Tuesday, April 28, 2026

Mensa Inequality Intermediate Algebra Problem

Revisiting Operations-Computations In Basic Tensor Algebra

We already know tensor algebra is a crucial first step in getting acquainted with tensor calculus and also getting a better grasp of differential geometry. Let's also begin by noting that just as in linear algebra, e.g.

Math Revisited: Looking At Linear Algebra

matrices factor into this.

I.e. every symmetric tensor will have the form (using dummy indices i, j):

a _{i j}=

(a₁₁ ……a₁₂………a ₁₃)

(a₁₂….. a₂₂……….a₂₃)

( a₁₃……a₂₃………a₃₃)

The anti-symmetric or skew symmetric tensor will have the form (note how the order of the dummy indices changes): a_ji=

(0 ……a₁₂………-a ₃₁)

(-a₁₂…..0……….a₂₃)

( a₃₁……- a₂₃……0)

Or effectively only three distinct components.

The most basic tensor of all is the unit tensor, idemfactor or identical dyadic. defined:

I = i^ i^ + j^ j^ + k^ k^ =

When I is multiplied by any other vector as a factor the vector is unchanged, i.e.

I × C = i^ C_x + j^ C_y + k^ C_z

The Kronecker delta, which we already saw in Introduction to Differential Geometry, Part 1, is:

d _ik =

(1….0…..0)

(0….1…..0)

(0….0… ..1)

Example 1: Compute: a_{i j}d i_j

If a_{i j} is a second order tensor with matrix:

Then we have:

And we let x_iand y_jbe first order (rank 1) tensors (i.e. vectors) respectively given by:

x_i= (2, 1, 4)

y_j= (3, 7, -1)

a)Find: a_ijx_i +a_ji

b)Find: ( a_{i j} - 2/3 d _ij)

Solutions:

a) From Mathcad matrix computations:

Diagonalizing tensors is analogous to obtaining the eigenvalues for a matrix in linear algebra. Hence, we need to extract the eigenvalue equations via diagonalization and obtain the distinct eigenvalues. Consider the object (solid tetrad) shown below for which we want to obtain the principal axis.

For which we define a _{i j} =

With T’ = A × I × A^t

Where t denotes the transpose. Then we obtain, T’ =

Which is to be diagonalized. Writing this out:

(15 - l…..0……..0)

(0…….11- l ....-3Ö2)

(0……..-3Ö2…8- l )

This leads to a cubic equation with triple roots which are:

l1 = 15, l2 = 5, and l3 = 14

Substituting l1 in the matrix we get:

Now find the remaining two eigenvalues, l2 and l3, to diagonaliize the matrix and obtain solutions in c _x, c _y and c _z.

Thereby obtain the principal axes in terms of: e^’_x, e ^’_y and e^’ _z .

Show in particular that:

e^’ _z = - 1/ Ö3 e1^ - 1/ Ö3 e2^ + 1/ Ö3 e3^

Solution:

Using the eigenvalue l2 = 5 we arrive at:

For which:

c _y = c _z / Ö 2 and

c = 0 e₁^ - c _z / Ö 2 e₂^ + c _ze₃^

Whence:

(c _z ²/ 2 + c _z ²) = 1 Þ c _z = Ö (2/ 3)

Further:

e₂^’ =

1/Ö 2 (Ö (2/ 3)) e₂^” + Ö (2/ 3) e₃^” = 1/ /Ö 3

e₂^” + Ö (2/ 3) e₃^”

For the eigenvalue l3 = 14:

We arrive at:

Yielding: c _y = - Ö 2 c _z and c _z = 1/ Ö 3

Þ c =

Ö 2 c _ze₂’’’ + c _z e₃^”’

= -Ö (2/ 3) e₃^” + Ö (1/ 3) e₃^”

Then e_j = å³ _{i= 1} a _{i j} e_j’ and: e_i’ = å³ _{j= 1} a _{i j} e_j

And:

e _x^” = - 1/ Ö 2 e ₁ - 1/ Ö 2 e ₂

So:

e _x^’ = e _x^” = 1/ Ö 2 e ₁ - 1/ Ö 2 e ₂ + 0

Similarly:

e _y^” = 1/ Ö 2 e^ ₁ + 1/ Ö 2 e^ ₂ + 0

And: e _z^” = e^ ₃ Þ e _y^’ =

1/ Ö 3 (1/ Ö 2) e^ ₁ + 1/ Ö 3 (1/ Ö 2) e^ ₂+ Ö (2/ 3) e₃^

= e^ ₁ / Ö 6 + e^ ₂ / Ö 6 + 2e^ ₃ / Ö 6

And: e _z^’ =

- 1/ Ö 3 (e^ ₁ ) - 1/ Ö 3 (e^ ₂) + 1/ Ö 3 (e^₃)

(The Principal axes are: e _x^’ , e _y^’ and e _z^’ )

Suggested Problem:

In a certain rectangular coordinate system, the directions of whose axes are given by the unit vectors i, j and k, the inertia tensor of an object is given by:

I = K x=

(1….0…..0)

(0….1…..1)

(0….1… ..1)

a) What are the principal moments of inertia of the object (the moments of inertia along the principal axis) relative to the origin of the above coordinate system?

b) What is the direction of the principal axis corresponding to the principal moment of inertia and equal to K?

c)If the origin of the above rectangular coordinate system is at the center of mass of the object and the total mass of the object is M, what is the change in the inertial tensor of the object if the rectangular coordinate system is displaced parallel to itself a distance r_o in the direction

(1/ Ö2)j + (1/Ö2)k?

Monday, April 27, 2026

Media's Inexcusable 'Fact Inflation' Of DC Hotel Incident Likely To Give Orange Narcissist A Boost In Polls

July, 2024, a bloody ear provided a useful campaign image

April 24, 2026 -Trump unbothered as sounds are heard from hotel lobby.

"There is a close relationship between Trump and violence — not just the attempts on his life but also the violence he’s unleashed on the world, the violence his ICE and Border Patrol agents have caused inside America, the violence he has incited among his followers. (A few of last night’s attendees were in Congress on January 6, 2021 when Trump’s thugs attacked the U.S. Capitol.)

Trump’s violence has resulted in thousands of deaths and injuries. That is no justification for last night’s attack, of course, but it is part of what he has wrought in America. He has changed the script in Washington." - Robert Reich, Thoughts on Last Night's Correspondents' Dinner. Smirkingchimp.com

"It appears the charge of attempting to kill the president is there for headlines. How do you prosecute the charge if he was never in the presence of the president? Does there not have to be the opportunity to commit the crime?" - WAPO comment

"I guess it never occurred to anyone that Cole Allen wasn't gunning for the orange menace. Credible reports indicate the Secret service agent was a victim of friendly fire".- WAPO comment

"He never made it to the same floor as tRump, nor even within eyesight of tRump. You can charge him with weapon possession and making threats but that is about it. I'm sure Trump is eating this up. Now he can brag about surviving three assassination attempts to his MAGA friends.".- WAPO comment

Sadly, and especially in parlous and chaotic times, the truth and facts often become a first casualty- by distortion, exaggeration or omission. Such is the case in the American media following a shots fired incident Saturday night at the Washington Correspondents dinner. (Trump, in fact, told Nora O'Donnell in a '60 Minutes' interview he "wasn't worried" during the entire incident.)

At least Freed Zakaria's CNN program and Velshi on MSNOW delivered factual takes on the events. Each also played an interview of Assistant AG Todd Blanche on Meet the Press where he stated clearly it is "believed the gunman targeted officials in the Trump administration." Not Trump himself, though that later got conflated into "targeted Trump, officials."

Because it was 'officials' - like House Speaker Mikey Johnson, seen running in the hotel lobby. Let's also note, as Velshi pointed out, "the gunman was on a different floor". That is, a different floor from Trump and the correspondents' dinner.

But when one tuned into Fox and Friends the first words heard, from a studio couch full of pie-eyed bone heads was:

"This man has ice in his veins! This is the third assassination attempt on him and he's survived."

Errr...no, hate to disabuse you but when the threat is actually not even on the same floor and had already been neutralized by metal detectors and Secret Service agents it's not actually a "third assassination attempt". To pass that threshold there has to be some degree of greater proximity. So, we have to lean on the FT's Edward Luce who wrote Sunday:

"After an assassination attempt in Pennsylvania in July, 2024 a bloodied Trump pumped his fist and shouted ‘fight, fight, fight’. Saturday night’s incident at the Washington Hilton provided no such moment.

So this time round any sympathy wave is likely to be more limited. The biggest difference is that the alleged gunman, Cole Allen, got nowhere near Trump. Though the cable news channels built it up as another near miss, Allen was never in the ballroom. Loud cracking sounds were heard but the president was not that close to danger. Yet he will try to make political hay of the incident."

Oh, indeed he will do that - in fact DID that! Only hours later the orange refuse and first-time felon president - and insurrectionist - didn't waste time blowing his own horn, i.e. according to The Washington Post:

Trump says he sees shootings as a reflection of his impact

I.e.

"President Donald Trump said Saturday that he views his repeated brushes with violence as a sign of his historic significance and is determined not to let the dangers affect him."

Historical significance? Yeah sure, as the first felon president and the first to incite an insurrection and still get elected- and then pardoned the insurrectionists afterward. Yet there is the senile, sleepy old turd bragging on himself rising to the level of a JFK, RFK or MLK in the Post,

“I’ve studied assassinations, and I must tell you, the most impactful people, the people that do the most ... they’re the ones that they go after,” Trump told reporters at the White House soon after a shooting suspect was apprehended. “And I hate to say I’m honored by that, but I’ve done a lot.”

Oh yes, you have done a lot. Launched illegal tariffs that crippled free trade and put enormous cost burdens on companies large and small, mounted ICE goon attacks in Blue cities "rounding up law-abiding migrant workers in Home depot parking lots" (WSJ's Barton Swaim's words from 4/18 column, p. A11), not to mention launching an unprovoked war on Iran which he doesn't know how to extricate himself from. And which is blowing a hole in Americans' pocket books, especially the soaring gas prices. Last but not least there are his attacks on the rule of law itself - turning the DOJ into his personal engine of retribution. (Going after James Comey, Lewis Powell, Letitia James and now former CIA director John Brennan.)

Yeppers, you've definitely "done a lot", only in the wrong national direction. Luce again has this orange renegade pegged (ibid.):

"Last week his approval rating hit a personal low of below 40 percent in a clutch of polls."

Noting also that assassination has played a central role in his own attack on Iran in so-called ‘Operation Urgent Fury’ when an ordered deadly strike took out the Grand Ayatollah Ali Khamenei on Feb. 28, i.e.

“Either way, Trump’s early zeal for assassinating senior Iranians is what is shaping his political future.”

The only sure prediction is that Trump will now build his $400 million ballroom."

Which indeed is likely to be his main 'win' this time around.

Oh, and his poll numbers are cratering. Now, suddenly for the first time in ten years he attends the White House Correspondents' dinner - but there isn't the usual roasting of the prez. No, the comedians have been banned and in their place a "mentalist"(Oz Pearlman) performs woo woo stuff for the Orange Fuhrer and the first lady, Melania, aka 'the Sphinx'.

So no there is no proof Saturday night's incident was a "3rd assassination attempt", as I noted. But you can bet your sweet bippy that the Foxites and the rest of the Trump TV propaganda orbit (including NEWSMAX, and OAN) will inflate it to those proportions. Why? Because it will serve their propaganda purposes. Of course, Trump himself, despicable liar and grifter that he is, played it to the hilt, saying (WaPo piece, ibid.):

"Yes, this is a dangerous job but I'm happy to do it."

Of course, he's happy to do it given how he's turned the government into his personal ATM, this according to the latest issue of FORBES, (April/ May p. 18)

Pardon me, then, if I'm not impressed by the goings on or the performative drama (Secret Service agents hustling Dotard and wife out), given the shooter was actually stopped at the screening station not anywhere inside the dinner venue itself.
But I will bet that the incident spikes Trump's polls among the MAGA cult so he at least gets into the low 40s again. However, I am hoping what I regard as the higher IQ segment of the populace doesn't bite. Especially in bestowing a new level of divine providence on the bloodiest thug to ever occupy the presidency.
See Also:

Donald Trump

Trump Distracts From Gunman’s Epstein Motivations To Pile On Jimmy Kimmel

by Sarah K. Burris | April 28, 2026 - 5:23am | permalink

— from Alternet

President Donald Trump lashed out at ABC late-night host Jimmy Kimmel about a joke he made the night before a man with a gun showed up at the White House Correspondents Association dinner, where Trump was set to speak.

First lady Melania Trump came out against Kimmel's joke on Monday as well, attempting to link Kimmel to the assassination attempt. Mrs. Trump didn't comment on the joke directly after it was made, which was filmed on Thursday and aired Friday evening at 11:35 p.m. EDT. It was 60-plus hours later that Mrs. Trump came out against the joke.

Kimmel's joke was: "Look at Melania, so beautiful. Mrs. Trump, you have a glow like an expectant widow.”

» article continues...

And:

And:

WHCD shooting exposed MAGA media's secret social media operation - Salon.com

excerpt:

“It’s time to build the ballroom,” Acting Attorney General Todd Blanche posted on X, attaching a letter demanding that the National Trust for Historic Preservation drop its lawsuit blocking the project. Former Attorney General Pam Bondi — who Trump had fired just weeks earlier — chimed in, writing, “We MUST have the Ballroom completed to protect POTUS and his guests.” On Fox News, House Speaker Mike Johnson, R-La., suggested the ballroom’s planned “7-inch thick glass” will make it “a very safe environment to do events like that.” Even Democratic Sen. John Fetterman of Pennsylvania fell in line.

The speed and uniformity of the response would be striking under any circumstances, even after Trump himself had laid the groundwork for using security concerns to justify construction of the White House ballroom just a few weeks before Saturday night’s events. But in the aftermath of a violent and potentially deadly incident, it’s perhaps the clearest example of a media apparatus that behaves more like a professional marketing operation.

MAGA influencers also used the shooting’s chaotic aftermath as cover to demand Jimmy Kimmel’s head over jokes he delivered days before the shooting during a bit he billed as his alternative version of the Correspondents’ Dinner speech.

And:

Robert Reich

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In the News

Sunday Thought: Last Night’s Correspondents Dinner

by Robert Reich | April 26, 2026 - 5:21am | permalink

— from Robert Reich's Substack

Friends,

For as long as I can remember, the White House Correspondents dinner was where the Washington press corps and Washington officials basked in each other’s celebrity.

Last night’s ended abruptly with gunshots, Secret Service officers screaming at attendees to “get down,” Trump and other officials being rapidly ushered out of the ballroom, plates crashing and chairs falling, and general pandemonium.

Last night, the celebrities became normal people feeling panic and fear.

Most of the time, Washington is a stage on which actors take on roles and dress up for assigned parts. I remember wearing an uncomfortable tuxedo to the White House Correspondents Dinner, trying to make pleasant conversation with people who had skewered me that very morning.

» article continues...

And:

In the News
Teacher of the Month Turned Alleged Gunman — What We Know So Far About Cole Tomas Allen
by Roxanne Cooper | April 27, 2026 - 4:46am | permalink
— from Alternet
Law enforcement has identified the suspect in Saturday night's shooting at the White House Correspondents' Dinner as Cole Tomas Allen, 31, of Torrance, California. According to multiple reports, Allen allegedly charged a security checkpoint at the Washington Hilton around 8:36 p.m., armed with a shotgun, a handgun, and multiple knives.
Allen fired at a Secret Service agent, but the officer survived because he was wearing a bulletproof vest. Allen was taken into custody at the scene without officers firing on him and was later transported to a hospital for evaluation.
Allen is now facing federal charges including assault on a federal officer and using a firearm during a crime of violence. He is scheduled for arraignment on Monday, April 27, in federal court.
» article continues...