The Role Of Functionals in Fitting A Curve To A Polynomial

 In numerical analysis the functional - in the form:

F(a,b,c) =  å ⁴ _n =1  (ax_i  ² +  bx_i   +  c   -  y_i)  ² 

Often plays a key role in fitting a curve to a polynomial of the form:

f(x) = ax ² +  bx  +  c 

Generally starting with being given a table, e.g.

We are then tasked with fitting a polynomial of form a x ² +  bx  + c  to the set of points,   (x₁ , y_1 ,  x₂ , y_{2  ,} x₃ , y_{3   ,}  x₄ , y₄ )say as displayed in the  table:

To proceed we  must choose a, b, c to   minimize the functional, such that:

F(a,b,c) =  å ⁴ _n =1  (ax_i  ² +  bx_i   +  c   -  y_i)  ² 

This can be done by solving the linear system of equations:

  ¶f/¶a = 0,        ¶f/¶b = 0,  ¶f/¶c = 0                                                                                                          

 Computation based on the respective values for  (x_i,  y_i) are shown below:

[1] ¶f/¶a =  2 (ax_i  ² +  bx_i   +  c   -y_i)x_i ²  = 0

Then:

(-2,1): 32a -  16b + 8c - 8 = 0

(-1, 1): 2a    - 2b   +2c + 2 = 0

(0, 2):   0

(1, 1): 2a  + 2b + 2c - 2  = 0

            ------------------------

Summing: 36a - 16b +12c - 8 = 0

[2] ¶f/¶b =  2 (ax_i  ² +  bx_i   +  c   -y_i)x_i   = 0

(-2,1 ): - 16a + 8b - 4c + 4 = 0

(-1, -1):  -2a  - 2b - 2c -2  = 0

(0, 2):      0

(1, 1):     2a  + 2b  + 2c  - 2 = 0

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

     Sum:   16a + 12b  - 4c = 0

[3] ¶f/¶c =  2 (ax_i  ² +  bx_i   +  c   -y_i)   = 0

(-2, 1):  8a - 4b + 2c - 2 = 0

(-1, -1): 2a - 2b + 2c + 2 = 0

(0, 2): 2c - 4 = 0

(1, 1): 2a -4b + 8c - 6 = 0

 -----------------------------

Sum:  12a - 4b + 8c - 6 = 0

From the 3 sums we arrive at a 3 x 3 system given by:

36a - 16b +12c - 8 = 0

16a + 12b  - 4c = 0

12a - 4b + 8c - 6 = 0

Written in matrix form:

The values for a, b, c can then either be found using Gaussian elimination with pivotal condensation or matrix inversion. We choose the latter as it is more straightforward and amenable to a good computing program.  Using Mathcad, for example, we obtain:

Thus: a = 0.25, b = 0.55, c = 0.65

The unknowns a, b and c can also be computed using the Gaussian elimination calculator below. 

Gaussian elimination calculator (onlinemschool.com)

Simply insert the original array values for M and the values for v - and see all the steps that would normally be needed and the solutions for a, b and c.

The calculator will also perform a check of the Mathcad result.

Suggested Problem:

 Fit a second degree polynomial  to the ordered pairs in the table below:

Hence, find a, b and c for the polynomial: ax_i ² +  bx_i   +  c 

Solution To Mensa Algebra Problem

 

 Which digits n (0 - 9) have a power x (greater than one) that is a sequence if the digit n?

n^x  =   n n n . . . n (x > 1)

Ans. Between 0 - 9, only 0 and 1 have a power x greater than 1 that is a sequence of the digit n.

E.g.  0 ^x  = 0 for any x

1 ^x  = 1 for any x

Try 2:

nnn. . . n = 222...2 = 222/2 = 111 (not a power of 2)

3) nnn. . . n = 333...3 = 333/3 = 111

If divisible by 3 is also divisible by 37 which is not a power of 3.

4) nnn. . . n = 444...4 = 444/4 = 111

which is not a power of 4.

5)nnn. . . n = 555... 5  = 555/5 = 111

which is not a power of 5.

6) nnn. . . n = 666... 6  = 666/6 = 111

which is not a power of 6.

7) nnn. . . n = 777... 7 

But the last 2 digits of  7 ^x  (x >1)are always:

49 (e.g  7 ² =  49),  43 (e.g  7 ³ =  343 ), or 01 (e.g.  7 ⁴ =  2,401) 

But never 77.

8)  nnn. . . n = 888... 8  = 888/8 = 111

which is not a power of 8.

9) nnn. . . n = 999... 9  = 999/9 = 111

But if 111...1 is divisible by  9 (when the number of 9s is a multiple of 9) then it is also divisible by 12, 345, 679 (which is not a power of 9)

Weaponizing Food : Dotard Goes "Great Gatsby" - While Refusing Court's Demands to Provide SNAP Benefits

 

              Trump dines at 'Great Gatsby' dinner as SNAP benefits are cut for millions ("A little party never killed anyone".)

Trump starves children, the elderly and disabled, while holding lavish “Great Gatsby” parties for his rich friends, more concerned with his $400 million ballroom and gilded bathroom renovations. We watch him stand idly by, unconcerned and annoyed, as a man collapsed in the Oval Office.  How much of this will Americans take? - WaPo Comment

Nope, you cannot make this shit up, folks, Traitor Trump actually barked out earlier today (on his pathetic Untruth Social)  that federal food benefits provided through the Supplemental Nutrition Assistance Program (SNAP) will go out only when Democrats vote to end the ongoing government shutdown.  Does this fucking dirtbag even grasp he’s putting the health and lives of 42 million Americans at risk? Oh wait. Of course not. The orange fungus could care less. He wants to entertain ‘Great Gatsby’ type dining fests at Mar-a-Lago e.g. see the top image.

"Let 'em eat cake, or sawdust!"

While the needy have to live on scraps or starve while they scratch for whatever they can at food banks, since this fleabag old fart wants to weaponize food. Something the Nazis did in 1932-33 to the Jews in Germany in an effort to drive them from the country.

In a post on Truth Social, Bonespurs wrote that SNAP benefits "will be given only when the Radical Left Democrats open up government, which they can easily do, and not before!"

Trump's comment comes a day after administration lawyers told a federal court in Rhode Island that it would tap into a contingency fund to issue partial food stamp payments for November. The Justice Department said in a filing that the Department of Agriculture would provide states with information necessary for calculating the benefits due to each eligible household, which would allow states to then begin disbursements.

Agriculture Secretary Brooke Rollins said on X that her agency sent that guidance to states Tuesday morning, but warned it would be a "cumbersome process" that could result in a weeks-long delay in food stamp payments going out recipients.

 If there was any question of whether Donnie Bonespurs Trump  is not demented, i.e. was feeling the pressure of the first test of his power, popularity and policies during his second term as key elections are underway, it was demolished hours ago. He remains the verminous vile swine he always was from the time he came down the escalator at Trump Tower 10 years ago.

How so?  The first felon president has spent the morning sending a torrent of social media posts attacking Democrats, stoking fears about Republicans losing the midterms, ranting about local elections and disparaging voters. What about the food needs of 42 million including 20 million kids? He doesn’t give a rat’s ass?  All he cares about is getting his giant ball room constructed and holding even bigger events to enrich himself and his grifting family.  Oh, and getting more retribution on his political enemies.

Here's a clue for this orange fucker. Today's elections are going to deliver a first  notice that sentient citizens are sick and tired of his Hitler strongman antics and are sending a message a House takeover by the Dems is on its way.  You got it: The tales of the Democrats in their graves and buried are premature. The elections today will show why.

See Also:

Supreme Court Ruling on SNAP Benefits Leaves Families in Limbo - The New York Times

Excerpt:

Millions of low-income families around the country confronted new delays and disruptions to their food stamp benefits on Saturday, after a late-night Supreme Court order allowed the Trump administration to continue withholding some funding for the nation’s largest anti-hunger program.

Only one day earlier, states including Massachusetts, New York, Pennsylvania and Oregon had started sending full benefits to the roughly one in eight Americans who receive aid each month, seeking to put an end to weeks of uncertainty and spare the poorest Americans severe financial hardship.

But the process appeared to grind to a halt on Friday night. The Supreme Court granted an emergency request by the Trump administration to pause an order issued by a federal judge, who had required the White House to fully fund the Supplemental Nutrition Assistance Program, or SNAP.

The decision once again upended food stamps, which are funded by the federal government and administered by states. It marked the latest turn in a weekslong battle waged by local leaders and nonprofits, which have filed multiple lawsuits to ensure that about 42 million low-income Americans would not lose their ability to buy groceries during the federal shutdown.

And:

Judge Orders Trump Administration to Fully Fund SNAP Benefits This Month - The New York Times

Excerpt:

A federal judge ordered the Trump administration on Thursday to fund food stamps in full for roughly 42 million low-income Americans, after admonishing the government for delaying aid under the nation’s largest anti-hunger program during the shutdown.

But the Justice Department almost immediately told the court that it would appeal the ruling. The move renewed fears that the poorest Americans would not receive their full benefits to purchase groceries this month, leaving many at risk of imminent and severe financial hardship.

Judge McConnell attributed the inappropriate delay, in part, to an attempt by President Trump and his aides to disrupt the program “for political reasons.” He also pointed to public comments by Mr. Trump, who said at one point that he would halt all food stamp payments until Democrats struck a deal to end the shutdown, now in its sixth week.

And:

‘He’s Coming Unhinged’: Trump Goes Off On an Imaginary Enemy He Never Names, Calls Ugly, and Brags About Winning a Fight That Only He Remembers

Excerpt:

President Donald Trump has found a new enemy — though even he won’t say who it is.

In a furious late-night post, Trump hurled insults at an unnamed figure he called “a criminal,” “a lowlife,” and “ugly inside and out,” before bizarrely bragging that he had already “beaten him badly.” The tirade appeared to reference a years-old political fight that still consumes him.

Only later did it become clear who Trump was talking about: President Joe Biden — the man who actually defeated him in 2020.

Trump has never gotten over losing the 2020 presidential election or the Jan. 6 investigation that followed. He and his cronies have repeated ad nauseum, over and over for five years now, that the 2020 election was “rigged” or “stolen,” which it was not.

And:

Trump hosted 'Gatsby'-themed party at Mar-A-Lago on eve of SNAP cutoff

Excerpt:

President Donald Trump hosted a Great Gatsby-themed party at Mar-A-Lago on Oct. 31, as multiple federal judges ruled the administration could not stop funding food aid amid the ongoing government shutdown.

The party was labeled "A Little Party Never Killed Nobody" according to multiple media attendees. The title was drawn from a song on the soundtrack of the 2013 movie adaptation of "The Great Gatsby."

Attendees included Secretary of State Marco Rubio, Ivanka and Tiffany Trump, and their husbands, Jared Kushner and Michael Boulos, according to ABC News.

Guests were seen mimicking "Roaring 20's" era attire, a period just before the Great Depression that historians note for its staggering income inequity. History.com notes that in 1928 the top 1% of families received 23.9% of all pretax income and about 60% of families made less than the income level the Bureau of Labor Statistics classified as the minimum livable income for a family of five.

And:

Donald Trump

‘Not What We Voted For’: Midwestern Trump Country Is Suffering From MAGA Economy

by Alex Henderson | November 3, 2025 - 6:26am | permalink

— from Alternet

In 2024, Donald Trump became the first president in U.S. history to win an election despite four criminal indictments — one of which resulted in a conviction on 34 counts — and two impeachments. And the thing that did the most to get Trump past the finish line was the economy. Trump campaigned on affordability, promising to lower prices "on Day 1" and narrowly winning the popular voting by roughly 1.5 percent.

But Trump's steep new tariffs, according to Paul Krugman, Robert Reich and many other economists, are making the United States less, not more, affordable.

In an article published by The Guardian on November 2, journalist cpoints to Jeffersonville, Ohio as an example of a midwestern town where tariffs are having a negative impact.

» article continues...

And:

Donald Trump

The President of The United States is Criminally Insane

by John Hamilton | November 2, 2025 - 5:44pm | permalink

And should be removed from office immediately. Section 4 of the 25th Amendment to the U.S. Constitution provides the legal framework for removal of a president who is unfit or unable to discharge his or her duties. It authorizes the vice president, along with a majority of the principal officers of the administration (the Cabinet) to submit a declaration of unfitness to the President pro tempore of the Senate and the Speaker of the House of Representatives.

This of course won't happen. The vice president and the Cabinet are all partners in crime with their boss. That makes them at least criminals without conscience, and likely criminally insane as well.

» article continues...

Solutions To Angular Measurements In Astronomy Problems: Practical Astronomy Focus

  1) Two observers using cross staffs obtain zenith distances from their respective locations of   φ = 45 degrees, and  φ =  35 degrees. How far apart in latitude (L)  are their locations?

Solution:

In each case the latitude is obtained from L = (90 - φ) 

So: L1 = (90 - φ1)    and  L2 = (90 - φ2) 

Then the difference in latitude is just the difference in zenith distances or:

L2  - L1 =  φ2   -   φ1  =   45 deg -  35 deg = 10 deg


2) Consider the system Epsilon Ursae Majoris which semi-major axis subtends an angle of  2½" and for which the parallax of the system is 0."127. Find the  semi-major axis in astronomical units. (Hint: p"  = 1/d)

Solution:

We have for the semi-major axis of a binary system whose plane is in the observer's visual plane:

a = (a" x d)

where: a"  =  2 ½"    and:   p = 0."127

But  p"  = 1/d (from hint)     so:  d = 1/ p"  =    1/ 0."127  = 7.87 pc

Then:   a =  (2 ½"     x  7.87 pc)  =  19.67 A.U.


3)  What telescope magnification would be required to observe the planet Uranus as a disk 2 arcminutes in diameter?

Solution:

Uranus is 4"  or 4 arcsec in diameter, unmagnified.

2 arcminutes  =   2 x (60 arcsec) = 120 "  since 60 arcsec = 1 arcmin

Then:  M  =  120"/ 4"  =    30x

Chern Surfaces and the Wave Model Of The Atom

                                                        

                                                                 Fig. 1: Assorted Polyhedra  

                         Fig. 2. Chern surfaces and class number

Chern surfaces, developed for use in global differential geometry, don't exactly roll off the tongue or conjure up cocktail table discussions, but they have immense importance in math as well as in physics. Developed by Shiing-Shen Chern, the Chern class number to be associated with a given surface was originally conceived as the number of times a closed surface could be wrapped around another closed surface. Before getting to a physics application, let's consider the math origins.

Basically these can be traced to the polyhedral theorem of Leonard Euler. The Table shown with representative polyhedra (Fig. 1) can serve as a basis for a more concrete appreciation. Starting from the left a distinct polyhedron is identified (e.g. tetrahedron) then simply illustrated in a no frills way, then its vertices (V), edges (E) and faces (F) identified before giving the "Euler characteristic" E in the final column on the right. This is given by the simple form: V - E + F.

Thus, the tetrahedron with V = 4, E = 6 and F = 4 will have characteristic:

V - E + F = (4) - (6) + (4) = (-2) + (4) = 2

As one can see from the table, Euler's theorem basically asserts the same number, 2, results for all cases in which his formula is applied - whether for regular or irregular convex polyhedra. Obviously, the more faces, vertices, edges added the more the particular surface is smoothed out to approach a sphere (e.g. with infinite edges, faces) then one has the Euler theorem transferring to the Gauss- Bonnet theorem, formulated and proven in the 19th century.

Now, a simple physics application of the Chern surface and class number is to Bohr "quantization", such as depicted in the accompanying graphic (Fig. 2) for two cases: left - where the quantization is not satisfied (i.e. the outer wave surface doesn't complete itself an integral number of times around a given Bohr orbit, and right - the case where it does, leading to the quantum number (n) being identified with the integer number of waves completed, in this case 4.

The beauty is that the radius is scaled into n (standing) waves of de Broglie wavelength l_D . A visual reference for this wave-orbiting electron atom can be represented as shown in the same right side of the graphic, with the de Broglie wavelength spanning the distance between successive "humps", and emphasizing that an integral number of such wavelengths form the circumference of the atomic orbit, as required by 2π r = n l_D . 

By extension of the concept of these standing waves, but for different n and r, one can arrive at the probabilistic wave model of the atom.   The wave model from Bohr’s uses the following quantized relationship:

m vr  = nh/ 2p  = L

where L is the angular momentum.  Re-arranging:

h/ mv =  2p r/ n = l_D    or  2p r =  n l_D

Showing the circumference is scaled into n (standing) waves of wavelength l_D   as shown below (Fig. 3):

Fig. 3 : Standing waves for a hydrogen atom

  This wave-orbiting electron atom still has a radius r,  but with waves each separated by one de Broglie wavelength , l_D.

     Thereby  an integral number of such wavelengths form the circumference of the atomic orbit, as required by  the condition:  2p r =  n l_D.

  For example, in the case of hydrogen the first three of these cloud-wave regions are shown in Fig. 4.

Fig. 4: Hydrogen orbitals

Let us ‘zoom in’ on the more spherical n= 1 configuration, and the probability for the electron in this space as depicted in Fig. 4 below:

Fig. 5: The n = 1 electron orbital for hydrogen

    This diagram more than any other dispenses with the notion that hydrogen electron occupies a definite position. Instead, it’s confined someplace within a “cloud” or probability space (b) but that probability can be computed as a function of the Bohr radius (a_o = 0.0529 nm).  The probability P_1s for the 1s orbital is itself a result of squaring the “wave function” for the orbital.  If the wave function is defined y (1s) = 1/Öp (Z/ a_o) exp (-Zr/ a_o), and the probability function is expressed:

P = ½y (1s) y (1s) *½

     Where y (1s) * is the complex conjugate, then the graph shown in Fig. 4 is obtained. Inspection shows the probability of finding the electron at the Bohr radius is the greatest, but it can also be found at distances less than or greater than  0.0529 nm.

Chern surfaces, as we see in this post, have a wide applicability including to quantum mechanics.

See Also:

Brane Space: An Introduction to Quantum Mechanics (3)

Documentary 'Age Of Disclosure' Confirms Jerry's Every Revelation That UAP Are Real And Here

 

                            Jerry : Insisted Govt Is Concealing Alien Bodies

   Ret. U.S. Navy Commander David Fravor: UAP are real alien craft

The  new movie “Age of Disclosure” – directed by Dan Farah - documents “an 80-year global cover-up of non-human intelligent life and a secret war among major nations to reverse-engineer advanced technology of non-human origin. As introduced by Farah in the opening segment of Friday night's "Real Time" it features testimony from 34 members of the U.S. Government, military and intelligence communities, the film exposes the profound stakes for the future of humanity.”

All of which coincides with what my late AF brother Jerry had been asserting for decades, and what I myself have been posting on this blog for well over 15 years.  E.g.

Brane Space: A Perspective on the UFO

• Newly Released Pentagon UFO Files - Not A Laughing Matter...
• Why Harvard's Top Astronomer (Avi Loeb) Is Wrong About UFOs - And Indeed Is Doing UFO Research Himself

For a separate interview (from Bill Maaher's) about the film see:

The Age of Disclosure Director Dan Farah on His UFO Documentary About Aliens and UAPs

The film also echoes what Lou Elzondo has been saying since the original UAP videos emerged in 2017, and also in his book, Imminent.

Farah, a debut feature director, spent three years making “The Age of Disclosure” in secrecy, seeking out sources with direct knowledge of the government’s work around UAPs (unidentified anomalous phenomena, the more formal term for UFOs). Among the interviewees are high-ranking politicians from both the Democratic and Republican parties, such as secretary of state Marco Rubio and senator Kirsten Gillibrand, as the congressional hearings about UAPs and the proposed UAP Disclosure Act have seen major bipartisan support.

“The Age of Disclosure” has had significant word-of-mouth momentum behind it as it should, given the nature of the content. Indeed,  this momentum and publicity has spread all year even without a public distribution plan.  The initial trailer alone quickly reached more than 20 million views across YouTube and social media when it was dropped in January leading up to the film’s premiere at SXSW (South by Southwest) in March.

As might be expected, the trolls, naysayers and pseudo-skeptics came out of the woodwork and appeared in some media - like The Hollywood Reporter.  The complaint being that the interviews of the experts appeared "too certain, too sure of themselves" and "Where's the doubt?"  T0 which I say we've been dealing with the skeptics and would-be debunkers for years, hell decades - and I myself have written numerous posts taking their balderdash down, e.g.

• Astrophysicist Adam Frank Gets It All Wrong On UAP...

Physics Today Book Reviewer Kate Dorsch Is As Clueless About UFOs As Neil DeGrasse Tyson

• Are Laser Plasma Holographs Created By "Foreign Adveraries" Really The UFOs Reported By Navy Pilots?...

Anthropocentric Parochialism Defines SETI Scientist Seth Shostak's Skepticism Over Aliens & UAP 

Holman Jenkins Jr. In Latest WSJ Piece Shows UFO Topics Are Way Beyond His Pay Grade 

Nimitz Pilots Knew That UFOs They Recorded Were Not " Supernatural " - But How To Break That To NY Times' Ross Douthat?

A New UFO Book Review (WSJ) - But Still Inaccurate, Incomplete, Wanting In Insight..

So the skeptic objections have been covered from just about every angle but still come up bupkis.  If you doubt that then read any or all of the posts above.

Dan Farah. meanwhile, deserves major kudos for his efforts - having directed and produced “The Age of Disclosure” via his Farah Films banner. Relentless Releasing serves as distributor.  I plan to see the film as soon as it comes out November 1st on Amazon Prime.

See the new trailer below.

The Age of Disclosure - Official Trailer | IMDb

See Also: 

• WSJ: Pentagon Disinformation Fueled America's UFO Mythology? More Codswallop for Snookered Journalists...

And:

Age of Disclosure director Dan Farah gets dead serious about UFOs with Bill Maher - Psicoactivo #672

And:

A Halloween “Real Time With Bill Maher” Put UFOs in the Spotlight - InsideHook

Excerpt:

Another element that the two men discussed was whether or not it was coincidental that UFO sightings increased after humanity developed nuclear bombs. “We’re still a violent species,” Farah said. As for where they might be hiding, Farah had some ideas there as well, saying, “80% of the ocean hasn’t been explored.”

One especially unnerving subject was whether UAPs could manipulate nuclear weapons remotely. “If they are going to be able to turn off our nuclear missiles, I say they can’t get here soon enough,” Maher said. 

That was one of two very emphatic statements uttered during this conversation. The other was a question posed by Farah: “If you were an advanced non-human species, would you want us showing up on your front lawn?”

And:

Congress Orders U.F.O. Records Released but Drops Bid for Broader Disclosure - The New York Times (nytimes.com)

And:

What The Pentagon Prelim UFO Report Isn't Saying: 'The Aliens Are Here, We Just Don't Want To Admit It."

And:

• Pentagon Brigadier General Doubles Down On Whitewash of UFOs, But The Facts Are Too Compelling To Bury...

And:

Brane Space: Jerry's Further Revelations About Past UFO Files And The Role of FOIA Releases - In A 1990 Group Discussion In FLA (brane-space.blogspot.com)