Wednesday, August 26, 2026

Introducing A New Series: Reviewing Algebra II (Part One Algebraic Fractions)

 Algebra II can be said to be the basis for successfully moving on to more advanced mathematics, whether Calculus, differential equations, linear algebra, numerical analysis or abstract algebra. It certainly is also the foundation for getting anywhere in the STEM field, and especially the quantitative sciences like astronomy, plasma physics, as well as astrophysics and quantum physics. Finally, delving into this background will also equip readers to work more of the Mensa algebra puzzles, problems!

 With that in mind, I decided to start this brief  (not exhaustive) series in Algebra II review with algebraic fractions - often as important in Calculus problems, i.e. introducing partial fractions in a problem,  as in Algebra. Future topics will include: solving exponential equations, radical-surd equations, complex & imaginary equations and solving simultaneous equations.

Basically, an algebraic fraction consists of one algebraic expression divided by another. If numerator and denominator contain similar factor (e.g. x, y, z) the algebraic fraction can be simplified. We say the algebraic fraction has been reduced to lowest terms if both numerator and denominator have no common factors.

Example:   Simplify:

(x + 2) 2  / x2  - 4

Factor numerator and denominator:

(x + 2) (x + 2) /  (x + 2) (x - 2)

Cancel out  (x + 2):

Þ   

(x + 2) / (x - 2)

Example 2:  Simplify:

x2  - 3x - 10/ 2 x2 +  8x + 8

Factor numerator and denominator:

(x + 2) (x - 5)/ 2(x + 2) (x + 2) =  x - 5/ 2 (x + 2)  

Note that a negative sign applied to an algebraic fraction can be considered either in the numerator or denominator.  Expressed in a general form:

- x / y  =  x/ - y

Example 3:  Simplify:

 a2  - 4/ 2 - a

Factor the numerator, cancel common factor:

a2  - 4/ 2 - a  =   (a + 2) (a - 2)/ - (a - 2) = (a + 2)/ -1

(Applying the negative sign rule)

Þ   

 - (a + 2)  OR:    - a - 2


Example 4: Simplify:

2x2  - 7 x + 6 / 3 + x  - 2x2

Factor numerator and denominator:

Þ   

(2x -  3) / (x - 2) /  (3 - 2x) (1 + x)  =

(2x -  3) / (x - 2) / - (2x - 3) (1 + x)

Note change in denominator from (3 - 2x) to - (2x - 3)

We can then cancel the  (2x - 3) factors in numerator and denominator:

Þ   

- (x -  2) / 1 + x   =  2 - x/ 1 + x


Practice Problems:  

Simplify each of the following:

1) x2  - y2y2  - x2

2)  27 a 3 x2y5  / 36 a 2 x4y4

3) 2a 1/2 (x + y)/  5 x a 3/2

4) (x -1) (2 - x) (3 - x)/ (1 - x) (x - 2) (x - 4)

Yes, The Data Shows We're Now Living In the Climate of 2037 - Thanks To The Ramping Up El Niño

 

                     Totally barren, hothouse world in 2106: no water, unrelenting heat, fires

                                                                         
                       David Suzuki - who predicted the first 'Year of No seasons' by 2040.


In an extensive NY Times Saturday article (Europe’s Great Rivers Are Dying. The Continent Is In for a Shock) we learned:

"All across Europe, shipwrecks and skeletons are rising from the deep.

Near Prahovo, Serbia, Nazi ships scuttled in 1944 now lie broken-backed on sandbanks. In Croatia, a 1930s cargo ship protruded from the water. On the newly exposed riverbed of the Danube in Bulgaria, what local experts say is most likely part of the skeleton of an ancient mammoth was found scattered across the sand, like discarded bones after a giant’s picnic.

Much of Europe is in the middle of a severe drought, and major rivers have been reduced to a fraction of their usual size. Europe’s second-longest river, the Danube, is now facing record lows, recently measuring just 9 inches deep in Budapest. Part of Italy’s mighty Po River is also at a record summertime low. And the Rhine, a major outlet for trade flowing from the Alps to the North Sea, is now shallower than at any time since records began."

Prompting the question: what is going on?  Clearly, it can't be normal given just 9 years ago - when we visited Budapest- the Danube's level was more than ample to convey barges and boats, e.g.


  This was when we enjoyed Hungarian fish soup at a cafe on the river

 

Meanwhile, in the Sunday Denver Post the lead headline was: 

"Drought has Colorado Burning Up" - 

referring to the dozens of fires that have raged in the state this year, including many still going on. The biggest the Aspen Acres fire, only 72% contained.  And, last but certainly not least, there are the record breaking temperatures this summer raising havoc around the globe. True, the abnormal  El Niño is playing a role, but it's also acting as a climate time machine giving us a preview of what it will be like in 2037.

That’s because this El Niño is temporarily pushing global temperatures to levels that might not otherwise be reached for another decade. According to an analysis by climate scientist Zeke Hausfather that was shared with The Washington Post, there’s around a 95 percent chance that 2027 will become Earth’s hottest year on record — soaring past the last record in 2024 — because the combination of El Niño and projected climate change (owing to increased CO2 concentrations) could spike global temperatures to around 1.76 degrees Celsius (3.17 degrees Fahrenheit) above preindustrial averages. This value would be so far above the warming trend of the past decade that it could bring global temperatures to levels not otherwise expected under that trend until around 2037.

Recall  (from my Feb., 2017 post) researcher and author David Suzuki, in his book It's A Matter of Survival (Harvard University Press, 1991) co-authored with Anita Gordon, predicted the first 'Year of No Seasons’ by 2040. Bear in mind Suzuki based his projections on 1991 data and the hope we'd be able to keep the warming increment to less than 2C by 2100. With the advent of Trump and massive return to fossil fuels, this isn’t going to happen.

Suzuki and Gordon wasted no time bitch-slapping the complacent reader from the first words of Chapter One: Beyond Your Worst Nightmare:

"A.D. 2040- If we were to give this year a name it would be Despair. This is the hopeless world we have left our children and grand children. Where once our lives were measured and enriched by the cycle of the seasons, there is now only searing heat and the certainty it will get hotter. Seasons exist only in the nostalgic longing of those of us old enough to remember the richness of life......”

On p. 8 of the same chapter, Suzuki and Gordon go on to link the total disappearance of the seasons worldwide by 2040 to global mean temperatures 5 C (9F) above what they were in 1991. Given that Carl Sagan once forecast an increment of 6C as adequate to trigger the runaway Greenhouse, Suzuki's numbers would put us right on the cusp. Once that begins, and the methane is also released from melting permafrost and later CO2 is released from all the carbonate rocks in the world, we are for the high jump.

Bottom line: Even factoring in the effects of the El Niño it appears we are getting a preview of what the climate conditions will be like in 2037, with forecasts already made (as far back as 1991 by Suzuki) that it will either be in the first year of no seasons or close to it - say be 2040. 

But more current research indicates the previews of future decades may also be feasible, and they are not sanguine. Specifically, in January 2021, a paper published in Frontiers in Conservation Science laid out the dismal details. According to the authors, we are already traveling down the path of a sixth mass extinction. The primary factors driving it include: loss of biodiversity, climate disruption, overconsumption, and population growth.  These are not future threats but happening now, pushing our civilization toward its end.

The fact is that two serious warnings have already been published: one by David Suzuki and Anita Gordon in 1991  in their popular book, It's A Matter of Survival.  The other appears in a paper in a scientific journal, written in careful academic language, and it says something no headline wants to print: 

Civilization as we know it may not survive this century.

And here is the sentence that should have stopped the world: the process may already be irreversible. Even if every emission ceased tomorrow, the climate emergency would continue. The machine is already running and will continue doing so.  I already elaborated on this process in multiple previous posts.

Even if all CO2 emissions halted it would not make a dent in the long term accumulations already in the atmosphere.  The process may be described like a series with terms being added, viz: to describe the CO2 content now in the atmosphere,  and added henceforth. We initiate the series with n= 1 (for 1926), viz.  CO2( 2026) =   x  1  + x  2 +  x  3 +   x  4 +.............+  x  1 00 

Terminating at the last term 100 years later. Here each ‘x’ denotes the CO2 burden added for each year in succession.

Thus, the CO2 effect for a given year is not just for that year, but rather inclusive of the cumulative additions for all the years - starting up to 100 years before.  Needless to say this series continues 100 years from now, into the future. 
Using this series model it is possible to capture climate 'snapshots' decade by decade, degree by degree, all the way to 2100.

The starting point is not some distant future. It is today. ( CO2( 2026))

TODAY – GLOBAL TEMPERATURE + 1°C

Look around at the effects:. Earthquakes, floods, hurricanes, bushfires, and a global pandemic have battered humanity within a single stretch of recent years, 2016 to 2o26.  Further, our ecologically unsustainable way of living makes more deadly pandemics almost guaranteed.  The planet’s surface has already warmed by 1 degree Celsius. The top 100 meters of the ocean have heated by 0.33 degrees, which sounds insignificant until one sees the body count: roughly 30 percent of the world’s coral reefs are already dead and dragging marine biodiversity down with them in a chain reaction. 

Add to this the ice sheets are shrinking.  China has been able for the first time to send ships across the Arctic to save time in global delivery, given the extent of melted ice at the top of the world.  Meanwhile, global sea levels have climbed 20 centimeters. Floods - like those now inundating Indiana and Ohio- strike more often, creeping into coastal communities. And in the animal kingdom, populations of mammals, birds, amphibians, reptiles, and fish have crashed by almost 70 percent.

Add to that, pollution and consumerism poisoning the soil, water, and air that sustains us. Let's also add one more detail making everything worse: the richest 1 percent now holds half the world’s wealth, and that staggering inequality keeps stalling the drastic measures that could still change the ending.

Add a half a degree more in 10 years, and the ocean starts to empty. At 1.5 degrees of warming, 70 to 90 percent of all coral reefs die, taking roughly 25 percent of marine life with them. The fish will be dead next as well as marine mammals like dolphins, leaving only jellyfish.

On land, the rising heat and accompanying drought will destroy potential harvests, threatening the loss of millions of tons of maize, wheat, and soybeans. The people who depend on fishing and farming face famine first, and they do what humans have always done when the food runs out. They leave, triggering waves of mass migration.  We will be in much the same position as in 1877 when an aberrant El Nino was responsible for the deaths of 50 million, i.e.

https://www.msn.com/en-us/news/other/disaster-that-killed-50-million-people-in-1877-likely-to-happen-again-in-2026/vi-AA23P2PE?ocid=socialshare


Scientists Warn That Civilization Could Collapse by 80 Years From Now

 

Thirty years on Earth’s population reaches 10 billion mouths, just as the planet’s ability to feed them buckles. The coral reefs and their ecosystems stand at the edge of total annihilation,

                                    Dying coral reefs in 2016

 Meanwhile, beyond the oceans, thousands of other species begin sliding into extinction. The warning is no longer a paper in a journal., it's the evening news, every night. Worse and worse news: Ochre skies redolent with smoke and ash, from prolonged blazes, heat indices hitting 135 degrees in most cities, lost crops, power grids collapsing - as well as rivers, lakes dried up.

80 years from now, by 2106, the seas swallow the coastlines, driving hundreds of millions of people inland. Nations strain and snap under the flood of refugees, and mass migration ignites conflicts, possibly wars. Food and clean water grow so scarce that the human population itself begins to collapse. The catastrophe strikes the poorest regions first, then spreads everywhere, feeding civil wars and global chaos.

What would the planet actually look like in that final act?  Humanity’s monument would be ruins and garbage: endless stretches of abandoned civilization: empty mounds of concrete and mountains of plastic that will not break down for thousands of years. The cities dying the moment maintenance stopped. Without workers running the pumps, the subways of a metropolis like New York drown within hours.

The energy sector turns from lifeline to threat, as unattended nuclear plants leak, and potentially implode like Chernobyl, poisoning everything around them. Scientists Warn That Civilization Could Collapse by 80 Years From NowFarmland falls to swarming locusts and other pests the moment the pesticides stopped. Then, year by year, the buildings themselves would surrender: glass shattering first, metal rusting and crumbling next, while stone endures, weathering slowly across millennia like tombstones for a species.

Earth itself will need millions of years to heal. And it would. Eventually, it would grow back greener, wilder, and more diverse than ever. Just without us in the picture. Scientists still disagree about how soon, how severe, and whether collapse could have been avoided. But no one will deny the reality.

Just look out your window at the fires, the floods, the heat records falling year after year, then ask yourself: Does it really sound so far off? Not if El Niño is already giving us a preview of life in 2037.

The warning has been on the table since 2021. The choice between drastic action and collapse by 2100 is still ours, but not for much longer.

 See Also:

by Bill McKibben | August 15, 2026 - 4:44am | permalink

— from The Crucial Years


Last time I marked the fifth anniversary of this newsletter with a remarkably heavy burst of analysis—a “where do we stand” summary that let me speak at the highest possible level about our past, present, and future. So of course I’m worn out, and perhaps you as well. We’re going to proceed less systematically this time—I’m going to show you some numbers that have arrived in the last few days, and try to make sense of each of them, with only slight regard for how exactly they fit together.

And it’s all a useful reminder that though the climate and energy crises have everything to do with psychology and sociology and economics and political science and theology and love and art and winter, they are rooted in quantifiable physical fact, which cannot be wished away and which must be faced resolutely.

So, to begin:

» article continues...

And:

Scientists warn end of civilization in 'decades' with just 10 percent chance of survival

And:

UN Climate Framework Withdrawal By Trumpers Puts The Planet On Path To Runaway Greenhouse Effect

And:

Colorado River projected to deliver a third of normal water supplies

And:


 


Tuesday, August 25, 2026

Mensa 3D Circle Geometry Problem

 



An interesting problem which combines geometry (in 3 dimensions) with algebra.


The Problem:  A cube has a side of length s. A blue circle is inscribed on one face of the cube. Another red circle is circumscribed about a neighboring face.

Find the minimal distance between any point  on the first circle and any point on the second circle.

Solution To Triatomic Molecule Problem

 Problem:

For each of the frequency solutions (h 1  etc. )applied to the linear tri-atomic molecule, give the relation between the normalized amplitudes: A1, A2 and A3 and the angular frequency.

Solution:

The three solutions for the angular frequency were already given in the Aug. 18 blog post, from the secular determinant:

(k - w 2 m    …-k…   . .0)

(-k …..2 k - w 2 m  ..  -k)

(0…. ….-k….. k - w 2 m)


I.e.

w1  =  0,     w2  = Ö(k/ m),   w3  = Ö{ k/ m (1 + 2m/ M)}


The amplitudes from these solutions are then obtained (as shown in the post) from the Lagrange tensor:

h 1  =   A1 exp (iw t)

 h 2  =   A2 exp (iw t)

h 3  =   A3 exp (iw t)

Monday, August 24, 2026

What If A Dem President Made Nice With A Communist Dictator And Threatened To Bomb A Mid-East Ally?

 

               Kim plays Trump to the hilt back in 2019, gets nukes in return

Hwasong - 17 North Korean ICBM capable of hitting U.S.

Trump acting in American interests? Is that a joke? His Communist buddy in North Korea is sending soldiers to Russia for the invasion of Ukraine and weapons to Iran. If a Dem were doing that, magas would be screaming 'treason'.- WaPo Comment

Trump made the biggest unforced error in American Presidential history by attacking Iran senselessly and incoherently. He’s the most incompetent President in American history. Our enemies couldn’t be happier with him. The sooner he’s gone, the better off we’ll all be. .- WaPo Comment


After seeing Trump's antics last week- or more accurately, treasonous betrayals - such as giving North Korean dictator what he wants and threatening to bomb a Mideast ally (Oman). One must ask how the media (or the GOP)  would react if a Democratic president pulled such stunts. I will tell you:  A thunderous roar would erupt across the land, calling for immediate impeachment, or outright removal under the 25th amendment. 

Histrionic? Hysterical? Consider: Trump decided last week to scale back military exercises with South Korea, even as he described North Korea's communist dictatorship (100 times more malignant than Fidel Castro's) as "unthreatening and respectful" (WSJ, 8/18, p. A14). This even as Pyongyang is deploying ICBMs aimed at California (see above image of one of them, the Hwasong -17).

 How much more evidence does the rightist media (like FOX, OAN, the WSJ) need before they recognize either a traitor inhabits the White House, or a raving lunatic? All I see across the board is fuming and fidgeting over ‘democratic socialists’ (like me) and meanwhile a 5-time draft dodging Jan. 6th insurrectionist betrays allies as he rekindles bromance with the North Korean Dictator Kim Jong Un. Which began even earlier this year with a freeze on $14 billion of military aid to Taiwan - to make North Korean partner China happy.

No, you cannot make this shit up. It is as real as a heart attack and our nation is in this traitor (or lunatic's) maw.  As former Nation Security advisor John Bolton wrote in his Op-Ed Wednesday (Pyongyang’s Threat Rises, With an Assist From Trump, p. A13)

"President Trump put North Korea and its rogue nuclear-weapons program back in the spotlight on Sunday. The president ordered the Defense Department “to substantially reduce” this week’s annual U.S.-South Korean military field exercises to defend against a potential attack by the North. He made the same unilateral concession in his first term, a mistake that is even worse now.

At the 2018 Singapore summit to discuss Pyongyang’s nuclear-weapons program, North Korean leader Kim Jong Un criticized the annual defense maneuvers. As I wrote later, Mr. Trump agreed “that the exercises were provocative and a waste of time and money.” He decided that “there would be no exercises as long as the two sides were negotiating in good faith.” Essentially repeating these points in his Sunday social-media post, Mr. Trump wrote that the exercises were “costly” and “totally inappropriate and hostile” to North Korea, which has been “unthreatening and respectful” during his presidency."

Last weekend's WSJ editorial was even more blunt but still reluctant to follow the logic that this infernal orange fart either has to be a traitor or a certifiable loon - in either case unfit to hold the office he occupies.  Bolstering Bolton's Op-ed  the WSJ editors laid into his act of appeasement to North Korean Dictator Kim Jong Un ('Trump Romances Kim Jong Un Again, Aug. 18, p. A14) writing: 

"President Trump is nothing if not impulsive, and his latest brainstorm is to rekindle his courtship of North Korean dictator Kim Jong Un.  He announced Sunday evening - where else? - on his social media that he ordered the Pentagon to "substantially reduce military exercises with South Kirea" which were set to start on Monday.

Mr. Trump cited his 'very good relationship' with Mr. Kim, adding that any military exercises 'send a signa that is totally inappropriate and hostile to a country that as long as Donald J. Trump has been President has been unthreatening and respectful."

Which last words must have had the WSJ  editors gagging as they wrote them, given what actually transpired is that Kim had his minions constructing nuclear test facilities even as he was wining, dining and charming the orange buffoon during his last visit in his 1st term.  The clever subterfuge - based on the insight one only had to feed Trump's ego to blind him- enabled the North to get its nuclear weapons much faster and also make great strides toward developing an ICBM  (Hwasong-17) to deliver them. Not to mention submarine-launched missiles as well as land based.  Indeed, Kim fired a missile off just days before the planned U.S. - South Korean exercises, e.g.

Why has North Korea fired a missile before US-South Korea drills? | Weapons News | Al Jazeera

Which maybe explains why the WSJ editors were none too happy about Trump's moves writing:

"To the contrary pulling back from the exercises sends a message of weakening deterrence and readiness against a regime that has a history of threats and aggression against the South.  The North had protested as usual against the exercises and Mr. Trump's order is an act of appeasement toward Mr.Kim.

And what did he get for his appeasement and obeisance to Kim Jong Un?According to a Aug. 20, WSJ lead story (North Korea rejects Trump’s drills-cut overture as an empty gesture, p. A1)

“North Korea gave an icy response to President Trump’s decision to end joint Washington-Seoul military exercises early, calling it an ineffectual goodwill gesture toward the Kim Jong Un regime.

The drill reductions don’t merit interest or comment because nothing has fundamentally changed, said Kim Yo Jong, the dictator’s younger sister, in a Wednesday evening statement carried by North Korean state media."

Of course, a predictable response, given Kim & Co. already manipulated everything they needed out of Trump by feeding his ego. Plus they've wrangled plenty of technology and  equipment from new buddy boy Putin. (WSJ, 'Trump Tries Old Ploy To Woo Kim Jong Un', Aug. 18, p. A6).  Bottom line, Kim already got everything he needs from Trump and doesn't want any more. Oh, except finally being recognized as another nuclear state. (If Trump accedes you can be certain Iran won't relent until it also joins the nuclear club.)

Now  add to all that Captain Bonespurs' other outburst, after he vowed to "bomb the shit out of Oman" (a powerful Middle East ally) if they "get in the way" of his machinations.  According to The Journal (p. A1, Aug. 18, 'Frustrated Over War, Trump Threatens Ally Oman'):

"Trump's frustration with his inability to end the nearly six-month war with Iran boiled over Monday into again threatening to bomb Oman, the latest U.S. partner to draw the president's ire over his view that the country is doing little to resolve the conflict.

"I don't think they behaved very well, but we'd handle them very easily like we did the others", Trump said Monday in the Ova Office. Earlier he had told FOX News in a phone interview "If Oman gets in the way we'll bomb the s---t out of them."

THIS is how the leader of the free world acts? Like a 5-year old in the throes of a hissy fit because he doesn't perceive an ally "behaves very well"?  Unaware the frustrations have all been derived from his own TACO behavior - claiming a great peace deal with Iran one day, then retracting it and vowing' hellfire and missiles' the next, then standing down again.  (Because he knows he can get the DOW to shoot back up.)

Having deduced Trump is unstable and untrustworthy in any talks we learned Iran's Revolutionary Guard "took full advantage of the ceasefire period and enhanced its capabilities, accelerated missile production and improved missile accuracy" (WSJ, 8/20, 'Iran's Plan To Escalate the War').

Iran also knew Trump's missile & drone stock was so low he no longer had enough to rain 'fire and brimstone' even if he wanted to. Apart from which the Iranians concluded Trump was now merely marking time and every further threat was hollow.  Indeed, anyone with a brain knows Iran effectively won from day one, with the closing of Hormuz after Trump and Netanyahu initiated the attack:

 "He lost the war on Day 1" — Paul Krugman Exposes Iran War Failure

 Hence, it's paved the way for an 'L' which Dotard has yet to accept. The rest as they say is history, and with it the end of Trump's delusions of American Empire, e.g.

The Professor Who Predicted the Iran War: This Is How America's Era Ends - And It’ll Only Get Worse

So the Orange traitor and fungal rotting renegade got zero, nada, for his obeisance to the North Korean dictator and will get nothing but bupkis from Iran or Oman, nor from Canada now - which started a trade war instead of accepting his 50% tariffs. But woe betide us if we take our attention off this dingbat.  As the WSJ editorial ends:

 "Frustrated by his decline in the polls and the lack of a clear victory against Iran, Mr. Trump is likely to become even more volatile in his foreign policy moves. Without the late Lindsey Graham to warn him away from bad decisions will other Republicans in Congress and inside the Administration speak up?"

I would wager here no more than you guys have, and at least to the extent of removing this batshit Orange Turd by the 25th amendment.  A Democratic president pulling the same shit - say like a President Gavin Newsome - would definitely have been impeached and removed by now. So why not Trump? Why is this senile fart untouchable?

See Also:

by John Feffer | August 20, 2026 - 4:28am | permalink

— from Foreign Policy In Focus

In a fit of pique, Donald Trump recently scaled back the big annual military exercise between the United States and South Korea. He would have canceled it altogether if the 11-day Ulchi Freedom Shield training hadn’t already started. The U.S. president cited as the reasons for his action the cost of the exercise and that it was “totally inappropriate and hostile” toward North Korea.

Trump also rationalized his show of disrespect for South Korea—or, at least, for the South Korean military—by complaining that Seoul hadn’t given him enough support for the U.S.-Israeli war against Iran. Back in April, at a press conference where he lambasted NATO for not sending warships to help open up the Strait of Hormuz, Trump said, “Who else didn’t help us? South Korea.”

Of course, neither did North Korea. In fact, Pyongyang quickly sided with Tehran after the initial attacks, denouncing the “illegal and reckless act of aggression” and adding that “the growing acts of U.S. hegemony witnessed by the international community are a clear example of its destructive role in undermining world peace and stability and the grave consequences that follow.”

» article continues...

And:

by Heather Digby Parton | August 24, 2026 - 4:55am | permalink

— from Salon

Donald Trump has always believed himself to be an expert at what he calls “nuclear,” due to the fact that his uncle, John Trump, was a physicist who taught at MIT. The president has made it clear on numerous occasions that his uncle’s expertise transferred to him due to the Trump family’s superior genes.

It’s too bad Dr. Trump died in 1985, because the president could use a refresher course on the subject. He certainly hasn’t learned anything new by virtue of the Oval Office.

We may never fully know what led Trump to start his war with Iran, which has been going on for nearly six months. Whether he was feeling his oats after the successful kidnapping of Venezuelan President Nicolás Maduro or by Israel’s assurances that the Iranian regime was so weak it would crumble quickly under American bombardment, Trump decided to make the move on impulse without having a specific goal or strategy other than “winning.”

» article continues...

And:

U.S. withdraws from military exercises with South Korea set for next month - The Washington Post

And:

Brane Space: Make No Mistake, Kim Got The Better Of Dotard In The Singapore "Deal"

Phase Spaces In Plasma Physics - The Klimontovich Equation & The Liouville Equation (Part 1 Basics)

In addition to the Klimontovich equation, e.g.

 Examining Aspects Of The Klimontovich Equation Used In Plasma Physics

Plasma physicists also work with the Liouville equation, which provides an exact description of a plasma.  However, like the Klimontovich equation it is of no direct use, instead giving us a starting point for the development of approximate statistical theories.  These latter begin with distinguishing the behavior of particles from the behavior of systems. Specifically the Klimontovich equation describes the former, while the Liouville equation describes the latter.

To fix ideas, consider a basic basic 'system' of one charged particle. How do we go about its description? Let us say we measure the particle's position in a coordinate system using x1   then the orbit of the particle - designated  X1(t) is the set of positions xi occupied by the particle at consecutive times t. Similarly, in velocity space one can denote the orbit of the particle as Vi(t)  which is the set of velocities of the particle at consecutive times t. But these are measured in a coordinate system vi  for velocity space.  Then we end up with a phase space denoted: (x1v1)  = ( x1 , y1 , z1vxi, vyi,  vzi)

Meanwhile, the density of systems in this phase space is:

N(x1 ,v1 ,t) = d (x X1 (t)) d (v-  V1 (t))

Next, we consider a system of two particles. Particle 2 now has coordinate axes (x2, v2) which overlay the (x1, v1) coordinate axes.  In addition the orbit X1 (t),  V1 (t) (i.e. of particle 1), is measured with respect to the (x1, v1) coordinate axes while the orbit X2 (t),  V2 (t) (i.e. of particle 2) is measured with respect to the (x2, v2) coordinate axes

Plasma statistics, basically, reference a 6-D phase space.  But in Liouville's Theorem the approach is very different. Specifically, it is applicable to a 6No  - D phase space, where No is the number of particles in the system.  For a one particle system then No  = 1  so the system phase space is just 6D.   I.e. the system is represented by only one phase space point.

For a 2-particle system (No = 2) so the phase space is:

6 (2) D = 12D

And again there is only one phase space point for the system. In general for an No -particle system, though with 6No  - D phase space, there is still only one phase point. 

Recall the number density for a 1-particle system is written:

N = d (X  -    X1 (t)) d (V  -   V1 (t))

And for a 2-particle system:

N( X1X,VV) =

d (X  -   X1 (t)) d (V  -  V1 (t))d (X  -    X2 (t)) d (V  -   V2 (t))

And in general:

N( Xi.... X,Vi ....V) = å No i=1      d (X  -   Xi (t)) d (Vi  -   Vi (t))

If no source or sink exists in the phase space then the number density must be conserved.  The conservation of N in the 6No - D phase space leads to:

 N / t +  å No i=1  Ñ x  (X · N) +å No i=1  Ñv  (V · N) = 0 

Where:

 Ñ x i  ·(X · N)  =   X' · Ñ x i  N +   N Ñ x i  ·X'i 

And:

Ñv i  (V · N)= V'i · Ñv i  N + Ñ v i  · V'

If we then identify:   X' = V

We arrive at:

 V' =   q/ m [ E (Xi ) + V  x  B( X)

which is just Newton's 2nd law of motion (F = ma) in an EM -field.  

Suggested Problems:  

1. (a)Write the phase space dimension for a 3-particle system.

(b) Write out the number density for this system.

2. We saw  (x1, v1)  = ( x1 , y1 , z1, vxi, vyi,  vzi)  is a 6-dimensional phase space.  What would be the number of dimensions for the 12- D phase space  (x1, v1x2, v2 )?

Write them all out in a bracket.

3.  Show how V' =   q/ m [ E (Xi ) + V  x  B( Xi )

translates to Newton's 2nd law of motion.

(Hint: You need to incorporate the Lorentz force.)