Tuesday, September 29, 2026

Solutions to Part 3- Looking Again At Fresnel Diffraction

 1) Without plotting a Cornu spiral, find the hypothetical value of Dv   for such a spiral plotted for the diffraction pattern of a single slit of  width 0.80mm, assuming Fresnel zone parameters a = 40.0 cm and b = 50.0 cm with red light  of 6400 Å.  From this result obtain the relative intensity I.


Soln.

Dv =   S Ö{ 2 (a + b)/  abl  }  = 

0.08 cm Ö{ 2 (40 cm + 50 cm )/ (40cm) (50 cm) 6.4   x  10 -5  cm } 

Dv = 3.0      

And  from table of Fresnel integrals:

 (Dx)  =    0.6058 – 0.0000 = 0.6058

(Dy)  =    0.4963 – 0.0000 = 0.4963

A2   =   (Dx) 2  +   (D y) 2

I    » A2   =   (0.6058) 2    +   (0.4963) 2

 I    »   0.613


2) A student is given the top section of a Cornu spiral to analyze for an exam:


Using this, obtain the relative amplitude for the particular diffraction pattern.  From this find the relative intensity.

Soln.

From the diagram, on inspection of v along the spiral,  we see:

(D v)   =   (1.4 - 0.9)  =  0.5

Obtain the  (Dx,  Dy)  limits from the table of Fresnel integrals for these values, e.g.

(Dx)  =   0.5431 -   (0.7648) =  (-0.222)

(Dy)  =  0.7135  -   0.3398   =   0.374

A2   =   (Dx) 2  +   (D y) 2

The relative amplitude is:

A =  Ö (Dx) 2  +   (D y) 2

=    Ö (-0.222) 2  +   (0.374) 2

=   0.435  

And the relative intensity is:

I    » A2   = (0.435) 2    = 0.189 

3) Plot the graph for a Cornu spiral for a single slit diffraction pattern at intervals of Dv = -0.10 to 3.0 and Dx from 0 to 0.90. On the graph draw a chord from x = 0.40 to x = 0.75 and from this estimate Dv along the Cornu arc.  Hence or otherwise obtain the relative amplitude and the intensity.

Soln.  

The  portion of the Cornu spiral of interest is shown below:


This is extracted from the full spiral graphed below:


From the given information and the graph segment (upper right quadrant of Cornu spiral) we obtain: 




Monday, September 28, 2026

Mark Elias Warned Us: 6 Conservo Supremes' Latest Bogus Ruling Is A "Tragedy For Democracy" - If We Allow It


                                    "When I say 'bow and scrape' you will do it!'


 One should never underestimate the desperation of a deranged autocrat to hold onto total, unchecked power at any cost. For example, Donnie Dotard, earlier this month announced that every American adult citizen would be paid a “dividend” of $5,000 should the Republicans perform well in the midterms. The payout would cost an estimated $1.2 trillion.  The bond market went berserk and sent the 10 year Treasury yield past 5% for the first time in nearly 2 decades. 

Then, probably realizing the move had felony written all over it, the Orange Turd did the next thing that came to his feral brain – do another “emergency’ plea to the Supremes to deliver his voter verification executive order. Sure enough the corrupt six obliged and delivered his wish, e.g.

Supreme Court Allows Federal Voter Identification Tool Ahead of Midterms - The New York Times

Meanwhile, the headline in The Financial Times for Saturday said it all

US Supreme Court lets Trump vet voters’ immigration status

But not delving into how this pernicious piece of judiciary Judas mimicry could cripple Dem voting levels in the midterms using the 'SAVE citizenship' database- a misnomer if ever there was one. How so? It's predicated on a fraudulent purpose; i.e. "to check whether any voter is a U.S. citizen".  Basically, a problem in search of any miniscule facts or evidence.  This is given the instances of voter fraud, as noted by the Brennan Center for Justice are roughly between 0.0003% and  0.0025%. (See links below).  The stated purpose could have as well been" "to check whether any voter is a citizen or a vampire."

So it is just a ruse from this foul Six to enable massive voter purges using a 25 year old data bank created under Bush Jr. (designed to help with eligibility for federal benefits  - NOT  identification of voting fraud), to the ability of renegade states to challenge any election results they don't like. Not to mention - in terms of the first - the probability of 81% errors in tagging a legitimate citizen voter as a fraudulent pretender.  Case in point: In Nevada, officials just debunked the Trump Administration’s false claims that over 15,900 voters identified by DHS were “noncitizens or ineligible.” Their audit proved that all of the voters provided by DHS were in fact citizens and valid voters. 

Worst of all, there is the despicable use of private Social Security individual data to add to their nutso 'SAVE' system. The fact this misbegotten system could now be used to harass and even remove legitimate voters has been called a "tragedy for democracy" by Mark Elias (election lawyer for Democracy Docket). Who added the reason for Trump & Co. for doing this is to "stave off the wave they know is coming in the midterms".

To see more go on the likely effects go to the Deadline White House opening segment from Friday: 

ALERT: Trump-aligned Supreme Court justices hand Trump STUNNING voter suppression win

Another aspect: The cowardly crew of 6 who rendered this emergency ruling left it unsigned. But we damned well know who they are: the same unelected misfits that believe in the wacko 'unitary executive' barf that basically makes Trump a King.

Jamelle Bouie, ‘Why Does Trump Think He Can Act Like A King? (NY Times, 9/26), hits the nail on the head with this Court of anti-Constitutional Trump supplicants:

"Trump believes the court is contradicting itself. It told him he is criminally immune and has far-reaching authority to exercise his powers, but then it wants to limit those powers. His anger is understandable, since this court is very much trying to have it both ways. The conservative justices want to write their vision of unitary executive power into the Constitution and somehow limit this particular president’s ability to use that power for his personal and political ends.

Put a bit differently, when Trump’s assertions of executive authority serve the ideological and institutional interests of the Roberts court, the conservative justices are happy to affirm them. When those assertions cut against those interests, the justices are more inclined to reject them.

This dynamic is why I think it is shortsighted to clap for this Supreme Court for its occasional moves against Trump. Those judgments exist in a context where the court has all but blessed an autocratic, imperial presidency

Trump's clear derangement and malignant narcissism - which forms the basis for his feral energy -  is what drives him to incessantly "work the refs" (6 conservo Supremes) as Jamie Raskin noted in his clip from Deadline Whitehouse Friday.  Thus, having repeatedly been given the benefit of the corrupt 'shadow docket' Traitor Trump has come to expect it as the perfect end around any lower judicial rulings he hates - like the blockage (in congress and courts) of his nefarious SAVE act. So now, when stopped at one level, he automatically runs to his servile Supreme Six to give him succor.

As well as feeding his frenzy of nonstop lies (mail ballots = frauds) while undermining our election integrity via bogus shadow docket rulings. The most degenerate aspect of the latest shadow docket SC stunt? 

"The high court's action wasn't a final ruling in the case. Instead it allows the government to continue with its project while legal proceedings stretch beyond the midterms". (WSJ, Sept. 26-27, p. A4)

This despite: "A federal judge had ruled this summer that the expanded database must be disbanded because it violated laws that prohibit the government from sharing Social Security records and other personal information in its possession."

So, in other words, a purely performative judicial jerkoff to placate the orange pestilence and enable the Trumper Nazis to break the law and try to rig the midterms.  It may work, if too many legitimate voters in too many states are terrified to exercise their voting franchise. 

This particularly applies to any naturalized U.S. citizens. Now is not the time to become fearful irrespective of how many goons Trump dispatches to the polls to act as "observers."

See also:

Brane Space: Mark Elias Was Correct: Supremes Don't Merit A "Gold Star" 

And:

Briefing_Memo_Debunking_Voter_Fraud_Myth.pdf

And:

Non-Citizens Are Not Voting. Here Are the Facts. | Brennan Center for Justice

And:

Heritage Fraud Database: An Assessment | Brennan Center for Justice

And:

by Sarah K. Burris | September 25, 2026 - 5:12am | permalink

— from Salon

Speaking to Politico's Thursday "Playbook," one Republican operative explained the mood is souring in the party.

“The mood is very bad right now,” a GOP operative told Politico. They are one of the many Republicans working on multiple races this cycle. “The polling is dropping like a rock, the map is expanding, and the knives are out among consultants."

They said there is a possibility they could save themselves from the bloodbath, but it would mean a significant shift.

"There is a world where we win everything if folks can turn it around and there is a world where this midterm is a historical blow out. People are very on edge," the operative said.

» article continues...

And:

by Robert Becker | September 27, 2026 - 4:43am | permalink

Electoral, then political payback kicks off the most lasting legal, moral, and knowledge-based assault against discredited MAGA betrayal of Constitutional government

Once upon a time, in a fit of pique induced by punitive fates, a boastful nation contracted a terrible, internal sickness – infected by fundamentalism, racism, wishful thinking, and ignorant reductionism. Not thanks to a foreign invasion, economic disaster, global pandemic, not even a looming asteroid – just the searing contradictions of a country too big for its britches or too unenlightened to act as prudent world leader. Thus it came to pass that a regressive 2024 minority rose up because they just couldn’t stand things as they were, certainly all prior presidents but one. The rampage, however deviant, demanded recognition, then control, then domination, as if a strongman autocratic theocracy was just what highly skeptical founders envisioned.

» article continues...

Friday, September 25, 2026

Reviewing Algebra II (Part Four): Manipulating Complex Numbers, Solving Complex Equations

 Fortunately, there are a number of previous blog posts which will be of use in solving complex or imaginary problems. These are given below:

 Taking Complex Roots (1)

Solution of Complex Roots of Unity Problem

Solutions to Complex Roots Problems

More Complex Roots

Obtaining Complex Conjugates

Brane Space: Introducing Complex Functions (1)

We can begin with a few practice exercises in manipulating complex expressions, the aim being to simplify each:

a) (5 + 2i) (3 + 4i) 

b) (2 + 6i)2

(c)Express:  (5 + 10i)/ (½   -   2i/3 ) in the form a + bi

The solutions follow in order:

 (a): (5 + 2i) (3 + 4i) =5 (3 + 4i) + 2i (3 + 4i) 

In which we are just using the distributive law.

Then:  15 + 20i + 6i +  8i2 = 15 + 26i + 8(-1)

= 15 + 26i - 8  Or:   7 + 26i (Simplest form)  

 (b): (2 + 6i)2   =   (2 + 6i) (2 + 6i)

Using the distributive law again:

 2(2 + 6i)  + 6i(2 + 6i) =  4 + 12i + 12i +  36 i2

= 4 + 24i  + 36 (-1) =  -32 + 24i (Simplest form) 


(c)(5 + 10i)/ (½   -   2i/3 ) 

  Þ

(5 + 10i) (6)/ (½   -   2i/3 ) (6)  

= 30 + 6o i/ 3 - 4i

 Þ

30 + 6o i (3 + 4i) / 3 - 4i (3 + 4i)  =

90 +  120i  + 180 i  + 240 2 /  (32   +  42  ) =

-150 + 300i/ 25   =  -6  + 12i (Simplest form) 


Finally, consider solving a complex equation:

x2   +  12 =  6x  

for which we need to find the complex roots.

Solution:

Begin by writing the equation in standard form:

x2  - 6x  + 12 = 0

This is easily seen to be a quadratic equation for which we will need to use the quadratic formula:


The key aspect to bear in mind is that the discriminant (term inside the radical sign) will always be a negative term for a complex equation. In this case, given a =1, b = -6, c = 12 we have:

 b 2  - 4ac    =  36 - 48  =  -12

Substituting values for the full formula:

x =  6 +  Ö(36 - 48)/  2(1) 

=

6 +  Ö -12/  2  =  6  +   Ö (4)/Ö (3)/ Ö (-1)/  2 

Þ

6 +   2i Ö (3) /  2 =

 3 + i Ö (3) 


Selected problems:

1)  Write: (2 + 3i)/ (1 + 2i) in the form a + bi

2) Show the product of a complex number a + bi and its conjugate (a - bi) is a real number. (You can use the result from (1).

3) Find the product: (3 - 5i) (2 + i)

4)  Write in the form a + bi:


Solve each of the following:

5)   x2   +  32  = 0  for all roots  

6) x4   - 1  = 0

7) x2   - 4x  + 8 = 0

8) x2   + 10x + 29 = 0

9) 4x2   - 4x  + 5 = 0

10) 4x - 7  = x2