Wednesday, August 31, 2022

Solving Differential Equations Using Clairaut's Equation

 Consider the equation:

  p2   +2xp - 2y = 0

How would you solve it for y, i.e. obtain general solutions for y but with no p in the answer?  Note that it cannot be solved in direct algebraic fashion.  For one thing there are 3 variables (x, y, p) but only one equation.  The only algebraic step is the first in which we write:

y  =   ½ p2   + xp 

Then, we need to differentiate with respect to x:

dy/dx = p =   p  dp/dx  + x dp/dx +   p   

But note: 

p  dp/dx  + x dp/dx  = (p + x) dp/dx = 0 

 It follows from this that either:

dp/dx = 0   or  p + x = 0

Hence:   p = c    or p =  -x  

Eliminate p between the last equations and the soln. for y yields:

y =  ½ c2   + xc

y =   ½ x2    -    x2  =  -½ x2  

Both of which satisfy the original DE.  Note, however, that the latter solution contains no arbitrary constant and therefore is not the general solution.  It is, however, a singular solution.

Fortunately, a method exists to solve differential equations which do not contain x or y explicitly.  These are of the form:

y = px + f(p)

The technique was first developed by Alexis Clairaut (1713- 1765) and is called Clairaut's equation.  For the foregoing equation if we differentiate with respect to x we get:

p  =  p + [x +  df/dp] p'

Or:   [x +  df/dp] p'  =  0

Then either:

i) p' = dp/dx = 0

or  (letting df/dp = f'(p):

ii) x + f'(p) = 0  

The solution of eqn. (i) is:  p = c (a constant)

After substituting this into the original equation:

y = cx +  f(c) 

 which is the general solution to Clairaut's equation.  In effect to obtain the general solution to Clairaut's equation it is only necessary to replace p by the arbitrary constant c.

Now consider eqn. (ii). If we solve this for x we obtain:

 (iii) x = - f'(p)

Considering this in tandem with Clairaut's equation we get:

(iv) y = f(p) - p f'(p)

The equations (iii) and (iv) yield parametric equations for x and y in terms of the parameter p since f(p) and f'(p) are known.  Note: If f(p) is not a linear function of p and not a constant then equations (iii) and (iv) form a solution of Clairaut's equation which can't be obtained from y = cx +  f(c)  and is therefore a singular solution.

Example problem: Solve

y = px + 2p

 Clairaut equation so we can write a general solution directly:

y = cx + 2c2

So the singular solution is found in parametric form:

f '(p)= df/ dp = 4p 

And by eqn. (iii):

x = - 4p  


y = px + 2p2

= -4p2  +  2p=  -  2p2

We can then eliminate the parameter p whereby:

y= -2p2    =  -2 (-x/4)2     =  - x/ 8

Suggested Problems for Math Mavens:

1) Solve:  p2 x - y  = 0

2) Solve: p2 + 2y  - 2x  = 0

As GOP Vows All Out "Investigations" Biden's Team Needs To Prepare If Voters Get Mesmerized By Misinformation


    One of biggest threats to democracy - Doug Mastriano.  This Toad wants to eliminate 8.7 m voter records and have all Pennsylvania citizens re-register.  The maggot also helped send busloads of insurrectionists to the Capitol on January 6th and marched with them.

 "A party that celebrates Hungary’s illiberal Viktor Orban, embraces Russian propaganda and grovels before Turkish strongman Recep Tayyip Erdogan cannot complain when it is held accountable for its right-wing nationalism and authoritarian hero-worship."-  Jennifer Rubin, The Washington Post, 'Biden Channels Harry Truman'

According to a WSJ piece yesterday (p. A4,  GOP Plans Probes if It Wins House):

"House Republicans are preparing across-the board investigations of the Biden administration if they take control next year, with an ascendant pro-Trump wing pushing GOP leadership to take an aggressive approach when the new Congress convenes.

Lawmakers say planned probes would focus on the southern border, where illegal crossings have surged and agents by mid-August had made a record 1.82 million arrests since the start of the fiscal year on Oct. 1; the foreign business dealings of President Biden’s son, Hunter; and the origins of the Covid-19 virus and pandemic policies that shut down workplaces—all hot-button issues for many conservatives.

Also on the list are the Justice Department’s operations under Attorney General Merrick Garland, who infuriated many GOP lawmakers when he authorized the application for a warrant to seize records with more than 700 pages of classified material that former President Donald Trump took to his home in Florida.


The potential GOP probes could cast an even wider net, taking aim at some of the Biden administration’s top officials and creating a tense political backdrop for the 2024 election. For many Republicans, one goal is to attempt to push Biden officials out of office."

And most alarming:

"At a Conservative Political Action Conference event, Mr. Jordan cast his planned probes as helping to:  'frame up the 2024 race, when I hope and I think President Trump is going to run again. And we need to make sure that he wins'.”

This, of course,  is a warning shot to sentient voters over what the future portends if these power mad reprobates take the House. It is also a 'call to arms' (voting arms, not Molotovs or ARs) for serious voters to protect this nation from thugs, Trump cultists and conspiracy mongers  who have zero interest in responsible governance. The Reep loser cult cannot be allowed to prevail, thereby setting up for a return of the traitor demagogue Trump in 2024.  Apart from that, we know that letting the Reeps control congress will set off the demolition of any coherent and competent governance.  Voters need to grasp the GOP is no longer a political party with definite policy positions, but a deformed personality CULT.  They must not allow themselves to be mesmerized into believing the GOP can be a responsible alternative for governance.

We've actually known even before the publication of this WSJ piece (in the regular news section, not Op-ed) that a long list of House Republicans,  from House Minority Leader Kevin McCarthy to Rep. Jim Jordan of Ohio to House Republican Conference Chair Elise Stefanik   plan to wreak havoc if they should regain power in the midterms.  This will be via dozens of nutso, bogus, 'payback' investigations. All sucking time and energy from genuine governance, while eating up taxpayer money. 

Pissed over high prices at the grocery? The last thing you want to do is make your life a hundred times worse by giving political power to these misfits and Trump idolaters in the GOP cult.  Women will see efforts initiated to outlaw abortion nationally, not just in red states. Young college grads - and other older loan recipients - will see aggressive efforts to use a Supreme Court ruling (WVa v. EPA) to dismantle the Biden student loan forgiveness executive action. Oldsters will see attempts to activate Sen. Rick Scott's plan to defund Social Security by slating it as a budget item every year, hence funding optional.  Oh, and struggling families will see their food stamps eligibility disappear as Reeps use the monies for more tax cuts for the rich.

As evident in the WSJ piece, nearly all of these GOP losers, traitors, grifters and imps  have vowed vengeance on everything Dem- from the January 6th committee - which exposed their venality and obeisance to Trump- to Dr. Fauci and even President Biden.  Given Dr. Anthony Fauci alone likely saved a half million lives via his staunch and capable Covid 19 policies, I want to recommend he diss and blow off any subpoenas issued by the likes of lowlifes such as Jim Jordan, Kevin McCarthy, Elise Stefanik or other Trump traitors, asslickers and seditionists.  Why? Because they won't be worth the paper printed on.

To pick on Fauci, as McCarthy and other imps have vowed, is to further prove the GOP's veering into mindlessness and insanity. (See the Mother Jones' piece, 'SWAG and Circuses' e,g.

Trump Merch, Rabid Fans, Disgraced Ex-Officials: Inside the Right-Wing Conference Circuit – Mother Jones

Part of the special issue on 'How The GOP lost Its Mind':

Let's be clear the deranged Right has sown numerous memes and tropes on how Fauci "misfired", gave bad advice or was otherwise derelict as the pandemic struck.  This is all B.S.  Despite Fauci's best efforts to inform Trump before pressers the demented Dotard inevitably messed up. He simply was unable to absorb complex information or communicate it. Worse, the orange imp would often take any kernel of real science given, convert it to wildly inaccurate remarks then spin it as some kind of national policy.  Dr. Fauci and colleagues would then often have to go into protective mode. See, for example:

by Heather Digby Parton | March 31, 2020 - 7:41am | permalink

Meanwhile bombastic blowhard Jim Jordan and Minority leader McCarthy especially resent Speaker Nancy Pelosi’s January 6th  select committee, which Trump toady Elise Stefanik has denounced as an “illegitimate partisan sham.”  This misinformed virago also claimed soon after the committee convened that the January 6th traitors were merely "expressing their political beliefs."  Uh no, you are not doing that if you defecate in Statuary Hall, carry nooses to hang lawmakers (especially Pence) and bring bear spray and clubs to disable DC cops.  Oh, and let's not forget about the AR-15s reported over police radios.

Had Stefanik and her traitor cohort been paying more attention (or been actual patriots instead of seditionists)  they'd see that J6  “sham” has presented a mountain of damning evidence against former President Donald Trump and his MAGA allies, including former White House aide Cassidy Hutchinson’s bombshell testimony, on June 28, that Trump knew some of his supporters were armed on January 6, 2021 and wanted to march with them to the U.S. Capitol Building anyway.

But regardless of what else the J6 special committee’s hearings reveal, a preponderance of 'pukes is enraged that the committee exists in the first place. Why wouldn't they be as it has exposed the Repukes' fascist underpinnings and especially their behavior as more like a personality cult than proper political party.  For this reason they yearn to get payback in terms of bogus subpoenas, bogus investigations and even efforts at bogus impeachments of Biden and even VP  Kamala Harris.  And given this, all such subpoenas issued by these rats - should they get unearned power - ought to be blown off as refuse, garbage.  

The very notion of putting Hunter Biden's laptop in the same political frame as the January 6th insurrection is a highlight for the logical crime of false analogy.  The mere mention of Trump's illegal NARA files seizures in the same sentence as Hillary's emails  (in a burst of GOP cockeyed 'What aboutism?') is an insult to every sentient American's intelligence. Hillary's emails, after all, contained material already published in numerous media venues, such as drone strikes.  Trump's seized National Archives files which we now know included SCI Top Secret documents  - were kept with magazines at his unsecured Mar-a-Lago hacienda.  

God only knows how many bad actors, foreign threats already had access to them. Or which staff may have photographed them. Don't buy it? Think it's all histrionics? Go back to 2019.  Yujing Zhang, a 32-year-old Chinese woman, was arrested after evading security at Mar-a-Lago. She was carrying five cell phone SIM cards, a hard drive, nine USB thumb drives including one primed with malware, and a device that could detect electronic signals. She had two Chinese passports, and investigators found a device for detecting hidden cameras when they searched her hotel room.

These pro-Trump Turds are so beset by power mania they hope that beating the "inflation" drum endlessly will fuck just enough voters' brains to sneak them back into the House majority. Journalists Alayna Treene and Jonathan Swan, in a report published by Axios on July 6, actually telegraphed some ways in which House Republicans are hoping to take revenge against the January 6 committee if they regain control of the U.S. House of Representatives in the 2022 midterms.  The WSJ piece yesterday confirmed it and worse, showed it's a setup for preparing for a Trump return in 2024.

On All In last night the background and plans of another GOP degenerate were highlighted, this time Doug Mastriano- see image at top-  running against Josh Shapiro for PA Governor. As noted, if this slime wit wins he plans to remove 8.7 million voting records left from 2020 and have all those citizens register again. In addition, he plans an anti-abortion dictate that will prevent any procedures even in case of rape, incest or ectopic pregnancy. This Trump-sucking traitor also helped organize busloads of insurrectionists into D.C. on Jan. 6, 2021. That the cockroach copuld even be polling a 44% compared to Shapiro at 47% is appalling. It shows too many in the Keystone state either aren't paying attention or have imbibed the Trump kool aid.

Now that we know what they plan to do, and how they intend to use their power, voters have the obligation to prevent it on November 8th.  Just as Trump was prevented in November, 2020 from gaining a 2nd disastrous term, the Trump cult itself must be prevented from gaining a majority in either House OR Senate in 70 days!  Biden is right when he prefigured his speech tomorrow evening saying that "democracy is on the ballot."  People had better believe it.  

If mass stupidity or political blindness does seize voters on Nov. 8, or too many Dems, patriotic independents sit out, it may herald a GOP takeover of congress.  Never mind. It will be by a derelict loser cult which merits no respect.  Hence NONE of their bogus, threatened investigations deserves to be honored. Blow them all off, Dems! And tell them to eat shit and go to hell!

See Also:

by Joan McCarter | August 17, 2022 - 7:56am | permalink


The immediate reaction from Republicans to finding that their hero had been storing extremely sensitive government documents in his Mar-a-Lago basement was not concern about national security, but to declare war on the Justice Department and Democrats.

With democracy already having a tenuous hold, the Republicans fury at their leader being subject to the rule of law like everyone else, this year’s midterms elections are everything. Not only could a Republican Congress block every legislative effort from the Biden administration, and block every judicial or executive nominee, and tie up the administration in dozens of phony hearings, they could gut the department and punish individual Justice employees.


by Jeff Cohen | August 30, 2022 - 4:52am | permalink


by Thom Hartmann | August 31, 2022 - 6:00am | permalink


by Joan McCarter | August 30, 2022 - 6:50am | permalink

— from Daily Kos


The predicted rout by Republicans over Democrats in the House this midterm is becoming less of a sure thing by the day. New polling from both CBS and NBC shows that support for Republicans in the generic balloting is falling as potential voters continue to be motivated by the U.S. Supreme Court’s decision to end federal abortion rights and the ensuring battles in the states...

The CBS News tracking poll has Republicans on track to win 226 seats, down four from last month. NBC News polling reinforces that, pointing to persuadable voters as the difference. It gives Democrats a 3-point advantage with those voters, 40%-37%, a flip from the first half of the year when Republicans were consistently up generically by 6 points (39%-33%). That goes along with an uptick in President Joe Biden’s approval rating.


by Meaghan Ellis | August 3, 2022 - 7:42am | permalink

Monday, August 29, 2022

The Artemis Mission - Back To The Moon - But A Leap Forward In Space Exploration


        Artemis mission sling shot trajectory to round the Moon and back

                         Artemis on its Cape Kennedy launch pad earlier today.

Why on Earth go back to the Moon when we already went there?  Why should NASA repeat what it did a half-century ago, especially because astronauts will not actually step on the Moon for several years, and by that time, NASA will have spent about $100 billion?  Many reasons (Btw, the launch is postponed at least until Friday because of discovery of a fuel leak in the SLS rocket engine.)

First of all the Artemis lunar program is not simply a do-over of the Apollo moon landings from 1969-72.  Artemis itself is larger and more powerful (more thrust, at 8.8 million lbs.) than the Saturn V rocket that helped hurl Apollo astronauts to the Moon.  Further, Artemis' objective is to lay the ground work for a lunar base as a stepping stone to Mars - perhaps by the late 2030s.  It will also help establish a new space station ('Gateway') orbiting the Moon, which will aid that objective.

In Greek mythology, Artemis was the twin sister of Apollo. The program’s first step will be the upcoming test flight of the moon rocket, known as the Space Launch System, with the Orion capsule on top where astronauts will sit during future missions. 

The flight planned for this morning will feature mannikins wired up with sensors so that comparisons can be made to what actual humans would endure.  The flight plan - see graphic from Wall Street Journal - will feature a 70% more spacious (compared to Apollo's)  Orion capsule, and swing around the Moon before returning to Earth.    The total mission to last 42 days. This unmanned test flight is designed strictly to expose any issues with the spacecraft before putting people on board.  That includes a test of the heat shield given the capsule will be re-entering at a significantly higher velocity.

What really excites me about Artemis is the chance - I say chance, given I don't know how much longer I'll live -  to see the first woman and person of color step foot on the Moon. If I can keep the cancer at bay long enough perhaps, I will see this historic step.

But there are clouds gathering to do with cost overruns and funding with which to reckon. Appearing in a CBS Morning interview Aug. 24, NASA Inspector General Paul Martin is not convinced the mission will land on the Moon with a human crew before late 2026.  This contradicting NASA Director Bill Nelson who argued (in the same segment) that 2025 "is doable."  But according to Martin - see the clip below from his interview:

NASA's inspector general projects agency will spend $93 billion on Artemis through 2025 - YouTube

 “The culture of optimism results in unrealistic costs and projections."

 He cites two prime reasons for the delay: 1) the Technological complexity of the missions, and 2) Poor project planning.  Already there is a cost overrun of $40 billion and Martin estimates the total cost to be $93 billon through 2025.  Now, while this sounds like a lot let’s bear in mind each year the Pentagon gets $800 billion.  And it still can’t account (in spending) for $2.2 trillion amassed during the 1990s. Despite such financing mischief, one reads on page A17 of the WSJ op-eds today a piece ('Restore Reagan’s Military ‘Margin of Safety’) from -  Roger Zakheim of the Reagan Institute.  He actually wrote:

"To meet this moment, we need defense investment along the lines of what the Reagan administration pursued: roughly 5% to 6% of gross domestic product annually. To those who say we can’t afford a buildup without sacrificing our prosperity, Reagan’s response from four decades ago still rings true: “Our government must stop pretending that it has a choice between promoting the general welfare and providing for the common defense. Today they are one and the same.”

 I can promise anyone reading this post that if lawmakers gave in to this insanity - as they might if voters are dumb enough to put Reeps back in charge of the House - you can kiss Artemis' Moon landing mission goodbye. As well as Social Security, Medicare and Medicaid. 

The first four Artemis launches will cost $4.1 billion each which is not sustainable in any environment with the current level of defense spending.  As Paul Martin pointed out in his interview with Mark Strassmann:


At $4,1 b per launch you’re going to be limited to only one launch every other year. That cadence of launches will not  achieve NASA’s mission of having a permanent presence on the lunar surface.

Meanwhile, Charles Duke Jr., a former NASA astronaut who spent more than 20 hours on the lunar surface during the 1972 Apollo 16 mission- said the coming launch represents a change for an agency that had become risk-averse.  This is especially after the Challenger and Columbia Space Shuttle disasters.

Artemis for sure will carry its own unique risks, but that is the nature of space exploration and crossing new frontiers in general.  Moreover, it provides for the capacity of our species to move beyond its home world.  As Isaac Asimov said in one of his lectures:  "It makes no sense to keep all our 'eggs' on this one small planet which could be annihilated with one major asteroid impact.  Ask the dinosaurs."

So Paul Martin may be correct in his assessment of the costs, but I say if we can afford to spend $800 billion a year on defense we can afford to spend a small fraction of that ensuring an opportunity to give Homo Sapiens a path toward exploring space. And even - perhaps settling new worlds.

See Also:

by Brandon Gage | August 29, 2022 - 6:00am | permalink


Solutions To Abstract Algebra Problems

 (1) By the Euclidean (division) algorithm there exist numbers q, r ∈  Z

Such that < r < n,   and a is of the form:

a = q n + r

By definition of   [a ]  mod n  =   [a]

 r     [a] ,   And:   r = a + (-q) n

  (2) Let Z be the integers.  The ideal:
           (5)  =   {5 j:  j   Z }

Show all the congruence classes with respect to this ideal.    

(Hint:  [a] =  {a + j: j ∈ [(5)}  = {a +  5j:    Z } )

The congruence classes are:

[a]   =  {a + j: j  ∈  (5)}  =  {a + 5j:   Z }

[0]  =  {0 + 5 j: j  ∈  Z}  =  { 5j:   Z }  =  (5)

[1] =  {1  + 5 j: j  ∈  Z}   =  {1, 6, -4, 11, -9}

[2]  =  {2  + 5 j: j  ∈  Z}   =  {2, 7, -3, 12, -8}

[3]  =  {3  + 5 j: j  ∈  Z}   =  {3, 8, -2, 13, -7}

[4]  =  {4  + 5 j: j  ∈  Z}   =  {4, 9, -1, 14, -6}

[5]  =  {5  + 5j: : j  ∈  Z}   =   5 (j+ 1) : j  ∈  Z}   =  [5] = [0]

3) Using set notation we may define:

·  B  =  {x ·  y: x ∈ A,  y ∈ B}

+  B  =  =  {x  y:  x ∈ A,  y ∈ B}

4) Take S as the set of integers, Z. Let the ideal I = (2) so that S / I =  Z 2     Thence or otherwise, find:

a (a)     [0]     b)   [1]     c) S/ I  =  Z 5       


Take S =    Z.  I = (2)  so that S / I =  Z 2    

a) [0] =  {0, + 2, + 4, + 6…..}  =  I = (2) 

b) [1]  =   {1, 3, -1, 5, -3, 7….}

c)If S / I  =  Z 5       

S/I =  {[0], [1], [2], [3], [4]}

5 ) Show every ring S has two ideals: S itself and {0}.

Every ring S always has at least two subrings, namely S and the zero ring,  S 0.

Further, if S is also a  field then the only ideals in S are S and {0}.  Further:

 if 1    I  then I = S. (Let I = S be an ideal,  I = {0} is also an ideal. )

 (In a field F, the only ideals are 0 and F)

Basically, an ideal of a ring S is an additive subgroup a  of  S with the property that:
a    S    and a    a    imply  ra    a  .  Clearly then the set containing the single element 0 and the set containing the whole ring S are ideals.

6)  Is Q [x] /x2  - 6 x  + 6 a field? Why or why not? 

Ans.   YES.  x2  - 6 x  + 6   is a maximal ideal of    Q [x] 

7)Let (G) and (H) be groups. Then a homomorphism of (G) into (H) is a map of the sets G and H which has the following property:  f(x o y) = f(x) o f(y)

 Let G = (0, 1, 2, 3) for the operation (+) which is addition in Z4

Let H = (2, 4, 6, 8) for the operation (·) which is multiplication in Z10

Prepare the respective tables for the isomorphism and give specific examples in terms of the function φ, i.e. show specific mappings.  (Where: φ(x) φ(y) = φ(xy) for example)


The addition table for Z4 is:

+ /----0 ----1 ---2 ----3
0 --- 0------1 ---2 ----3

1 -----1 -----2 ---3---- 0

2 -----2 -----3--- 0 ----1

3----- 3 -----0 ---1---- 2

The multiplication table in Z10

-·--/----6 ----2 -----4 ----8
6 --- ---6------2 --- 4 ----8

2 ------2 ------4 ----8----6

4 ------4 ------8--- -6 ---2

8----- 8 -----6 ----2---- -4

Since 6 · 6 = 6 then the identity element in  Z10  is 6.. 

So φ(x) φ(x) = φ(x2) = (φ6)φ(6) =φ( I)

Then the mapping isomorphisms between the tables will be:

Φ: 0 -> 6 (identity to identity element)

Φ: 2 -> 4

Φ: 3 -> 8

Φ: 1 -> 2

So each of the numbers on the left side of the arrow is mapped to the corresponding number on the right side, and this is done using the two tables such that each element of Z4  is mapped to each element of Z10