Solutions to Complex Functions Problems

Solutions:

(1)   Let:  z = x + iy, and z* = x – iy

Adding:   z + z* = (x + iy) + (x - iy) = 2x

We see: x = (z + z*)/2

Subtracting:

(z – z*) = [x + iy – x + iy] =  i2y

Then:  y = (z – z*)/ 2i

We can now formulate the function f(z,z*):

f(z,z*) = 2[(z + z*)/2]²  + i2[(z – z*)/ 2i]²

2)  Rewrite:  z² + z – 3   in polar form:

z² = r² exp(i2(q)) = r² (cos (2q) + isin(2 q))

z = r exp(i(q)) = r(cos(q) + isin(q)

so:

 z² + z = r² (cos (2q) + isin(2 q)) +  r(cos(q) + isin(q)

Collecting like terms in i and simplifying:

f(z) =  r²(cos (2q) + r(cos(q)) + i{sin(2q) + sin(q)} – 3

so:

 iv(r, q) =  i{sin(2q) + sin(q)}

and 

v (r, q) =  {sin(2q) + sin(q)}

while:

u(r, q) =  r² (cos (2q) + r(cos(q)) – 3

3)     8x ² + 3iy  -  4    = 8y   – 4iy

 Write real function as:

 8x ²    =   4

So:   x ²    =   4/ 8   =  1/ 2

Therefore:  x =  Ö( 1/2)  =   1/ Ö2

=  Ö2/ 2

And for complex part:  

8 y   =   - 3iy  - 4iy   =   - 7 iy

y  =   -7 i/  8

Sorry, Peggy Noonan, A Trump Indictment Is NOT "Below Us"

 

                                                                   

                       Trump wields bat on DA Bragg in image from his Truth Social Site

"The weekend has arrived, and Donald Trump, contrary to his predictions, has not been indicted. He has nonetheless used this possibility to make himself the center of attention of both the legal and political worlds, offering a window into his campaign strategy while highlighting the dangers he poses to the stability of the country."  Dan Batz, Washington Post, 'Indictment or no, Trump’s strategy is the same: Attack and threaten', Friday

The potential indictment of Traitor Trump - who has now even shown himself to be a rogue inciter to violence (in his latest rhetoric against Manhattan DA Alvin Bragg, )  has caused millions of minds to shatter.  How vile was the rhetoric? The orange fungal scum actually threatened "death and destruction" if indicted by Manhattan DA Bragg and bleated about the prosecutors (with an accompanying image of him holding a baseball bat near DA Bragg's head - like a classic Alabama Klansman about to go on the attack): 

"They are HUMAN SCUM!” 

and  raged in another post actually asking why Bragg “refuses to do the right thing and ‘call it a day?’”

Making one wonder if WSJ columnist Peggy Noonan saw any of this before writing her latest piece of fishwrap: 

The Wrong Indictment Against Trump

 bleating:  

"One of the marks of personal maturity is a sense of proportion. A healthy democracy has a gracious sense of the rightness and wrongness of things, and is alive to symbols and signs. Is this, perhaps, the wrong indictment to bring?

....Charging him in the Stormy Daniels case is below us— not below him, but us. The subject matter is below us. The nature of the charges is below us. The players in the drama aren’t people of import who stand for big things."

In the ideal world, of course, we do always opt for import and "big things" especially in the indictment and prosecution of a former president. We also opt for "proportion" and exhibiting a "rightness and wrongness of things" in pursuing a case. However, not all things are equal or proportionate in the case of a slimeball and cockroach like Donald Trump. Who already nearly took this country down in an insurrection and is likely to take the Reep nomination with or without an indictment - putting our nation once more in peril.  

What? Does Peggy expect us all to turn a deaf ear and blind eye to Trump's incitements to violence? As WaPo columnist Dan Batz noted (see top quote), in the past such Trump rhetoric wasn't taken seriously enough, often dismissed as the usual Trump bluster. But after the cockroach's lies about the 2020 election and the storming of the U.S. Capitol - as well as undermining voting in this country - that is no longer possible. Only a low IQ moron, MAGA zombie or inbred cretin would claim the recent remarks are "histrionic" or "just a joke".  Also, what's clear, as Batz noted, is that the orange swine's primary modus operandi will be "attack, attack, attack."  The media - unlike in 2016  must be smart enough not to give him free air time.  In particular, it needs to warn, like Batz the "dangers he poses to the stability of the country."

As I've written in previous posts, if my Revolutionary War ancestors were still around they'd have hung him already for treason.  But that isn't going to happen and it is equally likely that in the end - despite all the media hype, hoopla and hysterics, no one - neither in Manhattan, Georgia, or with the DOJ (Special Counsel) will bring an indictment that sticks, meaning Trump ends up in prison for 5 years or more. Which would be the optimal outcome short of him croaking from a stroke, brain aneurysm or heart attack.  Also not likely given the roach - despite being overweight and gorging on high calorie, processed crap - seems to thrive on it and go on. Just like the roach he is.

To be sure, psychiatrist Bandy X. Lee already warned that indicting Trump for a lesser crime could fuel his manic MAGA mutts into a frenzy to at least propel his renomination as Reepo standard bearer for 2024. And of that I have little doubt- and neither did Bill Maher Friday night on Real Time -,saying it would provide "jet fuel" for his candidacy.  Maybe.  But it's no reason to totally ignore an indictment if he indeed broke the law.  Bottom line: a venomous, vicious swine like Trump needs to be taken down by someone, somewhere.  And if this is the only indictment that does it, so be it. (See link to Karl Rove's WSJ column below.)

This leads to Noonan's claims that we need "big charges" - and the ones at issue: Trump paying hush money to Stormy Daniels, are "below us".  No they are not below us, if indeed they entail misdirection of campaign funds in violation of campaign finance laws.

 We  now know campaign finance laws were broken, based on the thousands of records obtained from Michael Cohen's residence in the April 9, 2018 warrant search conducted by the FBI, e.g.  hard copy financial, legal documents, seized electronic devices, audio recordings made by Cohen, emails,  text messages sent over encrypted applications, records retrieved from the corporation referenced in the charging documents as "Corporation One" and records from the media company also therein referenced.  

And as Cohen told the House Committee in February, 2019, , he didn't expect to just have his word taken at face value, but wanted members to examine the documents he brought - including a signed check for $35,000 with Trump's signature. This was to pay Cohen back for using his own $130,000 from a home equity line of credit. 

Last but not least, there is the documented testimony of witnesses involved in the transactions for illegal payoffs, like Trump accountant Allen Weisselberg,  e.g.

Clearly then, there is a case to indict Trump for breaking the law, like his consigliere Cohen did.  As Lanny Davis put it way back in August, 2019:

"If those payments were a crime for Michael Cohen, then why wouldn’t they be a crime for Donald Trump?" 

Can Noonan be ok with this? Probably not but that's not an excuse to not go after the rat given also the dispositive evidence against him. What's that? As Lanny Davis explained to a puzzled CBS Early Morning cast (Aug. 22, 2018) Trump's lawyers themselves delivered recorded statements (in depositions) saying that Trump had paid Cohen to pay off Daniels.  Hence, those recorded statements qualified as dispositive evidence. And, if needed to indict, they need to be reported to the grand jury.  

Look, I am not a fan of nailing the Turd on a lesser "paper" crime but if that's what it takes to sully his brand for independent and other voters in 2024 I am all for it.  This is a point already made by Karl Rove in a recent WSJ column.

Trump’s Indictment Won’t Win Him Votes

The recent threatening letter sent to DA Bragg "I am going to kill you" and Trump's clear role in provoking it needs to be taken seriously.  This cockroach cannot be permitted to sully the highest office again! Indict the slime if even on a lesser charge if that's what it takes to derail his sorry ass. 

Too many, like Peggy Noonan and former Trump AG Bill Barr (warning in a recent WSJ op-ed about the "left" overstepping), forget that former VP Dem candidate John Edwards (in the 2004 campaign) was similarly charged with the crime of using campaign funds to conceal an affair. It literally blew up his political career.  A disreputable maggot like Donald Trump deserves no less.  

And as NY Times columnist Charles Blow put it in a recent column: 

"Indictment and prosecution of Trump is not the problem. Donald Trump himself is."  

Bingo!

See Also:

Skewering The Moral Mush Of Sasha Mudd: "Trump Is A Human With Dignity"

And:

Donald Trump

‘Can’t’ Even Commit Financial Fraud Anymore’: Jon Stewart Questions Rule of Law When It Comes to Trump

by Maya Boddie | March 27, 2023 - 6:24am | permalink

— from Alternet

And:

Donald Trump

Donald Trump Knows a Perp Walk Is Good TV

by Amanda Marcotte | March 24, 2023 - 7:31am | permalink

— from Salon

While the world waits with bated breath for a Donald Trump indictment, reporters on the Mar-a-Lago palace intrigue beat say that no one claims more anticipation than Trump himself.

And:

• Like It Or Not - House Oversight Committee Hearing Shows Michael Cohen Was The One With Moral Standing...

Solutions To Simpler Problems In Special Relativity (March 17 post)

Solutions:

1) We have: t = t’/ [1 - v²/c²] ^½

But the proper time is defined such that:

t' = t/2   or t'/ t = ½

Then:

[1 - v²/c²] = (t'/ t)²

and:

v²/c² = 1 - (t'/ t)²

v² = c²[1 - (t'/ t)²]

so:

v = c[1 - (t'/ t)²]^½ = c[1 - 0.5²]^½ = c[0.75]^½ = 0.866c

2)  The proper time t' = 3600 s

Since v = 300 m/s = (10^-6) c and hence v/c << 1 we need the form:

 

 t = t’/ [1 + v²/2c²] t = 3600s/ [1 +     (10 ^-12)c²/2c²]

 

Since the numerator is only slightly larger than 1, the time t will be:

3600 s/(1.000000000001)= 3600.0000000018

= 3600 + 1.8 x 10^-9 s

or very slightly longer than one hour.

3)(a)The proper time t' applies to the muon's reference frame.

So:   t = t’/ [1 - v²/c²] ^½   and t' = t [1 - v²/c²] ^½

where v = 0.99 c and v² = (0.99c)² = 0.98c²

Then: t' = t [1 - 0.98c²/c²]^½ = t [0.02] ^½ = t(0.14)

recall distance travelled = 4.6 km = 4600 m

To get t' we need to find t first, e.g.

t = 4600m/ (2.97 x 10⁸ m/s) = 1.55 x 10^-5 s

Then: t' = (1.55 x 10^-5 s) (0.14) = 2.1 x 10^-6 s

b) The distance traveled in its frame is just the proper length, L'  so:

L' = 4600 m [1 - v²/c²]^½ = 4600m (0.02)^½

L' = 4600 m (0.14) = 644 m

4)  The proper time t' = 2.6 x 10^-8 s

t = t’/ [1 - v²/c²] ^½    and v = 0.95c

so:

t = (2.6 x 10^-8 s)/ [1 - (0.95c)²/c²] ^½

t = (2.6 x 10^-8 s)/ [0.0975]^½ = (2.6 x 10^-8 s)/ 0.312

t = 8.3 x 10^-8 s

Oumuamua Is Just An "Ordinary Comet" - Harvard Astronomer Avi Loeb Disagrees - And He's Right To Do So

 

                                                                              

               Oumuamua (artist rendition)  and Harvard's Avi Loeb who suspected an alien artifact

Readers may recall that nearly six years ago, an object roughly the size of a football field baffled planetary scientists as it traversed our solar system. The irregular shape and motion of the object, dubbed ‘Oumuamua, even led one Harvard astronomer (Avi Loeb) to conjecture an artificial, extraterrestrial origin, e.g.

Astronomer Avi Loeb Says Aliens Have Visited, and He's Not Kidding - Scientific American

But this has now been challenged by research published Wednesday in the journal Nature,  e.g.

Acceleration of 1I/‘Oumuamua from radiolytically produced H2 in H2O ice | Nature

The Cornell postdoc Darryl Seligman concluding:

"Oumuamua is a typical comet that expelled gas in an odd way as it traveled through our solar system. We’ve gone through every weird, crazy, possible theory— ideas that stretch the imagination to match all of these observed things, and then we figure out that it is just the simplest thing you could possibly imagine, which is just a water-rich comet,”

Astronomers first spotted ‘Oumuamua at an observatory in Hawaii in October 2017. It passed by the sun at speeds of nearly 196,000 miles an hour. The body’s trajectory indicated that it arrived from distant stars, having likely traveled for many millions of years before arriving in our system.  

Observatories around the globe had barely weeks to gather enough data to try to identify the nature of the interloper before it continued on to the greater Milky Way. No satisfactory explanation or hypothesis was ever provided though Harvard's Avi Loeb conjectured it was most likely an artifact from an extraterrestrial civilization. This emerged in his co-authored paper appearing in   Astrophysical Journal Letters  in November.  2020.   

The blowback from peers was as fierce as if he'd co-authored a paper on UFOs as opposed to a trans-stellar object or artifact  powered by solar radiant energy. For example, Paul M. Stutter, an astrophysicist at Ohio State University,  shortly after the paper was published,  tweeted    

 'Oumuamua is not an alien spaceship, and the authors of the paper insult honest scientific inquiry to even suggest it,"

Whoa!  Says who?  Honest scientific inquiry would, in fact, include all reasonable hypotheses including the extraterrestrial one - just as I included in my paper published in the Journal of the Royal Astronomical Society of Canada e.g.

Transient Optical Phenomena of the Atmosphere - a Case Study

Concerning an object reported by a founder member of the Barbados Astronomical Society.   Let me also add it isn't impossible for a genuine scientist - astronomer, physicist or other -  to stretch his mind to get beyond the "ET phobia".  After all the late planetary astronomer Carl Sagan  - once a debunker of the idea of UFOs as extraterrestrial craft-  did finally come around to accepting the validity in a one-on-one with Northwestern University astronomer J. Allen Hynek, e.g.

https://setiathome.berkeley.edu/forum_thread.php?id=76926

Excerpt:  "The pillar of modern space science Dr. Carl Sagan revealed to Dr. J. Allen Hynek, that he knew UFOs were real but could not talk publicly about the matter and possibly risk the loss of academic funding."

Perhaps the best explanation for all this recent dodging, dismissing and aversion by so-called scientists was provided in a paper published some years ago in the journal Political Theory by Alexander Wendt and Raymond Duvall, viz.

Sovereignty and the UFO - Alexander Wendt, Raymond Duvall, 2008

 Therein they noted the phenomenon of the UFO (or any ET hypothesis)  tends to be rejected as real because it comes up against the human concept of state sovereignty.  In effect, to admit the reality of the UFO as an alien craft would directly challenge this comforting codswallop, engendering enormous cosmic uncertainty and rejection of the fake techno superiority meme.   

 This is also exactly why "stigmas"  arise associated with UFO research or claims/admissions of their reality.  In a word, those  skeptical responses amount to an elaborate defense mechanism for egocentric humans, including too many astronomers.  

Let's now return to the Nature paper and the take of another co-author, Jennifer Bergner, an astrochemist at the University of California, Berkeley. According to Jennifer, "Oumuamua is in fact a comet that was restructured during its long journey".

The object, according to her, was born in another solar system but made its way into ours. In that other solar system it was a "normal" comet made of ice, but then changed its identity over its long sojourn.   How did it change? Well, as the comet traveled through interstellar space toward our part of the Milky Way, it was struck by cosmic rays. The radiation from those cosmic rays then separated the hydrogen from some of the comet’s ice-forming water molecules. That hydrogen then became trapped inside tiny pockets within the ice deep inside Oumuamua.   But we're not done yet.   

Bergner goes on to argue in the paper that once ‘Oumuamua approached the Sun, the heat from our star rearranged its icy structure.  This led to the collapse of those tiny ice pockets and then the formation of channels within the ice through which trapped gas escaped to the comet’s surface and accelerated it.

Why resort to this contrived explanation? Well because Oumuamua was being propelled by forces beyond the gravity of the Sun or planets.   Therefore a "non-gravitational" hypothesis was needed.  It's also known that  nongravitational forces are observed in typical comets.  Thus, the closer a comet gets to the sun and heats up, the more of its ice transitions into gas—creating its tail while changing its motion in space. This process is known as outgassing.  But is it really enough to generate the acceleration observed? I'm not convinced and neither is Avi Loeb.  

The Harvard astronomer posited in multiple papers (e.g. see above) and a book that its odd acceleration could be because it is a piece of alien technology. (Not necessarily inhabited, in fact, unlikely to be). Dr. Loeb also has said he takes issue with the authors’ characterization of ‘Oumuamua as a "typical comet.'  Noting:

 “It must be a very unusual comet if it is a comet at all,” 

I totally agree, especially given the convoluted explanation by the authors of how it altered its original "normal" comet form in transiting to our solar system. As one who always looks at the simpler of of 2 or more competing hypotheses first, Dr. Loeb's beats the Nature authors' version hands down.

Dr. Loeb said the topic of extraterrestrial intelligence remains taboo among his astronomy colleagues. And he is correct, for the reasons I gave earlier. Which is a pity because they all are blinded to the Occam's Razor alternative of Loeb's simpler hypothesis which their emotions cause them to avoid.

Thankfully, Dr. Loeb said he continues to explore the idea that ‘Oumuamua is, for example, a piece of alien technology that harnesses the pressure of sunlight to move.

 

 See Also:

Why Harvard's Top Astronomer (Avi Loeb) Is Wrong About UFOs - And Indeed Is Doing UFO Research Himself 

Looking Again At The Korteweg de Vries Equation

 The  Korteweg de Vries (KdV) equation:

(- v _o  + c _s   + v) ¶  v / ¶x’  -  m ¶ ² v / ¶x²’  +  a ¶ ³ v / ¶x³ =  0

is a well-known example of a soliton equation admitting nonlinear superposition, see e.g. a graphical representation of a soliton solution here:

In the KdV equation c _s is the ion sound speed, and v _o the  electron thermal speed.  In the form shown, note the appearance of the dissipative term:

 (m ¶ ² v / ¶x²)  

and for a soliton to evolve into a shock a dissipative mechanism is needed.  In the more common situation steepening of the wave balances dispersion and we obtain a wave form such as shown in the graphic

  On integrating once, and excluding the shock evolution term (in m) ,  we obtain:

a  d² V/ dx²   -    (v _o   -   c _s ) v  -   v ²/ 2   =  0 

Which has the same mathematical form as Newton's 2nd law of motion, e.g. 

 m x" =  F(x)  =   -  ¶ x V(x)  

Where V(x) is the potential energy. With some further manipulation we find:

a d² V/ dx²   =   -  ¶ v  V' (x)  =

-  ¶ v [(c _s   -  v _o) v ²/ 2   +     v³/ 6 ]

For a particle 0f mass a   moving under a potential field given by the quantity in brackets.  We call this quantity the pseudo potential and designate it:

 F (v) =  [(c _s   -  v _o) v ²/ 2   +     v³/ 6 ]

Which is also known as the Sagdeev potential.  It is left to the industrious reader to do a simple plot of  F (v) vs.  v,   with   v _max   shown on v -axis.

For the criteria on  F  to obtain a soliton solution we have:

i)   ¶ F / ¶ V ] _{v= 0}  =  0;  ¶ ² F/ ¶ v²   <   0

ii)  F   <   0,   For   0   <   v   <   v _max

iii)  d F / d V ] _{v= v max}  >  0

We  note here that two graphs of  F   vs.  v are possible,  one for  c _s   -  v _o   >   0,  the other for  c _s   -  v _o   <   0.  Since we demand only a localized wave form then it will always be the latter form used, i.e. in further analyses.  One such is to obtain a soliton solution for the KdV equation:

a  d² V/ dx²   -    (v _o   -   c _s ) v  -   v ²/ 2   =  0 

This may be solved exactly i.e. with c _s   -  v _o   <   0.   The procedure is then to multiply the KdV by v'  and then integrate to obtain:

a/ 2  (v' ²)  =  (v _o   -   c _s ) v ²/ 2  -    v³/ 6

And we choose the constant of integration to be zero because we want v' = 0  when  v = 0.   Working through the process the final solution is found to be:

v   =   3 (v _o   -   c _s ) sech ²  [(v _o   -   c _s  / 4 a) ½     x' ]

Suggested Problems:

1)  Plot a graph of  F   vs.  v  for the case c _s   -  v _o   <   0 and indicate the position of v _max  on the graph.

2)  Integrate  the KdV equation:

a  d² V/ dx²   -    (v _o   -   c _s ) v  -   v ²/ 2   =  0 

And show how the soliton solution:

v   =   3 (v _o   -   c _s ) sech ²  [(v _o   -   c _s  / 4 a) ½     x' ]

Is obtained.