Saturday, February 15, 2025

Asteroid Sample Retrieval Mission Rivals That Of Japan - But Can NASA Keep Up With DOGE Lurking In Background?

 

Asteroid Bennu - from which samples were taken

The National Aeronautics and Space Administration’s Osiris-Rex mission  -launched back in 2016 - has since returned from the asteroid Bennu, with a sample laden with amino acids.  (Those are the building blocks of life.)  The achievement can now be said to rival the earlier Hayabusa mission of Japan:

One Of A Kind Hayabusa 2 Probe To Asteroid Makes Amino Acid Discovery - After Dropping Sample In Australian Outback

The NASA -sponsored craft scraped the surface of the asteroid in 2020, scooping up just over 2 ounces (roughly the amount of a bar of soap) of rock and dust which included sodium salts highlighted in the photo underneath the asteroid's image.  The craft then departed Bennu and swung past Earth 3 years later, dropping a sealed capsule containing the material in the Utah desert.  The capsule and its landing spot are shown below:

                                   Bennu capsule - landed in Utah desert with sample

In the manner of asteroid approach and sample retrieval the mission actually resembled the earlier Japanese mission, captured in the graphic below:

Japanese asteroid sample retrieval mission - resembling NASA's Bennu mission

In the case of Bennu, a glob of rubble that formed 4.5 billion years ago, researchers examining the recovered sample found 14 of the 20 amino acids that make up proteins.  They also found all five molecules that comprise RNA and DNA in mineral deposits of salty brine on the asteroid surface.  That sample is shown below:

                           Sodium salts portion of retrieved Bennu asteroid sample

The findings were published last Wednesday in two papers:

Abundant ammonia and nitrogen-rich soluble organic matter in samples from asteroid (101955) Bennu | Nature Astronomy

And:

An evaporite sequence from ancient brine recorded in Bennu samples | Nature

The process of analysis itself was done at the NASA Goddard Spaceflight Center in Greenbelt, MD.  There, tiny chips of the asteroid sample were ground into a fine powder, then made into a 'Bennu tea'  by boiling it in water.  This allowed the researchers to extract and identify the organic compounds - this according to Jason Dworkin a co-author of a paper published in Nature Astronomy. 

The question now - among all of us who pursue astronomy and space - is how long before such missions become extinct in the new DOGE era of axing agencies.  Will DOGE find the billions allocated to NASA too "wasteful". If so, will DOGE and Musk's coder Muskrats take a wrecking ball to the agency and let all its researchers join the breadlines?  We will have to wait and see, but the early signs - like the destruction of the Dept. Of Education- do not look promising.


See Also:

Asteroid fragments upend theory of how life on Earth bloomed

And:

NASA's Asteroid Bennu Sample Reveals Mix of Life's Ingredients

Friday, February 14, 2025

Solutions to Partial Differential Equations Via Lagrange Method

 1)  Find two integrals (solutions) for:

dx/z2    =  dy/ x( z - y) = dz/ x

           Solution:

Treat two equations:

A)  dx/z2    =  dz/ x

B)  dy/ x( z - y) = dz/ x


For (A)  rewrite, separating variables:

x dx =     zdz

 Then the solution from integration is:

½   x2 =   1/3  (3 ) + c1


For (B),  note that a multiplication by x yields an equation containing only y and z:

dy =  (z - y) dz

Or:

dy/ dz   + y  = z   

Which is a 1st order linear equation with solution:

y = z - 1  + c2 exp (-z)

Then we arrive at the two final results:   

e z (y + 1 - z)  = c2

and:

3x2  -   2 3 = c3

 

2)  Find two integrals of:

dx/x   =  dy/ y  = dz/ xy


  Solution:

Note the first equation: dx/x   =  dy/ y    has for its solution:

y = c1 x

Substitute this solution into the denominator of dz to obtain:

dx/x   =  dz/ c1x2

or:  c1x dx = dz

Integration yields:

½ c1x2    =   z  + c2

Or:  2z    =  yx + c2

Then we obtain:

y = c1 x    and 2z    =  yx + c2

 

Thursday, February 13, 2025

Why Any Aliens We Might Encounter On Other Worlds (Or This One) Will Emphatically Not Be "Human-like"

 

                                Reptilian alien postulated by several xenobiologists

                             'Grey' alien of the type reported to be piloting UAP craft


In an interview with BBC's Science Focus magazine some years ago, Simon Conway Morris - an evolutionary paleobiologist at the institution's Department of Earth Sciences - declared that researchers can "say with reasonable confidence" that human-like evolution has occurred in other locations around the Universe.  The core of Morris' belief comes from the theory of convergent evolution. It claims that life will repeatedly evolve in different ways due to similar pressures or starting points.

According to Encyclopedia.com: "Convergent evolution is the process by which unrelated species independently evolve similar traits due to having to adapt to similar environments or ecological niches."   

We can all agree that this mechanism has played an essential role in shaping the Earth's biodiversity but Conway Morris insists it can also lead lead to intelligent alien life (but like human, or human-like)  on other planets!   

Of course, this is total nonsense, the reasons for which I will get to below. 

For now let's note that convergent evolution has only been observed on Earth, for obvious reasons. It is a response to evolutionary pressure, the need of an organism to survive. This may occur under conditions similar across different species (such as living in certain climates), or it can be driven by unique factors depending on what type of life form.  

But Conway Morris goes even further and claims that scientists can say with "reasonable confidence" that human-like creatures have evolved outside our solar system.  He bases this on "decades spent studying the theory of convergent evolution."   But studying a theory is not proof that it applies everywhere. We are not talking here of a biological cosmological principle after all.  

According to one science writer on the same topic:

"Suppose you think of the entire Universe as one enormous ecosystem. The Universe is a prominent place so that some aliens will have brains and bodies resembling humans. Don't you think?"


No,  I don't think any aliens will have "human" like brains or bodies and here's why. On Earth biological evolution for humans is an algorithmic process, e.g.



 The top diagram displays a realistic algorithmic development of two hominid families, called A and A' - initially. Note here the primary differentiating characteristic is not a uniform linking in a linear sequence, but rather a TREE-like branching. The tree-like branching fits the evidence we have that while a 'shoot' of the evolutionary lineage starts off, it often quickly hits a dead end, then stops. Others continue to develop before they branch off...or don't.



Meanwhile, each specific algorithmic hominid offshoot in Fig. 2 was evolved dependent on the ecology and environmental conditions at that time, as well as atmosphere, climate etc.

Just look at the most elemental level of the human genome which is also packed with up to 8 percent of virus DNA, as recent research has shown, e.g.

Ancient Viruses Are Buried in Your DNA - The New York Times (nytimes.com)

These ancient viruses for sure will be found on no other planet because no other planet will have the exact same conditions for them to have originated.  (Or at least the probability of doing so will be next to zero.)  Hence, we can reckon that proportion of the genome will not be replicated. We can, indeed, trace that back to the origin of life on Earth in abiogenesis, e.g.

There is also little doubt the Krebs cycle  (for aerobic metabolism) referenced above, i.e.

CH2 COOH-coenzyme A +  2 H2O ->   2 CO2  +8 (e- + H+) + coenzyme A

 Cannot be replicated on any other worlds for the simple reason that the assorted coenzyme (A) needed to make it work will not be found in any reactions on those worlds, again  because the primitive atmospheric, ecological conditions would not have been the same.   

Even assuming life on any other world arises, how will it adapt to its conditions?  Plant-like aliens are unlikely because photosynthesis doesn't encourage complex survival strategies. The need to pursue diverse food sources favors mobile life. If such life evolved in a thick atmosphere, it's likely to be a horizontal creature. A thinner atmosphere would favor the most vertical animals. 

Go on then to the g-force present on a given planet. The size and shape of the hypothetical  alien would be determined by gravity, ambient density, and source of energy (spectral type for central star or sun). While two legs and two arms are more efficient than four legs (or no legs)  say for terrestrial advanced organisms,  a blob--like physique - analogous to a giant amoeba -  might adapt better to a high g world. So  appear nothing like a human. If the creature was also silicon-based and not carbon -based, the differences would be even more dramatic and be more like a giant 'rock'.

But to be more or less exactly like a human in appearance would again assume - at each algorithmic step on the alien planet - an adaptation was made by that emerging life form that matched the algorithmic step on our Earth.  Primitive hominids on Earth grew opposable thumbs so they could better grasp objects. Monkeys developed prehensile tails for the same reason. All our ape cousins and ourselves have eyes tuned to the visible end of the electromagnetic spectrum.

But these sets of  consecutive adaptations are simply not that likely given the differences in conditions on different worlds, say for the assay found so far on those discovered by the Kepler  telescope, e.g.. 

Brane Space: How Really "Earth -like" Is The Newly Discovered Planet Kepler 186f? (brane-space.blogspot.com)


If life had occurred on a completely different world, with different temperatures, gravity and topography then it is a no brainer that flora and fauna must have evolved differently- whether plant. fish, amphibian, reptile or mammal. And if that other world had a completely different chemical composition, so would any higher life, assuming it was even possible to extend the algorithmic processes that far - as seen in Fig. 2. All life on Earth is carbon-based, but that wouldn't be the case elsewhere. Life forms could be silicon-based or iron-based or anything else at all.

 A further thought:  consider that 95 percent of the universe (mainly as dark matter, dark energy) is not perceptible by our senses or technology then it's most likely that's where the actual aliens are-  and explains why we haven't found any meeting our demands for existence yet.

The main problem we have in regard to  processing (or searching for)  alien life is that our assumptions -  like those of Conway Morris - are way too anthropomorphic. Thus, a physicist like Michio Kaku could - in an appearance on CBS Morning some 5 moths ago-   essentially dismiss actual aliens piloting the UFOs reported by military pilots, e.g.  

Watch USS Nimitz ‘Tic Tac’ UFO: Declassified Video Clip | HISTORY Channel

 because the "dramatic changes in acceleration would have been too much to survive".    But Prof. Kaku makes such assertions precisely because he is assuming human-like aliens with the exact same biological peculiarities and limits as terrestrial humans.

I just think this is a huge error, for him as well as Conway Morris - to assume any  human biological or biochemical aspects -  in putative  "thought experiments" concerning extraterrestrial life.  Including such life as may actually have already reached our Earth - via UAP incursions, e.g.

Brane Space: Astrophysicist Adam Frank Gets It All Wrong On UAP- UFOs

See Also:

Brane Space: Hawking's Alien Conquest Scenario - By Seeding Earth With Parasites & Viruses - Should Be On Everyone's Radar

And:

Yes, Reaching Out For Alien Contact May Be Humanity's Worst Mistake - According To Michio Kaku

Wednesday, February 12, 2025

How Does Musk's View Of Efficiency (in DOGE) Compare with the Pareto Model Advanced By Friederich von Hayek?

 




Economics Professor Herman Daly of the University of Maryland once noted that the most formidable problem in our society is the judicious sharing of its resources. This sharing is an ethical problem that demands solution. In this context of judicious sharing, Daly's view (and my own) is that - while people are not created equally- the Earth's resources exist to be shared equally. Indeed, the Utilitarian economist Jeremy Bentham once noted that the happiness of a given society is maximized when its utilities are maximized. That means ALL of its people, not just a few at the top, able to exercise their talents to their maximum potential. In the words of Bentham:

"The more nearly the actual proportion approaches to equality, the greater will be the mass of happiness'

Because resources, i.e. money, health care, property, ensure one's talents can realize the choices to be used in the most able ways. Then, Bentham argued, the greatest happiness is assured because the distribution of the resources to support the development of talents are nearly equal. If 10 million men or women wish to become fulfilled artists or poets, then they can, or if they wish to be writers, or builders.

No one "owns" the Earth's resources though multi-billionaire Elon Musk comes close to believing he does, given he's worth $378 billion. The problem is that such magnitude of wealth and control of resources distorts and degrades thinking, making one believe he is more powerful in ownership terms than he is. Musk's exercise of his DOGE wrecking ball mechanism to subdue and destroy the federal bureaucracy is an example of this.

Musk's wealth and in particular since he's used it to leverage power to act on Trump's behalf, has left him shortsighted in terms of how he's now diminishing the lives of millions of other humans who share this planet.  Just the elimination of U.S.A.I.D. which Musk - in his warped billionaire mind calls "evil" - may well lead to the deaths of over 8 million arising from famine in sub-Sahara Africa in the coming months.

Enter now Vilfredo Pareto who along with his Italian elitist compadre Gaetano Mosca sought to erect an economic model to negate Marx surplus value theory.  They saw Marx's postulated unpaid labor as a direct threat to the elites at the time. In modern parlance, Marx's "surplus value" amounted to economic waste for the wealthiest, since they were denied access. Thus, Pareto and Mosca reasoned that something had to be done to quash this idea in its infancy, and so they developed their Elite theory. The risk they saw was that Marxism as defined would deliver too much proportion of resources to the masses by way of compensating what was originally unpaid labor. 

After all, according to Elite theory and its Pareto Distribution basis, the mass of people had to be kept poor if the few elites were to maximize their own efficiencies of wealth and resources.  It was a zero sum game, after all, though later Elite theory propaganda would try to say zero sum didn't apply if the little sheeple just got off their duffs and worked harder! Then, they might get more money or resources.

This, ironically, is exactly the same philosophy behind Musk's DOGE puppets and coders raiding one agency after another and putting them through their "wood chippers" - by either letting workers go or freezing capital (resources).

 The idea behind Pareto efficiency thus had to be NOT to make the masses richer, but to make them poorer. Poor masses couldn't make trouble, since they would have to work longer and harder to even get the bare necessities. And they'd likely have precious little time to stick their noses into politics.  At least not in any comprehensive learning format.

Pareto and Mosca - like Elon Musk-  also saw the need at the time to develop a massive economic propaganda industry to sow memes that would destroy Marxism. (In Elon Musk's case using his 'X" to sow gigabytes of misinformation to destroy democracy and empower a traitor- felon resident like Trump.) They suggested use of the term "evil", or "inhuman" and this was improved and enhanced as other elites took up the message. 

For example, Friederich von Hayek later waxed on about the "road to serfdom" as being deprived of the freedom to spend. Hayek, like other capitalists and purveyors of Elite theory, saw that if rampant consumerism could be incited and promoted, Marxism's tenets would become redundant. A person enmeshed in whatever his consumerist fancy desired, whether a fresh bottle of rum or a new ipad, wouldn't give two mites about some long-haired radical commie and his theories of labor!

Enter then Pareto's infamous summary of his mode of economics which underlay the Pareto Distribution and by extension, Pareto-style economics:

"Assume a collectivity made up of a wolf and a sheep. The happiness of the wolf consists in eating the sheep, that of the sheep in not being eaten. How is this collectivity to be made happy?"

Pareto concluded the only happy ending was for the wolf to eat the sheep. If the collective had ten wolves and 200 sheep, the only happy ending was for ALL the sheep to be devoured by the wolves. In this way utility was maximized along with efficiency. The sheep would get their fill of the grass while the wolves held out for a time, but in the end all that sheep meat would be consumed by the wolves. The smaller set of the more powerful faction of the collective owned and merited the killing and eating of the larger ("mass") sheep.

In like manner, Musk's concept for DOGE is to have the "sheep" (all the federal bureaucrats) 'devoured' by his 'wolves' (efficiency hawks, coders) such that the end result is all the federal agencies be put through the "wood chipper" and left with only minimal resources, workers ("sheep").

This template - for the Pareto model and DOGE -  translates directly to the basis for Elite economic theory. That is, only a very select few - an extreme minority - in any given society - often belonging to the most elite classes (e.g. billionaires) , are fit to govern that society and moreover, partake of the bulk of its resources and wealth. In a word, the elite are also the "elect" - predestined by the process of alleged "social Darwinism" to attain the pinnacle and be served by the masses in the society. These masses, in addition, to be kept docile must be forced to adhere to an economic efficiency model that redistributes their resources to the elites.

Example:  Recall that former Fed Chairman Alan Greenspan went on record in an appearance before congress (in 2003) as asserting that "Social Security benefits need to be cut to pay for Bush’s tax cuts."  What on Earth was the man thinking? Well, he's thinking on the basis of Pareto efficiency!

 Social Security payments, especially with COLAs, do everything the Fed Chairman didn’t want. They pour money into the economy, but not via productive labor or market investments, returns. People receive their checks merely by existing and breathing day to day -  having paid into the system with FICA deductions. Even then, they receive far more in benefits than actually paid in, making a total mess of "utils" earned. In a way, the Social security recipient (in the eyes of this Pareto-riguer bunch) are like a rent subsidized couple with their "consumer surplus". Worse, the S.S. COLAs increase the non-productive payments each year, one reason why – back in 1997 – Greenspan demanded an artificially much lower COLA increase than had originally been proposed.

Greenspan had surmised – correctly in terms of Pareto Efficiency – that the Bush tax cuts accomplished the same for the well off. The tax cuts poured more money into their hands each year, and actually allowed them to keep more of their higher valued dollars and their associated utility. Greenspan obviously reasoned, based on his 2003 Senate Banking Committee testimony, that it made more fiscal sense to let these wealthy keep their ill-gotten gains from the tax cuts, than to preserve Social Security and reward the unproductive (and mostly non-investors) with unearned compensation. In terms of both magnitude of the wealthy's "utils" and greater worth of each $1 they got, it was more Pareto Efficient!


Appreciating the extent to which typical billionaire capitalists (like Musk or Trump) use Pareto efficiency (and the Pareto distribution) is key to understanding how economic elites secure and preserve their advantages.  It is also why they want no part of democratic socialist prescriptions that would level the playing field.   The Reeps' general objection to Biden's American Rescue Plan (ARP) is of the same memetic piece as Greenspan's abhorrence of Social Security:  They don't want the lower economic strata of Americans getting any capital or resources they have not earned via work or investment decisions.   We can therefore call the GOP's ARP objectors the allies of Vilfredo Pareto's wolves.

In like manner, the DOGE destruction of federal agencies designed to help the masses (i.e. not in the top 1%)  whether in terms of health security (CDC, FDA) or financial protection (CFPB) or from weather disasters (FEMA) is seen as "inefficient" because the workers in those agencies are viewed as unproductive - given they are not making more profits for the capitalists. Unlike the Pareto capitalists the DOGE lot are staging an open coup d'etat against America,.

See Also:


And:

by Sam Pizzigati | February 12, 2025 - 6:16am | permalink

— from Inequality.org

A classic coup d’état has guns. Uniformed men run wild seizing government agencies and claiming control over what government does and who government serves.

But in our new cyber age, the Yale historian Timothy Snyder reflected this past week, a coup can unfold without any armed overthrow. We can have “a couple dozen young men go from government office to government office, dressed in civilian clothes and armed only with zip drives.”

These young men, operating upon “vague references to orders from on high,” can gain access to basic computer systems and “proceed to grant their Supreme Leader” effective power over just about everything that government does.

The historian Snyder is, of course, describing America’s current reality. He’s calling this reality a coup — and so are countless other defenders of America’s democratic faith.

» article continues...

And:

by Sarah van Gelder | February 13, 2025 - 6:53am | permalink

As President Donald Trump tests the limits of manufactured crisis and chaos, he claims a mandate from the American people. But his razor-thin electoral victory tells a different story. Voters didn’t ask for an illegal takeover of government offices, a freezing of funds for needed services, sending our immigrant neighbors to camps at Guantanamo, or aggression against our allies. Yes, we wanted change. But as multiple polls show, a clear majority were seeking relief from unaffordable prices, real economic hardship, and inequality, not an authoritarian takeover.

The Trump administration’s initial barrage of orders will make life worse for the most vulnerable — especially immigrants and transgender people — but soon enough for everyone else in the non-billionaire community. The rapid roll out of these policies is right out of the fascist playbook, designed to overwhelm and demobilize the public.

How can we regain our footing and our strength? We need to not only stop the roll out of policies that threaten to make life worse for ordinary people, but we need to keep focusing on the changes we the people are actually looking for. We need to demand the changes we voted for. If we do that, we can stay grounded during the turmoil, resist the chaos, and build the power to create an authentically populist future.

» article continues...

And:

by Tim Karr | February 13, 2025 - 6:11am | permalink

On February 10, the Electronic Privacy Information Center, or EPIC, filed suit for damages against the Trump-Musk-DOGE cartel. The lawsuit, which EPIC filed before the U.S. District Court for the Eastern District of Virginia, calls for damages on behalf of tens of millions of government workers and Americans resulting from the administration’s illegal breach of personal privacy and its threat to national security.

“These basic security failures have resulted in the unlawful disclosure of personal data—including social security numbers and tax information,” reads the complaint.

EPIC is claiming the data incursion—among many other violations—is illegal under the Privacy Act of 1974. “Plaintiffs have a constitutional right to the privacy of their information… Defendants have violated and continue to violate that right by unlawfully disclosing extremely personal information about plaintiffs and millions of others to unchecked actors in violation of law,” the complaint states.

» article continues...

And:
by Amanda Marcotte | February 12, 2025 - 6:29am | permalink

— from Salon

Donald Trump, his shadow president Elon Musk, and the so-called Department of Government Efficiency are waging a highly illegal rampage through the federal budget. Among the first wave of intended victims are thousands of scientists across the country who depend on federal grants and loans to fuel their research. The administration is trying to unilaterally slash billions of dollars from the National Institutes of Health (NIH) and the National Science Foundation (NSF). To defend this, Musk claims it's a "ripoff" when grant funding goes to pay for salaries, lab space and equipment, even though no one can conduct scientific research without these baseline necessities.

"Trump and King Musk framed this as a shot at baddies like Harvard and Yale, but this absolutely destroys public schools and every state has those and they all are doing scientific research," historian Erik Loomis explained at Lawyers, Guns and Money. Even Republican senators are having trouble concealing their panic, as public universities are economic centers in many red states.

» article continues...

And:

by Thom Hartmann | February 12, 2025 - 6:10am | permalink

— from The Hartmann Report

A new poll from CBS News finds that most Americans agree with Trump’s policies of gutting protections for women and Black people while arresting and deporting Brown people here without documentation. Roughly two-thirds of respondents called him “tough,” “energetic,” “focused” and “effective.”

They think they are free.

White people approving of Trump’s actions believe marginalizing minorities, gutting the federal government, and attacking (and suing) media outlets and reporters will help them, and some of them probably will. A return of white supremacy and the marginalization of women in the workplace and politics, for example, may produce some small benefits for a few white men.

But there’s a price to be paid in what Americans like to refer to as freedom.

But there’s a price to be paid in what Americans like to refer to as freedom.

For Trump to continue to reverse over 100 years of progress on civil, women’s, and workers’ rights, he’s eventually going to confront pushback from those very groups. He’s going to have to deny working people their right to organize into unions. He’s going to have to restrict the ability of poor and middle-class people to rise above their economic stations through education.

All of these things will eventually harm the middle class, and he’s already started on every one of them.

But it’s even worse than that: as our freedoms are taken away, and our government is turned into a one-party state, governance that assists the majority of the people begins to vanish. The freedom to speak out goes away. Eventually, society turns cold, as Trump’s authoritarianism hardens the national Zeitgeist.

» article continues...

And:

Monday, February 10, 2025

Solving Cauchy-Riemann Partial Differential Equations

Cauchy-Riemann Partial Differential Equations are at the core of many problems in mathematical physics. What are the Cauchy- Riemann equations?  

Given an equation of form:


f(z) = f(x + iy)   = u(x,y) + iv(x,y)
 
which is taken to be differentiable at the point z 0 = x+ iy0  - then the partial derivatives of u and v exist at the point (x,  y0 ) and satisfy the equations:
 
a)  u/  x  =   v/  y   and   b)   v/  x  =  -   u/  y  

Which are the Cauchy- Riemann equations. If such condition is met then the function is said to be analytic in the region, Ã‚.  

An additional condition concerns whether the function is harmonic. If it is, then u and v also have continuous 2nd partial derivatives. Then, we can differentiate both sides of (a) with respect to x, and (b) with respect to y to obtain:

1)  2 u/  x2  =     2 v/  x   y    and

2)   2 u/  y2  =     2 v/  y   x  

 
From which a special example is obtained:

  2 u/  x2  =    2 u/  y2       or     2 u/  x2  +   2 u/  y2       = 0


Which is known as Laplace’s equation.

Similarly, we can perform an analogous process for v (differentiating both sides of (a) with respect to y and (b) with respect to x) to arrive at:

 2 v/  x2  +   2 v/  y2       = 0

If then this equation is satisfied, v is harmonic.

We next look at the function.

  f(z) = (x– y2) + i2xy

So that: u(x,y) =  x2 -  y2

 
And v(x,y) =  2x y

 
Next, we check to see if the eqns. are analytic


Take  u/  x    =   2x    and:   v/  y    =   2x

 
Since:   u/  x  =   v/  y     then u(x.y) is analytic

 
Now check the other function, v(x.y):     v/  x  =  2y

And we see:  -   u/  y   =  - (-2y) = 2y

 
So that v(x,y) is analytic..

  If f(z) is analytic everywhere in the complex plane it is said to be an entire function.  Now, check to see whether the functions are harmonic.

For u(x,y) we need:  2 u/  x2  +   2 u/  y2       = 0

Since:  u/  x    =   2x, then    2 u/  x2  =   2
 
Since:  u/  y   =   -2y then   2 u/  y2      -2 

Then:  2 u/  x2  +   2 u/  y2       = 2 + (-2) = 0

 For v(x.y): we need  2 v/  x2  +   2 v/  y2       = 0

 Since:  v/  x  =  2y then   2 v/  x2  =   2

Since:  v/  y    =   2x   then   2 v/  y2      = 2


Then:  2 v/  x2  +   2 v/  y2         = 2 + 2 = 4

 
 Evidently, v(x,y) isn't harmonic since the sum of the 2nd partials doesn't equal zero.

 
Suggested Problems:

1)     Given the function: u(x,y) = x3 – 3xy2
 
Show the function is harmonic on the entire complex plane.
 
2)     Given the function: u(x.y) = exp(-x) [x sin y – y cos y]

a)     Show u(x,y) is harmonic

b)     Find v(x,y) such that f(z) = u + iv is analytic

 
3) Let f(z) = exp(x) cos(y) + i(exp(x)sin(y) = u(x,y) + iv(x,y)

a) Determine if the function is analytic for both u and v.

b) Determine if the function is harmonic for both u and v.