Friday, June 30, 2023

Solutions To Numerical Analysis Problems (2)

1)Find the linear approximation to:

 f(x) =  Ö (1 + x) at x = 3


 We evaluate the first two terms of the Taylor series for f at a = 3:

f(a) + f ’(a) × (x – a) 

For which f '(x) =   ½ (1 + x) -1/2

f(3) = 2

f '(3) =   ½ (1 + 3) -1/2

=     ½  (1/Ö (4))  =   ½ (½)  =  1/4

Then: f(x) =  2 + 1/4  (x - 3) = 2 + x/4 - 3/4

= 5/4  + x/4

At x = 3.2 we obtain:

Ö (1 + x)  =  Ö (1 + 3.2) »  5/4  + 3.2/4

=  1.250 + 0.800 = 2.050  

Which differs from the true value: 

Ö 4.2  »  2.04939   

By less than one one thousandth

2)  Show that in general:

(1 +  x)k  » 1 +  kx

For any k, provided x  »  0


Check at two different values for k:

Rem: Approximation is good for any x »  0

a) k = -1 and we will get (-x) instead of x, so:

(1 +  xk  =   (1 +  x-1 

 1/ (x - 1) = (1 +  x)-1  »  1 + (-1) (-x) =  1 + x

Try x= 0.01, then:

 (1 +  x)-1   = 1.01

(1 + kx) = 1.01

But if larger x is tried, e.g. x = 0.1 we see:

(1 +  x)-1   = 1.11   vs. 1.1 

b) We use k = 1/3  and   x4   instead of x:

(1 +  x1/3  =    3Ö( 1 +  x)  =  (1 +  x41/3

»  1 + 1/3 (x4) »  1 + 1/3 (x4)  »  1 + 5 x4/ 3

Try x= 0.50, then:

(1 +  x1/3  =   0.883

(1 + kx) =  1 + 1/3 (x4»  0.896

But if smaller x is tried, e.g. x = 0.01, we get equal values (1)

Thursday, June 29, 2023

Solar Max Peak Expected To Be Earlier & More Violent Than Originally Forecast

 Astronomers know that the Sun exhibits an 11-year average sunspot cycle. This means that every eleven years - on average, sunspot numbers reach a peak. This period of increased activity, known as the solar maximum, is also a potentially perilous time for Earth, which gets bombarded by solar storms that can disrupt communications, damage power infrastructure, harm some living creatures (including astronauts) and send satellites plummeting toward the planet.

From a distance, the Sun may seem calm and steady. But zoom in, and our home star is actually in a perpetual state of flux, transforming over time from a uniform sea of fire to a chaotic jumble of volatile plasma and back again in a recurring cycle. In the lead-up to each gargantuan reversal, the Sun amps up its activity: belching out fiery blobs of plasma, growing dark planet-size spots and emitting streams of powerful radiation.

Superimposed on the 11-year cycle is a 22-year cycle, for reversal of magnetic polarities to occur in the "leading" spot.  The 22-year cycle means that 22 years must elapse before leader spots again have negative polarities at sunspot maximum. In the intervening, or 11-year maximum, the leader spots will all have positive polarity.  Of course, the frequency of sunspots in different (and succeeding) cycles is subject to change. In the diagram below such variations in sunspot numbers are shown over time from 1750, occurring as cycles:

These two complementary aspects and effects (defined as a and  dynamos)  are illustrated below:

Comparison of the a - w  dynamos (a  on right)

Note the expansion and direction of the reference field line for the w- effect,  and conversely the compressed nature and field line directions for the a-  effect.  Given below are the four primary equations pertaining to magnetic induction :


Within these equations lies the key to why the level of eruptions and disruptions of a given cycle may vary from a prior one. In particular, the azimuthal component of the magnetic induction (j) and the poloidal component , or B p = (Br ,  j)  =   Ñ X A, where A is the vector potential, may determine the degree of magnetic twist. This in turn can determine the level of activity for the cycle. 

Sunspots themselves are formed when strong magnetic fields rise up from the convection zone, a region beneath the photosphere that transfer energy from the interior of the Sun to its surface. At the surface, the magnetic fields concentrate into bundles of flux elements or tubes, which prevent the hot rising plasma from reaching the surface.  

When two such large (and magnetically complex) spots come into close proximity there is the potential for what is called "polarity intrusion" and ensuing explosive eruptions, e.g. when magnetic vector magnetographs detect this sort of aspect:

The important point to note is that as the deformation progresses the magnetic gradient dramatically increases.  The magnetic gradient is written simply as:

grad B = [+B N - (-B N)] / x

 where the numerator denotes the difference in the normal  components of the magnetic field B N (between opposite polarities (+, -)  of the active region as measured by vector magnetograph and the denominator denotes the linear (x) scale separation between them.  As x decreases, as it will deform the neutral line whereupon the gradient spikes and this signals probable flare conditions, namely for a  d -class sunspot given it is most likely to feature such gradient change.

A number of solar physicists now believe the next cycle will see such indices and hence be exceptional in violent eruptions, to the extent of disrupting electronic communications and possible even power grids. Also, originally most solar physicists believed the current cycle would peak in 2025, however this may not be so. A bumper crop of sunspots, solar storms (as well as auroral substorms) suggest the maximum may arrive by the end of this year at the earliest. Many also warn we earthlings are poorly prepared for the worst (X-ray flare, and CME outbursts).

See Also:

10 signs the Sun is gearing up for its explosive peak — the solar maximum


New Solar Research Confirms Why Delta Sunspots Are More Flare Worthy Than Other Magnetic Classes


 A Threat To U.S. Power Grids? Solar X-Ray Flares Presage Active Solar Cycle 25 And Potential Earlier CMEs As Well As Aurora


Ensemble Modeling of Coronal Mass Ejections - A Superior Means Of Prediction? The Jury Is Still Out

WSJ Troll Barton Swaim Gaslights Again With Claims Of GOP "Imaginary Threats To American Left"

                                           Barton Swaim - Hard to change old troll habits.

 When  WSJ rightist writer Barton Swaim screamed two years ago about "elitists" and the "expert class" e.g. 

WSJ's Barton Swaim Attacks The "Expert Class" -

I more or less took it with a grain of salt given the guy is a known Trumpie Troll. His latest screed ('The American Left's Fantastic Threats', p. A11) takes journalistic gaslighting to new heights, trying to portray the "American Left" (a misnomer given there is no unilateral "left") as a gaggle of effete paranoids. That is, scaredy cats who are simply inventing a hodgepodge of threats from the Right - from its patented white supremacy, to the yen to cut or kill Social Security, to banning abortion access nationwide. But how plausible are Swaim's claims? Let's take a deep dive look into his WSJ piece.  He writes:

"Mr. Biden is worried about book bans. The American Library Association recently claimed in a report that 2,571 books were “challenged” in American libraries last year. These challenges the ALA calls “attempted book bans,” nearly all of which involve a request by a patron that a public library or school library remove a book from its shelves because it is obscene or otherwise offensive."

Swaim makes light of the book bans, but none other than young poet laureate Amanda Gorman isn't laughing. After her book, The Hill We Climb was challenged by one kook (Daily Salinas) in Florida, it was relegated to "restricted grade" use at Bob Graham Elementary. Effectively this was a ban given students who once may have seen it on library shelves, now didn't, and wouldn't even know of its existence to request it. This basically applied to all the book removals and restrictions which Swaim treats lightly. Let's also be aware that the rubric "obscene or otherwise offensive" is a catchall used by the culture warrior Right to restrict any book that touches on sex, LGBTQ issues or even racial violence. In other words, it is censorship plain and simple,  no matter how they choose to spin it.  According to PEN America, 565 books were banned in Florida schools in the 2021-22 school year.

Swaim goes on to carp about the Left fretting over endless "horribles" marking the Trump years but as is his wont tiptoes over the worst: the vile incitement of the Jan. 6, 2021 insurrection. And the torching of our democracy with his vile lies - i.e. that he actually "won" the 2020 election.  Thereby setting his maniac traitor zombies off on their domestic terror binge, e.g.  

Should Trump Be Held Accountable For Fomenting Terror Strike On Capitol? Yes - No Less Than the 9/11 Terrorists

Of course, the viral infection of white supremacy was at the core of much of this deranged treason, but Swaim belittles that too as little more than "tilting at windmills" e.g.

"The preoccupation with white supremacy in progressive media— an outgrowth of claims about “systemic racism,” “white privilege” and “implicit bias”—has the advantage of unfalsifiability. The plain reality is that racism, except in its progressive “antiracist” varieties, has no purchase in American life. Tilting at the windmill of white supremacy is evidence of delusion."

One may be excused for wondering how he pulls this excrement  ("evidence of delusion") out of his ass given what we've beheld since Trump was catapulted into power with the help of the Russians in 2016.  But let's give Swaim the benefit of the doubt that his memory just suffered some temporary lapses while he was scribbling his bunkum. So we can begin by reminding him of the "Unite the Right" march in Charlottesville in August,  2017 which saw hundreds parading Nazi style across the U VA campus bellowing "Jews will not replace us!" E.g.

Then there were also the attacks by white supremacists during the 2 day event especially when Antifa members sought to protect elderly black people who had been on campus for a church get together and were set upon by racist thugs:

Or the marchers the next day in Charlottesville who paraded with all their racist (Nazi, KKK etc. ) flags and outerwear, e.g.

Evidence of delusion? Racism has no purchase in American life?  How about Swaim's brain just runs off the reservation and that fact has no purchase with what he writes in his WSJ columns

Or how about the Nazis that showed up here in Colorado Springs to voice support for the Nazi Charlottesville imps and Trump's support for these "very fine people" e.g.

What can we also say about Swaim's delusional claim regarding progressive media's obsession with "systemic racism" or "white privilege"?  Maybe we can give the guy a break and imagine he just never read the actual news story on page A3 of the same issue where one reads:

"Strict residential zoning laws have a deeply unsavory origin. In the early 20th century, many cities enacted racial zoning policies that forbade Black people from buying in white neighborhoods....Since passage of the 1968 Fair Housing Act, racial segregation in housing has fallen by 30%.."

But more than 3 in 10 U.S. urban areas are still subject to a form or redlining, or restricting available minority housing to areas located near chemical factories, oil refineries,  waste dumps or similar unsavory, unhealthy zones.

Swaim then goes on to assert, in respect of Social Security:

"Some of these unnervings arise intermittently and briefly—the idea, for example, that the GOP would “cut” Social Security if given the opportunity, an outcome that has as much a chance of happening as I have of earning an award from the American Library Association.."

Which has to make one wonder what MJ edibles he's consuming with his brewskies, or if  (less generously) he's vying with Trump to be the biggest Reep liar in these United States.  Maybe Swaim simply didn't do sufficient due diligence in his research, failing to note all the Reepo legislator deficit hawks who've gone on the record in their plans to cut Social Security, e.g.

But at least Swaim does redeem himself in a minor way near the end of his twaddle after initially claiming the left is hysterical regarding the GOP "dictating what healthcare decisions women can make".  This is when he finally admits:

"At the outset I asked the reader to leave aside Mr. Biden’s line about Republicans “dictating what healthcare decisions women can make.” Even if all the other threats mentioned earlier were make-believe, the attempt to regulate abortion is real. "

Jeezum, thanks Swaim for admitting at least one GOOPr threat is real and not a figment of our (progressive) imaginations. Sheesh! Will Swaim ever change? Not bloody likely after seeing this chestnut at the very end:

"May I suggest to my friends on the left that they permit themselves the happiness of ignoring imaginary threats?"

Yeah, well, maybe Swaim, when you cease soft soaping the real threats and gaslighting us into letting our guard down. Oh, and maybe lay off the MJ candy before submitting your next pile of WSJ fish wrap.

See Also:

WSJ Troll Barton Swaim Implodes Again With Torrent Of Lies About Trump, Right Wingers and The Insurrection 

Tuesday, June 27, 2023

My Deep Sub Dive To See A Long Lost Ship Wreck - No, Not The Titanic (Includes Photos)

 One of our most memorable marine adventures occurred in April, 2016 in Barbados when we took the mini-sub Atlantis III to a depth of 155 feet to witness a wreck from over 65 years earlier.  The photos below - taken during the dive - show some of what we saw:

                             Atlantis III ready to get under way.

                                Captain Griffiths prepares to take her down
                                  Inside the sub - peering through portholes
                                          Some of the marine life we saw near coral reef
                                     Coming up on the wreck 155' below ocean surface

                                   Another view of ship wreck from some 65 years ago

                             Yet another view of the ship wreck

The discovery of wreckage from the Titan submersible last Thursday on the North Atlantic seabed reminded us of our own much shallower (but arguably much safer) sub dive.  The difference is we had no qualms about going down given the Atlantis III  had already made hundreds of such depth dives compared to less than 20 for the Titan. Also, the pressure at the depth we settled at (about 130 feet) was vastly less than for the submersible, e.g. 1,911 lbs. per square inch, vs. nearly 6,000 lbs. per square inch.  So the Titan's 5 passengers were in vastly greater jeopardy with a far more suspect vessel.  (See first link under 'See Also' - below).

 Those five occupants likely were lost the first day and barely 6 hrs. into the dive - not even halfway to the final depth.  This based on high tech U.S. Navy sonar (developed after WWII)  which captured an "acoustic anomaly" at the time, e.g.

The US Navy detected an implosion Sunday and told rescuers, an official tells CNN | CNN

And from which point all communication was lost. In other words the extended 4- day search saga was more a case of eyeball-grabbing performative TV drama than actual hopeful purpose, i.e. in rescuing anyone.

In the era of 24-hour news, few events grabbed the public imagination (and attention) quite as firmly as a real-time "rich people-in-peril story".   The sinking of the migrant ship Adriana in the Mediterranean, not so much. Too many perishing at once, and besides they weren't doing a deep sea dive to get to the Titanic.

Beyond all that, the prospect of these extreme "bucket list", high cost ($259k each)  tourist adventures calls into question their innate wisdom - no matter how much the  adrenalin junkies want them.

And it’s hard to imagine a more extreme or unpleasant peril than being suddenly imploded nearly 1 1/2 miles deep in the north Atlantic in a craft the size of a minivan.  In this case he design of the Titan was also suspect. Thus, the elongated shape - which would be fine in an actual submarine - like we were in -  is undesirable for mega-deep dives,  given pressure loads increase in the midsections.  This increases fatigue and invites disaster after repeated deep dives - which increase stress and fatigue.

The sad aspect is the Titan's passengers were likely doomed from the moment they set foot into the tiny submersible - built using carbon fibers instead of more enduring steel like in real subs. They'd have been better off going to Barbados and taking the Atlantis III to see the ship wreck we visited. But alas, they were obsessed with the 111-year old mystique of Titanic

Now we learned last night (NBC News), the U.S. plans to launch a full maritime investigation into the disaster. It ought to begin with why the early 'acoustic anomaly' was basically ignored in favor of a multi-million $ search. When the assorted powers-that-be knew it was all over- except for the final location of the imploded debris field.  Respect for the deceased and dearly departed?  What about those hundreds of migrants lost on their sinking vessels? Anyone shedding tears for them?

See Also:

Son’s Warning About Safety of 'Titan' Sub Saved Him and His Father (


'That could've been us,' say father and son who pulled out of doomed Titan trip out of safety concerns | CNN


by Sonali Kolhatkar | June 28, 2023 - 7:22am | permalink


A hundred and eleven years ago, a ship considered to be “unsinkable” fell to the bottom of the ocean after hitting an iceberg. More than a thousand people died as the Titanic sank into the icy waters of the Atlantic Ocean, among them the engineer who had warned against cutting corners and safety standards.

The parallels between the Titanic and the Titan—the tiny headline-garnering submersible that was lost while trying to view the sunken ship’s remains—would be laughably striking were it not for the fact that lives were lost.

Whereas the White Star Line company was focused on building luxury liners for the very wealthy, OceanGate, the company that owned the doomed submersible, runs a marine tourism business for people willing to pay $250,000 to dive more than 12,000 feet into the ocean inside a cramped tube to see the ruins of the Titanic.

 OceanGate CEO Stockton Rush, who was piloting the submersible and is presumed dead along with his four passengers, once boasted about “breaking rules” to engineer a craft made of carbon fiber instead of steel to withstand the pressures of the deep ocean.


by Dan La Botz | June 24, 2023 - 5:58am | permalink

— from Foreign Policy In Focus


The sinking of two vessels—the Andrianna, filled with hundreds of desperate migrants, and the Titan, with a handful of multi-millionaires—provides a vivid picture of the world today. All drowning deaths in the ocean are tragic, and one has to sympathize with the families who have lost loved ones. Yet these events also dramatically demonstrate global economic inequality and injustice.

Those who died on the Titan have names. Stockton Rush, chief executive and founder of OceanGate, was the pilot of the Titan. Hamis Harding was a British businessman, chairman of Action Aviation based in Dubai, and an explorer. Paul-Henri Nargeolet was director of underwater research for RMS Titanic, Inc., an American firm that owns the rights to the wreck of the Titanic. And finally, Shazad Dawood and his 19-year-old son Sulem Dawood were scions of one of Pakistan’s wealthiest families.


by Amy Goodman and Denis Moynihan | June 26, 2023 - 5:45am | permalink