field theory - Gauge symmetry is not a symmetry?

I have read before in one of Seiberg's articles something like, that gauge symmetry is not a symmetry but a redundancy in our description, by introducing fake degrees of freedom to facilitate calculations.

Regarding this I have a few questions:

1. Why is it called a symmetry if it is not a symmetry? what about Noether theorem in this case? and the gauge groups U(1)...etc?



2. Does that mean, in principle, that one can gauge any theory (just by introducing the proper fake degrees of freedom)?


3. Are there analogs or other examples to this idea, of introducing fake degrees of freedom to facilitate the calculations or to build interactions, in classical physics? Is it like introducing the fictitious force if one insists on using Newton's 2nd law in a noninertial frame of reference?

riddle - Best company EVER!

My first twice flies you
My second twice puzzles you
My third twice measures you
My fourth twice tests you

ASCII from my first,

Shaped like my first,
Anchored on my first,
Misses 1 of my first

Catching a loss,
I become my head;
Taking two more,
A triangle instead

When filled as so,
My third row's a scheme;
And then, my end,

A French magazine

My first remains quite solitary;
Like I too it has matches nary
My products made can be quite scary;
When defective, effects may vary

Who am I?

Answer

I think you are:

ACME - the generic company from Looney Tunes.

My first twice flies you
My second twice puzzles you
My third twice measures you
My fourth twice tests you

Taking each letter twice:

AA = American Airlines (who 'fly you')
CC = Cryptic crossword (which 'puzzles you')
MM = Millimetre (which 'measures you')
EE = Entrance exam (which 'tests you')

ASCII from my first,
Shaped like my first,
Anchored on my first,
Misses 1 of my first

This describes the drawing of a letter A ('Shaped like my first') using component letters ('from my first'), like this:

    A
   C C
  M   M

 E     E 

Note it is 'anchored' on two A's, but is missing a character to draw the horizontal line across the middle ('1 of my first').

Catching a loss,
I become my head;
Taking two more,
A triangle instead

If we 'catch a loss' - add an L - then the shape becomes a complete A ('my head'):

    A
   C C

  M L M
 E     E 

If it then 'takes two more' in the bottom row, we get 'a triangle instead':

    A
   C C
  M L M
 E L L E 

When filled as so,
My third row's a scheme;

And then, my end,
A French magazine

An MLM strategy (from row 3 of the triangle) is a multi-level marketing scheme. Elle (row 4) is 'a French magazine'.

My first remains quite solitary;
Like I too it has matches nary
My products made can be quite scary;
When defective, effects may vary

Back to a typical riddle! 'A' is the indefinite article for a singular ('solitary', with 'matches nary') object. And in Looney Tunes the ACME products tend to be traps ('can be quite scary') used by Wile E. Coyote for catching his nemesis, Roadrunner. But do they ever work??! No! ('defective, effects may vary')

As for the title:

Not only is ACME a company, but the word 'acme' means 'peak', 'top' or 'best'...

science - 45x45 Word Search

You'll need to be a serious fan of these puzzles to find over 300 of the things hidden in this jumble. There are no clues as this is test of knowledge as well as determination. Once you get started it's easy enough to find a list of what you are searching for.

Hint:

When all the parts are spread before you on the table you will know each by number.

If you would like an alternative grid with the correct spelling of Aluminium and copyable text:

i1u105muinehtuRCm8S7muidanaVmuiboiN3N al7mPbs5KrmuillahT80yDnegyxO0L711Bae6 NeTum31u1B57r112mMyBO5m352019muoC89nd rkmiu6mur1iu313utro85muinatiTzut3T6iG Bcuminu6Rofsn3iHur9mCuimuidibuRieFld8 Siidpoi55lhimmocomTumigm69m64Fm6dtNo4 mNnaotd3uTTplurnBumiudrgu44u7u2bTniI5 umeCrpoS2m1osetkmiuviioMSienitatsAIu4 iulHuyh2puH1MoHhutnocrb1n1smLdP921Umm niedErRBmi391ghIidirnIa0e4yobso193SuN ocSlAKRhurnoneXPnatee2e3gdnDrLuS9inib timomuinietsniEmitalr7SZomo40p8Ul5enN uruG6eF4nnR7OlmumslFwA8erPgmm2siEetfd leim103mot26AumnumPPassStyruuTcyrnsaF Pmru3mOuli57imuelrCFLaH2inAiio79DigHm pAtinusioe2cSuidAaRar523NomnnM406rndu uenhoi9rPMi4nibb0DgPlP611muei3o16ouMi Unutbc5uCnu1orry2lmm1i391iigtU58muTen HinirlCCrRP1rael68upm1fB0trtcmmLulSnu fmULaarem2hCIBToaiimuM3o1nonAuumiFAit Io1sCCpmu9nealCMtGmui23UrAhe4iiubr8rp mr0mDo1uiesenagnaMsiduLu0nTo0nlire3oe uB6uC1Wiln801iemmmOtrCGo5miR1aetevmlN i36i7stleo10MpuNuumeoeu8HuvuVrbptluh4 n77r0Cllkd21nigmiiunfsm90i2umUoetiiC2 oLma19ayra8ubAmLnhihrEuC7vA4SmNsYSs94 crum91breRnr57uvitdceSiUae45iuPn85e5L rmia00oeBUE935imtenehmmuileH1idu1mnr8 iumS10CBpN4817tucmaTturtheXcmtmn7ugeR ZiyJmuinilodaGniaoc7uie99diBucuUmiape bddmuissatoPRboltrScRrF28ne8iolBurMpm SaoE46muinbuD4rloPNSAe63ee7fsnaiiuPou NleumuiseaCPo7taro6Y9CFsdM7Rsut8llaCi elNmuinamreG14SGP5b11mr01171ann2ul6Ar oaKmuicnarFmuirhoBf14AdaeL8LHUa1heI6t nPCu72ZnegordyH62C8Y5munahtnaLTTTTr4t RaRadiummuiromreviL4ZincTdCmuimorhC6Y

SPOILER:

The names, numbers, and symbols of each element is included. Some overlap completely.

Answer

I have found the names, symbols, and atomic numbers for the first 118 elements in the periodic table. $118*3=354$ so I presume these are the "over 300" hidden things mentioned by OP.

I used the small grid with the plain text that I could copy. My search preferred to find items in the top left so, although there could be multiple instances of a symbol or number, the highest and furthest left option will be highlighted. To be fair, this could cause some confusion if, for instance, the same two cells are highlighted for both 17 and 71. However, I can assure you that you will find each answer highlighted.

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Element Names

Element Symbols

Element Atomic Numbers

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On an unrelated note, only 242 of the 1,369 characters in the word search were left unused in my results. This gives a usage rate of about $82\%$. That could probably be increased by a more intelligent search that favored all characters being used instead of just the first one it finds. That's some tightly packed information.

energy - Boundedness of general relativity Hamiltonian

When one consider a lagrangian and construct hamiltonian, we expect to be bounded below. While looking to the Hamiltonian formulation of general relativity, I have difficulties to see how it can be bounded.

1. 
   
   

   How this can be shown?

   
   


2. 
   

   Is it related to the positive energy theorem?

   
   

electromagnetism - Clarification on factors of $c$ in the Lorentz' force

I was told that Lorentz' force is given by

$${\bf F}= q{\bf v} \times {\bf B}.$$

But I have read that it is given by $${\bf F}= \frac{q}{c}{\bf v} \times {\bf B}.$$

Why have these two relations different forms if they represent the same force? Thanks for any help!

Answer

Both relations you mention are completely equivalent, the only difference being the system of units in which they are expressed. Every system of units $A$ is consistent with any other system of units $B$ as long as you yourself are consistent in their usage and correctly transform everything between $A$ and $B$ when desired. So the factor of $c$ does not constitute a conflict, which can be seen when the equations are transformed. This is shown in this useful section on the wikipage of the Lorentz force.

Looking at the full Lorentz force expression (of which your expressions are a special case with no electric field), the first one you mention, $$\vec{F} = q\left(\vec{E} + \vec{v}\times\vec{B}\right),$$ is expressed in SI units.$^1$ The second relation, $$\vec{F} = q\left(\vec{E} + \frac{1}{c}\vec{v}\times\vec{B}\right),$$ is expressed in Gaussian units. So both relations are equally valid, as long as you use the correct expression consistent with any other expressions - meaning you should at all times stay within the same system of units. Consistency is key.

The SI way of writing these kinds of electromagnetic expressions is steadily gaining popularity and most new books adopt this system of units, but Gaussian units have long been dominant. You will therefore mostly see those units used in older books. Also in literature these units are still widely used.

Transformations between Gaussian units and SI units require some extra care since - as you may have noticed on the wikipage I linked - they are not simple (dimensionless) rescaling transformations. As a consequence it is possible for a dimensionless equation in SI units (Gaussian units) to yield a non-dimensionless equation when transformed into Gaussian units (SI units).

One example is when we consider Gauss's law in Gaussian units divided by the free charge density: $$(1/\rho)\vec{\nabla}\cdot\vec{E} = 4\pi.$$ The quantity on the left-hand side is dimensionless in Gaussian units, but not in SI units, where it is $$(1/\rho)\vec{\nabla}\cdot\vec{E} = 1/\epsilon_0.$$ So you have to watch out for that when transforming your equations. Dimensional analysis may therefore also yield seemingly different or contradicting results, but there is no problem if you remember the conventional differences and, again, stay consistent.

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$^1$ Note that the expression for the Lorentz force also looks like this in natural units, which is another widely used system of units. Here the units are chosen such that certain natural constants such as the speed of light $c$ have a numerical value of 1. It is then common practice to omit those constants from all equations, for sake of simplicity.